UBC Theses and Dissertations

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UBC Theses and Dissertations

Binary (e,2e) spectroscopy and momentum space chemistry Cook, John P.D. 1981

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BINARY ( e , 2 e ) S P E C T R O S C O P Y a n d MOMENTUM S P A C E CHEMISTRY by JOHN P . D . COOK B . S c . ( H o n s ) , M c G i l l U n i v e r s i t y , 1976 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF DOCTOR OF P H I L O S O P H Y i n T H E F A C U L T Y OF GRADUATE S T U D I E S ( D e p a r t m e n t o f C h e m i s t r y ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d : T H E U N I V E R S I T Y OF B R I T I S H COLUMBIA N o v e m b e r 1981 (c) J o h n P . D . C o o k In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of £t4£ A U S T T g y The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 2.7" r\fOV£<K(l€£ 11 ABSTRACT Binary ( e , 2 e ) spectroscopy i s an intermediate energy electron scattering coincidence technique measuring the binding energy and spherically-averaged momentum di s t r i b u t i o n s of in d i v i d u a l valence electrons in small gaseous molecules. Momentum space chemistry i s a term used to refer to the study of the at t r i b u t e s of molecular o r b i t a l s in the momentum space representation, rather than the usual position space representation. The r e l a t i o n between the two spaces i s the Fourier Transform. This thesis discusses experimental measurements and the o r e t i c a l c a l c u l a t i o n of the binding energy spectra and/or momentum d i s t r i b u t i o n s of. H 2S, COS, C0 2, NO and 0 2 in d e t a i l . It also attempts to bring into the ken of ordinary chemistry concepts and p r i n c i p l e s for dealing with momentum-space molecular o r b i t a l density functions, which are e s s e n t i a l to the understanding of the nature of momentum d i s t r i b u t i o n s . In order to i l l u s t r a t e this,. s p e c i f i c examples of th e o r e t i c a l momentum and charge density maps for several molecules are discussed. S i g n i f i c a n t new understanding of the el e c t r o n i c structure of these molecules is attained. I l l T h e d e s i g n , c o n s t r u c t i o n , a n d p r e l i m i n a r y t e s t i n g o f a new b i n a r y ( e , 2 e ) s p e c t r o m e t e r i n c o r p o r a t i n g a m u l t i - c h a n n e l p l a t e d e t e c t o r f o r i m p r o v e d d a t a c o l l e c t i o n e f f i c i e n c y i s p r e s e n t e d . F i n a l l y , s ome p r o p o s i t i o n s f o r f u t u r e d i r e c t i o n s o f s t u d y a r e p u t f o r w a r d . C . E . B r i o n T h e s i s s u p e r v i s o r i v CONTENTS A b s t r a c t i i C o n t e n t s i v F i g u r e s i * T a b l e s x i i F o r e w o r d x i i i A c k n o w l e d g e m e n t s x v D e d i c a t i o n x i x C h a p t e r 1 I n t r o d u c t i o n A n d T h e o r y 1 1.1 T h e ( e , 2 e ) R e a c t i o n 4 1.2 T h e B i n a r y ( e , 2 e ) P r o c e s s 8 1.2.1 T h e b i n a r y ( e , 2 e ) f o r m f a c t o r 9 1.2.2 T h e momentum d i s t r i b u t i o n 14 1.2.3 T h e v a l i d i t y o f t h e PWIA 18 1.3 B i n a r y ( e , 2 e ) i n C o m p a r i s o n w i t h O t h e r M e t h o d s ..19 1.3.1 B i n d i n g e n e r g y s p e c t r a 19 1.3.2 Momentum d i s t r i b u t i o n s 21 1.4 T h e o r e t i c a l T r e a t m e n t s . . . 22 1.4.1 B i n d i n g e n e r g i e s 22 1.4.2 Momentum d i s t r i b u t i o n s ..30 C h a p t e r 2 Momentum S p a c e C h e m i s t r y 31 2.1 T h e F o u r i e r T r a n s f o r m D e f i n i t i o n 33 2.2 T h e F T S y m m e t r y P r o p e r t y 34 2.3 T h e R e c i p r o c i t y p r i n c i p l e . . . 36 2.4 T h e O n e - d i m e n s i o n a l W a v e f u n c t i o n P r o j e c t i o n Q . . . 4 3 2.4.1 C l a s s 0: C e n t r o s y m m e t r i c s y s t e m s .45 2.4.2 C l a s s I : One l o b e 45 2 . 4 . 3 C l a s s - I I : Two l o b e s ......... 45 2.4.4 C l a s s I I I : T h r e e l o b e s 46 2 . 4 . 5 F o u r l o b e s 50 2.5 T h e B o n d i n g P r i n c i p l e 50 2.5.1 H y d r i d e s ( A H n ) 51 2.5.2 D i a t o m i c m o l e c u l e s ( A 2 , A X ) 58 2 . 5 . 3 L i n e a r s y m m e t r i c t r i a t o m i c s ( A X 2 ) 67 2.6 B o n d O s c i l l a t i o n . 71 2.7 T h e S p h e r i c a l A v e r a g e 75 C h a p t e r 3 E x p e r i m e n t a l 77 3.1 T h e V a c u u m S y s t e m 77 3.2 T h e S p e c t r o m e t e r 79 3.2.1 T h e e l e c t r o n g u n •. 80 3.2.2 T h e b e am s t e e r i n g u n i t 8 3 3.2.3 T h e g a s c e l l 84 3.2.4 T h e e l e c t r o n l e n s e s 8 5 3 . 2 . 5 T h e c y l i n d r i c a l m i r r o r s e g m e n t " a n a l y s e r s . . . . 8 6 3.2.6 T h e c h a n n e l t r o n s 90 3.2.7 T h e a n g l e - s c a n n i n g s y s t e m 93 3.3 C o n s t r u c t i o n a n d M a t e r i a l s 93 3.4 S i g n a l P r o c e s s i n g 9 5 3.4.1 C h a n n e l t r o n c o u p l i n g 95 3.4.2 P u l s e a m p l i f i c a t i o n a n d d i s c r i m i n a t i o n 97 vi 3 . 4 . 3 C o i n c i d e n c e d e t e c t i o n 100 3.4.4 S i g n a l - a v e r a g i n g d a t a a c q u i s i t i o n s y s t e m . . . . 1 0 5 3.5 T h e I n s t r u m e n t a l R e s p o n s e F u n c t i o n 108 3.5.1 T h e q - 0 - 0 s u r f a c e . . 1 1 5 3.5.2 C o m p a r i s o n o f 4 0 0 e V v s 1 2 0 0 e V o p e r a t i o n 118 3.6 D a t a A n a l y s i s 122 C h a p t e r 4 H y d r o g e n S u l p h i d e 125 4.1 E x p e r i m e n t a l R e s u l t s l 2 6 4.2 C a l c u l a t i o n s 134 4 . 3 D i s c u s s i o n 136 4.3.1 B i n d i n g e n e r g y s p e c t r a o f H 2 S 136 4.3.2 M o m e n t u m d i s t r i b u t i o n s 142 4 . 3 . 3 MFS s t r u c t u r e 147 4.3.4 T r e n d s i n t h e A X n h y d r i d e s a n d r a r e g a s e s . . . 1 4 8 ' 4.4 H 2 S D e n s i t y M a p s 158 4.5 C o n c l u s i o n s 159 C h a p t e r 5 C a r b o n y l S u l p h i d e 161 5.1 I n t r o d u c t i o n 161 5.2 R e s u l t s a n d D i s c u s s i o n 163 5.3 C o n c l u s i o n s 172 C h a p t e r 6 C a r b o n D i o x i d e 173 6.1 I n t r o d u c t i o n 173 6.2 E x p e r i m e n t a l 174 6.3 C a l c u l a t i o n s 186 v i i 6.4 MFS S t r u c t u r e o f C 0 2 187 6.5 Momentum D i s t r i b u t i o n s a n d B o n d i n g i n C 0 2 190 6.5.1 T h e 3<r{g} a n d 2e{u] o r b i t a l s 192 6.5.2 T h e 4<<{g} o r b i t a l . . . 1 9 4 6.5.3 T h e 3o{u] a n d 1 i r { u } o r b i t a l s 196 6.5.4 T h e 1 rr{g} o r b i t a l 198 6.6 C o n c l u s i o n s 2 0 0 C h a p t e r 7 N i t r i c O x i d e a n d O x y g e n 2 0 2 7.1 N i t r i c O x i d e 2 0 2 7.1.1 T h e NO i n n e r v a l e n c e o r b i t a l s . . . 2 0 8 7.1.2 T h e NO o u t e r v a l e n c e o r b i t a l s 2 0 9 7.2 O x y g e n 2 1 2 7.2.1 T h e 0 2 <r{g} O r b i t a l s 2 1 3 7.2.2 T h e 2<<{u} O r b i t a l 2 1 9 7.2.3 T h e 1tr{u} A n d 1 rr {g} O r b i t a l s 2 1 9 7.3 C o n c l u s i o n s 221 C h a p t e r 8 T h e New MCP S p e c t r o m e t e r 2 2 2 8.1 I n t r o d u c t i o n 2 2 2 8.2 M C P s ( M u l t i c h a n n e l p l a t e a r r a y s ) 2 2 3 8.3 P o s i t i o n - s e n s i t i v e S i g n a l C o l l e c t i o n 2 2 5 8.4 T h e New S p e c t r o m e t e r D e s i g n 2 3 5 8.4.1 The. v a c u u m s y s t e m 2 3 5 8.4.2 T h e e l e c t r o n g u n 237 8.4. 3 T h e beam s t e e r i n g u n i t . . . 2 3 7 8.4.4 T h e g a s c e l l 2 3 7 v i i i 8.4.5 T h e c y l i n d r i c a l m i r r o r a n a l y s e r . . 2 3 8 8.4.6 T h e c o n i c a l l e n s 2 4 0 8.5 T h e New B i n a r y ( e , 2 e ) MCP D e t e c t o r 2 4 3 8.5.1 C o n s t r u c t i o n 2 4 3 8.5.2 S i g n a l p r o c e s s i n g 2 4 5 8.5.3 M C P / c o m p u t e r i n t e r f a c e 2 4 7 8.5.4 T h e c o m p u t e r s y s t e m 2 5 0 8.6 P r e l i m i n a r y T e s t i n g 2 5 3 8.6.1 T h e v a c u u m s y s t e m 2 5 3 8.6.2 T h e e l e c t r o n g u n 2 5 3 8.6.3 T h e a n a l y s e r 2 5 4 8.6.4 MCP t e s t i n g 2 5 4 C h a p t e r 9 C l o s i n g R e m a r k s 2 5 8 A p p e n d i x A G l o s s a r y 2 6 7 A.1 S y m b o l s a n d U n i t s 2 6 7 A.2 A b b r e v i a t i o n s 271 A. 3 T e r m s 2 7 2 A p p e n d i x B D e f i n i t i o n s A n d D e r i v a t i o n s 2 7 5 B. 1 M omentum D e n s i t y M a p s f r o m LCAO-MO W a v e f u n c t i o n s 2 7 5 B.2. M o m e n t u m D i s t r i b u t i o n s f r o m LCAO-MO W a v e f u n c t i o n s .278 A p p e n d i x C M i s c e l l a n e o u s 2 8 3 A p p e n d i x D R e f e r e n c e s 2 8 5 i x F I G U R E S C h a p t e r 1 1.1 K i n e m a t i c s o f t h e ( e , 2 e ) r e a c t i o n 6 1.2 S c a t t e r i n g g e o m e t r i e s 16 1.3 G r e e n ' s f u n c t i o n p r o p a g a t o r s 25 C h a p t e r 2 2.1 D e n s i t y maps o f a t o m i c o r b i t a l s 39 2 . 2 O n e - d i m e n s i o n a l p r o j e c t i o n s 47 2 . 3 CH« d e n s i t y maps . 52 2 . 4 CH„ 1 t 2 L C A O - M O s c h e m a t i c d i a g r a m 53 2 . 5 N H 3 d e n s i t y maps. 54 2 . 6 H 2 0 d e n s i t y maps 55 2 . 7 HF d e n s i t y maps 56 2 . 8 B o n d i n g r e g i o n s 59 2 . 9 H 2 d e n s i t y maps 60 2 . 1 0 N 2 d e n s i t y maps 61 2.11 CO d e n s i t y maps . . . 6 2 2 . 1 2 O v e r l a p o f F T w a v e s 69 2 . 1 3 N 2 c o r e o r b i t a l d e n s i t y maps 73 2 . 1 4 B o n d o s c i l l a t i o n i n t h e 0 3 3 a , MO 74 C h a p t e r 3 3.1 S c h e m a t i c d i a g r a m o f t h e s p e c t r o m e t e r 7$ 3 . 2 C y l i n d r i c a l m i r r o r a n a l y s e r 87 3 . 3 C h a n n e l t r o n p u l s e s h a p e s a n d C F D r e s u l t s 91 3.4 Channeltron pulse decoupling c i r c u i t s 96 3.5 Schematic diagram of electronics 98 3.6 Typical time spectrum from spectrometer 103 3.7 The c o l l i s i o n volume 109 3.8 The q-©-# surface at 4l5eV 116 3.9 The q-e-# surface at 1215eV 117 3.10 Mott-scattering cross-section at 415eV .........119 3.-11 Mott-scatter ing cross-section at 12l5eV 120 Chapter 4 4.1 Binding energy spectrum of H2S 128 4.2 MFS structure in H2S 129 4.3 Momentum d i s t r i b u t i o n s of H 2S 130 4.4 Correlation diagrams 149 4.5 Density maps for H2S .- 151 Chapter 5 5.1 Binding energy spectra of COS 164 5.2 Calculated IPs and pole strengths of COS 165 Chapter 6 6.1 C0 2 Binding energy spectrum 178 6.2 C0 2 Deconvolution Binding Energy Spectra 179 6.3 C0 2 MDs and density maps 180 6.4 C0 2 MFS momentum d i s t r i b u t i o n s 183 6.5 Momentum d i s t r i b u t i o n at l9.4eV 184 6.6 Summed A+B momentum d i s t r i b u t i o n s 1?5 xi Chapter 7 7.1 NO momentum d i s t r i b u t i o n s 204 7.2 Theoretical DZ-NSO-CI density maps 205 7.3 Theoretical GTO-UHF density maps 206 7.4 0 2 MDs And Density Maps 214 Chapter 8 8.1 The multi-channel plate 228 8.2 The new spectrometer 229 8.3 The vacuum support system 230 8.4 Signal processing schematic 231 8.5 Signal processing timing diagram 232 8.6 Test c i r c u i t for MCP 233 Appendix C C.1 Molecular geometries' and contour map planes 284 x i i TABLES 2.1 Examples of p r o j e c t i o n s 48 2.2 C o n s t r u c t i v e and d e s t r u c t i v e i n t e r f e r e n c e 70 4.1 H 2S b i n d i n g e n e r g i e s 137 4.2 E n e r g i e s and peak p o s i t i o n s f o r some h y d r i d e s ...152 5.1 I P s and p o l e s t r e n g t h s f o r COS 166 6.1 S t a t e and o r b i t a l l a b e l l i n g 176 6.2 Bonding r e g i o n s i n C 0 2 177 8.1 MCP i n t e r f a c e c o n t r o l l i n e s 234 B.1 R a d i a l forms R { n l } ( r ) and P { n l } ( p ) 281 B. 2 S p h e r i c a l harmonics Y{lm}(e,*) 282 C. 1 V a l e n c e E l e c t r o n i c S t r u c t u r e 283 F O R E W O R D I d e s i r e d t o d o t h i s f o r my own s a t i s f a c t i o n , a n d I h a d l i t t l e h o p e t h a t o t h e r p e o p l e w o u l d b e i n t e r e s t e d i n t h i s w o r k . . . Due t o t h e l i m i t a t i o n s o f t h e c o m p u t e r h a r d w a r e u s e d t o p r i n t t h e t e x t , n o s u b s c r i p t o r s u p e r s c r i p t o r b o l d f a c e l e t t e r s a r e a v a i l a b l e , a n d o n l y a l i m i t e d s e t o f G r e e k c h a r a c t e r s c a n b e u s e d . T h e r e f o r e , a l l s u b s c r i p t a n d some s u p e r s c r i p t l e t t e r s a r e r e p r e s e n t e d w i t h i n c u r l y b r a c k e t s : f o r i n s t a n c e Y{lm } ( e , $ ) d e n o t e s t h e c o m p l e x s p h e r i c a l h a r m o n i c f u n c t i o n o f t h e p o l a r a n g l e s 9 a n d 0 , w i t h a n g u l a r a n d m a g n e t i c q u a n t u m n u m b e r s 1 a n d m, a n d 2<r{u} i s u s e d t o d e n o t e t h e a*(2s) a n t i b o n d i n g o r b i t a l o f N 2 w i t h i t s u n g e r a d e s y m m e t r y ; s c a l a r q u a n t i t i e s a r e i n d i c a t e d b y a s y m b o l ' x ' , v e c t o r s b y a n u n d e r s c o r e d s y m b o l ' r _ ' , a n d m a t r i c e s b y a d o u b l y - u n d e r s c o r e d s y m b o l ' E / . A f u l l g l o s s a r y o f m a t h e m a t i c a l t e r m s a n d n o m e n c l a t u r e u s e d i n t h i s w o r k • i s i n c l u d e d a m ong t h e a p p e n d i c e s t o e l i m i n a t e a n y p o s s i b l e c o n f u s i o n . M a n y d i f f e r e n t m o l e c u l e s a r e d i s c u s s e d i n t h i s w o r k , a n d a s t h e i r e l e c t r o n i c m o l e c u l a r o r b i t a l t e r m s y m b o l s a r e d i f f i c u l t t o r e m e m b e r , a t a b l e o f v a l e n c e e l e c t r o n i c s t r u c t u r e h a s b e e n i n c l u d e d i n A p p e n d i x C a s a r e f e r e n c e . x i v T h r o u g h o u t t h i s t h e s i s a n a t t e m p t i s made t o s t a n d a r d i z e o n a t o m i c u n i t s w h e r e v e r a p p r o p r i a t e . T h i s m e a n s t h a t a l l d i m e n s i o n s w h i c h d e s c r i b e m o l e c u l a r p r o p e r t i e s a s s u m e t h i s s y s t e m o f u n i t s , w i t h t h e n o t a b l e e x c e p t i o n o f o r b i t a l e n e r g i e s , w h e r e e l e c t r o n v o l t s a r e u s u a l l y u s e d . R e f e r e n c e s t o a r t i c l e s i n t h e s c i e n t i f i c l i t e r a t u r e a r e e n c l o s e d i n c u r l y b r a c k e t s . T h e c o m p l e t e c i t a t i o n a n d t h e s i s p a g e c r o s s - r e f e r e n c e s may b e f o u n d i n A p p e n d i x D. F i n a l l y i t s h o u l d be n o t e d t h a t t h e s u b s t a n c e o f C h a p t e r s 4 a n d 5 h a v e a l r e a d y a p p e a r e d i n t h e l i t e r a t u r e { B r i o n ( 1 9 7 8 b ) , C o o k ( 1 9 8 0 ) , C o o k .(1 981 ) } . C h a p t e r 6 w i l l s h o r t l y b e s u b m i t t e d f o r p u b l i c a t i o n i n f o r m s i m i l a r t o t h a t a p p e a r i n g h e r e . M u c h o f t h e c o n t e n t o f C h a p t e r 7 i s d i s c u s s e d i n a p a p e r a c c e p t e d f o r p u b l i c a t i o n i n t h e n e a r f u t u r e , a l s o . E a c h o f t h e s e c h a p t e r s may be r e a d m o r e o r l e s s i n d e p e n d e n t l y o n c e C h a p t e r s 1 a n d 2 a r e d i g e s t e d . C h a p t e r s 3 a n d 8 may b e u n d e r s t o o d a f t e r r e a d i n g C h a p t e r 1. XV ACKNOWLEDGEMENTS ' I d o n ' t know h a l f o f y o u h a l f a s w e l l a s I s h o u l d l i k e ; a n d I l i k e l e s s t h a n h a l f o f y o u h a l f a s w e l l a s y o u d e s e r v e . ' S c i e n t i f i c r e s e a r c h c a n s e l d o m t h e s e d a y s b e d o n e b y o n e p e r s o n a l o n e , a n d , e v e n t h o u g h we g r a d u a t e s t u d e n t s c a l l o u r t h e s e s ' o r i g i n a l r e s e a r c h ' , t h e g r o u n d o u t o f w h i c h o u r i d e a s g e r m i n a t e h a s b e e n t r o d b y many o t h e r p e o p l e . A s t h e r e a d e r c o n s i d e r s t h i s d i s s e r t a t i o n h e s h o u l d b e a r i n m i n d t h a t i t c o u l d n o t h a v e b e e n a c h i e v e d w i t h o u t t h e h e l p o f t h e s e p e o p l e . F o r many h o u r s o f f r u i t f u l d i s c u s s i o n s a b o u t c h e m i s t r y , p l u s many m o r e h o u r s s p e n t i n t h e i r c o m p a n y o u t s i d e t h e l a b , I m u s t e x p r e s s my a p p r e c i a t i o n a n d h i g h r e g a r d f o r Adam a n d I l o n a H i t c h c o c k . F o r t h e i r p a t i e n c e , i d e a s , c o o p e r a t i o n a n d g o o d c h e e r may I t h a n k , t h e s t a f f o f t h e D e p a r t m e n t o f C h e m i s t r y T e c h n i c a l S h o p s , e s p e c i a l l y J o e S a l l o s , H a r o l d Chow, M i k e H a t t o n , B r i a n G r e e n e , J o e S h i m , B r i n P o w e l l , a n d B i l l H e n d e r s o n . E x t r a o r d i n a r y m e n t i o n s h o u l d b e made o f t h e c o n t r i b u t i o n o f D r . E d w a r d P. Gomm t o t h e d e s i g n , c o n s t r u c t i o n , a n d m a i n t e n a n c e o f t h e s p e c t r o m e t e r s u s e d i n x v i t h i s t h e s i s w o r k . ( T h e d e g r e e i s h o n o r a r y , a n d w a s a w a r d e d b y o u r r e s e a r c h g r o u p o n b e h a l f o f a l l t h o s e w i t h whom D r . Gomm h a s w o r k e d i n h i s 21 y e a r s , t o d a t e , a t UBC.) T h e s t a f f a n d s e r v i c e s o f t h e UBC C o m p u t i n g C e n t r e a r e a l s o g r a t e f u l l y a c k n o w l e d g e d . M a n y t h a n k s t o m e m b e r s o f t h e C h e m i s t r y F a c u l t y f o r h e l p f u l d i s c u s s i o n s , e s p e c i a l l y D r . D. C h o n g a n d h i s g r o u p , D r . S n i d e r , a n d D r . J . A . R . C o o p e , a n d f r o m P h y s i c s , D r . M. M c M i l l a n . M u c h o f t h i s w o r k w a s s u p p o r t e d w i t h p o s t - g r a d u a t e f e l l o w s h i p s f r o m UBC a n d N S E R C , a n d w i t h a p r e - d o c t o r a l f e l l o w s h i p f r o m t h e K i l l a m F o u n d a t i o n . F o r t h e r e l i e f f r o m l a b o r a t o r y t e a c h i n g d u t i e s a t t h e v e r y l e a s t , a n d , m o r e i m p o r t a n t , f o r t h e d i s t i n c t i o n o f b e i n g s e l e c t e d f o r t h e s e a w a r d s , I e x p r e s s my a p p r e c i a t i o n t o t h e s e i n s t i t u t i o n s . I s h o u l d l i k e t o t h a n k e s p e c i a l l y D r . R. M i k u l a a n d B e n C l i f f o r d f o r t h e i r h u m o u r a n d a s s i s t a n c e i n my r e s e a r c h a n d t e a c h i n g d u t i e s a t UBC. P e r h a p s a n h o n o u r a b l e m e n t i o n i n t h i s c a t e g o r y s h o u l d g o t o S . F . B . P i c k e t t who i s a f a s c i n a t i n g s p e c i m e n o f w h a t a p p e a r s t o b e a new s p e c i e s o f g r a d u a t e s t u d e n t . I t i s a p l e a s u r e t o a c k n o w l e d g e my g o o d f o r t u n e i n f i n d i n g D r . C h r i s B r i o n f o r a r e s e a r c h s u p e r v i s o r . H i s c o n t r i b u t i o n s i n c a r r y i n g o u t t h e s e s t u d i e s i n b i n a r y ( e , 2 e ) s p e c t r o s c o p y w e r e i n v a l u a b l e , a s w e r e h i s c o m m e n t s f r o m a x v i i c r i t i c a l r e a d i n g o f t h i s t h e s i s . Q u o t a t i o n s t h r o u g h o u t t h i s t h e s i s a r e t a k e n f r o m t h a t i n e s t i m a b l e w o r k 'The L o r d o f t h e R i n g s ' b y P r o f e s s o r J . R . R . T o l k i e n ( A l l e n a n d U n w i n ) , w i t h t h e e x c e p t i o n o f t h a t o f C h a p t e r 8 w h i c h i s t a k e n f r o m 'The H o b b i t ' , b y t h e same a u t h o r . P a r t i c u l a r c o n t r i b u t i o n s t o s p e c i f i c a s p e c t s o f t h i s t h e s i s a r e g i v e n b e l o w . C h a p t e r 4 H y d r o g e n S u l p h i d e " T h e c o n t r i b u t i o n s o f D r . M.F. G u e s t i n m a k i n g a v a i l a b l e h i s o p t i m i s e d - b a s i s f o r H 2 S a n d D r . W.R. R i c h a r d s f o r h e l p w i t h u s i n g t h e ATM0L2 p a c k a g e o n t h e O x f o r d U n i v e r s i t y c o m p u t e r a r e g r a t e f u l l y a c k n o w l e d g e d . D r . W. D o m c k e a n d h i s c o - w o r k e r s w e r e h e l p f u l i n t h e i r c o r r e s p o n d e n c e c o n c e r n i n g m a n y - b o d y e f f e c t s . I s h o u l d a l s o l i k e t o t h a n k D r . G. Z e i s s f o r p r o v i d i n g me w i t h t h e r e s u l t s o f h i s d i f f u s e G a u s s i a n b a s i s c a l c u l a t i o n s . I am i n d e b t e d t o D r . I . S u z u k i f o r t h e i d e a f o r t h e c o r r e l a t i o n d i a g r a m s . D r . A n d r e w H a m n e t t o f O x f o r d U n i v e r s i t y p r o v i d e d e x p e r t a s s i s t a n c e i n t h e c a l c u l a t i o n s a n d i n t h e d i s c u s s i o n o f d - f u n c t i o n c o n t r i b u t i o n s i n H 2 S , a n d p l e a s a n t h o s p i t a l i t y d u r i n g my v i s i t t o O x f o r d , w h i c h i s m u c h a p p r e c i a t e d . x v i i i C h a p t e r 5 C a r b o n y l S u l p h i d e T h i s w o r k was c a r r i e d o u t i n c o l l a b o r a t i o n w i t h D r s . W. v o n N i e s s e n , W. D o m c k e , a n d L . S . C e d e r b a u m , who d i d t h e t h e o r e t i c a l w o r k i n c o m p u t i n g t h e m a n y - b o d y G r e e n ' s f u n c t i o n i o n i z a t i o n p o t e n t i a l s . T h e e x p e r i m e n t a l d e t e r m i n a t i o n o f t h e COS d i p o l e ( e , 2 e ) s p e c t r u m w a s d o n e b y D r . M.G. W h i t e a n d M r . K.T. L e u n g . C h a p t e r 6 C a r b o n D i o x i d e I am g r a t e f u l f o r t h e a s s i s t a n c e o f D r . D.P. C h o n g a n d c o - w o r k e r s i n d o i n g t h e G a u s s i a n 76 c a l c u l a t i o n s a n d f o r p r o v i d i n g me w i t h t h e r e s u l t s o f t h e HAM/3-CI c a l c u l a t i o n . T h e c o r r e s p o n d e n c e o f D r . A. G i a r d i n i - G u i d o n i i n t h e m a t t e r o f t h e momentum d e n s i t y m a p s w a s v e r y h e l p f u l . C h a p t e r 7 N i t r i c O x i d e a n d O x y g e n T h e e x p e r i m e n t a l r e s u l t s i n t h i s s e c t i o n w e r e r e c o r d e d a t F l i n d e r s U n i v e r s i t y b y D r . C . E . B r i o n , P r o f . E . W e i g o l d , a n d c o w o r k e r s . C a l c u l a t i o n s o f NO a n d 0 2 w a v e f u n c t i o n s w e r e g e n e r o u s l y p r o v i d e d b y D r . A . B . K u n z a n d M r . K. B e d f o r d . C h a p t e r 8 T h e New S p e c t r o m e t e r I w i s h t o a c k n o w l e d g e h e l p f u l d i s c u s s i o n s w i t h D r . P r o f . M . J . v a n d e r W i e l a n d D r . S. D a v i e l . x i x T o My P a r e n t s Sam w e n t r e d t o t h e e a r s a n d m u t t e r e d s o m e t h i n g i n a u d i b l e , a s h e c l u t c h e d t h e b o x a n d b o w e d a s w e l l a s h e c o u l d . 1 C H A P T E R 1 I N T R O D U C T I O N AND THEORY ' A h ! ' s a i d G a n d a l f . ' T h a t i s a v e r y l o n g s t o r y . . . I w i l l r i s k a b r i e f t a l e . . . ' T h i s d i s s e r t a t i o n i s a n e x p l o r a t i o n o f t h e e l e c t r o n i c s t r u c t u r e a n d momentum s p a c e c h e m i s t r y o f some s m a l l g a s e o u s m o l e c u l e s , a s r e v e a l e d b y t h e i r b i n d i n g e n e r g y s p e c t r a a n d momentum d i s t r i b u t i o n s o b t a i n e d f r o m t h e o r e t i c a l m o d e l s a n d b i n a r y ( e , 2 e ) s p e c t r o s c o p i c m e a s u r e m e n t s . T h e t e r m ' e l e c t r o n i c s t r u c t u r e ' r e f e r s t o t h e e n e r g i e s a n d w a v e f u n c t i o n s o f t h e e l e c t r o n s i n t h e m o l e c u l e , a n d i s g e n e r a l l y u n d e r s t o o d b y a l l c h e m i s t s . 'Momentum s p a c e c h e m i s t r y ' i s o n t h e o t h e r h a n d a v e r y r a r e l y e n c o u n t e r e d t e r m a n d n e e d s a m p l i f i c a t i o n . W h i l e a p a r t i c l e may b e c o m p l e t e l y d e s c r i b e d ( i n d e p e n d e n t o f t i m e ) b y i t s w a v e f u n c t i o n i n t e r m s o f t h e t h r e e o r t h o g o n a l s p a t i a l c o o r d i n a t e s , i t i s a l s o t r u e t h a t t h i s w a v e f u n c t i o n may b e r e c a s t i n t h e t h r e e o r t h o g o n a l momentum c o o r d i n a t e s : b o t h r e p r e s e n t a t i o n s a r e e q u a l l y v a l i d , a n d n e i t h e r c o u l d b e s a i d t o t a k e p r e c e d e n c e o v e r t h e o t h e r , b u t f o r p r a c t i c a l 1/Introduction and Theory 2 purposes we have come to use the former almost exclusively. The use of the term 'momentum space chemistry' therefore denotes the electronic structure and bonding behaviour of atoms and molecules as viewed in the momentum-space representation. The investigation of this f i e l d i s not only a novel and interesting exercise in i t s e l f , but is also v i t a l to the understanding of momentum d i s t r i b u t i o n s , and constitutes a large part (Chapter 2) of this thesis. Whenever one can est a b l i s h the energies of a l l the free electrons in an ionizing c o l l i s i o n with some target, one has the a b i l i t y to measure the 'binding energy' of the ionized p a r t i c l e , namely: that minimum energy necessary to remove the p a r t i c l e from the target. This is not quite the same thing as the energy that the ionized p a r t i c l e had when i t was bound in the molecule, but i s closely related to i t , and is an important piece of information. Binding energy spectra, both for their i n t r i n s i c value and in comparison with theory, also form a large part of this thesis (Chapters 4-6). The theoretical c a l c u l a t i o n of binding energy spectra has made great progress in the l a s t decade, esp e c i a l l y in the many-body Green's "function method. Much of the experimental work presented here complements these theoretical predictions, sometimes revealing their inadequacy and sometimes showing impressive agreement. The term 'momentum d i s t r i b u t i o n ' , l i k e 'momentum space chemistry', i s seen uncommonly in the mainstream of 1 / I n t r o d u c t i o n a n d T h e o r y 3 c h e m i s t r y , u s u a l l y e n j o y i n g u s e o n l y i n C o m p t o n s c a t t e r i n g a n d s o l i d s t a t e c h e m i s t r y c i r c l e s . I t m e a n s a l m o s t e x a c t l y w h a t i t s a y s : t h e d i s t r i b u t i o n o f momentum i r r e s p e c t i v e o f d i r e c t i o n , o r i n o t h e r w o r d s , a p r o b a b i l i t y d i s t r i b u t i o n o f t h e m a g n i t u d e o f a p a r t i c l e ' s momentum. T h e a b i l i t y t o m e a s u r e o r b i t a l - s e l e c t i v e momentum d i s t r i b u t i o n s i s t h e f e a t u r e w h i c h d i s t i n g u i s h e s a n d e l e v a t e s b i n a r y ( e , 2 e ) s p e c t r o s c o p y i n r e l a t i o n t o o t h e r t e c h n i q u e s , w h i c h a l l o w o n l y s u c h i n d i r e c t p r o b e s o f t h e e l e c t r o n i c s t r u c t u r e a s b i n d i n g e n e r g i e s , C o m p t o n p r o f i l e s , b r a n c h i n g r a t i o s , F r a n c k - C o n d o n e n v e l o p e s , s h a p e r e s o n a n c e s , a n d s o o n : t h e momentum d i s t r i b u t i o n i s t h e c l o s e s t t h a t c h e m i s t r y h a s y e t come t o m e a s u r i n g t h e w a v e f u n c t i o n s o f t h e e l e c t r o n s i n t h e i r m o l e c u l a r o r b i t a l s . ( O n e s h o u l d o f c o u r s e a l w a y s b e a r i n m i n d t h a t t h e m o l e c u l a r o r b i t a l i s n o t a p h y s i c a l p h e n o m e n o n , b u t i s o n l y a c o n v e n i e n t c o n c e p t a r i s i n g o u t o f t h e o n e - e l e c t r o n s e l f - c o n s i s t e n t f i e l d a p p r o x i m a t i o n o f t h e N - e l e c t r o n w a v e f u n c t i o n , y e t i s a d e q u a t e f o r m u c h o f t h e d i s c u s s i o n i n t h i s w o r k . A l s o , i t i s i n f a c t t h e p r o b a b i l i t y d e n s i t y , o r t h e s q u a r e d m o d u l u s o f t h e w a v e f u n c t i o n w h i c h i s a m e a s u r a b l e q u a n t i t y , n o t t h e w a v e f u n c t i o n i t s e l f . ) T h e e x p e r i m e n t a l m e a s u r e m e n t , t h e o r e t i c a l c a l c u l a t i o n , a n d d i s c u s s i o n o f momentum d i s t r i b u t i o n s f o r m t h e m o s t i m p o r t a n t p a r t o f t h i s t h e s i s ( C h a p t e r s . 2 , 4 - 7 ) . T h o u g h I h a v e b e e n s p e a k i n g o f b o t h t h e o r e t i c a l a n d e x p e r i m e n t a l i n v e s t i g a t i o n s i n t o b i n a r y ( e , 2 e ) s p e c t r o s c o p y , 1 / I n t r o d u c t i o n a n d T h e o r y 4 t h i s t h e s i s m u s t f i n a l l y b e c o n s i d e r e d t o be a n e x p e r i m e n t a l w o r k . I s h a l l s p e n d c o n s i d e r a b l e t i m e d i s c u s s i n g t h e i n s t r u m e n t u s e d t o r e c o r d t h e d a t a ( C h a p t e r 3) a n d t h e d e s i g n , c o n s t r u c t i o n , a n d p r e l i m i n a r y t e s t i n g o f a new s p e c t r o m e t e r i n c o r p o r a t i n g s i g n i f i c a n t t e c h n o l o g i c a l a d v a n c e s ( C h a p t e r 8 ) . 1.1 T h e ( e , 2 e ) R e a c t i o n I n t h e f o l l o w i n g d i s c u s s i o n s i t s h o u l d b e n o t e d t h a t n o a t t e m p t h a s b e e n made t o b e r i g o r o u s a n d c o m p l e t e i n t h e d e r i v a t i o n s a n d e q u a t i o n s . T h i s i s t h e r e a l m o f t h e s c a t t e r i n g t h e o r i s t , a n d a s a n e x p e r i m e n t a l c h e m i s t I a c c e p t t h e i r f i n d i n g s a n d s h a l l a t t e m p t t o d e m o n s t r a t e t h e p r i n c i p l e s a n d a p p r o x i m a t i o n s o f t h e t h e o r y w i t h o u t d e l v i n g i n t o t h e c o m p l e x m a t h e m a t i c a l f r a m e w o r k . T h e t h e o r y h a s b e e n e x t e n s i v e l y d i s c u s s e d i n t h e l i t e r a t u r e { N e u d a c h i n ( 1 9 6 9 ) , H o o d ( 1 9 7 3 ) , M c C a r t h y ( 1 9 7 6 a ) , M c C a r t h y ( 1 9 7 6 b ) , W e i g o l d ( 1 9 7 8 ) } . ( e , 2 e ) e l e c t r o n i m p a c t s p e c t r o s c o p y i s a s c a t t e r i n g e x p e r i m e n t i n w h i c h a n i n c i d e n t e l e c t r o n , w i t h e n e r g y E 0 a n d momentum v e c t o r k 0 , c o l l i d e s w i t h a t a r g e t s y s t e m M w h i c h i n i t i a l l y h a s N - e l e c t r o n s a n d t o t a l e n e r g y E ( M ) a n d i s a s s u m e d t o b e a t r e s t i n t h e l a b o r a t o r y f r a m e . T h e i n c i d e n t 1 / I n t r o d u c t i o n a n d T h e o r y 5 e l e c t r o n i s made s u f f i c i e n t l y e n e r g e t i c t o i o n i z e a b o u n d e l e c t r o n f r o m M g i v i n g r i s e t o t w o o u t g o i n g e l e c t r o n s o f e n e r g i e s E , a n d E 2 a n d m o m e n t a k^ a n d k2 r e s p e c t i v e l y , l e a v i n g b e h i n d a n ( N - 1 ) - e l e c t r o n s y s t e m M* + w i t h t o t a l e n e r g y E ( M * + ) . T h e t a r g e t i s u s u a l l y i n i t i a l l y i n i t s e l e c t r o n i c g r o u n d s t a t e , b u t a f t e r t h e c o l l i s i o n t h e ( N - 1 ) - e l e c t r o n s y s t e m n e e d n o t n e c e s s a r i l y b e i n i t s i o n g r o u n d s t a t e , h e n c e t h e a s t e r i s k i n t h e n o t a t i o n M* +. T h e r e a c t i o n c a n b e w r i t t e n : ( 1 . 1 ) e 0 ( E o , k 0 ) + M ( N , E ( M ) ) — > M * + ( N - 1 , E ( M * + ) ) + e , ( E 1 ( k , ) + e 2 ( E 2 , k 2 ) T h e ( e , 2 e ) s p e c t r o m e t e r i s a r r a n g e d s o t h a t t h e k i n e m a t i c s o f a l l t h e f r e e e l e c t r o n s - e n t e r i n g a n d l e a v i n g t h e s c a t t e r i n g r e g i o n ( i . e . t h e e n e r g i e s a n d t r a j e c t o r i e s E 0 , E, , E 2 , k 0 > i i i a n d k 2 ) a r e c o m p l e t e l y d e t e r m i n e d ( t o w i t h i n some e x p e r i m e n t a l u n c e r t a i n t y ) . A t t h i s p o i n t i t c a n b e s e e n t h a t t h e ( e , 2 e ) t e c h n i q u e c a n m e a s u r e t h e b i n d i n g e n e r g i e s o f t h e e l e c t r o n s o f t h e t a r g e t . C o n s e r v a t i o n o f e n e r g y r e q u i r e s t h a t : ( 1 . 2 ) E 0 + E ( M ) = E ( M * + ) + E , + E 2 s o o n e c a n d e f i n e € a s : ( 1 . 3 ) e = E ( M * + ) - E ( M ) = E 0 " E , - E 2 w h e r e e i s u n d e r s t o o d a s t h e b i n d i n g e n e r g y o f t h e e l e c t r o n 1 / I n t r o d u c t i o n and Theory e 0 , ko F i g u r e 1.1 K i n e m a t i c s of the (e,2e) r e a c t i o n 1 / I n t r o d u c t i o n a n d T h e o r y 7 t h a t i s i o n i z e d f r o m t h e t a r g e t , o r m o r e p r e c i s e l y , t h e d i f f e r e n c e i n t h e e n e r g i e s o f t h e N - e l e c t r o n t a r g e t b e f o r e c o l l i s i o n a n d t h e ( N - 1 ) - e l e c t r o n t a r g e t a f t e r t h e c o l l i s i o n . T h e r e f o r e t h e m e a s u r e m e n t o f t h e ( e , 2 e ) s i g n a l a s a f u n c t i o n o f E 0 s h o w s t h e s p e c t r u m o f f i n a l ( N - 1 ) - e l e c t r o n s t a t e e n e r g i e s o f t h e t a r g e t r e l a t i v e t o t h e N - e l e c t r o n g r o u n d s t a t e . T h i s h i s t o r i c a l l y h a s b e e n c a l l e d a b i n d i n g e n e r g y s p e c t r u m s i n c e i n e a r l i e r d a y s i t was f o u n d t h a t , i n s i m p l e s y s t e m s a n d o v e r a l i m i t e d e n e r g y r a n g e , i o n i z a t i o n o f a n e l e c t r o n f r o m a n o r b i t a l p r o d u c e d o n e f i n a l i o n s t a t e , a n d s o , o n e p e a k i n t h e e n e r g y s p e c t r u m . H o w e v e r , s i n c e we now know t h a t i n many i n s t a n c e s m o r e t h a n o n e f i n a l i o n s t a t e c a n r e s u l t f r o m t h e i o n i z a t i o n o f o n e e l e c t r o n , p e r h a p s t h e t e r m ' b i n d i n g e n e r g y s p e c t r u m ' s h o u l d b e r e t i r e d , a n d be r e p l a c e d w i t h t h e m o r e a c c u r a t e t e r m ' f i n a l s t a t e e n e r g y s p e c t r u m ' . T h e t e r m ' s e p a r a t i o n e n e r g y ' h a s a l s o b e e n u s e d i n t h i s c o n t e x t { M c C a r t h y ( 1 9 7 6 a ) } . One o f t h e a s s e t s o f t h e ( e , 2 e ) t e c h n i q u e i s t h a t b i n d i n g e n e r g y s p e c t r a a r e n o t l i m i t e d i n t h e i r r a n g e , u n l i k e H e - I P E S s p e c t r a , a n d a r e a b l e t o e x p l o r e t h e r e g i o n b e y o n d 2 1 . 2 e V , o p e n i n g u p m u c h new s t r u c t u r e . 1 / I n t r o d u c t i o n a n d T h e o r y 8 1.2 T h e B i n a r y ( e , 2 e ) P r o c e s s One now g o e s o n t o p l a c e some r e s t r i c t i o n s o n t h e k i n e m a t i c s o f t h e g e n e r a l ( e , 2 e ) p r o c e s s . T h e e x p e r i m e n t i s r e q u i r e d t o d e t e c t o n l y t h o s e c o l l i s i o n s w h e r e : ( 1 ) T h e i n t e r a c t i o n b e t w e e n t h e i n c i d e n t e l e c t r o n a n d t h e e l e c t r o n t o b e i o n i z e d i s t h e s t r o n g e s t i n t e r a c t i o n i n t h e c o l l i s i o n e v e n t ; ( 2 ) T h e i n c o m i n g a n d o u t g o i n g e l e c t r o n s d o n o t i n t e r a c t s i g n i f i c a n t l y w i t h t h e o t h e r p a r t i c l e s i n t h e t a r g e t s y s t e m . T h i s i s k n o w n a s t h e ' b i n a r y e n c o u n t e r c o n d i t i o n ' , a n d g i v e s r i s e t o t h e name ' b i n a r y ( e , 2 e ) ' f o r t h i s e x p e r i m e n t . T h e f i r s t c o n d i t i o n i s m e t i n t h e ' s y m m e t r i c ' s c a t t e r i n g g e o m e t r y w h e r e : ( 1 . 4 ) E , = E 2 = 1 / 2 ( E 0 - € ) ; ©,=62 = 45° T h i s g e o m e t r y m e a n s t h a t t h e i n c i d e n t e l e c t r o n a n d t h e b o u n d e l e c t r o n w i l l p a s s ( t h i n k i n g i n c l a s s i c a l m e c h a n i c s f o r t h e m o m e n t ) c l o s e t o e a c h o t h e r , a n d t h e r e f o r e t h a t t h e i n t e r a c t i o n b e t w e e n t h e m i s s t r o n g . T h e s e c o n d c r i t e r i o n c a n be o b t a i n e d b y e n s u r i n g t h a t t h e i n c i d e n t a n d e x i t e l e c t r o n s a r e o f s u f f i c i e n t l y h i g h e n e r g y a s t o be u n d i s t u r b e d t o a n y s i g n i f i c a n t e x t e n t b y t h e o t h e r p a r t i c l e s i n t h e s y s t e m f r o m t h e i r e n e r g i e s a n d t r a j e c t o r i e s s p e c i f i e d i n t h e f i r s t c r i t e r i o n . 1 / I n t r o d u c t i o n a n d T h e o r y 9 1.2.1 T h e b i n a r y ( e , 2 e ) f o r m f a c t o r P u t t i n g t h i s i n t o a m a t h e m a t i c a l f o r m u l a t i o n r e q u i r e s some d e f i n i t i o n s : t a k e t h e i n i t i a l s t a t e w a v e f u n c t i o n o f t h e e n t i r e s c a t t e r i n g s y s t e m * t o b e t h e p r o d u c t o f t h e N - e l e c t r o n t a r g e t g r o u n d s t a t e * 0 { N } a n d t h e f r e e i n c i d e n t e l e c t r o n s t a t e w a v e f u n c t i o n X 0 ( k o , r 0 ) : ( 1 . 5 ) * = * 0 ( N } X 0 ( k 0 , r 0 ) S i m i l a r l y t h e t o t a l f i n a l s t a t e w a v e f u n c t i o n 4> i s t a k e n a s t h e p r o d u c t : ( 1 . 6 ) * = * { N - 1 } X . U ^ r , ) X 2 ( k 2 , r 2 ) w h e r e * { N - 1 } i s t h e ( N - 1 ) - e l e c t r o n f i n a l i o n s t a t e , a n d X, a n d X 2 a r e t h e o u t g o i n g f r e e e l e c t r o n w a v e f u n c t i o n s . T h e a m p l i t u d e M f o r a s y s t e m t o g o f r o m i t s i n i t i a l s t a t e * t o i t s f i n a l s t a t e * i n t h e i n f l u e n c e o f some i n t e r a c t i o n p o t e n t i a l V i s ( u s i n g D i r a c b r a - k e t n o t a t i o n ) : ( 1 . 7 ) M = < * | V | • > S i n c e t h e o n l y i n t e r a c t i o n a l l o w e d i s t h a t b e t w e e n t h e i n c i d e n t a n d b o u n d e l e c t r o n s , t h e ' i n t e r a c t i o n p o t e n t i a l r e d u c e s t o . t h e C o u l o m b p o t e n t i a l b e t w e e n t h e s e t w o e l e c t r o n s . T h i s i s w r i t t e n a s T ( r , , r _ 2 ; r , r 0 ) : t h e a m p l i t u d e f o r t w o e l e c t r o n s s t a r t i n g a t £ a n d r 0 , t o s c a t t e r v i a t h e r " 1 p o t e n t i a l t o p o i n t s r , a n d r 2 . Now t h e a m p l i t u d e f o r t h e b i n a r y ( e , 2 e ) p r o c e s s i s g i v e n b y : . — 1 / I n t r o d u c t i o n a n d T h e o r y 10 ( 1 . 8 ) M = < X 1 ( k , , r 1 ) X 2 ( k 2 , r 2 ) • { i , N - l } x | T ( r , , r 2 ; r , r 0 ) | * 0 { N } X 0 ( k 0 , r 0 ) > N e x t , t h e f o l l o w i n g a p p r o x i m a t i o n s w i l l b e made o n t h e a b o v e e q u a t i o n : ( 1 ) T h e N - e l e c t r o n t a r g e t w a v e f u n c t i o n w i l l " b e e x p r e s s e d a s a n a n t i s y m m e t r i z e d p r o d u c t o f N o n e - e l e c t r o n m o l e c u l a r o r b i t a l w a v e f u n c t i o n s , a n d t h e ( N - 1 ) - e l e c t r o n i o n s t a t e w a v e f u n c t i o n b y a s i m i l a r p r o d u c t , l e a v i n g o u t t h e i t h MO f r o m w h i c h a n e l e c t r o n w a s i o n i z e d : ( 1 . 9 ) *0(N) = A n * { j } ( r ) j * { i , N - 1 ) = A n * { j } . ( r ) w h e r e A i s t h e a n t i s y m m e t r i z a t i o n o p e r a t o r a n d * { j } ( r _ ) i s a o n e - e l e c t r o n m o l e c u l a r - o r b i t a l w a v e f u n c t i o n . I f n e c e s s a r y , c o n f i g u r a t i o n i n t e r a c t i o n may b e i n c o r p o r a t e d i n b o t h t h e i n i t i a l a n d f i n a l s t a t e s . ( 2 ) T h e i m p u l s e a p p r o x i m a t i o n i s made i n t h e t r e a t m e n t o f t h e c o l l i s i o n b e t w e e n t h e i n c i d e n t e l e c t r o n a n d t h e b o u n d e l e c t r o n . I n o t h e r w o r d s , i t i s a s s u m e d t h a t t h e c o l l i s i o n e v e n t t a k e s v e r y n e a r l y z e r o t i m e a n d t h e r e f o r e i s s e p a r a b l e f r o m t h e m o t i o n o f o t h e r p a r t i c l e s i n t h e t a r g e t . ( 3 ) T h e f r e e e l e c t r o n w a v e f u n c t i o n s i n a n d o u t o f t h e 1 / I n t r o d u c t i o n a n d T h e o r y s c a t t e r i n g r e g i o n a r e t r e a t e d a s p l a n e w a v e s . T h i s a s s u m e s t h a t t h e f r e e e l e c t r o n s d o n o t i n t e r a c t s i g n i f i c a n t l y w i t h t h e e l e c t r o n s o f t h e t a r g e t o t h e r t h a n w i t h t h e e l e c t r o n b e i n g i o n i z e d , a n d t h a t t h e y a r e n e g l i g i b l y d i s t o r t e d b y t h e p o t e n t i a l o f t h e n u c l e i . T h e w a v e f u n c t i o n o f a f r e e e l e c t r o n w i t h momentum k i s w r i t t e n a s : i k_. r X ( k , r ) = e A p p r o x i m a t i o n s 2 a n d 3 a b o v e c o m p r i s e w h a t i s k n o w n a s t h e p l a n e w a v e i m p u l s e a p p r o x i m a t i o n ( P W I A ) . U n d e r t h e s e a p p r o x i m a t i o n s t h e a m p l i t u d e i s s e p a r a b l e : ( 1 . 1 0 ) |-M | 2 = |T| 2 |C| 2 |F| 2 w h e r e | C | 2 = | < # { i , N - 1 ) | * { i , N - 1 ) > | 2 T h e i n f o r m a t i o n o f i n t e r e s t t o c h e m i s t s i s c o n t a i n e d i n t h e l a s t f a c t o r , w h i c h i s w r i t t e n i n f u l l a s : ( 1 . 1 1 ) | F | 2 = | j d r e 1 3 ' £ * { i } ( r ) | 2 w h e r e C[ = k, + k 2 ~ Jio T h e s i g n i f i c a n c e o f g i s u n d e r s t o o d b y t h e f o l l o w i n g a r g u m e n t : i f t h e N - e l e c t r o n t a r g e t s y s t e m i s a s s u m e d t o be a t r e s t a n d t h e i t h e l e c t r o n h a s a momentum p_ a t t h e i n s t a n t 1 / I n t r o d u c t i o n a n d T h e o r y o f c o l l i s i o n , t h e n t h e t o t a l momentum o f t h e ' c o r e ' ( t h e r e m a i n i n g p a r t i c l e s o f t h e t a r g e t ) m u s t b e -p_. I f t h e c o r e i s a s s u m e d t o be s o m a s s i v e t h a t i t s m o t i o n i s u n a f f e c t e d b y t h e ( e , 2 e ) c o l l i s i o n , t h e n c o n s e r v a t i o n o f momentum r e q u i r e s t h a t : ( 1 . 1 2 ) k 0 = k i + k 2 _ £ o r 2 = k, + k 2 - k 0 = g I n o t h e r w o r d s g i n e q u a t i o n 1.11 i s i d e n t i c a l t o t h e momentum o f t h e b o u n d e l e c t r o n a t t h e i n s t a n t o f i o n i z a t i o n . T h i s m e a n s t h a t e q u a t i o n 1.11 i s t h e F o u r i e r T r a n s f o r m o f t h e i t h p o s i t i o n - s p a c e m o l e c u l a r o r b i t a l i n t o i t s momentum s p a c e r e p r e s e n t a t i o n . ( 1 . 1 3 ) * { i } ( p _ ) = ( 2 t r ) a n d r - 1 2 « £ d r e + { i } ( r ) ( 1 . 1 4 ) | M [ 2 = | T | 2 | C | 2 | * { i } ( 2 ) | 2 G e n e r a l l y 2 a n c * 2 w i H be u s e d i n t e r c h a n g e a b l y , a l t h o u g h g c a r r i e s t h e c o n n o t a t i o n o f b e i n g e x p e r i m e n t a l l y d e t e r m i n e d , a n d 2 °f b e i n g t h e i n d e p e n d e n t v a r i a b l e o f a t h e o r e t i c a l w a v e f u n c t i o n . T h e g e n e r a l name f o r T i s t h e ' h a l f - o f f - s h e l l M o t t s c a t t e r i n g f a c t o r ' a n d i s s o m e t i m e s w r i t t e n < r { M o t t } . M o t t s c a t t e r i n g i s n a m e d a f t e r i t s o r i g i n a t o r N . F . M o t t , a n d d e n o t e s t h e s c a t t e r i n g b e h a v i o u r o f a n y t w o i d e n t i c a l f r e e p a r t i c l e s i n t h e C o u l o m b p o t e n t i a l i n c l u d i n g e x c h a n g e I / I n t r o d u c t i o n a n d T h e o r y 13 e f f e c t s . ' H a l f - o f f - s h e l l ' , o r m o r e f u l l y , ' h a l f - o f f t h e e n e r g y s h e l l ' , r e f e r s t o t h e f a c t t h a t o n e e l e c t r o n i s n o t a c t u a l l y f r e e , b u t h a s a b i n d i n g e n e r g y e, a l t h o u g h i t i s t r e a t e d a s i f i t w e r e a f r e e p a r t i c l e . T i s a n e x a c t l y k n o w n f u n c t i o n a n d i s g i v e n b y t h i s e q u a t i o n : ( 1 . 1 5 ) T = 1 2rrr, 4*" 2irn e -1 K'-K|-« + U'+KI-* " |JC'-K|' 2 |K'+K| " 2 c o s ( n l n { |K'+K| 2 | K ' - K | - 2 } ) w h e r e r, = ( 2 K ' ) - 1 ; K = 1 / 2 ( k 0 + g ) ; jc' = 1 / 2 ( k 1 - k 2 ) . P l o t s o f | T | 2 a s a f u n c t i o n o f 9 a n d <t> a t 4 0 0 e V a n d 1 2 0 0 e V a r e g i v e n i n f i g u r e s 3.10 a n d 3 . 1 1 . | C | 2 i s t h e r e l a x a t i o n f a c t o r d e s c r i b i n g t h e o v e r l a p o f t h e r e m a i n i n g ' n o n - i n t e r a c t i n g e l e c t r o n w a v e f u n c t i o n s i n t h e t a r g e t , a n d i n t h e f r o z e n o r b i t a l a p p r o x i m a t i o n i s a s s u m e d t o b e u n i t y . S i n c e t h e t a r g e t i s u s u a l l y a g a s e o u s s p e c i e s a n d a t t h e r m a l e n e r g i e s i t s o r i e n t a t i o n i s c o m p l e t e l y r a n d o m o n e m u s t r e p l a c e | * { i } ( p _ ) | 2 b y i t s s p h e r i c a l a v e r a g e : ( 1 . 1 6 ) | * { i } ( p ) | 2 = ( 4 r r ) - 1 J dn | * { i } (p_) | 2 = F { i } ( q ) w h e r e p = | p j , a n d dn = sineded*. By s i m p l e t r i g o n o m e t r y i t i s p o s s i b l e t o d e f i n e q a s a f u n c t i o n o f s c a t t e r i n g a n g l e s a n d e n e r g i e s , w h i c h a r e t h e 1 / I n t r o d u c t i o n a n d T h e o r y q u a n t i t i e s d e t e r m i n e d b y t h e e l e c t r o n s o u r c e a n d d e t e c t o r s i n t h e s p e c t r o m e t e r . F o r t h e s y m m e t r i c s c a t t e r i n g g e o m e t r y : ( 1 . 1 7 ) q = { ( 2 k c o s e - k 0 ) 2 + (2ksin© s i n 1 / 2 0 ) 2 } w h e r e k = k v = k 2 ; © = © 1 = © 2 . T h e f u l l e q u a t i o n f o r t h e b i n a r y ( e , 2 e ) d i f f e r e n t i a l c r o s s - s e c t i o n f o r s c a t t e r i n g o f t w o e l e c t r o n s o f e n e r g y E, = 1 / 2 ( E 0 - € ) i n t o s o l i d a n g l e s dn,dn2 i s : ( 1 . 1 8 ) A 5 * = ( 2 r r ) a N { o c c } S { i } k, k 2 | T | 2 F { i } ( q ) dE 1dn 1dn 2 k 0 T h e f a c t o r k , k 2 / k 0 i s a k i n e m a t i c n o r m a l i z a t i o n f a c t o r ; N { o c c } g i v e s t h e o c c u p a n c y o f t h e o r b i t a l b e i n g i o n i z e d ; S { i } i n c l u d e s a l l e f f e c t s w h i c h a l t e r t h e r e l a t i v e i n t e n s i t y o f t h e m e a s u r e d s t r u c t u r e , b u t d o n o t c h a n g e i t s s h a p e : t h e s e i n c l u d e r e l a x a t i o n e f f e c t s , p o l e s t r e n g t h s ( d i s c u s s e d i n s e c t i o n 1 . 4 . 1 ) , a b s o r p t i o n e f f e c t s , a n d s o o n , b u t n o t e x p e r i m e n t a l f a c t o r s l i k e g a s d e n s i t y , r e s o l u t i o n , e t c . 1.2.2 T h e momentum d i s t r i b u t i o n I f t h e s p e c t r o m e t e r i s a r r a n g e d s o t h a t b o t h a n a l y s e r s a r e f i x e d a t ©=45° a n d o n e a n a l y s e r i s a l l o w e d t o r e v o l v e i n t h e <f> d i m e n s i o n ( s e e f i g u r e 1 . 2 ) , t h e n t h e f u n c t i o n a l f o r m o f < r { M o t t } i s m u c h s i m p l i f i e d a n d i s n e a r l y c o n s t a n t a s a f u n c t i o n o f <t> a n g l e , a n d a l a r g e r a n g e o f p may b e a c c e s s e d : 1/Introduction and Theory 15 (1.19) | T | 2 = 1 | k 0 - J i | - a _ 2 j r n _ 2rr * 2 j r r i e -1 2irT I = tr{|/2ksin© / c o s 0 + 1 }" 1 |k0-k.|-a = {k2 + k 0 2-2kk 0cos©}- 2 In t h i s arrangement the binary (e,2e) cross-section, in the PWIA> i s proportional to the spherical average of the squared momentum-space wavefunction of the molecular o r b i t a l selected by the energy conservation rules. This is c a l l e d the symmetric non-coplanar scattering geometry, and is the technique used to obtain the experimental results presented in t h i s d i s s e r t a t i o n . Another usual scattering geometry i s the symmetric coplanar arrangement where the azimuthal angle is fixed at </>=0°, and e , the polar angle of the two analysers, is the scanned parameter (figure 1.2). However here the cross-section i s a strong function of both |T|2 and the form factor F{i}(q). This geometry has more application in investigating the physics of the c o l l i s i o n process, but is less useful for probing the momentum d i s t r i b u t i o n (and thus exploring, momentum space chemistry) as, in order to get the form factor out, the behaviour of the incident and scattered waves must be accurately approximated.- The measurement of the binary (e,2e) cross-section in the symmetric non-coplanar geometry (or the c a l c u l a t i o n of thi s function) plotted against q defines the term 'momentum d i s t r i b u t i o n ' (MD), used in thi s d i s s e r t a t i o n . This function 1 / I n t r o d u c t i o n and Theory 16 F i g u r e 1.2 B i n a r y (e,2e) s c a t t e r i n g g e o m e t r i e s : (a) Symmetric c o p l a n a r (b) Symmetric n o n - c o p l a n a r 1 / I n t r o d u c t i o n a n d T h e o r y i s s o c l o s e l y r e l a t e d t o t h e p o s i t i o n - s p a c e m o l e c u l a r o r b i t a l w a v e f u n c t i o n t h a t i t i s v e r y u s e f u l i n s t u d y i n g t h e e l e c t r o n i c s t r u c t u r e o f a t o m s a n d m o l e c u l e s , a n d a l s o i n j u d g i n g t h e q u a l i t y o f t h e o r e t i c a l e l e c t r o n i c w a v e f u n c t i o n s . T h e momentum d i s t r i b u t i o n , s i n c e i n p r i n c i p l e i t i s u n i q u e t o t h e i o n i z i n g m o l e c u l a r o r b i t a l , i s a l s o a d e f i n i t i v e t o o l i n t h e a s s i g n m e n t o f p e a k s i n b i n d i n g e n e r g y s p e c t r a { H o o d ( 1 9 7 6 b ) } . W h e r e i o n i z a t i o n f r o m a g i v e n g r o u n d s t a t e m o l e c u l a r o r b i t a l p r o d u c e s m o r e t h a n o n e f i n a l i o n s t a t e , t h e b i n a r y ( e , 2 e ) t e c h n i q u e i s i n p r i n c i p l e c a p a b l e o f i d e n t i f y i n g t h e o r i g i n o f e a c h f i n a l s t a t e p e a k ( a s e a c h p e a k w i l l show a momentum d i s t r i b u t i o n c h a r a c t e r i s t i c o f a p a r t i c u l a r g r o u n d s t a t e o r b i t a l ) { H a m n e t t ( 1 9 7 7 ) , C o o k ( 1 9 8 0 ) , M c C a r t h y ( 1 9 7 6 a ) } . S i n c e t h e momentum d i s t r i b u t i o n i s a u n i q u e f i n g e r p r i n t o f t h e o r b i t a l , t h e b i n a r y ( e , 2 e ) m e t h o d p r o v i d e s m o r e d e t a i l e d i n f o r m a t i o n o n t h e i d e n t i t y o f p e a k s t h a n d o e s p h o t o e l e c t r o n s p e c t r o s c o p y . I n t h i s r e g a r d t h e momentum d i s t r i b u t i o n i n f o r m a t i o n i s i n m o s t c a s e s m o r e s p e c i f i c t h a n t h a t p r o v i d e d b y t h e a r g u m e n t s ' u s e d i n P E S b a s e d o n v a r i a t i o n s i n t h e r e l a t i v e i n t e n s i t y o f « a n d rr o r b i t a l s w i t h p h o t o n e n e r g y { S i e g b a h n ( 1 9 6 9 ) , R a b a l a i s ( 1 9 7 7 ) } . I t m u s t f i r s t b e made c l e a r , h o w e v e r , u n d e r w h a t c o n d i t i o n s t h e PWIA a p p r o x i m a t i o n s a r e v a l i d a n d w h e r e t h e y b r e a k d o w n . 1 / I n t r o d u c t i o n a n d T h e o r y 18 1.2.3 T h e v a l i d i t y o f t h e PWIA I t h a s b e e n e s t a b l i s h e d b y s e v e r a l s t u d i e s { M c C a r t h y ( 1 9 7 6 a ) , M c C a r t h y ( 1 9 7 6 b ) , C a m i l l o n i ( 1 9 7 7 ) , C a m i l l o n i ( 1 9 7 8 ) } t h a t g e n e r a l l y f o r v a l e n c e e l e c t r o n s a n i n c i d e n t e n e r g y o f 4 0 0 e V ( p l u s t h e b i n d i n g e n e r g y c ) i s n e c e s s a r y t o e n s u r e t h a t t h e m e a s u r e d f o r m f a c t o r i s e q u i v a l e n t t o t h e s h a p e o f t h e momentum d i s t r i b u t i o n , t h o u g h n o t n e c e s s a r i l y o f t h e c o r r e c t i n t e n s i t y r e l a t i v e t o t h e momentum d i s t r i b u t i o n s o f o t h e r m o l e c u l a r o r b i t a l s i n t h e t a r g e t . T h i s m e a n s t h a t t h e f r e e e l e c t r o n v e l o c i t i e s a r e h i g h e n o u g h t h a t t h e i r m o t i o n s a r e v e r y n e a r l y u n d i s t u r b e d b y t h e p o t e n t i a l o f t h e n u c l e i a n d t h e o t h e r e l e c t r o n s o f t h e s y s t e m , a n d t h e r e f o r e , t h a t p l a n e w a v e s a r e a s u f f i c i e n t l y a c c u r a t e m a t h e m a t i c a l r e p r e s e n t a t i o n o f t h e f r e e e l e c t r o n w a v e f u n c t i o n . I n o r d e r t o g e t r e a s o n a b l y c o r r e c t r e l a t i v e i n t e n s i t i e s i t h a s b e e n f o u n d n e c e s s a r y t o go t o i n c i d e n t e n e r g i e s o f l O O O e V o r m o r e . T h i s h a s a d v a n t a g e s a n d d i s a d v a n t a g e s o v e r 4 0 0 e V o p e r a t i o n w h i c h a r e d i s c u s s e d i n C h a p t e r 3. I n d e t e r m i n i n g t h e s h a p e o f a momentum d i s t r i b u t i o n , b r e a k d o w n o f t h e PWIA i s m o s t l i k e l y t o o c c u r f o r t h e i n n e r v a l e n c e a n d c o r e o r b i t a l s , a s t h e s e a r e s p a t i a l l y d e e p e r i n s i d e t h e m o l e c u l e w h e r e t h e i n c i d e n t a n d o u t g o i n g p a r t i c l e w a v e s a r e m o r e l i k e l y t o be d i s t o r t e d b y t h e f i e l d o f t h e n u c l e i . I f s u c h a b r e a k d o w n o c c u r s , i t u s u a l l y t a k e s t h e f o r m o f a d i s t o r t i o n o f t h e momentum d i s t r i b u t i o n i n t h e 1 / I n t r o d u c t i o n a n d T h e o r y 19 r e g i o n q > 1 . 5 a 0 ~ 1 a s t h i s r e g i o n i s s o m e w h a t m o r e s e n s i t i v e t o t h e f o r m o f t h e R - s p a c e w a v e f u n c t i o n n e a r t h e n u c l e i . ( T h i s i s a c o n s e q u e n c e o f t h e R e c i p r o c i t y p r o p e r t y o f t h e F o u r i e r T r a n s f o r m r e l a t i o n s h i p o f R - s p a c e a n d P - s p a c e , f u r t h e r d e s c r i b e d i n C h a p t e r 2.) 1.3 B i n a r y ( e , 2 e ) i n C o m p a r i s o n w i t h O t h e r M e t h o d s A s t h e b i n a r y ( e , 2 e ) r e s u l t s w i l l b e c o m p a r e d w i t h o t h e r t y p e s o f s p e c t r o s c o p i c d a t a , a s h o r t d e s c r i p t i o n o f some o f t h e m i s i n c l u d e d h e r e . 1.3.1 B i n d i n g e n e r g y s p e c t r a T h e r e a r e many o t h e r t e c h n i q u e s c a p a b l e o f m e a s u r i n g t h e b i n d i n g e n e r g i e s o f e l e c t r o n s : p h o t o e l e c t r o n a n d p h o t o a b s o r p t i o n s p e c t r o s c o p y i n t h e i r many f o r m s , d i p o l e ( e , 2 e ) s p e c t r o s c o p y a n d e l e c t r o n e n e r g y - l o s s s p e c t r o s c o p y , a n d o t h e r s . T h e o n e m o s t c l o s e l y r e l a t e d t o b i n a r y ( e , 2 e ) i s , n o t s u r p r i s i n g l y , d i p o l e ( e , 2 e ) s p e c t r o s c o p y { I n o k u t i ( 1 9 7 1 ) , B r i o n ( 1 9 7 5 ) , B r i o n ( 1 9 7 7 ) , B r i o n ( 1 9 7 8 a ) , B r i o n ( 1 9 8 1 ) } . A s t h e e x p e r i m e n t a l w o r k i n t h i s t h e s i s i s o f t e n d i s c u s s e d i n r e l a t i o n t o d i p o l e ( e , 2 e ) r e s u l t s I s h a l l b r i e f l y d e s c r i b e t h i s t e c h n i q u e . T h e d i p o l e a n d b i n a r y ( e , 2 e ) r e a c t i o n s a r e s i m i l a r i n 1 / I n t r o d u c t i o n a n d T h e o r y 20 t h a t t h e r e i s o n e e l e c t r o n i n c i d e n t o n a t a r g e t , t h e t a r g e t i s i o n i z e d b y t h e p a s s a g e o f t h i s e l e c t r o n , a n d t w o e l e c t r o n s e x i t : t h e y d i f f e r i n t h a t t h e o n e r e q u i r e s f a s t f o r w a r d s c a t t e r i n g k i n e m a t i c s , a n d t h e o t h e r s y m m e t r i c s c a t t e r i n g . ' F a s t f o r w a r d s c a t t e r i n g ' m e a n s t h a t t h e i n c i d e n t e l e c t r o n h a s v e r y h i g h e n e r g y ( E 0 > 3 k e V ) , a n d , i n c o l l i s i o n w i t h t h e t a r g e t , s u f f e r s o n l y a r e l a t i v e l y s m a l l l o s s o f e n e r g y ( E ) t o t h e t a r g e t a n d i t s d i r e c t i o n r e m a i n s ( i n t h e i d e a l c a s e ) u n a l t e r e d . Wherea'S s y m m e t r i c s c a t t e r i n g i m p l i e s a b i n a r y e n c o u n t e r c o l l i s i o n - t h e c l o s e s t p o s s i b l e b e t w e e n t h e i n t e r a c t i n g e l e c t r o n s - f a s t f o r w a r d s c a t t e r i n g i m p l i e s a l a r g e i m p a c t p a r a m e t e r , w h e r e i n t h e i n t e r a c t i o n b e t w e e n i n c i d e n t a n d t a r g e t e l e c t r o n s t a k e s p l a c e o v e r a l a r g e d i s t a n c e . T h e e l e c t r i c f i e l d f e l t b y t h e t a r g e t i n s u c h a c o l l i s i o n i s , t o a v e r y g o o d a p p r o x i m a t i o n , t h e same a s t h a t i n d u c e d b y t h e a b s o r p t i o n o f a p h o t o n o f e n e r g y E. When t h e e n e r g y l o s s i s s u f f i c i e n t , a n e l e c t r o n may be i o n i z e d f r o m t h e t a r g e t w i t h s e l e c t i o n r u l e s w h i c h a r e i d e n t i c a l t o t h e i r o p t i c a l c o u n t e r p a r t s , a n d t h e c r o s s - s e c t i o n i s r e l a t e d t o t h e p h o t o i o n i z a t i o n c r o s s - s e c t i o n b y a s i m p l e k i n e m a t i c r e l a t i o n . A s w i t h b i n a r y ( e , 2 e ) , t h e k i n e m a t i c s o f a l l f r e e e l e c t r o n s a r e d e t e r m i n e d b y t h e s p e c t r o m e t e r , a n d b i n d i n g e n e r g y s p e c t r a a r e o b t a i n e d b y s c a n n i n g t h e e j e c t e d e l e c t r o n e n e r g y s p e c t r u m . H o w e v e r , t h e p e a k i n t e n s i t i e s i n s u c h a s p e c t r u m a r e d e t e r m i n e d p a r t l y b y t h e d i p o l e m a t r i x e l e m e n t a n d a r e o n l y i n d i r e c t l y r e l a t e d t o b i n a r y ( e , 2 e ) p e a k i n t e n s i t i e s ; s u c h f a c t o r s a s 1 / I n t r o d u c t i o n a n d T h e o r y 21 o c c u p a t i o n n u m b e r , f i n a l s t a t e C I c o e f f i c i e n t s , r e l a x a t i o n f a c t o r s - a n y t h i n g n o t d i r e c t l y i n v o l v e d i n t h e e - e i n t e r a c t i o n - w i l l o f c o u r s e b e t h e same f o r b o t h m e t h o d s . T h e f u l l d i p o l e ( e , 2 e ) c r o s s - s e c t i o n i s g i v e n b y { I n o k u t i ( 1 9 7 1 ) , B r i o n ( 1 9 8 1 ) , H a m n e t t ( 1 9 7 6 ) } : ( 1 . 2 0 ) d £ = 2 k { n } 1 d f ( K ) d n E k 0 K 7 d E w h e r e E a n d K a r e t h e e n e r g y l o s s a n d momentum t r a n s f e r r e s p e c t i v e l y o f t h e i n c i d e n t e l e c t r o n t o t h e t a r g e t , k { n } a n d k 0 a r e t h e m o m e n t a o f t h e e j e c t e d a n d i n c i d e n t e l e c t r o n s , a n d d f ( K ) / d E i s t h e g e n e r a l i z e d o s c i l l a t o r s t r e n g t h f o r t h e i o n i z a t i o n e v e n t . I n t h e l i m i t o f v e r y s m a l l momentum t r a n s f e r , K 2 - > 0 , t h e g e n e r a l i z e d o s c i l l a t o r s t r e n g t h r e d u c e s t o t h e o p t i c a l o s c i l l a t o r s t r e n g t h d f 0 / d E . 1.3.2 M o m entum d i s t r i b u t i o n s T h e r e i s n o t e c h n i q u e o t h e r t h a n b i n a r y ( e , 2 e ) s p e c t r o s c o p y w h i c h i s c a p a b l e o f d e t e r m i n i n g momentum d i s t r i b u t i o n s o f i n d i v i d u a l e l e c t r o n s i n m o l e c u l e s . T h e r e a r e s e v e r a l t e c h n i q u e s f o r m e a s u r i n g t o t a l m o l e c u l a r momentum d i s t r i b u t i o n s - C o m p t o n s c a t t e r i n g a n d p o s i t r o n a n n i h i l a t i o n a r e e x a m p l e s - b u t t h e s e a r e c r u d e t o o l s , s e n s i t i v e t o i n d i v i d u a l m o l e c u l a r o r b i t a l momentum d i s t r i b u t i o n s o n l y i n s o f a r a s t h e y a f f e c t t h e t o t a l N - e l e c t r o n d i s t r i b u t i o n . 1 / I n t r o d u c t i o n a n d T h e o r y 22 1.4 T h e o r e t i c a l T r e a t m e n t s M a n y t y p e s o f t h e o r e t i c a l c a l c u l a t i o n s a r e c o m p a r e d w i t h t h e e x p e r i m e n t a l r e s u l t s . T h e f o l l o w i n g t w o s e c t i o n s g i v e b r i e f d e s c r i p t i o n s o f t h e m . 1.4.1 B i n d i n g e n e r g i e s G i v e n a n LCAO-MO-SCF w a v e f u n c t i o n t h e s i m p l e s t a p p r o x i m a t i o n f o r t h e e l e c t r o n b i n d i n g e n e r g i e s i n a m o l e c u l e i s K o o p m a n s ' t h e o r e m { K o o p m a n s ( 1 9 3 3 ) } : t h e i o n i z a t i o n p o t e n t i a l s a r e a s s u m e d t o be e q u a l t o t h e o r b i t a l e n e r g i e s w h i c h a r e t h e e i g e n v a l u e s o f t h e o n e - e l e c t r o n F o c k o p e r a t o r : ( 1 . 2 1 ) I P = - e { i } T h e t h e o r e m i g n o r e s r e l a x a t i o n a n d c o r r e l a t i o n e f f e c t s , b u t . i n t h e o u t e r v a l e n c e o r b i t a l s o f a l l b u t t h e m o s t e l e c t r o n e g a t i v e e l e m e n t s t h e t w o e f f e c t s a r e f o u n d e m p i r i c a l l y t o n e a r l y c a n c e l . T h e a c c u r a c y o f t h e s e o r b i t a l e n e r g i e s d e p e n d s o n t h e s i z e o f t h e F o c k s p a c e , a n d t h e t h e o r e m g i v e s e n e r g i e s t o s e c o n d o r d e r i f t h e F o c k s p a c e i s l a r g e ( i . e . i f t h e q u a l i t y o f t h e w a v e f u n c t i o n a p p r o a c h e s t h e H a r t r e e - F o c k l i m i t ) . T h e r e i s s t i l l , n e v e r t h e l e s s , o n l y o n e i o n i z a t i o n p o t e n t i a l p r e d i c t e d p e r g r o u n d s t a t e m o l e c u l a r o r b i t a l , w h i c h i s g r o s s l y i n a c c u r a t e f o r some s y s t e m s . T h e r e v e l a t i o n o f 1 / I n t r o d u c t i o n a n d T h e o r y 23 c o m p l e x m a n y - e l e c t r o n s t r u c t u r e i n t h e i o n i z a t i o n o f c e r t a i n e l e c t r o n s o f c e r t a i n m o l e c u l e s , a n d t h e b r e a k d o w n o f K o o p m a n s ' t h e o r e m w h i c h p r e d i c t s o n l y t h e c e n t r o i d o f s u c h s t r u c t u r e , i s d i s c u s s e d i n some d e t a i l i n t h e f o l l o w i n g c h a p t e r s . T h e m a n y - b o d y G r e e n ' s f u n c t i o n m e t h o d (MBGF) { C e d e r b a u m ( 1 9 7 5 a ) , C e d e r b a u m ( 1 9 7 5 b ) , C e d e r b a u m ( 1 9 7 7 ) } i s o n e w h i c h g o e s b e y o n d K o o p m a n s ' t h e o r e m a n d w i l l b e s h o w n t o be s u c c e s s f u l i n a c c o u n t i n g s e m i - q u a n t i t a t i v e l y f o r m u c h o f t h e c o m p l e x s t r u c t u r e i n t h e i o n i z a t i o n o f i n n e r v a l e n c e e l e c t r o n s . T h e o n e - e l e c t r o n p i c t u r e a s s u m e s t h a t i o n i z a t i o n o f a v a l e n c e e l e c t r o n f r o m a m o l e c u l e w i l l r e s u l t i n o n l y o n e f i n a l i o n s t a t e ( i g n o r i n g e f f e c t s l i k e s p i n - o r b i t s p l i t t i n g ) , a n d c o n s e q u e n t l y g i v e r i s e t o o n l y o n e p e a k i n t h e b i n d i n g e n e r g y s p e c t r u m . F o r many m o l e c u l e s { C e d e r b a u m ( 1 9 7 7 ) } t h e o n e - e l e c t r o n p i c t u r e i s o b v i o u s l y b r e a k i n g d own t o a v e r y s i g n i f i c a n t e x t e n t , a n d i o n i z a t i o n o f e s p e c i a l l y t h e m o r e s t r o n g l y b o u n d i n n e r v a l e n c e e l e c t r o n s g i v e s r i s e t o a h o s t o f f i n a l i o n s t a t e s d i s t r i b u t e d i n e n e r g y f r o m 20 t o 5 0 e V . T h i s i s c a l l e d ' M u l t i p l e F i n a l S t a t e ( M F S ) s t r u c t u r e ' . No s i n g l e o n e o f t h e s e s t a t e s h a s t h e r e l a t i v e l y l a r g e t r a n s i t i o n a m p l i t u d e a t t r i b u t a b l e t o a c l e a r l y i d e n t i f i a b l e , s i n g l e p a r t i c l e ' p a r e n t ' i o n s t a t e . T h i s i s d i f f e r e n t f r o m t h e s i t u a t i o n i n c o r e i o n i z a t i o n w h e r e t h e r e i s a h i g h i n t e n s i t y p e a k c l e a r l y i d e n t i f i a b l e a s a p a r e n t , w i t h a n u m b e r o f m u c h l e s s i n t e n s e 1 / I n t r o d u c t i o n a n d T h e o r y 24 s a t e l l i t e s a r o u n d i t . T h e MBGF m e t h o d i s b a s e d o n d i a g r a m m a t i c p r o p a g a t o r t h e o r y { M a t t u c k ( 1 9 6 7 ) , T h o u l e s s ( 1 9 6 1 ) } : t h e o n e - e l e c t r o n p r o p a g a t o r G( r , t ; r ' , t ' ) i s t h e a m p l i t u d e f o r a p a r t i c l e c r e a t e d i n a s y s t e m a t p o i n t £ a t t i m e t t o p r o p a g a t e v i a a l l p o s s i b l e p a t h s t o p o i n t r ' a t t i m e t ' . One s u c h p a t h i s t h a t w h e r e t h e p a r t i c l e i s f r e e , w i t h e n e r g y 6 0 , a n d g o e s s t r a i g h t f r o m r t o r_' w i t h o u t i n t e r a c t i o n w i t h a n y p o t e n t i a l : t h i s i s d e s c r i b e d b y t h e f r e e p a r t i c l e p r o p a g a t o r G 0 ( r , t ; r ' t ' ) a n d i s r e p r e s e n t e d d i a g r a m m a t i c a l l y i n f i g u r e 1 . 3 a . I f t h e f r e e e l e c t r o n i s p e r t u r b e d b y some p o t e n t i a l V, t h e n a n o t h e r p a t h i s f o r t h e e l e c t r o n t o p r o p a g a t e t o p o i n t r , , i n t i m e t 1 f i n t e r a c t w i t h t h e p o t e n t i a l ( e . g . a n o t h e r e l e c t r o n i n t h e s y s t e m ) , a n d t h e n p r o p a g a t e t o r ' a t t i m e t ' . D i a g r a m m a t i c a l l y t h i s i s r e p r e s e n t e d i n f i g u r e 1 .3b. Y e t a n o t h e r p a t h w a y w o u l d b e s c a t t e r i n g t w i c e o f f t h e p o t e n t i a l b e f o r e p r o p a g a t i n g t o r ' , t ' ( f i g u r e 1 . 3 c ) , a n d s o o n : ( 1 . 2 2 ) G ( r , t ; r ' , t ' ) = G 0 ( r , t ; r ' , t ' ) . + J d t ! G 0 ( r , t ; r 1 , t , ) V G 0 ( r , , t , ; r ' , t ' ) +JJdt 1dt 2 G 0 ( . r , t ; r , , t , ) V G 0 ( r , , t , ; r 2 , t 2 ) V G 0 ( r 2 , 1 2 , r ' , t ' ) + ... T h e i n f l u e n c e o f t h i s p o t e n t i a l i s s a i d t o c o n v e r t t h e s i m p l e f r e e p a r t i c l e o f e n e r g y e 0 , i n t o a q u a s i - p a r t i c l e o f e n e r g y €. When t h e F o u r i e r t r a n s f o r m f r o m t h e t i m e t o t h e 1 / I n t r o d u c t i o n a n d T h e o r y 25 F i g u r e 1.3 G r e e n ' s f u n c t i o n p r o p a g a t o r s : ( a ) F r e e p a r t i c l e : G = G 0 ( b ) S i n g l e i n t e r a c t i o n : G = G 0 + G 0 V G 0 ( c ) D o u b l e i n t e r a c t i o n : G = G o + G 0 V G 0 V G o 1 / I n t r o d u c t i o n a n d T h e o r y 26 e n e r g y r e p r e s e n t a t i o n i s t a k e n , t h e f o l d e d p r o d u c t s b e c o m e s i m p l e p r o d u c t s : ( 1 . 2 3 ) G U ) = G 0 U ) + G 0 U ) V G 0 U ) + G 0 U ) V G 0 U ) V G 0 U ) + . . . = G 0 U ) + G o U ) V G ( u ) = G 0 U ) { 1+VGo U ) + ( V G 0 U ) ) 2 + ( V G 0 U ) ) 3 + . . . } a n d t h e f o r m o f G U ) i s U - € + i 6 ) ~ 1 w h e r e e i s t h e p a r t i c l e ' s ' e n e r g y . I t i s s e e n t h a t t h e p r o p a g a t o r G U ) h a s a p o l e a t t h e e n e r g y o f t h e p a r t i c l e . T h e i m a g i n a r y p a r t o f t h e e n e r g y i 6 g i v e s t h e q u a s i - p a r t i c l e a f i n i t e l i f e t i m e , i f 6 i s ' n o t z e r o . C o r r e s p o n d i n g l y t h e f o r m o f G 0 U ) i s ( u - 6 0 ) _ 1 w h e r e t 0 i s t h e e n e r g y o f t h e p a r t i c l e w h e n i t i s u n p e r t u r b e d b y t h e p o t e n t i a l . T h e s e c o n d f a c t o r i n e q u a t i o n 1.23 i s a g e o m e t r i c s e r i e s w h i c h i s e q u a l i n t h e l i m i t t o ( 1 - V G 0 ) ~ 1 . S u b s t i t u t i n g t h i s b a c k ' g i v e s : G U ) = G 0 U ) ( 1 - V G o U ) } " 1 o r ( 1 . 2 4 ) G - ' U ) = G 0 - 1 U ) - V = o - € 0 - V T h i s i s a s i m p l e v e r s i o n o f t h e D y s o n e q u a t i o n : i f G U ) h a s a p o l e a t t h e e n e r g y o f t h e q u a s i - p a r t i c l e € t h e n G _ 1 U ) m u s t h a v e a n o d e o r z e r o p o i n t h e r e , a n d s o t h e e n e r g y o f t h e q u a s i - p a r t i c l e i s f o u n d b y l o o k i n g f o r t h e p l a c e w h e r e : (1 . 2 5 ) u - € 0 = V 1 / I n t r o d u c t i o n a n d T h e o r y 27 T h i s s i t u a t i o n a p p l i e s e q u a l l y t o t h e p r o p a g a t i o n o f a q u a s i - h o l e s t a t e - i . e . how t h e h o l e c r e a t e d b y t h e i n s t a n t a n e o u s r e m o v a l o f a p a r t i c l e f r o m a s y s t e m p r o p a g a t e s . I n s t a n t a n e o u s r e m o v a l o f a n e l e c t r o n i s a f a i r l y g o o d d e s c r i p t i o n o f t h e ( e , 2 e ) i o n i z i n g e v e n t , a n d s o t h e t e c h n i q u e f i n d s a p p l i c a t i o n h e r e . T h e r e a r e d i f f e r e n c e s o f c o u r s e : t h e t o t a l l y f r e e q u a s i - h o l e i s r e p l a c e d b y t h e H a r t r e e - F o c k l i m i t a p p r o x i m a t i o n ( i . e . K o o p m a n s ' t h e o r e m ) a n d t h e s i m p l e p o t e n t i a l V i s r e p l a c e d b y j l ( u ) , t h e ' s e l f - e n e r g y ' m a t r i x w h i c h d e s c r i b e s t h e e f f e c t , b e y o n d t h e HF l i m i t , o f a l l t h e o t h e r e l e c t r o n s on t h e p r o p a g a t i o n o f t h e q u a s i - h o l e ; b u t t h e b a s i c m e t h o d o f s e a r c h i n g f o r z e r o e s o f G " 1 ( o ) b y s o l v i n g u]_-e_=l(u) i s s t i l l t h e s a m e , e x c e p t t h a t i s now a m a t r i x o f t h e e i g e n v a l u e s o f t h e o n e - e l e c t r o n F o c k o p e r a t o r : ( 1 . 2 6 ) G ' 1 (o) = " 1 - 4 ~ J^u) I t i s i n t h e c a l c u l a t i o n o f Z(u) w h e r e t h e g r e a t e s t d i f f i c u l t i e s a r i s e , a n d w h e r e m o s t o f t h e e f f o r t s o f C e d e r b a u m a n d c o w o r k e r s h a v e b e e n e x p e n d e d . T h e y i n c l u d e a l l d i a g r a m s o f s e c o n d a n d t h i r d o r d e r i n t h e i r c o m p u t a t i o n o f E(u), a n d some d i a g r a m s o f f o u r t h o r d e r . W h e r e t h e f i n a l i o n s t a t e e n e r g y s t r u c t u r e i s v e r y c o m p l e x t h e t w o - p a r t i c l e - o n e - h o l e T a m m - D a n c o f f a p p r o x i m a t i o n ( 2 p h - T D A ) { S c h i r m e r ( 1 9 7 8 ) } i s m a d e : o n l y d i a g r a m s w i t h t w o h o l e l i n e s a n d o n e p a r t i c l e l i n e a r e a l l o w e d i n _E(u). 1 / I n t r o d u c t i o n a n d T h e o r y 28 T h e G r e e n ' s f u n c t i o n may a l s o b e w r i t t e n i n t h e f o r m o f p r o p a g a t i o n o f p a r t i c l e s t a t e s i n t i m e : - i ( E { N - 1 } - E { N } ) ( t - f ) ( 1 . 2 7 ) G ( t , t ' ) = e x E < * 0 { N } | a * { k } | * { i f N - 1 } > < * { i r N - 1 } | a { j } | * 0 { N } > i w h e r e a * { k } a n d a { j } a r e t h e c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s f o r p a r t i c l e s i n s t a t e s k a n d j r e s p e c t i v e l y . T h i s d e s c r i b e s t h e p r o p a g a t i o n o f a h o l e c r e a t e d i n o r b i t a l j a s i t i n t e r a c t s w i t h t h e r e m a i n i n g e l e c t r o n s , a n d i n t h e e n e r g y r e p r e s e n t a t i o n i s : ( 1 . 2 8 ) G U ) = E < * 0 { N } j a * { k ] | » { i , N - 1 } > < * { i , N - 1 } I a { j } | * 0 { N } > i u + E { N - 1 } - E { N ] - i 6 T h i s a g a i n s h o w s t h a t G U ) w i l l h a v e p o l e s a t t h e i o n i z a t i o n p o t e n t i a l s o f t h e s y s t e m , b u t a l s o t h a t t h e s t r e n g t h o f s u c h p o l e s ( d e t e r m i n e d n u m e r i c a l l y f r o m s o l u t i o n s o f e q u a t i o n 1.26) i s g i v e n b y : ( 1 . 2 9 ) p { n } = E | x { i , n } | 2 i w h e r e x { i , n } = < * { n , N - 1 } | a { i } | * 0 ( N ) > T h i s p o l e s t r e n g t h i s a f a c t o r i n t h e b i n a r y ( e , 2 e ) i n t e n s i t y . x { i , n } i s c a l l e d t h e ' h o l e s t a t e a m p l i t u d e ' . T h e d i a g o n a l a p p r o x i m a t i o n h a s b e e n made i n e q u a t i o n 1 . 29: i n t e r f e r e n c e t e r m s o f t h e t y p e x * { i , n } x { j , n } a r e i g n o r e d , 1 / I n t r o d u c t i o n a n d T h e o r y 29 a l t h o u g h e x p e r i m e n t a l v e r i f i c a t i o n o f t h i s i n t e r f e r e n c e e f f e c t h a s r e c e n t l y b e e n f o u n d { B r a d s h a w ( 1 9 8 0 ) } f o r -a c e t y l e n e . When t h e o n e - p a r t i c l e i o n i z a t i o n p i c t u r e i s v a l i d a n d r e l a x a t i o n e f f e c t s a r e s m a l l , t h e n a l l t e r m s i n t h e s u m m a t i o n v a n i s h e x c e p t f o r i = n , a n d t h e r e i s o n e m a i n p o l e w h o s e s t r e n g t h a p p r o a c h e s u n i t y . When r e l a x a t i o n a n d c o r r e l a t i o n e f f e c t s b e c o m e i m p o r t a n t t h e n t h e s i n g l e m a i n p o l e s p l i t s i n t o many p o l e s o v e r w h i c h t h e t o t a l i n t e n s i t y i s d i s t r i b u t e d . I t s h o u l d b e n o t e d t h a t t h e c a l c u l a t e d e n e r g i e s a n d i n t e n s i t i e s o f t h e n u m e r o u s i n n e r v a l e n c e l i n e s c a n n o t b e e x p e c t e d t o b e q u a n t i t a t i v e l y a c c u r a t e o w i n g t o t h e l i m i t a t i o n s i n h e r e n t i n t h e 2 p h - T D A a p p r o a c h a s w e l l a s t o t h e u s u a l l y s i g n i f i c a n t b a s i s s e t d e p e n d e n c e o f t h e r e s u l t s i n t h i s e n e r g y r e g i o n { C e d e r b a u m ( 1 9 8 0 ) } . T h i s t e c h n i q u e h a s r e c e n t l y h a d m u c h s u c c e s s i n m o d e l l i n g b i n d i n g e n e r g y s p e c t r u m s t r u c t u r e w h i c h a r i s e s w h en i o n i z a t i o n f r o n o n e g r o u n d s t a t e o r b i t a l l e a d s t o many f i n a l i o n s t a t e s . F o r a c o m p r e h e n s i v e t r e a t m e n t o f t h i s t e c h n i q u e t h e r e a d e r i s d i r e c t e d t o t h e i r e x c e l l e n t r e v i e w a r t i c l e { C e d e r b a u m ( 1 9 7 7 ) } a n d t o o t h e r a r t i c l e s { D o m c k e ( 1 9 7 9 ) , S c h i r m e r ( 1 9 7 7 ) , C e d e r b a u m ( 1 9 7 8 ) , C e d e r b a u m ( 1 9 8 0 ) } . O t h e r m e t h o d s o f c o m p u t i n g I P s b e y o n d K o o p m a n s ' t h e o r e m w i l l a l s o b e c i t e d , b u t a s t h e s e a r e m o r e f a m i l i a r t o u s , I s h a l l n o t d e s c r i b e t h e m h e r e . 1 / I n t r o d u c t i o n a n d T h e o r y 30 1.4.2 M omentum d i s t r i b u t i o n s A l m o s t a l l t h e w a v e f u n c t i o n s u s e d t o c o m p u t e t h e o r e t i c a l momentum d i s t r i b u t i o n s f o r c o m p a r i s o n w i t h e x p e r i m e n t a r e t h e r e s u l t o f LCAO-MO-SCF R o o t h a a n - H a r t r e e - F o c k c a l c u l a t i o n s . T h e q u a l i f y i n g f a c t o r i n s u c h w a v e f u n c t i o n s i s t h e s i z e a n d f l e x i b i l i t y o f t h e b a s i s s e t u s e d i n t h e c a l c u l a t i o n . O w i n g t h e t h e e n e r g y o p t i m i z a t i o n c r i t e r i o n ( v a r i a t i o n a l p r i n c i p l e ) u s e d i n t h e u s u a l t y p e o f . LCAO-MO-SCF c a l c u l a t i o n , t h e r e s u l t i n g m o l e c u l a r o r b i t a l w a v e f u n c t i o n s s o m e t i m e s p o o r l y r e p r e s e n t t h e t r u e s p a t i a l d i s t r i b u t i o n o f e l e c t r o n d e n s i t y { H o o d ( 1 9 7 6 a ) , H o o d ( 1 9 7 7 ) , G o d d a r d ( 1 9 7 8 ) } e x c e p t i n t h e n u c l e a r r e g i o n s w h e r e t h e m a j o r c o n t r i b u t i o n t o m o l e c u l a r e n e r g i e s i s m a d e . I t i s , h o w e v e r , t h e m o r e s p a t i a l l y e x t e n d e d r e g i o n s o f t h e c h a r g e d i s t r i b u t i o n w h i c h a r e o f p r i m e i m p o r t a n c e i n t h e u n d e r s t a n d i n g o f c h e m i c a l b o n d i n g , momentum d i s t r i b u t i o n s , a n d r e a c t i v i t y . I t i s p a r t o f t h e a i m o f t h i s d i s s e r t a t i o n t o o f f e r t o t h e r e s e a r c h e r s who c a r r y o u t s u c h c a l c u l a t i o n s some d i r e c t i o n i n t h e i r c h o i c e o f b a s i s s e t w h i c h w i l l p r o d u c e n o t o n l y g o o d e n e r g i e s , b u t a l s o g o o d w a v e f u n c t i o n s ; t h e o n e d o e s n o t a l w a y s i m p l y t h e o t h e r . G o o d w a v e f u n c t i o n s i n t u r n a l l o w t h e a c c u r a t e c a l c u l a t i o n o f d i p o l e m o m e n t s , p o l a r i z a b i l i t i e s { Z e i s s ( 1 9 7 9 ) } a n d s o o n . 31 C H A P T E R 2 MOMENTUM S P A C E C H E M I S T R Y . . . i n s e c r e t c h a m b e r s w i t h e r e d men c o m p o u n d e d s t r o n g e l i x i r s o r i n h i g h t o w e r s a s k e d q u e s t i o n s o f t h e s t a r s . A l m o s t a l l o f m o d e r n c h e m i s t r y i s c o n c e p t u a l i z e d i n p o s i t i o n s p a c e ( R - s p a c e ) : we t h i n k a b o u t t h e ' l o c a t i o n ' o f a p a r t i c l e , o r t h e s p a t i a l ' s h a p e ' o r ' s i z e ' o f a n o r b i t a l , o r t h e ' s t r u c t u r e ' o f a m o l e c u l e . H o w e v e r , i n d e a l i n g w i t h t h e r e s u l t s o f b i n a r y ( e , 2 e ) s p e c t r o s c o p y o n e r e q u i r e s some f a c i l i t y i n t h i n k i n g i n momentum s p a c e ( P - s p a c e ) - w h i c h d o e s n o t come e a s i l y w h e n o n e i s f i r m l y r o o t e d , b o d i l y a n d c o n c e p t u a l l y , i n p o s i t i o n s p a c e . T h i s c h a p t e r i s a n e x p l o r a t i o n o f t h e p r o p e r t i e s o f m o m e n t u m - s p a c e w a v e f u n c t i o n s , a n d a t t e m p t s t o p r e s e n t some s i m p l e r u l e s f o r e s t i m a t i n g t h e s h a p e o f o r b i t a l s i n P - s p a c e f r o m a k n o w l e d g e o f t h e i r R - s p a c e c o u n t e r p a r t s a n d o f t h e F o u r i e r T r a n s f o r m r e l a t i o n s h i p . M o s t o f t h e p r i n c i p l e s o f momentum s p a c e c h e m i s t r y h a v e a l r e a d y b e e n b r i e f l y o u t l i n e d i n t h e l i t e r a t u r e { C o u l s o n ( 1 9 4 4 ) , E p s t e i n ( 1 9 7 3 ) , E p s t e i n ( 1 9 7 7 ) } , b u t i t i s a p p r o p r i a t e t o r e s t a t e a n d a m p l i f y t h e m h e r e , f o r t h e y a r e 2/Momentum S p a c e C h e m i s t r y 32 n o t g e n e r a l l y k n o w n o r u n d e r s t o o d , a n d a l s o s u c h d i s c u s s i o n s h a v e u s u a l l y b e e n a p p l i e d t o C o m p t o n s c a t t e r i n g o r p o s i t r o n - a n n i h i l a t i o n r e s u l t s , n o t b i n a r y ( e , 2 e ) . T h e f i g u r e s u s e d t o i l l u s t r a t e t h i s c h a p t e r a r e c o n t o u r m a ps o f t h e e l e c t r o n a n d momentum d e n s i t y , />, e v a l u a t e d i n a p l a n e : />{i}(p_) = * * { i } ( E ) * { i } ( 2 ) ( 2 . 1 ) / > { i } ( r ) = * * { i } ( r ) * { i } ( r ) F o r m o s t d i a g r a m s 9 c o n t o u r l i n e s a r e s h o w n , s p a c e d a t f r a c t i o n s 0 . 1 , 0 . 2 , 0.3 ... 0.9 o f t h e m aximum d e n s i t y , a n d i n some c a s e s t h r e e m o r e c o n t o u r s a r e a d d e d a t f r a c t i o n s 0 . 0 1 , 0 . 0 2 , a n d 0 . 0 5 . A s m a l l n u m e r a l 9 o r 12 i n t h e f i g u r e i n d i c a t e s t h e n u m b e r o f c o n t o u r s . T h i s i s n o t t h e b e s t way t o d e p i c t d e n s i t y f u n c t i o n s : i t w o u l d b e p r e f e r a b l e t o p l o t c o n t o u r s w h i c h e n c l o s e v a r i o u s f r a c t i o n s o f t h e t o t a l d e n s i t y , b u t s u c h c o m p u t e r a l g o r i t h m s w e r e n o t a v a i l a b l e w h en t h e s e p r o g r a m s w e r e w r i t t e n . T h e s i g n o f t h e w a v e f u n c t i o n i s i n d i c a t e d w h e r e n e c e s s a r y o r a p p r o p r i a t e . A l l R - s p a c e . w a v e f u n c t i o n s p r e s e n t e d h e r e a r e r e a l f u n c t i o n s . H o w e v e r , d u e t o t h e n a t u r e o f t h e F o u r i e r T r a n s f o r m , t h e c o r r e s p o n d i n g P - s p a c e f u n c t i o n may be c o m p l e x . S i g n s a r e o n l y i n c l u d e d i n t h e P - s p a c e d i a g r a m s when b o t h t h e r e a l a n d i m a g i n a r y p a r t s o f t h e f u n c t i o n c h a n g e s i g n . 2/Momentum S p a c e C h e m i s t r y 33 S c h e m a t i c d i a g r a m s s h o w i n g t h e o r i e n t a t i o n o f t h e c o n t o u r map p l a n e a r e g i v e n i n A p p e n d i x C. A l l p l a n e s i n c l u d e t h e o r i g i n . A l m o s t a l l t h e d i a g r a m s a r e o n t h e same s c a l e ( s p a n n i n g 5 a t o m i c u n i t s i n b o t h d i r e c t i o n s ) , e x c e p t t h e R - s p a c e m aps f o r C 0 2 ( f i g u r e 6 . 3 ) , a n d t h e P - s p a c e maps f o r t h e N 2 1*{g} a n d 1«{u} o r b i t a l s ( f i g u r e 2 . 1 2 ) a n d t h e 0 3 3 a , o r b i t a l ( f r g u r e 2 . 1 3 ) . 2.1 T h e F o u r i e r T r a n s f o r m D e f i n i t i o n T h e r e l a t i o n s h i p b e t w e e n t h e p o s i t i o n - s p a c e r e p r e s e n t a t i o n o f a w a v e f u n c t i o n , * ( r _ ) , a n d i t s m o m e n t u m - s p a c e c o u n t e r p a r t , *(p_)f i s t h e w e l l - k n o w n u n i t a r y F o u r i e r T r a n s f o r m ( F T ) { D i r a c ( 1 9 5 8 ) } : - 3 / 2 f - i £ - I ( 2 . 2 ) • ( £ ) = (2rr) I d r e * ( r ) a n d i t s i n v e r s e : - 3 / 2 f i p _ . r ( 2 . 3 ) * ( r ) = ( 2 T T ) d 2 e * ( p ) P o s i t i o n s p a c e ( R - s p a c e ) r e f e r s t o t h e s p a c e o f t h e t h r e e o r t h o g o n a l d y n a m i c a l v a r i a b l e s . r = ( x , y , z ) : momentum s p a c e ( P - s p a c e ) r e f e r s t o t h e c o r r e s p o n d i n g s p a c e o f 2 = ( p ( x } , p { y } , p { z } ) , t h e t h r e e o r t h o g o n a l momentum d i r e c t i o n s . 2/Momentum S p a c e C h e m i s t r y 34 2.2 T h e F T S y m m e t r y P r o p e r t y T h e F o u r i e r T r a n s f o r m p r e s e r v e s a l l a s p e c t s o f t h e s y m m e t r y o f t h e R - s p a c e w a v e f u n c t i o n i n P - s p a c e . I n a d d i t i o n t h e F T i n t r o d u c e s ( i f i t i s n o t a l r e a d y p r e s e n t ) t h e s y m m e t r y e l e m e n t i , ( s y m m e t r y o n i n v e r s i o n t h r o u g h t h e P - s p a c e o r i g i n ) . I n v e r s i o n s y m m e t r y i s n e c e s s a r y i f t h e p a r t i c l e d e s c r i b e d b y t h e w a v e f u n c t i o n i s t o h a v e n o n e t t r a n s l a t i o n a l m o t i o n . P r e s e r v a t i o n o f s y m m e t r y m e a n s t h a t a l l R - s p a c e n o d a l p l a n e s ( s e e d e f i n i t i o n , A p p e n d i x A ) w i l l a l s o a p p e a r i n P - s p a c e . N o t e , h o w e v e r , t h a t n o d a l s u r f a c e s ( s e e a l s o d e f i n i t i o n , A p p e n d i x A) w i l l n o t n e c e s s a r i l y b e p r e s e r v e d u n l e s s t h e y a r e a l s o s y m m e t r y e l e m e n t s . A n e x a m p l e o f a n o d a l s u r f a c e w h i c h i s p r e s e r v e d i n t h e F T i s t h e s p h e r i c a l n o d a l s u r f a c e o f a h y d r o g e n - l i k e 2 s o r b i t a l , s i n c e a t o m s o b v i o u s l y h a v e s p h e r i c a l s y m m e t r y . A n e x a m p l e o f a n o d a l s u r f a c e t h a t i s n o t p r e s e r v e d i s t h a t o f a <r* o r b i t a l o f a h e t e r o n u c l e a r d i a t o m i c m o l e c u l e ( f i g u r e 7 . 2 ) . A s t r a i g h t f o r w a r d c o n s e q u e n c e o f t h e r o t a t i o n a l i n v a r i a n c e o f t h e F T i s t h a t a t o m i c o r b i t a l s w i l l h a v e t h e same a n g u l a r d e p e n d e n c e i n b o t h r e p r e s e n t a t i o n s a n d t h a t o n l y t h e r a d i a l d e p e n d e n c e w i l l d i f f e r : ( 2 . 4 ) * { n l m } ( r ) = N R { n l } ( r ) Y { l m } ( e , * ) ( 2 . 5 ) * { n l m } ( E ) = N P { n l } ( p ) Y { l m } ( e , * ) 2/Momentum S p a c e C h e m i s t r y 35 w h e r e N i s a n o r m a l i z a t i o n f a c t o r , R { n l } ( r ) a n d P { n l } ( p ) g i v e t h e r a d i a l d e p e n d e n c e o f t h e w a v e f u n c t i o n , Y { l m } ( e , 0 ) i s t h e s p h e r i c a l h a r m o n i c f u n c t i o n o f a n g l e , a n d n , 1, a n d m a r e r e s p e c t i v e l y t h e p r i n c i p a l , a z i m u t h a l a n d m a g n e t i c q u a n t u m n u m b e r s . T h i s m e a n s t h a t s - o r b i t a l s a r e s p h e r i c a l , t h a t p - o r b i t a l s h a v e t h e c h a r a c t e r i s t i c ' d u m b b e l l ' s h a p e a n d s o o n , i n b o t h s p a c e s ( s e e f i g u r e 2 . 1 ) . ( N o t e t h a t ' r a d i a l ' i s u s e d t h o u g h o u t t h i s t h e s i s i n t h e s e n s e o f ' t h e m a g n i t u d e o f t h e c o o r d i n a t e v e c t o r ' ( r o r 2 ) ; i t i s n o t i n t e n d e d t o r e f e r t o ' t h e r a d i a l c o m p o n e n t o f t h e momentum o f a p a r t i c l e ' i n t h e s e n s e o f t h e c o m p o n e n t o f t h e p a r t i c l e ' s m o t i o n t o w a r d s o r a w a y f r o m a p o i n t i n R - s p a c e . ) T h e r a d i a l f u n c t i o n s a r e r e l a t e d b y t h e s p h e r i c a l B e s s e l t r a n s f o r m : A n o b s e r v a t i o n t h a t c a n b e made i m m e d i a t e l y f r o m t h i s i s t h a t o n l y a n s - o r b i t a l ( 1 = 0 ) w i l l h a v e n o n - z e r o a m p l i t u d e a t p= 0 . T h i s a r i s e s b e c a u s e o n l y t h e 1=0 B e s s e l f u n c t i o n h a s n o n - z e r o i n t e n s i t y a t p r = 0 . A l l o t h e r 1^0 B e s s e l f u n c t i o n s h a v e z e r o a m p l i t u d e a t t h e o r i g i n . H e n c e t h e o r i g i n o f t h e n o m e n c l a t u r e ' s - t y p e ' a n d ' p - t y p e ' i n r e f e r r i n g t o t h e t y p e o f momentum d i s t r i b u t i o n m e a s u r e d o n a t o m s . T h i s ' s - t y p e ' a n d ' p - t y p e ' n o m e n c l a t u r e i s a l s o a p p l i e d l o o s e l y t o m o l e c u l e s t o r e f e r t o d i s t r i b u t i o n s w h i c h a r e m a i n l y s - t y p e o r m a i n l y p - t y p e b u t w i t h a s m a l l a d m i x t u r e o f t h e o t h e r t y p e . S u c h s m a l l c o n t r i b u t i o n s w h i c h a r i s e o u t o f t h e ( 2 . 6 ) P { n l } ( p ) = 2 ( - i ) 2/Momentum S p a c e C h e m i s t r y 36 l o w e r e d s y m m e t r y i n m o l e c u l e s c o m p a r e d t o a t o m s mean t h a t m a i n l y s - t y p e d i s t r i b u t i o n s may h a v e i n c r e a s e d i n t e n s i t y a t m e d i u m t o h i g h q , a n d t h a t m a i n l y p - t y p e d i s t r i b u t i o n s may h a v e some i n t e n s i t y a t q = 0 , r e l a t i v e t o a t o m i c d i s t r i b u t i o n s . T h e a n a l y t i c a l f o r m o f S l a t e r - t y p e a n d G a u s s i a n a t o m i c o r b i t a l r a d i a l f u n c t i o n s a r e p r e s e n t e d i n d e t a i l i n A p p e n d i x B. T h e s e r a d i a l f o r m s a r e q u i t e s i m i l a r i n b o t h s p a c e s , a s i s s e e n i n f i g u r e 2 . 1 . A n s - o r b i t a l h a s t h e f a m i l i a r m o n o t o n i c a l l y d e c r e a s i n g f o r m i n R - s p a c e a n d P - s p a c e , a n d a p - o r b i t a l h a s z e r o a m p l i t u d e a t t h e o r i g i n a n d a maximum a t some p o i n t r e m o v e d f r o m t h e o r i g i n . 2.3 T h e R e c i p r o c i t y p r i n c i p l e A b a s i c p r o p e r t y o f t h e F o u r i e r T r a n s f o r m i s t h a t a d i l a t a t i o n o f a d i m e n s i o n i n o n e s p a c e c a u s e s a c o n t r a c t i o n o f t h e c o r r e s p o n d i n g d i m e n s i o n i n t h e o t h e r s p a c e a n d v i c e v e r s a . T h i s d e f i n e s t h e R e c i p r o c i t y p r i n c i p l e . A c o n s e q u e n c e o f t h i s p r i n c i p l e i s f i r s t s e e n i n c o m p a r i n g s i z e s ( a s s u m i n g r o u g h l y s i m i l a r r a d i a l f o r m s ) o f o r b i t a l s i n R - s p a c e a n d P - s p a c e : ( 1 ) C o r e o r b i t a l s , w h i c h a r e t i g h t l y c o n f i n e d s p a t i a l l y a r o u n d t h e a t o m i c c e n t r e s ( r < . 1 a 0 ) w i l l 2/Momentum S p a c e C h e m i s t r y 37 e x t e n d i n P - s p a c e t o l a r g e v a l u e s o f momentum ( q > 5 a 0 ~ 1 ) . ( 2 ) V a l e n c e o r b i t a l s , w h i c h a r e r e l a t i v e l y d i f f u s e i n R - s p a c e ( r > 2 a 0 ) h a v e f a i r l y c o m p a c t ( q < 2 a 0 ~ 1 ) d i s t r i b u t i o n s i n P - s p a c e . T h e R e c i p r o c i t y p r i n c i p l e m a n i f e s t s i t s e l f i n s e v e r a l d i f f e r e n t m a t h e m a t i c a l f o r m s . One i s t h e a b o v e d i l a t i o n / c o n t r a c t i o n p r o p e r t y . A n o t h e r i s t o c o n s i d e r t h a t t h e F o u r i e r T r a n s f o r m p r o j e c t s o u t t h e s p e c t r u m o f c o n s t i t u e n t s i n u s o i d a l f r e q u e n c i e s i n a f u n c t i o n . A s h a r p l y p e a k e d R - s p a c e w a v e f u n c t i o n ( a s i n c o r e o r b i t a l s ) i s s a i d t o c o n t a i n l a r g e h i g h - f r e q u e n c y ' c o m p o n e n t s , a n d b y t h e d e B r o g l i e r e l a t i o n : ( 2 . 7 ) p = hv/c t h i s m e a n s a l s o l a r g e h i g h - m o m e n t u m c o m p o n e n t s . T h e c o n v e r s e a r g u m e n t may b e made f o r b r o a d R - s p a c e w a v e f u n c t i o n s ( a s i n v a l e n c e o r b i t a l s ) . A t h i r d way o f l o o k i n g a t t h e R e c i p r o c i t y p r o p e r t y i s t o c o n s i d e r t h e f o r m o f t h e q u a n t u m m e c h a n i c a l momentum o p e r a t o r P: ( 2 . 8 ) P = - i V T h i s i m p l i e s t h a t t h e momentum o f a p a r t i c l e i s r e l a t e d t o t h e g r a d i e n t o f i t s w a v e f u n c t i o n . T h u s a s p a t i a l l y b r o a d w a v e f u n c t i o n w i t h a g e n e r a l l y s h a l l o w g r a d i e n t e m p h a s i z e s 2/Momentum S p a c e C h e m i s t r y 38 l o w v a l u e s o f momentum, a n d v i c e v e r s a . ( N o t e t h a t <rj> i s z e r o d u e t o t h e i n v e r s i o n s y m m e t r y o f P - s p a c e w a v e f u n c t i o n s , b u t <p> w i l l r e f l e c t t h e r a d i a l e x t e n t o f * ( p ) . ) T h e a b o v e c o n c e p t s a r e , o f c o u r s e , i n a c c o r d w i t h o u r b a s i c c h e m i c a l i n t u i t i o n a b o u t t h e n a t u r e o f o r b i t a l s : we s a y t h a t e l e c t r o n s i n t i g h t l y b o u n d , h i g h l y e n e r g e t i c o r b i t a l s a r e f a s t - m o v i n g p a r t i c l e s , a n d t h a t m o r e l o o s e l y b o u n d e l e c t r o n s a r e r e l a t i v e l y s l o w - m o v i n g . T o i l l u s t r a t e t h i s , f i g u r e 2.1 s h o w s c o n t o u r maps o f a p p r o x i m a t e a t o m i c o r b i t a l s { C l e m e n t i ( 1 9 7 4 ) } i n p o s i t i o n s p a c e a n d momentum s p a c e f o r h y d r o g e n a n d some a t o m s o f t h e f i r s t r o w o f t h e p e r i o d i c t a b l e . A n i m p o r t a n t , b u t s o m e t i m e s m i s u n d e r s t o o d , a s p e c t o f t h e F T r e l a t i o n s h i p i s t h a t t h e r e i s n o o n e - t o - o n e m a p p i n g o f p o i n t s i n R - s p a c e i n t o P - s p a c e . One c a n n o t s a y t h a t c h a n g i n g t h e a m p l i t u d e a t a g i v e n p o i n t i n R - s p a c e w i l l n e c e s s a r i l y p r o d u c e a c h a n g e a t a r e l a t e d p o i n t i n P - s p a c e . One c a n o n l y r e f e r i n t h e m o s t g e n e r a l way t o r e g i o n s o f t h e t w o s p a c e s a s b e i n g r e l a t e d , a n d t h i s i s d u e t o t h e n a t u r e o f w a v e f u n c t i o n s : t h e l a r g e r r e g i o n s a r e g e n e r a l l y s h a l l o w f u n c t i o n s , a n d s o t e n d t o c o n t r i b u t e t o l o w momentum, a n d n e a r t h e n u c l e i t h e R - s p a c e w a v e f u n c t i o n i s u s u a l l y r a p i d l y c h a n g i n g , t h u s c o n t r i b u t i n g t o h i g h momentum r e g i o n s . T h e o n l y way t o t h i n k o f c o r r e s p o n d e n c e s b e t w e e n p o i n t s i n R-a n d P - s p a c e i s t h a t a m p l i t u d e a t a p o i n t i n P - s p a c e i s a r e s u l t o f R - s p a c e a m p l i t u d e i n a r e a s w h i c h a r e s e p a r a t e d b y 2/Momentum Space C h e m i s t r y 39 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY CHARGE DENSITY He1s F i g u r e 2.1 Momentum d i s t r i b u t i o n s and momentum and charge d e n s i t y maps f o r atomic o r b i t a l s \ 2/Momentum S p a c e C h e m i s t r y SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY CHARGE DENSITY N2s r-2 0 2s h 2 F i g u r e 2.1 c o n t i n u e d 2/Momentum S p a c e C h e m i s t r y 41 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY CHARGE DENSITY 02p F2s F i g u r e 2.1 c o n t i n u e d 2/Momentum S p a c e C h e m i s t r y 42 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY CHARGE DENSITY Ne 2s F i g u r e 2.1 c o n t i n u e d 2/Momentum S p a c e C h e m i s t r y 43 n r r / r , w h e r e n = 0 , 1 , 2 , 3 . . . T h i s i s t h e b a s i s f o r b o n d o s c i l l a t i o n d i s c u s s e d f u r t h e r i n s e c t i o n 2.6 a n d f o r b o n d i n g i n p o l y a t o m i c s , i n s e c t i o n 2 . 5 . 3 . 2.4 T h e O n e - d i m e n s i o n a l W a v e f u n c t i o n P r o j e c t i o n Q B e c a u s e i t i s a t i m e - c o n s u m i n g a n d e x p e n s i v e p r o c e s s t o c o m p u t e momentum d e n s i t y m aps e x a c t l y f r o m t h e m o l e c u l a r o r b i t a l c o e f f i c i e n t s , i t i s d e s i r a b l e t o b e a b l e t o f o r m some i d e a o f t h e s h a p e o f t h e momentum d e n s i t y f r o m a k n o w l e d g e o f t h e R - s p a c e f u n c t i o n a n d f r o m some f a m i l i a r i t y w i t h t h e e f f e c t s o f t h e F o u r i e r T r a n s f o r m . T o t h i s e n d I s h a l l c l a s s i f y s e v e r a l t y p e s o f w a v e f u n c t i o n s a c c o r d i n g t o t h e i r s y m m e t r y a n d n o d a l s t r u c t u r e , a n d i l l u s t r a t e t h e i r a p p e a r a n c e i n t h e R- a n d P - s p a c e r e p r e s e n t a t i o n s . D e f i n e t h e f u n c t i o n Q ( r ' ) a s t h e p r o j e c t i o n o f t h e t o t a l t h r e e - d i m e n s i o n a l R - s p a c e w a v e f u n c t i o n > ( r ) o n t o a l i n e p a r a l l e l w i t h t h e v e c t o r r ' : ( 2 . 9 ) Q ( r ' ) = (p U ( r ) ; r ' ) T h e F o u r i e r T r a n s f o r m o f t h e r e s u l t i n g o n e - d i m e n s i o n a l f u n c t i o n Q ( r ' ) g i v e s t h e a m p l i t u d e Q ( p ' ) o f t h e m o m e n t u m - s p a c e w a v e f u n c t i o n a l o n g a l i n e p a s s i n g t h r o u g h t h e P - s p a c e o r i g i n w i t h t h e same d i r e c t i o n a s v e c t o r r ' : 2/Momentum S p a c e C h e m i s t r y 44 f - i p ' r ' ( 2 . 1 0 ) Q ( p ' ) = ( 2 r r ) - 1 / 2 \ d r ' e Q ( r ' ) T h i s f u n c t i o n i s u s e f u l , b e c a u s e t h e r e a r e o n l y a s m a l l n u m b e r o f d i f f e r e n t c l a s s e s o f t h e f u n c t i o n Q, w h i c h c a n e a s i l y b e w o r k e d o u t . I t i s t h e n u s u a l l y p o s s i b l e t o s k e t c h , o r v i s u a l i s e m e n t a l l y t h e e n t i r e momentum d e n s i t y i f o n e k n o w s t h e f o r m o f | Q ( p ' ) | 2 a l o n g t w o o r t h r e e l i n e s . I t i s a s s u m e d t h a t Q ( r ' ) i s e v e r y w h e r e r e a l ; h o w e v e r , i f t h i s i s s o , Q ( p ' ) i s n o t n e c e s s a r i l y a l s o e v e r y w h e r e r e a l , b u t may h a v e a n i m a g i n a r y p a r t , a n d f o r t h i s r e a s o n I s h a l l l o o k a t t h e m a g n i t u d e | Q ( p ' ) | 2 . A s t h e b i n a r y ( e , 2 e ) f o r m f a c t o r a n d momentum d e n s i t y maps a r e a l s o s q u a r e d f u n c t i o n s , t h i s i s n o t i n a p p r o p r i a t e . F i g u r e 2.2 s h o w s s c h e m a t i c a l l y t h e v a r i o u s c l a s s e s o f p r o j e c t i o n s . G a u s s i a n f u n c t i o n s h a v e b e e n u s e d t o s i m u l a t e t h e p r o j e c t i o n s . T h e s c a l e f o r Q ( r ' ) i s a r b i t r a r y ; t h e s c a l e f o r | Q ( p ' ) | 2 i s s y m m e t r i c a b o u t t h e m i d p o i n t w h e r e p=0. T a b l e 2.1 i s a s u m m a r y o f e x a m p l e s o f e a c h c l a s s o f p r o j e c t i o n . When w o r k i n g o u t t h e f o r m o f Q ( r ' ) o f a v a l e n c e o r b i t a l i t i s p e r m i s s i b l e t o i g n o r e t h e v e r y l o w r r e g i o n w h e n t h e w a v e f u n c t i o n h e r e i s r a p i d l y c h a n g i n g d u e t o c o n t r i b u t i o n s f r o m i n n e r s h e l l a t o m i c o r b i t a l s . B y t h e R e c i p r o c i t y p r i n c i p l e t h e s e w i l l a f f e c t o n l y t h e v e r y l a r g e p r e g i o n s w h i c h a r e u s u a l l y n o t d i s p l a y e d i n t h e momentum d e n s i t y maps i n t h i s w o r k . 2/Momentum S p a c e C h e m i s t r y 45 2.4.1 C l a s s 0: C e n t r o s y m m e t r i c s y s t e m s A l l a t o m i c o r b i t a l s a n d s i n g l e - c e n t r e , c o m p l e t e l y n o n - b o n d i n g m o l e c u l a r o r b i t a l s h a v e t h e same s h a p e i n R - s p a c e a n d P - s p a c e ( S y m m e t r y p r o p e r t y ) . T h e i r r a d i a l f o r m s a r e s l i g h t l y d i f f e r e n t . 2 .4.2 C l a s s I : One l o b e T h i s i s t h e s i m p l e s t c a s e : Q ( r ' ) i s p u r e s - t y p e ( i . e . s y m m e t r i c a b o u t i t s p o i n t o f m a x i m u m a m p l i t u d e ) , f a l l s o f f m o n o t o n i c a l l y o n e i t h e r s i d e , a n d i s e v e r y w h e r e t h e same s i g n . | Q ( p ' ) | 2 i s a s i m i l a r l y s h a p e d f u n c t i o n . I t i s f o u n d i n a t o m i c 1s S TOs a n d a l s o i n t h e < r - o r b i t a l s o f a s y m m e t r i c l i n e a r s y s t e m s p e r p e n d i c u l a r t o t h e b o n d a x i s . C l a s s I i s d i f f e r e n t f r o m c l a s s 0 i n t h a t i t c a n b e u s e d t o r e p r e s e n t o r b i t a l s w h i c h a r e m a i n l y s - t y p e , b u t a l s o e l o n g a t e d o r c o n t r a c t e d i n some d i m e n s i o n , a s , f o r i n s t a n c e , t h e < r - b o n d i n g MO i n a h o m o n u c l e a r d i a t o m i c . 2 . 4 . 3 C l a s s I I : Two l o b e s T h e n e x t s i m p l e s t c a s e , c l a s s I I , c o m p r i s e s t h o s e p r o j e c t i o n s Q ( r ' ) w h e r e t h e r e a r e t w o l o b e s o f p o s i t i v e a n d n e g a t i v e a m p l i t u d e , s e p a r a t e d b y a n o d a l p o i n t . T h i s c l a s s b r e a k s i n t o t w o s u b c l a s s e s , H i a n d H a . C l a s s H i c o m p r i s e s t h e s y m m e t r i c c a s e ( h e n c e t h e n o t a t i o n H i . ) w h e r e t h e w a v e f u n c t i o n h a s a n o d a l p l a n e p e r p e n d i c u l a r t o t h e 2/Momentum S p a c e C h e m i s t r y 46 p r o j e c t i o n a x i s . H e r e | Q ( p ' ) | 2 i s z e r o a t p=0 a n d r i s e s o n e i t h e r s i d e t o g i v e t w o e q u a l a r e a s o f momentum d e n s i t y . T h i s s u b c l a s s i s e x e m p l i f i e d i n 2 p S T O s a n d i n t h e <s* MOs o f h o m o n u c l e a r d i a t o m i c m o l e c u l e s . C l a s s I l a r e f e r s t o t h e a s y m m e t r i c c a s e ( n o t a t i o n I l a ) , w h e r e t h e p o s i t i v e a n d n e g a t i v e l o b e s a r e d i f f e r e n t . T h e d i s t i n g u i s h i n g f e a t u r e i n | Q ( p ' ) | 2 i s t h a t t h e r e i s now some d e n s i t y a t p=0. T h i s i s how t h e momentum d e n s i t y maps o f h e t e r o n u c l e a r a n d h o m o n u c l e a r d i a t o m i c m o l e c u l e s may be t o l d a p a r t . H o m o n u c l e a r MOs w i l l a l l be c l a s s I o r H i . H e t e r o n u c l e a r MOs w i l l b e s o m e w h e r e b e t w e e n c l a s s 0 a n d I l a . E x a m p l e s o f c l a s s I l a a r e f o u n d i n t h e m o s t l y - a n t i b o n d i n g a* MOs o f h e t e r o n u c l e a r d i a t o m i c s , a n d a l s o i n t h e o u t e r a , MOs o f b e n t A H 2 m o l e c u l e s . 2.4.4 C l a s s I I I : T h r e e l o b e s F o l l o w i n g t h e t r e n d f r o m c l a s s e s I t o I I t h e n e x t l o g i c a l e x t e n s i o n i s t o t h r e e l o b e s : i . e . Q ( r ' ) h a s t w o n o d a l p o i n t s s e p a r a t i n g t h r e e a r e a s o f a m p l i t u d e o f o s c i l l a t i n g s i g n . I n c l a s s I I I t h e r e a r e f o u r s u b c l a s s e s . C l a s s 111 i -i r e f e r s t o t h e c a s e w h e r e t h e p o s i t i v e a n d n e g a t i v e p a r t s o f t h e p r o j e c t i o n e x a c t l y c a n c e l i n t h e o v e r l a p w i t h t h e z e r o f r e q u e n c y ( p = 0 ) F T w a v e ; t h i s c a s e w i l l h a v e n o a m p l i t u d e a t p = 0 . T h i s s u b c l a s s o c c u r s i n t h e 1 t 2 o r b i t a l o f C H „ , t a k e n 2/Momentum S p a c e C h e m i s t r y g u r e 2 . 2 S c h e m a t i c d i a g r a m s o f Q - p r o j e c t i o n s . o n e - d i m e n s i o n a l 2/Momentuin Space C h e m i s t r y 48 T a b l e 2.1 Examples of Q p r o j e c t i o n s C l a s s 0 C l a s s I C l a s s H i C l a s s H a C l a s s I I I i , C l a s s I l i a , C l a s s 111i 2 C l a s s I I I a 2 C l a s s I V i C l a s s IVa A l l atomic o r b i t a l s Non-bonding o r b i t a l s t h a t a r e e s s e n t i a l l y a t o m i c A l l s atomic o r b i t a l s cr-bonding o r b i t a l s H 2 1*{g}, N 2 2*{g}, C 0 2 3«y{g} A l l p atomic o r b i t a l s D i a t o m i c <r{u} o r b i t a l s N 2 2tf{u}, C 0 2 2tf{u} D i a t o m i c rr{u} o r b i t a l s N 2 1ir{u}, NO 2rr, 0 2 1rr{u}, C 0 2 1 i r { u } , C e r t a i n h y d r i d e o u t e r v a l e n c e o r b i t a l s : NH 3 1e, 3 a H 2 0 3a,, 1b 2, 2a, HF 3<r, 2c Asymmetric d i a t o m i c rr o r b i t a l s p a r a l l e l t o bond a x i s : NO 2ir CHft 1 t 2 o r b i t a l on a l i n e p a r a l l e l t o H-atoms of l i k e s i g n , a l l o r b i t a l s of d-type symmetry Outer a, MOs i n H 20 and H 2S i n H-H d i r e c t i o n rr* o r b i t a l s 0 2 1tr{g}, C 0 2 1ir{g} Symmetric t r i a t o m i c (AX 2) «{g} o r b i t a l s : C 0 2 4«r{g} H i g h e r h e t e r o n u c l e a r d i a t o m i c <r o r b i t a l s : N0.5«r H i g h e r AX 2 c o r b i t a l s C 0 2 3tf{u} H i g h e r * o r b i t a l s of asymmetric t r i a t o m i c s (ABC) 2/Momentum S p a c e C h e m i s t r y 49 i n t h e d i r e c t i o n j o i n i n g H a t o m s o f l i k e s i g n , a n d i n MOs o f d - t y p e s y m m e t r y ( t w o p e r p e n d i c u l a r p l a n e s o f s y m m e t r y ) i n a n y d i r e c t i o n e x c e p t t h o s e o f t h e n o d a l p l a n e s . I n a l l t h e o t h e r s u b c l a s s e s t h e a m o u n t s o f t h e p o s i t i v e a n d n e g a t i v e l o b e s a r e n o t e q u a l a n d s o | Q ( p ' ) | 2 a l w a y s h a s n o n - z e r o d e n s i t y a t p=0. C l a s s I l i a , r e f e r s t o t h e c a s e w h e r e t h e t o t a l a m p l i t u d e i n t h e m i d d l e l o b e o f Q ( r ' ) o u t w e i g h s t h e . sum o f t h e o u t e r o n e s , a n d t h e p r o j e c t i o n may be s y m m e t r i c o r a s y m m e t r i c a b o u t t h e m i d p o i n t . T h e r e s u l t i n g | Q ( p ' ) | 2 i s s i m i l a r t o I H i , e x c e p t t h a t t h e r e i s d e n s i t y a t p = 0 . E x a m p l e s a r e t h e 3 a , a n d 5 a , MOs o f H 2 0 a n d H 2 S r e s p e c t i v e l y , t a k e n i n t h e d i r e c t i o n j o i n i n g t h e H a t o m s . C l a s s 1 1 1 i 2 c o n c e r n s t h e s i t u a t i o n s w h e r e a m p l i t u d e i n t h e t w o o u t e r l o b e s o f Q ( r ' ) i s g r e a t e r t h a n i n t h e m i d d l e , a n d t h e f u n c t i o n i s s y m m e t r i c a b o u t i t s m i d p o i n t . One f i n d s t h a t { Q ( p * ) | 2 h a s t h r e e p e a k s s e p a r a t e d b y n o d a l p o i n t s . E x a m p l e s o f t h i s c l a s s i n c l u d e t h e 4<r{g} o r b i t a l a n d t h e ( u n o c c u p i e d ) 2rr{u} o r b i t a l o f C 0 2 . C l a s s I I I a 2 i s t h e a s y m m e t r i c c a s e o f 1 1 1 i 2 • H e r e | Q ( p ' ) | 2 s h o w s t h a t t h e n o d e s o f t h e c l a s s 1 1 1 i 2 P - s p a c e f u n c t i o n a r e f i l l e d i n . C l a s s I I I a 2 i s s e e n i n t h e 5a MO o f n i t r i c o x i d e ( s e e C h a p t e r 7 ) . 2/Momentum S p a c e C h e m i s t r y 50 2 . 4 . 5 F o u r l o b e s T h e n u m b e r o f p o s s i b l e s u b c l a s s e s i n c r e a s e s w i t h t h e n u m b e r o f n o d e s i n Q ( r ' ) ; h o w e v e r , f o r t h e p u r p o s e s o f t h i s t h e s i s , o n l y t w o a r e i m p o r t a n t . S u b c l a s s I V i d e s c r i b e s t h e s y m m e t r i c c a s e w h e r e f o u r p e a k s a r e s e e n i n | Q ( p ' ) | 2 a n d w h e r e t h e d e n s i t y a t p=0 i s z e r o . T h e a m o u n t s o f i n t e n s i t y i n t h e t w o i n n e r | Q ( p ' ) | 2 p e a k s r e l a t i v e t o t h e o u t e r t w o i s i n v e r s e l y p r o p o r t i o n a l t o t h e a m o u n t o f i n t e n s i t y i n t h e i n n e r l o b e s o f Q ( r ' ) r e l a t i v e t o t h e o u t e r o n e s . C l a s s I V i i s e x e m p l i f i e d i n t h e 3 t r { u } o r b i t a l o f C 0 2 . When t h e s y m m e t r y i n s u b c l a s s I V i i s r e d u c e d , - t h e n o d e s i n | Q ( p ' ) | 2 f i l l i n , g i v i n g t h e s u b c l a s s I V a . When Q ( r ' ) i s made v e r y a s y m m e t r i c , e v e n t u a l l y | Q ( p ' ) | 2 w i l l s t a r t t o l o o k l i k e o n e o f t h e o t h e r a - t y p e c l a s s e s . 2.5 T h e B o n d i n g P r i n c i p l e We h a v e s e e n t h a t a t o m i c o r b i t a l s , w h e t h e r p l o t t e d i n R - s p a c e o r P - s p a c e , l o o k a l m o s t t h e same e x c e p t f o r t h e r e c i p r o c a l s i z e r e l a t i o n s h i p . N e x t , I s h a l l c o n s i d e r t h e e f f e c t o f b o n d i n g b e t w e e n a t o m s a s v i e w e d i n momentum s p a c e . A s m o s t o f t h e c a l c u l a t i o n s d o n e i n t h i s w o r k a r e o f 2/Momentum S p a c e C h e m i s t r y 51 LCAO-MO-SCF w a v e f u n c t i o n s I s h a l l u s e t h e c o n c e p t s a n d n o m e n c l a t u r e o f MO t h e o r y i n t h i s c h a p t e r . A d i s c o n c e r t i n g f e a t u r e o f momentum s p a c e i s t h a t , u n l i k e p o s i t i o n s p a c e , t h e r e i s n o w h e r e t o p l o t t h e p o s i t i o n s o f t h e n u c l e i . T h i s m a k e s l i f e c o n c e p t u a l l y d i f f i c u l t b e c a u s e , a s c h e m i s t s , we a r e u s e d t o h a n g i n g t h e e l e c t r o n d e n s i t y o n t h e n u c l e a r g e o m e t r y s k e l e t o n . I t w o u l d s e e m , a t f i r s t t h e n , t o mean a s e r i o u s l o s s o f i n f o r m a t i o n i n g o i n g t o P - s p a c e , b u t i t w i l l b e s h o w n l a t e r t h a t i n f a c t t h e n u c l e a r g e o m e t r y i n f o r m a t i o n i s s t i l l p r e s e n t i n t h e P - s p a c e r e p r e s e n t a t i o n , b u t i n a s o m e w h a t o b s c u r e d f o r m . 2.5.1 H y d r i d e s ( A H n ) T h e m o l e c u l a r o r b i t a l s o f h y d r i d e s ( A H n ) { B a s c h ( 1 9 7 2 ) } ( f i g u r e s 2 . 3 , 2 . 5 - 7 ) a r e s i m i l a r i n c h a r a c t e r t o t h e . a t o m i c o r b i t a l s o f t h e c o n s t i t u e n t h e a v y a t o m , a n d s o t h e momentum d e n s i t y m a p s w i l l a l s o b e s i m i l a r t o t h o s e o f t h e n s a n d n p a t o m i c o r b i t a l s ( f i g u r e 2 . 1 ) . I n p a r t i c u l a r , t h e a d d i t i o n o f t h e h y d r o g e n 1s f u n c t i o n s w i l l c a u s e t h e s p h e r i c a l d i s t r i b u t i o n o f t h e h e a v y a t o m A ( n s ) o r b i t a l t o c o n t r a c t i n t h e d i r e c t i o n s o f t h e A-H b o n d s . T h e c o m b i n a t i o n w i t h t h e A ( n p ) o r b i t a l s w i l l g e n e r a l l y s h r i n k t h e momentum d e n s i t y ( a s t h e P - s p a c e MOs a r e now l a r g e r ) , a n d , w h e n e v e r t h e s y m m e t r y o f t h e new m o l e c u l a r o r b i t a l i s l o w e r t h a n t h e o r i g i n a l a t o m i c n p 2/Momentum Space Chemistry CH 4 MOMENTUM DENSITY CHARGE DENSITY Figure 2.3 CH„ momentum and charge density maps. 2/Momentum Space Chemistry Figure 2.4 Schematic diagram of the constituent atomic o r b i t a l s of the CH, 1t 2 MO. 2/Momentum S p a c e C h e m i s t r y 54 NH3 MOMENTUM DENSITY CHARGE DENSITY F i g u r e 2.5 N H 3 momentum a n d c h a r g e d e n s i t y m a p s . 2/Momentum Space C h e m i s t r y H2Q MOMENTUM DENSITY CHARGE DENSITY 1 b i xz -r*-!—r -2 -1 n — i — r 3ai 1b, 2a i F i g u r e 2.6 H 20 momentum and charge d e n s i t y maps. 2/Momentum S p a c e C h e m i s t r y 56 HF MOMENTUM DENSITY CHARGE DENSITY F i g u r e 2.7 HF momentum a n d c h a r g e d e n s i t y m a p s . 2/Momentum S p a c e C h e m i s t r y 57 o r b i t a l , t h e P - s p a c e n o d a l p l a n e w i l l b e f i l l e d i n t o some e x t e n t . I l l u s t r a t i v e d e n s i t y maps f o r f i r s t r o w h y d r i d e s a r e g i v e n i n f i g u r e 2 . 3 , a n d 2 . 5 - 2 . 7 , a n d f o r H 2 S i n f i g u r e 4 . 5 . T h e i n n e r s - t y p e a , MOs ( A ( 2 s ) + H , ( 1 s ) + H 2 ( 1 s ) ) a r e a l l C l a s s I . T h e h i g h e r p - t y p e o r b i t a l s a r e C l a s s H i i f t h e n o d a l p l a n e i s s t i l l p r e s e n t , o r C l a s s H a i f i t h a s d e g e n e r a t e d t o a n o d a l s u r f a c e . T h e m e t h a n e 1 t 2 o r b i t a l s h o w s a n i n t e r e s t i n g d i f f e r e n c e f r o m t h e m o m e n t a l s o f t h e o t h e r f i r s t r o w h y d r i d e s . T h e l a t t e r e i t h e r p r e s e r v e o r d e s t r o y t h e n o d a l p l a n e i n f o r m i n g t h e m o l e c u l a r o r b i t a l . T h e m e t h a n e 1 t 2 momentum d e n s i t y f a l l s i n b e t w e e n : i t h a s t w o n o d a l l i n e s ( l i n e s o n w h i c h t h e d e n s i t y i s z e r o ) w h i c h b i s e c t t h e H-C-H a n g l e b e t w e e n H a t o m 1s f u n c t i o n s o f o p p o s i t e s i g n . M omentum c o n t o u r s u r f a c e s f o r t h i s o r b i t a l w i l l l o o k l i k e a r o u n d b a l l o o n w h i c h h a s b e e n p o k e d i n o n f o u r s i d e s ( f i g u r e 2 . 3 ) . I t was a g r a t i f y i n g e x e r c i s e t o w o r k o u t w h a t t h e 1 t 2 d e n s i t y map m u s t l o o k l i k e a c c o r d i n g t o t h e a b o v e p r i n c i p l e s a n d t h e Q p r o j e c t i o n s i n s e c t i o n 2 . 4 , a n d t h e n t o c o n f i r m i t b y c o m p u t i n g t h e d e n s i t y m a p s . T h i s o r b i t a l i s C l a s s I H i , i n t h e d i r e c t i o n j o i n i n g t w o H - a t o m s w i t h 1s AOs o f l i k e s i g n ( s e e t h e s c h e m a t i c r e p r e s e n t a t i o n o f t h e 1 t 2 MO, f i g u r e 2 . 4 ) a n d C l a s s H i i n t h e d i r e c t i o n o f t h e c e n t r a l c a r b o n 2 p a t o m i c o r b i t a l . 2/Momentum S p a c e C h e m i s t r y 58 2.5.2 D i a t o m i c m o l e c u l e s ( A 2 , A X ) T h i s s e c t i o n i s i l l u s t r a t e d w i t h t h e w a v e f u n c t i o n s o f H 2 a n d CO { B a s c h ( 1 9 7 2 ) } a n d N 2 { K u n z } . G o i n g b a c k t o b a s i c c h e m i s t r y , o n e d e s c r i b e s t h e s i m p l e s t c a s e o f a a b o n d i n g o r b i t a l b e t w e e n t w o i d e n t i c a l a t o m s , A, s e p a r a t e d b y d i s t a n c e R, a s h a v i n g o c c u r r e d w h e n e l e c t r o n d e n s i t y m o v e s f r o m i t s a t o m i c d i s t r i b u t i o n a r o u n d t h e n u c l e i i n t o t h e b o n d i n g r e g i o n b e t w e e n t h e n u c l e i , d e p i c t e d i n f i g u r e 2 . 8 . G e n e r a l l y t h i s p r o d u c e s a m o l e c u l a r o r b i t a l c e n t e r e d m i d w a y b e t w e e n t h e a t o m s a n d i s e l o n g a t e d a l o n g , a n d c o n s t r i c t e d p e r p e n d i c u l a r t o , t h e b o n d a x i s : i n f a c t , t h e f a m i l i a r ' l o z e n g e - s h a p e d ' <s b o n d i n g o r b i t a l . T h e r e f o r e , b y t h e R e c i p r o c i t y p r i n c i p l e , t h e P - s p a c e d e n s i t y w i l l a l s o b e m o r e o r l e s s s - t y p e b u t d i l a t e d p e r p e n d i c u l a r t o t h e b o n d a x i s a n d c o n t r a c t e d p a r a l l e l t o t h e b o n d d i r e c t i o n . T h i s i s i l l u s t r a t e d b y t h e 1*{g} o r b i t a l o f t h e H 2 e l e c t r o n d e n s i t y map ( f i g u r e 2.9) w h e r e t h e <r MO i s f o r m e d f r o m t h e g e r a d e c o m b i n a t i o n o f t w o 1s a t o m i c o r b i t a l s , a n d , t o a l e s s e r e x t e n t > i n t h e 2 * { g } o r b i t a l o f N 2 ( f i g u r e 2 . 1 0 ) . S u c h o r b i t a l s p r o j e c t a s C l a s s I ( s e c t i o n 2 . 4 . 2 ) . H i g h e r * - b o n d i n g o r b i t a l s s u c h a s t h e N 2 3<y{g} ( f i g u r e 2 . 1 0 ) d o n o t d i s p l a y s u c h c o n t r a c t i o n s . T h e y h a v e a s i g n i f i c a n t c o n t r i b u t i o n o f 2 p a t o m i c o r b i t a l s w h i c h p u t s d e n s i t y i n t o t h e a n t i b o n d i n g r e g i o n s a n d i n c r e a s e s t h e 2/Momentum Space C h e m i s t r y 59 Bonding A n t i b o n d i n g A A r~A A n t i b o n d i n g Bonding A n t i b o n d i n g X~| A hX A n t i b o n d i n g F i g u r e 2.8 Bonding and a n t i b o n d i n g r e g i o n s i n d i a t o m i c ( A 2 , AX) and t r i a t o m i c ( A X 2 , AXY) m o l e c u l e s . \ 2/Momentum S p a c e C h e m i s t r y 60 H 2 MOMENTUM DENSITY CHARGE DENSITY F i g u r e 2 . 9 H 2 momentum a n d c h a r g e d e n s i t y maps f o r t h e 1*{g} a n d t h e ( u n o c c u p i e d ) 1 « { u } M O s . 2/Momentum Space Chemistry Figure 2.10 N 2 momentum and charge density maps for the valence MOs. 2/Momentum S p a c e C h e m i s t r y CO MOMENTUM DENSITY CHARGE DENSITY g u r e 2.11 CO momentum a n d c h a r g e d e n s i t y m a p s . 2/Momentum S p a c e C h e m i s t r y 63 ' p s e u d o - a n g u l a r ' momentum ( s e e d e f i n i t i o n b e l o w ) . T h e w a v e f u n c t i o n i s now a t h r e e - l o b e d s t r u c t u r e w i t h n o d a l s u r f a c e s n e a r t h e n i t r o g e n a t o m s , w h i c h o v e r l a p s d e s t r u c t i v e l y w i t h F T w a v e s o f a p e r i o d a p p r o x i m a t e l y t w i c e t h e i n t e r n u c l e a r s p a c i n g , a n d c o n s t r u c t i v e l y w i t h F T w a v e s o f s o m e w h a t s h o r t e r p e r i o d . T h i s g i v e s r i s e t o t h e t w o n o d a l s u r f a c e s i n t h e N 2 3 c { g } momentum d e n s i t y map a t a b o u t p | | = + 0 . 7 a o _ 1 a n d t h e a r e a s o f momentum d e n s i t y a t p| | = + 1 . 4 a 0 " 1 . T h i s h i g h p | | d e n s i t y i s a r e s u l t o f t h e i n c r e a s e d a n t i b o n d i n g c h a r a c t e r a n d d e c r e a s e d b o n d i n g c h a r a c t e r o f t h e 3<r{g} o r b i t a l r e l a t i v e t o t h e 2<r{g}. T h e N 2 3 t f { g } o r b i t a l f a l l s i n t o C l a s s 1 1 1 i 2 • T h e n e x t c a s e i s t h e i r - b o n d i n g MO, a s e x e m p l i f i e d i n t h e N 2 1 i r { u } o r b i t a l . E v e r y o n e i s f a m i l i a r ( f i g u r e 2.10) w i t h t h e t w o b a n a n a - s h a p e d l o b e s o f t h i s o r b i t a l , f o r m e d f r o m t h e u n g e r a d e c o m b i n a t i o n o f N 2p{x} a n d 2p{y} a t o m i c o r b i t a l s . T h i s o r b i t a l l o o k s l i k e a n a t o m i c p o r b i t a l t h a t h a s b e e n s t r e t c h e d i n t h e b o n d d i r e c t i o n . A g a i n t h e r e f o r e , b y t h e R e c i p r o c i t y p r i n c i p l e , t h e P - s p a c e r e p r e s e n t a t i o n o f t h i s o r b i t a l s h o u l d , i n c o m p a r i s o n w i t h t h e a t o m i c N2p o r b i t a l , s h o w a l m o s t t h e same d e n s i t y d i s t r i b u t i o n p e r p e n d i c u l a r t o t h e b o n d , b u t w i t h a c o n t r a c t i o n i n t h e d i s t r i b u t i o n p a r a l l e l t o t h e b o n d , w h i c h i n f a c t i t d o e s . T h e 1rr{u} w a v e f u n c t i o n i s p r o j e c t i o n c l a s s H i . A n t i b o n d i n g o r b i t a l s a r i s e w i t h t h e c o m b i n a t i o n o f l i k e a t o m i c o r b i t a l s o f o p p o s i t e s i g n , s u c h t h a t a n o d a l p l a n e i s 2/Momentum Space Chemistry 64 introduced between the two centres, pushing electron density out of the bonding region. The molecular o r b i t a l has a 'pseudo-angular momentum' greater than the individual atomic o r b i t a l s : i . e . i f the AOs are two s- o r b i t a l s , the ungerade combination produces an MO which looks l i k e a p - o r b i t a l . On going to momentum space the nodal plane is preserved, and we get a p-type P-space o r b i t a l whose r a d i a l extent is inversely related to the bond length of the molecule. Thus the c h a r a c t e r i s t i c of **(2s) antibonding o r b i t a l s i s the complete removal of density from the perpendicular momentum di r e c t i o n , and a p||{max} which r e f l e c t s a convolution of the internuclear distance and the ra d i a l extent of the constituent atomic momentum density. This is an instance where the nuclear geometry information is apparent in the P-space density function, though in an obscure form. The N 2 2tf{u} o r b i t a l i s in Class H i . I have introduced a concept 'pseudo-angular momentum' here, which i s a hand-waving idea at best: what I am trying to refer to i s the increasing complexity of the nodal plane or nodal surface structure of molecular o r b i t a l s , going from the inner MOs to the outer ones. This tendency has p a r a l l e l s in the nodal structure of atomic o r b i t a l s which becomes more i n t r i c a t e with increasing p r i n c i p a l quantum number, however the analogy i s not exact because, by d e f i n i t i o n , molecules have more than one centre, and the term 'pseudo-angular momentum' to represent t h i s concept i s the best that can be 2/Momentum S p a c e C h e m i s t r y 65 d e v i s e d . F o r e x a m p l e , t h e d i f f e r e n c e b e t w e e n t h e 2 * { g } , 2<r{u} a n d 3<y{g} MOs o f N 2 ( f i g u r e 2 . 1 0 ) i s m a i n l y o n e o f ' p s e u d o - a n g u l a r momentum'; t h e n u m b e r o f n o d a l s u r f a c e s i s i n c r e a s i n g f r o m n o n e , t o o n e , t o t w o . A n o n - b o n d i n g o r b i t a l ( t h a t i s n o t n o n - b o n d i n g b y v i r t u e o f e q u a l a m o u n t s o f b o n d i n g a n d a n t i b o n d i n g c h a r a c t e r ) i s a t r i v i a l c a s e i n P - s p a c e a s i n R - s p a c e : t h e d e n s i t y w i l l b e v e r y c l o s e t o t h a t o f t h e c o n s t i t u e n t a t o m i c o r b i t a l , m o d i f i e d o n l y b y b o n d o s c i l l a t i o n e f f e c t s i n t h e c a s e o f d e g e n e r a t e n o n - b o n d i n g o r b i t a l s o n t w o o r m o r e c e n t r e s ( s e e s e c t i o n 2 . 6 ) . I n b e t w e e n t h e s e t w o e x t r e m e s o f b o n d i n g a n d a n t i b o n d i n g l i e t h e m o l e c u l a r o r b i t a l s o f h e t e r o n u c l e a r d i a t o m i c m o l e c u l e s w h e r e t h e m o l e c u l a r o r b i t a l s a r e c o n s t i t u t e d f r o m d i s s i m i l a r a t o m i c o r b i t a l s . a- a n d i r - b o n d i n g MOs a r e now c o n s t i t u t e d f r o m u n e q u a l a m o u n t s o f d i s s i m i l a r a t o m i c o r b i t a l s w i t h t h e r e s u l t t h a t t h e y h a v e some n o n - b o n d i n g c h a r a c t e r , a n d r e s e m b l e m o r e c l o s e l y t h e i r c o n s t i t u e n t a t o m i c o r b i t a l s . T h i s m e a n s t h e r e i s a r e d u c t i o n i n t h e b o n d i n g e f f e c t s d e s c r i b e d f o r t h e h o m o n u c l e a r MOs a b o v e : ( 1 ) B o n d i n g o r b i t a l s t h a t a r e m a i n l y <r(2s) o r r r ( 2 p ) s h o w l e s s c o n t r a c t i o n i n t h e b o n d a x i s d i r e c t i o n . E x a m p l e s a r e t h e 3<r ( c l a s s I ) a n d 1 IT ( c l a s s H i ) o r b i t a l s o f CO ( f i g u r e 2 . 1 1 ) a n d NO ( f i g u r e s 7.2 a n d 7 . 3 ) ; 2/Momentum S p a c e C h e m i s t r y 66 ( 2 ) c r - a n t i b o n d i n g o r b i t a l s t h a t a r e e s s e n t i a l l y o f < r * ( 2 s ) c h a r a c t e r now h a v e t h e p||=0 n o d a l p l a n e f i l l e d i n a l i t t l e . T h i s i s s e e n i n t h e 4<r o r b i t a l s o f CO a n d NO w h i c h a r e C l a s s H a i n t h e b o n d d i r e c t i o n a n d C l a s s I i n t h e p e r p e n d i c u l a r d i r e c t i o n . T h e r e a s o n f o r t h e p=0 i n t e n s i t y i s t h a t t h e r e a r e n o l o n g e r e q u a l p o s i t i v e a n d n e g a t i v e p a r t s o f t h e R - s p a c e w a v e f u n c t i o n , a n d s o t h e o v e r l a p w i t h t h e z e r o - f r e q u e n c y F T w a v e i s n o t e x a c t l y z e r o ; ( 3 ) < r - b o n d i n g o r b i t a l s t h a t a r e p r e d o m i n a n t l y <r(2p) n o l o n g e r s h o w n o d a l s u r f a c e s i n P - s p a c e f o r t h e same r e a s o n a s t h e <j-*(2s) c a s e . E x a m p l e s a r e a g a i n f o u n d i n t h e 5<r MOs o f CO a n d NO. T h e . f i n a l g e n e r a l i z a t i o n f o r b o n d i n g e f f e c t s i n homo-a n d h e t e r o n u c l e a r d i a t o m i c s may be s t a t e d a s f o l l o w s : (1) T h e b o n d i n g r e g i o n i n R - s p a c e i s t h e i n t e r a t o m i c r e g i o n ; b o n d i n g c h a r a c t e r i s i n d i c a t e d b y i n c r e a s e d d e n s i t y i n t h i s r e g i o n . T h e a n t i b o n d i n g r e g i o n i s a t t h e e n d s o f t h e m o l e c u l e : a n t i b o n d i n g c h a r a c t e r i s l i k e w i s e i n d i c a t e d b y d e n s i t y i n t h e s e r e g i o n s ; ( 2 ) T h e b o n d i n g d i r e c t i o n i n P - s p a c e i s t h e p l a n e p e r p e n d i c u l a r t o t h e b o n d a x i s ; b o n d i n g e f f e c t s a r e i n d i c a t e d b y a c o n t r a c t i o n o f d e n s i t y t o w a r d t h i s p l a n e , r e l a t i v e t o t h e a t o m i c d e n s i t y . A n t i b o n d i n g e f f e c t s a r e i n d i c a t e d b y a d i s p l a c e m e n t o f d e n s i t y f r o m t h i s p l a n e t o h i g h e r p | | r e g i o n s . 2/Momentum S p a c e C h e m i s t r y 67 T h e d i s c u s s i o n o f ? r * ( 2 p ) MOs i s d e f e r r e d t o C h a p t e r 7. 2.5 . 3 L i n e a r s y m m e t r i c t r i a t o m i c s ( A X 2 ) T h e s i t u a t i o n f o r t h e s e h e a v y a t o m t r i a t o m i c s i s m o r e c o m p l i c a t e d t h a n f o r t h e p r e v i o u s s y s t e m s , b e c a u s e o f t h e i n c r e a s e d n u m b e r o f v a l e n c e e l e c t r o n s . T h e b o n d i n g r e g i o n s . ( f i g u r e 2 .8) a r e now t h e t w o a r e a s b e t w e e n t h e a t o m s , a n d t h e a n t i b o n d i n g r e g i o n s a r e o u t s i d e t h e e n d a t o m s o f t h e m o l e c u l e ; a n o r b i t a l w h i c h p l a c e s d e n s i t y o n t h e e n d a t o m s a n d n o n e o n t h e m i d d l e a t o m may be c o n s i d e r e d m o s t l y n o n - b o n d i n g , s i n c e t h e e n d a t o m s a r e f a r e n o u g h a p a r t a s t o h a v e l i t t l e i n t e r a c t i o n . T h e R - s p a c e w a v e f u n c t i o n s o f s u c h s y s t e m s h a v e s y m m e t r y ' g e r a d e ' , o r ' u n g e r a d e ' , d e p e n d i n g o n w h e t h e r t h e y c h a n g e s i g n o n i n v e r s i o n t h r o u g h t h e c e n t r a l a t o m c o o r d i n a t e s , a n d a r e c o r n d e p e n d i n g o n w h e t h e r t h e y c h a n g e s i g n o n r e f l e c t i o n i n a <y{v} p l a n e . T h e u n d e r s t a n d i n g o f b o n d i n g i n A X 2 s y s t e m s h i n g e s o n t h e f a c t t h a t t h e momentum d e n s i t y f u n c t i o n f o r a n L C A O w a v e f u n c t i o n c a n b e b r o k e n u p i n t o o n e - c e n t e r a n d t w o - c e n t e r p a r t s : ( 2 . 1 1 ) P{i](g) = PI(E) + fi2(£) T h e e x p l i c i t f o r m f o r t h e s e i s g i v e n i n A p p e n d i x B. T h e o n e - c e n t e r p a r t i s j u s t t h e sum o f t h e i n d i v i d u a l b a s i s s e t 2/Momentum Space Chemistry 68 P-space density functions and i s everywhere p o s i t i v e . For an AX2 system aligned in the z-direction the one-centre part w i l l be a combination of 2s and 2p{z} density, leading to a roughly spherical momentum density function, but elongated in the p|| di r e c t i o n and centred at the P-space o r i g i n . The two-center part i s the sum of a l l the interference terms between two atomic centers, and i s not r e s t r i c t e d to positive sign. As with any interference e f f e c t , there are constructive and destructive components which vary as a function of frequency and interatomic separation. Therefore, electron amplitude at two (or more) areas separated by r. reinforces the atomic one-center momentum density near point 2=2njr/r (n = 0,1,2...) in P-space i f the areas are of the same sign, or near point p_= (2n+1 ) rr/r (n=0,1,2...) i f these areas are of o s c i l l a t i n g sign ( i . e . +|~| +|- and so on). The atomic density is likewise reduced near p_= (2n+1 ) t r / r where there i s electron density of the same wavefunction sign separated by r, and near p_=2njr/r for o s c i l l a t i n g signs. This i s i l l u s t r a t e d schematically in figure 2.6 and summarised in Table 2.2. Referring again to the' Reciprocity p r i n c i p l e , i f the R-space bonding regions are on the bond axis to either side of the central atom, and the antibonding regions are outside either end of the molecule, then in P-space antibonding density w i l l reinforce low momentum regions, and bonding density reinforces high momentum regions. -Non-bonding 2/Momentum Space C h e m i s t r y 6 9 SAME SIGN ALTERNATING SIGN F i g u r e 2.12 Schematic diagrams of the o v e r l a p of d i f f e r e n t f r e q u e n c y components of the F o u r i e r T r a n s f o r m w i t h w a v e f u n c t i o n s of d i f f e r e n t c h a r a c t e r . 2/Momentum S p a c e C h e m i s t r y 70 T a b l e 2 . 2 C o n s t r u c t i v e a n d d e s t r u c t i v e i n t e r f e r e n c e Same s i g n O s c i l l a t i n g s i g n C o n s t r u c t i v e Q=2T\V/V_ p_= (2n+1 ) r r / r D e s t r u c t i v e 2= (2n+1 ) rr/r_ 2 = 2 n f f / £ n = 0 , 1 , 2 . . . T a b l e 2 . 2 C o n s t r u c t i v e a n d d e s t r u c t i v e i n t e r f e r e n c e 2/Momentum S p a c e C h e m i s t r y 71 d e n s i t y w i l l f a l l s o m e w h e r e i n b e t w e e n . I n t h e c a s e o f s y m m e t r i c A X 2 s y s t e m s t h e d e s t r u c t i v e p a r t s o f t h e i n t e r f e r e n c e t e r m c a n b e s t r o n g e n o u g h t o e x a c t l y c a n c e l t h e o n e - c e n t r e p a r t a t c e r t a i n v a l u e s o f p | | , r e s u l t i n g i n n o d a l p l a n e s . H o w e v e r , i n a s y m m e t r i c A X Y s y s t e m s t h i s w i l l n o l o n g e r o c c u r ( a s i t u a t i o n s i m i l a r t o h o m o n u c l e a r a n d h e t e r o n u c l e a r d i a t o m i c s ) , a n d t h e o n l y n o d a l p l a n e s w i l l b e t h o s e i n t h e ir o r b i t a l s . T h e e x a c t v a l u e s o f momentum c a n b e w o r k e d o u t k n o w i n g t h e s y m m e t r y o f t h e o r b i t a l a n d t h e g e o m e t r y o f t h e m o l e c u l e . T h e d e t a i l e d c a s e f o r C 0 2 i s p r e s e n t e d i n s e c t i o n 6.5. I t m i g h t b e n o t e d t h a t t h e s e a r g u m e n t s c a n a l s o b e a p p l i e d t o t h e b o n d i n g i n d i a t o m i c m o l e c u l e s , t o r a t i o n a l i z e t h e s h a p e o f t h e P - s p a c e d e n s i t y i n t h e 3 * { g } o r b i t a l s o f N 2 a n d 0 2 . 2.6 B o n d O s c i l l a t i o n F u r t h e r d e v e l o p m e n t o f t h e i d e a o f c h a r g e s e p a r a t i o n r e i n f o r c i n g c e r t a i n c o m p o n e n t s o f t h e F T w a v e l e a d s t o t h e c o n c e p t o f b o n d o s c i l l a t i o n { L e v i n ( 1 9 7 5 ) , C o u l s o n ( 1 9 4 4 ) } . T h e momentum d i s t r i b u t i o n w i l l , a s a r e s u l t o f t h e f o u r 2/Momentum S p a c e C h e m i s t r y 72 s i t u a t i o n s i n T a b l e 2 . 2 , h a v e w e a k f r i n g e s g o i n g o u t t o l a r g e p , s e p a r a t e d b y 2n/R w h e r e R i s t h e v e c t o r b e t w e e n t w o n u c l e i . T h i s p r o d u c e s a s i n u s o i d a l m o d u l a t i o n o f t h e F T a m p l i t u d e i n t h e R d i r e c t i o n . T h i s e f f e c t i s a n a l o g o u s t o o p t i c a l i n t e r f e r e n c e f r i n g e s i n t h e w e l l - k n o w n d o u b l e s l i t e x p e r i m e n t . T h e s i n u s o i d a l m o d u l a t i o n a r i s e s f r o m t h e f a c t o r e x p ( i p _ . R { A J } ) b e t w e e n p a i r s o f a t o m s A , J ('see e q u a t i o n B . 8 ) . B o n d o s c i l l a t i o n e f f e c t s a r e m o s t n o t i c e a b l e w hen t h e t w o a r e a s o f c h a r g e d e n s i t y a r e w e l l - l o c a l i s e d a n d d i s t i n c t f r o m e a c h o t h e r . E x c e l l e n t e x a m p l e s a r e t h e c o r e MOs o f h o m o n u c l e a r d i a t o m i c s ( s e e e x a m p l e N 2 1 t f { g } , 1<r{u} { K u n z } , f i g u r e 2 . 1 3 ) . H e r e t h e m a g n i t u d e o f t h e o s c i l l a t i o n i s a s l a r g e a s t h e p=0 d e n s i t y . I n v a l e n c e MOs w h e r e a l a r g e p a r t o f t h e d e n s i t y i s l o c a t e d i n t h e b o n d i n g r e g i o n , t h e m o d u l a t i o n s d u e t o b o n d o s c i l l a t i o n a r e m u c h w e a k e r , a n d may o n l y b e s e e n b y p l o t t i n g MD maps w i t h v e r y l o w c o n t o u r v a l u e s o u t t o l a r g e m o m e n t a ( e . g . f i g u r e 2 . 1 4 ) . V a l e n c e b o n d o s c i l l a t i o n s may b e c o m e s t r o n g e r i f t h e r e a r e s e v e r a l a t o m s a t s i m i l a r i n t e r a t o m i c s p a c i n g s , w h e r e u p o n s e v e r a l o s c i l l a t i o n s o f t h e F T w a v e a r e r e i n f o r c e d , w i t h c o n s e q u e n t i a l i n c r e a s e i n m o d u l a t i n g a m p l i t u d e . T h e n u c l e a r g e o m e t r y i s p r e s e r v e d i n t h e P - s p a c e r e p r e s e n t a t i o n i n t h e p r e s e n c e o f b o n d o s c i l l a t i o n s s i n c e t h e s p a c i n g b e t w e e n s u c c e s s i v e p e a k s o f t h e b o n d o s c i l l a t i o n d e n s i t y i s i n v e r s e l y p r o p o r t i o n a l t o t h e i n t e r a t o m i c d i s t a n c e o f t h e t w o c e n t r e s . I n a p o l y a t o m i c s y s t e m e a c h 2/Momentum Space Chemistry 73 N2 MOMENTUM DENSITY CHARGE DENSITY Figure 2.13 N 2 inner s h e l l U{g} and U{u} momentum and charge density maps, showing bond o s c i l l a t i o n . 2/Momentum Space C h e m i s t r y 74 F i g u r e 2.14 0 3 3a, o r b i t a l momentum and charge d e n s i t y maps showing e x t e n s i v e bond o s c i l l a t i o n . There a r e 13 c o n t o u r s i n the P-space map a t f r a c t i o n s .5, .2, .1, .05, .02, and so on down t o .00005 of the maximum d e n s i t y . 2/Momentum S p a c e C h e m i s t r y 75 p a i r o f a t o m s w i l l g i v e r i s e t o i t s own b o n d o s c i l l a t i o n s e r i e s , a n d s o t h e n u c l e a r g e o m e t r y i s a p p a r e n t i n m u c h t h e same way a s t h e c r y s t a l s t r u c t u r e i s a p p a r e n t f r o m a n X - r a y c r y s t a l l o g r a p h y s t u d y . A n e x a m p l e o f t h i s i s t h e 3 a , MO o f 0 3 ( o z o n e ) { B a s c h ( 1 9 7 2 ) } s h o w n i n f i g u r e 2 . 1 4 . I t w i l l b e n o t e d t h a t a s i d e e f f e c t o f t h e b o n d o s c i l l a t i o n p h e n o m e n o n i s a n a p p a r e n t e l o n g a t i o n o f t h e P - d e n s i t y p e r p e n d i c u l a r t o t h e b o n d a x i s . T h i s e f f e c t d o e s n o t i n d i c a t e t h a t t h e MO i n q u e s t i o n n e c e s s a r i l y h a s s i g n i f i c a n t b o n d i n g c h a r a c t e r . An e x t r e m e e x a m p l e i s N 2 < r { l s } o r b i t a l : we know t h i s c o r e MO t o be n o n - b o n d i n g , b u t t h e r e a p p e a r s t o be a v e r y s t r o n g e l o n g a t i o n i n d i c a t i n g b o n d i n g . H o w e v e r , t h e o v e r a l l d e n s i t y i s s t i l l m o r e o r l e s s s p h e r i c a l l y d i s t r i b u t e d a b o u t t h e P - s p a c e o r i g i n , i n d i c a t i n g a n o n - b o n d i n g MO. 2.7 T h e S p h e r i c a l A v e r a g e A s g a s e o u s t a r g e t s a r e u s e d i n t h e b i n a r y ( e , 2 e ) e x p e r i m e n t a s p h e r i c a l a v e r a g e o f t h e t h r e e - d i m e n s i o n a l momentum d e n s i t y i s o b s e r v e d o n a c c o u n t o f t h e r o t a t i o n a l m o t i o n o f t h e t a r g e t m o l e c u l e s . T h i s n a t u r a l l y o b s c u r e s i n f o r m a t i o n i n t h e momentum d e n s i t y , b u t t h i s i s n o t s u c h a h a n d i c a p a s m i g h t be t h o u g h t s i n c e t h e o b s c u r e d a n g u l a r f o r m 2/Momentum S p a c e C h e m i s t r y 76 i s o f t e n d i c t a t e d b y t h e s y m m e t r y o f t h e o r b i t a l a n d we c a n o f t e n t i m e s w o r k o u t w h a t t h a t a n g u l a r f o r m m u s t b e . T h e s p h e r i c a l a v e r a g e d o e s t e n d t o d i s t o r t t h e r e l a t i v e a m o u n t s o f v a r i o u s a n g u l a r momentum c o m p o n e n t s o f t h e t o t a l d e n s i t y f u n c t i o n i n f a v o u r o f t h e l o w e r 1 c o n t r i b u t i o n s . A n e x a m p l e o f t h i s i s t h e 4<r o r b i t a l o f NO ( s e e f i g u r e s 7.2 a n d 7.3) w h e r e t h e momentum d e n s i t y map s h o w s a m a i n l y p - t y p e < r * ( 2 s ) o r b i t a l w i t h a s m a l l a m o u n t o f d e n s i t y (<10 p e r c e n t o f t h e maximum) a t p = 0 : t h e s p h e r i c a l l y - a v e r a g e d momentum d i s t r i b u t i o n ( f i g u r e 7 . 1 ) s h o w s m u c h m o r e i n t e n s i t y a t q = 0 , r e l a t i v e t o t h e m a x i m u m . T h e s p h e r i c a l a v e r a g e a l s o t e n d s t o d i s t o r t t h e d e p t h o f m i n i m a i n n o d a l a r e a s : a n o d a l o r n e a r - n o d a l s u r f a c e i n a l i n e a r m o l e c u l e a t f o r e x a m p l e p||=0.5 w i l l b e f i l l e d i n t o some e x t e n t b y d e n s i t y a t pj_>0.5 w h e n t h e s p h e r i c a l a v e r a g e i s t a k e n . A g o o d e x a m p l e o f t h i s i s t h e 3 * { g } MO o f 0 2 ( f i g u r e 7 . 2 c ) . 77 CHAPTER 3 EXPERIMENTAL ...they did not understand or l i k e machines more complicated than a forge bellows, a water m i l l , or a hand loom... This chapter describes the physical construction of the spectrometer, the vacuum support systems, the electronics of the spectrometer control and signal processing and data a c q u i s i t i o n systems, operating and c a l i b r a t i o n procedures, and data analysis. The basic system has been reported in the l i t e r a t u r e {Hood (1977)}. Further modifications incorporated during the present work are indicated in the following text. 3.1 The Vacuum System As with a l l electron spectrometers the instrument can only function in high vacuum (<l0-"torr; 1torr=1mmHg). Accordingly, the spectrometer i s housed in a vacuum chamber of about 0.1m3 volume. This chamber is pumped by a 6-inch Varian VHS-4 o i l d i f f u s i o n pump to attain a base pressure of 3 / E x p e r i m e n t a l 78 F i g u r e 3.1 Schematic diagram of the b i n a r y (e,2e) s p e c t r o m e t e r used t o o b t a i n most of the e x p e r i m e n t a l r e s u l t s i n t h i s t h e s i s . Legend: CMA C y l i n d r i c a l m i r r o r a n a l y s e r segment C C h a n n e l t r o n ZL E l e c t r o s t a t i c zoom l e n s FC Faraday cup beam dump T T u r n t a b l e EC End c o r r e c t o r s SP2 Spray p l a t e 2 GC Gas c e l l Q Quad e l e c t r o n d e f l e c t o r s EL E i n z e l l e n s A Anode G G r i d F F i l a m e n t 3 / E x p e r i m e n t a l 79 about 5 x 1 0 ~ 7 t o r r ; Convalex d i f f u s i o n pump o i l i s used. The d i f f u s i o n pump exhaust i s pumped away by a Welch Duo-seal 1402 r o t a r y m e c h a n i c a l pump. The system can be pumped down t o o p e r a t i n g p r e s s u r e i n about 3 h o u r s , but i t i s customary t o w a i t o v e r n i g h t b e f o r e commencing o p e r a t i o n , t o a l l o w the chamber w a l l s and s p e c t r o m e t e r s u r f a c e s t o outga s . The ambient p r e s s u r e i s measured by a Veeco i o n i z a t i o n gauge mounted on the t o p of the s p e c t r o m e t e r chamber. The p o r t l e a d i n g t o the gauge i s s h i e l d e d w i t h a m e t a l p l a t e t o e l i m i n a t e i n t e r f e r e n c e from the l a r g e numbers of e l e c t r o n s produced by the i o n gauge f i l a m e n t . A l l m e c h a n i c a l j o i n t s i n the vacuum h o u s i n g a r e s e a l e d w i t h V i t o n 0 - r i n g g a s k e t s , or e l s e s o l d e r e d or welded permanently. Two p l e x i g l a s - c o v e r e d v i e w i n g p o r t s on the s i d e and t o p of the vacuum chamber a l l o w o b s e r v a t i o n of the s p e c t r o m e t e r i n o p e r a t i o n . 3.2 The Spectrometer The s p e c t r o m e t e r i t s e l f ( f i g u r e 3.1) c o n s i s t s of an e l e c t r o n beam s o u r c e , a beam s t e e r i n g u n i t , a gas c e l l , and two e l e c t r o n a n a l y s e r s each made up of a t h r e e - e l e m e n t e l e c t r o n l e n s . a n d a c y l i n d r i c a l m i r r o r segment e l e c t r o n energy a n a l y s e r w i t h c h a n n e l t r o n e l e c t r o n d e t e c t o r s , and a m e c h a n i c a l a n g l e - s c a n n i n g system. 3 / E x p e r i m e n t a l 80 3.2.1 T h e e l e c t r o n g u n T h e e l e c t r o n g u n u s e d f o r m o s t o f t h i s w o r k i s a C l i f t r o n i c s C E 5 A H u n i t d e s i g n e d t o s u p p l y a n e l e c t r o n beam o f i n t e r m e d i a t e e n e r g y ( 1 0 0 - 2 0 0 0 e V ) . T h e g u n c o n s i s t s o f s e v e r a l e l e m e n t s : ( 1 ) A t h e r m i o n i c e m i t t e r , u s u a l l y a t u n g s t e n w i r e h a i r p i n o r r i b b o n l o o p ; ( 2 ) A g r i d , i m m e d i a t e l y n e x t t o t h e f i l a m e n t , w i t h a s m a l l a p e r t u r e (0.5mm) t h r o u g h w h i c h t h e e x t r a c t e d e l e c t r o n s p a s s ; ( 3 ) A n a n o d e 1mm f r o m t h e g r i d , o f s i m i l a r c o n s t r u c t i o n ; ( 4 ) A t h r e e - e l e m e n t e i n z e l l e n s a f t e r t h e a n o d e . T h e f i l a m e n t i s e l e c t r i c a l l y h e a t e d w i t h a DC c u r r e n t t o i t s o p e r a t i n g t e m p e r a t u r e ( 1 0 0 0 - 1 5 0 0 ° C ) a t w h i c h p o i n t e l e c t r o n s c a n b e e x t r a c t e d e a s i l y b y a n e l e c t r i c f i e l d . T h i s e x t r a c t i o n f i e l d i s d e f i n e d b y t h e s m a l l a p e r t u r e s i n t h e g r i d a n d a n o d e , a n d b y t h e r e l a t i v e e l e c t r i c p o t e n t i a l o n t h e t h r e e e l e m e n t s . N o r m a l l y t h e f i l a m e n t i s m a i n t a i n e d a t - E 0 v o l t s ( i . e . i f a 4 0 0 e V b e a m i s d e s i r e d t h e n t h e f i l a m e n t i s s e t t o - 4 0 0 v o l t s w i t h r e s p e c t t o t h e g a s c e l l w h i c h i s a t g r o u n d p o t e n t i a l ) . T h e g r i d i s u s u a l l y s e t t o a p o t e n t i a l c l o s e t o t h a t o f t h e f i l a m e n t . T h e f i l a m e n t / g r i d c o n f i g u r a t i o n now d e f i n e s a r e l a t i v e l y f i e l d - f r e e r e g i o n a t t h e t i p o f t h e f i l a m e n t w h e r e a s p a c e c h a r g e o r ' c l o u d ' o f f r e e e l e c t r o n s b u i l d s u p . T h e a n o d e i s s e t s e v e r a l h u n d r e d 3 / E x p e r i m e n t a l 81 v o l t s p o s i t i v e w i t h r e s p e c t t o t h e g r i d a n d , b e c a u s e o f t h e s m a l l s i z e o f t h e a p e r t u r e s i n t h e g r i d a n d a n o d e , t h i s d e f i n e s a v e r y s h a r p n e e d l e - s h a p e d e x t r a c t i o n f i e l d i n t o w h i c h t h e s p a c e c h a r g e f l o w s . T h e s h a r p e r t h i s e x t r a c t i o n f i e l d c a n b e m a d e , t h e m o r e c o l l i m a t e d w i l l b e t h e r e s u l t i n g e l e c t r o n b e a m . T h e p h y s i c a l l i n e - u p a n d v o l t a g e s o f t h e f i l a m e n t / g r i d / a n o d e a s s e m b l y i s c r u c i a l t o t h e p r o d u c t i o n o f a g o o d b e a m . B y ' g o o d ' h e r e I mean a w e l l - c o l l i m a t e d b eam, w i t h h i g h f l u x a n d o f s m a l l e n e r g y s p r e a d . T h e f i l a m e n t - g r i d s p a c i n g i s e s p e c i a l l y c r i t i c a l : b y t r i a l a n d e r r o r a s e p a r a t i o n o f 0.2-0.3mm w a s f o u n d t o g i v e g o o d , r e p r o d u c i b l e r e s u l t s . T h e e i n z e l l e n s i n t h e g u n a l l o w s t h e o p e r a t o r t o c o r r e c t a n y s m a l l d i v e r g e n c e s i n t h e beam e m a n a t i n g f r o m t h e a n o d e . I n t h r e e - e l e m e n t e i n z e l l e n s e s t h e f i r s t a n d l a s t e l e m e n t s a r e a t t h e same p o t e n t i a l ( i n t h i s c a s e , a t g r o u n d p o t e n t i a l ) a n d t h e m i d d l e e l e m e n t p o t e n t i a l i s a d j u s t e d s o a s t o o b t a i n t h e d e s i r e d f o c u s s i n g e f f e c t . W i t h t u n g s t e n r i b b o n l o o p f i l a m e n t s d e p e n d i n g o n t h e g a s p r e s s u r e a n d t y p e o f g a s u s e d , 4 0 0 e V b e a m s o f 2 0 - l 0 0 » # A c o u l d b e o b t a i n e d , c o l l i m a t e d s o w e l l t h a t 98 p e r c e n t o f t h e beam f l u x w o u l d p a s s t h r o u g h a' 2mm a p e r t u r e i n a s p r a y p l a t e 2 0cm r e m o v e d f r o m t h e g u n . W i t h t h o r i a t e d t u n g s t e n w i r e h a i r p i n f i l a m e n t s t h e beam w a s l e s s g o o d ( 1 0 - 5 0 M A , 90-95 p e r c e n t f o c u s s e d t h r o u g h t h e s p r a y p l a t e ) b u t t e n d e d t o l a s t 5 0 - 1 0 0 p e r c e n t l o n g e r b e f o r e b u r n i n g o u t . F a c t o r s t e n d i n g t o p r o l o n g t h e l i f e o f t h e f i l a m e n t a r e : ._. • - " 3 / E x p e r i m e n t a l 82 (1) Running a t lower t e m p e r a t u r e s ( i . e . lower f i l a m e n t c u r r e n t s ) ; (2) U s i n g t u n g s t e n w i r e f i l a m e n t m a t e r i a l ( 3 ) U s i n g r e l a t i v e l y i n e r t gases a t low p r e s s u r e s . F a c t o r s which tend t o s h o r t e n the f i l a m e n t l i f e a r e : (1) Poor p h y s i c a l l i n e - u p of the f i l a m e n t or h i g h f l u x r e q u i r e m e n t s which n e c e s s i t a t e h i g h f i l a m e n t t e m p e r a t u r e s ; (2) U s i n g t u n g s t e n r i b b o n f i l a m e n t m a t e r i a l ; ( 3 ) Running r e a c t i v e or c o r r o s i v e gases a t h i g h p r e s s u r e s . The beam produced by the gun w i l l have a f i n i t e energy s p r e a d . T h i s i s due t o the Boltzmann d i s t r i b u t i o n of e l e c t r o n e n e r g i e s i n the f i l a m e n t m a t e r i a l which i s heated t o h i g h t e m p e r a t u r e , and t o the c h a r a c t e r i s t i c s of the space charge g e n e r a t e d between the f i l a m e n t and the g r i d . F a c t o r s t e n d i n g t o reduce the energy s p r e a d of the e l e c t r o n beam a r e : (1) Lower f i l a m e n t t e m p e r a t u r e s ; (2) Having the f i l a m e n t and the g r i d a t s i m i l a r p o t e n t i a l s . F a c t o r s a f f e c t i n g the beam f l u x , energy and c o l l i m a t i o n a r e complex and i n t e r r e l a t e d , so i t i s not a s t r a i g h t f o r w a r d m a tter t o d i s c u s s t h e s e i n any s i m p l e way. R e c e n t l y i t has been found t h a t f o r 1200eV o p e r a t i o n a 3 / E x p e r i m e n t a l 83 h a i r p i n t u n g s t e n f i l a m e n t mounted so t h a t the t i p p r o t r u d e s i n t o the g r i d - a n o d e space g i v e s good beams up t o 1 0 0 M A . In t h i s mode of o p e r a t i o n the e l e c t r o n s emerging from the anode i s q u i t e d i v e r g e n t , and one r e l i e s on the e i n z e l l e n s t o c o l l i m a t e the beam. The e l e c t r o n gun i s surrounded by a l o o s e w i r e mesh which i s b i a s e d a t the cathode ( f i l a m e n t ) p o t e n t i a l . The mesh p r e v e n t s unwanted e l e c t r o n s p o u r i n g out of the back of the gun from s t r a y i n g near the a n a l y s e r s . 3.2.2 The beam s t e e r i n g u n i t The beam s t e e r i n g u n i t i s i n t e n d e d t o c o r r e c t f o r r e s i d u a l magnetic f i e l d s and m e c h a n i c a l m i s a l i g n m e n t as t h e beam t r a v e l s from the gun t o the gas c e l l . The u n i t c o n s i s t s of two s p r a y p l a t e s , a Faraday cup beam dump, and two s e t s of e l e c t r o s t a t i c beam d e f l e c t o r s , a l l mounted c o a x i a l l y w i t h the e l e c t r o n gun. Spray p l a t e s a r e s i m p l y p l a t e s w i t h s m a l l a p e r t u r e s t h r o u g h which the beam p a s s e s , c o n n e c t e d t o a microammeter. I f any p a r t of the beam, t h r o u g h magnetic f i e l d d e f l e c t i o n or poor a l i g n m e n t or f o c u s s i n g , s h o u l d s t r i k e the s p r a y p l a t e , then the r e s u l t i n g c u r r e n t can be obs e r v e d and the beam c h a r a c t e r i s t i c s c o r r e c t e d . S i n c e t h e r e i s one s p r a y p l a t e b e f o r e the gas c e l l and one a f t e r , i t i s p o s s i b l e t o ensure t h a t the beam passes down the a x i s of the gas c e l l i n s p i t e of r e s i d u a l magnetic f i e l d s which may t e n d to d e f l e c t i t . The Faraday cup s e r v e s t o c o l l e c t the 3 / E x p e r i m e n t a l 84 u n s c a t t e r e d beam and p r e v e n t s i t from c a u s i n g i n t e r f e r e n c e i n p e n e t r a t i n g the a n a l y s e r s . The Faraday cup i s a l s o connected t o the microammeter so the t o t a l i n c i d e n t beam c u r r e n t can be m o n i t o r e d . 3.2.3 The gas c e l l The gas c e l l i t s e l f i s s i m p l y a tube b l o c k e d o f f a t both ends, mounted c o a x i a l l y w i t h the e l e c t r o n gun. A p e r t u r e s a r e made i n the ends f o r the beam t o pass t h r o u g h , and s l o t s a r e c u t around the m i d d l e f o r the s c a t t e r e d e l e c t r o n s t o pass out of the c e l l . Where the s l o t s a r e open t o the vacuum chamber the s e open a r e a s a r e c o v e r e d w i t h s l i d i n g m e t a l shim b l i n d s . T h i s a g a i n i s i n t e n d e d t o p r e v e n t n o i s e from s t r a y e l e c t r o n s and t o make the c e l l more n e a r l y g a s - t i g h t . A s m a l l h o l e i s made t o admit the gas. The gas c e l l p e r m i t s a l o c a l i n c r e a s e i n the gas d e n s i t y i n the s c a t t e r i n g r e g i o n , w i t h o u t h a v i n g t o f l o o d the whole vacuum chamber t o a c h i e v e the d e s i r e d t a r g e t d e n s i t y . I t has been e s t i m a t e d {Hood (1977)} t h a t the gas p r e s s u r e i n s i d e the c e l l i s a p p r o x i m a t e l y one t o two o r d e r s of magnitude g r e a t e r than the ambient vacuum chamber p r e s s u r e . 3 / E x p e r i m e n t a l 8 5 3.2.4 The e l e c t r o n l e n s e s The t h r e e - e l e m e n t e l e c t r o n l e n s e s which t r a n s p o r t the e l e c t r o n s from the s c a t t e r i n g r e g i o n t o the a n a l y s e r s have two f u n c t i o n s : (1) To f o c u s t h o s e s c a t t e r e d p a r t i c l e s w i t h the c o r r e c t energy and t r a j e c t o r y on t o the a n a l y s e r ' e n t r a n c e a p e r t u r e ; (2) To r e t a r d the e n e r g i e s of the s c a t t e r e d p a r t i c l e s by some f i x e d a d j u s t a b l e amount E { r e t a r d } . Three-element zoom l e n s e s have the p r o p e r t y t h a t d i f f e r e n t r e t a r d r a t i o s may be o b t a i n e d w i t h o u t a l t e r i n g the f o c a l p o s i t i o n s of the l e n s . Each l e n s has a beam d e f l e c t i o n u n i t i m m e d i a t e l y b e f o r e the a n a l y s e r e n t r a n c e a p e r t u r e t o c o r r e c t f o r r e s i d u a l magnetic f i e l d and m e c h a n i c a l m i s a l i g n m e n t e f f e c t s . Each l e n s has an e n t r a n c e a p e r t u r e which s e r v e s as an a n g u l a r s t o p p r e v e n t i n g the e n t r a n c e of a l a r g e number of unwanted e l e c t r o n s . The l e n s e s a r e mounted a t a p o l a r a n g l e of 45°. T h i s p a r t i c u l a r a n g l e i s chosen f o r two r e a s o n s : (1) B i n a r y e n c o u n t e r t h e o r y d i c t a t e s t h a t the c l o s e s t e-e c o l l i s i o n produces s c a t t e r e d e l e c t r o n s a t a p p r o x i m a t e l y t h i s p o l a r a n g l e ; (2) More p a r t i c u l a r l y , i t i s p o s s i b l e t o measure down t o l o w e s t q o n l y near a 45° p o l a r a n g l e . To be p r e c i s e , the l o w e s t measurable q f o r an i n c i d e n t energy of 400eV and a r e p r e s e n t a t i v e b i n d i n g energy of 15 eV i s 0.1a o"V, and o c c u r s a t *=0° and 9=44.6°. The 3 / E x p e r i m e n t a l 86 measured q r i s e s r a p i d l y i f the p o l a r a n g l e i s changed even s l i g h t l y (see f i g u r e s 3.8 and 3.9). T h i s t o p i c i s t r e a t e d more f u l l y i n s e c t i o n 3.5. 3.2.5 The c y l i n d r i c a l m i r r o r segment a n a l y s e r s The e l e c t r o n a n a l y s e r s a r e c y l i n d r i c a l m i r r o r a n a l y s e r (CMA) segments. A c y l i n d r i c a l m i r r o r a n a l y s e r i s d e f i n e d as two i n f i n i t e l y l o n g c o a x i a l m e t a l c y l i n d e r s ( f i g u r e 3.2). They have the p r o p e r t y t h a t c h a r g e d p a r t i c l e s emanating from a p o i n t (A) w i t h i n the i n n e r c y l i n d e r can, i f they are of the c o r r e c t energy and t r a j e c t o r y , be f o c u s s e d on a n o t h e r p o i n t (B) w i t h i n the i n n e r c y l i n d e r by c o r r e c t l y a p p l y i n g e l e c t r i c p o t e n t i a l s t o the two e lements. Inasmuch as i n f i n i t e l y l o n g c y l i n d e r s a r e u n a t t a i n a b l e by mere m o r t a l s , the u s u a l p r a c t i c e i s t o t r u n c a t e the c y l i n d e r s somewhere beyond p o i n t s A and B, and p l a c e c o r r e c t i n g elements i n the ends of the c y l i n d e r s t o m a i n t a i n the o r i g i n a l e l e c t r i c f i e l d . S i n c e i n the p r e s e n t a p p l i c a t i o n i t i s a l s o i m p r a c t i c a l t o have f u l l y c y l i n d r i c a l a n a l y s e r s , they have a l s o been t r u n c a t e d i n the a z i m u t h a l d i r e c t i o n t o a l l o w room f o r the gas c e l l and l e n s e s : hence the name CMA segments. Elements are a l s o p r o v i d e d t o c o r r e c t f o r the a z i m u t h a l t r u n c a t i o n . One of the a n a l y s e r s i s f i x e d , and the o t h e r i s mounted on a r o t a t i n g t u r n t a b l e t o p e r m i t the v a r i a t i o n of the a z i m u t h a l s c a t t e r i n g a n g l e <t>. As shown by e q u a t i o n 1.16 and f i g u r e s 3.8 and 3.9 t h i s i s c l o s e l y r e l a t e d t o 3 / E x p e r i m e n t a l 87 F i g u r e 3.2 Schematic diagram of a c y l i n d r i c a l m i r r o r segment a n a l y s e r and t r a j e c t o r y of an a n a l y s e d e l e c t r o n . 3/Exper i m e n t a l 88 s c a n n i n g q. Because p a r t i c l e s must have the c o r r e c t energy t o pass from p o i n t A t o p o i n t B the CMAs se r v e as energy s e l e c t o r s . The c o r r e c t energy of the e l e c t r o n s t o pass t h r o u g h the a n a l y s e r i s termed the pass energy, E { p a s s } . C o r r e c t o p e r a t i o n of the s p e c t r o m e t e r r e q u i r e s t h a t : (3.1) E, = E 2 = 1 / 2 ( E 0 - e ) = E { r e t a r d } + E{pass} The n e c e s s a r y v o l t a g e s a r e s u p p l i e d by v a r i o u s DC power s u p p l i e s , a d j u s t a b l e by the o p e r a t o r . The a n a l y s e r s a r e d e s i g n e d t o have f i r s t and second o r d e r f o c u s s i n g : (3.2) dz = d 2 z = 0 de d e 2 where 6 i s now the a n a l y s e r e n t r a n c e a n g l e , and z i s the a x i a l component of the d i s t a n c e between p o i n t s A and B ( f i g u r e 3.2). S i n c e t h e s e p o i n t s were chosen t o be on the a x i s of the a n a l y s e r , the e n t r a n c e a n g l e must be 42.3° t o o b t a i n second o r d e r f o c u s s i n g ( e x p l a i n i n g the 2.7° t i l t t o the a n a l y s e r s seen i n f i g u r e 3.1) a c c o r d i n g t o t h e o r e t i c a l c a l c u l a t i o n s of CMA parameters { R i s l e y (1972)}. These c a l c u l a t i o n s a l s o g i v e the d i s p e r s i o n f o r such a case as: (3.3) D = E{pass} dz = 5.6 dE{pass} ( i n u n i t s of the i n n e r c y l i n d e r r a d i u s ) . I f the e n t r a n c e and 3 / E x p e r i m e n t a l 8 9 e x i t a p e r t u r e s a r e both of 1mm d i a m e t e r , t h i s means a r e s o l u t i o n o f : (3.4) R = AE{pass} = 0.01 E { pa s s} T h e r e f o r e , f o r a pass energy of 50eV, and an energy s p r e a d of about 0.8eV f o r the i n c i d e n t e l e c t r o n s (from e l a s t i c s c a t t e r i n g measurements), the o v e r a l l r e s o l u t i o n f o r the s p e c t r o m e t e r s h o u l d be: y 2 R = (.5 2 + , 5 2 + ,8 2) =1.07 (The r e s u l t of the c o n v o l u t i o n of s e v e r a l G a u s s i a n peaks t o g e t h e r i s another G a u s s i a n whose w i d t h (FWHM) i s the square r o o t of the sum of the squares of the i n d i v i d u a l w i d t h s . ) In a c t u a l f a c t the r e s o l u t i o n i s g e n e r a l l y s l i g h t l y worse than t h i s , g e n e r a l l y between 1.2-1.6eV depending on the f i l a m e n t , because the f i g u r e R=0.01 does not take i n t o account the spread i n e n t r a n c e a n g l e s , which, a l t h o u g h the a n a l y s e r has second o r d e r f o c u s s i n g , does have an e f f e c t on the energy r e s o l u t i o n . I t might be noted here t h a t the b e s t energy r e s o l u t i o n a t t a i n e d i n a b i n a r y (e.,2e) experiment i s a 0.06eV FWHM measurement on argon { W i l l i a m s (1978)} but t h i s r e q u i r e d l o n g a c q u i s i t i o n t i m e s . W i t h the e l e c t r o n l e n s e s as t r a j e c t o r y s e l e c t o r s we now have a system which w i l l c h a r a c t e r i z e the p o s i t i o n s and momenta of the s c a t t e r e d p a r t i c l e s emerging from the gas 3 / E x p e r i m e n t a l 90 c e l l t o w i t h i n some e x p e r i m e n t a l e r r o r . S i n c e the energy and t r a j e c t o r y of the i n c i d e n t beam i s a l s o known, t h i s e x p e r i m e n t a l arrangement p e r m i t s the measurement of the b i n a r y (e,2e) c r o s s - s e c t i o n , as a f u n c t i o n of b i n d i n g energy (e) and momentum ( q ) . 3.2.6 The c h a n n e l t r o n s E l e c t r o n s which s u c c e s s f u l l y pass t h r o u g h the a n a l y s e r are c o l l e c t e d i n the c h a n n e l t r o n s ( c h a n n e l t r o n = c h a n n e l e l e c t r o n m u l t i p l i e r ) . These a r e s e m i c o n d u c t i n g g l a s s tubes w i t h a secondary e m i s s i o n c o e f f i c i e n t s i g n i f i c a n t l y g r e a t e r than u n i t y . A b i a s of s e v e r a l thousand v o l t s i s a p p l i e d over the l e n g t h of the tube. When an e l e c t r o n from the a n a l y s e r s t r i k e s the i n n e r s u r f a c e of the tube i t r e l e a s e s a shower of s e v e r a l secondary e l e c t r o n s from the g l a s s . Because of the b i a s on the tube t h e s e secondary e l e c t r o n s a r e a c c e l e r a t e d toward the p o s i t i v e end of the tube and e v e n t u a l l y they i n t u r n s t r i k e the w a l l s of the tube r e l e a s i n g many more showers of e l e c t r o n s . S i n c e each e l e c t r o n produces more than one secondary e l e c t r o n t h e r e i s a net a m p l i f i c a t i o n of the s i n g l e i n c i d e n t e l e c t r o n i n t o a p u l s e of 1 0 7 — 1 0 8 e l e c t r o n s a t the o t h e r end of the tube. T h i s p u l s e can now be a m p l i f i e d and p r o c e s s e d by c o n v e n t i o n a l e l e c t r o n i c c i r c u i t s . The p u l s e has the f o l l o w i n g c h a r a c t e r i s t i c s : (1) -10 t o -50mV p u l s e a m p l i t u d e when c a p a c i t a t i v e l y 3 / E x p e r i m e n t a l 91 F i g u r e 3.3 Schematic diagram of c h a n n e l t r o n p u l s e shapes and d i f f e r e n t t y p e s of d i s c r i m i n a t o r r e s p o n s e s : (a) Poor t e r m i n a t i o n of c a b l e s r e s u l t i n g i n r i n g i n g on the s i g n a l , and t h r e s h o l d - c r o s s i n g d i s c r i m i n a t i o n w i t h a t t e n d a n t t i m i n g u n c e r t a i n t y . (b) C l e a n c h a n n e l t r o n s i g n a l and a c c u r a t e t i m i n g from c o n s t a n t f r a c t i o n d i s c r i m i n a t i o n . 3 / E x p e r i m e n t a l 92 c o u p l e d (0.05*/F) i n t o a 50n l o a d . N o i s e on t h i s s i g n a l i s l e s s than 5mV. (2) Near-Gaussian p u l s e shape - 1Ons FWHM. The c h a n n e l t r o n i s housed i n s i d e a metal s h i e l d t o p r e v e n t i n t e r f e r e n c e from s t r a y e l e c t r o n s . Care i s t a k e n t o p r e v e n t a r c i n g of the h i g h v o l t a g e (3.5-4kV) c h a n n e l t r o n b i a s t o nearby l o w - v o l t a g e p o i n t s . C h a n n e l t r o n s a r e u s u a l l y c u r v e d t o p r e v e n t secondary i o n feedback. A gas m o l e c u l e which happens t o be i o n i z e d by a c o l l i s i o n w i t h a secondary e l e c t r o n would be a c c e l e r a t e d towards the e n t r a n c e of the c h a n n e l t r o n , where i t c o u l d t r i g g e r a n o ther b u r s t of secondary e l e c t r o n s . C u r v i n g the c h a n n e l t r o n means t h a t the i o n s t r i k e s the w a l l w e l l b e f o r e i t r e a c h e s the e n t r a n c e . . The c h a n n e l t r o n e l e c t r o n m u l t i p l i e r d e t e c t o r s ( o r i g i n a l l y M u l l a r d B419AL) have been r e p l a c e d by a s i m i l a r but s m a l l e r s e t ( M u l l a r d B318AL). T h i s r e p l a c e m e n t , combined w i t h c l o s e r impedance matching a t the p u l s e p i c k o f f p o i n t on the c h a n n e l t r o n o u t p u t , has improved the t i m i n g r e s o l u t i o n from about 12ns t o 7-8ns (FWHM). W i t h somewhat l o n g e r d a t a a c q u i s i t i o n times (3-4 days per o r b i t a l ) than had p r e v i o u s l y been used, and improved t i m i n g , a s i g n i f i c a n t improvement i n the o v e r a l l s i g n a l - t p - n o i s e r a t i o was o b t a i n e d , compared w i t h work p r e v i o u s t o the H 2S r e s u l t s . 3 / E x p e r i m e n t a l 93 3.2.7 The a n g l e - s c a n n i n g system An e l e c t r o m e c h a n i c a l s e r v o l o o p system i s used t o s e t the a n g l e of the moving a n a l y s e r . The servo a m p l i f i e r compares a r e f e r e n c e v o l t a g e from the d a t a a c q u i s i t i o n c o n t r o l system w i t h a scan v o l t a g e from a p o t e n t i o m e t e r m e c h a n i c a l l y l i n k e d t o the a n a l y s e r t u r n t a b l e , and g e n e r a t e s an AC m o t o r - d r i v i n g output w i t h the c o r r e c t phase t o t u r n the motor i n the d i r e c t i o n which makes the d i f f e r e n c e between the r e f e r e n c e and scan v o l t a g e s l e s s . Once the two v o l t a g e s a r e the same the s e r v o a m p l i f i e r output drops t o z e r o and the motor s t o p s . C o n s i d e r a b l e e f f o r t was expended t o make t h i s system r e l i a b l e and a c c u r a t e . P r e s e n t l y the a n g l e i s r e s e t t a b l e t o +0.2° A s p e c i a l s i l e n c e r u n i t was d e v i s e d t o t u r n o f f the motor once the a n g l e was s e t . P r e v i o u s l y , c h a t t e r i n the motor was v e r y n o i s y , and caused l o o s e n i n g and wear i n the m e c h a n i c a l p a r t s of the system. 3.3 C o n s t r u c t i o n and M a t e r i a l s The s p e c t r o m e t e r i s c o n s t r u c t e d a l m o s t e n t i r e l y out of b r a s s and aluminum because magnetic f i e l d s must be reduced t o a minimum, and because th e s e m a t e r i a l s are cheap, and easy t o machine. The vacuum h o u s i n g i s surrounded by a 3 / E x p e r i m e n t a l 94 M - m e t a l s h i e l d (a m a t e r i a l of h i g h p e r m e a b i l i t y ) t o reduce the e f f e c t s of the e a r t h ' s magnetic f i e l d . Where h i g h t e m p e r a t u r e s or m e c h a n i c a l s t r e n g t h r e q u i r e i t , s t a i n l e s s s t e e l i s sometimes used: i f b r a s s i s heated t o h i g h t e m p e r a t u r e s the z i n c i n i t w i l l e v a p o r a t e . E l e c t r i c a l i n s u l a t i o n i s a c h i e v e d w i t h s a p p h i r e b a l l s , T e f l o n , c e r a m i c , and Macor (a machinable c e r a m i c ) . E l e c t r i c a l c o n n e c t i o n s t h r o u g h the vacuum h o u s i n g a r e p r o v i d e d by o c t a l ( V a r i a n Eimac) and HV-BNC (Ceramaseal) s e a l e d f e e d t h r o u g h s s o l d e r e d i n t o b r a s s f l a n g e s which i n t u r n a r e b o l t e d onto the b a s e p l a t e and s e a l e d w i t h 0 - r i n g s . A l l s u r f a c e s (except f o r the e l e c t r o n gun) which d e f i n e an e l e c t r i c p o t e n t i a l d i r e c t i n g the motion of e l e c t r o n s w i t h i n the s p e c t r o m e t e r a r e c o a t e d w i t h benzene s o o t . T h i s i s s i m p l y a form of amorphous carbon and h e l p s t o : ( 1 ) M a i n t a i n an e q u a l p o t e n t i a l over the e n t i r e s u r f a c e of a g i v e n s p e c t r o m e t e r element; ( 2 ) Reduce the secondary e m i s s i o n c o e f f i c i e n t of the s u r f a c e m a t e r i a l so t h a t e l e c t r o n s which s t r i k e the s u r f a c e s do not bounce o f f or produce f u r t h e r secondary e l e c t r o n s which c o n t r i b u t e t o n o i s e i n the s p e c t r o m e t e r . A p e r t u r e s i n the e l e c t r o n l e n s and i n the spray p l a t e s a r e made of molybdenum, which a l s o has a low secondary e m i s s i o n c o e f f i c i e n t , f u r t h e r r e d u c i n g n o i s e w i t h i n the system. 3 / E x p e r i m e n t a l 95 3.4 S i g n a l P r o c e s s i n g The f o l l o w i n g s e c t i o n s d e s c r i b e the p r o c e s s i n g of the raw c h a n n e l t r o n p u l s e s , i n o r d e r t o e x t r a c t the (e,2e) event r a t e . F a i r l y s o p h i s t i c a t e d e l e c t r o n i c c i r c u i t r y i s r e q u i r e d t o p r o c e s s the c h a n n e l t r o n s i g n a l s , and t o d e t e c t t r u e c o i n c i d e n c e s ( i . e . two c h a n n e l t r o n p u l s e s t r i g g e r e d by s c a t t e r e d e l e c t r o n s from the same (e,2e) e v e n t ) . 3.4.1 C h a n n e l t r o n c o u p l i n g The c h a n n e l t r o n p u l s e s a r e c a p a c i t a t i v e l y d e c o u p l e d from the h i g h v o l t a g e b i a s and t r a n s m i t t e d t o the f i r s t s t a ge of a m p l i f i c a t i o n by 50n c o a x i a l c a b l e (RG58/U, RG174/U) (see f i g u r e 3.4). T h i s a m p l i f i c a t i o n stage i s done by ORTEC 9301 F a s t P u l s e P r e a m p l i f i e r s ( F P P ) . I t i s v e r y i m p o r t a n t t h a t the s i g n a l c a b l e be p r o p e r l y t e r m i n a t e d i n o r d e r t o ensure t h a t the s i g n a l q u a l i t y i s not degraded due t o impedance mismatches which cause ' r i n g i n g ' ( f i g u r e 3.3). S e v e r a l c i r c u i t s have been t r i e d o u t , and the most used ones ar e shown i n f i g u r e 3.4. C i r c u i t A has the advantage t h a t o n l y two f e e d t h r o u g h s and two l e n g t h s of c a b l e a r e r e q u i r e d , but has the d i s a d v a n t a g e t h a t the l i n e s a r e not p r o p e r l y t e r m i n a t e d , and hence g e n e r a t e s a l a r g e amount of r i n g i n g on the p u l s e s . Best r e s u l t s a r e o b t a i n e d w i t h c i r c u i t B. A l l f o u r ends of the two s i g n a l c a b l e s are t e r m i n a t e d i n 50fi r e s i s t i v e 3 / E x p e r i m e n t a l F i g u r e 3.4 C h a n n e l t r o n p u l s e d e c o u p l i n g c i r c u i t s (a) C i r c u i t A: improper t e r m i n a t i o n of c a b l e s (b) C i r c u i t B: c o r r e c t t e r m i n a t i o n of c a b l e s 3 / E x p e r i m e n t a l 97 l o a d s , and a l l f o u r ends of the o u t e r s h i e l d elements a r e c a r e f u l l y grounded t o reduce p i c k u p from e x t e r n a l s o u r c e s of e l e c t r i c a l n o i s e , and from the o t h e r c h a n n e l t r o n s . ( I t s h o u l d be noted t h a t a p u l s e i s t r a n s m i t t e d e q u a l l y w e l l when a p p l i e d t o e i t h e r the i n n e r element or the o u t e r s h i e l d of a c o a x i a l c a b l e . Thus i f the o u t e r s h i e l d i s not grounded at both ends i t w i l l a c t as an e x c e l l e n t antenna.) The time c o n s t a n t of the t e r m i n a t o r r e s i s t o r and d e c o u p l i n g c a p a c i t o r i s chosen t o be much l o n g e r than the c h a n n e l t r o n p u l s e : t h i s means t h a t the s i g n a l w i l l not be d i s t o r t e d by c a p a c i t a t i v e d i f f e r e n t i a t i o n t o any s i g n i f i c a n t e x t e n t . 3.4.2 P u l s e a m p l i f i c a t i o n and d i s c r i m i n a t i o n A schematic diagram of the s i g n a l p r o c e s s i n g e l e c t r o n i c s i s g i v e n i n f i g u r e 3.5. The f i r s t stage of a m p l i f i c a t i o n i s done by the FPPs: these i n a sense are s i m p l y g l o r i f i e d f u s e s because they i s o l a t e the e x p e n s i v e and s e n s i t i v e main e l e c t r o n i c s from the d e s t r u c t i v e e f f e c t s 'of v e r y h i g h v o l t a g e t r a n s i e n t s which r e s u l t from breakdown of the c h a n n e l t r o n b i a s v o l t a g e i n s i d e the s p e c t r o m e t e r . I t i s cheaper and q u i c k e r t o r e p l a c e components i n the preamps than i n main a m p l i f i e r s , i n the event of a r c - o v e r s . They a l s o s e r v e as l i n e d r i v e r s f o r the c o a x i a l c a b l e s l e a d i n g t o the main a m p l i f i e r s : the weak s i g n a l from the c h a n n e l t r o n s r e q u i r e s a m p l i f i c a t i o n b e f o r e i t can be t r a n s m i t t e d w i t h o u t i n t e r f e r e n c e from 3 / E x p e r i m e n t a l F i g u r e 3.5 Schematic d i a g r a m of s i g n a l - p r o c e s s i n g e l e c t r o n i c s Legend: FPP F a s t p u l s e preamps SCA S i n g l e - c h a n n e l a n a l y s e r TFA T i m i n g f i l t e r amps SS Spectrum s c a n n e r ' CFD Constant f r a c t i o n PC Program C o n t r o l d i s c r i m i n a t o r TAC Time t o a m p l i t u d e c o n v e r t e r 3 / E x p e r i m e n t a l 99 ground l o o p s or p i c k u p over c a b l e s t h a t may be up t o 3m l o n g . T h i s way the c a b l e l e n g t h from the c h a n n e l t r o n s t o the f i r s t a m p l i f i c a t i o n stage i s o n l y about 30cm. The main stage of a m p l i f i c a t i o n i s done by ORTEC 454 Timing F i l t e r A m p l i f i e r s (TFA). These a r e f a s t r i s e time 50P. a m p l i f i e r s w i t h a d j u s t a b l e g a i n up t o 20:1. The g a i n of each a m p l i f i e r i s s e t so t h a t the p u l s e h e i g h t d i s t r i b u t i o n s of each c h a n n e l a t the a m p l i f i e r output a r e r o u g h l y e q u a l , compensating f o r unequal c h a n n e l t r o n g a i n s . They a r e a l s o equipped w i t h a c t i v e f i l t e r networks t o a d j u s t the r i s e and f a l l time c o n s t a n t s of the output p u l s e , but thes e a r e u s u a l l y not used as i t i s d e s i r a b l e t o keep the p u l s e as shar p as p o s s i b l e . A l o t of r i n g i n g or o s c i l l a t i o n on the p u l s e may be reduced by a d j u s t i n g the f a l l time c o n s t a n t , but t h i s i s a 'bandaid' s o l u t i o n a t b e s t , and i t i s a d v i s a b l e t o l o c a t e and e l i m i n a t e the source of the problem. The main a m p l i f i e r s a r e f o l l o w e d by ORTEC 463 Constant F r a c t i o n D i s c r i m i n a t o r s (CFD). The f u n c t i o n of these u n i t s i s t o p r o v i d e a v e r y s h a r p , u n i f o r m l y timed and shaped ou t p u t p u l s e whenever t h e r e i s a v a l i d i n p u t p u l s e . ' V a l i d ' i n t h i s case means t h a t the i n p u t p u l s e a m p l i t u d e must exceed a c e r t a i n a d j u s t a b l e t h r e s h o l d b e f o r e the u n i t w i l l r espond and produce an o u t p u t p u l s e . T h i s t h r e s h o l d f e a t u r e i s used t o r e j e c t n o i s e below a c e r t a i n l e v e l . The t i m i n g of the o u t p u t p u l s e i s c r i t i c a l t o t h i s e xperiment i n p r o v i d i n g a p r e c i s e i n d i c a t i o n of the r e l a t i v e times of the p u l s e s on 3 / E x p e r i m e n t a l 100 each l i n e . Simple t h r e s h o l d - c r o s s i n g d i s c r i m i n a t i o n i s not adequate f o r t h i s a p p l i c a t i o n , as i n s p e c t i o n of f i g u r e 3.2 w i l l show: p r o d u c t i o n of the output p u l s e i s i n i t i a t e d when a m p l i t u d e of the i n p u t p u l s e c r o s s e s a g i v e n t h r e s h o l d v o l t a g e . I t i s seen t h a t the r e l a t i v e t i m i n g of the output p u l s e i s s e n s i t i v e t o the a m p l i t u d e of the i n p u t p u l s e . I f the r i s e time of the p u l s e i s 4-5ns t h i s means an u n c e r t a i n t y i n the t i m i n g of the output p u l s e of a s i m i l a r amount. The o n l y way t o reduce t h i s u n c e r t a i n t y i s t o s e t the t h r e s h o l d c l o s e t o z e r o , but then l o w - l e v e l n o i s e s t a r t s t o t r i g g e r the d i s c r i m i n a t o r . Constant f r a c t i o n d i s c r i m i n a t i o n i s the method chosen t o s o l v e t h i s problem. In t h i s t e c h n i q u e the output p u l s e i s produced when the i n p u t waveform reaches a f i x e d f r a c t i o n of the t o t a l p u l s e h e i g h t . T h i s g i v e s the t i m i n g r e s u l t shown i n f i g u r e 3.3, and i s n e a r l y independent of the p u l s e h e i g h t . 3.4.3 C o i n c i d e n c e d e t e c t i o n The main t a s k o f . t h e s i g n a l p r o c e s s i n g e l e c t r o n i c s i s t o d e t e c t c o i n c i d e n c e s i n the two streams of e l e c t r o n s e n t e r i n g the c h a n n e l t r o n s . A v a l i d (e,2e) event produces two e l e c t r o n s which r e a c h the c h a n n e l t r o n s a t the same time ( t 1 = t 2 , or At=0) and t h e r e f o r e i n i t i a t e s two s i m u l t a n e o u s p u l s e s i n the CFD ou t p u t l i n e s . However t h e r e i s an o t h e r s c a t t e r i n g event which produces an i d e n t i c a l response i n the 3/Exper i m e n t a l 101 c h a n n e l t r o n s . T h i s can happen i f two u n r e l a t e d e l e c t r o n s i n the i n c i d e n t beam s i m u l t a n e o u s l y c o l l i d e w i t h two gas-m o l e c u l e s w i t h i n the c o l l i s i o n volume p r o d u c i n g (by some p r o c e s s of i n e l a s t i c s c a t t e r i n g or i o n i z a t i o n , e t c . ) two u n r e l a t e d s c a t t e r e d e l e c t r o n s : one has the n e c e s s a r y momentum k, t o pass through one a n a l y s e r , and the o t h e r e l e c t r o n k.2 through the o t h e r a n a l y s e r . O b v i o u s l y t h i s a c c i d e n t a l c o i n c i d e n c e s i g n a l a t the c h a n n e l t r o n s w i l l be i n d i s t i n g u i s h a b l e from a t r u e c o i n c i d e n c e which s i g n i f i e s a r e a l (e,2e) e v e n t . The problem then i s how t o remove the background of a c c i d e n t a l c o i n c i d e n c e s and measure o n l y the t r u e c o i n c i d e n c e r a t e . The answer l i e s i n the f a c t t h a t the a c c i d e n t a l c o i n c i d e n c e r a t e i s o n l y the s p e c i a l case where the t i m i n g of the two i n c i d e n t e l e c t r o n s i s s i m u l t a n e o u s ( A t = t , - t 2 = 0 ) . The r a t e f o r the o b s e r v a t i o n of two random c h a n n e l t r o n p u l s e s h a v i n g a non-zero time s e p a r a t i o n ( A t = T ) must be the same as the At=0 r a t e . A l s o , the r a t e of non-(e,2e) c o i n c i d e n c e s i s not a f u n c t i o n of b i n d i n g energy or a z i m u t h a l s c a t t e r i n g a n g l e . T h e r e f o r e , i f the r a t e of o b s e r v a t i o n of p u l s e p a i r s of time s e p a r a t i o n At=0 and At=r can both be measured, i t f o l l o w s t h a t the t r u e (e,2e) r a t e i s g i v e n by: (3.5) Rate{e,2e} = Rate{At = 0} - Rate{At = -r} Rate{e,2e} i s termed the ' t r u e c o i n c i d e n c e rate',Rate{At=0} 3 / E x p e r i m e n t a l 102 i s c a l l e d the ' t o t a l c o i n c i d e n c e r a t e ' , and Rate{At=r} i s the r a t e of random p u l s e p a i r s . ' A c c i d e n t a l c o i n c i d e n c e r a t e ' r e f e r s t o those c o i n c i d e n c e s where At=0, but the two e l e c t r o n s do not come from a v a l i d (e,2e) e v e n t . The r e j e c t i o n of a c c i d e n t a l c o i n c i d e n c e s i s done e l e c t r o n i c a l l y w i t h a t i m e - t o - a m p l i t u d e c o n v e r t e r (ORTEC TAC 467), two s i n g l e c h a n n e l a n a l y s e r s (ORTEC SCA 406), a c o a x i a l c a b l e d e l a y l i n e , and a l o g i c c i r c u i t or computer s o f t w a r e ( f i g u r e 3.5). A TAC has two i n p u t s , START and STOP: one CFD i s c o n n e c t e d t o the START i n p u t and the o t h e r CFD through the d e l a y l i n e t o the STOP i n p u t . A v a l i d p u l s e a t the START i n p u t s t a r t s an i n t e r n a l c l o c k i n the TAC a t time t , . A subsequent p u l s e a t the STOP i n p u t s t o p s t h i s c l o c k a t time t 2 and t r i g g e r s the c o n v e r s i o n of t h i s i n t e r n a l measured time i n t o an o u t p u t p u l s e whose a m p l i t u d e i s p r o p o r t i o n a l t o the time s e p a r a t i o n A t . The t r u e c o i n c i d e n c e s do not a l l o c c u r e x a c t l y a t At=0, but r a t h e r t h e r e i s a f i n i t e time w i d t h , 6T, a s s o c i a t e d w i t h t h i s s i g n a l due t o s l i g h t changes i n p r o p a g a t i o n t i m e s of the s i g n a l s t h r o u g h the s p e c t r o m e t e r and e l e c t r o n i c s ; a d e l a y i s used i n t h e STOP l i n e t o s h i f t the t r u e c o i n c i d e n c e s i g n a l c o m p l e t e l y i n t o the range of the TAC, o t h e r w i s e t r u e c o i n c i d e n c e s where the STOP p u l s e comes s l i g h t l y b e f o r e the START p u l s e w i l l not be d e t e c t e d by the TAC. With M u l l a r d B318AL c h a n n e l t r o n s 6r i s about 6-8ns. A h i s t o g r a m of the TAC output i s shown i n f i g u r e 3.6. A 3 / E x p e r i m e n t a l 103 CO O O 6T • V '• • • . • • SCA1 S C A 2 A t F i g u r e 3.6 T y p i c a l time spectrum showing placement of SCA windows. 3 / E x p e r i m e n t a l 104 f l a t background i s o b s e r v e d , due t o the random p u l s e p a i r s , and superimposed on t h i s i s a peak due t o the t r u e c o i n c i d e n c e s i g n a l . The p o s i t i o n of t h i s peak i s r e l a t e d t o the c u m u l a t i v e d i f f e r e n c e s i n p r o p a g a t i o n t i m e s of the two s i g n a l s . The w i d t h of the peak i s a f u n c t i o n of the t i m i n g spread due t o the d i s p e r s i o n of p a r t i c l e s i n the a n a l y s e r s , n o i s e i n the system, and j i t t e r i n the CFDs. The TAC output p u l s e i s f e d t o the two SCAs. An SCA g e n e r a t e s an output p u l s e o n l y i f the i n p u t p u l s e has an a m p l i t u d e w i t h i n a v o l t a g e window w i t h a d j u s t a b l e upper and lower bounds. One SCA i s s e t t o span the c o i n c i d e n c e peak, and the o t h e r has the same w i d t h , but i s p o s i t i o n e d on the f l a t background (see f i g u r e 3.6). The d a t a a c q u i s i t i o n system i s a r r a n g e d so t h a t a p u l s e from SCA1 w i l l increment a c o u n t e r r e g i s t e r , and a p u l s e from SCA2 w i l l decrement the c o u n t e r . T h i s i s how e q u a t i o n 3.5 i s implemented i n the da t a a c q u i s i t i o n e l e c t r o n i c s : i t can be seen t h a t the time-ave r a g e d p u l s e r a t e from SCA2 c a n c e l s o f f the a c c i d e n t a l c o i n c i d e n c e r a t e i n SCA1 so t h a t o n l y the t r u e c o i n c i d e n c e s accumulate i n the c o u n t e r . In f a c t , as f i g u r e 3.6 shows, the r a t i o of the SCA window w i d t h s need not be u n i t y , but can be some i n t e g e r n g r e a t e r than u n i t y , as l o n g as an e q u a l count n i s added t o the spectrum upon r e c e i p t of an SCA1 p u l s e . - T h i s m o d i f i c a t i o n improves the s t a t i s t i c a l a c c u r a c y of the spectrum s i n c e more measurements of R a t e { A t = T } a r e made. The 3 / E x p e r i m e n t a l s t a t i s t i c a l a c c u r a c y of the f i n a l spectrum i s : v2 (3.6) tf = N{At=0} + N { A t = T } n where N{At=0} i s the number of c o u n t s from SCA1, N { A t = T } i s the number of c o u n t s from SCA2, and n i s the r a t i o of the window w i d t h s . " C l e a r l y , i n c r e a s i n g n reduces a. For most of the measurements i n t h i s t h e s i s , n was s e t t o 15. 3.4.4 S i g n a l - a v e r a g i n g d a t a a c q u i s i t i o n system Two d a t a a c q u i s i t i o n systems were used: the f i r s t i s a c o m b i n a t i o n of a s p e c i a l l y b u i l t 1 6 - b i t up/down c o u n t e r r e g i s t e r , a commercial N i c o l e t 1072 s i g n a l a v e r a g e r , and ORTEC scan c o n t r o l u n i t s ( t h i s system was a l r e a d y i n p l a c e a t the b e g i n n i n g of the work f o r t h i s t h e s i s ) ; the second was d e s i g n e d d u r i n g the t h e s i s work around a PDP LSI 11/03 microcomputer. The d a t a a c q u i s i t i o n system must p e r f o r m s e v e r a l f u n c t i o n s : (1) S t o r a g e space f o r a m u l t i - c h a n n e l h i s t o g r a m must be p r o v i d e d ; (2) There must be an a c t i v e c h a n n e l where the t o t a l count can be incremented by n on r e c e i p t of a p u l s e from SCA1, or decremented i n the case of a p u l s e from SCA2; (3) There must be p r o v i s i o n whereby the a c t i v e c h a n n e l 3 / E x p e r i m e n t a l 106 can be stepped t h r o u g h the h i s t o g r a m and r e s e t , c o n c u r r e n t l y w i t h s t e p p i n g and r e s e t t i n g a r e f e r e n c e v o l t a g e which i s used t o c o n t r o l the scanned parameter; (4) The c o n t e n t s of the h i s t o g r a m must be d i s p l a y e d g r a p h i c a l l y f o r o b s e r v a t i o n , and t h e r e s h o u l d be a method f o r making permanent g r a p h i c a l and n u m e r i c a l r e c o r d s of the d a t a . The term ' s i g n a l a v e r a g i n g ' denotes the p r o c e s s of r e p e t i t i v e accumulated s c a n n i n g of a spectrum, where the r e s u l t s of a scan are added t o the p r e v i o u s s c a n s . T h i s i s done f o r two r e a s o n s : (1) I f the s i g n a l i n t e n s i t y i s comparable w i t h , or l e s s than the n o i s e i n a measurement, the n o i s e w i l l c a n c e l out over many scans because i t i s a random phenomenon, but the s i g n a l , though weak, i s p r e s e n t a l l the t i m e , and s h o u l d r e i n f o r c e i t s e l f w i t h every scan. W i t h a l o n g enough c o l l e c t i o n time i t w i l l e v e n t u a l l y be v i s i b l e over the n o i s e ; (2) A l t h o u g h t h e r e might be a l a r g e change i n the s i g n a l r a t e w i t h time over the c o u r s e of a r u n , t h i s w i l l not a f f e c t the f i n a l spectrum s i n c e d u r i n g any s i n g l e scan the change i n i n t e n s i t y w i l l be i n s i g n i f i c a n t . The N i c o l e t 1072 s i g n a l a v e r a g e r i s a d a t a a c q u i s i t i o n and s t o r a g e d e v i c e w i t h the f e a t u r e s d e s c r i b e d above. I t has 3 / E x p e r i m e n t a l 107 a hardware memory of I 0 2 4 x 1 6 - b i t words, d i v i d e d i n t o 4x256-word segments. There i s a p u l s e c o u n t i n g c i r c u i t r y and e x t e r n a l c h a n n e l advance and r e s e t l i n e s . Outputs a r e p r o v i d e d f o r a c o n t i n u o u s X-Y o s c i l l o s c o p e d i s p l a y , p o i n t p l o t t e r , and t e l e t y p e . The up/down c o u n t e r i n t e r f a c e s between the two SCAs and the N i c o l e t . The i n t e r n a l 1 6 - b i t r e g i s t e r i s incremented by a p u l s e from SCA1 and decremented by a p u l s e from SCA2. R e c e i p t of a c h a n n e l advance p u l s e i n d i c a t e s the end of the da t a c o l l e c t i o n p e r i o d f o r a g i v e n a c t i v e c h a n n e l and t r i g g e r s the readout of the f i n a l c o n t e n t s of the i n t e r n a l r e g i s t e r i n t o the N i c o l e t and c l e a r s the r e g i s t e r . T h i s readout t a k e s the form of a s e r i a l p u l s e t r a i n t o the N i c o l e t p u l s e i n p u t l i n e , and a s i m u l t a n e o u s h i / l o l o g i c l e v e l i n t o the N i c o l e t a d d / s u b t r a c t l i n e . The number of p u l s e s i s e q u a l t o the a b s o l u t e magnitude of the accumulated c o u n t , and the s t a t e of the a d d / s u b t r a c t l i n e g i v e s the s i g n . T h i s r e s u l t i s added t o the accumulated c o u n t . Then a d e l a y e d v e r s i o n of the c h a n n e l advance p u l s e s t e p s the N i c o l e t t o the next c h a n n e l . The N i c o l e t 1072 i s c a p a b l e of f u n c t i o n i n g ( v i a p l u g - i n u n i t s ) as a p u l s e h e i g h t a n a l y s e r as w e l l as a s i g n a l a v e r a g e r . In t h i s mode the TAC output i s c o n v e r t e d t o a h i s t o g r a m of f r e q u e n c y vs p u l s e h e i g h t . The N i c o l e t a l s o p r o v i d e s some p r i m i t i v e d a t a r e d u c t i o n f a c i l i t i e s t o add and smooth s p e c t r a , s u b t r a c t backgrounds, and so on. 3 / E x p e r i m e n t a l 108 The microcomputer-based d a t a a c q u i s i t i o n system i s d e s c r i b e d i n d e t a i l i n Chapter 8. A l l t h a t need be s a i d here i s t h a t a l l the b a s i c f u n c t i o n s of the N i c o l e t system a r e emulated, and, due t o the v e r s a t i l i t y and p o w e r f u l s o f t w a r e of a computer system, many t e d i o u s d a t a r e d u c t i o n and a n a l y s i s p r o c e d u r e s a r e automated. Data i s semi-permanently s t o r e d on f l o p p y d i s k - e t t e medium, a l o n g w i t h a l l parameters p e r t i n e n t t o the run. Permanent g r a p h i c a l and t e x t u a l r e c o r d s a r e made on a g r a p h i c s p r i n t e r . The computer-based system a l l o w s more f l e x i b i l i t y i n d a t a a c q u i s i t i o n . For i n s t a n c e i t i s p o s s i b l e , s i n c e the computer can c o n t r o l both i n c i d e n t energy and s c a t t e r i n g a n g l e s i m u l t a n e o u s l y , t o r e c o r d s e v e r a l b i n d i n g energy s p e c t r a and momentum d i s t r i b u t i o n s c o n c u r r e n t l y and so ma i n t a i n - the c o r r e c t r e l a t i v e a b s o l u t e i n t e n s i t y of a g i v e n s t r u c t u r e w i t h r e s p e c t t o i t s f e l l o w s . 3.5 The I n s t r u m e n t a l Response F u n c t i o n The s p e c t r o m e t e r i s s e n s i t i v e o n l y t o those e l e c t r o n s emerging from the gas c e l l which have p a r t i c u l a r p r o p e r t i e s . We do not observe an i d e a l (e,2e) c r o s s - s e c t i o n of a s i n g l e m o l e c u l e , but r a t h e r some r e s u l t which a l s o has c o n v o l u t e d i n a r e s o l u t i o n f u n c t i o n w i t h s e v e r a l d i m e n s i o n a l i t i e s : 3 / E x p e r i m e n t a l 109 F i g u r e 3.7 The c o l l i s i o n volume r e s u l t i n g from the o v e r l a p of h y p o t h e t i c a l c y l i n d e r s and cones r e p r e s e n t i n g the i n c i d e n t beam volume and the a c c e p t a n c e cones of the l e n s e s . \ 3 / E x p e r i m e n t a l 110 ( 1 ) The s p a t i a l c o l l i s i o n volume as d e f i n e d by the o v e r l a p of the i n c i d e n t e l e c t r o n beam, and the ac c e p t a n c e cones of the l e n s e s ( f i g u r e 3 . 7 ) . T h i s o v e r l a p d e t e r m i n e s the .mean 9 and <t> a n g l e s , and the e f f e c t i v e 9 and <t> r e s o l u t i o n . I t i s v e r y i m p o r t a n t t o u n d e r s t a n d t h a t t h i s volume i s not n e c e s s a r i l y a c o n s t a n t f u n c t i o n of the a z i m u t h a l s c a t t e r i n g a n g l e . The q u e s t i o n w i l l be e l a b o r a t e d below; (2) The f i n i t e energy r e s o l u t i o n which i s a f u n c t i o n of the energy s p r e a d of the i n c i d e n t beam, and the d i s p e r s i o n of the two a n a l y s e r s ; ( 3 ) The o v e r a l l t i m i n g u n c e r t a i n t y which a r i s e s from d i f f e r e n c e s i n a n a l y s e r t r a n s i t t i m e s , c h a n n e l t r o n r e s p o n s e , and e l e c t r o n i c p r o c e s s i n g d e l a y s ; T h i s does not n o r m a l l y e n t e r i n t o the response f u n c t i o n s i n c e one e f f e c t i v e l y i n t e g r a t e s over At by s e t t i n g SCA1 t o span the e n t i r e c o i n c i d e n c e peak; (4) I n c i d e n t beam f l u x : t h i s i s almost never an a b s o l u t e l y c o n s t a n t f u n c t i o n of i n c i d e n t energy; (5) Gas d e n s i t y : i t i s assumed t h a t the sample gas i s homogeneously d i s t r i b u t e d t hroughout the gas c e l l ; (6) L e n s / a n a l y s e r t r a n s m i s s i o n f u n c t i o n : t h i s i s not n e c e s s a r i l y p e r f e c t l y c o n s t a n t ' as a f u n c t i o n of a z i m u t h a l a n g l e . The o p t i m i z a t i o n of the s p e c t r o m e t e r o p e r a t i o n must tak e i n t o account the f o l l o w i n g p o i n t s (numbered t o r e f e r t o 3 / E x p e r i m e n t a l the same t o p i c s as above): (1) In o r d e r t o ensure t h a t the c o l l i s i o n volume does not a l t e r s i g n i f i c a n t l y w i t h a z i m u t h a l s c a t t e r i n g a n g l e the mean dia m e t e r of the e l e c t r o n beam ( i t p r o b a b l y has a semi-Gaussian c r o s s - s e c t i o n ) must be made s i g n i f i c a n l y l e s s than the d i a m e t e r of the l e n s a c c e p t a n c e cones a t t h e i r i n t e r s e c t i o n . I f t h i s i s done then the c o l l i s i o n volume w i l l be a t h i n c y l i n d e r whose shape changes o n l y v e r y s l i g h t l y a t the ends, as the moving a n a l y s e r r e v o l v e s . A l s o , i f t h e r e i s any d r i f t i n the p o s i t i o n of the beam, the a c c e p t a n c e cone of the moving l e n s , or m i s a l i g n m e n t i n the p h y s i c a l c o n s t r u c t i o n of the s p e c t r o m e t e r , such t h i n g s w i l l have l i t t l e e f f e c t on the c o l l i s i o n volume. The o n l y d i s a d v a n t a g e t o t h i s type' of o p e r a t i o n i s t h a t a f a i r l y l a r g e © a n g l e i s subtended over the l e n g t h of the c o l l i s i o n volume, which must be ta k e n i n t o account i n comparing t h e o r e t i c a l momentum d i s t r i b u t i o n s w i t h e x p e r i m e n t . I f however, i n the i n t e r e s t s of good ©-and ^ - r e s o l u t i o n , the a c c e p t a n c e cones a r e made v e r y sharp then s e v e r a l problems w i l l r e s u l t : i t w i l l be d i f f i c u l t t o keep the moving l e n s a c c e p t a n c e cone c e n t r e d on the beam; the a l i g n m e n t of the l e n s e s i n the ©-dimension i s a l s o now more c r i t i c a l ; c h a n g i n g c o l l i s i o n volume w i t h 54-angle w i l l be u n a v o i d a b l e , u n l e s s h i g h e r i n c i d e n t energy i s used t o compress the d i s t r i b u t i o n i n t o a s m a l l #-range, whereupon the 3 / E x p e r i m e n t a l r e l a t i v e ^ - r e s o l u t i o n i s back t o where i t was b e f o r e ; and f i n a l l y , the d a t a r a t e w i l l be d i m i n i s h e d . W ith the o p t i m i z e d c o l l i s i o n volume the ©-resolution w i l l be worse than the ^ - r e s o l u t i o n . T h i s w i l l have the g r e a t e s t e f f e c t i n p-type d i s t r i b u t i o n s near q=0, where the s h a r p minimum w i l l be f i l l e d i n t o some e x t e n t . Because t h e r e i s u n c e r t a i n t y i n t h i s r e s o l u t i o n f a c t o r , and a l s o because of the u n c e r t a i n t y i n the mean © v a l u e a r i s i n g from the use of the d e f l e c t o r s i n the l e n s e s , I am u n w i l l i n g t o draw f i r m c o n c l u s i o n s about the shape of the d i s t r i b u t i o n or i t s agreement w i t h t h e o r y i n the r e g i o n q<0.4a o' 1; (2) Where the b i n d i n g energy spectrum i s s i m p l e enough ( i . e . the peaks a r e few and w e l l - s e p a r a t e d ) i t i s d e s i r a b l e t o degrade the energy r e s o l u t i o n t o the p o i n t where the b i n d i n g energy peaks b e g i n t o merge t o g e t h e r . T h i s w i l l a f f o r d the h i g h e s t p o s s i b l e d a t a r a t e , y e t s t i l l a l l o w the momentum d i s t r i b u t i o n s t o be d e t e r m i n e d w i t h o u t o v e r l a p from o t h e r s t a t e s . For complex b i n d i n g energy s p e c t r a one can t r y t o r e s o l v e the o u t e r v a l e n c e peaks, but i n the p r e s e n t s t a t e of the a r t , i t i s not p r a c t i c a l t o t r y t o r e s o l v e the i n n e r v a l e n c e r e g i o n as t h i s i s almost always v e r y i n t r i c a t e . In any c a s e , i t i s not p r a c t i c a l t o t r y t o improve the o v e r a l l energy r e s o l u t i o n t o much beyond a f a c t o r 1.5 t i m e s the i n c i d e n t beam energy s p r e a d . The energy response f u n c t i o n i s o p t i m i z e d by r e d u c i n g the 3 / E x p e r i m e n t a l i n c i d e n t beam energy from E 0 + e t o E, and o b s e r v i n g the energy d i s t r i b u t i o n of e l a s t i c a l l y s c a t t e r e d e l e c t r o n s i n each a n a l y s e r . The l e n s f o c u s c o n t r o l and the l e n s d e f l e c t o r s were used t o make t h i s d i s t r i b u t i o n as sh a r p and as n e a r l y symmetric as p o s s i b l e , but i t i s now r e a l i s e d t h a t the use of the d e f l e c t o r s i n the v e r t i c a l p l a n e d i s t o r t s the e f f e c t i v e Q a n g l e . I t i s recommended t h a t t h e s e two elements be s h o r t - c i r c u i t e d t o the a n a l y s e r i n n e r element; (3) I t i s d e s i r a b l e t o reduce the o v e r a l l t i m i n g s p r e a d so as t o be a b l e t o i n c r e a s e the r a t i o of the w i d t h s of the SCA1 and SCA2 windows, and improve the s t a t i s t i c s of the measurement. T h i s can be done by the f o l l o w i n g : n a r r o w i n g the ac c e p t a n c e cone of the e l e c t r o n l e n s e s and r a i s i n g the pass energy i n the a n a l y s e r s (where f e a s i b l e ) so t h a t the d i s p e r s i o n i s l e s s and the l e n s / a n a l y s e r t r a n s i t t i me s p r e a d i s t h e r e b y r e d u c e d ; u s i n g s m a l l e r c h a n n e l t r o n s w i t h s h a r p e r p u l s e o u t p u t s ; and i m p r o v i n g t h e q u a l i t y of the s i g n a l d e c o u p l i n g and t r a n s m i s s i o n between the c h a n n e l t r o n s and the preamps so t h a t o n l y a bare minimum of n o i s e and r i n g i n g i s imposed on the s i g n a l . I t i s i m p o r t a n t t o check from time t o time t h a t the c o i n c i d e n c e peak has not d r i f t e d out of the SCA1 window; (4,5) Changes i n the i n c i d e n t beam f l u x , as a f u n c t i o n of energy, and gas d e n s i t y , as a f u n c t i o n of t i m e , can 3 / E x p e r i m e n t a l be a ccounted f o r by n o r m a l i z i n g the scan time t o the s i n g l e s s c a t t e r i n g r a t e i n the moving a n a l y s e r , and by the s i g n a l - a v e r a g i n g d a t a a c q u i s i t i o n method. One must be c a r e f u l when u s i n g n o r m a l i z a t i o n as i t can obscure p o o r l y - o p t i m i z e d f a c t o r s i n the i n s t r u m e n t a l response f u n c t i o n : f o r i n s t a n c e , d r i f t s i n the c o l l i s i o n volume due t o p o o r l y - t u n e d beam and l e n s e s , magnetic f i e l d changes, or c h a r g i n g e f f e c t s ; or c h a n g i n g background count r a t e s due to a n i s o t r o p i c s t r a y e l e c t r o n s ; (6) The l e n s / a n a l y s e r t r a n s m i s s i o n f u n c t i o n i s an u n l i k e l y s o u r c e of t r o u b l e , but i t can change s l i g h t l y i f magnetic f i e l d f l u c t u a t i o n s cannot be r i g o r o u s l y e l i m i n a t e d . The moving a n a l y s e r i s e s p e c i a l l y s u s c e p t i b l e t o t h i s , and a l s o t o a v a r y i n g background of s t r a y e l e c t r o n s w i t h i n the vacuum chamber. T h i s i s why »»-metal s h i e l d i n g must be used, and measures t a k e n t o c o n t r o l or e l i m i n a t e a l l s o u r c e s of s t r a y e l e c t r o n s . W h i l e such s t r a y e l e c t r o n s do not a f f e c t the c o i n c i d e n c e count r a t e s i n c e they do not a r i s e from (e,2e) e v e n t s , they can g i v e r i s e t o a s l o p i n g background by a l t e r i n g the random p u l s e - p a i r r a t e as a f u n c t i o n of a n g l e . 3 / E x p e r i m e n t a l 3.5.1 The q-e-c* s u r f a c e F i g u r e s 3.8 and 3.9 shows a r e p r e s e n t a t i o n of e q u a t i o n 1.17 where q i s p l o t t e d as a f u n c t i o n of the © and <t> s c a t t e r i n g a n g l e s f o r E 0 v a l u e s of 400 eV and 1200 eV. A h y p o t h e t i c a l b i n d i n g energy of 15eV i s used. I t i s seen t h a t a t 1200eV the momentum d i s t r i b u t i o n i s compressed t o lower * - a n g l e and a l s o t h a t q i s a sh a r p e r f u n c t i o n of ©, than a t 400eV. T h e r e f o r e a t the h i g h e r energy i t i s r e l a t i v e l y more im p o r t a n t t o know p r e c i s e l y the e f f e c t i v e mean © and <t> a n g l e s i n c o n v e r t i n g the t j - s c a l e t o the q - s c a l e , and the ©- and q - r e s o l u t i o n i n comparing experiment w i t h t h e o r y . For a l l of the work i n t h i s t h e s i s a r e s o l u t i o n c o n v o l u t i o n program was used t o f o l d the e x p e r i m e n t a l a n g u l a r r e s o l u t i o n and Mott s c a t t e r i n g f a c t o r i n t o the t h e o r e t i c a l momentum d i s t r i b u t i o n . T h i s program ( w r i t t e n by Dr s . A. Hammett and S.T. Hood) ap p r o x i m a t e s the f i n i t e a n g u l a r a c c e p t a n c e cone of the l e n s w i t h an average over s e v e r a l r a y s of s l i g h t l y d i f f e r i n g ©- and cj-angle. The number and d i r e c t i o n of these r a y s were d e t e r m i n e d e m p i r i c a l l y by comparison of a good t h e o r e t i c a l w a v e f u n c t i o n w i t h experiment f o r h e l i u m . U n f o r t u n a t e l y , the h e l i u m 1s momentum d i s t r i b u t i o n i s not v e r y s e n s i t i v e t o the a n g u l a r r e s o l u t i o n , and the p a r a m e t e r i z a t i o n s h o u l d a l s o have been done on the argon 3p momentum d i s t r i b u t i o n . J u d g i n g by the 3 / E x p e r i m e n t a l F i g u r e 3.8 Contour map of q as a f u n c t i o n of 9 and * E o=400eV and e=15eV. Co n t o u r s a r e a t i n t e r v a l s of 0.1a o 3 / E x p e r i m e n t a l F i g u r e 3.9 Contour map of q as a f u n c t i o n of © and * a t E o=l200eV and c=15eV. C o n t o u r s a r e a t i n t e r v a l s of 0 . 1 a o _ 1 . 3 / E x p e r i m e n t a l H 2S r e s u l t s the e s t i m a t e d a n g u l a r r e s o l u t i o n i s p r o b a b l y worse than i t r e a l l y i s : the e x p e r i m e n t a l p-type momentum d i s t r i b u t i o n s a r e c o n s i s t e n t l y s h a r p e r than c a l c u l a t i o n ; however, the o p p o s i t e seems t o be the case f o r C 0 2 . 3.5.2 Comparison of 400eV vs 1200eV o p e r a t i o n The c h o i c e of i n c i d e n t energy E 0 i s an i m p o r t a n t f a c t o r i n r u n n i n g the s p e c t r o m e t e r . The s e v e r a l arguments f o r p r e f e r r i n g 400eV or 1200eV o p e r a t i o n a r e g i v e n h e r e : (1) S i g n a l / n o i s e r a t i o : the b i n a r y (e,2e) c r o s s - s e c t i o n f a l l s o f f as E~ 3/ 2 w i t h i n c r e a s i n g energy (see e q u a t i o n s 1.18 and 1.19), whereas the i n e l a s t i c s c a t t e r i n g c r o s s - s e c t i o n f o r p r o d u c t i o n of s i n g l e p a r t i c l e s of energy E , = E 2 = % ( E 0 - e ) { I n o k u t i (1971)} f a l l s o f f f a s t e r as E" 2. For t h i s reason i t i s d e s i r a b l e t o o p e r a t e a t h i g h e r energy so as t o i n c r e a s e the s i g n a l / n o i s e r a t i o . T h i s does mean a lower d a t a r a t e , but t h i s can be improved by i n c r e a s i n g the c o l l i s i o n r a t e ( i . e . h i g h e r beam f l u x and gas d e n s i t y ) ; (2) A n g u l a r r e s o l u t i o n : as demonstrated i n the p r e v i o u s s e c t i o n , the a n g u l a r r e s o l u t i o n must be improved g o i n g t o h i g h e r energy, i n o r d e r t o m a i n t a i n the e q u i v a l e n t q - r e s o l u t i o n . As w i t h any type of r e s o l u t i o n improvement t h i s goes hand-in-hand w i t h a l o s s i n d a t a r a t e . A l o s s i n r e l a t i v e a n g u l a r 3 / E x p e r i m e n t a l F i g u r e 3.10 Contour map of <r{Mott} as a f u n c t i o n of e and <t> a t E o=400eV and €=l5eV. Contour v a l u e s a r e l a b e l l e d on the diagram. 3 / E x p e r i m e n t a l 120 50-! 40-1 3 0 -cp _ 2 0 -10-4 0 2.8 3.0 2.6 2.4 40 ~ 1 J 2 2.0 1.8 2.2 ^ I r 44 Q 46 -6 1.6x10 1215eV 48 50 F i g u r e 3.11 Contour map of <*{Mott} as a f u n c t i o n of 9 and * a t E o=1200eV and €=15eV. Contour v a l u e s a r e l a b e l l e d on the diagram. 3 / E x p e r i m e n t a l 1 r e s o l u t i o n a f f e c t s p r i m a r i l y the d e t e r m i n a t i o n of p-type momentum d i s t r i b u t i o n s near q=0, and makes i t d i f f i c u l t t o t e l l the s/p c h a r a c t e r of the d i s t r i b u t i o n . T h i s s/p c h a r a c t e r i s sometimes v e r y r e v e a l i n g as t o the asymmetry of the m o l e c u l a r o r b i t a l , and the o b s c u r i n g of t h i s r e g i o n i s s i g n i f i c a n t l o s s of i n f o r m a t i o n ; (3) E l e c t r o n beam and l e n s -eff e c t s : i t i s much e a s i e r t o e x t r a c t and c o n t r o l a 1200eV beam than a 400eV one, and f o r t h i s reason one can p r o l o n g the f i l a m e n t l i f e by r u n n i n g i t c o o l e r . I t i s , however, h a r d e r f o r the l e n s t o d e c e l e r a t e 600ev e l e c t r o n s down t o the pass energy, compared t o 200eV e l e c t r o n s , w i t h o u t l o s s of s i g n a l . The l e n s m a g n i f i c a t i o n may a l s o change w i t h d i f f e r e n t d e c e l e r a t i o n r a t i o s ; (4) The Mott s c a t t e r i n g f a c t o r becomes more n e a r l y c o n s t a n t as a f u n c t i o n of ?j-angle w i t h h i g h e r i n c i d e n t energy E 0 . While t h i s i s not a problem a t e i t h e r energy because t h i s f a c t o r i s e x a c t l y known, i t means the measured d i s t r i b u t i o n a t h i g h e r energy (assuming p e r f e c t r e s o l u t i o n f o r the moment) i s a s l i g h t l y c l o s e r r e f l e c t i o n of the s p h e r i c a l l y - a v e r a g e d momentum d e n s i t y ; (5) The PWIA i s a b e t t e r a p p r o x i m a t i o n of the a c t u a l s c a t t e r i n g p r o c e s s a t 1200eV, than a t 400eV, and r e l a t i v e i n t e n s i t i e s w i l l be more a c c u r a t e a t the h i g h e r energy. 3 / E x p e r i m e n t a l 122 A l l t h e s e c o n s i d e r a t i o n s suggest t h a t w i t h l i m i t e d a n g u l a r r e s o l u t i o n perhaps a good compromise would be o p e r a t i o n a t 800eV. 3.6 Data A n a l y s i s The raw data p r o v i d e d by the s p e c t r o m e t e r u s u a l l y needs v e r y l i t t l e i n the way of m a n i p u l a t i o n b e f o r e i t i s p r e s e n t a b l e . Data i s c o l l e c t e d i n one of t h r e e modes: (1) A time spectrum ( f i g u r e 3.6) i s a h i s t o g r a m of the count of TAC p u l s e s s e p a r a t e d i n t o b i n s ( N i c o l e t memory c h a n n e l s ) a c c o r d i n g t o t h e i r a m p l i t u d e , which i s d i r e c t l y p r o p o r t i o n a l t o the p u l s e - p a i r time s e p a r a t i o n A t . No f u r t h e r p r o c e s s i n g i s n e c e s s a r y , except t o put the p o i n t s on the c o r r e c t time s c a l e ; (2) A b i n d i n g energy spectrum can be o b t a i n e d as a spectrum of the t r u e (e,2e) event count as a f u n c t i o n of the s e p a r a t i o n energy e a t which the c o u n t s were ac c u m u l a t e d , w i t h f i x e d <t> a n g l e . (3) An a n g u l a r c o r r e l a t i o n i s the spectrum of the t r u e (e,2e) event count over a range of a z i m u t h a l s c a t t e r i n g a n g l e s w i t h f i x e d b i n d i n g energy. B i n d i n g energy s p e c t r a may a l s o be p r e s e n t e d as they are r e c o r d e d , except t o c a l i b r a t e the energy s c a l e ( u s u a l l y 3 / E x p e r i m e n t a l 123 a g a i n s t a c c u r a t e PES s p e c t r a ) . The d a t a may be t h r e e - p o i n t smoothed, which i s a p r o c e s s of g e n e r a t i n g a new spectrum from the o l d , where the count i n the i t h c h a n n e l of the new spectrum i s a w e i g h t e d average of the i t h c h a n n e l i n the o l d spectrum and the two c h a n n e l s on e i t h e r s i d e of i t . T h i s reduces the s c a t t e r of the p o i n t s and makes s t r u c t u r e e a s i e r to see, but i t s h o u l d be used c a r e f u l l y as where the s t a t i s t i c s a r e v e r y poor, or the number of p o i n t s i n a g i v e n s t r u c t u r e i s s m a l l , i t may d i s t o r t peak h e i g h t s or i n t r o d u c e a r t i f a c t s i n t o the d a t a . A n g u l a r c o r r e l a t i o n s must be c a l i b r a t e d t o the c o r r e c t i> s c a t t e r i n g a n g l e , checked t h a t they a r e symmetric about 0=0° (as they s h o u l d be i n the symmetric non-coplanar geometry a c c o r d i n g t o e q u a t i o n 1.17) and then c o n v e r t e d t o the q - s c a l e . T h i s type of d a t a may a l s o be t h r e e - p o i n t smoothed i f the s t r u c t u r e i s weak and the s t a t i s t i c a l e r r o r l a r g e . G e n e r a l l y i f the SCA windows a r e s e t c o r r e c t l y t h e r e s h o u l d be no background on the s i g n a l , but i f t h e r e i s , i t i s p e r m i s s i b l e t o s u b t r a c t o f f a background o n l y when i t i s a b s o l u t e l y c e r t a i n where the t r u e z e r o of i n t e n s i t y i s . When the f i n a l i o n s t a t e e n e r g i e s a r e too c l o s e t o be r e s o l v e d e x p e r i m e n t a l l y i t i s not p o s s i b l e t o measure the a n g u l a r c o r r e l a t i o n d i r e c t l y as t h e r e w i l l be some c o n t r i b u t i o n from o t h e r peaks c l o s e by. In such a case the a n g u l a r c o r r e l a t i o n s must be found by d e c o n v o l u t i n g ( u s i n g some s o r t of p e a k - f i t t i n g a l g o r i t h m ) the a r e a s of peaks from 3 / E x p e r i m e n t a l s e v e r a l b i n d i n g energy s p e c t r a made a t d i f f e r e n t <t> a n g l e s . Knowing the r e l a t i v e peak a r e a s of a l l the peaks and the a n g u l a r c o r r e l a t i o n of a t l e a s t one w e l l - s e p a r a t e d peak one can g e n e r a t e the o t h e r a n g u l a r c o r r e l a t i o n s . T h i s method i s used i n the study on C0 2 (Chapter 6 ) . 1 25 CHAPTER 4 HYDROGEN SULPHIDE Sam s n i f f e d the a i r . 'Ugh! That s m e l l ! ' he s a i d . ' I t ' s g e t t i n g s t r o n g e r and s t r o n g e r . ' B i n a r y (e,2e) s t u d i e s a re now a v a i l a b l e on many Group I V - V I I h y d r i d e s , and on the i s o e l e c t r o n i c r a r e gases. S t u d i e s i n c l u d e CH„ {Weigold (1976)}, NH 3 {Hood (1976a)}, PH 3 {Hamnett (1977)}, H 20 {Hood (1977), Dixon (1977)}, the p r e s e n t work on H 2S { B r i o n (1978b), Cook (1979), Cook (1980)}, HF and HC1 { B r i o n (1979), S u z u k i (1980a), B r i o n (1980)}, HBr and HI { B r i o n } , and the r a r e gases He, Ne, A r , K r , and Xe {Weigold (1973), McCarthy (1976a), McCarthy (1976b), W e i g o l d (1975)}. The p r e s e n t e x p e r i m e n t a l and c o m p u t a t i o n a l work on H 2S was c a r r i e d out w i t h the f o l l o w i n g o b j e c t i v e s : (1) To o b t a i n the b i n a r y (e,2e) b i n d i n g energy spectrum and the momentum d i s t r i b u t i o n s of the v a l e n c e s h e l l m o l e c u l a r o r b i t a l s of H 2S; (2) To o b t a i n a r e p r e s e n t a t i v e s e t of c a l c u l a t i o n s r a n g i n g i n q u a l i t y from the • s i m p l e s t approximate methods t o those g i v i n g r e s u l t s near the H a r t r e e - F o c k 4/Hydrogen S u l p h i d e l i m i t , i n o r d e r t o i n v e s t i g a t e the r e q u i r e m e n t s n e c e s s a r y f o r a b a s i s s e t which can not o n l y g i v e a c c e p t a b l e e n e r g i e s but a l s o a d e q u a t e l y reproduce the v a l e n c e m o l e c u l a r o r b i t a l momentum d i s t r i b u t i o n s . In t h i s a s p e c t of the study the importance of d - o r b i t a l s , a s u b j e c t of c o n s i d e r a b l e c o n t r o v e r s y i n r e c e n t y e a r s , w i l l be a s s e s s e d ; (3) To study any MFS s t r u c t u r e i n an attempt t o un d e r s t a n d i t s o r i g i n . In view of t h e o r e t i c a l many-body e f f e c t p r e d i c t i o n s {Domcke (1978)} t h i s i n v o l v e s i n v e s t i g a t i o n of the b i n d i n g energy spectrum up t o a t l e a s t 32eV. P r e v i o u s s t u d i e s have not l o o k e d above about 25eV; (4) To compare and c o n t r a s t the r e s u l t s f o r H 2S w i t h t h o s e f o r o t h e r h y d r i d e s and r e l a t e d systems. 4.1 E x p e r i m e n t a l R e s u l t s The i n c i d e n t energy used was 400eV and the o v e r a l l FWHM energy r e s o l u t i o n i s about 1.6eV. The maximum t r u e c o i n c i d e n c e count r a t e on peak 1 (2b,) i s a p p r o x i m a t e l y 0 . 1 s - 1 . The H 2S sample gas was purchased from the Matheson Co. and was used w i t h o u t f u r t h e r p u r i f i c a t i o n . No e v i d e n c e of i m p u r i t i e s i s seen i n the b i n d i n g energy s p e c t r a . 4/Hydrogen S u l p h i d e 127 The H 2S b i n d i n g energy s p e c t r a a r e p r e s e n t e d i n f i g u r e 4.1. The lower energy p o r t i o n from 8-l8eV was r e c o r d e d a t 0=12°. The r e m a i n i n g segments spanning l8-34eV were r e c o r d e d a t t h r e e d i f f e r e n t a z i m u t h a l a n g l e s (0=0,5,11°) and have each been t h r e e - p o i n t smoothed once. A b r i e f e x p l o r a t i o n above 34eV showed no o b s e r v a b l e s t r u c t u r e . The energy s c a l e was c a l i b r a t e d on the l0.48eV 2b, peak of the h i g h - r e s o l u t i o n p h o t o e l e c t r o n spectrum {Turner (1970), K a r l s s o n (1976)}. The l8-34eV r e g i o n i s compared w i t h the r e s u l t of many-body c a l c u l a t i o n s i n f i g u r e 4.2 (see f o l l o w i n g d i s c u s s i o n ) . The d a t a shows t h r e e w e l l - d e f i n e d peaks below 18eV (1,2,3 i n f i g u r e 4.1, T a b l e 4.1). These were.determined t o have p-type momentum d i s t r i b u t i o n s . Above t h i s p o i n t the q u a l i t y of the d a t a drops due t o the low i n t e n s i t y of the s t r u c t u r e and the l i m i t e d s t a t i s t i c a l a c c u r a c y . N o n e t h e l e s s i t i s p o s s i b l e t o i d e n t i f y a v e r y weak peak (4) a t 19.4eV-which was de t e r m i n e d t o have an s-type momentum d i s t r i b u t i o n by measuring the r e l a t i v e (e,2e) c r o s s s e c t i o n a t t h r e e a z i m u t h a l a n g l e s (0=0,12,24°). Two p a r t i a l l y r e s o l v e d peaks (5,6) of moderate i n t e n s i t y were obs e r v e d a t 22.0 and 23.4eV, h a v i n g s-type momentum d i s t r i b u t i o n s ( a g a i n checked a t t h r e e a z i m u t h a l a n g l e s 0=0,12,24°). F i n a l l y a br o a d , low i n t e n s i t y s t r u c t u r e was observe d spanning 25-34eV and i s c o n s i d e r e d t o c o n t a i n f o u r or more major peaks (7,8,9,10). Measurements of the b i n d i n g energy spectrum a t t h r e e a n g l e s ( f i g u r e 4.1) and of the a n g u l a r c o r r e l a t i o n s a t t h r e e e n e r g i e s ( f i g u r e 4.3) show 4/Hydrogen S u l p h i d e 128 A ± I S N 3 i N I 3 A I 1 V 1 3 U F i g u r e 4.1 B i n d i n g energy spectrum of H 2S 4/Hydrogen S u l p h i d e 129 E N E R G Y CeV) F i g u r e 4.1 B i n d i n g energy s p e c t r a of H 2S. F i g u r e 4.2 (a) B i n d i n g energy spectrum of the MFS r e g i o n of H 2S a t #=0°. A l e a s t - s q u a r e s f i t of seven G a u s s i a n peaks i s shown: (b) Green's f u n c t i o n c a l c u l a t i o n of the MFS s t r u c t u r e , w i t h 0.7eV added t o the t h e o r e t i c a l e n e r g i e s . The t h e o r e t i c a l l i n e s a r e c o n v o l u t e d w i t h G a u s s i a n peaks of 1.'7, 2.0, and 2.5eV FWHM. \ \ 4/Hydrogen S u l p h i d e 130 A 1 I S N 3 1 N I 3 A I 1 V 1 3 U F i g u r e 4.3 Momentum d i s t r i b u t i o n s of the v a l e n c e o r b i t a l s of H 20 and H 2S. The m o l e c u l e name, o r b i t a l l a b e l and e x p e r i m e n t a l energy f o r each measurement are g i v e n i n the box. T h e o r e t i c a l momentum d i s t r i b u t i o n s f o r H 20 a r e : ( l ) SZ; (2) 1 1/ 2Z; (3) HF; (4) DG; and f o r H 2S a r e : (1) CNDO; (2) SZ; (3) SZ+3d; (4) DZ; (5) DZ+3d; (6) HF; (7) DG. 4/Hydrogen S u l p h i d e 131 A 1 I S N 3 1 N I 3AI1VH3U F i g u r e 4.3 c o n t i n u e d apiud-rns ua6oap^H/t> 4/Hydrogen S u l p h i d e ( w i t h i n the l i m i t s of the s t a t i s t i c s ) t h a t t h i s s t r u c t u r e i s p r e d o m i n a n t l y s - t y p e . The a n g u l a r c o r r e l a t i o n s were measured over a range of a z i m u t h a l a n g l e s +45 t o -20° i n s t e p s of 1.5°. The d i s p l a y e d momentum d i s t r i b u t i o n s ( f i g u r e 4.3) a r e shown l e a s t - s q u a r e s n o r m a l i z e d t o the c a l c u l a t e d momentum d i s t r i b u t i o n s which have had the e x p e r i m e n t a l a n g u l a r r e s o l u t i o n f o l d e d i n . I t s h o u l d be noted t h a t the low momentum r e g i o n s of the c a l c u l a t e d momentum d i s t r i b u t i o n s a r e p a r t i c u l a r l y s e n s i t i v e t o the v a l u e s of the a n g u l a r r e s o l u t i o n chosen f o r the f o l d i n g - i n p r o c e d u r e . T h e r e f o r e , i n comparing c a l c u l a t i o n and exp e r i m e n t , a t t e n t i o n i s p a r t i c u l a r l y f o c u s s e d on the r e g i o n above q=0.4a o' 1. T h i s i n c l u d e s the p o s i t i o n of the maximum of the d i s t r i b u t i o n ( i n the case of p-type d i s t r i b u t i o n s ) and the b e h a v i o u r of the curv e above about q=0.4a o- 1. In f i g u r e 4.3 the r e s u l t s of p r e v i o u s work on H 20 {Hood (1977)} a re p r e s e n t e d i n the same format as t h a t used f o r H 2S in- o r d e r t o f a c i l i t a t e comparisons between r e s u l t s f o r the two m o l e c u l e s l a t e r i n the d i s c u s s i o n . F i g u r e 4.3 a l s o shows the e x p e r i m e n t a l r e s u l t s f o r the m u l t i p l e f i n a l s t a t e s t r u c t u r e i n H 2S above l9eV. S i n c e a l l c a l c u l a t e d d i s t r i b u t i o n s a r e almost i d e n t i c a l , o n l y the r e s u l t s of c a l c u l a t i o n H 2S-HF a r e shown ( f o r d e t a i l s see f o l l o w i n g s e c t i o n ) . The s t a t i s t i c s a r e poor on d i s t r i b u t i o n s 4/Hydrogen S u l p h i d e 134 7 and 9 because these peaks a r e v e r y weak i n the b i n d i n g energy spectrum. 4.2 C a l c u l a t i o n s The c a l c u l a t i o n s on H 2S (shown i n - f i g u r e 4.3) e i t h e r c a r r i e d out by us or o b t a i n e d from the l i t e r a t u r e and o t h e r s o u r c e s a r e l i s t e d h e r e : (1) CNDO: a s i m p l e CNDO/2 c a l c u l a t i o n u s i n g a min i m a l b a s i s s e t of S l a t e r - t y p e o r b i t a l s (STOs), augmented by d - f u n c t i o n s . The 3s, 3p, and 3d f u n c t i o n s a l l have the same exponent; (2) SZ: a s i n g l e - z e t a a b - i n i t i o c a l c u l a t i o n u s i n g STOs {C l e m e n t i (1963)}. T h i s c o m p u t a t i o n was done w i t h the ATMOL/2 package a t O x f o r d U n i v e r s i t y ; (3) SZ+3d: a s i n g l e - z e t a ab i n i t i o c a l c u l a t i o n but augmented w i t h d i f f u s e d - f u n c t i o n s { H i l l i e r (1970)} , a l s o done on the ATMOL/2 package; (4,5) DZ, DZ+3d: s i m i l a r t o SZ and SZ +3d except t h a t the b a s i s s e t used was of d o u b l e - z e t a q u a l i t y i n a l l but t h e d - f u n c t i o n s { C l e m e n t i (1964)}; (6) HF: an a c c u r a t e c a l c u l a t i o n a p p r o a c h i n g the H a r t r e e - F o c k l i m i t u s i n g a t r i p l e - z e t a e q u i v a l e n t GTO b a s i s i n c l u d i n g d - f u n c t i o n s {Guest (1976)}; (7) DG: a c a l c u l a t i o n u s i n g a b a s i s s e t of Ga u s s i a n 4/Hydrogen S u l p h i d e f u n c t i o n s (GTOs), not i n c l u d i n g d - f u n c t i o n s , but i n c o r p o r a t i n g some ve r y d i f f u s e low-exponent f u n c t i o n s { Z e i s s } . The exponents were o b t a i n e d by l e a s t - s q u a r e s f i t t i n g of the GTO s e t t o the r e s u l t s of atomic c a l c u l a t i o n s . These atomic c a l c u l a t i o n s were done w i t h a d o u b l e - z e t a q u a l i t y b a s i s s e t , and f o u r GTOs were f i t t e d t o each atomic o r b i t a l , but w i t h o u t the u s u a l r ~ 1 w e i g h t i n g of the f i t t i n g f u n c t i o n . The absence of the r ~ 1 w e i g h t i n g means the GTO b a s i s i s more a c c u r a t e i n d u p l i c a t i n g the d o u b l e - z e t a r e s u l t i n the o u t e r s p a t i a l r e g i o n s of the atomic o r b i t a l s . The c a l c u l a t i o n s f o r H 20 d e s c r i b e d i n a p r e v i o u s paper a r e summarized here f o r r e f e r e n c e {Hood (1977)}. ( 1 ) SZ: a s i n g l e - z e t a b a s i s c a l c u l a t i o n w i t h STO f u n c t i o n s , of the same q u a l i t y as H 2S~SZ, c a l l e d ' b a s i s 1 ' i n {Hood (1977)}; (2) 1 V 2 Z : a mixed b a s i s c a l c u l a t i o n ; w i t h oxygen 2p l e v e l s d e s c r i b e d by d o u b l e - z e t a f u n c t i o n s , and r e m a i n i n g atomic o r b i t a l s by s i n g l e - z e t a f u n c t i o n s , l a b e l l e d ' b a s i s 2' i n {Hood'( 1977)}; (3) HF: a f u l l d o u b l e - z e t a c a l c u l a t i o n augmented by 3d . f u n c t i o n s on oxygen, and 2s, and 2p p o l a r i z a t i o n f u n c t i o n s on hydrogen. The oxygen 2p l e v e l s a re d e s c r i b e d by t r i p l e - z e t a f u n c t i o n s , c a l l e d ' b a s i s 4* i n {Hood (1977)}; (4) DG: a GTO b a s i s c a l c u l a t i o n s i m i l a r t o t h a t of 4/Hydrogen S u l p h i d e H 2S-DG { Z e i s s } . 4.3 D i s c u s s i o n The m o l e c u l a r o r b i t a l t h e o r y e l e c t r o n i c c o n f i g u r a t i o n f o r H 2S i s : ( 1 a , ) 2 ( 2 a , ) 2 ( 1 b 2 ) 2 ( 3 a , ) 2 ( 1 b , ) 2 ( 4 a , ) 2 ( 2 b 2 ) 2 ( 5 a , ) 2 ( 2 b , ) 2 w i t h the l a s t f o u r o r b i t a l s c o n s t i t u t i n g the v a l e n c e s h e l l . 4.3.1 B i n d i n g energy s p e c t r a of H 2S The b i n d i n g energy spectrum of H 2S has a l r e a d y been p a r t i a l l y i n v e s t i g a t e d by PES methods u s i n g b oth UV {Turner (1970)} and X-ray {Siegbahn (1972)} l i g h t s o u r c e s . These s t u d i e s a l l show c o n s i s t e n t r e s u l t s f o r the b i n d i n g e n e r g i e s of the e l e c t r o n s i n the t h r e e outermost o r b i t a l s . The f i r s t band i n the He-I p h o t o e l e c t r o n spectrum r e p o r t e d by Turner w i t h a v e r t i c a l I P of l0.48eV has a v i b r a t i o n a l e nvelope c h a r a c t e r i s t i c of the removal of a non-bonding e l e c t r o n , and may be u n e q u i v o c a l l y a s s i g n e d t o the 2b, ( l o n e - p a i r ) o r b i t a l on s u l p h u r . LCAO-MO-SCF c a l c u l a t i o n s g i v e t h i s o r b i t a l a l most e x c l u s i v e l y s u l p h u r 3p{x} ( o u t - o f - p l a n e ) c h a r a c t e r (see geometry diagram, Appendix C ) . The next band w i t h a v e r t i c a l IP of 13.25eV i s c l e a r l y 4/Hydrogen S u l p h i d e Table 4.1 C a l c u l a t e d and e x p e r i m e n t a l H 2S b i n d i n g e n e r g i e s . Peak E x p t 1 E x p t 2 E x p t 3 1 2b, 10.5 10.48 2 5a, 13.5 13.35 3 2b 2 15.3 15.25 4 19.4 19.4 -5 22.0 22.0 -6 23.4 23.4 -7 4a, 1 1 27 -8 29 -9 31 -10 33 — Peak SZ" SZ+3d 5 DZ 6 DZ+3d 7 HF 8 DG 9 MBGF 1 0 1 2b, 10.33 9.95 10.75 10.46 1 0.49 12. 60 9.61 2 5a, 12.87 13.05 13.42 13.70 13.65 15. 12 12.39 3 2b 2 15.74 15.57 16.22 16.20 16.21 17. 57 1 5.44 4 21 .33 5 23.65 6 26. 1 1 7 4a, 26.71 26.05 27.38 26.72 26.87 30. 33 29.00 8 29. 16 9 29.45 10 31 .34 1 P r e s e n t work : c e n t r o i d of MFS s t r u c t u r e i s a t 25eV. 2 Hig h - r e s o l u t i o n He-II PES 3 He-I PES 4-10 These r e s u l t s a r e d e s c r i b e d i n s e c t i o n 4.4. 11 E n e r g i e s f o r t h e s e l e v e l s i n the p r e s e n t work ar e from the f i t t e d r e s u l t s . 4/Hydrogen S u l p h i d e s t r o n g l y bonding, as e v i d e n c e d by i t s v i b r a t i o n a l envelope which e x h i b i t s a l o n g p r o g r e s s i o n i n the v2 or bending normal mode. C a l c u l a t i o n s f o r t h i s 5a, o r b i t a l put e l e c t r o n d e n s i t y l a r g e l y i n the c e n t r a l r e g i o n between the t h r e e atoms, thus a c c o u n t i n g f o r the a n g l e - s t a b i l i z i n g n a t u r e of t h i s MO. T h i s o r b i t a l i s c o n s t i t u t e d of b a s i c a l l y the s u l p h u r 3p{z} o r b i t a l ( i n - p l a n e , a l o n g the Cfc°v} a x i s ) t o g e t h e r w i t h major c o n t r i b u t i o n s of the r e q u i r e d symmetry (a,) from the hydrogens. The t h i r d o r b i t a l , d e s i g n a t e d 2 b 2 , w i t h a v e r t i c a l IP of 15.3eV i s perhaps the most s t r o n g l y bonding of a l l . I t s v i b r a t i o n a l envelope i s e x t e n s i v e , and c o r r e s p o n d s t o s t i m u l a t i o n of the v,, or symmetric s t r e t c h i n g mode. T h i s c o r r e l a t e s w e l l w i t h c a l c u l a t i o n s which g i v e t h i s o r b i t a l m a i n l y S-H <y-bonding c h a r a c t e r brought about by o v e r l a p of the l o b e s of a s u l p h u r 3p{y} o r b i t a l ( i n - p l a n e , p a r a l l e l t o hydrogen atoms) w i t h each hydrogen 1s. T h e r e f o r e e l e c t r o n d e n s i t y i s p l a c e d a l o n g the bonds. Few PES measurements have been made of the b i n d i n g energy of the deepest v a l e n c e s h e l l MO (4a,) s i n c e i t l i e s beyond the range of the He-I l i n e . A l o w - r e s o l u t i o n He-II spectrum has been r e p o r t e d { R a b a l a i s (1977)} which shows a broad peak i n the r e g i o n of 22eV w i t h l i t t l e d i s c e r n i b l e v i b r a t i o n a l s t r u c t u r e , but u n f o r t u n a t e l y the b i n d i n g energy s c a l e does not ex t e n d above 25eV. The v a l e n c e s h e l l XPS spectrum of H 2S {Siegbahn (1972)} a l s o shows a peak a t 22eV, 4/Hydrogen S u l p h i d e 139 w i t h some p o s s i b l e i n d i c a t i o n of s t r u c t u r e above 22eV, which the a u t h o r s d i d not c o n s i d e r s i g n i f i c a n t . A peak a t 19eV a s c r i b e d t o a n i t r o g e n i m p u r i t y i s a l s o shown. However, i n view of the w i d t h of the 19eV peak, the i m p u r i t y assignment i s q u e s t i o n a b l e . The s i t u a t i o n has been p a r t i a l l y c l a r i f i e d by the h i g h - r e s o l u t i o n He-II spectrum of H 2S {Maier, B r i o n (1978b), Domcke (1978)}. T h i s spectrum shows t h a t the broad peak a t 22eV i s i n f a c t composed of two major peaks at 22.0 and 23.4eV. A v e r y broad low i n t e n s i t y peak i s a l s o i n d i c a t e d a t l 9 . 4 e V . T h i s l 9 . 4 e V f e a t u r e i s t o o broad and a l s o too f a r removed from the l 8 . 6 e V 2s<r{u} band of n i t r o g e n t o be a s c r i b a b l e t o a n i t r o g e n i m p u r i t y . T h i s peak a t l 9 . 4 e V i s of r e l a t i v e l y low i n t e n s i t y and t h e r e f o r e cannot be seen a t the s e n s i t i v i t y of the l o w - r e s o l u t i o n He-II spectrum. The 4a, o r b i t a l i s not s t r o n g l y bonding as i n d i c a t e d by c a l c u l a t i o n which a s c r i b e t o i t m a i n l y s u l p h u r 3s c h a r a c t e r w i t h o n l y a s m a l l a, symmetry c o n t r i b u t i o n from the hydrogens. Thus the 4a, MO has c o n s i d e r a b l e atomic c h a r a c t e r . A r e c e n t l y - r e p o r t e d d i p o l e (e,2e) b i n d i n g energy spectrum of H 2S { B r i o n (1978b)} has i n d i c a t e d broad peaks c e n t e r e d around 22.5 and 29 eV. R e c e n t l y t h e r e has been a growing r e a l i z a t i o n of the im p o r t a n t r o l e of many-body e f f e c t s i n the i o n i z a t i o n of i n n e r v a l e n c e o r b i t a l s . There have been some a t t e m p t s t o p r e d i c t t h e o r e t i c a l l y the m u l t i p l e f i n a l s t a t e , or shake-up, s t r u c t u r e . of H 2S. A r e c e n t c a l c u l a t i o n u s i n g r a t h e r l i m i t e d 4/Hydrogen S u l p h i d e 140 VB-SCF-CI methods {Chipman (1978)} i n c o r p o r a t i n g o n l y f o u r e x c i t e d c o n f i g u r a t i o n s p l u s the ground s t a t e , c l a i m s t o reproduce the e n e r g i e s of the e x p e r i m e n t a l l y o b s e r v e d 19.4 and 22.0eV peaks. At the time t h i s c a l c u l a t i o n was c a r r i e d out the e x t e n t of the shake-up s t r u c t u r e i n H 2S was not f u l l y r e a l i z e d , and, i n view of the r e s u l t s t o be p r e s e n t e d l a t e r i n t h i s paper, the l i m i t e d t o t a l w a v e f u n c t i o n chosen f o r the work i s p r o b a b l y i nadequate t o d e s c r i b e a l l the s t r u c t u r e we have now ob s e r v e d . N e v e r t h e l e s s w i t h t h i s i n mind the r e l a t i v e e n e r g i e s p r e d i c t e d a r e more r e a s o n a b l e when one c o n s i d e r s t h a t the e x p e r i m e n t a l XPS peak a t 22eV d i s c u s s e d by Chipman i s i n f a c t two peaks as shown by t h i s work and the He-II spectrum. The p r e d i c t e d s p a c i n g of 4.5eV i s c o n s i s t e n t w i t h the assignment of the 23.4 peak as the 'p a r e n t ' , r a t h e r than the one a t 22.0eV, even though the l a t t e r i s more i n t e n s e . The Green's f u n c t i o n t h e o r e t i c a l t e c h n i q u e i s more comprehensive i n the sense t h a t t h e r e a re not the r e s t r i c t i o n s on the c a l c u l a t i o n t h a t a r e imposed by the method of Chipman, and f u r t h e r m o r e i n t h a t a more f l e x i b l e b a s i s s e t i s used. The MBGF c a l c u l a t i o n p r e d i c t s t h r e e major w e l l - s e p a r a t e d peaks i n the r e g i o n 21-26eV, w i t h e x t e n s i v e lower i n t e n s i t y s t r u c t u r e e x t e n d i n g up t o around 32eV ( f i g u r e 4.2b). A l l of the peaks above 19eV a r e p r e d i c t e d t o be a s s o c i a t e d w i t h i o n i z a t i o n from the 4a, m o l e c u l a r o r b i t a l and t h e r e f o r e would be e x p e c t e d t o show the same momentum d i s t r i b u t i o n . The t h r e e peaks between 21-26eV a r e n i c e l y r e f l e c t e d i n the h i g h - r e s o l u t i o n He-II 4/Hydrogen S u l p h i d e 141 spectrum {Domcke (1978)} i f one a l l o w s a s h i f t of about 1eV t o lower energy i n the t h e o r e t i c a l energy s c a l e . The c a l c u l a t i o n s a l s o g i v e r e a s o n a b l e e s t i m a t e s of the r e l a t i v e i n t e n s i t i e s of t h e s e t h r e e peaks. Peaks 1,2 and 3 of the b i n a r y (e,2e) H 2S b i n d i n g energy spectrum ( f i g u r e 4.1) show r e a s o n a b l e agreement of peak p o s i t i o n s w i t h p r e v i o u s He-I PES work {Turner (1970), Goddard (1978)}. The r e l a t i v e i n t e n s i t i e s of these t h r e e l o w e s t energy peaks a r e s i m i l a r t o t h o s e i n H 20 {Hood (1977)}. In the r e g i o n of l9-25eV t h i s work shows c l o s e agreement w i t h the l o w - r e s o l u t i o n He-II spectrum w i t h r e s p e c t t o peak e n e r g i e s , i n t e n s i t i e s and l i n e shapes. Both s p e c t r a bear out the MBGF p r e d i c t i o n s of t h r e e peaks i n t h i s energy range, as d i s c u s s e d above and shown i n f i g u r e 4.2. F i n a l l y , the f i r s t e v i d e n c e f o r e x t e n s i v e MFS s t r u c t u r e i n the r e g i o n 25-34eV i s r e p o r t e d . A g a i n , the Green's f u n c t i o n c a l c u l a t i o n s have p r e d i c t e d s t r u c t u r e i n t h i s r e g i o n which seems t o show some c o r r e l a t i o n s w i t h the p r e s e n t work ( f i g u r e 4.2), but i t s h o u l d be noted t h a t the c a l c u l a t i o n s are q u i t e s e n s i t i v e t o b a s i s s e t changes. I have enumerated s e v e r a l bands (7,8,9,10, f i g u r e 4.1) i n t h i s r e g i o n , where t h e r e appear t o be a t l e a s t f o u r major peaks. I t i s p o s s i b l e (see f i g u r e 4.2a) t o o b t a i n a good l e a s t - s q u a r e s f i t of f o u r G a u s s i a n peak shapes (1.7eV FWHM) t o t h i s s t r u c t u r e , though t h i s s h o u l d not be taken as the o n l y p o s s i b l e d e c o n v o l u t i o n of the d a t a . The assignment of peaks i n t h i s r e g i o n i s 4/Hydrogen S u l p h i d e d i f f i c u l t t o make out due t o the r e l a t i v e l y poor energy r e s o l u t i o n and the low i n t e n s i t y of t h i s s t r u c t u r e . The r e s u l t s a r e c o n c l u s i v e o n l y i n t h a t they c o n f i r m the MBGF p r e d i c t i o n of some s o r t of e x t e n s i v e MFS s t r u c t u r e of a, symmetry. C l e a r l y h i g h e r r e s o l u t i o n work i s r e q u i r e d here f o r comparison w i t h the many-body Green's f u n c t i o n c a l c u l a t i o n s . 4.3.2 Momentum d i s t r i b u t i o n s The momentum d i s t r i b u t i o n s a r e shown i n f i g u r e 4.3. For the 2b, and 5a, MOs of H 2S the g r e a t e s t improvement i n t h e f i t of c a l c u l a t e d c u r v e s o c c u r s i n g o i n g from a s i n g l e - z e t a t o a d o u b l e - z e t a q u a l i t y b a s i s s e t . The a d d i t i o n of d - f u n c t i o n s t o e i t h e r the SZ or the DZ c a l c u l a t i o n produced no s i g n i f i c a n t changes i n the d i s t r i b u t i o n and f o r t h i s reason t h e s e r e s u l t s were not d i s p l a y e d i n the f i g u r e s . S i g n i f i c a n t improvement over the DZ r e s u l t i s o n l y o b t a i n e d w i t h the HF l i m i t c a l c u l a t i o n , but t h i s does not produce as g r e a t a change as g o i n g from SZ t o DZ. The SZ and DZ r e s u l t s on the 2b 2 MO show v e r y s i m i l a r d i s t r i b u t i o n s , i n c o n t r a s t t o the p r e v i o u s two MOs and here the HF l i m i t c a l c u l a t i o n g i v e s the g r e a t e s t improvement over both SZ and DZ i n the f i t t o ex p e r i m e n t . Both SZ and DZ bases p r e d i c t a d i s t r i b u t i o n somewhat more s p a t i a l l y e xtended than i s i n f a c t measured. The f a c t t h a t i n H 2S and 4/Hydrogen S u l p h i d e 143 H 20 the HF l i m i t c a l c u l a t i o n i s s u p e r i o r t o e i t h e r SZ or DZ may be due t o the presence of hydrogen 2p p o l a r i z a t i o n f u n c t i o n s . As d e s c r i b e d i n s e c t i o n 4.1 the b 2 MO i s p r i m a r i l y an S-H (or O-H) bonding o r b i t a l w i t h d e n s i t y d i s t r i b u t e d a l o n g the bond axes. T h e r e f o r e the a b i l i t y of the hydrogen p o l a r i z a t i o n f u n c t i o n s t o d i r e c t d e n s i t y i n t o t h e s e r e g i o n s and improve the o v e r l a p w i t h the s u l p h u r 3p{y} (or oxygen 2p{y}) atomic o r b i t a l s h o u l d be p a r t i c u l a r l y marked i n t h i s MO. There has been c o n s i d e r a b l e c o n t r o v e r s y d u r i n g the past few y e a r s over the r o l e t h a t S-3d o r b i t a l s p l a y i n the bonding i n s u l p h u r compounds. H a r t r e e - F o c k c a l c u l a t i o n s of the b i n d i n g e n e r g i e s of t h e s e d - o r b i t a l s i n atomic e x c i t e d s t a t e s such as s 2 p 3 d ( 5 D ) su g g e s t e d t h a t the d - o r b i t a l energy was f a r too h i g h f o r i t t o p l a y a s i g n i f i c a n t r o l e . However, an i n c r e a s i n g body of e x p e r i m e n t a l e v i d e n c e {Cruikshank (1968)} has p o i n t e d t o an a c t i v e i n v o l v e m e n t of t h e s e d - o r b i t a l s i n the bonding under c e r t a i n c i r c u m s t a n c e s . The most c l e a r - c u t c a s e s a r i s e when h i g h l y e l e c t r o n e g a t i v e elements a r e bonded t o s u l p h u r and an e x t e n s i v e s e r i e s of c a l c u l a t i o n s on S 0 2 {Rothenberg (1970)} showed t h a t u n l e s s S-3d o r b i t a l s were i n c l u d e d , q u a n t i t a t i v e l y r e a s o n a b l e r e s u l t s c o u l d not be o b t a i n e d . In the case of S 0 2 , f u r t h e r e v i d e n c e i s p r o v i d e d by the p h o t o e l e c t r o n spectrum. One c o m b i n a t i o n of 0-2p o r b i t a l s t r a n s f o r m s as a 2 and s h o u l d t h e r e f o r e be a c c u r a t e l y non-bonding i f S-3d o r b i t a l s are 4/Hydrogen S u l p h i d e u n i m p o r t a n t {Ratner (1971)}. In f a c t , as the spectrum shows, the 2 A 2 bond e x h i b i t s c o n s i d e r a b l e bonding c h a r a c t e r { H i l l i e r (1971)}. The p o s i t i o n - w i t h r e g a r d t o H 2S i s l e s s c l e a r . By comparing the c o n t r i b u t i o n s of d - o r b i t a l s t o the bond s t r e n g t h s i n S 0 2 and H 2S, i t was c o n c l u d e d {Rothenberg (1970)} t h a t they were an o r d e r of magnitude l e s s i m p o r t a n t i n the l a t t e r c a s e . Support f o r t h i s c o n c l u s i o n has come from some r e c e n t c a l c u l a t i o n s {Von N i e s s e n (1977)} who found t h a t o n l y v e r y d i f f u s e d - o r b i t a l s had any impact on the o r b i t a l energy. I t has been suggested t h a t i f the s and p bases a r e e x h a u s t i v e l y d e s c r i b e d , no d - o r b i t a l c o n t r i b u t i o n w i l l be i m p o r t a n t i n any m o l e c u l e . A s i d e from the e v i d e n c e of the S 0 2 spectrum quoted above, t h i s statement a l s o can be q u e s t i o n e d on the b a s i s of i t s c o r o l l a r y . I f a v e r y poor b a s i s i s s e l e c t e d , we s h o u l d expect s u b s t a n t i a l improvement i n the o r b i t a l d e s c r i p t i o n by employing d - o r b i t a l s . In f a c t , the r e s u l t s of our p r e s e n t work suggest t h a t , a t l e a s t f o r H 2S, t h i s i s not so. The poor d e s c r i p t i o n of the 2b, o r b i t a l found i n a minimum b a s i s s e t c a l c u l a t i o n i s not s i g n i f i c a n t l y improved by the a d d i t i o n of d - o r b i t a l s t o the s u l p h u r atom but i s b e t t e r e d by making the s/p b a s i s more f l e x i b l e . Where the s/p b a s i s i s not e x t e n s i v e , but d - o r b i t a l s have been i n c l u d e d , one s h o u l d not n e c e s s a r i l y a s s i g n any i n t r i n s i c c h e m i c a l importance t o the e f f e c t of 4/Hydrogen S u l p h i d e the d - f u n c t i o n s {Kwart (1977)}. R a t h e r , any changes here a r e due t o the s l i g h t a d d i t i o n a l f l e x i b i l i t y which t h e s e f u n c t i o n s i n t r o d u c e i n t o the b a s i s used i n m o d e l l i n g the m o l e c u l a r o r b i t a l w a v e f u n c t i o n . Changes i n t o t a l e l e c t r o n d e n s i t y i n H 20 and H 2S as a r e s u l t of a d j u s t i n g the b a s i s s e t have been s t u d i e d { B i c e r a n o (1977)}. In t h e i r c a l c u l a t i o n s the d e n s i t y d i f f e r e n c e p l o t s on H 2S show the g r e a t e s t changes between the s i n g l e - z e t a and d o u b l e - z e t a b a s i s s e t s . S m a l l e r changes oc c u r on g r a d u a l l y i m p r o v i n g the b a s i s t h r o u g h a d d i t i o n of s u c c e s s i v e l y e x t r a s- and p- f u n c t i o n s , then d - f u n c t i o n s , then hydrogen 2 p - f u n c t i o n s , and f i n a l l y e x t r a p o l a r i z a t i o n f u n c t i o n s t o g i v e a near-HF l i m i t r e s u l t . C a l c u l a t i o n of the d e n s i t y d i f f e r e n c e p l o t s on H 20 u s i n g analogous b a s i s s e t s shows t h a t the e f f e c t of d - f u n c t i o n s , though r e l a t i v e l y s m a l l i n both c a s e s , i s as l a r g e or even g r e a t e r i n H 20 than i n H 2S. C e r t a i n l y d - o r b i t a l s i n H 20 can have v i r t u a l l y no e x p l i c i t c h e m i c a l importance and t h e i r e f f e c t i n H 20 and H 2S seems o n l y t o make the s/p b a s i s s l i g h t l y more f l e x i b l e . These r e s u l t s , t o g e t h e r w i t h t h o s e of the p r e s e n t work, suggest t h a t r e a s o n a b l e w a v e f u n c t i o n s f o r H 20 r e q u i r e a v e r y e x t e n s i v e , c a r e f u l l y chosen s/p b a s i s , and f o r H 2S an s/p b a s i s somewhat l e s s f l e x i b l e . I f t h i s i s done no d - f u n c t i o n s need be i n c l u d e d . E x a m i n a t i o n of T a b l e 4.1 shows t h a t a d d i t i o n of d - f u n c t i o n s produces no s i g n i f i c a n t s y s t e m a t i c changes i n 4/Hydrogen S u l p h i d e the o r b i t a l e n e r g i e s . The s m a l l changes seen here are not m e a n i n g f u l , but r e s u l t o n l y from a l l o w i n g the SCF c a l c u l a t i o n more 'space' w i t h i n which t o v a r i a t i o n a l l y m i n i m i z e the t o t a l energy. One s h o u l d not t h e r e f o r e c o n c l u d e t h a t the w a v e f u n c t i o n i n the bonding r e g i o n has a u t o m a t i c a l l y been improved on a d d i t i o n of e x t r a f u n c t i o n s , w i t h o u t l o o k i n g f o r s y s t e m a t i c changes i n o r b i t a l e n e r g i e s , l o w e r i n g of the t o t a l e nergy, or s i g n i f i c a n t a l t e r a t i o n s of d e n s i t y p l o t s or momentum d i s t r i b u t i o n s . Of p a r t i c u l a r note i s the performance of the d i f f u s e G a u s s i a n b a s i s (DG) c a l c u l a t i o n i n r e p r o d u c i n g the e x p e r i m e n t a l d i s t r i b u t i o n i n both H 20 and H 2S. The f i t of the c u r v e s i s c o n s i s t e n t l y as good a s , and sometimes m a r g i n a l l y b e t t e r than the d o u b l e - z e t a r e s u l t s . The method of o p t i m i z i n g G a u s s i a n exponents, as d e s c r i b e d above, has been used w i t h some s u c c e s s t o c a l c u l a t e o t h e r q u a n t i t i e s , such as d i p o l e moments and p o l a r i z a b i l i t i e s { Z e i s s (1979)} which a r e more c r i t i c a l l y dependent on the s p a t i a l d i s t r i b u t i o n of charge. The advantage of the d i f f u s e b a s i s c a l c u l a t i o n l i e s i n the f a c t t h a t a l t h o u g h i t i s e s s e n t i a l l y s i n g l e - z e t a i n c o m p l e x i t y , ' i t i s e f f e c t i v e l y d o u b l e - z e t a i n the q u a l i t y of i t s d e s c r i p t i o n of the atomic o r b i t a l w a v e f u n c t i o n s . The c o m p u t a t i o n i s t h e r e f o r e s i m p l i f i e d . Due t o the l i n e a r w e i g h t i n g and i n c o r p o r a t i o n of v e r y d i f f u s e f u n c t i o n s i t i s f l e x i b l e i n the ( e , 2 e ) - s e n s i t i v e bonding r e g i o n s , but on the o t h e r hand, i t i s f a i r l y r i g i d near the 4/Hydrogen S u l p h i d e 147 n u c l e a r c e n t r e s , and so w i l l not reproduce o r b i t a l e n e r g i e s v e r y a c c u r a t e l y . 4.3.3 MFS s t r u c t u r e A l l major s t r u c t u r e s i n the H 2S b i n d i n g energy spectrum beyond the t h r e e o u t e r v a l e n c e IPs have been shown e x p e r i m e n t a l l y (see f i g u r e 4.3) t o e x h i b i t s-type momentum d i s t r i b u t i o n s , c o n f i r m i n g the Green's f u n c t i o n p r e d i c t i o n s t h a t a l l MFS s t r u c t u r e i s a s s o c i a t e d w i t h i o n i z a t i o n of the 4a, MO. The d i f f e r e n t SCF c a l c u l a t i o n s of the 4a, momentum d i s t r i b u t i o n a r e a l l v i r t u a l l y i d e n t i c a l ( w i t h t h e e x c e p t i o n of the v e r y poor CNDO r e s u l t ) and f o r t h i s reason o n l y the HF l i m i t r e s u l t i s shown i n f i g u r e 4.3. T h i s ease of d e s c r i p t i o n i n d i c a t e s the l a r g e atomic c h a r a c t e r of the 4a, MO. Some attempt was i n i t i a l l y made, on e m p i r i c a l and s e m i - q u a n t i t a t i v e grounds, t o a s s i g n the MFS s t r u c t u r e t o a c o m b i n a t i o n of d i s c r e t e i o n i z a t i o n s and e x c i t a t i o n s , as has been done f o r i n n e r s h e l l XPS s p e c t r a { A l l a n (1972)}, but no c o n v i n c i n g argument c o u l d be made f o r any such assignment. T h i s s u p p o r t s the view t h a t t h i s s t r u c t u r e i n v o l v e s , i n the v a l e n c e s h e l l , a complex rearrangement or m i x i n g of s t a t e s which a r e not r e l a t e d t o the o c c u p i e d and u n o c c u p i e d l e v e l s of the ground s t a t e i n any s i m p l e way. T h e r e f o r e no i o n s t a t e can be s i n g l e d out as 'parent' on the b a s i s t h a t i t s e l e c t r o n i c s t r u c t u r e i s ve r y s i m i l a r , except f o r one m i s s i n g v a l e n c e e l e c t r o n , t o the ground s t a t e s t r u c t u r e . In the 4/Hydrogen S u l p h i d e c o n f i g u r a t i o n i n t e r a c t i o n p i c t u r e t h i s would mean t h a t the w a v e f u n c t i o n f o r each f i n a l i o n s t a t e c o n t a i n s a number of i m p o r t a n t c o n f i g u r a t i o n s , none of which dominate s t r o n g l y over the o t h e r s . For these r e a s o n s I have g e n e r a l l y used the term ' m u l t i p l e f i n a l s t a t e ' (MFS) t o d e s c r i b e the s t r u c t u r e a s s o c i a t e d w i t h the i n n e r v a l e n c e o r b i t a l s , r a t h e r than terms such as 'shake-up' and 'shake-down' or ' s a t e l l i t e s t r u c t u r e ' . These terms can s t i l l be used i n the d i s c u s s i o n of i n n e r s h e l l s p e c t r a as measured by ESCA or XPS t e c h n i q u e s from which they o r i g i n a l l y a r o s e . Because of the l a r g e s e p a r a t i o n i n energy between the i n n e r s h e l l h o l e and the v a l e n c e l e v e l s , t h e r e i s v e r y l i t t l e m i x i n g of s t a t e s , and the p i c t u r e i n c o r p o r a t e d i n the 'sudden' a p p r o x i m a t i o n t h e o r y {Aberg (1967)} i s l i k e l y t o be more r e a s o n a b l e . 4.3.4 Trends i n the AXn h y d r i d e s and r a r e gases The v a l e n c e e l e c t r o n i c s t r u c t u r e of the group V-VII h y d r i d e s (AHn) i s g r o s s l y s i m i l a r i n t h a t t h e r e e x i s t s a p-type m a i n l y non-bonding, low-energy ' l o n e - p a i r ' o r b i t a l , f o l l o w e d by one or two sometimes degenerate h i g h e r - e n e r g y , m a i n l y bonding p-type MOs, and one s t i l l h i g h e r s-type MO of e s s e n t i a l l y atomic c h a r a c t e r . In o r d e r t o study the r e l a t i o n s h i p s among e x p e r i m e n t a l MDs of the v a r i o u s h y d r i d e m o l e c u l e s and r a r e gas atoms I have made c o r r e l a t i o n diagrams ( f i g u r e 4.4) of the v a l u e s of q{max} and q{ 1/ 2max} a g a i n s t IP V 2 . q{max} i s the momentum a t 4/Hydrogen S u l p h i d e 149 F i g u r e 4.4 C o r r e l a t i o n diagrams of q{max} and q{V 2max} a g a i n s t I P 1 / 2 . S o l i d symbols denote q{max} and open ones g i v e q{ 1/ 2max}. S o l i d l i n e s j o i n the p o i n t s of the same m o l e c u l e , and e r r o r e s t i m a t e s a r e g i v e n by the e l l i p s e s . 4/Hydrogen S u l p h i d e F i g u r e 4.4 c o n t i n u e d 4/Hydrogen S u l p h i d e 151 H2S MOMENTUM DENSITY CHARGE DENSITY 4.5 Momentum and charge d e n s i t y maps f o r H 2S from the H 2S-DG w a v e f u n c t i o n . 4/Hydrogen S u l p h i d e T a b l e 4.2 E n e r g i e s and v a l u e s 1 of q{max} and qV 2{max} f o r the h y d r i d e m o l e c u l e momentum d i s t r i b u t i o n s . H 2S r e s u l t s a r e from t h i s work. Sources f o r the o t h e r m o l e c u l e s a r e g i v e n a t the b e g i n n i n g of Chapter 4. ( M o l e c u l e ) ( O r b i t a l q{max}/q{ 1/2max} IP) CH, 1 t 2 .6 14.2 2a, .5 24 NH 3 3a, .50 11.0 1e .68 16.5 2a, .7 32 H 20 1b, .59 12.6 3a, .66 14.7 1b 2 .72 18.4 2a, .7 32 HF 1n .70 16.1 3c .75 19.9 2c .7 39.7 Ne 2p .75 22 2s .7 49 PH 3 5a, .45 10.6 2e .55 13.4 4a, .45 27 H 2S 2b, .45 10.5 5a, .52 13.5 2b 2 .63 15.3 4a, .45 26 HCl 2rr .55 12.8 be .6 16.5 4* .5 30 Ar 3p .65 15.8 3s .6 36 HBr 4rr .5 11.8 8c .6 15.6 Ic .45 29 Kr 4p .6 14.2 4s .55 32 • HI 6TT .45 10.8 11* .55 14.2 10* .3 25 Xe 5p .5 12.3 5s .4 28 1 I t i s o n l y the innermost, h i g h e s t energy o r b i t a l of a g i v e n m o l e c u l e i n t h i s t a b l e which i s s-type and f o r which q{ 1/ 2max} v a l u e s a r e g i v e n . A l l o t h e r s a r e q{max} v a l u e s f o r p-type MDs. 4/Hydrogen S u l p h i d e which the MD reaches i t s maximum i n a p-type d i s t r i b u t i o n ( s o l i d symbols, f i g u r e 4.4); q{ 1/ 2max} i s the momentum at which an s-type d i s t r i b u t i o n has f a l l e n t o h a l f i t s q=0 i n t e n s i t y (open symbols, f i g u r e 4.4). I d e a l l y one would l i k e t o p l o t the e x p e c t a t i o n v a l u e <q>, but the c o m b i n a t i o n of a n g u l a r r e s o l u t i o n e f f e c t s , s t a t i s t i c a l u n c e r t a i n t y and an i n s u f f i c i e n c y of p o i n t s a t h i g h q do not p e r m i t the e x t r a c t i o n of <q> by i n t e g r a t i o n over q of momentum d i s t r i b u t i o n s so f a r p r e s e n t e d i n the l i t e r a t u r e . q{max} and q{ 1/ 2max} a r e p r o p o r t i o n a l t o <q> but w i t h d i f f e r e n t c o n s t a n t s of p r o p o r t i o n a l i t y . A l s o t h i s p r o p o r t i o n a l i t y may be e x p e c t e d t o change s l i g h t l y w i t h s m a l l changes i n the s/p c h a r a c t e r of the momentum d i s t r i b u t i o n . The p l o t s a r e made a g a i n s t I P 1 / 2 s i n c e one has i n Koopmans' theorem: -I P = e = T + V T = <p 2>/2 where T i s the k i n e t i c energy of the e l e c t r o n . I t i s not un r e a s o n a b l e t o suppose t h a t the p o t e n t i a l energy f e l t by e l e c t r o n s i n each of the t h r e e o u t e r p-type MOs of a g i v e n system i s a p p r o x i m a t e l y the same and so a r o u g h l y l i n e a r r e l a t i o n between <q> (as r e f l e c t e d by q{max}) and I P 1 2 would not be s u r p r i s i n g . T h i s r e l a t i o n would not c a r r y over t o the i n n e r s-type o r b i t a l as the p o t e n t i a l energy of t h i s o r b i t a l i s d i f f e r e n t , and q{ 1/ 2max} r e f l e c t s <q> d i f f e r e n t l y . 4/Hydrogen S u l p h i d e Due t o s t a t i s t i c a l u n c e r t a i n t y and a n g u l a r r e s o l u t i o n e f f e c t s i n the momentum d i s t r i b u t i o n s , and t o the n e c e s s i t y of e x t r a p o l a t i n g s-type d i s t r i b u t i o n s back t o q= 0 , t h e r e i s s i g n i f i c a n t p o s s i b i l i t y f o r e r r o r i n a s c e r t a i n i n g the v a l u e s of q{max} and q{ 1/ 2max}. V a l u e s of IP 1/ 2 are more p r e c i s e , except where the c e n t r o i d of an MFS band must be computed i n o r d e r t o e s t i m a t e the o n e - e l e c t r o n I P . A c o n s e r v a t i v e e s t i m a t e of t h e s e e r r o r s i s i n d i c a t e d by e l l i p s e s i n f i g u r e 4.4. V a l u e s of q{max}, q{ 1/ 2max} and IP a r e l i s t e d i n T a b l e 4.2. S e v e r a l t r e n d s i n t h e s e c o r r e l a t i o n diagrams may be d i s c e r n e d : (1) As e x p e c t e d from the d i s c u s s i o n above the p o i n t s f o r the o u t e r p-type MOs of a g i v e n m o l e c u l e (where they are not degenerate) g e n e r a l l y f a l l on a l i n e which has v e r y n e a r l y the same s l o p e i n a l l the h y d r i d e s . T h i s i n d i c a t e s t h a t the b e h a v i o u r of the p o t e n t i a l energy of t h e s e o r b i t a l s i s , i f not a c o n s t a n t i n each m o l e c u l e , a t l e a s t f o l l o w i n g a s y s t e m a t i c t r e n d i n d i f f e r e n t MOs of d i f f e r e n t m o l e c u l e s ; (2) The p o i n t s g e n e r a l l y move t o h i g h e r energy g o i n g from l e f t t o r i g h t a l o n g the rows. T h i s i s i n a ccordance w i t h the i n c r e a s i n g charge of the heavy atom n u c l e u s ; 4/Hydrogen S u l p h i d e 155 (3) S i m i l a r l y , and f o r the same r e a s o n , the p o i n t s move up goi n g r i g h t a l o n g the rows, r e f l e c t i n g the i n c r e a s e d P-space r a d i a l e x t e n t , and hence, by the R e c i p r o c i t y p r i n c i p l e , the d e c r e a s i n g s p a t i a l s i z e of the MOs. T h i s i s a l s o seen i n the atomic o r b i t a l s of the heavy atoms i n f i g u r e 2.1; (4) P o i n t s f o r the second row h y d r i d e s and r a r e gases (and subsequent rows i n groups V I I and V I I I ) a re always below and t o the l e f t of the c o r r e s p o n d i n g p o i n t s i n the p r e c e d i n g row. Again t h i s r e f l e c t s the i n c r e a s e d s i z e and lowered energy e x p e c t e d when the v a l e n c e s h e l l i s made up of atomic o r b i t a l s whose p r i n c i p l e quantum number i s i n c r e a s i n g . These t r e n d s show a n i c e c o n s i s t e n c y i n a l l the systems p r e s e n t l y s t u d i e d . The o n l y e x c e p t i o n s a r e the HCl 5<r and Xe 5s p o i n t s , which appear t o be out of l i n e w i t h the c o r r e s p o n d i n g p o i n t s of the o t h e r systems i n those groups. No e x p l a n a t i o n i f o f f e r e d f o r these d i s c r e p a n c i e s a p a r t from the e r r o r s i n h e r e n t i n the d e t e r m i n a t i o n s as d e s c r i b e d above. In the case of c e r t a i n h y d r i d e s (NH 3, H 20, HF, PH 3, H 2S and HCl) the momentum d i s t r i b u t i o n s have been computed from w a v e f u n c t i o n s of v a r y i n g q u a l i t y of b a s i s s e t s . A g a i n , s e v e r a l t r e n d s may be no t e d : (1) The i n n e r s-type MD of a l l thes e h y d r i d e s i s 4/Hydrogen S u l p h i d e w e l l - a p p r o x i m a t e d by even the s i m p l e s t b a s i s s e t c a l c u l a t i o n s . T h i s i n d i c a t e s the e s s e n t i a l l y atomic c h a r a c t e r of the s e o r b i t a l s ; (2) Of the o u t e r p-type MDs i t i s always the out e r m o s t , l o w e s t - e n e r g y MD which i s h a r d e s t t o c a l c u l a t e . In the second row h y d r i d e s a w a v e f u n c t i o n a p p r o a c h i n g the HF l i m i t i s r e q u i r e d t o f i t the da t a c l o s e l y , and i n the f i r s t row h y d r i d e s even t h i s i s not adequate. In a l l c a s e s t h e c a l c u l a t i o n s p r e d i c t an MD extended t o h i g h e r q than i s a c t u a l l y measured: the t r u e e l e c t r o n d e n s i t y i s t h e r e f o r e s p a t i a l l y more extended than i s p r e d i c t e d t h e o r e t i c a l l y ; (3) The r e m a i n i n g h i g h e r - e n e r g y p-type MDs r e q u i r e a near-HF l i m i t w a v e f u n c t i o n t o c o r r e c t l y approximate t h e i r shape i n the f i r s t row h y d r i d e s , w h i l e t h e i r second row c o u n t e r p a r t s can be c l o s e l y f i t t e d w i t h a d o u b l e - z e t a w a v e f u n c t i o n . That a g i v e n q u a l i t y b a s i s s e t performs b e t t e r f o r second row h y d r i d e s than f o r t h e i r f i r s t row c o u n t e r p a r t s i s l i k e l y due t o two f a c t o r s : f i r s t of a l l , a g i v e n q u a l i t y b a s i s s e t i s more f l e x i b l e i n the second row s i m p l y because t h e r e a r e more f u n c t i o n s , and second, the s h i f t i n e l e c t r o n d e n s i t y d u r i n g m o l e c u l e f o r m a t i o n i s l e s s extreme i n the second row and l e s s f l e x i b i l i t y i s r e q u i r e d of the b a s i s s e t . By t h i s argument i t might be supposed the HF, w i t h the g r e a t e s t e l e c t r o n e g a t i v i t y d i f f e r e n c e between the 4/Hydrogen S u l p h i d e 157 c o n s t i t u e n t atoms, would show the g r e a t e s t d i s c r e p a n c y between e x p e r i m e n t a l and t h e o r e t i c a l MDs, and so i t would i f the number of hydrogen atoms were the same f o r a l l the h y d r i d e s . However H 20, though h a v i n g l e s s e l e c t r o n e g a t i v i t y d i f f e r e n c e between the atoms, does have two H atoms which r e q u i r e s t h a t the b a s i s s e t have more p o l a r i z a t i o n f u n c t i o n s t o d i r e c t d e n s i t y i n t o the bonding r e g i o n s , whereas - HF, w i t h o n l y one H atom, does not have t h i s r e q u i r e m e n t . T h i s p r o b a b l y a c c o u n t s f o r the o b s e r v a t i o n t h a t the H 20 2b, MD shows the g r e a t e s t d i s c r e p a n c y between experiment and t h e o r y . The e x t e n t of the MFS s t r u c t u r e a s s o c i a t e d w i t h the i n n e r a, or s o r b i t a l i n a l l the systems d i s c u s s e d i n t h i s s e c t i o n i n c r e a s e s g o i n g down the groups. The f i r s t row h y d r i d e s and neon show no MFS s t r u c t u r e beyond a s l i g h t asymmetry of the s i n g l e b i n d i n g energy peak. W i t h subsequent rows the amount of s t r u c t u r e i n c r e a s e s u n t i l i n HI and Xe t h i s s t r u c t u r e spans 20-30eV. T h i s t r e n d a r i s e s out of the i n c r e a s i n g a v a i l a b i l i t y of l o w - l y i n g v i r t u a l s t a t e s (which reduces the energy denominator i n the f o r m u l a {Cederbaum (1977), Chong (1974)} f o r the i n t e n s i t y of t h e s e l i n e s ) , and t h e r e i s a l s o a g r e a t e r d e n s i t y of such v i r t u a l s t a t e s of d i f f e r e n t 1 quantum number t o which e x c i t a t i o n s may o c c u r . T h i s i n t u i t i v e r a t i o n a l i z a t i o n a l s o e x p l a i n s the e x i s t e n c e of MFS s t r u c t u r e i n the f i r s t row d i a t o m i c {Schirmer (1977)}: t h e r e i s now a l o w - l y i n g n* v i r t u a l l e v e l 4/Hydrogen S u l p h i d e which i s not p r e s e n t i n the h y d r i d e s . 4.4 H 2S D e n s i t y Maps D e n s i t y maps computed from the H 2S-DG w a v e f u n c t i o n are shown i n f i g u r e 4.5. S e v e r a l t h i n g s a r e i m m e d i a t e l y o b v i o u s about th e s e maps. The e f f e c t of the s u l p h u r atom c o n s t i t u e n c y i s seen i n the l a r g e s i z e of the R-space d e n s i t y maps compared t o the H 20 maps ( f i g u r e 2.6) and i n the presence of nodal s u r f a c e s between the n=2 s u b s h e l l and the n=3 v a l e n c e s h e l l . By the R e c i p r o c i t y p r i n c i p l e the H 2S momentum d e n s i t y maps t h e r e f o r e show a l e s s extended f u n c t i o n than do the H 20 maps. The n=2 s u b s h e l l i s not seen i n the momentum d e n s i t y maps as i t would appear a t v e r y l a r g e p, a g a i n a c c o r d i n g t o the R e c i p r o c i t y p r i n c i p l e , and has l i t t l e e f f e c t i n the p < 2 . 5 a 0 _ 1 v a l e n c e r e g i o n . The H 2S 2b, o r b i t a l shows the f a m i l i a r p-type d e n s i t y f u n c t i o n and i s unremarkable except f o r the c h a r a c t e r i s t i c s n oted i n the p r e v i o u s p a r a g r a p h . The 2b, w a v e f u n c t i o n f a l l s i n t o Q - c l a s s H i . The S3p{z}+H,1s+H 21s c o n s t i t u e n c y of the 5a, o r b i t a l means the n o d a l p l a n e i n t h i s o r b i t a l i s reduced t o a n o d a l s u r f a c e . The Q - p r o j e c t i o n taken i n the y - a x i z d i r e c t i o n f a l l s i n t o C l a s s I l i a , ( f i g u r e 2.2) and e x p l a i n s the 4/Hydrogen S u l p h i d e 159 s t r u c t u r e of the P-space d e n s i t y on the p{x} a x i s . In the z - d i r e c t i o n the p r o j e c t i o n i s C l a s s H a and g i v e s r i s e t o the p a r t i a l p-type c h a r a c t e r of the MD. The 2 b 2 o r b i t a l m a i n t a i n s the <y{v} p l a n e of symmetry, and so r e t a i n s the p{y}=0 n o d a l p l a n e , and f a l l s i n t o c l a s s H i . The presence of the hydrogen 1s f u n c t i o n s extends the R-space d e n s i t y , and so by the R e c i p r o c i t y p r i n c i p l e s h r i n k s the momentum d e n s i t y . A c a r e f u l e x a m i n a t i o n of the 2b, and 2b 2 P-space d e n s i t y maps w i l l show the l a t t e r i s s l i g h t l y s m a l l e r than the former. F i n a l l y , the 4a, o r b i t a l maps show t h a t t h i s MO i s m a i n l y of s u l p h u r 3s c h a r a c t e r ( c l a s s I ) , as i t i s v e r y n e a r l y s p h e r i c a l . The e f f e c t of the H i s f u n c t i o n s i s o n l y t o s l i g h t l y d i s t o r t t h i s s p h e r i c i t y . 4.5 C o n c l u s i o n s The r e s u l t s i n d i c a t e t h a t , i n o r d e r t o reproduce r e a s o n a b l y w e l l the momentum d i s t r i b u t i o n s (and t h e r e f o r e the s p a t i a l l y extended p a r t of the w a v e f u n c t i o n ) i n f i r s t and second row h y d r i d e s , the i n c o r p o r a t i o n of d - f u n c t i o n s i n t o the b a s i s s e t f o r H a r t r e e - F o c k c a l c u l a t i o n s i s much l e s s i m p o r t a n t than the p r o v i s i o n of a s u f f i c i e n t l y f l e x i b l e s e t of s and p f u n c t i o n s . For f i r s t row h y d r i d e s t h i s s e t 4/Hydrogen S u l p h i d e s h o u l d be of a t l e a s t t r i p l e - z e t a q u a l i t y and f o r the second row a t l e a s t d o u b l e - z e t a w i t h perhaps e x t r a s and p f u n c t i o n s i n the v a l e n c e s h e l l of the heavy atom. I f d - f u n c t i o n s a r e t o be added the s/p b a s i s s h o u l d f i r s t be e s s e n t i a l l y complete. In o t h e r words, a s i n g l e - z e t a b a s i s w i t h d - f u n c t i o n s i s not l i k e l y t o g i v e m e a n i n g f u l r e s u l t s . The b i n a r y (e,2e) measurements of bo t h b i n d i n g energy s p e c t r a and momentum d i s t r i b u t i o n s have p r o v i d e d new and d e f i n i t e e v i d e n c e of s t r o n g e l e c t r o n c o r r e l a t i o n s e f f e c t s i n the i o n i z a t i o n of the 4a, i n n e r v a l e n c e o r b i t a l of H 2S. These f i n d i n g s s t r o n g l y support r e c e n t t h e o r e t i c a l p r e d i c t i o n s u s i n g the many-body Green's f u n c t i o n t e c h n i q u e . An i n t e n s e group of m u l t i p l e f i n a l i o n s t a t e peaks was p r e d i c t e d t o a r i s e from 4a, i o n i z a t i o n , as has now been o b s e r v e d . F u r t h e r m o r e , the measured momentum d i s t r i b u t i o n s c o n f i r m the o r i g i n of the m u l t i p l e f i n a l s t a t e s t r u c t u r e . 161 CHAPTER 5 CARBONYL SULPHIDE 'Thank you, Sam' he s a i d i n a c r a c k e d w h i s p e r . 'How f a r i s t h e r e t o go?' 5. 1 I n t r o d u c t i o n In t h i s c h a p t e r f o u r s t u d i e s on the COS.valence s h e l l b i n d i n g energy spectrum a r e c o n s i d e r e d : (1) The t h e o r e t i c a l spectrum, as computed by the 2ph-TDA many body Green's f u n c t i o n t e c h n i q u e ; (2) The v a l e n c e X-ray p h o t o e l e c t r o n (ESCA) spectrum (1254eV), p r e v i o u s l y r e p o r t e d { A l l a n (1972)}; (3) The b i n a r y (e,2e) spectrum, o b t a i n e d a t two a z i m u t h a l a n g l e s U = 0 ° and *=15°) {Cook (1981)}; (4) The d i p o l e (e,2e) spectrum a t an e q u i v a l e n t •photon' energy of 40eV {White (1980)}. The ESCA and d i p o l e (e,2e) e x p e r i m e n t a l s p e c t r a both g i v e i n t e n s i t i e s p r o p o r t i o n a l t o the o p t i c a l o s c i l l a t o r 5/Carbonyl S u l p h i d e 162 s t r e n g t h s f o r the t r a n s i t i o n s a t t h e i r r e s p e c t i v e photon e n e r g i e s . The d i f f e r e n c e between them i s t h a t the d i p o l e (e,2e) work uses low energy ' v i r t u a l photons', whereas the ESCA r e s u l t was o b t a i n e d w i t h h i g h energy X - r a d i a t i o n . The c a l c u l a t i o n y i e l d s a spectrum of p o l e s t r e n g t h s which do not i n c l u d e the d i p o l e m a t r i x element, and s h o u l d t h e r e f o r e o n l y be.compared w i t h p h o t o e l e c t r o n s p e c t r a a t the h i g h energy l i m i t where the d i p o l e m a t r i x element i s e f f e c t i v e l y c o n s t a n t . At the lower 'photon' e n e r g i e s used i n the d i p o l e (e,2e) work the d i p o l e m a t r i x element i s e x p e c t e d t o show s t r o n g energy dependence over the range of b i n d i n g e n e r g i e s s t u d i e d . I n t e n s i t i e s i n the b i n a r y (e,2e) spectrum a r e p r o p o r t i o n a l t o the momentum d e n s i t y i n the ground s t a t e m o l e c u l a r o r b i t a l from which i o n i z a t i o n o c c u r r e d , a t the p a r t i c u l a r v a l u e s of energy and momentum s e t by the e x p e r i m e n t a l c o n d i t i o n s . The i n t e n s i t y d i s t r i b u t i o n i n the o v e r a l l b i n d i n g energy spectrum w i l l t h e r e f o r e be i n g e n e r a l q u i t e d i f f e r e n t from t h a t o b t a i n e d i n p h o t o e l e c t r o n s p e c t r o s c o p y (which g e n e r a l l y probes much h i g h e r momentum components). The many-body Green's f u n c t i o n t e c h n i q u e i s d e s c r i b e d i n Chapter 1. The i n i t i a l a b - i n i t i o w a v e f u n c t i o n f o r COS used f o r the C and 0 atoms a (9s5p/4s2p) d o u b l e - z e t a b a s i s se t {Huzinaga (1965)}, augmented by a s e t of d - f u n c t i o n s w i t h o{d}(C)=0.6, a{d}(O)=0.8, and f o r the S atom the 5/Carbonyl S u l p h i d e 163 (I2s9p/6s4p) b a s i s s e t of V e i l l a r d { V e i l l a r d (1968)}, augmented by a s e t of d - f u n c t i o n s w i t h a{d}(S)=0.5. The t o t a l SCF energy r e s u l t i n g from t h i s a b - i n i t i o c a l c u l a t i o n done a t the e x p e r i m e n t a l e q u i l i b r i u m geometry u s i n g the program system MUNICH { D i e r c k s e n , D i e r c k s e n (1974)} was -51u.1140au. In the 2ph-TDA c a l c u l a t i o n the o c c u p i e d v a l e n c e o r b i t a l s and the 11 v i r t u a l o r b i t a l s l o w e s t i n energy were i n c l u d e d . The r e s u l t s of the many-body c a l c u l a t i o n s a r e g i v e n i n Table 5.1. 5.2 R e s u l t s and D i s c u s s i o n The e l e c t r o n i c c o n f i g u r a t i o n of COS i s : { c o r e } 1 " ( 6 * ) 2 ( 7 t f ) 2 ( 8 * ) 2 ( 9 t f ) 2 ( 2 T r ) ' , ( 3 T r ) ' , of which the 6a and le l e v e l s a r e c a l l e d the i n n e r v a l e n c e o r b i t a l s , and the 8tf,9tf,2rr and 3ir l e v e l s the o u t e r v a l e n c e o r b i t a l s , by reason of t h e i r d i s t i n c t l y d i f f e r e n t i o n i z a t i o n b e h a v i o u r which w i l l be e l a b o r a t e d upon below. The c a l c u l a t e d p o l e s t r e n g t h s a re shown i n f i g u r e 5.2, w h i l e more d e t a i l e d i n f o r m a t i o n i s shown i n Table 5.1. D e f i n i t i o n s of terms were g i v e n i n s e c t i o n 1.4.1. F i g u r e 5.1 shows a comparison of the v a r i o u s e x p e r i m e n t a l s p e c t r a w i t h the c a l c u l a t i o n . To f a c i l i t a t e t h i s , comparison the 5/Carbonyl S u l p h i d e 164 20 30 40 BINDING ENERGY eV F i g u r e 5.1 B i n d i n g energy s p e c t r a of COS: (a) T h e o r e t i c a l spectrum v i a the Green's f u n c t i o n t e c h n i q u e (b) the ESCA spectrum (c) b i n a r y (e,2e) spectrum a t *=0° (d) b i n a r y (e,2e) spectrum a t #=15° (e) d i p o l e (e,2e) spectrum a t 40eV energy l o s s 5/Carbonyl S u l p h i d e 1 65 CO CD CO-CO J P . CM C M CO-t t D D D D CO CM CD 00 N- CO i- CM CO "<fr IT) CO CO O O co- CM-co co CO in- to in- 5fJ CM—I CM CO-LO-CM-IfH C M J O CO o o cr LU z L U LP in d o d H10N3dlS 310d F i g u r e 5.2 R e s u l t s of the Green's f u n c t i o n c a l c u l a t i o n of the COS i o n i z a t i o n p o t e n t i a l s and p o l e s t r e n g t h s . 5/Carbonyl S u l p h i d e 166 T a b l e 5.1 V e r t i c a l I P s , h o l e s t a t e i n t e n s i t i e s , and p o l e s t r e n g t h s f o r COS. F i n a l I P 1 Hole P o l e 2 s t a t e ( s p e c t r a l s t a t e s t r e n g t h f e a t u r e ) i n t e n s i t y p{n} |x {2p} (X) 10. 53 0. 004 (A) 14. 89 0. 745 17. 46 0. 015 (Q) 22. 22 0. 070 26. 55 0. 016 27. 15 0. 035 {6s} (B) 15. 31 (C) 17. 29 -21 . 35 -22. 64 -24. 1 1 0. 001 25. 24 -25. 84 0. 002 (R, 28. 36 0. 007 S) 29. 17 -29. 27 -29. 91 -30. 1 2 0. 001 32. 57 0. 009 32. 70 0. 005 32. 80 -34. 83 0. 074 35. 1 7 0. 066 35. 30 0. 248 36. 13 0. 116 36. 34 0. 018 36. 55 0. 049 (T) 37. 1 1 0. 010 37. 34 0. 016 37. 96 0. 001 38. 63 -38. 86 0. 046 39. 75 0. 056 39. 95 0. 011 |x ( 3 p } | 2 0. 920 0. 004 0. 002 0. 003 0. 004 0. 001 I * { 7 s } | 2 I * {8s 0. 015 - 0. 795 0. 030 0. 001 0. 014 0. 011 0. 099 -0. 313 -0. 105 0. 001 0. 016 0. 048 0. 060 0. 005 0. 006 0. 01 1 0. 019 0. 004 0. 1 35 -0. 003 0. 001 0. 004 0. 006 0. 022 0. 001 0. 002 -0. 002 -- 0. 001 0. 002 0. 008 0. 001 0. 001 0. 005 0. 002 0. 002 -0. 010 0. 001 0. 011 -0. 002 0. 008 0. 002 0. 001 0. 924 0. 749 0. 017 0. 073 0. 020 0. 036 I * {9s} j I 2 0. 876 0. 891 0. 016 0. 811 0. 004 0. 035 0. 003 0. 028 0. 006 0. 106 - 0. 313 0. 001 0. 109 0. 004 0. 071 0. 001 0. 065 0. 004 0. 021 0. 001 0. 024 0. 001 0. 140 0. 001 0. 014 0. 001 0. 016 0. 001 0. 024 - 0. 076 0. 0.01 0. 069 - 0. 249 0. 001 0. 127 - 0. 020 - 0. 056 -• 0. 012 0. 005 0. 023 - 0. 012 0. 003 0. 013 - 0. 046 0. 001 0. 060 0. 005 0. 016 1 L a b e l l i n g as i n f i g u r e 5.2 2 The v a l u e s of p{n} a r e shown i n f i g u r e 5.1 and a r e d e f i n e d as the sum of the h o l e s t a t e i n t e n s i t i e s a c c o r d i n g t o e q u a t i o n s 1.28 and 1.29. 5/Carbonyl S u l p h i d e 167 c a l c u l a t e d p o l e s t r e n g t h spectrum has been c o n v o l u t e d w i t h a 1.3eV-FWHM Ga u s s i a n peak shape, and 0.7eV has been added t o a l l c a l c u l a t e d e n e r g i e s t o a l i g n the s p e c t r a . The d i p o l e and b i n a r y (e,2e) s p e c t r a have each been t h r e e - p o i n t smoothed once. A l l f o u r e x p e r i m e n t a l s p e c t r a show t h r e e s t r o n g peaks i n the range lO-20eV (X,A,B,C), and a weak s e r i e s of broad s t r u c t u r e s e x t e n d i n g over the range 20-45eV (P,Q,R,S,T). The c a l c u l a t e d spectrum i s i n good agreement w i t h t h i s : T a ble 5.1 shows t h a t the o n e - e l e c t r o n p i c t u r e h o l d s q u i t e w e l l f o r i o n i z a t i o n from the 8<r,9tf,2ir and 3ir o u t e r v a l e n c e o r b i t a l s , l e a d i n g t o the s t r o n g peaks between 10 and 20eV, the 9* - 1 and 2IT" 1 s t a t e s b e i n g u n r e s o l v e d . The i n t e n s i t y a s s o c i a t e d w i t h i o n i z a t i o n from each of t h e s e o r b i t a l s i s l a r g e l y c o n c e n t r a t e d i n one l i n e , and o n l y 10-20 per c e n t of the i n t e n s i t y i s spread over s a t e l l i t e l i n e s . For the 6a and la i n n e r v a l e n c e o r b i t a l s , on the o t h e r hand, a d i f f e r e n t b e h a v i o u r i s found, b o t h e x p e r i m e n t a l l y and t h e o r e t i c a l l y : the t h e o r e t i c a l i n t e n s i t y from t h e s e o r b i t a l s i s s p r e a d over numerous l i n e s , and t h i s i s r e f l e c t e d i n the weak, extended s t r u c t u r e between 20 and 45eV i n the e x p e r i m e n t a l s p e c t r a . No s i n g l e l i n e c a r r i e s more than 30 per c e n t of the i n t e n s i t y a s s o c i a t e d w i t h a s i n g l e o r b i t a l , and one i s f a c e d w i t h a complete breakdown of the m o l e c u l a r o r b i t a l p i c t u r e of i o n i z a t i o n {Domcke (1979), Schirmer (1977), Cederbaum (1978), Cederbaum (1980)}. From f i g u r e s 5.1 and 5/Carbonyl S u l p h i d e 5.2, the b a s i c d i f f e r e n c e between the o u t e r v a l e n c e p a r t (lO-20eV) and the i n n e r v a l e n c e p a r t (20-45eV) i s e v i d e n t . The r e s u l t s f o r the 2Z* symmetry i o n s t a t e s (6e~ 1 , le' 1 , Be" 1 , 9e' 1 ) e x h i b i t a remarkably mixed c h a r a c t e r i n most of the f i n a l i o n s t a t e s ( T a b l e 5.1) - most of the s t a t e s d e r i v e t h e i r i n t e n s i t y from more than one ground s t a t e o r b i t a l . Even the two outermost 2 E + s t a t e s r e p r e s e n t i n g the 8<r and 9<r o r b i t a l s are not p ure. The s a t e l l i t e l i n e s of t h e s e o u t e r v a l e n c e o r b i t a l s a r e seen ( f i g u r e 5.1) t o be i n t e r m i n g l e d w i t h the i n n e r v a l e n c e l i n e s . One can no l o n g e r d i s t i n g u i s h between o u t e r v a l e n c e s a t e l l i t e l i n e s and " ' C I - s t a t e s ' r e p r e s e n t i n g the i n n e r v a l e n c e o r b i t a l s . The mixed c h a r a c t e r of the 2E + s t a t e s l e a d s t o v a r i a t i o n s i n the i n t e n s i t y of i n d i v i d u a l l i n e s w i t h v a r y i n g photon energy. T h i s e f f e c t i s c l e a r l y seen i n the v a r i a b l e 'photon' energy d i p o l e (e,2e) s p e c t r a of COS {White (1980)}. The 0=0° and *=15° b i n a r y (e,2e) r e s u l t s show a g e n e r a l d i m i n u t i o n of i n t e n s i t y over the whole 20-45eV MFS r e g i o n , w i t h i n c r e a s i n g <t>. T h i s i n d i c a t e s t h a t the whole s t r u c t u r e a r i s e s from ground s t a t e o r b i t a l s t h a t have p r e d o m i n a n t l y s - c h a r a c t e r , thus r u l i n g out major c o n t r i b u t i o n s from the 2rr and 3tr o r b i t a l s which would be l a r g e l y 'p-type'. However, on c l o s e r i n s p e c t i o n one sees t h a t the shapes of the two s p e c t r a a r e not i d e n t i c a l i n t h i s r e g i o n , which means t h a t t h e r e i s some weak 'p-type' c h a r a c t e r i n the s t r u c t u r e : i n 5/Carbonyl S u l p h i d e 169 p a r t i c u l a r t h e r e i s a f e a t u r e a t 23.5eV (Q) seen i n a l l the s p e c t r a which i s r e l a t i v e l y s t r o n g e r a t 0=15° than at <f>=0° compared t o the r e s t of the MFS s t r u c t u r e . T h i s i s i n e x c e l l e n t agreement w i t h the t h e o r e t i c a l p r e d i c t i o n of a f a i r l y s t r o n g 'p-type' 2rr s a t e l l i t e a t 22.22eV (22.92eV i n f i g u r e 5.2, s i n c e 0.7eV i s added t o the t h e o r e t i c a l energy s c a l e ) . A s i m i l a r e f f e c t i s p r e d i c t e d i n CS 2, N 20 and ( t o a l e s s e r e x t e n t ) i n C 0 2 {Domcke (1979)}. The e x i s t e n c e of a l o w - l y i n g and l o c a l i z e d ir* o r b i t a l i s r e s p o n s i b l e f o r t h i s e f f e c t . However, the l a r g e r e l a t i v e i n t e n s i t y of f e a t u r e Q i n the ESCA spectrum s u g g e s t s t h a t i t may i n v o l v e a s u p e r p o s i t i o n of rr- and <y-type l i n e s , due t o the low n/e i n t e n s i t y r a t i o s p r e d i c t e d by c a l c u l a t i o n { A l l a n (1972)}. On the b a s i s of the t h e o r e t i c a l spectrum we can a s s i g n t h r e e r e g i o n s of the MFS s t r u c t u r e : the f i r s t i s the 23.5eV peak (Q) due t o a 2n s a t e l l i t e , mentioned above; the second i s a broad band from 25 t o 33eV (R,S) which the c a l c u l a t i o n i n d i c a t e s t o be p r i m a r i l y le i n o r i g i n , though w i t h a s m a l l a d m i x t u r e from the 8* o r b i t a l which may account f o r the s l i g h t d i f f e r e n c e i n the shape of t h i s band between <t> v a l u e s of 0° and 15°; the t h i r d r e g i o n (T) i s a n o t h e r broad band (33-40eV) due almost e x c l u s i v e l y t o i o n s t a t e s o r i g i n a t i n g i n the 6e o r b i t a l . The c a l c u l a t i o n does not p r e d i c t the v e r y weak s h o u l d e r a t 20.5eV (P) which seems t o be p r e s e n t i n a l l the e x p e r i m e n t a l s p e c t r a . A c o n s i d e r a t i o n of r e l a t i v e i n t e n s i t i e s i n the b i n a r y (e,2e) s p e c t r a s u g g e s t s some 5/Carbonyl S u l p h i d e 170 p-type c h a r a c t e r f o r t h i s s t r u c t u r e (see f i g u r e 5.1). What i s i n t r i g u i n g about t h i s MFS s t r u c t u r e i s t h a t the r e l a t i v e i n t e n s i t i e s a t *=0° and *=15° f o r the 6e and le bands suggest t h a t they have s i m i l a r momentum d i s t r i b u t i o n s , as mentioned above - m a i n l y s - t y p e . From the atomic o r b i t a l c o m p o s i t i o n {Basch (1972)} and from the c a l c u l a t e d momentum d i s t r i b u t i o n { G i a r d i n i (1977)} of the 2*{u} m o l e c u l a r o r b i t a l i n C0 2 (see { G i a r d i n i (1977)} and a l s o s e c t i o n 6.5) one might expect the a n a l o g o u s le o r b i t a l i n COS t o have a p p r e c i a b l e p - c h a r a c t e r . A p p a r e n t l y c h a n g i n g one oxygen atom t o a s u l p h u r , thus r e d u c i n g the D^h} symmetry t o C{<»v}, i s enough t o i n t r o d u c e a p p r e c i a b l e s - c h a r a c t e r i n t o the o r b i t a l . T h i s argument i s s u p p o r t e d by an a n a l y s i s of the p o l e s t r e n g t h s i n T a b l e 5.1 which shows the le m i x i n g w i t h the o t h e r e o r b i t a l s . An a l t e r n a t i v e e x p l a n a t i o n of c o u r s e i s t h a t the 6e and le l i n e s a r e not s e p a r a t e d i n t o two f a i r l y d i s t i n c t bands as suggested by the c a l c u l a t i o n , but r a t h e r t h a t they i n t e r m i n g l e so e x t e n s i v e l y t h a t o n l y the sum of the 6e and le momentum d i s t r i b u t i o n s can be d e t e c t e d . Making the rough a p p r o x i m a t i o n t h a t the d i p o l e m a t r i x element f a c t o r i n the peak i n t e n s i t i e s may be c o n s i d e r e d c o n s t a n t i n the v a l e n c e r e g i o n under d i s c u s s i o n , the ESCA s p e c t r a l i n t e n s i t i e s may be compared d i r e c t l y w i t h the t h e o r e t i c a l p o l e s t r e n g t h s (which do not i n c l u d e the d i p o l e m a t r i x e l e m e n t ) . F i v e r e g i o n s of i n t e n s i t y can be i d e n t i f i e d i n the ESCA spectrum, c e n t e r e d a t 20.5 ( P ) , 23.5 (Q), 5/Carbonyl S u l p h i d e 171 27 ( R ) , 30 (S) and 36eV ( T ) . The l a t t e r f o u r a r e reproduced i n the (0.7eV s h i f t e d ) t h e o r e t i c a l spectrum w i t h o n l y s m a l l d i f f e r e n c e s of e n e r g i e s and i n t e n s i t i e s . The c o r r e c t way t o compare the b i n a r y (e,2e) s p e c t r a i n the MFS r e g i o n w i t h the t h e o r e t i c a l work would be t o i n t e g r a t e the b i n d i n g energy spectrum over momentum b e f o r e c o mparison. However, as i n the MFS r e g i o n the shape of the spectrum i s f a i r l y c o n s t a n t w i t h a z i m u t h a l a n g l e , t h e r e i s some j u s t i f i c a t i o n f o r comparing the r e l a t i v e l y more i n t e n s e 0=0° spectrum w i t h the c a l c u l a t i o n and the ESCA spectrum. Doing so, one sees e s s e n t i a l l y the same f i v e bands (P,Q,R,S,T) as i n the ESCA spectrum and t h i s i s a l s o c o m p a t i b l e w i t h the t h e o r e t i c a l p r e d i c t i o n s . The l a r g e d i f f e r e n c e i n the r e l a t i v e i n t e n s i t y of the 23.5eV f e a t u r e (Q) i s due t o i t s s i g n i f i c a n t 'p-type' c h a r a c t e r , as e x p l a i n e d e a r l i e r . The b i n a r y (e,2e) s p e c t r a e x t e n d t o h i g h e r energy than the o t h e r e x p e r i m e n t a l d a t a and beyond the h i g h e s t c a l c u l a t e d f i n a l s t a t e s shown i n f i g u r e 5.1 and T a b l e 5.1. A p p r e c i a b l e MFS i n t e n s i t y i s p r e s e n t beyond 40eV. A l t h o u g h the c a l c u l a t i o n i n d i c a t e s some i n t e n s i t y i n t h i s r e g i o n , no q u a n t i t a t i v e p r e d i c t i o n s can be made due t o l i m i t a t i o n s i n the method {Cederbaum (1980)}, as s t a t e d e a r l i e r . The d i p o l e (e,2e) spectrum s e r v e s t o c o n f i r m the p resence of dense s t r u c t u r e e x t e n d i n g up t o a t l e a s t 40eV; 5/Carbonyl S u l p h i d e however, as e x p l a i n e d e a r l i e r , the i n t e n s i t i e s a re not r e l a t e d i n any s i m p l e way t o the o t h e r s p e c t r a . The spectrum a l s o c o n f i r m s the presence of the peaks a t 20.5 (P) and 23.5eV (Q), but does n o t , of c o u r s e , show s t r u c t u r e above 40eV s i n c e t h i s i s the l i m i t of the e q u i v a l e n t photon energy. 5.3 C o n c l u s i o n s E x p e r i m e n t a l measurements of the e l e c t r o n b i n d i n g energy spectrum once a g a i n c o n f i r m the Green's f u n c t i o n t e c h n i q u e p r e d i c t i o n s of complete breakdown of the o n e - e l e c t r o n p i c t u r e of i o n i z a t i o n i n the i n n e r v a l e n c e r e g i o n . The i n t e n s i t y of i o n i z a t i o n from t h e 6e and le MOs i s s p r e a d over a wide energy range from 22 t o 45eV, w i t h no c l e a r l y i d e n t i f i a b l e main peaks. \ 1 73 CHAPTER 6 CARBON DIOXIDE He l o o k e d back, and then he l o o k e d up; and he was amazed t o see how f a r h i s l a s t e f f o r t had brought him. 6.1 I n t r o d u c t i o n Carbon d i o x i d e i s a 2 2 - e l e c t r o n m o l e c u l e of D{o°h} symmetry. The e l e c t r o n i c s t r u c t u r e i s : ( c o r e ) 6 ( 3 f f { g } ) 2 ( 2 t f { u } ) 2 ( 4 t f { g } ) 2 ( 3 t f { u } ) 2 ( l f f { u } ) M l f f { g } ) ' t The c o r r e s p o n d e n c e between the v a r i o u s n o t a t i o n s used i n t h i s c h a p t e r i s g i v e n i n T a b l e 6.1. The b i n d i n g energy spectrum of C0 2 i s w e l l known i n the PES He-I r e g i o n {Turner (1970)} where the f o u r o u t e r v a l e n c e s t a t e s appear a t l3.8eV ( X ) , 17.6eV ( A ) , 18.1eV (B) and l9.4eV ( C ) . The i n n e r s t a t e s -have p r e v i o u s l y been obse r v e d u s i n g 6/Carbon d i o x i d e 174 XPS { A l l a n (1972)}, d i p o l e (e,2e) s p e c t r o s c o p y {Domcke (1979)}, and b i n a r y (e,2e) s p e c t r o s c o p y { G i a r d i n i (1977)}: a broad band i s seen i n the range 30-42eV i n a l l c a s e s . Both the 2<*{u} and the 3<r{g} s t a t e s c o n t r i b u t e t o the i n t e n s i t y i n t h i s r e g i o n , which i s d i s t r i b u t e d over many l i n e s , w i t h no c l e a r l y i d e n t i f i a b l e main peak f o r e i t h e r s t a t e . T h i s broad spectrum of f i n a l i o n s t a t e s a r i s i n g from the two i n n e r v a l e n c e o r b i t a l s has been p r e d i c t e d u s i n g many-body Green's f u n c t i o n s c a l c u l a t i o n s {Domcke (1979)}, and i s a common f e a t u r e of the b i n d i n g energy s p e c t r a of many s m a l l m o l e c u l e s beyond 20eV. The p r e v i o u s b i n a r y (e,2e) study { G i a r d i n i (1977)} was done a t 1200eV impact energy w i t h an energy r e s o l u t i o n (2.5eV FWHM) somewhat lower than i n t h i s work, and an a n g u l a r r e s o l u t i o n c o n s i d e r a b l y b e t t e r . Due t o the l i m i t e d energy r e s o l u t i o n the A, B, and C peaks were not s e p a r a t e d , and o n l y the momentum d i s t r i b u t i o n of the X - s t a t e was o b t a i n e d d i r e c t l y . 6.2 E x p e r i m e n t a l The i n c i d e n t energy i n the p r e s e n t study i s 400eV and the e x p e r i m e n t a l energy r e s o l u t i o n i s 1.2eV (FWHM). Wide range b i n d i n g energy scans i n c l u d i n g the i n n e r v a l e n c e 6/Carbon d i o x i d e r e g i o n s a r e shown i n f i g u r e 6.1. W i t h t h i s energy r e s o l u t i o n we a r e a b l e t o p a r t i a l l y r e s o l v e the A, B, and C peaks t o an e x t e n t t h a t a l l o w s us t o e s t i m a t e the a n g u l a r c o r r e l a t i o n s f o r the s e p a r a t e o r b i t a l s by d e c o n v o l u t i o n of the b i n d i n g energy s p e c t r a . These were o b t a i n e d over the range 10 t o 25eV at a s e r i e s of a z i m u t h a l a n g l e s (0=0-48°) and are shown i n f i g u r e . 6.2. The d e c o n v o l u t i o n was done w i t h a l e a s t - s q u a r e s G a u s s i a n peak f i t t i n g program. F i r s t the X - s t a t e peak a l o n e i n a g i v e n b i n d i n g energy spectrum was f i t t e d , t o o b t a i n the e x p e r i m e n t a l energy r e s o l u t i o n f o r t h a t spectrum. Then the f u l l spectrum was f i t t e d u s i n g f i x e d peak w i d t h s and f i x e d peak s e p a r a t i o n s . The f i x e d peak w i d t h s were c a l c u l a t e d from the X - s t a t e e x p e r i m e n t a l r e s u l t and the Franck-Condon en v e l o p e s o b t a i n e d from h i g h r e s o l u t i o n PES measurements {Turner (1970)}. ( T y p i c a l b i n d i n g energy s p e c t r a used i n the d e c o n v o l u t i o n are shown i n f i g u r e 6.2.). Once the r e l a t i v e peak a r e a s of the f o u r peaks were found, the A, B, and C a n g u l a r c o r r e l a t i o n s were g e n e r a t e d by m u l t i p l y i n g the X - s t a t e a n g u l a r c o r r e l a t i o n s by the peak a r e a r a t i o s (A/X, B/X, C/X). The- r e s u l t s of the d e c o n v o l u t i o n a r e g i v e n i n f i g u r e 6.3. The d i r e c t l y - m e a s u r e d a n g u l a r c o r r e l a t i o n s f o r the i n n e r v a l e n c e r e g i o n are g i v e n i n f i g u r e 6.4. 6/Carbon d i o x i d e 176 T a b l e 6.1 S t a t e and o r b i t a l nomenclature i n C 0 2 . S t a t e Ground s t a t e Ion s t a t e Ion h o l e number MO l a b e l s t a t e 1 M g } X ( 1 ir { g } ) " 1 2 M u } A O t r { u } ) - 1 3 3<r{u} B ( 3 * { u } ) - 1 4 4 * { g } C ( 4 * { g } ) - 1 5 2<r{u} MFS ( 2 * { u } ) - 1 6 3<r{g} MFS ( 3 * { g } ) " 1 6/Carbon d i o x i d e 177 T a b l e 6.2 Bonding, non-bonding and a n t i b o n d i n g r e g i o n s i n C 0 2 (atomic u n i t s ) (see a l s o T a b l e 2.2). R-space P-space r e g i o n r e g i o n *{u} *{g} *{g} Bonding 2-3 2-3 1-1.5 Non-bonding 4.5 1 .4 0.7 A n t i b o n d i n g 6 1 .0 0.5 6/Carbon d i o x i d e 178 (1ng) (lnu)(3ou)(4og) CO2 X A B C ? P Q 1 ' ' • I ' 1 '' 1 11 1—"—r—1—j—1—1—1—1—]—r—1—I—1—r—1—1—i—1—1 1 \ i 10 15 20 25 30 35 40 BINDING ENERGY feV) F i g u r e 6.1 F u l l C 0 2 b i n d i n g energy spectrum a t #=0" (a) i n c l u d i n g r e s u l t s of MBGF c a l c u l a t i o n (b) i n c l u d i n g r e s u l t s of HAM/3-CI c a l c u l a t i o n 6/Carbon d i o x i d e 179 F i g u r e 6.2 T y p i c a l b i n d i n g energy s p e c t r a used i n the d e c o n v o l u t i o n p r o c e d u r e . 6/Carbon d i o x i d e 180 o> oz oo oz- o>- o f az oo oz- of-02 01 00 01- 02- 02 01 OT) OX- 02-F i g u r e 6.3 E x p e r i m e n t a l and t h e o r e t i c a l momentum d i s t r i b u t i o n s , and t h e o r e t i c a l momentum and charge d e n s i t y maps f o r C 0 2 . S o l i d l i n e s a r e the DZ+3d-G76 r e s u l t s and the dashed l i n e s a r e the DZ-SB r e s u l t s . SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION TJ (-•• C >-» 0) o o D n-D C fD a 6/Carbon d i o x i d e 182 F i g u r e 6.3 c o n t i n u e d 6/Carbon d i o x i d e co 2 LU f-Z LU > _ l LU r r C02 P (MFS) y—\ 3 3 e V ^ ^ C N \ • \ \ \ • 0 • • \ \ • • v : • •»-•— ~~ 1 1 1 ' 1 00 0.5 1.0 q 1.5 i 2.0 t CO 2 LU LU > LU r r b. Q (MFS) 3 8 e V 1 i — 1 — 1 — 1 — 1 — 1 — * — 1 — / 0 0 0.5 1.0 q 15 2.0 F i g u r e 6.4 E x p e r i m e n t a l and t h e o r e t i c a l momentum d i s t r i b u t i o n s a t 33eV and 38eV i n the MFS r e g i o n . 6/Carbon d i o x i d e 184 F i g u r e 6.5 D i r e c t l y - m e a s u r e d momentum d i s t r i b u t i o n a t 19.4eV. 6/Carbon d i o x i d e 185 F i g u r e 6.6 The summed A+B e x p e r i m e n t a l and t h e o r e t i c a l momentum d i s t r i b u t i o n s . 6/Carbon d i o x i d e 186 6.3 C a l c u l a t i o n s C a l c u l a t i o n s of the b i n d i n g energy spectrum of C 0 2 have been done w i t h : (1) MBGF: the 2ph-TDA many-body Green's f u n c t i o n method {Domcke (1979)}; (2) HAM/3-CI: the s e m i - e m p i r i c a l HAM/3-CI method {Chong}. The Green's f u n c t i o n method i s d e s c r i b e d b r i e f l y i n s e c t i o n 1.4.1-. T h i s method i n c l u d e s e x t e n s i v e f i n a l s t a t e c o r r e l a t i o n but not ground s t a t e c o r r e l a t i o n . The HAM/3 c a l c u l a t i o n i s based on the t r a n s i t i o n - s t a t e of V 2 - o c c u p a t i o n method { A s b r i n k (1977)}. The method r e l i e s on e x t e n s i v e p a r a m e t e r i z a t i o n from many PES measurements on s m a l l m o l e c u l e s . C o n f i g u r a t i o n i n t e r a c t i o n has been i n c o r p o r a t e d by Chong and c o - w o r k e r s , and i n t h i s case i n c l u d e s the 84 l o w e s t energy s i n g l y - and d o u b l y - e x c i t e d s t a t e s above the ground i o n i c s t a t e . The method a c c o u n t s f o r e x t e n s i v e ground s t a t e and f i n a l s t a t e c o r r e l a t i o n . Note' t h a t t h e s e two t h e o r e t i c a l models g i v e ( a p a r t from the b i n d i n g e n e r g i e s ) o n l y the s p e c t r o s c o p i c f a c t o r of the f i n a l i o n s t a t e i n q u e s t i o n and do not i n c l u d e o t h e r f a c t o r s i n the d i p o l e or b i n a r y (e,2e) c r o s s - s e c t i o n f o r m u l a e . T h i s must be remembered when comparing c a l c u l a t i o n s w i t h e x p e r i m e n t a l s p e c t r a . The r e s u l t s a r e shown i n f i g u r e 6.1; 6/Carbon d i o x i d e 187 note t h a t the o c c u p a t i o n number has not been i n c l u d e d i n the t h e o r e t i c a l l i n e s . Two HF-SCF c a l c u l a t i o n s were chosen t o compare w i t h the e x p e r i m e n t a l work on the momentum d i s t r i b u t i o n s : (1) DZ-SB: an LCAO-MO-SCF c a l c u l a t i o n {Basch (1972)} of e s s e n t i a l l y d o u b l e - z e t a q u a l i t y ; (2) DZ+3d-G76: a s i m i l a r c a l c u l a t i o n done w i t h the G a u s s i a n 76 package u s i n g the 4-31G* s t a n d a r d b a s i s s e t , i n c l u d i n g d - f u n c t i o n s . E x p e r i m e n t a l and t h e o r e t i c a l momentum d i s t r i b u t i o n s , and t h e o r e t i c a l momentum and p o s i t i o n d e n s i t y maps from the DZ-SB c a l c u l a t i o n a r e shown i n f i g u r e s 6.3 and 6.4. In a l l c a s es shown i n these f i g u r e s , the p l a n e of the diagram c o n t a i n s the a x i s of the m o l e c u l e o r i e n t e d a l o n g the v e r t i c a l a x i s of the diagram. 6.4 MFS S t r u c t u r e of C0 2 The p r e s e n t r e s u l t s agree w e l l w i t h the p r e v i o u s b i n a r y (e,2e) study { G i a r d i n i (1977)} on the v a l e n c e b i n d i n g energy spectrum of C0 2 when the l a r g e d i f f e r e n c e i n energy r e s o l u t i o n i s taken i n t o a c c o u n t , and a l s o w i t h p r e v i o u s XPS { A l l a n (1972)} and d i p o l e (e,2e) work {Domcke (1979)} i n r e s p e c t of peak e n e r g i e s . I t has been n o t e d b e f o r e 6/Carbon d i o x i d e 188 ( s e c t i o n 5.2) t h a t a l t h o u g h the i n t e n s i t i e s i n b i n a r y (e,2e) s p e c t r o s c o p y and X-ray s p e c t r o s c o p y a r e governed by d i f f e r e n t r e l a t i o n s h i p s , the r e l a t i v e shapes of the MFS s t r u c t u r e (30~42eV) are q u i t e s i m i l a r . T h i s i s r e a s o n a b l e s i n c e the d i p o l e m a t r i x element i n the XPS c r o s s - s e c t i o n i s almost c o n s t a n t w i t h b i n d i n g energy and i f , c o i n c i d e n t a l l y , the 2<r{u} and 3<r{g} o r b i t a l s have s i m i l a r magnitudes of the d i p o l e m a t r i x element and b i n a r y (e,2e) form f a c t o r a t t h e i r r e s p e c t i v e s c a t t e r i n g k i n e m a t i c s then the s p e c t r a s h o u l d be s i m i l a r . The b i n a r y and d i p o l e (e,2e) c r o s s - s e c t i o n s a r e a l s o d i f f e r e n t but s i m i l a r i t i e s a r e not n e c e s s a r i l y e x p e c ted between the MFS s t r u c t u r e i n these two c a s e s s i n c e d i p o l e (e,2e) s p e c t r o s c o p y i s done c l o s e r t o t h r e s h o l d where the d i p o l e m a t r i x element goes through l a r g e v a r i a t i o n s w i t h b i n d i n g energy. Even so t h e r e i s s t i l l a rough correspondence between the two s p e c t r a e x c e p t around 27eV. Here the d i p o l e spectrum shows two m o d e r a t e l y i n t e n s e peaks ( f e a t u r e s I , I I of f i g u r e 1 i n {Domcke (1979)}) which appear o n l y v e r y weakly i n the b i n a r y and XPS s p e c t r a . These a r e p r e d i c t e d t o be 1 T T { U } s a t e l l i t e s by both c a l c u l a t i o n s but i t i s not p o s s i b l e t o check t h i s assignment by measuring the b i n a r y (e,2e) a n g u l a r c o r r e l a t i o n as the s t r u c t u r e i s t o o weak. The c a l c u l a t i o n s g i v e f a i r l y good r e s u l t s i n the o u t e r v a l e n c e r e g i o n , except t h a t the l i n e s a r e a l l too low by about 1.2eV i n the MBGF r e s u l t and by about 0.6eV i n the 6/Carbon d i o x i d e 189 HAM/3-CI r e s u l t . In f i g u r e 6.1 the t h e o r e t i c a l e n e r g i e s have not been s h i f t e d , u n l i k e f i g u r e s 4.2 and 5.1. Both c a l c u l a t i o n s on the b i n d i n g energy spectrum of C0 2 p r e d i c t t h a t the b u l k of the MFS s t r u c t u r e (30-42eV) a r i s e s m a i n l y from the two i n n e r v a l e n c e o r b i t a l s (2a{u},3*{g}) w i t h a s m a l l a d m i x t u r e from the o u t e r v a l e n c e s t a t e s , and t h a t the s e v e r a l f i n a l i o n s t a t e peaks from each i n n e r v a l e n c e o r b i t a l a r e so e x t e n s i v e l y i n t e r m i n g l e d t h a t t h e r e i s a t p r e s e n t no p o s s i b i l i t y of r e s o l v i n g them. T h i s i s i n c o n t r a s t t o the case of COS (see Chapter 5) where the f o u r o r b i t a l s mix s i g n i f i c a n t l y i n the MFS r e g i o n but t h e r e a r e broad bands of the s t r u c t u r e which a r e p r e d i c t e d t o be from m a i n l y one o r b i t a l or a n o t h e r . Of thes e two c a l c u l a t i o n s on C0 2 the HAM/3 method g i v e s c o n s i d e r a b l y b e t t e r agreement ( f i g u r e 6.1) which i s not s u r p r i s i n g , as t h i s method i s e x t e n s i v e l y p a r a m e t e r i z e d , whereas the MBGF c a l c u l a t i o n i s c o m p l e t e l y a b - i n i t i o . Indeed such i s the e x c e l l e n c e of the agreement i n t h i s and o t h e r c a s e s t h a t , i n view of the v e r y low c o s t , one might c o n s i d e r abandoning the f a r more e x p e n s i v e Green's f u n c t i o n method were i t not f o r the f a c t t h a t a t t h i s time the p a r a m e t e r i z a t i o n has o n l y been done f o r the atoms of the f i r s t row (up t o Z=10). In comparison the Green's f u n c t i o n method i s g e n e r a l f o r a l l atoms. I t must a l s o o f c o u r s e be r e a l i z e d t h a t t h e r e i s a l a r g e h idden c o s t i n the HAM/3 p a r a m e t e r i z a t i o n . p r o c e d u r e , which i s l i k e l y t o be d i f f i c u l t f o r the more complex second row 6/Carbon d i o x i d e 190 atoms. 6.5 Momentum D i s t r i b u t i o n s and Bonding i n C0 2 T h i s s e c t i o n c a r r i e ' s on the i d e a s i n t r o d u c e d i n s e c t i o n 2.5.3 on AX 2 m o l e c u l e s . The two c a l c u l a t i o n s used t o compute momentum d i s t r i b u t i o n s g i v e v e r y s i m i l a r shapes f o r the MDs, w i t h the p o s s i b l e e x c e p t i o n of the 4<r{g} o r b i t a l . The i n c l u s i o n of d - f u n c t i o n s i n the DZ+3d-G76 c a l c u l a t i o n has l i t t l e e f f e c t on the w a v e f u n c t i o n ( t h e c o e f f i c i e n t s a r e a l l l e s s than 10 per c e n t of the s- and p - f u n c t i o n c o e f f i c i e n t s ) . As was suggested i n Chapter 4, improvements t o the s e w a v e f u n c t i o n s s h o u l d f i r s t be concerned w i t h the f l e x i b i l i t y of the s/p b a s i s , and then w i t h i n c l u d i n g d - f u n c t i o n s . The momentum d i s t r i b u t i o n s computed from the mod e r a t e l y s o p h i s t i c a t e d d o u b l e - z e t a w a v e f u n c t i o n s a r e a l l good a p p r o x i m a t i o n s t o the e x p e r i m e n t a l f a c t ( w i t h the p o s s i b l e e x c e p t i o n of the 4<y{g} o r b i t a l d i s c u s s e d b e l o w ) . Some of the su c c e s s of these w a v e f u n c t i o n s i s due t o the f a c t t h a t the geometry of the m o l e c u l e , which i s a c c u r a t e l y known, d i c t a t e s the form of the w a v e f u n c t i o n t o a g r e a t e r e x t e n t than i n s i m p l e r m o l e c u l e s l i k e the h y d r i d e s (Chapter 2,4). Tab l e 6.2 summarizes the l o c a t i o n of the bonding (and 6/Carbon d i o x i d e 191 non-bonding and a n t i b o n d i n g ) r e g i o n s of the C 0 2 m o l e c u l e and i n d i c a t e s where P-space d e n s i t y might be e x p e c t e d t o a r i s e . As t h i s t a b l e and f i g u r e 6.3 demonstrate, the bonding r e g i o n s a re l o c a l i s e d i n c e r t a i n known r e g i o n s governed by the C0 2 geometry, and the momentum d i s t r i b u t i o n must r e f l e c t c l o s e l y the s e p a r a t i o n d i s t a n c e s of thes e r e g i o n s . T h e r e f o r e i t c o u l d have been p r e d i c t e d , b e f o r e d o i n g any c a l c u l a t i o n s , t h a t the momentum d i s t r i b u t i o n s w i l l have i n t e n s i t y , or the absence of i n t e n s i t y , a t c e r t a i n v a l u e s of q, s i m p l y because C0 2 has t h r e e atoms, spaced a t 2.25a 0, and t h e r e f o r e i t s bonding and a n t i b o n d i n g r e g i o n s a r e s e p a r a t e d by c e r t a i n d i s t a n c e s ; the o n l y degree of freedom then l e f t t o the SCF c a l c u l a t i o n i s the e s t i m a t i o n of the amount of d e n s i t y t o appear i n thos e r e g i o n s , and they a re c l e a r l y d o i n g a f a i r l y good j o b of t h i s a l s o ( w i t h the p o s s i b l e e x c e p t i o n of the 3<r{g} and 2<r{u} o r b i t a l s which cannot be ob s e r v e d d i r e c t l y , and the 4*{g} o r b i t a l which w i l l be d i s c u s s e d b e l o w ) . T h i s a c c o u n t s f o r the f a c t t h a t the peak p o s i t i o n s (q{max}) of a l l t he e x p e r i m e n t a l momentum d i s t r i b u t i o n s a r e q u i t e a c c u r a t e l y p r e d i c t e d by the c a l c u l a t i o n s , as i s not the case f o r the h y d r i d e s . 6/Carbon d i o x i d e 192 6.5.1 The 3<j{g} and 2e{u] o r b i t a l s The d e n s i t y maps i n f i g u r e 6.3f shows t h a t the innermost o r b i t a l i s e{q] symmetry (no n o d a l p l a n e a t p||=0; c l a s s I ) and the s t r o n g c o n t r a c t i o n i n the p|| d i r e c t i o n i n d i c a t e s t h a t i t exte n d s over the whole m o l e c u l e , and i s t h e r e f o r e bonding. T h i s cannot be c o n f i r m e d e x p e r i m e n t a l l y as i t i s not yet p o s s i b l e t o observe v i b r a t i o n a l s t r u c t u r e i n the i n n e r v a l e n c e r e g i o n . (There i s one e x c e p t i o n t o t h i s t h a t I know o f : a weak s a t e l l i t e of the C 2 E + { u } s t a t e of N 2 + around 25eV has an o b v i o u s v i b r a t i o n a l s e r i e s which has been obser v e d { A s b r i n k (1974)} w i t h He-II PES and c a l c u l a t e d {Domcke (1975)} by the Green's f u n c t i o n t e c h n i q u e . ) The <r{u} n a t u r e of the next o r b i t a l i s i n d i c a t e d i n the presence of the n o d a l p l a n e p||=0. The momentum d e n s i t y i s c l a s s H i and peaks a t p||{max}=0.6 i n d i c a t i n g non-bonding c h a r a c t e r (Table 6.1) but t h e r e i s a l s o s i g n i f i c a n t d e n s i t y a t p||>0.7 which s u g g e s t s some o v e r a l l bonding c h a r a c t e r . A g a i n , u n f o r t u n a t e l y , i t i s not p o s s i b l e t o t e s t t h i s s u p p o s i t i o n a g a i n s t the shape of the Franck-Condon e n v e l o p e . An e s t i m a t i o n of t h i s s t r u c t u r e v i a a p o t e n t i a l energy s u r f a c e c a l c u l a t i o n would be u s e f u l h e r e , and a l s o i n the 3<r{g} c a s e . Both t h e o r e t i c a l c a l c u l a t i o n s of the s e two o r b i t a l s g i v e v e r y n e a r l y the same r e s u l t : t h i s i s why o n l y one cu r v e i s shown i n t h e i r MDs i n f i g u r e 6.3e,f. 6/Carbon d i o x i d e The momentum d i s t r i b u t i o n of the l a r g e peak a t 38eV i s p r e d i c t e d by both c a l c u l a t i o n s t o be due t o i o n i z a t i o n from the 3tf{g} and 2<*{u} s t a t e s i n r o u g h l y e q u a l p r o p o r t i o n s , ( f i g u r e 6.1), and i n f a c t a l e a s t squares f i t of the 3<*{g} and 2<i{u} t h e o r e t i c a l momentum d i s t r i b u t i o n s t o the e x p e r i m e n t a l p o i n t s y i e l d s a good f i t i n the r a t i o of 1:1.3. The momentum d i s t r i b u t i o n measured a t 33eV however shows ( f i g u r e 6.4) c o n s i d e r a b l e i n t e n s i t y a t l a r g e q which cannot be accounted f o r by a c o m b i n a t i o n of 3<r{g} and 2<r{u} a l o n e . S e v e r a l l e a s t - s q u a r e s f i t s of c o m b i n a t i o n s of t h e o r e t i c a l MDs t o the e x p e r i m e n t a l p o i n t s a r e shown i n f i g u r e 6.4 ( o n l y the DZ+3d-G76 r e s u l t i s shown, as the o t h e r r e s u l t i s s i m i l a r ) : a c o m b i n a t i o n of 3«{g} and 2*{u} i n the r a t i o 1:5.7 ( s h o r t - d a s h e d l i n e ) c l e a r l y does not account f o r the l a r g e q i n t e n s i t y ; a c o m b i n a t i o n of the 3*{g} and 4<y{g} o r b i t a l s i n the r a t i o 1:4.9 f i t s a l i t t l e b e t t e r ( l o n g - d a s h e d l i n e ) as the 4*{g} o r b i t a l has c o n s i d e r a b l e i n t e n s i t y around q=1.0, but s t i l l does not account f o r a l l the i n t e n s i t y above t h i s p o i n t ; a b e t t e r r e s u l t s t i l l comes w i t h a l e a s t - s q u a r e s f i t of the 3<*{g} and 1 ir{g} t h e o r e t i c a l d i s t r i b u t i o n s i n the r a t i o 1:3.2 ( s o l i d l i n e ) , which i s i n disagreement w i t h the MBGF c a l c u l a t i o n s which p r e d i c t v e r y l i t t l e i n t e n s i t y due t o the 1 TT{g} or 4<y{g} o r b i t a l s , but i s i n e x c e l l e n t agreement w i t h the HAM/3-CI p r e d i c t i o n (see f i g u r e 6.1). Other c o m b i n a t i o n s of t h e o r e t i c a l momentum d i s t r i b u t i o n s were f i t t e d t o t h i s measurement, but as they 6/Carbon d i o x i d e 194 r e s u l t e d i n n e g a t i v e c o e f f i c i e n t s , they a r e p a t e n t l y not v a l i d . 6.5.2 The 4«{g} o r b i t a l The e x p e r i m e n t a l momentum d i s t r i b u t i o n s f o r the f i r s t t h r e e o u t e r v a l e n c e o r b i t a l s were o b t a i n e d by the d e c o n v o l u t i o n procedure d e s c r i b e d i n the e x p e r i m e n t a l s e c t i o n . T h i s p r o c e d u r e was d i f f i c u l t due t o the c l o s e s p a c i n g of the b i n d i n g energy peaks and the r e s u l t s a r e c o r r e s p o n d i n g l y of r a t h e r l i m i t e d s t a t i s t i c a l a c c u r a c y p a r t i c u l a r l y i n the case of the s e p a r a t e A and B momentum d i s t r i b u t i o n s . However they do p e r m i t c e r t a i n c o n c l u s i o n s t o be drawn. In f i g u r e 6.5 i s a l s o shown the e x p e r i m e n t a l and t h e o r e t i c a l momentum d i s t r i b u t i o n s f o r the 1ir{u} and 3<*{g} o r b i t a l s summed t o g e t h e r . I t i s seen t h a t the e x p e r i m e n t a l p o i n t s form a r e l a t i v e l y smooth cu r v e f o r t h i s c o m b i n a t i o n , as do the p o i n t s i n the s e p a r a t e 4<r{g} d i s t r i b u t i o n . T h i s i s some i n d i c a t i o n t h a t the d e c o n v o l u t i o n p r o c e d u r e .gives r e a s o n a b l y good r e s u l t s i n s e p a r a t i n g the 4<y{g} s t a t e peak from the envelope and t h a t t h e . m o d u l a t i o n f e a t u r e observed i n t h i s o r b i t a l i s genuine. The s t a t i s t i c a l a c c u r a c y of the s e p a r a t e d 1ir{u} and 3<r{u} momentum d i s t r i b u t i o n s i s s i g n i f i c a n t l y l ower than f o r the 4»{g} d i s t r i b u t i o n , as i s t o be exp e c t e d c o n s i d e r i n g t h e i r b i n d i n g e n e r g i e s . 6/Carbon d i o x i d e 195 The 4<y{g} o r b i t a l i s an o u t e r v a l e n c e o r b i t a l where i t i s p o s s i b l e t o check the bonding c h a r a c t e r deduced from the d e n s i t y maps a g a i n s t the e x p e r i m e n t a l Franck-Condon e n v e l o p e s . The e{q] symmetry of t h i s o r b i t a l ( c l a s s 111i 2) i s seen i n the d e n s i t y a t p||=0, and i n d i c a t e s ( i n t h i s case) some bonding c h a r a c t e r . The l a r g e d e n s i t y i n the l o b e s a t p||{max}=0.9 i n d i c a t e s a l s o a s i g n i f i c a n t a n t i b o n d i n g c o n t r i b u t i o n from the charge d e n s i t y o u t s i d e the oxygen atoms. These two c o n t r i b u t i o n s tend t o b a l a n c e each o t h e r o u t , l e a v i n g a m a i n l y non-bonding o r b i t a l , as c o n f i r m e d by the absence of v i b r a t i o n a l s t r u c t u r e on the He-I PES b i n d i n g energy peak {Turner 1970)}. The t h e o r e t i c a l momentum d i s t r i b u t i o n o n l y a g r e e s w i t h experiment ±0 the e x t e n t t h a t the p o s i t i o n s of the maxima are p r e d i c t e d c o r r e c t l y , but the i n t e n s i t i e s of these f e a t u r e s a r e n o t . There may be s e v e r a l reasons why t h i s i s so: the d e c o n v o l u t i o n of the t h r e e peaks may not be a c c u r a t e i n the low q r e g i o n , a l l o w i n g some i n t e n s i t y from peak B t o l e a k i n t o the peak C, but i n view of the good agreement of the o t h e r two d e c o n v o l u t e d MDs w i t h t h e o r y t h i s p r o b a b l y does not account f o r a l l the q=0 i n t e n s i t y ; t h e r e may a l s o be ground s t a t e c o r r e l a t i o n between the 4<r{g} and the 3*{g} which would tend t o augment the q=0 i n t e n s i t y ; or i t may j u s t be a f a i l u r e of the d o u b l e - z e t a b a s i s t h a t the SCF p rocedure cannot p r o p e r l y e s t i m a t e two MO w a v e f u n c t i o n s when they a r e of the same symmetry. Measurements on 0 2 6/Carbon d i o x i d e {Suzuki (1980b)} i n d i c a t e t h e r e i s s i g n i f i c a n t c o r r e l a t i o n i n the analogous case of the 3*{g} and 2e{q] o r b i t a l s , which i s i n s u pport of the second argument above. An attempt was made t o measure the 4«r{g} momentum d i s t r i b u t i o n d i r e c t l y , but t h i s proved i n c o n c l u s i v e ; the r e s u l t shows ( f i g u r e 6.5) an s-type d i s t r i b u t i o n w i t h no e v i d e n c e of the hump a t q=1.0. Very l i k e l y the B s t a t e i s f i l l i n g i n the q=0.6 r e g i o n . F i n a l l y , i t may be p o s s i b l e t h a t the C0 2 4<y{g} i s j u s t a d i f f i c u l t o r b i t a l w a v e f u n c t i o n t o c a l c u l a t e w i t h the l i m i t e d f l e x i b i l i t y of these b a s i s s e t s - t h e r e i s a s i g n i f i c a n t d i f f e r e n c e between the two t h e o r e t i c a l c u r v e s , even though the b a s i s s e t s a r e q u i t e s i m i l a r . 6.5.3 The 3*{u} and 1rr{u} o r b i t a l s The 3«{u} o r b i t a l i s i n t e r e s t i n g i n t h a t i t s MD shows a q{max} a t 0.4 - s u r p r i s i n g l y low c o n s i d e r i n g t h a t the a t o m i c C2p and 02p o r b i t a l s have t h e i r p{max} at 0.6 and 0.8 r e s p e c t i v e l y ( f i g u r e 2.1). The r e ason f o r t h i s i s n e a t l y e x p l a i n e d by the a n t i b o n d i n g d e n s i t y a t p||=0.4 (Table 6.2) seen i n the momentum d e n s i t y maps ( c l a s s I V i ) . The 3<y{u} o r b i t a l i s a n o t h e r s i t u a t i o n where the bonding and a n t i b o n d i n g c o n t r i b u t i o n s r o u g h l y c a n c e l , l e a v i n g l i t t l e v i b r a t i o n a l s t r u c t u r e i n the PES spectrum {Turner 1970)}". There i s p r o b a b l y some, s m a l l net a n t i b o n d i n g tendency from t h i s o r b i t a l . 6/Carbon d i o x i d e The g r e a t e r p a r t of the momentum d e n s i t y i n the 1ir{u} o r b i t a l ( c l a s s H i ) i s i n the p||=0 p l a n e , i n d i c a t i n g t h a t the v o r b i t a l e xtends over the whole m o l e c u l e , as i s o b v i o u s from the charge d e n s i t y map. T h i s o r b i t a l i s r e s p o n s i b l e f o r the major p a r t of the bonding i n C0 2 and the PES spectrum shows an e x t e n s i v e Franck-Condon envelope f o r t h i s peak. The agreement between experiment and t h e o r y i s good on both th e s e o r b i t a l s , e s p e c i a l l y i n r e s p e c t of the q{max} v a l u e s . In the 3<r{u} the s m a l l e r m o d u l a t i o n around q=1.5 i n the t h e o r e t i c a l c u r v e i s not seen i n the e x p e r i m e n t a l p o i n t s . Here a g a i n the e x p l a n a t i o n of t h i s d i s c r e p a n c y may l i e i n the inadequacy of the b a s i s s e t s , or i n c o r r e l a t i o n between the 3<*{u} and 2*{u} o r b i t a l s . For a time t h e r e was some u n c e r t a i n t y i n the e n e r g i e s of the 1rr{u} and 3*{g} s t a t e s i n C0 2 {Turner ( 1 9 7 0 ) } : d i f f e r e n t c a l c u l a t i o n s p r e d i c t e d d i f f e r e n t o r d e r i n g of t h e s e s t a t e s . The problem was r e s o l v e d when the PES spectrum was found t o show t h a t the h i g h e r energy s t a t e (17.6eV) must be the 1rr{u} o r b i t a l because of i t s extended v i b r a t i o n a l e n v e l o p e . The p r e s e n t C 0 2 r e s u l t s a r e i n agreement w i t h t h i s a s s i g n m e n t : the e x p e r i m e n t a l momentum d i s t r i b u t i o n s p l a i n l y show the q{max} f o r the 1ir{u} MD a t 0.7 i s h i g h e r than t h a t f o r the 3<j{g} MD a t 0.4, which i s i n agreement w i t h the c a l c u l a t i o n s . That t h i s good agreement i s seen i s e v i d e n c e of the s u c c e s s of the d e c o n v o l u t i o n p r o c e d u r e , and some encouragement t o suppose t h a t the 4<r{g} q=0 i n t e n s i t y i s not 6/Carbon d i o x i d e an e x p e r i m e n t a l a r t i f a c t . 6.5.4 The 1 rr{g} o r b i t a l Because C0 2 has enough e l e c t r o n s t o p o p u l a t e a ir{g} l e v e l , u n l i k e any of the systems s t u d i e d so f a r i n t h i s t h e s i s , an i n t e r e s t i n g f e a t u r e i s seen i n the momentum d i s t r i b u t i o n of t h i s outermost o r b i t a l : q{max} i s a t the u n u s u a l l y h i g h v a l u e of 0 . 9 . The reason f o r t h i s i s o b v i o u s on i n s p e c t i o n of the momentum d e n s i t y map ( c l a s s I l l i , ) : the gerade c o m b i n a t i o n of 2p{x} and 2p{y} oxygen atomic o r b i t a l s means t h e r e are two p e r p e n d i c u l a r n o d a l p l a n e s which push q{max} from i t s atomic p o s i t i o n of 0 . 7 a o - 1 a l o n g the pj_ a x i s , out t o 0.9 between the n o d a l p l a n e s . T h i s e f f e c t w i l l be seen i n a l l o r b i t a l s which have t h i s 'pseudo d - o r b i t a l ' symmetry: f o r i n s t a n c e the NO 2rr and 0 2 1 ir{g} o r b i t a l s where p{max} o c c u r s a t 0.8 and 0.9a o" 1 r e s p e c t i v e l y (Chapter 7 ) . In o t h e r words, an a n t i b o n d i n g r r * - o r b i t a l i s e s s e n t i a l l y of s i m i l a r a n g u l a r form as an atomic d - o r b i t a l s i t u a t e d on the bond m i d p o i n t (see s e c t i o n 2.5.1 on 'pseudo-angular momentum'). T h i s case a l s o has p a r a l l e l s w i t h the ir-system of t r a n s - b u t a d i e n e (C„H 6) { T o s s e l l ( 1 9 8 1 ) } . The C„H 6 1a{u} and 1b{g} m o l e c u l a r o r b i t a l s have some s i m i l a r c h a r a c t e r i s t i c s t o the C 0 2 1ir{u} and 1n{g} m o l e c u l a r o r b i t a l s r e s p e c t i v e l y . A l l f o u r m o l e c u l a r o r b i t a l s have a nod a l p l a n e c o n t a i n i n g the atom c e n t r e s which c h a r a c t e r i z e s i r - s y s t e m s . The 1 ir{g} and 1b{g} m o l e c u l a r o r b i t a l s have 6/Carbon d i o x i d e 199 another n o d a l p l a n e i n the m i d d l e of the m o l e c u l e and p e r p e n d i c u l a r t o the atomic backbone. The most p r o b a b l e momenta a r e 0.6a o" 1 and 0 . 8 a o _ 1 f o r the C„H 6 1a{u} and 1b{g} m o l e c u l a r o r b i t a l s r e s p e c t i v e l y . The C0 2 1 TT{g} most p r o b a b l e momentum (0.9) i s g r e a t e r than t h a t f o r C„H 6 1b{g} because the C0 2 m o l e c u l e i s one atom s h o r t e r than C„H 6. By the R e c i p r o c i t y p r i n c i p l e t h i s means a momentum d e n s i t y of p r o p o r t i o n a l l y g r e a t e r e x t e n t i n the bond a x i s dimension i n C0 2 . S i m i l a r l y the C0 2 1ir{u} q{max} (0.7) i s h i g h e r than t h a t of the C„H 6 1a{u} o r b i t a l . T h i s emphasizes t h a t one must e x e r c i s e c a u t i o n i n a s s o c i a t i n g the energy of an o r b i t a l d i r e c t l y w i t h the most p r o b a b l e momentum of the d i s t r i b u t i o n . I t i s not n e c e s s a r i l y t r u e t h a t one s h o u l d expect the momentum d i s t r i b u t i o n t o move t o h i g h e r v a l u e s of q as one goes deeper i n t o the v a l e n c e s h e l l of the m o l e c u l e ( i . e . t o h i g h e r b i n d i n g e n e r g y ) . I t i s seen here t h a t i t i s p o s s i b l e f o r an e l e c t r o n t o have a momentum d i s t r i b u t i o n a t h i g h e r most p r o b a b l e q, but s t i l l be a t a lower b i n d i n g energy than a n o t h e r o r b i t a l . Of c o u r s e the b a l a n c e of the e l e c t r o n ' s energy i s i n the Coulomb p o t e n t i a l i n t e r a c t i o n s w i t h the o t h e r p a r t i c l e s i n the system, and s i n c e the 1 rr{g} i s an a n t i b o n d i n g o r b i t a l i t may be e x p e c t e d t o have s t r o n g Coulomb r e p u l s i o n terms which r a i s e i t s energy. p||{max} i s a t 0.6 f o r t h i s o r b i t a l i n d i c a t i n g non-bonding c h a r a c t e r . Because the momentum d e n s i t y f a l l s 6/Carbon d i o x i d e 200 o f f s t e e p l y w i t h i n c r e a s i n g p|| t h e r e i s l i t t l e bonding c h a r a c t e r , u n l i k e the 2<*{u} o r b i t a l , e x p l a i n i n g why the PES b i n d i n g energy peak has no v i b r a t i o n a l s t r u c t u r e . The e x p e r i m e n t a l MD f o r t h i s o r b i t a l shows i n t e n s i t y near q=0 which i s i n disagreement w i t h t h a t e x p e c t e d from the symmetry of the o r b i t a l (pseudo-d). T h i s i n t e n s i t y i s not l i k e l y t o be r e a l , and i s p r o b a b l y an a r t i f a c t due t o the combined e f f e c t s of a n g u l a r r e s o l u t i o n , s t a t i s t i c a l e r r o r , and u n c e r t a i n t y i n the e x a c t mean e s c a t t e r i n g a n g l e . 6.6 C o n c l u s i o n s C0 2 shows i n t e r e s t i n g v a r i e t y i n i t s m o l e c u l a r o r b i t a l s and momentum d i s t r i b u t i o n s . The geometry of the m o l e c u l e has a s t r o n g e f f e c t on the shapes of the MDs ( t h e s e a re q u i t e d i f f e r e n t from the MDs of atomic carbon and oxygen), and t h i s shape can be used t o e s t i m a t e r o u g h l y the r e l a t i v e b o n d i n g , non-bonding, and a n t i b o n d i n g c h a r a c t e r of the o r b i t a l s . The use of momentum and charge d e n s i t y maps i s v e r y r e v e a l i n g i n t h i s s t u d y . The c a l c u l a t i o n of the C 0 2 MO w a v e f u n c t i o n s does not b e n e f i t g r e a t l y from the a d d i t i o n of d - f u n c t i o n s , as was a l s o found f o r H 2S, and i n f a c t , can be done f a i r l y s u c c e s s f u l l y w i t h o n l y a d o u b l e - z e t a b a s i s , a l t h o u g h t h e r e 6/Carbon d i o x i d e 201 i s some e r r o r i n the e s t i m a t i o n of the 4<y{g} o r b i t a l . The HAM/3-CI method, of c a l c u l a t i n g b i n d i n g energy s t r u c t u r e has been shown t o g i v e e x c e l l e n t r e s u l t s f o r the case of C 0 2 . 202 CHAPTER 7 NITRIC OXIDE AND OXYGEN 'This m a t t e r winds i t s e l f ever i n new r i d d l e s ! ' B i n a r y (e,2e) s t u d i e s have been c a r r i e d out on NO and 0 2 by the F l i n d e r s U n i v e r s i t y (e,2e) group. To complement the e x p e r i m e n t a l r e s u l t s { B r i o n } , the momentum d i s t r i b u t i o n s , momentum d e n s i t y maps and charge d e n s i t y maps from s e v e r a l t h e o r e t i c a l m o l e c u l a r w a v e f u n c t i o n s have been computed here a t UBC. 7.1 N i t r i c Oxide The NO e l e c t r o n i c c o n f i g u r a t i o n (not i n c l u d i n g s p i n ) i s : C c o r e ) 4 ( 3 * ) 2 ( 4 * ) 2 ( 5 < r ) 2 ( l i r ) 4 ( 2 i r ) 1 E x p e r i m e n t a l r e s u l t s f o r the NO m o l e c u l e were done a t !200eV i n c i d e n t energy. B i n d i n g energy s p e c t r a a t two a n g l e s 7 / N i t r i c Oxide and Oxygen 203 were r e c o r d e d , as w e l l as a n g u l a r c o r r e l a t i o n s a t e n e r g i e s 9.5, 21, and 40.5eV: the s e e n e r g i e s c o r r e s p o n d t o the 2 i r ' \ 4 t f _ 1 , and 3*' 1 o r b i t a l vacancy i o n s t a t e s . A n g u l a r c o r r e l a t i o n s f o r the o t h e r peaks c o u l d not be r e c o r d e d s i n c e they a r e spaced t o o c l o s e l y t o be r e s o l v e d e n e r g e t i c a l l y . " The NO w a v e f u n c t i o n i s n o t o r i o u s l y d i f f i c u l t t o c a l c u l a t e . Two t h e o r e t i c a l w a v e f u n c t i o n s were s t u d i e d f o r NO: (1) DZ-NSO-CI: a d o u b l e - z e t a , n a t u r a l s p i n o r b i t a l , r e s t r i c t e d CI c a l c u l a t i o n {Kouba (1971)}; (2) GTO-UHF: a (9s5p/4s3p) c o n t r a c t e d GTO b a s i s , u n r e s t r i c t e d H a r t r e e - F o c k , s i n g l e - c o n f i g u r a t i o n c a l c u l a t i o n {Kunz}. The t o t a l e n e r g i e s of t h e s e w a v e f u n c t i o n s a r e ( H a r t r e e s ) : -129.2599 (DZ-NSO-CI), and -129.2065 (GTO-UHF). The l i m i t i n g H a r t r e e - F o c k energy has been e s t i m a t e d t o be -129.3443 H a r t r e e s { S i n a n o g l u (1966)}. The f i r s t c a l c u l a t i o n assumed the e x p e r i m e n t a l i n t e r n u c l e a r d i s t a n c e of 2.1747a 0, and the second p l a c e s the atoms 2.25a 0 a p a r t . The e x p e r i m e n t a l and t h e o r e t i c a l MDs a r e shown i n f i g u r e 7.1. The computed d e n s i t y maps a r e shown i n f i g u r e 7.2 and 7.3. S i n c e the a l p h a and beta MOs of the DZ-NSO-CI work a r e v e r y s i m i l a r ( e x c e p t f o r the 2 i r ) , o n l y the a l p h a ones a r e shown. I t was assumed t h a t the t a b u l a t e d b a s i s f u n c t i o n c o e f f i c i e n t s i n the DZ-NSO-CI w a v e f u n c t i o n r e f e r t o a b a s i s s e t where the p o s i t i v e l o b e of the a x i a l 2p b a s i s 7 / N i t r i c Oxide and Oxygen 204 F i g u r e 7.1 E x p e r i m e n t a l and t h e o r e t i c a l momentum d i s t r i b u t i o n s of NO. S o l i d l i n e s a r e the DZ-NSO-CI r e s u l t s and dashed l i n e s a r e the GTO-UHF r e s u l t s . 7 / N i t r i c Oxide and Oxygen 205 F i g u r e 7.2 Momentum and charge d e n s i t y maps from the DZ-NSO-CI w a v e f u n c t i o n ( s p i n o p a r t s o n l y ) . 7 / N i t r i c Oxide and Oxygen 2 0 6 F i g u r e 7 . 3 Momentum .and charge d e n s i t y maps from the GTO-UHF wavefunct i o n . MOMENTUM DENSITY CHARGE DENSITY MOMENTUM DENSITY CHARGE DENSITY 7 / N i t r i c Oxide and Oxygen 208 f u n c t i o n s and the oxygen atom a r e toward the same end of the m o l e c u l e ; the o r i e n t a t i o n of the 2p f u n c t i o n s i s not e x p l i c i t l y s t a t e d i n the paper, though I am a s s u r e d by the a u t h o r s and by o t h e r a u t h o r i t i e s t h a t t h i s assumption i s p r o b a b l y c o r r e c t . 7.1.1 The NO i n n e r v a l e n c e o r b i t a l s The DZ-NSO-CI c a l c u l a t i o n s u g g e s t s the innermost 3<r o r b i t a l has some bonding c h a r a c t e r , ( w i t n e s s the c o n t r a c t i o n i n t he bond d i r e c t i o n of the momentum d e n s i t y map) w h i l e the o t h e r c a l c u l a t i o n show non-bonding a t t r i b u t e s (the c o n t o u r l i n e s a r e n e a r l y c i r c u l a r ) . The momentum d i s t r i b u t i o n s i n d i c a t e t h a t t h i s e s t i m a t i o n of bonding c h a r a c t e r i n the DZ-NSO-CI c a l c u l a t i o n i s p r o b a b l y not c o r r e c t , as the f i t between c a l c u l a t i o n and t h e o r y i s not q u i t e as good as i n the GTO-UHF c a l c u l a t i o n . Our c h e m i c a l i n t u i t i o n i s i n a c c o r d w i t h t h i s : we c o n c e i v e of the i n n e r v a l e n c e o r b i t a l s as b e i n g ' a t o m i c - l i k e ' and showing l i t t l e bonding or a n t i b o n d i n g tendency. The s i t u a t i o n i s r e v e r s e d i n the 4<r o r b i t a l , where the DZ-NSO-CI MD c l e a r l y f i t s b e t t e r than the GTO-UHF: the former i n d i c a t e s t h a t the o r b i t a l , though n o m i n a l l y a n t i b o n d i n g , has some nonbonding c h a r a c t e r as seen i n the d e n s i t y around p=0, w h i l e the l a t t e r have almost no d e n s i t y a t low q, i n disagreement w i t h the e x p e r i m e n t a l r e s u l t . 7 / N i t r i c Oxide and Oxygen 209 The momentum d e n s i t y f o r the 3c o r b i t a l can be c o n s t r u c t e d m a i n l y from c l a s s I Q - p r o j e c t i o n s . The 4* momentum d e n s i t y i s c l a s s I l a i n the bond d i r e c t i o n , and the d i f f e r e n c e s between the two c a l c u l a t i o n s i s due t o the v a r y i n g asymmetric c o n t r i b u t i o n of 2s f u n c t i o n s on each c e n t r e . 7.1.2 The NO o u t e r v a l e n c e o r b i t a l s The outermost 2IT o r b i t a l i s an e x c e l l e n t example of the s e l e c t i v i t y of b i n a r y (e,2e) i n j u d g i n g the q u a l i t y of w a v e f u n c t i o n s which, examined under o t h e r c r i t e r i a - energy, f o r i n s t a n c e , or d i p o l e moment, may have l i t t l e t o d i s t i n g u i s h them. The two c a l c u l a t i o n s e s t i m a t e d i f f e r e n t l y the c h a r a c t e r of t h i s MO as almost non-bonding (DZ-NSO-CI) or n e a r l y f u l l a n t i b o n d i n g (GTO-UHF). The i m p o r t a n t f e a t u r e i s the amount of d e n s i t y i n the p||=0 p l a n e : i f t h i s i s not l a r g e then the p o s i t i o n of p{max} i n the MD w i l l move t o l a r g e r p compared w i t h the atomic o r b i t a l s , which i s the case f o r the GTO-UHF c a l c u l a t i o n . The DZ-NSO-CI c a l c u l a t i o n , on the o t h e r hand, does put c o n s i d e r a b l e d e n s i t y i n the pj_ p l a n e , w i t h the r e s u l t t h a t the v a l u e of p{max} does not change as much compared w i t h the atomic o r b i t a l s ( f i g u r e 2.1). T h i s k i n d of o r b i t a l f a l l s somewhere between c l a s s H i and c l a s s 111i 2 i n the bond a x i s d i r e c t i o n (see s e c t i o n 2.4) depending on whether i t i s more n e a r l y non-bonding or a n t i b o n d i n g r e s p e c t i v e l y . 7 / N i t r i c Oxide and Oxygen 210 Next c o n s i d e r the s p h e r i c a l l y - a v e r a g e d momentum d i s t r i b u t i o n s from the two c a l c u l a t i o n s : b o th d i s t r i b u t i o n s w i l l be p-type because of the n o d a l p l a n e , but due t o i t s s i g n i f i c a n t non-bonding c h a r a c t e r , the DZ-NSO-CI c a l c u l a t i o n has the l o w e s t p{max}, w i t h the GTO-UHF r e s u l t s i g n i f i c a n t l y h i g h e r . The e x p e r i m e n t a l r e s u l t shows i n a v e r y g r a t i f y i n g manner t h a t t he DZ-NSO-CI r e s u l t i s c o r r e c t , and the o t h e r d e f i n i t e l y i n e r r o r . Another e x p l a n a t i o n has been advanced {Suzuki (1980b)} t o account f o r the l a r g e p{max} i n the outermost MO of NO and 0 2 , which r e s t s on t h e f a c t t h a t i n b o t h c a s e s i t i s s i n g l y - o c c u p i e d . For both 0 2 and NO t h i s MO i s a rr* o r b i t a l o c c u p i e d by a s i n g l e ( u n p a i r e d ) e l e c t r o n ( t h e doubly degenerate rr* o r b i t a l s a r e each s i n g l y o c c u p i e d i n 0 2 ) . In t h a t d i s c u s s i o n {Suzuki (1980b)} of the 0 2 r e s u l t s i t was suggested t h a t the h i g h e r most p r o b a b l e momentum might be the r e s u l t ( v i a the R e c i p r o c i t y p r i n c i p l e ) of l e s s s p a t i a l e x t e n s i o n of the o r b i t a l a r i s i n g from the absence of i n t r a - o r b i t a l e l e c t r o n - e l e c t r o n r e p u l s i o n . The f a c t t h a t a c o r r e s p o n d i n g l y l a r g e most p r o b a b l e momentum has a l s o been ob s e r v e d f o r the d o u b l y - o c c u p i e d outermost ir* o r b i t a l of C 0 2 s u g g e s t s t h a t t h i s e f f e c t i s p r i m a r i l y due t o the ' d - l i k e ' p s eudo-angular momentum of the a n t i b o n d i n g (ir*) o r b i t a l s i n a l l t h r e e m o l e c u l e s . I n t r a - o r b i t a l c o r r e l a t i o n e f f e c t s can f o r the most p a r t be r u l e d out as a d i r e c t c o n t r i b u t i n g f a c t o r on two c o u n t s : 7 / N i t r i c Oxide and Oxygen (1) C o r r e l a t i o n e f f e c t s a r e g e n e r a l l y q u i t e s m a l l i n the ground s t a t e , a c c o u n t i n g f o r about 1eV of the t o t a l energy. T h i s i s not enough t o cause such a l a r g e s h i f t d i r e c t l y ; ( 2 ) ' I n t r a - o r b i t a l ' v e r s u s ' i n t e r - o r b i t a l ' c o r r e l a t i o n i s o n l y a c o n v e n i e n t a r t i f i c i a l p a r t i t i o n i n g of the t o t a l c o r r e l a t i o n between a l l the e l e c t r o n s , and t o s i n g l e out the former as the cause of a s p e c i f i c e f f e c t i n one o r b i t a l i s not v a l i d . That the DZ-NSO-CI w a v e f u n c t i o n does do a b e t t e r j o b than the GTO-UHF w a v e f u n c t i o n must be due m o s t l y (as the b a s i s s e t s a r e s i m i l a r ) t o the n a t u r a l - s p i n - o r b i t a l c o n f i g u r a t i o n - i n t e r a c t i o n t e c h n i q u e wherein c o r r e l a t i o n i s p r o p e r l y a c c o u n t e d f o r i n the whole m o l e c u l e (which i n t u r n a f f e c t s t he SCF p o t e n t i a l and the shape of the 2n NSO), r a t h e r than s o l e l y because the 2rr i n t r a - o r b i t a l c o r r e l a t i o n i s removed. The r e m a i n i n g t h e o r e t i c a l m o l e c u l a r o r b i t a l s , w h i l e they have not been checked e x p e r i m e n t a l l y due t o the c l o s e s p a c i n g of t h e i r f i n a l i o n s t a t e e n e r g i e s , a r e of academic i n t e r e s t as a d e m o n s t r a t i o n of bonding i n h e t e r o n u c l e a r d i a t o m i c s . The c a l c u l a t i o n s both p r e d i c t bonding r r - o r b i t a l s between the two atoms, and non-bonding i r - o r b i t a l s c e n t r e d m a i n l y on the oxygen. The bonding c h a r a c t e r i s r e f l e c t e d i n the amount of c o n t r a c t i o n of the momentum d e n s i t y i n the bond d i r e c t i o n . T h i s i n t u r n a f f e c t s the 7 / N i t r i c Oxide and Oxygen 212 s p h e r i c a l l y - a v e r a g e d MDs: a c o n t r a c t i o n of the d e n s i t y i n the p|| d i r e c t i o n means the MD s h i f t s t o lower q, as may be seen comparing f i g u r e 7.1 w i t h f i g u r e s 7.2 and 7.3 (not a l l the 1 xi MDs are shown i n f i g u r e 7.1 as they a r e o f t e n v e r y s i m i l a r ) . The g r e a t e s t d i f f e r e n c e i n the DZ-NSO-CI and GTO-UHF r e s u l t s i s i n the 5a MO: the DZ-NSO-CI r e s u l t p r e d i c t s r o u g h l y e q u a l bonding and a n t i b o n d i n g c o n t r i b u t i o n s t h a t g i v e r i s e t o MD i n t e n s i t y around q=0 and q=1.2. The GTO-UHF c a l c u l a t i o n p r e d i c t s somewhat more a n t i b o n d i n g c h a r a c t e r which i n c r e a s e s the t o t a l l y symmetric component i n t h i s o r b i t a l and so augments the q=0 i n t e n s i t y a t the expense of the q=1.2 r e g i o n . I t has been suggested {Kouba (1971)} t h a t the 5a o r b i t a l p r o b a b l y a c c o u n t s f o r most of the bonding i n t h i s m o l e c u l e , but e x a m i n a t i o n of the charge d e n s i t y maps i n d i c a t e s t h a t t h i s i s perhaps i n c o r r e c t i n l i g h t of the argument above, and t h a t the bonding a r i s e s m a i n l y from the 1 ir m o l e c u l a r o r b i t a l . 7.2 Oxygen The e l e c t r o n i c MO c o n f i g u r a t i o n of the 1 6 - e l e c t r o n 0 2 m o l e c u l e i s : ( c o r e ) " ( 2 t f { g } ) 2 ( 2 f f { u } ) 2 ( 3 t f { g } ) 2 ( 1 > r { g } ' t ( l T r { u } ) 2 7 / N i t r i c Oxide and Oxygen 213 A c a l c u l a t i o n on oxygen {Kunz} s i m i l a r t o the GTO-UHF c a l c u l a t i o n on NO i s d i s p l a y e d i n the form of charge and momentum d e n s i t y maps, and momentum d i s t r i b u t i o n s i n comparison w i t h e x p e r i m e n t a l b i n a r y (e,2e) r e s u l t s { S uzuki (1980b)} i n f i g u r e 7.4. The t h e o r e t i c a l momentum d i s t r i b u t i o n i s shown w i t h a s o l i d l i n e . S o l i d c i r c l e s i n d i c a t e 1200eV e x p e r i m e n t a l d a t a , and open c i r c l e s 400eV d a t a . The a n g u l a r r e s o l u t i o n i n the momentum d i s t r i b u t i o n measurements i s c l a i m e d t o be s u f f i c i e n t l y good t h a t i t i s unnecessary t o f o l d i n e x p e r i m e n t a l r e s o l u t i o n f a c t o r s i n t o the c a l c u l a t e d momentum d i s t r i b u t i o n s , and i n f i g u r e 7.4 no such f a c t o r s have been i n c l u d e d . The agreement between e x p e r i m e n t a l and t h e o r e t i c a l momentum d i s t r i b u t i o n s i s , i n g e n e r a l , v e r y good f o r a l l MOs w i t h i n the s t a t i s t i c a l a c c u r a c y of the measurements. 7.2.1 The 0 2 <r{g} o r b i t a l s There seems t o be e x t e n s i v e c o r r e l a t i o n , i n e i t h e r the i n i t i a l ground s t a t e or f i n a l i o n s t a t e s or b o t h , i n v o l v i n g the 2<y{g} and 3<r{g} 0 2 o r b i t a l s . T h i s produces a t l e a s t f o u r b i n d i n g energy peaks which a l l e x h i b i t an s-type momentum d i s t r i b u t i o n , but have v a r y i n g degrees of ' k i n k ' a t q = 0 . 6 a o _ 1 . Of the s e f o u r momentum d i s t r i b u t i o n s I have chosen the lo w e s t and h i g h e s t i n energy t o compare w i t h t h e o r y , as these a r e l i k e l y t o be 'purer' i n one s t a t e or the o t h e r , as indeed f i g u r e s 7.4c and 7.4e show: the 7 / N i t r i c Oxide and Oxygen 214 F i g u r e 7 . 4 a Momentum d i s t r i b u t i o n s and momentum and charge d e n s i t y maps f o r 0 2 : the 1w{g} o r b i t a l . 7 / N i t r i c Oxide and Oxygen 215 F i g u r e 7.4b Momentum d i s t r i b u t i o n s and momentum and charge d e n s i t y maps f o r 0 2 : the 1ir{u} o r b i t a l . 7 / N i t r i c Oxide and Oxygen 2 1 6 F i g u r e 7.4c Momentum d i s t r i b u t i o n s and momentum and charge d e n s i t y maps f o r 0 2 : the 3*{g} o r b i t a l . 7 / N i t r i c Oxide and Oxygen 217 F i g u r e 7.4d Momentum d i s t r i b u t i o n s and momentum and charge d e n s i t y maps f o r 0 2 : the 2*{u} o r b i t a l . 7 / N i t r i c Oxide and Oxygen 218 F i g u r e 7.4e Momentum d i s t r i b u t i o n s and momentum and charge d e n s i t y maps f o r 0 2 : the 2«{g} o r b i t a l . 7 / N i t r i c Oxide and Oxygen 2 1 9 agreement i s e x c e l l e n t , w i t h the s m a l l e x c e p t i o n of the p o s i t i o n of the ' k i n k ' which the c a l c u l a t i o n puts r a t h e r h i g h e r than the measurement. The p o s i t i o n of the k i n k i s dete r m i n e d by the l o c a t i o n of the P-space n o d a l s u r f a c e s , which i s i n t u r n d e t e r m i n e d by the r a t i o of s/p c h a r a c t e r i n the o r b i t a l : i f t h e r e i s i n f a c t some e x t r a 2s c h a r a c t e r coming i n from the 2<r{g} o r b i t a l t h i s may account f o r the d i s c r e p a n c y . 7.2.2 The 2<r{u} o r b i t a l I n s o f a r as may be judged, the agreement between t h e o r y and experiment i s s a t i s f a c t o r y , w i t h i n the s t a t i s t i c a l a c c u r a c y of the measurement. T h i s o r b i t a l i s of a n t i b o n d i n g c h a r a c t e r and i s d e s c r i b e d by the C l a s s H i Q - p r o j e c t i o n . 7.2.3 The 1ir{u} and 1n{g} o r b i t a l s As has- been mentioned e a r l i e r , oxygen i s another example of the s i t u a t i o n where p{max} i n c r e a s e s as € d e c r e a s e s , f o r two rt o r b i t a l s . A g a i n , the reason f o r the l a r g e p{max} ( 0 . 9 a o _ 1 ) i n the 1ir { g } , compared t o the 1ir{u} ( 0 . 7 a o ~ 1 ) i s the second n o d a l p l a n e , p||=0, which pushes momentum d e n s i t y t o l a r g e r p, a l t h o u g h pj_ i s almost the same. The reason why the c a l c u l a t i o n a g r e e s w e l l w i t h experiment h e r e , i n c o n t r a s t w i t h the NO 2 i r , i s t h a t the 0 2 1 TT{g} i s a pure a n t i b o n d i n g o r b i t a l , i n a homonuclear m o l e c u l e , w i t h no p o s s i b i l i t y of non-bonding c h a r a c t e r 7 / N i t r i c Oxide and Oxygen 220 coming i n . T h e r e f o r e , the c o e f f i c i e n t of the atomic 2 p o r b i t a l s on one oxygen w i t h r e s p e c t t o the o t h e r i s e x a c t l y the same ( t o w i t h i n a s i g n ) , f i x e d by the symmetry of the system, whereas i n the NO 2TT MO the r e l a t i v e amounts of 2 p AOs on each c e n t r e (and hence the r e l a t i v e n o n - b o n d i n g / a n t i b o n d i n g c h a r a c t e r , and the p o s i t i o n of p{max}) i s a v a r i a b l e , and l i k e l y v e r y s e n s i t i v e t o the q u a l i t y of the b a s i s s e t and the method of c a l c u l a t i o n t o boot. T h i s s i t u a t i o n s h o u l d be r e f l e c t e d i n the comparison of the o u t e r C 0 2 . l i r { u } and 1 ir{g} MOs w i t h the c o r r e s p o n d i n g COS 2?r and 3ir o r b i t a l s : t h e o r e t i c a l w a v e f u n c t i o n s f o r the C 0 2 o r b i t a l s a r e good, but I p r e d i c t t h a t the COs 3rr q{max} w i l l be lower than the C 0 2 1 rr{g} and a l s o d i f f i c u l t t o c a l c u l a t e . In g e n e r a l , where symmetry i s lowered by c h a n g i n g an atom but m a i n t a i n i n g s i m i l a r s t r u c t u r e (as i n 0 2 t o NO, N 2 t o HCN, C 0 2 t o COS), we may expect c a l c u l a t e d r e s u l t s u s i n g s i m i l a r q u a l i t y b a s i s s e t s t o be l e s s good i n the l e s s symmetric c a s e . In b o t h these c a s e s the agreement of measurement and t h e o r y i s good: the o n l y d i s c r e p a n c y i s t h a t t h e . t h e o r e t i c a l momentum d i s t r i b u t i o n s a r e a t v e r y s l i g h t l y h i g h e r momentum compared t o the e x p e r i m e n t a l p o i n t s ( i g n o r i n g the low q r e g i o n where the d i s c r e p a n c y may be due p a r t l y t o r e s i d u a l a n g u l a r r e s o l u t i o n e f f e c t s ) . T h i s has been noted b e f o r e i n the o u t e r v a l e n c e o r b i t a l s of o t h e r systems, and a g a i n means 7 / N i t r i c Oxide and Oxygen 221 t h a t the (9s6p/5s3p) b a s i s s e t , w h i l e a c c u r a t e f o r i n n e r , more a t o m i c - l i k e MOs, f a i l s t o account c o m p l e t e l y f o r the e l e c t r o n d e n s i t y s h i f t i n the o u t e r MOs when forming the m o l e c u l e . 7.3 C o n c l u s i o n s Charge and momentum d e n s i t y maps have been used t o r e f u t e p r e v i o u s s u p p o s i t i o n s as t o the o r i g i n of the h i g h q{max} i n the NO 2 tr and 0 2 1 ir {g} o r b i t a l momentum d i s t r i b u t i o n s . T h i s f e a t u r e a r i s e s m a i n l y out of the n* a n t i b o n d i n g c h a r a c t e r of t h e s e o r b i t a l s and not d i r e c t l y from the absence of i n t r a - o r b i t a l c o r r e l a t i o n . The d o u b l e - z e t a w a v e f u n c t i o n of Kunz has been shown t o g i v e good r e s u l t s i n the e s t i m a t i o n of the 0 2 MDs, but i n the case of NO a c o r r e l a t e d w a v e f u n c t i o n i s n e c e s s a r y t o account f o r the e l e c t r o n d e n s i t y s h i f t s i n f o r m i n g t h i s m o l e c u l e . 222 CHAPTER 8 THE NEW MCP SPECTROMETER They make no b e a u t i f u l t h i n g s , but they make many c l e v e r ones... I t i s not u n l i k e l y t h a t they have i n v e n t e d some of the machines t h a t have s i n c e t r o u b l e d the w o r l d . . . 8.1 I n t r o d u c t i o n U n t i l r e c e n t l y s p e c t r o m e t e r s of almost a l l d e s c r i p t i o n s i n c l u d e d i n some p a r t of t h e i r works a s l i t or a p e r t u r e f o l l o w e d by a p a r t i c l e d e t e c t o r of some s o r t . These d e t e c t o r s respond o n l y t o the presence or absence of s p e c i f i c k i n d s of p a r t i c l e s i n the a c t i v e r e g i o n of the d e t e c t o r . T h e r e f o r e i n o r d e r to. d e t e c t a spectrum as a f u n c t i o n of some parameter x one must have a d e v i c e t o p h y s i c a l l y d i s p e r s e the p a r t i c l e s i n p r o p o r t i o n t o t h e i r x v a l u e . I f a s l i t i s p l a c e d i n the d i s p e r s i o n p l a n e , then t h i s d e f i n e s a system which i s s e n s i t i v e t o a p a r t i c u l a r v a l u e of the parameter x, and can be scanned through a range 8/The New MCP S p e c t r o m e t e r 223 of x by moving the s l i t . The u n f o r t u n a t e a s p e c t of t h i s system i s t h a t the g r e a t p a r t of the s i g n a l t h a t does not pass t h r o u g h the s l i t i s s i m p l y thrown away. T h i s i s tremendously i n e f f i c i e n t s i n c e the p r o c e s s which i s p r o d u c i n g the p a r t i c l e s i s f o r the most p a r t g e n e r a t i n g unused s i g n a l . Some s p e c t r o m e t e r s have been b u i l t {Moore (1978)} t o make more e f f i c i e n t use of the s i g n a l . These u s u a l l y i n v o l v e u s i n g a r r a y s of d e t e c t o r s and m u l t i p l e s l i t s . However i t r a p i d l y becomes p r o h i b i t i v e i n terms of c o s t and s i g n a l p r o c e s s i n g t o use such methods, and perhaps o n l y one o r d e r of magnitude i n c r e a s e d e f f i c i e n c y can be a t t a i n e d . 8.2 MCPs ( M u l t i c h a n n e l p l a t e a r r a y s ) W i t h the advent of MCP t e c h n o l o g y i t now becomes p o s s i b l e t o have a s l i t l e s s d e t e c t o r and d e t e c t a l l the s i g n a l a l l the t i m e . T h i s d e v i c e can be used t o d e t e c t charged p a r t i c l e s which can be d i s p e r s e d i n a p l a n e as a f u n c t i o n of some parameter x. The reader may get a p i c t u r e of the p h y s i c a l c o n s t r u c t i o n of an MCP by i m a g i n i n g thousands of d r i n k i n g s t r a w s s t a n d i n g on end, packed t i g h t l y s i d e by s i d e , g l u e d 8/The New MCP Spectrometer 224 t o g e t h e r t o form a d i s k or r e c t a n g l e , and shrunk by a f a c t o r of 200. Next t r a n s m u t a t e the straws i n t o the s p e c i a l s e m i c o n d u c t i n g g l a s s used i n c h a n n e l t r o n s and the gl u e t o o r d i n a r y g l a s s and spray b oth f a c e s w i t h n i c k e l or g o l d t o make e l e c t r i c a l c o n t a c t : the r e s u l t i s an MCP. Of c o u r s e these d e v i c e s a r e not made by t h i s f a n c i f u l method, but r a t h e r by s o p h i s t i c a t e d new t e c h n i q u e s of g l a s s drawing and d o p i n g , which a c c o u n t s f o r t h e i r h i g h c o s t . An MCP works by e s s e n t i a l l y the same p r i n c i p l e as a c h a n n e l t r o n ( s e c t i o n 3.2.6). The d i f f e r e n c e i s t h a t an i n c i d e n t p a r t i c l e s t i m u l a t e s o n l y one of the ver y t i n y c h a n n e l s i n the d e v i c e s . The d e v i c e output i s t h e r e f o r e l o c a l i s e d a t the same p l a c e as the i n c i d e n t p a r t i c l e , e x cept t h a t t h e r e a r e now 1 0 3 — 1 0 4 output e l e c t r o n s f o r each i n c i d e n t p a r t i c l e . T h i s i s s t i l l t oo few e l e c t r o n s t o be d e t e c t e d e l e c t r o n i c a l l y , so the u s u a l p r a c t i c e i s t o cascade two MCPs i n s e r i e s , g i v i n g 1 0 6 — 1 0 7 e l e c t r o n s o u t p u t . MCPs were o r i g i n a l l y d e v e l o p e d as image i n t e n s i f i e r s f o r use i n m i l i t a r y n i g h t - v i s i o n b i n o c u l a r s and i n low l i g h t t e l e v i s i o n t u b e s . At f i r s t p r o d u c t i o n was l i m i t e d t o f a i r l y s m a l l 25mm di a m e t e r d i s k s , but w i t h improved t e c h n i q u e s 50 and 70mm diameter d i s k s a r e now a v a i l a b l e . Indeed so u s e f u l a r e t h e s e d e v i c e s t h a t one or two companies a r e m a r k e t i n g MCP a s s e m b l i e s complete w i t h mounting hardware and e l e c t r o n i c s i g n a l p r o c e s s i n g u n i t s . 8/The New MCP Spectrometer 2 2 5 MCPs f i n d a use i n s p e c t r o s c o p y wherever charged p a r t i c l e s can be d i s p e r s e d as a f u n c t i o n of s c a t t e r i n g a n g l e , energy, mass e t c . There are even MCPs now which w i l l d e t e c t photons of s u f f i c i e n t energy (UV or g r e a t e r ) . The most common use t o date of these d e v i c e s i n r e s e a r c h i s i n e l e c t r o n s p e c t r o m e t e r s . An MCP i s p l a c e d i n the d i s p e r s i o n p l a n e of the e l e c t r o s t a t i c a n a l y s e r , so t h a t e l e c t r o n s of a l a r g e range of e n e r g i e s can be d e t e c t e d c o n c u r r e n t l y . 8.3 P o s i t i o n - s e n s i t i v e S i g n a l C o l l e c t i o n A p a r t from t h e i r c o n s t r u c t i o n , the most d i f f i c u l t problem encountered i n the use of MCPs i s how t o c o n v e r t the p o s i t i o n of the out p u t p u l s e i n t o an e l e c t r o n i c s i g n a l which can be p r o c e s s e d t o r e v e a l the spectrum. There a r e p r o b a b l y almost as many ways of d o i n g t h i s as t h e r e a r e s p e c t r o m e t e r s u s i n g the d e v i c e s . Most a p p l i c a t i o n s use one of the f o l l o w i n g methods: (1) M u l t i p l e d i s c r e t e anodes:' a c o l l e c t o r p l a t e d i v i d e d up i n t o s m a l l e l e c t r i c a l l y - s e p a r a t e m e tal anodes i s p l a c e d next t o the output f a c e of the MCP. Each c o l l e c t o r anode i s connected t o i t s own p r e a m p / a m p l i f i e r / d i s c r i m a t o r e l e c t r o n i c s . P o s i t i o n i s decoded t o w i t h i n the a r e a of one anode by the -8/The New MCP Spectrometer 226 presence of a p u l s e on one of the output l i n e s . R e s o l u t i o n i s l i m i t e d t o the s i z e of the anodes; (2) Charge-coupled d i o d e CCD: the output e l e c t r o n shower i s c o n v e r t e d v i a a phosphor s c r e e n t o a b u r s t of photons which a r e o p t i c a l l y f o c u s s e d onto an a r r a y of p h o t o s e n s i t i v e d i o d e s . The d i o d e s b u i l d up a charge a c c o r d i n g to how much l i g h t f a l l s on them over a g i v e n time p e r i o d . At the end of the sample p e r i o d the d i o d e a r r a y i s read out s e r i a l l y i n t o some s o r t of s t o r a g e d e v i c e . R e s o l u t i o n l i m i t e d by the number of elements i n the photodiode a r r a y . C o i n c i d e n c e e x p e r i m e n t s cannot be done w i t h t h i s d e v i c e ; n e i t h e r can i n d i v i d u a l events be a c c e s s e d ; (3) Charge d i v i s i o n : a c o l l e c t o r made of a c o n t i n u o u s r e s i s t i v e s t r i p , or of - d i s c r e t e c o l l e c t o r elements c o n n e c t e d i n s e r i e s w i t h c a p a c i t o r s or r e s i s t o r s i s p l a c e d next to the MCP output f a c e . Leads a r e a t t a c h e d t o the ends of the s t r i p and two charge preamps co n n e c t e d t o t h e s e l e a d s . The output p u l s e l a n d s on the s t r i p and sees two p a r a l l e l p a t h s t o ground. The c u r r e n t down one s i d e of the s t r i p and t h r o u g h each preamp i s i n v e r s e l y p r o p o r t i o n a l t o the impedance of t h a t p a t h t o ground. The preamp output a m p l i t u d e V i s p r o p o r t i o n a l to the t o t a l charge p r e s e n t e d a t the i n p u t . I f the l e n g t h of the s t r i p i s L and the p u l s e l a n d s a d i s t a n c e 1 from one end, then t h a t d i s t a n c e i s o b t a i n e d from the preamp p u l s e a m p l i t u d e s by t a k i n g : 8/The New MCP Spectrometer 227 (8.1) 1 = Lv v+v' T h i s can be done by analogue d i v i d e r c i r c u i t s or by d i g i t i z i n g the p u l s e a m p l i t u d e s and d o i n g the c a l c u l a t i o n w i t h a d i g i t a l p r o c e s s o r . An advantage of t h i s method i s t h a t the r e s o l u t i o n w i t h a r e s i s t i v e s t r i p c o l l e c t o r i s t h e o r e t i c a l l y i n f i n i t e and o n l y l i m i t e d by the s i z e of the p o r e s i n the MCPs, and by the r e s o l u t i o n of the d i g i t i z e r . I f a c a p a c i t o r s t r i n g i s used then the r e s o l u t i o n i s n o r m a l l y l i m i t e d t o the s i z e of the d i s c r e t e e l ements; however, i t i s p o s s i b l e t o o b t a i n h i g h e r r e s o l u t i o n i f the output p u l s e i s a l l o w e d t o d e f o c u s s l i g h t l y and span two elements of the d i s c r e t e anode c h a i n . T h i s tends t o smooth out what would o t h e r w i s e be a s t a i r c a s e - l i k e response f u n c t i o n . Another advantage o f t h i s method over the CCD method i s t h a t the f i n a l computed 1 i s not s e n s i t i v e t o v a r i a t i o n s i n c h a n n e l g a i n a c r o s s the f a c e of the MCP. The a m p l i t u d e of the MCP output p u l s e would appear i n both the .numerator and the denominator of e q u a t i o n 8.1 and so c a n c e l s o u t . I have chosen the t h i r d method, charge d i v i s i o n t h r o u g h a c a p a c i t o r s t r i n g , f o r i m p l e m e n t a t i o n i n the new s p e c t r o m e t e r . T h i s i s a proven t e c h n i q u e {Van Hoof (1980)} and seemed t o be the one which was both s i m p l e i n concept and f a i r l y cheap i n e x e c u t i o n . I t i s a l s o the one which can 8/The New MCP Spectrometer 228 F i g u r e 8.1(a) E x p l o d e d view of the MCP assembly, (b) Schematic diagram of the MCP c i r c u i t . 8/The New MCP Spectrometer 229 F i g u r e 8.2 Schematic diagram ( t o s c a l e ) of the new b i n a r y (e,2e) s p e c t r o m e t e r . 8/The New MCP Spectrometer 230 F i g u r e 8.3 Schematic diagram of the vacuum support system f o r the new b i n a r y (e,2e) s p e c t r o m e t e r . Legend: IG i o n i z a t i o n gauge SV s o l e n o i d v a l v e W vent v a l v e TC thermocouple gauge GV gate v a l v e WB water b a f f l e 8/The New MCP Spectrometer 231 sna- 0 a. o cr LU • k— HI g cr LS111/03 ANALOG-DIGITAL CONVER1 SCALER 8^ LU 3 LT CJ DIGITAL 1 DIGITAL-ANALOG CONVER1 F i g u r e 8.4 B l o c k diagram of the s i g n a l - p r o c e s s i n g and computer e l e c t r o n i c s f o r the MCP o u t p u t s . 8/The New MCP Spectrometer 2 3 2 0 100 200ns 1 -J 1 1 t}-L 2 _ l _ 3 4 5 6 100 200 300 us -J 1 leH i • At CPi,2 CP3.4 G S A SH G S A SH 1.2 1.2 3.4 3,4 T R U E START TRUE STOP INHIBIT/I R E S E T f TAC S H 5 HOLD1 HOLD2 HOLD3 r delay 1 delay 2 delay 3 cpu processing time F i g u r e 8.5 Timin g diagram f o r the s i g n a l s i n the p r o c e s s i n g c i r c u i t s . 8/The New MCP Spectrometer 233 2JZ CD 2ZLI c: : 0 cn Q . O O CD jo a CD CO CO . a collector 2 - N V W -3. a O LD "1 O - b L«jk "Mi- V CO a £ CO O H i ' F i g u r e 8.6 I n i t i a l t e s t c i r c u i t f o r the MCP. 8/The New MCP Spectrometer T a b l e 8.1 MCP i n t e r f a c e c o n t r o l l i n e s 234 D e v i c e B i t Name/Function DRV1 1 DRVCSRO DRVCSR1 DRVCSR6 DRVCSR7 DRVCSR2-5,8-15 DRVINO DRVIN1 DRVIN2-15 DRVOUTO DRVOUT1 DRVOUT2,3 DRVOUT4-15 ADV11-A ADVCSRO ADVCSR6 ADVCSR7 ADVCSR8-11 ADVCSR14 ADVCSR15 ADVCSR1-5,12,13 ADVBUFO-11 AAV11-A DACAO-15 DACB,C0-15 DACDO-15 KWV11-A KWVCSRO KWVCSR1,2 KWVCSR3-5 KWVCSR6 KWVCSR7 KWVCSR8-15 KWVBUFO-15 P a r a l l e l 1 6 - b i t d i g i t a l I/O MCP i n t e r f a c e i n h i b i t / r e s e t BNC9010 r e m o t e / l o c a l E n a b l e i n t e r r u p t s E x t e r n a l event i n t e r r u p t f l a g (TAC TRUE STOP) not used SCA1 event SCA2 event not used BNC9010 pulse/DC X-Y p l o t t e r pen up/down not used BNC9010 d i g i t a l a m p l i t u d e word (BCD) 16-channel 1 2 - b i t m u l t i p l e x e d a n a l o g / d i g i t a l c o n v e r t e r Go b i t - s t a r t c o n v e r s i o n E n a b l e i n t e r r u p t on c o n v e r s i o n c o m p l e t i o n C o n v e r s i o n complete Channel s e l e c t 0-1 7 B E n a b l e e r r o r i n t e r r u p t E r r o r f l a g n o t . u s e d 1 2 - b i t s i g n e d c o n v e r s i o n r e s u l t 4-channel 1 2 - b i t d i g i t a l t o a n a l o g c o n v e r t e r R e f e r e n c e v o l t a g e f o r i n c i d e n t energy E 0 power s u p p l y X and Y d e f l e c t i o n s f o r scope d i s p l a y R e f e r e n c e v o l t a g e f o r a n g l e s c a n n i n g s e r v o l o o p R e a l - t i m e c l o c k Go b i t - s t a r t t i m e r Mode C l o c k r a t e I n t e r r u p t on o v e r f l o w C l o c k o v e r f l o w not used C l o c k count 8/The New MCP S p e c t r o m e t e r 235 e a s i l y be combined w i t h the n e c e s s a r y c o i n c i d e n c e d e t e c t i o n c i r c u i t r y . 8.4 The New Spectrometer D e s i g n The d e s i g n r e q u i r e m e n t s f o r the new s p e c t r o m e t e r , the vacuum system, the e l e c t r o n i c s and the computer system a r e g i v e n i n the f o l l o w i n g s e c t i o n s . 8.4.1 The vacuum system T r i p l e d i f f e r e n t i a l pumping was a p r i m a r y requirement f o r the new machine. The reasons a r e : (1) The s e n s i t i v i t y of the MCP t o n o x i o u s gases which can degrade the s p e c i a l s e m i c o n d u c t i n g g l a s s , and t o h i g h p r e s s u r e s of any gas which can l e a d t o a r c - o v e r s t h r o u g h the v e r y f i n e c h a n n e l s ; (2) I f r e a c t i v e gases a r e e x c l u d e d from the r e g i o n of the e l e c t r o n gun then o x i d e cathodes can be used i n s t e a d of t u n g s t e n f i l a m e n t s . Oxide c a t h o d e s a r e t h i n l a y e r s of barium o x i d e s on a m e t a l c o n t a c t , i n d i r e c t l y h e a t e d . These e l e c t r o n e m i t t e r s run c o o l e r than t u n g s t e n f i l a m e n t s , and so the energy s p r e a d of the r e s u l t a n t e l e c t r o n beam i s l e s s . Where a t u n g s t e n f i l a m e n t might have a AE of 0.7-0.9eV o x i d e cathodes have 8/The New MCP Spectrometer 236 0.3-0.5eV. D i f f e r e n t i a l pumping r e f e r s t o the t e c h n i q u e of d i v i d i n g a vacuum chamber i n t o s e v e r a l compartments w i t h o n l y a s m a l l a p e r t u r e c o n n e c t i n g them. I f a pumping s t a c k i s a t t a c h e d t o each compartment then one can m a i n t a i n p r e s s u r e d i f f e r e n t i a l s a c r o s s the a p e r t u r e s of up t o an o r d e r of magnitude. T h i s a l l o w s t h r e e compartments: one w i t h the gas a l l a t r e l a t i v e h i g h p r e s s u r e , and the o t h e r two w i t h the MCP and e l e c t r o n gun a t low p r e s s u r e . The gas chamber and the a n a l y s e r chamber both have 5 - i n c h pumping t h r o a t s l e a d i n g t o V a r i a n VHS-4 d i f f u s i o n pumps ( t h e same type as on the o l d machine). The gun chamber has a 2 1 / 2 - i n c h tube l e a d i n g t o a s m a l l 3 - i n c h d i f f u s i o n pump. A l l t h r e e pumping s t a c k s have gate v a l v e s and water t r a p s . L N 2 c o l d t r a p s , a re not n e c e s s a r y i n the d i f f u s i o n pump s t a c k s . A y 2 - i n c h gate v a l v e between the gun chamber and the gas chamber a l l o w s a c c e s s t o the e l e c t r o n gun, w h i l e s t i l l k e e p i n g the r e s t of the system pumped down. The d i f f u s i o n pumps a r e backed by two l a r g e r o t a r y pumps: one f o r the a n a l y s e r and gun pumps, and one f o r the gas chamber pump. Four p o r t s a r e p r o v i d e d f o r i o n gauges: one on each pumping s t a c k and one on t o p of the chamber. 8/The New MCP Spectrometer 237 8.4.2 The e l e c t r o n gun The same type of gun body i s used i n the new machine as i n the o l d ( C l i f t r o n i c s CE5AH), except t h a t a mounting f o r o x i d e c a thodes has been s u b s t i t u t e d f o r t u n g s t e n f i l a m e n t s . The beams from t h i s gun are not as good as the o t h e r i n the o l d machine (10-30»IA, 90 p er cen t f o c u s s e d ) but t h i s i s p a r t l y due t o the l o n g e r d i s t a n c e the beam must t r a v e l , and the f a c t t h a t i t must pass t h r o u g h a 1 / 2 - i n c h gate v a l v e . 8.4.3 The beam s t e e r i n g u n i t T h i s u n i t i s s i m i l a r i n a l l r e s p e c t s t o the one i n the o l d machine, except t h a t an e x t r a s e t of d e f l e c t o r s has been i n s t a l l e d between the gun and the gate v a l v e . The f i r s t s p r a y p l a t e a f t e r the gate v a l v e d e f i n e s the a p e r t u r e f o r d i f f e r e n t i a l pumping between the gas chamber and the gun chamber. The l a s t spray p l a t e and the Faraday cup are mounted i n the l e n s . 8.4.4 The gas c e l l A g a i n , s i m i l a r i n most r e s p e c t s t o the one i n the o l d machine, the new gas c e l l i s made i n two p a r t s : the lower h a l f i s f i x e d t o the t o p of the beam s t e e r i n g u n i t , and the upper h a l f t o the l e n s . The gap between the two h a l v e s a l l o w s the s c a t t e r e d e l e c t r o n s t o pass i n t o the l e n s . The gas c e l l d e f i n e s a f i e l d f r e e s c a t t e r i n g r e g i o n , and p e r m i t s 8/The New MCP Spectrometer 238 a l o c a l b u i l d u p of gas d e n s i t y . 8.4.5 The c y l i n d r i c a l m i r r o r a n a l y s e r In t h i s s p e c t r o m e t e r the a n a l y s e r i s a f u l l CMA mounted c o a x i a l l y w i t h the gas c e l l and the c o n i c a l l e n s . T h i s a n a l y s e r has almost p e r f e c t c y l i n d r i c a l symmetry, so a l l s c a t t e r e d e l e c t r o n s emerging from the c o l l i s i o n volume a t e=45° and w i t h energy E=E{retard}+E{pass} w i l l pass through the e x i t s l i t of the a n a l y s e r r e g a r d l e s s of a z i m u t h a l a n g l e . (The o n l y l a p s e s from p e r f e c t c y l i n d r i c a l symmetry a r e i n some s m a l l support columns i n the i n n e r c y l i n d e r , and pumping s l o t s c u t i n the i n n e r and o u t e r c y l i n d e r s . These pumping s l o t s have been c o v e r e d w i t h a 99 per cent t r a n s p a r e n t copper mesh t o p r e s e r v e the e l e c t r i c f i e l d . ) T h i s means t h a t (e,2e) e l e c t r o n s of a l l a z i m u t h a l a n g l e s w i l l be t r a n s m i t t e d t o the d e t e c t o r , and t h a t a l l . o r i e n t a t i o n s of a g i v e n a z i m u t h a l a n g l e w i l l a l s o be t r a n s m i t t e d . T h i s g i v e s an upper l i m i t t o - t h e improvement i n e f f i c i e n c y over the o l d machine of a f a c t o r of about 500 (assuming an ac c e p t a n c e h a l f - a n g l e f o r the o l d s p e c t r o m e t e r l e n s e s {Hood (1977)} of about 4°). Such an i n c r e a s e i n the data r a t e means t h a t d a t a a c q u i s i t i o n can be c u t from a p e r i o d of days down t o h o u r s . The s c a t t e r e d e l e c t r o n t r a j e c t o r i e s t h r ough the l e n s and a n a l y s e r d e s c r i b e a s u r f a c e which i s s i m i l a r t o t h a t of a f o o t b a l l ( A m e r i c a n - s t y l e ) . T h i s image may be h e l p f u l i n v i s u a l i z i n g 8/The New MCP Spectrometer 239 the w o r k i n g s of t h i s s p e c t r o m e t e r . I t i s not p o s s i b l e t o a c h i e v e even p e r f e c t f i r s t o r d e r f o c u s s i n g ( l e t a l o n e second o r d e r f o c u s s i n g ) due t o v a r i o u s p h y s i c a l c o n s t r a i n t s i n the d e s i g n of the s p e c t r o m e t e r ; however, as i s e x p l a i n e d i n the next s e c t i o n , good f o c u s s i n g a t t r i b u t e s a r e not n e c e s s a r i l y a h i g h p r i o r i t y i n t h i s a p p l i c a t i o n . S i n c e the a n a l y s e r e n t r a n c e a n g l e must be 45° (the same as the e s c a t t e r i n g a n g l e ) , t h i s r e q u i r e s t h a t the sum of the r a d i a l components of the d i s t a n c e s from the source and image f o c i t o the i n n e r c y l i n d e r be 3.4, and the a x i a l component of the d i s t a n c e between the f o c i be 9.2, both i n u n i t s of the i n n e r c y l i n d e r r a d i u s { R i s l e y (1972)}. Due t o the s i z e of the c o n i c a l l e n s and the MCP d e t e c t o r assembly, the minimum r a d i u s of the i n n e r c y l i n d e r i s 2.5 i n c h e s , which means t h a t the h e i g h t of the a n a l y s e r would have t o be at l e a s t 20 i n c h e s , and the r a d i u s of the o u t e r c y l i n d e r about 25 i n c h e s , which i s too l a r g e f o r our machine shops t o h a n d l e . E l e c t r o n s of the c o r r e c t energy a r e t r a n s m i t t e d from an a n n u l a r e n t r a n c e s l i t 2cm i n s i d e the i n n e r c y l i n d e r t o an a n n u l a r e x i t s l i t 10cm above the e n t r a n c e s l i t and 3cm i n s i d e the i n n e r c y l i n d e r . The r a d i u s of the i n n e r c y l i n d e r i s 2.5 i n c h e s , and the o u t e r c y l i n d e r 5 i n c h e s . These p r o p o r t i o n s f o r the CMA mean t h a t i t w i l l have a p p r o x i m a t e l y 0.5 per cent r e s o l u t i o n , i . e . AE{pass}/E{pass}=0.005 { R i s l e y (1972)}; s i n c e the MCP d e t e c t o r i s expected t o 8/The New MCP Spectrometer 240 improve the d a t a r a t e so much, t h e r e i s no reason not t o t r y f o r b e t t e r energy r e s o l u t i o n . The d e s i g n of the CMA means t h a t the p a r t i c l e beam w i l l pass a p p r o x i m a t e l y h a l f way between the elements of the CMA a t the p o i n t of l a r g e s t r a d i u s : t h i s h e l p s t o make the e l e c t r o n s i n s e n s i t i v e t o s u r f a c e e f f e c t s near the a n a l y s e r elements. The c h o i c e of the p o s i t i o n of the out p u t s l i t i s i n t e n d e d t o a l l o w . t h e a n a l y s e d e l e c t r o n s t o d i v e r g e s l i g h t l y b e f o r e s t r i k i n g the MCP. T h i s ensures t h a t the count r a t e per u n i t area of the MCP w i l l not exceed 1 count per ch a n n e l per second. Above t h i s r a t e the d e v i c e s t a r t s t o s a t u r a t e . 8.4.6 The c o n i c a l l e n s There i s n o t h i n g i n the l i t e r a t u r e on the d e s i g n of an e l e c t r o s t a t i c l e n s t o f o c u s an o b j e c t from a p o i n t on the beam a x i s t o an a n n u l a r image around the beam a x i s , where image and o b j e c t d e f i n e a 45° cone. L a c k i n g the s k i l l s t o c a l c u l a t e the s t r u c t u r e of such a b e a s t , and f o r want of any b e t t e r i d e a s , i t was d e c i d e d t o b u i l d a t h r e e - e l e m e n t c o n i c a l l e n s i n the same p r o p o r t i o n s as the o l d s p e c t r o m e t e r l e n s e s . T h i s d e v i c e would have some s o r t of t r a n s p o r t p r o p e r t i e s ( c e r t a i n l y not the same as the o l d l e n s e s ) which c o u l d be o p t i m i z e d by s u i t a b l y a d j u s t i n g the v o l t a g e a p p l i e d t o the mi d d l e element. I t was not un d e r s t o o d u n t i l a f t e r i t was b u i l t , however, t h a t t h i s s o r t of l e n s would a l s o d e f l e c t the s c a t t e r e d e l e c t r o n s i n t o a s l i g h t l y s t e e p e r or 8/The New MCP Spectrometer 241 s h a l l o w e r cone. C o n s i d e r : the e l e c t r i c f i e l d g r a d i e n t a t the l e n s gaps w i l l not be the same i n the t o p and bottom h a l v e s of the l e n s assembly (see f i g u r e 8.1) because the volume elements near the upper and lower gaps a r e d i f f e r e n t ; the r a d i a l d i s t a n c e s and p o l a r a n g l e s from the c e n t r a l a x i s a r e s l i g h t l y d i f f e r e n t . T h i s i m p l i e s t h a t the a c t i o n of the r e s u l t i n g e l e c t r i c f i e l d w i l l not be t h a t of a pure l e n s (as i n t h r e e element t u b u l a r l e n s e s ) but w i l l a l s o behave p a r t l y l i k e a p r i s m , d e f l e c t i n g the s c a t t e r e d e l e c t r o n s away from t h e i r o r i g i n a l t r a j e c t o r y i n the 9 d i m e n s i o n . In o r d e r t o compensate f o r t h i s , the upper and lower p a r t s of the m i d d l e l e n s element have been w i r e d i n t o a c i r c u i t s i m i l a r t o tho s e s u p p l y i n g the beam d e f l e c t o r v o l t a g e s ; t h e r e i s one p o t e n t i o m e t e r t o s e t the p o t e n t i a l midway between the two elements, and another t o s e t a p o s i t i v e and n e g a t i v e o f f s e t w i t h r e s p e c t t o the c e n t e r p o t e n t i a l , which i s t o be a p p l i e d d i r e c t l y t o the elements. In t h i s way i t s h o u l d be p o s s i b l e t o c o r r e c t f o r d e f l e c t i o n i n t r o d u c e d by the l e n s . In my view the be s t s o l u t i o n t o the l e n s problem (see a l s o s e c t i o n 3.5) f o r t h i s and o t h e r (e,2e) s p e c t r o m e t e r s would be t o use a s p h e r i c a l r e t a r d i n g f i e l d f o r the s c a t t e r e d e l e c t r o n s , w i t h no l e n s m a g n i f i c a t i o n e f f e c t or p o l a r a n g l e d e f l e c t i o n . Such a f i e l d c o u l d be e s t a b l i s h e d w i t h c o n c e n t r i c s h e l l s or g r i d s . When t h e r e a r e no e l e c t r o n o p t i c a l e f f e c t s the a n g u l a r a c c e p t a n c e and the e f f e c t i v e 9 a n g l e would be d e f i n e d by the p h y s i c a l c o n s t r u c t i o n a l o n e , 8/The New MCP Spectrometer 242 w i t h o u t h a v i n g t o worry about l e n s m a g n i f i c a t i o n or f o c u s s i n g e f f e c t s . ( I t would be imp o r t a n t t o reduce magnetic f i e l d s i n such a d e v i c e , i n orde r t o be a b l e t o do w i t h o u t d e f l e c t o r s i n the l e n s . ) R e t a r d i n g l e n s e s s e r v e a purpose which i s , i n f a c t , p a r t l y c o n t r a d i c t o r y t o our i n t e n t i n b i n a r y (e,2e) s p e c t r o s c o p y . In o t h e r t y p e s of s p e c t r o s c o p y l e n s e s were i n t e n d e d t o r e t a r d s c a t t e r e d e l e c t r o n s from t h e i r s c a t t e r e d energy down t o a lower energy where they c o u l d then be a n a l y s e d w i t h h i g h e r r e s o l u t i o n . They a l s o e n a b l e d one t o focu s s c a t t e r e d e l e c t r o n s from a l a r g e volume of space on t o the s m a l l e n t r a n c e a p e r t u r e of the l e n s ; the l a r g e r the c o l l i s i o n volume, the g r e a t e r the s i g n a l r a t e (gas and beam d e n s i t y b e i n g e q u a l ) which i s almost always a d e s i r a b l e t h i n g . In such t y p e s of s p e c t r o s c o p y i t i s u s u a l l y i n c o n s e q u e n t i a l t h a t t h i s l a r g e c o l l i s i o n volume i m p l i e s i n d e t e r m i n a t e s c a t t e r i n g a n g l e s , as a n g u l a r r e s o l u t i o n was not a p r i o r i t y . However, i n b i n a r y (e,2e) i t i s v e r y i m p o r t a n t t h a t the p o l a r s c a t t e r i n g a n g l e be s h a r p l y r e s t r i c t e d around ©=45° i n o r d e r t o a c c u r a t e l y determine q at low a z i m u t h a l s c a t t e r i n g a n g l e s (see f i g u r e 3.7), and i t i s a l s o i m p o r t a n t t o have good <f> r e s o l u t i o n . T h i s i s why one r e a l l y o n l y needs a r e t a r d i n g d e v i c e t o improve energy r e s o l u t i o n , and not a f o c u s s i n g d e v i c e w i t h i t s i n h e r e n t m a g n i f i c a t i o n of the image. In t h i s r e s p e c t the U n i v e r s i t y of M a r y l a n d s p e c t r o m e t e r d e s i g n {Moore (1978)} i s an 8/The New MCP S p e c t r o m e t e r 243 improvement, as i t has no l e n s e s . However, n e i t h e r does i t have a r e t a r d i n g d e v i c e , so energy r e s o l u t i o n may be l i m i t e d . 8.5 The New B i n a r y (e,2e) MCP D e t e c t o r 8.5.1 C o n s t r u c t i o n The MCP assembly d e s i g n i s shown i n e x p l o d e d view i n f i g u r e 8.1. Two 52cm MCPs ( M u l l a r d G25-501A) a r e used i n c hevron f o r m a t i o n , spaced 0.010 i n c h e s a p a r t , f o l l o w e d by the c o l l e c t o r spaced 0.010 i n c h e s from the output f a c e of the second p l a t e . The m i c r o c h a n n e l s a r e not e x a c t l y p e r p e n d i c u l a r t o the f a c e of the p l a t e , but a r e c a n t e d s l i g h t l y . Chevron f o r m a t i o n means t h a t the two p l a t e s a r e mounted so t h a t the c h a n n e l s of one p l a t e s l o p e i n the o p p o s i t e d i r e c t i o n t o the o t h e r . T h i s a t t e m p t s t o p r e v e n t secondary i o n feedback, i n i m i t a t i o n of the c u r v a t u r e of c h a n n e l t r o n s . E l e c t r i c a l c o n t a c t s t o the p l a t e s a r e made w i t h s i l v e r r i n g s , and i n s u l a t i o n i s done w i t h n y l o n or T e f l o n . The whole assembly i s g e n t l y sandwiched t o g e t h e r and s e c u r e d w i t h a s p r i n g c l i p . The d e t e c t o r must be e l e c t r i c a l l y d i v i d e d i n two i n o r d e r t o p e r f o r m the c o i n c i d e n c e d e t e r m i n a t i o n between the 8/The New MCP Spectrometer 244 s c a t t e r e d e l e c t r o n s . One s c a t t e r e d e l e c t r o n s t r i k e s one h a l f of the d e t e c t o r and the second s t r i k e s the o t h e r s i d e . With a f i n e s a n d b l a s t e r , the n i c k e l p l a t i n g of the output f a c e of the second p l a t e was c a r e f u l l y removed a l o n g a di a m e t e r t o e l e c t r i c a l l y s e p a r a t e the two s i d e s . C o n t a c t s T, and T 2 are made t o each h a l f : when an e l e c t r o n t r i g g e r s the d e t e c t o r t h i s produces as a momentary d e p l e t i o n of charge a t one of the n i c k e l c o n t a c t s and hence a p o s i t i v e p u l s e i n one of the T, or T 2 l i n e s . The c o l l e c t o r i s t o be a r i n g of d i s c r e t e anodes ( f i g u r e 8.1) d e s i g n e d t o l i n e up w i t h the i n t e r s e c t i o n of the t r a j e c t o r i e s of the e l e c t r o n s l e a v i n g the a n a l y s e r w i t h the i n p u t f a c e of the MCP d e t e c t o r . The anodes aire connected w i t h c a p a c i t o r s i n s e r i e s t o form two h a l f - c i r c l e s t r i n g s . C o n n e c t i o n s C 1,C 2,C 3, and C„ a r e made t o the f o u r ends. The two c o l l e c t o r s t r i n g s a r e a l s o a l i g n e d p h y s i c a l l y w i t h the two h a l v e s of the MCP output f a c e . Each c a p a c i t o r , has a r e s i s t o r w i r e d i n p a r a l l e l t o s l o w l y d r a i n away accumulated c h a r g e . The v a l u e s of the c a p a c i t o r s a r e 100 pF and the r e s i s t o r s 10 6 ohms, g i v i n g a time c o n s t a n t of RC/n=2.5ms w i t h n=40 d i s c r e t e anodes per s t r i p . T h i s i s w e l l i n excess of the i n i t i a l p r o c e s s i n g time of the c o l l e c t o r p u l s e s , so no d e g r a d a t i o n of the s i g n a l s h o u l d o c c u r . The c o l l e c t o r i s t o be made from a s u i t a b l y e t c h e d , t w o - s i d e d f i b e r g l a s s p r i n t e d c i r c u i t b o a r d . The d i s c r e t e anodes would be on the MCP s i d e of the PC board, and the 8/The New MCP Spectrometer 245 r e s i s t o r and c a p a c i t o r s on the o t h e r . Sundry components f o r p r o v i d i n g the MCP h i g h v o l t a g e b i a s , and f o r d e c o u p l i n g the-t i m i n g and c o l l e c t o r s i g n a l s w i l l a l s o be mounted h e r e . 8.5.2 S i g n a l p r o c e s s i n g The MCP assembly produces s i x p u l s e s when t h e r e i s a c o i n c i d e n c e : T 1 f T 2, C 1 f C 2, C 3 and C„. The T, and T 2 l i n e s c a r r y s h a r p p o s i t i v e p u l s e s f o r the c o i n c i d e n c e d e t e r m i n a t i o n which i s done u s i n g the same TAC/SCA method as i n the o l d machine (see s e c t i o n 3.4.3). L i n e s C ^ - C ^ c a r r y an amount of charge which depends on where the two e l e c t r o n s l a n d . I f one can d i g i t i z e the amount of these c h a r g e s then the a z i m u t h a l a n g l e between the two e l e c t r o n s can be computed a s : T h i s r e s u l t can be used as a p o i n t e r t o a s t o r a g e l o c a t i o n . The a c c umulated count a t t h a t l o c a t i o n i s then incremented or decremented a c c o r d i n g t o whether the c o i n c i d e n c e d e t e r m i n a t i o n has shown a t r u e c o i n c i d e n c e or a random p u l s e p a i r . In t h i s way a spectrum of the a z i m u t h a l d i s t r i b u t i o n of s i g n a l i s o b t a i n e d . I t w i l l be n e c e s s a r y t o m u l t i p l y t h i s spectrum by a ramp of the form: (8.2) t> a + c 3 c 3+ c „ C i + C 2 (8.3) 180° 180°-* 8/The New MCP Spectrometer 246 s i n c e t h e r e i s the p o s s i b i l i t y which i n c r e a s e s w i t h <t> t h a t the two (e,2e) e l e c t r o n s might both l a n d on the same h a l f of the MCP d e t e c t o r , and so would not be d e t e c t e d . The s i g n a l p r o c e s s i n g f o r the Q^-CB, o u t p u t s i s t o c o n s i s t of f o u r s e t s ORTEC 142AH charge preamps (CP) f o l l o w e d by ORTEC 485 G a u s s i a n Shaping A m p l i f i e r s (GSA). A charge p u l s e a p p e a r i n g a t the CP i n p u t i s c a p a c i t a t i v e l y d e c o u p l e d from the h i g h v o l t a g e l i n e and i n t e g r a t e d . The CP out p u t i s a p u l s e w i t h a r i s e time of the w i d t h of the i n p u t charge p u l s e and a f a l l time of about 500»iS. Data r a t e s of more than (500»«s)" 1 = 2000s _ 1 a r e not a problem s i n c e CP out p u t p u l s e s may p i l e up w i t h no l o s s of s i g n a l . The GSAs c o n t a i n s p e c i a l s h a p i n g networks which respond t o s t e p s i n the i n p u t v o l t a g e , such as are p r o v i d e d by the CPs. The GSA output i s a smooth, u n i f o r m l y shaped semi-Gaussian p u l s e 3»#s wide whose a m p l i t u d e i s d i r e c t l y p r o p o r t i o n a l t o the s i z e of the i n p u t s t e p . The use of the GSA means t h a t the system s h o u l d be i n s e n s i t i v e t o any change i n the preamp output r i s e time due t o cha n g i n g w i d t h s of the charge p u l s e , or d i s t o r t i o n of the charge p u l s e as i t t r a v e l s down the c a p a c i t o r c h a i n , and the shape and u n i f o r m i t y of the output p u l s e a l l o w i t t o be p r e c i s e l y d i g i t i z e d . 8/The New MCP Spectrometer 247 8.5.3 MCP/computer i n t e r f a c e The. i n t e r f a c i n g of the s i x . s i g n a l l i n e s p l u s many d a t a and c o n t r o l l i n e s between the s p e c t r o m e t e r and the computer i s done w i t h a custom b u i l t u n i t (MCP I n t e r f a c e ) d e s i g n e d and c o n s t r u c t e d i n the UBC shops t o my s p e c i f i c a t i o n s . A b l o c k diagram f o r the i n t e r f a c e i s g i v e n i n f i g u r e 8.4. The MCP I n t e r f a c e d a t a a c q u i s i t i o n l o g i c i s i d l e u n t i l i t i s 'woken up' by the presence of a h i g h l o g i c l e v e l s i m u l t a n e o u s l y i n the TAC TRUE START and TRUE STOP o u t p u t s . These two s i g n a l s i n d i c a t e t h a t the TAC has d e t e c t e d a c o i n c i d e n c e on the T, and T 2 l i n e s , and t h a t the computer may expect v a l i d s i g n a l s on the C ^ C , l i n e s . The i n t e r f a c e then does s e v e r a l t h i n g s : (1) An i n t e r r u p t f l a g i s s e t i n one of the computer's d i g i t a l I/O r e g i s t e r s . T h i s f l a g i n t e r r u p t s the CPU and t r a n s f e r s c o n t r o l t o a s o f t w a r e r o u t i n e t o p r o c e s s the (e,2e) e v e n t ; (2) The TAC i s i n h i b i t e d from p r o c e s s i n g any more START and STOP s i g n a l s u n t i l the i n t e r f a c e i s r e s e t by the computer; (3) An i n t e r n a l ' v a l i d event' l o g i c l e v e l i s s e t h i g h , which a l l o w s the f i v e s ample/hold u n i t s t o sample the TAC and GSA output p u l s e s when t h r e e i n t e r n a l t i m e r s i n d i c a t e t h e s e p u l s e s have reached t h e i r peak. The i n t e r f a c e l o g i c must s u p p l y a ' h o l d ' s i g n a l t o the 8/The New MCP Spectrometer 248 sample and h o l d (SH) c i r c u i t s a t the p r e c i s e i n s t a n t t h a t the TAC and 485 p u l s e s r e a c h t h e i r peak. N o r m a l l y the SH output ' t r a c k s ' the i n p u t , but when a ' h o l d ' s i g n a l i s r e c e i v e d , the output i s h e l d c o n s t a n t a t the v a l u e of the i n p u t a t t h a t i n s t a n t (see f i g u r e s 8.5 and 8.6). The TAC TRUE START output l e a d i n g edge i s used t o s t a r t an i n t e r n a l c l o c k K, , which measures out a f i x e d time i n t e r v a l . T h i s i n t e r v a l can be a d j u s t e d t o span the time the C, and C 2 GSA output p u l s e s t a k e t o r i s e t o t h e i r peaks. At the end of t h i s time a 'h o l d ' l o g i c l i n e i s s e t h i g h , and i f the ' v a l i d e vent' l i n e i s a l s o h i g h the SH c i r c u i t s SH, and SH 2 are s w i t c h e d from ' t r a c k ' t o ' h o l d ' . S i m i l a r l y , i n t e r n a l c l o c k K 2 t i m e s from the TRUE STOP l e a d i n g edge t o the peak of the C 3 and C 4 GSA o u t p u t s , p r o v i d i n g the ' h o l d ' s i g n a l f o r SH c i r c u i t s SH 3 and SH„. The ' h o l d ' s i g n a l f o r SH 5 (the TAC output SH c i r c u i t ) i s a l s o t i med by K 3 from the TRUE STOP l e a d i n g edge. The reason f o r h a v i n g s e v e r a l t i m e r s i s t h a t the T, and T 2 p u l s e s a r e s e p a r a t e d i n time by an amount At from z e r o up t o the range of the TAC, which may be s e v e r a l hundred nanoseconds. T h i s i s enough time f o r the GSA output t o be s i g n i f i c a n t l y below i t s peak v a l u e . The TAC output p u l s e of c o u r s e has a d i f f e r e n t o f f s e t from T, and T 2 than the GSA o u t p u t s , so i t must have i t s own t i m e r . The o u t p u t s of the f i v e SH u n i t s a r e co n n e c t e d t o a 16-channel, m u l t i p l e x e d , 1 2 - b i t v o l t a g e d i g i t i z e r board i n the computer, which would be c o n t r o l l e d by a s o f t w a r e 8/The New MCP S pectrometer 249 i n t e r r u p t s e r v i c e r o u t i n e . The SCA o u t p u t p u l s e s l a t c h e d by two f l i p - f l o p s i n the i n t e r f a c e and the r e s u l t i n g l o g i c l e v e l s are c o n n e c t e d t o two i n p u t l i n e s of a 1 6 - b i t d i g i t a l I/O board i n the computer. The e n t i r e " l o g i c system i n the MCP I n t e r f a c e can be r e s e t by c l e a r i n g a s p e c i a l b i t i n the d i g i t a l I/O b o a r d , and can then be e n a b l e d by a s s e r t i n g t h i s b i t r . The l o g i c system i s r e s e t and e n a b l e d a f t e r every i n t e r r u p t . In a d d i t i o n t o s i g n a l and d a t a l i n e s t h e r e a r e c o n t r o l l i n e s f o r s e t t i n g the v o l t a g e of the i n c i d e n t energy power s u p p l y , f o r s e t t i n g the a n g l e (of the a n g l e scan system i n the o l d machine) and f o r p r o d u c i n g a d i s p l a y of s p e c t r a on an X-Y o s c i l l o s c o p e . These f o u r v o l t a g e s a r e s u p p l i e d by a f o u r - c h a n n e l d i g i t a l - t o - a n a l o g c o n v e r t e r (DAC) from 1 2 - b i t numbers l o a d e d i n t o the DAC r e g i s t e r s by the computer. The c h a n n e l advance and r e s e t f u n c t i o n s of the o l d ORTEC 4610 Program C o n t r o l U n i t are emulated i n the computer-based system i n s o f t w a r e , u s i n g a r e a l - t i m e c l o c k d e v i c e . A f i x e d time base w i l l be a v a i l a b l e , and a l s o a f e a t u r e a l l o w i n g the o p e r a t o r t o t i e the d w e l l time per c h a n n e l t o the t o t a l s i n g l e s r a t e i n the two T, and T 2 c h a n n e l s . Sundry o t h e r output l i n e s on the d i g i t a l I/O board 8/The New MCP Spectrometer 250 i n c l u d e a p l o t t e r pen up/down b i t and s e v e r a l l i n e s t o c o n t r o l the output a m p l i t u d e of a remote-programmable p u l s e g e n e r a t o r ( B e r k e l e y N u c l e o n i c s 9010). Such a p u l s e g e n e r a t o r c o u l d be used (among i t s myriad o t h e r uses) t o do c o m p u t e r i z e d s e l f - t e s t and s e l f - c a i i b r a t i o n of the MCP p o s i t i o n d e c o d i n g system. There i s a l s o unused space i n the d i g i t a l I/O board which c o u l d be used t o read i n a number from a d i g i t a l v o l t m e t e r and so p r o v i d e s e l f - c a i i b r a t i o n of the i n c i d e n t energy power s u p p l y . The f u n c t i o n of a l l the l i n e s i n the computer i n t e r f a c i n g i s summarized i n Tab l e 8.1. The MCP i n t e r f a c e has been i n o p e r a t i o n s a t i s f a c t o r i l y on both s p e c t r o m e t e r s u s i n g a l l i t s f u n c t i o n s except the C,-C4 d i g i t i z i n g l i n e s . 8.5.4 The computer system The computer system i s based on a PDP LS111/03 p r o c e s s o r manufactured by the D i g i t a l Corp. The system p e r i p h e r a l s i n c l u d e the DEC VT100 v i d e o t e r m i n a l , the MDB DSD-440 d u a l f l o p p y d i s k e t t e d r i v e mass-memory, and an IDS-460 g r a p h i c s impact p r i n t e r and a l l t he i n t e r f a c e boards d e s c r i b e d above. The system was purchased from the T r a n s d u c t i o n Co. ( T o r o n t o ) , w i t h the e x c e p t i o n of the p r i n t e r . The s o f t w a r e runs under the DEC RT-11 o p e r a t i n g system: t h i s p r o v i d e s a foreground/background environment f o r user programs i n which i t i s p o s s i b l e t o have a 8/The New MCP Spectrometer 251 f o r e g r o u n d j o b ( c o n t r o l l i n g the s p e c t r o m e t e r and p e r f o r m i n g data a c q u i s i t i o n ) r u n n i n g c o n c u r r e n t l y w i t h a background j o b (which c o u l d . b e p e r f o r m i n g , f o r i n s t a n c e , d a t a a n a l y s i s or a v a r i e t y of o t h e r f u n c t i o n s ) . T h i s o p e r a t i n g system was a major f a c t o r i n the c h o i c e of the LSI 11/03 computer system, and means t h a t the computer, which r e p r e s e n t s a s i z e a b l e i n v e s t m e n t , i s not c o m p l e t e l y t i e d up when i t i s a c q u i r i n g d a t a , u n a v a i l a b l e f o r a n y t h i n g e l s e . RT-11 i n c l u d e s a complete set of u t i l i t i e s f o r c r e a t i n g , m a i n t a i n i n g , and m a n i p u l a t i n g f i l e s on d i s k , p l u s c o m p i l e r s f o r the FORTRAN and BASIC h i g h l e v e l languages and the MACRO assembly language. A p o r t f o l i o of s p e c i a l i z e d programs has been w r i t t e n f o r b i n a r y (e,2e) a p p l i c a t i o n s : these i n c l u d e an e x t e n s i v e l i b r a r y of p l o t s u b r o u t i n e s f o r the g r a p h i c s p r i n t e r , a f o r e g r o u n d (e,2e) d a t a a c q u i s i t i o n program, a background program which a l l o w s communication w i t h t h e .foreground and m a n i p u l a t i o n of d a t a f i l e s , d a t a p l o t t i n g programs, e t c . T h i s computer i s n o t - s u i t a b l e , however, f o r the programs which compute momentum d i s t r i b u t i o n s , d e n s i t y maps and G a u s s i a n l e a s t - s q u a r e peak f i t s : t h e s e r e q u i r e a h i g h - s p e e d , l a r g e memory, number-crunching p r o c e s s o r to.be c o n v e n i e n t , f o r which the Amdahl V7 i n the UBC Computing C e n t r e i s e n t i r e l y a p p r o p r i a t e . I t i s p o s s i b l e t o t r a n s f e r d a t a between the l a b and the Computing C e n t r e on 8 - i n c h f l o p p y d i s k e t t e s c o n f o r m i n g t o the IBM s t a n d a r d . 8/The New MCP Spectrometer 252 B i n a r y (e,2e) d a t a f i l e s s t o r e d on d i s k e t t e a r e s t r u c t u r e d i n f o u r 256x16-word b l o c k s : the f i r s t b l o c k c o n t a i n s l a b e l s and s p e c t r o m e t e r parameters p e r t i n e n t t o the r u n ; the second and t h i r d b l o c k s c o n t a i n the v a l u e s of the independent spectrum v a r i a b l e ( v o l t a g e , a n g l e or time) as 3 2 - b i t f l o a t i n g p o i n t numbers; the f o u r t h b l o c k c o n t a i n s the accumulated d a t a . The o p e r a t o r i s a b l e t o s p e c i f y the t ype of s c a n , and a l l the f i x e d scan parameters i n t e r a c t i v e l y a t the v i d e o t e r m i n a l . At p r e s e n t o n l y the t r u e c o i n c i d e n c e r a t e i s r e c o r d e d i n the d a t a f i l e , and o n l y a s i n g l e o p e r a t i n g mode a t a time i s a v a i l a b l e ( i . e . b i n d i n g energy spectrum, a n g u l a r c o r r e l a t i o n , t ime spectrum or s i n g l e (e,e) s c a t t e r i n g ) , but p l a n s a r e a f o o t t o modify the d a t a a c q u i s i t i o n program so t h a t the t r u e and a c c i d e n t a l c o i n c i d e n c e r a t e and the random p u l s e p a i r r a t e a r e r e c o r d e d s e p a r a t e l y ( t o a l l o w the c a l c u l a t i o n of the s t a t i s t i c a l a c c u r a c y of the measurement), and t o a l l o w c o n c u r r e n t d a t a a c q u i s i t i o n i n a l l modes. The advantage of the l a t t e r m o d i f i c a t i o n i s i n m a i n t a i n i n g the proper r e l a t i v e a b s o l u t e i n t e n s i t y of each b i n a r y (e,2e) s t r u c t u r e w i t h r e s p e c t t o i t s f e l l o w s , r a t h e r than h a v i n g t o compute r e n o r m a l i z a t i o n f a c t o r s from b i n d i n g energy s p e c t r a a f t e r the d a t a i s r e c o r d e d . 8/The New MCP Spectrometer 253 8.6 P r e l i m i n a r y T e s t i n g 8.6.1 The vacuum system A f t e r t r a c k i n g down one or two l e a k s the base p r e s s u r e measured on the l a r g e pump s t a c k s was 7 x l 0 " 8 t o r r , measured at the gun chamber i t i s 1 x l 0 _ 6 t o r r and on the t o p of the chamber i t i s 5 x l 0 ' 7 t o r r . When enough gas i s a d m i t t e d t o r a i s e the ambient p r e s s u r e one o r d e r of magnitude i n the gas chamber, the a n a l y s e r and the gun chamber p r e s s u r e s o n l y go up o n e - t e n t h t h i s amount, i n d i c a t i n g t h a t the d i f f e r e n t i a l pumping i s working s a t i s f a c t o r i l y . 8.6.2 The e l e c t r o n gun The i n s t a l l a t i o n d e s c r i b e d i n s e c t i o n 8.4.2 g i v e s an adequate beam a t 400 eV (10-20»<A, 90 per cen t f o c u s s e d ) . At l200eV i t improves t o 20-30//A and 95 per cen t f o c u s s e d . The gate v a l v e i s o l a t i n g the gun i s a g r e a t c o n v e n i e n c e : s i n c e o x i d e cathodes must not be exposed t o a i r once they a re a c t i v a t e d i t i s p o s s i b l e t o open the main chamber but s t i l l keep the gun under h i g h vacuum. 8/The New MCP Spectrometer 254 8.6.3 The a n a l y s e r I n i t i a l t e s t i n g of the a n a l y s e r was done w i t h two M u l l a r d B318AL c h a n n e l t r o n s mounted a t the a n a l y s e r e x i t s l i t a t an a z i m u t h a l a n g l e of 11°. The MCPs were w i r e d i n t o a c i r c u i t s i m i l a r t o f i g u r e 3.4b. E { r e t a r d } was s e t t o z e r o t o e l i m i n a t e problems w i t h d e f l e c t i o n of the s c a t t e r e d e l e c t r o n by the l e n s . The spectrum of 200eV e l e c t r o n s s c a t t e r e d e l a s t i c a l l y from h e l i u m w i t h 200eV pass energy s e t i n the a n a l y s e r gave an e l a s t i c peak of 0.5eV FWHM which i s a s i g n i f i c a n t improvement i n r e s o l u t i o n over the 0.8-1.0eV FWHM o b t a i n a b l e on the o l d machine. I t cannot be e s t a b l i s h e d w i t h t h i s method whether or not the a n a l y s e r t r a n s m i s s i o n i s i s o t r o p i c as a f u n c t i o n of <t>. 8.6.4 MCP t e s t i n g The f i r s t • t e s t s w i t h the MCP were made w i t h a s i m p l i f i e d c o l l e c t o r c o n s i s t i n g of two b r a s s h a l f - d i s k s mounted so as t o l i n e up w i t h the d i v i d i n g s t r i p on the MCP output f a c e . The b r a s s d i s k s s h o u l d c o l l e c t a l l the s i g n a l i n each h a l f of the s p e c t r o m e t e r , e s s e n t i a l l y y i e l d i n g a b i n d i n g energy spectrum i n t e g r a t e d over a n g l e . The f i r s t t e s t c i r c u i t w i t h t h i s c o l l e c t o r i s shown i n f i g u r e 8.5. In an attempt t o improve the c h a r a c t e r i s t i c s of the MCP p u l s e s a t the preamp T, and T 2 i n p u t s , a n o v e l s i g n a l f e e d t h r o u g h arrangement was d e v i s e d . I n d i v i d u a l 8/The New MCP S p e c trometer 255 m e t a l / c e r a m i c BNC f e e d t h r o u g h s were d i s p e n s e d w i t h , and s i n g l e l e n g t h s of c o a x i a l c a b l e were used i n s t e a d , r u n n i n g w i t h o u t a break s t r a i g h t t h r o u g h a f l a n g e on the b a s e p l a t e . Vacuum i n t e g r i t y was m a i n t a i n e d by c a r e f u l l y s e a l i n g the c a b l e s i n the f l a n g e w i t h h i g h - q u a l i t y , low vapour p r e s s u r e epoxy r e s i n g l u e . Combined w i t h c a r e f u l impedance matching and c a b l e t e r m i n a t i o n , t h i s arrangement v i r t u a l l y e l i m i n a t e s a l l r i n g i n g on the s i g n a l . I t i s not n e c e s s a r y t o use an impedance-matched c o l l e c t o r anode, as has been d e v i s e d by o t h e r groups. T h i s c i r c u i t d i d , however, g i v e huge c o u p l i n g p roblems, between the T, and T 2 s i g n a l s . T h i s was not u n d e r s t o o d f o r a l o n g t i m e , and many d i f f e r e n t m o d i f i c a t i o n s were t r i e d i n the attempt t o e l i m i n a t e t h i s problem. I t was f i n a l l y r e a l i z e d t h a t the MCP o u t p u t f a c e was o n l y a t v i r t u a l A C ground, not a t r u e AC ground, and a l s o t h a t coax c a b l e s do not t r a n s m i t the v o l t a g e a p p l i e d t o the c e n t r a l c o n d u c t o r a l o n e , but r a t h e r they t r a n s m i t the d i f f e r e n c e v o l t a g e between the i n n e r and o u t e r c o n d u c t o r s . I t i s seen from f i g u r e 8.6 t h a t the p o s i t i v e p u l s e g e n e r a t e d on the MCP output f a c e a t the i n s t a n t the e l e c t r o n shower l e a v e s i s c a p a c i t i v e l y c o u p l e d t o both coax s h i e l d s , and so produces a n e g a t i v e p u l s e a t the preamp i n p u t s which i s i n d i s t i n g u i s h a b l e from the normal MCP output p u l s e . The remedy was p a i n f u l l y s i m p l e : the coax s h i e l d s must be grounded at both ends, and the MCP o u t p u t f a c e must be 8/The New MCP Spectrometer 256 t i e d t o the AC s i g n a l ground p o t e n t i a l . That the new f e e d t h r o u g h arrangement and s i g n a l d e c o u p l i n g works was e s t a b l i s h e d by a seven-second o b s e r v a t i o n of the preamp output p u l s e shape on an o s c i l l o s c o p e . Then the h i g h v o l t a g e s t a r t e d b r e a k i n g down, and the r e s u l t i n g t r a n s i e n t s d e s t r o y e d the preamp t r a n s i s t o r s . Subsequent e x a m i n a t i o n of the s p e c t r o m e t e r showed t h a t the breakdown had o c c u r r e d i n s e v e r a l of the MCP c h a n n e l s , r e n d e r i n g the d e v i c e a t l e a s t t e m p o r a r i l y and p r o b a b l y permanently i n o p e r a t i v e . The o n l y way t o r e s u s c i t a t e the d e v i c e i s t o d i s c o n n e c t the f a u l t y c h a n n e l s e l e c t r i c a l l y from the r e s t of the d e v i c e and t h i s n e c e s s i t a t e s removing the n i c k e l c o n t a c t f i l m from those c h a n n e l s - a t i c k l i s h b u s i n e s s a t b e s t . I t has been l e a r n e d through p a i n f u l e x p e r i e n c e t h a t t h e s e MCP d e v i c e s are not as rugged as had been thought. I t i s recommended t h a t extreme c a r e be taken i n h a n d l i n g , s t o r i n g , and o p e r a t i n g t h e s e d e v i c e s . I t s h o u l d be noted t h a t the s h e l f - l i f e of MCP d e v i c e s i s not i n f i n i t e a t a t m o s p h e r i c p r e s s u r e , even when s t o r e d w i t h d e s s i c a n t . U n i d e n t i f i e d c r y s t a l l i n e growths w h i c h c o u l d not be removed by s o l v e n t appeared on the f a c e s of our d e v i c e a f t e r b e i n g s t o r e d f o r about a y e a r . I t i s l i k e l y t h a t the presence of t h e s e growths i n c o m b i n a t i o n w i t h f i n e d u s t p a r t i c l e s caused the breakdowns. Measures taken i n the c a r e of t h e s e d e v i c e s must i n c l u d e : 8/The New MCP Spectrometer 257 (1) S t o r a g e f o r l o n g p e r i o d s of time s h o u l d be i n a h y d r o c a r b o n - f r e e vacuum system ( i . e . a c o n t a i n e r w i t h O - r i n g s e a l s which can be ev a c u a t e d by a pump w i t h t r a p s f o r o i l v a p o u r s , and then c l o s e d o f f ) ; (2) C o n t a c t w i t h o r d i n a r y d u s t - l a d e n l a b o r a t o r y a i r s h o u l d be a v o i d e d as much as p o s s i b l e . The d e v i c e and i t s mounting can be c l e a n e d and remounted under d ry n i t r o g e n u s i n g g l o v e box t e c h n i q u e s . High p u r i t y s o l v e n t ( n - p r o p a n o l ) i s recommended f o r c l e a n i n g , and a d r y n i t r o g e n gas j e t can be used t o blow any r e m a i n i n g dust away; (3) Once i n s t a l l e d i n the s p e c t r o m e t e r the system s h o u l d be pumped down f o r s e v e r a l days i n o r d e r t o remove adsorbed water vapour from the MCP c h a n n e l s . The w a l l s of the c h a n n e l s r e p r e s e n t a c o n s i d e r a b l e s u r f a c e a r e a , and pumping w i t h i n the c h a n n e l s i s c o n s t r i c t e d due t o t h e i r e x t r e m e l y s m a l l d i a m e t e r . 258 CHAPTER 9 CLOSING REMARKS 'Have you thought of an ending?' 'Yes, s e v e r a l , and a l l are dark and u n p l e a s a n t , ' s a i d Frodo. The v a l u e of b i n a r y (e,2e) s p e c t r o s c o p y both i n i t s e l f i n p r o v i d i n g f a s c i n a t i n g and i n s t r u c t i v e i n s i g h t s i n t o the e l e c t r o n i c s t r u c t u r e of m o l e c u l e s , and as a y a r d s t i c k f o r j u d g i n g the q u a l i t y of t h e o r e t i c a l w a v e f u n c t i o n s has been demonstrated i n t h i s t h e s i s . I t has been shown t h a t the b i n a r y (e,2e) t e c h n i q u e i s c a p a b l e of making f i n e d i s t i n c t i o n s i n the shape of v a l e n c e o r b i t a l s of s m a l l m o l e c u l e s , and can be used t o g r e a t advantage i n t e s t i n g t h e q u a l i t y of c a l c u l a t e d w a v e f u n c t i o n s , where i t i s seen t h a t the t e s t of c a l c u l a t e d e n e r g i e s a g a i n s t e x p e r i m e n t a l ones cannot always be r e l i e d on as an a c c u r a t e i n d i c a t i o n of improvement i n the w a v e f u n c t i o n . The t e c h n i q u e i s y e t i n i t s i n f a n c y , w i t h much m o l e c u l a r s t r u c t u r e s t i l l u n e x p l o r e d . The work on H 2S, C 0 2 , NO, and 0 2 shows t h a t the momentum d i s t r i b u t i o n , w h i l e o b s c u r i n g some i n f o r m a t i o n because i t i s a s p h e r i c a l a v e r a g e , n e v e r t h e l e s s r e v e a l s more about the n a t u r e or c h a r a c t e r of the e l e c t r o n s of the 9 / C l o s i n g Remarks 259 m o l e c u l e than does any o t h e r e x p e r i m e n t a l probe. I t s h a r p l y d e l i n e a t e s the d i f f e r e n c e s between the many t y p e s of m o l e c u l a r o r b i t a l s : non-bonding, a t o m i c - l i k e MOs, v a l e n c e TT —, and i r * - bonding MOs i n d i a t o m i c s , t r * - a n t i b o n d i n g MOs i n t r i a t o m i c C 0 2 , and so on, and a l s o the d i f f e r e n c e between r e s u l t s of v a r i o u s methods of c a l c u l a t i n g the w a v e f u n c t i o n s f o r t h e s e systems. The t e c h n i q u e has r e v e a l e d h i t h e r t o unsuspected s t r u c t u r e i n the i o n i z a t i o n of i n n e r v a l e n c e e l e c t r o n s ( C h a p t e r s 4, 5 and 6) and has g r a t i f y i n g l y c o n f i r m e d the t h e o r e t i c a l p r e d i c t i o n s of the many-body Green's f u n c t i o n method put fo r w a r d by L.S. Cederbaum and co w o r k e r s . Owing t o the c o m p l e x i t y of t h i s s t r u c t u r e i t i s e v i d e n t t h a t a c c u r a t e c a l c u l a t i o n s i n t e n d e d t o p r e d i c t t h i s s t r u c t u r e s h o u l d i n f u t u r e u t i l i z e s i m i l a r t e c h n i q u e s g o i n g w e l l beyond the H a r t r e e - F o c k l i m i t . T h i s s o r t of d a t a i s l e s s e a s i l y o b t a i n e d from PES or XPS, and a l s o , when o b t a i n e d , i s l e s s i n f o r m a t i v e , l a c k i n g the momentum d i s t r i b u t i o n i n f o r m a t i o n which i s the s i g n a t u r e of the o r i g i n a t i n g o r b i t a l . The l i m i t a t i o n s of the t e c h n i q u e a r e t h r e e f o l d : (1) Energy r e s o l u t i o n r e s t r i c t s the d e t e r m i n a t i o n of momentum d i s t r i b u t i o n s t o those o r b i t a l s which can be r e s o l v e d i n the b i n d i n g energy spectrum, or d e c o n v o l u t e d w i t h adequate p r e c i s i o n . (2) A n g u l a r u n c e r t a i n t y and r e s o l u t i o n e f f e c t s d i s t o r t or obscure the momentum d i s t r i b u t i o n near ©=45°, near 9 / C l o s i n g Remarks 260 0=0°, and where t h e r e i s s h a r p l y c hanging i n t e n s i t y . T h i s makes i t d i f f i c u l t t o observe a c c u r a t e l y one of the more i n t e r e s t i n g r e g i o n s of the momentum d i s t r i b u t i o n around q=0. (3) T a r g e t s must be gaseous and s t a b l e , and i n f a i r l y p l e n t i f u l s u p p l y . T h i s r u l e s out v e r y r e a c t i v e gases, and ones where l a r g e q u a n t i t i e s cannot be produced at once. I t a l s o l i m i t s the measurement of the MDs of t r a n s i e n t s p e c i e s . I t i s hoped t h a t the new MCP s p e c t r o m e t e r (Chapter 8 ) , when comp l e t e d , w i l l reduce t h e s e l i m i t a t i o n s . S i n c e t h i s s p e c t r o m e t e r i s v a s t l y more e f f i c i e n t i t w i l l be f e a s i b l e t o s a c r i f i c e s i g n a l f o r b e t t e r energy r e s o l u t i o n . The d e t e c t o r has, i n p r i n c i p l e , i n f i n i t e a n g u l a r r e s o l u t i o n , and so i t remains o n l y t o ensure t h a t the p o l a r s c a t t e r i n g a n g l e i s a c c u r a t e l y known. The d i f f e r e n t i a l pumping s h o u l d a l l o w the use of n o x i o u s gases as t a r g e t s , but even so i t i s u n l i k e l y t h a t measurements on . t r a n s i e n t s p e c i e s w i l l be v i a b l e f o r some t i m e . There i s a wide range of m o l e c u l e s not y e t s t u d i e d . I t would be i n t e r e s t i n g t o c a r r y out b i n a r y (e,2e) s t u d i e s on the f o l l o w i n g : (1) Some t r a n s i t i o n m e t a l compounds, t o l o o k a t genuine d - e l e c t r o n s . VC1„ and TiCl„ would be f e a s i b l e , and perhap some of the s i m p l e t r a n s i t i o n - m e t a l c a r b o n y l s ( F e ( C O ) 6 , e t c . ) , as the d - o r b i t a l i s w e l l 9 / C l o s i n g Remarks 261 s e p a r a t e d e n e r g e t i c a l l y from the o t h e r o r b i t a l s . I t would be i n t e r e s t i n g t o see whether here the outermost d-type MO i s as hard t o t r e a t t h e o r e t i c a l l y as i s the outermost o r b i t a l i n the f i r s t - r o w h y d r i d e s ; (2) Systems where t h e r e i s more complex nodal s t r u c t u r e than j u s t j r * - a n t i b o n d i n g o r b i t a l s . Some of the h i g h e r u n s a t u r a t e d m o l e c u l e s , or systems w i t h more group VI and V I I atoms t o p o p u l a t e h i g h e r l e v e l s may f i t the b i l l h e r e ; (3) Systems w i t h a h i g h e r degree of asymmetry; (4) Systems where t h e r e might be a p o s s i b i l i t y of o b s e r v i n g bond o s c i l l a t i o n , f o r i n s t a n c e C 3 0 2 . In a d d i t i o n t o s t u d y i n g more i n t e r e s t i n g t a r g e t s , we need t o l o o k more c l o s e l y a t the i n f o r m a t i o n the momentum d i s t r i b u t i o n c a r r i e s . T h i s has been s t a r t e d w i t h the a u t o - c o r r e l a t i o n f u n c t i o n B ( r ) devel o p e d by the M a r y l a n d r e s e a r c h group { T o s s e l l (1981)}, and w i t h the q{max} and q{ 1/ 2max} vs I P 1 ^ 2 p l o t s i n t h i s t h e s i s . I t s h o u l d be p o s s i b l e t o measure <q> and <q 2> by i n t e g r a t i n g the measured momentum d i s t r i b u t i o n s d i r e c t l y : yjq q 2 F { i } ( q ) assuming t h a t *{Mott} had l i t t l e e f f e c t , and t h a t the r e s o l u t i o n - s e n s i t i v e q=0 r e g i o n would be w e i g h t e d out by the (9.1) ( q 9 / C l o s i n g Remarks 262 q 2 d q volume element f a c t o r . T h i s would then enable one t o o b t a i n ( i n the HF l i m i t and under Koopmans' a p p r o x i m a t i o n ) the k i n e t i c and p o t e n t i a l p a r t s of the o r b i t a l t o t a l energy a s : T = <p2>/2 (9.2) e = - I P V = € - T which s h o u l d a f f o r d f u r t h e r i n s i g h t i n t o m o l e c u l a r e l e c t r o n i c s t r u c t u r e . A r e c e n t a c c u r a t e d e t e r m i n a t i o n of the momentum d i s t r i b u t i o n of atomic hydrogen {Lohmann} where the w a v e f u n c t i o n i s e x a c t l y known a f f o r d s the o p p o r t u n i t y t o t e s t t h i s p r o p o s i t i o n . The measurement, done a t i n c i d e n t e n e r g i e s of 400, 800, and l200eV i n the range 0<q<1.5a 0' 1, shows an e x c e l l e n t f i t t o the c u r v e ( l + q 2 ) - " , which i s the f u n c t i o n a l form one e x p e c t s from the *=exp(-r) hydrogen atom w a v e f u n c t i o n . The e x p e c t a t i o n v a l u e of the n t h power of p i n t h i s w a v e f u n c t i o n i s g i v e n by: which p r e d i c t s <p>=8/3rr = 0.85 and <p 2>=1. N u m e r i c a l e v a l u a t i o n by t r a p e z o i d i n t e g r a t i o n of e q u a t i o n 9.1 w i t h the e x p e r i m e n t a l d a t a g i v e s the r e s u l t <q>=0.86 and <q 2>=0.98. 9 / C l o s i n g Remarks 263 To r e a c h t h i s r e s u l t i t was n e c e s s a r y t o add a t a i l t o the e x p e r i m e n t a l d a t a beyond q=1.5 of the form ( 1 + q 2 ) " " . As was found i n p r e v i o u s work { T o s s e l l (1981)} i t i s n e c e s s a r y t o have h i g h l y a c c u r a t e d a t a up t o a t l e a s t q=4.0 i n o r d e r t o get m e a n i n g f u l r e s u l t s from i n t e g r a t i o n s of t h i s k i n d . Measurements of adsorbed gases on s u r f a c e s have r e c e n t l y g a i n e d a l o t of a t t e n t i o n and one n a t u r a l l y wonders i f b i n a r y (e,2e) c o u l d be adapted t o such t a r g e t s . An indeed f a s c i n a t i n g and v a l u a b l e a b i l i t y would be the o b s e r v a t i o n of how the o r b i t a l s of a gas m o l e c u l e d i s t o r t when i n c o n t a c t w i t h a me t a l s u r f a c e . However, i n my o p i n i o n , i t i s not l i k e l y t h a t we s h a l l soon be a b l e t o s u f f i c i e n t l y d i s t i n g u i s h e n e r g e t i c a l l y the v a l e n c e e l e c t r o n s of the adsorbed gas m o l e c u l e , or atom, from the c o n d u c t i o n e l e c t r o n s of the me t a l s u b s t r a t e . T h i s l e a v e s us w i t h the co r e e l e c t r o n s : i t may be p o s s i b l e t o e s t i m a t e the s u r f a c e geometry of the adsorbed atoms by l o o k i n g f o r the s t r o n g bond o s c i l l a t i o n s t h a t w i l l be seen i n an extended, o r d e r e d a r r a y of adsorbed atom c o r e o r b i t a l s . The p l a n e wave a p p r o x i m a t i o n however i s c e r t a i n t o be inadequate f o r c o r e o r b i t a l s i f the u s u a l s c a t t e r i n g c o n d i t i o n s a r e used, and i t w i l l be n e c e s s a r y t o go t o v e r y h i g h i n c i d e n t energy (which w i l l l e a d t o a n g u l a r r e s o l u t i o n problems) or e l s e somehow account t h e o r e t i c a l l y f o r the d i s t o r t i o n of the f r e e - e l e c t r o n waves, i n o r d e r to o b t a i n m e a n i n g f u l d a t a . M u l t i p l e s c a t t e r i n g e f f e c t s may a l s o be a problem. The g r e a t 9 / C l o s i n g Remarks 264 advantage of the b i n a r y (e,2e) t e c h n i q u e over some o t h e r s u r f a c e methods, i t must be remembered, i s t h a t of o r b i t a l s e l e c t i v i t y , which means t h a t the a c t u a l s c a t t e r e r i s a much s i m p l e r t h i n g than i n , f o r i n s t a n c e , LEED (low energy e l e c t i o n d i f f r a c t i o n ) s t u d i e s , where whole atoms are the s c a t t e r e r s . The o n l y o t h e r comparable t e c h n i q u e s a re SEXAFS ( s u r f a c e extended X-ray a b o r p t i o n f i n e s t r u c t u r e ) , and these r e q u i r e an e l a b o r a t e d a t a a n a l y s i s p r o c e d u r e . I t may be t h a t the advantage of o r b i t a l s e l e c t i v i t y w i l l be found t o outweigh the d i f f i c u l t i e s . F i n a l l y , I hope t h a t t h i s work has been i n s t r u m e n t a l i n i n t r o d u c i n g , momentum space c h e m i s t r y (Chapter 2) t o the o r d i n a r y c h e m i s t , and t h a t , w i t h some p r a c t i c e we w i l l g a i n some ease and f a m i l i a r i t y w i t h momentum d e n s i t i e s and momentum d i s t r i b u t i o n s . The use of p o s i t i o n and momentum d e n s i t y maps has been shown t o be of g r e a t h e l p i n u n d e r s t a n d i n g the .shapes of momentum d i s t r i b u t i o n s . They found him a l l a l o n e i n h i s l i t t l e room. I t was l i t t e r e d w i t h papers and pens and p e n c i l s ; but B i l b o was s i t t i n g i n a c h a i r b e f o r e a s m a l l b r i g h t f i r e . He l o o k e d v e r y o l d , but p e a c e f u l , and s l e e p y . 26ka Leaves 265 and 266 missed i n page numbering 267 APPENDIX A GLOSSARY H o b b i t s d e l i g h t e d i n such t h i n g s , i f they were a c c u r a t e : they l i k e d t o have books f i l l e d w i t h t h i n g s they a l r e a d y knew, se t out f a i r and square w i t h no c o n t r a d i c t i o n s . A.1 Symbols and U n i t s c Exponent of a G a u s s i a n b a s i s f u n c t i o n , p r o p o r t i o n a l i t y 6 ( x - x 0 ) D e l t a f u n c t i o n 6T FWHM of the (e,2e) c o i n c i d e n c e peak i n a time spectrum, u n c e r t a i n t y i n time 6{mn} K r o n e c k e r d e l t a At Time d i f f e r e n c e e B i n d i n g energy K Momentum v e c t o r i n c e n t r e of mass c o o r d i n a t e s fiS M i c r o s e c o n d s v Frequency, v i b r a t i o n a l quantum number u Energy parameter of the Green's f u n c t i o n M U n i t s of e l e c t r i c a l r e s i s t a n c e (ohms) n(r) P o l a r a n g l e s of the v e c t o r r <f> A z i m u t h a l s c a t t e r i n g a n g l e , a z i m u t h a l c o o r d i n a t e * { J j } ( r ) , * { J j } (p_) The j t h b a s i s f u n c t i o n on atom J i n R- and P-space * The f i n a l s t a t e of the e n t i r e s c a t t e r i n g system * { i } ( r ) ,*{i}(p_) The i t h m o l e c u l a r o r b i t a l i n R- and P-scace * The i n i t i a l s t a t e of the e n t i r e s c a t t e r i n g system * 0 (N) Target i n i t i a l N - e l e c t r o n ground s t a t e ( S l a t e r d e t e r m i n a n t ) p { i } ( r } , P { i } ( E ) P r o b a b i l i t y d e n s i t y of the i t h m o l e c u l a r o r b i t a l i n R- and P-space I Summation 2 6 8 E ( o ) ~ S e l f - e n e r g y p a r t i n the energy r e p r e s e n t a t i o n r Time, d i p o l e m a t r i x element 9 P o l a r s c a t t e r i n g a n g l e , p o l a r c o o r d i n a t e * { i , N - l } T a r g e t f i n a l N-1 e l e c t r o n e x c i t e d s t a t e (one e l e c t r o n m i s s i n g from o r b i t a l i ) V Grad ° Degrees * Complex c o n j u g a t e a Index of atom a 0 D i s t a n c e i n atomic u n i t s a o " 1 Momentum i n atomic u n i t s a * { i } C r e a t i o n o p e r a t o r of a p a r t i c l e i n s t a t e i a { i } A n n i h i l a t i o n o p e r a t o r of p a r t i c l e i n s t a t e i au Atomic u n i t s b B a s i s f u n c t i o n index i n summations c Speed of l i g h t i n vacuum c { i J j } LCAO c o e f f i c i e n t of the j t h b a s i s f u n c t i o n on atom J i n the i t h m o l e c u l a r o r b i t a l e 2 . 7 1 8 2 8 1 8 2 8 . . . , charge on the e l e c t r o n K P l a n c k ' s c o n s t a n t / 2 i r i I n v e r s i o n symmetry element, square r o o t - 1 , m o l e c u l a r o r b i t a l index j B a s i s f u n c t i o n index i n summat i o n s j U H p r ) S p h e r i c a l B e s s e l f u n c t i o n k 0 Momentum v e c t o r of the i n c i d e n t (e,2e) e l e c t r o n k.1 i is 2 Momentum v e c t o r s of the two e x i t (e,2e) e l e c t r o n s 1 A n g u l a r quantum number, l e n g t h m E l e c t r o n mass, metres mm M i l l i m e t e r s n P r i n c i p a l quantum number n{A} Number of b a s i s f u n c t i o n s on atom A nm Nanometres ns Nanoseconds p|| Component of momentum p a r a l l e l t o the bond d i r e c t i o n p||{max} Maximum p r o b a b l e momentum i n the d i r e c t i o n p a r a l l e l t o the bond pj_ Component of momentum p e r p e n d i c u l a r t o the bond d i r e c t i o n 269 pjjmax} Maximum p r o b a b l e momentum i n the d i r e c t i o n p e r p e n d i c u l a r t o the bond P,e, * Momentum-space s p h e r i c a l p o l a r c o o r d i n a t e s p_,p Momentum v e c t o r and magnitude p{max} Most p r o b a b l e momentum p{x},p{y},p{z} Momentum-space C a r t e s i a n c o o r d i n a t e s g,q Momentum v e c t o r and magnitude q{ 1/ 2max} Momentum a t which the e x p e r i m e n t a l MD has f a l l e n t o h a l f i t s q=0 i n t e n s i t y i n an s-type d i s t r i b u t i o n q{max} Most p r o b a b l e momentum ( i n an e x p e r i m e n t a l d i s t r i b u t i o n ) r ,e,« P o s i t i o n - s p a c e s p h e r i c a l p o l a r c o o r d i n a t e s r_,r P o s i t i o n v e c t o r and magnitude s Seconds t Time t o r r U n i t of p r e s s u r e . 1torr=1mmHg x,y ,z P o s i t i o n - s p a c e C a r t e s i a n c o o r d i n a t e s x { i , n } Hole s t a t e a m p l i t u d e s of the Green's f u n c t i o n A Atom index i n summations, a n t i s y m m e t r i z a t i o n o p e r a t o r C O v e r l a p of n o n - i n t e r a c t i n g o r b i t a l s i n a t r a n s i t i o n p r o b a b i l i t y E Energy l o s s E(M* +) T o t a l energy of the t a r g e t i o n M*+ E(M') T o t a l energy of the t a r g e t m o l c u l e M E 0 Energy of the i n c i d e n t b i n a r y (e,2e) e l e c t r o n E, , E 2 E n e r g i e s of the two e x i t (e,2e) e l e c t r o n s E{pass} Energy an e l e c t r o n must have t o pass through on e l e c t r o s t a t i c a n a l y s e r E { r e t a r d } The r e d u c t i o n i n energy of an e l e c t r o n p a s s i n g through an e l e c t r o n l e n s F { i } ( q ) B i n a r y (e,2e) form f a c t o r G(o) Green's f u n c t i o n of energy J Atom index i n summations L Length M (e,2e) s c a t t e r i n g a m p l i t u d e , t a r g e t m o l e c u l e b e f o r e (e,2e) c o l l i s i o n M*+ Target i o n a f t e r c o l l i s i o n . N Number of atoms, number of e l e c t r o n s , n o r m a l i z a t i o n f a c t o r P Momentum o p e r a t o r P { n l } ( p ) R a d i a l p a r t of a P-space atomic o r b i t a l w a v e f u n c t i o n Q(r),Q(p) O n e - d i m e n s i o n a l w a v e f u n c t i o n p r o j e c t i o n R { n l } ( r ) R a d i a l p a r t of an R-space atomic o r b i t a l wavefunct i o n S { i } I n t e n s i t y f a c t o r s i n t h e b i n a r y (e,2e) c r o s s - s e c t i o n T U , , k 2 ; t , k 0 ) , T ( r ! , r 2 ; r , r 0 ) Mott s c a t t e r i n g a m p l i t u d e X 0(ko,r_)' . Free (e,2e) i n c i d e n t e l e c t r o n w a v e f u n c t i o n Y{lm}(©,0) Complex s p h e r i c a l harmonic f u n c t i o n s 1 U n i t m a t r i x 271 A.2 A b b r e v i a t i o n s ADC Analog t o d i g i t a l c o n v e r t e r AO Atomic o r b i t a l CFD Constant f r a c t i o n d i s c r i m i n a t o r CI C o n f i g u r a t i o n i n t e r a c t i o n CP Charge p r e a m p l i f e r DAC D i g i t a l ' t o a n a l o g c o n v e r t e r DZ D o u b l e - z e t a FPP F a s t p u l s e p r e a m p l i f i e r GSA G a u s s i a n shaping a m p l i f e r GTO G a u s s i a n - t y p e o r b i t a l HF H a r t r e e - F o c k LCAO L i n e a r c o m b i n a t i o n of atomic o r b i t a l s MBGF Many-body Green's f u n c t i o n MCP M u l t i c h a n n e l p l a t e MD Momentum d i s t r i b u t i o n MO M o l e c u l a r o r b i t a l NSO N a t u r a l s p i n o r b i t a l PWIA Plane-wave impulse a p p r o x i m a t i o n RHF R e s t r i c t e d H a r t r e e - F o c k SCA S i n g l e c h a n n e l a n a l y s e r SCF S e l f - c o n s i s t e n t f i e l d SH Sample and h o l d STO S l a t e r - t y p e o r b i t a l SZ S i n g l e - z e t a TAC Time t o a m p l i t u d e c o n v e r t e r TFA Timing f i l t e r a m p l i f i e r UHF U n r e s t r i c t e d H a r t r e e - F o c k 2ph-TDA T w o - p a r t i c l e - o n e - h o l e Tamm-Dancof f a p p r o x i m a t i o n 272 A.3 Terms A n g u l a r c o r r e l a t i o n The raw b i n a r y (e,2e) d a t a as a f u n c t i o n of s c a t t e r i n g a n g l e s e or <t>. Bond o s c i l l a t i o n The m a n i f e s t a t i o n of the m o l e c u l a r geometry i n P-space: at l a r g e p the momentum d e n s i t y i s tends t o a pure s i n u s o i d a l o s c i l l a t i o n , w i t h a p e r i o d i n v e r s e l y p r o p o r t i o n a l t o the bond l e n g t h . Bonding p r i n c i p l e A s e t of axioms which d e s c r i b e the m a n i f e s t a t i o n s of bon d i n g , non-bonding and a n t i b o n d i n g c h a r a c t e r i n an MO i n R-space and P-space. C o i n c i d e n c e The occurence of two ev e n t s w i t h i n a c e r t a i n time i n t e r v a l . E l e c t r o n d e n s i t y The d i s t r i b u t i o n of the p r o b a b i l i t y of f i n d i n g an e l e c t r o n a t a p o i n t i n R-space. Form f a c t o r B i n a r y (e,2e) form f a c t o r d e s c r i b i n g the d i s t r i b u t i o n of i n t e n s i t y as a f u n c t i o n of s c a t t e r i n g a n g l e . Green's f u n c t i o n A f u n c t i o n t h a t d e s c r i b e s the p r o p a g a t i o n i n space and time of a system of p a r t i c l e s . The energy r e p r e s e n t a t i o n of the Green's f u n c t i o n has p o l e s a t the e n e r g i e s of the p a r t i c l e s . Inner v a l e n c e e l e c t r o n s Those v a l e n c e e l e c t r o n s w h ich, when i o n i z e d , l e a d t o s e v e r a l f i n a l i o n s t a t e s w i t h no p a r t i c u l a r main p a r e n t peak, d i s t r i b u t e d i n energy above about 20eV. Momentum d e n s i t y The d i s t r i b u t i o n of t h e p r o b a b i l i t y of f i n d i n g an e l e c t r o n w i t h a c e r t a i n ( v e c t o r ) momentum. Momentum d i s t r i b u t i o n The e x p e r i m e n t a l a n g u l a r c o r r e l a t i o n o b t a i n e d i n the symmetric n o n - c o p l a n a r s c a t t e r i n g geometry, p l o t t e d a g a i n s t q; a l s o the s p h e r i c a l l y - a v e r a g e d t h e o r e t i c a l momentum d e n s i t y . 273 Mott s c a t t e r i n g The s c a t t e r i n g of two i d e n t i c a l p a r t i c l e s i n the Coulomb p o t e n t i a l , i n c l u d i n g e f f e c t s of exchange. MFS s t r u c t u r e That s t r u c t u r e i n a b i n d i n g energy spectrum, u s u a l l y above 20eV, which a r i s e s because i o n i z a t i o n of one i n n e r v a l e n c e o r b i t a l l e a d s t o m u l t i p l e f i n a l i o n s t a t e s of d i f f e r e n t e n e r g i e s . Nodal p l a n e A p l a n e of symmetry where the w a v e f u n c t i o n a m p l i t u d e and d e n s i t y i s z e r o . Nodal s u r f a c e A s u r f a c e of any shape where the w a v e f u n c t i o n a m p l i t u d e and d e n s i t y i s z e r o . Outer v a l e n c e e l e c t r o n s Those v a l e n c e e l e c t r o n s w hich, when i o n i z e d , l e a d t o one s t r o n g peak i n the s e p a r a t i o n energy spectrum, g e n e r a l l y below 20eV, and perhaps some v e r y weak peaks a t h i g h e r energy. P l a n e wave The w a v e f u n c t i o n of a f r e e p a r t i c l e exp( i k . r_) . P o l e s t r e n g t h The i n t e n s i t y of p o l e s i n the N - p a r t i c l e Green's f u n c t i o n , a f a c t o r i n the b i n a r y (e,2e) i n t e n s i t y . R e c i p r o c i t y p r i n c i p l e That a t t r i b u t e of the F o u r i e r T r a n s f o r m where a d i l a t a t i o n of a d i m e n s i o n i n R-space l e a d s t o a c o n t r a c t i o n of the di m e n s i o n i n P-space, and v i c e v e r s a R e l a t i v e a b s o l u t e i n t e n s i t y S i m i l a r t o ' r e l a t i v e i n t e n s i t y ' (q.v.) but the s c a l e b e a r s some known r e l a t i o n t o o t h e r s i m i l a r p l o t s . R e l a t i v e i n t e n s i t y The o r d i n a t e ' of a p l o t of the measured s i g n a l r a t e a g a i n s t some scanned parameter. The s c a l e b e a r s no r e l a t i o n s h i p t o a n y t h i n g e l s e . S e l f - e n e r g y The a l t e r a t i o n i n the energy of an i d e a l p a r t i c l e when i t e x e r t s a d i s t u r b i n g i n f l u e n c e on s u r r o u n d i n g i d e a l p a r t i c l e s which i n t u r n r e a c t back on the f i r s t p a r t i c l e t o change i t s energy and c o n v e r t i t i n t o a q u a s i p a r t i c l e . 274 Symmetric c o p l a n a r The b i n a r y (e,2e) s c a t t e r i n g geometry where the a z i m u t h a l s c a t t e r i n g a n g l e i s f i x e d a t 0° and the p o l a r a n g l e i s scanned. Symmetric non-coplanar The b i n a r y (e,2e) s c a t t e r i n g geometry where the p o l a r a n g l e i s f i x e d a t e=45° and the a z i m u t h a l a n g l e i s scanned. T h i s method g i v e s the momentum d i s t r i b u t i o n d i r e c t l y . W a v e f u n c t i o n The m a t h e m a t i c a l f u n c t i o n d e s c r i b i n g the a m p l i t u d e of a p a r t i c l e or system of p a r t i c l e s i n space and t i m e . 275 APPENDIX B DEFINITIONS AND DERIVATIONS 'Are you s t i l l awake?' Sam w h i s p e r e d . B.1 Momentum D e n s i t y Maps from LCAO-MO Wa v e f u n c t i o n s The w a v e f u n c t i o n f o r the i t h m o l e c u l a r o r b i t a l i s e x p r e s s e d as a l i n e a r c o m b i n a t i o n of S l a t e r - t y p e b a s i s f u n c t i o n s or of G a u s s i a n p r i m i t i v e s , t h u s : N n { j } (B.1) * { i } ( r ) = E E c { i J j } * { J j } ( r { j } ) J j where * { J j } ( r _ { J } ) i s the j t h b a s i s f u n c t i o n on c e n t r e J , N i s the number of atoms and n{J} i s the number of f u n c t i o n s on atom J . c { i J j } i s the c o e f f i c i e n t of the j t h b a s i s f u n c t i o n on atom J i n the i t h m o l e c u l a r o r b i t a l , and the c o o r d i n a t e v e c t o r r { J } i s r e f e r r e d t o atom J , not the o r i g i n of R-space. The F o u r i e r t r a n s f o r m of t h i s i s : -3/2 r " i p . r (B.2) + {i}(p_) = ( 2 i r ) \ d r e * { i } ( r ) I f we d e f i n e r=R{J} + r_{j} where R{J} i s the p o s i t i o n v e c t o r 276 of c e n t r e J where the j t h atomic o r b i t a l i s s i t u a t e d , t h e n : -3/2 N n { j } -ip_.R{j} (B.3) * { i } ( p j = (2ir) E I c { i J j } e J j (• - i p _ . r { j } x l d r { J } e * { J j } ( r { j } ) N n { j } -ip_.R{J} = E E c { i J j } e *{Jj}(p_) J j The i n t e g r a l i s j u s t the F o u r i e r t r a n s f o r m of the p o s i t i o n space atomic o r b i t a l w a v e f u n c t i o n . I t i s seen t h a t the momentum space m o l e c u l a r o r b i t a l i s a l i n e a r c o m b i n a t i o n of atomic momentals, but the n u c l e a r geometry i n f o r m a t i o n i s removed from the atomic o r b i t a l and now appears i n a phase f a c t o r exp(-ip_.R{ J } ) . I f the b a s i s f u n c t i o n s a re d e f i n e d a s : (B.4) * { J j } ( r ) = N{j} R { n { j } , l { j } } Y { 1 { j } , m { j } } ( Q { r } ) w i t h n=1,2,3...; 1=0,1...n-1; m=-l...+l and the f o l l o w i n g i d e n t i t y i s used {Messiah (1958)}: - i g . r oo +1 1 (B.5) e = 4ir E E ( - i ) j { l } ( p r ) 1=0 m=-l x Y { l m } ( n { 2 } ) Y * { l m } ( n { r } ) then the i n t e g r a l i n e q u a t i o n B.3 becomes: 277 00 +1 (B.6) E E O i ) 1=0 m=-l 2 \ r 2 d r j { l } ( p r ) R { n { j } , l { j } } f x Y{lm}(n{ 2 }) dn{r} Y* {lm} (n { r } ) Y{ 1 { j } ,m{ j }} (n{ r } ) = 6 { l , l { j } } 6{m,m{j}} = N{j} P { n { j } , l { j } } ( p ) Y{l{j},m{j}}(n{rj»}) where P { n l } = (-i) \2 l r 2 d r j { l } ( p r ) R { n l } ( r ) A c o n t o u r p l o t of the momentum d e n s i t y may be ge n e r a t e d d i r e c t l y from the w a v e f u n c t i o n as />{i}(p_) = **{ i } (p_) *{ i } (p_) , which i s the more s t r a i g h t f o r w a r d way, and, i n terms of computing c o n s i d e r a t i o n s f a s t e r and l e s s c o s t l y . I t i s a l s o p o s s i b l e t o expand the summation and p a r t i t i o n the d e n s i t y i n t o a sum of o n e - c e n t r e and t w o - c e n t r e c o n t r i b u t i o n s as f o l l o w s : (B.7) />{i}(p_) = N n{A} -ip_.R{A} E E c { i A a } e <»{Aa}(p_) A a N n{j } -p_.R{J} E E c { i J j } e * { J j } ( 2 ) J j The o n e - c e n t r e p a r t i s : 278 N n{A} n{A} ,(B.8a) /»! { i } (jg) = E E E c * { i A a } c { i A b } A a b x <**{Aa}(p_) *{Ab}(p_) The t w o - c e n t r e p a r t i s : N N n{A} n{ J} ip_.R{AJ} (B.8b) /> 2{i}(p_) = 2Re E I I E c*{iAa} c { i J j } e A J a j A<J x **{Aa } ( 2 ) *{Jj}(p_) where R{AJ}=R{A}-R{J} i s the R-space v e c t o r from atom A t o atom J . The momentum d e n s i t y p l o t s were g e n e r a t e d by e v a l u a t i o n of e i t h e r f u n c t i o n over a mesh of 100x100 p o i n t s i n a p l a n e i n the m o l e c u l e . E q u a t i o n B.8 i s g e n e r a l f o r any m o l e c u l e . A s i m i l a r d e r i v a t i o n p r e s e n t e d i n the l i t e r a t u r e { C a m i l l o n i (1979)} f o r l i n e a r d i a t o m i c m o l e c u l e s c o n t a i n s an e r r o r i n e q u a t i o n 16 of t h a t r e f e r e n c e : the s i g n of the exponent ip_.R{AJ} i s g i v e n i n c o r r e c t l y . B.2 Momentum D i s t r i b u t i o n s from LCAO-MO Wa v e f u n c t i o n s In o r d e r t o compare the t h e o r e t i c a l d e n s i t y w i t h e x p e r i m e n t , e q u a t i o n B.8 must f i r s t be s p h e r i c a l l y averaged: (B.9) F { i } ( p ) = ( 4 i r ) - 1 jdn />{i}(p_) T h i s g i v e s the r e s u l t { L e v i n (1975), Dey (1977)} used t o 279 compute momentum d i s t r i b u t i o n s from t h e o r e t i c a l LCAO-MO w a v e f u n c t i o n s : (B.10) F { i } ( q ) = F { a t } + F { i n t } where N n{A} n{A} ( B . l l a ) . F { a t } = ( 4 n ) " 1 E E E N{a} N{b} c*{iAa} c { i A b } A a b x p { n { a } l { a } } ( p ) . P { n { b } l { b } } ( p ) x 6{l{a}l{b}}6{m{a}m{b}} and, u s i n g i d e n t i t y B.4 a g a i n : N N n{A} n { j } (B.11b) F { i n t } = 2Re E E E E N{a} N{j} c*{i A a } c { i J j } . A J a j A<J x P { n { a } l { a } } ( p ) P { n { j } 1 { j } } ( p ) 00 1 x E i j { l } ( p R { A J } ) Y{lm}(n{R{AJ)}) lm x j dn{pj Y * { l m } ( f ) { e } ) Y*{l{a}m{a}}(n{£}) Y{ 1 { j }m{ j }} (n {pj The i n t e g r a l over t h r e e a n g u l a r momenta g i v e s : 1 0 (B.11c) m{a} (-) (21+1)(2l{a}+1)(21{j}+1) in fl l { a } l { j } \ 71 l { a } l { j } -m m{a} m{j}/ \0 0 0 In computed momentum d i s t r i b u t i o n s i n t h i s t h e s i s the 280 summation l=0-oo i s t r u n c a t e d a f t e r convergence a t 1=5. The a n a l y t i c a l forms of the r a d i a l f u n c t i o n s P { n l } ( p ) used i n the c o m p u t a t i o n s a r e found by s o l u t i o n of the s p h e r i c a l B e s s e l t r a n s f o r m { L e v i n (1975), Komarov (1976), K a i j s e r ( 1 9 77), E p s t e i n (1971)} and a r e g i v e n i n T a b l e B.1. The forms of the s p h e r i c a l harmonics a r e g i v e n i n T a b l e B.2. The m a t r i x t o t r a n s f o r m c a r t e s i a n s p h e r i c a l harmonics, i n which l i t e r a t u r e w a v e f u n c t i o n s a r e u s u a l l y e x p r e s s e d , i n t o complex s p h e r i c a l h a rmonics, which a r e r e q u i r e d i n e q u a t i o n s B.6 and B.11, i s the f o l l o w i n g : (B. 12) "Y{S} ' Y{x} Y{y} Y{z} Y { x 2 } Y { y 2 } = Y { z 2 } Y{xy} Y{xz} Y{yz}_ 1 0 0 0 0 0 0 0 -1 1 /2 /2 0 i i /2 /2 i/5 0 3 /5 0 3 /5 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 •J_ 3 ]_ 3 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 y/6 /6 0 -1 -1 /6 /6 0 0 0 0 ) 0 - i i /2 /2 0-1 1 0 0 /2 /2 0 i i 0 0 /2 /2 r Y 0 ° • Y i " 1 Y 2° Y ; 1 Y r 2 Y- " 2 281 T a b l e B.1 R a d i a l f u n c t i o n s R { n l } ( r ) and P { n l } ( p ) STOs % -y 2 5/2 P{1S} = 2 ir g ( p 2 + £ 2 ) - 2 5/2 y 2 - y 2 % 2 P{2s} = 2 " 3 " " rr " t ( 1 - f t / 3 p 2 ( p 2 + £ 2)-1 ) ( p 2 + e 2 ) " 2 9/2 -v 2 -y 2 7/; 2 P{2p} = i 2 3 rr t ( p 2 + C 2 ) " 3 ~ 1 / 2 ~ / / 2 9 / 2 ~ 3 2 - 1 2 _ 5 ^ P{3s} = 2 5 5 2 rr 2 c 2 ( 2c, 2 ( p 2 + c 2 ) 2-(p 2+£ 2) ') (p 2+£ 2) V 2 ~ 1/2 7/2 P{3p} = i 2 5 3" 1 5 2 ir 2 £ 2 ( l - 6 5 p 2 ( p 2 + £ 2)-1 ) ( p 2 + £ 2 ) ' 3 P{3d} = - 2 6 5 2 ir 2 s p 2 ( p 2 + £ 2 ) - 1 1 3 / 2 - s r R{1S} = 2 £ e "V 2 5 / 2 - s r R{2s},R{2p} = 2 3 Q re % _ V 2 7/2 _ £ r R{3s},R{3p},R{3d} = 2 2 3" 1 5 2 g 2 r 2 e GTOs % " % " % "P 2 / 4 a P{1S} = 2 rr c e % -V 2 "Va "3/« "P 2/4o P{2p} = i 2 3 rr o pe 7/« "3/« % - a r 2 R{ 1 s} = 2 rr o e ,3/a -3/» % - a r 2 R{2p} = 2 rr c re 282 T a b l e B.2 S p h e r i c a l harmonics Complex s p h e r i c a l harmonics C a r t e s i a n s p h e r i c a l harmonics Y 0° = (4tr)- 1/2 Y{s} = (4rr)- 1/2 Y 1 ° = Z I 2 Y{x} = /3 x /4ir r /Tir r Y, 1 = ~/3 x + i y Y{y} = /3 y_ v/8~ir r /4~Jr r Y r 1 = /3 x - i y Y{z} = /3 z /8ir r /4ir r 283 APPENDIX C MISCELLANEOUS 'Mercy!' c r i e d G a n d a l f . . . 'What more do you want t o know?' 'The names of a l l the s t a r s , and of a l l l i v i n g t h i n g s , and the whole h i s t o r y of M i d d l e - e a r t h and Over-heaven and of the S u n d e r i n g Seas,' laughed P i p p i n . T a b l e C.1 V a l e n c e m o l e c u l a r o r b i t a l e l e c t r o n i c s t r u c t u r e . MOs a r e o r d e r e d by energy; occupancy i s shown by the s u p e r s c r i p t ; a bar s e p a r a t e s o u t e r v a l e n c e from i n n e r v a l e n c e MOs where t h e r e i s a s h a r p d i s t i n c t i o n ; the symmetry i s g i v e n underneath the name. He ( 1 s ) 2 Ne ( 2 p ) 6 Ar ( 3 p ) 6 Kr U p ) 6 Xe ( 5 p ) 6 ( 2 s ) 2 TlsP TtsV TtsP" CH« ( V t 2 ) 6 NH 3 ( 3 a , ) 2 H 20 ( 1 b , ) 2 HF ( 1 IT ) T{d} ( 2 a , ) 2 C{3v} ( l e ) f t C{2v} ( 3 a , ) 2 ' " ( 2 a , ) 2 ( 1 b 2 ) 2 2 2 (2a, ) H 2S (2b,) C{2v} ( 5 a , ) 2  ( 2 b 2 ) 2 (4a 2 (3<r ) 2 (2c) 2 HCl C{oov} (2n)« (5c) 2 U * ) 2 HBr C{oov] (4n) " ( 8 t f ) 2 (le)2 HI C{»v} (6n)« ( I U ) 2 H 2 O s { g } ) 2 N 2 ( 3 f f { g } ) 2 0 2 d i f { g } ) 2 C 0 2 O ^ g } ) " D{<»h} D{coh} ( l i r { u } ) a D{«?h} ( l i r U } ) " Dfch} ( l i r { u } ) a ( 2 * { u } ) 2 ( 3 f f { g } ) 2 (3< r { u } ) 2 (2c{q})2 ( 2 t f { u j ) 2 ( 4 t f { q } ) 2 (2< r { g } ) 2 (2< r { u } ) 2 ( 3 * { g } ) 2 CO (5e)2 NO (2n) 1 COS ( 3 i r ) 8 C { * ° v } ( l i r ) « C{cov} ( 1 i r ) * C { » v } (2n ) 4 U7T7 (5e)2 (9*) 2 (3c)2 U c l 7 (8c)2 (3c)2 Tier* (6c)2 284 F i g u r e C . 1 Geometry and c o n t o u r map p l a n e s f o r s e l e c t e d m o l e c u l e s . 2 8 5 APPENDIX D REFERENCES . . . t h e r e l i e i n h i s hoards many r e c o r d s t h a t few can now r e a d , even of the l o r e - m a s t e r s , f o r t h e i r s c r i p t s and tongues have become dark t o l a t e r men. L i t e r a t u r e r e f e r e n c e s t o j o u r n a l a r t i c l e s are g i v e n a s : P r i n c i p l e a u t h o r surname ( y e a r ) t h e s i s pages F u l l names of a l l a u t h o r s J o u r n a l name volume page R e f e r e n c e s t o books a r e g i v e n a s : P r i n c i p l e a u t h o r surname (y e a r ) t h e s i s pages F u l l names of a l l a u t h o r s T i t l e P u b l i s h e r Aberg ( 1 9 6 7 ) 148 T Aberg Phys Rev 156 3 5 A l l a n ( 1 9 7 2 ) 1 4 7 , 1 6 1 , 1 6 9 , 1 7 4 , 1 8 7 CJ A l l a n U G e l i u s DA A l l i s o n G Johansson H Siegbahn and K Siegbahn J E l Spect J_ 131 A s b r i n k ( 1 9 7 4 ) 192 L A s b r i n k and C F r i d h Phys Scr 9 3 3 8 A s b r i n k ( 1 9 7 7 ) 186 L A s b r i n k C F r i d h and E L i n d h o l m Chem Phys L e t t e r s 5_2 63 Basch ( 1 9 7 2 ) 5 1 , 5 8 , 7 5 , 1 7 0 , 1 8 7 H Basch and LC Snyder M o l e c u l a r Wave F u n c t i o n s and P r o p e r t i e s W i l e y New York B i c e r a n o ( 1 9 7 7 ) 145 J B i c e r a n o DS M a r y n i c k and WN Lipscomb J Am Chem Soc 100 7 3 2 286 Bradshaw (1980) 29 AM Bradshaw W E b e r h a r d t HJ- L e v i n s o n W Domcke and LS Cederbaum Chem Phys L e t t 7_0 36 B r i o n 125,202 CE B r i o n p r i v a t e communication B r i o n (1975) 19 CE B r i o n R a d i a t Res 64 37 B r i o n (1977) 19 CE B r i o n A Hamnett GR Wight and MJ van der W i e l J E l Spect J_2 323 B r i o n (1978a) 19 CE B r i o n and KH Tan Chem Phys 34 141 B r i o n (1978b) 14,125,139 CE B r i o n JPD Cook and KH Tan Chem Phys L e t t e r s 59 241 B r i o n (1979) 125 CE B r i o n IE McCarthy IH S u z u k i and E W e i g o l d Chem Phys L e t t e r s 67 115 B r i o n (1980) 125 CE B r i o n ST Hood IH S u z u k i E Weigo l d and GRJ W i l l i a m s J E l Spect 2_1_ 71 B r i o n (1981) 19,21 CE B r i o n and A Hamnett Adv Chem Phys 45_ 2 C a m i l l o n i (1977) 18 R C a m i l l o n i A G i a r d i n i - G u i d o n i G M i s s o n i G S t e f a n i G T i r i b e l l i and D V i n c i g u e r r a AIP Conf Proc 36 205 C a m i l l o n i (1978) 18 R C a m i l l o n i A G i a r d i n i - G u i d o n i IE McCarthy and G S t e f a n i Phys Rev A J_7 1634 C a m i l l o n i (1979) 278 R C a m i l l o n i G S t e f a n i R F a n t o n i A G i a r d i n i - G u i d o n i J E l Spect 17 209 Cederbaum (1975a) 23 LS Cederbaum J Chem Phys 62 2160 Cederbaum (1975b) 23 LS Cederbaum J Phys B 8 290 Cederbaum (1977) 23,29,157 LS Cederbaum and W Domcke Adv Chem Phys 36 205 Cederbaum (1978) 29,167 LS Cederbaum W Domcke J Schirmer W von N i e s s e n GHF D i e r c k s e n and WP Kraemer J Chem Phys 69 1591 Cederbaum (1980) 29,167,171 LS Cederbaum W Domcke J Schirmer and W von N i e s s e n Phys S c r 2_1_ 481 Chipman (1978) 140 DW Chipman J E l Spect j_4 323 Chong 186 DP Chong p r i v a t e communication Chong (1974) 157 DP Chong FG H e r r i n g and D M c W i l l i a m s J Chem Phys 61 78 C l e m e n t i (1963) 134 E C l e m e n t i and DL Raimondi J Chem Phys 38 2686 C l e m e n t i (1964) 134 E C l e m e n t i J Chem Phys 40 1944 C l e m e n t i (1974) 38 E C l e m e n t i and C R o e t t i At Data and N u c l Data T a b l e s j_4 177 Cook (1979) 125 JPD Cook CE B r i o n and A Hamnett J E l Spect J_5 233 Cook (1980) 14,17,125 JPD Cook CE B r i o n and A Hamnett Chem Phys £5 1 288 Cook (1981) 14,161 JPD Cook MG White CE B r i o n W Domcke J Sc h i r m e r LS Cederbaum and W von N i e s s e n J E l Spect 22 261 • Coulson (1944) 31,71 CA Coulson and WE Duncanson Proc Camb P h i l Soc 4_0 190 and r e f e r e n c e s c o n t a i n e d t h e r e i n C r u i k s h a n k (1968) 143 DWJ C r u i k s h a n k and BC Webster I n o r g a n i c s u l p h u r c h e m i s t r y (ed G N i c k l e s s ) E l s e v i e r Amsterdam Dey (1977) 278 S Dey AJ Dixon KR Lassey IE McCarthy PJO Teubner E We i g o l d PS Bagus and EK V i i n i k k a Phys Rev A 15 102 D i e r c k s e n 163 GHF D i e r c k s e n and WP Rraemer MUNICH M o l e c u l a r Program System R e f e r e n c e Manual S p e c i a l T e c h n i c a l Report Max-Planck I n s t i t u t f u r P h y s i k und A s t r o p h y s i k D i e r c k s e n (1974) 163 GHF D i e r c k s e n Theor Chim A c t a 3_3 1 D i r a c (1958) 33 PAM D i r a c The P r i n c i p l e s of Quantum Mechanics O x f o r d U n i v e r s i t y P r e s s London Dixon (1977) 125 AJ Dixon S Dey IE McCarthy E W e i g o l d and GRJ W i l l i a m s Chem Phys 2± 81 Domcke (1975) 192 W Domcke and LS Cederbaum J Chem Phys 64 612 Domcke (1978) 126,139,141 W Domcke LS Cederbaum J Schirmer W von N i e s s e n and JP M a i e r J E l Spect _1_4 59 Domcke (1979) 29,167,169,174,186,187,188 W Domcke LS Cederbaum J S c h i r m e r W von N i e s s e n CE B r i o n and KH Tan Chem Phys 40 171 289 E p s t e i n (1971) 280 IR E p s t e i n Chem Phys L e t t e r s 9 9 E p s t e i n (1973) 31 IR E p s t e i n Acc Chem Res 6 145 E p s t e i n (1977) 31 IR E p s t e i n and AC Tanner Compton S c a t t e r i n g (ed B W i l l i a m s ) M c G r a w - H i l l I n t e r n a t i o n a l G i a r d i n i (1977) 170,174,187 A G i a r d i n i - G u i d o n i R T i r i b e l l i D V i n c i g u e r r a R C a m i l l o n i and G S t e f a n i J E l Spect J_2 405 Goddard (1978) 30,141 J Goddard and IG C s i z m a d i a J Chem Phys 68 2172 Guest (1976) 134 MF Guest and WR Ro d w e l l Mol Phys 32 1075 Hamnett (1976) 21 A Hamnett W S t o l l G Br a n t o n CE B r i o n and MJ van der W i e l J Phys B 9 945 Hamnett (1977) 17,125 A Hamnett ST Hood and CE B r i o n J E l Spect U_ 263 H i l l i e r (1970) 134 LH H i l l i e r and VR Saunders Chem Phys L e t t e r s 5 384 H i l l i e r (1971) 144 IH H i l l i e r and VR Saunders Mol Phys 22 193 Hood (1973) 4 ST Hood IE McCarthy PJO Teubner and E W e i g o l d Phys Rev A 8 2494 Hood (1976a) 30,125 ST Hood A Hamnett and CE B r i o n Chem Phys L e t t e r s 39 252 290 Hood (1976b) 17 ST Hood A Hamnett and CE B r i o n Chem Phys L e t t e r s 4J_ 428 Hood (1977) 30,77,84,125,133,135,141,238 ST Hood A Hamnett and CE B r i o n J E l Spect j j _ 205 Huzinaga (1965) 162 S Huzinaga J Chem Phys 42 1293 I n o k u t i (1971) 19,21,118 M I n o k u t i Rev Mod Phys _43 297 K a i j s e r (1977) 280 P K a i j s e r and VH Smith J r Adv Quant Chem j_0 37 K a r l s s o n (1976) 127 L K a r l s s o n L M a t t s s o n R J a d r n y T Bergmark and K Siegbahn Phys S c r i p t a J_3 229 Komarov (1976) 280 FF Komarov and MM Temkin J Phys B 9 L255 Koopmans (1933) 22 TA Koopmans P h y s i c a J_ 104 Kouba (1971) 203,212 JE Kouba and Y Ohrn I n t J Quant Chem 5 539 Kunz 58,72,203,213 AB Kunz and K B e d f o r d p r i v a t e communication Kwart (1977) 145 H Kwart and KG K i n g ' R e a c t i v i t y and s t r u c t u r e c o n c e p t s i n o r g a n i c c h e m i s t r y v o l 1 ' ' d - o r b i t a l s i n the c h e m i s t r y of s i l i c o n phosphorous and s u l p h u r ' S p r i n g e r B e r l i n L e v i n (1975) 71,278,280 VG L e v i n VG Ne u d a t c h i n AV P a v l i t c h e n k o v and YuF Smirnov J Chem Phys 63 1541 291 Lohmann 262 B Lohmann and E We i g o l d p r i v a t e communication M a i e r 139 JP M a i e r p r i v a t e communication Mattuck (1967) 24 RD Mattuck A Guide t o Feynman Diagrams i n the Many-body Problem M c G r a w - H i l l London McCarthy (1976a) 4,7,17,18,125 IE McCarthy and E W e i g o l d Phys Rep C 21_ 275 McCarthy (1976b) 4,18,125 IE McCarthy and E We i g o l d Adv Phys 25 489 Me s s i a h (1958) 276 A M e s s i a h Quantum Mechanics N o r t h - H o l l a n d Amsterdam Moore (1978) 223,242 JH Moore MA Coplan TL S k i l l m a n and ED Brooks Rev S c i I n s t r 49 463 Neudachin (1969) 4 VG Neudachin GA N o v o s k o l ' t s e v a YuF Smirnov S o v i e t P h y s i c s JETP 28 540 R a b a l a i s (1977) 17,138 J R a b a l a i s P r i n c i p l e of u l t r a v i o l e t p h o t o e l e c t r o n s p e c t r o s c o p y W i l e y New York R a t n e r (1971) 144 MA Ratner and JR Sa b i n J Am Chem Soc 93 3542 R i s l e y (1972) 88,239 JS R i s l e y Rev S c i I n s t r 43 95 Rothenberg (1970) 143,144 S Rothenberg and HF S c h a e f e r J Chem Phys 53 3014 292 Sc h i r m e r (1977) 29,157,167 J S c h i r m e r LS Cederbaum W Domcke and W von N i e s s e n Chem Phys 26 149 Schi r m e r (1978) 27 J S c h i r m e r and LS Cederbaum J Phys B J J _ 1889 Siegbahn (1969) 17 K Siegbahn C N o r d l i n g G Johansson J Hedman PF Heden K Harmin U G e l i u s T Bergmark L-0 Werme R Manne and Y Baer ESCA A p p l i e d t o Free M o l e c u l e s N o r t h - H o l l a n d Amsterdam Siegbahn (1972) 136,138 K Siegbahn C N o r d l i n g G Johansson J Hedman PF Heden K Harmin U G e l i u s T Bergmark L-0 Werme R Manne and Y Baer ESCA A p p l i e d t o Free M o l e c u l e s N o r t h - H o l l a n d Amsterdam S i n a n o g l u (1966) 203 0 S i n a n o g l u and C H o l l i s t e r J Am Chem Soc 88 13 Su z u k i (1980a) 125 IH S u z u k i CE B r i o n E W e i g o l d and GRJ W i l l i a m s I n t J Quant Chem J_8 275 S u z u k i (1980b) 195,210,213 IH S u z u k i E Weigold and CE B r i o n J E l Spect 20 289 Th o u l e s s (1961) 24 DJ T h o u l e s s The Quantum Mechanics of Many-Body Systems Academic P r e s s New York T o s s e l l (1981) 198,261,263 JA T o s s e l l JH Moore MA Coplan J E l Spect 22 61 Turner (1970) 127,136,141,173,175,195,196,197 DW Turner C Baker and CR B r u n d l e M o l e c u l a r p h o t o e l e c t r o n s p e c t r o s c o p y W i l e y I n t e r s c i e n c e Van Hoof (1980) 227 HA van Hoof and MJ van der W i e l J Phys E J_3 409 293 V e i l l a r d (1968) 163 A V e i l l a r d Theor Chim A c t a J_2 405 Von N i e s s e n (1977) 144 W von N i e s s e n LS Cederbaum W Domcke and GHF D i e r c k s e n J Chem Phys 66 4893 W e i g o l d (1973) 125 E W e i g o l d ST Hood and PJO Teubner Phys Rev L e t t e r s 30 475 We i g o l d (1975) 125 E W e i g o l d ST Hood and IE McCarthy Phys Rev A j j _ 566 Wei g o l d (1976) 125 E W e i g o l d S Dey A J Dixon IE McCarthy and PJO Teubner Chem Phys L e t t e r s 41 21 Wei g o l d (1978) 4 E W e i g o l d and IE McCarthy Adv At Mol Phys j_4 127 White (1980) 161,168 MJ White T Leung and CE B r i o n J E l Spect 23 127 W i l l i a m s (1978) 89 JF W i l l i a m s J Phys B J J _ 2015 Z e i s s 135,136 GD Z e i s s p r i v a t e communication Z e i s s (1979) 30,146 GD Z e i s s WR S c o t t N S u z u k i and DP Chong Mol Phys 37 1543 'Why, you have n e a r l y f i n i s h e d i t , Mr. Frodo!' Sam e x c l a i m e d . ' W e l l , you have kept a t i t , I -must say.' 'I have q u i t e f i n i s h e d i t , Sam,' s a i d Frodo. 

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