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Theoretical calculations on molecules and clusters composed of some heavier elements Head, John David 1978

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THEORETICAL CALCULATIONS ON MOLECULES AND CLUSTEfiS COMPOSED OF SOME HEAVIER ELEMENTS-BY JOHN DAVID HEAD B.Sc.(Hons.), U n i v e r s i t y C o l l e g e , London 1973. A THESIS SUBMITTED IH PARTIAL FULFILLMENT OF THE REQUIREMENTS FOE THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Chemistry) He accept the t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September 1978 © J o h n Davia Head, 1978 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 o f the r e q u i r e m e n t s f o r an advanced degree a t the 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 , !I agree t h a 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 t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f 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 r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g 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 g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department n f C H e H j - S T < V  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 DE-6 BP 75-51 1 E ABSTRACT In t h i s t h e s i s c a l c u l a t i o n s have been made with the X=>< s c a t t e r e d wave ( X t x C S E ) and complete n e g l e c t of d i f f e r e n t i a l o v e r l a p (CNDO) methods. A common theme i n v o l v e s the p a r a m e t r i s a t i o n of the CHDO method f o r the hea v i e r elements (e.g. atomic numbers g r e a t e r than 10), although an i n i t i a l study concerned the r e - p a r a m e t r i s a t i o n of the CNDO method f o r systems based on L i F - , I t i s i n d i c a t e d that the c o n v e n t i o n a l CNDO/2 parameters appear inadequate f o r i o n i c systems which i n v o l v e s u b s t a n t i a l r e - a l l o c a t i o n of charge from the normal atomic s t a t e s . For t h i s a p p l i c a t i o n , the CHDO method was repa r a m e t r i s e d by d i r e c t r e f e r e n c e to a p r e v i o u s l y p u b l i s h e d near-Hartree-Fock c a l c u l a t i o n on di a t o m i c L i F , and the s u i t a b i l i t y of the new CNDO parameters was assessed by c a l c u l a t i o n s on (LiF)^ and on c l u s t e r s c o n s i s t i n g of up 18 atoms. In independent c a l c u l a t i o n s , Evarestov and Lovchikov (1977) showed these new parameters have advantages over the other s e t s when u s i n g CNDO-type c a l c u l a t i o n s f o r i n v e s t i g a t i n g the band s t r u c t u r e o f L i F . Chapter 3 r e p o r t s new X^SW c a l c u l a t i o n s f o r some c l u s t e r s formed by s i l v e r atoms; these are f o r Ag n and Ag t o. which simulate the (111) s u r f a c e o f s i l v e r , A g ( 3 and Ag|c, which s i m u l a t e bulk s i l v e r and two forms o f Ag^I which are designed to i n v e s t i g a t e the a d s o r p t i o n of i o d i n e atoms on the (111) su r f a c e of s i l v e r . L o c a l and t o t a l d e n s i t y of s t a t e s c u r v e s are r e p o r t e d , and high c o o r d i n a t e atoms have been found to e x h i b i t s t r o n g l o c a l i s a t i o n O B the e l e c t r o n i c s t r u c t u r e . So f a r CNDO p a r a m e t r i s a t i o n s a re not w e l l developed f o r So f a r CNDO p a r a m e t r i s a t i o n s a r e not w e l l developed f o r c l u s t e r s of t r a n s i t i o n metal atoms, and i n c h a p t e r 4 new CNDO parameters f o r s i l v e r have been o b t a i n e d by comparing l o c a l and t o t a l d e n s i t y of s t a t e s and charge d i s t r i b u t i o n s from CNDO c a l c u l a t i o n s with those from the Xo^SS method f o r the &g 7 c l u s t e r - The new CNDO parameters have then been used f o r making CNDO c a l c u l a t i o n s on f a r t h e r s i l v e r c l u s t e r s , namely Ag t, & 9 | D * fig ,3 and Ag , and the r e s u l t s a re compared with data from the Xc < s a c a l c u l a t i o n s . These CNDO c a l c u l a t i o n s give d-band widths i n broad agreement with those from the X Sfl method. The most s i g n i f i c a n t d i f f e r e n c e i s t h a t the CNDO method g i v e s l e s s l o c a l i s a t i o n on c e n t r a l atoms with high c o o r d i n a t i o n numbers than i s found from the X « S i c a l c u l a t i o n s . I t i s suggested t h a t t h i s apparent d e f i c i e n c y of the CNDO c a l c u l a t i o n s may be l e s s s e r i o u s when the c l u s t e r s are being used f o r modelling p a r t of a s o l i d metal r a t h e r than f o r s p e c i f i c a l l y i n v e s t i g a t i n g the p r o p e r t i e s of s m a l l metal p a r t i c l e s . The f i n a l c hapter of t h i s t h e s i s p r e s e n t s c a l c u l a t i o n s on systems composed of the elements aluminium to su l p h u r . New c a l c u l a t i o n s have been made with the X <x SW method f o r the molecular c l u s t e r s A l 7 # S i s H ^ and P^, and comparisons made with experiment and with other c a l c u l a t i o n s where p o s s i b l e . However the main reason f o r making these c a l c u l a t i o n s has been to use the charge d i s t r i b u t i o n s and t r a n s i t i o n - s t a t e e n e r g i e s , along with i n f o r m a t i o n obtained p r e v i o u s l y by Salahub e t a l . f o r S3, f o r d e r i v i n g new CNDO parameters f o r the elements aluminium to s u l p h u r . C a l c u l a t i o n s u s i n g these parameters have been t e s t e d a g a i n s t a new XckSl c a l c u l a t i o n made here f o r an a.l l o c l u s t e r (which simulates the ( 1 1 1 ) surface of aluminium), and against X<*SH calculations made previously for P s, p ^ s s » SiH^, PH S, HjS and S0 2. This way of getting CNDO parameters seems broadly successful in these cases, and t h i s work thereby provides a firmer basis f o r using results from X°<S» cal c u l a t i o n s f o r extending the CNDO procedures systematically to a wider range of the heavier elements-V ABSTRACT TABLE OF CONTENTS LIST OF TABLES ... LIST OF FIGURES --ft C K N0 W LEDG EH EN TS . CHAPTER 1. INTRODUCTION 1 1.1 Density of States ................................. 2 1.2 Semi-infinite Solids 4 1.3 Calculations on Clusters ................. .......... 6 1-4 Outline of Thesis ..........-...----.-»--».»-.-.•-- 9 CHAPTER 2. QUANTUM THEORY OF MOLECULES AND SOLIDS ..... 12 2. 1 Introduction ...•.»•.•..».........».....•••».•• — ».» 12 2.2 Hartree-Fock Equations ............................ 15 2.3 moi Equations ..«....••.....«.......••.....»•.••»•» 17 2.4 Slater's Transition State Theory 22 2.5 Rootaaan's Equations ..................,-- ......... 23 2.6 The Ze r o - D i f f e r e n t i a l Overlap Approximation ....... 26 2.7 Parametrisation of the CNDO Method ................ 29 2.8 Applications of CNDO to LiF ....................... 39 2.9 Scattered Have Theory for Molecules and Clusters 45 52 2.11 Molecular Charge D i s t r i b u t i o n s .................... .54 TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v i i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x v i i v i CHAPTER 3. X<* SS CALCULATIONS ON CLUSTERS 0? SILVER ATO MS — — — — • — -,•— — -. — .^. — ,.••-•». — — •— *• — - — — — — • — —- — ** — 3-1 I n t r o d u c t i o n -..---.•»•-..».--»----•»--•-••-•-»-•--3.2 Study of I on Ag(111) 3.2.1 S p e c i f i c a t i o n of C a l c u l a t i o n s ................ 3.2.2 R e s u l t s ................................. ..-^ <. • 3.2.3 C o n c l u s i o n .•,»......,•••..•.,»»•.».•• ••»••»«--—'• 3.3 C a l c u l a t i o n s on Ag l o , Ag l 3 and A g w ............... 3.3.1 Method of C a l c u l a t i o n 3.3.2 R e s u l t s ..........-.-----..-.--------«--------3.3.3 D i s c u s s i o n 3.4 Summary ......»..<....•.....--......".......•«»•»•»» CHAPTER 4. CNDO CALCULATIONS ON CLUSTERS OF SILVER ATO MS .».,.-»--.» — — « — .- — «- -«,»« — •», — .,-- — «••- —<•* — — -••• ,-95 4.1 I n t r o d u c t i o n .....*...........-....-.-»-••---••-»«?---• /99 4.2 S e l e c t i o n o f Ag CNDO parameters 101 4.2-1 Comparisons between CNDO and X<*S8 c a l c u l a t i o n s : »-,.«-,-«.,»--•-•, «.........••.••:«•«> 102 4.2.2 Parameters and P r o p e r t i e s Examined 103 4.2.3 Parameter S e l e c t i o n .......................... 104 4.3 Comparisons with the X^ SI C a l c u l a t i o n s f o r Ag 7 ...114 4.4 Other S i l v e r C l u s t e r s ..........-.-......---..-...-117 4—4-1 A g ^  ............... — .- .....-.-... — -...'.».-••—-117 4-4.2 Ag l o • .,...'...•..--»•«..•-•-•.--••»••'- — •••-"*-•« • 1 19 4.4.3 Ag,3 -.--.-...,-.,...-......-..-..-------.-.--119 4.4.4 Ag^ 124 4.5 Concluding Remarks -127 56 56 63 63 64 85 86 87 88 88 97 CHAPTER 5. CALCULATIONS ON MOLECULES AND CLUSTERS FORMED BY SECOND BOH ELEMENTS 129 5.1 I n t r o d u c t i o n .................. ....................129 5.2 X(7<SW C a l c u l a t i o n s on Second Sow Elements .........130 5.2.1 A1-, and A l l c .........,..,-...-.--,.-,,---.--.131 5.2.2 S i^H . . . . . , — . — ,.— ....,..-. — —,,,,,., — — ,,,,135 5— 2. 3 • P^ ...,,,».. ,•- - ,-— , , «•••.— — 140 5.2-4 D i s c u s s i o n ..,,...,.,.,.,,,,.,,,,,..,,.,,,.,,.141 5.3 S p e c i f i c a t i o n of the CNDO C a l c u l a t i o n s on the Second Row Elements . . . . . . . . . . . . 1 4 2 5.4 S e l e c t i o n o f CNDO Parameters i . . . . * , , , , , , , , 1 4 4 5-5 CNDO Parameter S e l e c t i o n f o r each Second Row E l em en t a , . - . . . , , . . . , . ! . . , , . , , . . . , . , — 149 5-5.1 Alumini um 150 5.2.2' S i l i c o n — —.— ...— ••..••»•—..'•...,, .,,,«»,, 154 5.5.3 Phosphorus 158 5.5.4 S ulphur — • - , — , ..,»,,, , . ...»»•»-».,.•»••• 161 5.6 A p p l i c a t i o n s to Other Molecules 164 5.7 I n c l u s i o n of d - o r b i t a l s 171 5.8 Co n c l u d i n g Remarks . . . . . . . . 1 7 7 REFERENCES ... , - - - - - - , - , .180 v i i i LIST OP TABLES I. Comparison of one-electron energies (£c/Hartree) and dipole moments (^/D) f o r occupied valence s h e l l molecular o r b i t a l s of diatomic LiF (bond length 2.8877 a.u.) using CNDO/2, non-empirical SCF £83]}, a n d MPCNDO cal c u l a t i o n s -......---....41 I I . Values of o r b i t a l s of L i III . Comparison of one-electron energies (in Hartrees) for (LiF)^ using CNDO/2, non-empirical SCF £87] and MPCNDO methods of c a l c u l a t i o n ........................................... 43 IV. Comparison of bulk CNDO c a l c u l a t i o n s using CNDO/2, Sichel and Whitehead (SW) 164]. and MPCNDO against standard hand structure c a l c u l a t i o n s (a) £89J and (b) £ 9 0 J . , - , , 4 3 V. ,Xo<S8 one-electron energies (in fiydbergs) f o r the valence o r b i t a l s of the Ag 7I clus t e r s ............................ 65 VI. Xo<SH one-electron energies (in Bydbergs) f o r the valence o r b i t a l s of the Ag 7 c l u s t e r 65 VII. * P a r t i a l wave decomposition of sphere charges (s,p,d) and net charges ( q n t t ) for d i f f e r e n t atoms i n Ag T and Ag.,1 ... 69 VIII. Charge d i s t r i b u t i o n s and d-band widths i n l o c a l DOS f o r parameters and ^  deduced f o r valence and F i n LiF 41 i x d i f f e r e n t s i l v e r c l u s t e r s 89 IX-. C a l c u l a t e d 3d c o r e l e v e l s h i f t s f o r d i f f e r e n t atom types i n the s i l v e r c l u s t e r s ...................................... 95 X. CNDO parameters f o r s i l v e r .......-...,.,,...,.....,...105 XI- l i s t o f d i f f e r e n t parameters used i n the CNDO c a l c u l a t i o n s O H A 5 ~f * * • * * • • * • *«•**<•*•*•*•••:*• * * • *• * * • mm m m m m m mm mm m-m 107 XII- Energy q u a n t i t i e s ( i n eV) obtained from the d i f f e r e n t CNDO c a l c u l a t i o n s on Ag n 108 X I I I . V a l e n c e - s h e l l charge a l l o c a t i o n s on the d i f f e r e n t atoms of the Ag-, c l u s t e r f o r the v a r i o u s s e t s of CNDO parameters 10<J XIV. Comparison of valence s h e l l charge a l l o c a t i o n s of d i f f e r e n t atoms of the c l u s t e r s A g l o , k g , z and Ag a c c o r d i n g to the X c< SH method and CNDO method using the f i n a l parameters i n t a b l e X 121 XV- Valence s h e l l e n e r g i e s ( i n eV) c a l c u l a t e d f o r A l 7 and A l r o .............................. 132 XVI. T o t a l s, p po p u l a t i o n s f o r the valence s h e l l s o f A l a n d X XVII. Valence s h e l l energies (in eV) for S i s H ^ 137 XVIII. Total s, p populations f o r the valence s h e l l of S i s H ( x ™ • • „ * * • • • * • * • ' ? ' • • "138 XIX* Calculated energies (in eV) for P^ . and i o n i s a t i o n energies from photoelectron spectroscopy ..,,,......,,.,....,,.,...141 XX. The Slater 149], Burns £76] and Clement! and Baimondi £75] exponent values for 41, S i , P and S 148 XXI. CNDO/2 and CNDO/HH parameters f o r ftl. S i , P and S ...149 XXXI. Individual and t o t a l s populations for the valence s h e l l of P^ . • ••••••••••»*<»•••••'*••. <•••••••••••«•»•*••• 161 XXIII. Comparison of io n i s a t i o n energies (in eV) calculated with the CNDO/2, CNDO/HH and X o< sw procedures for SiH ,PH and HjS 167 XXIV. Total s, p, d populations f o r the valence s h e l l of P^s * * • * • •««••"»• m *m m mm • • * * *» * • m-m m> m m m m m'm mmm mm *••••*••«•••• mmmmmmkmmmm mm 170 XXV. Total s # p, d populations from d i f f e r e n t CNDO cal c u l a t i o n s on P 4 . - . . . i 172 XXVI. Total s, p, d populations f o r P 4 using d i f f e r e n t CNDO d- o r b i t a l parameters; the s, p parameters are the CNDO/HH x i values ---..-...-....--.-..----..-..-.......-..-.---.--.--174 XXVII. Energy of s e p a r a t i o n (in eV) between the lowest and h i g h e s t occupied v a l e n c e - l e v e l s f o r the molecules used to parametrise the second-row elements ...................... 178 x i i LIST OF FIGOBES 1- A comparison of the t o t a l DOS f o r N i l 3 c a l c u l a t e d by EH, CNDO and X o < S H and the c a l c u l a t e d bulk Ni DOS {reproduced from £1.3]) ... 10 2- Comparison of groups of c l o s e l y packed e i g e n v a l u e s c a l c u l a t e d with the CNDO/2 and MPCNDO methods f o r some arrangements o f L i and F atoms c o n s i d e r e d by Hayns |, 79 ], . . 44 3- The cu b o - o c t a h e d r a l c l u s t e r c o n t a i n i n g the 13 atoms ... 57 4. The Ag 7 c l u s t e r used t o model the Ag(1l1) s u r f a c e ; the f u l l c i r c l e s are f o r atoms (designated 1-4) i n the top Ag l a y e r and the dashed c i r c l e s are f o r atoms (5-7) i n the second Ag l a y e r — 61 5. (a) Deconvoluted H e l l photoemission spectrum of c l e a n Ag from t12 3 3; (b) l o c a l DOS f o r Ag(2,3,5} i n Ag as c a l c u l a t e d with the Xo<sw method 6. DOS c a l c u l a t e d f o r the A g 7 I F c l u s t e r 70 7. DOS c a l c u l a t e d f o r the ftgTIH c l u s t e r ..................71 8- DOS c a l c u l a t e d f o r the Ag_, c l u s t e r .................... 72 9. Atom p o s i t i o n s i n the wavefunction contour p l o t s ( f i g u r e s x i i i 10-19); atoms I , A g ( 1 ) # Ag {U) , and Ag(5) are i n the plane of the of the p l o t s 74 10. The 93a* wavefunction f o r A g 7 I F — 75 11. The 102a» wavefunction f o r A g 7 I c 76 12- The 113a" wavefunction f o r Ag^Ip 77 13. The 114a* wavefunction f o r Ag^Ip 78 14. The 115a» wavefunction f o r A g ^ I F ..................... 79 15, The 93a* wavefunction f o r A g 7 l u - - - - - - - - - - f « 8 0 16.. The 102a* wavefunction f o r i g ^ I ^ ----------,,,-,-,«,-- 81 17- The 113a* wavefunction f o r Ag_I u 82 18. The 114a* wavefunction f o r A g 7 I M ..................... 83 19- The 115a* wavefunction f o r &g-,IH -------------^------- 8 1* 20- DOS c a l c u l a t e d f o r the Ag v o c l u s t e r .................. 90 21. DOS c a l c u l a t e d f o r the Ag,^ c l u s t e r .................. 91 22. DOS c a l c u l a t e d f o r the k q ) 0 l c l u s t e r ,92 x i v 23. V a r i a t i o n o f the CNDO e l e v e l s i n the d-band of Ag-, with 24. L o c a l and t o t a l DOS f o r Ag^ c a l c u l a t e d with the X<*SW method and the CNDO method using the f i n a l parameters f o r s i l v e r i n t a b l e X ..... ,.,115 25. T o t a l DOS f o r Ag f o r each i r r e d u c i b l e r e p r e s e n t a t i o n A,, A^ and E c a l c u l a t e d with the X <x SW and with t h e CNDO method ... ......... 116 26- A comparison of the valence s h e l l energy l e v e l s c a l c u l a t e d f o r the Ag f a c l u s t e r with the X « S W method £101] and with the CNDO method using the f i n a l parameters f o r s i l v e r i n t a b l e X *•>,-•- .->-.- - ... - -- -- 118 27. L o c a l and t o t a l DOS f o r Ag , 5 c a l c u l a t e d with the X csSM method and the CNDO method using the f i n a l parameters f o r s i l v e r i n t a b l e X 120 28. L o c a l and t o t a l DOS f o r Ag, 3 c a l c u l a t e d with the X°e SW method and the CNDO method us i n g the f i n a l parameters f o r s i l v e r i n t a b l e X ........................................122 29. L o c a l and t o t a l DOS f o r Ag | £ j c a l c u l a t e d with the X ^  SM method and the CNDO method u s i n g the f i n a l parameters f o r s i l v e r i n t a b l e X .,.,,,.,.,,..,..,.....,.,,,,,.,.,..,,,,,125 XV 30. Local and t o t a l DOS calculated with the X SB method for a i . o . . . . . . — —,,.,,,,..-,,134 31. The l i n e a r r e l a t i o n s h i p between the X <* SS one-electron ener g i e s 6 y ^ 3 V / 0 and the t r a n s i t i o n state energies £-rs .... - 143 32. Comparison for &1 7 of one-electron energies calculated with the CNDO/2 and CNDO/AH schemes and t r a n s i t i o n state energies calculated with the X<* SH method .........................151 33. Plots of B , and 1/B fo r the variation of a single ftl parameter whilst the other parameters are fixed at the CNDO/HM values ..............-.--..--------..-..-..---.-------.-.-153 34- Plots of R^ , fi^ and 1/B f o r the variation of a single Si parameter whilst the other parameters are fixed at the CNDO/HM values ...................................................156 35- Comparison for Si^H,^ of one-electron energies calculated with the CNDO/2 and C8DO/HM schemes and t r a n s i t i o n state energies calculated with the X^vSW method .,,,,.,,.,,,,,,.157 36. Plots of R<£, R ^  and B for the variation of a single P parameter whilst the other parameters are fixed at the CNDO/HM values 159 37- Comparison for P^ of one-electron energies calculated with the CNDO/2 and CNDO/HM schemes and t r a n s i t i o n state energies x v i c a l c u l a t e d with the . X<* SW method .<,.-....,,,,...,,....,,,-160 38- Comparison f o r S a of o n e - e l e c t r o n e n e r g i e s c a l c u l a t e d with the CNDO/2, CNDO/HH and Xcx SB methods 163 39- Comparison f o r A l l 0 of o n e - e l e c t r o n e n e r g i e s c a l c u l a t e d with the CNDO/2, CNDO/HH and Xo<S8 methods 165 40- Comparison f o r P g of o n e - e l e c t r o n e n e r g i e s c a l c u l a t e d with the CNDO/2, CNDO/HH and X<xSH methods 166 41- Comparison of v a l e n c e - s h e l l energy l e v e l s f o r P^.Sg .,,169 42. Comparison of v a l e n c e - s h e l l energy l e v e l s f o r SO^ -,.,176 x v i i &CKNOBLEDGEHENfS I wish to thank Dr. K.A.B. M i t c h e l l f o r being my s u p e r v i s o r and f o r h i s v a r i o u s words of encouragement d u r i n g the course of my r e s e a r c h . I am extremely g r a t e f u l to Dr. L. Noodleman f o r the many i n t e r e s t i n g d i s c u s s i o n s we have had conc e r n i n g the XtxSH method. I would a l s o l i k e t o thank of the X « s a computer programs. I am g r a t e f u l f o r a U n i v e r s i t y of B r i t i s h Columbia Graduate F e l l o w s h i p (1975-1978) 1 CHAPTER 1 INTRODUCTION During the l a s t few years there has been an i n c r e a s e d i n t e r e s t i n the t h e o r e t i c a l methods which c o n t r i b u t e t o an understanding o f e l e c t r o n i c and chemic a l p r o p e r t i e s o f molecules, c l u s t e r s and s o l i d s formed by the h e a v i e r elements (e.g. elements with atomic numbers g r e a t e r than 10). T h i s i s important both as an area o f i n t r i n s i c i n t e r e s t , but i t i s a l s o r e l e v a n t t o many modern t e c h n o l o g i c a l p r o c e s s e s . For example, recent developments i n experimentation have produced many p r a c t i c a l approaches t o the study o f the a t o m i s t i c p r o p e r t i e s of s o l i d s u r f a c e s , and these technigues i n c l u d e low-energy e l e c t r o n d i f f r a c t i o n (LEED), Auger e l e c t r o n spectroscopy (AES), X-ray photoemission spectroscopy (XPS), i o n n e u t r a l i s a t i o n s pectroscopy (INS) and f i e l d i o n microscopy (FIM) I 1 ]. Such work i s now p r o v i d i n g i n f o r m a t i o n which i s fundamental to an understanding of such a r e a s as heterogeneous c a t a l y s i s , c o r r o s i o n , vacuum technology and e l e c t r o n i c d e v i c e s . However the i n t e r p r e t a t i o n o f many experiments and the development of models of chemical processes a re both g r e a t l y aided by the e x i s t e n c e of r o u t i n e methods f o r c a l c u l a t i n g e l e c t r o n i c wavefunctions and e n e r g i e s . F o r s u r f a c e s the t h e o r e t i c a l t e c h n i g u e s are e s s e n t i a l l y of two types; those which r e c o g n i s e the s u r f a c e as being i n f i n i t e i n two dimensions, and those which are concerned with the l o c a l a s p e c t s of the s u r f a c e bonding. The f i r s t type n a t u r a l l y i n v o l v e an e x t e n s i o n of s o l i d s t a t e t h e o r i e s t o i n c l u d e a d e s c r i p t i o n of the s e m i - i n f i n i t e s o l i d . F o r the second type, the methods of quantum chemistry 2 are used t o examine the l o c a l bonding, the s u r f a c e being modelled by a c l u s t e r o f atoms. However c l u s t e r s are a l s o important i n t h e i r own r i g h t . Indeed many t e c h n o l o g i c a l c a t a l y t i c processes i n v o l v e supported metal c l u s t e r s , and t h i s encourages the f u r t h e r development of t h e o r e t i c a l schemes which are a p p l i c a b l e t o the h e a v i e r atoms. The r e l e v a n t quantum chemical techniques are examined i n t h i s study. ,, T h i s chapter aims to p r o v i d e some p e r s p e c t i v e t o the c l u s t e r approach, r e l a t i v e t o the other t e c h n i q u e s d e r i v e d from s o l i d s t a t e theory, and to g i v e a b r i e f overview of the s u b j e c t matter o f t h i s t h e s i s . 1 - J D e n s i t y of S t a t e s The s t a r t i n g p o i n t f o r most quantum mechanical s t u d i e s of e l e c t r o n i c s t r u c t u r e i s the time independent Schrodinger equation £2] approaches t o the s o l u t i o n o f t h i s f o r complex systems are g e n e r a l l y made w i t h i n the Born-Oppenheimer approximation £ 3 ] , where the equ a t i o n f o r the e l e c t r o n i c motion i s s o l v e d f o r f i x e d n u c l e a r c o o r d i n a t e s . F u r t h e r , f o r polyatomic molecules and s o l i d s , i t i s g e n e r a l l y necessary t o invoke the independent e l e c t r o n approximation, and t h i s r e q u i r e s s o l u t i o n of a pseudo o n e - e l e c t r o n wave equation f o r the o n e - e l e c t r o n energy e i g e n v a l u e s 6^ and the cor r e s p o n d i n g o n e - e l e c t r o n wavefunctions F o r s o l i d s , t h i s r e s u l t s i n a very l a r g e ( e s s e n t i a l l y (1.1). 3 i n f i n i t e ) number of o n e - e l e c t r o n e n e r g i e s , and an important g u a n t i t y i s the d e n s i t y o f s t a t e s (DOS) f u n c t i o n n(£)- T h i s i s d e f i n e d so t h a t nC&) it g i v e s the number of s t a t e s with energy eigen v a l u e between 6 and 6 + . I f the o n e - e l e c t r o n f u n c t i o n i s expanded i n terms of l o c a l i s e d b a s i s f u n c t i o n s , such as atomic o r b i t a l s , a l o c a l DOS can be d e f i n e d as f o r the j*- - t h l o c a l f u n c t i o n at the i - t h l a t t i c e s i t e . In (1.2), £ i s the D i r a c d e l t a f u n c t i o n . I t i s o f t e n convenient to c o n s i d e r the l o c a l DOS f o r each atom by summing over a l l b a s i s f u n c t i o n s , then f o r the i - t h s i t e and the atomic l o c a l DOS f o r an i n f i n i t e ordered s o l i d composed of a s i n g l e atom type w i l l be independent of the atom's l o c a t i o n i n the s o l i d , whereas i n the r e g i o n of l a t t i c e d e f e c t s and s u r f a c e s the l o c a l DOS w i l l vary with atom p o s i t i o n . The DOS have an important r o l e i n r e l a t i n g e l e c t r o n i c s t r u c t u r e c a l c u l a t i o n s t o experimental data. Thus i n OV photoemission the number o f e l e c t r o n s found at a c e r t a i n b i n d i n g energy w i l l to a f i r s t approximation be p r o p o r t i o n a l to the DOS. However, i n so f a r as OV photoemission i s s u r f a c e s e n s i t i v e , i t should g i v e i n f o r m a t i o n on the s u r f a c e r a t h e r than the bul k DOS. In a c l u s t e r c a l c u l a t i o n , a d i s c r e t e energy l e v e l spectrum i s o b t a i n e d , although i f the number o f e l e c t r o n s i s l a r g e , or f o r r e l a t i n g t o p r o p e r t i e s of an i n f i n i t e o r s e m i - i n f i n i t e s o l i d , i t i s o f t e n u s e f u l to d e f i n e analogous DOS f u n c t i o n s . R e p l a c i n g the D i r a c d e l t a f u n c t i o n i n (1.2) by a Gaussian enables the c l u s t e r l o c a l DOS t o be d e f i n e d as 4 " ( £ , ' ^ = 1 5 . ' r_ (1.4), where ^ i s the charge a s s o c i a t e d with eigenvalue at the i - t h atom, and <r i s a broadening f a c t o r - For an or t h o g o n a l s e t of b a s i s f u n c t i o n s ^ , the atomic charges a r e r 1 where the summation i s over the b a s i s f u n c t i o n s l o c a l i s e d a t the i - t h atom, A t o t a l DOS ^ (&) f o r a c l u s t e r i s obtained when i n (1.4) i s r e p l a c e d by the number o f e l e c t r o n s a s s o c i a t e d with e i g e n v a l u e £ u - The l o c a l DOS w i l l vary f o r the d i f f e r e n t atoms i n a c l u s t e r . One t e s t of using a c l u s t e r f o r s i m u l a t i n g a s o l i d o r a s u r f a c e i s t o check t h a t atoms c o o r d i n a t e d s i m i l a r l y have comparable l o c a l DOS both i n the c l u s t e r and i n the s o l i d £ 4 j . 1 - 2 S e m i - i n f i n i t e S o l i d s The o n e - e l e c t r o n Schrodinger eguation f o r ordered s o l i d s i s s i m p l i f i e d by the i n c l u s i o n o f l a t t i c e symmetry- The p o t e n t i a l due to e l e c t r o n - e l e c t r o n and e l e c t r o n - n u c l e a r i n t e r a c t i o n s w i l l have the same p e r i o d i c i t y as t h e l a t t i c e , and t h i s e n a b l e s the i n t r o d u c t i o n o f c y c l i c boundary c o n d i t i o n s -A c c o r d i n g t o B l o c h * s theorem the wavefunctions can be w r i t t e n as where fi^ i s a t r a n s l a t i o n l a t t i c e v e c t o r , and k i s a v e c t o r expressed i n r e c i p r o c a l space £ 5 ] ; the energy l e v e l s o b t a i n e d are a f u n c t i o n o f the vect o r k. The technigues f o r c a l c u l a t i n g 5 the energy band structure of regular s o l i d s are well established, and are described i n a number of standard texts £ 6 , 7 J . So f a r , however, a general s o l i d state theory has not been developed for the treatment of l a t t i c e defects and surfaces. If a l a t t i c e defect results i n a small change i n the l a t t i c e p o t e n t i a l , as i n the case of a P impurity i n a S i l a t t i c e , the Bloch function of the perfect l a t t i c e should not be strongly modified so enabling a perturbation treatment to be used; one such procedure i s the " e f f e c t i v e mass theory" £ 8 ] , Where the l a t t i c e defects correspond to strong perturbations, such as a vacancy i n the diamond l a t t i c e , other technigues must be used £ 9 ] , For a regular surface the potential i s periodic i n two dimensions, and the p e r i o d i c i t y of the s o l i d i s l o s t i n the di r e c t i o n normal to the surface. In t h i s d i r e c t i o n , the Bloch functions e f f e c t i v e l y decay as they approach the surface, and correspondingly the surface layers produce surface states which decay i n t o the s o l i d - The most extensive c a l c u l a t i o n s to date including these features have been performed f o r s i l i c o n by Appelbaum and Hamann £10 j , with a f u l l y s e l f consistent pseudopotential-based method. The surface region i s modelled by n layers (where n i s between 2 and 5 ) , and wavefunctions for t h i s n-layer region obtained with two dimensional p e r i o d i c i t y are made to match at the n-th layer with the bulk wavefunctions. Appelbaum and Hamann extended t h i s work to treat the adsorption of H on s i l i c o n surfaces £1.1 #12]; !EED demonstrates readily that adsorbates are freguently, although not always, ordered on surfaces. 6 I t i s not cl e a r at present whether the procedure used by Appelbaum and Hamann can be applied to t r a n s i t i o n metals or other materials with l o c a l i s e d d - o r b i t a l s £13]. Surfaces of these materials have often been treated by consideration of a c r y s t a l "slab" or " f i l m " consisting of a number of atomic planes p a r a l l e l to the c r y s t a l face of i n t e r e s t , the number of atom layers considered i s around 10 i n . most cases. This procedure i s intermediate between the c l u s t e r and the s o l i d state approach. The treatment of surfaces of t r a n s i t i o n metals usually makes use of a multiple scattering formalism which can be regarded as an extension to two dimension of the KKH method 114] used i n three-dimensional s o l i d state theory, Methods that use r e a l i s t i c potentials have been proposed by Beeby £15], Kasowski £ 16], Kar and Soven £17], and Kohn £ 18]. 1.3 Calculations on Clusters Theoretical treatments for non-bordered adsorption are analogous to those for l a t t i c e defects, and the cl u s t e r method provides a chemical approach which focuses on the l o c a l aspects of the bonding. This can be achieved by considering the i n t e r a c t i o n of the adsorbate with a c l u s t e r of atoms from the s o l i d , t h i s c l u s t e r being chosen to represent the surface or the l a t t i c e environment of the s o l i d £9,13]. An important consideration i n the approximation concerns how large a c l u s t e r needs to be to adequately represent the substrate. Some in d i c a t i o n of the necessary cluster size i s given by 7 the work of Einstein £19] and Davenport et a l £20]. E i n s t e i n determined the l o c a l DOS fo r atoms i n the surface region at di f f e r e n t distances from an adsorbate. The substrate was modelled by a simple cubic s-band s o l i d , and the l o c a l DOS f o r the next-nearest-neighbour atoms were only weakly modified from the clean surface l o c a l DOS £ 19. ], Davenport et a l have used a model Hamiltonian to show that the atoms on a second layer inside a surface already have the l o c a l DOS resembling that of a hulk atom £20]. These two c a l c u l a t i o n s demonstrate the l o c a l nature of the bonding encountered i n adsorption, however a clu s t e r d i f f e r s from the above systems by haying many more surface atoms. Nevertheless, aside from their possible help i n inter p r e t i n g surface phenomena, cal c u l a t i o n s on c l u s t e r s should be useful both for the i r d i r e c t relevance to understanding metal p a r t i c l e s on a support £21], and for assessing protruding atoms on i r r e g u l a r surfaces £22]-The e l e c t r o n i c structures of c l u s t e r s are generally calculated by the standard technigues of quantum chemistry- A set of discrete energy l e v e l s are obtained, usually by procedures based on the Hartree-Fock eguations £23]- As i n the s o l i d state case, the el e c t r o n i c equations can be s i m p l i f i e d by the introduction of symmetry, however t h i s i s determined by the point group of the system being investigated £24]. In practice l i m i t a t i o n s are set on the siz e of the system that can be investigated; correspondingly the accuracy of the c a l c u l a t i o n s f o r representing s o l i d s w i l l depend upon the computer time available. Claims have been made for useful c l u s t e r s consisting of as few as 5 atoms, although larger c l u s t e r s are to be 8 preferred f o r simulating s o l i d s or surfaces; c l e a r l y f or routine analyses for the heavier elements i t then becomes necessary to use either a semi-empirical molecular o r b i t a l procedure, or the s e l f consistent f i e l d Xoc' scattered wave (Xo< SW) method £25,26 ]. The necessity f or the faster approximate methods becomes c l e a r when i t i s r e a l i s e d that a non-empirical a l l - e l e c t r o n c a l c u l a t i o n on a molecule consisting of 26 atoms, a l l atomic numbers being le s s than ten, required some 10 1* i n t e g r a l s to be evaluated and the calcu l a t i o n s took 192 hours on an IBM 360/195 machine £27]. , Two semi-empirical molecular o r b i t a l methods are commonly used for cl u s t e r calculations; these are the extended Huckel (EH) method developed by Hoffmann £28], and the complete neglect of d i f f e r e n t i a l overlap (CNDO) method of Pople and co-workers £29,30]. In pri n c i p l e the CNDO method should be better than the EH method insofar that i t includes electron-electron repulsions, i t represents an approximation to the Hartree-Fock equations, and i t i s an i t e r a t i v e procedure. Another p o s s i b i l i t y i s the X <*Sw method, which has been developed recently by Slater and Johnson and which i s more computationally tractable than the non-empirical methods £31], The I o(SB method appears to introduce le s s severe approximations than those occurring i n the semi-empirical methods, and much experience now indicates that the X<?<SW method i s p a r t i c u l a r l y successful at calculating one electron properties such as io n i s a t i o n energies. A review of some of these c a l c u l a t i o n s can be found i n Slater*s book £32] and i n an a r t i c l e by Johnson £33]; further examples w i l l be given i n 9 t h i s t h e s i s . The re s u l t s of semi-empirical c a l c u l a t i o n s often need to he treated with caution owing to the v a r i a b i l i t y i n the properties obtained by d i f f e r e n t parametrisations. For example, Messmer et a l £34,35] recently compared the bulk DOS for n i c k e l £36] with the t o t a l DOS for a N i l 5 c l u s t e r obtained by the EH £37], the CNDO £38], and the X<* S» met hods. , The r e s u l t s of these c a l c u l a t i o n s are reproduced i n figure 1, and th i s appears to demonstrate the success of the X^SS c a l c u l a t i o n . Further considerations of such comparisons w i l l be made i n t h i s thesis. 1,4 Outline of Thesis The purpose of t h i s thesis i s to assess the cl u s t e r approach further and to examine the f e a s i b i l i t y of using semi-empirical molecular o r b i t a l methods, e s p e c i a l l y the CNDO method. The basis of the t h e o r e t i c a l methods used i n t h i s work w i l l be presented i n the next chapter. In chapter 3 some Xcx'SW cal c u l a t i o n s are presented f o r c l u s t e r s of s i l v e r . This work started as an attempt to provide an understanding of the geometry formed by iodine atoms adsorbed on a Ag (111) surface, but before a sa t i s f a c t o r y analysis of that problem could be tackled with confidence i t turned out to be advantageous to make some more detailed analyses of di f f e r e n t c l u s t e r s i z e s and coordinations, and to consider the l o c a l DOS as well as the t o t a l DOS f o r the s i l v e r c l u s t e r s . A study of CNDO calculations on s i l v e r c l u s t e r s i s made i n 1 0 ENERGY (RYDBERGS) ENERGY (RYDBERGS) Figure 1 . A coaparison of the t o t a l DOS f o r N i i j c a l c u l a t e d by EH, CNDO and XKSW and the c a l c u l a t e d bulk Ni DOS (reproducad from r 1 3 D . 11 chapter 4. I t i s indicated that with c a r e f u l choice of CNDO parameters the semi-empirical method can provide useful information. The parametrisation was done with data from the X<* sw ca l c u l a t i o n s . This approach appears advantageous for systems where data from experiment or from near-Hartree-Fock calc u l a t i o n s are sparse or absent. In chapter 5, the idea of using X c< SS c a l c u l a t i o n s to parametrise the CNDO method i s extended to the second row elements a l , S i , P, and S. I t appears that t h i s procedure for parametrising the CNDO method should be extendible on a systematic basis to a wide range of the heavier elements. An advantage f o r exploratory studies i s that the re s u l t i n g CNDO cal c u l a t i o n s can be made for e l e c t r o n i c a l l y complex systems f o r r e l a t i v e l y modest expenditures on computing. 12 CHAPTER 2 QOANTUH THEORY OF MOLECULES AND SOLIDS 2-1 Introduction In t h i s chapter a review of the quantum chemical methods used i n t h i s thesis w i l l be given- The f i r s t part of t h i s chapter, up to section 2-4, deals with the s e l f consistent f i e l d theory, including the Hartree-Fock and X <* equations- The l a t t e r part of the chapter reviews the approximations used i n solving the Hartree-Fock and X equations, such as the l i n e a r combination of atomic o r b i t a l s (LCAO) expansion, as well as other aspects associated with approximate procedures, such as the zero d i f f e r e n t i a l overlap (ZDO) approximation, the se l e c t i o n of parameters for the CNDO method, the scattered wave formalism and the choice of parameters required i n the Xo< calc u l a t i o n s , , Since many chemists have some f a m i l i a r i t y with the Hartree-Fock equations and molecular o r b i t a l theory, extra emphasis w i l l tend to be given to properties associated with the Xc* equations. The t o t a l e l e c t r o n i c energy i n both the Hartree-Fock and X<* approaches i s obtained from the el e c t r o n i c Hamiltonian, which for n-electrons and N-nuclei i s where 2 - f i s the charge on nucleus P, r ? ^ i s the separation between the electrons p and q, and r p ^ i s the separation between electron p and nucleus P £2]. The Hamiltonian in (2.1) i s expressed i n atomic units (a. u.) , where 1 a-u. of 13 energy equals 27.2116 eV (one Hartree), and 1 a.u. of length o equals 0.529177 A (the Bohr radius). Another a.u. of energy frequently used in I o ( c a l c u l a t i o n s i s the Bydberg which equals h a l f a Hartree, however i n t h i s chapter only Hartree units w i l l be used. The three terms i n (2.1) give i n turn the summed elec t r o n i c k i n e t i c energy, the electron-nuclear a t t r a c t i o n s , and the electron-electron repulsion energy. For fixed nuclear positions, within the Born^Oppenheimer approximation I 3 }, the t o t a l energy i s " - ^ ' - K?a (2.2), where B ? i s the position vector of each nucleus, E i s the t o t a l e l e c t r o n i c energy obtained from solving the Schrodinger equation with the e l e c t r o n i c Hamiltonian i n (2.1), and the second term sums the nuclear-nuclear repulsions. The t o t a l e l e c t r o n i c energy i s given by E = ^ Arc J X 1 (2.3) , where the integration i s over a l l electron space and spin coordinates. Within the independent electron or o r b i t a l approximation the t o t a l e l e c t r o n i c wavefunction ^ i s represented by a Slater determinant, which aay be abbreviated by the diagonal term and where i n general ^ ^ f ^ i s the p-th normalised function for the g-th electron. .These spin o r b i t a l functions are usually written as a product of a s p a t i a l function and a spin function of either o< or ^ type. In the absence of spin o r b i t coupling e l e c t r o n i c wavefunctions are chosen to be eigenfunctions of the spin operators S 2 and S £ , In general t h i s may require a sum of determinants to describe the system, although i n t h i s thesis we w i l l be only concerned with single determinantal functions. Substituting (2-4) into (2.3) gives the t o t a l e l e c t r o n i c energy t ~ Hpp * ± - h : * * } (2.5), provided the spin o r b i t a l s i n (2.4) are a l l orthonormal. The terms in (2.5) are V - W * c ^ t o r ' ^ V X >> ^ ( 2* 6 > # (2.8) and (2.9). The one-electron i n t e g r a l H ^  sums the ki n e t i c energy and the energy of at t r a c t i o n to the N nuclei f o r an electron i n the spin o r b i t a l . The Coulomb integral. J p ^ measures the inter a c t i o n between the two e l e c t r o n i c charge d i s t r i b u t i o n s 1^* 1 ^ and ^ ^ ^ ^ i and the exchange i n t e g r a l K , which has i t s o r i g i n i n the antisymmetric nature of the e l e c t r o n i c wavefunction, measures the inte r a c t i o n between the two ele c t r o n i c d i s t r i b u t i o n s ^ ^ ) a n d ^ k ) o ^ ( - i ) . From the defining equations i t i s seen that K fj> equals J p p and t h i s provides a correction f o r the s e l f i n t e r a c t i o n energy of the electron 15 repulsion i n (2-5). Also introduces an energy s t a b i l i s a t i o n due to the p a r t i a l c o r r e l a t i o n of electrons with p a r a l l e l spin; Kp<^ i s zero f o r electrons with a n t i p a r a l l e l spin. 2.2 Hartree-Fock Equations The best one-electron wavefunctions i n (2.4) are obtained by minimising the energy expression (2.5), subject to maintaining the orthonormality of the o r b i t a l s ^ 139], Using the technigue of Lagrange* s undetermined m u l t i p l i e r s , the /^j-j> are obtained by solving F 1 1 j, = ^ f ^ (2-10), where f = U t o r ^ + g C ^ - k O (2.11), and the Coulomb and exchange operators, J ^  and respectively, are written as To CO = fof Vra ^  C ^ (2- 12) , and These equations were f i r s t derived by Fock £23] based on e a r l i e r work by Hartree £40], and are generally known as the Hartree-Fock equations. Equation (2.10) states that the best one-electron functions or o r b i t a l s are eigenfunctions of the Fock hamiltonian F. The operators J<^ and are evaluated from the one-electron functions , and (2.10) i s a pseudo-eigenvalue equation. The solution of an Hartree-Fock equation 16 i s started by choosing a t r i a l set of o r b i t a l s , from which a preliminary F can be evaluated and (2- 10) solved to give r a new set of o r b i t a l s . The c a l c u l a t i o n i s repeated i t e r a t i v e l y u n t i l the set of o r b i t a l s at the end of the cycle are indistinguishable from those used at the s t a r t , at that stage, the wavefunctions are s e l f consistent with th e i r own pot e n t i a l f i e l d . A complete solution of the Hartree-Fock equations minimises the energy f o r a t o t a l e l e c t r o n i c wavefunction expressed as a single determinantal function. Lower energies can of course be obtained by using a more complicated e l e c t r o n i c wavefunction, such as a l i n e a r combination of Slater determinants as used i n the configuration i n t e r a c t i o n scheme (.41]. These approaches are outside the scope of t h i s t h e s i s , however they do remind us that s t r i c t l y c o r r e l a t i o n corrections are needed with c a l c u l a t i o n s at the l e v e l of Hartree-Fock theory. The one-electron eigenvalue 6 j > i s given by where H^ , and K ^  are given i n eguations (2.6) - (2. 9) . A convenient interpretation of &^  i s provided by a theorem due to Koopraahs £42] which states that given the solutions of the Hartree-Fock equations, and i f these solutions are also stationary for an n-1 electron system, where one electron has been ionised from . Then f. sr P _ £ , (2-15) , where E ^ i s the energy expression f o r n-electrons given by (2-5), and E„._| i s the corresponding expression f o r the n-1 electron system- According to (2.15), a one-electron eigenvalue 17 equals the negative of the i o n i s a t i o n energy, provided the o r b i t a l s do not change on removing the electron from the system. As »ell as a correction associated with t h i s relaxation, a c o r r e l a t i o n correction must also be expected in r e l a t i n g calculated £y> values to measured i o n i s a t i o n energies £433. 2-3 Equations The Xoi equations provide a s t a t i s t i c a l approximation to the exchange potential i n the Hartree-Fock equations. The advantage of t h i s approximation i s that i t greatly s i m p l i f i e s the evaluation of the exchange terms as i t , r e s u l t s i n a l o c a l p o t e n t i a l f o r each electron. By contrast, i n the Hartree-Fock method there i s a d i f f e r e n t l y defined l o c a l p o t e n t i a l associated with each o r b i t a l , and t h i s greatly increases the computational e f f o r t f o r systems with large numbers of electrons. The Hartree-Fock t o t a l energy expression i n (2-5) can be rewritten as E - <S H P p -v E c -t E x , (2-16). where the Coulombic energy E i s p t ^ I (2.17) in terms of the one-electron Coulomb potential U and (2. 18) , 18 fCC) - £ , C^^- vCO (2-19), where n i s the occupation number of the molecular function The charge density operator >^ can be interpreted as an alt e r n a t i v e to the use of the Slater determinant; they are related by « ° J ~ ^ ( 2 - 2 0 ) , although yi? i s more general than a determinantal function £ 41 J . The exchange i n t e g r a l E x i n (2.16) i s a sum of contributions f o r each spin type £ 2 2 k *v + ( 2 - 2 1 ' or . i t ' f E x = 2 . C ^ ^ a x H F ^ C ^ + U x k F d ^ l ( 2 - 2 2 ) , where ptC*") i s defined analogously to (2.19) but the summation i s only over o r b i t a l s with spin up. The one-electron pot e n t i a l U K M 5^ i s given by rt-tf • = ~ and corresponding expressions are used to define £>b(<>} a n d ° K H f V 5 n e one-electron exchange pote n t i a l may formally be interpreted as being associated with an "exchange charge density" £32J This represents the Fermi hole, associated with each electron, which corresponds to a sphere of influence, from whicJh el e c t r o n i c charge i s excluded; s p e c i f i c a l l y i t prevents the coincidence of two electrons with p a r a l l e l spins- The 19 coincidence of electrons 1 and 2 gives a cancellation of the Coulomb s e l f interaction term, as can be seen from (v, i) « - p i (2-25). Integrating over a l l space and spin coordinates f o r electron 2 gives ' Sf>x*^>A ^ - -± ' (2.26), and t h i s indicates that the Fermi hole i s associated with a unit charge. These two properties of ^ are independent of the position of electron 1 and t h i s suggests that the exchange density may simply be described by '•^(v,a\ = - j o ^ l ^ C r l V | ^ (2-27), where f i s a r a d i a l function dependent only on the separation between electrons 1 and 2. Equation (2-25) requires that f(0) i s unity, and equation (2-26) r e s u l t s i n 4nc} ^ tocU = 1 (2.28). Substituting p \ into the one-electron exchange potential gives r r oO = - 4 T T A Z i \ ) \ 0 % ^(.-A (2>29), and on eliminating the range parameter a, t h i s expression becomes U * W F * ( . ^ - ~ C C 4 Y T ( > ^ ^ (2. 30), where Thus the one-electron exchange potential i s proportional to the t h i r d power of the charge density p - I t i s generally not possible to evaluate C e x p l i c i t l y by (2.31), and therefore C i s usually approximated via a scaling parameter c< so that one-^ / s > (2-31). 20 electron exchange potential i n (2.30) becomes U y ^ = " 5 oi ^ p t c O ' ] " 3 (2.32).. For a free electron gas, C can be determined d i r e c t l y from (2.31) and t h i s corresponds to <^  having the value 2/3. For atomic and molecular systems, of i s usually between 2/3 and 1, the choice of appropriate values i s discussed i n section 2-10. The X c * expression for the e l e c t r o n i c energy i s obtained by substituting eguations (2-17), (2.22) and (2.32) into (2.16). This gives Ak j C ^ x c , p i cO + ^ C 0 piv C,)3 a r c , (2. 33) , and application of the v a r i a t i o n a l method £39 ] leads to the one-electron eguations with a corresponding equation for spin down o r b i t a l s . The one-electron eigenvalue i s These X o< one-electron eguations are derived by an analogous argument to that used f o r deriving the Hartree-Fock equations. However, the ULU one-electron energy i s fundamentally d i f f e r e n t from the Hartree-Fock one-electron energy, and i s related to the Xo< t o t a l e l e c t r o n i c energy by the r e l a t i o n (2-36), where n p i s the occupation number of the orbital/1|-^ -Equation (2.34) i s the spin-dependent form of the Xck one-electron equations. A spin-independent onei-electron equation can be obtained i f the exchange i n t e g r a l E x , i n equation (2.22), i s written, instead of a sum of the two i n t e g r a l s over 21 space and either spin ^ or spin t , as a single i n t e g r a l over a l l space and spin. Thus * i i P f0 U * ^ ^ (2-37), and the one-electron exchange pote n t i a l i s now approximated as The X c< t o t a l energy has the same form as (2.33) = S- ? Vl P p H^CO lAcdUf, +~ f^'O U^(,)^x and the spin-independent one-electron equation i s (2.39) ; where the one-electron energy (z-^ i s s t i l l given by (2.36). A l l new c a l c u l a t i o n s reported i n t h i s thesis have been performed with the spin-independent Xoi. equations. One of the major differences between the X a n d Hartree-Fock methods i s that the o r b i t a l s i n the XC* method may be p a r t i a l l y occupied, t h i s being a consequence of expressing the X c < t o t a l energy E X o < i n terms of the density operator p These f r a c t i o n a l occupation numbers enable Fermi s t a t i s t i c s to be rigorously obeyed by the Xc* scheme [32]. Also since any system can be represented by a single density operator, the X cx t o t a l molecular energy w i l l be the sum of the free atom t o t a l energies i n the l i m i t of large internuclear separation. In contrast, the Hartree-Fock method would require several determinants to represent the free atom and i o n i c s i t u a t i o n s i n the separated atom l i m i t . A more complete review of the X<x formalism can be found i n references £25,32,44,45], 22 2-4 Slater's Transition State Theory Useful expressions for i o n i s a t i o n energies and e x c i t a t i o n energies of a system can be obtained within the X u theory by expressing Ey^. i n a Taylor s e r i e s as a function of occupation number n t , with a l l other occupation numbers fixed Considering e x p l i c i t expressions f o r t h i s series when the k-th o r b i t a l i s f i r s t occupied and then empty, about the intermediate state with n*^  = ,1/2 gives (2.42) , and E * ( » V - E M ( V V J ^ ( v t v i ^ £ « t M .45, ( , t V . . S H . 4 S r ^ . w | . (2.43). The energy f o r removing one electron from the k-th o r b i t a l i s where equation (2.36) was used to introduce 6 ^ . Because of the dominant Coulomb term, E ^ i s , to a good approximation, a quadratic function of n^. Therefore, the i o n i s a t i o n energy I fe of the k-th o r b i t a l can be approximated by the negative of the k-th energy eigenvalue, calculated for the state with h a l f an electron removed. This state defines a t r a n s i t i o n state. The t r a n s i t i o n state concept, o r i g i n a l l y proposed by Slater 146 ], provides an equivalent i n the X oi. framework to the use of 23 Koopmans' theorem within the Hartree-Fock method. Exc i t a t i o n energies can also he calculated using a t r a n s i t i o n state, where the Taylor expansions are made about the i n i t i a l and f i n a l states with half an electron i n each- The The t h i r d order terms have been calculated i n a number of cases and found to be small (e.g. ,0.1 eV) £47]. The t r a n s i t i o n state concept often provides calculated i o n i s a t i o n energies which are closer to experimental values than are those from the Hartree-Fock method with Koopmans* theorem- & possible advantage of the use of the t r a n s i t i o n state theory i s that the Xo< i o n i s a t i o n energy i s obtained from an o r b i t a l with an electron p a r t i a l l y removed, and t h i s enables some account of the relaxation e f f e c t s to be made. In many cases i t has been found that the t r a n s i t i o n state energies correspond to a uniform s h i f t downwards i n energy from the one-electron energies 6^. This provides a useful s i m p l i f i c a t i o n when one i s only interested i n the r e l a t i v e spacing of the l e v e l s rather than t h e i r absolute values. This point w i l l be elaborated on later i n the thesis-2-5 Boothaan«s Equations The normal method of solving the Hartree-Fock eguations f o r molecular systems i s to express the molecular o r b i t a l s ^ e x c i t a t i o n energy from o r b i t a l \ L to i s (2.45) . i n terms of basis functions 24 - <S C i f (2-46,.,, The y>ji* may be any convenient set of functions, but most often they correspond to atomic o r b i t a l functions centred at the nuclear positions and have the form of (2.47); the molecular o r b i t a l s are then expressed i n the l i n e a r combination of atomic o r b i t a l s (LCAO) scheme £48]. In (2.47) ^ c r , e , eft = R n t ( 0 Y^ c © , ^ <2-«7>< YLy^(9;<ft i s a spherical harmonic £39] and 8^ ( 1 - ^ i s a r a d i a l function. The l a t t e r can be expressed i n various ways, but in th i s thesis Slater type o r b i t a l s £49], given by have generally been used. In (2.48) n i s the p r i n c i p a l guantum number, and ^ the o r b i t a l exponent which takes into account the electron screening e f f e c t . As an alternative to (2.48) many non-empirical c a l c u l a t i o n s have used Gaussian r a d i a l functions £50], With the expansion (2.46), the molecular integ r a l s a r i s i n g i n the Hartree-Fock theory need to be expressed i n terms of atomic i n t e g r a l s . Thus for the overlap i n t e g r a l between two molecular o r b i t a l s one obtains where ^ <^o(>) <kx, (2.50). _ Similarly the one-electron core i n t e g r a l H ^ » the Coulomb i n t e g r a l J ^  and the exchange i n t e g r a l K : j are given by where o / 25 (2.52) , ^ ' - . S f j ^ S ^ ^ C ^ C ^ (2-53), and ^ - S £1 2 g c a i c t j ^ ( r , 1 . where ^ . 55) . For a closed s h e l l of 2n electrons, where each s p a t i a l function i s doubly occupied with one electron of ol spin and one of ^ spi n , the Hartree-Fock t o t a l e l e c t r o n i c energy becomes where i s the density matrix defxned by This i s related to the density operator >^ introduced in section 2.3, and i t i s used i n the population analysis, developed by Mulliken 15,1], which i s generally used within the LC&0 scheme f o r interpreting charge d i s t r i b u t i o n s i n molecules (see section 2.11 f o r further d e t a i l s ) -The condition f o r minimum energy E, subject to the constraint of the orthonormality of the molecular o r b i t a l s expressed by (2.46), i s given by the solution of S r f ^ - ^ s ^ o - ° . ( 2 - 5 8 ) ' where the matrix elements of the Fock operator F ^ i s given by These equations are generally c a l l e d the Boothaan equations £48], although they were also derived independently by Hall 26 152] . The Boothaan equations d i f f e r from the Hartree-Fock equations i n (2.10) in that they are algebraic rather than of d i f f e r e n t i a l form, but again they have to be solved i t e r a t i v e l y . Generally i n solving the Boothaan equations, p a r t i c u l a r l y f o r electronically-complex systems, a compromise has to be made between the accuracy required and the s i z e of the basis set used. The l e a s t accurate cal c u l a t i o n s are made with a minimal basis set comprised of atomic o r b i t a l s up to and including o r b i t a l s of the valence s h e l l for each atom (one function of type (2.47) f o r each o r b i t a l on the atom). Better r e s u l t s are expected with extended basis sets which include extra atomic o r b i t a l functions both inside and outside the valence s h e l l . However, the larger the basis set> the greater the number of two-electron i n t e g r a l s which are r e l a t i v e l y time-consuming to evaluate; their number increases as the fourth power of the basis s i z e . 2.6 The Ze r o - D i f f e r e n t i a l Overlap Approximation The z e r o - d i f f e r e n t i a l overlap (ZDO) approximation amounts to making the following approximations f o r i n t e g r a l s involving the atomic basis functions, thus " ^  (2.60) , f ^ l V ^ ly^lVX} ^ W (2.61), and £ CO l r l B focO A-c, - ^ V £ < 2 - ™ • This set of approximations was o r i g i n a l l y suggested by Parr 27 £53] f o r p i systems, and i t enables a large reduction i n the number of i n t e g r a l s to be evaluated i n the Roothaan eguations. Many of those inte g r a l s which are neglected have values close to zero, although errors are c e r t a i n l y introduced by t h i s approximation. Some authors have avoided i t s use f o r the one and two-centre electron-nuclear a t t r a c t i o n i n t e g r a l s £54,55], but the i n t e g r a l s which are most numerous, and are therefore most affected by the 2D0 approximation, are the two-electron i n t e g r a l s . The various l e v e l s of approximate s e l f consistent molecular o r b i t a l theory d i f f e r mainly in the extent to which the 2D0 approximation has been invoked f o r these electron repulsion i n t e g r a l s . When applied consistently to a l l two-electron i n t e g r a l s , the Focfc matrix elements between ^ on atom A and ^ on atom B become = - ^ ^ V ^ (2.64), . where V " J^" ^ [ - ' ' l ^ O -*«/r,„3^(,)aX, ,2.65), y - ^ t - ' ' ^ ' t o - f a t i t ; ( 2 _ 6 6 ) r and ^> - f ^ h ^ <2-67>-Equations (2.63) and (2.64), f o r a minimum basis set of Slater type o r b i t a l s for the valence s h e l l , form the basis of the complete neglect of d i f f e r e n t i a l overlap (CNDO) approximation. The CNDO method i s semi-empirical because some i n t e g r a l s are determined empirically, although others may be evaluated 28 a n a l y t i c a l l y . This i s discussed i n the next section i n r e l a t i o n to parametrisation schemes. A l e s s approximate s i m p l i f i c a t i o n to the Fock hamiltonian i s to retain monoatomic d i f f e r e n t i a l overlap i n the one-electron i n t e g r a l s . This enables some exchange terms to be treated l e s s crudely, and i t forms the basis of the intermediate neglect of d i f f e r e n t i a l overlap (INDO) method £56], Discussion of the INDO method and other related schemes which are l e s s approximate than CNDO can be found elsewhere 154-57 ]. Ideally properties calculated with semi-empirical methods are invariant to the various transformations, i n much the same way as the f u l l s e l f consistent f i e l d equations £ 29 ]. In practice, the semi-empirical methods should be invariant to transformations of the molecular coordinates and to transformations which generate hybrid o r b i t a l s ; that i s the calculated properties should be unchanged i f the molecular axes are redrawn or i f hybridised atomic o r b i t a l s are used instead of pure atomic o r b i t a l s . In the CNDO method these requirements are ensured by making the following approximations: f/y, I ^ * K g ( 2 . 6 8 ) R V ^ - V A D 77* ftB (2.69), and ^ M O = (2.70) , where cj)^ i s on A and ^  i s on B; equation (2-70) includes the Mulliken approximation £58]. With a l l these approximations, the Fock matrix as used i n 29 the CNDO method becomes ^ - ' i ? ^ X « , (2.72), where P & g i s the t o t a l electronic density associated with atom \% = *Zl *\> (2-73) , and the summation i s over a l l o r b i t a l s on B. The values of uu are made to depend on o r b i t a l type i n order, f o r example, to maintain the important energetic d i s t i n c t i o n between s and p o r b i t a l s . 2-7 Parametrisation of the CNDO Method The evaluation of the Fock matrix elements, as used i n the CNDO method (eguations (2.71) and (2.72)}, requires values for the following parameters: a) & M , the two-centre two-electron repulsion i n t e g r a l s ; h) » t n e e s s e n t i a l l y atomic term which gives the k i n e t i c energy of an electron plus the at t r a c t i o n to the nucleus on which ^ i s centred; c) , which measures the a t t r a c t i o n of an electron i n an o r b i t a l centred on A with the nuclear core at B; d) (2>° G , the bonding parameter used i n (2.70) ; e) , the overlap matrix elements. An i n i t i a l decision i n choosing values f o r these parameters depends on the goal of the CNDO c a l c u l a t i o n s . For 30 example, i n t h e i r early work, Pople and Segal attempted to f i n d parameter values that allowed for CNDO ca l c u l a t i o n s to reproduce the r e s u l t s of minimal basis non-empirical calculations £54], whereas Hiberg £59], concentrated on predicting experimental heats of formation* The CNDO/2 parametrisation has probahly become the most widely used scheme, and t h i s i s detailed i n the next paragraph f o r elements i n the f i r s t row of the periodic table (Li to P) £54J. The two-electron i n t e g r a l s Kftg # i a the CNDO/2 scheme, are evaluated a n a l y t i c a l l y using Slater type 2s o r b i t a l s , the exponent \ for each atom being chosen by Slater*s rules £49], The overlap integrals S ^ are also determined e x p l i c i t l y (but the angular components of the atomic o r b i t a l are included i n the evaluation). The core i n t e g r a l s are determined from where ^ i s on A, and (I^*A^)/2 i s the average of the io n i s a t i o n energy and electron a f f i n i t y for the ^  -th o r b i t a l obtained from atomic spectral data. The electron-nuclear a t t r a c t i o n parameter Vft& i s approximated by V f t B « - g B W (2-75), and the bonding parameter xs wrxtten as /* ( 2- 7 6'' where ^ i s an atomic quantity, and i s determined by comparison against accurate minimal basis non-empirical c a l c u l a t i o n s . The f i n a l CNDO/2 equations are and $.3 = jCs - V ^ ^ ( 2 . 7 8 ) . 31 The other parametrisation schemes f o r f i r s t row elements are usually based on equations (2-77) and (2.78), but they represent d i f f e r e n t methods of sel e c t i n g the parameters. For example, Sichel and Whitehead £60] obtain the atomic parameters Us^ , 0"^ and ^ from the valence state energies of Hinzre and J a f f e [61 J. This approach represents a more extensive analysis of the atomic spectral data, than performed f o r CNDO/2, with the f i n a l two-electron i n t e g r a l being obtained by averaging a l l the one-centre i n t e g r a l s . S i c h e l and Whitehead also suggested that the two-electron i n t e g r a l X(\g should be evaluated with the semi-empirical formulae of OJhno-Klopman 162], although the Hataga-Nishimoto (63] method gave almost comparable r e s u l t s . The bonding parameters R° were obtained by f i t t i n g to the binding energies of binary hydrides £64]. Boyd and Whitehead £65] improved on t h i s by choosing as the average of and , which were i n turn obtained by f i t t i n g to the experimental d i s s o c i a t i o n energies of the diatomic molecules A a and B^. Bene and Jaf f e £66] used the Pariser and Parr £67] formula, o r i g i n a l l y used i n p i molecular o r b i t a l theory, to obtain the two-electron i n t e g r a l that i s - Xft -Eft (2.79) , where 1^ and E ^  are f o r atom A the i o n i s a t i o n energy and electron a f f i n i t y respectively. The two-centre i n t e g r a l s ]^ f t B were calculated by simple e l e c t r o s t a t i c s , using the charge sphere approximation. An i n t e r e s t i n g feature of t h i s work i s the approximation introduced for the resonance i n t e g r a l , namely I C . + n * \ (2-80) , 32 where K i s set to 1 for sigma bonds and 0.585 fo r pi bonds. This enables a d i f f e r e n t i a t i o n between the strength of the sigma and pi bonds, which i s removed from the CNDO equations when r o t a t i o n a l invariance i s enforced. This procedure has only limited applications, for example to planar molecules where the d i s t i n c t i o n between the p ? o r b i t a l , forming a pi bonding system, and the p* and py o r b i t a l s , forming the sigma bonds, can be made unambiguously. Fischer and Kollmar 1.68] suggested a d i f f e r e n t approximation for the one-electron term v"ftg Y M - ^  t 0-<^W + « l i r \ z \'^] (2.81), where o( i s a parameter set egual to 0.22- If o( equals zero V f t g w i l l just be the CNDO/2 approximation given by equation (2.75), and i f o( i s one the equation approximates the d i r e c t 0 evaluation of ? f t g. Fischer and Kollmar obtain O^ y^  and fi^ from valence state i o n i s a t i o n energy, the bonding p a r a m e t e r u s e s a Wolfsberg-Helmholz type formula where k ^  i s some adjustable parameter, 1^ i s the valence state i o n i s a t i o n potential and depends on whether JA i s either an s or p o r b i t a l . Equation (2.82) r e s u l t s i n the hybridisation invariance condition being vi o l a t e d , however Fischer and Kollman consider that t h i s i s not an es s e n t i a l condition [69]. This approximation seems to provide the most successful CNDO parametrisation scheme fo r predicting heats of atomisation and equilibrium geometries. Santry and Segal £70] extended the CNDO equations to enable the treatment of molecules which include second row 33 elements. Using a basis set of 3s and 3p o r b i t a l s , the CNDO/2 eguations (2.77) and (2-78) need l i t t l e modification, extensions only become necessary when consideration of 3d or b i t a l s i s made. The simplest approach, which includes 3d o r b i t a l s , i s to keep the d-exponent egual to that used f o r the 3s and 3p o r b i t a l s , and t h i s i s help f u l f o r maintaining the hybridisation invariance conditions. Formally eguations (2.77) and (2.78) remain the same, but a new (I^+A^)/2 parameter i s required for the d-o r b i t a l s . However there i s some uncertainty whether the r a d i a l function of a 3d-orbital i s s i m i l a r to the r a d i a l functions of the 3s and 3p o r b i t a l s . Santry and Segal investigated t h i s s i t u a t i o n using a spd» basis set. This basis set requires the determination of three d i f f e r e n t types of two-electron i n t e g r a l s , depending on whether the o r b i t a l s involved are of sp or d character- The i n t e g r a l s required are *A8 = ^ 4* ^ V(1 ^ ^Cx\ dk, (2-83) , fr^'il & * c 0 < k f t c ^ V a ^ g c A <j>dg(^ A r c , ^ - < 2 - « 4 ) . , and - tt 4?* ^ ' ' r a cfdl LX) fa fa a-c, -fa l2-85)i It should be noted that does not necessarily equal <Sgfl . Santry and Segal evaluated these i n t e g r a l s e x p l i c i t l y by replacing each nd o r b i t a l by the corresponding ns o r b i t a l , but with the exponent of the d- o r b i t a l -The introduction of these new two-electron i n t e g r a l s requires the CNDO Fock equations to be modified, but instead of following the procedure of Santry and Segal i t i s convenient here to consider the more general el e c t r o n i c configuration 34 k St M s p d £71,72]- The motive f o r examining this configuration i s that the spd' basis w i l l be e s s e n t i a l for systems with t r a n s i t i o n elements where not only i s the d-exponent d i f f e r e n t from that of the sp o r b i t a l s , but the d - o r b i t a l w i l l have a di f f e r e n t p r i n c i p a l quantum number. Further, atoms of the t r a n s i t i o n elements w i l l have some occupancy of the d - o r b i t a l s , and t h i s contrasts with atoms of the second row elements where the d-orbitals are empty at l e a s t in the free atoms. The core integrals are redefined as U S S > "I ( T ^ f t ^ - O V ^ W A - M f t \ ^ (2.86), (2-87) (2.88) , where N=k+1. The nuclear a t t r a c t i o n for an s - o r b i t a l on A i s and for a d - o r b i t a l on A The diagonal Fock matrix elements, f o r an s or p - o r h i t a l becomes s and for the d - o r b i t a l (2-91) , (2. 92) , where the charge density i s f o r the s and p o r b i t a l s on A, 35 and P^ft i s the d-population on a . Equations (2.91) and (2.92) were o r i g i n a l l y obtained by Baetzold £71]; when M i s zero the equations reduce to the form suggested by Santry and Segal for the second row elements £70]. The off-diagonal Fock matrix elements given previously by (2.78) now also require modification where the two electron i n t e g r a l t i^g w i l l depend on whether ^ and ^ are s or d o r b i t a l s , the bond parameter ^ °^  can be written as where K i s some parameter, and the atomic quantity may or may not be chosen to depend on whether^ i s e i t h e r an s or d o r b i t a l . The manner i n which Santry and Segal £70] selected parameters f o r the second row CNDO equations was a simple extension of the CNDO/2 procedure used for f i r s t row elements, and the two-electron and overlap i n t e g r a l s were calcul a t e d d i r e c t l y from Slater type o r b i t a l s , using Slater exponents £49], The e l e c t r o n e g a t i v i t i e s (I^*a )/2 were obtained from spectral data. For the sp and spd basis c a l c u l a t i o n s , the atomic bonding parameters ^ were calculated using the relatio n s h i p ( ^ 1 (2.95) , where the f i r s t row element B has the same number of valence electrons as the second row element A. This approximation was required because, at the time, could not be ca l i b r a t e d 36 d i r e c t l y owing to the lack of accurate minimal basis non-empirical calculations f o r second row elements. Better agreement between the CNDO calcu l a t i o n s and experiment were obtained when the value of K i n the expression f o r the bonding parameter^ ^ , eguation (2.94), was fixed at 0.75 when either A or B i s a second row element Santry and Segal concluded, from comparisons with experiment, that better r e s u l t s were obtained f o r the CNDO scheme when the spd^ basis was used; the i n t e g r a l s were evaluated with f u = 01S (2.96), and the d-bonding parameter was taken as ^ U * \ ^ £ ( f l , + W j ^ ( A ) ) fa <2-»7).. Santry £73] l a t e r recalibrated the CNDO eguations for second row elements by performing comparisons with some non-empirical c a l c u l a t i o n s for small molecules such as H£S. Only a spd basis set was considered f o r these CNDO ca l c u l a t i o n s but diffe r e n t bonding parameters were allowed f o r the s and d o r b i t a l s , these parameters being fi x e d by d i r e c t reference to the more accurate ca l c u l a t i o n s . Improvements i n calcula t e d dipole moments were obtained when ad d i t i o n a l empirical f a c t o r s were introduced into the off-diagonal Fock matrix elements. Sichel and Whitehead £60] have also obtained parameters for second row elements following a natural extension of t h e i r analysis of the valence state energies of Hinze and Jaff e £61]. Sichel and Hhitehead gave parameters for elements up to the fourth row of the periodic table, although: they did not 37 consider d o r b i t a l s ; the extensions to include d functions i n thi s parametrisation scheme were made by Levison and Perkins As mentioned e a r l i e r , Baetzold £ 7 1 ] i n i t i a l l y proposed a CNDO treatment of tr a n s i t i o n elements. In t h i s approach, spectral data was used to obtain the (I^*A / l A ) /2 parameters, although the electron a f f i n i t i e s often had to be estimated. The two-electron i n t e g r a l s and the overlap matrix were evaluated e x p l i c i t l y using dementi's exponents £ 7 5 J , although some averaging of the Coulomb i n t e g r a l s was made to improve the calculated bond lengths. The atomic bonding parameter ^° was given a single value for each atom, with no d i s t i n c t i o n being made for the o r b i t a l type. The f i n a l parameter set was obtained by comparing calculated bond lengths of homonuclear molecules with experimental values. Blyholder £ 7 2 ] , using the same equations as Baetzold, attempted a more thorough derivation of parameters f o r n i c k e l . A c l u s t e r of s i x n i c k e l atoms arranged octahedrally was treated as a model of bulk n i c k e l . The CNDO parameters were treated as variables, subject to t h e i r values being p h y s i c a l l y reasonable. For (I^+A ) / 2 t h i s involved comparing with experimental spectral data, and the exponents were kept within the range of values suggested by the rules of Slater £ 49 ], Burns £ 7 6 ] , Clementi [ 7 5 J and Goutermann £ 7 7 ] . The f i n a l parameters were chosen so that the Ni c l u s t e r had calculated properties which agreed as cl o s e l y as possible with experimental values (e.g. d-band width, Fermi energy) for bulk n i c k e l . This study indicated that d i f f e r e n t bonding p a r a m e t e r s w e r e required for the s 38 and d o r b i t a l s . This l a t t e r requirement represents the major difference from the work of Baetzold where j&fi depends only on the atom, and not on the o r b i t a l type. Clack, Hush and landle £78] have also extended the CNDO method to treat t r a n s i t i o n metals of the t h i r d row in the periodic table. They have been p r i n c i p a l l y interested i n molecules which only contain a single t r a n s i t i o n metal atom, and t h e i r parametrisation scheme i s not as general as the procedures given by Baetzold and Blyholder. F i r s t l y , Clack et a l . assume that each t r a n s i t i o n element i s r e s t r i c t e d to a valence configuration of the form 3d"- 24s 2. This often indicates the ground state configuration of a t r a n s i t i o n element, yet there are exceptions (e.g. Cry. A more serious problem with the approach of Clack et a l . seems to be i n the form they use for the one-electron term 0 ^ f o r example whereas Baetzold, using equation (2.86), obtains for the configuration 3 d n - 2 4 s 2 = -i ( 1 , 4 ^ - 1 r<S - <2-99'• from consideration of the t o t a l atomic energy expressed i n the CNDO framework. Clack et a l . selected parameters i n a manner si m i l a r to the conventional CNDO/2 scheme for the fir s t - r o w o elements; thus and were calibrated against non-empirical minimal basis calc u l a t i o n s . The de f i c i e n c i e s i n the formalism, introduced by (2.98), do not become serious unless the molecules examined contain more than one atom with d functions. 39 2.8 Applications of CNDO to L i F The o r i g i n a l hope had been that the CNDO/2 parameters f o r the f i r s t 10 elements of the periodic table would be useful f o r molecules encompassing a wide range of chemical bonding s i t u a t i o n s . Most often t h i s hope has been upheld to a reasonable degree [54], although simple considerations of the electr o n i c charge d i s t r i b u t i o n s for some c l u s t e r s based on lith i u m f l u o r i d e [79,80] suggest that the conventional parametrisation gives too large a covalent character to L i F , both i n the diatomic and the c r y s t a l l i n e states. Near-flartree-Fock calculations on diatomic LiF indicate c l e a r l y that there i s e s s e n t i a l l y complete i o n i c character i n the sense L i + F ~ J. 81], However, a population analysis (see section 2.11, f o r the CNDO/2 ca l c u l a t i o n f o r LiF , using a bond length of 2-8877 a.u., shows net charges Li+°•••F _ 0••* and a dipole moment of 4.906 D. The l a t t e r value compares with 6.297 D from non-empirical calculations £ 82,83], the experimental value i s 6.284 D £84]. Somewhat s i m i l a r l y , for an 18-atom c l u s t e r considered by Hayns £79], where L i and F atoms are arranged i n a cubic array, CNDO/2 gives average net charges Li*°« 2 SF~<> N 2 S, and these values seem inconsistent with the very high i o n i c character generally associated with s o l i d LiF £85]. These considerations became apparent at about the time I started my research, and my i n i t i a l project £86] involved reconsidering the parametrisation of L i and F f o r i o n i c species. The fe e l i n g was that the conventional CNDO/2 parameters may be better suited to systems which are b a s i c a l l y more covalent and that some extra parametrisation could be 40 advantageous i n systems which involve substantial r e - a l l o c a t i o n of charge from the normal atomic states- I t i s convenient to present t h i s work here since i t forms the basis f o r many ideas l a t e r in this thesis-f a b l e I shows a comparison for diatomic L i F between CNDO and non-empirical c a l c u l a t i o n s of occupied one-electron energy l e v e l s and one-electron moments; the l a t t e r being defined as fo r a molecular o r b i t a l , the one-dimensional position vector r F has the F nucleus as o r i g i n and the L i nucleus as (2,877). One way of choosing CNDO parameters s p e c i f i c a l l y f o r L i and F in c r y s t a l l i n e LiF i s to use diatomic L i F as a reference. Bather than use exponents from Slater's rules 1493 for the free atoms, Clementi and Baimondi^s exponents [87] f o r the valence s h e l l appropriate to L i * and F~ (namely 0.8197 and 2-3792 respectively) were used. Aside from two other differences the CNDO/2 procedure has been followed completely. These other differences are, f i r s t l y , the l o c a l core matrix element 0 ^ has been parametrised d i r e c t l y , rather than use atomic spectral data i n equation (2.74), and, secondly, the bonding parameter ^  i s reparametrised. , These new parameters were chosen by minimising the function where € L and ^  are one-electron energies and dipole moments calculated with the new parameters; <£^SCt" and yt, l S c P are the corresponding quantities from the non-empirical s e l f - c o n s i s t e n t f i e l d c a l c u l a t i o n , The new parameters which minimise B f o r (2-100) , (2. 101) , 41 CNDO/2 SCF MPCNDO €jL_ £L fj. /<_ fj Mj-1TT -0.5580-0.5536 -0.4786-0.1721 -0.5139-0.1895 4^ -0.5729 0.0689 -0.5046 -0.0042 -0.5095 -0.0396 3 s- -1.4280 -0.1784 -1.3817 -0. 1707 -1.3691 -0.0764 Table I . Comparison of one-electron energies ( <£;/Hartree) and dipole moments (^ :/D) f o r occupied valence s h e l l molecular o r b i t a l s of diatomic L i F (bond length 2.8877 a.u.) using CNDO/2, non-empirical SCF £83], and MPCNDO cal c u l a t i o n s . 0 ^ 2 s L i -0.2463 -0. 1208 0.0308 F -6.9587 -6.1307 0.7676 Table I I . Values of parameters 0 ^ and deduced for valence o r b i t a l s of L i and F i n L i F . The units are hartrees. 42 diatomic LiF (bond length 2-8877 a.u.) are i n table I I , and the appropriate 6^and ^ are l i s t e d i n the th i r d column of table I. The use of the parameters i n table II defines what we c a l l here the molecule-parametrized CNDO ( i . e . MPCNDO) method; with these new parameters the t o t a l dipole moment of LiF i s calculated to be 6-350 D-Preliminary investigations suggested that these new parameters are he l p f u l i n making up for d e f i c i e n c i e s of the CNDO/2 method f o r other systems based on the monomer unit L i F . In table I I I the one-electron energy l e v e l s for the dimer L i F from CNDO/2, from HPCNDO and from non-empirical c a l c u l a t i o n s £87] are compared, and i t seems there are some improvements i n using the new parameters- The net charge at F i n (LiF)^ i s -0-74 according to MPCNDO rather than -0-32 using CNDO/2, suggesting a more r e a l i s t i c charge d i s t r i b u t i o n i s being predicted with the new parameters- Substantial differences i n d e t a i l are found between some res u l t s of Hayns using CNDO/2 f o r L i F c l u s t e r s and those using MPCNDO and some examples are indicated in the energy l e v e l diagram of figure 2-The new parameters i n table I I are not i n any s t r i c t sense optimal, i n part t h i s i s because of the a r b i t r a r y form of the function minimised and also the parameters obtained for the diatomic molecule apply to an atomic separation d i f f e r e n t from that i n the i o n i c l a t t i c e . Nevertheless these new parameters have recently been used i n a s o l i d state study by Evarestov and Lovchikov £88], and some of the comparisons these authors report are reproduced i n table IV. Evarestov and Lovchikov conclude that MPCNDO reproduces 43 CNDO/2 SCF HPCNDO -0.6-0.79. -0.5100 -0.5330 -0.6245 -0.5128 -0.5467 -0.6629 -0.5271 -0.5516 -0.6784 -0.5286 -0.5622 -0.6981 -0.5368 -0.5708 -0.7058 -0.5581 . -0.5839 -1.5327 -1.4189 -1.4072 -1.6379 -1.4231 -1.4407 Table I I I . Comparison of one-electron energies (in Hartrees) . for ( L i F ) 2 using CNDO/2, non-empirical SCF £87] and HPCNDO methods of calcu l a t i o n -band structure k-point CNDO/2 SW HPCNDO cal c u l a t i o n s (a) (b) X tl| L w L i e charge on atot (abs. val.) s-band width p-band width forbidden gap -72.21 -11.47 5. 19 -32.45 -10.25 31.91 -38.64 -10.33 11.01 . -12.24 - 1.34 -44.15 -16.66 6-30 -33.90 -12.16 -11.74 34.37 -31.10 -10-53 -10-36 22-89 -33.64 -11. 17 -10.65 11.97 -14.62 -12.92 4.08 -42. 31 -20. 18 -17.76 17.24 -30.81 -13.49 -11.77 13.52 -30.93 -10-62 -10-30 24-73 -32.97 -12. 10 -10.61 11.04 -14.28 -12.15 1.70 -42.83 -19.69 -17-11 8.46 0.75 41.40 2.02 16.66 0.99 1-52 0.37 42- 16 0.91 5.67 1.77 21.34 2-38 1 0-90 1.84 3. 52 22.96 Table IV. Comparison of bulk CNDO calculations using CNDO/2, Sichel and Whitehead (SW) £64] and HPCNDO against standard band structure calculations (a) £89] and (b) £90]. The energies are i n eV and the subscripts v and c on the k-point labels distinguish the valence and conduction band states. 44 Fiqure 2. Cciparison of groups of closely packed eiqenvalues calculated with the CNDO/2 and HPCNDO methods for soae arranqeaents of L i and F atoms considered by Hayns {79 1. 45 e x c e l l e n t l y the main features of the LiF band structure- The CNDO/2 parameters give too wide an anion s-band, and the Sichel and Whitehead parameters £60,64] give a valence band which i s too narrow and a gap between the valence and conduction band which i s too large. This section brings out that the CNDO formalism may have hel p f u l applications to i o n i c c r y s t a l s , and also that conventional parameters obtained f o r systems i n l a r g e l y covalent environments may not be suitable f o r some elements combined i o n i c a l l y . Also t h i s work demonstrates that the near-Hartree-Fock c a l c u l a t i o n s can be useful f o r c a l i b r a t i n g semi-empirical molecular o r b i t a l methods 2.9 Scattered Wave Theory for Molecules and Clusters While the Hartree-Fock eguations for molecules have usually been solved with the LCAO approximation, the solution of the Xo< equations f o r s i m i l a r systems have commonly used the scattered wave formalism- The r e s u l t i n g X scattered wave (Xc< SS) eguations represent an extension of the Korringa-Kohn-Eostoker (KKfi) method £14] used i n the c a l c u l a t i o n of energy bands i n bulk s o l i d s . The X<x equations have also been solved using an LCAO expansion, an example of such a scheme i s the discrete v a r i a t i o n a l method (DYM) developed by E l l i s £92,93]. However these LCAO procedures have not yet been applied to many systems and w i l l not be discussed in t h i s thesis. The scattered wave formalism depends on p a r t i t i o n i n g a 4 6 molecule into three regions:-I the atomic region, which i s the space enclosed by spheres centred at each nuclear position; II the intersphere region, which i s the region external to the atomic spheres but i n t e r n a l to the outersphere, which surrounds the whole molecule; III the outersphere region, which i s a l l the space outside the molecule. This p a r t i t i o n i n g allows the solution of the one-electron Schrodinger equation to be determined partly as a numerical function, t h i s avoids the necessity of having to adopt large basis set expansions which are encountered i n the LCAO approximation- The one-electron energies are determined from the matching conditions for the wavefunction and i t s f i r s t d erivative at the region boundaries. An outline f o r the non-overlapping sphere model i s given i n t h i s section; a more complete discussion can be found i n a review a r t i c l e by Johnson Consider a set of atomic spheres of r a d i i b^, b^, ... associated with atoms <*rjZ , ... and an outersphere of radius bt, chosen to be tangential to some atomic spheres so as to minimise the volume of the intersphere region. Let the nuclear coordinates be defined by the position vectors B^, JBg, --- and l e t the centre of the outersphere be at B 0- I f a position r e l a t i v e to the <s\ nucleus i s given by r ^ , then a general position r i s I = - + ( 2 . 1 0 2 ) . , fe write the vector from nucleus** to nucleus /§, as 47 R ^ - R i - E ^ (2-103).^ In section 2.3 the lowest X « t o t a l e l e c t r o n i c energy was obtained from a solution of the one-electron equation where the potential i s V - - £ - * -r f P ^ ^ - , (2.105).. In the scattered wave theory, the potential i n (2.104) i s approximated by a muffin t i n form. This requires the potential i n the regions I and III to be sp h e r i c a l l y averaged; such a potential for the «; sphere i s represented by (r^.). also the intersphere p o t e n t i a l i s replaced by the volume average . The s p h e r i c a l potential i n each region enables the wavefunction to be written as < • S o ? j ; „ f £ ) [ ^ „ ( r ^ , e t r ^ k , (2-106). where are p a r t i a l wave c o e f f i c i e n t s which need to be determined, Y L m are spherical harmonics, and £;r^) are numerical r a d i a l functions which are solutions to the r a d i a l Schrodinger equation at the centre > d * + + 2 fa ( r ^ - ^ & U ^ o (2.107). The wavefunction f o r the outersphere i s s i m i l a r l y • W f r V g ^ ^ C f c ^ W r . ^ ^ - ^ o ^ < 2 - ' 0 8 ) ' where fi° are^the r a d i a l functions for the outersphere, and are solutions to the r a d i a l equation with the same form as (2.107). Solutions to the one-electron equation ( " \ \ % C r ^ - o (2.109), for the intersphere region can be obtained as a multicentre expansion i n terms of Bessel-type functions, which f o r bound states (6<V-rr ) are 48 where the A5* are c o e f f i c i e n t s which need to be determined. f,\ . • r• \ (2-111) i s a modified Bessel function, - ^ (2.112) i s a modified spherical Hankel function of the f i r s t kind; and k- W -t)JK (2.113). The f i r s t term i n (2-110) represents an expansion i n terms of the function k<*> f o r each atomic centre and th i s f u n c t i o n rapidly decays with increasing r . The second term i n (2.110) represents an expansion from the outersphere boundary decaying towards the centre of the molecule. Bhen 6 , ^Vxis written i n a modified form, however the present discussion i s r e s t r i c t e d to bound states; the f i n a l eguations are s i m i l a r f o r both s i t u a t i o n s . So as to match wavefunctions at the region boundaries i t i s necessary to transform (2.110) to an expansion about a single centre £94]; i n terms of oi , the intersphere wavefunction becomes (?) - s v ^ < * ~ i 1 < where each angular component has the form (2.115, , with and where C - 2 ftL- <d + 2 flivsi? ( 2.117, 8 /J/-.' 2 ' * . ' 49 Suu'ft*s,e> » (L~ ^ r ^ ^ V x (2.118), C " I ^\ A r \H<' Cl _ , , (2.119), and 1 - ^ To be acceptable, the wavefunction and i t s f i r s t d erivative must both be continuous across a sphere boundary; t h i s matching of ^ and ^\when r ^ b ^ reguires ^ * + \ * > - r * £ * (2-121) fo r each lm, and the requirement of matching of the f i r s t derivatives gives 4 . ±& -V Wl i « - C t l ^  (2- "2, . Equations (2.121) and (2.122) lead to the condition « ,1 L t f c o r ' -wt - o < 2 - 1 2 3 ' -where r D * 1 i ^ _ __ L , i t A with the square brackets i d e n t i t y i n (2-124) being the wronskian. The f i n a l equations become CT-' rolj = s f t B sLL, ik*cor-C'(2-126>-These matching conditions are applied to each atomic sphere boundary and to the outersphere boundary; the l a t t e r gives S 2 A ^ . - ^ I b°L ^ T' A ° c V " ° ( 2 - 1 2 7 ) , where :SjJ i s symmetric and i s s t i l l given by (2.119) provided that r e a l spherical harmonics are used. Equations (2.125) and (2-127) give the X <=<S» secular 50 equation, however t h i s secular equation i s d i f f e r e n t from the usual eigenvalue problem encountered i n molecular o r b i t a l theory i n that the matrix elements have a non-linear dependence on the one-electron energy. Thus equations (2. 125) and (2.127) cannot be solved by the standard diagonalisation procedure, and instead a determinant has to be evaluated, with the one-electron energies given when A disadvantage of the scattered wave formalism i s that the f i n a l secular equation takes longer to solve than the l i n e a r eigenvalue problem, due to the necessity of having to evaluate e x p l i c i t l y several determinants- However, t h i s d i f f i c u l t y i s compensated with the ease i n which the matrix elements i n equations (2-125) and (2.127) can be computed; the l a t t e r i s due in part to these quantities depending only on r a d i a l functions £e.g. &f(a;b^) and k<*»(^<b^)J and their f i r s t d erivatives at the appropriate sphere r a d i i b*, and on "structure f a c t o r s " (e.g. k<* > (vk B^ ) Y L (R^) ] which are functions of the the interatomic vector B^- Moreover, there i s no need for four centre inte g r a l s of the type encountered i n the Boothaan equations- The s i z e of the secular equation i n the X^S8 method i s also smaller than that of the LCAO approach; for the former the order of the equations equals the t o t a l number of d i f f e r e n t spherical harmonics used i n regions I and I I I ; although, as i n many other applications of quantum chemistry, symmetry can be used to reduce the e f f e c t i v e s i z e of the secular equations. (2. 128) . 51 The procedure for carrying out X c ^ s i c a l c u l a t i o n s on molecules i s no* outlined. The i n i t i a l step involves obtaining the s p h e r i c a l charge d i s t r i b u t i o n of each constituent atom with a computer program developed by Herman and Skillman £95J. Prom a superposition of the atomic charge d i s t r i b u t i o n s , an approximation to the molecular potential i s calculated (including the Xo< exchange term). The poten t i a l i s now converted to the muffin t i n form for the appropriate sphere r a d i i . The number of spherical harmonics to be included are chosen; a common scheme includes the p a r t i a l waves 1-0 and 1 in the atomic spheres which represent l i g h t atoms, and up to 1=2 for the heavier atoms; the outersphere usually needs p a r t i a l waves with a maximum 1 value at lea s t two greater than the maximum value used i n the atomic spheres. The secular equation i s then fac t o r i s e d using the point group symmetry of the system being examined. A l i n e a r search i s used to solve (2.128), and, from the corresponding occupied one-electron states, a new potential i s derived. This pot e n t i a l i s now muffin t i n averaged, and the values of the improved one-electron energies can be obtained by perturbation theory; t h i s enables the zeros in (2.128) to be found by a bracketing technique which i s much more e f f i c i e n t than the time consuming li n e a r search procedure. This i s continued u n t i l s u f f i c i e n t l y small differences are found i n the potential between one i t e r a t i o n and the next. About 20 to 30 i t e r a t i o n s are generally needed. 52 2-10 Parameters in X*SW Calculations The X o< eguations derived i n section 2, 3# and the scattered wave formalism developed i n the previous section require the d e f i n i t i o n of two sets of parameters- These are the sc a l i n g parameter <^  f o r the exchange potential given by (2.38); and the r a d i i of the spheres used f o r p a r t i t i o n i n g the molecule i n t o the three regions. For a free electron gas o< has the value of 2/3, but for atoms and molecules the s i t u a t i o n i s considerably more complicated. The v a r i a t i o n a l method cannot be used to determine the values of p< since the e l e c t r o n i c energy E x < < i s l i n e a r l y dependent on o<, and no minimum i n E X ( K would be found on varying - However th i s l i n e a r dependence of enables o< to be chosen so that E a g r e e s with some standard quantity, and Schwarz £96] chose values of cx so that E x ^ matched the t o t a l atomic energies given by the Hyper-Hartree-Fock c a l c u l a t i o n s of Hann £97 J. This set of values of cx1 show a uniform trend from around 0.78 f o r atoms of low atomic number to about 0.70 f o r atoms of the f i r s t t r a n s i t i o n s e r i e s . Other procedures have been suggested f o r f i x i n g £32 3, hut i n molecular calculations the values of Schwarz have generally been used for the atomic regions. Intersphere and outersphere regions have mainly used values of which represent weighted averages of the atomic sphere values, where the weighting factors are i n the r a t i o s of the atomic numbers of the constituent atoms. The selection of the sphere r a d i i i s also important. The most d r a s t i c approximation i n the Xo<S8 method i s the volume averaging of the potential i n the intersphere region, and t h i s 53 suggests the spheres should be chosen to minimise t h i s volume, thereby reducing e r r o r s . In the non-overlapping sphere model the atomic spheres are chosen to be tangential, and f o r simple homonuclear systems the r a d i i are simply half the internuclear distance, additional c r i t e r i a are needed for heteronuclear systems. One approach would be to consider the sphere r a d i i as v a r i a t i o n a l parameters and to minimise the t o t a l e l e c t r o n i c energy £ 44 ]. However t h i s i s impractical from the computational viewpoint, and there are inherent e r r o r s i n the X °< t o t a l e l e c t r o n i c energy. The X c x t o t a l e l e c t r o n i c energy does not rigorously correspond to a t o t a l e l e c t r o n i c wavefunction, the connection i s l o s t when the exchange approximation i s introduced. The common method for s e l e c t i n g the sphere r a d i i of d i f f e r e n t atoms i s to use r a t i o s related to the Slater covalent r a d i i £7]-The Xo^S8 eguations are s t i l l v a l i d when the atomic spheres overlap £98], and the overlapping sphere model has the advantage of further reducing the volume of the intersphere region. A procedure described by Norman £99] has often been used for determining the sphere r a d i i i n the overlapping sphere scheme. The i n i t i a l molecular charge d i s t r i b u t i o n i s considered, and the radius of the sphere which contains a l l a p a r t i c u l a r atoms e l e c t r o n i c charge i s found ( i . e . 6 electrons for C, 9 f o r F, e t c . ) . The X <=<S'S ca l c u l a t i o n i s then performed with a set of uniformily reduced r a d i i , the absolute values of the r a d i i being chosen by the condition that the v i r i a l theorem V / 2 T - - 1 - (2.129) i s s a t i s f i e d , where V and T are the calculated t o t a l p o t e n t i a l 54 energy and t o t a l k i n e t i c energy respectively. The X ex. eguations for any system at equilibrium must obey t h i s condition for any value of c< £32]. I n . practice only a l i m i t e d number of calc u l a t i o n s are performed i n the search for the optimal degree of sphere overlap. The overlapping-spheres model seems to give better guantitative agreement with experiment (e.g. for i o n i s a t i o n energies) than calculations using the non overlapping sphere model [100]. 2.11 Holecnlar Charge Dis t r i b u t i o n s An important guantity from a c a l c u l a t i o n i s the f i n a l charge d i s t r i b u t i o n ; for example, t h i s i s needed i n considering the v a r i a t i o n of the l o c a l DOS i n a c l u s t e r . For the LCAO expansion, the ftulliken population analysis 1.51] i s generally used, where the charge on t h e ^ - t h atomic o r b i t a l for the i - t h energy l e v e l i s given as In the ZDO approximation Sy^ w i l l be zero unless yu = 0, and the charge becomes simply %hi - I c^l 2 (2.131). The wavef unctions encountered in the X^SH formalism have much i n common with the LCAO wavefunctions, i n so f a r as thej correspond to expansions of atomic o r b i t a l s , within the atomic regions. The differences are that the wavefunctions are expressed numerically in the X <XS¥ method, and there are in e v i t a b l e modifications at the sphere boundaries. The o r b i t a l -55 type expansion i s continued i n the intersphere region through the use of Bessel-type functions, and i n p r i n c i p l e a Hulliken s t y l e population analysis could be made. However, within an atomic sphere, the charge associated with a s p e c i f i c atomic o r b i t a l has often been defined as f o r the molecular o r b i t a l with the one-electron energy £_i. This type of population analysis ignores the charge i n the intersphere and outersphere regions; t h i s may be reasonable i f the amount of charge i n these regions i s small, although often the atomic populations are renormalised to give a corrected t o t a l e l e c t r o n i c charge i n each molecular o r b i t a l . This r e s u l t s i n an apportioning of the intersphere and outersphere charges to the atomic regions according to some approximate prescription. The charge d i s t r i b u t i o n s given by (2.131) and (2.132) are used i n equation (1.4) to obtain the l o c a l DOS for c l u s t e r s from the CNDO and the X^SH methods respectively. 56 CHAPTEB 3 I2<M CALCULATIONS OH CLUSTERS OF SILVER ATOMS 3,1 Introduction Several X <xs» cal c u l a t i o n s on t r a n s i t i o n metal c l u s t e r s have been reported recently, and a topic of considerable interest has involved assessing the s i m i l a r i t i e s between the el e c t r o n i c structures of the small metal c l u s t e r s and the corresponding bulk band structure £35,101], Messmer et a l . {35] have performed the most extensive study where they examined a variety of c l u s t e r s of copper, n i c k e l , palladium, and platinum, the largest c l u s t e r studied for each element consisted of 13 atoms arranged with cubo-octahedral geometry (see figure 3). The cubo-octahedron has O^ symmetry and corresponds to a central atom being surrounded by 12 atoms arranged as the near-neighbours of the face-centred-cubic l a t t i c e . Messmer et a l . concluded from t h i s study that the e l e c t r o n i c structure of c l u s t e r s with only 13 atoms show s i m i l a r i t i e s to the e l e c t r o n i c structures of the bulk. For example, i n the c a l c u l a t i o n on Cu ) 3 there i s a region where the energy l e v e l s are c l o s e l y spaced, with these l e v e l s being predominantly derived from the d-o r b i t a l s , and this region forms the c l u s t e r analogue of the d-band i n the bulk. This Cu^ d-band i s overlapped by l e v e l s of s and p character which are s t a r t i n g to form the basis of the bulk s,p band. The d^-band in Cu l S i s well below the highest occupied l e v e l , whereas i n each of the c l u s t e r s , Ni,^, Pd l 3 > and Pt\^ the Fermi l e v e l i s positioned near the top of the d-band, much as found in the bulk f o r these metals. Further, the 57 c l u s t e r s Ni,^ , Pd,^ and P t \ j show the trend of i n c r e a s i n g d-bard width which a g a i n i s s i m i l a r to the bulk s i t u a t i o n . F i g u r e 3. The cubo-octahedral c l u s t e r c o n t a i n i n q the 13 atoms. The 10 atcm c l u s t e r used t o model the (111) s u r f a c e i s obtained by renoving the atoms 1, U and 5. These c o n c l u s i o n s from XotSH c a l c u l a t i o n s on t r a n s i t i o n metal c l u s t e r s c o n t r a s t with r e s u l t s from some s e m i - e m p i r i c a l c a l c u l a t i o n s . For example, B a e t z c l d and Mack f 10 2 1 usinq the EH and CNDO methods f o r t r a n s i t i o n metal c l u s t e r s composed of up to 19 atoms, concluded that the e l e c t r o n i c s t r u c t u r e s of c l u s t e r s o f t h i s s i z e are d i f f e r e n t from those of the bulk; d i s c u s s i o n of the r e s u l t s of s e m i - e m p i r i c a l c a l c u l a t i o n s w i l l be given ir. l a t e r c hapters. Another aspect of the T<* SW study by flessmer et a l . [35] was t h a t these authors noted some s t a t e s in which symmetry r e s t r i c t s to involvement by the twelve outer atoms on the 13 atom c l u s t e r . For an s, p, d b a s i s , the 58 one-electron wavefunctions only include the c e n t r a l atom i n the i r r e d u c i b l e representations a ^  (s-orbital) , t (^ (p-orbitals) , ea { d 2 a , d ^ i ^ i ) and t o (dx>^,dV2.,d fc) , whereas the other i r r e d u c i b l e representations, such as a a , are only composed of 0 basis functions from the outer atoms. I t may be wondered to what degree these states r e l a t e to the bulk structure. To aid the interpretation of some photoelectron spectra, Bosch and Menzel £101] examined c l u s t e r s of Ni, Cu and Ag. In thi s work the largest c l u s t e r s considered consisted of s i x atoms arranged at the vertices of an octahedron. Again these authors noted s i m i l a r i t i e s to the bulk structures, e.g. the Fermi l e v e l for Nit, m S L S just below the top of the d-band, whereas i n Cu^ and Ag t the d-band was below the Fermi l e v e l . Bosch and Henzel*s calculations show some consistent trends with bulk c a l c u l a t i o n s . For example the d-band widths f o r N i L , Cu^ and Ag t are 2-0, 1.3 and 1.6 eV respectively, whereas bulk calculations for the corresponding elements give 4.9 £103], 3.3 £104] and 4.1 eV £105], s i m i l a r l y the separations between the top of d-band and the Fermi l e v e l are -0.1, 1,1 and 3.6 eV f o r N i t , Cu t and Ag^, and the values compare with the respective bulk values of -.02 £103], 1.8 £104] and 4.1 eV £ 105]. X e< sw c a l c u l a t i o n s have also been performed f o r t r a n s i t i o n metal c l u s t e r s so as to provide an i n t e r p r e t a t i o n of chemisorption experiments. For example, when CO i s adsorbed on the Ni(100) surface, photoelectron spectra show that i n addition to the main peak due to the Ni d-band, there are two new peaks P, and p^ at approximately 8 and 11 eV below the Fermi l e v e l £106], Batra and Bagus £107] modelled the CO 59 adsorption by using a Ni^CO) cluster with C^ 0 symmetry, where 4 Ni atoms are i n a square i n one la y e r , the other Ni i s on the 4-fold axis i n the layer below, and the CO l i e s along the 4-f o l d axis above the top layer with the C directed towards the Ni c l u s t e r . This c l u s t e r was chosen because LEED experiments suggested that CO adsorbs on the 4-fold s i t e of the Ni(100) surface £108]. As a r e s u l t of these c a l c u l a t i o n s , Batra and Bagus concluded that the P^ peak i s associated with the 1 TT and 5^ l e v e l of CO and the 2X peak i s predominantly associated with the 4 o r l e v e l , the t i l d a s are used to indicate that the CO le v e l s have been strongly perturbed due to inte r a c t i o n s with the substrate. This interpretation of CO adsorbed on Ni(100) i s supported by further angular resolved photoemission measurements £ 109 ]- Also Batra and Bagus reported that the same picture of the chemisorption bonding resulted from increasing the number of substrate atoms i n the c l u s t e r to nine. Similar X ^SJI calc u l a t i o n s have considered NO adsorbed on Ni(100) £110], 0 on S i (100) £ 101,111,112], 0 on Ag(100) and Ag(110) £101] , and 0 on Cu(100) £101]. A l l of these c a l c u l a t i o n s model the substrate with four or f i v e atoms and appear to give he l p f u l r e s u l t s f o r interpreting photoelectron spectra; the calculated l e v e l s associated with the adsorbate-substrate interactions could be matched up with new peaks, i n the photoemission spectrum. In another study, Niemczyk £ 112 ] investigated the adsorption of O, S, Se and Te on Ni (100). The Ni surface was modelled by the same Ni^-cluster as used by Batra and Bagus £107] and the chalcogen was placed on the 4-f o l d axis above the c l u s t e r . However, Niemczyk performed 60 several X°<SW calculations so as to obtain the binding energy curve as a function of the chalcogen distance above the Ni surface. Niemczyk did not obtain good guantitative agreement with the experimental chalcogen-Ni(100) distances from LEED [113], although consistent trends were found, flore recently L i and Connolly £ 1 1 4 ] reinvestigated the binding energy curve of oxygen on the Ni c l u s t e r using the X o! SH method, and these authors obtained a minimum i n the curve with the oxygen 1.42 a.u- above the surface which i s i n guite good agreement with the experimental value (1.70 a.u. £ 113]),, whereas Niemczyk obtained a distance of 1.00 a.u. This improved c a l c u l a t i o n of L i and Connolly includes corrections to the muffin-tin approximation £ 1 1 5 ] . Other studies of oxygen on Ni (100) have emphasised the high exposure regime where oxygen incorporates into the n i c k e l ; t h i s i s suggested by the photoemission spectrum which i s no longer dominated by the Ni d-band peak, but instead resembles that of bulk n i c k e l oxide £ 116]. Messmer, Tucker and Johnson £ 1 1 7 ] were able to provide an inte r p r e t a t i o n of the peaks i n the high oxygen exposure spectrum by performing an Xc* SH c a l c u l a t i o n on a NiO*°~ c l u s t e r . The N i O 1 0 - c l u s t e r involves a N i 2 + ion octahedrally coordinated by six O 2 - and t h i s corresponds to the l o c a l enviroment i n the bulk oxide. The energy l e v e l s calculated for NiO»o - matched peaks i n the photoemission spectrum f o r high oxygen exposure. The apparent success of r e l a t i v e l y small t r a n s i t i o n metal clusters f o r i n t e r p r e t i n g photoelectron spectra opens the question as to whether ca l c u l a t i o n s on small c l u s t e r s could be 61 Fiqure 4. The Aq-, cluster used to model the Aq(111) surface; the f u l l c i r c l e s are for atoms (designated 1-4) i n the top Aq layer and the dashed c i r c l e s are for atoms (5-7) i n the second Aq layer. Adsorption cf I i s considered at s i t e s F and H on the top s i l v e r layer. useful for predictinq qeometrical arranqeaents on surfaces. The work of Niemczyk indicates that whilst quantitative r e s u l t s may not be obtained, some q u a l i t a t i v e interpretation miqht be possible. In t h i s chapter the adsorption of iodine on the Aq(111) surface i s examined. This system has been studied experiaentally and in an analysis of LEED i n t e n s i t i e s from Aq (111) - (/3x J3) 30°-I, Forstaan, Berndt and Buttner f118l concluded that iodine adsorbs on those three-coordinate s i t e s (desiqnated F) which l i e over holes on the second s i l v e r layer, and which correspond to a continuation of the face-centred cubic (fee) structure. By contrast iodine does net adsorb on 62 the other set of three-coordinate s i t e s (designated H) which l i e over atoms i n the second s i l v e r layer and would give the ordering of the hexagonal-close packed (hep) arrangement for the top three layers- These two d i f f e r e n t adsorption s i t e s are i l l u s t r a t e d i n f i g u r e 4. The reasons for t h i s choice of adsorption s i t e pose an i n t e r e s t i n g problem i n s t r u c t u r a l surface chemistry. Fcrstmann> Berndt and Buttner stated that I adsorbs where i t "saturates the broken dangling bonds", although at present the s i g n i f i c a n c e of t h i s statement i s obscured by a dearth of knowledge about eith e r the bonding nature at the s i l v e r surface or the e f f e c t of adsorbates on the surface e l e c t r o n i c structure. In the study of iodine adsorption, only c a l c u l a t i o n s with iodine at the three-coordinate s i t e have been made. The Ag(111) surface i s represented by the Ag 7 c l u s t e r , shown in f i g u r e 4; t h i s i s the simplest c l u s t e r to d i s t i n g u i s h between the F and H s i t e s . In the next section, the Xc*'S¥ calc u l a t i o n s f o r Ag 7, Ag 7Ip and Ag-,1^  (the subscript on I indicates the p a r t i c u l a r adsorption s i t e ) are discussed £119]. Although the c a l c u l a t i o n s on these simple c l u s t e r s c o r r e c t l y predict that the iodine w i l l adsorb at the F s i t e , i t was concluded that larger c l u s t e r s need to be examined fo r a more complete understanding of the e l e c t r o n i c structure of the s i l v e r surface. In section 3.3 some calc u l a t i o n s on larger s i l v e r c l u s t e r s are presented, these clust e r s represent atoms i n the bulk and surface environments £ 120 J. 63 3 . 2 Studj of I on Ag (111) 3.2.1 Sp e c i f i c a t i o n of Calculations The X cx SU calculations on Ag 7, Ag-pip and Ag^I^ have been made with a l l the nearest-neighbour distances fixed at the value i n bulk s i l v e r (2-89 A) £121] and with I 2-25 A above the plane of the topmost Ag layer [118]- The Ag 7 c l u s t e r has the symmetry of the point group C^ v, and both Ag^I c l u s t e r s have Cs symmetry. The atomic potentials and charge d i s t r i b u t i o n s f o r each element were obtained from the Herman-Skillman computer program [95] using the u exchange parameters of Schwarz, namely 0 - 7 0 1 3 0 f o r s i l v e r and 0 - 7 0 0 2 2 f o r iodine [96]. The Ag and I o o atomic sphere r a d i i were fixed at 1-60 A and 1.?0 A respectively; these values are in the r a t i o s of the sphere r a d i i given by Norman's procedure [99]- (The s i l v e r represents an average over the ineguivalent atoms.) The centre of the outersphere was fixed f o r the Ag^ c l u s t e r at the i n t e r s e c t i o n of the C^ axis with the plane containing the Ag atoms l a b e l l e d 2, 3 and 5 (see figure 4), and f o r the Ag 7I c l u s t e r s the outersphere was centered at the middle of the four Ag atoms labelled 1-4. The outersphere radius was fixed at 4.87 A i n a l l ca l c u l a t i o n s , and the value f o r the intersphere and outersphere regions was taken equal to 0.70117. The cal c u l a t i o n s included p a r t i a l waves up to 1=2 i n the atomic regions and up to 1=4 i n the outersphere region. Large secular matrices are involved for the valence o r b i t a l s , and even when they are factorised by symmetry one s t i l l has f o r the 64 Ag rI c l u s t e r s , matrices of siz e 57x57 f o r a' and 40x40 for a" i r r e d u c i b l e representations; t h i s requires a l l the c a l c u l a t i o n s to be performed using double precision arithmetic. The non-line a r nature of the secular equation, discussed i n section 2 . 9 , coupled with the low symmetry of the Ag ?I c l u s t e r , and the large number of d-electrons, r e s u l t s i n a close spacing of energy l e v e l s . This requires care to be taken so as to ensure that i n the lin e a r search for zeros i n the secular determinant, pairs of energy l e v e l s are not overlooked. Jennings et a l . [ 1 2 2 J have encountered the same problem i n t h e i r non-self consistent c a l c u l a t i o n s on 13 atom t r a n s i t i o n metal c l u s t e r s with symmetry. For Ag-,, the number of d i f f e r e n t i r r e d u c i b l e representations i s larger ( a ( , a^ and e ) , and t h i s makes i t much easier to f i n d a l l the one-electron energy l e v e l s . A l l ca l c u l a t i o n s required about 20 i t e r a t i o n s f o r convergence. The v i r i a l r a t i o s were found to be 1 .000023 f o r Agy and 1 .000021 f o r both Ag 7I c l u s t e r s . 3- 2 . 2 Results The one-electron energies, for the occupied valence o r b i t a l s , of the Ag 7 1 and & g n c l u s t e r s are given i n tables V and VI, the highest occupied l e v e l for Ag_, (54e) has occupation number 3 . When these c a l c u l a t i o n s were i n i t i a l l y reported, the i r r e l i a b i l i t y was assessed by comparison of the l o c a l DOS with the Hell photoemission spectrum of the clean Ag surface 1 1 2 3 ] , as i n figure 5 . The calculated curve for atoms 2 , 3 and 65 Level 115 a« -0.398 -0.376 114 a» -0.488 -0.505 113 a» -0.521 -0.528 112 a« -0.615 -0. 624 111 a« -0.619 -0.630 110 a» -0.632 -0.634 109 a« -0.643 -0.646 108 a» -0.647 -0.650 107 a* -0.64 9 -0.661 106 a» -0.660 -0.666 105 a* -0.669 -0.668 104 a« -0.686 -0. 677 103 a* -0.690 -0.692 102 a* -0.700 -0.700 101 a» -0.701 -0.703 100 a* -0.706 -0.718 99 a* -0.729 -0.735 98 a» -0.731 -0.741 97 a» -0.740 -0.746 96 a» -0.745 -0.753 95 a« -0.765 -0.763 94 a* -0.774 -0.775 93 a» -0.780 -0.783 92 a» -1.215 -1.223 76 a" -0.345 -0.354 75 a" -0.512 -0,518 74 a" -0.615 -0.625 73 a" -0.627 -0.633 72 a" -0.631 -0.635 71 a" -0.633 -0.642 70 a" -0.641 -0.644 69 a" -0.643 -0.649 68 a" -0.645 -0.660 67 a" -0.653 -0.670 66 a" -0.662 -0.676 65 a" -0.682 -0.684 64 a" -0.698 -0.704 63 a" -0.706 -0.708 62 a n -0.718 -0.719 61 a" -0.731 -0.724 60 a" -0.742 -0.749 59 a" -0.752 -0.751 Table V. X<*SW one-electron energies (in Bydbergs) f o r the valence o r b i t a l s of the Ag-,1 c l u s t e r s . 66 Level Energy 43 a, -0.430 42 a, -0.614 41 a, -0.651 40 a, -0.685 39 a, -0.699 38 a, -0.738 37 a, -0.754 36 a, -0.772 35 a. -0.784 14 a a -0,643 13 a^ -0.644 12 a 2 -0.653 11 a a -0.719 54 e -0.367 53 e -0.628 52 e -0.642 51 e -0.652 50 e -0.659 49 e -0.665 48 e -0.680 47 e -0.685 46 e -0.705 45 e -0.722 44 e -0.732 43 e -0.749 42 e -0.759 Table VI. X<7<S8 one-electron energies (in fiydbergs) f o r the valence o r b i t a l s of the Ag-, c l u s t e r . 67 Fiqure 5. (a) Deconvoluted Hell photoemissicn spectru* of clean Aq from r 1 2 3 1 ; <b) l o c a l DOS for Aq ( 2 , 3,5) in Aq-> as calculated with the X«.sw method. The Fermi enerqy (Ef) i n (a) has been matched tc the calculated value i n (b) . 68 5, for the energy range -0*6 to -0*8 By, reproduces, i n broad respects, the two peak d-band (experimental band width 3.2 eV £123]), although e f f e c t s associated with o p t i c a l matrix elements are neglected i n t h i s comparison. Also other data from bulk band structures £124,125] and X-ray photoemission spectra (.126], indicates the r e l a t i v e heights of the peaks to be more nearly egual. Bosch and Menzel £101] i n t h e i r X<*rS8 ca l c u l a t i o n on the Ag^ c l u s t e r report a s i n g l e d-peak and a band width of 1.6 eV. The c a l c u l a t i o n for the Ag c l u s t e r gives a d-band width of 2.1 eV, composed of the states 35a ( to 53e, with the top of the d-band 3.6 eV below the Fermi l e v e l (corresponding to the 54e s t a t e ) . The l a t t e r agrees with the value reported by Bosch and Menzel, and i s 0.4 eV less than the experimental value £123]. Above the d-band, there are three energy l e v e l s , 42a, , 43a, and 54e, which are p r i n c i p a l l y of s,p character; they correspond to the high energy t a i l i n the photoemission spectrum. Charge d i s t r i b u t i o n s f o r the Ag_, and Ag ?I c l u s t e r s are reported i n table VII. The 5p populations for the atoms i n the Ag 7 c l u s t e r increase steadily (0.13,0.21,0.30) with increase in coordination number (3,4,5), although the s i l v e r 5s and 4d populations do not show any s i g n i f i c a n t trends, t y p i c a l values being 0.63 and 9.73 respectively. Overall the c a l c u l a t i o n s f o r the Ag-^  c l u s t e r indicate a net charge transfer from the 3-coordinate Ag (designated 1 in figure 4) to the 5-coordinate atoms Ag (2,3,5), and t h i s aspect w i l l be elaborated i n section 3.3. The binding energy for I i n s i t e F was found to be greater than that i n s i t e H by 0.38 eV. This observation seems to be 69 Table VII. P a r t i a l wave decomposition of sphere charges (s,p,d) and net charges ( g ^ i - ) for d i f f e r e n t atoms i n Ag-, and Ag 1I. Cluster Ag, I f Ag(1) Ag (2,3) Ag<4) Ag(5) Ag{6,7) I s 0.52 0.63 0.64 0.63 0.64 p 0. 13 0.30 0.21 0.30 0.21 d 9.72 9.73 9.73 9.73 9. 73 g **t- + 0.22 -0.08 0.00 -0.08 0.00 s 0.51 0.64 0.51 0.58 0.68 1. 87 p 0. 13 0.45 0.37 0.32 0. 19 4. 44 d 9. 72 9.75 9.77 9.74 9.72 0. 15 • 0.39 -0.08 + 0. 10 •0.10 •0. 15 -0. 7 2 s 0.47 0.62 0.58 0.70 0.63 1. 87 p 0.28 0.45 0.21 0.27 0.21 4. 42 d 9.75 9.74 9.74 9.72 9.73 0. 14 gwetr • 0.24 -0.08 •0.22 •0.04 + 0.17 -0. 72 In c a l c u l a t i n g qn*k , the intersphere and outersphere charge i s allocated to Ag and I i n the proportion 1 to 5. 7 consistent with the LEED analysis, for I adsorbed on the (111) surface of s i l v e r , but unfortunately i n the X <*S» ca l c u l a t i o n s f o r the c l u s t e r s , the p o s s i b i l i t y that the r e l a t i v e values are influenced by Ag(1) and Ag (4) having s u b s t a n t i a l l y d i f f e r e n t net charges (•0.22 and 0.00) i n Ag^ cannot be ruled out. This highlights a deficiency of t h i s simple c l u s t e r , since these two atoms are equivalent i n the actual (111) surface. In both A q 7 I F and Ag-,IK the electron populations on the Ag atom nearest to the I increase by an average of 0.15 electrons; this corresponds to the average increase for the 5p population, the 5s and 4d populations change on average by -0.02 and +0.-02 respectively. In figures 6, 7 and 8 the t o t a l and l o c a l DOS are LEAF 70 OMITTED IN PAGE NUMBERING, 70R Fiqure 6. DOS calculated for the hqnIF cluster. The broadeninq factor r=0.0075 used i n equation (1.4). The curves desiqnated by tr i a n q l e s and squares correspond to s and p components respectively, and these components have both been multiplied by a factor of 2. For the iodine atom the tr i a n q l e s , squares and hexagons qive the unsealed s, p and d components. loc 7 o E Energy (Ry) "7oF L E A F 71 O M I T T E D I N P A G E N U M B E R I N G . 71 ft Figure 7. DOS c a l c u l a t e d f o r the Ag-jln c l u s t e r . The broadening f a c t o r •* = 0.0075 used i n equation (1.4). The curves desiqnated by t r i a n g l e s and squares correspond to s and p components r e s p e c t i v e l y , and these components have bcth been m u l t i p l i e d by a f a c t o r of 2. For the i o d i n e atom the t r i a n q l e s , squares and hexagons g i v e the unsealed s, p and d components. 1 2 0 Atom 5 6 0 l i t Energy (Ry) 8 0 n 4 0 A Energy (Ry) Energy (Ry) 72 Figure 8. DO £ calculated for the hq-j cluster. The broadeninq factor •"=0.0075 used in equation (1.4). The curves desiqnated by trianqles and c i r c l e s correspond to s and p components respectively, and these components have both been multiplied by a factor of 2. The atoms are labelled as in table VIII with A beinq Ag(2,3,5), B beinq Aq(4,6,7) and C beinq Aq(1) i n fiqure 4. 73 presented for the A g 7 I p , A g 7 I H a n d Ag7 c l u s t e r s respectively. These figures do not show the 92a • l e v e l f o r the Ag-,1 c l u s t e r s , whose binding energy i s around 1.2 By. This l e v e l i s almost e n t i r e l y associated with the iodine 5s atomic o r b i t a l , and i t i s found not to i n t e r a c t to any s i g n i f i c a n t degree with the s i l v e r o r b i t a l s . These figures i l l u s t r a t e f o r the Ag^I c l u s t e r s that those atoms which are nearest to the iodine have the largest modifications to their DOS. The Ag^I l e v e l s 113a*, 114a* and 75a", with an approximate energy of -0.51 By, are predominantly composed of iodine 5p-orbitals, however the l o c a l DOS f o r iodine indicates that the 5p-orbitals do i n t e r a c t with the s i l v e r d-band. Contour maps for some one-electron wavef unctions of Ag-,1 p are plotted i n f i g u r e s 10-^ 14 i n the plane containing I, Ag{1), Ag (4) , and Ag(5) (the atom positions in the wavefunction plots are indicated by f i g u r e 9), the l e v e l s plotted are for a* symmetry since t h i s includes the two iodine p - o r b i t a l s which are symmetric to r e f l e c t i o n i n the <r^ plane of symmetry. The 93a' l e v e l (figure 10) which i s at the bottom of the d-band, shows strong bonding interactions between the 5p o r b i t a l at I and the 4d o r b i t a l at Ag(4) as well as 4d-4d interactions between Ag(1) and Ag(5)./ The 102 a* l e v e l (figure 11), which i s approximately at the centre of the d-band, also shows bonding interactions between the I 5p-orbital and the 4d-orbital on Ag(4), and some antibonding taking place between Ag(1) and Ag (5). Antibonding interactions occur between Ag(4) 4d and I 5p f o r the l e v e l s 113 a» (figure 12) and 114 a» (figure 13). These higher energy states are modified by contributions from the Ag 5s and 5p o r b i t a l s , and t h i s i s 74 Ag(1) Ag(2,3) Ag (4) Ag (5) Ag(6,7) Figure 9. Atom positions i n the wavefunction contour plots (figures 10-19); atoms I , Ag(1), Ag(4), and Ag (5) are i n the plane of the of the plots-i l l u s t r a t e d by the 115a* l e v e l (figure 14)-Equivalent wavefunctions f o r the Ag^I^ c l u s t e r are plotted i n figures 15-19- Bithin the a* set of l e v e l s the two I 5p o r b i t a l s i n t e r a c t p r i n c i p a l l y with Ag(1), although of course the t h i r d 5p o r b i t a l (which belongs to a") i n t e r a c t s with Ag(2) and Ag (3)- The 93a* l e v e l (figure 15) shows strong i n t e r a c t i o n s between I and Ag(1), just as i n the A g r I p case, and 4d-4d 75 Figure 10. The 93a 1 wavefunction f o r Aq-,Ir. The contour values are 1=0.004, 2=0.016 and 3=0.064, and the aton p o s i t i o n s are shown i n f i g u r e 9. 76 Fiqure 11. The 102a' wavefunction f o r Aq 7Ip. The contour values are 1=0.004, 2=0.016 and 3=0.064, and the atea p o s i t i o n s are shown in f i q u r e 9. 77 Figure 12. The 113a» wavefunction f o r Aq-j I p . The contour values are 1=0.004, 2=0.016 and 3=0.064, and the ato» p o s i t i o n s are shown in f i q u r e 9. 78 79 Figure 11. The 115a» wavefunction for A q ^ I p . The contour values are 1=0.004, 2=0.016 and 3=0.064, and the atca positions are shown in fiqure 9. 80 Fiqure 15. The 93a 1 wavefunction f o r Aq.,IH. The contour values are 1=0.004, 2=0.016 and 3=0.064, and the atoa p o s i t i o n s are shown in f i q u r e 9. 81 Fiqure 16. The 102a« wavefunction f o r Aq-,IH. The contour values are 1=0.004, 2=0.016 and 3=0.064, and the atom p o s i t i o n s are shown in f i q u r e 9. The i n * . r . f i q u r e 17. xue 113a' vavef u n c t i o n f o r Ao, = 0.004, 2=0.016 and 3=0.064 all f h . J c o n t o u r ™lues are in f i q u r e 9. ' 4 ' a n d t h e a t C B P o s i t i o n s are shown 8.3 Fiqure 18. 1-n nnu a' . T n ^ 0 ^ f ° r Aq^J«' T h e c o n t ° ^ values are ir f i q u r - T 3 = 0 * 0 6 4 ' a n d t h e a t 0 " Positions are shown 84 Fiqure 19. The 115a' wavefunction f o r Aq-jI H. The contour values are 1=0.004, 2=0.016 and 3=0.064, and the atoa p o s i t i o n s are shown in f i q u r e 9. r 85 interactions between Ag(4) and ftg (5) - The 1.02 .a* l e v e l (figure 16) i s somewhat unexpected i n that the I 5p-orbitals i n t e r a c t primarily with Ag(5) rather than with Ag(1). Antibonding interactions between I and Ag(1) are again shown f o r the 113a» (figure 17) and 114a» (figure 18) l e v e l s , and there i s an appreciable involvement by the Ag 5s and 5p o r b i t a l s i n the 115a* l e v e l (figure 19). Generally these plots of wavefunctions indicate interactions between I and the Ag to be very s i m i l a r for I i n either of the s i t e s designated F or H-3.2.3 Conclusion The charge d i s t r i b u t i o n s i n table VII e s t a b l i s h that the s i l v e r 4d and 5p o r b i t a l s , as well as the 5s, contribute to the net bonding, both at the clean surface and i n the presence of iodine. The contribution to the bonding by the 4d o r b i t a l s i s small, and i f the bonding only involved the s i l v e r 5s o r b i t a l s the choice of adsorption s i t e would depend es p e c i a l l y on non-bonded and Van der Saals i n t e r a c t i o n s . However, the involvement of s i l v e r 5p o r b i t a l s suggests that the adsorption s i t e may be determined by d i r e c t bonding factors perhaps involving hybridisation between the p and d o r b i t a l s [127]. For metals, t h i s model involves resonance between various structures involving p a r t i a l l y occupied hybrid o r b i t a l s . The appropriate hybrids f o r an atom in the fee l a t t i c e corresponds to p 3 d 3 , whereas fo r the hep l a t t i c e there can be a mixture of spd*, pd s and sd*. These d i f f e r e n t hybridisations suggest that d i f f e r e n t 86 atomic populations might be expected for the iodine near-neighbours at the d i f f e r e n t adsorption s i t e s ; however t h i s has not been found in t h i s work. S i m i l a r l y Brewer £128] has indicated e l e c t r o n i c configurations of the type (4d) *<>-* (5s) M 5 p ) ^ f o r bulk s i l v e r with n=0.7 for the hep arrangement of atoms and n>1.5 for the fee arrangement. These ele c t r o n i c configurations correspond to 2.4 and at l e a s t 4.0 bonding electrons per atom respectively, and are appreciably greater than the 1,2 and 1.3 bonding electrons found here for Ag 7 and Ag^I .The l a t t e r values may represent lower estimates because of the reduced coordination numbers i n the c l u s t e r s . This section demonstrates that, although c l u s t e r s were used whose sizes are comparable with, or greater than, those considered useful for interpreting photoelectron spectra, i t seems l i k e l y that larger c l u s t e r s w i l l be needed for establishing r e l a t i o n s , such as i n an extension of the Engel-Brewer co r r e l a t i o n £128], between geometrical and e l e c t r o n i c structure at surfaces. 3.3 Calculations on Ag . o , A g i v and Agn In t h i s section additional X^SH c a l c u l a t i o n s on s i l v e r clusters are presented £120] i n order to study further how the e l e c t r o n i c structures of s i l v e r c l u s t e r s r e l a t e to those for the surface and the bulk regions. The new c l u s t e r s examined are Ag iX) (C^ symmetry), Ag ^ (0 K) and A g M ( 0 K ) ; with the atoms in Ag being arranged to form a cubo-octahedron as shown in 87 figure 3. The Ag,0 c l u s t e r i s obtained when atoms 1, 4 and 5 are removed from Ag^ (see figure 3) ; the c e n t r a l atom i n Ag,0 (designated atom 13) has the same nearest neighbour enviroment as an atom at the (111) surface. The Ag(C) c l u s t e r i s formed from Ag)5> by adding one atom on the four-?fold axis above each cube face. The c e n t r a l atom i n both Ag,3 and Ag ^  have the appropriate coordination numbers and arrangements of near neighbours as i n bulk s i l v e r ; the A g H c l u s t e r also has the required number of next-nearest neighbours to the central atom. Interatomic distances and angles i n a l l these c l u s t e r s correspond to those appropriate for the bulk structure of s i l v e r £ 121 J. 3-3,1 Method of Calculation The c a l c u l a t i o n s follow the procedure described i n section 3.2-1- A l l s i l v e r atoms were given an atomic s i l v e r sphere radius which corresponds to a 6% increase over the value appropriate f o r the touching sphere model. The outersphere was always centred on the central atom, and given a radius so that i t was tangential to the spheres of the outer atoms. The standard Schwarz <X exchange parameter f o r s i l v e r (0.70130) was used f o r a l l regions of the c l u s t e r £96]. P a r t i a l waves through to 1=2 were included in the atomic regions, although higher p a r t i a l waves were used when a l l components 1<2 were absent on an atomic centre f o r a p a r t i c u l a r i r r e d u c i b l e representation. P a r t i a l waves through to at l e a s t 1=4 were used i n the 88 outersphere region. These ca l c u l a t i o n s yield nearly i d e a l v i r i a l r a t i o s , namely -2T/V i s 1.000039 for Ag , 1.000023 f o r Ag ( S and 1.000040 f o r Ag^. 3. 3. 2 He s u i t s Calculated DOS are given i n figures 8, 20, 21 and 22, and the s e l f - c o n s i s t e n t charge d i s t r i b u t i o n s and d-band widths are summarized i n table VIII- The symmetrically eguivalent atoms i n the s i l v e r c l u s t e r s are la b e l l e d according to t h e i r coordination numbers, these l a b e l s are l i s t e d i n table VIII. 3-3-3 Discussion As mentioned i n section 3-1, a number of recent c a l c u l a t i o n s have compared the t o t a l DOS for c l u s t e r s with bulk band structures and photoelectron spectra [35,101]. The observation of some generally consistent trends between d-band widths from cal c u l a t i o n s and from the experimental spectra has often been taken as evidence that the c l u s t e r used i s providing a good representation of the s o l i d , although s t r i c t l y photoelectron spectra from s o l i d s represent averages over l o c a l DOS f o r surface and bulk atoms. The l o c a l DOS are important also i n other contexts- Thus i n p r i n c i p l e they are s i g n i f i c a n t for chemical r e a c t i v i t i e s i n s o f a r as they determine o r b i t a l 89 Clu-ster ftg. Ag 10 Ag ( 3 Ag atom Nu miser Coordi- Valence s h e l l Sphere d-band type of nation charges charge width atoms number • " s P d (eV) a 3 5 0. 63 0.30 9.73 10.66 1.70 B 3 4 0. 64 0.21 9.73 10.58 1-71 C 1 3 0. 52 0. 13 9-72 10.36 0-58 Intersphere 2.83 Outersphere 0.09 a 1 9 0. 58 0.50 9.68 10.77 0.33 B 3 5 0. 63 0.23 9-69 10.55 1-01 C 6 4 0. 53 0. 16 9.65 10-34 1.20 Intersphere 4-92 Outersphere 0.64 a 1 12 0. 62 0.68 9.74 11-09 0.30 B 12 5 0. 55 0.21 9.62 10-37 1.51 Intersphere 6-43 Outersphere 1.02 a 1 12 0. 73 0.59 9.79 11. 18 0.35 B 12 7 0. 64 0.31 9.65 10.60 1.92 C 6 4 0. 42 0. 14 9.66 10.22 0.41 Intersphere 9.05 Outersphere 0.23 Table VIII. Charge d i s t r i b u t i o n s and d-band widths in l o c a l DOS f o r di f f e r e n t s i l v e r c l u s t e r s . The d-band width i s determined as the f u l l width at h a l f maximum height. overlap and energy resonance with chemisorbed species. Also, as w i l l be emphasized i n t h i s section, the l o c a l DOS bring out the d i s t i n c t i v e features of small c l u s t e r s . Figures 8, 20, 21 and 22 show for the s i l v e r c l u s t e r s that there are substantial differences between the t o t a l DOS and the various l o c a l DOS, and, as noted below, these differences are informative of fundamental aspects of the e l e c t r o n i c structure. A convenient reference point f o r assessing these differences i s provided by 90 Fiqure 20. DOS calculated for the Aq,© cluster. The broadeninq factor «r=0.0075 used i n equation (1.1). The curves desiqnated by trianq l e s and c i r c l e s correspond to s and p components respectively, and these components have both been multiplied by a factor of 2. 91 SOCH >aocH *5 •g'looH & Ag r a local DOS atom A BCh 9CH 6(H Ag„ loco) DOS atom B Fiqure 21. EOS calculated for the Aq,t cluster. The broadeninq factor ••=0.0075 used in equation (1.4). The curves desiqnated by trianqles and c i r c l e s correspond to s and p coaponents respectively, and these coaponents have bcth been a u l t i p l i e d by a factor of 2. 92 Fiqure 22. EOS calculated for the Aq n cluster. The broadeninq factor f=0.0075 used i n equation (1.4). The curves desiqnated by trianqles and c i r c l e s correspond to s and p coapcnents respectively, and these coaponents have bcth been a u l t i p l i e d by a factor of 2. 93 a recent s e l f c o n s i s t e n t - f i e l d c a l c u l a t i o n f o r a 3-layer Cu(100) thi n f i l m . In t h i s work. Gay et a l . £129] showed that the t o t a l DOS of the thin f i l m resembles cl o s e l y that calculated f o r bulk copper. Also the l o c a l DOS of the surface layer exhibits some band narrowing with respect to that of the central plane. The l o c a l DOS calculated for the s i l v e r c l u s t e r s generally show greater differences than those found by Gay et a l - ; i t i s also c l e a r that the l o c a l DOS f o r high coordination s i t e s do not approach the values expected for surface or bulk atoms of the extended l a t t i c e - Thus the d-band width from a photoemission study of s i l v e r i s 3.2 eV £ 123], while the l o c a l DOS for high coordination s i t e s are much narrower (e.g. those for the 12-coordinate atoms i n Ag,s a n <* A9 n a r e 0.30 and 0-35 eV respectively}- These l a t t e r features are associated with states that are largely l o c a l i s e d on the c e n t r a l atoms. For example, in Ag l 3 the two lowest valence states have symmetries t a ^ (-0.883 Ry) and e^ (-0-873 By), and over 8 of the 10 electrons belonging to these states are l o c a l i s e d on d-type functions on the central atom. These three states are a l l at higher binding energies than the lower edge of the main d-band. Such s h i f t s i n energy l e v e l s are i d e n t i f i e d most c l e a r l y by l o c a l DOS from self-consistent c a l c u l a t i o n s , although s i m i l a r tendencies can be detected i n other work, such as i n the l o c a l DOS reported by Jennings et a l . £122] with non-self-consistent c a l c u l a t i o n s and i n the t o t a l DOS calculated by Messmer et a l . £35] with se l f - c o n s i s t e n t c a l c u l a t i o n s . The s e l f - c o n s i s t e n t c a l c u l a t i o n s reported here are for 94 c l u s t e r s whose component atoms show appreciable ranges i n coordination numbers. Incomplete screening of the neighbouring ion cores by the valence electrons ensures that atoms with high coordination number are i n a region of lower p o t e n t i a l , r e l a t i v e to their lower coordinate neighbours, and t h i s r e s u l t s i n atoms with high coordination numbers having l o c a l i s e d states and net negative charges. This l o c a l i s a t i o n i s enhanced i n Ag , where the coordination number of the c e n t r a l atom i s 12, compared with A g l o , where the coordination number of the cen t r a l atom i s 9. I t i s apparent also that, i n t h i s context, l o c a l i s a t i o n reguires s i g n i f i c a n t differences i n coordination numbers for neighbouring atoms. Thus the l o c a l i s a t i o n i s greater at the c e n t r a l atom in Ag |^  than i n .Ag^ . The c e n t r a l atoms are 12 coordinate i n both cases, but the neighbouring atoms have coordination numbers 5 and 7 respectively. Two other s i g n i f i c a n t features should be noted. F i r s t , narrow features i n l o c a l DOS can be shown by atoms with low coordination numbers, and so correspond to the converse of the s i t u a t i o n f o r high coordination number discussed above. For example, the outer in-coordinate atoms i n Ag |6j have l o c a l DOS centred on the low binding energy side of the o v e r a l l d-band, and these correspond to l o c a l i s e d states of the "dangling bond" type. Second, the l o c a l DOS f o r the 5-coordinate atoms i n Ag,0 and Ag ] Z are appreciably d i f f e r e n t even though the neighbouring atoms are arranged i n the same way i n both cases. This emphasises that ensuring the correct l o c a l coordination i s not by i t s e l f necessarily s u f f i c i e n t for a cl u s t e r to give a good account of ele c t r o n i c structure i n a s o l i d . In r e a l s o l i d s , atoms with 95 high coordination numbers neighbour other atoms with high coordination numbers. For s i l v e r the bulk coordination number of 12 i s reduced only to 9 at the (111) surface, although i t i s reduced further for atoms at kink s i t e s on high index surfaces. Christmann and E r t l £130] have recently reviewed measurements of work function on stepped surfaces, and they provide evidence, f o r a stepped platinum surface, that edge atoms of low coordination number can be p o s i t i v e l y charged while the high coordination s i t e s are negatively charged. Such a conclusion i s consistent with the cl u s t e r c a l c u l a t i o n s and the arguments about screening which are presented here. atom type Cluster • a B c &g 7 0 0.05 -0.38 0 -0.24 -1.03 a g , 3 0 -1. 76 . -0 -1.26 -1.82 Table IX. Calculated 3d core l e v e l s h i f t s f o r d i f f e r e n t atom types i n the s i l v e r c l u s t e r s . The l e v e l s h i f t s (in eV) are r e l a t i v e to the high coordinate atoms i n each c l u s t e r , a p o s i t i v e s h i f t i ndicates a greater 3d binding energy than i n the reference atom. The atom types are the same as i n table VIII. 96 Although the comparisons made here highlight some d i f f i c u l t i e s i n representing s o l i d s , or parts of s o l i d s , fcy c l u s t e r s , small atom c l u s t e r s are important i n t h e i r own r i g h t and experimental information i s available f o r some systems. For example, Mason and Baetzold 1.131] have measured X-ray photoelectron spectra f o r s i l v e r evaporated, onto a carbon substrate, and with increasing coverages in the range 2.5x10 1 5 to 4x10** s i l v e r atoms cm-* they observed the development of a shoulder on the high binding energy side of the s i l v e r d-band. Mason and Baetzold believed for these coverages, on the basis of theories of the nucleation process, that most s i l v e r atoms belong to c l u s t e r s containing four or more atoms, that there i s a d i s t r i b u t i o n of c l u s t e r s i z e s , that the average s i z e increases with increasing coverage, and that the. c l u s t e r s are arranged to maintain minimal contact with the carbon substrate, the development of high-energy structure, found by Mason and Baetzold for increasing coverage, i s broadly consistent with the c a l c u l a t i o n s presented here i f there i s an increase i n cluster s i z e and a consequent increase i n coordination number. The increase i n high-energy structure with increasing c l u s t e r size i s seen i n the t o t a l DOS in figures 8, 20, 21 and 22, and these can be supplemented by the t o t a l DOS of an Ag b c l u s t e r obtained i n the c a l c u l a t i o n s by Bosch and Menzel £101]. At present no information seems to be available from experiments f o r c l u s t e r s i n the range Ag-, to Ag^ , however the spread i n the net charges would be expected to influence some properties i f these c l u s t e r s are studied eventually. For example, predictions of ESCA s h i f t s f o r the s i l v e r 3d core 9 7 l e v e l are given i n table IX; these s h i f t s are calculated from the ground state X SW eigenvalues. These cal c u l a t i o n s do not include relaxation, but that seems unlikely to change the impression that some of these s h i f t s should be large enough to be: measured. For most cl u s t e r s s h i f t s to higher binding energy with increasing coordination number are predicted. This tendency must be seen as the r e s u l t of two opposing e f f e c t s associated with the increased valence electron charge and with the increased number of surrounding ion cores f o r atoms having higher coordination numbers. The c a l c u l a t i o n s presented here indicate the second dominates i n determining 3d l e v e l s h i f t s . 3.4 Summary This chapter i l l u s t r a t e s some of the uses and l i m i t a t i o n s of c l u s t e r s for modelling l o c a l e l e c t r o n i c structure i n surface and bulk s i t u a t i o n s . It has been concluded that whilst the c l u s t e r approach has proved useful for interpreting changes in the surface e l e c t r o n i c structure, as indicated by changes i n photoemission spectra when atoms are adsorbed on a surface, c a l c u l a t i o n s on larger c l u s t e r s w i l l be needed to explain why an adsorbate favours a p a r t i c u l a r geometrical arrangement on the surface. Consideration of l o c a l DOS, rather than the t o t a l DOS, demonstrate that the i n d i v i d u a l atoms in a c l u s t e r have el e c t r o n i c structures appreciably d i f f e r e n t from the atoms in the bulk. This perhaps suggests that i n c a l c u l a t i o n s on c l u s t e r s more attention should be given to the l o c a l DOS, 98 rather than the t o t a l DOS which most workers have considered previously £37], especially since photoemission spectra e f f e c t i v e l y provide a measure of the average l o c a l DOS at a surface. I t i s cle a r that there i s s t i l l much work to be done i n determining the range of usefulness of c a l c u l a t i o n s on c l u s t e r s . 99 CHAPTER 4 CNDO CALCIILATIQNS ON CLUSTERS OF SILVER ATOMS 4.1 Introduction In t h i s chapter the s u i t a b i l i t y of the CNDO method f o r examining c l u s t e r s of t r a n s i t i o n metals i s investigated. The motive f o r t h i s study i s that, while the X<*S8 calc u l a t i o n s give helpful r e s u l t s , the computational expense for the CNDO method i s much l e s s , p a r t i c u l a r l y f o r e l e c t r o n i c a l l y complex systems. For example, the X o < SB c a l c u l a t i o n on Ag ^ , discussed in the l a s t chapter, reguired 45 minutes of computer time, with the Ok symmetry of the cl u s t e r being used to simplify the secular eguations, whereas the CNDO c a l c u l a t i o n , which did not include symmetry s i m p l i f i c a t i o n s , took only 25 minutes (times for both ca l c u l a t i o n s are for using an IBM 370/168 computer). X cx.S8 cal c u l a t i o n s on Ag,^ c l u s t e r s with lower symmetry would require much longer computer times; for systems with very low symmetry there are also complications associated with f i n d i n g near degenerate roots i n the secular equation (e.g. see section 3-2-1)- By contrast, the l a t t e r problem i s absent i n the CNDO method, and the CNDO ca l c u l a t i o n on any Ag (e, cl u s t e r would always reguire approximately 25 minutes of computer time. Exploratory c a l c u l a t i o n s which investigate reaction p r o f i l e s or which compare a series of d i f f e r e n t adsorption s i t e s w i l l most often involve systems of low symmetry, and r e l i a b l e semi-empirical schemes would be most advantageous. In order to explore the degree to which the CNDO method i s pot e n t i a l l y capable of treating c l u s t e r s of t r a n s i t i o n metals. 100 the parametrisation of the CNDO method i s reconsidered for s i l v e r [ 132J. Baetzold £71] made the f i r s t attempt to obtain CNDO parameters for s i l v e r atoms by u t i l i s i n g experimental data so that certain quantities, such as the equilibrium bond -length of diatomic s i l v e r , are reproduced. However, Baetzold»s o r i q i n a l parameters seemed to qive u n r e a l i s t i c a l l y narrow d-band widths, e.g. approximately 0.1 eV f o r c l u s t e r s composed of six s i l v e r atoms (this value i s estimated because Baetzold does not s p e c i f i c a l l y quote band widths) , whereas an X <*SH method cal c u l a t i o n on octahedral Ag^ qives the d-band width as 1.76 £101]. However a caveat must be added that I did not reproduce Baetzold's c a l c u l a t i o n s i n f u l l d e t a i l because I did not make an "averaging" of ce r t a i n two-electron i n t e g r a l s which Baetzold u t i l i s e s in h i s calculations. At t h i s stage we f e l t we should be doing b a s i c a l l y standard CNDO cal c u l a t i o n s . More recently Baetzold £133] has proposed the use of le s s contracted 4d functions f o r s i l v e r which r e s u l t i n more acceptable d-band widths. As emphasised i n chapter 2, the problem of getting CNDO parameters f o r atoms i n metal c l u s t e r s i s n o n - t r i v i a l i n the absence of accurate c a l c u l a t i o n s or extensive experimental data. Our parametrisation of the CNDO method for s i l v e r i s performed by making comparisons against the X^ SB c a l c u l a t i o n on the Ag-, c l u s t e r discussed i n chapter 3. The use of X cx SB calcul a t i o n s to c a l i b r a t e a semi-empirical procedure was i n i t i a l l y proposed for the extended Huckel method £34,35,134], and i t also represents an extension of the approach used for LiF i n section 2.8. The Ag 7 c l u s t e r , with C 3 v / symmetry, has 101 three set of ineguivalent atoms (one atom on the Cj axis and two sets of three atoms around the a x i s ) ; therefore Ag n has more properties f o r following the e f f e c t s of varying parameters than would a clus t e r of higher symmetry (e.g. an octahedral clu s t e r analogous to the Ni^ c l u s t e r used by Blyholder £ 72J) . The main part of t h i s chapter s t a r t s by describing the derivation of the CNDO parameters f o r s i l v e r by comparison with the X oi SW cal c u l a t i o n s f o r Ag-,. These new parameters are then used for making further CNDO calcu l a t i o n s on Ag c l u s t e r s , and comparisons are made with the re s u l t s from the other X* SW calc u l a t i o n s presented i n the previous chapter. 4.2 Selection of Ag CNDO parameters The s i l v e r c l u s t e r s are examined using the CNDO eguations (2.91)-(2.94) , with the Ag basis set chosen to consist of the 4d, 5s and 5p Slater type atomic o r b i t a l s . The Ag7 , Ag^ and Ag|c, c l u s t e r s contain an odd number of electrons. S t r i c t l y an open s h e l l treatment i s reguired f o r such systems, and i n some cases a sum of Slater determinants may be necessary; however to circumvent such a treatment here the density matrix P^ ,) i s expressed i n a spin independent form analogous to the density operator in the spin independent Xo<sw eguations. Thus the new form of the density matrix i s where n i i s the occupation number of the l e v e l s with energy X and c ^ c o ^ i s expressed as the average 102 where the summation involves appropriate c o e f f i c i e n t s f o r a l l the degenerate functions for the i - t h l e v e l . The summation in (4.2) i s of course redundant f o r non-degenerate l e v e l s , hut with partially-occupied degenerate l e v e l s i t ensures that the t o t a l charge d i s t r i b u t i o n retains the point group symmetry assumed for the nuclear framework. The CNDO eguations (2.91) and (2.92) assume the ground state e l e c t r o n i c configuration of 4d*°5s* to provide the H ^ and N^, values for s i l v e r . 4 .2.1 Comparisons between CNDO and X&fSjj c a l c u l a t i o n s In parametrising the CNDO method d i r e c t l y against r e s u l t s from Xo<! SM calc u l a t i o n s , i t i s necessary to take i n t o account cert a i n differences between the two methods. For example, as discussed i n chapter 2, the one-electron wavefunctions have diff e r e n t forms. However some support f o r d i r e c t l y comparing the population analyses, defined i n section 2.11, for the two di f f e r e n t methods was indicated from our previous study on P^ S^  which found close s i m i l a r i t i e s to ex i s t between the atomic charges given by the X c*SW method and the conventional Hulliken type populations from the CNDO method I 135.].. Further d e t a i l s on the P^S^ c a l c u l a t i o n w i l l be given i n chapter 5. , As pointed out i n section 2.3, probably a more fundamental difference between the Xc^SH and CNDO methods involves the one-electron eigenvalues. However, in t h i s chapter i t has been 103 supposed, even though d i f f e r e n t interpretations are applicable to these one-electron energies, that the re l a t i v e energies (e.g. band widths) from the two sets of ^ values can be compared d i r e c t l y . I t may seem more reasonable to compare the CNDO eigenvalues with the X ° < SB t r a n s i t i o n state energies as both of these quantities represent approximations to the negative of the ionisation enerqies, however the X o< SW tr a n s i t i o n energies are often related to the X c ^ s w one-electron eigenvalues by a uniform relaxation. For example, Bosch and Menzel £.101J obtained t r a n s i t i o n state energies for Ag clus t e r s which were uniformily relaxed to within ±0.3 eV; si m i l a r observations are also presented i n chapter 5 f o r molecules and clust e r s formed by some second-row atoms. tt.2.2 Parameters and Properties Examined The use of the CNDO equations (2-91)-(2.94) requires the s p e c i f i c a t i o n of the following parameters; a) the exponents = and ^  i n the Slater type o r b i t a l s which are used for evaluating the overlap and Coulomb i n t e g r a l ; b) the average of the i o n i s a t i o n energy and electron a f f i n i t y f o r each member of the basis set, i . e . ( I s * A s ) / 2 , The main properties used f o r comparing with the X SW cal c u l a t i o n s are: a) the d-band width ( A ^ ) , that i s the difference i n the (I p * a p ) / 2 and (Ij+A^/2; c) the bonding para 104 L S H S one-electron energies of the lowest (d ) and highest (d ) states which can be regarded as having mainly 4d character; b) the d-band shape, considered with reference to the l o c a l DOS and t o t a l DOS given by equation (1.4); c) the enerqy separations f o r the s-type energy l e v e l s of symmetry a (two states designated a, and a, to dist i n g u i s h the lower a and higher a states) and of symmetry e (designated e to emphasise that i t i s primarily based on s functions) such that 2 3 (»•*);• d) the atomic populations including the net al l o c a t i o n s to s, p and d functions. If the differences i n inter p r e t a t i o n regarding electron populations are neglected then numerical evaluations between the Xtxsw and CNDO cal c u l a t i o n s are d i r e c t l y a vailable f o r (a), (c) and (d), although v i s u a l inspection only i s used f o r (b). 4.2.3 Parameter Selection Because of the necessity f o r some v i s u a l inspection of DOS curves, no precise optimisation procedure could be used f o r comparing the r e s u l t s from the X<XSI method with those from the CNDO cal c u l a t i o n s ; instead a trial^and-error approach was used. The range over which the CNDO parameters were varied was guided i n part by the parameters used by Baetzold £71]. These 105 parameters, together with the f i n a l values are shown i n table X- For the exponents, Baetzold made use of dementi and £ 8 I n i t i a l F i n a l 1-35 1-25 3 . 6 9 2 . 2 5 (I s+A s ) / 2 4 . 2 6 6 -10 ( I ? + A? ) / 2 2 -39 4 . 6 5 (I^+A^)/2 8 -28 5 .23 - 1 . 0 0 - 0 . 7 5 - 1 - 0 0 - 4 . 4 5 Table X. CNDO parameters for s i l v e r . The i n t i a l parameters are from Baetzold £ 7 1 J . The f i n a l parameters are from comparisons with the X-<S« cal c u l a t i o n s on Ag7 and correspond to run 41 of table XI. The units of the (I *A^) /2 and jj, are ev. fiaimondi*s values £ 7 5 ] (viz- ^ s = 1 - 3 5 , 3 . 6 9 ) , but other plausible s t a r t i n g exponents are those of Slate r £ 4 9 ] ( ^ = 0 . 9 3 , = 2 .12) and Burns £76 ] ( j s = 1 .54 , = 3 . 0 5 ). In the search for improved CNDO parameters for 106 s i l v e r , CNDO cal c u l a t i o n s were made for Ag 7 with some 70 di f f e r e n t combinations of parameters. The f i r s t , group of runs led to an o v e r a l l view of the main e f f e c t s of varying each parameter; the search for new optimal values was then continued with the sets of cal c u l a t i o n s summarised i n tables XI, XII and XIII. For example i t became clear that the parameters £j and ^ mainly a f f e c t A d with only a s l i g h t modification of the d-band shape. Increasing ^ w i l l reduce the d - o r b i t a l overlap i n t e g r a l s , and as a consequence the d-band width J ^ d i s reduced; i n p r i n c i p l e there i s also some change i n the d-band shape but the l o c a l DOS f o r d i f f e r e n t ^ indicate that t h i s i s a r e l a t i v e l y weak e f f e c t . Increasing the magnitude of jS^ , increases the d-band width £^; indeed the bonding parameter i s found to have an almost l i n e a r r e l a t i o n s h i p with hj( whilst r e t a i n i n g the r e l a t i v e spacing of the energy l e v e l s . For example, i n figure 23, the v a r i a t i o n of the d-band one-electron energies which belong to the e i r r e d u c i b l e representation are plotted against ^ , whilst a l l the other parameters are f i x e d . Similar r e s u l t s have also been obtained for the other symmetry types, and t h i s observation i s a consequence of having e s s e n t i a l l y a f u l l y occupied set of d-o r b i t a l s f o r s i l v e r ; to a f i r s t approximation these d-orbitals can be factorised out of the secular equation. Further, since the d-orbitals are almost non-bonding on balance, the charge density matrix elements P j are close to zero, and the off-diagonal Fock matrix elements for the d-orbitals reduce approximately to 107 .SUr is U l W ' W $. fa 1 0-93 2 0.93 3 0.93 4 0.93 5 0.93 6 0.93 7 0.93 8 0.93 9 0.93 10 0.93 11 0.93 12 0.93 13 0.93 14 0.93 15 1.35 16 1.35 17 0.93 18 0.93 19 0.93 20 1.35 21 1.35 22 1.35 23 1.54 24 1.54 25 1.13 26 1. 13 27 1. 13 28 1. 13 29 1.35 30 1.35 31 1.35 32 1.35 33 1.35 34 1.35 35 1.35 36 1.35 37 1.35 38 1.35 39 1.25 40 1.25 41 1.25 2.12 4.26 2.35 4.26 2.45 4.26 2-25 4.26 2.25 6.20 2.25 7.10 2.25 7.10 2.25 4.26 2.25 6.30 2.25 2-22 2-25 4.26 2-25 4.26 2.25 4.26 2.25 4.26 3.69 4.26 2-75 4.26 2.25 4-26 2.25 5-40 2-25 5-00 2.25 4-26 2-25 5.34 2-25 5.34 2.25 4.26 2.25 4,26 2.25 4.26 2.25 4.26 2.25 5.56 2-25 4.4 6 2.25 5.34 2.25 5.34 2.25 5.34 2.25 5.34 2-25 5.34 2.25 5.34 2.25 5.34 2-25 5.34 2.25 6.10 2. 25 6. 1 0 2.25 6.10 2.25 6.10 2.25 6.10 2.39 8.28 2.39 8.28 2.39 8.28 2-39 8.28 2.39 8.28 2.39 8-28 2-39 8-28 2-39 8.28 4.43 8-28 0-35 8-28 2-39 8.28 2.39 8.28 2.39 8.28 2.39 5-23 2.39 8.28 2.39 8.28 2.39 5.23 2.39 5.23 2.39 5-23 2- 39 5.23 2.39 5.23 3- 39 5.23 2.39 5.23 2-39 5.23 2.39 5.23 2.39 5-23 3.79 5.23 2.79 4.23 3.39 5.23 3.39 5.23 3.90 5.23 3.39 5.23 3.39 5-23 3. 39 5- 23 4- 39 5.23 4.05 5-23 4.05 5-23 4.85 5.23 4.85 5.23 4.05 5.23 4.65 5.23 -1-00 -8.00 -1.00 -8.00 -1.00 -8.00 -1.00 -8.00 -1.00 -8.00 -1.00 -8.00 -2.00 -8.00 -2.00 -8.00 -2.00 -8.00 -2.00 -8.00 -1.00 -4,00 -1.00 -6.00 -1.00 -5.30 ^1.00 -5.30 -1.00 -15.00 -1.00 -15.00 -0.75 -5.30 -0.75 -5.30 -0.75 -5.30 -1.00 -5.30 -1.00 -5.30 -1.00 -5.30 -1.00 -5.30 -1.25 -5.30 -1.00 -5.30 -0.75 -5.30 -0.75 -5.30 -0.75 -5-30 -1.00 -4.80 -1.00 -4.30 -1-00 -4.30 -1.00 -4.45 -0.75 -4.45 -0.50 -4.45 -1.00 -4.30 -0.75 -4.45 -0.75 -4.45 -0.75 -4.45 -0.75 -4.45 -0.75 -4.45 -0. 75 -4.45 Table XI. L i s t of di f f e r e n t parameters used i n the CNDO ca l c u l a t i o n s on Ag . The units of (I +A^)/2 and ^  are eV. Bun LS HS number d d 1 -16.94 -12-35 2 -16.33 -13.54 3 -16.24 -14.02 4 -16.51 -13.03 5 -16.63 -13.15 6 -16.66 -13. 18 7 -16.59 -13.15 8 -16.60 -13.04 9 -16.61 -13. 18 10 -16.47 -12-97 11 -15.41 -13.67 12 -15.95 -13.34 13 - 15.76 -13-46 14 -12.80 -10.49 15 -18,52 -18.35 16 -17.27 -15.23 17 -12.82 -10.51 18 -12.89 -10.59 19 -12.87 -10.57 20 -13. 11 -10.66 21 -13,18 -10.69 22 -13.24 -10.72 23 -13.24 -10.75 24 - 13.24 -10.71 25 - 12.91 -10.54 26 -12-92 -10.58 27 -13. 04 -10.68 28 -12.03 -9.67 29 -13.06 -10.78 30 -12.87 -10.83 31 -12.87 -10.82 32 -12-93 -10.82 33 -12-91 -10.86 34 -12-93 -10-89 35 -12-89 -10.82 36 -12-93 -10.88 37 - 13. 0 1 -10.93 38 -13.06 -10.97 39 -12.95 -10.90 40 -12-93 -10.88 41 -12. 95 -10.90 X(*S8 -10-66 -8.55 LS HS S a a e -10-44 -7.04 -5.96 -10.41 -7.03 -5.94 -10.40 -7.02 -5.93 -10.42 -7.03 -5.95 -12.36 -8.39 -7- 14 -13.25 -9. 16 -7.90 -17.97 -10.76 -9.32 -15.21 -9.45 -8.42 -17.66 -11.49 -10.45 -13.18 -7.41 -6.38 -10.39 -7.01 -5.93 -10.40 -7.02 -5.93 -10.40 -7.02 -5.93 -10-41 -7.04 -5.96 -9.71 -6.92 -5.35 -9-67 -6.93 -5.39 -9.29 -6.52 -5.44 -10.48 -7.41 -6.27 -10.06 -7.07 -5. 95 -9.66 -6.94 -5.41 -10.66 -7.84 -6.38 -10-67 -8.01 -6.49 -9.31 -6.93 -5.37 -9.87 -7.18 -5.43 -10.06 -6.94 -5.57 -9.15 -6.58 -5.31 -10.45 -7.94 -6. 67 -9.35 -6.86 -5.59 -10.69 -8-02 -6.49 -10.69 -8.01 -6-48 -10.67 -8-13 -6-56 -10.69 -8.02 -6-48 -10-02 -7.75 -6-40 -9-34 -7.56 -6-39 -10-66 -8.30 -6-70 -10,00 -7.89 -6.47 -10.76 -8.52 -7- 18 -10,75 -8.69 -7-28 -10.83 -8.65 -7-29 -10.86 -8.48 -7. 17 -10.84 -8.60 -7.25 -8.35 -5.85 -4.99 A d A „ A 4.59 3.40 1.09 2.79 3.38 1.09 2.22 3-38 1-09 3-48 3-39 1.09 3.48 3.98 1.25 3.48 4.10 1.25 3-44 7.22 1.44 3.56 5-76 1-04 3-43 6-18 1-04 3.50 5.76 1.03 1- 74 3.38 1.08 2.61 3.38 1-09 2.30 3.38 1.09 2.31 3.36 1.09 0.17 2.79 1.56 2.04 2.75 1.54 2- 30 2.76 1-08 2.30 3.07 1.13 2.30 2.99 1- 13 2-45 2-71 1-54 2. 50 2.82 1.46 2-52 2.67 1.52 2.50 2-38 1-55 2.53 2-69 1.75 2.37 3.12 1.38 2.34 2.57 1.27 2-36 2-51 1.26 2.36 2-49 1.27 2.28 2.67 1.53 2.04 2.68 1.53 2.05 2.54 1.57 2-11 2-67 1.53 2.05 2-27 1.35 2.04 1.79 1.16 2.07 2,36 1.60 2-05 2-11 1.42 2.08 2.24 1.34 2.09 2.06 1.41 2.05 2.18 1.36 2.05 2.38 1.31 2.05 2.24 1.35 2. 11 2.51 0,86 fable XII. Energy guantities (in eV) obtained from the d i f f e r e n t CHDO calcul a t i o n s on Ag-,. The symbols used are defined i n the text. R u n n u m b e r 1 2 3 <4 5 6 7 8 9 1 0 1 1 1 2 1 3 m 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 0 2 5 2 6 2 7 2 8 2 9 3 0 3 1 3 2 3 3 3<t 3 5 3 6 3 7 3 8 3 9 0 0 0 1 A t c a A A t o a B A t o a C 0 . 5 1 1 0 . 5 0 0 0 . 4 9 7 0 . 5 0 K 0 . 7 6 0 0 . 8 1 9 0 . 5 5 7 0 . 3 8 1 0 . 3 8 5 3 7 9 1 9 3 4 9 8 4 9 6 5 0 2 7 0 6 7 2 8 6 0 8 0 . 7 8 8 0 . 7 4 1 0 . 7 3 2 0 . 8 5 8 0 . 7 7 3 0 . 7 4 9 0 . 7 0 6 0 . 6 5 1 0 . 7 4 9 0 . 7 5 0 0 . 7 3 1 0 . 7 5 1 0 . 7 4 4 0 . 6 7 7 0 . 7 4 4 0 . 8 0 2 0 . 8 5 8 0 . 5 7 7 0 . 7 0 4 0 . 8 2 1 0 . 7 1 0 0 . 6 9 3 0 . 8 0 5 0 . 7 3 0 0 . 5 7 8 0 . 5 7 6 0 . 5 7 6 0 . 5 7 7 0 . 2 4 1 0 . 1 6 0 0 . 5 0 2 0 . 7 3 6 0 . 7 4 9 0 . 7 3 2 0 . 5 7 6 0 . 5 7 6 0 . 5 7 6 0 . 5 8 1 0 . 3 0 0 0 . 3 1 7 0 . 4 4 7 0 . 2 1 1 0 . 2 7 4 0 . 3 2 8 0 . 1 9 9 0 . 3 2 5 0 . 2 6 5 0 . 3 4 8 0 . 4 4 0 0 . 3 0 2 0 . 3 3 2 0 . 3 5 8 0 . 3 2 1 0 . 3 1 5 0 . 4 2 0 0 . 3 1 8 0 . 2 1 4 0 . 1 2 0 0 . 5 6 2 0 . 3 4 0 0 . 2 1 3 0 . 3 6 9 0 . 4 1 2 0 . 2 3 8 0 . 3 5 5 9 . 9 7 4 9 . 9 B 7 9 . 9 9 0 9 8 2 9 8 0 9 7 9 9 7 4 9 7 8 9 6 2 9 8 5 9 . 9 9 5 9 . 9 8 9 9 . 9 9 1 9 . 9 8 1 9 . 9 9 4 9 . 9 5 5 9 . 9 8 2 9 . 9 8 0 9 . 9 8 1 9 . 9 4 6 9 . 9 4 1 9 . 9 2 5 9 . 9 3 5 9 . 9 2 9 9 . 9 6 6 9 . 9 6 8 9 . 9 5 0 9 . 9 5 0 9 . 9 3 7 9 . 9 4 7 9 . 9 3 8 9 . 9 4 4 9 . 9 4 9 9 . 9 5 4 9 . 9 2 7 9 . 9 3 7 9 . 9 3 5 9 . 9 1 5 9 . 9 2 8 9 . 9 4 5 9 . 9 3 3 T o t a l 1 1 . 0 6 3 1 1 . 0 6 3 1 1 . 0 6 3 1 1 . 0 6 4 1 0 . 9 8 0 1 0 . 9 5 8 1 1 . 0 3 3 1 1 . 0 9 6 1 1 . 0 9 6 1 1 . 0 9 6 1 1 . 0 6 3 1 1 . 0 6 4 1 1 . 0 6 4 1 1 . 0 6 5 1 1 . 0 0 0 1 1 . 0 0 0 1 1 . 0 3 7 1 0 . 9 7 9 1 0 . 9 9 7 1 1 . 0 0 6 1 0 . 9 9 8 1 1 . 0 2 3 1 0 . 9 4 9 1 0 . 9 8 3 1 1 . 0 5 6 1 1 . 0 1 9 1 1 . 0 3 5 1 1 . 0 3 8 1 1 . 0 0 9 1 1 . 0 6 4 1 1 . 0 3 6 1 1 . 0 0 5 1 0 . 9 6 5 1 0 . 9 3 1 1 1 . 0 6 5 1 0 . 9 8 3 1 0 . 9 6 9 1 0 . 9 9 5 1 1 . 0 3 3 1 0 . 9 8 8 1 1 . 0 1 8 0 . 5 5 6 0 . 5 4 8 0 . i 4 7 0 . 5 5 1 0 . 8 4 7 0 . 9 1 9 0 . 6 2 0 . 4 0 5 . 4 0 7 , 4 0 3 , 5 4 4 . 5 4 7 . 5 4 6 . 5 4 9 . 8 2 4 . 8 4 2 . 6 6 9 0 . 8 7 2 0 . 8 1 6 0 . 8 3 6 0 . 9 1 4 0 . 8 3 6 0 . 9 2 7 0 . 8 6 1 0 . 7 0 1 0 . 8 1 5 0 . 7 9 7 0 . 7 8 0 0 . 8 4 3 0 . 8 4 2 , 7 6 6 . 8 9 3 , 9 2 5 . 9 9 2 . 6 7 1 , 8 5 7 , 9 3 3 . 8 4 8 . 7 8 0 . 8 7 7 . 8 1 8 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 4 4 0 0 . 4 3 6 0 . 4 3 6 0 . 4 3 7 0 . 1 8 4 0 . 1 2 4 0 . 3 9 3 0 . 5 7 7 0 . 5 8 6 0 . 5 7 2 0 . 4 3 5 0 . 4 3 6 0 . 4 3 6 0 . 4 4 0 0 . 1 8 4 0 . 1 9 9 0 . 3 3 3 0 . 1 5 9 0 . 2 0 6 0 . 2 0 8 0 . 1 3 5 0 . 2 1 4 0 . 1 5 8 0 . 2 1 2 0 . 3 0 3 0 . 2 0 4 0 . 2 2 9 0 . 2 4 4 0 . 2 0 7 0 . 2 0 1 0 . 2 6 9 0 . 2 0 3 0 . 1 3 5 0 . 0 7 7 0 . 3 5 8 0 . 2 0 4 0 . 1 3 7 0 . 2 2 5 0 . 2 6 3 0 . 1 5 5 0 . 2 2 8 9 . 9 7 9 9 . 9 8 9 9 . 9 9 2 9 . 9 8 6 9 . 9 8 3 9 . 9 8 2 9 . 9 7 8 9 . 9 8 2 9 . 9 6 8 9 . 9 8 8 9 . 9 9 6 9 . 9 9 1 9 . 9 9 3 9 . 9 8 5 9 . 9 9 5 9 . 9 6 2 9 . 9 8 5 9 . 9 8 3 9 . 9 8 4 9 . 9 5 6 9 . 9 5 3 . 9 4 1 . 9 4 3 . 9 4 0 . 9 7 3 . 9 7 4 . 9 6 0 9 . 9 6 0 9 . 9 4 9 9 . 9 5 7 9 . 9 5 1 9 . 9 5 5 9 . 9 5 8 9 . 9 6 1 9 . 9 4 4 9 . 9 5 0 9 . 9 4 6 9 . 9 3 1 9 . 9 4 3 9 . 9 5 6 9 . 9 4 7 9 . 9 . 9 . 9 . 9 . 9 . T o t a l 1 0 . 9 7 5 1 0 . 9 7 5 1 0 . 9 7 5 1 0 . 9 7 5 1 1 . 0 1 5 1 1 . 0 2 5 1 0 . 9 9 0 1 0 . 9 6 3 1 0 . 9 6 3 1 0 . 9 6 3 1 0 . 9 7 5 1 0 . 9 7 5 1 0 . 9 7 5 1 0 . 9 7 4 1 1 . 0 0 4 1 1 . 0 0 4 1 0 . 9 8 7 1 1 . 0 1 5 1 1 . 0 0 6 1 1 . 0 0 1 1 1 . 0 0 2 1 0 . 9 9 1 1 1 . 0 2 8 1 1 . 0 1 3 1 0 . 9 7 7 1 0 . 9 9 4 1 0 . 9 8 7 1 0 . 9 8 5 1 0 . 9 9 9 1 1 . 0 0 0 1 0 . 9 8 6 1 1 . 0 0 7 1 1 . 0 1 8 1 1 . 0 3 1 1 0 . 9 7 1 1 1 . 0 1 1 1 1 . 0 1 6 1 1 . 0 0 5 1 0 . 9 8 7 1 1 . 0 0 8 1 0 . 9 9 4 0 . 5 9 2 0 . 5 d 9 0 . 5 H 8 0 . 5 9 0 0 . 9 0 1 0 . 9 d 0 0 . 6 6 7 0 . 4 2 2 0 . 4 2 2 0 . 4 2 1 0 . 5 8 7 0 . 5 8 8 0 . 5 8 8 0 . 5 8 7 0 . 8 8 3 0 . 8 9 9 0 . 7 1 3 0 . 9 2 4 0 . 8 6 5 0 . 8 8 7 0 . 9 5 0 0 . 8 6 6 1 . 0 1 7 0 . 9 2 2 0 . 7 2 5 0 . 8 5 3 0 . 8 2 1 0 . 8 0 5 0 . 8 8 7 0 . 8 9 0 0 . 8 0 3 0 . 8 9 2 1 . 0 0 0 1 . 0 9 7 0 . 7 1 0 0 . 9 3 2 1 . 0 0 4 0 . 9 1 4 0 . 8 2 2 0 . 9 5 1 0 . 8 6 3 0 . 3 0 9 0 . 3 0 4 0 . 3 0 3 0 . 3 0 6 0 . 1 2 7 0 . 0 8 6 0 . 2 8 0 0 . 4 1 4 0 . 4 2 3 0 . 4 1 0 0 . 3 0 2 0 . 3 0 3 0 . 3 0 3 0 . 3 0 7 0 . 1 0 8 0 . 1 1 9 0 . 2 2 6 0 . 1 0 7 0 . 1 3 8 0 . 1 2 8 0 . 0 8 6 0 . 1 3 6 0 . 1 0 4 0 . 1 3 9 0 . 1 9 5 0 . 1 2 6 0 . 1 4 5 0 . 1 5 * 0 . 1 2 9 0 . 1 2 4 0 . 1 6 8 0 . 1 2 5 0 . 0 8 3 0 . 0 4 9 0 . 2 2 4 0 . 1 2 5 0 . 0 8 6 0 . 1 3 9 0 . 1 6 1 0 . 0 9 5 0 . 1 4 0 9 8 4 9 9 2 9 9 4 9 8 9 9 8 7 9 8 6 9 8 3 9 8 7 9 7 8 9 9 1 9 9 7 9 . 9 9 4 9 . 9 9 4 9 . 9 8 9 9 . 9 9 6 9 . 9 7 1 9 8 9 9 8 7 9 8 8 9 6 6 9 6 3 9 5 4 9 5 1 9 5 0 9 . 9 8 1 9 . 9 8 1 9 . 9 7 1 9 . 9 7 1 9 . 9 6 0 9 . 9 6 7 9 . 9 6 3 9 . 9 6 5 9 . 9 6 6 9 . 9 6 7 9 . 9 5 8 9 . 9 6 0 9 . 9 5 6 9 . 9 4 5 9 . 9 5 8 9 . 9 6 6 9 . 9 6 0 T o t a l 1 0 . 8 8 5 1 0 . 8 U 5 1 0 . 8 8 5 1 0 . 8 8 5 1 1 . 0 1 4 1 1 . 0 5 2 1 0 . 9 3 1 1 0 . 8 2 2 1 0 . 8 2 2 1 0 . 8 2 2 1 0 . 8 8 6 1 0 . 8 8 5 1 0 . 8 8 6 1 0 . 8 8 3 1 0 . 9 8 8 1 0 . 9 9 0 1 0 . 9 2 8 1 1 . 0 1 9 1 0 . 9 9 2 1 0 . 9 8 1 1 0 . 9 9 9 1 0 . 9 5 7 1 1 . 0 7 1 1 1 . 0 1 2 1 0 . 9 0 1 1 0 . 9 6 0 1 0 . 9 3 6 1 0 . 9 3 0 1 0 . 9 7 6 1 0 . 9 8 0 1 0 . 9 3 4 1 0 . 9 8 2 1 1 . 0 4 9 1 1 . 1 1 3 1 0 . 8 9 2 1 1 . 0 1 7 1 1 . 0 4 6 1 0 . 9 9 9 1 0 . 9 4 0 1 1 . 0 1 3 1 0 . 9 6 3 X M S M 0 . 7 8 0 0 . 3 6 1 9 . 9 3 4 1 1 . 0 7 5 0 . 8 0 6 0 . 2 5 6 9 . 9 3 0 1 0 . 9 9 2 0 . 6 5 1 0 . 1 4 8 9 . 9 9 8 1 0 . 7 9 7 T a b l e X I I I V a l e n c e - s h e l l c h a r t j f c a l l o c a t i o n s o n t h e d i f f e r e n t a t o m s o f t h e A q - i c l u s t e r f o r t h e v a r i o u s s e t s o f C N D O p a r a m e t e r s . T h e C N D O c h a r g e s a r e o b t a i n e d b y t h e H u l l i k e n p o p u l a t i o n a n a l y s i s T511. I n t h e X w S H c a l c u l a t i o n , t h e i n t e r s p h u r e a n d o u t e r s p h e r e c h a r q e s w e r e r e a l l o c a t e d t o t h e s , p a n d d c o a p o n e n t s o f e a c h a t o a b y r e n o r a a l i s i n q e a c h a o l e c u l a r o r b i t a l a s d e t a i l e d i n s e c t i o n 2 . 1 1 . T h e a t o a s a r e l a b e l l e d a s i n t a b l e V I I I . 110 - • 60 H — i — - 4 0 - 8 0 0dL (eV) Fiqure 23. Variation of the CNDO e l e v e l s in the d-band of Aq^ with /S* . The other Aq parameters are l s = 0.93, $d = 2.25, (I s+A s)/2 = 4.26, (I ?*Ap)/2 = 2.39, (I d+A < i)/2 = 8.28 and (3, =-1.00. (Runs 4, 11, 12 and 13 in table XI.) 111 The diagonal elements F ^ are e s s e n t i a l l y constant and at a value which i s related to the d - o r b i t a l atomic energy. These observations ensure that the r e l a t i v e spacings of the d-levels are determined mainly by the overlap matrix s ^, and the absolute spacings by ^  . The net non-bonding c h a r a c t e r i s t i c of the d - o r b i t a l s gave the expectation that £ s and ^ s would only influence the s properties. As expected the parameter ^ s was found to influence properties d i r e c t l y involved with the s-functions, but not those involving d-functions; by contrast £ s also markedly aff e c t s both the d-band width b>\ and shape. The e f f e c t of ^ on the d-properties must be associated i n part with the non-zero overlap i n t e g r a l s between the r e l a t i v e l y d i f f u s e s-functions and certain d-functions on neighbouring atoms, although an analysis of the l o c a l DOS f o r contracted s-functions indicates that the dependence of the d-band on ^ i s also associated with the numerical values of the two-electron repulsion i n t e g r a l s . In general the parameter ^ a f f e c t s the spread of the s-states i n the sense that and £s 2 3increase . with • J ^ I, ,The s-exponent ^ a f f e c t s the s-levels in a more complicated manner; A t i decreases and £\.xz increases with i n c r e a s i n g ^ . The parameters of the type (I^ + A^)/2 relate to the atomic energies of the s, p and d o r b i t a l s ; the v a r i a t i o n of these parameters i s found to a l t e r the spacings between the l e v e l s of the appropriate o r b i t a l type. Also the separation between the atomic energy lev e l s determines the r e l a t i v e s, p and d charges. Thus increasing <I^+As)/2, which corresponds to a lowering i n energy of the s-electrons, whilst the other 112 parameters are fixed (e.g. runs 17 (with (I s+A s)/2 = 4.26).,. 19 (5.00), 18 (5.40) i n table XI) re s u l t s i n an increase of s charge r e l a t i v e to the p charge (atom A gives for run 17 (s=0.61, p=0.44), 19 (0.74, 0.27), 18 (0.79, 0.21) i n table XIII); correspondingly the reverse trend i s found when (I +A )/2 i s increased (e.g. consider runs 39, 41, 40). I n i t i a l calculations indicated that the Clementi and Baimondi£75] exponents produced d-bands which are too narrow; i t was immediately cle a r that ^ s and should have d i f f e r e n t values even though Baetzold £ 713 had kept these parameters equal. Such problems are i l l u s t r a t e d by run 15 where the Clementi and Baimondi s and d exponent values are used; the s properties and are comparable to those obtained in the X<*SH c a l c u l a t i o n whilst the CNDO d-band width i s only 0.17 eV (compared with 2.11 eV from X cx SB fo r Ag-?) even thouqh i s considerably larger than ^ s ( ^ = - 1 . 0 0 , ^ =-15.00). In the cal c u l a t i o n for run 16 (same parameters as 15 except ^ = 2-75) the d-band i s wider (2.04 eV) , but i t i s perhaps s t i l l r e l a t i v e l y narrow considering the large value used for ^  . Having got a broad view of the e f f e c t of varying parameters on the calculated energy l e v e l s and charge d i s t r i b u t i o n s , the following basic procedure was used f o r f i x i n g the f i n a l CNDO parameters. Buns 1 to 4 (table XI) investigate the eff e c t of varying ^ i n the region of Slater's £49] value; the comparisons of the l o c a l DOS calculated with the CNDO method against those from the X SW c a l c u l a t i o n showed that the best l o c a l DOS were obtained with ^ = 2.25 (run 4). The d-band width was fixe d by varying ^ , as i n runs 4, 11, 12 113 and 13 (with J S = 0.93, td = 2.25, (Ls*\s)/2 = 7.10, (I f + A p)/2 = 2.39, {I a+A a)/2 = 8.28 and ^ s =-1.00), and at t h i s stage ^  =-5.30 gave the most reasonable d-band width A Q ( . The values of the (I +A )/2 parameters were varied i n runs 1 to 14, although at the time the importance of on the d-band was not r e a l i s e d . Buns 1 to 14 indicated that (I^+A^)^ should be reduced to 5-23 (run 14) t h i s gave a sim i l a r overlap between the the s and p states i n the l o c a l DOS curves to that obtained from the X o<SW cal c u l a t i o n s . The s-exponent ^ s was varied i n runs 14, 20, 23 and 25; from both the l o c a l DOS and the amount of s, p character i n the d-band, the Clementi and Baimondi value f o r J S (1.35) was found to be guite s u i t a b l e . Buns 20, 21 and 22 (with J 5 = 1-35, = 2-25, {l^kA)/2 = 5.23, ^ s =-1.00, ^«t=-5.30) attempt tc improve A , 2 and the spacing between the s, p l e v e l s , and the charge d i s t r i b u t i o n on the Ag-? c l u s t e r by modifying the (I 3*A s)/2 and (I p+Ap)/2 parameters; and at t h i s stage run 22 ((I s+A s)/2 = 5-34, (I p*Ap)/2 = 3.39) matched closest to the X^SH ca l c u l a t i o n s , although, there i s l i t t l e difference from run 20 ((I s+A s)/2 = 4.26, (I p*A p)/2 = 2.39). Further consideration of the d-band width A.^ was made because run 22 gave too large a value ( A ( l = 2-52 eV compared with 2-11 eV from Xo/ SW) and jS>ol w a s re-examined; a f i n a l value of =-4.45 was suggested from runs 22, 29, 30 and 32- The parameter ^ s was further modified i n runs 32, 33 and 34 and the value (2>s =-0.75 (run 33) was found to give an improved l o c a l DOS i n the s-region as well as reasonable A,^ and A ^ values. At t h i s stage the trends i n the l o c a l DOS obtained from runs 14, 20, 23 and 25 were checked again, and i t was decided that 114 the l o c a l DOS match more clos e l y to the Ic^ Si r e s u l t s with parameters are those used in run 41 (table XI). 4.3 Comparisons with the X txSB Calculations f o r A C M In t h i s section we consider the degree of f i t that has been obtained between the JLa SB c a l c u l a t i o n f o r Ag7 and the CNDO c a l c u l a t i o n using the f i n a l parameters f o r s i l v e r i n table X, Comparing the CNDO results from run 41 i n tables XII and XIII with the X &( SB ca l c u l a t i o n one can see that a reasonably s a t i s f a c t o r y l e v e l of agreement has been reached; absolute energies are indicated by CNDO to be about 2,5 e? too low, but t h i s has not been considered too s i g n i f i c a n t in view of the differences i n int e r p r e t a t i o n , referred to i n section 4.2.1, between the one-electron energies from the two methods. The t o t a l and l o c a l DOS are compared in fi g u r e 24. A f a i r l e v e l of match i s apparent f o r the various l o c a l DOS, although the t o t a l DOS seems to correspond l e s s well. To look at that more c l o s e l y , figure 25 shows the t o t a l DOS for each of the in d i v i d u a l i r r e d u c i b l e representations. The CNDO curves for the A, and A^  states match up excellently with those from the X^ < SB method, although the agreement i s less good f o r the E states. The difference i s the X<=<SH method shows the E l e v e l s to be spread f a i r l y evenly throughout the d-band, whereas the CNDO 115 1 2 C - , Ag? local DOS atom A X a SW 7 5 5 0 tXh KOi I2CH 4 0 H 7 5 0 50CH 25CH Ag^ local DOS atom A CNDO Ag? local DOS atom B CNDO J A A Ag^ local DOS atom C CNDO -OB - 0 . 7 - Q 6 -Oi - Q 4 -QJ Energy (Ry) Ag, total DOS CNDO 1 A A - I J O - O S - O S - 0 . 7 - 0 6 - 0 5 Energy (Ry) Local and t o t a l and the CNDO table X. The correspond to components have were calculated equation (1.4) . Fiqure 24. DOS for Ag-j calculated •ethod usinq the f i n a l curves desiqnated by s and p components both been multiplied by with the X« SH method parameters for s i l v e r in trianql e s and c i r c l e s respectively, and these a factcr of 2. The DOS with a broadeninq factor a =0.0075 Ry in 1 16 -10 -0.9 -Q8 -07 -0.6 -05 -0.8 -07 -0.6 -05 -04 -03 Energy (Ry) Energy (Ry) Fiqure 25. Total D O S for Aq 1 for each i r r e d u c i b l e representation A, , kx and E calculated with the X « X S H and with the C N D O aethod. The D C S were calculated with a broadeninq factor <r = 0 . 0 0 7 5 By. 117 method indicates they are s p l i t into an upper group of seven l e v e l s and a lower group of f i v e l e v e l s . This p a r t i t i o n i n g i s maintained over considerable variations in parameters, and i t i s concluded that CNDO i s not capable of giving a closer account of the t o t a l DOS for the E representation as calculated with the X oc SH method. 4-4 Other S i l v e r Clusters 4. 4.1 Ag «. This octahedral c l u s t e r was treated previously with the X i^ SW method by Bosch and Menzel £ 101 J . Figure 26 compares the energy l e v e l s from the X <vSB c a l c u l a t i o n and a CNDO c a l c u l a t i o n using the f i n a l parameters i n table X. The l e v e l of agreement i s encouraging, although i t i s noted that the CNDO energy le v e l s show a tendency to group together whereas the Xc^SH l e v e l s correspond to a more evenly-spread spectrum. The same ordering of l e v e l s i s found except that CNDO gives the 1 t ) w state below le,, and 1tn , whereas the X cx SH method in d i c a t e s that I t,^ should be above the other two. The CNDO d-band i s s l i g h t l y broader than that from the X O J S B method: the energy difference between 1a,^ and-It, states i s 1 . 9 8 eV for CNDO and 1 . 7 3 eV from X o(SW. The 2a ^  and 3 t l l L l e v e l s , corresponding to 5s/5p states, are separated more i n the CNDO c a l c u l a t i o n . {Note, the numbering of the Ag t energy l e v e l s includes only the valence electrons, as i n the manner used by Bosch and Menzel 1 18 -0.5-CNDO X a S W -0.3 -0.6H h-0.4 -07A k-0.5 c r cu c UJ - 0 . 8 H - t 2 u •e u ;1u 2 g -09-I ' l u - Q 2 u - t a , - O l g - ' t g • Q 2 g -t2u •e u - < 2 u -e g - t 2 g J i g c r CP a; . c UJ -0.6 h-07 -1.0- L - 0 . 8 Fiqure 26. A comparison of the valence s h e l l enerqy lev e l s calculated for the Aqt c l u s t e r with the X « SV method f101] and with the CNDO method using the f i n a l parameters for s i l v e r i n table X. A s h i f t of 0.2 By i s indicated to qive a better correspondence between the two sets of enerqy le v e l s . 119 £101], whereas i n calcu l a t i o n s on other s i l v e r c l u s t e r s reported i n t h i s t h e s i s , the l e v e l numbering includes the core states.) .4.4.2 M ± o The DOS curves obtained with the CNDO and Xcyss cal c u l a t i o n s are shown i n figure 27. Although the t o t a l DOS c l e a r l y show a number of features i n common, differences are also apparent. The t o t a l d-band widths are 2-28 eV for CNDO and 2.74 eV from the X<^  SI! method. The best match for the l o c a l DOS appears to be for the ce n t r a l 9-coordinate atom, although CNDO indicates a pronounced peak at the top of the d-band. The strong l o c a l i s e d bonding on the ce n t r a l atom, noted previously from the X^ SW c a l c u l a t i o n , seems less well developed according to the CNDO c a l c u l a t i o n . The charge d i s t r i b u t i o n s calculated for t h i s c l u s t e r with the two methods are included i n table XIV- Some further discussions of results for Ag/G are included with those for Ag^ i n the following section. 4.4.3 Ac[j3 The l o c a l and t o t a l DOS obtained with the two ca l c u l a t i o n s are shown i n figure 28. As f o r Ag l 0 , there are differences i n shape, although some general s i m i l a r i t i e s are also apparent. In 120 axh UJ F i q u r e 27. l o c a l and t o t a l DOS for Ag,© calculated with the X«*Sw uethod and the CNDO method using the f i n a l parameters for s i l v e r in table X. The curves desiqnated by trianqles and c i r c l e s correspond tc s and p components respectively, and these components have both been multiplied by a factor of 2. The DOS were calculated with a broadening factor o- =0. 0075 Ey i n equation (1.1) . Number Coordi- Xo»sw charges CNDO charges C l u -s t e r Atom ot type atcms nation number s P d T o t a l s P d T o t a l Aq l o A 1 9 C. 58 0. 50 9.68 10.77 0. 44 0.68 9.88 10.99 B 3 5 0. 63 0.23 9.69 10. 55 0. 75 0. 31 9. 94 11.00 C 6 4 0. 53 0. 16 9.65 10.34 0.84 0.22 9. 95 11. 00 Intersphere 4. 92 Outersphere 0. 64 A 1 12 0. 62 0.68 9. 74 11. 09 0. 29 0.71 9. 86 10. 86 B 12 5 0. 55 0.21 9. 62 10.37 0.80 0.28 9.93 11.01 Inte r s p h e r e 6. 43 Outersphere 1. 02 *qn A 1 12 0. 73 0.59 9.79 11. 18 0. 42 0. 48 9. 89 10.79 B 12 7 0. 64 0.31 9.65 10.60 0.74 0.46 9. 91 11. 10 C 6 4 0. 42 0.14 9. 66 10. 22 0.64 0.25 9. 95 10. 84 Intersphere 9. 05 Outersphere 0. 23 Table XIV Comparison of valence s h e l l charqe a l l o c a t i o n s of d i f f e r e n t atoms of the c l u s t e r s Aq l w , Aq t i and hqx* accordinq to the X«*SW method and CNDO method u s i n g the f i n a l parameters i n t a b l e X. 122 I 2 0 0 1 f J 100 Ag, 3 local DOS atom A X a S W s 1 60-1 Aa.local DOS atom B X a S W Ag,, total DOS 3 X a SW I A A. -0.9 -08 -Q7 -0.6 -05 -04 -0.3 Energy (Ry) -1.0 -0.9 -0.8 -07 -06 -05 Energy (Ry) Fiqure 28. Local and t o t a l DOS for Aq,^ calculated with the X <*SW method and the CNDO method usinq the f i n a l parameters for s i l v e r in table X. The curves desiqnated by tr i a n q l e s and c i r c l e s correspond to s and p components respectively, and these components have both been multiplied by a factor of 2. The DOS were calculated with a broadeninq factor «" =0.0075 Py in equation (1.4) . 123 both c l u s t e r s , the states of lowest energy are less l o c a l i s e d on the c e n t r a l atom according to the CNDO c a l c u l a t i o n than they are i n the loi Sw ca l c u l a t i o n . For example, from the l a t t e r c a l c u l a t i o n f o r Ag,-^  the three lowest energy l e v e l s , namely 16t 2^, 15e^ and 1 aa^ » contribute 8.7 electrons to the c e n t r a l atom. By contrast t h i s i s reduced to about 5.5 electrons with the CNDO c a l c u l a t i o n , and 14a.a i s then indicated to be lower than the other two states. However, CNDO predicts these three leve l s to be closer to the main d le v e l s so causing a somewhat narrower t o t a l d-band (width 2.76 eV instead of 3.67 eV from the XcxfSW c a l c u l a t i o n ) . The large peak at the top of the d-band in the CNDO l o c a l DOS f o r the central atom involves the antibonding 20t^. and 18e^ l e v e l s , and corresponds to a s i m i l a r peak observed previously f o r Ag | 0. The CNDO method indicates the outer atoms i n Ag,s (designated atom type B) to have a l o c a l DOS which resembles quite c l o s e l y those f o r atoms of types B and C i n Ag V o . There i s a suggestion here that CNDO indicates more si m i l a r l o c a l DOS for d i f f e r e n t atoms in c l u s t e r s with s i m i l a r l o c a l environments than i s the case for the X ^ S H method. For the l a t t e r method, i t has been emphasised i n chapter 3 that l o c a l DOS are sen s i t i v e to the coordination numbers of neighbouring atoms as well as to that of the atom concerned; t h i s i s seen from figures 27 and 28 by comparing the l o c a l DOS f o r the B atoms i n Ag V o with that for the B atoms in Ag )5> . The most consistent trend i n the charge populations for the clusters Ag^, Ag l o and Ag,^ i s that 5p populations increase with coordination number. This i s c l e a r from the Xc^SW method 124 (table XIII and XIV) and i s mimicked well by the CNDO charge d i s t r i b u t i o n s . The correspondence between the Xo< SH and CNDO populations f o r the 5s and 4d functions i s les s regular, although the 4d populations are always close to 10 and the 5s populations are i n or around the range 0-5-0.8. 4. 4. 4 Agj/i The l o c a l and t o t a l DOS shown i n figure 29 f o r the two methods of cal c u l a t i o n again have general s i m i l a r i t i e s , although there are some differences i n d e t a i l p a r t i c u l a r l y for the central atom (A)- For example, the X c<SH c a l c u l a t i o n indicates the lowest l e v e l s to be 2 1 t ^ and 24e^, which contribute 7.44 d electrons on atom A, and the 22a ,^  state with 0.45 s electrons on t h i s atom; these three l e v e l s being s p l i t o f f from the main d-band. The CNDO c a l c u l a t i o n , on the other hand, indicates the lowest l e v e l to be the 22a,^ state, with 0.20 s electrons on atom A, while the 21t A^ and 24e^ l e v e l s are the next i n energy. The l a t t e r two states are now e f f e c t i v e l y part of the d-band and give only 2. 14 d electrons on atom A. The dominant peak i n the CNDO l o c a l DOS f o r the ce n t r a l atom i s due to the antibonding lev e l s 28e^, 26t^ » 2 9 e 3 a n d ^ t a ^ tfnich contribute 4-37 d electrons on atom A. This peak i s absent i n the l o c a l DOS from the XcxSH c a l c u l a t i o n , f o r which the corresponding l e v e l s have only 0.22 d electrons on A. The t o t a l d-band widths are 3.50 eV f o r X <xss and 3.46 eV f o r CNDO, although these values are reduced to 2-18 eV and 2-55 eV 125 Fiqure 29. Local and t o t a l DOS for Aq(q calculated with the X« S3 method and the CNDO method usinq the f i n a l parameters for s i l v e r in table X. The curves desiqnated by trianqles and c i r c l e s correspond to s and p components respectively, and these components have both been multiplied by a factor of 2. The DOS were calculated with a broadeninq factor g- =0.0075 By i n eq uation (1.4) . 126 respectively i f the " s p l i t - o f f " states are not included. The l o c a l DOS from the two sets of calcu l a t i o n s agree more c l o s e l y for the other atoms of the c l u s t e r . I t i s c l e a r the CNDO method d i f f e r s from the X <^  SW method, for these s i l v e r c l u s t e r s , in that the former gives more delocalised bonding. This difference contributes to the dif f e r e n t net charges on the central atom, as given by the X <^ SW and CNDO methods of ca l c u l a t i o n (table XIV), and to the differences in the l o c a l DOS. Those from the X c* SH cal c u l a t i o n s , f or the high coordination atoms (A) and the low coordination atoms (C), are characterised by sharp peaks, whereas the corresponding curves from CNDO show broader d i s t r i b u t i o n s of structure. As for the ce n t r a l atoms i n Ag | D and Ag,3 c l u s t e r s , the CNDO cal c u l a t i o n s again give, for the corresponding l o c a l DOS i n Ag l n , a dominant peak associated with an antibonding l e v e l at the top of the d-band. Although the d e t a i l s ace d i f f e r e n t , CNDO does indicate the trend, f i r s t observed for the X<^S8 c a l c u l a t i o n s , wherein electron binding energies appear on average to increase with increasing coordination number. The t o t a l DOS for Agic, from the X°<:SW calculations shows three small 5s/5p peaks above the d-band, the peak at highest energy being due to the overlap of the levels 28t^ and 26a ,^  . According to the CNDO ca l c u l a t i o n s the 28t 2 l e v e l i s at lower energy than 26a and i s near the 30e^ state; the remaining 5s/5p state, namely 3 9 t l H , i s close to the top edge of the d-band. 127 4.5 Concluding fiemarks This study started froa a r e a l i s a t i o n that CNDO calcul a t i o n s cn s i l v e r c l usters, with an early parametrisation for s i l v e r £71J, do not give an acceptable l e v e l of agreement with calculations made with the XoTSW method. The present work has shown that substantial improvements (as can be seen for the resul t s obtained from run 41 detailed i n XII and XIII) can apparently be made by parametrising the CNDO method with reference to r e s u l t s from X o(S9 c a l c u l a t i o n s f o r a p a r t i c u l a r c l u s t e r . These parameters have been tested by comparing against X o< sw ca l c u l a t i o n s for other c l u s t e r s of d i f f e r e n t s i z e s . Levels of agreement are varied, but at least the d-band widths are generally comparable. This study provides the f i r s t i n v e s t i g a t i o n of l o c a l DOS with the CNDO method; t h i s method of cal c u l a t i o n does not give the substantial l o c a l i s a t i o n at high coordinate atoms found i n the X c<rSW c a l c u l a t i o n s . Insofar as the l a t t e r observation can be taken as representative of re a l s i l v e r c l u s t e r s , the CNDO calcu l a t i o n s appear d e f i c i e n t . However, t h i s deficiency could possibly be les s serious when clusters are used for modelling surface or bulk properties of metals. As indicated in chapter 3, the aim of such work i s to simulate an actual surface region or bulk with a c l u s t e r which can be used to give a helpful account of some property. High computational costs often encourage the choice of small c l u s t e r s , and, i n t h i s regard, our experience f o r s i l v e r indicates that i t may be possible to use smaller c l u s t e r s with the CNDO method than with the X<*SW method for the purpose of getting DOS curves. For the X o< SW method, l o c a l DOS are more 128 sensitive to the coordination numbers of atoms which neighbour the p a r t i c u l a r atom of i n t e r e s t . 129 CHAPTER 5 CALCULATIONS ON HOLECU LES AND CLUSTERS FORMED BY SECOND ROJ ELEMENTS 5-1 Introduction In chapter 4, CNDO parameters for s i l v e r atoms were deduced by making use of X©< S» ca l c u l a t i o n s on clu s t e r s of s i l v e r atoms- A potential advantage of t h i s approach of deriving CNDO parameters from comparisons with the re s u l t s from the X^<S8 method i s that i t can be extended systematically and consistently to a wide range of elements- The purpose of thi s chapter i s to assess the r e l i a b i l i t y of thi s approach for the elements aluminium to sulphur £136] by comparing CNDO calcul a t i o n s which use new parameters with those from the scheme given previously by Santry and Segal £70]- Some support fo r the v a l i d i t y of t h i s approach would be provided i f r e s u l t s from the CNDO method with the new parameters turn out to be comparable with, or better than, those given by the previous CNDO scheme- I f t h i s approach appears broadly successful for these elements, which have already been investigated by CNDO schemes to a f a i r degree £60,70,73], then there would be a better basis f o r extending i t to situa t i o n s where CNDO procedures are at best poorly developed-The four elements considered here range from a free-electron metal, through a semi-conductor to molecular s o l i d s in the cases of phosphorus and sulphur, and these differences i n basic c h a r a c t e r i s t i c s influence the p a r t i c u l a r molecular clust e r s considered for parametrising the CNDO method- For 1 3 0 phosphorus and sulphur, the molecular forms P^ and S g seem natural choices. S i l i c o n i s represented by the molecule Sig-H^ since the l o c a l enviroment around the central atom corresponds to that in the s i l i c o n l a t t i c e ; also the presence of the hydrogen atoms l i m i t s e f f e c t s associated with broken covalent bonds. For aluminium the cl u s t e r s A l 7 and Al ! o are considered; these have been chosen to provide r e l a t i v e l y simple simulations of the (111) surface. For the parametrisations, i t seems preferable that a l l the X * SU cal c u l a t i o n s are at corresponding l e v e l s of refinement. A problem here i s that many X<*SW calc u l a t i o n s i n the l i t e r a t u r e have been made only i n the non-over lapping-spheres scheme, as i n the e a r l i e s t formulation of the method, although, at the present time, the X t X S W approach has been most successful with the overlapping-spheres version. For t h i s reason new Xc<Sl c a l c u l a t i o n s were made for the Al1 and A l ( 0 c l u s t e r s , and some e a r l i e r c alculations f o r Si sH,^ and P^ were refined; f o r S^ the r e s u l t s produced by Salahub et a l . were used £137]. These new Xo<SH cal c u l a t i o n s are presented i n the next section. The corresponding CHDO ca l c u l a t i o n s on the second row elements are presented i n the remainder of this chapter* 5.2 XoiSM Calculations on Second Sow Elements The X cX SH cal c u l a t i o n s reported here for Al^ , A l l o , Si^H,^ and P were performed using the same procedure as outlined in section 3.2.1. The only s i g n i f i c a n t difference f or the new 131 calc u l a t i o n s described i n t h i s chapter i s that the p a r t i a l saves sere r e s t r i c t e d through to 1=1 for the atomic spheres of the second row atoms and to 1=3 for the outersphere. Generally the atomic spheres have been allowed to overlap as extensively as possible without introducing u n r e a l i s t i c features such as negative intersphere charges i h some molecular o r b i t a l s and while maintaining good v i r i a l r a t i o s . The actual degree of sphere overlap i s s p e c i f i e d below for each p a r t i c u l a r system. 5. ?. 1 A l x and Alio The clu s t e r s kln and A l l o both have symmetry, and they have the same basic configuration as the Ag 7 and Ag l o c l u s t e r s discussed i n chapters 3 and 4; both of these seven and tea-membered cl u s t e r s correspond to fragments of the fee bulk structure. The X °< SH calculations on the aluminium c l u s t e r s were made with a l l nearest-neighbour distances fixed equal to the bulk value of 2.86 A £121]. For AI7, the atomic spheres had r a d i i 15% greater than those that just touch; the outersphere of radius 4.51 A was centred at the intersection of the Cj axis with the plane containing the three atoms designated A i n table XVI. For the XocSH c a l c u l a t i o n on A l l 0 , some negative intersphere charges required the sphere r a d i i to be reduced to a 61 increase over the contact radius; the outersphere was centred on the c e n t r a l atom of the 7-atom lay e r and had a radius of 4.38 A. The exchange parameter <^  was fixed at 0.72853 £ 96] through a l l regions of both c l u s t e r s . 132 A l 7 a l l x > Level* - ^ ^ s v o - ^ T S Level* - £y*Sod 5a 4-83 6-76 6a 4-74 1 I 3e 5-27 7. 16 4e 5- 34 4a 5.32 7. 15 5a 5. 45 i t 2e 6-99 8.96 4a 6-02 3a 7-46 9-49 1a 6-71 I 2 1e 9-39 11-43 3e 7.84 2 a 10.49 12-55 3a 8.05 i i •a, 13-22 15-39 2e 8.63 2a 10.38 1e 11- 16 . 1a a 13.54 Table XV. s s h e l l energies (in eV) calculated for A1-, and designates the X o( SS one -electron energies and e t r a n s i t i o n state energies. * The numbering scheme i s for the valence o r b i t a l s only. The valence-shell one-electron energies (6>«><su)) a r e reported i n table XV for A l n and A l ( Q where a l l o r b i t a l s , including those of the core, have been determined s e l f -consistently- For Al-,, the highest-occupied l e v e l (Sa^ i s 133 : lu s t e r Calculation it 1B <? P ^ B ^ c : " i l AI^  CNDO/2 1.17 1-85 1.21 1.77 1-35 1.66 X ^ S W 1.60 1.51 1-66 1.38 1.64 0-94 CNDO/HH 1-64 1.42 1.74 1.23 1.81 1.12 a l , Q CNDO/2 0.74 2.18 1.16 1.77 1.37 1.69 Xo(S8 1.59 2.20 1-76 1-41 1-74 1.04 CNDO/HM 1-11 2.08 1-68 1.34 1-74 •1-22 Table XVI. Total s, p populations f o r the valence s h e l l s of Al-? and Al i o -The atoms are designated A, B, C i n order of decreasing coordination number: for Al-7 they correspond to atoms with coordination numbers 5, 4 and 3 respectively, and for A l l o they are for coordination numbers 9, 5 and 4 respectively. singly occupied; i n A l / D the highest-occupied l e v e l (6a( ) has a one-electron energy which i s very close to that (-4-70 eV) of the f i r s t unoccupied l e v e l (5e)- Transition state energies (ers) f o r the occupied lev e l s of A l ^ are also reported i n table XV and the charge d i s t r i b u t i o n s f o r A l n and A l l o are included in table XVI. Local and t o t a l DOS from the X <xSW c a l c u l a t i o n on A l , o are shown i n figure 30- The v i r i a l r a t i o -2T/V i s 1.000354 for i l - , and 1.000864 for A l 1 0 . Previous XcXSB c a l c u l a t i o n s f o r aluminium have been made for c l u s t e r s with O k symmetry to represent bulk atoms [138J, Energy (eV) -10 -8 Energy (eV) Piqure 30 Local and t o t a l DOS with the sw method for Al,. . The curves desiqnated by tr i a n q l e s and c i r c l e s correspond to s and p components respectively. The broadening factor r =0.02 By used i n aquation ( 1 . 4 ) . 1 3 5 and for c l u s t e r s with symmetry for investigating the (100) surface and i t s adsorption of oxygen £ 139,140 J . Salahub and Hessmer £ 1 3 8 ] noted for some large c l u s t e r s , e.g. kl^ and A l 4 3 , a rough correspondence of the DOS with the dependence expected f o r a free-electron metal, where E i s measured from the bottom of the valence hand. He ce r t a i n l y f i n d an increase in the t o t a l DOS with energy f o r both A l 7 and A l l o , but, as expected for small c l u s t e r s , our r e s u l t s also bring out a substantial dependence on the l o c a l environment; t h i s i s apparent both i n the l o c a l DOS for A l \ 0 (see f i g u r e 30) and in the charge d i s t r i b u t i o n s (table XVI). Total sphere charges increase with coordination number, and the variations are associated e s p e c i a l l y with the populations of the p-functions. Indeed s to p promotion appears important for the bonding of the high-coordinate atoms in cl u s t e r s of aluminium, and t h i s r e l a t e s to analogous e f f e c t s i n corresponding c l u s t e r s of s i l v e r discussed i n section 3.3. The l o c a l i s a t i o n a f f e c t s are less marked for aluminium, compared with those i n the Ag clu s t e r s , i n part because of the greater diffuseness of the Al atomic o r b i t a l functions of the valence s h e l l . 5. 2.2 Si^-H ,^ C a r t l i n g et a l . £141] have previously performed an Xo<SB cal c u l a t i o n on S i ^ H i ^ , where t h i s molecule was used as a clust e r approximation f o r the s i l i c o n l a t t i c e . Both the calc u l a t i o n s by Ca r t l i n g et a l . and those presented here were 136 made for a model i n which the c e n t r a l S i atom i s surrounded tetrahedrally by the other four S i atoms, and the twelve H atoms are directed from the outer group of the four S i atoms towards where the next group of S i would be i n the actual s i l i c o n l a t t i c e . This molecule has T^ symmetry, and the S i - S i distances were fix e d at 2.35 A as i n the bulk S i l a t t i c e £142]. C a r t l i n g et a l . also took the Si-H distance as 2.35 A, whereas here the more r e a l i s t i c Si-H distance of 1.47 A i s used; t h i s corresponds to the Si-H distance usually found in silanes £142], The other differences i n the c a l c u l a t i o n s made here from those of C a r t l i n g et a l . , include using the over lapping-spheres version of the X <*SH method and using an exchange parameter «• appropriate to the H spheres. C a r t l i n g et a l * used the S i o< value of 0.72751 £96] for a l l regions i n the molecule, whereas here we take ^ =0.77725 i n the hydrogen spheres as suggested by Slater £ 143J- The atomic sphere r a d i i used are 1-29 A for Si and 0.80 A for H; these values were determined by Normals procedure £99] for a 10$ increase i n radius f o r spheres which just touch i n the s i l i c o n l a t t i c e . The outersphere was centred on the c e n t r a l Si atom with a radius equal to 3.96 A (so making i t just touch each of the H spheres). C a r t l i n g et a l . included vacancy spheres i n t h e i r c a l c u l a t i o n s , and i t seemed worthwhile to investigate t h e i r value i n t h i s case, p a r t i c u l a r l y i n view of some recent discussion about th e i r v a l i d i t y i n t h i s type of c a l c u l a t i o n £100]. Vacancy spheres, which can be envisaged as extra atomic spheres but with zero nuclear charge, enable parts of the 137 Level* Vacancy spheres included Vacancy spheres neglected ~~ ^  Xtx'SM 3t 5-70 5.78 7-92 n j 8-52 8.56 10-52 1e 8-78 8-78 10- 72 2t 8-94 8.99 10.88 2a 10.80 10.87 13.05 1t 2 13-42 13-46 15.56 1a I 14.43 14-49 16.53 Table XVII. Valence s h e l l energies (in eV) for Si^H, x . designates the X cx.su one-electron electron energies and 6 T 5 the t r a n s i t i o n state energies. 1 The numbering scheme i s f o r the valence o r b i t a l s only. intersphere region to be s p h e r i c a l l y averaged rather than subjected to a (presumably) l e s s accurate volume average. In these calculations the vacancy spheres were given the same r a d i i as the S i spheres and were positioned tetrahedrally around the c e n t r a l S i atom so that t h e i r centres, and the centres of the four surrounding Si spheres, mark the corners of a cube. The exchange parameter for s i l i c o n (<K =0.727.51). ••is also used i n the vacancy sphere, the outersphere and the intersphere 138 regions* Valence s h e l l energies and charges calculated with the Io(SU method for S i ^ f l ^ are shown i n tables XVII and XVIII respectively; a l l o r b i t a l s , including those of the core, have been determined s e l f - c o n s i s t e n t l y . I t i s clear that the vacancy spheres have a n e g l i g i b l e e f f e c t for t h i s molecule; the population of the vacancy sphere comes almost e n t i r e l y from the intersphere region. The v i r i a l r a t i o s are 1.000436 and 1.000439 when the vacancy spheres are included and neglected respectively. s Sic I s i c s Sio i P Sio s H CHDO/2 1. 05 3.09 1. 05 2. 54 1. 12 x<*ss 1. 41 2.74 1. 23 2. 43 1. 10 ODO/HM 1. 58 2.50 1. 23 2. 38 1. 12 Table XVIII. Total s, p populations f o r the valence s h e l l of S i ^ H ^ . Sic i s the c e n t r a l S i atom, and Sio i s the outer S i atom i n S i ^ f l ^ . Some s i g n i f i c a n t differences have been found from the r e s u l t s of C a r t l i n g et a l . £141], and these are presumably associated with the differences noted above between the two sets of c a l c u l a t i o n s . Surprisingly, C a r t l i n g et a l . report the 139 central Si sphere to contain 0.9 electron more than that i n the neighbouring S i spheres; our X^ ; SH c a l c u l a t i o n s indicate a difference of 0.15 electron. Also the energy l e v e l structure i s d i f f e r e n t from the two c a l c u l a t i o n s ; C a r t l i n g et a l . report the 3t^ l e v e l to be just below 1e whereas we f i n d It, , 1e and 2t^ to be close together and the 3t l e v e l nearly 3 eV higher i n energy. According to our X o<SH ca l c u l a t i o n s , the lowest l e v e l 1a, l a r g e l y involves interactions between s - o r b i t a l s on the c e n t r a l - and outer-Si atoms; 2a, i s concentrated e s p e c i a l l y on the s - o r b i t a l of the c e n t r a l - S i atom, and 1t^ corresponds to strong interactions between outer-Si s - o r b i t a l s and the neighbouring hydrogen 1s-orbitals. The band of l e v e l s 2 t a , 1e and It, involves b a s i c a l l y Si3p-H1s interactions while 3 t L corresponds to interactions between the c e n t r a l - S i 3p-orbitals and those on the outer-Si atoms. Although S i ( S i f l 5 ) ^ i s known £144-],. i t s photoelectron spectrum does not seem to have been measured so f a r . The trends in structure (and t h e i r assignments) f o r photoelectron spectra of l i n e a r s i l a n e s 1145] and permethylated s i l a n e s , including Si£Si(CH 3) 3 ] t146]J, seem broadly consistent with the calculated t r a n s i t i o n state energies i n table XVII- These l a t t e r values should be h e l p f u l for interpreting data from photoelectron spectroscopy f o r SiCSifl^)^. 140 5-2.3 P_i This molecular cluster has T^ symmetry, with a l l P-P distances equal to 2-21 I. The X*SH calcu l a t i o n s were made with atomic sphere r a d i i equal to 1-38 a (corresponding to a 25% increase over the r a d i i which just touch)- The outersphere, with a radius of 2.73 A, was centred at the centre of the tetrahedron. The exchange parameter was fixe d equal to 0-7262 0 in a l l regions. On taking c a l c u l a t i o n s to the s e l f -consistent l i m i t , the calculated valence s h e l l charges are 1.68 of s type and 2.80 of p type i n each P sphere; the outersphere and intersphere charges being 0.99 and 1.11 respe c t i v e l y . The v i r i a l r a t i o was 1.000158. Valence s h e l l energies from t h i s X©<S« c a l c u l a t i o n are reported i n table XIX, and comparisons are made with experimental io n i s a t i o n energies and with previous c a l c u l a t i o n s . S p e c i f i c a l l y f o r the l a t t e r we report t r a n s i t i o n state energies calculated i n the non-overlapping spheres version of the X ^  SB method I "WT ] and values using Koopmans* theorem (but reduced by a factor 0.92) from a non-empirical molecular o r b i t a l c a l c u l a t i o n with an s,p Gaussian basis set £148]. The present X<*S« ca l c u l a t i o n s appear to provide a sat i s f a c t o r y l e v e l of agreement • with the experimental io n i s a t i o n energies. 141 Level 2e 6t z 5a 1 5t 2 4a l This work Ionisation Bef.£ 147 ] Bef-i148] - £ x^su - £ TS energies* " e TS - 0 . 9 2 £ S C F 6.09 8.93 9-5, 9.9 10-44 9.63 7. 14 10.05 10.4, 10.6 11-42 10.26 8.54 11.47 11.87 12.27 11.46 12.89 15.90 15.2, 16-3,17.5 17-58 19.32 20.70 24.17 22.3 28.9 8 Table XIX. Calculated energies (in eV) for P^ . and io n i s a t i o n energies from photoelectron spectroscopy. 1 From r e f . £148] except 4a, i s from r e f . £149]; the multiple entries r e l a t e to Jahn-Teller s p l i t t i n g s of the i o n i c states. 5- 2. 4 Discussion The new X <*S8 cal c u l a t i o n s reported here relate to three neighbouring atoms i n the second row, and as expected the populations of the valence s h e l l p-functions increase with atomic number through the c l u s t e r s considered. For P^, the occupied l e v e l s show a clear separation into s-type o r b i t a l s (4a, and 5t^) and p-type o r b i t a l s (5a 4 to 2e) , and t h i s has been reported previously from molecular o r b i t a l c a l c u l a t i o n s on P-f 11*8] . a n d X ^ S8 c a l c u l a t i o n s on P 4S 3 £ 135j and Sg £137], For A l 7 and A l l o , on the other hand, there i s a greater 142 mixing of s- and p-cb a r a c t e r i s t i c s i n the i n d i v i d u a l molecular o r b i t a l s ; a somewhat intermediate s i t u a t i o n i s indicated for the s i l i c o n atoms i n S i s i i , x -Figure 31 indicates that the calculated X<*SW one-electron energies 6 ^ 5 ^ reported i n t h i s section for A l 7 , S i ^ H ^ and P^ show clos e l y l i n e a r relationships to the corresponding t r a n s i t i o n state energies 6-yS . This uniform relaxation, mentioned previously i n section 4.2.1, provides further j u s t i f i c a t i o n f o r using e i t h e r the one-electron eigenvalues or the t r a n s i t i o n state energies i n parametrising the CNDO method, p a r t i c u l a r l y when the parametrisation procedure emphasises differences i n o r b i t a l energies. Figure 31 also indicates that, whilst the relaxation i s uniform, the absolute s h i f t varies with the molecule. This relaxation i s a measure of the e f f e c t of removing half an electron from the system, and w i l l be largest for the l i g h t molecules. 5.3 S p e c i f i c a t i o n of the CNDO Calculations on the Second Row  Elements In the f i r s t instance, the CNDO calcu l a t i o n s made here f o r the elements aluminium to sulphur use only s and p functions i n the basis set, although the extension to include d-functions i s considered l a t e r (section 5.7). He followed the basic scheme detailed by Santry and Segal £70] for this basis set (see section 2.7), but fo r one minor difference. This concerns the expression f o r the resonance parameter 1U3 io H H 1 < -15 J e • -20H n 1 , — r -20 -15 .-IO - 5 € X a S W ( e V ) Fiqure 31. The l i n e a r r e l a t i o n s h i p between the X*SW one-electrcn enerqies **«*wand the t r a n s i t i o n state enerqies e-rc • The open c i r c l e s , cresses and dots are froa the calc u l a t i o n s on AL,, s i H„ and respectively. ^ x * 144 where the constant K i s now kept equal to unity; previously i t had been taken as 0-75 i f either A or B i s an atom of the second row. Our i n i t i a l choice of CTSDO parameters i s r e s t r i c t e d to four f o r each element: (i) the exponents \ % = ^ ? i n the Slater-type o r b i t a l s which are used for evaluatinq the overlap and Coulomb i n t e g r a l s ; ( i i ) the average of the io n i s a t i o n energy and electron a f f i n i t y f o r each member of the basis set, i . e . ( I s + i s ) / 2 and (Ip + A ?)/2 which are used i n the diagonal elements of the Fock matrix; and ( i i i ) the bonding parameters j2,s = ^ p which correspond to the terms on the rig h t hand side of equation (5-1) and which are used i n the off-diagonal elements of the Fock matrix. 5.4 Selection of CNDO Parameters In chapter 4 new CNDO parameters for s i l v e r were obtained by comparing One-electron energies and charge d i s t r i b u t i o n s from CNDO calculations on s i l v e r c l u s t e r s with the corresponding quantities calculated with the X SW method. The one-electron enerqies were compared v i s u a l l y throuqh t o t a l DOS curves, and the charqe d i s t r i b u t i o n s were compared by inspection of atomic populations and l o c a l DOS curves. In pr i n c i p l e these comparisons would be easier to make, and perhaps less subjective, with the provision of a numerical index which measures the correspondence of the one-electron 145 energy l e v e l s and the charge d i s t r i b u t i o n s from the two types of c a l c u l a t i o n . In section 2 . 8 new C N D O parameters f o r diatomic L i F were found by minimising a function of the type (5. 2) , where the 6^ and ^  represent one-electron energies and dipole moments from either the C N D O or reference c a l c u l a t i o n s f o r the i- t h molecular o r b i t a l . Such a function seemed s a t i s f a c t o r y f o r LiF, where the two types of properties compared make s i m i l a r contributions to the value of H, but further experience has indicated that considerable care i s needed f o r larger molecules since, with this type of function, either the charge d i s t r i b u t i o n or the energies can be p r e f e r e n t i a l l y emphasised in the comparison. In the work presented i n t h i s chapter, the one-electron charge d i s t r i b u t i o n s and the one-electron energy l e v e l s from the two sets of cal c u l a t i o n s are compared separately, and attempts have been made to assess those C N D O parameters which offer the best o v e r a l l agreement, a simple function f o r the charge d i s t r i b u t i o n i s V f fWi ^ ' ] , 5 . 3 , , where g^ and g ; represent populations (either t o t a l or al t e r n a t i v e l y the i n d i v i d u a l partial-wave populations) on the j-th atom for the i - t h molecular o r b i t a l from the C N D O and X <SH cal c u l a t i o n s respectively. The populations from the C N D O c a l c u l a t i o n s were derived using the analysis of Mulliken £ 5 1 J , and those from the X C K . S H c a l c u l a t i o n s were obtained by 146 a l l o c a t i n g the intersphere and outersphere charges i n the r a t i o s of the component atomic sphere charges as discussed in section 2-11. In eguation (5.3), the m and w^  are appropriate weight f a c t o r s ; i n t h i s study n- has been taken as the occupation number of the i - t h molecular o r b i t a l , but, as discussed further below, the w^  have been . treated i n various ways depending on the p a r t i c u l a r molecular system. Some support f o r the use of (5-3) i s provided by observations f o r P4 S^ £135] that the CNDO/2 and X<XSH cal c u l a t i o n s give s i m i l a r trends in charge d i s t r i b u t i o n s for corresponding molecular o r b i t a l s even though the ordering of the energy l e v e l s i s di f f e r e n t i n some cases. The simplest index for comparing the di s t r i b u t i o n of energy l e v e l s i s (5.4), with an analogous notation to those used above. Experience with the s i l v e r c l u s t e r s (chapter 4) indicated that variations i n some CNDO parameters (e.g. the values of (I^v+A/lA)/2) cause primarily a s h i f t i n energy l e v e l s , whereas changes i n other parameters (e.g* the values of ^  ' ) cause mainly a change i n scale* Such observations were confirmed i n the present work, and as a re s u l t i t was found to be convenient i n practice to use a modified form of the function in (5.4), namely 6 t (5.5) . For a set of CNDO parameters that give a perfect match to the Xo^SB energy l e v e l s , i t i s clear that S*E i s zero, the scale factor R i s unity and the energy s h i f t S i s zero. For matching 1 4 7 the CNDO and X oi Sw energy l e v e l s our objective has been to fin d CNDO parameters that allowed the closest approach to these l i m i t i n g values. Except for S^, the values of € L used rn O05>o these analyses are t r a n s i t i o n state energies; then both 6 c and t=i represent approximations to ion i s a t i o n energies. The determination of new CNDO parameters f o r the elements aluminium to sulphur involved searching for minima of the correspondence functions, such as ( 5 . 3 ) and ( 5 . 5 ) , by cal c u l a t i n g these functions f o r many combinations of CNDO input parameters. The basic approach started with the parameters given f o r s and p basis functions by Santry and Segal £ 7 0 ] . These authors used Slater*s exponents £ 4 9 ] , but the /f given by the schemes of Burns , £ 7 6 ] and Clement! and fiaimondi £ 7 5 ] are also presented i n table XX. The e f f e c t on the CNDO calculations of varying the exponents over the ranges indicated i n table XX was considered f i r s t ; t h i s suggested an improved value f o r the exponent ( ^ s = J ^ ) - With t h i s new value of ^ , jS> was then varied to minimise the correspondence functions, and the approach continued to vary ^ and |3 i t e r a t i v e l y . Next ( I 5 + A s ) / 2 was varied, and i f the modification from the o r i g i n a l CNDO value was substantial the values of ^ and ^  were reinvestigated* The value of (I^ - * A ) / 2 was kept egual to the value used by Santry and Segal. By t h i s means i f was hoped that the parameters derived here f o r the second-row atoms should maintain a broad consistency with the conventional parameters given for atoms of the f i r s t row i n the CNDO/2 scheme £ 5 4 ] . In cases where a range of CNDO parameters gave comparable o v e r a l l l e v e l s of correspondence to the XrxSS. r e s u l t s , the f i n a l 148 Al Si Slater \ 1- 37 1.38 1-60 1.82 Burns f s 1.32 1.48 1-75 1.97 j p 0.87 1-08 1.30 1.52 Clementi 5 s 1.34 1.59 1.84 2.09 and Haimondi 1.22 1.43 1.66 1.87 Table XX. The Slater £49j. Burns £76] and Clementi and Baimondi £75] exponent values for A l , S i , P and S. se l e c t i o n was completed by comparing calculated bond lengths with experimental values f o r some simple molecules. The parameter values found from t h i s procedure f o r the elements aluminium to sulphur are designated in the following discussions as CNDO/HH. Table XXI summarises the CNDO/HH and CNDO/2 parameters used i n calcu l a t i o n s with s, p basis sets. Further information f o r each reference molecular system i s given i n the next section, where comparisons are detailed between the X oi. SB, CNDO/2 and CNDO/HM ca l c u l a t i o n s . 149 Al Si P S CNDO/2 I 1-17 1.38 1-60 1.82 (I s+A s ) / 2 7.77 10.03 14.03 17.65 (I p *A ? ) /2 3.00 4-13 5. 46 6.99 -8.48 -9 .80 -11.30 -13.61 CNDO/HM I 1.23 1-59 1.80 2.04 { l s * \ )/2 7.77 10.0 3 13.24 19.07 d f +a f ) /2 3.00 4. 13 5-46 6. 99 IS -5 .00 -5 .25 -11.30 -13.61 Table XXI-CNDO/2 and CNDO/HM parameters f o r A l , S i , P and S. The units of (I^+A^/2 and ^  are eV. 5-5 CNDO Parameter Selection f o r each Second Bow Element Some considerations i n choosing c l u s t e r s or molecules, for parametrising the second row elements, are that the systems should require only modest amounts of computer time f o r performing the CNDO calcu l a t i o n s whilst providing s u f f i c i e n t properties f o r making the parametrisations. • For aluminium, cl u s t e r s were chosen i n analogy with some previously used f o r simulating aspects of metallic s i l v e r - Since phosphorus and 150 sulphur form molecular s o l i d s , the molecules and S 8 seem appropriate for parametrising these elements- S i l i c o n i s intermediate between aluminium on the one hand and phosphorus and sulphur on the other i n so f a r as s i l i c o n forms an extended l a t t i c e , with the diamond structure, whilst e x h i b i t i n g strong covalent bonding, As a resul t of the l a t t e r , low-rindex surfaces of s i l i c o n reconstruct, unlike those of aluminium £ 1 5 0 J - , T h i s suggests the parametrisation c a l c u l a t i o n s should be performed for a system in which each s i l i c o n atom i s 4 coordinate, and fo r t h i s reason Si^fl l z was chosen-5.5-1 Aluminium The f i n a l CNDO/HH parameters f o r aluminium were chosen af t e r performing 73 CNDO ca l c u l a t i o n s on the A l 7 c l u s t e r - The X cx SS and CNDO charge d i s t r i b u t i o n s were compared by the functions i n equation (5-3) evaluated as a sum of the separate s and p populations on each symmetrically inequivalent atom, with the w equated to the number of atoms i n each set. Enerqy l e v e l s from the CNDO/2, X*SW and CNDO/HH calcul a t i o n s are plotted i n fiqure 32 and the separation between the lowest and highest occupied l e v e l s from each ca l c u l a t i o n i s 26-54, 8-63 and 16,35 eV respectively. The separation between the 1a, and 5a, levels f o r CNDO/2 i s therefore 3,07 times greater than that for the X^SM c a l c u l a t i o n ; the corresponding r a t i o with the CNDO/HM parameters i s reduced to 1-89- In choosing the f i n a l CNDO/HH 151 O n - 5 J C N D O / 2 ' 5a , - I O H * - I 5 > -o cr LU LU - 2 0 -- 2 5 H - 3 0 ^ - 3 5 ^ le 2 a , Xa S W , 5 a | , = \ 4 a | 3 e 2e 3 a, le 2 a , C N D O / H M 5 a , 3e 4 a , 2e 3 a, I e 2 a , 1a F i q u r e 32. Comparison f o r Al-j of o n e - e l e c t r o n e n e r q i e s c a l c u l a t e d with the CNDO/2 and CNDO/HM schemes and t r a n s i t i o n s t a t e enerqies c a l c u l a t e d with the X« SW method. 152 parameters, the aim was to make the spread of the l e v e l s as comparable as possible with that from the Xo( SH method while preserving the same orderings of the lev e l s - The wide spread of the CNDO/2 l e v e l s r e l a t i v e to that from the X o< SH ca l c u l a t i o n highlights the problem of defining a suitable numerical index for measuring o v e r a l l improvements in the correspondence between the energy l e v e l s from the two sets of c a l c u l a t i o n s . Thus i t i s clea r that i n d i r e c t l y using equation (5-4), s t a r t i n g with the CNDO/2 parameters, the comparisons would be overdominated by the differences for the 1a( l e v e l s , and to a lesser extent the 2a y levels- These d i f f i c u l t i e s are reduced by including the scaling parameter 8 i n the function defined by equation (5.5). An indication of the se l e c t i o n of the CNDO/HM parameters i s given by figure 33, where and 8^ are plotted as one CNDO parameter i s varied at a time while the others are fixed at the CNDO/HM values. This figure i l l u s t r a t e s that the optimal parameters for the charge d i s t r i b u t i o n do not coincide with the parameters which provide the best one-electron energy l e v e l s . However, the chosen CNDO/HM parameters provide a compromise agreement for these two types of properties. Also shown i n figure 33 are the variation of the scale f a c t o r , plotted as 1/8. Ideally for perfect agreement i l equals unity. The values shown i n figure 33 demonstrate the d i f f i c u l t y i n making the calcul a t i o n s reproduce an energy spread comparable to the X c*' SH cal c u l a t i o n on AI7- . Analysis of the s and p atomic populations f o r each valence o r b i t a l from the d i f f e r e n t c a l c u l a t i o n s on A1-, shows 153 , Fiqure 33. P l c t s of R^ , Rt and 1/R for the variation of a s i r q l e i l parameter whilst the other parameters are fixed at the CNDO/HM values. The plots are for the variables: 5 in U) . (2 i n (b) and ( I j*A 4)/2 i n (c) . The open c i r c l e s i d e n t i f y the R^ curves (xlO), the crosses the RJ curves (x100) and the dots correspond tc the 1/R curves. 154 that the CNDO/HM ca l c u l a t i o n agrees with X c<S8 i n that the lower l e v e l s e s s e n t i a l l y involve s functions while the higher l e v e l s 4a,, 3e and 5a, are largely associated with p functions. By contrast the CNDO/2 ca l c u l a t i o n indicates a high degree of s and p mixing; t h i s has the e f f e c t of increasing the p populations at the expense of the s populations. Table X V I indicates that the CNDO/HM and X SB calculations are i n good agreement fo r the d i s t r i b u t i o n of the t o t a l populations i n A l 7 -5.2-2 S i l i c o n The f i n a l choice of CNDO/HM parameters f o r s i l i c o n resulted from running more than 70 CNDO ca l c u l a t i o n s on S i ^ H ^ with d i f f e r e n t combinations of parameters. So as to reduce the p o s s i b i l i t y of the hydrogen charge dominating the correspondence function B^ > the weight factors Wj i n (5.3) were taken equal to unity for each set of symmetrically equivalent atoms. The method of parametrisation used for s i l i c o n d i f f e r s s l i q h t l y from that for the other second row elements due to the heteronuclear nature of Si^H,^ - For homonuclear c l u s t e r s , such as Al-^, a uniform s h i f t of (I^+Ay^) /2 a l t e r s the absolute values of the one-electron energies whilst retaining the r e l a t i v e spacings between the levels- However i n Si^H^ , a uniform s h i f t of (I^*A^)/2 for s i l i c o n w i l l also modify the r e l a t i v e spacing of the energy le v e l s - This i s associated with the Si-H interactions- As a r e s u l t both (I<. *AS )/2 and (I p+Ap)/2 had to be investigated, although i t turned out that the o r i g i n a l 1 5 5 CNDO/2 value for (Ip*Ap)/2 could reasonably be maintained in the CNDO/HH scheme. These parameters were investigated by varying the separation between (I 3*A 5)/2 and (Ip+Ap)/2, but keeping th e i r average value constant, and by considering the eff e c t s of s h i f t i n g both (I3*ks)/2 and (Ip*& p)/2 parameters by the constant amounts. In a l l these CNDO cal c u l a t i o n s the hydrogen parameters were kept fi x e d a t t h e i r CNDO/2 values £54]. The choice of the CNDO/HM parameters i s indicated by figure 34, again i t i s apparent that d i f f e r e n t sets of optimal parameters are indicated by the charge and energy l e v e l d i s t r i b u t i o n s alone. The valence s h e l l energies from the various c a l c u l a t i o n s are plotted in figure 35. Although the highest-occupied to lowest-occupied separation i s greater from the CNDO/HM cal c u l a t i o n , compared with that from the XcxSW c a l c u l a t i o n , by a factor of 1.31, t h i s does represent an improvement over the CNDO/2 cal c u l a t i o n f o r which the corresponding fa c t o r i s 1.85. The CNDO/HM ca l c u l a t i o n reproduces the X <xSW re s u l t that the o r b i t a l s 2t^, 1e and 1t^ are associated with Si3p o r b i t a l s bonding with the hydrogen atoms, but the CNDO/2 ca l c u l a t i o n also includes the 2a ^  o r b i t a l i n t h i s category. The o v e r a l l atomic populations from the X OCSH c a l c u l a t i o n are reproduced f a i r l y well with the CNDO/HH procedure (table XVIII)* 156 4 0 + 2 0 + 4 0 + 2 0 + P l c t s of and 1/B Fiqure 34. for the variation of a sinqle Si I — • - - ~ — parameter "whilst the other parameters are fixed at the CNDO/HM values. The plots are for the variation of 5 i° (a) » /S i n (°) » the spread between (I S*A, )/2 and (I f*A p)/2 (whilst keepinq t h e i r average value equal to 7.C8) in (c) and a unifortr s h i f t of both (I s*A t)/2 and (I?»Aj,)/2 i n (d) ; the abscissae i n (c) and (d) qive the ( I s + A s)/2 value. The c i r c l e s i d e n t i f y the R*. curves (x5) , the crosses the R£ curves (x100) and the dots the 1/R curves. 157 0 CNDO/2 X a S W C N D O / H M - 5 -- I O H > C D LU LU 3 t : I t , l e 2 t 2 2 a , 3 t 2 2 t 2 2 a , 1 t , -20H 3 t : -It, l e ^ 2 t 2 2 a , I t ; -25H I t ; - 3 0 J Fiqure 35. Comparison f o r SiyH xx of o n e - e l e c t r o n e n e r q i e s c a l c u l a t e d with the CNDO/2 and CNDO/HH schemes and t r a n s i t i o n s t a t e e n e r q i e s c a l c u l a t e d with the X* SV method. 158 5. 5. 3 Phosphorus The high symmetry and the smaliness of P^  combine to make i t somewhat l e s s than i d e a l f o r parametrising the CNDO method since a l l the atoms are equivalent and the number of d i f f e r e n t enerqy l e v e l s i s r e s t r i c t e d . A consequence of the high symmetry for the function R^ i s that a l l the unique information i s contained in the s - o r b i t a l populations. A t o t a l of 9 8 CNDO cal c u l a t i o n s were performed on P. with d i f f e r e n t combinations of parameters. The values of the functions R^and R| (equations (5.3) and (5.5)) are plotted i n figure 36 f o r variations of i n d i v i d u a l parameters while the rest are held at the CNDO/HH values. The larger number of CNDO calc u l a t i o n s required f o r P^, r e l a t i v e to Al-, and S i ^ a , was a res u l t of an i n a b i l i t y to fin d minima i n the functions R^ and R^ simultaneously f o r a si n g l e combination of |S and (l s+A s)/2 values; t h i s r e s u l t s i n optimal parameters rather d i f f e r e n t from those of Santry and Segal £70]. The bond length i n P. i s calculated to be 2.14 A with the CNDO/HH parameters and therefore to be i n reasonable agreement with the experimental value of 2.21 A and this served as an ad d i t i o n a l check for determining the f i n a l CNDO/HM parameters; however the CNDO/2 parameters give an equilibrium bond length of 2.25 A. The d i f f i c u l t y of simultaneously f i x i n g ^ and (I s+A s)/2 has been encountered previously by Santry £73]. The CNDO/HM and CNDO/2 l e v e l structures shown i n figure 37 are in good agreement with that from the X°<SW c a l c u l a t i o n , although the valence band i s broader in both cases; by fac t o r s of 1.24 and 1.54 f o r the CNDO/HM and CNDO/2 ca l c u l a t i o n s 1 0 Fiqure 36 Plots of R^, R'4 and 8 for the variation of a sinqle P parameter whilst the other paraaeters are fixed at the C N D O H S values the plots are for the variation of 5 i n ( a ) , /S in (b) and ( I 4 * A S ) / 2 in (c) . The open c i r c l e s identify the R^ curves (xlO'), the crosses the R \ curves (xlO 3), and the dots the R curves. 160 - 5 C N D O / 2 X a S W C N D O / H M -\04 - I 5 -> C D >-o cr LU LU - 2 0 H - 2 5 ^ - 3 0 ^ le 2 t 2 2 a , It I e 2 t 2 2 a , le 2 t 2 2 a , 11. - 3 5 J Fiqure 3 7 . Comparison for P* of one-electrcn enerqies calculated with the CNDO/2 and CNDO/Hfl schemes and t r a n s i t i o n state enerqies calculated with the X«t SB method. 161 1a 2 a l e 1t 2t Total 1 l 2 2 CNDO/2 0. 19 0. 06 0. 0. 17 0- 04 1. 77 Xo<SW Q. 17 0. 04 0. 0. 20 0- 02 1. 73 CNDO/HM 0. 20 0. 05 0. 0. 20 0.02 1. 83 Table XXII. Individual and t o t a l s populations f o r the valence s h e l l of P y respectively* Information on the charge d i s t r i b u t i o n s from the di f f e r e n t c a l c u l a t i o n s i s given i n table XXII. 5. 5. 4 Sulphur Overlapping-spheres X USU c a l c u l a t i o n s have been reported recently f or t h i s molecule £137] and the r e s u l t s were kindly provided by Dr. D.fi. Salahub (Universite de Montreal) for our use. The D., symmetry of S. ensures that a l l the sulphur atoms are equivalent, and, as f o r P., t h i s r e s t r i c t s the amount of information for making the parametrisations, although for S^  the number of d i f f e r e n t energy l e v e l s i s greater. again simultaneous minima i n B, and fi* (equations (5.3) and (5.5)) could not be found on varying & and (I s+A 5)/2, and i t was 162 necessary to f i x the values of these parameters by checking the calculated values of the S-S bond length against the experimental value (2.07 i ) . The CNDO/HM parameters gave a bond length of 1.91 ft, and t h i s can also be compared with the CNDO/2 0 value of 2.14 ft. The functions and B*^  vary s i m i l a r l y to those shown previously for . the other systems; a t o t a l of 88 CNDO ca l c u l a t i o n s were performed on S %. The energy l e v e l s from the di f f e r e n t c a l c u l a t i o n s are plotted i n figure 38. The CNDO/HH ca l c u l a t i o n gives a good account of the separation of s—type and p-type molecular o r b i t a l s found with the X c< SH method ( i . e . the lev e l s l a ^ to 1b^ are predominantly s-type while the upper l e v e l s are predominantly p-type) # whereas the CNDO/2 c a l c u l a t i o n shows a degree of overlap between the s and p regions. Some differences in ordering of the l e v e l s i n the p regions are found with both the CNDO/2 and CNDO/HH cal c u l a t i o n s compared with the U S S cal c u l a t i o n . These differences can be associated i n part with the high density of states, although, of course, some errors are i n e v i t a b l e within the CNDO procedures; thus the approximations reguired f o r r o t a t i o n a l invariance tend to blur some differences between o r b i t a l s with cr and TT c h a r a c t e r i s t i c s . The t o t a l atomic populations f o r s functions are 1-82, 1.95 and 1.91 from the CNDO/2, X c k S w and CNDO/HH calc u l a t i o n s respectively 163 CNDO/2 - 5 -- I O H > o o r U J -z. LU - I 5 H -20-- 2 5 ^ 1 b 2 1 e 3 1 e -X a S W C N D O / H M 1 b 2 1 e 3 1 e 2 l e , l a , 3 a, / 3 e 3 • m ^ 2 b 2 \ 2 e , 2 a , 1 b z 1 e 3 1 e 2 - 3 0 H 1 e l e , 1a, - 3 5 J F i q u r e 38. Coaparison f o r S g of o n e - e l e c t r o n e n e r q i e s c a l c u l a t e d with th<=> CNDO/2, CNDO/HM and X««Sw methods. The X t SW r e s u l t s are frcm refer e n c e f 137 "J. 164 5-6 Applications to Other Molecules In t h i s section CNDO/HH ca l c u l a t i o n s , using the parameters of table XXI, are applied to a number of other molecules and molecular c l u s t e r s formed by the elements aluminium to sulphur; the objective i s to assess how the parameters derived for the pa r t i c u l a r reference molecules transfer to other molecules-F i r s t we check for some homonuclear clus t e r s which are larger than those used i n deriving the CNDO/HH parameters. Figure 39 compares valence s h e l l energy l e v e l s from the three types of cal c u l a t i o n on A l l o - The separation between the highest and lowest occupied l e v e l s i s much greater from CHDO/2 than from the X<<XSH c a l c u l a t i o n , and there are some differences i n the ordering of the higher le v e l s - The t o t a l energy separation from the CNDO/HM calcu l a t i o n i s intermediate, but the ordering and r e l a t i v e positions of the l e v e l s are i n better agreement with those from the X°<S8 method; the 5a ± and 4e l e v e l s are reversed but these two l e v e l s have very s i m i l a r energies according to both c a l c u l a t i o n s . The charge d i s t r i b u t i o n s i n the molecular o r b i t a l s agree better with the X «SH d i s t r i b u t i o n s than do those from the CNDO/2 c a l c u l a t i o n ; the t o t a l populations over the occupied l e v e l s are included i n table XVI- , Figure 40 compares valence-shell energy l e v e l s from the various calc u l a t i o n s f or the hypothetical P<g molecule- The eight phosphorus atoms are at the vertic e s of a cube and the P-P distance i s taken to be the same as i n P4 (2-21 A) £142], again the CNDO/HM scheme appears to give a better account of the Xc*S8 l e v e l structure £147] than does the CNDO/2 method-Next an assessment i s made of the a p p l i c a b i l i t y of the 165 -5H - I O H - I 5 H > >-CD Q: LU z : LU -20H - 2 5 ^ -30H - 3 5 J C N D O / 2 XaSW C N D O / H M — 2e 2a • 6 a | '5 a | - 4 e Ma, s l a 2 :3e 3 a , ^2e 2a , 1 e F i q u r e 39. Comparison f o r Al,© of o n e - e l e c t r o n e n e r q i e s c a l c u l a t e d with the CNDO/2, CNDO/HM and U SI methods. The t r a n s i t i o n s t a t e e n e r q i e s should be about 2 eV lower than the one - e l e c t r o n e n e r g i e s f r c n the X«*SW method. 166 - 5 -- I O H - I 5 -> ^ - 2 0 -> -o cr LU 2 - 2 5 -LU - 3 0 H - 3 5 ^ •40J C N D O / 2 X a S W C N D O / H M 2t 2 g ,1t *2t 2 u l a 2 u 2a l e , It 2 g I t , 1a ' 2 g _/1t 2 u ^ 2 t ( u - 2 a , g 1 e a It 2 g It l u l a , 2t 2 g . • 1 t 2 u C T lu — i e g 1 ° 2 u 1 t 2 g 1 t lu 1a Fiqure 40. Cctpariscn for P» of one-electrcn enerqies calculated with the CHDO/2, CNDO/HM and lot s i methods. The t r a n s i t i o n s t a t e enerqies should be about 3 eV lower than the one-electron enerqies from the X«*SW aethod (froa ref. [147}). 167 CNDO/HH parameters f o r the hydrides SiH^., PH 5 and H^S. The calculated energy l e v e l s are given i n table XXIII. Compared lolecule O r b i t a l CNDO/2 CNDO/HH X^Sl SiH It 15 .90 14 .63 11.97 * i 1a 2 2 . 7 7 21-11 17-70 i PH 2a 13 .62 13.99 10.61 3 1 1e 15 .90 15 .62 13.42 1a 2 5 . 5 4 24 .86 2 0 . 5 5 i H S l b 1 3 . 1 3 13 .85 11.7 2 1 2a 1 5 . 8 5 16 .04 13.8 1b 17 .02 16.51 15 .7 2 1a 2 7 . 5 0 28 .59 2 4 . 2 l Table XXIII. Comparison of i o n i s a t i o n energies (in eV) calculated with the CNDO/2, CNDO/HH and X ^ s « procedures f o r SiH^.,PH3 and HZS. The X< x s a i o n i s a t i o n energies f o r SiH are from reference £ 1 5 1 ] , f o r PH from £99J and for H2S from £ 1 5 2 ] , with the X<xSW calculations, the CNDO cal c u l a t i o n s show rather greater separations between the lowest and highest occupied l e v e l s . In the CNDO/HH ca l c u l a t i o n f o r H.S the separation 168 between 1bR and 2a^ seems too small, bat o v e r a l l the CNDO/HM calcul a t i o n s show reasonable l e v e l s of agreement f o r the energy l e v e l s of these hydrides. The charge d i s t r i b u t i o n s from the CNDO/HM scheme correspond very c l o s e l y to those from CHDO/2; i t i s s a t i s f y i n g that the atomic charges vary smoothly over the series of molecules, for example according to CNDO/HM the net charges on the H atoms are -0.134, -0-047 and +0.036 for SiH^.,PH3 and H£S respectively. The o v e r a l l s i m i l a r i t y i n the res u l t s from CNDO/HM and CNDO/2 for these hydrides may perhaps suggest that there i s a ce r t a i n f l e x i b i l i t y i n choosing the CNDO parameters, p a r t i c u l a r l y for molecules containing j u s t a single second-row atom* However, our experience with the larger c l u s t e r s suggests t h i s f l e x i b i l i t y i s reduced for molecules containing several second-row atoms. Valence-shell energies for P^  S 3 from s i x di f f e r e n t c a l c u l a t i o n s encompassing the Xc*SW, CNDO/2 and CNDO/HM procedures are given i n figure 41 and information on the atomic charges i s i n table XXIV. The X o< SH ca l c u l a t i o n (designated ft) includes d-waves i n the atomic spheres; some consideration of an extension of the basis set to include d-functions f o r the CNDO-type c a l c u l a t i o n s i s given i n the next section. The entries i n figure 41 and table XXIV designated B and D correspond respectively to the CNDO/2 and CNDO/HM calculations f o r an s, p basis* I t i s immediately apparent that the CNDO/HM scheme accounts better than the CNDO/2 procedure for the energy separation between the highest and lowest occupied valence l e v e l s given by the X <^ S8 c a l c u l a t i o n , although both CNDO calculations interchange the closely-spaced l e v e l s 14e and 169 > OJ or LU LU -icH - I 5 J - 2 C H - 2 5 --3CH ,l7e / 3 a 2 : / i7o, " I6e : _ l 5 e •-I6a, J 5 o , -I4e I3e I4a, l3o, I2e I2a, B J7e . / l 7 a , X l 6 e , 3o2 I I5e • I6a, I4e l5o, I3e I 4Q, ' -13a, 12 e I7e . ^ ' 7 0 , , - 3 o 2 • I6e . I5e ' 16a, I4e - 1 5 0 , >l3e. 14a, l3o, I2e D I7e .^170, :^ l6e ' : - ! 5 e , !^"3o 2 ^•160," I4e ' •—15a,. 13 e 14a, 13a, I2e / l 7 e ^170, ^ 3 a 2 S 6 e I5e 16a, I4e '. 15a, I3e 14a 13a, I2e / I 7e : - ! 7 a , : - l 6 e •\3a 2 N l 6 o , "—14 e ""-15 a, 13 e 14a 13a, I2e -35H l2o, 12a, 12 o. I2O, 12 a - 4 0 -Fiqure 11. Comparison of valence-shell enerqy levels for P ^ S j . The desiqnations are: A t r a n s i t i o n state enerqies frca the X ^ S i calculation includinq d-waves [135]; B one-electrcn enerqies from the CNDO/2 calcu l a t i o n with s, p basis; C as B but s, p, d basis; D one-electron enerqies from the CNDO/HM calculation with s, p basis; E as D bat s, p, d basis with At =/3k = /2P ; F as I except the values are -5 eV for P and -9 eV for S. Calculation q s P d q s -p „d <,  s o p o d •'Pa >Pa ''Pa •'Pb *Pb Vpb > S > S V s A 1.76 B 1.68 C 1.61 D 1. 75 E 1.72 P 1. 72 2.71 0.34 3.02 2.39 C.95 3.02 2.48 0.64 2.72 C.36 1. 81 2. 76 1. 72 3. 20 1. 65 2. 62 1. 78 3. 14 1. 74 2. 68 1. 76 2. 91 0. 24 2. 00 -1. 79 4. 39 0.71 1.75 1. 88 0.52 1.85 0.27 1.86 4. 07 0. 19 3.93 0. 36 4. 27 4. 03 0. 23 4. 12 0. 14 Table XXIV Total s, p, d populations for the valence s h e l l of P4 S j . The calculation designations are: A Xo«,sw calculation where d waves are included in the atomic spheres M34 1; B CNDO/2 calculation with s, p basis; C CNDO/2 calculation with s, p, d basis; D CNDO/HH calculation with s, p basis; E CNDO/HH calculation with s, p, d basis where &A~fi>t.-(lf I P CNDC/HM calculation where the £ 4 values are reduced, namely -5 eV for P and -9 eV for S. Pa corresponds to the apical and Pb corresponds to the basal P atom. 171 15a( , and they depress the 3a 2 l e v e l compared with the xdi SH ca l c u l a t i o n . The charge d i s t r i b u t i o n s i n the i n d i v i d u a l molecular o r b i t a l s show that the CNDO/HM ca l c u l a t i o n s , unlike the CNDO/2 calculations, follow the Xc< SB c a l c u l a t i o n i n characterising the o r b i t a l s as predominantly s-type and predominantly p-type £135J-, Associated with t h i s i s an observation from table XXI? that the ; CHDO/HM cal c u l a t i o n s indicate s l i g h t l y higher s populations than do the CNDO/2 calc u l a t i o n s . The experience here for P^S^ and the hydrides suggests that the CNDO/HM parameters extend s a t i s f a c t o r i l y to some molecules involving d i f f e r e n t types of atoms, even though these parameters were deduced primarily f o r homonuclear molecules. 5.7 Inclusion of d-orbitals For many years there have been freguent discussions about the extent of the involvement of d-o r b i t a l s i n the bonding of molecules containing second-row atoms £153,154]. Generally t h i s involvement seems greatest f o r atoms i n high valence states, but even for the low valence states (e.g* divalent S, t r i v a l e n t P) d - o r b i t a l s can be needed i n computational schemes either as polari s a t i o n functions or to take up de f i c i e n c i e s associated with other aspects of the ca l c u l a t i o n s . Here preliminary ca l c u l a t i o n s are reported for the inclus i o n of d-functions in the CNDO/HM scheme f o r which values are needed for the parameters (I^*A^)/2 and . Total s, p, d populations 172 obtained from CNDO/2 and CNDO/HM ca l c u l a t i o n s on P^ , are given in table XXV where the d-parameters were chosen by taking (S<A-|&s# l i = Is a a d (I^*A^)/2 =0-5 eV as o r i g i n a l l y suggested by Santry and Segal (section 2.7) for spd ca l c u l a t i o n s . Although Qs 2 -p Qd CNDO/2 1.77 CNDO/2 D 1.74 CNDO/HM 1.83 CNDO/HMD 1.80 Table XXV. Total s, p, d populations from d i f f e r e n t CNDO ca l c u l a t i o n s on P4. CNDO/2D and CNDO/HMD include d - o r b i t a l s i n the basis set with ^ = £ s . 5<* = L and ( I ^ A ^ / 2 = 0.5 eV. only rather small differences were found i n the energy l e v e l spectrum when the calcu l a t i o n s include and neglect d - o r b i t a l s (there i s e s s e n t i a l l y a uniform lowering of the one-electron energies by approximately 1..3 eV when d-functions are included), table XXV indicates substantial d-populations are possible. A s i m i l a r observation has also been made for P^S^ for which calculated information on the charge d i s t r i b u t i o n i s given i n table XXIV and information on the one-electron 3.23 2-53 0.73 3. 17 2.63 0.57 173 energies i s included i n figure 41. The cal c u l a t i o n s f o r P^S^ designated fl and D correspond to CNDO/2 and CNDO/HM calcul a t i o n s without d-functions respectively, whereas these functions are included i n the cal c u l a t i o n s designated C and E; the d-pararaeters f o r phosphorus are as stated above f o r P^  and those for sulphur are chosen i n an analogous manner with (I^+A d)/2 =0.713 eV. C corresponds to the CNDO/2 ca l c u l a t i o n and i t i s apparent that the d-populations are too large i n rel a t i o n to re s u l t s from the X c K S H c a l c u l a t i o n (designated A). The d-population i s reduced somewhat i n the CNDO/HM scheme, although i t i s s t i l l about twice that from the X t ^ S H c a l c u l a t i o n . Figure 41 i l l u s t r a t e s the effect of including d-orbitals on the one-electron energies f o r P^S^; the major changes occur f o r states with low binding energy which i s to be expected since the 3d-orbitals are closer i n energy to the 3p-orbitals than to the 3 s - o r b i t a l s . Similar si t u a t i o n s were found f o r the other molecules referred to in sections 5.5 and 5.6; thus usually the orderings of the energy l e v e l s are es s e n t i a l l y unchanged when d-functions are included within the CNDO/HM procedure, but the d-populations can be as large as 0.5 electron per atom. (The i n c l u s i o n of d-orbitals i n the CNDO/HM scheme does change somewhat the l e v e l ordering i n the p region f o r S g; but again t h i s i s not surpr i s i n g i n view of the r e l a t i v e l y high density of states f o r that region.) To assess the most appropriate way of reducing the d-o r b i t a l populations i n the CNDO scheme, a se r i e s of cal c u l a t i o n s were made for P^ i n which the d-parameters , (I^*A^)/2 and were varied; the res u l t i n g charge 174 (a) = (SS =-11-30 eV, Sdi = i 1.80 Qs Q9 Q cA. -5-50 1.81 2-95 0. 24 -2-50 1-81 2.84 0.35 -0-50 1.81 2.71 0. 48 0.50 1.80 2.63 0-57 1.50 1-80 2.51 0.68 lb) a A + hA)/2 =0.50 eV, c^A = = 1. 8 0 Q? 2 d -11.30 1.80 2.63 0.57 -9.00 1.81 2.75 0-44 -7-00 1 - 8 1 2-84 0-35 -5.00 1.81 2-92 0.27 (c) • Aj^/2 =0.50 eV, fa = Is' =-11-30 eV Qs Q ol 1.80 1.80 2-63 0- 57 1.35 1.71 2.25 1.03 Table XXVI. Total s, p, d populations f o r P 4 using d i f f e r e n t CNDO d - o r b i t a l parameters; the s, p parameters are the CNDO/HM values. 175 d i s t r i b u t i o n s are presented i n table XXVI. Santry 173 j has previously proposed that the d-populations should be reduced to more plausible values by reducing the ^  values. Table XXVI indicates that the d-population-can-be reduced eit h e r by giving (Ij+A^)/2 a negative value, or by reducing the absolute value of ^ ; the l a t t e r approach may possibly seem to be the more physically reasonable. Santry and Segal |_ 70J favoured the spd* l e v e l of cal c u l a t i o n s v i a =0.75 | s , but t h i s contrasts with the r e s u l t i n table XXVI that reducing ^ by t h i s amount can increase the d-populaticn. Accordingly we have kept = j*, and used values reduced to -5 eV f o r P and -9 eV f o r S. Table XXIV shows that t h i s CNDO/HH ca l c u l a t i o n for P^S^ (entry F) gives rather a good account of the X ex. SB populations, although the corresponding energy l e v e l structure (figure 41) i s seen to be changed only marginally from those f o r the cal c u l a t i o n s designated by D and E. Figure 42 compares the valence l e v e l structure f o r SO^ from a number of d i f f e r e n t c a l c u l a t i o n s . The X^SH ca l c u l a t i o n included d-waves i n the sulphur spheres £155 3, and i t i s clear that the in c l u s i o n of d-orbitals i n the CNDO ca l c u l a t i o n s a f f e c t s the ordering of the upper l e v e l s . Again the CNDO/HM calc u l a t i o n s appear somewhat more successful than the CNDO/2 cal c u l a t i o n at matching up the ordering and r e l a t i v e spacings found from the Xc*SH c a l c u l a t i o n ; the two CNDO/HH cal c u l a t i o n s which include d-orbitals (designated E and F i n fig u r e 42) seem equally successful i n t h i s regard although, compared with the X <x.sa r e s u l t s , the 3a, l e v e l seems too low and the 2bz l e v e l too high. 176 - I 5 --20H > >-o LX. LU LU - 2 5 -- 3 0 H - 3 5 H - 4 0 -- 4 5 ^ B D ,4a, •3b2 ' -lb, ' -3a, 2b 2 2a, lb 2 la, l a 2 4a, 3b 2 - 4a, — 3b 2 / 2 b 2 _ — lo 2 lb, __ —/lb i ^3 a, — "~^2b2 — 3a i 2a, — 2a, lb 2 la, lb2 la, la 2 4 a , 3b ? -lb, . -2b2 3a, : 2a, la, . 4 0 1 ^3b2 la 2 lb, 2b 2 3a, 2a lb; la, -3b 2 "la 2 lb, 2b 2 3a , 2a lb-la, Pigure 42. Comparison of valence-shell enerqy levels for SOj. The designations are: A t r a n s i t i o n state enerqies frcm the X *>SW calcul a t i o n includinq d-waves [155 1; B one-electron enerqies from the CNDO/2 ca l c u l a t i o n with s,p basis; C as B but s, p, d basis for S; D one-electron enerqies from the CNDO/HM calculation with s f p basis; E as D but s, p, d basis for S with fit F as E e x c e p t ^ =-9 eV. 177 5.8 Concluding Remarks Within the l i m i t s of our present experience, the CNDO/HM parameters given i n table XXI for the elements aluminium to sulphur seem at l e a s t as successful as the conventional CNDO/2 parameters f o r simple molecules containing a single second-row atom, and these parameters appear more successful for molecules containing a number of second-row atoms- Nevertheless we do not claim that the parameters i n table XXI are optimal for a l l situ a t i o n s for these elements. There could be some advantages in applying the procedures i n sections 5-4 and 5-5 i n a search for sets of parameters that give the best o v e r a l l account of energy l e v e l s and charge d i s t r i b u t i o n s f o r a range of molecules formed by each element. It i s possible that t h i s would lead to more even trends i n CNDO parameters than those reported in table XXI- The few ir r e g u l a r features i n the trends of CNDO/HM parameter values are perhaps not surp r i s i n g i n view of the fact that they were obtained from complex molecules of d i f f e r e n t types (e.g. metal, semiconductor, molecular s o l i d s ) . By contrast the CNDO/2 parameters come mainly from atomic spectral data and from non-empirical c a l c u l a t i o n s on simple molecules-, The energy differences between the highest and lowest occupied valence l e v e l s are given in table XXVII f o r Al , Si^H^ , P and S g , and the X <xsw c a l c u l a t i o n s show that these energy differences for A1-, and Si sH ,^  are approximately half those f o r P^  and Sg. In the CNDO/2 calculations, the separation energies between the highest and lowest occupied valence states are roughly 22 eV for each molecule. The CNDO/HM calcu l a t i o n s provide improvements over the CNDO/2 spread of valence-levels. 178 Al S i H P S 7 5 12 4 8 CNDO/2 26.54 16. 19 23.49 22.03 X*.SW 8.63 8.74 15.23 14.98 CNDO/HM 16.35 11.47 18.85 19-69 Table XXVII. Eaergy of separation (in eV) between the lowest and highest occupied valence-levels f o r the molecules used to parametrise the second-row elements. although the CNDO/HM energy separation f o r A1-, (16.35 eV) appears too large on the basis of the results from the X cx SW method. These l a s t two chapters demonstrate the use of X &< SW calcul a t i o n s f o r parametrising the CNDO method. I t seems that the approach used i n t h i s chapter could provide a help f u l route for getting CNDO parameters f o r systems of the heavier elements (e.g* t r a n s i t i o n metals) for which both suitable non-empirical calculations and suitable experimental data are sparse. Certainly a f t e r obtaining the new parameters, the CNDO calculations made i n t h i s thesis f o r c l u s t e r s of s i l v e r and for molecules of second-row elements do seem capable of highlighting some s a l i e n t features shown by the X^SW calc u l a t i o n s . Nevertheless, c e r t a i n differences are present between the X o<S8 and CNDO c a l c u l a t i o n s . For example i n chapter 179 4, the l o c a l i s a t i o n e f f e c t s i n the s i l v e r c l u s t e r s were observed to be less strong in the CNDO ca l c u l a t i o n s , and in t h i s chapter the energy spread of the the valence lev e l s i n the molecules composed of second-row elements i s usually larger i n the CNDO/HH cal c u l a t i o n s than those from the Xoc'SH procedure. Certainly the X »<SM method i s well established f o r th e o r e t i c a l analyses of systems of the heavier elements with r e l a t i v e l y high symmetries, but the study reported i n t h i s thesis suggests that t h i s method should also be h e l p f u l in guiding semi-empirical molecular o r b i t a l methods, such as CNDO, fo r applications to complex systems of r e l a t i v e l y low symmetries. 180 REFERENCES 1. 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