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UBC Theses and Dissertations

Study of the BH molecule Gagnon, Paul Joseph 1970

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A STUDY OF THE BH MOLECULE by  . ' •  ••  PAUL JOSEPH GAGNON B.Sc,  St. Francis  Xavier University,  1968  A T H E S I S SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.Sc. i n t h e Department of  .  .  Chemistry  We a c c e p t t h i s t h e s i s required  as c o n f o r m i n g  tothe  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA J u l y , 1970  In p r e s e n t i n g t h i s  thesis  in p a r t i a l  f u l f i l m e n t o f the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, the L i b r a r y s h a l l I  make i t f r e e l y a v a i l a b l e  f u r t h e r agree t h a t p e r m i s s i o n  for  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying o f t h i s  that  study. thesis  f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s  representatives.  It  i s understood that copying o r p u b l i c a t i o n  of this  t h e s i s f o r f i n a n c i a l gain s h a l l  written  permission.  Department of  CHEtN\\STRY  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Date  MX  Columbia  not be allowed without my  i  Abstract Antisymmetrized several  gerainal p r o d u c t w a v e f u n c t i o n s  limited basis  s e t s were c a l c u l a t e d  s t a t e o f BH a t R=2.329 a t o m i c wavefunctions  units.  more t h a n  In our case, t h i s  the best o v e r a l l  13-term,  "split  -25.14769 a t o m i c  units.  the v i r i a l  one-point  o f 'V/O.Ol  configuration-  A "contracted" double-zeta  basis  energy.  core",  w a v e f u n c t i o n was d e v e l o p e d  satisfy  these  improvement a c c o u n t s f o r  90% o f t h a t a c h i e v e d by a f u l l  yielded A  Extending  i n a l o w e r i n g i n energy  i n t e r a c t i o n wavefunction. set  f o r t h e ground  t o i n c l u d e Kapuy's " o n e - e l e c t r o n t r a n s f e r "  configurations resulted atomic  units.  with  configuration-interaction  and y i e l d e d  an e n e r g y o f  T h i s w a v e f u n c t i o n was t h e n made t o  theorem.  P a r r and W h i t e ' s method f o r  f o r c e c o n s t a n t c a l c u l a t i o n was a p p l i e d  scaled wavefunction with negative A s i m i l a r wavefunction  to the  results.  was p a r t i a l l y  optimized a t three  i n t e r n u c l e a r d i s t a n c e s f o l l o w e d by s c a l i n g w i t h f i x e d V a r i o u s p a r a b o l i c m o d e l s were u s e d and  t o f i t the v i r i a l  e n e r g i e s c o r r e s p o n d i n g t o each R v a l u e .  constants k  e  calculated  good and t h e e f f e c t Parabolic  expansions  R. forces  The f o r c e  f r o m t h e s e m o d e l s were u s u a l l y  very  o f s c a l i n g was shown t o be i m p o r t a n t . i n 1/R g a v e b e t t e r r e s u l t s  than  ii parabolas values.  i n R, compared  t o a q u i n t i c m o d e l and t o  experimental  iii  TABLE OF CONTENTS Page ABSTRACT  i  L I S T OF TABLES AND FIGURES  V  ACKNOWLEDGMENT  v i  CHAPTER I  INTRODUCTION  1  CHAPTER I I  GEMINALS  3  2-1  Concept  3  2-2  Computational Procedure  6  2- 3  Application  t o BH  9  EXTENDED GEMINALS  12  3- 1  Kapuy's T h e o r y  12  3-2  Kapuy's A p p l i c a t i o n  17  3- 3  Application  18  CHAPTER I I I  CHAPTER I V  ONE-POINT CALCULATION OF k  4- 1  Theory  4-2  Scaling Obtain  e  22 22  P r o c e d u r e Used t o f at R  4-3  Selection  4- 4  Scaling  CHAPTER V  t o BH  g  o f Wavefunction  and F o r c e C o n s t a n t R e s u l t s  24 26 31  V I R I A L SCALING  38  5- 1  Thorhallsson  38  5-2  W a v e f u n c t i o n s and Data  39  5-3  Numerical Analysis  42  a n d Chong's A p p r o a c h  BIBLIOGRAPHY APPENDIX I APPENDIX I I  V  L I S T OF TABLES AND  FIGURES  TABLE  Page  I  Geminal  Calculations  20  II  13-Term W a v e f u n c t i o n  27  III  D a t a f r o m Ohno  28  IV  Optimized Wavefunction  29  V  Wavefunction ¥  ^2  VI  O r b i t a l Exponents  VII  Scaling  Results  35  VIII  Force Constants  36  IX  Unsealed Data  40  X  S c a l e d Data  41  XI  Quintic  Polynomial F i t  43  XII  P a r a b o l i c Model R e s u l t s  45  XIII  Comparison  47  XIV  BH W a v e f u n c t i o n s  XV  Data from B a s i s  XVI  Coefficients  T  0  S  C  f o r 13-Term W a v e f u n c t i o n s  o f Data  33  48 Set II  o f 13-Term W a v e f u n c t i o n s  56 57  FIGURE I  E versus a  54  II  E v e r s u s £(H)  55  vi  ACKNOWLEDGMENT The  author wishes  t o thank  encouragement and h e l p f u l  D r . D. P. Chong f o r h i s  advice i n the preparation of t h i s  thesis. The received  a u t h o r a l s o acknowledges from t h e N a t i o n a l  the financial  Research  Council.  support  CHAPTER I INTRODUCTION The g e n e r a l purpose of t h i s work i s t o develop and, o r , t e s t s e v e r a l methods o f s e e k i n g i n f o r m a t i o n on the e l e c t r o n i c r e p r e s e n t a t i o n of m o l e c u l a r systems. i n v o l v e the  A l l calculations  ground s t a t e o f BH o n l y , and a Born-Oppenheimer  s p i n l e s s Hamiltonian.  Three main p r o j e c t s were undertaken:  t o c o n f i r m the u s e f u l n e s s o f Kapuy's extended geminal theory 11, 2 ] ; t o perform a one-point c a l c u l a t i o n o f k f o r c e constant) a c c o r d i n g t o P a r r and White  e  (quadratic  [3]; and to seek  a b e t t e r approach t o t h r e e - p o i n t c a l c u l a t i o n o f  k . e  Kapuy a p p l i e d h i s extended geminal theory t o the i r - e l e c t r o n s o f t r a n s - b u t a d i e n e w i t h good r e s u l t s  [2] .  Our  a p p l i c a t i o n o f extended geminal theory t o BH a t R=2.329 atomic units  [4] t r e a t s a l l e l e c t r o n s and s e v e r a l l i m i t e d b a s i s s e t s  are t r i e d .  P u l l c o n f i g u r a t i o n - i n t e r a c t i o n c a l c u l a t i o n s were  w i t h i n r e a c h and were used f o r comparison. every case support those o f Kapuy.  The r e s u l t s i n  In Chapter I I the b a s i c  geminal t h e o r y i s p r e s e n t e d , w h i l e the extended geminal c a l c u l a t i o n s are d e s c r i b e d i n Chapter I I I . P a r r and White perturbation at R  e  [3] developed a p u r e l y k i n e t i c - e n e r g y which i s used t o d e s c r i b e the m o l e c u l a r  -2p o t e n t i a l energy  f u n c t i o n o f d i a t o m i c molecules.  Using  their  method and experimental data they c a l c u l a t e the f o r c e constants k results.  e  and l  e  f o r many d i a t o m i c molecules w i t h e x c e l l e n t  S i n c e t h e i r method r e q u i r e s o n l y the k i n e t i c - e n e r g y  m a t r i x elements o f a g i v e n wavefunction  a t R , they e  suggest  t h a t a c t u a l c a l c u l a t i o n s be c a r r i e d out u s i n g l i m i t e d b a s i s s e t s o f atomic o r b i t a l s . a s u i t a b l e wavefunction theory.  In Chapter  IV we begin by seeking  on which t o apply Parr and White's  S p l i t t i n g the core o r b i t a l on boron i n t o two  o r b i t a l s f o r Ohno's 13-term c o n f i g u r a t i o n - i n t e r a c t i o n wavefunction  [5] g i v e s a l a r g e improvement i n energy.  p e r t u r b a t i o n s c a l i n g procedure " s p l i t c o r e " wavefunction calculated.  A  i s then a p p l i e d t o t h i s  before f o r c e constants a r e  The r e s u l t s a r e v e r y poor and i t i s concluded  t h a t a much l a r g e r b a s i s s e t i s r e q u i r e d . In  Chapter V a v i r i a l  s c a l i n g procedure,  t h a t used on L i H by T h o r h a l l s s o n and Chong a c o n f i g u r a t i o n - i n t e r a c t i o n wavefunction internuclear distances. f o r c e and energy constant k  e  similar to  [6], i s a p p l i e d t o  f o r BH a t three  V a r i o u s p a r a b o l i c models i n v i r i a l  a r e i n v e s t i g a t e d and v a l u e s f o r the f o r c e  a r e found t o be good.  Parabolic f i t s to scaled  data a r e found t o g i v e b e t t e r r e s u l t s than p a r a b o l i c f i t s t o unsealed d a t a . energy  Expansion  o f the d i a t o m i c molecular  potential-  f u n c t i o n i n 1/R r a t h e r than R l e d t o b e t t e r r e s u l t s i n  most c a s e s .  CHAPTER  II  GEMINALS 2-1  Concept The  Hartree-Fock  of  the  of  much u s e  energy  experimental  differences,  are  energy  often  of  the  either  state.  action  antiparallel of  parallel  principle  spins,  and a r e  antiparallel fundamental  the  is  energy  are thus  spin.  treatments  used  inherently  too  introduced  by means  is  which  kept  are  the  fact  interest  in  the  for.  by the  antisymmetry  described  better  than  i n chemical  inaccurate. a  binding.  electrons found to A l l  quantum mechanics Electronic  superposition  with  of be  of  one-electron  are  correlation of  inter-  Electrons  apart  is  in  energies  electrons  properly accounted  differtotal  coulombic  especially  correlation  in  the  better  that  not  between  of  a great  give  percent  energy  Correlation error  electrons,  i n molecular  of  these  there  Electronic  importance  difference  Thus,  to  1  usually interested  1 percent  not  about  energies  about  due  of  total  than  method.  is  pairs  spins  is  calculations  method  between  usually within  Unfortunately,  larger  Hartree-Fock  Hartree-Fock  one  states.  no  is  However,  such as  quantum-mechanical than  value.  i n themselves,  two s p e c t r o s c o p i c ences  energy  is  usually  configurations.  -4Although t h i s method can  l e a d t o r e s u l t s of any  desired  accuracy, the wavefunctions become extremely complicated even for  r e l a t i v e l y simple molecules.  the wavefunction i n c r e a s e s t o a s s i g n any given  A l s o , as the complexity of  i t becomes more and more d i f f i c u l t  p h y s i c a l s i g n i f i c a n c e t o the  importance o f  any  configuration. The  electron p a i r i s conceptually  a t t r a c t i v e t o chemists.  Molecular e l e c t r o n i c structure i s u s u a l l y described  i n terms  of i n n e r s h e l l s , bond p a i r s and  also  lone p a i r s .  I t has  been shown t h a t the most important c o r r e l a t i o n e f f e c t s are those i n v o l v i n g a p a i r o f e l e c t r o n s a t a time f o r e i t seems reasonable t o c o n s t r u c t two-electron functions (orbitals).  i n s t e a d of o n e - e l e c t r o n  i n t r a p a i r c o r r e l a t i o n and  The  have l i t t l e  There-  a method which uses  Such e l e c t r o n - p a i r f u n c t i o n s  C o r r e l a t i o n can  [7, 8 ] ,  could  functions emphasize  interpair correlation.  e a s i l y be admitted w i t h i n each p a i r  n a t u r a l tendency to make the p a i r f u n c t i o n s  function.  correspond  t o d i s t i n c t bond p a i r s , e t c . , a l l o w s the p a i r f u n c t i o n s be r e l a t i v e l y separated and  highly l o c a l i z e d .  For  to  example  two  bonds a t o p p o s i t e  ends of a long molecule or the  sigma  and  p i bonds i n a double bond, are cases i n which, t o a very  good approximation, the s p a t i a l p a r t s of the wavefunction d e s c r i b i n g d i f f e r e n t bonds do not The  high degree of i n v a r i a n c e  overlap. of bond p r o p e r t i e s  molecule t o molecule suggests t h a t p a i r - f u n c t i o n s w i l l  from be  -5transferable  f r o m one m o l e c u l e  approximation simplify  the  molecules, simpler a  because  the  one  of  c o u l d use  m o r e common b o n d s , on l a r g e  accuracy  to  a  good  would  greatly  good w a v e f u n c t i o n s the  Development of  calculations  much g r e a t e r  another  Such t r a n s f e r a b i l i t y  construction  systems.  few of  allow  [9].  to  results  for  large  obtained  good p a i r - f u n c t i o n s such as  systems  to  C-C or be  and s i m p l i c i t y than  from for  only  C - H , would  c a r r i e d out  with  is  at  possible  present. The  idea of  by P a u l i n g Actually,  [10].  case  example,  Usually pair  of  the  Refs.  would describe such as  functions  geminal theory  particular for  pair  [11,  (or  "separated  self-consistent 12,  13].)  system.  molecular wavefunction is  localized geminals.  individual  geminals  are  is  b u i l t from orthogonal  spin  o r b i t a l enters  in  a  of  to  the  The  that  called theory")  is  group  group i n a  (See,  function  molecule,  nature  fact  that  of  Slater  i n terms  u s i n g an  a  a  antisymmetrized  separated  d e s c r i p t i o n of  employed i n the  geminals.  group approach.  an  one-electron  manner,  originated  geminal approach  one g e m i n a l upon a n o t h e r  "self-consistent"  analogous  pair  m a i n t a i n e d by the  ants  The e f f e c t  In  l i n e a r i l y expressed  the  are  been  An e l e c t r o n  w r i t t e n as  product of  have  functions  a d i s t i n c t many-electron  a it-electron  geminals  seems t o  spin  of  if  a l l  determin-  orbitals,  no  more t h a n one  geminal.  is  account  taken  into  '.iterative  standard  method  SCF a p p r o a c h  [14].  -62-2  Computational  Procedure  The f o l l o w i n g treatment i s a c c o r d i n g t o Parks and P a r r [12]. The m o l e c u l a r wavefunction f o r a 2 n - e l e c t r o n system i s written a  (1,2)*  b  (3,4)* ( 5 , 6 ) . . . * ( 2 n - l , 2 n ) ] , c m  (2-1)  where each geminal I|K i s an antisymmetric f u n c t i o n o f the space and s p i n c o o r d i n a t e s o f the two e l e c t r o n s  involved.  The square b r a c k e t s r e p r e s e n t the normalized p a r t i a l  anti-  symmetrization o p e r a t o r which generates a completely a n t i symmetric  Y from the simple products o f the i n d i v i d u a l  antisymmetric The geminals are w e l l behaved and normalized t o u n i t y :  7/1^(1,2) There i s a complete  | dt dx =l for a l l i . 1  ii*  r  i 2 ' *"*  2  s e t o f s p i n o r b i t a l s which can be  p a r t i t i o n e d i n t o subsets r r  (2-2)  2  s u c  ^  a  l  , r  t  n  a  t  a  , r  2  t  n  e  a  3  ,  r b l  /  r  geminal i|>^ may  b2'  r  b3'  be  expressed i n terms o f S l a t e r determinants b u i l t from the subset i o r b i t a l s o n l y , f o r a l l i . V i l * i l C  +  C  That i s ,  i 2 * i 2 " - ' +  ( 2  -  3 )  where the C^j are c o n s t a n t s and the <|>^j are S l a t e r determinants b u i l t from the s p i n o r b i t a l s i n subset i . Geminals d e f i n e d i n the above manner are m u t u a l l y orthogonal,  / * J ( l , 2 ) * (1,4)^=0 for i j  ji  j,  (2-4)  -7and / / * * ( 1 , 2 ) ^ (l 2)dT dx =0 f  1  These o r t h o g o n a l i t y r e l a t i o n s Also,  2  for i ?  greatly  individual  is  normalized.  conditions  geminals,  fixed  for electron  Hamiltonian  nuclei a with  the t o t a l  charges  of a 2n-electron  e 2  /r  P n  ),  (2-6)  ^  H (S)=TU)+U (S),  (2-7)  U (e)=-^ (Z e /r^ ) .  (2-8)  N  a  N  the p o t e n t i a l  a  energy  2  a  of attraction  between  electron  £ and the bare n u c l e i and T(£) i s the k i n e t i c  operator  for electron  For earlier, energy  a"system  £ i f the other  described  the expectation  would  pair  were  value  f o r the t o t a l  energy  absent. discussed  electronic  be ±  ...m.  electrons  by the wavefunction  E=2? i +1/27?J~!$ where  of  2n  N  gives  system  a  N  N  the  Z e may b e w r i t t e n  2n  U (C)  wavefunction ¥  pairs.  H(l,2,...2n)=2|H (5)+l/22 < 5^1 5,n=i  where  calculations.  as d e f i n e d so f a r , s a t i s f y  of separability  The e l e c t r o n i c  the  and the n o r m a l i z a t i o n  i t follows that  Geminals,  (2-5)  simplify  from the orthogonality r e l a t i o n s  of  j .  ±  L  t h e sums a r e o v e r The q u a n t i t y i would have  (Jij-Kij ) ,  the d i s t i n c t  electron  1^ i s t h e e l e c t r o n i c  i f the other  p a i r s were  (2-9) pairs  energy  a,  b,  electron  absent.  -8-  I =//**(l,2)H°(l,2)* (l,2)01x^2, i  (2-10)  i  where H° (l,2)EH (l)+H (2) + (e /r N  The q u a n t i t y pair  i  J^j  is  the  and e l e c t r o n  3i±=JSft**(1,2)**(3,4) 1  3  [(e /r 2  K  i j  (2-11)  Coulomb r e p u l s i o n between  electron  j :  1 3  )+(e2/r  1 4  )+( 2/r e  i  j  1  f  2  3  a c o r r e s p o n d i n g exchange  2 3  )+(e /r 2  2 4  )] (2-12)  4  repulsion:  =////*J(l 2)*j(3,4) r  t(e /r  .X  2  +  (e /r 2  2  + (e /r 2  There are electronic  i  1 4  2 3  2 4  i  )* (l,3)* (2,4) i  j  ) * ( l , 4 ) i | > . (3,2) ] d T d T d x d T . i  of  of  starts  with  method  subsets  t h e n one s e e k s  the  because  Eq.  best  form o f  then minimizes  energy  (2-9)  a  of  c a n be  r  b  ,  ...r  geminals this  (2-13)  4  the  some s p e c i f i c  function  all  3  total  geminals.  spin orbitals  r ,  2  for minimization of  a system  nonoverlapping  the  1  ;  orthonormal one-electron  fixes  (1,4)  j  )* (4,2)*.(3,l)  two methods  energy  The f i r s t  )* (3,2)*  1 3  +(e /r  one  ) •  * (l,2)* (3 4)dT dT dT dT .  X The K - L J i s  1 2  3  1  X  total  pair  2  N  m  ,  set  r partitioned  E _£  pairs  but  m  is i ,  the  total  into  i n some s p e c i f i c  [ * * b . • • ip l • a  except one,  geminal.  way,  To do  m  say  T h i s may be  this, and  done  rewritten E^Em-i+Ei,  where  of  electronic  including their  (2-14) energy  interactions.  of  all  the  geminal  Then one  selects  -9a n o t h e r g e m i n a l , and v a r i e s t h e new g e m i n a l u n t i l i s minimized.  This procedure i s repeated  improvements a r e The the  best  until  i t s energy  no f u r t h e r  obtained.  s e c o n d method i s more g e n e r a l  and s t a r t s by  set of one-electron  and t h e b e s t  functions  seeking partition-  i n g o f them. The  f i r s t method i s u s e d  procedure  i s required  i n t h i s work b u t no  since there  i s only  bond g e m i n a l , w h i c h i s n o t o f f i x e d The  a c t u a l working equations  as p r e s e n t e d  by P a r k s and P a r r  representation. equivalent  Klessinger  form.  f o r geminal c a l c u l a t i o n s ,  and McWeeny  treatment of geminal theory;  only  approach.  geminal theory  and a l s o uses p e r t u r b a t i o n  generalized  techniques  to  theory.  A p p l i c a t i o n t o BH Our  t r e a t m e n t o f BH i n v o l v e d  containing  four o r b i t a l s .  four basis  The o r b i t a l s  s_ (2s on b o r o n ) , £ ( 2 p a on b o r o n ) , and h basis  an e x a c t l y  they use a  [1] d e s c r i b e s  d e v e l o p a method t o e x t e n d g e m i n a l  operator  [13] g i v e  density matrix  2-3  one g e m i n a l , t h e  [ 1 2 ] , a r e i n an  Kapuy  iterative  s e t I, the o r b i t a l s  Slater-type orbitals orbital orbital)  are:  k  ( I s on b o r o n ) ,  ( I s on h y d r o g e n ) .  on b o r o n a r e t h e s i n g l e - z e t a  o f C l e m e n t i and R a i m o n d i  exponent 5 o f h i s optimized  s e t s , each s e t  (also a single-zeta  to give  [15], but the  Slater-type  the lowest energy f o r the  In  -10-  antisymmetrized set  II i s l i k e  t o k.  geminal  product  s e t I, but  s_ has  I n s e t I I I , £ ( h ) = l and  (AGP)  is  i s , with  identical  fixed  linear  the o r b i t a l s  o f which i s o p t i m i z e d as Two  from  s_ and  is  the b e s t energy  bonding  and  (1-a ) 2  1 / 2  the  exponent  a:  a  ,  (2-15)  s - aa  ,  (2-16)  wavefunction*.  The  to hybrid b  i n t o K,  geminals  k,  b,  n,  B,  N and  and H  f o r t h e c o r e and  h are  symmetricly  respectively. lone p a i r s  are  simply  ^!=(KK),  (2-17)  T|) =(NN),  (2-18)  2  pair  symbols  i s described  represent S l a t e r determinants.  1  t h e AGP  2  wavefunction f  A G p  The  bond  by:  <J'3=C (BB)+C [ (BH) + (HB) Now  IV  n i s nonbonding.  orthonormalized  where t h e  171,  Basis set  parameter a i s a l s o o p t i m i z e d  f o r t h e AGP  Next, the o r b i t a l s  The  [16,  V2  b = as +  give  are  i n set I.  2  where t h e h y b r i d i z a t i o n  [17].  the e x c e p t i o n o f h,  h y b r i d s were f o r m e d  n =  on b o r o n  set of Clementi  coefficients  to set III with  Basis  been Schmidt o r t h o g o n a l i z e d  " c o n t r a c t i o n s " of the double-zeta that  wavefunction.  ]+C (HH). 3  i s written  =[i(» (l,2)T|> (3 4)* (5,6)] . 1  (2-19)  2  f  3  * O p t i m i z a t i o n o f a i s c a r r i e d o u t f o l l o w i n g any r e q u i r e d o p t i m i z a t i o n o f the hydrogen Is exponent.  (2-20)  -11-  The  form o f geminals  a n <  ^  g e m i n a l ty^ must be v a r i e d no  iteration  wavefunction interaction  *  2  ^  s  a  l  e  a  dy  f i x e d and  only  t o m i n i m i z e t h e e n e r g y , and  procedure i s required. i s equivalent  r  This  particular  t o a 3-term l i m i t e d  (CI) w a v e f u n c t i o n w i t h t h e  therefore AGP  configuration-  configurations  (j) = (KKNNBH) + (KKNNHB) ,  (2-21)  <J>= (KKNNBB) ,  (2-22)  1  2  4> = (KKNNHH) .  (2-23)  3  In actual  order to describe calculations  i n Appendix The Chapter  t h e methods g e n e r a l l y  i n the  a sample g e m i n a l c a l c u l a t i o n i s p r e s e n t e d  I.  r e s u l t s of calculations I I I f o r comparison  g e m i n a l and  used  full  CI  using  V^Q-p a r e p r e s e n t e d i n  w i t h the r e s u l t s of  calculations.  extended  CHAPTER I I I EXTENDED GEMINALS 3-1  Kapuy's The  linear  Theory  best geminal wavefunction  combination <F B  Rr  ¥  of antisymmetrized  G  =  geminal  A., i j , . . . 1 1D...1  -1 a r e n u m e r i c a l c o e f f i c i e n t s  where t h e A,-  XJ • • • X  from each  U  x  and  subset, the geminal  d  k  ¥. .  -,  e  being  the subset being designated  the p o s i t i o n of the s u b s c r i p t . O  Y..  X J • • • X  a s u b s c r i p t and  C  a  (3-1)  ,  contains a geminal d e s i g n a t e d by  products  13...1  f  f u n c t i o n V^GP  c a n be e x p r e s s e d a s  B G  Our  w r i t t e n as  BH  by  g r o u n d s t a t e waveand  i f a l l of  the  s u b s c r i p t s were n o t e q u a l t o 1 i t w o u l d r e p r e s e n t some excited  c o n f i g u r a t i o n of the  The  wavefunction  c o u l d be d e t e r m i n e d  by  *F  BG  system.  and  the c o r r e s p o n d i n g energy  solving  the s e c u l a r  E  1  in  the  i s the matrix of the t o t a l  B G  -. r e p r e s e n t a t i o n . x j• ... x  p r o p e r t i e s of the geminals, o c c u r o n l y between g e m i n a l differing  ( 3  Hamiltonian operator  F o r t u n a t e l y , due  nonvanishing matrix products * i j  i n n o t more t h a n two  t # <  x  subscripts.  B  G  )  equation  '"BG-^BG) ! - °' where H  (  a n  ^  t o the elements  1  ~  2 )  -13Kapuy  [1]  introduces p a r t i c l e  each of the N subspaces N where r ^ i s one such  2N = XL 21  /dcxr„Y (a) r *  1£l X  K  X  K  used  (a) ,  (3-3)  X  i n the Kth  geminal.  o p e r a t o r s commute w i t h e a c h o t h e r and w i t h  t o t a l p a r t i c l e number o p e r a t o r « / C = S N .  I t c a n be  all  any  k  antisymmetrized  combination  geminal  products  e i g e n v a l u e s N j = N = . . .=N =2.  o p e r a t o r s , the N result  the  system.  K  H w h i c h c o n t a i n s one-  and  H cannot  i s that  cannot  It i s just  of the N I f the  K  and t w o - p a r t i c l e  9  an e i g e n f u n c t i o n o f  t o the e i g e n v a l u e  2.  the Hamiltonian  eigenstates.  Any  and  comprises be  t o the eigenvalue  with  2N.  The  eigenstates of ^ c o r r e s p o n d i n g  to different  t h e o c c u p a t i o n numbers N-^,  ... N  2  N  be  eigenstates  s e t of the  i t s " o r t h o g o n a l complement".  N ,  have  e i g e n s t a t e o f H can  only a p a r t of these eigenstates o f « ^  completed  number  they  expanded i n terms o f a l l l i n e a r l y independent which belong  eigenstates.  i s complete the t o t a l p a r t i c l e  o p e r a t o r t ^ / commutes w i t h simultaneous  commutes  K  t h e b e s t p o s s i b l e a p p r o x i m a t e wave-  which belong  s e t {r}  linear  the exact e i g e n s t a t e of  f u n c t i o n w i t h t h e c o n s t r a i n t t h a t i t be all  shown t h a t  with  K  have s i m u l t a n e o u s be  the  arbitrary  S i n c e none o f t h e N  N  w i t h the Hamiltonian  The  and  o f them a r e e i g e n f u n c t i o n s o f t h e N 2  in  K  (subsets)  of the o r b i t a l s  R  All  K  number o p e r a t o r s N  a n  d  ^ij.,.1 should  Since  two  partitions  are a u t o m a t i c a l l y  of  -14orthogonal, linearly all  N  complement  eigenstates  partitions  of the set N ^ , N  . . .  £ 2 N , where  K  subspace  linearly  K.  n  2 N , when n > 2 N ; i s t h e number  of a l l  N ,2  eigenstates  there  2N,  eigenvalues . . . n  1, 2,  exist  o f e^°which  to  n K =  n  of one-electron  To a g i v e n p a r t i t i o n  independent  o r 0,  R  K  , . . .  2  The p o s s i b l e  N  1, 2 ,  can consist  of</jf*which c o r r e s p o n d  f o r w h i c h 1 ^ = ^ = . . .=N =2. are 0,  K  when n in  independent  possible  except of  the orthogonal  R  ,  functions  TT^N^  c a n be  orthogonalized. For are  suitable  building  blocks  introduced and defined  *N k K  =  < K*>-l/22: ( " l ) N  P  the group  functions  4*Nvk K  as  P  x  <  ^ .  >  <  v  .Kk ^ . . v  K X  r  (  1  )  r  K X <  ) - - -  2  r  K v (  N  K  )  '  (3-4) where  N =0 R  . . .  1, 2,  r  Group f u n c t i o n s  n  K  ; k = l , 2,  belonging  .. ' ^ N ^  to different  orthogonal  i n the strong  normalized  and mutually orthogonal /  The i j . . . l s  ^N ki' K  sense  N K  d  d N  antisymmetrized product _ /  SlN2-'.'% ' /nl\  N  1  \ A  !  N  2 ' - ' !  N  N  !  N1/2  (2N)1 / _ ^n \ 0  X  ' ^ ' *•*  subspaces  and w i t h i n  l l 2--d  7  K =  6  X  ' ' 2  N  sense:  (3-5)  functions —  P  ^  *Nli*N j /n^v KT» P  J  _  '( 3 - 6 )  2  5V '  =1,2,...(^, j = i , 2 , . . . ^ , . . . 1 = 1 , 2 , . . . ^ , c o n t a i n i n g one group f u n c t i o n from each o f t h e N form a complete  set such  T c a n be expanded  that  2N  subspaces,  the exact wavefunction of the  i n terms  K'  subspace  k l -  i-  system  =  are  every  i n the usual  V  of  them.  -15In  order  that  a l l the states  from t h e geminal ground s t a t e "simple Simple  excitation" excitation  ¥ by another Electron of  i n E q . (3-6) m a y b e d e r i v e d  Kapuy i n t r o d u c e s  and "electron  substitutes  transfer  the subspaces  simultaneously:  excitation to another, 5jT  . t"T  one o f the group f u n c t i o n s i n  transfers  *  -*• ijT  one e l e c t r o n  changing two group  . -*•  TINKI^D  functions  , *.. N  antisymmetrized product  with  t h e terms  elementary to  reach  K - l '  N  k  L+1'  X  excitations  1 1 < - < 1  functions  described  series  i n Eq.  (3-6)  g r o u p e d a c c o r d i n g t o t h e minimum number o f  the actual  ¥ = c<°>f  , k ^ l .  from one  Now t h e e x a c t w a v e f u n c t i o n ¥ c a n b e w r i t t e n a s a of  of  excitation".  b e l o n g i n g t o t h e same s u b s p a c e :  transfer  the ideas  (simple  + electron  transfer)  needed  term from the ground s t a t e  - g + 2CU>T<1>  ^  + Sc< >¥< > + ... Z , c < - > y < - . 2  2  2 N  2  2 N  +  (3-7) The they are  K  a n d N >2 a r e s t i l l  , Njr^r  R  may b e d e t e r m i n e d identified with  respectively,  i n principle, but for simplicity  the corresponding r  R  ^ and *  K  k  they  (1, 2, . . . N ) K  where  * ( l » 2 , . . . . N ) = - ( N ! ) l / 2 2. ( - i ) P r K k  unspecified so f a r ,  K  P  K  K X  (l)r  K X  (2) ...r  R v  (N ) K  , (3-8)  and where In  X<A<I.,<v a n d  '•'j^  K  E q . (3-7) i t i s t h e t e r m s  s u m m a t i o n s w h i c h may h a v e with  X,X,..,v=l,2,...n .  1«  A list  of the f i r s t  nonvanishing matrix  and second elements  o f them i s g i v e n b e l o w i n w h i c h o n l y t h e  2 )  -16f a c t o r s b e i n g c h a n g e d u n d e r e x c i t a t i o n a r e shown. A l l configurations  except  t h o s e due t o one s i m p l e e x c i t a t i o n may  have n o n v a n i s h i n g m a t r i x e l e m e n t s First  with  ^li .l« # >  summation: one *  simple  (l,2)  K l  excitation  4> (lr2)  -  Kk  for  a l l K and k  one  electron  (k^l);  transfer  (3-9) excitation  * (i,2)i|> (3,4) + r (l)i|> K l  L l  for second  K X  L l  (3-10)  a l l K, L ( K ^ L ) , X and 1;  summation:  two  simple  excitations  iJ> <i,2)!ji (3,4) Kl  Ll  •*  ^(1,2)^(3,4),  for  a l l K<L, k a n d l ( k , l ^ i ) ;  one  s i m p l e + one e l e c t r o n  ^  (2,3,4) ,  K l  (l,2)^  L l  (3,4)^  M l  (5,6)  (3-11)  transfer * r -(l)* KX  excitations L|l  (2,3,4)i|»  for  a l l K, L , M(K^L^M), X, 1 and m(m^i);  two  electron  iP  K ±  for  (1,2) t P  L l  transfer  (3,4)  -  (5,6), (3-12)  excitations  i|i (i,2,3,4), L l  a l l K, L ( K ^ L ) a n d 1;  ^(1,2)^(3,4)^(5,6) for  Min  (3-13) +  ^(1,3,4)^(2,5,6),  a l l K, L<M(K^L, M), 1 a n d m;  (3-14)  -17-  *  K l  (1.2)*  L l  (3,4)*  M l  (5,6)  ^(1,^(31*^(2,4,5,6),  *  f o r a l l K<L, M(M^K,L), X, X a n d m;  ^ ( 1 , 2 ) ^ ( 3 , 4 ) ^ ( 5 , 6 ) ^ ( 7 , 8 )  -  (3-15)  ^ ( 1 ) 1 ^ ( 2 , 3 , 4 ) ^ ( 5 ) ^ ( 6 , 7 , 8 ) , (3-16)  f o r a l l J<L, K<M(J,L^K,M), X/ k, X and m. In p r a c t i c a l determining finite  the b e s t p o s s i b l e geminal  t o terminate  Kapuy u s e d  electron  i n h i s work.  His results  Kapuy's  show t h a t t h e most  o f Y..  . come f r o m o n e -  Application  [2]  a p p l i e d h i s extended  perturbation Hamiltonian.  geminal  theory t o the  The c a l c u l a t i o n s a-ir separability,  f o l l o w i n g approximations:  involved second  order  t h e o r y a n d a G o e p p e r t - M a y e r and S k l a r Four  SCF o r b i t a l s were u s e d  t h e same o r b i t a l s were u s e d  o f which i s taken  to which a l l r e s u l t s  are referred.  product wavefunction  correlation  energy.  figurations  due t o two s i m p l e  as a s t a n d a r d  The  accounts  The t o t a l  as a s t a r t i n g  point  i n a complete CI c a l c u l a t i o n  by N e s b e t t h e r e s u l t  geminal  upfeto  •L X • • • Jt r a n s f e r c o n f i g u r a t i o n s (Eq. ( 3 - l 0 ) ) .  TT-electrons o f t r a n s - b u t a d i e n e .  and  ^ is a 11 • • • l  t h e s e r i e s o r use p e r t u r b a t i o n t h e o r y .  c o r r e c t i o n s t o the energy  Kapuy  the  product  Rayleigh-Schrfldinger perturbation theory  order  important  3-2  t h e s e t {r} o b t a i n e d by  one o f c o u r s e , b u t f o r s y s t e m s w i t h N l a r g e i t i s  necessary  second  calculations  (100%)  antisymmetrized  f o r 93.1% of the t o t a l  c o n t r i b u t i o n o f a l l conexcitations  i s 1.9%.  "One-electron  -18transfer" of  c o n f i g u r a t i o n s make a n i m p o r t a n t  6.2% m a k i n g t h e t o t a l  value. tions  of t h e i r 3-3  c o r r e l a t i o n 101.3% o f t h e s t a n d a r d  The i m p o r t a n c e o f " o n e - e l e c t r o n confirms  contribution  Kapuy's p r e v i o u s  transfer" configura-  estimate  [1] o f t h e m a g n i t u d e  c o n t r i b u t i o n i n t h e c a s e o f t h e Be atom.  A p p l i c a t i o n t o BH "One-electron  t r a n s f e r " c o n f i g u r a t i o n s have t h r e e  orbitals  f r o m one s u b s e t  and one s p i n o r b i t a l  subset.  In our case the subsets  transfer  labelled  equivalent  i n the extended  G+l ( g e m i n a l s p l u s  configurations).  c o n f i g u r a t i o n sfy^,<J> and 2  <f> 3  r  B, H.  geminal  one-electron  F o r our simple  t o a 7-term l i m i t e d  another  a r e K, K; N, N; B, H  These c o n f i g u r a t i o n s a r e i n c l u d e d calculations  from  spin  e x a m p l e , ^Q+I i  s  CI wavefunction with the  ( s e e E q s . (2-21) t o (2-23))  plus  <j> =  (KKHHNB) + (KKHHBN) ,  (3-17)  <J> =  (KKBBNH) +  (KKBBHN),  (3-18)  $  (NNBBKH) +  (NNBBHK),  (3-19)  (NNHHKB) + (NNHHBK).  (3-20)  4  5  =  c  D  4> = 7  Because o f t h e s m a l l b a s i s tions only  c a n be e a s i l y other  performed.  configurations  in  s e t s used, f u l l  CI c a l c u l a -  I n a d d i t i o n t o <j>^ t o <f>^,  the  are  <j> = 8  (KKBBHH) ,  (3-21)  <t> =  (NNBBHH) ,  (3-22)  9  <J> = (BBHHKN) + 10  (BBHHNK) .  (3-23)  -19-  These c o r r e s p o n d  to  "two-electron  extended geminal terminology. to ^ G I 2 +  (9  +  e m  i n a l s plus  transfer" configurations  In t h i s  case Y  one  and  two-electron  G+l  and  full  is  C I  in  equivalent  transfer  configurations). The the  r e s u l t s o f AGP,  four  E (AGP)  basis  ~ (CI) E  particular  sets •"- ^ s  t  basis  are i e  m  a  and is the  set  c a l c u l a t i o n s on  by not  Krauss  [19].)  to obtain  system but  an  to  example.  fraction  f i n Table  This  The set  f represents  The sets  are  and  little  BH.  (See  respectable  c o m p e n d i a by  However, t h e  the  of  i n atomic  units.  optimization  t o save computer t i m e , i s quite  for  "one-electron  energies  IV  difference  fraction  only  very  The  obtain  the  include  the  CI  compared Cade and  main p u r p o s e o f  This  sets  i s achieved  I i s over  0.9  by  fact  i n d i c a t e t h a t when one and  full  r e s u l t s i n Table [16,  the  with a l l four  CI  to Huo  this  energy  17],  that  G  +  i ,  work  for  basis  with  sets  u s e s more  including only  I show t h a t  [18]  the  becomes unmanageable  t r a n s f e r " c o n f i g u r a t i o n s , may  of Clementi  I.  improvement we  m  extended geminal wavefunction ^ electron  i n Table  e x c e l l e n t w a v e f u n c t i o n and  seems t o  extended b a s i s  c a l c u l a t i o n s with  s u p p o r t Kapuy's e x t e n d e d g e m i n a l t h e o r y  a better  used.  basis  i n order  from b a s i s  u  when we  configurations.  have been used  other  m  s e t ; and  Although small  energy  i  x  t h i s maximum.we o b t a i n transfer"  summarized  CI  the "one-  be  sufficient.  the  double-zeta  even w i t h c o n t r a c t i o n  (fixed  basis linear  -20-  TABLE I GEMINAL CALCULATIONS  A l l calculations R o f 2.329 a . u .  a r e c a r r i e d o u t a t an i n t e r n u c l e a r E n e r g i e s a r e i n atomic u n i t s .  Basis Set  I  S(H) ct  a  a  E  (AGP)  E  (G+1)  E  (CI)  f E a  b  distance  IV  II  III  1. 26  1.31  (1.0)  1.362  0. 47  0.50  0.52  0.507  -25. 07331  -25.07601  -25.06673  -25.10324  -25. 08589  -25.08541  -25.08971  -25.11575  -25. 08683  -25.08627  -25.09116  -25.11634  0. 930  0.916  0.941  0.955  ( E x p e r i m e n t a l ) = -25.29 a . u . Optimized Ref. [18].  f o r minimum E (  A  G  P  ),  except  the value i n parentheses.  -21-  coefficients is  [ 1 7 ] ) and w i t h o u t  f u r t h e r exponent  capable of g i v i n g s i g n i f i c a n t  zeta basis single-zeta compared  improvement o v e r a s i n g l e -  s e t . F o r example, a f u l l set gives  e c o n o m i c a l as w e l l  CI w i t h an  optimized  an e n e r g y o f -25.09034 a . u . f o r BH [20]  t o o u r -25.11634 a . u .  systems, c o n t r a c t e d  optimization,  T h i s means t h a t ,  double-zeta basis  f o r some  s e t s may be more  as capable o f g i v i n g lower  energies.  CHAPTER I V ONE-POINT CALCULATION OF k e 4-1  Theory The  approach used  i s t h a t o f P a r r and W h i t e -[3] w i t h  some o f t h e t e r m s r e d e f i n e d . The diatomic  Born-Oppenheimer e l e c t r o n i c  Hamiltonian  m o l e c u l e may be w r i t t e n i n t h e f o r m H = R~  t + R"  2  fora  ,  v,  1  (4-1)  where R i s t h e i n t e r n u c l e a r d i s t a n c e , and t and v a r e t h e k i n e t i c - e n e r g y and p o t e n t i a l - e n e r g y o p e r a t o r s confocal e l l i p t i c are  coordinates,  i n d e p e n d e n t o f R and t h e e i g e n v a l u e H¥  is  for unit  the conventional  R.  expressed  in  The t e r m s t and v  W(R) o f t h e e q u a t i o n  = W(R)Y  (4-2)  potential-energy  function f o r the  n u c l e a r m o t i o n o f t h e m o l e c u l e a s a f u n c t i o n o f R.  At the  equilibrium  will  internuclear separation the Hamiltonian H  where T— = t/R e '  z e  and V e  e  = e T  = v/R^. e  +  v  e  '  be  (4-3)  I f we i n t r o d u c e t h e  parameter y = R /R  = 1 + 3  e  (4-4)  we may w r i t e E q . (4-1); a s H = y  2  T  e  + u V . e  (4-5)  -23-  Dividing  Eq.  (4-5) by y we  obtain  H/y and  s u b s t i t u t i n g 1+3  for y  H/(l+3)= T According  t o Eq.  = u T  e  + V  e  may  W(R)/(l+3) where T as  g  i s the matrix  a perturbation  ,  e  (4-6)  gives  + 3T  (4-7) we  + V  e  = H  e  + 3T .  e  (4-7)  E  write  = W(R )  + 3T  e  of kinetic-energy  p a r a m e t e r we  ,  e  (4-8)  operator.  ¥ and W ( R ) / ( l + 8 )  expand  3  Taking as a  power s e r i e s i n 8 , + 8  Y =  + 8  2  ¥  ( 2 )  ...,  (4-9)  i  e  W(R)/(l+8')= oo  + 0^8 + u 2 8  0  2  +  ... .  (4-10)  Then 00 W(R)  = w  +  0  (w  S  k - 1  + wk)8k,  (4-11)  and w Taking W(R)  = W(R ).  0  (4-12)  e  t h e d e r i v a t i v e o f W(R)  = dW(R)/dR = d W ( R ) / d 8  .  with d8/dR  respect =  t o R,  (-R /R )2» 2  e  k8  (" .i  +  k  w k  ) »  k=l (4-13) we  find  t h a t when R=R . / e W(RJ = (-R /R )(L0 e e e 0 2  for  any w a v e f u n c t i o n  therefore, The  u  =  0  satisfying  e  + oO = 0, 1 the v i r i a l  (4-14) theorem;  -w-^.  force constants k  N  o  v  =  p r e d i c t e d by t h i s  p o t e n t i a l are  ( d W ( R ) / d R ) = 2.(o) +w )/R^ , 2  2  1  2  (4-15)  -24-  l  =  e  (d W(R)/dR ) = 3  [-6(o) +u> ) - ^ ( u ^ + o ^ ) ]/R  3  2  = m  e  =  (d W(R)/dR ) ,  =  [24(o) +a) )  4  Scaling  these force  e  k ]/R  2  3  e  theorem.  can c a l c u l a t e T .  satisfies  the v i r i a l  e  i s then p o s s i b l e  -36R2  kJ/R  4  ,  (4-16)  .  Used t o O b t a i n ¥ a t  i s performed  the v i r i a l  Let  -6R  (4-17)  c o n s t a n t s we r e q u i r e  therefore  it  3  the k i n e t i c -  only.  e  Procedure  The s c a l i n g satisfy  2  -12R| l  4  energy m a t r i x a t R 4-2  [-6(o) +w )  4  3  To c a l c u l a t e  ,  3  3  R  e  t o make t h e w a v e f u n c t i o n I n d o i n g s o we o b t a i n  R  £  and  A l s o , when t h e w a v e f u n c t i o n  t h e o r e m , E q . (4-14) i s s a t i s f i e d to calculate  and  the force constants.  us r e p r e s e n t t h e u n s e a l e d and s c a l e d w a v e f u n c t i o n o f  an N - e l e c t r o n d i a t o m i c m o l e c u l e by, Y  =  £ l ' E2'  Y (  £N'  R  (4-18)  )  and %  =  *( £i» n  n  E2'  ••• E N ' N  n  R  )  '  (4-19)  where r ^ i s t h e p o s i t i o n v e c t o r o f t h e > I t h e l e c t r o n , the  i n t e r n u c l e a r d i s t a n c e and n i s some a r b i t r a r y  R is  scale  factor. For mation  d i a t o m i c i m o l e c u l e s i n t h e Born-Oppenheimer the t o t a l  function ¥  energy  associated with the scaled  approxiwave-  i s [21] E(n,R) = n  2  T(l,nR) + n (l,nR) V  ,  (4-20)  -25where T(l nR)  = Tfr^, r  V(l,nR)  = V(rjy r  f  2  ,  ...  r ^ , nR) ,  ,  • • •"  (4-21)  and  If  we l e t o u r u n s e a l e d  distance  p=nR,  then  w a v e f u n c t i o n have  the unsealed E = T(l,p)  while  the scaled  energy  Let  n = 1 + X, E  internuclear  is  + V(l,p),  T(l,p)  2  + 2XT(l,p)  = E + X[2T(l,p)  (4-23)  + nV(l,p).  + X  T(l,p)  2  + V(l,p)]  = E + X[E + T ( l , p ) =  (4-22)  (4-24)  then  = T(l,p)  n  energy  an  > •  is  = n  Ev  nR  + X  + X  2  + V(l p) f  + XV(l,p)  T(l,p)  T(l,p) ]  2  (1+X)E + X ( 1 + X ) T ( 1 , p ) .  (4-25)  Therefore E /(1+X) n  Expanding  E /(1+X) n  (  0  )  E /(l+\) n  scaled  energy  E^ At  X=0,  WQ=E  (4-26)  and Y as a power s e r i e s  Y = y  the  = E + XT(l,p).  + XT  ( 1 )  = W  + XW + X W  Q  + X Y 2  ...  ( 2 )  2  X  i n X,  2  ,  (4-27)  . . . ,  (4-28)  c a n be w r i t t e n  = WQ + X ( W  Q  + Wj_) + X ( W 2  x  + W ) + 2  . . .  .  (4-29)  so CO  E To o b t a i n  = E +  n  2 . Xk (w  k-1  + W ) R  .  (4-30)  optimum s c a l i n g dE /dX n  = 0  .  (4-31)  -26Therefore co_  kX*'  k=l  1  (W. . + W. ) = 0 . ~ K  l  (4-32)  k  Optimum X i s o b t a i n e d by v a r y i n g X u n t i l Eq. (4-32) i s satisfied.  Now we can generate the s c a l e d wavefunction and  from i t the k i n e t i c - e n e r g y m a t r i x a t R . e  4-3  S e l e c t i o n o f Wavefunction The extended b a s i s s e t o f Ohno  starting point. boron  [5] was s e l e c t e d as a  The S l a t e r type o r b i t a l s used a r e :  ( l s , 2s, P Q * 2  B  2p  and 2p_) and on hydrogen  +  on  (ls ). A l l H  c o n f i g u r a t i o n s , i n which two e l e c t r o n s are r e t a i n e d i n the  ls_, o r b i t a l , are taken i n t o account t o o b t a i n a t h i r t e e n -  term s e t o f b a s i c f u n c t i o n s .  (See Table I I . )  To check  computer i n p u t a C I c a l c u l a t i o n i d e n t i c a l t o Ohno's case (b) [5],  except t h a t our l s  H  o r b i t a l was not o r t h o g o n a l i z e d ,  was performed and the same ground s t a t e energy  resulted.  (See Table I I I . ) Using the t o t a l e l e c t r o n i c energy as a c r i t e r i o n , the o r b i t a l exponents were o p t i m i z e d through one c y c l e i n the following order: ing  l s , 2s, 2pQ, (2p ) and l s . B  +  For r e s u l t -  H  o r b i t a l exponents and energy see Table I V . The improve-  ment i n energy on going from ( Q H N 0 ) e  t  o  E  (OPT)  x  s  n  o  t  t  o  °  significant. Now the I s o r b i t a l on boron was s p l i t 1st, and l s ^ .  i n t o two o r b i t a l s ,  The number o f S l a t e r determinants i n each term  -27-  TABLE  II  13-TERM WAVEFUNCTION  The l s  B  omitted  ls  core,  B  i n the  following  1.  (2s  2s* p  2.  (2s  2s  3.  (P  P~  4.  (2s Js p  5.  (P  6.  (2s  V~  7.  (2s  2s" p  8.  (P  P^" P  9.  (ls  10.  (2s  0  Ts  (P  0  Ts +  13.  (p  +  Is >  + (P  Ts" )  +  p  p  H  p ( l s  P^ P_  H  s  2s  pj  +  (P  Py" P_  p^)  +  +  2? p  pj R  +  (ls  pi)  +  +  2s)  Q  Ts  +  TS +  H  P7  }  pp  £T)  (P H  p_  H  2"s p_ p"""")  0  p_ p"7)  jT~) + ( l s (p  ls" )  H  p_ p~)  + (2s Ts* +  p"p  H  p_  PQ" P+ P~D PJ  H  (2s  Is p pj  0  H  2s" l s  0  (2s  +  +  (p  l  +  Q  (ls  P^"  0  p~j  pj  +  (2s "2s* l s  +  H  r  +  H  H  +  H  (p  (2s l s "  Ts ) H  s  PQ" p  + 12.  description.  H  Q  1S  +  11.  determinants,  p"^)  Q  R  2  H  the  ls„ Is) H n is TF )  0  0  occuring in a l l  H  2~s p_ p]"")  p_ p^")  h  <  l s  Po" P  H  P+  }  TABLE I I I DATA FROM OHNO  Orbital  Exponents  Boron ls  4.70136  B  2s 2p  Hydrogen l s  H  1.0038  1.30092 Q  1.30092  2p+ 2p_  •»  I n t e r n u c l e a r D i s t a n c e R = 2.32911 a . u . Total Electronic E  a  E n e r g y E( HNO) 0  ( E x p e r i m e n t a l ) = -25.29 a . u .  Ref.  [18].  a  =  "25.11045 a . u .  TABLE IV OPTIMIZED  Orbital  WAVEFUNCTION  Exponents  Boron ls  B  2s  Hydrogen  4.64915  l s  H  1.14662  1.33334  2p  Q  1.44977  2p  +  1.36771  2p_ I n t e r n u c l e a r D i s t a n c e R = 2.32911 a . u . Total Electronic E  a  Energy E (  0 p T  )  ( E x p e r i m e n t a l ) = -25.29 a . u . Ref. [18].  = -25.11980 a . u . a  -30of  the wavefunction  i n Table I I i s doubled.  o r i g i n a l determinant  the spin o r b i t a l  r e p l a c e d by t h e s p i n o r b i t a l avefunction  an a p p r o x i m a t i o n  values of the o r b i t a l ls  f i  are  B  V  (or  •  exponents  of the s p l i t  is  Starting  core  orbitals  These v a l u e s  by S i l v e r m a n , P l a t a s and  o f Ohno  o r b i t a l s other than the s p l i t energy  E^  s c  j  out using  this  [18] u s e d  (See T a b l e I I I . ) f o r  core o r b i t a l s .  of this  was -25.13333 a . u .  an e x t e n d e d  B  core wavefunction while r e t a i n i n g the  orbital  electronic  Ti*  T h i s makes o u r  t o a 39-term C I .  [ 2 2 ] . A c a l c u l a t i o n was c a r r i e d  original  Huo  Ts  exponents  c l o s e t o those suggested  13-term s p l i t  C  i s r e p l a c e d by  and l S g were 5.5 and 3.9 r e s p e c t i v e l y .  Matsen  ¥g  B  ls'g ( o r lSg) w h i l e t h e s p i n o r b i t a l  the s p i n o r b i t a l  w  l s  In each  split  core  The  total  wavefunction  T h i s i s q u i t e good.  Cade and  an i n t e r n u c l e a r d i s t a n c e o f 2.305 a . u . and  basis  set of Slater  o u t a SCF H a r t r e e - F o c k  type f u n c t i o n s t o c a r r y  calculation resulting  i n an e n e r g y o f  -25.13147 a . u . The  energy  exponents 5.46404  improved  t o -2 5.134 07 8 a . u . when t h e o r b i t a l  o f t h e l s and l s ^ o r b i t a l s were o p t i m i z e d t o B  and 3.87451 r e s p e c t i v e l y .  optimized together, that both exponents wavefunction g e n e r a l i t y we  Both  exponents  i s when optimum e n e r g y  were c h a n g e d by t h e same f a c t o r .  VQ Q S  i s quite suitable  start  were  was  reached  This  f o r our purpose  new  but f o r  a t t h e b e g i n n i n g a g a i n u s i n g t h e more  -31-  common i n t e r n u c l e a r d i s t a n c e o f 2.329 a.u. e x p o n e n t s by  Slater's  orbitals  l s  and  The and  Allen  equal  lower  a YB  than (TOSC)* E  i s lower individual  energy  of the  The  B  e  n  d  e  r  g°°d.  Harrison using  Harrison  a  n  a.u.  with  the  a ground s t a t e  Davidson  d  performed and  an  t h e sum  o f the ground  atoms i n v o l v e d , and  thus  [19]  functions resulting calculation energy  a  1123-term  internuclear  t o o b t a i n t h e v a l u e -25.26214  e x p o n e n t s and  term  state  a.u.  energies  of  i n c l u d e s some b i n d i n g  and  Force  rediagonalized  Constant  to ensure  elements i s n e g l i g i b l e , The  original  e n e r g i e s f o r the  i s g i v e n i n T a b l e VI w h i l e  coefficients  eigenvalue matrix  generated.  v  At present o n l y the  [24] y i e l d s  13-term w a v e f u n c t i o n s  Scaling  r  V.)  molecule.  corresponding 4-4  e  (See T a b l e  o f -25.1426 a.u.  d i s t a n c e o f 2.50  A summary o f o r b i t a l various  v  core  i n t e r n u c l e a r distance of  using natural o r b i t a l s  than  s  split  c a l c u l a t i o n using Gaussian-lobe  d i s t a n c e o f 2.336 a.u.  the  an  energy  Davidson  CI c a l c u l a t i o n  This  (TOSC)' ^  t o -25.1455 a.u.  o f B e n d e r and  f o r the  a 1 3 - t e r m VBCI c a l c u l a t i o n  t o o b t a i n , an  internuclear  energy  E  f u n c t i o n s and  a l s o performed an  energy,  [18] p e r f o r m e d  2.336 a.u.  [23], except  orbital  , then optimize a l l exponents.  resulting  Gaussian-lobe  and  H  rules  and  are given i n Appendix I I . Results  of the wavefunction  t h a t t h e v a l u e o f any then  the  t h e new  ground s t a t e  ^ipQgQ  was  o f f diagonal  eigenvector matrix energy  remained  was  the  -32-  TABLE V WAVEFUNCTION 4* TOSC  Order o f exponent o p t i m i z a t i o n : and Is,' 'BOrbital Starting  (2p ), +  1  S H  /  2  p , Q  2  S  '  1  S  B  Exponents Optimized  Values  Values  5.5  5.47771  3.9  3.84914  1.3  1.30378  1.3  1.33424  1.3  1.32475  2p_  1.3  II  ls  1.1  Is B  2s 2  P0  R  Internuclear  (arbitrary  selection)  1.21170  D i s t a n c e R = 2.329 a . u .  E  (Split  Core + T o t a l O p t i m i z a t i o n )  E  (Experimental)  = -25.29 a . u .  =  E ^ Q S C ) = -25.14769 a.u,  -33-  TABLE V I ORBITAL EXPONENTS FOR  ^OHNO  ''.OPT.  4.70136  K  2s  13-TERM WAVEFUNCTIONS  ^OSC  Y  TOSC  4.64915 5.46406  5.47771  3.87451  3.84914  1.30092  1.33334  1.30092  1.30378  2p  0  1.30092  1.44977  1.30092  1.33424  2p  ±  1.30092  1.36771  1.30092  1.32475  1.00038  1.14662  1.00038  1.21170  1 S  H  -E E  25.11045 a . u . (Experimental)  25.11980 a . u .  = -25.29 a . u .  25.134078 a . u .  25.14769 a . u .  -34same h o w e v e r .  Now  t h e k i n e t i c - e n e r g y o p e r a t o r and  e i g e n v e c t o r s were u s e d The  eigenvalues  perturbation results The  to calculate  the k i n e t i c - e n e r g y matrix.  a n d ' k i n e t i c - e n e r g y m a t r i x were t h e n u s e d  computer program t o c a l c u l a t e  are given i n Table  where V and  The  = 0.9993749  T are the p o t e n t i a l  Hamiltonian  (4-25).  t o see  ,  (4-33)  k i n e t i c - e n e r g y of  i s constructed according  i s diagonalized to obtain  V^OSC'  n  the  check  A  r j  T + V)c'=  = £'(2  case, Now  to the  lowest  energy,  the u s u a l k i n e t i c  and  (4-34)  c o n s t a n t s may  (4-17).  a  r  e  u s e o  T  t e r m s o f t h e p a r a m e t e r 8 we  Eq.  w h i l e T and  The  be  can  V are, i n  -  t  o  obtain  the  Expanding the wavefunction calculate  calculated  results  Y-posC  potential-energy matrices.  the e i g e n v e c t o r s of * 0SC  scaled kinetic-energy matrix.  force  was  0  where c ' i s t h e e i g e n v e c t o r o f t h e s c a l e d w a v e f u n c t i o n  this  to  that dE /d  corresponding  the  "^osc*  matrix  This matrix  and  e i g e n v e c t o r s of the s c a l e d wavefunction out  a  [21]  ground s t a t e wavefunction A new  in a  VII.  rf = -V/2T  carried  t o W^.  v a l u e o f n i s v e r y c l o s e t o t h a t o b t a i n e d by  s i m p l e r method  Eq.  new  t o w^.  a c c o r d i n g t o Eq.  are given i n Table V I I I .  in  The (4-15) t o  -35-  TABLE V I I SCALING RESULTS  k  W  k k  1  25.179196  2  -1.5153996 x 1 0 "  3  ( k-l w  +  W  k>  0.0315011 1  50.0553121  1.7958395 x IO""  0.0841320  1  4  -2.2055059 x I O "  1  -0.16386656  5  -2.6166944 x I O '  1  0.2055942  6  -2.9812845 x I O "  1  -0.2187541  7  3.1521269 x I O "  1  0.1195897  X  Q p T  n = R  g  (optimized  X) = -0.0006293  0.9993707 = R/  n  = 2.3304666 a . u .  -36-  TABLE V I I I  FORCE CONSTANTS  k  u. k  1  25.147705  2  -1.5159298 x 1 0 "  1  3  1.7968714 x 1 0 "  1  4  -2.2068003 x 1 0 "  1  5  2.6180703 x I O  6  -2.9820331 x I O "  1  7  3.1509448 x I O "  1  Calculated k  e  l m  a  - 1  Experimental  9.204849  0.1958  a  -23.712046  -0.53l9  a  e  e  61.049772  1.237.3  a  From t h e u n p u b l i s h e d r e s u l t s o f M u l l i k e n and Ramsy a s r e p o r t e d by Cade and Huo [ 1 8 ] .  -37Using  experimental  d a t a we s e e t h a t  io, + u) = 1/2 .2 k , 1 2 e = (0.5) (2.336 a . u . ) 0  e  = The  experimental  calculated basis  values  2  (0.19590) ,  0.5345 .  v a l u e o f O J i s -24.756. 2  for  set calculations  (4-35) I t seems t h a t t h e  and 0J3 a r e f a r t o o s m a l l . a r e doomed t o f a i l .  Good  Limited  results  m i g h t be o b t a i n e d u s i n g a much l a r g e r b a s i s s e t , s u c h as t h a t o f B e n d e r and D a v i d s o n [ 2 4 ] ,  CHAPTER V V I R I A L SCALING 5-1  T h o r h a l l s s o n a n d Chong's A p p r o a c h T h o r h a l l s s o n and Chong  of  expanding  force  the v i r i a l  [6] i n v e s t i g a t e d  the p o s s i b i l i t y  f o r c e F i n powers o f R.  The  virial  i s d e f i n e d as F(R)  = d E ( R ) / d R = -(2T+V)/R,  where V and T a r e t h e p o t e n t i a l  and k i n e t i c  equation holds only f o r exact wavefunctions, the v i r i a l  (5-1) energy.  This  b u t by a p p l y i n g  t h e o r e m t o an a p p r o x i m a t e w a v e f u n c t i o n  one c a n  make d E ( R ) / d R = F ( R ) . Thorhallsson  and Chong u s e d  a 10-term v a l e n c e  c o n f i g u r a t i o n - i n t e r a c t i o n wavefunction of  LiH.  f o r t h e ground  A t each o f the t h r e e R v a l u e s the o r b i t a l  were r o u g h l y o p t i m i z e d and s c a l i n g was p e r f o r m e d . of  finding  according  the c o r r e c t value of the s c a l i n g  ,P' = 3V(l,p)/3p  P derivatives V  p  were d e t e r m i n e d x.  exponents The method  t o LOwdin [21]  where p = nR, V  k +  state  parameter n i s  n = - [ V ( l , p ) + p V / ( l , p ) ] / [ 2 T ( l , p ) + pT  R  bond  v-  and T from  p  (l,p)],  (5-2)  P and Tp = 3 T ( l , p ) / 3 p .  The  t>  a r e t h e most d i f f i c u l t  t o e v a l u a t e and  the unsealed wavefunction  a t R^-x and  -39In F  t h i s p r e s e n t work we  i n R a n d 1/R  derived  find  All  scaled  f o r n derived  e x p a n s i o n s o f E and  and u n s e a l e d w a v e f u n c t i o n s  from our w a v e f u n c t i o n ¥  expression we  using  investigate  .  S i n c e Eq.  (5-2) i s t h e  from the c o n d i t i o n t h a t  3E(n,R)/3ri = 0,  i t more c o n v e n i e n t t o p e r f o r m t h e s c a l i n g  orbital  exponents  were m u l t i p l i e d  v a l u e s o f n, and t h e r e s u l t i n g  at fixed  by t h r e e d i f f e r e n t  e n e r g i e s when f i t t e d  to a  p a r a b o l a g i v e a minimum c o r r e s p o n d i n g t o t h e c o r r e c t of  Wavefunctions Three  1/R-^  - 1/R  and Data  internuclear distances are selected 2  = 1/1*2 ~  1 / / R  3*  T  h  e  w  each  of the s p l i t  described  f  u  n  c  t  i °  n  ^TOSC'  that  described  p o i n t , and t h e o r b i t a l  Two  above, t h e o t h e r i n v o l v e d  was p e r f o r m e d  individually  sets of calculations  taking  scaling  the mentioned  them.  The  scaling  a t e a c h i n t e r n u c l e a r d i s t a n c e by c a l c u l a t i n g  1  e n e r g y o f t h e w a v e f u n c t i o n as w e l l a s t h e e n e r g i e s g i v e n  when a l l o f t h e o r b i t a l 1±0.01.  orbital  exponents  Then a p a r a b o l i c  exponents  wavefunction. is  e  such  o u t , one s e t i n v o l v e d d a t a f r o m t h e w a v e f u n c t i o n s  w a v e f u n c t i o n s one s t e p f u r t h e r ,  of  v  c o r e a r e o p t i m i z e d once  internuclear distance.  were c a r r i e d  the  a  T a b l e V, i s u s e d a s a s t a r t i n g  exponents at  value  n.  5-2  in  R.  were m u l t i p l i e d  from the s c a l e d  g i v e n i n T a b l e s IX and X.  factor  f i t on e n e r g y g i v e s t h e s c a l e d  a t l o w e s t e n e r g y and t h e r e f o r e  Data  by a  the scaled  and u n s e a l e d w a v e f u n c t i o n s  -40-  TABLE IX UNSCALED DATA  *1 R  2.329 a . u .  *2  *  3  2.433688 a . u .  2.548230 a . u .  E  -25.147692 a . u .  -25.149526 a . u .  -25.148820 a . u .  V  -50.344143 a . u .  -50.373239 a . u .  -50.360489 a . u .  T  25.196451 a . u .  25.223713 a . u .  25.211669 a . u .  y  0.252673 a . u .  0.224906 a . u .  0.190736 a . u .  ls exponent  5.465763  5.479920  5.479919  ls"exponent  3.865785  3.860518  3.860635  /  F(R) E  -0.020936 a . u .  ( E x p e r i m e n t a l ) = -25.29 a . u .  y = d i p o l e moment  -0.03483 a . u .  -0.024664 a . u .  -41-  TABLE X SCALED DATA  f  l  *2  *3  R  2.329 a . u .  2.433688 a . u .  2.548230 a . u .  Scale Factor  1.00043246  0.99877845  1.00000008  E  -25.147697 a . u .  -25.149563 a . u .  -25.148928 a . u .  V  -50.365648 a . u .  -50.312475 a . u .  -50.258097 a . u .  T  25.217952 a . u .  25.162912 a . u .  25.109169 a . u .  y  0.252367 a . u .  0.225838 a . u .  0.192422 a . u .  F  -0.03016574 a . u . Orbital  -0.00548509 a . u .  +0.01560259 a . u .  Exponents  lSg  5.468127  5.473226  5.468635  lSg  3.867456  3.855802  3.852686  2s  1.304343  1.302187  1.301095  2p  Q  1.334822  1.3326151  1.331498  2p  +  1.325324  1.323133  1.3220237  2p_ ls„  H  1.212213  H  1.210209  n  1.209194  -42-  5-3  Numerical The  t h r e e e n e r g i e s and  are f i t t e d  to a q u i n t i c  the unsealed unsealed  Analysis  polynomial.  wavefunctions  data  the q u i n t i c  f o r c e s from  t o be  are too  fitted  the s c a l e d  The  f i t i s given i n Table  forces calculated  irregular  to a quintic  to permit  polynomial.  The  from  the  Data  from  XI.  S e v e r a l d i f f e r e n t p a r a b o l i c models a r e used f o r c e s or energies of both  wavefunctions  t h e s c a l e d and  to f i t v i r i a l  unsealed  wavefunctions.  v a r i o u s p a r a b o l i c models a r e d e s c r i b e d below.  Model A E(R)  = A + BR  + CR  (5-3)  F(R)  = dE/dR = B + 2CR  2  (5-4)  R  e  = -B/2C  (5-5)  k  e  = 2C  (5-6)  Model B E(p) where p = 1/R and  and  p  = A + B(p-p ) + C ( p - p ) 0  Q  = 1/R «  0  ,  Let X = E(p -h),  2  0  Z = E ( p + h ) , where h = P^P-^  =  P  Q  A = Y  2  3~ 2* P  ,  (5-7) Y = E(p ), Q  Then, (5-8)  B =  [Z-X]/2h  ,  (5-9)  C =  [X+Z-2Y]/[2h ]  ,  2  (5-10)  and F(R) Therefore  p -p e  = -[B+2C(p-  = -B/2C 0  and  k  )]/R . 2  P ( )  = 2Cp e  4  K e  .  (5-11)  -43-  TABLE XI  QUINTIC POLYNOMIAL F I T  Coefficients -25.149634 -0.934849 x 1 0 " 0.993244  x 10"  8  1  -0.131321 0.129005 0.255402 Calculated E  -25.149634 a.u.  e R  e  k e 1  a  e'  Experimental -25.29 a . u .  2.459919 a.u.  2.336 a . u .  0.198649  a.u.  0.1958 a . u .  a  -0.787925 a.u.  -0.5319 a . u .  a  a  From t h e u n p u b l i s h e d r e s u l t s o f M u l l i k e n and Ramsy a s r e p o r t e d by Cade and Huo [ 1 8 ] .  -44-  Model C F(R) = A + BR + C R  (5-12)  2  A t F=0, R  =  g  [-B±/B^ - 4ACJ/2C  ,  (5-13)  d F ( R ) / d R = B+2CR ,  (5-14)  and k  = +  [B+2CR 1 = + [±/B^ - 4 A C ] .  Therefore, take the p o s i t i v e  (5-15)  root.  Model D F(p) = A + B(p-p ) 0  where p = 1/R a n d p and Z = F ( p + h ) Q  B =  = 1/R  Q  2  .  where h = p2~ i p  [ Z - X ] / 2 h and C = p -  + C(p-  P ( )  )  2  ,  Let X = F(p -h), Q  =  p  [X+Z-2Y]/2h . 2  3~ 2* p  T  n  e  n  (5-16) Y =F(p ) Q  A = Y,  A t F ( R ) = 0, ,  (5-17)  ,2 d F ( R ) / d R = - [ B + 2 C ( p - p ) ]/R'.  (5-18)  e  p  0  = [-B±/B^ - 4AC]'/2C  and 0  At  o  e  , k  = -p  2  [B+(-B±/B^ - 4 A C ) ] ,  = -p  2  [±/B^ - 4 A C ] ,  t h e r e f o r e , take negative  root.  Table X I I gives t h e r e s u l t s o f each p a r a b o l i c calculation cubic  (5-19)  model  f o r b o t h s c a l e d and u n s e a l e d w a v e f u n c t i o n s .  force constant  l  e  calculated  from t h e p a r a b o l i c  The  models,  where p o s s i b l e , was v e r y p o o r e x c e p t f o r s c a l e d M o d e l C w h i c h gave t h e v a l u e  -0.471199.  -45-  TABLE X I I PARABOLIC MODEL RESULTS  Model  k  e  E  e  (a.u.)  R  e  (a.u.)  A unsealed  0. 216050  -25.149615  2.462430  A scaled  0. 213184  -25.149667  2.464954  B unsealed  0. 203903  -25.149589  2.458223  B scaled  0. 199896  -25.149639  2.460911  C unsealed  0. 282013  2.669423  C scaled  0. 198469  2.460473  D unsealed  0. 229515  2.689602  D scaled  0. 196769  2.460718  Quintic  0. 198649  -25.149634  2.459919  Experimental  0.1958  -25.29  2.336  -46The  quintic  f i t results  are used  as a s t a n d a r d  comparing the v a r i o u s p a r a b o l i c models. e n e r g i e s and polynomial Table  virial  from  the  fits  i n R to the  p r e d i c t e d by v a r i o u s m o d e l s .  Q  d i p o l e moments a r e c a l c u l a t e d  Virial  s c a l e d and  scaling  has  the r e s u l t s  than  the case  expansions  R l e a d s t o an  of going  from  Thus a F u e s p o t e n t i a l oscillator Borkmah  potential.  [25].  the q u i n t i c  The  from  i n an  Chong  o f energy  different  simple p a r a b o l i c  data. improved v a l u e  importance  o f T h o r h a l l s s o n and  Parabolic rather  unsealed  resulted  a l l p a r a b o l i c models.  gives  quintic  X I I I a l s o g i v e s t h e d i p o l e moments a t t h e s e The  in  R  f o r the d i f f e r e n t  R. e  forces calculated  Table XIII  for  of s c a l i n g  or v i r i a l  force in  for k , Q  b e t t e r than  This supports  1/R  except  an  in  D.  harmonic  the approach o f P a r r  Model C s c a l e d g i v e s the b e s t agreement  polynomial  results  but  the value  for 1  factor  calculated  g r e a t e r than  Thomson and  used given  Dalby  d i p o l e moments a r e t o o two,  [26]  the e x p e r i m e n t a l  i s .4997 ± 0.08  s m a l l by  seems  a  value given  to  i n t h i s work and  other sources  i n Table  XIV.  wavefunctions  by  a.u.  A summary o f e n e r g i e s c o r r e s p o n d i n g from  and  with  overly fortunate. The  e  confirms  s c a l e d model C t o s c a l e d m o d e l t o be  k  [6].  improved v a l u e  appears  for  wavefunctions is  TABLE  XIII  COMPARISON OF DATA  Quintic Polynomial Models  R p r e d i c t e d by v a r i o u s models  Parabolic F i t  e  E  (a.u.)  F(R)  (a.u.) y  (u) (a.u.)  A (u) A (s)  2.462430 2.464954  -25.149633 -25.149631  -0.000496 -0.000990  0.216710  B (u) B (s)  2.458223 2.460911  -25.149634 -25.149634  +0.000338 -0.000197  0.217926  Ci (u) C (s)  2.669423 2.460473  -25.149634 -25.149634  -0.053673 -0.000110  0.150276  D (u) D (s)  2.689602 2.460718  -25.143994 -25.149634  -0.066635 -0.000158  0.143108  Data P r e d i c t e d  (s) (a.u.)  0.217178  0.218318  0.218441  0.218372  by Q u i n t i c P o l y n o m i a l  2.459919 ) = unsealed s) = s c a l e d y = d i p o l e moment  u  y  -25.149634  0.0  0.217430  0.218590  -48-  TABLE XIV BH  WAVEFUNCTIONS  Wavefunction  CI  ( F r a g a a n d R a n s i l [20])  R  (a.u.)  2. 3 2 9  a  E  (a.u.)  -25.09034  1 3 - t e r m C I (Ohno [5])  2. 32911  -25.11018  10-term CI (Table I : i v )  2. 329  -25.11634  1 3 - t e r m C I ( T a b l e IV)  2. 32911  -25.11980  Hartree-Fock  2. 305*  -25.13147  VB ' ( H a r r i s o n [19])  2. 50*  -25.1455  13-term s p l i t  c o r e CI ( T a b l e V)  2. 329  -25.14769  13-term  c o r e CI ( T a b l e XI)  2. 460*  -25.14963  1 1 2 3 - t e r m C I (Bender a n d D a v i d s o n [ 2 4 ] )  2. 336  -25.26214  Experimental  2. 3 3 6  split  (Cade a n d Huo [18])  * Calculated R a b  c  b  -25.29  e  E x p e r i m e n t a l v a l u e r e p o r t e d by H e r z b e r g [ 2 7 ] . 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Van N o s t r a n d  -51APPENDIX I SAMPLE GEMINAL The  following description deals with basis setI I ,  (See T a b l e to  I.) b u t most o f t h e o p e r a t i o n s i n v o l v e d a l s o  the other basis The  p a r a m e t e r a was  optimized.  product  and 1.34.  a  energies  In t h i s  case  A t e a c h o f t h e 5(H) t h e  The f o l l o w i n g p r o c e d u r e  was  £ ( H ) , were u s e d  interaction wavefunction possible  Slater-type orbitals,  the exponents o f Clementi  the Is o r b i t a l  i n a fortran  used,  Diatom*.  t h e o n l y p u r p o s e was  A  [15]  value  fictitious  to calculate a l l  integrals.  o n e - e l e c t r o n a n d t h e 55 d i f f e r e n t  i n t e g r a l s were t h e n  those  language c o n f i g u r a t i o n -  computer program c a l l e d was  and R a i m o n d i  on h y d r o g e n has some s e l e c t e d  o v e r l a p , one and t w o - e l e c t r o n  different  *  1.30  the four s i n g l e - z e t a  on b o r o n h a v i n g  arrays  geminal  from  o u t a t e a c h o f t h e f o u r £(H) v a l u e s .  First  for  p was d e t e r m i n e d  a t three d i f f e r e n t values of £(H).  t h e v a l u e s were 1.26,  while  A G  f i t t o antisymmetrized  calculated  apply  sets.  v a l u e o f £(H) f o r *  parabolic  carried  CALCULATION  s t o r e d a s two a n d f o u r  The 10  two-electron dimensional  respectively.  D r . D. P. Chong, D e p a r t m e n t o f C h e m i s t r y , of B r i t i s h Columbia.  University  -52-  The  next  s t e p was t o S c h m i d t o r t h o g o n a l i z e s_ t o k, s = N[sf - S k ] ,  where N = l / ^ l - S k: by  and S i s t h e o v e r l a p i n t e g r a l  2  S = /k (1) s j ( l ) d T ^ . matrix  (1-1) between £ a n d  T h i s o r t h o g o n a l i z a t i o n was c a r r i e d p u t  multiplication, (k,  s , a , h) =  (k, £ , o, h ) A ,  (1-2)  where  A =  1 0 0 0  (-NS) N 0 0  0 0  0 0 0 1  1 0  (1-3)  N = 1.02242, a n d S = 0.20827. The  r e q u i r e d h y d r i d i z a t i o n was p e r f o r m e d w i t h m a t r i x B, (k,  b , n , h) =  (k, s , a , h ) B ,  (1-4)  where r  B =  1 0 0 0  0 fl ([11--ca^ )) / 0 1  2  /  2  9  1  0 (l-a ) / -a 0  2  1  2  0 o 0 1  S y m m e t r i c o r t h o g o n o r m a l i z a t i o n was c a r r i e d  (1-5)  out using  m a t r i x C, (K, B, N, H) = where C i s t h e m a t r i x integrals.  A~l/  (k, b , n , h ) C ,  (1-6)  and A i s t h e m a t r i x o f o v e r l a p  2  I f U i s the matrix  t h a t d i a g o n a l i z e s A,  U+AU = X ,  (1-7)  then UX" /^ 1  = A"" / . 1  2  (1-8)  -53By  these  alternative  t h r e e t r a n s f o r m a t i o n s we  have i n t r o d u c e d  an  basis,  = 2  r  ±  r' T  ,  (1-9)  j where T j ^ i s a m a t r i x this  new  element of the m a t r i x product  b a s i s the corresponding  are r e l a t e d  one  and  ABC.  two-electron  In  integrals  by: <r |h|r > i  j  <r.r.|g|r,r > = i ^ k s  = 2  ^>"\ 7 t  T  J i  <  r r  l  h  l  r  s  >  T  s j  '  (  I  "  h and  a r e t h e one  A  two-electron operators respectively.  computer program y i e l d e d w h i c h were t h e n energy.  Various  used  the completely  i n the geminal  Figure  sets of i n t e g r a l s ,  corresponding  E  E  (AGP) (CI)*  of  w  completely  e  T  r  ^  e  e  then c o e  used  of E versus  transformed  until  along w i t h the are given i n Table  of the  the  was a  and  corresponding  terms i n the bonding  c o e f f i c i e n t s o f terms i n V ± G+  XV.  short  £(H).  integrals  g  different  optimum a  i n p r o g r a m DIATOM t o o b t a i n E (  fficients  )  integrals  to  F i g u r e I g i v e s a sample p l o t o f E v e r s u s  II gives a plot  The  transformed  program t o o b t a i n  v a l u e s o f t h e p a r a m e t e r a, were u s e d obtained.  0  T* . T* . <r r l I g| r ' r >T., T . , (1-11) r i s] r s'^« t u tk u l '  where t h e p r i m e s d e s i g n a t e t h e o r i g i n a l o r b i t a l s , and and  1  G +  ]j  to and  geminal and  FIGURE I E VERSUS a H(?) =  E  1.34  (a.u.)  -25.0759  1 0.48  , 0.49  •  , 0.50  , 0.51  a  , 0.52  FIGURE I I  E VERSUS £(H)  -25.0754  H  -25.0755  E (a.u.)  -25.0760 1.26  1.30 5(H)  1.34  -56-  TABLE XV  DATA FROM BASIS SET I I  Orbital  Exponents  Boron  Hydrogen  Is  4.6795  2s  1.2881  2p  Is  E  (optimized)  1.2107  0  G+l  AGP C O E F F I C I E N T S  1.3125  C  x  «(>! 0.54252394 <J) 0.48625970 0.766514474 • 3 0.41595987 <t> -0.13886659 0.42129379 cf> -0.14870197 <j> 0.15156648 cj>° -0.16986274  CI  4>a  0.48473405  2  C  2  4  C  3  a = 0.50 R = 2.329 a . u .  3  10-1  5  -25.07601 a . u . -25.08541 a . u .  0.54239176 <t>2 0.48537033 <j) 0.41656740 <f>4 -0.12764533 -0.18499101 0.15135885 -0.16975390 <j/ -0.21502238 -0.30078238 0.10816345 10  10 10-1 1  -25.08627  x x x x x x x  10 -1 10"! 10 -1 10 -1 10*1  IO" IO"  3  2  APPENDIX I I TABLE XVI COEFFICIENTS OF 13-TERM WAVEFUNCTIONS  TERM  WAVEFUNCTION OHNO  OPT  *QSC  Y  0.71924579  X  10" 1  0.76587498  X  10" 1  -0.57602491  X  10" 2  0.65463450  X  10" 1  10" 1  -0.26711443  X  10" 1  X  10" 2  0.10012272  X  10" 1  0.93060592  X  10" 2  0.14028880  X  10" 1  10" 1  0.17170934  X  10~ 1  0.12314110  X  10" 1  X  10" 2  -0.74646173  X  10" 2  -0.17865829  X  10" 2  0.28109768  X  10" 1  0.14850164  X  10" 1  0.15692586  X  10" 1  io- 1  0.82458844  X  io- 1  0.45148265  X  10" 1  0.41494663  X  10" 1  io- 2  -0.93721254  X  io- 2  -0.51528255  X  10" 2  -0.50561302  X  10" 2  10"•1  10"•1  10"•1  1  0. 65160911  2  0. 21686080  3  -0. 31982327  4  0. 50163363  0.47944053  0.25833018  5  0. 11724651  0.11538376  0.60380664  X  10" 1  6  - 0 . 59753748  X  10"•1  -0.65619440  X  10"•1  -0.30337938  X  7  0. 13112438  X  10" 1  0.16147282  X  10"•1  0.68367720  8  0. 17893071  X  10" 1  0.20531238  X  io- 1  9  0. 34097752  X  10" 1  0.40209341  X  10  -0. 14793546  X  io- 1  -0.81108863  11  0.28422092  X  io- 1  12  0. 88044107  X  13  -0. 99948530  X  X  0.97622451  X  0.23936148 X  TOSC  10"•1  -0.14220806  0.34391515  X  0.1097724 X  10"•1  -0.15799579  X  10"•1  0.24332556  

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