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Experimental quantum chemistry by binary (e,2e) spectroscopy Leung, Kam Tong 1984

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EXPERIMENTAL QUANTUM CHEMISTRY BY BINARY (e,2e) SPECTROSCOPY by KAM TONG LEUNG B . S c , U n i v e r s i t y Of B r i t i s h C o l u m b i a , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF . THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES C h e m i s t r y D e p a r t m e n t 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 THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t 1984 © Kam Tong L e u n g , 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requ i rements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Co lumbia , I agree tha t the L i b r a r y 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 r e f e r e n c e and s tudy . I f u r t h e r agree tha t p e r m i s s i o n fo r e x t e n s i v e copy ing of t h i s t h e s i s f o r s c h o l a r l y purposes may be g ran ted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s unders tood tha t copy ing or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l l owed wi thout my w r i t t e n p e r m i s s i o n . Department of Chemis t ry The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver , Canada V6T 1W5 Da te : August 1984 i i A b s t r a c t The n o t i o n tha t chemis t s might b e n e f i t by l o o k i n g at mo lecu l a r o r b i t a l s and chemica l bonding phenomena from the complementary momentum-space p e r s p e c t i v e was f i r s t suggested by Cou lson and Duncanson some f o r t y years ago . With the development of b i na r y (e ,2e) spec t ro s copy in the l a s t decade, expe r imen ta l momentum d e n s i t i e s of i n d i v i d u a l o r b i t a l s can now be measured d i r e c t l y and t h i s has p r o v i d e d the f i r s t " r e a l " look at mo lecu la r bonding in the l a b o r a t o r y . B inary (e ,2e) spec t roscopy measures the b i n d i n g energy spectrum and the s p h e r i c a l l y averaged momentum d i s t r i b u t i o n us ing h igh energy e l e c t r o n impact i o n i z a t i o n and c o i n c i d e n c e d e t e c t i o n t e c h n i q u e s . The exper imenta l o r b i t a l momentum d i s t r i b u t i o n s not on l y have he lped to i d e n t i f y the symmetry (s-type or p-t y p e ) , o rder and nature of the c h a r a c t e r i s t i c o r b i t a l i n v o l v e d in the i o n i z a t i o n p r o c e s s , but a l s o have made i t p o s s i b l e to s t r i n g e n t l y e va l ua t e the q u a l i t y of ab-initio s e l f - c o n s i s t e n t - f i e l d wave func t i ons . The v a l e n c e - s h e l 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 of the noble gases and a number of sma l l mo lecu les i n c l u d i n g H 2 , C 0 2 , C S 2 , OCS and CF« have been measured us i ng a s t a t e - o f - t h e - a r t b i na r y (e ,2e) spec t romete r . An e x i s t i n g spect rometer has been m o d i f i e d to p rov i de h igh momentum and t i m i n g r e s o l u t i o n s as we l l as s u f f i c i e n t energy r e s o l u t i o n f o r r e s o l v i n g most of the s t r u c t u r e s of the s p e c i e s r e p o r t e d . New and d e f i n i t i v e r e s u l t s on the va lence-s h e l l e l e c t r o n i c s t r u c t u r e and o r b i t a l bonding p a t t e r n of these s p e c i e s have been o b t a i n e d . P o s s i b l e chemica l t r ends in the e l e c t r o n i c s t r u c t u r e and o r b i t a l d e n s i t i e s in the noble gas group and in the va l ence i s o e l e c t r o n i c l i n e a r t r i a t o m i c g roup : C 0 2 , CS 2 and OCS have been i n v e s t i g a t e d . Computer genera ted d e n s i t y contour maps and t h r e e - d i m e n s i o n a l o r b i t a l d e n s i t y v i s u a l i z a t i o n of t h e o r e t i c a l wave func t ions in both p o s i t i o n and momentum space are used to f a c i l i t a t e i n t e r p r e t a t i o n of the expe r imen ta l momentum d i s t r i b u t i o n s . T h i s d e n s i t y t o p o g r a p h i c a l approach i s i n s t r u m e n t a l in ex tend ing the present unde rs tand ing of momentum-space chemica l p r o p e r t i e s . In p a r t i c u l a r , such an approach has p r o v i d e d a new and complementary p i c t u r e of the c o v a l e n t bond in mo l ecu l a r hydrogen in momentum-space. The f i r s t expe r imen ta l e s t i m a t i o n of the s p h e r i c a l l y averaged bond d e n s i t y in momentum-space by b ina r y (e ,2e) spec t ro s copy i s a l s o a t t a i n e d . Table of Contents A b s t r a c t i i L i s t of Tab l e s v i i L i s t of F i g u r e s Acknowledgement x i i i Chapter I INTRODUCTION 1 1.1 MOMENTUM DISTRIBUTIONS, ORBITALS AND QUANTUM CHEMISTRY 1 1.2 BINARY (e ,2e) SPECTROSCOPY AND CONVENTIONAL METHODS 7 1.3 BRIEF HISTORICAL REMARKS 12 1.4 MAJOR APPLICATIONS 17 1.4.1 V a l e n c e - S h e l l B i n d i n g Ene rg i e s 18 1.4.2 O r b i t a l Symmetry 19 1.4.3 O r b i t a l O r d e r i n g 20 1.4.4 Assignment Of The Many-body S t a t es 20 1.4.5 S t r i n g e n t Tes t Of A b - i n i t i o Wavefunct ions 22 1.5 SCOPE OF THE THESIS 23 Chapter II THEORETICAL BACKGROUND 26 2.1 REACTION THEORY 26 2.1.1 S c a t t e r i n g K inemat i c s 27 2 .1 .2 D i f f e r e n t i a l C ross S e c t i o n . . 3 0 2 .1 .3 P lane Wave Impulse Approx imat ion 31 2 .1 .4 Ta rge t Har t ree-Fock Approx imat ion 34 2 .1 .5 V a l i d i t i e s Of The Approx imat ions 38 2.2 MOMENTUM-SPACE LCAO-MO WAVEFUNCTIONS 40 2.2.1 F o u r i e r T rans fo rm P r o p e r t i e s 43 2 .2 .2 V i r i a l P rope r t y 49 Chapter III EXPERIMENTAL METHOD 50 3.1 INTRODUCTION 50 3.2 BINARY (e ,2e) SPECTROMETER 53 3.2.1 E l e c t r o n Gun 55 3 .2 .2 Gas C e l l ; 57 3 .2 .3 Beam C o l l i m a t o r 58 3 .2 .4 Three-element Asymmetric Immersion Lens 59 3 .2 .5 Sec to r C y l i n d r i c a l M i r r o r Ana l y se r 62 3 .2 .6 S i n g l e Channel E l e c t r o n M u l t i p l i e r . . . 6 5 3 .2 .7 Computer S i m u l a t i o n Of The E l e c t r o n T r a j e c t o r y 68 3 .2 .8 Vacuum And M a t e r i a l S p e c i f i c a t i o n s 70 3.3 CONTROL ELECTRONICS 72 3.3.1 Power S u p p l i e s . . . . 7 3 3 .3 .2 T iming E l e c t r o n i c s . . . . 7 6 V 3.4 COMPUTER CONTROL 82 3.4.1 Parameters Scanning 84 3.4.2 Event Counting 86 3.5 OPERATION PROCEDURE AND SAMPLE HANDLING 87 3.6 PERFORMANCE 90, 3.6.1 R e s o l u t i o n s 90 3.6.2 Signal-To-Noise R a t i o 97 3.6.3 L i m i t a t i o n s 99 Chapter IV NOBLE GASES 101 4.1 INTRODUCTION 101 4.2 AN OVERVIEW 104 4.2.1 Helium 106 4.2.2 Neon 109 4.2.3 Argon 114 4.2.4 Krypton 119 4.2.5 Xenon 124 4.3 SATELLITE STRUCTURE 128 4.4 OUTER VALENCE ORBITAL ELECTRON MOMENTUM DISTRIBUTIONS 133 Chapter V TWO ELECTRON SYSTEMS: HELIUM AND MOLECULAR HYDROGEN ...146 5.1 INTRODUCTION 146 5.2 GROUND-STATE BINDING ENERGY SPECTRA 150 5.3 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTIONS 152 5.4 MOMENTUM-SPACE CHEMISTRY AND ORBITAL DENSITY TOPOGRAPHY 159 5.4.1 D e n s i t y Mapping Convention 159 5.4.2 He 1s Atomic O r b i t a l 163 5.4.3 H 2 1a O r b i t a l 166 5.4.4 H 2 1a= O r b i t a l 172 5.4.5 M o l e c u l a r D e n s i t y D i r e c t i o n a l R e v e r s a l In H 2 .176 5.4.6 M o l e c u l a r D e n s i t y O s c i l l a t i o n In H 2 180 5.4.7 Bond Formation In H 2 183 Chapter VI BOND DENSITY OF MOLECULAR HYDROGEN IN MOMENTUM SPACE ..190 6.1 INTRODUCTION 190 6.2 ESTIMATION OF THE ORBITAL MOMENTS AND NORMALIZATION OF THE MOMENTUM DISTRIBUTION 193 6.3 SPHERICALLY AVERAGED MOMENTUM-SPACE BOND DENSITY 199 6.4 DIRECTIONAL BOND DENSITY 202 6.4.1 Wavefunction Dependence 202 6.4.2 Dependence Of I n t e r n u c l e a r S e p a r a t i o n 207 Chapter VII CARBON DIOXIDE 213 7.1 INTRODUCTION 213 7.2 OUTER VALENCE BINDING ENERGY SPECTRUM 214 7.3 MOMENTUM DISTRIBUTIONS OF OUTER VALENCE ORBITALS 217 7.4 THEORETICAL ORBITAL MOMENTUM DISTRIBUTIONS AND ORBITAL DENSITY TOPOGRAPHY 221 Chapter VIII CARBON DISULPHIDE 232 8.1 INTRODUCTION 232 8.2 VALENCE-SHELL BINDING ENERGY SPECTRUM 234 8.3 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTIONS 242 8.3.1 Outer Valence O r b i t a l s 243 8.3.2 Inner Valence O r b i t a l s 252 8.4 SUMMARY 259 Chapter IX CARBONYL SULPHIDE 262 9.1 INTRODUCTION 262 9.2 VALENCE-SHELL BINDING ENERGY SPECTRUM 265 9.3 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTIONS 270 9.3.1*Outer Valence O r b i t a l s 271 9.3.2 Inner Valence O r b i t a l s 280 9.4 MANY-BODY STRUCTURE OF C0 2, CS 2 AND OCS 284 9.5 MOMENTUM-SPACE CHEMISTRY OF C0 2, CS 2 AND OCS ...287 9.5.1 Outer Valence Antibonding IT O r b i t a l s ...288 9.5.2 Outer Valence Bonding a O r b i t a l s 294 Chapter X CARBON TETRAFLUORIDE 298 10.1 INTRODUCTION 298 10.2 VALENCE BINDING ENERGY ELECTRONIC STRUCTURE ...300 10.3 ORBITAL MOMENTUM•DENSITIES 309 10.3.1 S p h e r i c a l l y Averaged Momentum D e n s i t i e s 313 10.3.2 D i r e c t i o n a l Momentum-Space Densi t y Topography In Two- And Three-Dimensions .." 328 Chapter XI CONCLUDING REMARKS 333 REFERENCES 339 APPENDIX A - BIBLIOGRAPHY OF BINARY (e,2e) SPECTROSCOPIC STUDIES 356 v i i L i s t of Tables 2.1 Summary of LCAO-MO wave func t ion and d e n s i t y equa t i ons in p o s i t i o n and momentum space 42 3.1 C o n f i g u r a t i o n of the microcomputer system 83 4.1 S a t e l l i t e s t r u c t u r e of Ne 111 4.2 S a t e l l i t e s t r u c t u r e of Ar 116 4.3 S a t e l l i t e s t r u c t u r e of Kr 122 4.4 S a t e l l i t e s t r u c t u r e of Xe 126 4.5 T o t a l a s s i gned s a t e l l i t e i n t e n s i t y r e l a t i v e to the parent fo r the ns i o n i z a t i o n of Noble Gases . 132 5.1 Comparison of t h e o r e t i c a l wave func t ions 161 5.2 Maximum a b s o l u t e v a lues of d e n s i t y d i f f e r e n c e f o r H 2 1a as a f u n c t i o n of R 187 9 6.1 O r b i t a l momentum moments ( i n atomic u n i t s ) of H 2 l a o r b i t a l 197 6.2 Maximum a b s o l u t e va lues of bond d e n s i t y as a f u n c t i o n of R 209 9.1 P-space p e r i o d i c i t i e s of momentum d e n s i t y o s c i l l a t i o n s and i n t e r n u c l e a r s e p a r a t i o n s of a n t i b o n d i n g u o r b i t a l s 293 10.1 Expe r imen ta l and t h e o r e t i c a l b i n d i n g e n e r g i e s and p o l e - s t r e n g t h s f o r C F , 307 v i i i L i s t of F i g u r e s 2.1 S c a t t e r i n g k inemat i c s of symmetric non-coplanar b i n a r y (e ,2e) r e a c t i o n 30 2.2 Mott 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 the symmetric non-coplanar k inemat i c s wi th E 0 = 1 2 l 5 . 6 e V , E ,=E 2 = 600eV and 6=45° 34 3.1 B lock diagram of the b i n a r y (e ,2e) s p e c t r o -system 52 3.2 Schematic of b i n a r y (e ,2e) spec t rometer 54 3.3 S i g n a l d e c o u p l i n g arrangement f o r the channe l t r on 68 3.4 Computer s i m u l a t i o n of e l e c t r o n t r a j e c t o r i e s f o r e l e c t r o n s p a s s i n g through the secondary e l e c t r o n o p t i c a l system w i th 600eV k i n e t i c e n e r g i e s 69 3.5 Power s u p p l i e s f o r the e l e c t r o n o p t i c a l e lements 74 3.6 T iming e l e c t r o n i c s and c o m p u t e r - c o n t r o l l e d da ta a c q u i s i t i o n system 77 3.7 T y p i c a l output pu l se-shapes f o r the p r e a m p l i f i e r ( top) and the t ime f i l t e r a m p l i f i e r (bottom) 79 3.8 T y p i c a l t ime spectrum fo r argon 3p i o n i z a t i o n 81 3.9 T y p i c a l t r a n s m i s s i o n f u n c t i o n f o r the secondary e l e c t r o n o p t i c a l system o p e r a t i n g w i th a lOOeV pass energy f o r the CMA 88 3.10 T y p i c a l b i n d i n g energy spectrum fo r he l ium Is i o n i z a t i o n at 0=0° 91 3.11 The c o l l i s i o n r e g i o n r e s u l t i n g from 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 acceptance cones of the l e n s e s 94 3.12 E f f e c t s of f o l d i n g d i f f e r e n t " e f f e c t i v e " angu la r spreads i n t o t h e o r e t i c a l wave funct ions 96 4.1 B i n d i n g energy spectrum fo r He 1s i o n i z a t i o n at 0=0° 107 4.2 Atomic momentum d i s t r i b u t i o n of He Is o r b i t a l . . . . 1 0 8 4.3 B i n d i n g energy spectrum fo r Ne at o>=0° and 1 0 ° . . . 110 4.4 Atomic momentum d i s t r i b u t i o n s f o r Ne 2p and 2s o r b i t a l s . . 113 4.5 Angu la r c o r r e l a t e d b i n d i n g energy spectrum f o r Ar 115 4.6 Atomic momentum d i s t r i b u t i o n s f o r Ar 119 4.7 Angu la r c o r r e l a t e d b i n d i n g energy spectrum fo r Kr 120 4.8 Atomic momentum d i s t r i b u t i o n s f o r Kr 124 4.9 Angu la r c o r r e l a t e d b i n d i n g energy spectrum f o r Xe 125 4.10 Atomic momentum d i s t r i b u t i o n s f o r Xe 128 ix 4.11 Comparison of va l ence b i n d i n g energy s p e c t r a of Noble Gases 1 30 4.12 Momentum d i s t r i b u t i o n s ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps f o r np o r b i t a l s of Noble Gases 134 4.13 Momentum d i s t r i b u t i o n s ( l e f t ) , momentum d e n s i t y ( cen t re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps fo r ns o r b i t a l s of Noble Gases 135 4.14 Comparison of extended range Har t ree-Fock s p h e r i c a l l y averaged momentum d i s t r i b u t i o n s f o r np (do t ted l i n e s ) and ns ( s o l i d l i n e s ) of Noble Gases 142 4.15 Comparison of extended range Har t ree-Fock s p h e r i c a l l y averaged p o s i t i o n d i s t r i b u t i o n s f o r np (do t t ed l i n e s ) and ns ( s o l i d l i n e s ) of Noble Gases 1 43 4.16 Comparison of the o r b i t a l s p a t i a l ex tent of the ns and np o r b i t a l s of Noble Gases i n p o s i t i o n and momentum space 145 5.1 B i n d i n g energy spectrum fo r He 1s i o n i z a t i o n at 0=0° 151 5.2 B i n d i n g energy spectrum fo r H 2 l a i o n i z a t i o n at 0=0° .? 152 5.3 Atomic momentum d i s t r i b u t i o n f o r He 1s o r b i t a l . . . 155 5.4 Comparison of a) expe r imen ta l mo lecu la r momentum d i s t r i b u t i o n s and b) c a l c u l a t e d momentum d i s t r i b u t i o n s f o r H 2 l a o r b i t a l 157 5.5 D e n s i t y contour maps of He 1s o r b i t a l ( t o p ) , the independent atom model ( cent re ) and d e n s i t y d i f f e r e n c e maps in momentum-space (LHS) and p o s i t i o n - s p a c e (RHS) 164 5.6 Contour maps of q u a s i - c l a s s i c a l d e n s i t y ( t o p ) , i n t e r a c t i o n d e n s i t y ( cen t re ) and the mo lecu l a r d e n s i t y (bottom) f o r H 2 1a i n momentum-space (LHS) and p o s i t i o n - s p a c e (RHS) 167 5.7 D e n s i t y contour maps of H 2 1a o r b i t a l ( t o p ) , the independent atom model ( cen t re ) and d e n s i t y d i f f e r e n c e maps in momentum-space (LHS) and p o s i t i o n - s p a c e (RHS) 169 5.8 Contour maps of q u a s i - c l a s s i c a l d e n s i t y ( t o p ) , i n t e r a c t i o n d e n s i t y ( cen t re ) and the mo lecu l a r d e n s i t y (bottom) f o r H 2 l a i n momentum-space (LHS) and p o s i t i o n - s p a c e (fciS) . .173 5.9 D e n s i t y con tour maps of H 2 l a o r b i t a l ( t o p ) , the independent atom model ( cent re ) and d e n s i t y d i f f e r e n c e maps i n momentum-space (LHS) and p o s i t i o n - s p a c e (RHS) : 175 5.10 Th ree-d imens iona l d e n s i t y s u r f a c e v i s u a l i z a t i o n of H 2 1a o r b i t a l in momentum-space (top) and p o s i t i o n ^ s p a c e (bottom) 177 X 5.11 Th ree-d imens iona l d e n s i t y s u r f a c e v i s u a l i z a t i o n of H 2 1tf o r b i t a l in momentum-space (top) and p o s i t i o n - s p a c e (bottom) 178 5.12 Extended range d e n s i t y contour maps of H 2 1a (LHS) and 1a (RHS) i n momentum-space ( top) and p o s i t ion-space (bottom) 181 5.13 D e n s i t y d i f f e r e n c e contour maps in p o s i t i o n and momentum space showing the dynamics of the fo rmat ion of the H 2 1a o r b i t a l as a f u n c t i o n of the i n t e r n u c l e a r s e p a r a t i o n R. 185 6.1 Comparison of the expe r imen ta l s p h e r i c a l l y averaged momentum d i s t r i b u t i o n of the H 2 1a o r b i t a l w i th the f i t t e d s e m i - e m p i r i c a l dens r t y f u n c t i o n s : OLDF, BSZ and TTP 196 6.2 S p h e r i c a l l y averaged momentum-space bond d e n s i t y of H 2 200 6.3 D e n s i t y d i f f e r e n c e (bond d e n s i t y ) maps in momentum and p o s i t i o n space as f u n c t i o n of the type of the H 2 1a wavefunct ion at e q u i l i b r i u m s e p a r a t i o n . . .203 6.4 Th ree-d imens iona l s u r f a c e p l o t of the d e n s i t y d i f f e r e n c e (bond d e n s i t y ) of H 2 in momentum and p o s i t i o n space 205 6.5 D e n s i t y d i f f e r e n c e (bond d e n s i t y ) maps in momentum and p o s i t i o n space as a f u n c t i o n of i n t e r n u c l e a r s e p a r a t i o n R 208 7.1 Outer va l ence b i n d i n g energy s p e c t r a f o r C 0 2 at 0=0° , 8 ° and 14° 216. 7.2 Mo l e cu l a r momentum d i s t r i b u t i o n s f o r C 0 2 ou te r va l ence o r b i t a l s 218 7.3 T h e o r e t i c a l o r b i t a l momentum d i s t r i b u t i o n s f o r C 0 2 e v a l ua t ed u s i n g the Snyder and Basch wave func t ion [SB72] 223 7.4 D e n s i t y contour maps f o r 1 ir and I T o r b i t a l s i n momentum-space ( l e f t ) and p o s i t i o n - s p a c e ( r i g h t ) 226 7.5 D e n s i t y contour maps f o r 3a , 4a , 2a and 3a o r b i t a l s i n momentum-space ( l e f t ) ancl p o s i t i o n -space - ( r ight ) 227 7.6 D e n s i t y contour maps f o r 2a , 1a and 1a o r b i t a l s i n momentum-space" ( le f t ) and p o s i t i o n -space ( r i g h t ) 228 7.7 Th ree-d imens iona l cons tan t d e n s i t y s u r f a c e p l o t s f o r C 0 2 Iff , 1 rr , 3a and 4a o r b i t a l s i n momentum-space ( top? and p o s i t i o n - s p a c e (bottom) 230 8.1 Outer va l ence b i n d i n g energy s p e c t r a f o r C S 2 at 0=0° , 4 ° , 8 ° , 12° and 14° 235 8.2 Va lence s h e l l b i n d i n g energy s p e c t r a f o r C S 2 at 0=0° and 8 ° 238 x i 8.3 T h e o r e t i c a l v a l e n c e - s h e l l b i n d i n g energy s p e c t r a f o r CS 2 240 8.4 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 ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the CS 2 2TT o r b i t a l 245 8.5 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 ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the C S 2 2-n o r b i t a l 247 8.6 M o l e c u l a r o r b i t a l momentum a i s t r i b u t i o n s ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the CS 2 5a o r b i t a l 249 8.7 M o l e c u l a r o r b i t a l momentum a i s t r i b u t i o n s ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the C S 2 6a o r b i t a l 251 8.8 Momentum d i s t r i b u t i o n ( l e f t ? measured at 21.2eV, momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of CS 2 4a o r b i t a l 254 8.9 Momentum d i s t r i b u t i o n ( l e f t ) measured at 33.1eV, momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the CS 2 5a o r b i t a l 255 8.10 Momentum d i s t r i b u t i o n measured at 24.3eV 257 8.11 Momentum d i s t r i b u t i o n measured at 27.5eV . .258 9.1 V a l e n c e - s h e l l b i n d i n g energy s p e c t r a f o r OCS at 0=0° and 8 ° 266 9.2 T h e o r e t i c a l v a l e n c e - s h e l l b i n d i n g energy s p e c t r a f o r OCS 268 9.3 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 ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the OCS 3TT o r b i t a l 272 9.4 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 ( l e f t ; ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the OCS 2it o r b i t a l 276 9.5 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 measured at 16.5eV ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the OCS 9a o r b i t a l 277 9.6 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 ( l e f t ) , momentum d e n s i t y ( cen t re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the OCS 8a o r b i t a l 279 9.7 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 ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the OCS la o r b i t a l 282 9.8 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 ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the OCS 6a o r b i t a l 283 9.9 Comparison of the v a l e n c e - s h e l l b i n d i n g energy s p e c t r a f o r C 0 2 [CB82a] , C S 2 ( chapter 8) and OCS at 0=0° 285 x i i 9.10 Comparison of momentum d i s t r i b u t i o n s ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the C 0 2 17r , CS 2 27r and OCS 3TT o r b i t a l s ? ? 289 9.11 Comparison of momentum d i s t r i b u t i o n s ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the C 0 2 4a , CS 2 6a and OCS 8a o r b i t a l s ? ? 295 10.1 Approximate mo lecu l a r o r b i t a l d iagram fo r the v a l e n c e - s h e l l of C F , 301 10.2 V a l e n c e - s h e l l b i n d i n g energy s p e c t r a f o r CF , a t 0=0° and 8 ° 303 10.3 Outer va l ence b i n d i n g energy s p e c t r a f o r C F , at 0=0° , 5 ° , 1 3 ° , 17° and 23° 304 10.4 T h e o r e t i c a l v a l e n c e - s h e l l b i n d i n g energy s p e c t r a f o r CF , c a l c u l a t e d us i ng the many-body 2ph-TDA G r e e n ' s f u n c t i o n method 306 10.5 M o l e c u l a r geometry of C F , and the d e f i n i t i o n of the contour p l ane 311 10.6 M o l e c u l a r momentum d i s t r i b u t i o n s ( l e f t ) and d e n s i t y v i s u a l i z a t i o n in momentum ( c e n t r e ) ' a n d p o s i t i o n ( r i g h t ) space f o r the C F , I t , o r b i t a l . .315 10.7 Momentum d i s t r i b u t i o n measured at l 7 .4eV ( l e f t ) and d e n s i t y v i s u a l i z a t i o n in momentum ( cent re ) and p o s i t i o n ( r i g h t ) space fo r the C F , 4 t 2 o r b i t a l . .316 10.8 Mo l e cu l a r momentum d i s t r i b u t i o n s ( l e f t ) and d e n s i t y v i s u a l i z a t i o n in momentum ( cent re ) and p o s i t i o n ( r i g h t ) space f o r the CF , 1e o r b i t a l 317 10.9 M o l e c u l a r momentum d i s t r i b u t i o n s ( l e f t ) and d e n s i t y v i s u a l i z a t i o n i n momentum ( cent re ) and p o s i t i o n ( r i g h t ) space f o r the CF , 3 t 2 o r b i t a l . .321 10.10 M o l e c u l a r momentum d i s t r i b u t i o n s ( l e f t ) and d e n s i t y v i s u a l i z a t i o n in momentum ( cent re ) and p o s i t i o n ( r i g h t ) space f o r the CF , 4a , o r b i t a l . .322 10.11 M o l e c u l a r momentum d i s t r i b u t i o n s ( l e f t ) and d e n s i t y v i s u a l i z a t i o n in momentum ( cen t re ) and p o s i t i o n ( r i g h t ) space fo r the C F , 2 t 2 o r b i t a l . .325 10.12 Mo l e cu l a r momentum d i s t r i b u t i o n s ( l e f t ) and d e n s i t y v i s u a l i z a t i o n i n momentum ( cent re ) and p o s i t i o n ( r i g h t ) space fo r the C F , 3a , o r b i t a l . .326 10.13 Th ree-d imens iona l 2% d e n s i t y s u r f a c e p l o t s of the 4 t 2 ( l e f t ) and 3a , ( r i g h t ) o r b i t a l s i n momentum-space 331 A.1 Chemica l r e f e r e n c e d iagram. 361 x i i i Acknowledgement I would l i k e to thank my r e s e a r c h s u p e r v i s o r , P r o f e s s o r C h r i s t o p h e r E. B r i o n , f o r the many t h i n g s tha t I have l e a rned from him both c o n s c i o u s l y and u n c o n s c i o u s l y . I t has indeed been my p r i v i l e g e and honor not on l y to be gu ided by h i s academic i n s i g h t s but a l s o perhaps more i m p o r t a n t l y to be i n f l u e n c e d by h i s approach to human r e l a t i o n s . I am most g r a t e f u l f o r h i s c o n t i n u a l suppor t throughout the course of t h i s work and e s p e c i a l l y f o r h i s encouragement and adv i c e at c r i t i c a l moments. I owe a s p e c i a l g r a t i t u d e to D r . John P.D. Cook f o r h i s i n v a l u a b l e a s s i s t a n c e d u r i n g the i n i t i a l phase of t h i s work. I wish a l s o to thank P r o f e s s o r s D .C . Chong, J . A . R . Coope, R .F . Sn ide r and L .S . We i l e r of the Chemis t ry Department and P r o f e s s o r P.W. Langhof f of Chemis t ry Department at Ind iana U n i v e r s i t y f o r many i l l u m i n a t i n g i n t e r a c t i o n s , d i s c u s s i o n s and p e r s o n a l a d v i c e . Many thanks are a l s o due to P r o f e s s o r M.J . van der W ie l and h i s co-workers of the FOM I n s t i t u t e f o r t h e i r a s s i s t a n c e i n the l e n s t r a j e c t o r y program and t h e i r ve ry k ind h o s p i t a l i t y d u r i n g my two week v i s i t to Amsterdam. I wish a l s o to thank D r . W. Lewis-Bevan f o r h i s i n t e r e s t and c a r e f u l p r o o f - r e a d i n g of t h i s t h e s i s . I would l i k e to g r a t e f u l l y acknowledge the s t a f f , members of the Chemis t ry Department f o r t h e i r t e c h n i c a l , s e c r e t a r i a l x i v and g r aph i c s u p p o r t . In p a r t i c u l a r , I w ish to thank c h i e f e n g i n e e r e r s B. Powel l (Mechanica l Shop) and J . S a l l o s ( E l e c t r o n i c Shop) not on l y fo r t h e i r i n t e r e s t and e f f i c i e n c y in job c o o r d i n a t i o n but a l s o fo r t h e i r i n c r e d i b l e t o l e r a n c e to my p e r s i s t e n c e in g e t t i n g every job done spon taneous l y . For the many e x c e l l e n t and i ns t an taneous r e p a i r and c o n s t r u c t i o n jobs of the many p o t e n t i a l t e c h n i c a l n igh tmares , I wish to express my deepest g r a t i t u d e to E. Gomm, J . Edwards and C . M c C a f f e r t y of the Mechan i ca l Shop and to B. Greene , P. Carpenda le and H. Chow of the E l e c t r o n i c Shop. For w i thout the t e c h n i c a l a s s i s t a n c e and enthus iasm of these p e o p l e , the work r e p o r t e d in t h i s t h e s i s would not be p o s s i b l e . I wish a l s o to thank A. Rees and B. Gray of the Main O f f i c e f o r t h e i r s e c r e t a r i a l adv i c e and s p e c i a l a s s i s t a n c e , and to P. Parsons and E. Jensen fo r t h e i r g r aph i c a s s i s t a n c e . The t e c h n i c a l s t a f f of the Computer Cen t re shou ld a l s o be acknowledged f o r t h e i r h e l p f u l c o n s u l t a t i o n s and s u g g e s t i o n s . F i n a l l y , I g r a t e f u l l y acknowledge the f i n a n c i a l a s s i s t a n c e of the N a t u r a l S c i ences and E n g i n e e r i n g Research C o u n c i l of Canada ( i n the form of a pos tg radua te s c h o l a r s h i p ) . Las t but by no means the l e a s t I wish to thank my pa r en t s f o r t h e i r unde r s t and ing and e v e r l a s t i n g suppor t in making a l l of t h i s p o s s i b l e . 1 Chapter I INTRODUCTION 1.1 MOMENTUM DISTRIBUTIONS, ORBITALS AND QUANTUM CHEMISTRY One of the major c h a l l e n g e s in quantum chemis t r y [PCR73, LCG77, C82, LP83] i s the search fo r the exact s o l u t i o n to the equa t ion of motion f o r e l e c t r o n s in atoms and m o l e c u l e s . The complex motion of the e l e c t r o n s in any atomic or mo l e cu l a r system ( i n a s t a t i o n a r y s t a t e ) i s c o n v e n i e n t l y d e s c r i b e d by the ( t ime- independent ) S ch rod inger equa t ion [P68, S72, A83, RC83] : [1 .1 ] H* = E*, where H i s the H a m i l t o n i a n . S ince t h i s equa t ion cannot be s o l v e d e x a c t l y f o r systems wi th more than one e l e c t r o n , i t e r a t i v e numer i ca l p rocedures [PB70, S72, W76, LP83, RC83] are used to ob t a i n the best approximate s o l u t i o n . The (norma l ized ) e i g e n - s o l u t i o n * i s commonly ob ta ined by the v a r i a t i o n a l method [PW35, S72, RC83] , which v a r i e s the wave func t ion parameters s t r i v i n g to o b t a i n the lowest t o t a l ene rgy . Once * has been de te rmined , the e x p e c t a t i o n va lue of any measurable p h y s i c a l obse rvab le can be c a l c u l a t e d by : 2 [1 .2] <0> = ;**(w) 0 ¥ ( w ) d w , where w can e i t h e r be the p o s i t i o n r or the momentum p. The quantum-mechanical ope ra to r O of the p h y s i c a l obse r vab l e i s chosen a c c o r d i n g to p h y s i c a l or chemica l i n t u i t i o n and o f t e n i n v o l v e s a p p r o x i m a t i o n . The many-e lec t ron wave func t i on , * , f o r atoms or mo lecu les i s u s u a l l y w r i t t e n as an t i s ymmet r i zed p roduc t s of one-e l e c t r on atomic or mo lecu l a r o r b i t a l s . i . e . [1 .3 ] * = A\ ^^2^3. • .. | , where A r ep re sen t s the an t i s ymmet r i z e r and ^ i s the it h o r b i t a l in the many-e lec t ron wave func t i on . The many-e lec t ron wave func t ion i s commonly o b t a i n e d u s i n g the s e l f - c o n s i s t e n t -f i e l d LCAO-MO method 1 [R51, D69, PB70, SB72, RC83] . A l though the o r b i t a l (or independent p a r t i c l e ) d e s c r i p t i o n [H28, M28, PW35, LP64, JS73, S073, VA75] of the many-e lec t ron wave func t ion i s on l y an a p p r o x i m a t i o n , i t has become one of the most important concepts i n modern quantum chemis t r y and has been found to be ext remely u s e f u l in i n t e r p r e t i n g a v a r i e t y of chemica l data [WM77]. Us ing q u a l i t a t i v e i n f o r m a t i o n on the o r b i t a l e n e r g i e s and e l e c t r o n d e n s i t i e s , gene ra l methods have been deve loped to p rov i de an unde rs t and ing of mo lecu l a r s t r u c t u r e s and shapes [B80] , LCAO stands fo r L i n e a r Combinat ion of Atomic O r b i t a l s and MO stands f o r Mo l e cu l a r O r b i t a l . 3 chemica l r e a c t i v i t i e s and bonding phenomena [K74] as we l l as o ther p h y s i c a l and chemica l p r o p e r t i e s [S72, L72a , SM72]. There are two gene ra l approaches to the expe r imen ta l e v a l u a t i o n of the q u a l i t y of the approximate wave func t i ons . The f i r s t i n v o l v e s compar ison of the e x p e c t a t i o n va lues of p h y s i c a l obse r vab l e s (equat ion 1.2) such as d i p o l e moment, p o l a r i z a b i l i t y , t r a n s i t i o n energy and p r o b a b i l i t y , and b i n d i n g e n e r g i e s , e t c . [S72, L72a , SM72] between those c a l c u l a t e d us i ng the approximate wavefunct ion and those measured by expe r imen ts . In the important case where the ope ra to r i s the H a m i l t o n i a n , the p h y s i c a l obse r vab l e i s the energy of the sys tem. T r a d i t i o n a l methods used to o b t a i n the energy i n f o r m a t i o n a s s o c i a t e d w i th v a r i o u s e l e c t r o n i c s t a t e s i n c l u d e p h o t o a b s o r p t i o n and p h o t o i o n i z a t i o n s p e c t r o s c o p i e s [B79] as we l l as h igh energy e l e c t r o n impact s p e c t r o s c o p i e s which s imu l a t e photon impact exper iments [HS&76, BH81, B82] , The e n e r g i e s of these e l e c t r o n i c s t a t e s , e s p e c i a l l y in the ou te r va l ence r e g i o n , are r e l a t e d to the o r b i t a l e n e r g i e s in the Koopmans' approx imat ion [K33] . Pho toabso rp t i on spec t roscopy [S67, B79] measures the d i f f e r e n c e s between ( o r b i t a l ) energy e i g e n v a l u e s , and through s e l e c t i o n r u l e s , some r e l a t i o n s between the quantum numbers of the i n v o l v e d p a i r s of s t a t e s . P h o t o e l e c t r o n spec t roscopy [S&69, TB&70, R77, B79, KK&81, E82 ] , on the o ther hand, measures the b i n d i n g e n e r g i e s of p h o t o e l e c t r o n s and hence the ene rg i e s of the i o n i c s t a t e s . In a d d i t i o n the i n t e n s i t i e s of the s t a t e s 4 observed in the pho toabso rp t i on and p h o t o e l e c t r o n s p e c t r a are r e l a t e d to the pho toabso rp t i on o s c i l l a t o r s t r e n g t h s and p a r t i a l i o n i z a t i o n o s c i l l a t o r s t r e n g t h s r e s p e c t i v e l y [B79] . The o s c i l l a t o r s t r e n g t h s are the p r o b a b i l i t i e s of the i n v o l v e d t r a n s i t i o n s and depend upon the photon energy (or e f f e c t i v e l y the photon momentum =h/X) [171, B79 ] . T h i s dependence of the p a r t i a l o s c i l l a t o r s t r e n g t h of a p a r t i c u l a r s t a t e on the photon energy can be used to i d e n t i f y tha t i o n i c s t a t e . Because of the l i m i t e d a v a i l a b i l i t y of con t inuous tunab le photon sources u n t i l the recent development of s ynch ro t ron r a d i a t i o n f a c i l i t i e s [K79, WD80] f o r photon-impact expe r imen ts , these measurements have been made p o s s i b l e by h igh energy e l e c t r o n impact s i m u l a t i o n [HS&76, BH81, B82] of pho toabso rp t i on and p h o t o e l e c t r o n s t u d i e s . I t shou ld be noted a l s o tha t the angu la r asymmetry parameters measured by angu la r r e s o l v e d p h o t o e l e c t r o n spec t roscopy [CA71, B79] can a l s o be used to i n d i c a t e the s i g n a t u r e of a p a r t i c u l a r s t a t e . A more d i r e c t approach to the e v a l u a t i o n of a b - i n i t i o wavefunc t ions i n v o l v e s the compar ison of the c a l c u l a t e d and measured p o s i t i o n (or charge) d e n s i t y ¥ * ( r ) * ( r ) and momentum d e n s i t y ¥ * ( p ) ¥ ( p ) . The d e n s i t y f u n c t i o n 2 p r o v i d e s an i n h e r e n t l y more d i r e c t q u a n t i t y f o r the e v a l u a t i o n of 2 The q u a n t i t y (r ) i / / . ( r ) d r i s the p r o b a b i l i t y of o b s e r v i n g the it h e l e c t r o n between p o s i t i o n r and r+dr . S i m i l a r l y the q u a n t i t y i//. (p)<//-(p)dp r ep re sen t s the p r o b a b i l i t y of obse r v i ng the ith e l e c t r o A moving wi th momentum between p and p+dp. 5 wave func t ions s i n c e there i s no ope ra to r i n v o l v e d . In the former app roach , approx imat ion to the mathemat ica l form of the ope ra to r r e p r e s e n t i n g a p a r t i c u l a r obse r vab l e i s o f t en n e c e s s a r y . Moreover , the compar ison between expe r imen ta l and c a l c u l a t e d wavefunct ion p r o p e r t i e s u s u a l l y depends upon which pa r t of the wave func t ion i s emphasized in the o p t i m i z a t i o n p r o c e d u r e . For i n s t ance i t has been found tha t the t h e o r e t i c a l wavefunct ion which g i v e s lower t o t a l energy does not n e c e s s a r i l y model the outer va l ence e l e c t r o n s c o r r e c t l y . The t o t a l p o s i t i o n - s p a c e wave func t ion * ( r ) and the t o t a l momentum-space wave func t ion ¥(p) are e q u i v a l e n t s o l u t i o n s w r i t t e n i n d i f f e r e n t r e p r e s e n t a t i o n s . They are r e l a t e d to each o ther by the F o u r i e r t r a n s f o r m , i . e . [1 .4 ] *(p) = (2TT ) - 3 / 2 / * ( r ) e x p ( - i p . r ) d r . T o t a l p o s i t i o n d e n s i t i e s [B75] can be measured u s i n g X-ray and e l e c t r o n d i f f r a c t i o n methods [BF80] which a re main ly l i m i t e d to the study of c r y s t a l s and s u r f a c e s . T o t a l momentum d e n s i t i e s [E75, W77a] can be d e r i v e d from the e x p e r i m e n t a l l y de termined Compton p r o f i l e s ob t a i ned us i ng X-ray and 7 - ray Compton s c a t t e r i n g [C71, W77b], h i g h energy e l e c t r o n impact i o n i z a t i o n [BF74] and p o s i t r o n a n n i h i l a t i o n [S75] t e c h n i q u e s . D e t a i l s of each of the i n d i v i d u a l methods are r e f e r r e d to in the c o r r e s p o n d i n g review a r t i c l e s . I t i s important to note tha t a l l of these c o n v e n t i o n a l methods 6 measure the e l e c t r o n d e n s i t y ( e i t h e r in p o s i t i o n or momentum space) due to a l l the e l e c t r o n s in the t a rge t sys tem, and cannot r o u t i n e l y p a r t i t i o n the t o t a l e l e c t r o n d e n s i t y i n t o i n d i v i d u a l o r b i t a l components. The r e c e n t l y i n t r o d u c e d techn ique of b i n a r y (e ,2e) spec t roscopy [MW76a] measures both the energy and momentum i n f o r m a t i o n a s s o c i a t e d w i th the e j e c t i o n of a s i n g l e e l e c t r o n us i ng h igh energy e l e c t r o n impact i o n i z a t i o n and c o i n c i d e n c e d e t e c t i o n t e c h n i q u e . The b i na r y (e ,2e) method can i n p r i n c i p l e p rov i de a f a i r l y complete "anatomy" of the t o t a l momentum d e n s i t y i n t o i t s i n d i v i d u a l o r b i t a l components. I t r ep r e sen t s one of the most power fu l methods f o r the study of e l e c t r o n i c s t r u c t u r e and in the i n v e s t i g a t i o n of the mo lecu l a r o r b i t a l bonding p a t t e r n . To d a t e , the techn ique has been used to p rov ide the most comprehensive s t u d i e s of o r b i t a l momentum d e n s i t i e s f o r both atoms and mo lecu les and has p r o v i d e d the p o s s i b i l i t y of c r i t i c a l e v a l u a t i o n of the a b - i n i t i o wavefunct ion e s p e c i a l l y in the c h e m i c a l l y i n t e r e s t i n g va l ence r e g i o n . T h i s has enhanced the unders tand ing of mo lecu l a r o r b i t a l concep ts and has p r o v i d e d a unique vantage p o i n t f o r o b s e r v i n g chemica l bonding phenomena in the l a b o r a t o r y . 7 1 .2 BINARY (e ,2e) SPECTROSCOPY AND CONVENTIONAL METHODS B ina ry (e ,2e) spec t roscopy [MW76a] (see a l s o appendix A) i s based on the f o l l o w i n g e l e c t r o n impact i o n i z a t i o n r eac t i o n : [1 .5 ] e + ( ion-e) e + ion + e ( E 0 , P o ) (E fp) (E 1 ( p , ) ( E 2 , p 2 ) Making use of the energy and momentum c o n s e r v a t i o n s , i t i s p o s s i b l e to sample the b i n d i n g energy E and momentum p of the bound e l e c t r o n be fo re the knockout r e a c t i o n by c o n t r o l l i n g the k i n e t i c e n e r g i e s and momenta of the i n c i d e n t e l e c t r o n ( l a b e l l e d by s u b s c r i p t 0) and the two ou tgo ing e l e c t r o n s ( l a b e l l e d by s u b s c r i p t s 1 and 2 f o r the s c a t t e r e d and e j e c t e d e l e c t r o n s r e s p e c t i v e l y ) . i . e . [1 .6a ] E = ( E 0 - E,) - E 2 ; E, = E 2 . [1 .6b] p = p 2 - (p0 - p , ) ; P o ~ P i » 0. Here the r e c o i l momentum of the ion i s i gnored (see chapte r 2 ) . The exper iment i n v o l v e s complete d e t e r m i n a t i o n of the s c a t t e r i n g k inemat i c s of the i n c i d e n t and ou tgo ing e l e c t r o n s and c o i n c i d e n t coun t i ng of the ou tgo ing e l e c t r o n s ( i . e . of (e ,2e) events ) under k inemat i c c o n d i t i o n s in which the momentum t r a n s f e r ( p 0~Pi) i s maximized and the ene rg i e s of the i n c i d e n t and ou tgo ing e l e c t r o n s are made s u f f i c i e n t l y 8 h i g h . The r e s u l t i n g h i s togram of c o i n c i d e n c e s y i e l d s the b i n d i n g energy spectrum or the o r b i t a l momentum d i s t r i b u t i o n depending upon the sampl ing parameter ( i . e . e i t h e r E or p, see a l s o chapte r 2 ) . It i s important to compare the i n f o r m a t i o n ob t a i ned by the new techn ique w i th those ob ta ined us i ng c o n v e n t i o n a l methods. I n fo rmat ion on i o n i z a t i o n ene rg i e s can be ob ta ined by p h o t o e l e c t r o n [S&69, TB&70, R77, B79, KK&81, E82] and d i p o l e (e ,2e) [HS&76, BH81, B82] s p e c t r o s c o p i e s . In the case of p h o t o e l e c t r o n and d i p o l e (e ,2e) s p e c t r o s c o p i e s the k inemat i c s can be w r i t t e n as P h o t o e l e c t r o n : 7 x [1 .7 ] V + ( ion-e) >• > + ion + e D i p o l e ( e , 2 e ) : e ' e ' ( E 0,p 0) (E,p) (E,,p,) ( E 2,p 2) In the d i p o l e l i m i t where the momentum t r a n s f e r i s m in im ized , i . e . P o ~ P i - > 0 f the c o r r e s p o n d i n g energy and momentum c o n s e r v a t i o n equa t i ons a r e : [1 .8a] E = ( E 0 - E,) - E 2 ; [1 .8b] p a p 2 . The q u a n t i t y E Q - E , i s the e f f e c t i v e photon energy ( i . e . = hv) i n the case of p h o t o e l e c t r o n spec t roscopy [B79] or i s the energy l o s s i n d i p o l e (e ,2e) spec t roscopy [B82] , By 9 combin ing equa t i ons 1.8a and 1.8b, one ge ts tha t the magnitude of the e l e c t r o n momentum becomes s imp ly ( in atomic u n i t s ) : [1 .8c] p * ( 2 [ ( E 0 -E 1 ) - E ] } 1 S ince the normal energy range in (X-ray) p h o t o e l e c t r o n spec t roscopy i s l00eV-1000eV [B79] , the co r r e spond ing e l e c t r o n momentum i s t h e r e f o r e i n the range 1 a 0 -1 0 a o " 1 . I t i s impor tant to no t e , however, tha t a v a r i a b l e photon energy i s necessa ry i n o rder to vary the momentum. The a v a i l a b i l i t y of t unab l e photon sources i s l i m i t e d on l y to s ynch ro t ron r a d i a t i o n [WD80] and photon s i m u l a t i o n t echn iques such as d i p o l e (e ,2e) spec t roscopy [B82] . A peak i n the b i n d i n g energy spectrum ( i . e . a h i s togram of (e ,2e) events as a f u n c t i o n of E at an e s s e n t i a l l y cons tan t momentum) measured by b i n a r y (e ,2e ) spec t roscopy co r responds to an ion s t a t e in the same way as tha t o b t a i n e d by p h o t o e l e c t r o n or d i p o l e (e ,2e) measurements [MUP78]. The enormous advantage i n the new t echn ique i s the e x t r a k inemat i c f l e x i b i l i t y which a l l ows the e l e c t r o n momentum to be v a r i e d i ndependen t l y . For each ion s t a t e , the momentum d i s t r i b u t i o n ( i . e . a h i s togram of (e ,2e) events as a f u n c t i o n of p) i s , to a c l o s e a p p r o x i m a t i o n , the square of the momentum-space o r b i t a l of the s t r u c k e l e c t r o n (see chapte r 2 ) . Such a momentum.prof i le a f f o r d s an i n s t a n t i d e n t i f i c a t i o n of the c h a r a c t e r i s t i c ho le 10 a s s o c i a t e d wi th the ion s t a t e . Moreover , the b i na r y (e,2e) r e a c t i o n samples the low momentum pa r t ( 0 a o " 1 - 3 a 0 " 1 ) of the o r b i t a l , which co r responds to the l a r g e ( p o s i t i o n ) s p a t i a l pa r t (the bonding r e g i o n ) , in c o n t r a s t to p h o t o e l e c t r o n spec t roscopy which probes on l y the l a r g e momentum reg ion c o r r e s p o n d i n g to the sma l l s p a t i a l pa r t of the o r b i t a l ( i . e . the r eg ion very c l o s e to the n u c l e i ) . Of paramount importance i s tha t b i n a r y (e ,2e) spec t roscopy i s s u f f i c i e n t l y s e n s i t i v e to p rov ide expe r imen ta l d e f i n i t i o n s fo r the o r b i t a l s best s u i t e d to form the b a s i s expans ion of the many-e lec t ron wave func t i on . The t r u e l y unique momentum d i s t r i b u t i o n a s s o c i a t e d w i th a p a r t i c u l a r s i n g l e p a r t i c l e s t a t e ( i . e . the o r b i t a l momentum d i s t r i b u t i o n ) p r o v i d e s d i r e c t expe r imenta l e v a l u a t i o n of the t h e o r e t i c a l a b - i n i t i o SCF LCAO-MO wavefunct ion in the c h e m i c a l l y s e n s i t i v e oute r r e g i o n s of va l ence o r b i t a l s . Other c o n v e n t i o n a l methods such as Compton s c a t t e r i n g type e x p e r i m e n t s 3 [W77b] can on l y sample t o t a l momentum d i s t r i b u t i o n s . The r e a c t i o n equa t i on fo r Compton s c a t t e r i n g ( i . e . a (7,7') r e a c t i o n ) can be r ep resen ted a s : 3 Compton s c a t t e r i n g type exper iments i n c l u d e X-ray and 7-ray Compton s c a t t e r i n g [C71, W77b] and h igh energy e l e c t r o n impact spec t roscopy (HEEIS) [BF74] . 11 Compton: 7 [1 .9 ] + ( ion-e) ( ion-e) HEEIS: e e' ( E 0 , p 0 ) (E,p) (E, ,p,) (E* ,p' ) where the c o n s e r v a t i o n laws d i c t a t e : [1 .10a] EQ ~ E = E' - E; [ 1 .10b] Po " Pi = P' " E s s e n t i a l l y Compton s c a t t e r i n g type exper iments can be c o n s i d e r e d as i n e l a s t i c s c a t t e r i n g of a h i g h energy photon or e l e c t r o n . The ( r e l a t i v i s t i c Dopp le r ) b roaden ing [W77b] of the s c a t t e r e d r a d i a t i o n i s r e l a t e d to the e l e c t r o n momentum d i s t r i b u t i o n . Because Compton s c a t t e r i n g type exper iments e s s e n t i a l l y i n v o l v e a s i n g l e - p a r t i c l e measurement ( i . e . the d e t e c t i o n of the photon or e l e c t r o n tha t s u f f e r s the i n e l a s t i c c o l l i s i o n ) , i t i s not p o s s i b l e to p a r t i t i o n the measured Compton p r o f i l e based upon o r b i t a l energy s e l e c t i o n . Consequent l y on l y the Compton p r o f i l e due to a l l of the e l e c t r o n s i s measurab le . Even in the s p e c i a l case where d o u b l e - p a r t i c l e measurements (such as u s i n g f l u o r e s c e n c e c o i n c i d e n c e t echn iques [FH72]) a re p o s s i b l e , the measured Compton p r o f i l e must be d i f f e r e n t i a t e d (o f ten u s i n g numer i ca l t e chn iques w i th v a r y i n g degrees of accuracy ) in o rder to d e r i v e the momentum d i s t r i b u t i o n . T h i s i s in marked c o n t r a s t to b i n a r y (e ,2e) spec t ro s copy which a l l ows d i r e c t 12 measurements of momentum d i s t r i b u t i o n s of i n d i v i d u a l o r b i t a l s . B ina ry (e ,2e) spec t roscopy i s t h e r e f o r e t r u e l y unique i n i t s c a p a b i l i t y of e v a l u a t i n g the e l e c t r o n i c wavefunct ion on an o r b i t a l - b y - o r b i t a l b a s i s in modern expe r imen ta l quantum c h e m i s t r y . 1.3 BRIEF HISTORICAL REMARKS The o r i g i n of b i n a r y (e ,2e) spec t roscopy can be t r a c e d back to two Nobel p r i z e winning exper iments in atomic p h y s i c s ; namely, the 1914 e l e c t r o n s c a t t e r i n g exper iment of Franck and He r t z [FH14] and the 1922 X-ray s c a t t e r i n g exper iment of Compton [C22 ] . The Franck and Her tz exper iment e s t a b l i s h e d the s i m i l a r i t y between e l ec t ron-a tom i n t e r a c t i o n s and photon-atom i n t e r a c t i o n s . T h i s exper iment subsequent l y gave r i s e to the l a t e r vas t development of e l e c t r o n spec t roscopy [MB52, TRK70, B H 8 1 ] . The Compton exper iment demonstrated not on l y the p a r t i c l e na ture of a h igh energy photon ( i n the wave-pa r t i c l e d u a l i t y of mat ter ) but a l s o the i n e l a s t i c s c a t t e r i n g of l i g h t . I t t r i g g e r e d the emergence of an a r r a y of Compton s c a t t e r i n g exper iments and demonstrated the importance of momentum d i s t r i b u t i o n s in the unders tand ing of e l e c t r o n i c s t r u c t u r e . A f t e r the i n i t i a l t h r u s t of the development in 1922-1936, d u r i n g which many important d i s c o v e r i e s conce rn i ng the Compton e f f e c t were made, the f i e l d l a y e s s e n t i a l l y "dormant" f o r f o r t y y e a r s , d e s p i t e 13 encouragement from the t h e o r e t i c a l community. In p a r t i c u l a r , Cou lson and Duncanson [CD41] had p u b l i s h e d a s e r i e s of papers in the e a r l y 1940s, s t r e s s i n g the importance of i n t e r p r e t i n g chemis t r y from the momentum-space p e r s p e c t i v e . The development of (e ,2e) exper iments was, however, r e a l l y i n s p i r e d by the important development of the (p,2p) (and a l s o the (7,27)) r e a c t i o n in nuc l ea r p h y s i c s . The 1 f e a s i b i l i t y of us ing the (e ,2e) r e a c t i o n to s tudy e l e c t r o n i c s t r u c t u r e was f ' i r s t noted by McCarthy et al . in 1959 [MJK59] in t h e i r t h e o r e t i c a l s tudy of a p p l y i n g the (p,2p) r e a c t i o n to i n v e s t i g a t e nuc leon s t r u c t u r e . T h i s , toge the r w i th the work by Math ies and Neudatch in in 1963 [MN63], l a i d the founda t i on of the (e ,2e) r e a c t i o n t h e o r y . Indeed, a vas t volume of j a rgon used nowadays by (e ,2e) s p e c t r o s c o p i s t s has come from nuc l ea r p h y s i c s t e r m i n o l o g y . The use of the term " ( e , 2 e ) " i t s e l f to i n d i c a t e e l e c t r o n impact i o n i z a t i o n r e a c t i o n and terms such as " s e p a r a t i o n energy " to mean b i n d i n g energy are j u s t two of many no t ab l e examples of the nuc l e a r p h y s i c s h e r i t a g e . The very f i r s t (e ,2e) measurement was r e p o r t e d by Ehrhard t et a l . ( i n West Germany) in 1969 fo r He [ES&69]. T h i s and subsequent works by Eh rha rd t el a l . were d i r e c t e d to i n v e s t i g a t e the s c a t t e r i n g r e a c t i o n mechanism at low ene rg i e s u s i n g asymmetric s c a t t e r i n g geometry . The second expe r imen ta l work on ( e , 2 e ) , r e p o r t e d by Amaldi et a l . ( in I t a l y ) a l s o in 1969 [AE&69], gave the f i r s t (e ,2e) b i n d i n g 1 4 energy spectrum fo r the carbon K-she l l of a t h i n f i l m at h igh e n e r g i e s . F o l l o w i n g t h i s p r e l i m i n a r y s t udy , C a m i l l o n i et al . ( i n I t a l y ) in 1972 [CG&72] r epo r t ed the f i r s t (e ,2e) angu la r c o r r e l a t i o n (or momentum d i s t r i b u t i o n ) f o r the carbon K and L s h e l l s of a t h i n f i l m . Both of these exper iments [AE&69, CG&72] made use of the h i g h energy cop l ana r symmetric geometry and c l e a r l y demonstrated the s e n s i t i v i t y of the momentum d i s t r i b u t i o n to the shape of the o r b i t a l wave func t i on . In 1973, Weigold et al . ( i n A u s t r a l i a ) [WHT73] r e p o r t e d the f i r s t (e ,2e) momentum d i s t r i b u t i o n s f o r separa te Ar 3p and 3s o r b i t a l s us ing the h i g h energy noncoplanar symmetric geometry . T h i s exper iment a l s o gave the f i r s t ev idence of i n t ense e l e c t r o n - e l e c t r o n c o r r e l a t i o n in the inner va l ence r eg ion fo r the i o n i z a t i o n of the Ar 3s e l e c t r o n s . The fo l l ow-up works by the A u s t r a l i a n group in the f i r s t mo lecu l a r (e,2e) exper iments f o r hydrogen [WH&73] and methane [HW&73] a l s o i n d i c a t e d the importance of e l e c t r o n c o r r e l a t i o n e f f e c t s in mo lecu l a r i o n i z a t i o n phenomena. The 1973 work on Ar [WHT73] has e s t a b l i s h e d the h i g h energy noncoplanar symmetric k inemat i c s as the most a p p r o p r i a t e s c a t t e r i n g geometry in s t r u c t u r a l d e t e r m i n a t i o n . These p i o n e e r i n g works of the atomic p h y s i c i s t s were l a t e r f o l l owed by two chemis t r y g roups : C . E . B r i on ( in Canada) ( t h i s group) in 1976 and M.A. Coplan and J . H . Moore ( in the U n i t e d S t a tes ) in 1978. Dur ing the i n f ancy of the (e ,2e) t echn ique much e f f o r t was d i r e c t e d towards the f i r m e s t ab l i shmen t of the 1 5 v a r i o u s r e a c t i o n approx imat ions [UWM75, CG&78, FM&78, DM&78] u s i n g nob le gases as t a r g e t s . I t was the input from the chemis t s which l e d to sys temat i c s t u d i e s of o r b i t a l momentum d i s t r i b u t i o n s which in tu rn a l l owed the study of chemica l t r ends and group r e l a t e d p r o p e r t i e s . To d a t e , over f o r t y atoms and sma l l mo lecu les have been i n v e s t i g a t e d by b i na r y (e ,2e) s p e c t r o s c o p y . These i n c l u d e a number of important s e r i e s of atoms and sma l l m o l e c u l e s . ( i ) Atoms 1) Carbon f i l m ; 2) Group V I I I : He, Ne, A r , Kr and Xe ; 3) One e l e c t r o n atom: H; 4) Open-she l l a toms: Na, K and C d . ( i i ) Mo l e cu l e s 1) F i r s t - r o w h y d r i d e s : HF, H 2 0 , NH 3 and CH „ ; 2) Second-row h y d r i d e s : HCI, H 2 S and PH 3 3) Hydrogen and hydrogen h a l i d e s : H 2 , HF, HCI, HBr and HI ; 4) Methane, methyl h a l i d e s and o ther m e t h y l - s u b s t i t u t e d compounds: CH«, C H 3 F , C H 3 C 1 , CH 3 B r , C H 3 I , C H 3 O H , CH 3 NH 3 and CH 3 CN; 5) E thy l ene and h a l o g e n - s u b s t i t u t e d e t h y l e n e s : C 2 H „ , C 2 H 3 F , C 2 H 3 C1 and C 2 H 3 B r ; 6) F l u o r o - s u b s t i t u t e d methane: 16 CH,,, C H 3 F , CHF 3 and C F , ; 7) The i s o e l e c t r o n i c s e r i e s : CO, N 2 and C 2 H 2 ; 8) The v a l e n c e - i s o e l e c t r o n i c s e r i e s : N 2 0 , C 0 2 , OCS and C S 2 ; 9) Open-she l l m o l e c u l e s : NO and 0 2 ; 10) Other sma l l o rgan i c m o l e c u l e s . A b i b l i o g r a h y of a l l the (e ,2e) works p u b l i s h e d to date i s g i ven in appendix A . D e t a i l s of each of the i n d i v i d u a l s t u d i e s are r e f e r r e d to in the c o r r e s p o n d i n g papers (see appendix A ) . Of a l l these s t u d i e s , the work of Lohmann and Weigo ld [LW81] on the H atom shou ld be p a r t i c u l a r l y noted s i n c e i t p r o v i d e s the f i r s t d i r e c t expe r imen ta l measurement of the momentum d e n s i t y where an exact s o l u t i o n to the S ch rod inge r equa t ion i s a v a i l a b l e . Of fundamental chemica l i n t e r e s t i s the recent work of Leung and B r i on [LB84a] (see a l s o chap te r 6) on mo lecu l a r hydrogen which p r o v i d e s the f i r s t expe r imen ta l e s t i m a t i o n of the bond d e n s i t y in mo l ecu l a r hydrogen . The r e l e vance of the (e ,2e) r e a c t i o n s as a unique power fu l t o o l not on l y fo r t e s t i n g d i f f e r e n t s c a t t e r i n g t h e o r i e s but a l s o f o r the e x t r a c t i o n of energy and momentum i n f o r m a t i o n of i n d i v i d u a l o r b i t a l s of atoms and mo lecu l es has been amply demons t ra ted . 1 7 1.4 MAJOR APPLICATIONS Two g e n e r a l types of e l e c t r o n i c s t r u c t u r a l i n f o rma t i on can be o b t a i n e d by b ina r y (e ,2e) s p e c t r o s c o p y , namely, the b i n d i n g energy spectrum (measured at an e s s e n t i a l l y f i x e d momentum) and the e l e c t r o n momentum d i s t r i b u t i o n (measured at a f i x e d b i n d i n g energy c o r r e s p o n d i n g to a p a r t i c u l a r ion s t a t e ) . In p r i n c i p l e , the t echn ique i s capab le of s tudy ing the f u l l i o n i z a t i o n e l e c t r o n i c spectrum i n c l u d i n g both the co re and va l ence r e g i o n s . Because of the very h igh ene rg i e s r e q u i r e d to s a t i s f y the k inemat i c approx imat ions (see chapte r 2) i n v o l v e d in the (e ,2e) b i n a r y encoun te r , knockout r e a c t i o n , a lmost a l l of the (e ,2e) works p u b l i s h e d to date are concerned on l y wi th the va l ence r e g i o n . I t shou ld be noted a l s o tha t because of the random o r i e n t a t i o n s of the gaseous t a r g e t s , the measured momentum d i s t r i b u t i o n co r responds to momentum d e n s i t y averaged over a l l the o r i e n t a t i o n s ( i . e . s p h e r i c a l a ve rag ing o c c u r s ) . D e s p i t e these l i m i t a t i o n s , important and i n f o r m a t i v e r e s u l t s can be ob t a i ned from these s p e c t r a . An overv iew of the more important a p p l i c a t i o n s i s g i ven below. These a p p l i c a t i o n s w i l l be f u r t h e r i l l u s t r a t e d i n g r ea t e r d e t a i l in i n d i v i d u a l s t u d i e s r e p o r t e d in the l a t e r c h a p t e r s . 18 1.4.1 V a l e n c e - S h e l l B i n d i n g E n e r g i e s B ina ry (e ,2e) spec t roscopy measures the complete v a l e n c e - s h e l l b i n d i n g energy spectrum of atoms and m o l e c u l e s . T h i s i s e s p e c i a l l y important in the inner va l ence r eg ion which i s i n a c c e s s i b l e to (UV) 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 . Ev idence of many-body s t a t e s i n the inner r e g i o n s of many atoms and m o l e c u l e s , such as f o r i n s t ance Ar [WHT73], CH„ [HW&73] and more r e c e n t l y methyl h a l i d e s [MC&84], have been uncovered us i ng the b i n a r y (e ,2e) method. These have p r o v i d e d d i r e c t expe r imen ta l c o n f i r m a t i o n of the breakdown of the Koopmans' approx imat ion [K33] and i n d i c a t e d the importance of e l e c t r o n c o r r e l a t i o n e f f e c t s i n i o n i z a t i o n phenomena. Fu r the rmore , the s p e c t r o s c o p i c f a c t o r (which i s d e f i n e d to be the p r o b a b i l i t y tha t an e i g e n s t a t e hav ing a p r i n c i p a l c o n f i g u r a t i o n of a ho le in the c h a r a c t e r i s t i c o r b i t a l , see chapte r 2) can be d e r i v e d from the i n t e n s i t i e s i n the b i n d i n g energy s p e c t r a . These p r o v i d e a d i r e c t i n d i c a t i o n of the ex ten t of an o r b i t a l c o n t r i b u t i o n in a p a r t i c u l a r 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 r o c e s s as we l l as f u r t h e r d e t a i l s of e l e c t r o n c o r r e l a t i o n e f f e c t s . 19 1 . 4 . 2 O r b i t a l Symmetry In g e n e r a l , two types of momentum d i s t r i b u t i o n s can be d i s t i n g u i s h e d a c c o r d i n g to where the maximum o c c u r s . The t o t a l l y symmetric " s - t y p e " d i s t r i b u t i o n has i t s maximum at p=0 wh i l e the nonsymmetric " p- t ype " d i s t r i b u t i o n has i t s maximum at nonzero p v a l u e . The symmetry of the o r b i t a l can be i d e n t i f i e d from i t s c h a r a c t e r i s t i c momentum d i s t r i b u t i o n . For i n s t a n c e , atomic ' s and d i a t om i c o r b i t a l s have t o t a l l y symmetric components in t h e i r wave func t ions and t h e r e f o r e the c o r r e s p o n d i n g momentum d i s t r i b u t i o n s f o l l o w the s-type behav i ou r . . Atomic p or d i a tom i c and TT o r b i t a l s , on the o ther hands , have p-type momentum d i s t r i b u t i o n s . Notab le examples are the momentum d i s t r i b u t i o n s of the oute r va l ence ns and np o r b i t a l s in the noble gases (chapter 4) and those of 7r and a o r b i t a l s of l i n e a r d i a t o m i c s and t r i a t o m i c s ( chapte rs 5-9). The symmetry of o r b i t a l s of po l ya tomic systems can be s i m i l a r l y i d e n t i f i e d . Moreover , in the case of a mixed " s-p t ype " o r b i t a l , the r e l a t i v e i n t e n s i t i e s between the maxima (due to s and p components) of the expe r imen ta l momentum d i s t r i b u t i o n can be used to probe the extent of s-p h y b r i d i z a t i o n . One example of a s-p type o r b i t a l i s the outermost sigma bonding o r b i t a l of the v a l e n c e - i s o e l e c t r o n i c t r i a t o m i c s e r i e s C 0 2 , CS 2 and OCS ( chapte rs 7-9 r e s p e c t i v e l y ) . 20 1.4.3 O r b i t a l O r d e r i n g In e s s e n c e , the momentum d i s t r i b u t i o n r e p r e s e n t s the f i n g e r p r i n t of an o r b i t a l . A l though the energy r e s o l u t i o n in b i na r y (e ,2e) spec t roscopy i s s t i l l not comparable to tha t of 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 , o r b i t a l o r d e r i n g can be e a s i l y i d e n t i f i e d by the c o r r e s p o n d i n g momentum p r o f i l e s measured at c a r e f u l l y chosen b i n d i n g e n e r g i e s . By comparing the shapes of the observed momentum d i s t r i b u t i o n s w i th those of the. t h e o r e t i c a l d i s t r i b u t i o n s of i n d i v i d u a l o r b i t a l s , one can unambiguously i d e n t i f y the p roper o r b i t a l o r d e r i n g . An example can be found i n OCS where an unambiguous i d e n t i f i c a t i o n between 2TT and 9a i s made us i ng the c o r r e s p o n d i n g expe r imen ta l o r b i t a l momentum d i s t r i b u t i o n s (see chap te r 9 ) . In formaldehyde the proper o r d e r i n g of the 5a , and 1b 2 o r b i t a l s was e a r l i e r i d e n t i f i e d by the angu la r v a r i a t i o n of the b i n a r y (e ,2e) b i n d i n g energy spectrum [HHB76b]. 1.4.4 Assignment Of The Many-body S t a t e s A c c o r d i n g to Koopmans' theorem [K33] , i o n i z a t i o n of an o r b i t a l can on l y r e s u l t in one t r a n s i t i o n or observed peak in the b i n d i n g energy spec t rum. T h i s independent p a r t i c l e i o n i z a t i o n (or one peak per o r b i t a l ) p i c t u r e g e n e r a l l y a p p l i e s q u i t e we l l to the oute r va l ence o r b i t a l s . However, in the case of inner v a l ence o r b i t a l s , many-body (or 21 s a t e l l i t e ) s t a t e s a r i s i n g from the i o n i z a t i o n of a s i n g l e o r b i t a l have been commonly obse r ved . Numerous examples can be found , f o r i n s t a n c e , i n the i s o e l e c t r o n i c CO [DD&77b, TM&82], N 2 [CS&76, WD&77] and C 2 H 2 [DM&77, CMT78, CM&79] s e r i e s , the va l ence i s o e l e c t r o n i c C 0 2 , CS 2 and OCS s e r i e s (see chap t e r s 7-9 r e s p e c t i v e l y ) and other methyl s u b s t i t u t e d compounds [CC&81, MBW81, MC&84]. T h i s breakdown of the independent p a r t i c l e p i c t u r e i s g e n e r a l l y a t t r i b u t e d to e l e c t r o n c o r r e l a t i o n e f f e c t s in both the i n i t i a l s t a t e s and/or f i n a l s t a t e s [MW76a], The major o r b i t a l c o n t r i b u t i o n of a many-body s t a t e can be ob ta ined from the momentum p r o f i l e sampled at the energy co r r e spond ing to the many-body s t a t e . A good example of t h i s advantage i s demonstrated in the heav i e r members of the noble gases : A r , Kr and Xe ; where the c h a r a c t e r i s t i c momentum d i s t r i b u t i o n s of the s a t e l l i t e s t a t e s in the inner v a l ence r eg i on are found to f o l l ow those of the c o r r e s p o n d i n g ns o r b i t a l s (see chapte r 4 ) . These s a t e l l i t e s can t h e r e f o r e be i d e n t i f i e d as a r i s i n g from the i o n i z a t i o n of the r e s p e c t i v e ns e l e c t r o n s . In f a c t , i t has been c o n s i s t e n t l y observed tha t s i g n i f i c a n t p o p u l a t i o n s p l i t t i n g s occur u s u a l l y on l y i n the inner va l ence o r b i t a l s . The c o r r e c t p r e d i c t i o n of these many-body s t a t e s has been the c e n t r a l ob j e c t of t h e o r e t i c a l methods such as the many-body G r e e n ' s f u n c t i o n method [CTY71, CD77] and the symmetry adapted c l u s t e r expans ion method [NH78]. The momentum d i s t r i b u t i o n s and the a s s o c i a t e d p o l e - s t r e n g t h s of these 22 many-body s t a t e s are i n s t r u m e n t a l in the e v a l u a t i o n of the q u a l i t y of these c a l c u l a t i o n s . 1.4.5 S t r i n g e n t Test Of A b - i n i t i o Wavefunctions A t h e o r e t i c a l ab-initio wave func t ion i s norma l l y ob t a i ned u s i n g the s o - c a l l e d s e l f - c o n s i s t e n t - f i e l d (SCF) p rocedure to min imize the t o t a l energy (the v a r i a t i o n a l method) [P68, S72, A83, RC83] . Such a procedure of energy m i n i m i z a t i o n o f t en underes t imates the c h e m i c a l l y s i g n i f i c a n t oute r ( l a r g e r) pa r t of v a l ence o r b i t a l s s i n c e these r eg ions do not c o n t r i b u t e s i g n i f i c a n t l y to the t o t a l ene rgy . The expe r imen ta l b i na r y (e ,2e) momentum d i s t r i b u t i o n s of the va l ence o r b i t a l s can thus be used as a s t r i n g e n t e x t r a t e s t f o r the q u a l i t y of the t h e o r e t i c a l wave func t ion s i n ce the method i s p a r t i c u l a r l y s e n s i t i v e to the low momentum ( l a rge r) pa r t of the o r b i t a l wave func t i on . The measurements made by b i n a r y (e ,2e) spec t roscopy have shown tha t the w ide l y used doub l e-ze t a q u a l i t y wave func t ions have been u n s u c c e s s f u l to p rov ide a s a t i s f a c t o r y agreement w i th the (e ,2e) data e s p e c i a l l y f o r the lone p a i r e l e c t r o n s in a number of sma l l m o l e c u l e s . Notab le examples are the f i r s t - r o w h y d r i d e s : HF [BH&80], H 2 0 [HHB77, DD&77a] and NH 3 [HHB76a, CS&76, TL&84] . The i n c l u s i o n of p o l a r i z a t i o n f u n c t i o n s i n the t h e o r e t i c a l wave func t ion (a l though h e l p i n g to lower the energy) does not n e c e s s a r i l y l e ad to a b e t t e r agreement w i th the momentum 23 d i s t r i b u t i o n s . Comparison of 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 s w i th the (e ,2e) da ta can t h e r e f o r e r e vea l i nadequac i e s in the t h e o r e t i c a l wave func t i on , e s p e c i a l l y in the v a r i a t i o n a l l y l e s s s e n s i t i v e outer va l ence p a r t . These expe r imen ta l r e s u l t s can then be used as feedback f o r f u r t h e r improvement of the t h e o r e t i c a l wave func t i ons . 1.5 SCOPE OF THE THESIS The above i n t r o d u c t o r y d i s c u s s i o n s of b i n a r y (e ,2e) spec t r o s copy are in tended to p r o v i d e a gene ra l overv iew of the f i e l d i n i t s b r i e f f i f t e e n yea r s of deve lopment . E x c e l l e n t rev iews [MW76a, GF&81, MW83] of the f i e l d can be found i n the l i t e r a t u r e (see Appendix A ) . In the f o l l o w i n g c h a p t e r s , the a u t h o r ' s c o n t r i b u t i o n s to the f i e l d w i l l be r e p o r t e d . More s p e c i f i c a l l y , b r i e f o u t l i n e s of the b a s i c k inemat i c and s c a t t e r i n g equa t i ons used i n noncoplanar symmetric b i na r y (e ,2e) r e a c t i o n s as w e l l as summaries of the major momentum-space concep ts are d i s c u s s e d i n chapte r 2. In chap te r 3, d e t a i l e d d i s c u s s i o n s of the c o n s t r u c t i o n of a second gene ra t i on b i n a r y (e ,2e) spec t rometer and i t s o p e r a t i o n are g i v e n . T h i s h igh momentum r e s o l u t i o n b ina r y (e ,2e) spec t rometer has been used f o r i n v e s t i g a t i n g the e l e c t r o n i c s t r u c t u r e of atoms as we l l as the bonding phenomena of a v a r i e t y of sma l l m o l e c u l e s . In p a r t i c u l a r , r e s u l t s f o r the noble gases are r e p o r t e d i n chapte r 4. The 24 momentum-space chemis t r y of the two e l e c t r o n sys tems : He and H 2 are i n v e s t i g a t e d in chap te r 5. Chapter 6 p r o v i d e s an expe r imen ta l e s t i m a t i o n of the s p h e r i c a l l y averaged bond d e n s i t y in mo lecu la r hydrogen . R e s u l t s f o r the va lence i s o e l e c t r o n i c l i n e a r t r i a t o m i c s e r i e s : C 0 2 , CS 2 and OCS are g i ven in chap te r s 7, 8 and 9 r e s p e c t i v e l y . F i n a l l y , the study of a t r u e l y t h r e e - d i m e n s i o n a l m o l e c u l e : C F 4 by b ina ry (e ,2e) spec t roscopy i s r e p o r t e d i n chapte r 10. In a l l of the expe r imen ta l works, two-d imens iona l d e n s i t y con tour maps and t h r ee-d imens iona l cons tan t d e n s i t y s u r f a c e p l o t s c a l c u l a t e d u s i n g t h e o r e t i c a l wave func t ions are used to a s s i s t i n t e r p r e t a t i o n of the observed f e a t u r e s i n the measured momentum d i s t r i b u t i o n s and to p rov i de gene ra l d i s c u s s i o n s of v a r i o u s momentum-space p r o p e r t i e s . These expe r imen ta l works ( chap te rs 4-10) have p r o v i d e d new d e f i n i t i v e r e s u l t s f o r the r e s p e c t i v e o r b i t a l momentum d i s t r i b u t i o n s and have f u r t h e r i l l u s t r a t e d the p r a c t i c a l i t y of b i n a r y (e ,2e) spec t roscopy as a probe f o r bonding and e l e c t r o n i c s t r u c t u r a l i n v e s t i g a t i o n s . A summary of t h i s t h e s i s and an ou t look on the f i e l d are g i ven i n chapte r 11. Un less o therw ise s t a t e d , atomic u n i t s (7r=m =e=l) are e used i n t h i s t h e s i s . The g r aph i c conven t ion f o r the d e n s i t y v i s u a l i z a t i o n i s g i ven in chapte r 5. As w i th the r e f e r e n c e s quoted i n t h i s c h a p t e r , r e f e r e n c e s in the rema in ing of t h i s t h e s i s are enc l o sed by square b r a cke t s i n the t ex t and are i n d i c a t e d us i ng a f i v e (or s i x ) c h a r a c t e r code composing of 25 the f i r s t c h a r a c t e r s of the l a s t names of the c o n t r i b u t i n g authors and the year of the p u b l i c a t i o n . ( D e t a i l s of the conven t ion are g iven in the R e f e r e n c e s . ) 26 Chapter II THEORETICAL BACKGROUND 2.1 REACTION THEORY The v a l i d i t y of a p a r t i c u l a r s c a t t e r i n g theory [MM65, MB69, M79] depends c r i t i c a l l y on the s c a t t e r i n g kinematics. There are b a s i c a l l y four d i s t i n c t kinematic regimes f o r the (e,2e) r e a c t i o n ; these are normally c l a s s i f i e d a c c o r d i n g to the momentum t r a n s f e r and energy i n v o l v e d . The asymmetric (e,2e) r e a c t i o n i n v o l v e s low momentum t r a n s f e r while the symmetric (e,2e) r e a c t i o n i n v o l v e s high momentum t r a n s f e r . The low energy asymmetric (e,2e) geometry was used i n the f i r s t (e,2e) experiment by Ehrhardt et al. [ES&69] f o r the i n v e s t i g a t i o n of the i n t e r a c t i o n mechanism [EJS80]. The high energy asymmetric (e,2e) kin e m a t i c s , a l s o known as d i p o l e (e,2e) spectroscopy, was used by van der Wiel and Brion [VB74] f o r an e f f e c t i v e s i m u l a t i o n of p h o t o e l e c t r o n measurements of d i p o l e o s c i l l a t o r s t r e n g t h s with the tunable v i r t u a l photon f i e l d of a f a s t e l e c t r o n [HS&76, BH81, B82]. The symmetric coplanar arrangement (both high and low energy) p r o v i d e s a s t r i n g e n t e v a l u a t i o n of v a r i o u s r e a c t i o n approximations (such as the o p t i c a l p o t e n t i a l model, d i s t o r t e d wave approximation, averaged e i k o n a l wave approximation and f a c t o r i z a t i o n approximation) [UWM75, MW76a, 27 CG&78, FM&78, SCG78]. F i n a l l y , the h igh energy symmetric noncop lanar k inemat i c s i s the most s u i t a b l e geometry fo r the i n v e s t i g a t i o n of e l e c t r o n i c s t r u c t u r e and i s the s c a t t e r i n g geometry used in the s t u d i e s r epo r t ed i n t h i s t h e s i s . The fundamental b i na r y (e ,2e) theory has been rev iewed e x t e n s i v e l y in the l i t e r a t u r e [FM73, M73, M75, MW76a, MW76b, MS80, M80] , Only a summary of the most r e l e v a n t equa t ions used in t h i s t h e s i s w i l l t h e r e f o r e be g i v e n . The f o l l o w i n g d i s c u s s i o n s shou ld be a p p l i e d s t r i c t l y to the h igh energy symmetric noncoplanar c o l l i s i o n s . 2 . 1 . 1 S c a t t e r i n g Kinematics In an (e ,2e) exper iment [MW76a, M80], the s c a t t e r i n g k i nema t i c s are comp le te l y de t e rm ined . A t a r g e t (atom or mo lecu le ) i s i o n i z e d by h igh energy e l e c t r o n impact u s i ng a f a s t i n c i d e n t e l e c t r o n w i th k i n e t i c energy E 0 and momentum p 0 . An e j e c t e d e l e c t r o n 1 w i th k i n e t i c energy E 2 and momentum p 2 emerges, a long wi th the s c a t t e r e d e l e c t r o n w i th k i n e t i c energy E, and momentum p, as w e l l as the ion w i th ion r e c o i l energy E_ . , and ion r e c o i l momentum q. In e f f e c t i t i s not p o s s i b l e to d i s t i n g u i s h the " e j e c t e d " e l e c t r o n from the " s c a t t e r e d " one. For conven i ence , one o f t e n a s s o c i a t e s the " f a s t e r " e l e c t r o n to be the " s c a t t e r e d " one i n s c a t t e r i n g t h e o r y . 28 [2 .1 ] M + e" > M + + e~ + e~. ( P o r E 0 ) ^ ' E r e C o i l ) ^ P " ^ < P a » E a ) N e g l e c t i n g the very sma l l ion r e c o i l energy , c o n s e r v a t i o n of energy r e q u i r e s t h a t : [2 .2 ] E = E 0 - (E, + E 2 ) , where E i s the b i n d i n g (or s e p a r a t i o n ) energy of the e j e c t e d e l e c t r o n . C l e a r l y , i f the k i n e t i c e n e r g i e s of the ou tgo ing e l e c t r o n s ( i . e . E, and E 2 ) a re kept c o n s t a n t , the b i n d i n g energy can be sampled by v a r y i n g E 0 . A l s o ( n e g l e c t i n g the very sma l l thermal mot ions of the t a r g e t be fo re the impact) c o n s e r v a t i o n of momentum r e q u i r e s t h a t : [2 .3a ] q = Po - ( P i + p 2 ) , where the momentum t r a n s f e r (of the i n c i d e n t e l e c t r o n to the i o n i z e d e l e c t r o n ) i s d e f i n e d as K = p 0 - P i . I f one c o n s i d e r s tha t the s c a t t e r i n g takes p l a c e in a c l o s e , b i n a r y encounter [MW76a, M80] s i t u a t i o n (a c o n d i t i o n which can be r e a l i z e d by maximiz ing the momentum t r a n s f e r ) , then the ion can be v i r t u a l l y regarded as a s p e c t a t o r . C l e a r l y , under these c o n d i t i o n s the momentum p of the bound e l e c t r o n to be i o n i z e d i n the (e ,2e) r e a c t i o n i s equa l in magnitude but o p p o s i t e i n s i g n to the ion r e c o i l momentum q. 29 [2.3b] p = p 2 - ( p 0 - P 1 ) , [2.3c] p = { [ 2 p 1 c 0 £ ? , - p o ] 2 J- [2p, s i n e , s i r . (0/2) ]2} 1 / 2 , where 0 = 7 r - ( 0 , - 0 2 ) . The symmetric noncop lanar s c a t t e r i n g k inemat i c s ( p , = p 2 , 0 ,=0 2 =45 O and 0 v a r i a b l e ) [MW76a, M80] used in the p resen t work and the d e f i n i t i o n s of the s o l i d ang les used in equa t i on 2.3c are g i ven i n f i g u r e 2 . 1 . Under the symmetric noncop lanar s c a t t e r i n g k inemat i c c o n d i t i o n s , the magnitude of the momentum p can be e f f e c t i v e l y sampled by v a r y i n g the r e l a t i v e az imutha l ang le 0. 2.1.2 D i f f e r e n t i a l Cross S e c t i o n Cons ide r an (e ,2e) r e a c t i o n on a system wi th N e l e c t r o n s in i t s e l e c t r o n i c ground s t a t e (whose wavefunct ion i s denoted by 4»Q) which l eaves a f i n a l ion i n an e l e c t r o n i c e i g e n s t a t e (denoted by 1 ) . The (e ,2e) d i f f e r e n t i a l c r o s s s e c t i o n 2 a p ^ P o » P i , P 2 ) i s p r o p o r t i o n a l to the a b s o l u t e square of the r e a c t i o n ampl i tude M F ( p 0 , p , , p 2 ) . In atomic u n i t s (*=me=e=1) the (e ,2e) d i f f e r e n t i a l c r o s s s e c t i o n can be w r i t t e n as [MW76a, M80]: 2 The (e ,2e) d i f f e r e n t i a l c r o s s s e c t i o n used here i s a l s o r e f e r r e d to as the t r i p l 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 d 3 a/dR,dS2 2 dE, [ES&69] or as the f i v e f o l d d i f f e r e n t i a l c r o s s s e c t i o n d 5 a / d O , d f i 2 d E , [MW76a] i n the l i t e r a t u r e . In t h i s t h e s i s the term t r i p l 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 i s p r e f e r r e d . Figure 2.1 - Scattering kinematics of symmetric non-cop l ana r b i n a r y (e ,2e) r e a c t i o n . 31 [2.4] o p ( p 0 , P i , p 2 ) = (27r)« 2 a v ( p 1 p 2 / p o ) | M p ( p 0 , P i ,p 2) | 2 , where L r ep r e sen t s an average over the i n i t i a l degenerate s t a t e s and a sum over the un reso l ved f i n a l s t a t e s . The square of the s c a t t e r i n g ampl i tude d e s c r i b e s the t r a n s i t i o n p r o b a b i l i t y of the i n i t i a l two-body system (the i n c i d e n t e l e c t r o n and the t a r g e t ) to the f i n a l three-body system (the s c a t t e r e d e l e c t r o n , the e j e c t e d e l e c t r o n and the r e s i d u a l i o n ) . [2 .5 ] M F(p 0,p,,p 2) = < X L X 2 * p " 1 | T ( E ) |*Q X 0 > , where the x ' s r ep resen t the wave func t ions of the i n c i d e n t and ou tgo ing e l e c t r o n s and T(E) r e p r e s e n t s the t r a n s i t i o n o p e r a t o r . A n t i s y m m e t r i z a t i o n i s assumed i m p l i c i t l y . 2.1.3 Plane Wave Impulse Approximation Under the h igh energy and maximized momentum t r a n s f e r k inemat i c c o n d i t i o n s , the i n c i d e n t e l e c t r o n knocks out a t a r g e t e l e c t r o n i n an e s s e n t i a l l y c l e a n manner. The i n c i d e n t , e j e c t e d and s c a t t e r e d e l e c t r o n s i n t e r a c t on l y very weakly w i th the r e s i d u a l i o n . The (e ,2e) c o l l i s i o n may then be c o n s i d e r e d as a c l o s e , b i na r y encoun te r , d i r e c t knockout r e a c t i o n . At s u f f i c i e n t l y h igh e n e r g i e s , the i n c i d e n t and the ou tgo ing e l e c t r o n wave func t ions can then be r ep re sen t ed 32 by p lane waves. (The p lane wave wave func t ion |p> of an e l e c t r o n of momentum p i s (2 i r ) " 3 / 2 exp ( i p . r ) . ) The t r a n s i t i o n ope ra to r T(E) becomes a three-body o p e r a t o r , depending on ly on the c o o r d i n a t e s of the two c o l l i d i n g e l e c t r o n s and the c e n t r e of mass of the r e s i d u a l i o n , and can be f u r t h e r approx imated by the e l e c t r o n - e l e c t r o n s c a t t e r i n g ope ra to r t ( E ) [MM65, MW76a, M80]. Under these c o n d i t i o n s , the s c a t t e r i n g ampl i tude becomes a p roduc t of the a n t i -symmetr ized e l e c t r o n - e l e c t r o n c o l l i s i o n ampl i tude and the o v e r l a p ampl i tude of the t a r g e t and ion s t a t e s ( in the momentum-space r e p r e s e n t a t i o n ) . i . e . where k'=(p,-p 2)/2; k=(p 0-p ) /2 ; and P = P i + p 2 - P o « The d i f f e r e n t i a l c r o s s s e c t i o n , in the p lane wave impulse a p p r o x i m a t i o n , i s t h e r e f o r e g i ven by : [2 .7 ] a F(p 0,p,,p 2) = ( 2 f f ) « ( p l P 2 / p o ) a M o t t I a v | < p ¥ £ 1 | * £ > | 2 . [2 .6 ] M F ( p 0 , P i , P 2 ) = <k'| t ( E ) |k> 1|*£>, The h a l f - o f f - s h e l l Mott s c a t t e r i n g c r o s s s e c t i o n , a M o t t ' i s g iven [MW76a] by : [2 .8a ] a Mot t = |<k'| t ( E ) |k>|2 , = ( 2 T r 2 ) - 2 [ 2 7 r a / ( e x p ( 2 7 r a ) - 1 ) ] { 1 / K « + 1 / K ' " - [ 1 / ( K 2 K ' 2 ) ] c o s [ a l n ( K 2 / K ' 2 ) ] } , 33 where K=|p 0-p 1|; K'=|p 0-p 2| and a=|p,-p 2|" 1. The Mott c r o s s -s e c t i o n v a r i e s r a p i d l y w i th 6 and t h i s must be taken i n t o account in the i n t e r p r e t a t i o n of the symmetric cop l ana r measurements [MW76a], In the noncoplanar symmetric geometry , a.... reduces t o : Mott [ 2 . 8 b ] a M o t f c = ( 2 7 r 2 ) - 2 { 2 7 r A / [ e x p ( 2 7 T A ) - 1 ] } ( 1 / K « ) , where A = l / { 2 p , s i n 0 , s i n [ (TT-o>)/2]} and K 2 = p 0 2 + P i 2 - 2 p o P i C O s 0 2 . A p l o t of a M 0 t t a s a f u n c t i o n of o> co r r e spond ing to E 0 =l215 .6eV and E 1 =E 2 =600eV i s g iven in f i g u r e 2 .2 . C l e a r l y f o r the range of <f> a c c e s s i b l e in the exper iment , the Mott c r o s s s e c t i o n i s v i r t u a l l y cons tan t ( v a r i a t i o n over the e f f e c t i v e expe r imen ta l range o>=0° to 35° i s l e s s than 2%). S ince a b s o l u t e c r o s s s e c t i o n s are not measured in the p resen t symmetric non-coplanar exper iment , the Mott term as w e l l as the cons tan t k inemat i c f a c t o r s in equa t ion 2.7 can be i g n o r e d . 2 . 1 . 4 Target Hartree-Fock Approximation The on l y important term in equa t ion 2.7 i s the modulus square of the o v e r l a p ampl i tude <P*p 1l*o> * H e r e aga in a n t i s y m m e t r i z a t i o n i s unders tood i m p l i c i t l y . The ground N s t a t e of the t a rge t wavefunct ion * Q and the it h e i g e n s t a t e of N- 1 the ion wavefunct ion * „ can each be expanded i n t o a set of 34 0 (degrees) F i g u r e 2.2 - Mott 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 the symmetric non-coplanar k i nema t i c s wi th E 0 = l 2 1 5 . 6 e V , E,=E 2=600eV and 6=45° . The normal range of <f> i n the exper iment i s shown by the shaded a r e a . 35 c o n f i g u r a t i o n s , $ [2 .9 ] [2 .10] The ion wavefunct ion i s regarded as an expans ion of a ho l e o r b i t a l , i / ^ f , coup led by the C lebsch-Gordon c o e f f i c i e n t s , C j s B ' w * t n a t a rge t c o n f i g u r a t i o n , to g i ve an ion s t a t e of symmetry s. The o v e r l a p amp l i tude then becomes: [2 .11] <P^"1|»S> - Z A B j a A t j B C j s B < p | ^ x * B | » A > . I f the same one e l e c t r o n p o t e n t i a l i s used f o r the ion as f o r the t a r g e t then o r t h o n o r m a l i t y of the c o n f i g u r a t i o n s g i v e s : [2 .12] < P * ^ 1 | ^ > - n s I A j a A t ^ A C j s A * j ( p ) , where n g i s the number of e q u i v a l e n t e l e c t r o n s (or the d imens ion of the symmetry group s ) . The momentum-space o r b i t a l t//j(p) i s g i ven by the F o u r i e r t r ans fo rm of the p o s i t i o n - s p a c e o r b i t a l \//j(r). i . e . [2 .13] tfj(p) = ( 2 i r ) - 3 ' 2 Jdr e x p ( - i p . r ) ^ ( r ) . In the s p e c i a l case when the Har t ree-Fock ground s t a t e N # n i s a good d e s c r i p t i o n of the t a r g e t s t a t e * n , i . e . a.^0 36 f o r A * 0 , the o v e r l a p ampl i tude becomes [MW76a]: t 2 . u a ] < P ^ _ 1 | * g > = n B Z . a ^ C . ^ . ( p ) ; and f o r a c l o s e - s h e l l t a r g e t , [2 .14b] < p ^ - 1 | * N > = nB\/*a0z^ t * 0 + . i p ) . Norma l l y , any set of independent p a r t i c l e o r b i t a l s , \pj, can be r e d e f i n e d so tha t f o r a g i ven ion s t a t e | * F 1> on l y one term i s s u f f i c i e n t i n the sum (equat ion 2 .14b ) . The a s s o c i a t e d o r b i t a l i s c a l l e d the c h a r a c t e r i s t i c o r b i t a l , i/>c, so t h a t : [2 .15] <P*iri|*S> - n s - 2 a 0 t ^ c ( p ) . The f i n a l e x p r e s s i o n i n the p l ane wave impulse and t a r g e t Har t ree-Fock app rox ima t ions f o r the d i f f e r e n t i a l c r o s s s e c t i o n i s g i ven by : [2 .16] o F ( p 0 , p 1 f p 2 ) = (2TT) • ( P i P 2 / P o ) o M o t t { C n s ( a O ) 2 S c O ] E a v l * j ( p ) | 2 ) ' F F The s p e c t r o s c o p i c f a c t o r S C Q = ( t c 0 ) 2 c a n ^ e r e 9 ^ r d e c * a s t n e p r o b a b i l i t y tha t the tth ion s t a t e c o n t a i n s the c o n f i g u r a t i o n w i th a ho l e o r b i t a l $ f in the t a r g e t . The d i f f e r e n t i a l 37 c r o s s s e c t i o n i s t h e r e f o r e s imply p r o p o r t i o n a l to Z a v j \£c (p) | 2 . In tho Born-Oppenheimer a p p r o x i m a t i o n , a mo lecu l a r wave func t ion i s a p roduc t of separa te e l e c t r o n i c , v i b r a t i o n a l and r o t a t i o n a l f u n c t i o n s . In cases where the f i n a l r o t a t i o n a l and v i b r a t i o n a l s t a t e s cannot be r e s o l v e d , the wave func t ion must be i n t e g r a t e d over the r o t a t i o n a l and the v i b r a t i o n a l s t a t e s . The d i f f e r e n t i a l c r o s s s e c t i o n i s g i ven by : [2 .17] o F ( p 0 , P i , p 2 ) tt JdO/d»v|^ c(p) | 2 . The v i b r a t i o n a l i n t e g r a l Jdv has been shown to be a c c u r a t e l y approx imated by assuming the n u c l e i a re f i x e d i n t h e i r e q u i l i b r i u m p o s i t i o n s [DM&75, MW76a]. The d i f f e r e n t i a l c r o s s s e c t i o n , i n the p lane wave impulse and t a r g e t Har t ree-Fock a p p r o x i m a t i o n s , i s t h e r e f o r e p r o p o r t i o n a l to the s p h e r i c a l l y averaged modulus square of the c h a r a c t e r i s t i c o r b i t a l in the momentum-space r e p r e s e n t a t i o n . i . e . [2 .18] a F ( p 0 , P , , p 2 ) - JdO|^ c (p ) | 2 . 38 2.1.5 V a l i d i t i e s Of The Approximations The f i n a l e x p r e s s i o n of the (e ,2e) (or t r i p l 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 ( equat ion 2.16) depends on the v a l i d i t i e s of the p lane wave impulse approx imat ion (PWIA) and the t a r g e t Har t ree-Fock approx ima t ion (THFA). The PWIA r e q u i r e s s c a t t e r i n g k inemat i c s w i th h igh momentum t r a n s f e r and h i g h energy . The noncop lanar symmetric s c a t t e r i n g k inemat i c s ensures maximized momentum t r a n s f e r . E a r l i e r , works [HM&73, HM&74, WHM75] on the PWIA have demonstrated i t s e s s e n t i a l v a l i d i t y f o r i n c i d e n t e n e r g i e s E 0 ^ 400eV. A l though the use of the more s o p h i s t i c a t e d s c a t t e r i n g approx imat ions such as the d i s t o r t e d wave impulse approx imat ion (DWIA) [UWM75, CG&78, FM&78, GF&80] and the averaged e i k o n a l approx ima t ion (EA) [FM&78, SCG79, CG&80, GF&80] has p r o v i d e d s i g n i f i c a n t improvement in the p r e d i c t i o n of the shape of the angu la r c o r r e l a t i o n i n cop l ana r (e ,2e) s t u d i e s , no obv ious d i f f e r e n c e s can be found in noncop lanar expe r imen t s , except i n the h i g h p r eg i on (pSM.Sao" 1 ) [DM&78, BH&80, BM&82], I t has , however, been sugges ted [MW76a, MW83] tha t i n o rder fo r the PWIA to p r e d i c t the c o r r e c t s-to-p i n t e n s i t y r a t i o (when a l l the c o r r e l a t i o n s a t e l l i t e s t a t e s are c o n s i d e r e d ) , an i n c i d e n t energy E o ^1200eV i s n e c e s s a r y . The use of h igher i n c i d e n t energy a l s o imposes more demanding requ i rements on the performance of the b i n a r y (e ,2e) spec t rome te r . At h ighe r i n c i d e n t e n e r g i e s the (e ,2e) c r o s s - s e c t i o n dec reases but the sma l l e r range of o> ( f o r a g i ven range of p) r e s u l t s in 39 improvement in s t a t i s t i c s due to a lower r e l a t i v e 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 . The sma l l e r <t> r ange , however, a l s o n e c e s s i t a t e s a sma l l e r acceptance ang le as we l l as b e t t e r des igned o p t i c s f o r the angu la r s e l e c t i o n (see chapte r 3) in order to ma in t a i n s u f f i c i e n t angu la r (or momentum) r e s o l u t i o n . A sma l l e r acceptance ang le w i l l undoubted ly cut down the count r a t e . A l l (e ,2e) works r epo r t ed i n t h i s t h e s i s are c a r r i e d out w i th E 0 equa l to l200eV p l u s the b i n d i n g energy . The p r e s e n t l y used i n c i d e n t energy appears to be a good compromise. The t a rge t Har t ree-Fock approx ima t ion (THFA) d e f i n e s a unique set of o r b i t a l s (the c a n o n i c a l ones) apar t from a u n i t a r y t r a n s f o r m a t i o n . The Har t ree-Fock o r b i t a l s f o r the ion ground s t a t e are in gene ra l not e x a c t l y the same as fo r the ion e x c i t e d s t a t e s or f o r the t a r g e t ( i . e . n e u t r a l ) ground s t a t e . The o v e r l a p ampl i tude has been c a l c u l a t e d in some cases [BS&74] and i s found to be over 90% f o r va l ence o r b i t a l s . T h i s o r t h o g o n a l i t y problem i s " e l i m i n a t e d " in the THFA by d e f i n i n g a l l s i n g l e - p a r t i c l e o r b i t a l s to be those of the t a r g e t ground s t a t e (see equa t i on 2 .12 ) . Such a t reatment has been found to be e s s e n t i a l l y adequate ( f o r the oute r va l ence o r b i t a l s ) in most of the (e ,2e) s t u d i e s to d a t e . In the case where e l e c t r o n c o r r e l a t i o n e f f e c t s are important [MU&74, DHW78], the more gene ra l e x p r e s s i o n (equat ion 2.11) must be used . Moreover , the ion wavefunct ion (equat ion 2.10) must be expanded to i n c l u d e 2-par t i c l e-1 -40 h o l e , 3 - p a r t i c l e - 2 - h o l e , e t c . p r o c e s s e s . In t h i s c a s e , the i n t e r p r e t a t i o n of the c r o s s s e c t i o n w i l l be complex [MW76a]. The r e s u l t i n g momentum d i s t r i b u t i o n c o n t a i n s an admixture of o r b i t a l momentum d e n s i t i e s and i s not c h a r a c t e r i s t i c of any s i n g l e o r b i t a l . The we igh t ing c o e f f i c i e n t s w i l l depend upon whether i n i t i a l or f i n a l s t a t e c o n f i g u r a t i o n a l i n t e r a c t i o n s (or both) a re impor tan t . 2.2 MOMENTUM-SPACE LCAO-MO WAVEFUNCTIONS T h e o r e t i c a l quantum chemis t r y [PCR73, LCG77, C82, LP83] has a lmost t o t a l l y r e l i e d upon the p o s i t i o n r e p r e s e n t a t i o n to fo rmu la te problems i n atomic and mo lecu l a r s t r u c t u r e . The momentum r e p r e s e n t a t i o n , a l t hough be ing an e q u i v a l e n t r e p r e s e n t a t i o n in atomic and mo lecu l a r p rob lems , i s g e n e r a l l y b e l i e v e d to l ead to g r e a t e r mathemat ica l d i f f i c u l t i e s . A l though the re have been some at tempts [PP29, F35 , L50 , L82] to use momenturn-space wave func t ions d i r e c t l y , these works have been l i m i t e d to some s imp le atomic systems main ly because of the c o m p l e x i t i e s i n v o l v e d in s o l v i n g the i n t e g r a l equa t i on of motion in momentum-space, which co r responds to the ( d i f f e r e n t i a l ) S ch rod inge r equa t ion in p o s i t i o n - s p a c e . A more p r a c t i c a l way of o b t a i n i n g the momentum-space wave func t ion i s t h e r e f o r e to f i r s t s o l v e the quantum mechan ica l problem in the more usua l p o s i t i o n - s p a c e and then app l y the F o u r i e r t r ans fo rm to the s o l u t i o n to o b t a i n the 41 momentum-space wavefunct ion [PP29] . In t a b l e 2 .1 , a summary of the b a s i c equa t ions of a mo lecu l a r wavefunct ion i n the LCAO-MO fo rma l i sm w r i t t e n in p o s i t i o n and momentum r e p r e s e n t a t i o n s i s g i v e n . The momentum-space wavefunct ion i s ob ta ined by F o u r i e r t r a n s f o r m a t i o n as ment ioned above . (See equa t ion 2 .13. ) D e t a i l e d d e r i v a t i o n s of these equa t ions can be found in r e f s . [E71, L72b, LN&75, KT76, MW76a, KS77, CB82a] . I t i s obv ious from t a b l e 2.1 tha t the F o u r i e r t r ans fo rm p re se r ve s the gene r a l LCAO-MO f o r m a l i s m ; i . e . a p o s i t i o n - s p a c e LCAO mo lecu l a r wavefunct ion can be expressed in momentum-space aga in as a l i n e a r combina t ion of momentum-space atomic wave func t i ons . The major d i f f e r e n c e i s the i n t r o d u c t i o n of a phase f a c t o r (which depends on the nuc l ea r p o s i t i o n s R ) i n t o the complex mo lecu la r o r b i t a l (MO) c o e f f i c i e n t s . There are many i n t e r e s t i n g r e l a t i o n s between the p o s i t i o n - s p a c e d e n s i t y and the c o r r e s p o n d i n g momentum-space d e n s i t y . These r e l a t i o n s are u s e f u l in u n r a v e l l i n g the o r i g i n s of many s t r u c t u r e s in the expe r imen ta l momentum d i s t r i b u t i o n s and in the t h e o r e t i c a l mo lecu l a r momentum d e n s i t y topography , namely, d e n s i t y [CS&79, CB82b] and d e n s i t y d i f f e r e n c e [HC68] maps. Many of the momentum-space n o t i o n s f i r s t d i s c u s s e d by Cou lson and Duncanson [CD41] in the e a r l y 4 0 ' s and l a t e r extended by E p s t e i n et al . [E73, E75, ET77] are indeed c e n t r e d upon F o u r i e r t r ans fo rm (FT) p r o p e r t i e s . Recen t l y the momentum-space p r o p e r t i e s of atoms Table 2.1 Summary of LCAO MO wavefunction and density equations ln position and momentum space • Wavefunction Position space a) Momemtum space a) Comment (I) molecular <l)_(r)=£c $ (r) orbital m - £ ma a - V (£. ) = ,^ Cma e x p (" i£-*^a )*a (£. ) L C A 0 M 0 formalism (II) atomic function (III) basis function (IV) radial part ^ O - ^ b i X a i C r - R a ) V M 0 <Da-AO AO is again a linear combination of basis function with correct symmetry X a * ( 0 - N „ i m R„ i ( r ) Y , m (Q) X a*(£> s , N„ 1 ™ p„ 1 (p> Yi ™ (°-) angular part of the a l -* "th™! V i a limi a l * "th"! nih basis function Is the same in both r-space and p-space where r^-r-R^ r ^ r - R j R n i l i ( r a ) : STF.GTF.etc. P n ! ( p ) - ( 2 / u ) 1 / 2 ( - i ) l i for P n t (p) see ref. J * * . J l 1 t K S 7 7 i - 1 N ° t e t h 3 t >h i i is the spherical Bessel function (V) density P m(r)=**(rH m(r) =PmC(L>+Pm<£> (VI) quasi- p f C r K I C ^ I 2 ! * ^ - ^ ) ! 2 classical a part (VII) interaction pj(r)- I C c ^ ^ U - R , ) ^ ^ ) a*b Pm (£- )° ( l 'm (2) < lT f l(£) p?W-XI<Wl 2l*.<*>l 2 orbital density decomposition one-centre part Pm(P)= I  claCmbe^~±2.'^%-la>>] two-centre part •a(£.)*b(£) a^ _r and JJ correspond to the position and momentum of the electron respectively. R^ refers to the equilibrium position of the atomic centre i . 43 have a l s o been examined by Smith and co-workers [HS&83, SWS84]. B r i e f l y , there are four major FT momentum-space c o n c e p t s : ( i ) symmetry i n v a r i a n c e w i th an automat ic i n v e r s i o n c e n t r e at the p-space o r i g i n ; ( i i ) i n ve r se s p a t i a l r e v e r s a l ; ( i i i ) p-space mo lecu l a r d e n s i t y d i r e c t i o n a l r e v e r s a l ; and ( i v ) p-space mo lecu l a r d e n s i t y o s c i l l a t i o n . E x t e n s i v e d i s c u s s i o n s of these have been g i ven in the l i t e r a t u r e [CD41, ET77, CB82a, CB82b, MB&83, LB83b] ; on l y a b r i e f summary w i l l t h e r e f o r e be g iven he re . In a d d i t i o n , another important p r o p e r t y ( f i r s t d i s c u s s e d by E p s t e i n and Tanner [ ET77 ] ) , which i s a d i r e c t consequence of the V i r i a l theorem, w i l l a l s o be ment ioned . T h i s . V i r i a l p r o p e r t y i s found to be u s e f u l in d i s c u s s i n g f e a t u r e s i n d e n s i t y d i f f e r e n c e (or bond d e n s i t y ) maps [HC68, LB83b] of the two e l e c t r o n systems (see chap t e r s 5 and 6 ) . 2 . 2 . 1 F o u r i e r T rans fo rm P r o p e r t i e s The momentum-space wavefunct ion i s o b t a i n e d by the F o u r i e r t r ans fo rm of the p o s i t i o n - s p a c e wave func t ion ( equat ion 2 .13 ) . Be fore d i s c u s s i n g the i n d i v i d u a l F o u r i e r t r ans fo rm p r o p e r t i e s , i t i s important to have a c l o s e r look at the F o u r i e r t r a n s f o r m i t s e l f . By w r i t i n g e x p ( - i p . r ) i n a power s e r i e s expans ion and r e p l a c i n g p w i th i t s c o r r e s p o n d i n g ( p o s i t i o n - s p a c e ) ope ra to r form V , the f o l l o w i n g r e l a t i o n i s o b t a i n e d , 44 [2 .19] ^ ( p ) = ( 2 T T ) - 3 / 2 / d r ( 1 - i ( V r . r ) - i ( V r . r ) 2 + ...)V/j ( r ) . The momentum-space (p-space) wave func t ion i n v o l v e s an i n t e g r a l of the sum of powers of the g r a d i e n t of the p o s i t i o n - s p a c e ( r-space) wavefunct ion over a l l space . In the case of atomic o r b i t a l s where l a r g e changes i n the r-space wave func t ion occur on l y near the n u c l e u s , the sma l l r pa r t of the r-space wavefunct ion c o n t r i b u t e s most s i g n i f i c a n t l y to the l a r g e p pa r t of the p-space wave func t ion (the i n ve r se s p a t i a l r e v e r s a l ) . In the case of m o l e c u l a r o r b i t a l s , however, the s p a t i a l r e v e r s a l r e l a t i o n , a l t hough s t i l l q u a l i t a t i v e l y c o r r e c t , must be a p p l i e d w i th c a u t i o n s i n ce any r eg i on where there i s a r a p i d change i n the r-space wave func t ion w i l l c o n t r i b u t e s i g n i f i c a n t l y to the h igh p pa r t of the p-space wave func t i on . Such r a p i d change can occur between atomic c e n t r e s , f o r i n s t a n c e a c r o s s a noda l p l a n e . A no t ab l e example of t h i s i s the C 0 2 ( l 7 r g) o r b i t a l [CB82b, LB84b] ( chapter 7 ) . I t shou ld be noted tha t t h i s p h y s i c a l a s s o c i a t i o n of the g r a d i e n t of the r-space wave func t ion w i th the p-space wavefunct ion i s a l s o h e l p f u l i n r a t i o n a l i z i n g the p-space mo lecu l a r d e n s i t y d i r e c t i o n a l r e v e r s a l p r o p e r t y . The l o n g i t u d i n a l change a c r o s s any r-space d e n s i t y lobe i s o b v i o u s l y l e s s r a p i d ( sma l l p pa r t ) than changes in the o ther p e r p e n d i c u l a r d i r e c t i o n ( l a rge p p a r t ) . The c o r r e s p o n d i n g d i r e c t i o n a l p r o p e r t i e s of the p-space d e n s i t y lobe w i l l be r e v e r s e d . 45 ( i ) Symmetry, P-space O r i g i n , And P-space V i r t u a l Boundary. A l l symmetry p r o p e r t i e s of the p o s i t i o n - s p a c e wave func t ions are p rese r ved under the F o u r i e r t r a n s f o r m . T h i s i s e v i den t from equa t ion ( I I I ) i n t a b l e 2.1 which i n d i c a t e s tha t the s p h e r i c a l harmonics (which determine the angu la r dependence of the atomic o r b i t a l s ) are i n v a r i a n t under the t r a n s f o r m a t i o n . In a d d i t i o n the momentum-space wave func t ion has i n v e r s i o n symmetry ( i . e . the e l e c t r o n has no net t r a n s l a t i o n a l motion in the cent re-of-mass f rame) . T h i s i n v e r s i o n symmetry i s aga in obv ious from t a b l e 2.1 and i s the r e s u l t of the phase s h i f t i n the MO c o e f f i c i e n t s . One important d i f f e r e n c e between the r-space and p-space LCAO-MO forma l i sms i s in the d i f f e r e n t ways of e x p r e s s i n g the l o c a l i t i e s of the d e n s i t y f u n c t i o n . In r-space these l o c a l i t i e s are expressed by r e f e r e n c i n g e l e c t r o n s to t h e i r own atomic c e n t r e s , i . e . , r^. In p-space, however, these d i r e c t r e f e r e n c e s are l o s t ( i . e . there are no nuc l ea r p o s i t i o n s in p-space) and the n u c l e a r p o s i t i o n v e c t o r s (R ) a i n t r oduce an e x t r a phase i n t o the MO c o e f f i c i e n t s (equat ion ( I ) , t a b l e 2 . 1 ) . The r e s u l t i n g momentum d e n s i t y f u n c t i o n has i n v e r s i o n symmetry i n p-space because of the l i n e a r combina t ion of the symmetric (wi th r e spec t to i n v e r s i o n ) p-space atomic o r b i t a l d e n s i t y f u n c t i o n s . However, the nuc l ea r geometry i n f o r m a t i o n i s not l o s t but i s r e t a i n e d in the i n t e r f e r e n c e te rm. The i n v e r s i o n symmetry in p-space can a l s o be regarded as a d i r e c t m a n i f e s t a t i o n of the i n v a r i a n c e 46 of the SCF wavefunct ion under t ime r e v e r s a l o p e r a t i o n . S ince e l e c t r o n s expe r i ence sharp a t t r a c t i v e p o t e n t i a l s near the nuc l ea r c e n t r e s ( thus i n c r e a s i n g t h e i r momenta) the nuc l ea r c e n t r e s in r-space a c t , by c o s m o l o g i c a l ana logy , l i k e " b l a c k h o l e s " , i . e . i n c r e a s i n g the momentum as the e l e c t r o n s f a l l towards them and c a u s i n g the c o r r e s p o n d i n g p-space d e n s i t y to i n f l a t e outward towards the p-space v i r t u a l boundary at p=°° (which co r r esponds to the e l e c t r o n momenta at the n u c l e i ) . The p-space o r i g i n , on the o the r hand, can be thought of as be ing l i k e a c o s m o l o g i c a l "wh i te h o l e " s i n c e the e l e c t r o n s w i th ze ro momentum ( i . e . f r e e e l e c t r o n s ) a re be ing pushed out i n t o the more a t t r a c t i v e h igh momentum r eg ion near the nuc l ea r c e n t r e s . The low momentum regime co r r esponds in r-space to the c h e m i c a l l y important va l ence r e g i o n . One can t h e r e f o r e p i c t u r e the e f f e c t of the F o u r i e r t r ans fo rm as " t u r n i n g the wavefunct ion i n s i d e o u t " , wi thout chang ing the symmetry p r o p e r t i e s , by i n f l a t i n g the c h e m i c a l l y l e s s i n t e r e s t i n g core ( i nne r r-space) r e g i o n ( a round , each nuc l e a r c en t r e ) towards the p-space v i r t u a l boundary and c o l l a p s i n g the c h e m i c a l l y s i g n i f i c a n t va l ence (outer r-space) r eg i on towards the p-space o r i g i n , where the b i na r y (e ,2e) t e chn ique i s most s e n s i t i v e . The b i n a r y (e ,2e) method t h e r e f o r e p r o v i d e s a new and unique vantage p o i n t from which to view the most c h e m i c a l l y s i g n i f i c a n t pa r t of the mo lecu l a r wave func t ion in chemica l bonding phenomena. 47 ( i i ) Inverse S p a t i a l R e v e r s a l . In a d d i t i o n to the i n v e r s i o n symmetry i n momentum-space o r b i t a l s , there i s a l s o ampl i tude i n v e r s i o n , namely, c o n t r a c t i o n of the r-space wavefunct ion co r r esponds to expans ion of the p-space wavefunct ion and v i c e v e r s a . T h i s i n v e r s e s p a t i a l r e v e r s a l a r i s e s from the pr dependence of the s p h e r i c a l Besse l f u n c t i o n in equa t i on ( IV ) , t a b l e 2 . 1 . T h i s p r o p e r t y has been demonstrated in many e a r l i e r works [CD41, ET77, CB82a, CB82b, LB83b] and i s we l l i l l u s t r a t e d in the s tudy of noble gases us i ng b i na r y (e ,2e) spec t roscopy [LB83a] ( chapter 4 ) . The p r o p e r t y can be r e a d i l y e x p l a i n e d us i ng the He isenberg u n c e r t a i n t y p r i n c i p l e , namely, a more l o c a l i z e d r-space wavefunct ion w i th a sma l l s p a t i a l u n c e r t a i n t y Ar w i l l r e s u l t in a l a r g e p-space u n c e r t a i n t y Ap, i . e . a more d i f f u s e p-space wave func t i on . ( i i i ) M o l e c u l a r Dens i t y D i r e c t i o n a l R e v e r s a l . The r e l a t i v e l o n g i t u d i n a l and t r a n s v e r s e s p a t i a l e x t e n s i o n s of lobes i n an o r b i t a l i n r-space are i n t e r changed in p-space and v i c e v e r s a [CD41, ET77, BC&82, CB82a, CB82b, MB&83, LB83b] . For i n s t a n c e , in the sigma bonding o r b i t a l of a d i a t o m i c mo lecu l e , the bonding d e n s i t y i s d i r e c t e d a long the i n t e r n u c l e a r a x i s i n r-space (the l o n g i t u d i n a l d i r e c t i o n ) whereas the momentum d e n s i t y of such a bonding o r b i t a l i s o r i e n t e d i n the t r a n s v e r s e d i r e c t i o n , i . e . p e r p e n d i c u l a r to the i n t e r n u c l e a r d i r e c t i o n . S i m i l a r concepts can be used to 48 d r a f t out the d i r e c t i o n s of l obes i n an a n t i b o n d i n g o r b i t a l . . In e s sence , the d e n s i t y d i r e c t i o n a l r e v e r s a l p rope r t y i s a combined e f f e c t of the symmetry p rope r t y and the i n ve r se s p a t i a l r e v e r s a l p r o p e r t y a p p l i e d s p e c i f i c a l l y to mo lecu la r o r b i t a l s . ( i v ) M o l e c u l a r D e n s i t y O s c i l l a t i o n s . The mo lecu l a r momentum d e n s i t y of a bonding ( an t ibond ing ) o r b i t a l e x h i b i t s c o s i n u s o i d a l ( s i n u s o i d a l ) modu la t ions w i th s p a t i a l p p e r i o d i c i t y of 27r/|R^-R a| in the b o n d - p a r a l l e l d i r e c t i o n [CD41, ET77, CB82a, CB82b, LB83b] . T h i s i s a d i r e c t consequence of the phase f a c t o r i n t r o d u c e d by the F o u r i e r t r a n s f o r m in the p-space LCAO-MO f o r m a l i s m . The modu la t ion e f f e c t s come from the complex c o e f f i c i e n t * product te rm, c m a C m b e x P ^ ~ ^ * ^ R b ~ R a ' * n t * i e i n t e r f e r e n c e d e n s i t y P^,(p) in equa t i on (V I I ) , t a b l e 2 . 1 . The p e r i o d i c i t y i s de termined by the p . f R ^ - R ^ term and the r e l a t i v e s i gn of * the p roduc t of MO c o e f f i c i e n t s C C . . (Note that C and ma mb ma f o r an a n t i b o n d i n g sigma o r b i t a l are o p p o s i t e i n s i g n , which i n t r o d u c e s an e x t r a 90° phase s h i f t . ) The s i g n i f i c a n c e of the o s c i l l a t i o n s i s tha t i n p r i n c i p l e the r e l a t i v e nuc l ea r s e p a r a t i o n s can be i n f e r r e d from the p-space p e r i o d i c i t i e s . 49 2 .2 .2 V i r i a l P rope r t y The fo rmat ion of a chemica l bond i s accompanied by an i n c r e a s e in the average momentum <p> [CD41, ET77] by t r a n s f e r r i n g momentum d e n s i t y at the p-space o r i g i n i n t o the h igh momentum r eg ion i n the t r a n s v e r s e d i r e c t i o n . T h i s i s a d i r e c t r e s u l t of the V i r i a l theorem ( fo r d i a t om i c mo lecu les in the Born-Oppenheimer a p p r o x i m a t i o n ) : [2 .20] 2<T> + <V> + RdE/dR = 0 where E, T and V are r e s p e c t i v e l y the t o t a l , k i n e t i c and p o t e n t i a l ene rg i e s and R i s the i n t e r n u c l e a r s e p a r a t i o n . For a system in i t s e q u i l i b r i u m nuc l e a r geometry , one has the s p e c i a l case where the average p o t e n t i a l energy i s the nega t i v e of tw ice the k i n e t i c energy , i . e . <V>=-2<T>. Hence any p roces s which lowers the t o t a l energy ( no t i ng tha t <E>=-<T>) w i l l r a i s e the average momentum <p> s i n c e <T>=^<p2>. T h i s t h e r e f o r e produces a gene ra l r e d i s t r i b u t i o n of the momentum d e n s i t y i n t o the h i g h momentum r e g i o n . The d i r e c t i o n of such d e n s i t y r e d i s t r i b u t i o n i s of course governed by the F o u r i e r t r ans fo rm p r o p e r t i e s ment ioned above. I n t e r e s t i n g l y , t h i s p r o p e r t y l eads to a redrawing of the b i n d i n g and a n t i b i n d i n g boundar ies in p-space and a l s o a somewhat d i f f e r e n t bonding concept i n momentum-space. T h i s p r o p e r t y w i l l be d i s c u s s e d f u r t h e r u s i n g d e n s i t y d i f f e r e n c e maps ( chapte rs 5 and 6 ) . 50 Chapter I I I EXPERIMENTAL METHOD 3.1 INTRODUCTION The f i r s t gene ra t i on v e r s i o n of- the U .B .C . b i n a r y (e ,2e) spec t rometer has been d e s c r i b e d by Hood et al. [HHB77] and i n f u r t h e r d e t a i l by Cook [C80] . The present (second gene ra t i on ) v e r s i o n of the b i na r y ( e r 2e ) spect rometer i s m o d i f i e d from the o r i g i n a l [HHB77, C80] to p rov ide major improvements i n momentum and t im ing r e s o l u t i o n s as we l l as in s i g n i f i c a n t l y b e t t e r s i g n a l - t o - n o i s e r a t i o . I t opera tes at a s i g n i f i c a n t l y h ighe r i n c i d e n t e l e c t r o n energy (>1200eV) than tha t used in the f i r s t gene r a t i on spec t rometer (>400eV) so as to ensure the v a l i d i t y of the r e a c t i o n approx imat ion ( i . e . p l ane wave impulse approx imat ion ) [MW76a]. It i s a l s o under the c o n t r o l of a d e d i c a t e d computer system, which a l l ows s e q u e n t i a l da ta c o l l e c t i o n and automat ic da ta n o r m a l i z a t i o n . The p resen t i n s t r u m e n t a t i o n r ep re sen t s a s t a t e - o f - t h e - a r t b i na r y (e ,2e) spec t romete r , which ope ra tes r o u t i n e l y w i th h igh momentum and t im ing r e s o l u t i o n s , f o r the i n v e s t i g a t i o n of o r b i t a l momentum d e n s i t i e s and the e l e c t r o n i c s t r u c t u r e of atoms and m o l e c u l e s . I t i s on l y p o t e n t i a l l y i n f e r i o r in terms of data c o l l e c t i o n e f f i c i e n c y to the m u l t i c h a n n e l p l a t e v e r s i o n tha t i s p r e s e n t l y under 51 development by Weigold and coworkers [CM&84] at F l i n d e r s U n i v e r s i t y . The b ina r y (e ,2e) spec t ro-sys tem can be d i v i d e d i n t o th ree major i n t e g r a t e d sub-systems, namely the spect rometer i t s e l f , the a s s o c i a t e d c o n t r o l e l e c t r o n i c s and an execu t i v e mic rocomputer . A s i m p l i f i e d b lock diagram o u t l i n i n g the b a s i c components of the sub-systems i s shown in f i g u r e 3 .1 . The spect rometer c o n s i s t s of one pr imary e l e c t r o n o p t i c a l system (PEOS) and two i d e n t i c a l secondary e l e c t r o n o p t i c a l systems (SEOSs). The PEOS p r o v i d e s a f i n e l y focussed i n c i d e n t e l e c t r o n beam fo r the h i g h energy e l e c t r o n impact i o n i z a t i o n df the gaseous t a r g e t s . The SEOSs then momentum ana l y se the ou tgo ing e l e c t r o n s a f t e r the (e ,2e) r e a c t i o n under the symmetric noncop lanar s c a t t e r i n g c o n d i t i o n s . The c o n t r o l e l e c t r o n i c s supp ly the power to the v a r i o u s o p t i c a l components in the spec t rometer and p rov i de the a c t u a l scan v o l t a g e s f o r the expe r imen ta l pa ramete rs : E 0 and <p. The c o i n c i d e n c e c i r c u i t r y r e g i s t e r s any c o i n c i d e n t occur rence of p u l s e s a r i s i n g from the two SEOSs. The microcomputer system per forms execu t i v e f u n c t i o n s such as t ime-averag ing the c o l l e c t e d data and c o n t r o l l i n g the scan sequence fo r the r ea l - t ime data a c q u i s i t i o n . Once v a r i o u s v o l t a g e s of i n d i v i d u a l e l e c t r o n o p t i c a l e lements have been o p t i m i z e d both in terms of r e s o l u t i o n and count r a t e , the ope ra to r on ly needs to input the gaseous t a r g e t s of i n t e r e s t i n t o the spec t rome te r , se t up the necessa ry scan sequences f o r the Gaseous Atoms or Molecules Momentum Analyse Outgoing Electrons SECONDARY ELECTRON OPTICS SYSTEM phi variable under vaccum and magnetic field free COLLISION CHAMBER Produce & Collimate Incident Electrons PRIMARY ELECTRON OPTICS SYSTEM Momentum Analyse Outgoing Electrons SECONDARY ELECTRON OPTICS SYSTEM Timing Electronics Beam Monitor Electron Optics Power Supplies P rog ram Power Supplies Digitize Count Signal t i m e -average Scan Eo or phi Binding Energy Spectra Momentum Distributions 01 INPUT BINARY (e,2e) SPECTROMETER CONTROL ELECTRONICS EXECUTIVE COMPUTER OUTPUT F i g u r e 3.1 - Block diagram of the b i n a r y (e,2e) s p e c t r o -system. 53 microcomputer and a l low s u f f i c i e n t time fo r the s p e c t r a to accumula te . The p resen t i n s t r u m e n t a t i o n r e q u i r e s a minimal amount of a t t e n t i o n and s e r v i c i n g from the ope ra to r fo r the o f t en t ime consuming data c o l l e c t i o n p r o c e s s . D e t a i l e d d i s c u s s i o n s of each of the major components in the b ina r y (e ,2e) spec t ro-sys tem are g i ven in the f o l l o w i n g s e c t i o n s . 3.2 BINARY (e ,2e) SPECTROMETER A schemat ic d iagram of the b i n a r y (e ,2e) spec t rometer i s g iven in f i g u r e 3 .2 . The spect rometer i s composed of a pr imary e l e c t r o n o p t i c a l system (PEOS) and two i d e n t i c a l secondary e l e c t r o n o p t i c a l systems (SEOSs), c o n t r o l l i n g the k inemat i c s of the i n c i d e n t e l e c t r o n beam and the ou tgo ing e l e c t r o n beams r e s p e c t i v e l y . The PEOS c o n s i s t s of an e l e c t r o n gun (EG), a gas c e l l (GC) and an e l e c t r o n beam c o l l i m a t o r . The SEOS each c o n s i s t s of a three-e lement asymmetric immersion l ens (A I L ) , a 135° s e c t o r 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) and a s i n g l e channe l e l e c t r o n m u l t i p l i e r (CEM). One of the SEOSs i s s t a t i o n a r y wh i le the o ther i s mounted on a r o t a t a b l e t u r n t a b l e (RT) d r i v e n by a servo-motor (SM) o u t s i d e the vacuum chamber, which a l l ows the r e l a t i v e az imu tha l ang le <j> to be v a r i e d . 54 0 VARIABLE STATIONARY F i g u r e 3.2 - Schematic of b i n a r y (e,2e) spectrometer. Legend: CEM - Channel E l e c t r o n M u l t i p l i e r , CMA -C y l i n d e r M i r r o r A n a l y s e r , AIL - Asymmetric Immersion Lens , GC - Gas C e l l , EG - E l e c t r o n Gun; EL - E i n z e l L ens , A - Anode, G - G r i d , C - Cathode , FC - Faraday Cup, EC - End C o r r e c t o r , RT - Ro t a t ab l e T u r n t a b l e ; A1 to A8 - A p e r t u r e s , P1 to P3 - Spray P l a t e s , D1 to D4 - D e f l e c t o r s . 55 3.2.1 E l e c t r o n Gun The i n c i d e n t e l e c t r o n s a re produced by the rmion i c em i s s i on u s i n g a commerc ia l i n t e r m e d i a t e energy (100-2000eV) e l e c t r o n gun ( C l i f t r o n i c s CE5AH). I t c o n s i s t s of a th ree-element cathode l ens and a three-e lement e i n z e l l e n s . The three-e lement cathode l e n s [D55, M67, G72] r e p r e s e n t s the s i m p l e s t v e r s i o n of a t he rm ion i c source and c o n s i s t s of an e m i s s i v e cathode (C ) , a f o c u s s i n g c o n t r o l g r i d (G) and an a c c e l e r a t i n g anode (A ) . In the p resen t ar rangement , the cathode i s made of a t h o r i a t e d tungs ten f i l amen t (Goodfe l low m e t a l , W/Th 0 .6%, 0.125mm d i a . ) and i s bent i n t o a V-shaped h a i r p i n . The f i l amen t i s e l e c t r i c a l l y heated by a DC cu r r en t of 2.2-2.6Amps. w i th a nomina l o p e r a t i n g f i l amen t tempera ture of 1700-1900K. When the e l e c t r o n s are r e l e a s e d from the m e t a l , they are a lmost s t a t i o n a r y w i th an energy spread of 0 .2-0 .8eV . The nomina l l i f e - t i m e of a f i l amen t o p e r a t i n g under t h i s c o n d i t i o n and a 1 0 " 3 To r r p r e s su re of i n e r t gases i s 50-80 days . These e l e c t r o n s can be e x t r a c t e d by an a c c e l e r a t i n g p o t e n t i a l ( u s u a l l y a few hundred eV) between the f i l amen t and the anode. F o c u s s i n g of the e l e c t r o n s i s a ch i eved by a p p l y i n g a sma l l p o t e n t i a l to the g r i d of o p p o s i t e p o l a r i t y w i th r e spec t to tha t of the a c c e l e r a t i n g (anode) p o t e n t i a l . Because the g r i d i s so c l o s e to the f i l a m e n t where the e l e c t r o n s are moving s l o w l y , a s l i g h t v a r i a t i o n i n the g r i d p o t e n t i a l can s i g n i f i c a n t l y a l t e r bo th the shape and the g r a d i e n t of the l o c a l p o t e n t i a l 56 d i s t r i b u t i o n near the ca thode . T h i s c r i t i c a l l y a f f e c t s both the i n t e n s i t y and the focus of the produced e l e c t r o n beam. Such a l o c a l p o t e n t i a l d i s t r i b u t i o n i s de te rmined p r i m a r i l y by three pa ramete rs : the c a t h o d e - g r i d s e p a r a t i o n , the ape r tu r e s i z e of the g r i d and the g r i d p o t e n t i a l . The o p t i m i z e d c a t h o d e - g r i d s e p a r a t i o n (ob ta ined by t r i a l - a n d -e r r o r ) f o r the V-shaped f i l amen t i s 0.25-0.40mm. The q u a l i t y of an e l e c t r o n gun i s u s u a l l y r a t e d by the b r i g h t n e s s 1 and the d i ve rgence of the produced e l e c t r o n beam. The b r i g h t n e s s R can be w r i t t e n [G72] a s : [3 .1 ] R - ( jV ) /T , where j i s the c u r r e n t d e n s i t y a t the f i l a m e n t , V i s the a c c e l e r a t i n g p o t e n t i a l i n e l e c t r o n v o l t s and T i s the f i l a m e n t t empera tu re . I t i s c l e a r that the b r i g h t n e s s can be i n c r e a s e d by a p p l y i n g e i t h e r a h ighe r a c c e l e r a t i n g p o t e n t i a l or a l a r g e r c u r r e n t through the f i l a m e n t . I t shou ld be n o t e d , however, tha t a l a r g e r a p p l i e d c u r r e n t w i l l have the adverse e f f e c t of i n c r e a s i n g the o p e r a t i n g t empera tu re , which w i l l not on l y dec rease the b r i g h t n e s s but a l s o cause a f a s t e r burn-out r a t e ( i . e . a s h o r t e r l i f e - t i m e ) . Beam d i ve rgence can be min imized by p h y s i c a l l y l i m i t i n g the e m i t t i n g s u r f a c e a rea so tha t the emi t ted e l e c t r o n s a l l t r a v e l c l o s e l y 1 B r i g h t n e s s i s d e f i n e d as the beam cu r r en t per u n i t a rea per u n i t s o l i d a n g l e . 57 p a r a l l e l to the mean beam d i r e c t i o n . The use of a V-shaped f i l amen t to g i ve a f i n e p o i n t has proven to produce an e l e c t r o n beam wi th a sma l l e r d i ve rgence [M67]. Beam d i ve rgence can be f u r t h e r c o n t r o l l e d by u s i n g a supp lementa l three-e lement e i n z e l l ens (EL) [AR72a]. The two outer e lements of the l e n s are at the same p o t e n t i a l (g round) . Only the p o t e n t i a l of the c e n t r a l element i s v a r i e d to produce the f o c u s s i n g a c t i o n . As the e l e c t r o n passes through the l e n s , i t i s f i r s t d e c e l e r a t e d (or a c c e l e r a t e d ) from the outer p o t e n t i a l to the c e n t r a l p o t e n t i a l and then a c c e l e r a t e d (or d e c e l e r a t e d ) back to the ou te r p o t e n t i a l . The k i n e t i c energy of an e l e c t r o n e x i t i n g the anode i s t h e r e f o r e u n a f f e c t e d by the e i n z e l l ens ( i . e . no r e t a r d a t i o n or a c c e l e r a t i o n ) . Only f o c u s s i n g of the e l e c t r o n beam i s ob t a i ned w i th the e i n z e l l e n s . 3.2.2 Gas C e l l The e l e c t r o n beam produced by the e l e c t r o n gun (EG) must be a l i g n e d w i th the a x i s of the gas c e l l (GC) u s i n g an e l e c t r o n beam c o l l i m a t o r to c o r r e c t f o r any sma l l p h y s i c a l misa l ignment between the e l e c t r o n gun and the gas c e l l as we l l as to compensate f o r any beam d i s t o r t i o n caused by r e s i d u a l magnet ic f i e l d (see s e c t i o n 3 . 2 . 8 ) . The gas c e l l i s a b rass c y l i n d e r (15.9mm o .d . ) e n c l o s e d at the bottom and the top by two spray p l a t e s : P2 and P3 r e s p e c t i v e l y . Annular 58 s l o t s in the gas c e l l a l l ow the passage of the ou tgo ing e l e c t r o n s i n t o the secondary e l e c t r o n o p t i c a l sys tems. S l i d i n g b rass shims are a t t a c h e d to the s i d e s of the movable l ens cone (see l a t e r ) to b l i n d o f f any open a reas not covered by the l e n s cones . Gas i s admi t t ed i n t o the gas c e l l through a sma l l ho l e (1mm d i a . ) in the gas c e l l . The p resen t semi-enc l o sed arrangement a l l ows the " l o c a l " dynamic gas p r e s su re i n s i d e the gas c e l l to be at l e a s t two o rde r s of magnitude h ighe r than the ambient vacuum chamber p r e s s u r e . The d i r e c t i o n of the i n c i d e n t e l e c t r o n beam i s d e f i n e d by the ape r t u r e s i z e s of the spray p l a t e s P2 (2.5mm d i a . ) and P3 (1.5mm d i a . ) . These d e f i n i n g a p e r t u r e s are m e c h a n i c a l l y a l i g n e d to unambiguously d e f i n e the a x i s of the gas c e l l . The maximum g e o m e t r i c a l en t rance h a l f - a n g l e (wi th r e spec t to the a x i s of the gas c e l l ) f o r an e l e c t r o n to pass through both P2 and P3 i s app rox ima te l y ± 1 . 3 ° . 3 .2 .3 Beam C o l l i m a t o r The beam c o l l i m a t o r c o n s i s t s of two s e t s of quadrupo le d e f l e c t o r s (D1, D2) , a set of th ree spray p l a t e s (P1, P2, P3) and a Faraday cup ( FC ) . The spray p l a t e s as w e l l as the Faraday cup are j o i n e d to sepa ra te micro-ammeters to monitor the c o l l e c t e d c u r r e n t at each p o i n t of the i n c i d e n t beam path i n d e p e n d e n t l y . Beam c o l l i m a t i o n i s a ch i e ved by man ipu l a t i ng the v o l t a g e s of the d e f l e c t o r e lements of the quadrupo le 59 d e f l e c t o r s to min imize the c u r r e n t c o l l e c t e d by the spray p l a t e s ( P i , P2 and P3) and to s i m u l t a n e o u s l y maximize the c u r r e n t c o l l e c t e d by the Faraday cup ( FC ) . In p r a c t i c e both the g r i d v o l t a g e and the focus v o l t a g e ( c e n t r a l vo l t age of the three-e lement e i n z e l l ens ) of the e l e c t r o n gun are a l s o ad ju s t ed d u r i n g the beam c o l l i m a t i o n p r o c e s s . T y p i c a l l y a c u r r e n t of 50uk i s c o l l e c t e d at the Faraday cup fo r a 1200eV i n c i d e n t beam energy . The nominal c u r r e n t ( in uk) r a t i o fo r FC:P3:P2:P1 in t h i s case i s 5 0 : 1 : 1 : 0 . 1 . I t shou ld be noted tha t i t i s ext remely important to have a w e l l a l i g n e d i n c i d e n t e l e c t r o n beam s i n c e the p o l a r ang les of the ou tgo ing e l e c t r o n beams are dependent on the i n c i d e n t beam d i r e c t i o n . Moreover , an o f f - a x i s i n c i d e n t beam w i l l cause v a r i a t i o n of the s i n g l e s count r a t e w i th the r e l a t i v e az imutha l ang le <t> because of the change in volume of the e f f e c t i v e c o l l i s i o n space . T h i s w i l l r e s u l t in s p u r i o u s v a r i a t i o n s in the i n t e n s i t y of the measured momentum d i s t r i b u t i o n . 3.2.4 Three-element Asymmetric Immersion Lens The s c a t t e r e d and e j e c t e d e l e c t r o n s ( a f t e r the (e ,2e) r e a c t i o n ) emerge from the gas c e l l each w i th a k i n e t i c energy of ^ ( E 0 ~ b i n d i n g e n e r g y ) , where E 0 i s the i n c i d e n t e l e c t r o n energy . In the p resen t c a s e , these ou tgo ing e l e c t r o n s have k i n e t i c e n e r g i e s of 600eV fo r an i n c i d e n t e l e c t r o n energy of E o >1200eV. A three-e lement asymmetr ic immersion l ens (AIL) 60 i s used to r e t a r d the ou tgo ing e l e c t r o n s so tha t a lower pass e n e r g y 2 of the e l e c t r o n energy a n a l y s e r can be used in order to improve the energy r e s o l u t i o n . A second f u n c t i o n of the AIL i s to t r a n s p o r t and focus the emerging e l e c t r o n s from the c o l l i s i o n c en t r e ( in the gas c e l l ) to the en t rance ape r tu r e of the energy a n a l y s e r . In the p resen t a p p l i c a t i o n , the three-e lement AIL f ocusses and r e t a r d s the ou tgo ing e l e c t r o n s from 600eV to lOOeV, which i s the normal pass energy of the a n a l y s e r . A s imple three-e lement asymmetr ic immersion l e n s (AIL) [H69, HK70, AR72b, HR76, R78] r e q u i r e s seven g e o m e t r i c a l s p e c i f i c a t i o n s , i . e . 3 w a l l - t h i c k n e s s e s , 2 element spac ings and 2 ape r t u r e s i z e s , each w i th r e spec t to the t h i r d ape r tu r e s i z e . The r e l e v a n t g e o m e t r i c a l parameters of the th ree-element e l e c t r o s t a t i c c y l i n d r i c a l l ens are g i ven in f i g u r e 3.2 (see a l s o f i g u r e 3 .4 ) . A l l the f o c a l p r o p e r t i e s of the three-e lement AIL are determined by two v o l t a g e r a t i o pa ramete r s : V3/V1 and V2/V1, where V 1 , V2 and V3 are the a p p l i e d v o l t a g e s of the e n t r a n c e , c e n t r a l and e x i t e lements of the AIL r e s p e c t i v e l y . Us ing the vo l t age r a t i o V2/V1 to c o n t r o l the f o c u s s i n g p r o p e r t i e s and the o the r V3/V1 to change the energy of the image, i t i s p o s s i b l e to keep the image d i s t a n c e f i x e d f o r a chosen ob jec t d i s t a n c e even when 2 The pass energy of an e l e c t r o n a n a l y s e r i s commonly used to r e f e r to the mean ent rance k i n e t i c energy of an e l e c t r o n beam be ing t r a n s p o r t e d from the en t rance to the e x i t of the e l e c t r o n energy a n a l y s e r . 61 the energy of the image i s v a r i e d . T h i s "zooming" p rope r t y i s important in the p resen t a p p l i c a t i o n i f a d i f f e r e n t r e t a r d a t i o n r a t i o ( i . e . V1/V3) i s d e s i r e d because the ob jec t to image s e p a r a t i o n i s p h y s i c a l l y f i x e d in the spec t rome te r . The r e l a t i o n s between the ob j ec t to image d i s t a n c e s fo r a p a r t i c u l a r l ens system are shown by the c o r r e s p o n d i n g P-Q cu rves [HR76] f o r v a r i o u s v o l t a g e r a t i o s V2/V1 and m a g n i f i c a t i o n s . The three-e lement AIL used i n the p resen t a p p l i c a t i o n has been c o n s t r u c t e d us i ng a unique d e s i g n . The c y l i n d e r s used f o r the l ens s t a c k s are made of b rass and are a c c u r a t e l y a l i g n e d u s i n g h idden t e f l o n tube s p a c e r s . T e f l o n r i n g s are used f o r p r o v i d i n g the important spac ings between l ens e l emen t s . Two angu la r s e l e c t i o n p l a t e s w i th d e f i n i n g a p e r t u r e s A1 (A5) and A2 (A6) are used to p h y s i c a l l y l i m i t the g e o m e t r i c a l acceptance ang le of the l e n s . In the p resen t a r rangement , the d e f i n i n g a p e r t u r e s A1 (A5) and A2 (A6) are 2.0mm d i a . and 1.0mm d i a . r e s p e c t i v e l y . The maximum g e o m e t r i c a l acceptance h a l f - a n g l e of the l ens i s t h e r e f o r e ±0 .8 ° w i th r e spec t to the a x i s of the l e n s . A set of X-Y p a r a l l e l p l a t e s e l e c t r o s t a t i c d e f l e c t o r s D3 (D4) [G72] are used to make minor c o r r e c t i o n to the ent rance ang le of the energy a n a l y s e r after the angu la r s e l e c t i o n . The ape r t u r e A3 (A7) co r r esponds to the i d e a l image p lane of the l ens and i s at the c o - a x i s of 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 ( s e c t i o n 3 . 2 . 5 ) . The complete l ens assembly i s c a r e f u l l y mounted on 62 the inner element of the 1 3 5 ° - s e c t o r 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) at an i n c l i n a t i o n ang le of 4 2 . 3 ° . The 4 2 . 3 ° i n c l i n a t i o n ang le i s used to take advantage of the second o rder a x i s - t o - a x i s f o c u s s i n g e f f e c t of the CMA [R72] ( s e c t i o n 3 . 2 . 5 ) . The p o l a r ang le 0 i s d e f i n e d m e c h a n i c a l l y to be the ang le between the a x i s of the gas c e l l and the a x i s of the A I L . T h i s ang le ( 4 5 ° ) i s a c c u r a t e l y a ch i e ved by mounting the complete CMA and l ens assembly on to a 2 . 7 ° s l o p i n g s l a b . The t h i c k n e s s of the s l a b i s p r e c i s e l y c o n s t r u c t e d so tha t the axes of the two l enses i n t e r c e p t wi th the a x i s of the gas c e l l . The CMA assembly and the s l a b are then p o s i t i o n e d a c c u r a t e l y onto the s u p p o r t i n g p l a t e u s i ng dowe l l p i n s . 3 .2 .5 Sec to r 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 Any p a i r of e l e c t r o d e s which c r e a t e an e l e c t r o s t a t i c f i e l d a c t i n g t r a n s v e r s e l y to the e l e c t r o n ins tan taneous v e l o c i t y g i ve the e f f e c t of an e l e c t r o s t a t i c p r i sm [EE70, G72 ] , I f the d e f l e c t i o n f i e l d i s weak, the pr imary l ens a c t i o n on a quas i - ch romat i c e l e c t r o n beam i s to s t ee r i t i n t o a new d i r e c t i o n , w i thout much deg rada t i on i n the convergence of the beam. Such weak e l e c t r o s t a t i c d e f l e c t i o n i s the main f u n c t i o n of the quadrupo le and p a r a l l e l p l a t e d e f l e c t o r s [EE70, G72] d i s c u s s e d in the e a r l i e r s e c t i o n s . I f , on the o the r hand, a s t r ong e l e c t r o s t a t i c d e f l e c t i o n f i e l d i s used to o b t a i n a l a r g e angu la r d e f l e c t i o n , a h igh energy 63 d i s p e r s i o n can be . e f f e c t e d f o r the purpose of energy (or momentum) a n a l y s i s of e l e c t r o n s from a quas i - ch romat i c beam. More common e l e c t r o s t a t i c energy a n a l y s e r s in e l e c t r o n impact and r e l a t e d s t u d i e s i n c l u d e (segments o f ) c y l i n d r i c a l , s p h e r i c a l or even t o r o i d a l condensers [EE70, G72, S73] . In the p resen t a p p l i c a t i o n , a 135° s e c t o r 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) [ZKK66, A71, A72, R72, DL77, AY78] i s used to energy ana l yse the ou tgo ing e l e c t r o n s in each c h a n n e l . The important g e o m e t r i c a l d imens ions of the CMA are g i ven i n f i g u r e 3 .2 . Because of the f i n i t e l e n g t h and c u t -o f f on the s i d e s of the CMA e lements , the imper f ec t non-i n f i n i t e l y long and i ncomp le te . 2n c y l i n d r i c a l l y symmetric d e f l e c t i o n f i e l d must be c o r r e c t e d . The e f f e c t of g e o m e t r i c a l t r u n c a t i o n in the s e c t o r CMA and the r e s u l t i n g end f i e l d e f f e c t s are compensated fo r by u s i n g l o g a r i t h m i c a l l y spaced end c o r r e c t o r s (EC) a t both the top and the bottom as w e l l as the s i d e s of the CMA. A d d i t i o n a l s i d e s h i e l d s are used to prevent any s t r a y e l e c t r o n s from e n t e r i n g i n t o the a n a l y s e r . The ent rance s l i t A3 (A7) w i th an ape r t u r e of 2.5mm d i a . i s ma in l y used to i s o l a t e the f i e l d of the l e n s d e f l e c t o r s D3 from that of the CMA. I t i s a l s o used to cover any exposed t e f l o n i n s u l a t o r used fo r mounting the l ens d e f l e c t o r s in order to prevent cha rg i ng of the i n s u l a t o r . The s i z e of the ape r t u r e in the e x i t s l i t A4 (A8) i n the 6 d i r e c t i o n (2.0mm) de te rmines the maximum p o s s i b l e A0 acceptance whereas tha t in the <t> d i r e c t i o n has no marked 64 e f f e c t on the f o c u s s i n g p rope r t y and i s t h e r e f o r e doubled ( i . e . 4.0mm) to take advantage of the i n c r ea se in count r a t e . The s e c t o r CMA i s norma l l y opera ted wi th a lOOeV pass energy . One i nhe ren t advantage of the CMA i s the double f o c u s s i n g p r o p e r t y . 3 In p a r t i c u l a r , Zashkvara et al. [ZKK66] showed tha t f o r a x i s - t o - a x i s f o c u s s i n g , e l e c t r o n s w i th energy E and en t rance ang le 0=42.3° are f ocussed at z=6.12r w i th an a p p l i e d p o t e n t i a l V=0 .763E ( l n ( R / r ) ) , where r and R are the r a d i i of the inner and ou te r c y l i n d e r s of the CMA r e s p e c t i v e l y . From the T a y l o r s e r i e s fo r Az one can show [ZKK66, R72, S73] tha t f o r o ther ang les 0±A0 and energy E±AE, the geometr i c a b e r r a t i o n as r ep resen ted by Az i s : [3 .2 ] Az « 5.6rAE/E - 1 5 . 4 r ( A 0 ) 3 + 10 .3r (AE/E )A0 . N e g l e c t i n g the second and h ighe r o rder te rms, one ge ts tha t the r e l a t i v e energy r e s o l u t i o n , i . e . AE/E, f o r t h i s case i s 1.4% f o r r=25.4mm and a 2mm e x i t ape r t u r e in the t he t a d i r e c t i o n (which d e f i n e s the maximum p o s s i b l e A z ) . D e t a i l e d study of the des i gn parameters fo r the CMA w i th double f o c u s s i n g was g i ven by R i l e y [R72] . I t i s however not p o s s i b l e f o r the s e c t o r CMA employed here to a ch i eve the 3 The n-th order f o c u s s i n g i s commonly r e f e r e d to the case where the n-th d e r i v a t i v e of the image d i s t a n c e w i th respec t to the en t rance ang le of the CMA i s z e r o . In the p resen t case doub le f o c u s s i n g means both the f i r s t and second d e r i v a t i v e s are z e r o . 65 double f o c u s s i n g e f f e c t . The ob j e c t to image s e p a r a t i o n in the p resen t des ign (Hood et al . [HHB77]) i s l a r g e r ( i . e . 7 .00r ) than the op t ima l s e p a r a t i o n ( i . e . 6 .12r ) r e q u i r e d fo r a x i s - t o - a x i s double f o c u s s i n g [R72] . As a r e s u l t , sma l l v o l t a g e s a p p l i e d to the Y - d e f l e c t o r s (D3) between the l ens assembly and the CMA are necessa ry to a l l ow s l i g h t a l t e r a t i o n of the en t rance ang le as w e l l as to r e o p t i m i z e the f o c u s s i n g p r o p e r t y . The a p p l i e d vo l t age co r responds to a change of approx imate l y 0 . 4 ° i n the en t rance a n g l e . A more d e t a i l e d d i s c u s s i o n of the energy r e s o l u t i o n ob ta ined by the p resen t o p t i c a l system i s g i v en in s e c t i o n 3 . 6 . 1 . I t shou ld be noted tha t the d e f l e c t o r s D3 come a f t e r the angu la r s e l e c t i o n a p e r t u r e s A1 and A2 and thus do not a f f e c t the mean va lue of the s c a t t e r i n g p o l a r ang le 6 sampled by the CMA. 3.2.6 S i n g l e Channel E l e c t r o n M u l t i p l i e r E l e c t r o n s w i th the c o r r e c t pass energy a re t r a n s p o r t e d from the en t rance s l i t to the e x i t s l i t by the d e f l e c t i o n a c t i o n of the CMA. A s i n g l e channe l e l e c t r o n m u l t i p l i e r (CEM) (Mu l l a rd B318AL) [Mu75, Mu76, W79] i s used to amp l i f y any d e t e c t e d e l e c t r o n i n t o a workable c u r r e n t p u l s e , which i s used as input fo r the t im ing e l e c t r o n i c s ( s e c t i o n 3 . 3 . 2 ) . The CEM c o n s i s t s of a ho l low g l a s s tube w i th the i n t e r n a l r e s i s t i v e s u r f a c e p rocessed to have a h igh secondary em i s s i on 66 c o e f f i c i e n t [Mu75, Mu76], When an i n c i d e n t e l e c t r o n en t e r s the input s i d e and c o l l i d e s w i th the w a l l , s e v e r a l secondary e l e c t r o n s are p roduced . These secondary e l e c t r o n s are a c c e l e r a t e d down the tube by a h igh b i a s p o t e n t i a l between the input s i d e and the output s i d e . Such a m u l t i p l i c a t i o n p rocess i s repeated many t imes u n t i l the output s ide i s r e a ched . T y p i c a l e l e c t r o n ga in i s 1 0 8 at a b i a s vo l t age of 3keV. The CEMs used are manufactured curved in order to e l i m i n a t e ion feedback [Mu75, Mu76], The e l e c t r o n p u l s e s are c a p a c i t i v e l y decoup led us ing a 0 . 0 0 2 2 i > t F c a p a c i t o r . The r e s u l t i n g c u r r e n t p u l s e i s fed to a p r e a m p l i f i e r ( ou t s ide the vacuum chamber) u s i ng a p r o p e r l y g round-sh i e l ded c o a x i a l c a b l e w i th a 500 c h a r a c t e r i s t i c impedence (RG174U). The l e n g t h of the c o a x i a l c a b l e used i s made as shor t as p o s s i b l e to p reven t unnecessary deg rada t i on of the p u l s e ampl i tude as w e l l as to e l i m i n a t e p o s s i b l e s i g n a l p i c k - u p s . A matching r e s i s t o r (470) i s used fo r the proper t e r m i n a t i o n of the c o a x i a l c a b l e at the r e c e i v i n g end ( i . e . the input s i d e of the p r e a m p l i f i e r ) . Proper t e r m i n a t i o n of the s i g n a l c a b l e i s ext remely v i t a l in e l i m i n a t i n g the problem of r i n g i n g [C72, C75 ] . The CEM as w e l l as the d e c o u p l i n g c a p a c i t o r are mounted i n s i d e a p r o p e r l y s h i e l d e d b rass h o u s i n g . Care has been taken to comp l e t e l y s h i e l d the whole CEM and s i g n a l c a b l e arrangement to e l i m i n a t e any c r o s s t a l k [C72, C75] between the two s i g n a l c a b l e s . A schemat ic d iagram of the s i g n a l d e c o u p l i n g system 67 i s shown in f i g u r e 3 .3 . Fu r the r d e t a i l s of the t i m i n g e l e c t r o n i c s , which r e q u i r e the genera ted c u r r e n t p u l s e s a f t e r the s i g n a l d e c o u p l i n g p rocess as i n p u t s , a re d e s c r i b e d i n s e c t i o n 3 . 3 . 2 . 3 .2 .7 Computer S i m u l a t i o n Of The E l e c t r o n T r a j e c t o r y In f i g u r e 3.4 some t y p i c a l e l e c t r o n t r a j e c t o r i e s through the secondary e l e c t r o n o p t i c a l system are shown. The t r a j e c t o r i e s are c a l c u l a t e d us i ng computer codes [V80] o r i g i n a l l y deve loped by H.A. van Hoof at the FOM I n s t i t u t e , Amsterdam. D e t a i l s and the p r i n c i p l e of the charge d e n s i t y method [MW73, HR76] of the t r a j e c t o r y program are g i ven i n r e f e r ence [V80] . In f i g u r e 3.4 the k i n e t i c energy of an e l e c t r o n t r a n s v e r s i n g the it h l ens element i s denoted by T^ wh i l e the c o r r e s p o n d i n g a p p l i e d v o l t a g e i s denoted by V\ . Under the p resen t s c a t t e r i n g c o n d i t i o n s , the k i n e t i c energy of the e l e c t r o n e n t e r i n g the three-e lement AIL ( i . e . T , ) i s 600eV and tha t of the e l e c t r o n e x i t i n g the AIL ( i . e . T 3 ) i s lOOeV, which i s a l s o the pass energy of the CMA. The c o r r e s p o n d i n g v o l t a g e s (wi th r e spec t to ground) a p p l i e d on the f i r s t and t h i r d l e n s e lements are t h e r e f o r e (V,=) OeV and (V 3=) -500eV r e s p e c t i v e l y . The inner c y l i n d e r of the CMA i s at the same p o t e n t i a l as the t h i r d l ens element of the AIL ( i . e . -500eV) . In e f f e c t , the re are on l y two c o n t r o l v o l t a g e s , i . e . tha t of the c e n t r a l element of AIL (V 2 ) and of 68 channeltrons 1Mfl vacuum baseplate preamps decoupling 3kV F i g u r e 3.3 - S i g n a l d e c o u p l i n g arrangement f o r the c h a n n e l t r o n . F i g u r e 3.4 - Computer s i m u l a t i o n of e l e c t r o n t r a j e c t o r i e s f o r e l e c t r o n s p a s s i n g through the secondary e l e c t r o n o p t i c a l system w i th 600eV k i n e t i c e n e r g i e s . 70 the outer c y l i n d e r of the CMA (V f t ) , which are be ing used to c o n t r o l the t r a j e c t o r y through the secondary e l e c t r o n o p t i c a l system (SEOS). The e f f e c t of the X-Y d e f l e c t o r s between A2 and the CMA i s sma l l and i s i gnored in the p resen t computer s i m u l a t i o n . Two p o s s i b l e t r a j e c t o r i e s r e p r e s e n t i n g the l i m i t i n g c a se s " of the e l e c t r o n t r a n s i t pa ths through the SEOS are shown in f i g u r e 3 .4 . The computed v a l ues of V 2 and V f l f o r the p resen t case ( i . e . f o r T 3 =l00eV ) co r r e spond very c l o s e l y to the o p t i m i z e d v a l ues used in the exper iment . It i s e v i den t from f i g u r e 3.4 tha t the three-e lement AIL i s c l e a r l y a convergent l e n s . In a d d i t i o n , f o r a cons tan t en t rance^ ang le and pass energy , e l e c t r o n s w i th d i f f e r e n t e n e r g i e s are focussed at a d i f f e r e n t image p o s i t i o n by the CMA. 3 .2 .8 Vacuum And M a t e r i a l S p e c i f i c a t i o n s The spect rometer i s mounted on an aluminum base p l a t e (43cm d i a . ) and ope ra tes under vacuum in a removable aluminum c y l i n d r i c a l vacuum chamber (40cm d i a . , 40cm he igh t ) O-r ing s e a l e d at the bottom by the base p l a t e and at the top by a removable l i d (43cm d i a . ) . I t i s evacuated by an a l k y l a t e d d i p h e n y l e ther (Santovac 5) o i l d i f f u s i o n pump (NRC VHS-4), which i s fore-pumped by a b e l t d r i v e n r o t a r y pump (Sergeant 4 In the p resen t case the maximum ent rance ang le i s taken to be h a l f of the maximum p o s s i b l e g e o m e t r i c a l acceptance a n g l e . 71 Welch 1402). B rass f l a n g e s are used to house h igh vacuum o c t a l connec to r s and h igh v o l t a g e c o n n e c t o r s , and are b o l t e d w i th O-r ing s e a l s to the feedthroughs of the base p l a t e to f a c i l i t a t e e l e c t r i c a l c o n n e c t i o n s o u t s i d e the vacuum. Two c i r c u l a r p l e x i - g l a s s windows b o l t e d wi th O-r ing s e a l s to the top l i d and to the s i d e of the vacuum chamber a re used fo r v i ew ing the spec t rometer i n o p e r a t i o n as w e l l as fo r read ing the angu la r • s c a l e on the t u r n t a b l e (without r e l e a s i n g the vacuum) f o r s c a l e c a l i b r a t i o n p u r p o s e s . The base p res su re of t h i s chamber i s 5 X 1 0 " 7 T o r r . The normal o p e r a t i n g dynamic p r e s s u r e , as measured by an i o n i z a t i o n gauge (Veeco RG75K) mounted on the top l i d , i s 5 x 1 0 " 5 To r r w i th a long term s t a b i l i t y b e t t e r than 5%. It has been e s t ima ted from the r e l a t i v e response of the s i n g l e s count r a t e as a f u n c t i o n of p r e s s u r e tha t the a c t u a l dynamic p re s su re i n s i d e the gas c e l l i s a t l e a s t two o rde r s of magnitude h igher than the ambient p r e s s u r e . No l i q u i d n i t r o g e n c o o l i n g t r a p i s necessary to ach ieve the p resen t vacuum s t a t u s . A lmost a l l of the components of the e l e c t r o n o p t i c s (both the p r imary or the secondary e l e c t r o n o p t i c a l systems) are c o n s t r u c t e d of b r a s s . In o rder to min imize the e f f e c t of back s c a t t e r i n g , a l l the s l i t s are made of molybdenum. In a d d i t i o n the i n t e r n a l s u r f a c e s of the o p t i c a l components, except those of the AILs and the e l e c t r o n gun, are benzene sooted to f u r t h e r reduce the e f f e c t of secondary emiss ion from s u r f a c e s . The end c o r r e c t o r s in the CMA are 72 e l e c t r i c a l l y i n s u l a t e d from each o the r u s i n g sapph i re b a l l s as we l l as ny lon screws and washers . Components in the l ens assembly as we l l as in the c o l l i m a t o r are i n s u l a t e d us ing t e f l o n s p a c e r s . Attempt i s made to shadow a l l the i n s u l a t o r m a t e r i a l from the e l e c t r o n beam paths so as to e l i m i n a t e any c h a r g i n g prob lem. E l e c t r i c a l c onnec t i ons from the o p t i c a l e lements to the o c t a l connec to r s in the chambers are made us i ng c o l o r coded t e f l o n coa ted w i res and Bundy c o n n e c t o r s . Such arrangement a l l ows qu i ck and easy s e r v i c i n g of the o p t i c s i n the spec t rome te r . The t u r n t a b l e on which one of the secondary e l e c t r o n o p t i c a l systems i s mounted i s suppor ted by nonmagnetic s t a i n l e s s s t e e l b a l l s . No magnet ic m a t e r i a l i s used i n s i d e the vacuum chamber. The e a r t h ' s magnet ic f i e l d (0.5G) i s reduced to 5mG u s i n g an e x t e r n a l mu-metal s h i e l d to enc l o se the vacuum chamber. 3.3 CONTROL ELECTRONICS The c o n t r o l e l e c t r o n i c s ma in ly c o n s i s t s of v a r i o u s DC power s u p p l i e s fo r the o p t i c a l components in the spec t rometer and f o r the a c t u a l scan v o l t a g e s f o r the expe r imen ta l pa ramete r s . Par t of the e l e c t r o n i c s (the t i m i n g e l e c t r o n i c s ) i s used f o r the c o i n c i d e n c e d e t e c t i o n of (e ,2e) e v e n t s . 73 3.3.1 Power S u p p l i e s The s t a b l i l i t y of the o p e r a t i n g c o n d i t i o n s of the e l e c t r o n o p t i c s depends upon many f a c t o r s . The s t a b i l i t i e s of the l o c a l r e s i d u a l magnet ic f i e l d and the o p e r a t i n g gas p r e s su re as w e l l as the v a r i o u s power s u p p l i e s of the o p t i c a l e lements can a f f e c t c r i t i c a l l y the s c a t t e r i n g dynamics . Measurements shou ld on l y be made when the re i s no obse rvab le f l u c t u a t i o n in the gas p r e s s u r e , s i n c e a d i f f e r e n t p r e s su re w i l l not on l y d i r e c t l y i n f l u e n c e the count r a t e but a l s o cause a d i f f e r e n t c o n t a c t p o t e n t i a l , which i n tu rn a f f e c t s the e f f e c t i v e p o t e n t i a l s a p p l i e d to v a r i o u s o p t i c a l e lements and consequen t l y the f o c u s s i n g p r o p e r t y of the o p t i c s . More i m p o r t a n t l y , the p o t e n t i a l s of o p t i c a l components must be s u p p l i e d by s t a b l e power s u p p l i e s to a v o i d any problem caused by v o l t a g e d r i f t s . F i g u r e 3.5 shows a d e t a i l e d schemat ic of power s u p p l i e s used fo r v a r i o u s e l e c t r o n o p t i c a l components. Power s u p p l i e s f o r on l y one of the two secondary e l e c t r o n o p t i c a l system are shown. 5 The a v a i l a b l e v o l t a g e and c u r r e n t ranges of the power s u p p l i e s are i n d i c a t e d . The i n c i d e n t e l e c t r o n energy , E 0 , i s scanned by chang ing a h i g h v o l t a g e power supp ly (F luke 412B). The v a r i a b l e p o r t i o n of the output v o l t a g e range (0-200eV) of t h i s power supp ly can be m o d i f i e d by a programmable r e s i s t o r 5 A s i m i l a r arrangement has a l s o been made to the other secondary o p t i c a l sys tem. 74 CMA computer CEM 0 to 3.5 kV CMA OUTER 0 to-200V LENS DEFL. -50 to *50V LENS 0 to -60V a CMA INNER 0 to -3.5kV QUAD DEFL. -135 to*135V QUAD DEFL. -135 to* 135V ELECTRON GUN EL -90to*260V A -90to*260V G -90 to *260V FILAMENT 0 to 10A.10VDC INCIDENT ENERGY oto-3.5kv F i g u r e 3 . 5 - Power s u p p l i e s f o r the e l e c t r o n o p t i c a l e l emen t s . The vo l t age (and c u r r e n t ) range of each u n i t i s i n d i c a t e d . Under normal o p e r a t i o n c o n d i t i o n s the CEM, CMA INNER and INCIDENT ENERGY h i g h v o l t a g e power s u p p l i e s are set a t 3.0keV, 500eV and 1200eV r e s p e c t i v e l y . A programmable range of 200eV f o r the INCIDENT ENERGY power supp ly i s c o n t r o l l e d by the computer sys tem. 75 c o n t r o l l e d by the DC vo l t age l e v e l of a d i g i t a l - t o - a n a l o g u e v o l t a g e conve r t e r (DAC) i n s i d e the computer system ( s e c t i o n 3 .4 ) . The power s u p p l i e s f o r the o p t i c a l components of the e l e c t r o n gun are f l o a t e d on top of t h i s i n c i d e n t e l e c t r o n energy power s u p p l y . The i n c i d e n t e l e c t r o n undergoes two s tages of a c c e l e r a t i o n , i . e . from the f i l amen t to the anode and from the anode to the gas c e l l . The gas c e l l i s ma in ta ined at ground p o t e n t i a l . The c u r r e n t s c o l l e c t e d by the Faraday cups and the th ree spray p l a t e s are i n d i v i d u a l l y mon i to red by micro-ammeters. The v o l t a g e of the inner c y l i n d e r of the CMA i s s u p p l i e d by a second h i g h v o l t a g e power supp ly (F luke 415B). The power s u p p l i e s of a l l the o ther o p t i c a l components of the secondary e l e c t r o n o p t i c a l system are f l o a t e d on top of t h i s power s u p p l y . A t h i r d h i g h v o l t a g e power supp ly (Hewlet t-Packard 6516A), m o d i f i e d to extend i t s range up to 3500V, i s used to p rov i de the b i a s p o t e n t i a l f o r the channe l e l e c t r o n m u l t i p l i e r s . The r e l a t i v e az imu tha l ang le i s v a r i e d by r o t a t i n g the t u r n t a b l e u s i n g a servo-motor d r i v e n by a se rvo a m p l i f i e r [C72] . The r e f e r e n c e v o l t a g e s (±24V) of the servo a m p l i f i e r a re p r o v i d e d by a NIM (Nuc lear Instrument Module) b i n power supp ly (ORTEC 401A) . The v a r i a b l e vo l t age of the se rvo a m p l i f i e r i s s u p p l i e d by a second DAC in the computer system (see s e c t i o n 3 .4 ) . C a l i b r a t i o n of the dynamic v o l t a g e ranges of the i n c i d e n t e l e c t r o n energy as w e l l as tha t of the angu la r s c a l e must be per formed r o u t i n e l y . E x t e r n a l t e s t p o i n t s are i n c o r p o r a t e d 76 in the v a r i o u s power s u p p l i e s f o r check ing tha t no vo l t age d r i f t o c cu r s du r i ng the p rog res s of the exper iment . 3 .3.2 T im ing E l e c t r o n i c s E l e c t r o n s wi th the c o r r e c t pass e n e r g i e s are de t e c t ed and a m p l i f i e d by the CEMs. The c u r r e n t p u l s e s coming from the movable and s t a t i o n a r y CEMs must be ana l y sed fo r c o i n c i d e n t occur rence i n o rde r to be counted as an (e ,2e) even t . A s i n g l e de lay t im ing method i s used [MW76a]. F i g u r e 3.6 shows the t im ing e l e c t r o n i c s and the computer c o n t r o l l e d data a c q u i s i t i o n system. In the s i n g l e de l a y t i m i n g method, the CEM p u l s e s are f i r s t a m p l i f i e d and conve r t ed i n t o v o l t a g e p u l s e s w i th a f a s t p r e a m p l i f i e r (ORTEC 9301) . These v o l t a g e p u l s e s are f u r t h e r a m p l i f i e d u s i n g a t im ing f i l t e r a m p l i f i e r (ORTEC 454 ) . T y p i c a l p u l s e shapes of these v o l t a g e p u l s e s ob t a i ned a f t e r the p r e - a m p l i f i c a t i o n and time shap ing a m p l i f i c a t i o n under normal o p e r a t i o n are shown in f i g u r e 3 .7 . I t shou ld be noted tha t a c o n s i d e r a b l e r e d u c t i o n in the number of secondary r e f l e c t i o n peaks has been ach i eved by a p p r o p r i a t e t e r m i n a t i o n of the t r a n s m i s s i o n l i n e s at the r e c e i v i n g ends as d i s c u s s e d in s e c t i o n 3 . 2 . 6 . In p r i n c i p l e t e r m i n a t i o n of t r a n s m i s s i o n l i n e s at both the send ing and the r e c e i v i n g ends [C75] shou ld be used fo r very l ong c a b l e s . In the p resen t c a s e , the c o a x i a l c a b l e s used are made as shor t as p o s s i b l e ( l e s s than 0.5m) and consequen t l y on l y (can EO scan ' 0 Display Graphic Line Floppy Disk Drive CRT Terminal Printer Prog. P.S. motor I/O DAC REAL-TIME CLOCK F i g u r e 3.6 - Timing e l e c t r o n i c s and c o m p u t e r - c o n t r o l l e d data a c q u i s i t i o n system. Legend: CEM - Channel E l e c t r o n M u l t i p l i e r , CFD - Constant F r a c t i o n D i s c r i m i n a t o r , TAC - Time to Ampl i tude C o n v e r t e r , SCA - S i n g l e Channel A n a l y s e r ; CPU - C e n t r a l P r o c e s s i n g U n i t , ADC - Analog to D i g i t a l C o n v e r t e r , DAC -D i g i t a l to Analog Conve r t e r . 78 t e r m i n a t i o n at the r e c e i v i n g end i s n e c e s s a r y . 6 The r e s i d u a l r e f l e c t i o n s in the p r e a m p l i f i e r pu l se are p robab l y due to the sma l l impedence mismatch between the a v a i l a b l e r e s i s t o r (470) used f o r t e r m i n a t i o n and the c h a r a c t e r i s t i c impedence of the c o a x i a l s i g n a l c ab l e (RG174U, 50S2) . The magnitude of the r e s i d u a l r e f l e c t i o n s can be f u r t h e r reduced by choos ing an a p p r o p r i a t e RC time f i l t e r f o r the time f i l t e r a m p l i f i e r . As demonstra ted in f i g u r e 3 .7 , c o r r e c t cho i ce of the RC cons tan t in the t i m i n g f i l t e r a m p l i f i e r can s i g n i f i c a n t l y reduce the magnitude of the secondary r e f l e c t i o n s wi thout a f f e c t i n g the r i s e - t i m e of the f i r s t peak. Moreover , i t can a l s o e l i m i n a t e s p u r i o u s no i s e in the s i g n a l c a b l e s in the expe r imen ta l a r e a . Only the f i r s t peak of the r e s u l t i n g f a s t ( r i s e - t i m e <2nsec) v o l t a g e pu l s e i s used f o r t i m i n g pu rposes . A l l the secondary r e f l e c t i o n s are b l o cked u s i n g cons tan t f r a c t i o n d i s c r i m i n a t o r s (ORTEC 463) by s e t t i n g the vo l t age d i s c r i m i n a t o r l e v e l c o n s i d e r a b l y h ighe r than the maximum v o l t a g e magnitude of the s t r o n g e s t secondary r e f l e c t i o n s . D i s c r i m i n a t i o n us ing the cons tan t f r a c t i o n method [Or80] has the advantage tha t no t ime j i t t e r i s p o s s i b l e even fo r pu l s e s w i th v a r i a b l e pu l se he i gh t d i s t r i b u t i o n s . Slow NIM-standard 7 l o g i c p u l s e s are output from the d i s c r i m i n a t o r s to the ra te-meters (HARSHAW NR-10) f o r m o n i t o r i n g the count r a t e . The 6 In f a c t , t e r m i n a t i o n at both ends of the c o a x i a l c ab l e s causes even more r e f l e c t i o n s than t e r m i n a t i o n at j u s t the r e c e i v i n g end . 7 NIM s tands f o r Nuc lea r Instrument Module . 79 P r e A m p . O u t p u t P u l s e S h a p e 0 . 2 H .4 . 6 -. 8 -.o-.2 -A m p . O u t p u t P u l s e S h a p e 0 2 -4 -6 - " 8 -1 0 -1 2 -v V o l t a g e D i s c r im ina to r L e v e l r i se t ime ^ 2 n s e c . F i g u r e 3 . 7 - T y p i c a l output pulse-shapes f o r the p r e a m p l i f i e r ( top) and the t ime f i l t e r a m p l i f i e r (bot tom) . C o n s i d e r a b l e r e d u c t i o n of secondary r e f l e c t i o n s i n the output pu l se-shape of the t ime f i l t e r a m p l i f i e r i s a ch i e ved by employ ing an a p p r o p r i a t e RC t ime c o n s t a n t . 8 0 f a s t NIM-standard l o g i c pu l se output from one of the d i s c r i m i n a t o r s i s used as a " s t a r t " p u l s e to a t ime-to-ampl i tude conve r t e r (TAC-ORTEC 467) . The output from the o ther d i s c r i m i n a t o r i s t ime de l ayed wi th a l e n g t h of c o a x i a l c ab l e to p rov ide the " s t o p " pu l s e to the TAC. The TAC produces a vo l t age pu l s e w i th he igh t p r o p o r t i o n a l to the t ime d i f f e r e n c e between s u c c e s s i v e " s t a r t " and " s t o p " p u l s e s . A h i s tog ram of the r e s u l t i n g pu l s e he igh t d i s t r i b u t i o n , which i s r e f e r r e d to as the t ime spec t rum, i s shown in f i g u r e 3 .8 . The t r u e c o i n c i d e n c e s can be e x t r a c t e d by us i ng two s i n g l e channe l a n a l y s e r s (SCA-ORTEC 406A and ORTEC 467) to pass p u l s e s w i th he igh t o c c u r r i n g i n s i d e the windows l a b e l l e d COINC and RAND in f i g u r e 3 .8 . A s i n g l e channe l a n a l y s e r f u n c t i o n s e s s e n t i a l l y l i k e a " d o u b l e - l e v e l " d i s c r i m i n a t o r as i t ou tpu t s a l o g i c p u l s e on l y when the he igh t of the input pu l se f a l l s between the two d i s c r i m i n a t o r ( lower and upper) l e v e l s . The r a t i o of the widths of the random to c o i n c i d e n c e windows, x, i s r e l a t e d to the number of t rue c o i n c i d e n c e s N _ P T T „ by the f o l l o w i n g r e l a t i o n : [3 .3 ] N, TRUE = N COINC - N, RAND A , with the s t anda rd d e v i a t i o n : [3 .4 ] AN, TRUE = (N COINC + N. RAND / X2 ) 1 / 2 . 8 1 i r 1 i | | r o o i fwhm = 5nsec come. rand. 0 i r 40 f " T \"tr \"-'-'\ •-•V 80 120 160 TIME DELAY (nsec) 200 F i g u r e 3.8 - T y p i c a l t ime spectrum f o r argon 3p i o n i z a t i o n . L o c a t i o n s of the COINCidence and RANDom SCA windows in the t ime spectrum are i n d i c a t e d . 82 C l e a r l y , a l a r g e x w i l l improve the o v e r a l l s t a t i s t i c a l a ccu racy in N T R U E « A window width r a t i o of (x=) 8 i s used i n the s t u d i e s r epo r t ed i n t h i s t h e s i s . The outputs from both SCAs as w e l l as ou tputs from the TAC are f ed i n t o a computer i n t e r f a c e un i t f o r f u r t h e r p r o c e s s i n g . 3 .4 COMPUTER CONTROL The major p e r i p h e r a l s [Di78] of the microcomputer system ( D i g i t a l PDP11/03) are l i s t e d in t a b l e 3 .1 . (See a l s o f i g u r e 3 .8 . ) A r ea l - t ime da ta a c q u i s i t i o n so f tware package was deve loped to r e t r i e v e and s to r e c o u n t i n g i n f o r m a t i o n . Four one-d imens iona l a r r a y s are used f o r sepa ra te s to rage of the scann ing exper imenta l parameters ( E 0 or 4>), the c o i n c i d e n c e N C O I N C t h e r a n ^ o m NRAND a n c ^ t h e t ^ m e h i s t o g r a m s . The t rue c o i n c i d e n c e n T R T J E h i s tog ram ( i . e . the expe r imen ta l c r o s s s e c t i o n ) i s then ob t a i ned from the c o i n c i d e n c e and random h i s tograms us ing equa t i on 3 .3 . Only one of the two expe r imen ta l parameters i s scanned at a t ime wh i l e the o the r i s kept unchanged. When the scan parameter i s the i n c i d e n t e l e c t r o n energy E 0 (=1200eV+binding energy i n the p resen t c a s e ) , the r e s u l t i n g t r u e c o i n c i d e n c e spectrum i s c a l l e d the b i n d i n g energy spectrum (measured at a f i x e d 0 ) . On the o ther hand, when the scan parameter i s the r e l a t i v e az imutha l ang le <t>, the r e s u l t i n g t rue c o i n c i d e n c e spectrum i s c a l l e d the angu la r c o r r e l a t i o n (at a f i x e d E 0 or b i n d i n g e n e r g y ) . A 83 Tab le 3.1 C o n f i g u r a t i o n of the microcomputer sys tem. Dev i ce Company Model LSI 11/03 CPU Memory (32k) Rea l-t ime C lock D/A Conve r te r A/D Conver te r P a r a l l e l I/O S e r i a l I/O CPU E n c l o s u r e System Mon i to r V ideo Te rm ina l Dual 8" F loppy D i sk D r i v e C o n t r o l l e r Dot Ma t r i x P r i n t e r Reference Manuals D i g i t a l MDB Systems Data T r a n s l a t i o n Data T r a n s l a t i o n Data T r a n s l a t i o n MDB Systems MDB Systems MDB Systems D i g i t a l Data Systems Des ign I n t e g r a l Data Systems M7270 MSC 4601 DT2769 DT2766-BR DT2762-SE-BR MLSI-DRV11C #40346 Rev A MLSI-DLV11 #40320 Rev B MLSI-SMU #40341 VT100-AA (upgraded to VT125) DSD-440 L11-2A A4432 Rev 2 IDS-460 D i g i t a l 84 c o n v e r s i o n from the az imutha l ang le <f> to the momentum p us ing equa t i on 2.3c genera tes the momentum d i s t r i b u t i o n (measured at a f i x e d b i n d i n g e n e r g y ) . One major advantage of the separa te s to rage of the c o i n c i d e n c e , random and time s p e c t r a i s to a l l ow more a c cu ra t e c a l c u l a t i o n of the s t a t i s t i c a l e r r o r s u s i ng equa t ion 3 .4 . The random spectrum a l s o p rov i des a s c a l e l i n e a r i t y check on the random c o i n c i d e n c e (or background n o i s e ) . The t ime spectrum i s of a f i x e d time range ' (200nsec) and the t ime v a r i a b l e i s ob t a i ned by d i g i t i z i n g the TAC o u t p u t . The t ime spectrum a l l ows a c l o s e m o n i t o r i n g of the presence of any spu r i ous no i se or c r o s s t a l k between s i g n a l l i n e s . Two major f u n c t i o n s of the microcomputer system and i t s a s s o c i a t e d r e a l - t i m e data a c q u i s i t i o n so f tware are summarized in the f o l l o w i n g s e c t i o n s . 3 .4.1 Parameters Scanning The expe r imen ta l independent v a r i a b l e s , i . e . the i n c i d e n t e l e c t r o n energy E 0 and the r e l a t i v e az imu tha l angle <p, a re set u s i ng a four channe l d i g i t a l - t o - a n a l o g u e conve r t e r (DAC) [ D i 7 8 ] . The DAC c o n v e r t s a d i g i t a l v a lue i n t o an analogue v o l t a g e (0-10V), which i s then used as the input for c o n t r o l l i n g the programmable power supply f o r E 0 or as a v a r i a b l e v o l t a g e l e v e l f o r the s e r v o - a m p l i f i e r f o r c o n t r o l of <p ( f i g u r e 3 .6 ) . Once the d i g i t a l - t o - a n a l o g u e c o n v e r s i o n s are 85 completed and the expe r imen ta l parameters ( E 0 and 4>) are set a c c o r d i n g l y , a coun t-un t i 1-ove r f l ow sequence i s i n i t i a t e d in the programmable r ea l - t ime c l o c k [Di78] w i th a p r e se t c l o c k -ra te and c l o c k - c o u n t . The ove r f low b i t of the c o n t r o l s t a t u s word of the c l o c k board i s then checked p e r i o d i c a l l y . T h i s p rocedure i s used fo r c o n t r o l l i n g the dwe l l - t ime per scan c h a n n e l . Any event that o ccu r s w i t h i n t h i s dwe l l - t ime i s counted and s t o r e d in the a p p r o p r i a t e a r r a y element c o r r e s p o n d i n g to the scann ing parameter . A new set of expe r imen ta l v a l ues f o r E 0 and <j> are used f o r the DAC channe l s on encounter of c l o c k over f low and the whole scan procedure i s repea ted u n t i l the scan sequence i s i n t e r r u p t e d by the ope ra to r or on response to the p re se t scan coun t . It i s p o s s i b l e to set up a scan procedure which per forms s e q u e n t i a l angu la r c o r r e l a t i o n scans at a set of b i n d i n g e n e r g i e s i n t e r l e a y i n g l y or v i c e v e r s a , i . e . b i n d i n g energy scans at a set of r e l a t i v e az imutha l a n g l e s . Such an i n t e r l e a v i n g scan procedure a l l ows automat i c s e l f -n o r m a l i z a t i o n between s p e c t r a w i t h i n the set wi thout pe r fo rm ing complex c o r r e c t i o n s due to p re s su re v a r i a t i o n and f l u c t u a t i o n s i n o the r expe r imen ta l c o n d i t i o n s . 86 3.4.2 Event Coun t ing Once a c o i n c i d e n t event ( r e g a r d l e s s of whether i t i s a random or t r ue event ) has been r e g i s t e r e d by the TAC, the TAC ( vo l t age ) ou tpu ts must be d i g i t i z e d to l o c a t e the a p p r o p r i a t e b i n number, c o r r e s p o n d i n g to a s p e c i f i c de l a y t ime , f o r count i n c r e m e n t a t i o n . The i n t e r f a c e u n i t ( f i g u r e 3.6) i s used to t e m p o r a r i l y " i n h i b i t " the TAC from a c c e p t i n g more i npu t s and to " h o l d " the TAC output s i g n a l s u f f i c i e n t l y l ong f o r the a n a l o g u e - t o - d i g i t a l c o n v e r t e r (ADC) [Di78] to complete the d i g i t i z a t i o n , which t y p i c a l l y takes 30itsecs. On c o m p l e t i o n , the i n t e r f a c e " r e s e t s " the TAC f o r f u r t h e r s i g n a l p r o c e s s i n g . Outputs from the two SCAs are used f o r i n c rement ing the number of counts i n the c o i n c i d e n c e and random a r r a y e l ements , each c o r r e s p o n d i n g to the r e s p e c t i v e scan parameter . When the chosen channe l dwe l l t ime e x p i r e s , the con t en t s of these a r r a y e lements a re s t o r e d . The next set of expe r imen ta l parameters are then used and the whole p rocedure i s r epea ted as d i s c u s s e d above ( s e c t i o n 3 . 4 . 1 ) . The computer a l s o p r o v i d e s o ther s e r v i c e s i n c l u d i n g p e r i o d i c da ta p r i n t o u t , r o u t i n e update of r eco rded s p e c t r a on f l o p p y d i s k e t t e s and d i s p l a y of the spectrum on e i t h e r a g r aph i c t e r m i n a l ( D i g i t a l VT125) or an X-Y s cope . P e r i o d i c da ta p r i n t o u t s are used to moni tor the s t a b i l i t y of the spec t romete r and to c o r r e c t f o r any s p u r i o u s no i s e due to power surges d u r i n g the o f t e n l eng thy scann ing p r o c e s s . 87 3.5 OPERATION PROCEDURE AND SAMPLE HANDLING There are three modes of o p e r a t i o n : the n o n c o i n c i d e n t e l a s t i c s c a t t e r i n g s c a n , and the c o i n c i d e n t b i n d i n g energy and angu l a r c o r r e l a t i o n s cans . The n o n c o i n c i d e n t e l a s t i c s c a t t e r i n g scan i s used f o r i n i t i a l t un ing and o p t i m i z a t i o n of the v o l t a g e s e t t i n g s of the secondary e l e c t r o n o p t i c a l systems (SEOSs) be fo re the c o i n c i d e n t (e ,2e) exper iments ( i . e . b i n d i n g energy or angu la r c o r r e l a t i o n scans ) are pe r fo rmed . In t h i s e l a s t i c mode, the i n c i d e n t energy E 0 i s se t to be i d e n t i c a l to E, ( i . e . 600eV). The SEOSs are a d j u s t e d to momentum and energy ana l y se on l y the e l a s t i c a l l y s ca t t e r ed* e l e c t r o n s w i th k i n e t i c e n e r g i e s of 600eV. F i g u r e 3.9 shows a t y p i c a l t r a n s m i s s i o n f u n c t i o n of the SEOS f o r an e l e c t r o n e n t e r i n g w i th 600eV k i n e t i c energy a f t e r the e l a s t i c c o l l i s i o n w i th a gaseous t a r g e t . The v o l t a g e s of the c e n t r a l e lements and the X-Y d e f l e c t o r s D3 (D4) f o l l o w i n g the AIL are man ipu l a t ed to maximize the amp l i tude and to min imize the FWHM of the peak wi thout i n t r o d u c i n g any asymmetry to the o v e r a l l peak-shape. The v o l t a g e of the oute r element of the CMA i s used to c e n t r e the peak at the c o r r e c t mean k i n e t i c energy (600eV) . T h i s p rocedure must be per formed i n d i v i d u a l l y f o r both secondary e l e c t r o n o p t i c a l sys tems. A f t e r the e l a s t i c s c a t t e r i n g s cans , the i n c i d e n t energy i s r e s e t to i t s o r i g i n a l v a lue ( i . e . 1200eV p l u s the b i n d i n g e n e r g y ) . The i n c i d e n t e l e c t r o n beam i s r e o p t i m i z e d as d i s c u s s e d i n s e c t i o n 3 . 2 . 3 . The b i n d i n g energy and the 88 Transmission Function o o 596 598 6 0 0 6 0 2 6 0 4 Kinetic Energy (eV) F i g u r e 3.9 - T y p i c a l t r a n s m i s s i o n f u n c t i o n f o r the secondary e l e c t r o n o p t i c a l system o p e r a t i n g with a lOOeV pass energy f o r the CMA. A Gaussian f u n c t i o n ( s o l i d l i n e ) with a 1.6eV FWHM g i v e s an e x c e l l e n t f i t to the data. 89 angu la r c o r r e l a t i o n scans can then be i n i t i a t e d wi th the a p p r o p r i a t e scan sequences in the computer sys tem. It shou ld be noted tha t the b i n d i n g energy scan i s ob t a i ned by scann ing E 0 at a f i x e d r e l a t i v e az imutha l angle 0. The angu la r c o r r e l a t i o n scan i s ob ta ined by scann ing 4> wh i l e keeping E 0 c o n s t a n t . The i n c i d e n t energy i s r e l a t e d to the b i n d i n g energy by equa t ion 2.2 and the momentum p i s r e l a t e d to <j> by equa t ion 2.3c under the p resen t noncop lanar symmetric s c a t t e r i n g c o n d i t i o n (chapter 2 ) . A l l chemica l s used in the s t u d i e s r e p o r t e d in t h i s t h e s i s were of h i g h p u r i t y (>99%) and were ob t a i ned commerc i a l l y from Matheson except f o r CS 2 which was s u p p l i e d by Matheson-Coleman & B e l l . No impur i t y was d e t e c t e d i n the b i n d i n g energy spec t rum. For C S 2 f a p p r o p r i a t e f reeze-thaw c y c l e s were per formed to e l i m i n a t e any absorbed a i r c o n t a i n e d i n the l i q u i d sample be fo re i n t r o d u c t i o n i n t o the gas c e l l . The gas h a n d l i n g system fo r sample i n t r o d u c t i o n c o n s i s t s of a p r e s su re r e d u c t i o n v a r i a b l e leak v a l v e ( G r a n v i l l e - P h i l l i p s 203) and v a r i o u s vacuum compat ib l e v a l v e s . Hea t ing tape i s used to ma in t a i n the whole gas h a n d l i n g system at a s u f f i c i e n t l y h igh temperature (40-60 °C ) to prevent condensa t ion and b l o c k i n g of the leak va l ve due to c o o l i n g caused by expans ion in the leak v a l v e . Each of the gas c y l i n d e r s i s equ ipped w i th the proper gas r e g u l a t o r to c o n t r o l the input p r e s s u r e . 90 3.6 PERFORMANCE The per formance of an e l e c t r o n spec t rometer i s u s u a l l y r a t ed by s e v e r a l f a c t o r s . The more important ones i n c l u d e r e s o l u t i o n s , s i g n a l - t o - n o i s e r a t i o , data c o l l e c t i o n e f f i c i e n c y and f l e x i b i l i t y in the range of a p p l i c a t i o n s . 3.6.1 Resolutions A good measure of the energy r e s o l u t i o n can be ob ta ined from the b i n d i n g energy spectrum f o r He 1s i o n i z a t i o n (see f i g u r e 3 .10 ) . Gauss ian and L o r e n t z i a n l i n e-shapes are f i t t e d to the expe r imen ta l He 1s b i n d i n g energy spec t rum. The e x c e l l e n t f i t of the Gauss ian l i n e-shape to the expe r imen ta l spectrum i n d i c a t e s tha t the energy l i n e-shape f u n c t i o n i s of a Gauss ian type w i th a FWHM of 1.6eV. I t a l s o j u s t i f i e s the use of Gauss ian f u n c t i o n s f o r the d e c o n v o l u t i o n procedure used i n the da ta a n a l y s e s r e p o r t e d in t h i s t h e s i s . The Gauss ian l i n e-shape i s due to the e f f e c t of the t r a n s m i s s i o n f u n c t i o n s of the secondary e l e c t r o n o p t i c a l sys tems, which dominates any e f f e c t of n a t u r a l l i n e - s h a p e . I t i s c l e a r from f i g u r e 3.9 tha t the t r a n s m i s s i o n f u n c t i o n i t s e l f can a l s o be c l o s e l y r ep re sen t ed by a Gauss ian f u n c t i o n of 1^ ,.6eV FWHM. Under the p resen t symmetric ar rangement , the o v e r a l l energy r e s o l u t i o n i s s imp ly the l a r g e r of the two s i n g l e energy r e s o l u t i o n s as measured in the e l a s t i c s exper iment . i . e . 91 O LU CO co ~ co O c o c = O -Q LU DC LU Li. • • He 1 2 0 0 e V 0=0' fwhm = 1.6 eV , l o r e n t z i a n • " g a u s s i a n 18 T 21 24 27 30 BINDING ENERGY (eV) 33 F i g u r e 3.10 - T y p i c a l b i n d i n g energy spectrum f o r helium 1s i o n i z a t i o n at 0=0 ° . The Gauss ian f u n c t i o n ( s o l i d l i n e ) w i th a 1.6eV FWHM g i v e s the best f i t t o the expe r imen ta l d a t a . 92 [3 .5 ] AE = (AE PEOS 2 + A E SEOS 2 ) 1 / 2 + 6e, = (AE f i l amen t 2 + AE CMA 2 ) 1 ' 2 + fie, where fie i s ze ro i f i d e n t i c a l l i n e-shapes are ob t a i ned by both SEOSs and there i s no d i f f e r e n c e in the energy p o s i t i o n between the co r r e spond ing Gauss ian peaks . The c o n t r i b u t i o n of the PEOS to AE i s ma in ly due to the thermal spread of the t he rm ion i c source ( i . e . the f i l a m e n t ) , which i s e s t ima ted to be 0.8eV i n the p resen t c a s e . The energy d i s p e r s i o n of the three-e lement AIL in the SEOS i s g e n e r a l l y sma l l compared to tha t of the CMA. The r e s o l u t i o n fo r the CMA i s 1.4% of the pass energy and i s t h e r e f o r e 1.4eV fo r a pass energy of 100eV. C l e a r l y l ower ing the pass energy of the CMA w i l l i n gene ra l improve the r e s o l u t i o n but on l y at the expense of count r a t e . Without the use of energy monochromation of the i n c i d e n t e l e c t r o n beam, i t i s u n l i k e l y tha t the o v e r a l l energy r e s o l u t i o n of the p resen t i n s t r u m e n t a t i o n can be improved beyond 1.0eV FWHM. The angu la r r e s o l u t i o n f u n c t i o n i s i n gene ra l complex and depends upon many f a c t o r s i n c l u d i n g b a c k l a s h of the t u r n t a b l e , angu la r d i v e rgence of the i n c i d e n t beam, the en t rance and e x i t a p e r t u r e s and the f o c a l p r o p e r t i e s of the three-e lement AILs as w e l l as the e x i t a p e r t u r e and the a b e r r a t i o n of the CMA. I t i s i m p r a c t i c a l , and very d i f f i c u l t , t o determine a c c u r a t e l y the e f f e c t s of a l l the c o n t r i b u t i n g f a c t o r s . An angu la r c o n v o l u t i o n p rocedu re , 93 f i r s t d i s c u s s e d by Hood et al. [HHB77] (see a l s o F r o s t and Weigold [FW82a]} has t h e r e f o r e been employed to i n c l u d e the e f f e c t s of the angu la r r e s o l u t i o n in 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 s be fo re compar ison w i th the expe r imen ta l r e s u l t s . To a f i r s t a p p r o x i m a t i o n , the p r imary c o n t r i b u t i n g f a c t o r s a re the en t rance and e x i t a p e r t u r e s of the AILs as these a p e r t u r e s d e f i n e g e o m e t r i c a l l y the maximum acceptance ang les of the l e n s e s . The c o l l i s i o n r eg i on i s d e f i n e d as the o v e r l a p volume between the i n c i d e n t e l e c t r o n beam and the two acceptance cones of the l e n s e s . The s i z e of the acceptance cones a re d e f i n e d by the maximum acceptance a n g l e . (See f i g u r e 3 .11 . ) Consequen t l y , the sma l l e r i s t h i s o v e r l a p volume, the sma l l e r i s the angu la r r e s o l u t i o n . ( I f the width of the i n c i d e n t e l e c t r o n beam or the d e f i n i n g a p e r t u r e s of the l e n s e s are i n f i n i t e l y s m a l l , then the o v e r l a p volume becomes the i d e a l c o l l i s i o n point.) Any (e ,2e) event o c c u r r i n g i n s i d e t h i s o v e r l a p volume would c o n t r i b u t e to the s i g n a l . I f one i gnores a l l the nonsymmetric c o l l i s i o n s , then symmetric c o l l i s i o n s can on l y occur from c o l l i s i o n c e n t r e s c o n f i n e d on the l a r g e s t c r o s s - s e c t i o n of the o v e r l a p volume. I t i s t h e r e f o r e reasonab le to average over the momentum d i s t r i b u t i o n s w i th momentum s c a l e s c o r r e s p o n d i n g to the set of p o l a r and az imutha l ang les d e f i n e d by p o i n t s on t h i s a r e a . The s i z e of the a rea i s d e f i n e d by the effective s i z e s of the a p e r t u r e s : ( f u l l - a n g l e ) A0 and Ac). The e f f e c t s of v a r y i n g e i t h e r A0 or Ao> on t y p i c a l p-type and s-type momentum 94 INCIDENT ELECTRON BEAM F i g u r e 3.11 - The c o l l i s i o n r e g i o n r e s u l t i n g from 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 acceptance cones of the l e n s e s . The shaded a rea i s d e f i n e d by the " e f f e c t i v e " angu la r spreads ( f u l l ang le ) A0 and A#. Adapted from f i g u r e 3.7 i n r e f e r e n c e [C80] . 95 d i s t r i b u t i o n s are shown in f i g u r e 3 .12. C l e a r l y , e n l a r g i n g these e f f e c t i v e a p e r t u r e d imens ions tends to i n c r e a s e the i n t e n s i t y of the p-type d i s t r i b u t i o n i n the low momentum r e g i o n . To determine the e f f e c t i v e ape r t u r e s i z e , the momentum d i s t r i b u t i o n of Ar 3p c a l c u l a t e d u s i n g the Ha r t r ee-Fock wavefunct ion [CR74] i s c o n v o l u t e d w i th v a r i o u s comb ina t ions of Ad and A<t> v a l u e s . The r e s u l t i n g convo lu t ed d i s t r i b u t i o n s are then compared wi th the expe r imen ta l d i s t r i b u t i o n f o r Ar 3p. Under the p resen t ar rangement , an e x c e l l e n t f i t i s o b t a i n e d f o r ( f u l l - a n g l e ) A0=1.2 ° and ( f u l l -ang le ) A # = 2 . 0 ° , which then d e f i n e the e f f e c t i v e ape r tu r e d imens ions f o r subsequent (e ,2e) s t u d i e s . The u n c e r t a i n t y i n the momentum c o r r e s p o n d i n g to these angu la r u n c e r t a i n t i e s i s =*0 .1a o " 1 . I t shou ld be noted tha t v a r y i n g the mean the t a ang le of the ou tgo ing e l e c t r o n s (away from 0=45° ) would not change the o v e r a l l shape of the t h e o r e t i c a l p-type d i s t r i b u t i o n . However, i t has the e f f e c t of s h i f t i n g the expe r imen ta l momentum d i s t r i b u t i o n toward the o r i g i n of the momentum s c a l e . T h i s appears to r a i s e the momentum d i s t r i b u t i o n in the low momentum r e g i o n . An i n c o r r e c t 6 ( i . e . not the o p t i m a l va lue ) would t h e r e f o r e produce a s i m i l a r expe r imen ta l o b s e r v a t i o n to that which would occur i f the momentum r e s o l u t i o n i s poor ( i . e . l a r g e Ad and A<f>). I t i s t h e r e f o r e ex t reme ly important to have h i g h p r e c i s i o n a l ignment of the l e n s e s at 0=45° w i th r e spec t to the a x i s of the gas c e l l ( s e c t i o n 3 . 2 . 4 ) . In the p resen t arrangement , 96 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION Ar 3s J Ac9 = 0.6 0 J e=45° J 0.5 1.0 1.5 P (o.u.) 0.5 1.0 1.5 P (o.u.) 2.0 2.5 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION 2 -O I ' l l 1 1 1 1 1 1 Ar 3p _ A<£=l.0° / \ e = 4 5 ° J A6(°) IK 1-8 "1 1.4 iL i.o - ^ 0 . 6 , 0 . 2 1 1 1 1 L 1 1 , 1 1 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION 0.0 0.5 1.0 1.5 P (o.u.) 2.0 2.5 q 1 9 to JD •2- o C O Z o l/l ^ in V) o £ ^ u S ° c ™ 0) 0.0 T — i — i — i — i — i — i — r A 5 (°) Ar 3s •0.6,0.2 A<t=l.0° 1 O „ 1.4 6 = 45° 1.8 0.5 1.0 1.5 P (o.u.) 2.0 2.5 Figure 3.12 - Ef f e c t s of fold i n g d i f f e r e n t " e f f e c t i v e " angu l a r spreads ( f u l l ang le ) AS and A<p i n t o the Ar 3p and Ar 3s momentum d i s t r i b u t i o n s c a l c u l a t e d u s i n g Har t ree-Fock wavefunct ions [ C R 7 4 ] . 97 the very low minimum observed at the momentum o r i g i n of the expe r imen ta l Ar 3p momentum d i s t r i b u t i o n ( f i g u r e 4 . 6 ) con f i rms the h igh accuracy in the p h y s i c a l a l ignment of the l e n s e s . F i n a l l y , the t im ing r e s o l u t i o n i s shown to be app rox ima te l y 5nsec FWHM ( f i g u r e 3 .8 ) . The sma l l t ime spread i s due to the sma l l v a r i a t i o n in the t r a n s i t t ime of an e l e c t r o n p a s s i n g through the CMAs as we l l as the d i f f e r e n c e in a m p l i f i c a t i o n t ime in the CEMs. The l a t t e r f a c t o r shou ld be much sma l l e r than the former f a c t o r because of the much s m a l l e r p h y s i c a l d imens ion of the CEMs. 3 .6 .2 S igna l-To-No i se R a t i o The expe r imen ta l t rue c o i n c i d e n c e count r a t e i s of course r e l a t e d to the t r i p l 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 (chapter 2 ) . i . e . [3 .6 ] N TRUE = n K d ' a / d G i d O a d E , ) AO, A0 2 AE , , where n i s the number of t a r g e t s p e c i e s i n the c o l l i s i o n r eg i on and I i s the number of i n c i d e n t e l e c t r o n s per second . The expe r imen ta l random c o i n c i d e n c e count r a te i s g i ven by : [ 3 ' 7 ] NRAND = N l N ' T ' 98 where T i s the width of the c o i n c i d e n c e t ime window. The s i n g l e s ( e l a s t i c s ) count r a t e s N, and N 2 a re d e f i n e d by: [3 .8 ] N, - n l ( d ' o / d O t d E , ) AS^AE , . The s i g n a l - t o - n o i s e r a t i o i s t h e r e f o r e g i ven by : [ 3 ' 9 ] N T R U E / N R A N D " ( m r A B ) - i ( d ^ / d n ^ O j d E , ) / ( d 2 a / d 0 1 d E , ) 2 , where AE=AE 1 =AE 2 . I t shou ld be noted tha t the s i n g l e s count r a t e i n c r e a s e s w i th the number of t a r g e t e l e c t r o n s p o s s i b l e to be e x c i t e d or e j e c t e d from the t a r g e t s p e c i e s . Consequent l y the s i g n a l - t o -n o i s e r a t i o becomes worse fo r the inner l e v e l s of h i g h Z t a r g e t s . I gno r ing the t a r g e t dependent t r i p l e and double c r o s s s e c t i o n s , the s i g n a l - t o - n o i s e r a t i o N__.TTC,/N _ . . „ _ . * 3 TRUE RAND l/ (n lTAE ) and the t rue c o i n c i d e n c e count r a t e N^^uE " ( n l )AO ,AQ 2 AE . C l e a r l y the bes t o p e r a t i n g c o n d i t i o n i s a compromise between the s i g n a l - t o - n o i s e r a t i o and the t rue c o i n c i d e n c e count r a t e . I n c r e a s i n g the s o l i d ang les w i l l i n c r e a s e the t rue c o i n c i d e n c e count r a t e at the expense of the angu la r r e s o l u t i o n . I t i s a l s o obv ious tha t i t i s advantageous to have a sma l l e r w id th f o r the t ime window because a s m a l l e r t ime width w i l l improve the s i g n a l - t o - n o i s e r a t i o w i thout a f f e c t i n g the count r a t e . 99 3.6.3 Limitations The more p e r t i n e n t l i m i t a t i o n s of the p resen t b i na r y (e ,2e) spec t ro-sys tem i n c l u d e da ta c o l l e c t i o n e f f i c i e n c y and f l e x i b i l i t y in s tudy ing c o r r o s i v e or s h o r t - l i v e d c h e m i c a l s . Under normal ambient sample p r e s su re and o p e r a t i n g c o n d i t i o n s , the ( nonco inc i den t ) s i n g l e s count r a t e of the p resen t spect rometer i s 100-1000Hz depending upon the t a r g e t . The co r r e spond ing c o i n c i d e n t count r a t e i s of the o rder of 0 .1Hz . A t y p i c a l spectrum w i th r easonab le s t a t i s t i c s no rma l l y r e q u i r e s 1-3 days to accumu la te . As such , the e f f i c i e n c y of the spec t rometer must be improved i n o rde r to extend the b ina r y (e ,2e) method to the s t u d i e s of s h o r t - l i v e d s p e c i e s such as r a d i c a l s and nega t i ve or p o s i t i v e i o n s . One approach to improv ing the e f f i c i e n c y i s by means of m u l t i c h a n n e l and angu l a r s e n s i t i v e d e t e c t i o n t echn iques p r e s e n t l y be ing i n v e s t i g a t e d i n t h i s l a b o r a t o r y (work not r e p o r t e d in t h i s t h e s i s ) . Such development r e q u i r e s a d i f f e r e n t e l e c t r o n o p t i c a l system as w e l l as s i g n a l d e c o u p l i n g methods. I t shou ld be noted tha t improv ing the da ta r a t e (us ing m u l t i c h a n n e l d e t e c t i o n techn ique ) i s a l s o a p r e r e q u i s i t e in upgrad ing the energy r e s o l u t i o n us ing e l e c t r o n energy monochromat ion. In the p resen t ar rangement , the spec t romete r ' i s a l s o not s u i t a b l e f o r s t u d y i n g c o r r o s i v e and r e a c t i v e gases . T h i s i s because long exposure of the spec t rometer to these types of gases w i l l cause i n s t a b i l i t y i n the p o t e n t i a l s of the o p t i c a l e lements due to c o n t i n u a l 100 metal s u r f a c e d e t e r i o r a t i o n , which c o n t i n u o u s l y a f f e c t s the e f f e c t i v e con t a c t p o t e n t i a l s . T h i s problem can be a l l e v i a t e d by d i f f e r e n t i a l l y pumping d i f f e r e n t s e c t i o n s of the spec t rome te r , in p a r t i c u l a r the gas c e l l . Du r ing the l a t e r s tage of t h i s Ph .D. work, a qu i ck m o d i f i c a t i o n (not r e p o r t e d i n t h i s t h e s i s ) has been made so that the s e c t i o n below the gas c e l l ( i . e . the e l e c t r o n gun s e c t i o n ) i s s e p a r a t e l y pumped. T h i s has he lped to extend the l i f e - t i m e of the f i l amen t by o p e r a t i n g the f i l amen t at a much lower ambient p r e s s u r e . T h i s arrangement has been used to study water (not r e p o r t e d i n t h i s t h e s i s , see [BL&84]) which would be p r o b l e m a t i c wi thout such arrangement due to the we l l known s t rong s u r f a c e a b s o r p t i o n c h a r a c t e r i s t i c of water . The p resen t p a r t i a l l y d i f f e r e n t i a l pumping arrangement i s i n tended f o r the study of c o r r o s i v e compound such as f l u o r i n e . The next gene r a t i on b i n a r y (e ,2e) spec t rometer can t h e r e f o r e be e n v i s i o n e d to i n c o r p o r a t e m u l t i c h a n n e l angu l a r s e n s i t i v e d e t e c t i o n and d i f f e r e n t i a l pumping t e c h n o l o g i e s . 101 Chapter IV NOBLE GASES 4.1 INTRODUCTION The study of atoms, in p a r t i c u l a r the nob le gases , has p l a yed an important r o l e i n the " e v o l u t i o n " of b i na r y (e ,2e) s p e c t r o s c o p y . S ince a l l the noble gases are gaseous at room temperature and a l s o because of t h e i r r e l a t i v e chemica l i n e r t n e s s and c l o s e d s h e l l e l e c t r o n i c s t r u c t u r e , they are the i d e a l t e s t t a r g e t s f o r s tudy ing the (e ,2e) i o n i z a t i o n t h e o r i e s . The cop l ana r angu la r c o r r e l a t i o n s t u d i e s which have been r e p o r t e d f o r He [CG&78, FM&78], Ne, Ar and Xe [UWM75, FM&78] were ma in ly in tended to t e s t the v a l i d i t y of v a r i o u s approx imat ions (p lane wave impulse a p p r o x i m a t i o n , d i s t o r t e d wave impulse a p p r o x i m a t i o n , e t c . ) i n the s c a t t e r i n g p r o c e s s . Abso lu t e t r i p l 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 s have a l s o been ob ta ined f o r He [SCG78, WK&81], Ne [SCG79], Ar [KPH81], and Xe [GFT803 . Most of these works s t u d i e d the a p p l i c a t i o n of the (averaged) e i k o n a l approx imat ion in the (e ,2e) p r o c e s s . Other t h e o r e t i c a l works on i o n i z a t i o n t h e o r i e s i n c l u d e those by Madison et al . [MC77, ML813 and C a m i l l o n i et al. [ C G & 8 0 3 . Of more r e l e vance to the p resen t work are e a r l i e r s t u d i e s of noble gases us i ng the noncoplanar s c a t t e r i n g k i n e m a t i c s : He [HM&73, DMW763, Ne [DM&78], Ar 1 02 [WHT73, HM&74, MUP78], Kr [WHM75, FG&81], and Xe [WHM75, HHB77, GF&79, FG&81]. In most of these s t u d i e s a great dea l of emphasis was p l a c e d on s t r u c t u r a l de t e rm ina t i on ( i . e . the importance of the wave func t ions in p r e d i c t i n g the measured (e ,2e) c r o s s s e c t i o n ) . However, most of these works were o f t e n l i m i t e d by the energy and/or momentum r e s o l u t i o n s compared w i th p r e s e n t l y a v a i l a b l e performance in b i na r y (e ,2e) spec t r ome te r s . I t shou ld be noted tha t b i na r y (e ,2e) s t u d i e s by Dixon et al. [DMW76] and more r e c e n t l y by Cook et al . [CM&84] have a l s o i n v e s t i g a t e d the r o l e of ground s t a t e c o r r e l a t i o n s in the n=2 [DMW76] and n=2 and 3 [CM&84] s t a t e s of H e + . O b s e r v a t i o n s of r e l a t i v i s t i c e f f e c t s i n the Xe 5 p 3 / 2 and 5 p i / 2 o r b i t a l s have a l s o been made r e c e n t l y by Cook et al. [CMW84]. I o n i z a t i o n in the ou te r v a l e n c e - s h e l l of the noble gases has a l s o been s t u d i e d w i th 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 : He [CKM71, KW72, SFC74, WA&80, WS80], Ne [CKM71, WK72, SFC74, WK74], Ar [CKM71, SFC74, AWS<78a, SH81 ] , Kr [ SFC74 ] and Xe [G74, SH77, W77c, AW&78b, HP78, SH81]. A l l of these data show s i g n i f i c a n t s a t e l l i t e 1 s t r u c t u r e s in the ns b i n d i n g 1 The pa ren t and s a t e l l i t e s t a t e d e s c r i p t i o n s are used here to d i s t i n g u i s h the most i n t ense i o n i c s t a t e from the o t h e r s . There i s no other i n tended p h y s i c a l d i s t i n c t i o n between the v a r i o u s i o n i c s t a t e s produced by many-body e f f e c t s and d e s c r i b e d i n a p p r o p r i a t e i o n i z a t i o n t h e o r i e s . In some c a s e s , f o r i n s t a n c e in the i o n i z a t i o n of inner va l ence mo lecu la r o r b i t a l s [CD77], the i o n i z a t i o n spectrum may c o n t a i n s e v e r a l r e l a t i v e l y i n t ense p o l e s , none of which maybe overwhelmingly dominant and thus no c l e a r l y d e f i n e d " p a r e n t " e x i s t s . 103 energy s p e c t r a of the heav i e r members of the nob le gases . A l t hough the r e l a t i v e i n t e n s i t i e s of these s a t e l l i t e s t a t e s are in gene ra l d i f f e r e n t from those found in the (e ,2e) study [MW76a, MUP78] due to the d i f f e r i n g fundamental na ture of the two p r o c e s s e s , the energy p o s i t i o n s p rov i de u s e f u l r e f e r e n c e s f o r the (e ,2e) s tudy because of the somewhat b e t t e r energy r e s o l u t i o n p r e s e n t l y a t t a i n a b l e in 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 . A number of ab-initio c a l c u l a t i o n s have been c a r r i e d out p e r t a i n i n g to s a t e l l i t e s t r u c t u r e and e l e c t r o n c o r r e l a t i o n s [CKM71, MS76, MKS78, DL79, L81 , DL82a, DL82b] . Recent m u l t i c o n € i g u r a t i o n c a l c u l a t i o n s by D y a l l and L a r k i n s [DL82a, DL82b] show that a l t hough good r e l a t i v e energy agreements can be o b t a i n e d , such c a l c u l a t i o n s are s t i l l inadequate in the p r e d i c t i o n of r e l a t i v e i n t e n s i t i e s between s a t e l l i t e s t a t e s . Other s i m i l a r c o n f i g u r a t i o n i n t e r a c t i o n (CI) c a l c u l a t i o n s on Kr [FG&81] show s i m i l a r d i s c r e p a n c i e s in i n t e n s i t y p r e d i c t i o n . A s ys t emat i c s tudy of the 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 of a l l the v a l ence o r b i t a l s of the nob le g a s e s : He, Ne, A r , " Kr and Xe i s g iven below. The p r e sen t measurements r ep resen t the f i r s t comprehensive study of a l l the noble gases on the same spec t rometer under i d e n t i c a l i n s t r umen ta l c o n d i t i o n s . The s i g n i f i c a n c e of l i t e r a t u r e SCF wave func t ions in p r e d i c t i n g the measured momentum d i s t r i b u t i o n s us i ng the p lane wave impulse 1 04 approx imat ion (PWIA) i s a l s o i n v e s t i g a t e d . E l e c t r o n c o r r e l a t i o n e f f e c t s and the breakdown of the independent p a r t i c l e p i c t u r e a re d i s c u s s e d and compared w i th the recent t h e o r e t i c a l work of D y a l l and L a r k i n s [DL82a, DL82b] as we l l as w i th p h o t o e l e c t r o n s p e c t r o s c o p i c d a t a . The measured momentum d i s t r i b u t i o n s of the noble gases a re compared wi th c a l c u l a t i o n s us i ng wave func t ions of Har t ree-Fock (HF) and l e s s e r q u a l i t y . The t r e n d s down the nob le gas group are d i s c u s s e d w i th the a i d of the b i n d i n g energy s p e c t r a as we l l as the momentum and p o s i t i o n d e n s i t y maps c a l c u l a t e d us ing HF q u a l i t y wave func t i ons . The observed behav iour i s c o r r e l a t e d w i th the known e l e c t r o n i c s t r u c t u r e and p e r i o d i c p r o p e r t i e s of the noble gases . 4.2 AN OVERVIEW The b i n d i n g energy s p e c t r a at s e v e r a l az imutha l ang les at E o=1200eV f o r the noble gases He, Ne, A r , Kr and Xe are p r e sen ted i n f i g u r e s 4 . 1 , 4 . 3 , 4 . 5 , 4.7 and 4.9 r e s p e c t i v e l y . For A r , Kr and Xe b i n d i n g energy s p e c t r a at s e v e r a l <j> ang les are o v e r l a y e d on an approx imate momentum (p) s c a l e to p rov ide a b e t t e r p e r s p e c t i v e of the v a r i a t i o n of the r e l a t i v e i n t e n s i t y of the s a t e l l i t e s t r u c t u r e s w i th r e spec t to p. (The p s c a l e i s on l y approximate s i n ce the 0 to p conve r s i on ( equa t ion 2.3c) depends s l i g h t l y on the scann ing b i n d i n g energy and in the above c a s e s , the i n d i c a t e d s c a l e s are on l y 105 exact f o r a b i n d i n g energy of 30eV.) The energy s c a l e s of the b i n d i n g energy s p e c t r a a re no rma l i zed on the p h o t o e l e c t r o n data [CKM71, SFC74] f o r the i o n i z a t i o n of the outermost o r b i t a l s ( i . e . the ( n p ) " 1 s t a t e s ) . The raw data (b lack dots ) are o v e r l a y e d in the r eg i on of ns i o n i z a t i o n w i th i n d i v i d u a l deconvo lu t ed Gauss ian l i n e shapes (dot-dash l i n e s i n f i g u r e s 4 . 5 , 4.7 and 4.9) w i th a FWHM c o r r e s p o n d i n g to the o v e r a l l i n s t r u m e n t a l energy r e s o l u t i o n ( l . 6 e V FWHM). The sums of the i n d i v i d u a l Gauss ians are r ep r e sen t ed by s o l i d l i n e s in f i g u r e s 4 . 3 , 4 . 5 , 4.7 and 4 . 9 . No d e c o n v o l u t i o n i s a t tempted f o r the p o r t i o n of the data w i th i n s u f f i c i e n t s t a t i s t i c s (h igh energy s i de ) a l t hough f u r t h e r s t r u c t u r e s are i n d i c a t e d c o r r e s p o n d i n g to h ighe r e x c i t e d ion s t a t e s . A l though a b s o l u t e t r i p l 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 s are not measured, r e l a t i v e n o r m a l i z a t i o n i s ma in ta ined in each se t of measurements. F i g u r e s 4 . 2 , 4 . 4 , 4 . 6 , 4.8 and 4.10 show the e x p e r i m e n t a l l y measured s p h e r i c a l l y averaged momentum d i s t r i b u t i o n s of the nob le gases at s e v e r a l s e l e c t e d b i n d i n g e n e r g i e s . The i n s t r umen ta l momentum r e s o l u t i o n (Ap =* 0 . 1 a o " 1 FWHM) has been convo lu t ed i n t o the c a l c u l 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 ob ta ined us i ng C lement i and R o e t t i wave func t ions of d i f f e r e n t q u a l i t i e s [CR74] . These are Har t ree-Fock ( s o l i d l i n e ) , doub l e-ze t a (dot-dashed l i n e ) and s i n g l e - z e t a (do t-shor t dashed l i n e ) . For the np d i s t r i b u t i o n , the c a l c u l a t i o n s are no rma l i zed at the maximum 106 of the expe r imenta l d i s t r i b u t i o n and fo r the ns d i s t r i b u t i o n , the c a l c u l a t e d d i s t r i b u t i o n s are a rea no rma l i zed i n the range 0 to 1 . 5 a 0 " 1 . C l e a r l y on l y shape compar ison can be made w i th the p resen t r e l a t i v e measurement. Note tha t the " s i t t i n g " expe r imen ta l b i n d i n g e n e r g i e s (not the t h e o r e t i c a l Ha r t r ee-Fock I . P . ' s ) are i n d i c a t e d in the f i g u r e s . T h i s i s of p a r t i c u l a r importance i n the case of ns i o n i z a t i o n in A r , Kr and Xe where m u l t i p l e f i n a l ion s t a t e s occur due to many-body e f f e c t s in the i o n i z a t i o n p r o c e s s . 4.2.1 Helium F i g u r e 4.1 shows the b i n d i n g energy spectrum of He. T h i s spectrum i s s i g n i f i c a n t because i t not on ly p r o v i d e s a measure of the expe r imen ta l i n s t r umen ta l energy r e s o l u t i o n , but a l s o j u s t i f i e s the use of a Gauss ian ( r a the r than a L o r e n t z i a n ) l i n e shape f o r the s t r u c t u r e d e c o n v o l u t i o n p rocedu re , which i s p a r t i c u l a r l y important in the i n t e r p r e t a t i o n of the heav i e r members of the g roup . The Gauss ian shape i s due to the e f f e c t of the e l e c t r o n ana l y se r t r a n s m i s s i o n f u n c t i o n ( s e c t i o n 3 .6 .1 ) which dominates any e f f e c t of n a t u r a l l i n e shape . The ze ro background con f i rms the accu racy of the method used fo r s u b s t r a c t i n g the a c c i d e n t a l c o i n c i d e n c e s ( s e c t i o n 3 . 3 . 2 ) . The momentum d i s t r i b u t i o n of He 1s e l e c t r o n i s shown in f i g u r e 4 . 2 . There i s e x c e l l e n t agreement between the exper iment and the 107 18 21 24 27 30 33 BINDING ENERGY (eV) F i g u r e 4.1 - Bindi n g energy spectrum f o r He Is i o n i z a t i o n at 0=0 ° . The Gauss ian and L o r e n t z i a n l i n e shapes are i n d i c a t e d by s o l i d and dot-dashed l i n e s r e s p e c t i v e l y . 108 F i g u r e 4 .2 - Atomic momentum d i s t r i b u t i o n of He Is o r b i t a l . T h e o r e t i c a l d i s t r i b u t i o n s are c a l c u l a t e d from C lement i and R o e t t i wave func t ions [CR74], The Har t ree-Fock and doub l e-ze t a t h e o r e t i c a l d i s t r i b u t i o n s are i n d i s t i n g u i s h a b l e from each o t h e r . 109 Har t ree-Fock c a l c u l a t i o n as has a l s o been seen in e a r l i e r work [HM&73]. T h i s a l s o c o n s t i t u t e s an important c o n s i s t e n c y check of the i n s t r u m e n t a l f u n c t i o n and a l s o the c o n v o l u t i o n of the momentum r e s o l u t i o n 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 . 4 . 2 . 2 Neon F i g u r e 4.3 shows the b i n d i n g energy s p e c t r a of Ne at two d i f f e r e n t az imutha l ang l e s (0=0° and 10 ° ) which co r respond approx ima te l y to momentum va lues at the maxima of the 2s and 2p momentum d i s t r i b u t i o n s r e s p e c t i v e l y ( f i g u r e 4 . 4 ) . The 0=0° spectrum emphasizes i o n i z a t i o n of an s (1=0) o r b i t a l wh i l e the 0=10° spectrum emphasizes the p (1=1) o r b i t a l . In a d d i t i o n to the two main t r a n s i t i o n s which co r respond to the ( 2 p ) _ 1 and ( 2 s ) " 1 s t a t e s , a t h i r d peak at 59.5eV i s a l s o found at 0 = 0 ° . In t a b l e 4.1 the energy p o s i t i o n s and the r e l a t i v e i n t e n s i t i e s of 2s s t r u c t u r e as determined by the p resen t (e ,2e) s tudy are compared w i th o ther e a r l i e r (e ,2e) [MW76a, DM&78], XPS [CKM71, SFC74] and o p t i c a l emiss ion data [P71] . A recent t h e o r e t i c a l c a l c u l a t i o n by D y a l l and L a r k i n s [DL82a, DL82b] i s a l s o i n d i c a t e d . Very b r i e f l y , the m u l t i c o n f i g u r a t i o n shake model of D y a l l and L a r k i n s [DL82a, DL82b] i n c l u d e s e l e c t r o n c o r r e l a t i o n e f f e c t s u s i n g a c o n f i g u r a t i o n a l i n t e r a c t i o n (CI) method where a l i m i t e d number of r e l e v a n t c o n f i g u r a t i o n s are 110 • mm C D _D D z g o LU CO CO CO o DC O LU oc LU LL LL Q 11 a6 -62 -3.8 -1.4 OD 11 r B6 -62 -3.8 -1.4 0.0 2P • " • 4 • " • * » " 4 Ne 1200 e V 0 = 10° T I I I I I I I | 2 s r ~1 0 = oc — i 1 1 1 1 1 1 1 1 1 6 12 18 24 3 0 36 42 4 8 54 6 0 6 6 BINDING ENERGY (eV) F i g u r e 4.3 - B i n d i n g energy spectrum fo r Ne a t 0=0° and 1 0 ° . Leas t squared Gauss ian d e c o n v o l u t i o n of the expe r imen ta l data i s i n d i c a t e d by s o l i d l i n e . Table 4.1 S a t e l l i t e structure of Ne. Peak* Dominant Energy (eV) Relative Intensity Configuration Binary (e,2e) XPS Optical Theory Binary (e,2e) XPS Theory This work a b c d e This work a b c f 1 2s2p« 48. 5 48.5 48 .5 48 .5 48 . 5 48 9 100 100 100 100 100 2s'2p 43s 55 8 55 .9 55 9 58 . 1 2 16 1 2 2s !2p 43d 59. 5(4) 60 59 .8 59 .5 59. .5 61 . 1 6(2) 4(1) 1 5 2 2s'2p 44d 63.7 1 * : see figure 4.3; a : reference [MW76a]; b : reference [SFC74], photon energy 1487eV; c : reference [CKM71], photon energy 132.2eV; d : reference [P71]; e : reference [DL82a]; f : reference [DL82b]. 112 i n c l u d e d . A s i n g l e i n i t i a l c o n f i g u r a t i o n i s chosen and s a t e l l i t e s appear ing i n the p r e d i c t e d spectrum are ob ta ined by c o n f i g u r a t i o n i n t e r a c t i o n of the f i n a l i o n i c s t a t e s wi th the same symmetry. The f r ozen co re approx imat ion i s i m p l i c i t in the c a l c u l a t i o n and the c a l c u l a t i o n i s i n t ended fo r h igh energy photon expe r imen ts . E x c e l l e n t agreement in the energy of the ( 2 s ) " 1 t r a n s i t i o n i s found between the present and the e a r l i e r (e ,2e) expe r imen t s , as w e l l as w i th XPS and o p t i c a l d a t a . The r e l a t i v e i n t e n s i t i e s in (e ,2e) cannot be r e l a t e d d i r e c t l y to those in XPS. E v i d e n t l y , the t h e o r e t i c a l i n t e n s i t i e s agree b e t t e r w i th the XPS data at 1487eV [SFC74] than tha t at . 132.3eV [CKM71] photon e n e r g i e s . F a i r agreement i s found between the two (e ,2e) measurements of the r e l a t i v e s a t e l l i t e i n t e n s i t y f o r the 2 s 2 2 p " 3 d c o n f i g u r a t i o n . The s a t e l l i t e w i th 2 s 2 2 p " 3 s c o n f i g u r a t i o n at 56eV p r e d i c t e d by theory and found in XPS [CKM71, SFC74] and o p t i c a l emiss ion [P71] da ta i s not observed in the b i n a r y (e ,2e) spectrum in the p resen t work nor i n tha t of Dixon et al . [DM&78], The momentum d i s t r i b u t i o n s of the 2p and 2s e l e c t r o n s are shown in f i g u r e 4 . 4 . I t can be seen tha t f o r both o r b i t a l s the SZ f u n c t i o n i s g r o s s l y inadequate whereas the HF g i v e s an e x c e l l e n t d e s c r i p t i o n of the momentum d i s t r i b u t i o n . The DZ f u n c t i o n r e s u l t i s i n d i s t i n g u i s h a b l e from HF in the case of the 2s o r b i t a l and i s a s l i g h t l y worse f i t f o r the 2p o r b i t a l . 113 O Q0 05 1.0 15 2D 25 00 05 1.0 15 20 25 F i g u r e 4.4 - Atomic momentum d i s t r i b u t i o n s f o r Ne 2p and 2s o r b i t a l s . The i n d i c a t e d ene rg i e s are the s i t t i n g b i n d i n g e n e r g i e s . The Har t ree-Fock and d o u b l e - z e t a [CR74] d i s t r i b u t i o n s f o r Ne 2s a re i n d i s t i n g u i s h a b l e from each o t h e r . 114 4 .2 .3 Argon The angu la r c o r r e l a t e d b i n d i n g energy spectrum of argon i s shown in f i g u r e 4 . 5 . The f i r s t peak (at 15.6eV) co r responds to i o n i z a t i o n s of the 3p e l e c t r o n s . The peak at 29.8eV and the s a t e l l i t e s t r u c t u r e at h i ghe r energy i s due to i o n i z a t i o n of the 3s e l e c t r o n s . Gauss ian d e c o n v o l u t i o n i s per formed to i d e n t i f y the more i n t ense s a t e l l i t e s t a t e s . The i n t e n s i t i e s of these s t a t e s g e n e r a l l y dec rease from the maximum at 0=0° as <f> i n c r e a s e s , i n d i c a t i n g an o v e r a l l s-type angu la r dependence. Tab l e 4.2 g i v e s the energy p o s i t i o n s and the r e l a t i v e i n t e n s i t i e s of the parent and s a t e l l i t e ( 3 s ) " 1 t r a n s i t i o n s ob t a i ned by the (e ,2e) [MW76a, W78], XPS [CKM71, SFC74] and o p t i c a l [N73] expe r imen t s . E x c e l l e n t agreement in energy p o s i t i o n s i s o b t a i n e d between the v a r i o u s d i f f e r e n t expe r imen t s . A l s o shown are the m u l t i c o n f i g u r a t i o n shake model c a l c u l a t i o n s of D y a l l and L a r k i n s [DL82a, DL82b] as w e l l as a more d e t a i l e d CI c a l c u l a t i o n by M i t r o y [M83] which i n c l u d e s cont inuum e f f e c t s and a l s o i n i t i a l and f i n a l s t a t e c o r r e l a t i o n s . There i s a d i s c r e p a n c y of app rox ima te l y 1eV between most of the c a l c u l a t e d ene rg i e s and the expe r imen ta l r e s u l t s . I t i s e v iden t tha t the c a l c u l a t i o n s by M i t r o y [M83] g i v e s the best agreement w i th the r e l a t i v e i n t e n s i t i e s measured by b i n a r y (e ,2e) s p e c t r o s c o p y . The c a l c u l a t e d r e l a t i v e i n t e n s i t i e s by D y a l l and L a r k i n s [DL82a, DL82b] appear to be in b e t t e r agreement w i th the XPS measurements than w i th the (e ,2e) d a t a . 115 10 14 18 22 26 30 34 38 42 48 50 BINDING E N E R G Y (eV) F i g u r e 4 .5 - Angular c o r r e l a t e d b i n d i n g energy spectrum f o r A r . The momentum, p, s c a l e i s approx imate . I n d i v i d u a l Gauss ian l i n e shapes (dot-dashed l i n e s ) a re l e a s t squared f i t t e d to the spec t rum. The sum of the Gauss ians i s r ep re sen t ed by the s o l i d l i n e . Table 4.2 S a t e l l i t e s t r u c t u r e of Ar. Peak* Dominant Energy (eV) - Relative Intensity Configuration Binary (e,2e) XPS Optical Theory Binary (e,2e) XPS Theory This work a b c d e f h This work a b c d g h 1 3s3p« 29 3(1) 29 3( 1) 29 3 29 3(1) 29 2 29 22 28 78 28 95 100 100 ioo 100 100 100 100 2 3s'3p 44s 36 8(4) 36 7 37 4(2) 36 6 36 48 37 28 37 51 7(3) 5 3(2) 3 2 2 3 3s'3p«3d 38 6(1) 38 6( 1) 38 6 38 7(1) 38 4 38 56 39 34 39 72 32(3) 40(3) 41 19(2) 19 13 27 4 3s'3p*4d 41 2(2) 41 2(2) 41 2 41 2(2) 40 6 41 19 41 77 42 37 20(3) 22(3) 25 6 9 9 12 5 3s'3p'5s 3s'3p'5d 43 5(2) 43 4(1) 43 4 42 1 42 41 42 43 90 06 12(3) 10(3) 1 1 2 2 * : see f i g u r e 4.5; a : reference [MW76a]; b : reference [W78]; c : reference [SFC74], photon energy 1487eV; d : reference [CKM71], photon energy 132.2eV; e : reference [N73]; f : reference [DL82a]; g : reference [DL82bj; h : reference [M83]. 1 17 There are a l s o some sma l l d i f f e r e n c e s in r e l a t i v e i n t e n s i t i e s between the p resen t noncop lanar (e ,2e) data and those of the cop l ana r (e ,2e) da ta of W i l l i a m s [W78], The r e l a t i v e i n t e n s i t i e s of the f i r s t th ree s a t e l l i t e s ob ta ined by W i l l i a m s are g e n e r a l l y h ighe r as compared w i th those in the p resen t s tudy . In the case of the 3 s 2 3 p f t 3 d s t a t e 2 the d i f f e r e n c e i s q u i t e l a r g e . The p resen t s tudy , however, shows good agreement w i th the p r e v i o u s (e ,2e) measurement of McCarthy and Weigo ld [MW76a]. With a poore r energy r e s o l u t i o n , the weak s t r u c t u r e at 36.8eV was not r e p o r t e d by these au thors [MW76a] and i t i s l i k e l y tha t the r e p o r t e d peak at 38.6eV a c t u a l l y co r responded to the f i r s t two e x c i t e d i o n i c s t a t e s . The sum of the i n t e n s i t i e s of the f i r s t two s t a t e s i n the p resen t s tudy (38%) i s i n e x c e l l e n t agreement w i th the r e p o r t e d [MW76a] i n t e n s i t y (40%). The sum d e r i v e d from W i l l i a m s ' da ta i s , however, 47%. The t h i r d s a t e l l i t e r e p o r t e d by W i l l i a m s i s a l s o more i n t ense compared to both noncoplanar (e ,2e) r e s u l t s . These d i s c r e p a n c i e s may be the r e s u l t of d i f f e r e n t s c a t t e r i n g c o n d i t i o n s , s i n ce W i l l i a m s used the cop l ana r s c a t t e r i n g k i nema t i c s w i th r a the r low impact energy (E o>300eV or E 1 =E 2 =150eV) where the v a r i a t i o n of the c r o s s s e c t i o n due to the Mott s c a t t e r i n g f a c t o r i s more s e v e r e . Under these low energy c o n d i t i o n s , i t i s we l l 2 The c o n f i g u r a t i o n g i ven to a s t a t e in the p resen t work i m p l i e s the dominant c o n f i g u r a t i o n in the CI e x p a n s i o n . 118 known [MW76a] tha t the impulse approx imat ion i s l e s s adequate and i t has been found tha t r e l a t i v e i n t e n s i t i e s a re m o d i f i e d a c c o r d i n g l y . In p a r t i c u l a r i t has been found tha t a l t hough c o r r e c t momentum d i s t r i b u t i o n s are g i ven at E o =400eV [MW76a] the r e l a t i v e i n t e n s i t i e s of d i f f e r e n t peaks are not in c o n f o r m i t y w i th PWIA c a l c u l a t i o n s . Even at 1200eV impact energy s i g n i f i c a n t d i f f e r e n c e s have been observed between PWIA c a l c u l a t i o n s and exper iment p a r t i c u l a r l y f o r inner v a l ence mo lecu l a r o r b i t a l s s i m i l a r to the s o r b i t a l s in the noble gases [BH&80, BM682], To d e f i n i t i v e l y a s s i g n some of the most prominent s a t e l l i t e s as a r i s i n g from the 3s ho l e s t a t e , momentum d i s t r i b u t i o n s have been measured at 38.5eV and 41.1eV ( f i g u r e 4 . 6 ) . T y p i c a l s-type behav io r i s ob ta ined i n each c a s e . As wi th he l i um and neon, Har t ree-Fock ( i n d i s t i n g u i s h a b l e from DZ) wave func t ions can be used to reproduce the expe r imen ta l d i s t r i b u t i o n s much b e t t e r than the SZ wave func t ions f o r both 3p and 3s e l e c t r o n s . 4 . 2 . 4 K r y p t o n F i g u r e 4.7 shows the angu la r c o r r e l a t e d b i n d i n g energy spectrum of K r . S i m i l a r r e s u l t s f o r k ryp ton were a l s o determined r e c e n t l y by Fuss et al . [FG&81] The ang les at which the b i n d i n g energy scans were taken are s l i g h t l y d i f f e r e n t i n the presen t case but n e v e r t h e l e s s i t i s apparent 119 F i g u r e 4.6 - Atomic momentum d i s t r i b u t i o n s f o r Ar. The s-type d i s t r i b u t i o n s at 2 9 . 2 , 38.5 and 41.1eV c o r r e s p o n d to i o n i c s t a t e s l a b e l l e d 1, 3 and 4 i n f i g . 4.5 r e s p e c t i v e l y . The HF and DZ d i s t r i b u t i o n s f o r Ar 3s are i n d i s t i n g u i s h a b l e from each o t h e r . 1 2 0 Kr 1200 eV 10 14 18 22 26 3 0 3 4 38 4 2 4 6 SO BINDING E N E R G Y (eV) F i g u r e 4 .7 - Angular c o r r e l a t e d b i n d i n g energy spectrum f o r K r . The momentum, p, s c a l e i s approx imate . I n d i v i d u a l Gauss ian l i n e shapes (dot-dashed l i n e s ) a re l e a s t squared f i t t e d to the spec t rum. The sum of the Gauss i ans i s r ep re sen t ed by the s o l i d l i n e . 121 that the two se t s of s p e c t r a are in e x c e l l e n t agreement. As in the case of argon 3s , the re are c o n s i d e r a b l e e l e c t r o n c o r r e l a t i o n e f f e c t s in the 4s i o n i z a t i o n of k r y p t o n , r e s u l t i n g i n s i g n i f i c a n t p o p u l a t i o n s p l i t t i n g of the s ho l e ambng s e v e r a l f i n a l i o n i c s t a t e s . The energy p o s i t i o n s and s a t e l l i t e i n t e n s i t i e s are summarized in t a b l e 4 . 3 . E x c e l l e n t agreement in ene rg i e s are found in a l l expe r imen t s . There i s good agreement between the s p e c t r a ob t a i ned in the b i na r y (e ,2e) work and those r e p o r t e d by Fuss et al . [FG&81], In a d d i t i o n to the two s a t e l l i t e s r e p o r t e d p r e v i o u s l y [FG&81], the presence of s a t e l l i t e s t a t e s at 36.7eV and 39.1eV i s i n d i c a t e d by the Gauss ian d e c o n v o l u t i o n p r o c e d u r e . With r e f e r e n c e to the m u l t i c o n f i g u r a t i o n c a l c u l a t i o n s of D y a l l and L a r k i n s [DL82a, DL82b] , these two s t a t e s are t e n t a t i v e l y a s s i g n e d to the 4 s 2 4 p * 5 d c o n f i g u r a t i o n and an admixture of 4 s 2 4 p " 6 s and 4 s 2 4 p * 6 d c o n f i g u r a t i o n s r e s p e c t i v e l y . O b v i o u s l y b e t t e r r e s o l u t i o n and improved s t a t i s t i c s are r e q u i r e d to f u r t h e r d e f i n e these inner va l ence s a t e l l i t e s t r u c t u r e s . I t i s i n t e r e s t i n g to note tha t the c a l c u l a t i o n s by D y a l l and L a r k i n s are in no b e t t e r agreement w i th the measured (e ,2e) s a t e l l i t e i n t e n s i t i e s than are the r e s u l t s of a l a r g e s e r i e s of ground s t a t e and f i n a l i o n i c s t a t e CI c a l c u l a t i o n s r e p o r t e d by Fuss et al. [FG&81], The c a l c u l a t i o n s of Fuss et al . [FG&81] took i n t o account of the cont inuum but used on l y a l i m i t e d number of c o n f i g u r a t i o n s . However b e t t e r agreement w i th the XPS data i s ob t a i ned by D y a l l and L a r k i n s . Table 4.3 S a t e l l i t e structure of Kr. Peak* Dominant Energy (eV) Relative Intensity Configuration ; Binary (e,2e) XPS Optical Theory Binary (e,2e) XPS Theory This work a b c a d This work a b a e 1 4s4p" 27 3(2) 27 5(1) 27 5 27 51 27 5 26 48 100 100 100 100 100 2 4s'4p*5s 32 2(5) 32 1(2) 32 4 32 07 32 1 32 29 11(6) 15(4) 9(2) 4 3 3 4s J4p'4d 34 0(2) 33 9(1) 34 0 33 93 33 9 34 22 47(6) 43(4) 25(3) 22 20 4 4s !4p'5d 36 7(3) >35 5 >35 5 36 46 27(6) 59(7) 56 13 5 4s'4p'6s 4s«4p*6d 39 1 37 37 40 64 15(6) 2 5 * : see figure 4.7; a : reference [FG&81]; b : reference [SFC74], photon energy 1487eV; c : reference [MSP69]; d : reference [DL82a]; e : reference [DL82bj. 123 Momentum d i s t r i b u t i o n s at s e v e r a l " s i t t i n g " b i n d i n g e n e r g i e s a re g i ven in f i g u r e 4 . 8 . The f i r s t and the second peaks i n the b i n d i n g energy spectrum co r r e spond to the i o n i z a t i o n of the 4p and the p r i n c i p a l ho le s t a t e f o r the removal of 4s e l e c t r o n s r e s p e c t i v e l y . The o the r momentum d i s t r i b u t i o n s s p e c t r a show that the s a t e l l i t e s at 33.8eV and 36.2eV are c l e a r l y due to the 4s e l e c t r o n . The Har t ree-Fock ( i n d i s t i n g u i s h a b l e from DZ) wave func t ion c l o s e l y p r e d i c t s the measured momentum d i s t r i b u t i o n s . 4 .2 .5 Xenon F i g u r e 4.9 shows the angu la r c o r r e l a t e d b i n d i n g energy spectrum of Xe . Tab le 4.4 compares the p resen t (e ,2e) data w i th o the r r e l a t e d expe r imen ta l d a t a . Two Gauss i ans are f i t t e d to the f i r s t ( 5 P ) " 1 peak to account f o r the w e l l known (1.3eV) s p i n - o r b i t s p l i t t i n g of 2 P 3 / 2 a n £ 3 2 P i / 2 s t a t e s of X e + . Gauss ian d e c o n v o l u t i o n was used to decompose the s t r u c t u r e above 28eV i n t o four CI s t a t e s as suggested by D y a l l and L a r k i n s ' c a l c u l a t i o n s [DL82a, DL82b] . The p resen t b i n d i n g energy spectrum ( f i g u r e 4.9) i s in g e n e r a l agreement w i th the low r e s o l u t i o n data of the F l i n d e r s U n i v e r s i t y group [MW76a] and i t i s a l s o in a c co rd w i th the h ighe r r e s o l u t i o n O . O e V FWHM) spectrum ob ta ined by Hood et al. [HHB77]. Ex t r a unass igned s t a t e s in the 24 to 26eV r e g i o n (which are l a b e l l e d 1 and 2 by Hood et al.) a re c l e a r l y p resen t in the 124 single-zeta P (Qo"1) F i g u r e 4.8 - Atomic momentum d i s t r i b u t i o n s f o r Kr. The s-type d i s t r i b u t i o n s at 2 7 . 6 , 33.8 and 36.2eV co r r e spond to i o n i c s t a t e s l a b e l l e d 1, 3 and 4 i n f i g . 4.7 r e s p e c t i v e l y . The HF and DZ d i s t r i b u t i o n s fo r Kr 4s are i n d i s t i n g u i s h a b l e from each o t h e r . 125 t "s Xe 1200 eV V / BINDING E N E R G Y (eV) F i g u r e 4 .9 - Angular c o r r e l a t e d b i n d i n g energy spectrum f o r Xe . The momentum, p, s c a l e i s approx imate . I n d i v i d u a l Gauss ian l i n e shapes (dot-dashed l i n e s ) a re l e a s t squared f i t t e d to the spec t rum. The sum of the Gauss i ans i s r ep resen ted by the s o l i d l i n e . Table 4.4 S a t e l l i t e structure of Xe. Peak* Dominant Energy (eV) Relative Intensity Configuration Binary (e,2e) XPS Optical Theory Binary (e,2e) XPS Theory This work a b c d e This work a b c f 1 5s5p« 23 .2(1) 23.2(1) 23 .4 23 .4 23, .37 21 t . 36 100 100 100 100 100 ? 24 .e 6(2) ? 25 .2 7(2) 2 5s'5p'6s 27 .7(3) >28. 1 28 .0 27 .6 28 . 13 27, .49 26(8) 194(4) 25(4) 11(3) 6 3 5s'5p*5d 28 .9(1) 29 .0 29. .0 28 . 85 28 ,83 67(8) 52(4) 62(7) 24 4 5s»5p'6d 31 .2(3) 31 .4 30. .91 23(5) 20(3) 16 5 5s«5p'7s 32 .8 31 . 93 6(2) 3 5s'5p«7d 33. . 1 33 .8 32 . 12 15(5) 4(2) 6 to * : see fi g u r e 4.9; a : reference [MW76a]; b : reference [G74], photon energy 1487eV; c : reference [SFC74], photon energy 1487eV; d : reference [HP78]; e : reference [DL82aj; f : reference [DL82b]. 127 form of a shou lder on the h igh energy s i de of the main ( 5 s ) " 1 peak. S i m i l a r s a t e l l i t e s are a l s o r epo r t ed i n the h i g h r e s o l u t i o n XPS data of G e l i u s [G74] . I t i s p o s s i b l e tha t these s t a t e s are the r e s u l t of i n i t i a l s t a t 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 s . Momentum d i s t r i b u t i o n s are measured at s e v e r a l b i n d i n g ene rg i e s ( f i g u r e 4 .10 ) . E x c e l l e n t agreement between Ha r t r ee-Fock (and DZ) momentum d i s t r i b u t i o n s and the expe r imen ta l ones are obse r ved . The s-type momentum d i s t r i b u t i o n s at 29.0eV and 33.5eV c o n f i r m tha t the dominant c o n f i g u r a t i o n s 5 s 2 5 p " 5 d and 5s 2 5p*7s/7d are e x c i t e d i o n i c s t a t e s a r i s i n g from the 5 s " 1 ho le s t a t e . The o b s e r v a t i o n of the deep minimum in the ext remely sharp momentum d i s t r i b u t i o n of the 5p e l e c t r o n i s in c o n f i r m a t i o n of the h igh angu la r r e s o l v i n g power of the p resen t spec t rome te r . 4.3 SATELLITE STRUCTURE To summarize, there i s g e n e r a l l y good agreement between the p resen t (e ,2e) da ta and o ther expe r imenta l da ta in the energy p o s i t i o n s of s a t e l l i t e s t a t e s above the main ( n s ) " 1 parent t r a n s i t i o n s . The r e l a t i v e i n t e n s i t i e s of these t r a n s i t i o n s in (e ,2e) and d , e ) (or p h o t o e l e c t r o n ) exper iments cannot be compared d i r e c t l y because of the d i f f e r e n t i o n i z a t i o n c o n d i t i o n s . The i n t e n s i t i e s of most of these s a t e l l i t e s r e l a t i v e to tha t of the parent (ns) " 1 ho l e 128 00 05 UO 15 2D 25 P («0 F i g u r e 4.10 - Atomic momentum d i s t r i b u t i o n s f o r Xe. s-type d i s t r i b u t i o n s at 2 3 . 6 , 29.0 and 33.5eV co r r e spond to i o n i c s t a t e s l a b e l l e d 1, 3 and 4 i n f i g . 4 .9 r e s p e c t i v e l y . The HF and DZ d i s t r i b u t i o n s f o r Xe 5s are i n d i s t i n g u i s h a b l e from each o t h e r . 129 s t a t e o b t a i n e d by (e ,2e) s t u d i e s are 1 to 3 t imes l a r g e r than those o b t a i n e d by (7,e) s t u d i e s . However, i t shou ld be noted tha t the re i s c l o s e r agreement in the case of xenon between (e ,2e) and (y,e) measurements. I t i s a l s o i n t e r e s t i n g to note in t a b l e s 4.1 to 4.4 tha t fo r 2 S symmetry, the n s 2 n p * n ' d (n '=n+1,n+2, e t c . ) s a t e l l i t e s a re much more i n t ense than the n s 2 n p * n ' s s a t e l l i t e s . There i s some c o n f u s i o n in some of the ass ignments by Spears et al . [SFC74] , For i n s t a n c e the ass ignment of the s a t e l l i t e a t 32.2eV of Kr to the " 5 p ( 2 P 0 ) shake-up" c o n f i g u r a t i o n [SFC74] does not agree w i th the measured s-type momentum d i s t r i b u t i o n and shou ld be a s s i g n e d to a 4 s 2 4 p " 5 s dominant c o n f i g u r a t i o n as i n d i c a t e d i n t a b l e 4 . 3 . A l s o the v a r i a t i o n of s a t e l l i t e i n t e n s i t i e s w i th momentum in the angu la r c o r r e l a t i o n b i n d i n g energy s p e c t r a suggests a " g l o b a l " s-type angu la r b e h a v i o u r , r e i n f o r c i n g the c o n c l u s i o n tha t the s a t e l l i t e s t r u c t u r e i s ma in ly the r e s u l t of f i n a l i o n i c s t a t 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 a s s o c i a t e d w i th the ( n s ) " 1 h o l e . There are s e v e r a l i n t e r e s t i n g t r ends in the b i n d i n g energy s p e c t r a of the nob le gases , a l l of which can be unders tood w i t h i n the framework of the quantum chemis t r y of atoms. F i g u r e 4.11 shows the 0=0° b i n d i n g energy s p e c t r a of the noble gases grouped toge the r to emphasize some of these chemica l t r e n d s . One s t r i k i n g f e a tu r e i s the o v e r a l l i n c r e a s e i n s a t e l l i t e i n t e n s i t i e s as compared to the parent 130 F i g u r e 4.11 - Comparison of valence b i n d i n g energy s p e c t r a of Noble Gaeses . The i o n i c s t a t e s i d e n t i f i e d by Gauss ian d e c o n v o l u t i o n a re l a b e l l e d in each of the s p e c t r a . See a l s o t a b l e s 4 . 1 - 4 . 4 . 131 i n t e n s i t i e s as one moves down the g roup : i . e . from Ne to Xe . Tab l e 4.5 shows the r a t i o of the t o t a l i n t e n s i t y of the deconvo lu t ed s a t e l l i t e s t a t e s to the i n t e n s i t y of the parent s t a t e . T h i s i s a lower l i m i t s i n ce the re w i l l be some f u r t h e r c o n t r i b u t i o n from h ighe r l y i n g e x c i t e d and cont inuum s t a t e s . The i n c r ease in s a t e l l i t e i n t e n s i t y p roceed ing down the group i s a l s o observed i n XPS data and p r e d i c t e d by D y a l l and L a r k i n s ' c a l c u l a t i o n [DL82a, DL82b] . The onset of the i n c r e a s e r e a l l y beg ins w i th Ar and not w i th Ne. T h i s i s c o n s i s t e n t w i th the idea tha t the n s 2 n p " n ' d c o n f i g u r a t i o n s are more important than the n s 2 n p a n ' s s e r i e s because of the f a c t tha t the re i s no d o r b i t a l in the n=2 s h e l l . The second n o t i c e a b l e t r end w i th i n c r e a s i n g atomic number i s the dec rease i n energy s e p a r a t i o n i n the ( n s ) " 1 m a n i f o l d between the f i r s t e x c i t e d ion s t a t e and the parent ion ( n s ) " 1 s t a t e . T h i s i s aga in c o n s i s t e n t w i th the f a c t tha t the energy spac ings between s h e l l s decrease w i th i n c r e a s i n g p r i n c i p a l quantum number. The dec rease in energy spac ings r e s u l t s i n more c o n f i g u r a t i o n m ix ing and f u r t h e r enhances the o v e r a l l s a t e l l i t e i n t e n s i t i e s . 132 Tab l e 4.5 T o t a l a s s i gned s a t e l l i t e i n t e n s i t y r e l a t i v e to the parent f o r the ns i o n i z a t i o n of Noble Gases . Spec i e s T o t a l a s s i g n e d Sa t . I n t . / Pa r en t I n t . * (e ,2e) XPS Theory T h i s work a b Ne 0.06 0.03 0.04 Ar 0.70 0.28 0 .28** Kr 1.00 0.34 0.43 Xe 1.31 1.07 0.55 a : r e f e r e n c e [SFC74] , photon energy l487eV; b : r e f e r e n c e [DL82b] . * : Only the measured peaks shown in t a b l e s 4 .1-4 .4 are taken i n t o c o n s i d e r a t i o n . The c o n t r i b u t i o n from h ighe r e x c i t e d and cont inuum s t a t e s i s not i n c l u d e d . * * : compare a l s o 0.43 ( r e f e r ence [M83], which does not i n c l u d e i n t e n s i t y from peak 5 - t a b l e 4.2) and 0.40 ( r e f e r ence [W79]). 1 33 4.4 OUTER VALENCE ORBITAL ELECTRON MOMENTUM DISTRIBUTIONS In o rde r to f u r t h e r e x p l o r e the e x i s t e n c e of any chemica l t r e n d a s s o c i a t e d w i th the momentum d i s t r i b u t i o n of the oute r va l ence o r b i t a l s of the noble gases , the momentum d i s t r i b u t i o n s of np and ns s e r i e s in the momentum range from 0 to 2 . 5 a 0 " 1 , are shown a long w i th the c o r r e s p o n d i n g Ha r t r ee-Fock d i r e c t i o n a l momentum d e n s i t y and p o s i t i o n (or charge) d e n s i t y con tour maps i n f i g u r e s 4.12 and 4.13 r e s p e c t i v e l y . The arrows on the momentum d i s t r i b u t i o n s i n d i c a t e p ( i . e . the momentum at which the p r o b a b i l i t y d i s t r i b u t i o n i s at i t s maximum he igh t ) f o r the np s e r i e s and Pi/2max ( i . e . the momentum at which the d i s t r i b u t i o n i s h a l f of i t s maximum) f o r the ns s e r i e s . A l l the o ther n o t a t i o n s are i d e n t i c a l to those in the p r e v i o u s momentum d i s t r i b u t i o n ( f i g u r e s 4 . 2 , 4 . 4 , 4 . 6 , 4.8 and 4 . 1 0 ) . The d e n s i t y maps show the " d i r e c t i o n a l " p r o b a b i l i t y of an o r b i t a l i n both momentum and p o s i t i o n - s p a c e . In p a r t i c u l a r , the p resen t momentum contour maps sample p r o b a b i l i t y momentum d e n s i t y ( ^ j(p ) ^ j(p) where ^ j(p) i s the jth o r b i t a l of the Har t ree-Fock wave func t ion in the momentum r e p r e s e n t a t i o n ) i n the p =0 p l a n e , and s i m i l a r l y the p o s i t i o n d e n s i t y maps sample the c o r r e s p o n d i n g p o s i t i o n d e n s i t y i n the r =0 p l a n e . The contour v a l u e s are l o g a r i t h m i c , namely, at 0 .2 , 0 . 4 , 0 . 6 , 0 . 8 , 2, 4, 6, 8, 20, 40, 60 and 80% of the maximum d e n s i t y . Note tha t the range of the p o s i t i o n d e n s i t y maps f o r the ns s e r i e s ( f i g u r e 4 .13 , r i g h t hand s i d e ) i s from -0.5 to 0.5 a 0 134 SPHERICALLY AVERAGED z UJ i P ( « 0 MOMENTUM DENSITY -2 -1 0 p y 1 2 0.01 0.1 10 \ t | | l l | J m m 1 -2 -1 0py1 2 aoi ai IO 8 i -2 -1 0 p yl 2 aoi ot io 8 -2-1 Op 1 2 aoi ai io POSITION DENSITY -2 -1 0,7 1 2 Q010.1 10 F i g u r e 4.12 - Momentum d i s t r i b u t i o n s ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps fo r np o r b i t a l s of Noble Gases . The arrows i n the momentum d i s t r i b u t i o n s i n d i c a t e p max 135 SPHERICALLY AVERAGED MOMENTUM DENSITY MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY 0.010.1 10 F i g u r e 4.13 - Momentum d i s t r i b u t i o n s . ( l e f t ) , momentum d e n s i t y ( cen t re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps f o r ns o r b i t a l s of Noble Gases . The arrows i n the momentum d i s t r i b u t i o n s i n d i c a t e P 1^2 136 wh i le the range f o r a l l the o ther maps in f i g u r e s 4.12 and 4.13 i s from -2.5 to 2.5 a . u . in both p o s i t i o n and momentum space . A long w i th the d e n s i t y maps, l i n e p r o j e c t i o n s ( a l so w i th l o g a r i t h m i c i n t e n s i t y ) of the d e n s i t y s u r f a c e s are shown to g i ve some idea of the o v e r a l l one d imens iona l d e n s i t y p r o j e c t i o n a long a c e r t a i n a x i s . The l i n e p r o j e c t i o n p l o t s are u s e f u l i n p r o v i d i n g an immediate i n d i c a t i o n of the r e l a t i v e d e n s i t y magnitude a s s o c i a t e d wi th each e q u i d e n s i t y contour l i n e in the sampl ing p l a n e . In the diagrams the z a x i s (shown by the dash l i n e in the d e n s i t y maps of f i g u r e s 4.12 and 4.13) i s chosen f o r the p r o j e c t i o n . There are s e v e r a l noteworthy f e a t u r e s i n the two f i g u r e s . The Har t ree-Fock wave func t i on , which g i v e s the lowest t o t a l ene rgy , i s the best of the th ree wave func t ions f o r d e s c r i b i n g the s p h e r i c a l l y averaged momentum d i s t r i b u t i o n (see l e f t hand s i d e s of f i g u r e s 4.12 and 4 .13 ) . The doub le-ze t a r e s u l t i s a lmost as good as the Har t ree-Fock and i s in f a c t i n d i s t i n g u i s h a b l e from Har t ree-Fock in the case of the ns o r b i t a l s . The s i n g l e - z e t a i s inadequate p a r t i c u l a r l y in the case of the np d i s t r i b u t i o n s . In the f a i r l y broad d i s t r i b u t i o n s such as those of Ne the d i f f e r e n c e s between the c a l c u l a t e d d i s t r i b u t i o n s of d i f f e r e n t q u a l i t y a re g r e a t e r . I t i s a l s o i n t e r e s t i n g to note tha t the b e t t e r the q u a l i t y of the wave func t i on , the sma l l e r a re P m a x and P 1/2max* A l though the momentum d i s t r i b u t i o n s of Har t ree-Fock (and to a l e s s e r ex ten t the doub le-ze ta ) wave funct ions are in 137 e x c e l l e n t agreement w i th the expe r imen ta l d i s t r i b u t i o n s , the re remain some sma l l d i s c r e p a n c i e s e s p e c i a l l y at momenta ^ 1 . 5 a 0 ~ 1 i n s p e c t r a of the heav i e r members of the g roup . These sma l l d i f f e r e n c e s are p a r t i c u l a r l y obv ious i n the Kr 4s and Xe 5s s p e c t r a . I t i s - i n t e r e s t i n g to note tha t t h i s d i s c r e p a n c y tends to i n c r ease down the g roup . For i n s t ance the Ne 2p and Ne 2s d i s t r i b u t i o n s are in q u i t e good agreement w i th the r e s p e c t i v e c a l c u l a t i o n s up to ** 2 . 2 a 0 " 1 whereas the Xe 5p and Xe 5s t a i l s appear at app rox ima te l y 1 . 5 a 0 " 1 . P r e v i ous d i s t o r t e d wave c a l c u l a t i o n s by Dixon et al. [DMW76] suggested the inadequacy of the p lane wave impulse a p p r o x i m a t i o n . The t r end seen i n the p resen t s tudy i s c o n s i s t e n t wi th the idea of Dixon et al. tha t d i s t o r t i o n of the e l e c t r o n waves i s more severe in the h i g h momentum r e g i o n . T h i s d i s t o r t i o n a p p a r e n t l y depends upon the s p a t i a l ex tent of the o r b i t a l i t s e l f . Another p o s s i b l e cause of the d i s c r e p a n c y i s the inadequacy of the Har t ree-Fock wave func t ion i t s e l f . The importance of c o n f i g u r a t i o n i n t e r a c t i o n s and r e l a t i v i s t i c e f f e c t s on the momentum d e n s i t y in the h i g h momentum r eg ion s t i l l awa i ts f u r t h e r t h e o r e t i c a l i n v e s t i g a t i o n . These e f f e c t s may a l s o be p a r t l y r e s p o n s i b l e f o r the sma l l d i s c r e p a n c y between the c a l c u l a t i o n s and the exper iments in the very low p r e g i o n . There i s a l s o a p o s s i b i l i t y that the range of v a l i d i t y of the r e a c t i o n model w i l l va ry depending on the na ture of the t a r g e t s p e c i e s . Perhaps the most n o t i c e a b l e t r e n d of the np and ns 1 38 momentum d i s t r i b u t i o n s i s the i n c r e a s e in momentum p r o b a b i l i t y c o n t r a c t i o n wi th p r i n c i p a l quantum number. T h i s can be seen q u i t e c l e a r l y w i th the i n c r e a s i n g l y sharper momentum d i s t r i b u t i o n go ing down the group and the d e c r e a s i n g Pm-,v a n c ^ P i / o m „ „ ' S i m i l a r e f f e c t s have been seen in the i s o e l e c t r o n i c hydrogen h a l i d e s : HF, HCI, HBr and HI [BH&80, BM&82]. T h i s behav io r can be unders tood in the l i g h t of the F o u r i e r t r ans fo rm (FT) p r o p e r t i e s . As d i s c u s s e d i n chap te r 2, the four commonly r e f e r r e d to FT e f f e c t s are the i n ve r se s p a t i a l r e l a t i o n , the e x i s t e n c e of an i n v e r s i o n c en t r e at the o r i g i n of the momentum-space, mo l e cu l a r d e n s i t y d i r e c t i o n a l r e v e r s a l and mo lecu l a r d e n s i t y o s c i l l a t i o n e f f e c t s [CD41, ET77, CB82a, CB82b, MB&83]. In the case of atoms, on l y f i r s t two are r e l e v a n t . The p r o p e r t y of s p a t i a l r e v e r s a l i s fundamental in the i n t e r p r e t a t i o n of the momentum-space chemis t r y [CD41, ET77] both f o r atoms and f o r mo lecu les s i n c e the b a s i c LCAO-MO fo rma l i sm i s i d e n t i c a l i n the two H i l b e r t spaces and t h i s w i l l be f u r t h e r e l a b o r a t e d below. Momentum d e n s i t y maps and p o s i t i o n d e n s i t y maps are a l s o shown in f i g u r e s 4.12 and 4.13 in o rder to f a c i l i t a t e the unde rs t and ing of the i n ve r se s p a t i a l p r o p e r t y between p o s i t i o n and momentum space . Be fore d i s c u s s i n g t h i s p r o p e r t y , i t shou ld be noted tha t the s p h e r i c a l harmonics are i n v a r i a n t under the F o u r i e r t r ans fo rm (see t a b l e 2 . 1 ) . The angu la r dependence of the atomic wave func t ion i s t h e r e f o r e p r e se r ved i n both spaces and on l y the r a d i a l pa r t of the 139 wave func t ion i s a f f e c t e d . Hence a t o t a l l y symmetric f u n c t i o n such as the ns o r b i t a l has the same shape i n both momentum and p o s i t i o n space (see f i g u r e 4 . 1 3 ) . One has to be c a r e f u l , however, in making the compar ison s i n c e the somewhat d i f f e r e n t appearance i n the np maps i s l a r g e l y due to the i n ve r se s p a t i a l r e l a t i o n and the r e l a t i v e prominence of l obes (nodes) in p o s i t i o n and momentum space . The i n ve r se s p a t i a l r e l a t i o n says that a c o n t r a c t i o n of p r o b a b i l i t y d e n s i t y in one space w i l l l e ad to an expans ion i n the o ther space and v i c e v e r s a . (Th is s p a t i a l r e v e r s a l r e l a t i o n i s a l s o a s s o c i a t e d i n d i r e c t l y w i th the bond d i r e c t i o n a l r e v e r s a l e f f e c t i n m o l e c u l e s . ) The t r end i n the momentum d i s t r i b u t i o n s i s a d i r e c t consequence of t h i s i n v e r s e s p a t i a l r e l a t i o n . I t i s perhaps e a s i e r to see t h i s phenomenon in the np s e r i e s . O r b i t a l d e n s i t y expans ion i n p o s i t i o n - s p a c e from Ne 2p to Xe 5p i s demonstrated i n the o v e r a l l contour expans ion in the p o s i t i o n d e n s i t y maps and p a r t i c u l a r l y i n the a s s o c i a t e d z-ax i s p o s i t i o n d e n s i t y p r o j e c t i o n p l o t s ( f i g u r e 4 . 1 2 ) , where the r e d i s t r i b u t i o n of the np p r o b a b i l i t y d e n s i t y over (h-1) l obes i s c l e a r l y i l l u s t r a t e d . (Each lobe i s l a b e l l e d in the p r o j e c t i o n p l o t s . ) That the p o s i t i o n d e n s i t y expans ion i s a s s o c i a t e d w i th the c o r r e s p o n d i n g c o n t r a c t i o n in the momentum-space can be most c l e a r l y seen i n f i g u r e 4 .12 , c en t r e co lumn. In c o n t r a c t to the p o s i t i o n d e n s i t y maps, the secondary l obes i n the momentum d e n s i t y are much too weak to be seen when 140 compared to the pr imary ones even i f a l og d e n s i t y s c a l e of th ree decades i s used . These secondary l obes occur o u t s i d e the range of the momentum d e n s i t y maps in f i g u r e 4.12 but w i l l be d i s c u s s e d below (see f i g u r e 4 . 1 4 ) . The m a j o r i t y of the momentum d e n s i t y of the np o r b i t a l i s t h e r e f o r e c o n c e n t r a t e d in the f i r s t l o b e , in c o n t r a s t to the i n c r e a s e d s i g n i f i c a n c e of the " h i g h e r " secondary lobes in p o s i t i o n -space . For the ns s e r i e s , the i n v e r s e s p a t i a l p r o p e r t y i s not as immediate ly obv ious in the d e n s i t y maps. A l s o i t shou ld be noted tha t the he l ium 1s o r b i t a l does not l i e i n the gene ra l t r end which i s not s u r p r i s i n g due to the unique nature of the n=1 s h e l l . I t i s c l e a r in the z - a x i s d e n s i t y p r o j e c t i o n p l o t that the ns p o s i t i o n d e n s i t y of the nob le gases i s d i s t r i b u t e d over n l o b e s , w i th the f i r s t (n-1) l obes hav ing the most s i g n i f i c a n t r e l a t i v e i n t e n s i t i e s (note the l o g s c a l e on f i g u r e 4 . 1 3 ) . In the p o s i t i o n d e n s i t y maps, the r eg ion from -0.5 to 0.5 a 0 i s shown i n o rde r to emphasize the most s i g n i f i c a n t l obes which " s h r i n k " c o n s i d e r a b l y i n t o the o r i g i n w i th i n c r e a s i n g p r i n c i p a l quantum number. With t h i s l i m i t e d range , the l a s t l obes i n Ar 3s and Kr 4s and the l a s t two l obes in Xe 5s (the wing of the 4th lobe i s j u s t v i s i b l e at the extremes of the p r o j e c t i o n p l o t ) cannot be s een . The o v e r a l l expans ion of p o s i t i o n d e n s i t y can be observed from f i g u r e 4.13 reasonab l y we l l in the l i m i t e d range of the p r o j e c t i o n p l o t s i f on l y the f i r s t (n-1) l obes are 141 c o n s i d e r e d . To f u r t h e r i l l u s t r a t e the idea of p o s i t i o n d e n s i t y expans ion ve r sus momentum d e n s i t y c o n t r a c t i o n , extended range l o g a r i t h m i c p l o t s of the s p h e r i c a l l y averaged ( i . e . r a d i a l ) momentum d e n s i t y and p o s i t i o n d e n s i t y f u n c t i o n s (both c o n s t r u c t e d from Har t ree-Fock wave func t ions [CR74]) a re shown i n f i g u r e s 4.14 and 4.15 r e s p e c t i v e l y . The s o l i d l i n e s co r r e spond to the r a d i a l f u n c t i o n s of ns o r b i t a l s wh i l e the dash l i n e s co r r e spond to the r a d i a l f u n c t i o n s of np o r b i t a l s . The range of the r a d i a l momentum d e n s i t y f u n c t i o n i s 0 - 2 5 a o _ 1 f o r a l l the members except f o r Xe where the range i s doub led to 5 0 a 0 " 1 in o rde r to i n c l u d e a l l 5 l obes of the ns o r b i t a l . For the r a d i a l p o s i t i o n d e n s i t y f u n c t i o n s , a somewhat sma l l e r range (0-2 .5a 0 ) i s s u f f i c i e n t to show a l l the lobe and noda l s t r u c t u r e s . One s t r i k i n g f e a t u r e found i n these r a d i a l d e n s i t y f u n c t i o n s i s tha t wh i l e the re are s i g n i f i c a n t p o r t i o n s of the p o s i t i o n d e n s i t y d i s t r i b u t e d over the ou te r l o b e s , the momentum d e n s i t y i s ma in l y c o n c e n t r a t e d i n the f i r s t lobe (note l o g a r i t h m i c s c a l e s ) . An obv ious example i s found in the case of neon 2s . I t i s c l e a r ( f i g u r e 4.14) tha t the second lobe of the Ne 2s momentum d e n s i t y does not c o n t r i b u t e s i g n i f i c a n t l y . T h i s i s in c o n t r a s t to the r e l a t i v e l y much l a r g e r c o n t r i b u t i o n of the second lobe to the r a d i a l d e n s i t y i n p o s i t i o n - s p a c e ( f i g u r e 4 . 1 5 ) . As e x p l a i n e d e a r l i e r t h i s c o n c e n t r a t i o n of the d e n s i t y i n momentum-space i n the f i r s t l o b e , co r responds to a c o n t r a c t i o n wh i le the 142 o MOMENTUM ( a i l . ) F i g u r e 4.14 - Comparison of extended range Hartree-Fock s p h e r i c a l l y averaged momentum d i s t r i b u t i o n s f o r np (do t ted l i n e s ) and ns ( s o l i d l i n e s ) of Noble Gases . The np and ns d i s t r i b u t i o n s e x h i b i t n-1 and n l obes r e s p e c t i v e l y . 1 4 3 -4 -6 0.0 0.4 0.8 1.2 1.6 2D 2A 21 •— , , , , 0.0 0.4 0.8 1.2 1.6 2.0 7A POSITION (a.u.) F i g u r e 4 .15 - Comparison of extended range Har t ree-Fock s p h e r i c a l l y averaged p o s i t i o n d i s t r i b u t i o n s f o r np (do t t ed l i n e s ) and ns ( s o l i d l i n e s ) of Noble Gases . The np and ns d i s t r i b u t i o n s e x h i b i t n-1 and n lobes r e s p e c t i v e l y . 1 44 sp read ing of the p o s i t i o n d e n s i t y over a l l the lobes i s r e l a t e d to the r e ve r se b e h a v i o r , i . e . an e x p a n s i o n . F i g u r e 4.16 shows the r a d i a l p o s i t i o n and momentum c o r r e s p o n d i n g to an e n c l o s e d 95% p r o b a b i l i t y , which can be v i r t u a l l y regarded as the s i z e of the o r b i t a l . The r e l a t i v e i n c r ea se (from Ne to Xe) in in the ns and np o r b i t a l i s c l e a r l y seen to be a s s o c i a t e d wi th a c o r r e s p o n d i n g r e l a t i v e decrease i n Pg5%' T h i s unambiguously demonstrates the i n ve r se s p a t i a l r e l a t i o n . I t can a l s o be noted t h a t , as i s o f t e n invoked i n s c r e e n i n g arguments , the ns o r b i t a l s are more p e n e t r a t i n g ( l o c a l i z e d ) in p o s i t i o n - s p a c e than t h e i r np c o u n t e r p a r t s wh i l e the r e ve r se i s t r ue i n momentum-space. Another i n t e r e s t i n g f e a t u r e i s the l i n e a r i t y of the o r b i t a l s p a t i a l ex ten t as a f u n c t i o n of the p r i n c i p a l quantum number s t a r t i n g from argon (n=3). (See f i g u r e 4 .16 . ) T h i s l i n e a r i t y i n d i c a t e s tha t the re i s a cons tan t l i n e a r o r b i t a l expans ion ( c o n t r a c t i o n ) s t a r t i n g from the n=3 s h e l l in r-space (p-s p a c e ) . The f a c t tha t neon d e v i a t e s from the gene r a l l i n e a r t r end i s expec ted s i n c e the re i s no nd s u b s h e l l i n the f i r s t -row (n=2) e l ements . He l ium i s a l s o expec ted to be d i f f e r e n t aga in because of i t s un ique e l e c t r o n i c s t r u c t u r e . E l e c t r o n s c r e e n i n g e f f e c t s in neon and argon are t h e r e f o r e d i f f e r e n t from He and a l s o Kr and Xe which have nd e l e c t r o n s . 1 45 q DO D w q i - ID c <D X L d O 9 D D. CO £ q ID o i o r —s pace He Ne Ar Kr Xe 0 2 4 6 Principal Quantum Number F i g u r e 4.16 - Comparison of the o r b i t a l s p a t i a l extent of the ns and np o r b i t a l s of Noble Gases i n p o s i t i o n and momentum space . The s p a t i a l ex ten t r e p r e s e n t s an e n c l o s e d 95% p r o b a b i l i t y and i s e f f e c t i v e l y the s i z e of the o r b i t a l . 146 Chapter V TVO ELECTRON SYSTEMS: HELIUM AND MOLECULAR HYDROGEN 5.1 INTRODUCTION The s tudy of two e l e c t r o n systems ( i . e . He and H 2 ) has a t t r a c t e d a c o n s i d e r a b l e amount of a t t e n t i o n not on l y in the expe r imen ta l d e t e r m i n a t i o n of t h e i r r e s p e c t i v e momentum d i s t r i b u t i o n s [E70, BW73, HM&73, MU&74, DM&75, DMW76, L77 , WM&77, MC&81] but a l s o i n the t h e o r e t i c a l i n v e s t i g a t i o n of t h e i r e l e c t r o n i c wave func t ions and r e l a t e d p r o p e r t i e s [W33, KR60, D62, LH67, BS72, L S78 ] , S i nce there i s on l y one type of o r b i t a l i n the ground s t a t e of the two e l e c t r o n systems in the s imp le mo lecu la r o r b i t a l (MO) p i c t u r e , i . e . 1s in He and 1Og i n H 2 , the t o t a l e l e c t r o n momentum d i s t r i b u t i o n s d e r i v e d from the exper imenta l Compton p r o f i l e s ob ta ined us i ng h igh energy photon Compton s c a t t e r i n g t echn iques [C71] and h i g h energy e l e c t r o n impact spec t roscopy [BF74] shou ld be e s s e n t i a l l y the same 1 as the o r b i t a l momentum d i s t r i b u t i o n s measured d i r e c t l y by the b i n a r y (e ,2e) method. The two e l e c t r o n systems t h e r e f o r e p r o v i d e a unique t e s t i n g ground f o r d i r e c t compar ison of d i f f e r e n t expe r imen ta l t e chn iques in 1 The Compton exper iment sees a l l e x c i t e d s t a t e s in a d d i t i o n to ground s t a t e . However the l a t t e r i s overwhelmingly the dominant c o n t r i b u t i o n . 147 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 . There are .many measurements of the Compton p r o f i l e s of He and H 2 u s ing X-ray and 7-ray Compton s c a t t e r i n g t e chn iques and h igh energy e l e c t r o n impact spec t r o s copy (HEEIS) [E70, BW73, L 7 7 ] . Measurements of the o r b i t a l momentum d i s t r i b u t i o n s us i ng b i n a r y (e ,2e) spec t ro s copy have a l s o been made f o r He [HM&73, MU&74, DMW76] and H 2 [DM&75, WM&77, MC&81]. In the case of He, i t has been found tha t the re i s good agreement between the momentum d i s t r i b u t i o n s observed by d i f f e r e n t b i n a r y (e ,2e) g roups . However, the re i s a major d i s c r e p a n c y in the r e p o r t e d measurements f o r the momentum d i s t r i b u t i o n of H 2 1 p g « I n p a r t i c u l a r , the recent h i g h momentum r e s o l u t i o n measurement of H 2 1 b y M i g d a l l et al. [MC&81] i s i n s i g n i f i c a n t d isagreement w i th the e a r l i e r (e ,2e) d e t e r m i n a t i o n s by We igo ld et al. [WM&77] and Dey et al. [DM&75]. A f u r t h e r independent measurement at h i gh impact energy i s c a l l e d f o r and t h i s has p r o v i d e d s t i m u l u s f o r the p resen t work. It shou ld be noted tha t ve ry r e c e n t l y the one e l e c t r o n system H 1s has been i n v e s t i g a t e d by b i n a r y (e ,2e) spec t ro s copy [LW81] g i v i n g a r e s u l t in e x c e l l e n t agreement w i th the exact s o l u t i o n of the Sch rod inge r equa t i on f o r the hydrogen atom. No such measurement has been r epo r t ed to date u s i n g Compton s c a t t e r i n g . There are numerous wave func t ion c a l c u l a t i o n s f o r the two e l e c t r o n sys tems, e s p e c i a l l y in H 2 because of i t s fundamental s i g n i f i c a n c e in bonding t h e o r y . The t h e o r e t i c a l atomic 148 momentum d i s t r i b u t i o n of He Is c a l c u l a t e d from the Ha r t r ee-Fock wavefunct ion [CR74] has been shown to be in e x c e l l e n t agreement w i th the r epo r t ed momentum d i s t r i b u t i o n s [HM&73, LB83a] (chapter 4 ) . I t shou ld be noted tha t a l though the Har t ree-Fock wavefunct ion i s adequate f o r the He + ground s t a t e , a c o n f i g u r a t i o n a l i n t e r a c t i o n (CI) wave func t ion i s r e q u i r e d f o r the e x c i t e d n=2 t r a n s i t i o n s [MU&74, DMW76, CM&84]. In the case of H 2 , a c c u r a t e t h e o r e t i c a l d e t e r m i n a t i o n s of i t s e l e c t r o n i c ground s t a t e wavefunct ion have to r e s o r t to c o n f i g u r a t i o n a l i n t e r a c t i o n methods i n o rde r to r ep resen t adequa te l y s p a t i a l e l e c t r o n c o r r e l a t i o n e f f e c t s i n t h i s d i a tom i c mo lecu le [MWY60, FR61, DW66, LH67] . In t h i s r e s p e c t , H 2 p r o v i d e s the s i m p l e s t t e s t mo lecu le f o r a p p l y i n g CI methods. Weigo ld et al. [WM&77] have i n v e s t i g a t e d ground s t a t e c o r r e l a t i o n s in H 2 u s i ng b i na r y (e ,2e) s p e c t r o s c o p y . The r e s u l t s which a re a very s e n s i t i v e probe f o r the wave func t ion were compared w i th c a l c u l a t i o n s u s i n g a CI wave func t ion [MWY60]. In a d d i t i o n , H 2 p r o v i d e s a unique p ro to t ype cova l en t bond fo r the s tudy of chemica i b i n d i n g and bond f o r m a t i o n . E a r l i e r works by B e r l i n [B51] and Roux el al . [BD&60, RC62] have f o cussed upon the use of p o s i t i o n d e n s i t y d i f f e r e n c e d i s t r i b u t i o n s to i n v e s t i g a t e bonding phenomena in d i a t o m i c s . The s tudy of p o s i t i o n d e n s i t y and d e n s i t y d i f f e r e n c e maps to r e l a t e concep ts of b i n d i n g and a n t i b i n d i n g has been f u r t h e r extended by Bader et al . [BH67, BC68] through the use of f o r c e s a c t i n g upon the 149 n u c l e i and the Hellmann-Feynman theorem. (See a l s o Deb [D73] . ) A somewhat d i f f e r e n t approach i s the work of F e i nbe rg et a l . [FRM70, FR71] , which examines the behav iour of the k i n e t i c and p o t e n t i a l energy upon bond fo rmat ion and the r o l e of the V i r i a l theorem in bond f o r m a t i o n . In the p resen t work, the momentum d i s t r i b u t i o n of H 2 1 i s measured and i s compared w i th e x i s t i n g b i n a r y (e ,2e) measurements as we l l as w i th t h e o r e t i c a l d i s t r i b u t i o n s of d i f f e r e n t q u a l i t y r ang ing from minimal b a s i s , doub l e-ze t a b a s i s [SB72] , l i m i t e d Har t ree-Fock [FR61] , CI [MWY60] and the o p t i m i z e d va lence c o n f i g u r a t i o n [DW66]. D i r e c t compar ison of the momentum d i s t r i b u t i o n s of the two e l e c t r o n systems w i th those d e r i v e d from the expe r imen ta l Compton p r o f i l e s of Lee [L77] i s used to compare the two t echn iques and a s se s s t h e i r v i a b i l i t i e s s i n ce each depends on a d i f f e r e n t r e a c t i o n model . In a d d i t i o n , a d i f f e r e n t q u a l i t a t i v e view of co va l en t chemica l b i n d i n g and bond fo rmat ion of H 2 1 i s p resen ted . through the use of momentum d e n s i t y [CS&79, BC&82, CB82a, CB82b, MB&83] and momentum d e n s i t y d i f f e r e n c e maps [HC68, ET77] to extend momentum-space chemica l concep ts (chapter 2 ) . In p a r t i c u l a r the momentum d e n s i t y d i f f e r e n c e map f o r H 2 la y g i v e s a c l e a r p i c t u r e of the na ture of a s i n g l e c o v a l e n t bond in momentum-space. ! 150 5.2 GROUND-STATE BINDING ENERGY SPECTRA F i g u r e s 5.1 and 5.2 show the b i n d i n g energy s p e c t r a of the He 1s and H 2 1 e l e c t r o n s r e s p e c t i v e l y . The He b i n d i n g energy spectrum p r o v i d e s a d i r e c t measure of the i n s t r u m e n t a l energy r e s o l u t i o n f u n c t i o n . A Gaussian l i n e shape with a FWHM=1.6eV was found to p r o v i d e an e x c e l l e n t f i t to the data (see a l s o f i g u r e 4.1). With the determined i n s t r u m e n t a l energy r e s o l u t i o n f u n c t i o n and the experimental width of 2.35eV FWHM of the H 2 1o g peak ( f i g u r e 5.2), the Franck-Condon width of the H 2 ground i o n i c s t a t e i s determined to be 1.7eV. T h i s i s i n e x c e l l e n t agreement with the high r e s o l u t i o n PES works of Asbrink [A70] and Turner and May [TM66]. (The present study i s l i m i t e d only to the ground s t a t e s of the two e l e c t r o n systems because of the extremely weak i n t e n s i t i e s of the e x c i t e d i o n i c s t a t e s [HM&73, MU&74, DMW76, WM&77].) 5.3 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTIONS F i g u r e s 5.3 and 5.4 (a and b) show the momentum d i s t r i b u t i o n s of He 1s and H 2 1(7^ e l e c t r o n s r e s p e c t i v e l y . Since the momentum d e n s i t y (p(p)) i s r e l a t e d to the Compton p r o f i l e ( J ( p ) ) by the f o l l o w i n g e q u a t i o n : [5.1] p(p) = -(27TP)- 1 dJ(p)/dp, 151 34.0 Binding Energy (eV) F i g u r e 5.1 - B i nd ing energy spectrum f o r He Is i o n i z a t i o n at 0=0 ° . The s o l i d l i n e co r r e sponds to a Gauss i an l i n e s h a p e wi th a FWHM of 1.6eV (the i n s t r u m e n t a l energy r e s o l u t i o n ) . 1 52 F i g u r e 5.2 - B i nd ing energy spectrum f o r H 2 l a i o n i z a t i o n at 0=0 ° . The s o l i d l i n e co r r e sponds to a Gauss ian l i n e shape w i th a FWHM of 2.3eV. 1 53 the momentum d i s t r i b u t i o n s can be ob t a i ned by d i f f e r e n t i a t i n g the f i t t e d exper imenta l Compton p r o f i l e s 2 r epo r t ed by Lee u s i n g h i g h energy e l e c t r o n impact spec t roscopy (HEEIS) [ L77 ] . These d e r i v e d r e s u l t s are p re sen ted in both s p e c t r a as open squares i n f i g u r e s 5.3 and 5.4 . 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 c a l c u l a t e d from wave func t ions of d i f f e r e n t q u a l i t y are a l s o compared w i th the expe r imen ta l d a t a . 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 ( s p h e r i c a l l y averaged) have been c o n v o l u t e d wi th the momentum r e s o l u t i o n f u n c t i o n of the b i n a r y (e ,2e) spect rometer ( chapter 3) to f a c i l i t a t e d i r e c t compar ison w i th exper iment . For He, atomic wave func t ions of Ha r t r ee-Fock , doub l e-ze t a and s i n g l e - z e t a q u a l i t i e s of C lement i and R o e t t i [CR74] a re used . For H 2 , f i v e wave func t ions of q u a l i t y r ang ing from min imal b a s i s 3 to the extended Har t ree-Fock are used [HL&80, SB72, FR61, MWY60, DW66], A b r i e f d e s c r i p t i o n of these wave func t ions i s g i ven i n t a b l e 5 .1 . The method of c a l c u l a t i n g the s p h e r i c a l l y averaged momentum d i s t r i b u t i o n s f o r CI type wavefunct ions f o r H 2 1 o-g and H 2 1 ou o r b i t a l s has been d i s c u s s e d by Weigo ld et al . [WM&77]. A s i m i l a r p rocedure i s used i n the p resen t c a l c u l a t i o n . I t shou ld be noted tha t M i g d a l l et al . [MC&81] 2 In the h i g h energy e l e c t r o n impact s p e c t r o s c o p i c (HEEIS) work [L77] Lee l e a s t - s q u a r e d f i t t e d the expe r imen ta l Compton p r o f i l e s to a sum of d i f f e r e n t i a b l e po l ynomina l f u n c t i o n s . The r e s u l t i n g expe r imen ta l parameters (see t a b l e s 7 and 8 of r e f . [L77]) a re used to c a l c u l a t e the momentum d i s t r i b u t i o n s . 3 The min imal b a s i s i s the s t anda rd ST0-3G b a s i s of the Gauss ian 76 package [HL&80], 1 54 have a l s o compared t h e i r r e s u l t w i th a wide range of SCF wave func t i ons . The HEEIS data and the t h e o r e t i c a l d i s t r i b u t i o n s are a rea no rma l i z ed to the (e ,2e) data from p = 0 - 1 . 5 a o " 1 f o r both He and H 2 ( f i g u r e s 5.3 and 5.4b r e s p e c t i v e l y ) . C l e a r l y on l y shape compar ison i s p o s s i b l e in the p resen t r e l a t i v e expe r imen ta l measurements. In the atomic c a s e , i . e . the He 1s momentum d i s t r i b u t i o n ( f i g u r e 5 .3 ) , e x c e l l e n t agreement between the (e ,2e) data and the t h e o r e t i c a l Har t ree-Fock [CR74] d i s t r i b u t i o n i s obse r ved . The doub l e-ze t a c a l c u l a t i o n i s a lmost i n d i s t i n g u i s h a b l e from the Har t ree-Fock r e s u l t but the s i n g l e - z e t a wave func t ion i s inadequate in p r e d i c t i n g the expe r imen ta l d i s t r i b u t i o n . A l though there i s g e n e r a l l y good agreement between the HEEIS data and the (e ,2e) d a t a , the HEEIS d i s t r i b u t i o n appears to be very s l i g h t l y narrower than both the p resen t (e ,2e) r e s u l t and the Har t ree-Fock t h e o r e t i c a l d i s t r i b u t i o n . T h i s d i s c r e p a n c y may be the r e s u l t of a no i s y Compton p r o f i l e in the low momentum r e g i o n , which as p o i n t e d out by Lee [L77] r e q u i r e s a f i v e term po l ynom ia l f i t and some smoothing in order to produce a good f i t . Another p o s s i b i l i t y i s the breakdown of the impulse approx ima t ion [BS72, LS78 ] . Whi le there i s g e n e r a l l y very good agreement between the expe r imen ta l measurements of the He Is momentum d i s t r i b u t i o n and the Har t ree-Fock t h e o r y , i t shou ld be noted tha t l i m i t a t i o n s in the l a t t e r have been demonstrated by measurements of n=2 t r a n s i t i o n s in He i o n i z a t i o n by McCarthy 155 S P H E R I C A L L Y A V E R A G E D M O M E N T U M D I S T R I B U T I O N *c >» D 15 O to c cu CP > i 1 1 1 1 1 1 1 r He 1 s ( 24. 6 e V ) 0.0 0.5 1.0 1.5 2.0 P ( a - u . ) F i g u r e 5.3 - Atomic momentum d i s t r i b u t i o n f o r He Is o r b i t a l . The open squares (n) g i ve the momentum d i s t r i b u t i o n d e r i v e d from the Compton s c a t t e r i n g (HEEIS) da ta [L77], The HF and DZ [CR74] t h e o r e t i c a l d i s t r i b u t i o n s are i n d i s t i n g u i s h a b l e from each o t h e r . 156 et al . [MU&74] and Dixon et a l . [DMW76] us i ng b i na r y (e ,2e) s p e c t r o s c o p y . For the mo lecu la r momentum d i s t r i b u t i o n of H, l a ; a 9 s i m i l a r g e n e r a l l y good agreement i s a l s o found between the Compton s c a t t e r i n g (HEEIS) and (e ,2e) da ta ( f i g u r e s 5 . 4 a , b ) . The HEEIS d i s t r i b u t i o n i s as in the case of he l ium very s l i g h t l y narrower than the b i na r y (e ,2e) momentum d i s t r i b u t i o n . From f i g u r e 5.4a i t can be seen that the p resen t (e ,2e) data i s a l s o in e x c e l l e n t agreement w i th the e a r l i e r (e ,2e) measurements of Weigold and coworkers [DM&75, WM&77] wh i l e the re i s s i g n i f i c a n t d isagreement w i th the recent (e ,2e) measurement of M i g d a l l et a l . [MC&81]. The expe r imen ta l (e ,2e) d i s t r i b u t i o n s of Weigo ld et a l . [WM&77] and M i g d a l l el a l . [MC&81] a re p o i n t no rma l i z ed at t h e i r r e s p e c t i v e maxima to the p resen t (e ,2e) d i s t r i b u t i o n . (Area n o r m a l i z a t i o n of these da ta does not change the c o n c l u s i o n be low. ) The l a t t e r work [MC&81] g i v e s a s i g n i f i c a n t l y broader momentum d i s t r i b u t i o n ( p ^ / 2 m a x ~ ^ * ~ 1 ^ than tha t ob ta ined both i n the p resen t work and in the e a r l i e r [DM&75, WM&77] b i na r y (e ,2e) works, which a l l have p 1 / 2 m a x a ' ^ * " 1 * I t : c a n a l s o be seen tha t the momentum d i s t r i b u t i o n d e r i v e d from the d i f f e r e n t i a t e d Compton p r o f i l e [L77] (see the case of He above) i s in good agreement wi th both the p resen t and e a r l i e r [DM&75, WM&77] b ina r y (e ,2e) d e t e r m i n a t i o n s . There are s e v e r a l i n t e r e s t i n g f e a t u r e s a s s o c i a t e d wi th 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 s of the H 2 10g o r b i t a l . Relative Intensity (arbitrary units) a 3 io o c to 3 i-i cr fl> (0 n 3 •-• cr Ul cr c • C 3 * rt H " Q j I O >-*• 3 tn r> tn ri- o n 3 o cr oi n c n ac (-•• w M o o D a -» cn Q O uO 0) rh 3 o a oi i-i ^ cr cr •-«• ^ (D r r X 0) O tJ • i—1 I-I o <-•  C 3 i—• » Q) 9 r t rf 0> 01 a M 3 3 o o 3 M ID n 3 O r t C C M 3 0) 1 • • r-Le Th a. « to i a i o o o — o o 3 , , rk CO « n < o ro r o Relative Intensity (arbitrary units) o b 0.0 2.5 5.0 -i 1 1 r 7.5 10.0 T 1 — i r i s — I O c r > DO XI O m 2 £ -to cr> CD X ) O m Z.9 I 158 (See f i g u r e .5 .4b. ) F i r s t , whereas the min imal b a s i s (MBS) [HL&80] and the doub l e-ze t a (DZ) [SB72] wave func t ions both g i ve broader momentum d i s t r i b u t i o n s compared to the expe r imen ta l one, the v a r i a t i o n a l l y b e t t e r q u a l i t y CI wave func t ions of McLean et al . (HF) [MWY60] and of Das and Wahl (Ext-HF) [DW66] both p r e d i c t s l i g h t l y narrower d i s t r i b u t i o n s . The somewhat more p h y s i c a l s i n g l e c o n f i g u r a t i o n b a s i s , which c o n s i s t s of 1s , 2s and 2pa f u n c t i o n s , of Fraga and R a n s i l (Ltd-HF) [FR61] g i v e s the best agreement w i th the expe r imen ta l (e ,2e) d i s t r i b u t i o n . T h i s seems to support the a s s e r t i o n tha t a v a r i a t i o n a l l y s u p e r i o r wave func t ion does not necessarily p r o v i d e the best d e s c r i p t i o n of the momentum d i s t r i b u t i o n . Such o b s e r v a t i o n has a l s o been made in an e a r l i e r mo lecu l a r (e ,2e) study [BC&82]. F i n a l l y , t he re i s a sma l l d i s c r e p a n c y between the (e ,2e) data and the b e t t e r q u a l i t y t h e o r e t i c a l d i s t r i b u t i o n s in the range p > 1 . 5 a p " 1 . A s i m i l a r d i s c r e p a n c y has a l s o been observed in the noble gases [MW76a, LB83a] ( chapter 4 ) . T h i s i s p robab l y due to the breakdown of the PWIA in the h igh momentum reg ion in the (e ,2e) r e a c t i o n model and to the l ack of r e l a t i v i s t i c and e x p l i c i t r e l a x a t i o n c o r r e c t i o n s in the wave func t i ons . To summarize, the re i s g e n e r a l l y good agreement between the p resen t (e ,2e) momentum d i s t r i b u t i o n and the b e t t e r q u a l i t y 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 [FR61, MWY60, DW66] as w e l l as w i th the expe r imen ta l (e ,2e) da ta of Weigo ld 159 et al . [DM&75, WM&77] and the Compton p r o f i l e (HEEIS) data of Lee [L77] fo r the H 2 1 Og o r b i t a l . The p resen t compar ison between exper imenta l (e ,2e) data and the HEEIS data shows tha t the HEEIS d i s t r i b u t i o n s of Lee may be somewhat l i m i t e d by the accu racy of the po l ynomia l f i t to o b t a i n the momentum d i s t r i b u t i o n s from the expe r imen ta l Compton p r o f i l e s and tha t the (e ,2e) data i s g e n e r a l l y l i m i t e d by the v a l i d i t y of PWIA above p = 1 . 5 a 0 ~ 1 . There i s , n o n e t h e l e s s , very good agreement between the (e ,2e) da ta and the HEEIS data at l e a s t in the low momentum reg ion and i t i s c l e a r tha t the data are r easonab l y c o n s i s t e n t . S ince the Compton p r o f i l e exper iments can p r o v i d e the t o t a l momentum d i s t r i b u t i o n s and the b i na r y (e ,2e) method i s most s e n s i t i v e to i n d i v i d u a l va l ence o r b i t a l d i s t r i b u t i o n s , i t seems f e a s i b l e tha t by combin ing da ta from the two expe r imen t s , one may o b t a i n the momentum d i s t r i b u t i o n s of the co re e l e c t r o n s i n o ther f u t u r e s t u d i e s of more complex sys tems. 5.4 MOMENTUM-SPACE CHEMISTRY AND ORBITAL DENSITY TOPOGRAPHY 5.4.1 D e n s i t y Mapping Convention The wave func t ions used in the p resen t s tudy of d e n s i t y maps are the Har t ree-Fock atomic wave func t ions of C lement i and R o e t t i [CR74] f o r He and the o p t i m i z e d va lence c o n f i g u r a t i o n mo lecu l a r wave func t ions of Das and Wahl [DW66] 160 f o r H 2 . Both of these are c o n s i d e r e d to be " v a r i a t i o n a l l y s u p e r i o r " compared to the o ther wave func t ions (see t a b l e 5 .1 ) . The Das and Wahl wavefunct ion (Ext-HF) , a l though s l i g h t l y l e s s s a t i s f a c t o r y in p r e d i c t i n g the observed momentum d i s t r i b u t i o n , i s chosen f o r the p resen t t o p o g r a p h i c a l study of chemica l bonding phenomena because of i t s o p t i m i z a t i o n p rocedure [DW66] and i t s a v a i l a b i l i t y at s e v e r a l i n t e r n u c l e a r s e p a r a t i o n s . Un less o therw ise s t a t e d , the contour va lues of the d e n s i t y maps (both p o s i t i o n and momentum d e n s i t y ) a re l o g a r i t h m i c ; r ang ing from 0 .6 , 0 . 8 , 2, 4, 6, 8, 20, 40, 60 to 80% of the maximum d e n s i t y v a l u e . The contour v a lues fo r the d e n s i t y d i f f e r e n c e maps are a l s o l o g a r i t h m i c and are ± 0 . 6 , ±0 .8 , ±2, ±4, ±6, ±8, ±20, 40, 60 and 80% of the a b s o l u t e maximum d e n s i t y d i f f e r e n c e v a l u e . The nega t i v e contour l i n e s are i n d i c a t e d by dash l i n e s . The p lane of the contour map i s d e f i n e d d i r e c t i o n a l l y by two u n i t v e c t o r s . In the p resen t case t h e . (0 ,0 ,1 ) v e c t o r co r responds to the z a x i s which i s p a r a l l e l to the i n t e r n u c l e a r d i r e c t i o n and the (0 ,1 ,0 ) v e c to r co r responds to the y a x i s , the bond-perpend i cu l a r d i r e c t i o n . The range of the momentum and p o s i t i o n i s from -2.5 a . u . to 2.5 a . u . in both v e c to r d i r e c t i o n s . A long w i th the d e n s i t y maps, l i n e d e n s i t y p r o j e c t i o n p l o t s p a r a l l e l to the two axes are a l s o p resen ted in o rder to b e t t e r i l l u s t r a t e the gene ra l changes in shape and ampl i tude of the r e l a t i v e d e n s i t y p r o f i l e s a long the axes . The d o t t e d l i n e s a c r o s s the contour 161 Tab le 5.1 Comparison of t h e o r e t i c a l wave func t i ons . O r b i t a l Wavefunct ion Q u a l i t y T o t a l Energy ( ha r t r ee ) Reference He 1s SZ S i n g l e - z e t a -2.8476562 C l emen t i -R o e t t i [CR74] DZ Doub le-ze ta -2.8616726 C l emen t i -R o e t t i [CR74] HF Har t ree-Fock -2.8616799 C l emen t i -R o e t t i [CR74] MBS Min ima l b a s i s gauss i an se t -1.1167 Gauss ian 76 ST0-3G [HL&80] DZ Doub le-ze ta gauss i an set -1.1266 Snyder-Basch [SB72] Ltd-HF S i n g l e - c o n f i g u r a t i o n : 1s, 2s , 2 p a STO b a s i s -1.1335 F r a g a - R a n s i l [FR61] HF 5 c o n f i g u r a t i o n s C I , Har t ree-Fock q u a l i t y -1.1672 McLean et a l . [MWY60] Ext-HF Op t im ized va l ence c o n f i g u r a t i o n , e x t e n d e d Har t ree-Fock q u a l i t y -1.1698 Das-Wahl [DW77] 162 maps i n d i c a t e where the p r o j e c t i o n s are t a k e n ; f o r i n s t ance the l i n e d e n s i t y p l o t on the top of the contour map i s a l ong the " h o r i z o n t a l " (0 ,1 ,0 ) d o t t e d l i n e . Contour v a lues r e l a t i v e to the maximum d e n s i t y va lue ( l i n e a r s c a l e ) are i n d i c a t e d on the s i d e s of these p r o j e c t i o n p l o t s . The d i f f e r e n c e d e n s i t y i s d e f i n e d in the p resen t study to be the d i f f e r e n c e between the mo lecu l a r (or a tomic ) o r b i t a l d e n s i t y and the averaged d e n s i t y of the independent atoms at a p a r t i c u l a r i n t e r n u c l e a r geometry ( the independent atoms mode l , IAM), i . e . [5 .2a] Ap = 2p{MO} - z | J t o i n c 1 p(atom}, where p{atom} i s the one e l e c t r o n d e n s i t y of the independent atoms. [5 .2b] Ap{He1s} = 2p{He1s} - (p{H1s}+p{H1s}) R=0a o , [5 .2c ] Ap{H 21o } = 2p{H 2 1a } - (p{H1s}+p{H1s}) R = 1 . 4 a 0 , y y where R i s the ( e q u i l i b r i u m ) i n t e r n u c l e a r s e p a r a t i o n , which i s " v a r i e d i n the study of the bond fo rmat ion of H 2 . (See s e c t i o n 5 .4 .7 . ) The wave func t ion of the atomic hydrogen 1s o r b i t a l i s e x a c t . In the f o l l o w i n g s e c t i o n s , the F o u r i e r t r ans fo rm (FT) and V i r i a l p r o p e r t i e s ( chapter 2) w i l l be used to i n v e s t i g a t e some of the most prominent f e a t u r e s in the d e n s i t y f u n c t i o n s 163 in the two e l e c t r o n systems wi th s p e c i a l focus on chemica l b i n d i n g in momentum-space. The nature of the d e n s i t y r e d i s t r i b u t i o n accompanying chemica l bonding i s we l l i l l u s t r a t e d by c o n s i d e r i n g the fo rmat ion of a " v i r t u a l " bond i n the s imp le atomic system d i s c u s s e d below. 5.4.2 He Is Atomic O r b i t a l The he l ium 1s atomic wave func t ion (on the b a s i s of the electronic s t r u c t u r e a lone ) can be c o n s i d e r e d to be a s p e c i a l case of the two e l e c t r o n H 2 mo l e cu l a r wave func t i on , namely when the i n t e r n u c l e a r s e p a r a t i o n between the two H atoms, R, i s z e r o . F i g u r e 5.5 shows the d e n s i t y maps of the He 1s atomic wave func t ion and tha t of the independent atoms model (IAM) wave func t ion and t h e i r d i f f e r e n c e d e n s i t y (bond d e n s i t y ) maps in both momentum and p o s i t i o n - s p a c e . S e ve ra l i n t e r e s t i n g f e a t u r e s can be seen i n these maps. ( i ) Only the symmetry and the i n v e r s e s p a t i a l r e v e r s a l p r o p e r t i e s app l y i n the case of atoms. The shape of momentum d e n s i t i e s of He 1s and of the independent H 1s at R=0 are c l e a r l y i d e n t i c a l to tha t of t h e i r c o r r e s p o n d i n g p o s i t i o n d e n s i t i e s . The on l y d i f f e r e n c e i s , in a c co rd w i th the i n v e r s e s p a t i a l r e v e r s a l r e l a t i o n , an i n ve r se we igh t ing v a r i a t i o n i n the s p a t i a l ex ten t of these d e n s i t y f u n c t i o n s i n d i f f e r e n t spaces . T h i s i s i l l u s t r a t e d in the d e n s i t y maps 164 MOMENTUM DENSITY POSITION DENSITY -2.0 -1.0 0.0 1.0 2.0 0.5 1.0 -2.0 -1.0 0.0 1.0 2.0 0.5 1.0 MOMENTUM DENSITY POSITION DENSITY -2.0 -1.0 0.0 1.0 2.0 0.5 1.0 MOMENTUM DENSITY DIFFERENCE -2.0 -1.0 0.0 1.0 2.0 0.5 1.0 POSITION DENSITY DIFFERENCE -2.0 -1.0 0.0 1.0 2.0 -«.5 0.0 He -2 H -2.0 -10 0.0 to 2.0 O.S to F i g u r e 5 .5 - D e n s i t y con tour maps of the He Is o r b i t a l ( t o p ) , the independent atom model ( cen t re ) and d e n s i t y d i f f e r e n c e maps (bottom) in momentum-space (LHS) and p o s i t i o n space (RHS). Negat i ve d e n s i t y c o n t o u r s are r ep r e sen t ed by dashed l i n e s . 165 and p r o j e c t i o n p l o t s where a c o n t r a c t i o n of r-space o r b i t a l d e n s i t y from the IAM case ( f i g u r e 5 .5 , c e n t r e ) to He case ( f i g u r e 5 .5 , top) l eads to a concomi tant expans ion i n p-space d e n s i t y . S i m i l a r behav io r has a l s o been observed i n the r e s t of the nob le gas group [LB83a] ( chapter 4 ) . ( i i ) A c o n t r a c t i o n of the r-space d e n s i t y accompanies the fo rma t ion of the s t a b l e atom from independent atom d e n s i t i e s because of the a s s o c i a t e d dec rease i n p o t e n t i a l energy of the e l e c t r o n s . Such d e n s i t y r e d i s t r i b u t i o n i s best i l l u s t r a t e d by the r-space d e n s i t y d i f f e r e n c e map ( f i g u r e 5 .5 , bottom RHS). In p-space, however, the c o r r e s p o n d i n g t r a n s f e r of d e n s i t y i s from the p-space o r i g i n to the ou te r h i g h momentum r e g i o n . T h i s i s a d i r e c t consequence of the V i r i a l r e l a t i o n s i n c e the l ower ing of the p o t e n t i a l energy upon the fo rmat ion of a s t a b l e bound system (a v i r t u a l bond) co r r esponds to an i n c r e a s e i n the k i n e t i c energy . In the case of He Is the k i n e t i c energy i s r a i s e d by t r a n s f e r r i n g d e n s i t y from the p-space o r i g i n to the h i g h momentum r eg ion i s o t r o p i c a l l y . T h i s i s a m a n i f e s t a t i o n of what has been r e f e r r e d to as a "whi te h o l e " e f f e c t (see chapte r 2 ) , i n which the d e n s i t y i s pushed out from the p-space o r i g i n in o rder to a ch i e ve a h ighe r k i n e t i c energy and hence a more s t a b l e sys tem. T h i s v i r t u a l bond i n atomic He has the shape of a "ho l l ow b a l l " ( f i g u r e 5 .5 , bottom LHS) . 166 5.4.3 H 2 l o „ O r b i t a l g For m o l e c u l e s , the d e n s i t y f u n c t i o n p can be decomposed OC i n t o a q u a s i - c l a s s i c a l p a r t , p^ , which i s j u s t the sum of one c e n t r e (atomic) te rms , and an i n t e r a c t i o n p a r t , p 1 . OC I ( R e c a l l : p=p +p , see t a b l e 2 . 1 ) . Such a p a r t i t i o n of p i s u s e f u l in unders tand ing bonding phenomena s i n c e the q u a s i -c l a s s i c a l pa r t i s p o s i t i v e over a l l space , the key i n c l a s s i f y i n g an o r b i t a l in momentum-space as bond ing , a n t i b o n d i n g or nonbonding , h inges upon the i n t e r a c t i o n d e n s i t y . Such a d e n s i t y decompos i t i on i n momentum and p o s i t i o n space f o r the H 2 1 a g o r b i t a l i s shown in f i g u r e 5 .6 . The q u a s i - c l a s s i c a l d e n s i t y can be regarded as d e n s i t y due to IAM ( independent atoms model) wave func t ions except OC w i th m o d i f i e d o r b i t a l exponen ts . As such , p v (r ) i s seen ( in f i g u r e 5 .6 , top RHS) s imp ly as two p a r t i a l l y o v e r l a p p i n g s p h e r i c a l charge c l ouds c o n c e n t r a t e d around the two H n u c l e i s epa ra t ed by the e q u i l i b r i u m s e p a r a t i o n of 1 . 4 a 0 . The sma l l v a l l e y seen in the b o n d - p a r a l l e l p r o j e c t i o n p l o t i s caused by summing the t a i l s of the two independent atomic d e n s i t i e s . As expec ted f o r a bonding o r b i t a l , p*(r) ( f i g u r e 5 .6 , c en t r e RHS) shows a s i g n i f i c a n t a ccumu la t i on of charge ( p o s i t i v e o ve r l ap ) i n the i n t e r n u c l e a r b o n d - p a r a l l e l r e g i o n . The independent nature of e l e c t r o n s in the q u a s i - c l a s s i c a l d e n s i t y f u n c t i o n i s perhaps bes t i l l u s t r a t e d i n p-space ( f i g u r e 5 .6 , top LHS) . The F o u r i e r t r a n s f o r m of two independent r-space s-type atomic f u n c t i o n s sepa ra ted by R 167 MOMENTUM DENSITY "*9 — I f " I — 1 1 1— I ™ ' ' (0.1.0) • > -2.0 -1.0 0.0 1.0 2.0 0.5 1.0 MOMENTUM DENSITY ' (0.1.0) MOMENTUM DENSITY • (o.u) j -2J> -10 OJ) to 2D 0-5 to F i g u r e 5.6 - Contour maps of ( t o p ) , I n t e r a c t i o n d e n s i t y d e n s i t y (bottom) f o r H 2 l a and p o s i t i o n space (RHS). POSITION DENSITY , C T 9 ^ ^ ^ ^ ^ -2.0 -1.0 0.0 1.0 2.0 0.5 1.0 POSITION DENSITY l C T 9 p' ^ ^ ^ ^ ^ ^ -2.0 -1.0 0.0 1.0 2 0 0.5 1.0 POSITION DENSITY H 2  , C T 9 -2.0 -tO OJ) VO 2.0 0.5 10 Q u a s i - C l a s s i c a l d e n s i t y ( cen t re ) and the mo lecu l a r in momentum-space (LHS) 168 g i v e s two p-space s-type atomic f u n c t i o n s on top of each o ther in momentum-space to genera te the i s o t r o p i c d e n s i t y d i s t r i b u t i o n as shown (see chap te r 2 ) . The i n t e r n u c l e a r geometry i n f o r m a t i o n i s absent in the p-space q u a s i - c l a s s i c a l d e n s i t y and can on ly be r ecove red by c o n s i d e r i n g the p-space i n t e r a c t i o n ( i n t e r f e r e n c e ) d e n s i t y (see f i g u r e 5 .6 , c en t r e LHS and a more complete d i s c u s s i o n l a t e r i n v o l v i n g f i g u r e 5 .11 ) . T h i s i n t e r a c t i o n momentum d e n s i t y shows an e l l i p s o i d a l d e n s i t y accumu la t ion e l onga t ed i n the bond-p e r p e n d i c u l a r d i r e c t i o n . The i n t e r f e r e n c e e f f e c t s t h e r e f o r e r e ve r se the bonding d e n s i t y c o n c e n t r a t i o n from the l o n g i t u d i n a l to the t r a n s v e r s e d i r e c t i o n i n p-space . T h i s i n t e r a c t i o n d e n s i t y seems to dominate the overall d e n s i t y appearance ( f i g u r e 5.6, bottom LHS) . F i g u r e 5.7 shows the d e n s i t y and d e n s i t y d i f f e r e n c e maps of the H 2 1tfg o r b i t a l in r-space and p-space. The IAM p-space d e n s i t y ( f i g u r e 5 .7 , c e n t r e LHS) i s , of c o u r s e , i d e n t i c a l . t o tha t of f i g u r e 5 .5 , c en t r e LHS. In a d d i t i o n t o the symmetry and i n v e r s e s p a t i a l r e v e r s a l FT p r o p e r t i e s ment ioned e a r l i e r f o r the atomic c a s e , the d e n s i t y d i r e c t i o n a l r e v e r s a l p rope r t y i s c l e a r l y observed f o r the momentum and p o s i t i o n d e n s i t i e s ( f i g u r e 5 .7 , t o p ) . In t h i s c a s e , c o n c e n t r a t i o n of d e n s i t y in the l o n g i t u d i n a l d i r e c t i o n in r-space co r responds to d e n s i t y c o n c e n t r a t i o n i n the t r a n s v e r s e d i r e c t i o n in p-space . T h i s d e n s i t y r e v e r s a l p r o p e r t y a r i s e s from c o n s t r u c t i v e i n t e r f e r e n c e of the FT 169 F i g u r e 5.7 - D e n s i t y con tour maps of the H 2 l a o r b i t a l ( t o p ) , the independent atom model ( cent re ) and d e n s i t y d i f f e r e n c e maps (bottom) in momentum-space (LHS) and p o s i t i o n space (RHS). Nega t i ve d e n s i t y con tou r s are r ep resen ted by dashed l i n e s . 170 waves w i th the d e n s i t y l obes l o c a t e d around the atomic c e n t r e s . The d e n s i t y d i f f e r e n c e maps ( f i g u r e 5 .7 , bottom) g i ve the quantum-mechanical , p i c t u r e s of a co va l en t sigma chemica l bond in p o s i t i o n and momentum space . A chemica l bond in r-space can be thought of as an accumu la t i on of d e n s i t y ( f r a c t i o n a l charge) i n the i n t e r n u c l e a r r e g i o n . The amount of d e n s i t y t r a n s f e r r e d i n t o the r-space b i n d i n g r eg ion p r o v i d e s an a t t r a c t i v e e l e c t r o n i c - n u c l e a r f o r c e to coun te rba l ance the r e p u l s i v e n u c l e a r - n u c l e a r r e p u l s i o n [BH67, BC68, D73] fo r b i n d i n g the mo lecu le t o g e t h e r . (Zero net f o r c e i s o b t a i n e d at the e q u i l i b r i u m i n t e r n u c l e a r s e p a r a t i o n , R=1 .4a 0 . ) T h i s i s c l e a r l y i l l u s t r a t e d in the p o s i t i o n d e n s i t y d i f f e r e n c e map ( f i g u r e 5 .7 , bottom RHS) in which d e n s i t y i s t r a n s f e r r e d from the r eg ion o u t s i d e the atoms (nega t i ve d i f f e r e n c e ) i n t o the r eg i on between them ( p o s i t i v e d i f f e r e n c e ) . ' A complementary view of a chemica l bond i s shown by the momentum-space d e n s i t y d i f f e r e n c e map ( f i g u r e 5 .7 , bottom LHS) . In p-space , the chemica l bond can be d e s c r i b e d p h y s i c a l l y by "a bone in a donut " p i c t u r e . A sigma chemica l bond i n p-space i s r ep re sen t ed by a r e d i s t r i b u t i o n of the d e n s i t y from the b o n d - p a r a l l e l h i g h momentum r eg ion (the "bone " pa r t ) i n t o a bond-pe rpend i cu l a r annu la r h i g h momentum r e g i o n (the " d o n u t " ) . • Note tha t the d e n s i t y i s c y l i n d r i c a l l y symmetric about the bond a x i s d i r e c t i o n , i . e . the (0 ,0 ,1 ) 171 a x i s . T h i s d e n s i t y r e d i s t r i b u t i o n in p-space can aga in be e x p l a i n e d q u a l i t a t i v e l y us ing the V i r i a l r e l a t i o n . I f the r-space d e n s i t y maps and t h e i r a s s o c i a t e d p r o j e c t i o n p l o t s of the 1Og o r b i t a l are compared w i th those of the independent atoms, i t i s found tha t the mo lecu l a r o r b i t a l i s i n f a c t s l i g h t l y c o n t r a c t e d , which i s the r e s u l t of the l a r g e r o r b i t a l exponents . (Th i s a l s o accounts fo r the nega t i ve t a i l seen in the p r o j e c t i o n p l o t s of the r-space d i f f e r e n c e map in the b o n d - p a r a l l e l d i r e c t i o n . ) T h i s c o n t r a c t i o n w i l l lower the p o t e n t i a l energy s u f f i c i e n t l y so tha t subsequent l o c a l i z a t i o n of charge i n t o the r-space b i n d i n g r eg i on to ba lance the nuc l ea r r e p u l s i o n f o r c e (a p rocess which makes the p o t e n t i a l energy l e s s nega t i ve ) w i l l not des t roy the o v e r a l l s t a b i l i t y of the sys tem. The i n i t i a l c o n t r a c t i o n i n r-space , which drops the p o t e n t i a l energy , co r responds to a d e n s i t y t r a n s f e r p rocess in p-space which w i l l i n c r e a s e the k i n e t i c energy a c c o r d i n g l y . The l o c a l i z a t i o n s t ep in r-space w i l l cause a decrease i n the b o n d - p a r a l l e l component of the k i n e t i c energy s i n ce the " f i l l i n g i n " of the b i n d i n g r eg ion lowers the g r a d i e n t of the wave func t i on . T h i s decrease t h e r e f o r e l eads to a c o r r e s p o n d i n g dec rease in the b o n d - p a r a l l e l component in <p> and hence in the momentum d e n s i t y in the b o n d - p a r a l l e l d i r e c t i o n . The i n c r e a s e in k i n e t i c energy must t h e r e f o r e r e l y upon the bond-perpend i cu l a r components. The r e s u l t i s a decrease i n the b o n d - p a r a l l e l p-space d e n s i t y (negat i ve d i f f e r e n c e , see f i g u r e 5 .7 , bottom LHS) to r a i s e the 172 t r a n s v e r s e p-space d e n s i t y ( p o s i t i v e d i f f e r e n c e ) . The "whi te h o l e " e f f e c t , i . e . the d e p l e t i o n of the d e n s i t y at the p-space o r i g i n , can a l s o be unders tood as f o l l o w s . I t shou ld be noted tha t in r-space the t r a n s f e r of d e n s i t y i s from the end r eg ions of the system (where the e l e c t r o n s expe r i ence weaker p o t e n t i a l ) i n t o the i n t e r n u c l e a r r e g i o n . T h i s m i g r a t i o n of charge d e n s i t y in p o s i t i o n - s p a c e t h e r e f o r e co r r esponds to the removal of d e n s i t y near the p-space o r i g i n . 5.4.4 H 2 l o u O r b i t a l F i g u r e 5.8 shows the q u a s i - c l a s s i c a l and i n t e r a c t i o n d e n s i t i e s of the H 2 Io o r b i t a l i n both momentum and p o s i t i o n space . The q u a s i - c l a s s i c a l d e n s i t i e s ( f i g u r e 5.8, top) a re s i m i l a r to those in f i g u r e 5 .6 . As e x p e c t e d , the d i s t i n c t i o n between a bonding and an a n t i b o n d i n g o r b i t a l i s determined by the i n t e r a c t i o n d e n s i t i e s ( f i g u r e s 5.6 and 5.8, c e n t r e ) . Whereas the symmetric f u n c t i o n s of the bonding o r b i t a l i n r-space produce charge accumula t ion between n u c l e i ( p o s i t i v e o v e r l a p ) , the an t i s ymmet r i c f u n c t i o n s of the a n t i b o n d i n g o r b i t a l g i v e r i s e to charge d e p l e t i o n in the b i n d i n g r eg ion (nega t i ve ove r l ap ) and charge c o n c e n t r a t i o n in the a n t i b i n d i n g r eg ion o u t s i d e the n u c l e i . The c o r r e s p o n d i n g i n t e r a c t i o n momentum d e n s i t y of the a n t i b o n d i n g o r b i t a l ( f i g u r e 5 .8 , c en t r e LHS) , which i s ob ta ined from d e s t r u c t i v e 173 MOMENTUM DENSITY POSITION DENSITY -2.0 -1.0 0.0 1.0 2.0 0.5 1.0 -2.0 -1.0 0.0 1.0 2.0 0.5 1.0 MOMENTUM DENSITY POSITION DENSITY 9U l-1-1+l--i--l^-<-h-l->-+-v-]J+-l--J--2.0 -1.0 DO 1.0 2.0 -0.5 0.0 -2.0 -1.0 0.0 1.0 2.0 -0.5 0.0 MOMENTUM DENSITY POSITION DENSITY F i g u r e 5.8 - Contour maps of Q u a s i - C l a s s i c a l d e n s i t y ( t o p ) , I n t e r a c t i o n d e n s i t y ( cen t re ) and the mo lecu l a r d e n s i t y (bottom) f o r H 2 1o i n momentum-space (LHS) and p o s i t i o n space (RHS). 174 i n t e r f e r e n c e of FT waves wi th the an t i s ymmet r i c d e n s i t y l o b e s , has a somewhat s i m i l a r appearance to the r-space i n t e r a c t i o n d e n s i t y ( f i g u r e 5.8, c en t r e RHS). D e n s i t y i s removed from the p-space o r i g i n to the a n t i b i n d i n g bond-OC I p a r a l l e l h i gh momentum r e g i o n . The sums of p* and p in both spaces g i ve the o v e r a l l d e n s i t i e s ( f i g u r e 5 .8 , bottom) of the a n t i b o n d i n g o r b i t a l , which has a s i m i l a r appearance to an atomic p o r b i t a l ( chapter 4 ) . The o v e r a l l d e n s i t i e s are c l e a r l y seen to obey the symmetry and i n v e r s e s p a t i a l r e v e r s a l FT r e 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 maps of the 1 o r b i t a l a re shown in f i g u r e 5.9 (bot tom) . The independent atoms model d e n s i t i e s ( f i g u r e 5 .9 , c en t r e ) a r e , of c o u r s e , i d e n t i c a l to those in f i g u r e 5 .7 . In r-space , an " a n t i b o n d " i s c h a r a c t e r i z e d by a d e p l e t i o n of charge in the i n t e r n u c l e a r b i n d i n g r e g i o n and an accumula t ion i n the a n t i b i n d i n g r eg i on at the ends of the m o l e c u l e . On the o ther hand, the a n t i b i n d i n g r eg ion in p-space i s the b o n d - p a r a l l e l h i gh momentum r e g i o n ( f i g u r e 5 .9 , bottom LHS) and the b i n d i n g r eg i on ( f i g u r e 5 .7 , bottom LHS) i s the bond-pe rpend i cu l a r h i g h momentum r e g i o n , as d e f i n e d above . I t i s a l s o apparent why the 1 a y o r b i t a l i s a n t i b i n d i n g . In p-space , the t r a n s f e r of d e n s i t y from the p-space o r i g i n i n t o the h i g h momentum b o n d - p a r a l l e l r eg ion co r r e sponds i n r-space to a charge t r a n s f e r from the low p o t e n t i a l r eg i on to the h igh p o t e n t i a l ( i . e . more nega t i v e V) r eg ion (the V i r i a l 175 MOMENTUM DENSITY POSITION DENSITY MOMENTUM DENSITY POSITION DENSITY MOMENTUM DENSITY DIFFERENCE POSITION DENSITY DIFFERENCE F i g u r e 5.9 - D e n s i t y contour maps of the H 2 l a o r b i t a l ( t o p ) , the independent atom model ( cent re ) and d e n s i t y d i f f e r e n c e maps (bottom) in momentum-space (LHS) and p o s i t i o n space (RHS). Negat i ve d e n s i t y con tou r s are r ep resen ted by dashed l i n e s . 176 r e l a t i o n ) . T h i s r e s u l t s in an o v e r a l l p o l a r i z a t i o n of charge around the atomic c e n t r e s at the expense of charge between the n u c l e i ( f i g u r e 5 .9 , bottom RHS). The n u c l e a r - n u c l e a r r e p u l s i v e f o r c e i s t h e r e f o r e g r e a t l y enhanced because of the poor e l e c t r o n i c s h i e l d i n g of the two n u c l e i . The r e s u l t i s an uns t ab l e sys tem. 5.4.5 M o l e c u l a r D e n s i t y D i r e c t i o n a l Re ve r s a l In H 2 F i g u r e s 5.10 and 5.11 g i ve a t h r e e - d i m e n s i o n a l v i s u a l i z a t i o n " of the H 2 1 a g and 1 o r b i t a l s r e s p e c t i v e l y in both momentum and p o s i t i o n space . The range of the momentum and p o s i t i o n of one s i d e of the " d e n s i t y cube" (which c o n t a i n s the d e n s i t y su r f a ce ) i s -2.5 to 2.5 a . u . except f o r the f i n a l s u r f a c e s on the r i g h t hand s i de of f i g u r e s 5.10 and 5.11 which are shown at 33% r e d u c t i o n ( i . e . the range has been t r i p l e d ) . The i n d i c a t e d d e n s i t y v a l ues of the ( cons tan t ) d e n s i t y s u r f a c e s c o r r e s p o n d , from l e f t to r i g h t , to 80, 20, 2, 0.2 and 0.0008% of the maximum d e n s i t y . Two of the th ree o r thogona l v e c t o r s which d e f i n e the d e n s i t y cube are a l s o shown and co r respond to the same v e c t o r d i r e c t i o n s as those of the contour maps (see f i g u r e s 5 .6-5 .9 ) . For the H 2 1o g o r b i t a l , the mo lecu l a r d e n s i t y r e v e r s a l p r o p e r t y i s c l e a r l y i l l u s t r a t e d by the s e r i e s of cons tan t * D e t a i l s of the t h r ee-d imens i ona l o r b i t a l v i s u a l i z a t i o n procedure are g i ven i n r e f . [W74]. (See a l s o chapte r 10.) MOMENTUM DENSITY 1crf p= 5.5 E-1 p= 1.4 E-1 p= 1.4 E-2 *p= 1.4 E-3 o o o o o o (0,1,0) (0.1,0) (0.1,0) (0,1,0) POSITION DENSITY 1cr p= 5.5 E-6 o o 4-3 (0,1,0) F i g u r e 5.10 - Three-dimensional d e n s i t y s u r f a c e v i s u a l i z a t i o n of H 2 1a o r b i t a l i n momentum-space (top) and p o s i t i o n space (bottom). The d e n s i t y values of the su r f a c e s correspond to 80, 20, 2, 0.2 and 0.0008% of the maximum d e n s i t y v a l u e . M O M E N T U M D E N S I T Y l e x , P O S I T I O N D E N S I T Y 1 c r u p = 3.0 E-1 p = 7.6 E-2 p = 7.6 E-3 p= 7.6 E-4 T . + (0.1.0) o o o o o o I I (0.1.0) (0.1.0) (0.1.0) p= 3.0 E-6 o o (0,1.0) F i g u r e 5.11 - Three-dimensional d e n s i t y s u r f a c e v i s u a l i z a t i o n of H 2 1a o r b i t a l in momentum-space (top) and p o s i t i o n space (bottom). The d e n s i t y v a l ues of the su r f a ces cor respond to 80, 20, 2, 0.2 and 0.0008% of the maximum d e n s i t y v a l u e . 179 d e n s i t y su r f a ce p l o t s , ( f i g u r e 5 .10 ) . I t can be seen that the r-space d e n s i t y of the bonding o r b i t a l i s o r i e n t e d e l l i p s o i d a l l y w i th the major a x i s in the i n t e r n u c l e a r (the (0 ,0 ,1 ) v e c to r ) d i r e c t i o n , whi le t he•co r r e spond ing p-space d e n s i t y i s d i r e c t e d p e r p e n d i c u l a r to the i n t e r n u c l e a r a x i s . I t can a l s o be seen i n r-space ( f i g u r e 5 .10, bottom) tha t the shape of the d e n s i t y s u r f a c e changes from e l l i p s o i d a l to near s p h e r i c a l as the d e n s i t y va lue becomes s m a l l e r ( i . e . the d e n s i t y boundary s u r f a c e becomes l a r g e r ) . In p-space, on the o ther hand, the d e n s i t y changes from s p h e r i c a l (h igh d e n s i t y su r f a ce ) to e l l i p s o i d a l (low d e n s i t y s u r f a c e ) . T h i s i s in a c c o r d , of c o u r s e , w i th the i n ve r se we igh t ing behav io r (the i n v e r s e s p a t i a l r e v e r s a l p rope r t y ) of the d e n s i t y f u n c t i o n in momentum and p o s i t i o n space as d i s c u s s e d above. There i s , however, no d i r e c t one-to-one cor respondence between the d e n s i t y su r f a ce in r-space and tha t in p-space; namely, a p a r t i c u l a r d e n s i t y s u r f a c e in r-space cannot be a s s o c i a t e d w i th a c o r r e s p o n d i n g one in p-space and v i c e v e r s a . F i g u r e 5.11 g i v e s the t h r ee-d imens iona l v i s u a l i z a t i o n of the H 2 1 <?u o r b i t a l . I t can be seen , aga in tha t the p-space d e n s i t y l obes are s i g n i f i c a n t l y e l onga ted in the bond-perpend i cu l a r d i r e c t i o n as the boundary s u r f a c e i s e n l a r g e d . T h i s e l l i p s o i d a l growth i s a l o g i c a l consequence of the d e n s i t y o s c i l l a t i o n s (see below) which appear in the b o n d - p a r a l l e l d i r e c t i o n f o r both the 1a and 1 o y o r b i t a l s . Another important FT f e a tu r e as demonstrated in the 180 s u r f a c e p l o t s fo r the 1o and 1a o r b i t a l s i s that the g u m a n i f e s t a t i o n s of the n u c l e i are d i f f e r e n t in p o s i t i o n - s p a c e and in momentum-space. In r-space the nuc l e a r p o s i t i o n s (shown by c r o s s e s (+) on the p o s i t i o n d e n s i t y s u r f a c e p l o t s in f i g u r e s 5.10 and 5 .11, lower LHS) can be i d e n t i f i e d d i r e c t l y by the h ighe r d e n s i t y s u r f a c e . In p-space, the nuc l ea r s e p a r a t i o n i s man i f e s t ed by d e n s i t y o s c i l l a t i o n s g i v i n g l obes at l a r g e momentum in the b o n d - p a r a l l e l d i r e c t i o n in the low d e n s i t y s u r f a c e p l o t s (see upper RHS of f i g u r e s 5.10 and 5.11 and in the d e t a i l e d d i s c u s s i o n s be low) . The th r ee-d imens iona l v i s u a l i z a t i o n s of the o r b i t a l d e n s i t y are t h e r e f o r e very u s e f u l in the i l l u s t r a t i o n of all of the FT p r o p e r t i e s d i s c u s s e d above. 5 .4.6 Mo l e cu l a r D e n s i t y O s c i l l a t i o n In H 2 Extended range momentum and p o s i t i o n d e n s i t y maps of H 2 1 a g and 1 o u o r b i t a l s a re shown in f i g u r e 5 .12. The d e n s i t y con tou r s l o g a r i t h m i c a l l y span s i x decades s t a r t i n g from 0 .9% , 0 .5% , 0 .1% , 0 .09%, 0 .05%, 0 .01%, e t c . down to 0.000001% of the maximum contour v a l ues i n o rde r to emphasize the weak modu la t ion f e a t u r e s a r i s i n g from the i n t e r f e r e n c e terms in the momentum-space d e n s i t y (see chapte r 2 ) . The range i s from -12.5 to 12.5 a . u . in both v e c to r d i r e c t i o n s . Note a l s o the l o g a r i t h m i c s c a l e (9 decades) in the a x i s p r o j e c t i o n p l o t s . The extended range momentum d e n s i t y maps c l e a r l y 181 MOMENTUM DENSITY MOMENTUM DENSITY POSITION DENSITY POSITION DENSITY F i g u r e 5.12 - Extended range d e n s i t y con tour maps of H 2 1a (LHS) and 1a (RHS) i n momentum-space (top) and p o i i t i o n space (Bottom). The contour values l o g a r i t h m i c a l l y span s i x decades s t a r t i n g from 0 . 9 , 0 . 5 , 0 . 1 , e t c . down to 0.000001% of the maximum d e n s i t y . 182 i l l u s t r a t e the e f f e c t of ampl i tude modu la t ions a r i s i n g from 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 s of the FT waves w i th the r-space d e n s i t y l obes around each of the atomic c e n t r e s [CB82a] , I t can a l s o be seen from the b o n d - p a r a l l e l a x i s p r o j e c t i o n p l o t s tha t the bonding o r b i t a l momentum d e n s i t y o s c i l l a t e s c o s i n u s o i d a l l y wh i le the a n t i b o n d i n g o r b i t a l d e n s i t y o s c i l l a t e s s i n u s o i d a l l y , i . e . 90° out of phase , due to o p p o s i t e s i g n s of the MO c o e f f i c i e n t s . The obv ious cons tan t p e r i o d i c i t y in both of these d e n s i t y o s c i l l a t i o n s can be ob ta ined from the b o n d - p a r a l l e l momentum d e n s i t y p r o j e c t i o n p l o t s and i s found to be 4 . 4 9 a 0 " 1 (= 27r/bond l e n g t h ) . 5 Of course the mo lecu l a r geometry i s directly shown in the p o s i t i o n - s p a c e maps ( f i g u r e 5 .12, bottom) where the nuc l e a r p o s i t i o n s are shown by c r o s s e s (+). I t i s noteworthy tha t the momentum d e n s i t y o s c i l l a t i o n e f f e c t i n the 1 a y o r b i t a l i s more severe and more extended than tha t of the 1a o r b i t a l . T h i s i s not s u r p r i s i n g s i n c e the o v e r l a p y of FT waves w i th two we l l d e f i n e d d e n s i t y l obes in the l a o r b i t a l shou ld be more complete than the c o r r e s p o n d i n g o v e r l a p w i th the p a r t i a l l y merged lobes i n the 1o g o r b i t a l (see f i g u r e 5.12, bottom and f i g u r e 5 .7 , upper RHS). T h i s i s because the s i g n i f i c a n t bonding d e n s i t y in the r eg i on between 5 In H 2 the e q u i l i b r i u m i n t e r n u c l e a r s e p a r a t i o n (bond l eng th ) i s 1 . 4 a 0 . T h i s o s c i l l a t i n g phenomenon has a l s o been r e f e r r e d to as "bond o s c i l l a t i o n " [ ET77 ] . The r e l a t i o n between momentum d e n s i t y and p o s i t i o n d e n s i t y i s ana logous to tha t between an X-ray d i f f r a c t i o n p a t t e r n and the c o r r e s p o n d i n g c r y s t a l s t r u c t u r e . 183 the l obes l eads to i n t e r f e r e n c e c a n c e l l a t i o n . I t i s c l e a r tha t i t i s u n l i k e l y tha t these d e n s i t y o s c i l l a t i o n s c o u l d be observed e x p e r i m e n t a l l y in the s p h e r i c a l l y averaged momentum d i s t r i b u t i o n s of H 2 because of the very sma l l amp l i tudes of the m o d u l a t i o n s . (Note tha t even f o r o r i e n t e d H 2 the f i r s t o s c i l l a t i o n i s a l r e a d y at l e a s t four decades lower in i n t e n s i t y ) . S i nce the b i na r y (e ,2e) exper iment i s most u s e f u l and s e n s i t i v e i n the c h e m i c a l l y i n t e r e s t i n g low momentum ( l a rge p o s i t i o n ) r e g i o n , the o b s e r v a t i o n of the d e n s i t y o s c i l l a t i o n s i s of marg ina l i n t e r e s t i n the study of chemica l bonding and r e a c t i v i t y . However the l a r g e momentum r e g i o n i n which the d e n s i t y o s c i l l a t i o n s occur i s of importance in the s t r i n g e n t e v a l u a t i o n of the (e ,2e) r e a c t i o n theory of mo lecu les s i n c e even i n the case of atoms apparent d e v i a t i o n s from the PWIA have been observed i n momentum d e n s i t i e s above ==• 1-1.5 a 0 " 1 [MW76a, LB83a] ( chapter 4) u s i ng impact e n e r g i e s below 1200 eV. 5.4.7 Bond Format ion In H 2 At t h i s p o i n t , i t i s i n s t r u c t i v e to c o n s i d e r the b i n d i n g and a n t i b i n d i n g r e g i o n s in r-space and p-space f o r the s i n g l e c o v a l e n t sigma bond of the hydrogen m o l e c u l e . I t i s i n t e r e s t i n g to note tha t in momentum-space the fo rmat ion of a s t a b l e or uns t ab l e system ( e i t h e r atomic ( f i g u r e 5.5) or mo l e cu l a r ( f i g u r e 5.7)) r e s u l t s in d e p l e t i o n s of d e n s i t y in 184 the low momentum r eg ion around the p-space o r i g i n (the so-c a l l e d "whi te h o l e " ) . A d d i t i o n of d e n s i t y to the bond-p e r p e n d i c u l a r h igh momentum reg ion has the e f f e c t of i n c r e a s i n g the t o t a l k i n e t i c energy (and hence lower ing the t o t a l energy by the V i r i a l r e l a t i o n ) and causes the fo rmat ion of a s t a b l e chemica l bond. Accumula t ion of d e n s i t y in the b o n d - p a r a l l e l h igh momentum r eg ion i s l e s s e f f e c t i v e i n i n c r e a s i n g the k i n e t i c energy s i n c e the p a r a l l e l component i s sma l l e r than the p e r p e n d i c u l a r components [FRM70, FR71] . T h i s a l s o causes outward charge p o l a r i z a t i o n i n r-space ( in the i n t e r n u c l e a r d i r e c t i o n ) which i n c r e a s e s the nuc l e a r -nuc l ea r r e p u l s i o n f o r c e , l e a d i n g to an uns tab l e sys tem. In o rder to f u r t h e r i n v e s t i g a t e the fo rmat ion of a s t a b l e chemica l bond, p o s i t i o n and momentum d e n s i t y d i f f e r e n c e maps at a number of i n t e r n u c l e a r s e p a r a t i o n s are shown in f i g u r e 5 .13. The wavefunct ions used are those of Das and Wahl [DW66], which a re r e p o r t e d as a f u n c t i o n of i n t e r n u c l e a r s e p a r a t i o n . The i n t e r n u c l e a r s e p a r a t i o n s are 8, 6, 4, 2, 1.4 and 1a 0 r e s p e c t i v e l y and are so i n d i c a t e d on the maps. The ranges of a l l momentum d e n s i t y d i f f e r e n c e maps are the same (from - 2 . 5 a 0 ~ 1 to 2 . 5 a 0 " 1 ) in both d i r e c t i o n s . The ranges of the p o s i t i o n d e n s i t y d i f f e r e n c e maps are d i f f e r e n t depending on the nuc l ea r s e p a r a t i o n s . Be fore d i s c u s s i n g these d e n s i t y d i f f e r e n c e maps, one must keep i n mind that compar ison of the r e l a t i v e nega t i v e and p o s i t i v e amp l i tudes can on l y be made w i t h i n a s i n g l e d e n s i t y map and i t s 185 POSITION DENSITY DTFERENCE R.Bcfc POSITION DENSITY nrrcRtwcc -«.o -ao ao 3.0 6.0 -as ao MOMENTUM DENSITY DIFFERENCE - * . ' 6 ' « ' J:O -ts (O.U» = 6t POSITPW DENSITY DIFFERENCE 0.5 00 MOMENTUM DENSITY DIFFERENCE A A. *n O B. Q CN Q o o - •<a^^^--— > p T o S -2.0 -1.0 OX) 1.0 2D O51.0 c. Rc4o o ^ ^ ^ ^ -4.0 -2J0 00 2D4J0 -0.5 ao MOMENTUM DENSITY DIFFERENCE C. 10.1.0) > •2.0 -1.0 OD 1.0 2D OS ID -2D -1.0 OD 1.0 2D 051.0 POSITION DENSITY DIFFERENCE d. R=2 0 o .' • ''/ - -i - *»»\ \ I i i i i i POSITION DENSITY DirrERENCE -2D -1.0 OD ID JD O S 1.0 MOMENTUM DENSITY DIFFERENCE e. R = 1.4 POSITION DENSITY DITEERENCE -2D -ID OD ID 2D O S ID MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE F. -2D -ID OD ID 2D Oi ID -2D -ID OD ID 2D OJ ID , W W ; -2D -ID OD ID »D -OJ ao F i g u r e 5.13 - D e n s i t y d i f f e r e n c e contour maps i n p o s i t i o n and momentum space showing the dynamics of the fo rma t ion of H 2 1o o r b i t a l as a f u n c t i o n of the i n t e r n u c l e a r s e p a r a t i o n R. Nega t i ve d e n s i t y con tou r s are i n d i c a t e d by dashed l i n e s . 186 a s s o c i a t e d p r o j e c t i o n p l o t s . The a b s o l u t e maximum d e n s i t y va lue in each d i f f e r e n c e map i s i n d i c a t e d in t a b l e 5.2 in o rder to g i ve some idea of the g l o b a l change upon a t t a i n i n g the e q u i l i b r i u m s e p a r a t i o n . The most consp i cuous p a t t e r n seen in the s e r i e s of p o s i t i o n and momentum d e n s i t y d i f f e r e n c e maps i s the g radua l t r a n s f e r of d e n s i t y i n the r e s p e c t i v e space from i t s a n t i b i n d i n g r eg i on i n t o i t s b i n d i n g r eg i on as the i n t e r n u c l e a r s e p a r a t i o n approaches the e q u i l i b r i u m v a l u e . The r-space topograph i c sequence of bond fo rmat ion sugges ts tha t the mechanism of bond fo rmat ion can be viewed as o r b i t a l c o n t r a c t i o n accompanied by d e n s i t y l o c a l i z a t i o n i n t o the b i n d i n g r e g i o n . The o r b i t a l c o n t r a c t i o n i n r-space i s i l l u s t r a t e d by the bond a x i s p r o j e c t i o n p l o t s fo r R=8 and 6 a 0 ( f i g u r e s 5.13a and b, note the s c a l e change) where r e l a t i v e l y l a r g e d e n s i t y d e p l e t i o n around each of the atomic c e n t r e s i s c l e a r l y obse r ved . The d e n s i t y l o c a l i z a t i o n i n t o the b i n d i n g r eg i on can be seen from the p o s i t i o n d e n s i t y d i f f e r e n c e maps and t h e i r a s s o c i a t e d p r o j e c t i o n p l o t s f o r R=4, 2, and l . 4 a 0 ( f i g u r e s 5 .13c , d and e) where the re i s an i n c r e a s i n g l y s i g n i f i c a n t charge accumula t ion between the atomic c en t r e s as R d e c r e a s e s . T h i s i s a l s o i n d i c a t e d by the r e l a t i v e magnitudes of the p o s i t i v e and nega t i v e segments in the bond-p a r a l l e l p r o j e c t i o n p l o t s . In p-space, the gene r a l t r a n s f e r of d e n s i t y from the b o n d - p a r a l l e l h igh momentum ( a n t i b i n d i n g ) r eg i on i n t o the 187 Tab l e 5.2 Maximum a b s o l u t e v a lues of d e n s i t y d i f f e r e n c e fo r H 2 1a as a f u n c t i o n of R. " R Maximum d e n s i t y d i f f e r e n c e ( a . u . ) ( a 0 ) r-space p-space 1 0.284 -0.571 1.4 0.115 -0.251 2 0.049 0.252 4 -0.023 0.505 6 -0.006 0.222 8 -0.005 0.131 188 bond-pe rpend i cu l a r h i g h momentum (b ind ing ) r e g i o n i s c l e a r l y o b s e r v e d . I t i s i n t e r e s t i n g to f i n d tha t the i n i t i a l o r b i t a l c o n t r a c t i o n in r-space (note s c a l e change between f i g u r e s 5.13a and b) not on l y r e s u l t s in momentum d e n s i t y s p a t i a l expans ion ( f i g u r e s 5.13A and B) but a l s o causes o s c i l l a t i o n s of the d i f f e r e n c e d e n s i t y between the p o s i t i v e and nega t i ve l o b e s ( e s p e c i a l l y a l ong the bond a x i s , see f i g u r e 5.13B) . Such o s c i l l a t i o n must not be confused w i th the p r e v i o u s l y d i s c u s s e d d e n s i t y o s c i l l a t i o n e f f e c t which a r i s e s p u r e l y from F o u r i e r t r a n s f o r m a t i o n of the r-space LCAO-MO wave func t i on . The e f f e c t of r-space l o c a l i z a t i o n of d e n s i t y i n t o the r eg ion between the atomic c e n t r e s i s c l e a r l y i l l u s t r a t e d by the r-space a x i s p r o j e c t i o n p l o t s where the re i s a t r a n s f e r of d e n s i t y from the a n t i b i n d i n g r eg i on (nega t i ve d e n s i t y d i f f e r e n c e ) i n t o the b i n d i n g r eg i on ( p o s i t i v e d e n s i t y d i f f e r e n c e ) . In p-space t h i s co r responds to a l o s s of d e n s i t y i n the f r e e e l e c t r o n r eg i on around the p-space o r i g i n . See f i g u r e s 5.13B, C, D and E. As the atoms approach each other the nega t i ve l obes are seen to pene t r a t e deeper and deeper i n t o the p-space o r i g i n a l ong the bond-p a r a l l e l d i r e c t i o n u n t i l the e q u i l i b r i u m geometry i s reached where the t o t a l e f f e c t of c o n t r a c t i o n and l o c a l i z a t i o n g i v e s r i s e to the "bone in a donut " p i c t u r e of chemica l bonding in momentum-space d i s c u s s e d above . I t can be seen tha t the bond-pe rpend i cu l a r p-space p r o j e c t i o n p l o t changes drastically go ing from R=2a 0 ( f i g u r e 5.13D) to R=1.4a 0 189 ( f i g u r e 5.13E) where the d e n s i t y d i f f e r e n c e has changed s i g n from p o s i t i v e to nega t i v e around the p-space o r i g i n . Changes in t h i s r e g i o n w i l l be f u r t h e r d i s c u s s e d i n the f o l l o w i n g c h a p t e r . F u r t h e r c l o s e r approach of the atoms at R=1a 0 i n c r e a s e s the r e l a t i v e d e n s i t y i n the h i g h momentum b o n d - p a r a l l e l r eg i on ( f i g u r e 5.13F) at the expense of d e n s i t y in the bond-p e r p e n d i c u l a r r e g i o n . The c o r r e s p o n d i n g p o s i t i o n d e n s i t y d i f f e r e n c e map ( f i g u r e 5 .13f ) shows tha t charge accumu la t i on between atomic c e n t r e s becomes s a t u r a t e d , which causes an apparent c o n c e n t r a t i o n of charge around the atomic c e n t r e s . Such a p o l a r i z a t i o n e f f e c t i n c r e a s e s the 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 s and t h e r e f o r e l e ads to i n s t a b i l i t y . A f u r t h e r decrease in R w i l l e v e n t u a l l y fuse the two l obes toge the r (see b o n d - p a r a l l e l p r o j e c t i o n p l o t ) to form a he l i um atomic system ( f i g u r e 5 .5 ) . Such a change w i l l f u r t h e r i n c r ea se the d e n s i t y i n the bond p a r a l l e l h i g h momentum r e g i o n , l e a d i n g to the "ho l low b a l l " arrangement of the d e n s i t y d i f f e r e n c e of a s t a b l e he l i um atom in p-space . The ho l low b a l l i s aga in a s t a b l e system as d i s c u s s e d above . (See f i g u r e 5.5. ) 190 Chapter VI BOND DENSITY OF MOLECULAR HYDROGEN IN MOMENTUM SPACE 6.1 INTRODUCTION The ground s t a t e e l e c t r o n i c wavefunct ion of mo lecu l a r hydrogen has been the sub j e c t of many expe r imen ta l i n v e s t i g a t i o n s i n c l u d i n g both b i n a r y (e ,2e) spec t roscopy [WM&77, LB83b] and Compton s c a t t e r i n g [ L77 ] . In gene ra l Compton s c a t t e r i n g exper iments sample the momentum d e n s i t y due to a l l the e l e c t r o n s of the t a r g e t . B ina ry (e ,2e) spec t roscopy [MW76a], on the o the r hand, samples s e l e c t i v e l y the momentum d e n s i t y of i n d i v i d u a l o r b i t a l s and p r o v i d e s a d i r e c t and s e n s i t i v e expe r imen ta l e v a l u a t i o n of mo lecu l a r o r b i t a l wave func t i ons . In the case of H 2 r the t o t a l momentum d e n s i t y i s to a ve ry good approx imat ion tha t of the 1a y o r b i t a l . The g e n e r a l l y good agreement of the momentum d i s t r i b u t i o n of mo lecu l a r hydrogen observed by the two ^ d i f f e r e n t t e chn iques has been demonstra ted in the p r e ced ing chap te r (see a l s o r e f . [ LB83b] ) . The ground s t a t e wave func t ion of H 2 has a l s o been i n v e s t i g a t e d by many e l a b o r a t e t h e o r e t i c a l a b - i n i t i o c a l c u l a t i o n s . The hydrogen molecu le i s a l s o the s i m p l e s t t e s t molecu le f o r c o n f i g u r a t i o n a l i n t e r a c t i o n methods. Moreover , i t i n v o l v e s the s i m p l e s t ( cova l en t ) chemica l bond and i s thus s u i t a b l e 191 f o r the fundamental s t u d i e s of chemica l b i n d i n g and e l e c t r o n i c s t r u c t u r a l p r o p e r t i e s . E a r l i e r works by B e r l i n [B51] , Roux and C o r n i l l e [RC62] and Bader et al. [BH67, BC68] i n v o l v e d p o s i t i o n (charge) d e n s i t y d i f f e r e n c e ('bond d e n s i t y ) maps and the f o r c e concep t . Other works by Bader and P res ton [BP69] and by Fe inbe rg et al. [FRM70, FR71] , examined the behav iour of the k i n e t i c and p o t e n t i a l e n e r g i e s upon bond fo rmat ion and the r o l e of the V i r i a l theorem t h e r e i n . The bond d e n s i t y i n e i t h e r p o s i t i o n or momentum-space i s d e f i n e d (as in the p r e ced ing chap te r ) to be the d e n s i t y d i f f e r e n c e between the mo l e cu l a r d e n s i t y (p{H 2 1o-g}) and the d e n s i t y due to independent atoms (the independent atom model d e n s i t y , p{lAM}) at p o s i t i o n s c o r r e s p o n d i n g to the mo lecu l a r n u c l e a r geometry, i . e . [6 .1a ] Ap = 2p{H 2 Io J - p{IAM} at R, y [6 .1b ] p{lAM} =p{H 1s} + p{H -is} at R, where R i s the i n t e r n u c l e a r s e p a r a t i o n and p denotes the s i n g l e e l e c t r o n d e n s i t y . I t i s p o s s i b l e to o b t a i n the " e x p e r i m e n t a l " ( s p h e r i c a l l y averaged) momentum-space bond d e n s i t y u s i n g the expe r imen ta l momentum d i s t r i b u t i o n s of the H 2 1Og o r b i t a l and e i t h e r the expe r imen ta l or the exact t h e o r e t i c a l momentum d i s t r i b u t i o n of the 1s o r b i t a l f o r the H atom. The momentum d i s t r i b u t i o n f o r the 1s o r b i t a l of atomic hydrogen has been determined r e c e n t l y by Lohmann and Weigo ld 192 [LW81] u s i n g b ina r y (e ,2e) s p e c t r o s c o p y . The measured r e s u l t i s found to reproduce the exact s o l u t i o n of the Schrod inger equa t ion in momentum-space. The on l y d i f f i c u l t y i n v o l v e d in o b t a i n i n g the bond d e n s i t y i s the n o r m a l i z a t i o n of the measured s p h e r i c a l l y averaged momentum d e n s i t y of H 2 1 because the noncoplanar symmetric (e ,2e) exper iment measures ( in most cases ) on l y r e l a t i v e c r o s s s e c t i o n s [MW76a]. In the p resen t s tudy , t h i s problem i s s o l v e d by employ ing a numer i ca l p ro cedu re . The " e x p e r i m e n t a l " momentum-space bond d e n s i t y thus ob t a i ned i s compared w i th t h e o r e t i c a l c a l c u l a t i o n s us i ng H 2 wave func t ions of d i f f e r e n t q u a l i t y i n c l u d i n g Extended Har t ree-Fock (Ext-HF) [DW66], L i m i t e d Har t ree-Fock (Ltd-HF) [FR61] , Doub le-Zeta (DZ) [SB72] and Min ima l B a s i s Set (MBS) [HL&80]. In a d d i t i o n the dependence of the bond d e n s i t y on the q u a l i t y of the wave func t ion i s a l s o examined by d i r e c t i o n a l momentum-space and p o s i t i o n -space d e n s i t y d i f f e r e n c e maps [CD41, B51, RC62, BH67, BC68, HC68, BP69, ET77, LB83b, R83] . The study (chapter 5) of the bonding i n H 2 at l a r g e s t ep spac i ngs between i n t e r n u c l e a r s e p a r a t i o n s of R=8 and 1a 0 has i n d i c a t e d tha t dramat ic changes in momentum-space bond d e n s i t y occur between R=2 and 1 a 0 . The t o p o g r a p h i c a l s tudy of the momentum-space bond d e n s i t y i n H 2 i s now extended to e x p l o r e in d e t a i l the c r i t i c a l range of R=2 to 1a 0 u s i n g the extended Har t ree-Fock q u a l i t y wavefunct ion of Das and Wahl [DW66]. The p resen t s tudy p r o v i d e s a complementary look at chemica l bonding 193 phenomena of H 2 in momentum and p o s i t i o n space and f u r t h e r i l l u s t r a t e s ' momentum-space chemica l concepts [CD41, ET77, CS&79, CB82b, MTC82, LB83b] . A th ree d imens iona l r e p r e s e n t a t i o n i s shown f o r the H 2 1 s i n g l e c o v a l e n t bond in momentum-space. 6.2 ESTIMATION OF THE ORBITAL MOMENTS AND NORMALIZATION OF THE MOMENTUM DISTRIBUTION In the presen t work, the s p h e r i c a l l y averaged momentum-space bond d e n s i t y of H 2 1 i s ob ta ined by equa t ion 6.1 u s i n g the e x p e r i m e n t a l l y determined momentum d i s t r i b u t i o n of the H 2 1Og o r b i t a l ( chapter 5) and the exact atomic hydrogen 1s momentum d i s t r i b u t i o n [LW81]. The expe r imen ta l momentum d i s t r i b u t i o n of the H 1s o r b i t a l has been shown to be i n e x c e l l e n t agreement w i th the exact s o l u t i o n (squared) of the Sch rod inge r equa t ion in momentum-space [LW81]. The procedure f o r n o r m a l i z a t i o n of the r e l a t i v e s p h e r i c a l l y averaged momentum d e n s i t y of the H 2 1a o r b i t a l to g i ve an a b s o l u t e d e n s i t y i s o u t l i n e d below. B r i e f l y , an a n a l y t i c a l f u n c t i o n i s f i t t e d to the expe r imen ta l s p h e r i c a l l y averaged momentum d i s t r i b u t i o n us ing a square r e s i d u a l m i n i m i z a t i o n method [L72c , M81]. The n t h o rder momentum moments of the jt h o r b i t a l , < p n > j , a re then e va l ua t ed from the f i t t e d f u n c t i o n u s i n g s tandard numer i ca l i n t e g r a t i o n t echn iques [ L72c ] . 1 94 [6 .2a] < p n > j = 4ir ; { P j (p) } p n + 2 d p , where (Pj (p) } i s the s p h e r i c a l l y averaged momentum d i s t r i b u t i o n of the it h o r b i t a l . The expe r imen ta l d i s t r i b u t i o n and the o p t i m i z e d a n a l y t i c a l d e n s i t y f u n c t i o n can be no rma l i zed by the z e r o t h order moment. In the case of H 2 , one h a s : [6 .2b] < P o > 1 0 g = occupancy number of 1 o r b i t a l = 2. Three s e m i - e m p i r i c a l a n a l y t i c a l f u n c t i o n s a re used to approximate {p(p)} of the H 2 1 a g o r b i t a l . These a r e : ( i ) O r b i t a l L o c a l D e n s i t y F u n c t i o n a l (OLDF) d e n s i t y : {p(p)} = K ( p 2 / 2 + a 1 p + a 2 ) " 3 ; ( i i ) Best S i n g l e Zeta (BSZ) d e n s i t y : {p(p)} = K (5 5 ) ( $ 2 +p 2 ) - \-( i i i ) Two Term Po l ynomia l (TTP) d e n s i t y : (p(p)} = K 1 ( 7 , 2 + P 2 > - ' ' + K 2 ( 7 2 2 + p 2 ) - 5 . The l i n e a r parameters (K, K, and K 2 ) a re n o r m a l i z a t i o n c o n s t a n t s . Both the l i n e a r (K ' s ) and the n o n l i n e a r parameters ( a ' s , $ and 7 ' s ) a re o p t i m i z e d by f i t t i n g to the expe r imen ta l momentum d i s t r i b u t i o n in the numer i c a l p rocedure [M81]. The OLDF d e n s i t y form used here i s tha t g i ven by Pathak et al. [PPG82] and shou ld s t r i c t l y be used f o r the 195 t o t a l momentum d e n s i t y of . atoms in the l o c a l d e n s i t y f u n c t i o n a l approx imat ion [ L79 ] . The BSZ (Best S i n g l e Zeta) momentum d e n s i t y co r responds to the square of the F o u r i e r t r a n s f o r m of a p o s i t i o n - s p a c e 1s S l a t e r - T y p e - O r b i t a l [KT76] w i th the $ va lue o p t i m i z e d to g i ve the best f i t to the expe r imen ta l momentum d i s t r i b u t i o n . The two term po l ynomia l f u n c t i o n (TTP) i s used by Lee [L77] in h i s s tudy of the t o t a l momentum d e n s i t y of H 2 by h i g h energy e l e c t r o n impact s p e c t r o s c o p y . I t shou ld be noted that the two term po l ynomia l f u n c t i o n i s e s s e n t i a l l y the BSZ f u n c t i o n w i th an a d d i t i o n a l h igh order te rm. F i g u r e 6.1 shows the expe r imen ta l momentum d i s t r i b u t i o n of the H 2 1a o r b i t a l (see a l s o f i g u r e 5.4) a l ong w i th the f i t t e d a n a l y t i c a l d e n s i t y f u n c t i o n s . I t i s obv ious tha t a l l th ree s e m i - e m p i r i c a l d e n s i t y f u n c t i o n s g i ve an e x c e l l e n t f i t to the expe r imen ta l d a t a . The BSZ and TTP d e n s i t i e s are i n d i s t i n g u i s h a b l e from each o the r wh i le the OLDF d e n s i t y i s on l y s l i g h t l y d i f f e r e n t from the BSZ and TTP d e n s i t i e s in the low and h i g h momentum r e g i o n . Tab l e 6.1 shows the o r b i t a l momentum moments c a l c u l a t e d u s i n g the o p t i m i z e d d e n s i t y f u n c t i o n s . O r b i t a l moments e v a l u a t e d d i r e c t l y from ab-initio SCF wave func t ions of d i f f e r e n t q u a l i t y [HL&80, SB72, FR61, DW66] are a l s o g i v e n . These moments a re compared w i th those r epo r t ed by Lee [ L77 ] . C l e a r l y , t he re i s an o v e r a l l good agreement between the " e x p e r i m e n t a l " o r b i t a l moments ( l e s s good f o r the h ighe r 196 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION Momentum (a.u.) F i g u r e 6.1 - Comparison of the expe r imen ta l s p h e r i c a l l y averaged momentum d i s t r i b u t i o n of H 2 1a o r b i t a l w i th the f i t t e d s e m i - e m p i r i c a l d e n s i t y f u n c t i o n s : OLDF, BSZ and TTP. The BSZ and TTP f u n c t i o n s are i n d i s t i n g u i s h a b l e from each o t h e r . Tab le 6.1 O r b i t a l momentum moments ( in atomic u n i t s ) of H 2 1o o r b i t a l . Moment " E x p e r i m e n t a l " * Theory** Lee+ OLDF BSZ TTP Ext-HF Ltd-HF DZ MBS <p 2> 6.36 7.56 7.62 8.24 7.86 7.20 6.92 8 .32 (0 .25 ) <p 1> 2.62 2.96 2.96 3.18 3.02 3.02 2.96 3 .08(0 .02) <p°> 2.00 2.00 2.00 2.00 2.00 2.00 2.00 1.99(0.02) <p~ 1 > 2.46 1 .96 1 .92 1 .82 1 .82 1 .84 1 .90 1 .86 (0 .02 ) <p" 2> 2.86 2.62 2.50 2.26 2.26 2.26 2.40 2 .34 (0 .09 ) * The accuracy i s expected to be about ±7%. * * The accuracy i s about ±4%. Note that the v a r i o u s wave func t ions a r e : Ext-HF [DW66], Ltd-HF [FR61], DZ [SB72] and MBS [HL&80]. + The moments are eva lua ted by Lee [L77] us ing a s i m i l a r two term po l ynomia l s o p t i m i z a t i o n of the Compton s c a t t e r i n g d a t a . U n c e r t a i n t i e s are shown in b r a c k e t s . 198 moments <p 2>) and the t h e o r e t i c a l ones . The s em i-emp i r i c a l BSZ and TTP moments are q u i t e c l o s e to each o ther w i th the OLDF moments be ing n o t i c e a b l y d i f f e r e n t . In p a r t i c u l a r , the OLDF d e n s i t y s i g n i f i c a n t l y underes t imates the average o r b i t a l moment <p>. T h i s d e f i c i e n c y i s due to the form of the OLDF d e n s i t y , which in gene ra l w i l l l e a d to a broader t o t a l momentum d i s t r i b u t i o n even f o r atoms [PPG82]. The OLDF moments show that the r e p o r t e d form of the OLDF f u n c t i o n [PPG82] i s not adequate to r ep resen t the momentum d e n s i t y of sma l l mo lecu l es l i k e hydrogen . I t i s a l s o e v iden t from t a b l e 6.1 tha t the " e x p e r i m e n t a l " moments are s l i g h t l y d i f f e r e n t from the t h e o r e t i c a l ones . The es t ima ted averaged o r b i t a l momenta <p> are g e n e r a l l y s l i g h t l y lower than the t h e o r e t i c a l ones . T h i s suggests tha t the expe r imen ta l H 2 l a o r b i t a l i s y l e s s s p a t i a l l y d i f f u s e ( in p o s i t i o n - s p a c e ) than tha t p r e d i c t e d by the more s o p h i s t i c a t e d a b - i n i t i o wave func t i ons . I t shou ld be noted tha t the momentum d i s t r i b u t i o n measured at a p a r t i c u l a r b i n d i n g energy i s in gene ra l r e l a t e d to the o v e r l a p form f a c t o r [MW76a]. In the t a r g e t Har t ree-Fock app rox ima t i on [MW76a], the form f a c t o r can be s i m p l i f i e d to the o r b i t a l momentum d e n s i t y . I t i s a l s o p o s s i b l e that sma l l d i f f e r e n c e s i n the e s t ima ted o r b i t a l momenta <p> between the exper iment and t h e o r i e s may be a t t r i b u t e d to the inadequacy of the t a r g e t Har t ree-Fock a p p r o x i m a t i o n . The r e s u l t s of b i n a r y (e ,2e) spec t roscopy i n v o l v e v i b r a t i o n a l a ve rag ing over the f i n a l ion s t a t e [MW76a], The e f f e c t s of t h i s a ve rag ing 199 on the compar ison of e x p e r i m e n t a l l y de termined momentum d i s t r i b u t i o n s and those c a l c u l a t e d at a s i n g l e R va lue are expected to be n e g l i g i b l e as demonstrated by the work of Dey et al . [DM&75] on H 2 and D 2 . 6 . 3 SPHERICALLY AVERAGED MOMENTUM-SPACE BOND DENSITY F i g u r e 6.2 shows the s p h e r i c a l l y averaged momentum-space bond d e n s i t y of H 2 . The r i g h t hand s e c t i o n of f i g u r e 6.2 shows the d e t a i l between 0 . 5 a o " 1 and 1 . 5 a o _ 1 . The cons t an t s o p t i m i z e d in the o r b i t a l moment e s t i m a t i o n procedure are used to no rma l i ze the expe r imen ta l s p h e r i c a l l y averaged momentum d i s t r i b u t i o n of the H 2 1 o r b i t a l ( chapter 5 ) . The Independent Atom Model d e n s i t y i s ob ta ined from the exact s o l u t i o n f o r the H 1s o r b i t a l . Equa t ion 6.1 i s then used to c a l c u l a t e the bond d e n s i t y . T h e o r e t i c a l bond d e n s i t i e s c o r r e s p o n d i n g to d i f f e r e n t q u a l i t y wave funct ions of the H 2 1Og o r b i t a l a re a l s o g i ven f o r compar i son . S e v e r a l i n t e r e s t i n g f e a t u r e s are appa ren t . F i r s t , the s p h e r i c a l l y averaged momentum-space (p-space) bond d e n s i t y c o n s i s t s of a nega t i ve we l l and a p o s i t i v e wing t a i l i n g o f f to p=°° w i th the z e r o - c r o s s i n g p o i n t at p = * 0 . 8 a o " 1 . A l though the d i r e c t i o n a l i n f o r m a t i o n i s l o s t a f t e r the s p h e r i c a l a v e r a g i n g , one can see tha t a sigma bond i s c h a r a c t e r i z e d by a t r a n s f e r of momentum d e n s i t y ( f r a c t i o n a l c u r r e n t ) from the low momentum reg ion to the h igh momentum r e g i o n . In Momentum (a.u.) F i g u r e 6.2 - S p h e r i c a l l y averaged momentum-space bond d e n s i t y of H 2 . An exploded view of the bond d e n s i t y in the range 0.5 to 1.5 a 0 _ 1 i s g i ven to i l l u s t r a t e the p o s i t i v e "wing" of the bond d e n s i t y . 201 a d d i t i o n , the v a r i a t i o n a l l y l e s s s u p e r i o r DZ [SB72] and MBS [HL&80] wave funct ions have "deeper w e l l s " ( i . e . more nega t i ve minima) at the p-space o r i g i n and s l i g h t l y sharper "w ings " ( p o s i t i v e maxima) than the more s o p h i s t i c a t e d ones [FR61, DW66]. Fu r the rmore , the " e x p e r i m e n t a l " bond d e n s i t y l i e s between the L i m i t e d Har t ree-Fock and the Double-Zeta q u a l i t y bond d e n s i t i e s at low momentum. I t i s , however, in c l o s e r agreement w i th the L i m i t e d Har t ree-Fock curve near the momentum o r i g i n . I t shou ld be noted (chapter 5) tha t the expe r imen ta l s p h e r i c a l l y averaged momentum d i s t r i b u t i o n of the H 2 1 <7g o r b i t a l g i v e s g e n e r a l l y good agreement w i th t h e o r e t i c a l d e n s i t y f u n c t i o n s c a l c u l a t e d us ing the L i m i t e d Har t ree-Fock and b e t t e r q u a l i t y wave func t i ons . The momentum-space bond d e n s i t y shows the d i f f e r e n c e s between d i f f e r e n t wave func t ions more c l e a r l y than s t r a i g h t f o r w a r d compar ison of the momentum d i s t r i b u t i o n s (see f i g u r e 5 .4b ) . The momentum-space bond d e n s i t y i s t h e r e f o r e an a l t e r n a t i v e way to e va lua te wave func t ions by compar ing w i th expe r imen ta l (e ,2e) d a t a , at l e a s t in the s p e c i a l case of the hydrogen molecu le where the bond d e n s i t y can be r e a d i l y d e r i v e d . 202 6.4 DIRECTIONAL BOND DENSITY 6.4.1 Wavefunct ion Dependence Except where o therw ise s t a t e d , the d e n s i t y contour v a l ues of the d i r e c t i o n a l bond d e n s i t y ( d e n s i t y d i f f e r e n c e ) maps used throughout the p resen t s tudy co r r e spond to 80, 60, 40, 20, ±8, ±6, ±4, ±2, ±0.8 and ±0.6% of the a b s o l u t e maximum d e n s i t y d i f f e r e n c e v a l u e . Contours of nega t i v e d e n s i t y d i f f e r e n c e are shown as dashed l i n e s . The d i r e c t i o n s of the c u t t i n g p lane are d e f i n e d by two v e c t o r s ; i . e . the b o n d - p a r a l l e l OL i n t e r n u c l e a r ( 0 ,0 ,1 ) v e c t o r and the bond-p e r p e n d i c u l a r (0 ,1 ,0 ) v e c t o r . The p r o j e c t i o n p l o t s on the r i g h t hand s i d e and on the top of the contour maps show the r e l a t i v e change in magnitude of the bond d e n s i t y f u n c t i o n a l ong and p e r p e n d i c u l a r to the i n t e r n u c l e a r a x i s (do t ted l i n e s ) r e s p e c t i v e l y . The p o s i t i o n and momentum are in atomic u n i t s . F i g u r e 6.3 shows the d i r e c t i o n a l d e n s i t y d i f f e r e n c e maps c o r r e s p o n d i n g to H 2 l a wave func t ions of d i f f e r e n t q u a l i t y y r ang ing from Extended Har t ree-Fock (Ext-HF) [DW66], L i m i t e d Har t ree-Fock (Ltd-HF) [FR61] , Double-Zeta (DZ) [SB72] and Min ima l B a s i s Set (MBS) [HL&80]. The e q u i l i b r i u m i n t e r n u c l e a r s e p a r a t i o n ( l . 4 a 0 ) has been used fo r the c a l c u l a t i o n s shown in f i g u r e 6 . 3 . I t shou ld be noted tha t because of the nature of the DZ and MBS wave func t i ons , the contour v a lues f o r the momentum-space (p-space) bond d e n s i t y POSITION DENSITY DIFFERENCE POSITION DENSITY DIFFERENCE POSITION DENSITY DIFFERENCE POSITION DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE F igu re 6.3 - Dens i t y d i f f e r e n c e (bond d e n s i t y ) maps i n momentum and p o s i t i o n space as a f u n c t i o n of the type of the H 2 1 cr wavefunct ion at e q u i l i b r i u m s e p a r a t i o n . Contours of negat i ve d e n s i t y d i f f e r e n c e are shown as dashed l i n e s . 204 ( d e n s i t y d i f f e r e n c e ) maps have been extended to i n c l ude ±0 .4 , ± 0 . 2 , - 0 . 0 8 , - 0 . 0 6 , - 0 . 0 4 , and -0.02% of the absolute maximum d e n s i t y d i f f e r e n c e v a l u e . I t i s e v iden t tha t a l l four wave func t ions p rov ide a q u a l i t a t i v e l y s i m i l a r p i c t u r e f o r the sigma bond. In g e n e r a l , the sigma bond in H 2 in p o s i t i o n - s p a c e ( r-space) can be a s s o c i a t e d wi th the f a m i l i a r "hamburger" p i c t u r e ; namely, d e n s i t y ( f r a c t i o n a l charge) a ccumu la t ion in the i n t e r n u c l e a r r eg i on and d e n s i t y d e p l e t i o n at the end r e g i o n s of the molecu le [BC68] , In momentum-space (p-space) a complementary view of the sigma bond i s o b t a i n e d as d i s c u s s e d in the p r e ced ing c h a p t e r . In t h i s "bone i n a donut " (or t o rus ) mode l , momentum d e n s i t y ( f r a c t i o n a l c u r r e n t ) i s d e p l e t e d th rough , the p-space o r i g i n a l ong the i n t e r n u c l e a r d i r e c t i o n (the "bone" pa r t ) and i s l o c a l i z e d a n n u l a r l y in the h igh momentum bond-perpend i cu l a r r eg i on (the " d o n u t " ) . S i m i l a r ideas have a l s o r e c e n t l y been p re sen ted by Ramirez [R83] . I t shou ld be noted that the c y l i n d r i c a l symmetry of the sigma bond in r-space i s p r e se r ved i n p-space . T h i s i s , of c o u r s e , a d i r e c t consequence of the F o u r i e r t r ans fo rm p r o p e r t i e s d i s c u s s e d p r e v i o u s l y [ET77, LB83b] . A t h r ee-d imens iona l s u r f a c e p l o t of the bond d e n s i t i e s genera ted us i ng the Ext-HF wave func t ion [DW66] i s shown in f i g u r e 6.4 to b e t t e r v i s u a l i z e the nature of the H 2 1 b o n d in both p-space and r-space . The bond d e n s i t i e s f o r s u r f a c e s shown co r respond to ±2% of the maximum bond d e n s i t y va lues i n the r e s p e c t i v e M O M E N T U M D E N S I T Y D I F F E R E N C E P O S I T I O N D E N S I T Y D I F F E R E N C E p= 2.3 E--3 ,0,1) o (0,1,0) F i gu re 6.4 - Three-d imens iona l su r f ace p l o t of the d e n s i t y d i f f e r e n c e (bond d e n s i t y ) of H 2 in momentum and p o s i t i o n space . The bond d e n s i t y s u r f a c e s cor respond to ±2% of the maximum bond d e n s i t y v a l u e s . P o s i t i v e bond d e n s i t y su r f a ce s are shaded. 206 spaces . A l though a l l four wave funct ions show the same gene ra l d e n s i t y t o p o g r a p h i c a l f e a t u r e s (see f i g u r e 6 . 3 ) , there are some important d i f f e r e n c e s , e s p e c i a l l y between the Ltd-HF and DZ. For the Ext-HF and the Ltd-HF bond d e n s i t i e s both the r-space and p-space bond d e n s i t i e s r e s p e c t i v e l y appear to be q u i t e s i m i l a r . The on l y n o t i c e a b l e d i f f e r e n c e i s in the end r eg ions of the "bone" • p a r t . The major d i f f e r e n c e occu r s between the Ltd-HF and the DZ bond d e n s i t i e s . The pronounced change in the na ture of the wave func t ion can be seen in the r-space bond d e n s i t y whereby the bond-pe rpend i cu l a r e l l i p s o i d a l p o s i t i v e lobe of the Ltd-HF (and Ext-HF) becomes more s p h e r i c a l i n the DZ (and MBS). T h i s can be seen more c l e a r l y in the b o n d - p a r a l l e l p r o j e c t i o n p l o t s , which show the appearance of four we l l d e f i n e d p o s i t i v e maxima f o r the DZ (and MBS) i n s t e a d of the f l a t p o s i t i v e " p l a t e a u " fo r the L t d -HF (and Ex t-HF ) . The nega t i ve lobe of the DZ not on l y has a r e l a t i v e l y sma l l e r ampl i tude but a l s o i s f u r t h e r away from the r-space o r i g i n . Even more dramat i c d i f f e r e n c e s can be seen in the p-space d e n s i t y d i f f e r e n c e maps. (Note tha t more contour l i n e s w i th sma l l e r bond d e n s i t y v a l ues are i n c l u d e d f o r the DZ and MBS bond d e n s i t i e s i n f i g u r e 6 . 3 ) . The p-space bond of the DZ (and a l s o MBS) i s dominated by the nega t i v e "bone" s t r u c t u r e , which appears a l s o to be c o n s i d e r a b l y broader than tha t of the Ltd-HF (and Ext-HF) one . The p o s i t i v e "donu t " s t r u c t u r e can on l y be seen when 207 r e l a t i v e contour va lues below 0.1% of the maximum ampl i tude are used . In the ccse of the MBS wave func t i on , i t s r-space bonding s t r u c t u r e i s even more inadequate as shown by the b o n d - p a r a l l e l p r o j e c t i o n p l o t . The p-space "donu t " s t r u c t u r e i s a l s o s l i g h t l y l a r g e r compared to the DZ one. 6.4.2 Dependence Of I n t e r n u c l e a r S epa r a t i on F i g u r e 6.5 shows the d i r e c t i o n a l bond d e n s i t y ( dens i t y d i f f e r e n c e ) in both p-space and r-space c o r r e s p o n d i n g to the Har t ree-Fock q u a l i t y wavefunct ion of Das and Wahl [DW66] as a f u n c t i o n of i n t e r n u c l e a r s e p a r a t i o n R. E a r l i e r work by Bader and Chandra [BC68] r e p o r t e d the r-space bond d e n s i t y maps e va l ua t ed us i ng the same Ext-HF wave func t ion [DW66] as a f u n c t i o n of R. The p resen t work s t u d i e s the p r o g r e s s of bond fo rmat ion i n p-space over the c r i t i c a l s e p a r a t i o n range of 1 . 0a o ^R^2 .0a o . A p r e l i m i n a r y s tudy of bond fo rmat ion over a more e x t e n s i v e range and wider spac i ng of s e l e c t e d R va lues has been g i ven i n the p r e ced ing c h a p t e r . I t shou ld be noted tha t compar ison can on l y be made w i t h i n a s i n g l e d i f f e r e n c e map s i n c e contour v a lues are r e l a t i v e to the maximum bond d e n s i t y . The maximum bond d e n s i t i e s i n the contour p l anes are shown in t a b l e 6.2 to g i ve some i n d i c a t i o n of the g l o b a l change as the H atoms approach each o t h e r . Bond fo rmat ion as r ep r e sen t ed by t h i s s e r i e s of sigma bond d e n s i t y maps f o r H 2 i n v o l v e s the t r a n s f e r of d e n s i t y 208 POSITION DENSITY DIFFERENCE POSITION DENSITY DIFFERENCE POSITION DENSITY DIFFERENCE POSITION DENSfTY DIFFERENCE -2.0 -10 0.0 to 2.0 0.S to -2.0 -10 0.0 to 2.0 0.* 10 -2.0 - t o 0.0 10 2.0 0.4 to -2.0 - t o 0.0 to 2.0 O.S 1.0 MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE -2.0 - t o 0.0 to 2.0 O i to -2.0 - t o 0.0 10 2.0 -O.SO.O -2.0 - t o 0.0 to 2.0 -O.50-0 -2.0 -10 0.0 to 20 -0.50.0 POSITION DENSITY DIFFERENCE POSfTION DENSfTY DIFFERENCE POSITION DENSITY DIFFERENCE POSmON DENSITY DIFFERENCE -2.0 -LO 0.D LO U U U -2.0 - t o 0.0 to 2.0 0.5 LO -2.0 -LO 0.0 10 2.0 0.5 10 -2.0 -10 0.0 10 2.0 O.S 10 MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE -2J> -UJ OX tO U -OJ 00 -20 -LO 0J> 10 2L0 - 0 4 0 . 0 -2.0 -10 OD LO 2 0 -040.0 -2.0 -to 0 4 LO 2.0 -040.0 F i g u r e 6 .5 - D e n s i t y d i f f e r e n c e (bond d e n s i t y ) maps i n momentum and p o s i t i o n space as a f u n c t i o n of i n t e r n u c l e a r s e p a r a t i o n R. Contours of nega t i v e d e n s i t y d i f f e r e n c e are shown as dashed l i n e s . 209 Tab le 6.2 Maximum a b s o l u t e va lues of bond d e n s i t y as a f u n c t i o n of R. R ( a 0 ) Maximum bond d e n s i t y r-space p-space 2.0 0.0451 0.192 1.8 0.0593 -0.0769 1.6 0.0792 -0.122 1.5 0.0930 -0.197 1 .4 0.113 -0.276 1.3 0. 142 -0.354 1.2 0. 178 -0.433 1.0 0.281 -0.585 * R and the bond d e n s i t y are i n atomic u n i t s . Note tha t the e q u i l i b r i u m i n t e r n u c l e a r s e p a r a t i o n i s 1.4 a 0 . 210 from the a n t i b i n d i n g r eg i on to the b i n d i n g r e g i o n . The r-space b i n d i n g r eg ion i s , of c o u r s e , the sma l l r ( i . e . i n t e r n u c l e a r ) r eg ion w i th d e n s i t y extended p r e f e r e n t i a l l y in the bond-pe rpend i cu l a r d i r e c t i o n wh i le the a n t i b i n d i n g r eg ion i s at l a r g e r ou t s i de the n u c l e i ex tend ing p r e f e r e n t i a l l y in the b o n d - p a r a l l e l d i r e c t i o n . In c o n t r a s t the l o c a t i o n s of the b i n d i n g and a n t i b i n d i n g r e g i o n s i n p-space are r e v e r s e d . S p e c i f i c a l l y , the b i n d i n g r e g i o n in p-space i s the annu la r (bond-perpend i cu l a r ) h i g h p r e g i o n wh i l e the a n t i b i n d i n g r eg ion i s c e n t r e d at low p and d i s t r i b u t e d p r e f e r e n t i a l l y a long the b o n d - p a r a l l e l d i r e c t i o n . Between R=2.0a o and the e q u i l i b r i u m s e p a r a t i o n at 1 . 4 a 0 , the g r adua l charge accumu la t i on i n the r-space b i n d i n g r eg i on i s p a r t i c u l a r l y obv ious as shown in the b o n d - p a r a l l e l p r o j e c t i o n p l o t s . F u r t h e r c l o s e r approach of the H atoms causes charge s a t u r a t i o n i n the i n t e r n u c l e a r r e g i o n , c a u s i n g the p i l e - u p of d e n s i t y a t the maxima seen p a r t i c u l a r l y c l e a r l y a t R = 1 . 0 a o . A much more dramat ic change can be seen i n the p-space d e n s i t y d i f f e r e n c e . At R=2.0a o the p-space bond d e n s i t y i s i n f a c t somewhat s i m i l a r in shape to i t s r-space c o u n t e r p a r t . In the R range of 2.0 to 1 . 4 a 0 , the p e n e t r a t i o n of the nega t i ve lobe of the p-space bond d e n s i t y i n t o the p o s i t i v e lobe a l ong the bond a x i s r e s u l t s in the fo rma t ion of the b i n d i n g "donu t " s t r u c t u r e . T h i s d ramat i c change ( in marked c o n t r a s t to the r-space p i c t u r e s which on l y change m a r g i n a l l y ) i s remarkably obv ious in both the b o n d - p a r a l l e l 21 1 and the bond-perpend i cu l a r p r o j e c t i o n p l o t s . Most no tab le i s the change between R=1.8 and 1 . 6 a 0 . Below the e q u i l i b r i u m i n t e r n u c l e a r d i s t a n c e ( l . 4 a 0 ) c l o s e r approach of the H atoms r e s u l t s in the t r a n s f e r of momentum d e n s i t y back i n t o the a n t i b i n d i n g b o n d - p a r a l l e l h igh p r e g i o n . The e n c l o s u r e of the nega t i ve lobe by the p o s i t i v e lobe genera tes the uns tab l e o v a l s t r u c t u r e at R = 1 . 0 a o . The d i f f e r e n c e i n d e n s i t y r e l o c a t i o n i n r-space and p-space i s in a c co rd w i th the V i r i a l p r o p e r t y ( s e c t i o n 2 .2 .2 ) [ET77, LB83b] . The fo rmat ion of a s t a b l e system must be accompanied by the lower ing of the t o t a l ene rgy , or e q u i v a l e n t l y by the r a i s i n g of the k i n e t i c energy . The k i n e t i c energy (T) can be i n c r e a s e d more e f f e c t i v e l y by t r a n s f e r r i n g the d e n s i t y i n t o the h igh momentum bond-p e r p e n d i c u l a r r eg i on because the p a r a l l e l component ( T p a r a ) of the k i n e t i c energy of a d i a tomic i s sma l l e r than the p e r p e n d i c u l a r component ^ T v e r t ^ [BP69, FRM70, FR71] . The phenomeno log ica l change in p-space bond d e n s i t y upon bond fo rmat ion i s i n d i c a t i v e of t h i s V i r i a l r equ i r ement . In t h i s r e g a r d , Bader and P res ton [BP69] have shown e a r l i e r tha t the d i f f e r e n c e between the bond-perpend i cu l a r and b o n d - p a r a l l e l components of the k i n e t i c energy , ^ T v e r t ~ T p a r a ^ ^ ' reaches i t s maximum in H 2 between R=2.0a o and 1 . 4 a 0 . The r e s u l t s of the p resen t i n v e s t i g a t i o n s are e n t i r e l y c o n s i s t e n t w i th the t h e o r e t i c a l a n a l y s i s of k i n e t i c energy d e n s i t y in H 2 g i ven by Bader and P res ton [BP69]. The p r e s e n t l y r epo r t ed 212 expe r imen ta l s t u d i e s of the d i s t r i b u t i o n of bond d e n s i t y in momentum-space are a l s o c o n s i s t e n t wi th the p r e d i c t i o n s made i n the p i o n e e r i n g t h e o r e t i c a l work of Cou lson and Duncanson [CD41] i n 1941 conce rn ing the e l e c t r o n momentum d i s t r i b u t i o n i n a s i n g l e bond. F i n a l l y , the p-space bond d e n s i t y maps complement the r-space bond d e n s i t y maps to p rov i de a more complete bonding p i c t u r e . Namely, the fo rmat ion of a s t a b l e sigma bond in mo lecu l a r hydrogen can be regarded as a t r a n s f o r m a t i o n of the " s l ow " charge moving w i th a low momentum a long the bond a x i s at the ends of the molecu le to " f a s t " charge moving w i th a h i g h momentum p e r p e n d i c u l a r to the bond^ a x i s in the i n t e r n u c l e a r space of the m o l e c u l e . C l e a r l y bond fo rmat ion i s m a n i f e s t e d , in the p resen t s t udy , much more d r a m a t i c a l l y in momentum-space than in p o s i t i o n -space . S ince measurements are p o s s i b l e in p-space but thus f a r not in r-space , the use of momentum-space concepts and b i n a r y (e ,2e) spec t roscopy promises new v i s t a s of chemica l bond ing both in t h e o r e t i c a l and exper imenta l work. 213 Chapter VII CARBON DIOXIDE 7.1 INTRODUCTION A h i g h momentum r e s o l u t i o n ( 0 . 1 a o " 1 FWHM) noncoplanar b i n a r y (e ,2e) measurement of the momentum d i s t r i b u t i o n s of the ou te r va l ence o r b i t a l s of C 0 2 i s r e p o r t e d . 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 C 0 2 can be w r i t t e n a s : O o ) 2 O o ) 2 ( 2 a ) 2 ( 3a ) 2 ( 2a ) 2 (4o ) 2 ( 3a ) 2 ( 1 TT ) * ( 1 * ) ' I * g . Two e a r l i e r b i na r y (e ,2e) s t u d i e s [GT&77, CB82a] of the oute r va l ence o r b i t a l momentum d i s t r i b u t i o n s of C 0 2 have been r e p o r t e d . The measurement by G i a r d i n i - G u i d o n i et al. [GT&77], a l t hough w i th comparable momentum r e s o l u t i o n ( 0 . 1 a o " 1 FWHM), i s l i m i t e d by the r a the r low energy r e s o l u t i o n (2.6eV FWHM). As a r e s u l t , on l y the momentum d i s t r i b u t i o n of the 1 ir^ o r b i t a l was sampled d i r e c t l y . A l s o because of the cop l ana r s c a t t e r i n g geometry employed, the o r b i t a l momentum d i s t r i b u t i o n s a re somewhat obscured by the Mott s c a t t e r i n g c r o s s s e c t i o n , which v a r i e s r a p i d l y w i th the p o l a r ang le t h e t a . The more recen t noncoplanar measurements r e p o r t e d by Cook and B r i on [CB82a] used h ighe r energy r e s o l u t i o n (1.3eV FWHM) but were l i m i t e d by the r a the r low 214 momentum r e s o l u t i o n ( 0 . 4 a 0 ~ 1 FWHM). Low momentum r e s o l u t i o n s e v e r e l y l i m i t s o b s e r v a t i o n of d e t a i l s i n the measured o r b i t a l momentum d i s t r i b u t i o n s [CBH80] such as ( i ) minima at p=0, f o r i n s t a n c e , i n the C 0 2 1flg o r b i t a l and ( i i ) the more complex s t r u c t u r e of momentum d i s t r i b u t i o n s f o r o r b i t a l s made up of symmetric (s-type) and nonsymmetric (p-type) components as fo r example in the C 0 2 4a o r b i t a l . As in a l l the e a r l i e r 9 s t u d i e s [GT&77, CB82a] , the p resen t expe r imen ta l energy r e s o l u t i o n (1.6eV FWHM) i s i n s u f f i c i e n t to r e s o l v e the c l o s e l y i n g A ( 1 W u ) " 1 and B ( 3 o u ) " 1 s t a t e s . However, d i r e c t measurements of the momentum d i s t r i b u t i o n s of the X ( l 7 r ) ~ 1 g and C ( 4 a g ) " 1 s t a t e s a re p o s s i b l e (without the use of the d e c o n v o l u t i o n method in r e f . [CB82a]) by j u d i c i o u s c h o i c e of the s i t t i n g b i n d i n g e n e r g i e s . With the improved momentum r e s o l u t i o n , the p resen t 1200 eV measurements p rov i de a more d e t a i l e d p i c t u r e of the outer va lence o r b i t a l momentum d i s t r i b u t i o n s of C 0 2 and r e s o l v e e a r l i e r u n c e r t a i n t i e s p a r t i c u l a r l y in the low momentum reg ion of the momentum d i s t r i b u t i o n of the 1 IT o r b i t a l . g 7.2 OUTER VALENCE BINDING ENERGY SPECTRUM The b i n d i n g energy spectrum of C 0 2 has been s t u d i e d i n d e t a i l by p h o t o e l e c t r o n spec t roscopy [TB&70], d i p o l e (e ,2e) spec t ro s copy [BT78] and b i n a r y (e ,2e) spec t roscopy [GT&77, CB82a] and a l s o t h e o r e t i c a l l y by the G r e e n ' s f u n c t i o n method 215 [ D C & 7 9 ] . In the p resen t work, l i m i t e d range b i n d i n g energy s p e c t r a i n the oute r va l ence r eg ion measured at <t>=0°, 8 ° and 14° a re shown in f i g u r e 7 . 1 . I n d i v i d u a l Gauss ian l i n e-shapes (dashed l i n e s ) c o r r e s p o n d i n g to the X ( l 7 T g ) ~ 1 , A ( 1 I T U ) ~ 1 , B ( 3 a u ) _ 1 and C ( 4 0 g ) ~ 1 s t a t e s are o v e r l a y e d in the s p e c t r a . The sum of the i n d i v i d u a l Gauss ians i s r ep resen ted by the s o l i d l i n e . The b i n d i n g energy p o s i t i o n s ( l 3 . 8 e V f o r 1 f l g , l 7 . 6 e V f o r 1 ir , l 8 . 1 e V fo r 3 a and l 9 . 4 e V f o r 4 a ) and the u u 9 c o r r e s p o n d i n g Franck-Condon widths used in the d e c o n v o l u t i o n p rocedu res were o b t a i n e d from 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 s p e c t r o s c o p i c data [TB&70]. The l a b e l l e d arrows ( i , i i , i i i ) i n the 0=14° spectrum i n d i c a t e the s i t t i n g b i n d i n g ene rg i e s s e l e c t e d f o r the angu la r c o r r e l a t i o n measurements (see f i g u r e s 7 . 2 ( i ) , ( i i ) and ( i i i ) r e s p e c t i v e l y ) . C l e a r l y , the energy r e s o l u t i o n does not a l low a c l e a r s e p a r a t i o n of the A and B s t a t e s . However, one can i n f e r some q u a l i t a t i v e f e a t u r e s about the symmetries of the i o n i z e d o r b i t a l s from the i n t e n s i t y v a r i a t i o n of the Gauss ian deconvo lu ted peaks in the th ree s p e c t r a . The r e l a t i v e i n t e n s i t y d i s t r i b u t i o n s of the th r ee outermost o r b i t a l s , i . e . the 1 Tr . 1 TT and 3 a , a l l g u u r e f l e c t the c h a r a c t e r i s t i c (nonsymmetric) p-type d i s t r i b u t i o n s wi th maxima away from p=0. The 4ag o r b i t a l , on the o the r hand, has i t s maximum in the #=0° spec t rum, which i n d i c a t e s a symmetric s-type component in the d i s t r i b u t i o n . In a d d i t i o n , the re i s s i g n i f i c a n t i n t e n s i t y i n the c/»=14° spectrum which sugges ts an important nonsymmetric 216 1 1 1 1 T T 1 1 1 I 1 1 1 r O 00 CO, 1200eV (fi — 14 d e g . o q d c 3 >» o 15 v. O C « to tyi O c 0) r.o i8.o Binding Energy (eV) 20.0 '22.0 24.0 10.5 12.0 14.0 16.0 18.0 20.0 22.0 24.0 Binding Energy (eV) M.O 16.0 18.0 20.0 Binding Energy (eV) v*-—— 2Z\0 24.0 F i g u r e 7.1 - Outer va l ence b i n d i n g energy s p e c t r a f o r C 0 2 at 0=0 ° , 8 ° and 1 4 ° . The l a b e l l e d arrows i n the #=14° spectrum i n d i c a t e the s i t t i n g b i n d i n g e n e r g i e s where the momentum d i s t r i b u t i o n s are measured ( see . f i g u r e 7 . 2 ) . 217 c o n t r i b u t i o n . 7.3 MOMENTUM DISTRIBUTIONS OF OUTER VALENCE ORBITALS F i g u r e 7.2 shows the momentum d i s t r i b u t i o n s sampled at th ree d i f f e r e n t b i n d i n g e n e r g i e s as i n d i c a t e d by the arrows in f i g u r e 7 .1 . Note tha t the s i t t i n g e n e r g i e s (not the v e r t i c a l I . P . ' s ) are i n d i c a t e d i n the s p e c t r a . T h e o r e t i c a l SCF momentum d i s t r i b u t i o n s c a l c u l a t e d from the doub l e-ze t a b a s i s of Snyder and Basch [SB72] ( s o l i d l i n e s ) and the s tandard 4-31G* b a s i s of the GAUSS76 package [HL&80] (dashed l i n e s ) as w e l l as an extended Gauss ian b a s i s w i th p o l a r i z a t i o n f u n c t i o n s of Langhof f [L84a] (dash-dot ted l i n e s ) a re g i ven f o r compar i son . In f i g u r e 7 . 2 ( i i ) 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 due to an admixture of 1 ir^ and 3 a y o r b i t a l s i s p r e s e n t e d . The mix ing c o e f f i c i e n t s of the i n d i v i d u a l o r b i t a l components ( i . e . 0.42 f o r 1 i r u and 0.58 f o r 3o u ) a re o b t a i n e d from the deconvo lu ted #=8° b i n d i n g energy spectrum a t 16.7eV ( f i g u r e 7 .1 ) . The t h e o r e t i c a l d i s t r i b u t i o n s are a l l i n d i v i d u a l l y area no rma l i zed to the exper iment i n the momentum range 0 to 1 . 5 a 0 _ 1 . G e n e r a l l y good agreement between the exper iment and the theory i s o b s e r v e d . The doub l e-ze t a q u a l i t y wave func t ions of Snyder and Basch [SB72] appear to be m a r g i n a l l y s u p e r i o r to both the 4-31G* GAUSS76 wave func t ions [HL&80] and the extended Gauss i an b a s i s wave func t ions [ L84a ] . The p resen t Figure 7.2 - Molecular momentum d i s t r i b u t i o n s f o r C0 2 outer va lence o r b i t a l s . 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 are eva lua ted us ing the Snyder and Basch b a s i s [SB72] ( s o l i d l i n e s ) , the 4-31G* GAUSS76 b a s i s [HL&80] (dashed l i n e s ) and the extended Gauss ian b a s i s of Langhoff [L84a] (dash-dot ted l i n e s ) . 219 measurements are c o n s i s t e n t w i th those r epo r t ed e a r l i e r [GT&77, CB82a] when the l a r g e d i f f e r e n c e s i n energy and momentum r e s o l u t i o n s are taken i n t o a c coun t . S ince d i r e c t compar ison wi th the momentum d i s t r i b u t i o n s of the cop l ana r work r e p o r t e d by G i a r d i n i - G u i d o n i et al. [GT&77] i s d i f f i c u l t because the v a r y i n g Mott s c a t t e r i n g c r o s s s e c t i o n d i s t o r t s the momentum d i s t r i b u t i o n s i n cop l ana r s t u d i e s , compar ison i s on l y made w i th the 400eV noncoplanar measurement of Cook and B r i on [CB82a] . It shou ld be noted tha t the lower accu racy of the momentum d i s t r i b u t i o n s r epo r t ed by Cook and B r i on [CB82a] of the l7r u, 3 a u and 40g o r b i t a l s i s due to the l i m i t e d s t a t i s t i c a l accuracy of the d i f f i c u l t d e c o n v o l u t i o n procedure [CB82a]. T h i s e a r l i e r measurement [CB82a] i s a l s o s e r i o u s l y l i m i t e d by the low angu la r r e s o l u t i o n of the spec t rometer which r e s u l t e d in the measured p-type d i s t r i b u t i o n s hav ing c o n s i d e r a b l e i n t e n s i t i e s at .p=0 [CBH80]. For the outermost nonbonding 1 tt^ o r b i t a l ( f i g u r e 7 . 2 ( i ) ) , the re i s a p p r e c i a b l e i n t e n s i t y in the p=0.2 to 0 . 5 a o " 1 r eg ion not accounted fo r by any of 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 s . A s i m i l a r o b s e r v a t i o n was a l s o made by Cook and B r i on [CB82a] in the e a r l i e r low momentum r e s o l u t i o n measurement at 400eV. However the p resen t measurement c l e a r l y r u l e s out any p o s s i b i l i t y of symmetric components in the wave func t ion s i n c e the observed momentum d i s t r i b u t i o n goes to zero at p=0 w i t h i n expe r imen ta l e r r o r . The maximum of the expe r imen ta l d i s t r i b u t i o n a l s o appears to 220 be at lower momentum (p ^ 0 . 8 a o " 1 ) when compared to the max theory ^ P m a x " 1 * " ' ^ ' These o b s e r v a t i o n s suggest an inadequacy in the b a s i s s e t s of the t h e o r e t i c a l wave funct ions f o r t h i s o r b i t a l . S i m i l a r behav iour i s a l s o found in the outermost 2ir o r b i t a l of the i s o e l e c t r o n i c mo lecu le N 2 0 [MFW82]. For the 17r u and 3 a y o r b i t a l s , e x c e l l e n t agreement between the exper iment and the theory i s observed as f a r as the admixture d i s t r i b u t i o n at 16.7eV ( f i g u r e 7 . 2 ( i i ) ) i n d i c a t e s . There i s a l s o good gene ra l agreement between t h i s admixture d i s t r i b u t i o n and a s i m i l a r d i s t r i b u t i o n r epo r t ed by Cook and B r i on (see f i g u r e 7 of r e f . [CB82a]) when d i f f e r e n c e s caused by very d i f f e r e n t momentum r e s o l u t i o n s are a l l owed f o r . There i s , however, a s t r i k i n g d i f f e r e n c e between the present r e s u l t and the e a r l i e r noncoplanar measurements [CB82a] i n the r e l a t i v e i n t e n s i t y r a t i o s of the symmetric and nonsymmetric components in the 4<Tg o r b i t a l ( f i g u r e 7 . 2 ( i i i ) ) . The symmetric (s-type) component accounts f o r the maximum at p^O wh i l e the nonsymmetric (p-type) component i s r e s p o n s i b l e f o r the second maximum at p = * 1 . 0 a o " 1 . The p resen t measurement shows s i g n i f i c a n t l y b e t t e r agreement w i th the t h e o r e t i c a l 4 t 7g d i s t r i b u t i o n s than does the 400eV measurement [CB82a] , T h i s i n d i c a t e s tha t the d e c o n v o l u t i o n method used in r e f . [CB82a] was not s u f f i c i e n t l y a c cu ra t e in the low momentum reg ion in tha t i t e v i d e n t l y a l l owed par t of the 3 a u i n t e n s i t y to mix in w i th the 4 a ^ i n t e n s i t y . A more j u d i c i o u s cho i c e of the s i t t i n g b i n d i n g energy f o r the 4 a 221 o r b i t a l in the p resen t work ( f i g u r e 7.1) a l s o ensures tha t any o v e r l a p of the n e i g h b o r i n g (3o )~1 s t a t e i s e f f e c t i v e l y e l i m i n a t e d . I t i s c l e a r tha t 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 c a l c u l a t e d u s i n g the Snyder and Basch wave func t ion [SB72] g i v e s a b e t t e r f i t to the exper iment than tha t c a l c u l a t e d us ing the 4-31G* wavefunct ion [HL&80] or the extended Gauss ian b a s i s of Langhof f [L84a] a l t hough a l l th ree t h e o r e t i c a l d i s t r i b u t i o n s c l e a r l y underes t imate the r a t i o of s-to-p components in t h i s o r b i t a l . It i s i n t e r e s t i n g to note a l s o tha t d e s p i t e the importance of the p o l a r i z a t i o n f u n c t i o n s in h e l p i n g to lower the t o t a l energy in the v a r i a t i o n a l method, the a d d i t i o n of more p o l a r i z a t i o n f u n c t i o n s as in the 4-31G* and the extended Gauss ian b a s i s wave func t ions does not n e c e s s a r i l y produce b e t t e r agreement w i th the expe r imen ta l momentum d i s t r i b u t i o n s . In the case of the 4<7g o r b i t a l , i n c l u s i o n of more p o l a r i z a t i o n f u n c t i o n s in f a c t g i v e s a b igge r d i s c r e p a n c y . 7.4 THEORETICAL ORBITAL MOMENTUM DISTRIBUTIONS AND ORBITAL DENSITY TOPOGRAPHY F i g u r e 7.3 shows the s p h e r i c a l l y averaged momentum d e n s i t i e s of both the va l ence and the co re o r b i t a l s of C 0 2 c a l c u l a t e d us ing the doub l e-ze t a q u a l i t y wave funct ions of Snyder and Basch [SB72]. The o r b i t a l s have been grouped i n t o th ree c l a s s e s , namely, the co re (1o , 1o and 2a ), the 222 va l ence sigma (3a , 2a , 4a and 3a ) and the va l ence p i (1 IT g u g u u and l 7 T g ) o r b i t a l s . The t h e o r e t i c a l SCF o r b i t a l ene rg i e s are a l s o g i ven in f i g u r e 7 .3 . I t can be seen , i n a c co rd w i th the gene r a l chemica l i n t u i t i o n , tha t the momentum d i s t r i b u t i o n s of the va l ence o r b i t a l s are c o n c e n t r a t e d i n the low momentum r e g i o n (p<3 .0a o ~ 1 ) wh i l e the momentum d i s t r i b u t i o n s of the co re o r b i t a l s are c l e a r l y much more extended i n t o the h igh p r e g i o n , and c o n v e r s e l y , more c o n s t r i c t e d i n p o s i t i o n - s p a c e . 1 Another very i n t e r e s t i n g f e a tu r e c o n c e r n i n g the co re ( 1 a n d 1o u> o r b i t a l s i s the o b s e r v a t i o n of ve ry s t r o n g d e n s i t y o s c i l l a t i o n s [CD41, ET77, CB82a, CB82b] . The p e r i o d i c i t i e s of the modu la t ions man i f e s t the " h i d d e n " i n f o r m a t i o n of the ( e q u i l i b r i u m ) mo lecu l a r geometry. I t shou ld , be noted tha t the p e r i o d i c i t i e s observed in the momentum d i s t r i b u t i o n s shou ld on l y be rough ly cons tan t due to the f a c t tha t the momentum d e n s i t i e s have been s p h e r i c a l l y a ve raged . The s p h e r i c a l l y averaged momentum d e n s i t i e s are p resen ted here in 1 The momentum-space (p-space) wave func t ion i n v o l v e s an i n t e g r a l of the sum of powers of the g r a d i e n t of the p o s i t i o n - s p a c e ( r-space) wavefunct ion over a l l space (see chap te r 2 ) . In the case of atomic o r b i t a l s where a l a r g e change i n the r-space wave func t ion o c cu r s on l y near the n u c l e u s , the sma l l r pa r t of the r-space wavefunct ion c o n t r i b u t e s most s i g n i f i c a n t l y to the l a r g e p p a r t of the p-space wavefunct ion (the i n ve r se s p a t i a l r e v e r s a l r e l a t i o n ) . In the case of mo lecu l a r o r b i t a l s , however, the s p a t i a l r e v e r s a l r e l a t i o n , a l t hough s t i l l q u a l i t a t i v e l y c o r r e c t , must be a p p l i e d wi th c a u t i o n s i n c e any r e g i o n where the re i s a r a p i d change in the r-space wave func t ion w i l l c o n t r i b u t e s i g n i f i c a n t l y to the h igh p pa r t of the p-space wave func t i on . Such r a p i d change can occur between atomic c e n t r e s , f o r i n s t a n c e a c r o s s a noda l p l a n e . A c l e a r example of t h i s i s the C 0 2 (1*0 o r b i t a l . 223 T r 1 r - i i r-VALENCE PI 1 1rrg 14.707 (eV) ITT. 20.105 3.5 . 4.0 4.5 C CD Q E ~c CD q co . C N • VALENCE SIGMA 3 3cru 20.116 4 4<rg 21.692 5 2cru 41.327 6 3crg 42.755 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Momentum (a.u.) F i g u r e 7.3 - T h e o r e t i c a l o r b i t a l momentum d i s t r i b u t i o n s f o r C 0 2 e v a l ua t ed us i ng the Snyder and Basch wave func t ion [SB72] . The i n d i c a t e d e n e r g i e s are the t h e o r e t i c a l o r b i t a l e n e r g i e s . 224 o rder to demonstrate the p o s s i b i l i t y of d i r e c t expe r imen ta l o b s e r v a t i o n of these d e n s i t y o s c i l l a t i o n e f f e c t s us ing a gaseous t a r g e t . The expe r imen ta l i n v e s t i g a t i o n of these d e n s i t y o s c i l l a t i o n e f f e c t s i s , however, t e c h n i c a l l y not f e a s i b l e at p resen t because of the very h igh o r b i t a l ene rg i e s of these core o r b i t a l s , which r e q u i r e a ve ry h i g h impact energy . in order to s a t i s f y the (e ,2e) r e a c t i o n model [MW76a]. The a lmost degenerate o r b i t a l e n e r g i e s of the 1 a y and 1 o r b i t a l s a l s o do not a l l ow o b s e r v a t i o n of i n d i v i d u a l momentum d i s t r i b u t i o n s wi th the p r e s e n t l y a v a i l a b l e energy r e s o l u t i o n . I t shou ld a l s o be p o i n t e d out tha t the measurement of the momentum d i s t r i b u t i o n s of the inner va lence sigma o r b i t a l s ( 2 o u and 3 0 g ) are not p o s s i b l e w i th the p resen t energy r e s o l u t i o n because the broad band in the inner v a l ence r eg ion i s composed of numerous i n t e rm ixed many-body l i n e s a r i s i n g from the i o n i z a t i o n s of these o r b i t a l s . T h i s many-body s t r u c t u r e in the inner va l ence r e g i o n has been observed e x p e r i m e n t a l l y [DC&79, CB82a] and a l s o p r e d i c t e d t h e o r e t i c a l l y by the G r e e n ' s f u n c t i o n s c a l c u l a t i o n s [DC&79]. F i n a l l y , f i g u r e 7 . 3 demonst ra tes one gene ra l f e a t u r e of momentum d i s t r i b u t i o n s a r i s i n g from the p a r i t y of the sigma o r b i t a l s in mo lecu les w i th symmetry. Namely, the momentum d i s t r i b u t i o n s of o r b i t a l s are c h a r a c t e r i z e d by maxima at p=0 (s-type) wh i l e those of o u o r b i t a l s have maxima at p*0 (p- t ype ) . T h i s d i f f e r e n c e in the appearance of the momentum d i s t r i b u t i o n s of o and o o r b i t a l s p r o v i d e s b ina r y 225 (e ,2e) spec t roscopy wi th a symmetry p rob ing advantage over c o n v e n t i o n a l s p e c t r o s c o p i e s . T h i s was a l s o e a r l i e r p o i n t e d out in the case of CO and N 2 by C a m i l l o n i et al . [CS&79]. These f e a t u r e s , seen in the s p h e r i c a l l y averaged momentum d e n s i t i e s , can be f u r t h e r i l l u s t r a t e d wi th d i r e c t i o n a l o r b i t a l d e n s i t y contour maps c a l c u l a t e d us i ng t h e o r e t i c a l wavefunct ions of doub l e-ze t a q u a l i t y [SB72] in momentum-space (p-space) and p o s i t i o n - s p a c e ( r-space) ( f i g u r e s 7 . 4-7 .6 ) . , The d e n s i t y maps of the va l ence o r b i t a l s of C 0 2 have been d i s c u s s e d e a r l i e r by Cook and B r i on [CB82a, CB82b] . In the present work, the range of the d e n s i t y maps has been extended to 4 atomic u n i t s to e x h i b i t a d d i t i o n a l (weak) f e a t u r e s which occur i n the h igh momentum r e g i o n . In a d d i t i o n , a x i s p r o j e c t i o n p l o t s [LB83b] are i n c l u d e d to show the r e l a t i v e changes in ampl i tude a long the p r o j e c t i o n l i n e s . Contour v a lues have a l s o been s e l e c t e d so as to extend l o g a r i t h m i c a l l y over th ree decades r e l a t i v e to the maximum d e n s i t y v a l u e s . These p rov i de b e t t e r p e r s p e c t i v e s i n t o the o f t e n sma l l but important f e a t u r e s a s s o c i a t e d w i th some of the momentum-space chemica l concep ts [CD41, ET77, CB82b, LB83b] . The d e n s i t y mapping conven t i on used in the present work i s the same as tha t g i ven i n chapte r 5 ( s e c t i o n 5 . 4 . 1 ) . As d i s c u s s e d in the p r e c e d i n g c h a p t e r s , the re are four gene ra l momentum-space p r o p e r t i e s a r i s i n g e s s e n t i a l l y from the F o u r i e r t r ans fo rm of p o s i t i o n - s p a c e wave func t ions [CD41, ET77, LB83b] . In the case of C 0 2 , the symmetry i n v a r i a n c e 226 F i g u r e 7 . 4 - D e n s i t y contour o r b i t a l s in momentum-space ( r i g h t ) . maps f o r 1TT and ln-d e f t ) and p o s i t i o n - s p a c e 227 MOMENTUM DENSITY POSITION DENSITY -3.2 -1.6 0.0 1.6 3.2 0.5 1.0 -3.2 -1.6 0.0 16 3.2 0.5 1.0 MOMENTUM DENSITY POSITION DENSITY -3.2 -1.6 0.0 1.6 3.2 0.5 1.0 -3.2 -1.6 0.0 16 3.2 0.5 1.0 MOMENTUM DENSITY POSITION DENSITY -3.2 -1.6 0.0 L6 3J 0.5 IO -3.2 -1.6 0.0 U 5.2 0.5 1.0 MOMENTUM DENSITY POSITION DENSITY -3.2 -US 0.0 1.6 SJ 0.5 1.0 -SJ -1.6 0.0 16 3.2 0.5 1.0 F i g u r e 7 . 5 - D e n s i t y con tour maps f o r 3a , 4 a 3a o r b i t a l s in momentum-space ( l e f t ) and po s p i c e ( r i g h t ) . F i g u r e 7.6 o r b i t a l s ( r i g h t ) . - D e n s i t y con tour maps f o r 2a , l a and l a in momentum-space ( l e f t ) and p o s i t r o n - s p a c e 229 p r o p e r t y i s very c l e a r l y demonstrated by the va l ence o r b i t a l s (both the 7r and a o r b i t a l s ) where the p-space and r-space wave func t ions (and t h e i r d e n s i t i e s ) possess i d e n t i c a l symmetry elements such as r e f l e c t i o n p l a n e s . The i nve r se we igh t ing p rope r t y can a l s o be seen in the va l ence o r b i t a l d e n s i t y maps and t h e i r a s s o c i a t e d p r o j e c t i o n p l o t s , which i n d i c a t e tha t the p-space d e n s i t y lobes are c o n c e n t r a t e d in the sma l l p r eg ion wh i l e the r-space d e n s i t y l obes are l o c a t e d in the l a rge r r e g i o n . The mo l e cu l a r d e n s i t y d i r e c t i o n a l r e v e r s a l i s i l l u s t r a t e d by the f a c t that i n d i v i d u a l p-space d e n s i t y l obes are e l onga ted i n the bond-p e r p e n d i c u l a r d i r e c t i o n wh i l e the r-space l obes are p o i n t i n g i n the b o n d - p a r a l l e l d i r e c t i o n . F i n a l l y , the mo l e cu l a r p-space d e n s i t y o s c i l l a t i o n e f f e c t i s perhaps bes t demonstrated by the s t r ong lobe modu la t ions a l ong the i n t e r n u c l e a r a x i s in the core 1 a u and 1 o r b i t a l s ( f i g u r e 7 .6 ) . The weak d e n s i t y o s c i l l a t i o n s in va lence o r b i t a l s are shown by the appearance of secondary lobes in the p-space d e n s i t y maps of the outer va l ence o r b i t a l s ( 1 i r „ , 17r , 3a and 4c , see f i g u r e s 7.4 and g u u g 3 7.5 r e s p e c t i v e l y ) where d e n s i t y con tou rs below 1% of the maximum d e n s i t y va lues are necessa ry f o r r e v e a l i n g these s t r u c t u r e s . These momentum-space p r o p e r t i e s can be f u r t h e r v i s u a l i z e d by us ing th ree d imens iona l cons t an t d e n s i t y s u r f a c e p l o t s (see s e c t i o n 5 . 4 . 5 ) . In f i g u r e 7 .7 , the cons t an t s u r f a c e s wi th d e n s i t y v a l ues c o r r e s p o n d i n g to 0.2% of the r e s p e c t i v e maximum d e n s i t y va lues f o r the four MOMENTUM DENSITY p= 8.5 E-4 Q (0,1,0) p= 2.1 E-3 p- 1.5 E-3 o (0,1,0) I77:. (0,1,0) (0,1,0) POSITION DENSITY p- 4.2 E-4 a (0,1,0) 3cr u p- 1.2 E-3 o o" (0,1,0) p= 6.2 E-4 (0,1,0) 4crn (0,1,0) to o F i g u r e 7.7 - Three-dimensional constant d e n s i t y s u r f a c e p l o t s fo r C 0 2 17r , 17r , 3a and 4a o r b i t a l s i n momentum-space (cop) a*nd p o s i t i o n - s p a c e (bot tom) . The d e n s i t y va lues of the su r f a ces co r r espond to 0.2% of the r e s p e c t i v e maximum d e n s i t i e s . The range of the d e n s i t y cube i s from -4.0 a . u . to 4.0 a . u . i n each of the th ree d i r e c t i o n s (same as tha t of the d e n s i t y maps). 231 outermost o r b i t a l s a re shown. C l e a r l y , the o r b i t a l d e n s i t y c f the it symmetry o r b i t a l s d i f f e r from that of the o symmetry -o r b i t a l s by the presence of the "noda l l i n e s " in the i n t e r n u c l e a r d i r e c t i o n . The o the r symmetry e lements such as r e f l e c t i o n (and noda l ) p l anes a re e a s i l y seen . A l s o , complex secondary d i s k - l i k e l obes of the p-space d e n s i t i e s r e f l e c t the mo lecu l a r d e n s i t y o s c i l l a t i o n and d i r e c t i o n a l r e v e r s a l p r o p e r t i e s i n momentum-space. 232 Chapter V I I I CARBON DISULPHIDE 8.1 INTRODUCTION The va l ence e l e c t r o n i c s t r u c t u r e of CS 2 has been i n v e s t i g a t e d by s e v e r a l t e chn iques i n c l u d i n g p h o t o e l e c t r o n [TB&70, AG&72, PW74, SD&79, HD&80, CKG82, K83 ] , Penning i o n i z a t i o n [BY77] and d i p o l e (e ,2e) [CWB81] s p e c t r o s c o p i e s , a l l of which have p r o v i d e d v a l u a b l e i n f o r m a t i o n on the v a l e n c e - s h e l l b i n d i n g energy spectrum of C S 2 . In a d d i t i o n , the v a l e n c e - s h e l l b i n d i n g energy spectrum of CS 2 has a l s o been c a l c u l a t e d us i ng the many-body 2ph-TDA G r e e n ' s F u n c t i o n (GF) method [SD&79] and more r e c e n t l y u s i ng the symmetry adapted c l u s t e r c o n f i g u r a t i o n i n t e r a c t i o n (SAC-CI) method [N83] . These t h e o r e t i c a l works [SD&79, N83] have shown e x t e n s i v e p o p u l a t i o n s p l i t t i n g s in the inner va l ence i o n i z a t i o n r eg ion of C S 2 , i n d i c a t i n g a s i g n i f i c a n t breakdown of the s i n g l 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 . In the p resen t work, the b i n d i n g energy s p e c t r a in the 2-48eV r eg ion measured at two r e l a t i v e az imu tha l ang les (#=0° and 8 ° ) a re r e p o r t e d . R e l a t i v e i n t e n s i t i e s of b i n d i n g energy s p e c t r a o b t a i n e d at s e v e r a l r e l a t i v e az imutha l ang l e s by b i n a r y (e ,2e) spec t roscopy are p r o p o r t i o n a l to the momentum p r o b a b i l i t y d e n s i t i e s at the c o r r e s p o n d i n g b i n d i n g e n e r g i e s . 233 The r e l a t i v e i n t e n s i t i e s in these s p e c t r a can t h e r e f o r e be used to i d e n t i f y a spec t s of the symmetry and c h a r a c t e r of the i o n i z e d o r b i t a l s a s s o c i a t e d w i th the p a r t i c u l a r energy bands e s p e c i a l l y i n the inner va l ence r e g i o n . T h i s has a l s o been used to unambiguously a s s i g n the i d e n t i t y and o r d e r i n g of mo lecu l a r o r b i t a l s in mo lecu l e s such as H 2 CO [HHB76b]. Moreover i f the momentum d i s t r i b u t i o n sampled at a g i ven b i n d i n g energy i s i n t e g r a t e d over the momentum, the r e s u l t i n g momentum-integrated i n t e n s i t y i s p r o p o r t i o n a l to the s p e c t r o s c o p i c p o l e - s t r e n g t h and thus can be compared d i r e c t l y w i th t h e o r e t i c a l r e s u l t s o b t a i n e d by the GF [SD&79] or the SAC-CI [N83] methods. Such a d i r e c t compar ison of measured and c a l c u l a t e d i n t e n s i t i e s i s not p o s s i b l e in the case of p h o t o e l e c t r o n s t u d i e s because of the energy dependent d i p o l e mat r ix element [MUP78]. In the p resen t work, momentum d i s t r i b u t i o n s measured at s e v e r a l b i n d i n g e n e r g i e s are compared w i th 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 c a l c u l a t e d us i ng wavefunct ions of the GAUSS76 431G b a s i s [HL&80] and the extended Gauss ian b a s i s of Langhof f [ L84b] . The momentum-i n t e g r a t e d i n t e n s i t i e s of the c o r r e s p o n d i n g momentum d i s t r i b u t i o n s are used to compare d i r e c t l y w i th the p r e d i c t e d i n t e n s i t i e s of the t h e o r e t i c a l b i n d i n g energy s p e c t r a [SD&79, N83] . The measured momentum d i s t r i b u t i o n s p r o v i d e the f i r s t d i r e c t look at the mo lecu la r bonding o r b i t a l s a s s o c i a t e d wi th the C=S bond in C S 2 . These d i s t r i b u t i o n s are compared w i th those of the co r r e spond ing va l ence o r b i t a l s in C 0 2 [LB84b] 234 ( chapter 7 ) . D i f f e r e n c e s in the o r b i t a l momentum d i s t r i b u t i o n s of CS 2 and those of C 0 2 a re i n t e r p r e t e d u s i n g the r e s p e c t i v e c a l c u l a t e d o r b i t a l d e n s i t y contour maps i n p o s i t i o n and momentum space . 8.2 VALENCE-SHELL BINDING ENERGY SPECTRUM Carbon d i s u l f i d e i s a 38-e l e c t ron system wi th D ^ p o i n t group symmetry. The ground s t a t e e l e c t r o n i c c o n f i g u r a t i o n of C S 2 i s : ( c o r e ) 2 2 ( 5 a ) 2 ( 4 a ) 2(6a ) 2(5a ) 2 (2TT )«(27r )«. y u y u u y Two s e t s of angu l a r c o r r e l a t e d b i n d i n g energy s p e c t r a are r e p o r t e d . F i g u r e 8.1 shows the ou te r va l ence (7-20eV) b i n d i n g energy s p e c t r a measured at c/»=0°, 4 ° , 8 ° , 12° and 1 6 ° . B i n d i n g energy s p e c t r a ex tend ing i n t o the inner va l ence r e g i o n (up to 48eV) measured at <t>=0° and 8 ° a re shown i n f i g u r e 8 .2 . The four oute r va l ence s t a t e s : X(27Tg)~ 1, A(27r B(5o, ) _ 1 and C ( 6 o „ ) " 1 as w e l l as the D s t a t e are u u g i d e n t i f i e d i n both f i g u r e s 8.1 and 8.2 by a Gauss ian d e c o n v o l u t i o n p r o c e d u r e . The energy p o s i t i o n s ( X ( l 0 . 0 7 e V ) , A ( l 2 . 8 5 e V ) , B ( l 4 . 4 7 e V ) , C ( l 6 . l 8 e V ) and D(17.05eV) ) and the c o r r e s p o n d i n g Franck-Condon widths used i n the d e c o n v o l u t i o n p rocedure a re o b t a i n e d from h i g h r e s o l u t i o n PES da ta [PW74, SD&79, HD&80, CKG82]. The sums of the i n d i v i d u a l Gauss ians 235 ifi = 16 d e g . C S 2 ! 2 0 0 e V 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 Binding Energy (eV) - i 1 r— | | [""f = ' 2 d e g . X (2-rr,)"1 A (2TTU)-' , B (5aJ' tfrM,T C ( 6 < r , r ' 6.0 %.0 10.0 12.0 14.0 16.0 18.6 20.0 22.0 Binding Energy (eV) —i 1 r T 1 1 1 ifi = B d e g . 6.0" 8".0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 Binding Energy (eV) - i 1 1 1 r 6.0 g.O 10.0 12.0 14.0 16.0 18.0 20.0 22.0 Binding Energy (eV) 6.0" 4.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 Binding Energy (eV) F i g u r e 8.1 - Outer va l ence b i n d i n g energy s p e c t r a f o r CS at <t>=0°, 4 ° , 8 ° , 12° and 1 6 ° . L eas t-squa res Gauss ian f i t s of the the expe r imen ta l da ta are i n d i c a t e d by the s o l i d l i n e s . The l a b e l l e d arrows in the </>=8° spectrum i n d i c a t e the s i t t i n g b i n d i n g e n e r g i e s where the momentum d i s t r i b u t i o n s a re measured. 2 3 6 (clotted l i n e s ) are r ep resen ted by s o l i d l i n e s . . In g e n e r a l , the re i s e x c e l l e n t o v e r a l l agreement between the Gauss ian c o n v o l u t e d enve lopes and the expe r imen ta l d a t a . The e x i s t e n c e of a c o n t r i b u t i o n from the D s t a t e at l7 .05eV (which escaped a t t e n t i o n in the e a r l i e r XPS [AG&72] and d i p o l e (e ,2e) [CWB81] works) i s a l s o q u i t e e v i den t i n the s p e c t r a ( f i g u r e s 8.1 and 8 . 2 ) . The r e l a t i v e i n t e n s i t y v a r i a t i o n s of i n d i v i d u a l deconvo lu ted (Gauss ian) peaks w i th r espec t to the az imu tha l ang le <p depend on the momentum d i s t r i b u t i o n s (MDs) of the a s s o c i a t e d c h a r a c t e r i s t i c o r b i t a l s . I t i s c l e a r from f i g u r e 8.1 tha t the X ( 2 » J " 1 , A ^ T ^ ) " 1 and B ( 5 o u ) " 1 s t a t e s a l l have maximum i n t e n s i t i e s at p^O, i n d i c a t i n g a g e n e r a l p-type dependence in the MDs of the r e s p e c t i v e o r b i t a l s . The C ( 6 0 g ) ~ 1 s t a t e , on the o ther hand, has maximum i n t e n s i t y at p^O ( i . e . in the 0=0° spec t rum) . The c o r r e s p o n d i n g MD t h e r e f o r e has a symmetric (s-type) component. The h i g h r e s o l u t i o n PES data [PW74, SD&79, HD&80, CKG82] i n d i c a t e the presence of two many-body s t a t e s ( s a t e l l i t e s ) in the ou te r va l ence r e g i o n , namely at 14.07eV and l 7 . 05eV . The i n t e n s i t y of the D s t a t e appears to vary in a s i m i l a r f a s h i o n to tha t of the A(2#T ) _ 1 s t a t e . T h i s i s in good q u a l i t a t i v e agreement wi th the symmetry adapted c l u s t e r c o n f i g u r a t i o n i n t e r a c t i o n (SAC-CI) c a l c u l a t i o n [N83], which p r e d i c t s a 2ir^ po le next to the pr imary 6a po l e (see f i g u r e 8 . 3 ) . The o ther ou te r va lence many-body s t a t e at 14.07eV i s on l y 237 m a r g i n a l l y obse rvab le i n h igh r e s o l u t i o n PES works [SD&79, HD&80] and w i l l not be c o n s i d e r e d in the p resen t work because of i t s ve ry sma l l i n t e n s i t y . I t shou ld be noted tha t t h i s many-body s t a t e i s p r e d i c t e d in the many-body G r e e n ' s f u n c t i o n c a l c u l a t i o n [SD&79] but not in the SAC-CI c a l c u l a t i o n [N83]. The long range angu la r c o r r e l a t e d b i n d i n g energy spectrum ( f i g u r e 8.2) shows e x t e n s i v e many-body s t r u c t u r e s i n the inner va l ence r eg i on (>20eV) r a the r than j u s t two peaks which would be expected from the s i n g l e p a r t i c l e i o n i z a t i o n model f o r the ( 4 a u ) _ 1 and (Ba^) " 1 p r o c e s s e s . These e x t e n s i v e s t r u c t u r e s were a l s o observed i n the e a r l i e r XPS [AG&72] and d i p o l e (e ,2e) [CWB81] s t u d i e s . 1 Three prominent o v e r l a p p i n g bands in the 22-34eV r e g i o n can be observed in the 0=0° spectrum ( f i g u r e 8 . 2 ) . The many-body s t a t e s above 22eV have h ighe r r e l a t i v e i n t e n s i t i e s at 0=0° than at 0=8° , i n d i c a t i n g a gene r a l s-type behav io r of the momentum d i s t r i b u t i o n s a s s o c i a t e d w i th these i o n i c s t a t e s . The s t r u c t u r e between 20eV and 22eV ( f i g u r e 8.2) however appears to have an a p p r e c i a b l e p-type c o n t r i b u t i o n . T h e o r e t i c a l v a l e n c e - s h e l l b i n d i n g energy s p e c t r a have been c a l c u l a t e d by Sch i rmer et al. [SD&79] u s i ng the many-1 I t shou ld be noted tha t s p e c t r a o b t a i n e d by d i f f e r e n t t e chn iques w i l l have d i f f e r e n t g l o b a l appearances due to the d i f f e r e n t dependences of the s p e c t r a l i n t e n s i t y upon the expe r imen ta l k inemat i c c o n d i t i o n s . For a d i s c u s s i o n of these d i f f e r e n c e s see r e f . [CW&81]. 238 o CO o CD q c q 3 C M 2 o 2 ° i 1 1 r C S 2 1 200eV <p = 8 deg. 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.01 45.0 50.0 Binding Energy (eV) 570 1 10.0 15.0 2*0* 0 25.0 30.0 35.0 40.0 45.0 50.0 Binding Energy (eV) F i g u r e 8.2 - Va lence s h e l l b i n d i n g energy s p e c t r a f o r CS 2 a t 0=0° and 8°. L e a s t - s q u a r e s G a u s s i a n f i t s of the the e x p e r i m e n t a l d a t a i n the o u t e r v a l e n c e r e g i o n a r e i n d i c a t e d by the s o l i d l i n e s . The l a b e l l e d a r rows i n the 0=8° spectrum i n d i c a t e the s i t t i n g b i n d i n g e n e r g i e s where the momentum d i s t r i b u t i o n s a r e measured. 239 body 2ph-TDA G r e e n ' s f u n c t i o n (GF) method and more r e c e n t l y by N a k a t s u j i [N83] u s i n g the symmetry adapted c l u s t e r c o n f i g u r a t i o n i n t e r a c t i o n (SAC-CI) method. (The SAC-CI spectrum p resen ted here co r responds to the p o l e - s t r e n g t h spectrum c a l c u l a t e d by N a k a t s u j i u s i ng b a s i s II in r e f . [N83]. ) The p o l e s c a l c u l a t e d a c c o r d i n g to the t h e o r i e s are shown in f i g u r e 8.3 and have been convo lu t ed wi th Gauss ians of the expe r imen ta l i n s t r u m e n t a l width ( l . 7 e V FWHM) to generate the i n d i c a t e d o v e r a l l enve lope to enable a r easonab le compar ison w i th the expe r imen ta l b i n d i n g energy s p e c t r a ( f i g u r e 8 . 2 ) . Momentum-integrated i n t e n s i t i e s of momentum d i s t r i b u t i o n s (see below) measured at s e v e r a l s e l e c t e d b i n d i n g e n e r g i e s ( s o l i d squares in f i g u r e 8.3) are no rma l i z ed ( r e l a t i v e to each o ther ) to the Gauss ian c o n v o l u t e d enve lope at 9.8eV. I t shou ld be noted tha t the i n d i c a t e d po l e a s s i gnmen t s , e s p e c i a l l y in the inner va l ence r eg ion of the c a l c u l a t e d s p e c t r a ( f i g u r e 8 . 3 ) , each co r r esponds to the dominant c o n f i g u r a t i o n c o n t r i b u t i n g to tha t p o l e . There i s g e n e r a l l y e x c e l l e n t agreement in the oute r va l ence r eg i on between the measured b i n d i n g e n e r g i e s ( f i g u r e 8.2.) and those c a l c u l a t e d by the SAC-CI method [N83] ( f i g u r e 8 . 3 ) . Good agreement i n the ou te r va l ence r eg ion between the measured s p e c t r a and the GF spectrum [SD&79] can a l s o be o b t a i n e d i f the second band ( i . e . the A, B and C enve lope) i s s h i f t e d by app rox ima te l y 0.8eV to h igh energy . T h i s and 240 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 Binding Energy (eV) ~T I I I I I I I I I I I I I 1 1 1 1 r oq d -c O J i t f O CN c s . SAC CI 1 277-g 2 2TTU 3 5cru 4 6crg 5 4cru 6 5cra 6 6 i i i i ^ i ^ i i i i i i i i i i i 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 Binding Energy (eV) F i g u r e 8.3 - T h e o r e t i c a l v a l e n c e - s h e l l b i n d i n g energy s p e c t r a f o r C S 2 . The p o l e s c a l c u l a t e d by the 2ph-TDA G r e e n ' s f u n c t i o n method [SD&79] (top) and by the symmetry adapted c l u s t e r c o n f i g u r a t i o n i n t e r a c t i o n method [N83] (bottom) are c o n v o l u t e d w i th Gauss i ans of the i n s t r umen ta l w i d t h . 241 s i m i l a r d i s c r e p a n c i e s (see l a t e r ) in the GF c a l c u l a t e d spectrum are o f t e n a t t r i b u t e d to the i nhe ren t l i m i t a t i o n s of the two p a r t i c l e - h o l e approx imat ion as w e l l as l i m i t a t i o n in the q u a l i t y of the employed b a s i s set [SD&79]. The experimental p o l e - s t r e n g t h s , as approx imated by numer i ca l i n t e g r a t i o n of the momentum d i s t r i b u t i o n s measured at the r e s p e c t i v e ene rg i e s in the inner va l ence r e g i o n , a re a l s o in good agreement (at l e a s t below 30eV) w i th the Gauss ian convo lu t ed enve lopes of both the SAC-CI and the " s h i f t e d " GF s p e c t r a (see f i g u r e 8 . 3 ) . The inner va l ence r e g i o n s of both t h e o r e t i c a l s p e c t r a [SD&79, N83] show an i n t ense i n t e r m i x i n g of (most ly ) the (4a , , ) " 1 and ( 5 a ) - 1 p o l e s . S i m i l a r u g i n t e r m i n g l i n g of p o l e s has a l s o been observed and p r e d i c t e d in the c o r r e s p o n d i n g inner v a l ence b i n d i n g energy s p e c t r a of C 0 2 [DC&79, N83] , In the o v e r a l l inner va l ence r e g i o n , on l y q u a l i t a t i v e agreement between exper iment ( f i g u r e 8.2) and l i t e r a t u r e c a l c u l a t i o n s [SD&79, N83] ( f i g u r e 8.3) i s found fo r C S 2 . In p a r t i c u l a r , the many-body s t r u c t u r e s in both the GF and SAC-CI c a l c u l a t e d s p e c t r a appear to be r e s t r i c t e d to the 20-30eV r e g i o n , i n c o n t r a s t to the c o n s i d e r a b l e i n t e n s i t y d i s t r i b u t i o n of many-body s t a t e s observed above 30eV in the expe r imen ta l s p e c t r a ( f i g u r e 8 . 2 ) . The expe r imen ta l energy p o s i t i o n s and p o l e - s t r e n g t h s below 30eV in the inner va lence r eg i on ( f i g u r e 8.3) are in b e t t e r agreement w i th the GF enve lope than wi th the SAC-CI enve lope . The SAC-CI enve lope above 20eV must be s h i f t e d towards the low energy s i de by 242 app rox ima te l y 1.2eV in o rder to o b t a i n a reasonab le agreement w i th the expe r imen ta l p o l e - s t r e n g t h s e s t ima ted at 21.2eV and 24 .3eV . I t i s important to note a l s o tha t such a downward s h i f t of the SAC-CI spectrum i s necessa ry i n order fo r the p o l e s (as a s s i g n e d by the c a l c u l a t i o n s ) to be c o n s i s t e n t w i th the momentum d i s t r i b u t i o n s measured at the co r r e spond ing e n e r g i e s (see l a t e r ) . No such s h i f t s are r e q u i r e d in the GF spec t rum. Momentum d i s t r i b u t i o n s measured i n the inner va l ence r eg i on can be used i n t h i s way to i d e n t i f y the o r b i t a l c h a r a c t e r of the many-body s t a t e s and to s u b s t a n t i a t e t h e o r e t i c a l ass ignments of the complex many-body s t a t e s . 8.3 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTIONS F i g u r e s 8 .4-8.9 show the momentum d i s t r i b u t i o n s (MDs) measured at s e l e c t e d s i t t i n g b i n d i n g e n e r g i e s of 9.8eV, 12.2eV, 14.4eV, 16.7eV, 21.2eV and 33.1eV r e s p e c t i v e l y . T h e o r e t i c a l MDs c a l c u l a t e d u s i n g the GAUSS76 431G wave func t ion [HL&80] (dashed l i n e s ) and the extended Gauss ian b a s i s wave func t ion of Langhof f [L84b] ( s o l i d l i n e s ) a re he igh t no rma l i z ed at the maximum fo r p-type d i s t r i b u t i o n s ( f i g u r e s 8 .4 , 8 .5 , 8.6 and 8.8) and are a rea no rma l i zed in the range 0-1.5 a 0 " 1 f o r s-type or sp-type d i s t r i b u t i o n s ( f i g u r e s 8.7 and 8 . 9 ) . I t shou ld be noted tha t the i n d i c a t e d e n e r g i e s i n the f i g u r e s a re not the v e r t i c a l I . P . ' s and co r r e spond on l y to the s i t t i n g b i n d i n g e n e r g i e s ( s e l e c t e d so 243 as to min imize the o v e r l a p from n e i g h b o r i n g s t a t e s see f i g u r e s 8.1 and 8 . 2 ) . A l though a b s o l u t e c r o s s s e c t i o n s are not measured, r e l a t i v e n o r m a l i z a t i o n i s ma in ta ined between the v a r i o u s expe r imen ta l MDs. In a d d i t i o n , momentum-space (p-space) d e n s i t y and p o s i t i o n - s p a c e ( r-space) d e n s i t y contour maps of i n d i v i d u a l o r b i t a l s c a l c u l a t e d us i ng the GAUSS76 431G wavefunct ion are a l so ' p r e s e n t e d . The d e n s i t y mapping conven t ion i s the same as that used in chap te r 5 ( s e c t i o n 5 . 4 . 1 ) . 2 8.3.1 Outer Va lence O r b i t a l s The s i t t i n g b i n d i n g ene rg i e s at which the c o r r e s p o n d i n g MDs are measured have been j u d i c i o u s l y chosen in o rder to min imize c o n t r i b u t i o n s from the n e i g h b o r i n g s t a t e s . I t i s c l e a r from f i g u r e s 8.1 and 8.2 tha t the MDs measured at 9.8eV, 12.2eV and 14.4eV can be a s s i g n e d e s s e n t i a l l y to the 27Tg, 2 7 r u and 5 o u o r b i t a l s r e s p e c t i v e l y . The MD sampled at l 6 . 7eV however c o n t a i n s s i g n i f i c a n t c o n t r i b u t i o n s from both the C and D s t a t e s because the sma l l n a t u r a l s e p a r a t i o n 2 B r i e f l y , the contour p l ane i s d e f i n e d by two o r thogona l d i r e c t i o n a l v e c t o r s w i th the (0,0,1) v e c t o r i n d i c a t i n g the b o n d - p a r a l l e l i n t e r n u c l e a r d i r e c t i o n . Contour l i n e s of d e n s i t y v a l ues c o r r e s p o n d i n g to 80, 60, 40, 20, 8, 6, 4, 2, 0.8, 0.6, 0.4 and 0.2% of the maximum d e n s i t i e s are shown. P r o j e c t i o n p l o t s of the d e n s i t y f u n c t i o n a long the i n d i c a t e d (dashed) l i n e s are used to show c l e a r l y the r e l a t i v e change in magnitude of the d e n s i t y f u n c t i o n s . The momentum and p o s i t i o n a re in atomic u n i t s . 244 between the two s t a t e s i s beyond the i n s t rumen ta l energy r e s o l u t i o n . The exper imenta l and t h e o r e t i c a l MDs of the o r b i t a l , a long w i th the c o r r e s p o n d i n g d e n s i t y maps are shown in f i g u r e 8 .4 . The p o s i t i o n d e n s i t y map c l e a r l y i n d i c a t e s the "nonbond ing" ( l one-pa i r ) c h a r a c t e r of the o r b i t a l which c o n s i s t s main ly of the S(3p , 3p ) atomic o r b i t a l s . The x y s l i g h t d i s t o r t i o n of the p o s i t i o n d e n s i t y near the p lane i s due to the nega t i v e o v e r l a p of the c o n t r i b u t i n g S(3p , 3p ) atomic o r b i t a l s . The c o r r e s p o n d i n g momentum x y d e n s i t y map a l s o shows a pseudo-d type main s t r u c t u r e w i th weak secondary l obes in the h igh p r e g i o n . There i s f a i r agreement between the expe r imen ta l MD and those c a l c u l a t e d by the extended Gauss ian b a s i s wave func t ion [L84b] and by the GAUSS76 431G wavefunct ion [HL&80] in the r e g i o n p > 0 . 5 a o " 1 . The t h e o r e t i c a l MD c a l c u l a t e d u s i n g the GAUSS76 431G b a s i s i s s l i g h t l y s h i f t e d to h ighe r momentum. In the r e g i o n p < 0 . 5 a o - 1 , the expe r imen ta l MD l i e s c l o s e r to the o r i g i n than the c a l c u l a t e d MDs. A s i m i l a r d i s c r e p a n c y i s a l s o observed in the MD of the c o r r e s p o n d i n g C 0 2 17r o r b i t a l [LB84b] y ( chapter 7 ) . T h i s sugges ts tha t the C S 2 2n^ o r b i t a l , l i k e the C 0 2 1ir o r b i t a l , may in f a c t be somewhat more d i f f u s e ( i . e . more extended i n r-space) than tha t mode l l ed by e i t h e r of the t h e o r e t i c a l wave func t i ons . A s i m i l a r d i s c r e p a n c y between the expe r imen ta l MD of the 2ir o r b i t a l ( f i g u r e 8.5) and the c a l c u l a t e d MDs i s a l s o SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION M O M E N T U M DENSITY POSITION DENSITY Figure 8 .4 - Molecular o r b i t a l momentum d i s t r i b u t i o n s ( l e f t ) , momentum d e n s i t y ( cent re ) and p o s i t i o n d e n s i t y ( r i g h t ) maps of the CS ? 2n o r b i t a l . 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 e va l ua t ed us ing wavefunct ions of the extended Gauss ian b a s i s [L84b] ( s o l i d l i n e s ) and the GAUSS76 431G b a s i s [HL&80] (dashed l i n e s ) . 246 observed in the r eg ion below O . S a o " 1 , d e s p i t e good agreement in the h i ghe r momentum r eg ion at l e a s t up to 1 . 5 a 0 _ 1 . The 2JTu o r b i t a l , as i s c l e a r l y e v i den t from the p o s i t i o n d e n s i t y map, r e p r e s e n t s the bonding ( p o s i t i v e ) o v e r l a p between S(3p , 3p ) and 0(2p , 2