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Positron-atom interaction Pai, David Mieng 1975

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P O S I T R O N - A T O M  I N T E R A C T I O  by  DAVID MIENG PAI B . S c , University of B r i t i s h Columbia, 1973  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  In the Department of Physics • i''  We accept t h i s t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October, 1975  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  for  of  the  requirements  freely  available  for  t h a t p e r m i s s i o n for e x t e n s i v e copying o f  this  representatives. thesis for  It  financial  The  irhys <  of  gain s h a l l  Date  c$  U n i v e r s i t y of B r i t i s h Columbia  2075 Wesbrook Place Vancouver, Canada V6T 1W5  SO  / 5~ 7  that  this  thesis or  i s understood that copying or p u b l i c a t i o n  written permission.  Department  for  r e f e r e n c e and study.  s c h o l a r l y purposes may be granted by the Head of my Department  by h i s of  agree  fulfilment  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make it I  in p a r t i a l  not  be allowed without my  ii Abstract Any attempt to describe accurately the l i f e time of positrons i n atoms or the low energy phase shifts for positron-at<bm scattering r e l i e s on an adequate treatment of electron-positron correlations. thesis  e  In this  an interaction model which treats such correlations i s presented.  This model has other desirable features.  It i s non-phenomenologlcal,  amenable to calculation even for complicated atoms, and applicable to calculations of both bound and scattering states of positron.  Following  the presentation of the theoretical aspect of t h i s model* methods which can be used to carry out the prescribed calculations are discussed.  At  the end of t h i s thesis, results obtained from applying t h i s model to positron-helium system axe presented.  In particular, the calculated  total elastic scattering cross sections are compared with experimental measurements.  iii Table of Contents Page H i i i iv  Abstract Table of Contents Acknowledgement Chapter 1  1  Introduction  Chapter 2 Theory and Model of Interaction  Chapter 3  Chapter 4  4  2.1.  Physical Consideration of Interaction  4  2.2.  Model of Interaction  7  Methods of Calculation  19  3.1.  Solution of Electron Equation  19  3.2.  Calculation of V(r p ) and 3 . 2 . 1 . Reductions of Integrals i n VC^p) and Mz to Sums of Matrix Elements between Basis Functions 3 . 2 . 2 . Evaluation of Matrix Elements between Basis Functions  24 24 28 35  Results  4?  Bibliography Appendix A  Derivation of ( 3 . 6 )  49  Appendix B  Derivation of 7p£  a n d  5  Appendix C  Derivation of 7pt  a n (  2  56  i Vpl  Appendix D Alternative Method on Deriving Vf \ and p% v  58  Appendix E  Electron Wavefunction  61  Appendix F  Matrix Elements "between Molecular Orbital $  69  Appendix G  V/(r ,&)and S(<>,$,U)  72  Appendix H  S  75  P  @) as a Function of Positron Energy  ) Acknowledgement I wish to take this opportunity to express my sincere gratitude to my research supervisor, Dr. E.W. Vogt, for his continuous patience and encouragement i n guiding me during the entire course of this research.  It would have been d i f f i c u l t to complete the project had  he not given me the help that I received. I am also grateful to Dr.D.M. Schrader of Marquette University for h i s many Invaluable contributions.  The project would have been a  d i f f i c u l t task without his participation. Finally I want to thank Dr. P.H.R. Orth who set up the computer programme BISON f o r us.  *  The authors* of BISON are also appreciated.  A.G. Wahl, P.J. Bertoncini, and R.H. Land of Argonne National Lab.  1  Chapter 1 The  interaction  interesting  and  of  Introduction  positrons  important  with  matter  is  phenomenon because the  an  manner i n  which a p o s i t r o n a n n i h i l a t e with an e l e c t r o n can o f t e n  reveal  t h e e l e c t r o n i c s t r u c t u r e of the  wonder  that  p o s i t r o n has  material.  It  is  no  been used s u c c e s s f u l l y t o probe s o l i d s and  1  l i q u i d s . Of  more  interaction  fundamental  interest,  is  the  of p o s i t r o n s with atomic gases because t h i s  kind  o f process i s dominated by a p o s i t r o n gas  atom  i.e.  assuming the s c a t t e r i n g l e n g t h  average  at  a  time,  interatomic  r e v e a l the  spacing  interaction  understanding the  i n gas.  mainly of two  kinds.  on  an is  mechanisms  studies  interacting  of  more complicated  Experimental are  instead  however,  assembly much  less  Studies which  with  one  of atoms, than  the  of t h i s k i n d are  valuable  can in  situations.  p o s i t r o n one-atom i n t e r a c t i o n s  One  is  the  life  time  spectrum  measurement of p o s i t r o n s i n gases, u s u a l l y as a f u n c t i o n of temperature and a p p l i e d e l e c t r i c f i e l d . S i g n i f i c a n t amount of 2-5  measurement has however  to  been made on r a r e gases  use  life  without ambiguity the  time  . It i s  difficult  spectrum measurements t o deduce  s c a t t e r i n g or  the  annihilation  cross  s e c t i o n s which c h a r a c t e r i z e the i n t e r a c t i o n . A d i f f e r e n t k i n d of experiment i s to measure the a t t e n u a t i o n  of  p o s i t r o n s of a  6-7 d e f i n i t e energy i n gases unt'il  recently  a  method  . T h i s experiment was was  found  on  how  not  possible  to o b t a i n mono  energetic  positrons.  direct  than the l i f e  cross  sections,  there  are  The  attenuation  measurement  time measurement i n y i e l d i n g  but u n c e r t a i n t y  some  still  unresolved  i s more  the  total  e x i s t s . At t h e moment,  discrepancies  among  the  measurements. Most  theoretical  investigations  on  positron  system are r e s t r i c t e d t o e l a s t i c s c a t t e r i n g calculations. interaction  Inelastic involves  scattering,  excitation  of  positronium formation, i s d i f f i c u l t the  meaningful  elastic  which the  or  one-atom  bound  means  state  t h a t the  atom  or  real  t o t r e a t - So f a r , most o f  scattering  calculations  were  for  3-10  simple atoms. Helium atom attention  because  , in  particular,  e x p e r i m e n t a l data were r e a d i l y  Agreements between c a l c u l a t i o n s and debatable.  Some  of  the  an  much  available.  measurements  are  still  c a l c u l a t i o n s a r e phenomenological,  o t h e r s are v a r i a t i o n a l . R e g r e t t a b l l y , giving  received  a c c u r a t e and s o l v a b l e  none shows  promise  method f o r more c o m p l i c a t e d  atoms. For bound s t a t e c a l c u l a t i o n s , Schrader c o n s t r u c t e d elaborate and  self-consistent-field  correct,  complicated  the  theory  is  in  theory**.  difficult  to  However be  an  complete  applied  to  systems.  T h i s t h e s i s p r e s e n t s a model f o r the i n t e r a c t i o n between a  positron  desirable elastic  and  a  features. scattering  single  atom.  This  F i r s t of a l l , i t i s  model  has  applicable  several to  both  and bound s t a t e c a l c u l a t i o n s , though not  3 to  inelastic  scattering.  phenomenological.  Secondly,  There i s no  parameter to be  experimental data. T h i r d l y , i t r e t a i n s picture phase  of  the  shifts  applicable)  positron.  or  to  binding be  positron liJte  and  near the  model  is  atoms. I t was for  the  is  t h i s idea  was  correlations  of  the  very  The  Lastly,  particle scattering (if  i t accounts between  to  and  the  the  most  behavior  r e s u l t s are  of  importantly, complicated  helium atom as a t e s t i n g  obtained  ground  compared i n t h i s  a v a i l a b l e experimental data.  any  between used  against  positron  Fourthly,  sensitive  be p o i n t e d out to  not  T h i s i s important because p r o c e s s  positron.  t h e s i s with p r e s e n t l y  correlations  energy  amenable to c a l c u l a t i o n , even f o r  model.  i s applicable  single  T h i s treatment e n a b l e s  decided to use  I t should  is  varied  range s p a t i a l c o r r e l a t i o n  electrons,,  annihilation  electrons the  the  the  easily calculated.  f o r the c r u c i a l s h o r t  it  system the  by  that t h e of  i d e a behind t h i s model  particles  p a r t i c l e s are  Vogt  to  spatial  important. In  calculate  between nucleons i n a n u c l e u s  where  12  the  fact,  effect  of  4 Chapter 2  2»/\  Theory and Model of I n t e r a c t i o n  Physical Consideration It  is  worthwhile  between a p o s i t r o n  of I n t e r a c t i o n  to describe  i n words the i n t e r a c t i o n  and an atom b e f o r e g i v i n g t h e mathematical  treatment of the model. F i r s t , c o n s i d e r  the  situation  when  the p o s i t r o n i s f a r enough from the atom t h a t the o n l y e f f e c t the  aton  can f e e l i s a s l o w l y  p o s i t r o n changes i t s field to  varying  position,.  electric field  Specifically,  the  as t h e  electric  v a r i a t i o n i s so slow t h a t t h e e l e c t r o n s have ample time  rearrange  themselves  as  the p o s i t r o n moves about.  This  s i t u a t i o n i s analogous to the case o f a molecule i n which t h e electrons  can  almost  instantaneously. far  follow  up  the  nucleus  motion  T h e r e f o r e when the p o s i t r o n i s s u f f i c i e n t l y  away from the atom, i t i s j u s t i f i e d t o use the ' a d i a b a t i c  a p p r o x i m a t i o n ' , i . e . c a l c u l a t e the e l e c t r o n motion as i f t h e positron the  i s s t a t i o n a r y . T h i s approximation g r e a t l y s i m p l i f i e s  problem. The r e s u l t i s that the p o s i t r o n  polarized  atom.  The  positron's  distance  degree  predominantly  distance  from the atom other  little  effect.  It  o f p o l a r i z a t i o n depends on t h e  from the atom. In f a c t ,  becomes  is  i n t e r a c t s with a  the  interaction  a d i p o l e - c h a r g e type because a t l a r g e multipoles  therefore  of  the  atom  give  expected that, the p o s i t r o n z  experiences a p o t e n t i a l where  oC  is  having  the  asymtotic  form  -^  the d i p o l e p o l a r i z a b i l i t y o f t h e atom. T h i s i s  the  w e l l known long  range p o l a r i z a t i o n p o t e n t i a l .  Now what i f the p o s i t r o n i s not f a r away enough from t h e atom f o r the a d i a b a t i c approximation t o be words, c o n s i d e r on  the  longer is  atom  follow  the  In  other  the s i t u a t i o n when the e f f e c t o f the p o s i t r o n varies  so  swiftly  up t h e p o s i t r o n ' s  problem,  t h a t the e l e c t r o n s can no  motion i n s t a n t a n e o u s l y .  c a s e when the p o s i t r o n  the e l e c t r o n cloud* similar  valid?  i s c l o s e by o r i n f a c t i n s i d e  C o n s i d e r f o r the the  This  interaction  moment  a  simpler  but  between a p o s i t r o n and a  s i n g l e e l e c t r o n . As can be found i n any mechanics t e x t t h a t a two  body i n t e r a c t i o n problem can always be  the  concept  of  reduced  mass  into  S p e c i f i c a l l y , the  wavefunction  ^(C)  relative  between  positron  motion  the  a  simplified  one  body  which  using  problem.  describes  and  the  the  electron  s a t i s f i e s a one body Schrodinger e q u a t i o n .  (2.1)  ahere the  / x i s the reduced mass of t h e system. I t e q u a l s 1/2 electron  mass  f o r the  system  being  considered.  where £ i s the r e l a t i v e c o o r d i n a t e the  of The  between  p o s i t r o n and t h e e l e c t r o n , g i v e s t h e p r o b a b i l i t y d e n s i t y  f o r f i n d i n g the e l e c t r o n words, one can s o l v e coordinate  relative  Y  apart  f o r the to  the  from the p o s i t r o n . I n o t h e r  electron  wavefunction  i n the  p o s i t r o n as i f the p o s i t r o n i s  infinitely  heavy and the e l e c t r o n mass i s  only i n t e r e s t e d one can always  where m g i s  i n the c o r r e c t solve  relative  the a l t e r n a t i v e  but with a charge o f  1/2 e  if  equation  the  instead  this  means t h a t  positron  of  the  adiabatic  approximation  is  by the atom because the e l e c t r o n s  close  rearrange themselves  However f a s t  is  seem  by using the i d e a d e s c r i b e d  with a s i n g l e  electron  not  was mentioned  valid  that  the  to  the  for a positron electron  can be e a s i l y  treating  positron  and  p o s i t r o n charge positron,  1/2 e.  however,  and o t h e r e l e c t r o n s from  0  electron positron  to  1 e.  stationary For e l e c t r o n s  such  so  the e f f e c t i v e Therefore,  that  charge  may  effective  close  to  with the range  interaction  the p o s i t r o n i s  stationary  at  calculated  the  nucleus anywhere  one can perhaps approximate the  one  interacting  u s i n g the  because of i n t e r a c t i o n s  wavefunction by t r e a t i n g as  not  motion.  wavefunction,  near by the p o s i t r o n ,  the  can no  electrons,  l e a s t the p a r t the  when  q u i c k l y t o the p o s i t r o n ' s  the p o s i t r o n may  to  1 e.  come back to the o r i g i n a l problem. I t  longer  the  stationary,  Now  positron  If  ^(C) #  which d e s c r i b e s t h e e l e c t r o n motion r e l a t i v e  t h e p o s i t r o n can be o b t a i n e d as  knows  wavefunction  the e l e c t r o n mass. In words,  wavefunction  that  reduced by h a l f .  the  with  the  but  with  some e f f e c t i v e charge l e s s charge depends on the way,  as  far  as  1 e.  than  positron's  the  distance  electron  which  the  T h i s i s the present  value  is  special  e l e c t r o n wavefunction has  given  the  diatomic  molecule  been w e l l  studied.  main i d e a of our model. I t i s a p p r o p r i a t e  the  this  concerned,  now  to  mathematics of the model. Some of the i d e a s  can  then be more p r e c i s e l y s t a t e d . A l l g u a n t i t i e s are  of  from the atom. T h i s  motion  positron-atom system resembles a for  The  in  Atomic  («v=e=n=1)  Unit  from  here  unless  on  specified  otherwise.  2-.2  Model of The  Interaction  approximate  positron-atom ^((tv]  wavefunction  system  •> •, £0rp))U(rp). The  explicitly  by the  is  taken  electron  electron  used to  be  to  describe  the  product  form  the  wavefunction ^  coordinates  i s expressed  Ifi  (i=1,2,... Z) #  where Z i s the atomic number. I t a l s o depends on the Tp  coordinate t o g e t h e r with positron  ,although @(rp)  builds into  correlations.  The  though a f u n c t i o n of only motion  because  depends on The the  it  parametrically.  is  the e l e c t r o n  Ifp  the  positron  T h i s dependence  important  electron-  wavefunction  , does depend on  determined  from  a  the  l i (Tp)  ,  electron  p o t e n t i a l which  configuration.  e l e c t r o n wavefunction  equation  ^  positron  ; Pp > (lOp)  ^  s  defined  by  8  \  \«t  u  i*<  '»  tat  j>i M J  js,  tip  /  ^V; : k i n e t i c energy o p e r a t o r o f t h e i - t h e l e c t r o n ^js. : i n t e r a c t i o n  between  the  nucleus  and  the  i—th  electron - ~ : i n t e r a c t i o n between t h e i - t h and the j - t h e l e c t r o n ,  nj«lri-ri|  u  6(Vo)—-•: interaction Tip r  v  positron  The  nucleus,  as  between  the  i - t h electron  i n the context of present method, Hp-| £V"Tp|  usual,  is  treated  as  p r o c e s s o f i n t e r a c t i o n . , I t can c o n v e n i e n t l y origin @(rp)  for  electrons  given.  charge  positron move  short  behavior  is  treated  It  be  used  as t h e  The r a t i o n a l behind t h e term  was  stated  that  i t i s the  as  quasi  stationary  when  the  about. I t should be determined by t h e e x t e n t  of  can f o l l o w up the  positron's  motion.  e l e c t r o n s near t h e p o s i t r o n , t h e : s o c a l l e d  range c o r r e l a t i o n , i s very s e n s i t i v e t o t h i s parameter.  S p e c i f i c a l l y , a t fi = £p, ^ cusp  „  stationary i n the  of the p o s i t r o n seen from t h e e l e c t r o n s i f  t o which t h e e l e c t r o n s The  £p  and  i s already  effective the  ITi  and t h e  condition  satisfies  the  electron-positron  which  can  be  interaction describe (2.1)  derived  terms  except  ideally  o r (2.2),  easily  the  @  from  the  (2.3)  by i g n o r i n g a l l  electron-positron  one.  To  s h o r t range c o r r e l a t i o n , a c c o r d i n g t o  should  be  set  1/2  to  i r r e s p e c t i v e the  p o s i t r o n ' s d i s t a n c e from the atom. More i m p o r t a n t l y ,  however,  Q should d e s c r i b e an a c c u r a t e e l e c t r o n wavefunction not only i n the r e g i o n c l o s e to the p o s i t r o n , but everywhere. In o t h e r words,  @«p)  is  average way i n ifp  position  are d e s c r i b e d inside  intended  the  sense  t o account f o r c o r r e l a t i o n s i n an that  at  a  particular  both the long and t h e s h o r t range c o r r e l a t i o n s by t h i s s i n g l e parameter. When the p o s i t r o n  the e l e c t r o n c l o u d ,  £  other  hand,  the p o s i t r o n i s s u f f i c i e n t l y f a r away from the atom, (3  should  equal t o 1 s i n c e only long  range  correlation  t h e r e . In between these  two extreme c a s e s ,  average  and  between  long  hoped t h a t the important badly  is  should be c l o s e t o 1/2 because  s h o r t range c o r r e l a t i o n dominates t h e r e . On the when  positron  represented  short  exists  (3 i s some s o r t o f  range c o r r e l a t i o n s . I t i s  s h o r t range c o r r e l a t i o n i s not t o o  t h i s way. The c o r r e l a t i o n s would of c o u r s e  be b e t t e r t r e a t e d i f one c o u l d c o n s t r u c t a theory not only a f u n c t i o n o f  using  (3 a s  Tp , but a l s o of H p  Now c o n s i d e r the p o s i t r o n  wavefunction  U(rp)  .  It i s  d e f i n e d by  (IT P + V  \  +  Vcrp))U(rp)=e Uccp) P  (2  r  .5)  10  . 2 :the p o s i t r o n k i n e t i c energy o p e r a t o r -^-V p  :the  p o s i t r o n nucleus i n t e r a c t i o n  'P  y(fp):th.e e f f e c t i v e p o t e n t i a l experienced by the due  t o the  electrons  6p:the p o s i t r o n This  positron  energy  e q u a t i o n shows that the p o s i t r o n i s t r e a t e d as a s i n g l e  p a r t i c l e moving i n an e f f e c t i v e p o t e n t i a l . T h i s treatment t h e advantage of c a l c u l a t i n g positron-atom shifts  easily.  In  case  binding  energy of the  one  scattering  has  phase  i s i n t e r e s t e d i n bound s t a t e ,  p o s i t r o n can a l s o be r e a d i l y c a l c u l a t e d  provided  VOtf i s n e g a t i v e enough. There i s no p r e s e t boundary condition general  ( K^-^oo) on  the p o s i t r o n wavefunction. The  enough f o r e i t h e r e l a s t i c s c a t t e r i n g or  model i s  bound  state  calculations. The  roles  But  how  exactly  of  \J(Yp)  .  played  by  @(rP)  and  the  The  trial  ideal  wavefunction  product  Specifically,  which  energy.  wavefunction ^/Ul  t h i s c o n d i t i o n s i n c e i t i s not an H a m i l t o n i a n H.  a r e now  explained.  are they determined? Consider f i r s t  i n t e r a c t i o n should have a d e f i n i t e case,  V(r p )  eigenstate  the  case  describes  In  the  does not of  the  an  present satisfy total  11  - He + H +0-?M.£^-p-V(r ) P  where and  He  and  (2.5).  (2.6)  p  Hp are r e s p e c t i v e l y the H a l m i i t o n i a n i n (2.3)  Therefore  H(^U)=£Cr )¥u + e P W+^^^^  (2.7)  P  Unfortunate  i t is  that  does  not  give  an  independent of p o s i t r o n and e l e c t r o n c o o r d i n a t e s at  a l l to  one  can ask i s  treatment  f i n d such a wavefunction).  such  the e l e c t r o n coordinate.  whether that  i t is  to  question  formulate  is  independent  of  the  positron  words, one asks  ^ * m ^ U ) ^ * Y z . . . ^ l AUOrpj where over  A  , a constant,  a l l the  equivalent  as  the  energy o f the system averaged over a l l  coordinates In other  (if possible  An immediate  possible  energy  electron demanding  (2.8)  i s the energy o f t h e system coordinates.  This  averaged  condition  is  the energy o f the system, averaged  over a l l the e l e c t r o n c o o r d i n a t e s , t o remain c o n s t a n t  as  the  12 positron  moves  about.  I t i s indeed a p h y s i c a l l y  reasonable  c o n d i t i o n and should be i n c o r p o r a t e d , i f p o s s i b l e , treatment.  One sees from  J V H (vpU)dV,  - «*S = { £ C r P ) + e  the e l e c t r o n  (v-0<rp))<^|^|*>  p +  u  wavefunction  i.e. 0 ^ 1 ^ ; 1  normalized,  the  (2.7) t h a t  *  Hhere  into  =  (2.9)  J  i s taken  t o be p r o p e r l y  • C o n d i t i o n (2.8) can t h e r e f o r e be  satisfied i f  X.£ r > + e H»-P< P>><¥l£^^^  (2.10)  R  C  One  P  P  r e a l i z e s immediately  K/i^p)  determining specified.  In  experienced  , assuming  other  by  t h a t t h i s equation can be  the  a t the moment t h a t A  words,  the  positron  due  chosen v i a the d e s i r a b l e c o n d i t i o n although  in  this  case  not  a  to  matter  where  i s . Such  distribution,  rather  than  e l e c t r o n s . T h i s i s indeed a r e a s o n a b l e  c a n be  potential  the  total  remains scheme  with  energy,  e i g e n v a l u e but an constant is in  e q u i v a l e n t as assuming that the p o s i t r o n i n t e r a c t s electron  for  t h e e l e c t r o n s c a n be  definite  over the e l e c t r o n c o o r d i n a t e s , positron  effective  that  average  the  used  the  with  no fact the  individual  approximation  as i t  has  been  used  well  i n other  situations.  F o r example,  H a r t r e e ' s s e l f - c o n s i s t e n t - f i e l d e q u a t i o n f o r e l e c t r o n s i n an atom  can be d e r i v e d by u s i n g the s i m i l a r c o n d i t i o n t h a t t h e  t o t a l energy  averaged  over a l l e l e c t r o n  coordinates,  except  the one under c o n s i d e r a t i o n , i s a c o n s t a n t . Consider t o t a l energy, the  now  the value  averaged  positron-atom  of A  - I t i s defined as the  over a l l the e l e c t r o n c o o r d i n a t e s , o f  system.  I t i s impossible t o c a l c u l a t e A  f o r any p o s i t r o n p o s i t i o n o t h e r than f o r the simple when the p o s i t r o n i s f a r away from t h e atom. there  i s almost  equals  £  + 0  no  where So  6p  interaction,  that  case,  t h e t o t a l energy  simply  i s the atom energy.  one p r e f e r s , t h e product ^RJ  i s an  In  situation  Mathematically, i f  exact  eigenfunction a t  large  Tp  energy  of the system as given by (2.7), o r t h e average  since  there  i s no i n t e r a c t i o n . One can see t h a t  as given by ( 2 . 9 ) i n keeping becomes Vp^  fcico^-tQ-p  e and V  large energy  To . of  coordinates, \jQTp)  since  with  the l a s t  the d e f i n i t i o n f o u r terms,  , which d e s c r i b e i n t e r a c t i o n s ,  Notice  that  the system,  1'^ £(rp)-*£,  0  averaged  i s independent  over  -—r-^ -  energy  of A  ,  » Vp^S? ,  a l l vanish  at  . Therefore the t o t a l a l l the  electron  of t h e p o s i t r o n ' s p o s i t i o n i f  i s c a l c u l a t e d by (2.10) with  A=£o+ep, i.e.  V(rp)= 8Crp)-eo+(t-<?(rP))<^|r^|^>+f  (2.11)  14 For c a l c u l a t i o n containing  U  Fortunately electron  feasibility which,  this  one may  after  term  about  i s real  identically  where  they  V  ^  C3--^Yp^UlJ  ^;  functions  p  where the symbol  0  . Equation  (2.14)  P  •—• (bar) means  to  average  over  a l l  the  c o o r d i n a t e s . For example,  A( P^  ^  R  f  t  —d V . ^ - ^ r  To show the consequence of choosing can  and £p  i c r ) - £ + (i-^(r ))A(rp')+ §  P  one  JTi  of  (2.13)  be c o n v e n i e n t l y w r i t t e n as  \J(r )-  electron  d e f i n e t h r e e terms:  a l l are  (2.11) can then  the  (2.12)  v  P  J B  VC^p) .  (see p. 23) and n o r m a l i z e d .  v  A-£^j,  term  because  < £'iYP^YpU>=<^i7p^>-v u=^7p(<: i'i^>)-ypa=o For s i m p l i c i t y ,  the  a l l , i s determined from  vanishes  wavefunction ^  wonder,  rewrite  equation  (2.15)  2  V^p)  (2.7) using  a c c o r d i n g t o (2.8), V^p)  as g i v e n by  (2.14).  H ^ U ) = (£• + e ) P  *(i-^c.r ))(A- A ) ^ U + ( 8 - 8 ) ^ l  U  (2.16)  This  equation  ^fU  wavefunction averaging energy  is  shows an  that  the  eigenfunction  trial in  the  same  average  product  the  sense  over the e l e c t r o n c o o r d i n a t e s . T h i s way  in  the  sense i s kept a t the  of  total  constant  8o ^P •  value  +  How first as  clearly  @Cp)  should  be determined? I t was  i n t r o d u c e d i n the  p l a c e to d e s c r i b e as an a c c u r a t e e l e c t r o n  wavefunction  p o s s i b l e when the p o s i t r o n i s t r e a t e d as g u a s i s t a t i o n a r y  relative  to the e l e c t r o n motion. In other words,  be determined such t h a t the t r i a l wavefunction the  §(rp)  ^Ui  resembles  t r u e energy e i g e n s t a t e as much as p o s s i b l e . One  which can measure the extent true eigensate  ^Li  to which  guantity  approximates  (2.17)  2  Eo  be the  value  the  is  H 2 ( P<*P>) S < ¥ 1 1 |(H - Eo) 1 ^ U >  where  should  i s the t r u e energy of the system, a l r e a d y shown to £o ^P  .  +  The  (Mz)'  value  measures  2  the  energy  makes about the t r u e energy. S p e c i f i c a l l y M2 can  spread be w r i t t e n  as  co  M = Z K E i l ^ U > | ( E i - E ) + { KEIvyU>| (E-Eb)'dE 2  2  2  where energy  2  0  [Ei) 's a r e £;  the  ( E; < E  0  bound  ) ; and  states, |E>  's a r e  i f  there  is  the  continuum  (2.18)  any,  of  states  of  energy E . T h i s e x p r e s s i o n c l e a r l y shows  that  non-vanishing  2 M2 i s due UEl^U)! to  be  to  the u n d e s i r a b l e c o e f f i c i e n t s  i f (MAJVs  i s t o be maximized.  >  l^PU)  better  approximates  write  component The  oh lEo^ ,  @("p)  by  down H2 e x p l i c i t l y i n terms o f  ought i.e.  M2 i s ,  smaller  the true eigenstate  t h e r e f o r e r e a s o n a b l e t o determine To  M2  where Ei,E * Eo . I t i s e a s i l y seen t h a t minimized  (CEol^UM  I^EtlSPU^ and  the  |E(^> . I t i s  minimizing  Ma •  and Ut , one c a n  use Eq. (2.16) which shows how H o p e r a t e s on ^ L l •  M = < ^ U I H -2HEo + Eo 2  2  I^U>  = <^uiH |^u>-E <uia> 2  2  0  = <H(^U)|H(^U)>-Eo<U\U> = ^U(r P ,^r p ))| 2 S(r P f ^(rP))dVP  (2.19)  where  S ( ^ = 0-^ ( 2  A )+2(i-^(A6-AB)+(^-B ) 2  2  - r 2 ( l - ^ ) ( ^ - A C ) + 2(6C-8C) + (C5--C2)  The terms A Their  7  , AB,  B, AC, z  (2.20)  B £ , and C~* a r e n o t d e f i n e d y e t .  symbols however s h o u l d i n d i c a t e c l e a r l y  what they a r e .  For example, r e f e r i n g t o (2.13) and (2.15) one can see t h a t  17  AB*$mXiifpXTi^)^' Equation to  (2.21,  know the p o s i t r o n ' s wavefunction Wp)  determined  from  comes from  minimizing  p r o c e s s which  f  o  r  that  Tp  values  in  of  obtained  can  6 depends  ( s i n c e c*  depends on 14  to  find  be determined  computation.  \Afp,(3)  for  minimum from  f o r f i n d i n g mimimum S  on  U  . The i d e a that  M2  and  «.  In  minimizing $ a t , it  was  found  follow  up  on Q.p  which  factor  the e l e c t r o n s  the p o s i t r o n ' s motion. T h e r e f o r e , i n c a s e o f  s c a t t e r i n g the h i g h e r the i n i t i a l (30Tp)  @(*p0 depends  ) i s p h y s i c a l l y r e a s o n a b l e . The  @ , as d e s c r i b e d , measures the extent t o  lower  actual  e x t e n s i v e i t e r a t i o n i s r e g u i r e d to take c a r e of the  f a c t that  can  which  (3 at each Tp i n d i c a t e t h a t no  i s necessary  „ Furthermore,  no  0(fp)  i s dep endent on  difficulty  series  (3(rp)  words,  each Tp  , which supposedly i s  M 2 « T h e r e f o r e t h e r e seems t o be a l o o p  the  a  i n t e g r a t i o n over other  Ui  has  f o r the case of e l a s t i c s c a t t e r i n g of p o s i t r o n s  helium atom,  SCrpj^jU)  VW  . But  suggests  Fortunately, from  H 2 one  (2.19) s t a t e s t h a t i n order to minimize  everywhere  g i v e s a rough i d e a f o r how  is  p o s i t r o n energy  expected.  (3ttp) should  The  Qp i s ,  the  following figure  vary with  .  18  Figure 1  The to Sp in  energy 17.7 was the  range f o r e l a s t i c  eV.  Over t h i s s m a l l  found. final  calculations.  A l l the chapter  positron-helium range,  no  scattering i s  variation  claims  made h e r e  using  actual  will  numbers  he  of  $(fp)  0  with  substantiated obtained  from  19  Chapter 3  The  methods  which  were  c a l c u l a t i o n s prescribed presented  in  Methods of C a l c u l a t i o n  this  used  by the  chapter.  to  carry  model Most  out the necessary  f o r helium  atom  are  t e c h n i q u e s used a r e a l s o  a p p l i c a b l e t o the more c o m p l i c a t e d atoms. F o r c l a r i t y ,  many  d e t a i l e d d e r i v a t i o n s a r e g i v e n i n t h e appendices.  3. ,1  Solution of Electron The f i r s t  tasX i s t o s o l v e f o r t h e e l e c t r o n  ^({tt}-, r ; §(r \) f  Equation  from  r  Eg. (2.3) f o r a p a r t i c u l a r  wavefunction TP  and (3 .  There e x i s t w e l l documented computer programmes t o s o l v e  this  p a r t i c u l a r e g u a t i o n s i n c e , as mentioned, i t i s i d e n t i c a l t o the  e l e c t r o n wave equation o f a d i a t o m i c molecule except t h e 13  factor  ^  . A programme by the name BISON  study a f t e r a few m o d i f i c a t i o n s The  method  BISON  uses  t o solve  SCF  combination  atomic o r b i t a l s ) method  the ground  of  (self-consistent-field) 14  Slater  this  (2.3) i s t h e H a r t r e e -  s t a t e of a c l o s e s h e l l system, a  in  were made on i t t o accomodate  Fock-Bdothaan s 1  was used  determinant  LCAO  (linear  . Specifically, for t h e wavefunction i s  approximated  as  of  one  electron  wavefunctions  (commonly c a l l e d molecular o r b i t a l s ) , i . e .  20  §,(r.)t(i)  $,fr.)i(l) 4Vr,)t(i)  CZ!)  (3.1)  where  i s t h e i - t h o r b i t a l , each being doubly o c c u p i e d by  e l e c t r o n s of s p i n t that  m  which,  and s p i n V . I t should  although  can  electron-electron serious  associated strongly  still  in  be  handled by BISON, does make  Fortunately,  terms  defect with  since  rare  on  gases  which  ^  satisfy  does  interests,  which  are  correlations,  but  correlations.  &(vp,p)  eigenvalue  Hartree-Fock E q u a t i o n . T h i s  treat  motion, a r e not expected t o be  of the assumed form shown by  an  not  T h i s i s however not  electron-electron  v a r i a t i o n a l p r i n c i p l e that Cj? *s  orbitals  present  the p o s i t r o n  dependent  Subjecting  of  correlations properly.  i n d e e d on e l e c t r o n - p o s i t r o n  the  out  o f main i n t e r e s t do form c l o s e s h e l l systems. The method  of approximating ^  a  pointed  has a more complicated form f o r an open s h e l l system  other c a l c u l a t i o n s d i f f i c u l t . are  be  (3.1) t o t h e  be minimum, one f i n d s equation,  equation  can  be  the  that  so c a l l e d  approximately  solved  by  the  Roothaan  Expansion  molecular o r b i t a l s are expressed as some  basis  orbitals),  functions  (usually  Method linear  in  the  in  which  combinations form  of  of  atomic  i.e.  § i = i:C;t,F*  (3.2)  where the F » s are the basis functions; and the associated  coefficients.  Using  this  c's  are  the  approximation,  the  Hartree-Fock Eguation can be reduced to a set of f i r s t linear  the  equations.  The  coefficients  i t e r a t i o n u n t i l reaching  then  solved  by  self-consistency.  The basis functions type o r b i t a l s centered  are  order  uses are the normalized S l a t e r  BISON  either  on  the  nucleus  or  on  the  positron and defined as  F „ , 1 . , , 1 ( r , 9 . ? ) = - | | p r "'e n  <3.3>  I  where  Vjlm ^ >^)  coordinate  e  (r,  i  s  t  n  e  usual  spherical  harmonics.  The  i s defined according to which center the  function resides on as shown below.  22  Figure 2  One can see t h a t  (fa.Qa.^a)  refers  b a s i s f u n c t i o n s on the nucleus, on  the  for  accuracy  center  with  one  this used  p r a c t i c a l computing c o s t as BISON  f e a t u r e t h a t b a s i s f u n c t i o n s can has  obtained  o r b i t a l s . F o r t u n a t e l y , one can o b t a i n  been used s u c c e s s f u l l y t o many d i a t o m i c The  of  i s f o r those  depends on t h e number o f b a s i s f u n c t i o n s  t o expand the molecular reasonable  coordinate  (Y~b,9b,*Pb)  and  the p o s i t r o n . The degree o f accuracy  way d e f i n i t e l y  has  to  great  advantage.  It  systems. reside  on  either  i s t h a t the e l e c t r o n  wavefunction near the p o s i t r o n , i n p a r t i c u l a r  the  electron-  positron  can  be e a s i l y  cusp  represented those  condition  stated  in  (2.4),  by b a s i s f u n c t i o n s on the p o s i t r o n ,  on the n u c l e u s .  describe  satisfactorily  positron  correlation  In other  but  not  by  words, i t can be d i f f i c u l t t o  the important s h o r t i f restricted  range  electron-  i n using b a s i s f u n c t i o n s  9 only  on the n u c l e u s .  I f so, as i n some p r e v i o u s  work  , many  23 basis functions have to be then used. It should therefore recognized  that t h i s model not only accounts for the c r u c i a l  short range electron-positron c o r r e l a t i o n (3(*"p)  but  t  also  in  actual  in  theory  two  f  integrals of  center  basis  functions  problem can be l e f t  there  accomodate  the  case  to BISON to take care of.  needs  the  and  V(*p)  Fortunately, as w i l l be shown, t h i s  For helium atom, the s i t u a t i o n since  by  give r i s e to two center  which can not be done a n a l y t i c a l l y as i n  one center i n t e g r a l s .  as  c a l c u l a t i o n by using basis  functions on the p o s i t r o n . However, for evaluating H2  be  two  be  only  one  electrons.  is  particularly  molecular  According  simple  orbitals  to  (3.1),  to the  electron wavefunction i s then expressed as  (3.4)  m= ^<$(rO<$(r )(t(i)U2)-Ul)t(2)) 2  Ac  this  is  a  singlet  antiparallel. orbital ^ positron _  _  in  a  since  diatomic  the  =  energy is  molecule,  the  spins  are  molecular  should have r o t a t i o n a l symmetry about the nucleusaxis.  In other words, g5 *s dependence on the angle im<p  shown i n F i g . 2, should be of the form Q. |m| dependent,  positron  interacts  with m=0 being the lowest.  necessary  . The Assuming  with a helium atom i n i t i a l l y at  ground state (thus t o t a l o r b i t a l angular then  electron  _  Y ( T a = Trb) t  the  &s  state  (but not sufficient)  momentura=0),  to choose m=0 for  it  its is to  24 approach  Tp . The s t a t e  a ground s t a t e atom a t l a r g e  non-degenerate.  One  has  therefore  a  close  m=0  is  s h e l l system.  I n c i d e n t a l l y , other r a r e gases which w i l l be i n v e s t i g a t e d the  future  form  s h e l l a l s o . To ensure <3?  close  c o r r e c t dependence on <P then  t h e Fn*o$  be  , the b a s i s f u n c t i o n s  o r b i t a l s . The ones chosen  a r e g i v e n i n the next  chapter  together  c o e f f i c i e n t s f o r v a r i o u s Ifp and  3.2  \J(r )  C a l c u l a t i o n of As  mentioned  in  order  to  A, A ,B, E*,AB,C^,AC,  and  molecular  which  5  2,  there  (3(n>)  orbitals  BC.  the  obtained  Since  are  ^  a  s e r i e s of  wavefunction V(*p)  is  ,  ^  namely  expressed  l i n e a r combinations to sums  of  by  of b a s i s matrix  between the b a s i s f u n c t i o n s . T h i s s e c t i o n d e s c r i b e s  f o r each i n t e g r a l what these matrix elements  a r e and how  are evaluated.  3.2. 1  should  f o r t h i s study  are  and  f u n c t i o n s , these i n t e g r a l s can be reduced elements  the  Mz.  Chapter  calculate  (3.3)  with  i n t e g r a l s t o be e v a l u a t e d from the e l e c t r o n in  having  <>  and  p  in  they -  Reduction of I n t e g r a l s i n V(Vp)  and  Ms  to  Sums  of  Matrix Elements Between-Basis F u n c t i o n s The  first  step  is  to s i m p l i f y the i n t e g r a l s t o be i n  terms of molecular o r b i t a l s . only  one  molecular  orbital  For as  helium  atom,  i n (3.4). For  contains convenience.  25  .1  denote  by <£> , and  <£(!f ) by $  . One then has  2  •2  where  Vb » as i n F i g . 2, i s the d i s t a n c e from  p o i n t to the p o s i t r o n *  By using t h e  same  the i n t e g r a t i o n  technigue,  i t i s  shown i n Appendix A t h a t  A =e<$i^j$w<$i-^i$> 2  ^ "  ^T7p~  £  l  ->r  7K  p  .9/w_r  '  v  T>r  i  p  2  v r  L  ik'  1  P  I  I  -all  P  r ^e afy rs«hep afy ^sme^ty u P  p  , A / -a^ i i  -al>v -aU i  AC= 2<« l g^p> TFp"0" ,  1  +  P  i \ Z , / , / <)$ i i  1  P  r *ep A r ^ u '  ~>U i  J  ^<&y-»U_L?U  ^ J^p_ _*p >ep *p p /-8  r  _ ! _  IT  A / i ~d& i I -a^v i au i *U 2.  * _JJ>_!L!f*  u  7pti^pi»<Pp XT All  the  u n d e r l i n e d terms above w i l l be shown to be e q u a l t o  z e r o . S i n c e the molecular o r b i t a l ^ of  basis  simplified  functions  as  i s a linear  combination  i n (3.2), the terms i n (3.6) can  f u r t h e r i n t o sums of matrix elements  between  be the  b a s i s f u n c t i o n s . I t i s simple t o see t h a t  <$i^-»$>=z:'z:c Cj<F«^iFj> i  (3-7)  <$l^$> THCrCj<Fll-felFj> s  However,  the  ones i n v o l v i n g d e r i v a t i v e s with r e s p e c t t o the  p o s i t r o n c o o r d i n a t e , i . e . those with B or C p r e s e n t , are so  straight  forward  because the c o e f f i c i e n t s c; 's a r e  not also  f u n c t i o n s of p o s i t r o n ' s c o o r d i n a t e . For example  It  i s simpler  coefficients,  , by  however,  for  symmetry  a x i s ) , are independent of the One  simply  or  (no  since  preferred  positron's  the  nucleus-positron angular  position.  has  ^$__rc-25-  ^ -rc'^'  13 9)  27 The case f o r Vp £  i s a little  ?<  V $= £ P  2  more c o m p l i c a t e d .  Ifrp^Fi + Ci V Fi * 2  P  Again, because  Z v C- • 7 Fi] P  P  (  3.io)  * 0 # one has  V CiVpFi = ^  ^  P  (3.11)  Therefore  ^J^lfcfR  where (3.8),  C*  +C i ^ i +eCi'^)  , c" denote  ,  (3.9), and (3.12),  (3.12)  V C; p  2  i t i s simple  reduce  a l l the  between  the b a s i s f u n c t i o n s . One has  <*i  -k  =  \ irp *36p • rp "^6ip~  CiC  T  By u s i n g  (but t e d i o u s )  to  terms i n (3.6) i n t o sums of matrix elements  J' ( F ;  I -4- w l > • ?T ,_!__?$_, j _ 2 $ _ \  respectively.  T  r  2- 4J  'i CiC  J  £f  1Fj>+  < F (1  Ci<;i <Fi  -re- w 8 >  .r• /_!_l£l I . L 2 f L \ 1  J  '-^ ^  fp ">6p  rp •jBp '  28  VpSwfcp^V ^pSi.&p ^p/-4-4-  _ !?_S??L*i ^vp^V  rp<u* V  lHi_J  C-C • (—!  /_!_?_&I _ !  »t  V "  i  «B  ^ IrpSthfrp * V  vp^V^>  J  j  ^  l £ k  r t»6p * V p  *X  ^  J  V  5  ? ^  *p r v tt»»i»*V >d  p  <FiIFj >*Z^CiCj <Ff I V p l F j > - 2 Z I C C j V F , - 1 ^  *b  i j  i J  *  +EEc Cj<^|2g I  ^1^4?* ^FCC3'<ft1  >+zrt  b  c,^Fii|B>  ^*  ' j  Ci e,<2Bi-2&> ,  +?£t  CiCj<<Fi1  .+rife<i<^iVB>+4r^Ht.i|s>+4rp v^i^)(3.i3j f  where the u n d e r l i n e d terms i n (3.6) are again u n d e r l i n e d . The guestion  3..2..2  now i s how t o e v a l u a t e these matrix elements.  E v a l u a t i o n of Matrix  * "'P  Elements Between B a s i s  Functions  29  An the  i n s p e c t i o n of the terms i n  matrix  operators and  vp  elements  involve  with r e s p e c t  (3.13) shows t h a t  two  act  on  of  to  find  difficulty  can  be  anticipated  from  differently Yp and Vp  those  found  to The  that  the  coordinate  is  and  ^  the ones on the  c a s e , V -f  . I t can  p  r  P  denotes the ones  positron. Consider  be expressed  -&6p ~ r \-aiQ» ae p  >ea  P  p  P  to F i g . 1, one  D  sees t h a t Ifa , the  , does not depend on any  = 0  P  on  first  )  -a«p *«P / a  radial  positron coordinate,  -^r-=0  by  as  »<fe >e  r $*ep»fy - rsio©pV^ra *«P "** -aGa a<? p  the n u c l e u s r e a c t  on the p o s i t r o n under o p e r a t i o n s  . To d i s t i n g u i s h them, l e t 4-  the s i m p l e r  of f  fact  positron  b a s i s f u n c t i o n s r e s i d e on  from  nucleus,  Refering  these  completion.  the  Yp  r a t h e r than e x p l i c i t .  Conceivably,  the  how  b a s i s f u n c t i o n s . T h i s i n f a c t was  dependence of b a s i s f u n c t i o n s on the implicit,  namely  out  be the major task i n c a r r y i n g t h i s model to  of  differential  to the p o s i t r o n c o o r d i n a t e ,  . i t i s t h e r e f o r e necessary  operators  types  most  P  coordinate i.e.  (3.15)  30 However,  the  angular  coordinate  of £  by the n u c l e u s - p o s i t r o n  axis. I t i s  the  of  angular  radial  position  , ( &a,9ai ) # i s d e f i n e d  therefore  the p o s i t r o n  dependent  on  (Op.^pJr but not t h e  p o s i t i o n of the p o s i t r o n Vp . Hence  Cos 6a Sin ©a  3% oos ©a •aCpp ~ S i n e *  where the dependence of (0a, % Appendix B.  ) on  cosqsias^&p+cosep  ( ©p <P ) f  P  (3.16)  is  derived  in  Eg. (3.14) can t h e r e f o r e be w r i t t e n as  - 0 *9a  (3.17)  "a ©a  where  the  i s independent of 9a  fact that (  been used. I t i s  now  straight  forward  to  ( s i n c e m=0) find  .  has By  definition  ^ Using  T  " r tr P  ?  ^  "*• r p &p p  l  n  (3.16), i t i s found  that  r ?ep* P  a  v^stfep *<p* f  (3.18)  =  Vp  How difficult  Vp  and  and  operates  t o f i n d . The  t h e o r i g i n of q^s itself.  (3.19)  Now  but  ,  ( fi^Sb, Vb  however,  )  is  the  i s more  certainly  and  ...  more d i f f i c u l t  £ , as shown i n Appendix C, one  By  that  positron  has the s i t u a t i o n t h a t a l l three of  *Pb are f u n c t i o n s of rp , Qp ,  similar  ^  c o m p l i c a t i o n a r i s e s from the f a c t  coordinate  one  on  going  \%, 6b  t  through  c o n s i d e r a t i o n s as f o r  f i n d s that  4-^rp = c o e a i - Tb S ^^eb 4. S  »  i  ri  «  *  fl^  ^p?  =  ^  -  n  ^  vJ  v  where, a g a i n ,  I  m=0  (  ^ " +•  f u n c t i o n s of the  functions  -  ri.\  ^ b ( 3  ^  .20)  i s assumed.  b a s i s f u n c t i o n s are d e f i n e d  basis  i  \  V  +  is" apparent t h a t most  weakness, Vogt  ri  -^^b'-^Wb)  difficulties  above method of t r e a t i n g Vp and  the  are  COS 6b  a  ^ o . ^ cos61, 3 $  ^9  ^ p l W  It  b  Vp  1  associated  l i e i n the f a c t  with r e s p e c t t o c o o r d i n a t e s  positron coordinate.  with  In  view  that which  of  proposed an a l t e r n a t i v e approach i n which are transformed  (by r o t a t i o n of  this the  coordinate)  32 into  a new  space,  c o o r d i n a t e system  i . e . independent  Such  angular  f u n c t i o n s can  be  dependence  on  the  explicit,  and  system)  is  case  of  respect out.  ^  .  This  operator  V  ,  2  is  terms that  /T7 PI—! X  ,  and  differential  proved  to  be  using  be  now  (3.6)  P  ^S- S /  (D..12) w i l l simplified straight  or  are  »e '' ^  this  a  >rf  of  1 r sin9 p  which  P  additive  o f )T ) p  operators  not  F  with  carried  in treating  the  v*p ^  can  Vp +  and  For  to  the  on  directly  ,  identical  in  the  to  (3.20)  same  obtained  (3.20).  It  i f the  last  further. t o show t h a t  indeed the  afy ' ' <• rp  a l l  positron's  However, t h e one  be r e d u c e d  forward  (3.13) ' a r e sums  but  is  coordinate  simple  useful  the  (9p,<2Pp)  the  be  method.  (3.19) i s o b t a i n e d .  that  they  -arp i rp  where  r' i s independent  (0.12 ) , i s s i m i l a r  is  in  way,  basis  of  p o s i t r o n c o o r d i n a t e can  as  i n i t can It  where  £  i s that the  v e c t o r i n new  independent of  the  coordinate  (position  case  simplified  expected  term  the  on  form  . i t i s shown i n A p p e n d i x D how  p  expression p  |T  approach  quickly  for V ^  that  This  to the  the  in  position.  depends  result  p o s i t r o n ' s angular  ( r-t'-lfp r  dependence  in  fixed  positron's angular  c o o r d i n a t e . The  expressed  in  the  understandably,  completely  coordinate  be  of  transformation,  positron's  whose o r i e n t a t i o n ••• i s  z e r o as  vanish  P  matrix  >rpli^p 7? > P  since,  underlined  claimed.  following  *e  the  <• ? '> r  their  Notice elements,  r  P  *e  7 P  '  integrands  33 (= <Pa= %)  depend on the angle  sintycosCp as can be seen from How The  are  In  (3.17),  remaining  this  study,  now  is  or  (3.19), and (3.20).  the other matrix elements  only problem  itself.  i n the form of sin<P, cos<P,  i n (3.13) e v a l u a t e d ?  the  actual  integration  s i n c e the b a s i s f u n c t i o n s used c a n  r e s i d e e i t h e r on t h e nucleus o r on the p o s i t r o n , some o f t h e matrix  elements  are  two c e n t e r i n t e g r a l s which can o n l y be  e v a l u a t e d n u m e r i c a l l y . F o r t u n a t e l y , i t i s not us  necessary  t o do any a c t u a l i n t e g r a t i o n f o r the f o l l o w i n g r e a s o n . I t  t u r n s out t h a t a S l a t e r type o r b i t a l a f t e r a c t e d o p e r a t i o n s s p e c i f i e d i n (3.17), be  for  expressed  multiplied elements <Fi\Fj>,  as  by in  ^•  on  by t h e  (3.19), and (3.20) can always  a sum of S l a t e r type o r b i t a l s , o r o r b i t a l s o  r  (3.13)  (Fil-^r-lFj>  «  In  other  words,  can be f i n a l l y broken and/or < F i l - J j r l F p  a l l matrix  down i n t o sums o f  which a r e t h e k i n d s  of  matrix element t h a t BISON can e v a l u a t e . To  verify  the  correctness  of  t h e c a l c u l a t i o n scheme  d e s c r i b e d i n t h i s s e c t i o n , i t i s a p p r o p r i a t e t o note t h a t t h e f o l l o w i n g two r e l a t i o n s h i p s a r e indeed s a t i s f i e d seen from the r e s u l t s presented i n Chapter  as  can  be  4.  <*l^>--°  <4 I V p ' * > - - - < ^ > - < ^ ' i ^ - < 7 ^ 4 l ^ j | > >  these  two  e q u a t i o n s are o b t a i n e d from  ( 3  -  2 1 >  -^-<<£l<£> and Vjf{4?|<|>> * P  \  u s i n g the f a c t t h a t  i s r e a l and  normalized.  35 Chapter 4  Results  Presented i n t h i s chapter are the  c a l c u l a t i o n s prescribed  i n t e r a c t i o n . The elastic  scattering  with e x p e r i m e n t a l proceeding Unit  ,  by the  WP)  obtained cross  r e s u l t s o b t a i n e d from model f o r  is  then  sections  results  and  all  positron-helium  used  to  calculate  which are then compared  other  calculations.  r e c a l l t h a t a l l q u a n t i t i e s are  Before  g i v e n i n Atomic  unless s p e c i f i e d otherwise. In  this  eighteen  study,  positron  calculations  were  carried  out  p o s i t i o n s ranging from .3 t o 6.5  for  . Outside  t h i s range, c a l c u l a t i o n i s not n e c e s s a r y . For  Yp s m a l l e r  .3  nucleus-positron  ,  the  interaction  repulsion.  f o r Tp  As  is  dominated  by  l a r g e r than 6.5  ,one  simply has  than  a  well  known d i p o l e - c h a r g e i n t e r a c t i o n as d e s c r i b e d  i n Chapter 2.  At  each p o s i t r o n  investigated  in  p o s i t i o n , a s e r i e s of  o r d e r to f i n d First  (i( p) and of  results  wavefunction f o r v a r i o u s expressed  by  a  were  thus \/(*p) .  r  set  @  t o be  jfp  molecular  presented i s the (3  and  orbital  *  Recall  <|>  which  combination of S l a t e r type o r b i t a l s FnAo^ the  nucleus  functions 9ioe 1.133  »  *s)  used are a n <  or the  on  the  *fcioi.*33»T h i s s e t was  e r r o r b a s i s . S p e c i f i c a l l y , the study  ;  performed  on  the  (He H)  -f o l0  that is  is a linear  centered e i t h e r  positron  following  electron  ($'s).  The  on  basis  i.83tf/"f»o» 3.18*7 * £a»» i.6oa7 #  chosen on  basis system  a semi t r i a l  functions J  used  and in  were adopted as  a the  :  36  initial  set  differs  since  from  the  (2.3)  electron  %  of the  u n t i l r e a c h i n g minimum *p =  .5  using  a d m i t t e d l y , may and  $ .  in  at  Appendix  find  C' By  lp-**"p  and  and  the  two  out  The o b t a i n e d s e t , fp  of r  lr* and  .  p  Notice  that  for  wavefunctions  a l s o have t o be e v a l u a t e d i n o r d e r  the  scheme d e s c r i b e d i n S e c t i o n 3.2,  e l e c t r o n wavefunction.  specified in  carried  the o b t a i n e d £( p,§) and  are  E  was  to  C* »  using  matrix  H)  varied  f o r other choices  i n t e g r a l s p r e s c r i b e d by the model were  of  (He  then  Yp under c o n s i d e r a t i o n ,  positron position  fp + STp  of  were  (3 assumed t o be 1/2.  c o e f f i c i e n t s f o r each c h o i c e of each  <ft 's  T h i s procedure  not be the best one  Given  equation  o n l y by the f a c t o r Ci . To accomodate t h i s  d i f f e r e n c e , the parameter  at  wave  calculated  various  from  P r e s e n t e d i n Appendix F are t h e  elements  (between  the  molecular  (3.13) f o r each c h o i c e o f  lT and ^ p  the  values  orbital $ .  Note  c o n d i t i o n s s p e c i f i e d i n (3.21) are indeed  )  that  satisfied.  T h i s e s t a b l i s h e s c e r t a i n c r e d i b i l i t y to the methods  used  in  calculation.  The  values of \ / ( p , ^ )  G, can be e a s i l y Appendix two  and  r  F.  calculated  I t i s now  from  o O ^ ^ J j ) , given i n Appendix the  matrix  elements  necessary t o e x t r a c t ^(^p)  s e t s of v a l u e s a c c o r d i n g to the c r i t e r i o n t h a t  defined Chapter  in  from Mz  these  »  (2.19), i s a minimum. As mentioned a t t h e end  2, i t i s d i f f i c u l t  t o c a r r y out t h i s p r o c e s s  in  as of  because  37 of  -the  inter-dependence  , UU^p)  V(^ ,^) P  take,  however,  minimum of  , P p) (r  is  to  v a r i e s most s i g n i f i c a n t l y This  be e v a l u a t e d to  # and Mz • A  for various  find  the l e a s t  (£(rp) a l i t t l e procedure  with  Qp  (3 than any  . The  then  be  during  £p  are  the i t e r a t i o n  TVT^I^y i°  c  Z  the  one  The  presented procedure,  in  \Jl  can  until  of  fact  S(rp,(i)  should  selffound as  T-r^r-rrand dominates  determine  does ^(Vp")  determined from M i • N o t i c e  be  a  Incidentally,  (0-17.7 eV). How to  be  This  iCfp,^)  i n Appendix H.  .  give a  reaching  the terms i n v o l v i n g  i n using  then  i t e r a t i o n s , i t was  values  since i t  to  with.  were n e g l e c t e d s i n c e s-wave s c a t t e r i n g  however, j u s t i f y  which  which enables  started  f o r the e l a s t i c s c a t t e r i n g energy range one,  can  (3(Yp) from t h e  which w i l l probably  iterated  p  of  one  o t h e r term i n &  obtained  $(^,(3)  ($(r ) s t a y e d c o n s t a n t .  function  step  2  d i f f e r e n t from  can  SC^p,^, U) ,  2 ( I-@KAB - A8 ) + (B*-S )  +  c o n s i s t e n c y . In f a c t , a f t e r o n l y two that  first  r i s e t o an approximate V(fp)  gives  used  the q u a n t i t i e s  o b t a i n an approximate  A* - ** )  (I -  among  from  G & A p p e n d i x B t h a t , at each  rp  occurs  same  A  nearly  at  the  , the minimum of (5  \/(p) and  . I n o t h e r words,  % (@) = 0 if  = 0 everywhere. I t i s t h e r e f o r e h i g h l y probable = 0 when  —"STpT"  =  that  everywhere, s i n c e  h I t i s p l o t t e d i n F i g . 3 how  & v a r i e s with  (4.1) at  three  38 positron  positions.  The e s s e n t i a l i d e a of t h e model i s w e l l  i l l u s t r a t e d by t h i s graph. N o t i c e t h a t , f o r each is  a  definite  expected,  S  v a l u e of ($ f o r which  Tp  ,  there  i s a minimum. As  t h i s v a l u e i s n e a r l y 1/2 when p o s i t r o n i s c l o s e  by  o r i n s i d e the atom, and 1 when f a r away. I t i s i n t e r e s t i n g t o see  that  $  f o r t h e c h o i c e o f ^=1 i s h i g h e r than  rp=.5 and 2.?-.. In approximation  other  (^=1)  words,  using  straight  ^=0 a t  adiabatic  i s probably even worse than t r e a t i n g no  p o l a r i z a t i o n a t a l l ( Q=0)  when  positron  i s close  by  or  i n s i d e the atom. Plotted  in  Fig. 4  i s the  Appendix H, N o t i c e t h a t over the range  (0-17.7  eV) ,  n e c e s s a r y . By u s i n g  no  curve  §(*p) e x t r a c t e d from  elastic  variation  of  scattering ^ (r ) p  energy Gp  with  ^C«p) and Appendix 6, the p o t e n t i a l  can be s p e c i f i e d . P l o t t e d i n  Fig. 5  i s the  net  is ^J(Vp)  effective  2 potential  ( V(V )  interacting findings,  )  p  experienced  with a helium atom. I n the  potential  is  potential  does  a  agreement  positron i n with  noted,  not have the expected  however,  V(p,@»0 r  (2.14),,  theorectically  gives  a  t  • large  Tp  becomes  a l l multipoles  of  that  dipole potential  form --^--^ (oc =1.38 f o r helium atom) a t l a r g e ITp . to  previous  found t o be not deep enough t o  support any bound s t a t e . I t should be this  by  According  Sirfi-Zo a  long  which range  16  interaction  .  The  net  p o t e n t i a l should t h e r e f o r e approach  the expected  d i p o l e form -4r-br s i n c e  other  multipoles  give  39 little  effect  at  Tp  large  .  In the  p r e s e n t c a s e , however,  because the •£ • s used were a l l s - o r b i t a l s which are of  expanding long  obtained range  £(r )  range p o l a r i z e d e l e c t r o n  consequently f a i l e d t o  p  polarization  (V^p,^- ") "f^ ) ^ 0  4  potential  s  effect.  For  i n the  s i l O W I i  experienced  wavefunction.  account  for  comparison,  same graph. I t  by the  incapable  any  the would  p o s i t r o n i f the  influenced  at a l l by the  two  curves can  be a t t r i b u t e d t o p o l a r i z a t i o n s caused  positron. Fig.  For r e f e r e n c e ,  positron. Difference  the  long  potential be  the  electrons  not  The  were  between by  the the  s i z e of helium atom i s shown i n  6. Osing the o b t a i n e d  i 1.38  r e s u l t s are  p l o t t e d i n F i g . 7.  which i s c o n s i d e r e d the  1 0  on  the  same  graph  scattering cross section  (T = In our  a  A  . The  (J  corrected  t o p a r t i a l wave X =5  s c a t t e r i n g phase s h i f t s up  calculated  shown  (long range p a r t ^  --^-jpf),  result  N/Op)  for  most r e l i a b l e c a l c u l a t i o n i s comparison.  can be c a l c u l a t e d  to  were  Humberston's  Total  elastic  from  J s " Z(2fc+0 Sirf&jt  case, c o n t r i b u t i o n s  to  (J*  (4.2)  from h i g h e r p a r t i a l  waves  (JL>5) are not i n c l u d e d s i n c e they are n e g l i g i b l e . P l o t t e d i n Fig.  8  i s the c a l c u l a t e d  below the f i r s t graph,  (J" as a f u n c t i o n  i n e l a s t i c threshold  r e s u l t s from two  (17.7  of p o s i t r o n eV) •  e x p e r i m e n t a l groups are  On  the  energy same  presented^.  ttO D i s c r e p a n c i e s among t h e i r measurements a r e  still  unresolved  a t the moment. Results  obtained  from  i n t e r a c t i o n i s encouraging. the  this  model f o r p o s i t r o n - h e l i u m  With only f i v e b a s i s  functions,  c a l c u l a t e d e l a s t i c s c a t t e r i n g cross s e c t i o n s are i n f a i r  agreement  with  definitely  be  experimental improved  by  results. using  more  Calculations and  better  f u n c t i o n s . More e x t e n s i v e c a l c u l a t i o n s f o r helium  can. basis  w i l l be  p r o g r e s s . Argon atom w i l l be i n v e s t i g a t e d a f t e r w a r d s .  in  io l  .ooolL Fig. 3  & ) a t three positron positions for  6p«" 7 eV.  Fig. 4  (30^)  obtained from minimizing &(l"p,(3)  Fig. 6 Size of helium atom, I.e. electron radial probability density as a function of radius.  Radius i n Bohr  45  and f p a r t i a l waves. Dotted l i n e s are Humberston's r e s u l t s (flef. 10).  Fig, 8  Total elastic positron-helium scattering cross section. Experimental points are from Ref. 6 (dotted error bars) and Ref, 7 (solid error bars).  47  BIBLIOGRAPHY  1)  Stewart, A.T. and Roellig, L.O., Eds., 1967  2)  Lee, G.F., Orth, P.H.R., and Jones, G. 1969. Phys. Lett., A, 28, 674.  3)  Orth, P.H.R. and Jones, G. 1969. Phys. Rev. 183, 7.  4)  Leung, C.Y. and Paul, D.A.L. I969.  5)  Coleman, P.G., G r i f f i t h , T.G., Heyland, G.R., and Killeen, T.L. 1975. J. Phys. Bi Atom. Molec Phys., Vol. 8, No. 10 173^»  f  "Positron Annihilation".  J. Phys. B, 2, 1278.  e  6.a)  Jaduszliwer, B. and Paul, D.A.L. 1973« Canadian Journal Physics, 51. 1565.  6. b)  Jaduszliwer, B. and Paul, D.A.L. 1974. Canadian Journal Physics, 52, 1C47.  7. a)  Canter, K.F., Coleman, P.G., G r i f f i t h , T.C., and Heyland, G.R. 1972. J. Phys. B» Atom. Molec. Phys., Vol. 5» LI67.  7.b)  Canter, K.F., Coleman, P.G., G r i f f i t h , T.C., and Heyland, G.R. 1973. J. Phys. Bi Atom. Molec. Phys., Vol. 6, L201. 1971. Phys., Rev., A, 3 , 1335-42.  8)  Houston, S.K. and Drachman, R.J.  9)  Aulenkamp, H., Heiss, P., and Wichmann, E. 1974. Z. Phys., 268, 213-5.  lO.a)  Humberston, J.W., 1973* L305-8.  J . Phys. Bi Atom. Molec. Phys., Vol. 6,  lO.b)  Humberston, J.W. 1973* L286-9.  J« Phys. B» Atom. Molec. Phys., Vol. 7,  10.c)  Campeanu, R.I. and Humberston, J.W. 1975* Phys., Vol. 8, No. 11, L244.  J . Phys. Bt Atom. Molec.  11)  Schrader, D.M. 1969. Phys. Rev., Vol. 1, No. 4, 1070.  12)  Vogt, E.W. and Lascoux, J. 1957. Phys. Rev. 107, 1028.  13)  Wahl, A.C., Bertpncini, P.J., and Land, R.H. 1968. "BISONt A FORTRAN Computer System f o r the Calculation of Analytic Self-consistent-field Wavefunctions, Properties, and Charge Densities f o r Diatomic Molecules", published as an ANL report, ANL-7271. /  48 1951  Revs. Mod. Phys. 23, 69.  14)  Roothaan, C.G.J.  15)  Peyerimhoff, S.  16)  Kleinman, C.J., Hahn, Y., and Spruch, L.  1965  J. Chem. Phys. 43, 998. 1968 Phys. Rev. l65» 53.  49 Appendix A Derivation of (3.6)  4- <$' i  =4<<£l  - <$'i **">•< **ift*'>  = -<<£|V*$>  • f < *q?| V ^>+<$'|<$ ><^Vp^>+2<^|V $y^l7p$ ? > ,  =  V  p  p  SO  + 2<?P$,|$'>-<?p$VPl*2>+2<7p^l^><7p$,l?p^> 7° T  ^N>r  »  .  *  =°  ar ^ r^eb r >e " 'rsJntv^rpO^ I  p  p  p  p  *  ,  /]  = ^r(2<Vp $|Vpi$>+2^lVp $>% 2  2  c  ^ *rP I rpSj-^Hp ^ >ifp 'ipSihepjep'  + / _ ! N  HplVrp~*§p'MpBp»p"*V  7  rPsih6p  I_!__>#v/ 1  rsi«ep ^ p  '  vFt  »  r sjnbpx(p r s»eP xp,/J J l  p  P  50  , v»<& 13$ J , , « "^rp C  ^  -atp '  1  >r  1 p  v  r a6 > r ->ep p  r >©p ' P  i ' ^\ , /  P  T  *  7  P  v  V sin8p p  -»r ' ^ e ^ f y '  ilL i_j  1  *<P  P  V rpSin&p->>9' P  i ^ V ip^y**?..  v  p  =^f£Vip^ + < T ^ * * * * ' W < # ™ >  ,  9  /  _ i  V | i _ J *®\(  ( i S B t ^  1  *  u  1  f  1  51  A8 <$ ^l(-^--t )TJVp^V> ,  S  p  < * ' I Vp'  I  * <$ 4 |  Vpl$*> * 2 < * * l  l  1  . - <* *'ll  pJI ^  ,  |r,  + p  r>  « + »r * .  p  ?  p  >ITp  u  + < $ $ I (-r7 - -7-)| T Ie>^ *  U  +  p  *  P  — 3 L ^ > _J  +2<$|-! r  p*»*p »fy  l U JL  V»»&p *<PP Ll  <4vpW)|-Vp(^J>.v a-[x  gas  P  ( $ ' V P **+ $V '$'+  ^  1  P  #y $V  * I  £*7 $'>-7pU-Q-  P  P  = ^ ^ V p ^ ' l Y p ^ + Kvp^'lyp^^-VpU-n -  p  <^'<!> |(^^)|-?p^)>.ypU j  AC-  =  v f>.  <  V  P  * I •j>*-p >Kp >  u  ^ P*I r ^ e p r » e u V  ?  p  p  P  v p d5'>]  52  Appendix B Let  and VpV  Derivation of  (Tp, ©p^p) denote the positron coordinate.  I t i s defined  with respect t o a system centered on the nucleus and f i x e d i n o r i e n t a t i o n . The coordinate systems of -f  and  ^  ,  ( r a ©a,T) t  and  (rb,©bJ<P)  ,  are defined with respect t o the positron-nucleus a x i s as shown below.  For purpose of i l l u s t r a t i o n , i t i s convenient, but with no l o s s of generality, t o orient x  a  (or  ) i n the 9  P  direction.  The diagram  below defines the symbols used i n t h i s and the f o l l o w i n g appendix.  The present aim i s t o f i n d out how ©a and <P vary with changes of 0p and  Qpp .  Consider f i r s t  ©a .  Since  I - fa Sin &  a  ,  r cos©a ~ |  . But JN r cos ©a C«e)(- cos <*>)  a  - ^  = -cos<p.  d  &  P  Similarly,  Consider now the case of <P  SmQp $.n e  P  which i s much more complicated.  Since  (ra - Ta cos 6a fp ) - e P = £ cos <p d ©  p  u ^ -  where  At  •-  •  P  r  a  ,  P  cos (Oa - cos  ( Ta - r  d  d  P  ©p  P  -^f  Therefore _  becomes  q>e ) C fp + d9 e )] • f 9 P  P  P  - dQ  fp]  P  , the above expression can be s i m p l i f i e d t o  cos9a f ) • e  a  H.-»Y - j ^ p  - fa cos 6 a CoS<p  (ra-raCos0aCp)-©p  Keep f i r s t order i n  d <D  -dip--  has already shown t o be  © +d8  [jTa -  "^ f j -  P  - fa cos 9  dQ  p  a  C*>S9a CP ) • | P ] = - Ta cos ©a  (fa-fa  cos  Can use s i m i l a r technique f o r  -d_5L.  .  Since  dQpp  (ra - Ta cos e r ) • Gp = 2 coscp a  P  -|^[cra-r3co oa rP).eP7^  cos  S  where  has already shown to be  T  _  - OP  2  *a cos ©a sin<p $<" ©p  ClQPp At  r  <Pp + d<p , p  [ra - ra  (Ha -  T cos6a fp) P -e  a  becomes  C05(ea+ sf'»epsin<pd<pp)Crp+ sine d<p £i )]. p  p  p  [e  P +  cosepdov^p]  54  Keep f i r s t order i n  d<p  , the above expression can be s i m p l i f i e d t o  p  C Ta - fa cosQa r ) . Q p - Pa sin ©a Sin<p Cos Sp d <Pp p  Therefore,  For  -f , the operator «*  ctr  rP  P  cos 0a Tp) • <§] = - r sin 0 sin <p cos Op  ( fa -  P  V  p  a  a  can therefore be written as  =o rP I  dQ ~ p  <jea+  ?«.ea  2<p  J  The above expression can be simply reduced t o Eq. ( 3 . 1 7 ) by using the f a c t that  -p  i s independent of 9  since  m=0. "2 C  It i s now straight forward t o derive the expression f o r V  p  i- .  By d e f i n i t i o n ,  But  = efore  s i  "^  - f£i!  t ^f  p  ">*<P ^ _  f o rm = 0  o r M t a l s  .  2  Appendix G  Derivation of V $ and "Vp ^ p  1^ v a r i e s with  Consider f i r s t how  r *=r b  2  a  where  +r  -2  2 P  rr a  P  ^ , © , and % P  .  Since  cos ©a  and &a stay constant as Y~ changes, one has P  2 T  b  ^  -rjpp- =  For  ^ 2r -2r p  cos©a  a  -7=-fr - fa cos 9a) = P  , since Va * i d  .  9  b  Yp stay constant as ©p changes, one has  a  where '  cos  ^ =. -cosSP  a  shown i n Appendix B.  s  Therefore  7) Dp  = -rar sineaeos<P_ p  = r Sin 0 sio 9 Sin Qp  -^5-  Similarly,  b  p  Consider now how Qb changes  H N r - 2r 2  P  where  b  b  ^  +  36b _ -sin 8b  ?r p  r  b  Since  p  Tp changes, one has  2 r p - 2 r p c o s e  It i s shown above that  Op , and <Pp .  Yp ,  r cos 9b  stays constant as o - 2 r  r p S | r i e b C o s ( p  b  ^  - J * ; a Cos 9h  +  2r r s.neb^ b  p  , therefore  b  56  For  ^  fa stays constant as 0p changes, one has  • , since  where  = - r sin0b  ^  p  tos<p  as shown above.  Therefore  = -coscpC-^-cxjseb - 0 Similarly,  Q  Z  b  .  r  The case for <P has already been considered in Appendix B.  The results  are  r  For  $  , the operator  b  3  r 5i^©p P  c-.nto  ^q?p"  0  • j>_-i ^.(pf  , C^^Qb -a  f < ? , n 9 K 2  »•  Y  can therefore be written as  (-^i'2  J _ J L _ =-coS<pfsin© -  _j  Vp  ' b Sm t?b  b Sin ©b  2r  b  r -j© J  b  p  b  T  l  r  b  s i  n  I Cos9b "a 1  a  Ll_l+cos(pf—J  r ae  b  '  ^_ 9  b  1 »' aq?J ^ r  ' c°sfo  aop " r  p  s;«e  The above expression i s reduced to Eq. (3.2l) for m=0 orbitals.  cas9p  B  p  b  To  2  derive V  p  using the above expression i s a straight forward but very  tedious process.  It i s not necessary to go through the detailed steps.  The result for m=0 orbitals i s Vp  %~ ^  +  $  - -pr  LSl,0  b  - ^  b  - t  y ^ - H  7  ^  -  ¥  -  _  J  *  si*e  P  57  2r  Appendix D  2  Alternative Method on Deriving V * and V ^P  Consider f i r s t the case of  .  P  Since -f i s defined with respect  to a coordinate system which r o t a t e s with positron's angular p o s i t i o n , i t i s inconvenient t o apply any operator with respect t o positron's Now we wish t o express I  coordinate on f  .  In a coordinate system  f i x e d i n space.  I t i s convenient t o choose one with the same o r i e n t a t i o n  as (x ,y ,z ). The thing t o do therefore i s t o r o t a t e (x ,y ,z ) t o p *-p p a -a -a coincide with (x ,y ,z ). How does -f transform under such a r o t a t i o n ?  -P -P "P  In the present case, since the -f 's are a l l m=0: o r b i t a l s , one can use the well known Addition Theorem f o r s p h e r i c a l harmonics t o express £ i n the new coordinate system.  ft ( c o s e , ) -  Specifically,  1  ^£Y, (e,<D)Y,* cep,^  (D.1)  m  where, as shown i n the diagram below, (9,<P) r e f e r s t o the new coordinate system (x,y,z) which has the same o r i e n t a t i o n as ( x ^ . y ^ z ^ ) and i s completely independent  of the positron p o s i t i o n .  itnin  Since  Y  {0  ( 9a") =  (cos 8a)  58  one  has  ?  R  where  n  (  n J l 0  r,©a) = R„(0(  «  a  ?m  stands f o r the r a d i a l part of -f  can be simply applied t o -f  (D.2)  ) £ Y (9,<P) Y ( e < P ) 8m  .  P)  P  The. operator  since only the term  V  Ye^p,^ i n m  now  p  e  (D.2)  needs be operated, i . e .  ^  '  r  m-  which i s i d e n t i c a l t o the one derived i n Appendix B. A s i m i l a r transformation f o r <j can be c a r r i e d out.  Now l e t  (x,y,z) stand f o r the new coordinate system which has the same o r i e n t a t i o n as (Xp,yp,Zp) but centered on the p o s i t r o n , as shown i n the diagram below.  | %  In the new coordinate system, again by using Addition Theorem, ^  can  can be expressed as  9n Cr ,e )= fl0  b  b  Rn(r)^_) £ L  "  Ve,<P)Y2m(ir-e ,n <p ) p  +  p  ( .L) D  m---f.  Unlike the case of "f , however, i t i s s t i l l d i f f i c u l t t o apply the  59  operator Vp on $ because the new coordinate ( T, & , 9 on the positron p o s i t i o n .  ) s t i l l depends  Because of the simple r e l a t i o n s h i p  (x=x'-x ,y ,z ) and * p , etc.)' between the two coordinate systems, (x p'-p*—p' x_  (x,y,z) , i t i s convenient t o consider V  i n Cartesian Coordinate.  9 % ( . X ^> *p,Vp.,Zp) where the dependence  <fy now i s of the form  Since  2 p  =  v  on x^p e t c . i s e x p l i c i t , one has  d*p  where and  ?  d «*p = -1  -4xr--  <  ~i  7*  "  a^p  -37  +  -»x  P  -a?  **P  should be distinguished. Since  a n  0*p  v^o;  -3*p  +  =. ^Xp  —  O  Xp  , Eq. (D.5) i s reduced t o  Tip  d^p Operate by  dVp  1  ~  -ay  (D.6)  -ay  p  again, one gets  ^y  ^yp  1  >*p  -a-/p  1  Similar expressions can be obtained f o r y^and z^ components.  (D.7) Since  <^ can be written i n the form  6= ZI [9.^.) 9  a  (i2p)]  as i n d i c a t e d by Eq. (D.4),  '-I  »(p  vp ^ 2  (D.8)  M  becomes  iy  aVp" " "a* 1  Hp'  (D.9)  60  The operator^ Vr  i  s  invariant under r o t a t i o n , i . e .  ^7 - ^rb r  ,  hence  L[h  ^J>}m  m  = I>r t«,L 2  b  Furthermore, since  Therefore, V # 2  P  ^ P  2  ^  $ - \ (n-9 2  Qp)  P  =  Jt(pp),  (D.10) one has  i s simplified to  = P  If2)[SV9,-?p90i» '  (3.12)  rr>  This expression i s i d e n t i c a l t o the one i n Appendix G except the l a s t term which i s d i f f i c u l t t o be s i m p l i f i e d f u r t h e r .  r  Appendix E.  0,3 0,00  r  0,20  0,50 0,00  o.as 0,50  0,53 0,60 0,70  0,299  02,660099763  0,100 •2,860100431 — o , 3 o r ~ '•2,860100699"  •3,485136043 "3,484697071 •3,464257619•3,816117943 0,299 •S.B156II2271 0,100 ~ 0, 3 0 1 ' • 3 , 8 1 4 9 4 6 1 2 9 ~ • 4 , 160015183 0,299 • 4 , 159038671 0,300 • "0,301 •— 4 , 1 5 8 0 6 1 4 5 9 " •4,116516681 0,299 •4,115190411 0.300 •4,134261599" -0,301 • 4,516122523 0,299 •4,514640281 0,300 " " © . S O I — •4,511557709 " 0,299 •4,698834971 0,300 •4,697390631 0,301 •0,6459«5979~ 0,299 »4,6'84664171 0,300 -4,883051251 0,301 — • 4 , 8 8 1 4 1 7 8 1 9 " 0,299 •5,265774673 0,300 •5,263800961 ' 0,301 ' ' • S , 2 6 1 6 2 6 7 8 9 ~ 0,299  0,100 0.301  •2.860100443•2,860101010 •2,860100413 •3,537919650" •3,536846946 •3,535774456 • 3,679808608"•3,678496945 =1,677185882 '"•3,823824474— •3,822265965 •3,820708483 "'•3,96997082«•3,968157729 0 •3,966346125 -•4,I|62«9J550,«5" 0 0* i •> v tf• 4 , 1 1 6 1 7 4 0 9 1 •4,l|4|00821 0, S 0 2 •4,268659857~ O.Sff •4,266315019 0 •4,263972708 0 '"•4,421200188"0,55" •4,418578555 0,300 •4,415960019 0,502 " " •4,575866252— "O.OOS • 4,572960800 0,500 •4,570059063 0,502 -«;75~ " 0 , 4 9 8 " " • 5 0 5 2 5 « B « « 0 ~  a  «  "oTio-  I  •0,967360546 •0,967360546 —•0,967380546•0,951751561 •0,951777240 -•0,951800335•0,947878432 •0,947838222 -•0;947797B92" •0,938152156 •0,938066575 " -0,917981675•0,930892312 •0,910795735 -•0,910700446•0,921864450 •0,921768823 •0,921675202•0,910881165 •0,910801114 -S0,910728572 •0,897681861 •0,897645946 • 0,897614416 " " •0,862970514 _ • 0 -, 6 6 3 1. 4 5 2 2 5 •0,863327338 _  —SO-,967380546"•0,967380546 •0,967380546 •0,950400571•0,950416249 •0,950431865  —i0,9«528322r•0,945293250 •0,945303370 -i"0,-93915S769•0,919163711 •0,919171869 ""•0,932635016" •0,912644165 •0,912651666 "•0,925140257" •0,925151578 •0,925167356 "SO,916893111— •0,9169112S1 •0,916933922 -•0,907917506— •0,907946667 •0,907976511 "~i0,B9823912B— •0.S982794S6 •0,898320320 -•0786527SS9T -  Electron Wavefunction  0,097699510 0,097699530 0.097699S30 0 o o9 n077Q9?2<5i 0 n9 o 0 ., 0 0,090740188 " 0,090687965 0,090034985 0,089942266 ~070B9S49831 0,091929751 0,091790014 "0,091650709 0,091881006 0,091715188 -"0,091550250 0,096501781 0.096310798 "0,096118109 0,099799130 0.099S77171 -0.09935607B 0,103768775 0.101S1690S 0,103265199 0,113752679 0 , 1 11436524 0,113120233  —0,0976995300,097699530 0,097699530 0,0767180800,078629323 0,078541106 -0,075752061— 0,075643622. 0,075535878 0,073133719— 0,073001112 0,072871167 ""0,0708631860,070708177 0,070554012 0,066940219— 0,068758040 0,068577094  00,150174021 •0,154174021 » 0 7 1 5 4 l 74021" 0,140159661 3 0,119752085 0,139345513" 0,279116729 0,278600611 07278085186~ 0,425288912 0,42461808] 0,4219869210,501119878 fl,5005«4S84 0,499848092" 0.579S61422 0,578725721 0,577885555" 0,660118125 0,659352177 -0,658382016"0 ,•7 4»4 0« 1 4" 8 «1 1" 0.742882B01 "0,741744271" 0,923054415 0,921387451 ~0;919707802"  •0;154174023" •0.1S417U023 •0,154174021 0 , 121009893" 0,120181516 0,119355148 —0,177310091 0,176134838 0,175161814 "O";234457070~ 0,233330232 0,212205887 "0,292381241" 0,291098752 0,269819046 ~0,1510f05060,149566956 0,148110504 "-OV06736220r~ - 0 . 4 1 0 2 7 0 5 6 0 0,067150756 0,408667228 0,066940745 0,407067317 - 0 , 0 6 6 1 2 6 1 5 8 — ""070700852190,065883232 0,468318104 0.065641954 0,466554721 "0,065227773— -0,530176897— 0,064951249 0,528444651 0,064676596 0,526516488 0,064499212— 0.7J1323359 _  _  0,000000000 0,000000000 0,-000000000" •0,271170174 •0,270935482 —•0.270502425" •0,396396609 •0,39589585} -»0,395396UJ•0,535197538 •0,534581839 -•0,533965628" •0,610226378 •0,609521577 -.0,608818910"" •0,689421057 •0,68860135} •0,687777869 •0,773248332 •0,772270014 •0,771285322•0,662164181 •0,66116754}  0,000000000 0,000000000 0"iO000O00OO •0,031836640 •0,031872044 —•0,031905318 "0,046161584 •0,046218182 — r,04631 48Bl" •0,063686793 •0,063606052 -"•0,061924583 o0,073737419 •0,073875622 -•0,074012706" •0,084721731 oO,084877565 •0,065029817 oO,096712507 •0,096896740 •0iO9705B639 •0,109066243 •0,110052460 " • 0 , 8 5 9 9 6 1 1 3 4 " "•0,110215401 •1,060679420 •0,140421673 •1,056733080 •0,140543205 - » r , 0 5 & 7 6 8 i n r - "oO-iT4065B7V5  0,000000000 ~ 0,000000000' 0,000000000 0,000000000 0,000000000 0,000000000 —•0,245355373— -.07041697675 •0.24454929B •0,041b59106 •0,243745795 •0,0416)9732 - • 0 , 2 9 7 2 1 3 6 8 4 "" " • 0 , 0 5 1 4 3 1 0 1 1 " •0,296269614 "0,051397550 •0,29S32B32S "0,051163072 " •0,350709786""" " * 0 , 0 6 1 7 8 2 8 5 4 •0,349617850 •0,061752769 •0,346526975 •0,061721490 " • 0 , 4 0 5 7 7 2 1 4 9 — "•0,072712845 •0,404523291 •0,072704667 •0,403277637 •0,072675069 - i 0 7 f l 6 2 3 2 3 9 0 8 "10,084258727 •0,460909954 • 0.0642H222 •0,459499795 •0,064202022 " • 0 , 5 2 0 2 8 3 2 2 1 — ' •0,096316469 •0,518696929 •0,096308611 •0,517114860 •0,096278812 " • " 0 7 5 7 9 5 6 1 6 7 ? — "voTToaoaoffOTi •0,577798781 •0,108911444 "0,576038568 •0,108680261 ~V0",6O007«722"•07122043381— •0,638125999 •0,122012834 •0,636182411 •0,121979810 —  •0,828735287"  "3V75"6"JI«6"491  9{acortftjcortb>  O O O —• 3 - « « l / < | / t ^ « * 0 ( f l  O — CO OO rt  O O O 1D3 fl B3» K O O l O - O 3 O O O O O 9r Ir t i M n j H o m w n j •-. _ o eo OOOiniAinfl4)J)NNKff 0> :o  <0 <0 3 3 3  o  *4 o o co — L.  o o  r » r\j — r u rt 3 o so o-  oirnii/inosa^  90H^r*M.4>ov* O O O LH J~i o A ^ I N N N C O N ^ n t o o - a -o m o o o o o o o o o o o o o » d ifl d m m  » O O O Q O O O O  O O O O O O O O O O O O O "  0  o o o p  t o a > c * o » o * o o o  o p O O D o o o o o o i n 9 9 9 t o »*- t n r o HI HI -*» i*- •»» *s »• O O O A j t f l o } 0 « O 9 3 yi^> 4crgao>B o o o r t ~ « * « c 0 n j 1/t 49 M AI so r* — w K N O l/l * A i O -A rtj tO 0> ^> KT —•rtr » o o • « Al IM K f» O O O J i f l i / U H ) 9 « co —• — A3 i dHI o* O O O f t t f t l A I W U t r - —• o o m m n i m co N M t r u IA|A<M>C> O O O A I M N N n i m e> o o> AJ  o o r t 7 w m r . MI s> o> . « m u i m o o o x> — » o i v m ™ « u j „ n i o >» — »  • • t n rt co •*» o> r » o> •no j ^ s »/» AI -*»rto> AJftlIP (A 9 « CO —» • *  rt  o tft  A  [  n  I  l  > o o o o o o o o o < 0 » » • 8  l  — —— U » « o » « o ( . « i n » . , o  AI * *  o o  ~ t\~\ ~ ~  ~  o o o a k o o a a  9 I A o- I A 9 w a> N 3 o  t  m  o  1  o  M  M  " m i m n n n n n n o ' o o o a ' a s a o  <*o  O O O O O  O O O O O O O O O O O O O  o o o o o o o o o o o o o  O O O O O «*Oo3AIAl| - * GO AJ 00  !  rt 9 - * r t 9 4) A » » » OO CD N K K  L L  *  O"-  r-  » > > >  fD" o p  N in o o  1/1i n  HI HI  i  J) * • 9 rf| *• m i n i ' 9 -O O 9 41 <n > i n -o o> K t o>co <o • o co -o m t n CO « 03 3 <4ft Ort o >rt0 co so co o*o> 3 4 4 0 0 <o • K X O •ortrtrtrtH V 3 O C* 0> 0> O" o* rt ru 1  ' O  ~  O O O O O O O O l  O O O O O O <  I  "  B  .  o  >o o In  =o 3 ru co o> m HI >inMonini so « N cr 4 0|> OC>Of\|N oO o o <o r » r>- .43 »/> i n o o ru o* t o co m HI r»» K> ««"t rg o c * c * -o i n i n r t H i M M sft)<yfU eo ao'co Hi M M C> O* O 1 U l t / t u t i n 3 3 3 3 3 3 HI tn i n i n m o o o o o O O O O O O O o o o oo oo ooo o o o « o o o o o o o o IO  OO  •A , I •) M  < (O (O « 4  C II  HI  i i n -o co 9  N N N O O O O ^ * ( \ | ( M r u » 0 ' C > U , I L f l U »  « o -o <o < «  I  _  v N m o o >o c i N r u o -4 O -O 9 r t r o> » n j - o « =x o o o r». r - ^> 4> I O O « 3 »»» O f\J S l> N r 9 4> m 3 O o 9 r u m cor*» >o r*\ AI •>* co r*. to • > i n o» o> o* HI r u f\ **\ -a ~o VQ • —• o o o - * n j u> « -a o *O* O* rt n i r a n i f t i f M — rti • I • O O O O O O O O O O Do Oo o o o o p o o p  1rtSO 4  O >0 O <\| (O l 9 i t 9 (O OO  _  > D 9 I Aft!9 <A  OOom»«-AOKinO4»^f«.4>0>-«4l9< t n t n i n « - w r t M r o « * * ~ ^ e ? 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IA4fUOIAOtOOrt-r\|*.NO o> o» m r 0 > 9*N. «o-(il *-9inocOAIOO o O 4> « ort— on t> ru—>4)rt9rtHtir> O O ( tnt>rtNCOCOe>l7>IAAIOrt4*W L * - *rtr>4 > r * > 9 o r u s 9 i « - > r t r > r t c o r t c o O O 4 o* 9 9 9 o* c o c o r t r t A i c o r - » r - > A i SO 4> to co l^>0 OOAllV)AliAiniANKh>0 ,  v  a  . j ..  i n o i ^ i n o t n u ' t o t n i n o t n i n o t n i n o i n i n c e> o o o> o o o > o o a o o o > o o o - o o o > < O r ^ r ^ o r ^ r ^ 4 ) r ^ r > . - o r ^ ^ > o * > . r ^ 4> **» i * . 4> r  O 0 o o o  i  4»<-o—or».—*rurto rrtr> rtrt<omrt9.omr>co9 I CO o ini/ir>«*ivin,u>4comi/i I o o r> o co«-«rtin3>*->4>w«i/) > -O 4><o I n X M O O x t h o n i f f M i M ' r> r> 4>4)9IAiArt99AJrtrt > o oo I-**r— 4 > 4 1 4 } 4 > 4 > 4 1 4 1 4 > 4 } O O O O o O o o O o O oO oO oO oO o o o o o o c  T  t > o n » r t a » o > r M K r t n j u n a L _ ««*4i/)>«ni«irirtoAtON«i-*j) 9rto-omAio>Airtrtrt-*r>io*« N«3f>iM<Aomnoo*4rur>90' N O - r y cjsruinojio-a 9 r u o> * * * - I ' , » S 0 9 9 9 0 0 0 l A 4 1 0 0 > 0 0 9 9 9 9 9 9 999rtrtrtr>>cOC0 4>o-oo<o4<io««>o4inir)iA  1 I  o  >  O O O O P O 4  ...............  o o o o «  (\icgrycoffo-oNcorvj-onj(Dt>Nrt 9cort9tn Ai-OAiru»r>r>9rtosoookrtrtin OOOOffUJU'0(\J«INN(>M/loo<lW w o o A rUi->.«M^4n)p*«-)9 9 o 9 9 9 4>0-fMD4>rtAJ-J^)9-««00ru990in 9 9 9- rj itr0t «i s- > rt9-/ioor\ii/iK9«->0'0>r> - . « - N r - ( o I\J rv ^ K N O O xrvjrvii/i^su'i 4 K . « - * - «N• «r \ J i A t D r > A | t r t - T E 0 _ c 0 A J 4 ) O 9 C 0 9990iS(aacO(ONNNKSN««4ooo •«rtrtrto«-.-«aocoo>in-04>rtrtrt t n i A i n 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 * - . * * ~ - > > - « - * - « o o o o o o o o o u^iAininiriiAiAink/ti/iLnu-iirtirtiAiA  o o o o o o o o o o o o o p o  >  o o o o o «  —  p  —  l»  ru  >4>904>904>904>k O 41 9> > 0 0 * - O 0 0 * ' 0 0 0 ^ 0 0 t > 0 0 0 k  —  rtrtirtrtrtrt rtrtrtrtrtrtrtrtrtrtrtrtrtr  i I  u  9 9 4 9 9 "9 9  L 1•  I  L  4.100 4,106 4,094 4,100 4,106 0,094" 4,100 4,106  0,75 "17 1TB  8.5  0,00_  4.493 "4,500 4,507 4,493 "4,500 4,507 0.50 4,493 "4,500 4,507 0.55 4,493 -4.500 4,507 0,60 4,493 "4,500 4,507 0,65 4,491 4,500 4,507 0,70 4,493 ~",500 4,507 0.75 4,493 —4,500" 4,507 4,493 1.00 "4,500" 4,507 _  0,60  •!,20232(420 •3,201814530 •3,2273686/8 •3.226B23IS0 -3,226279138 •3,350391728 -3,349657170 •3,348924968  •2,860100975 — •2,860100983•2,860100986 •3,060605175 — •3,060291318" •3,059978457 •3,082890715 « 3 , 0 8 2 5 4 IB68" •3,08219(1117 •3,105179685 — •1,100795778•3,104411087 -1,127472555 •3.127051538-3,126615847 •3,149769895 — •3,149315705•3,148862957 •3,172072425 •1,171582988" •3,171095107 •3,19438|04S • 3 , 193856278"" •1,193313177 •3,306072285 •3,305168748" •3,304667487  -  •0,966326160 -0,9*6337744 •0,966176487 •0,966189744 .0,966202864 i0,965015035 •0,965041149 •0,965066995  0,095983443 0,096001440 0,095838312 0,095857946 0,095877170 "07094960049" 0,094989458 0,095018576  0,967180546 0,967380546 0,967180546 0,967098564 0,967102249 0,967105888 0,967061837 07 9 6 7 0 6 7 9 9 4 0,967072099 0,967025053 0,967029738 0,967034364 0.9669B152B 0,966986806 0,966992017 0,966932400 0.966938347" 0,966944218 0,966876561 0,966883269 0,966889891 0,966812565 0,966820145" 0.966B27630 0,966269661 0,966284613 0,966299372  0,097699530 "070976995300,097699510 0,097094718 0,0971022740,097109723 0,097051923 "070^70601)5—" 0,097068095 0,097006959 0.097015718" 0,097024375 0,096959482 0,096968902"" 0,096978211 0,096909036 0,096919158" 0,096929159 0,096855038 "0,096865910" 0,096876651 0,096796733 0,096808413" 0,096819952 0,096388970 0,096406243" 0,096423301  •0,154174021 0,000000000 0,000000000 —SO,"! 5 * 1 7 4 0 2 1 070 00 OOO 0 0 0 " 07O0O000000 • 0 , 154174023 0,000000000 0,000000000 •0,152737460 •0,007085224 •0,006060518 —SO", 1 5 2 7 5 5 0 4 5 " " S 07 0 0 7 0 1 5 7 4 5""— • 0 , 0 0 6 0 2 8 2 2 7 •0,152772429 •0,006986599 •0,005996044 =0,152630631 •0,007869700 •0,006496596 —S07152649685" D7007B14225" ~SO;00646T352 •0,152668519 •0,007759127 •0,006426254 •0,152517728 •0,008739921 •0,006949739 — • 0 , 1 5 2 5 3 8 3 3 5 " "•0,008677821" — » 0 ; 0 0 6 9 1 1 4 5 1 •0,152558700 •0,008616147 •0,006873328 "0,152397745 •0,009710653 •0,007421231 "'•0,152420001" • 0 , 0 0 9 6 4 1 1 9 5 " ""•070073797B1 •0.I5244199S •0,009572218 •0,007318515 •0.152269419 •0,010800237 •0,007912517 ""•0,152293440" •0,010722563- "•0,007867777 •0,152317174 •0,010645428 •0,007823240 •0,152131144 •0,012031737 •0,008425267 "SO,152157060" SD7O0B377096 "0,152182674 •0,006329150 •0,011658529 •0,151980835 •0,008961430 •0,013434549 ""•0,152008821 " " • 0 , 0 1 3 3 3 7 1 9 8 ""•0,006909677 •0,152036468 • 0,006858169 •0,013240524 •0,150907384 •0,012096032 •0,024900907 •0,150950009" "•0,024721331 "•0,012023405 •0,150992107 "0,011951144 •0,024542966  0,097699530 0,097699530 0,097699530" 0,097603402 0,097605316 0,097607194 0,097597803 0,097599837 0,097601832" 0,097591876 0,097594037 0,097596156" 0,097578782 0,097581221 0,0975836TB0,097563390 0,097566156 0,097568869" 0,097544389 0,097547557 ,v975506"6r"  •0,154174021 "0,154174023 "iiO,154174023" -0.153934796 •0,153939438 "•0,153943994 •0,153920202 •0,15392S1S0 "^0,153930006" •0,153904664 •0,153909958 -•0,153915132•0,153870169 •0,153876165 "S0",153882007~ •0,153629245 •0,153836096 "•0,153842815" •0,153776314 •0,153786226 -50,l5379T9ffS—  •0,967380546 •0,967380546 ~~iD, 967380546" •0,967335784 •0,967116682 ">0,967337562 •0,967330589 •0,967331592 "•0,967332576 •0,967324772 •0,967325895 ,967326995 ,967310798 ,967312208 ,967313590" ,967292511 ,967294297 ,967296048 ,967267445 -9fc7?h07U9  •0,150052957 •0,150095943 "0,149681805 •0,149710905 •0,149777541 •"07147176700" •0,147449585 •0,147521701  -0,018014851 •0,017905112 •0,020259928 •0,020136350 •0,020013487 • 07037364283" •0,037141097 •0,036919105  •0,011508297 •0,011455360 •0,012340326 "0,012282939 •0,012225763 "5"0;016879851 •0,016797995 •0,016716458  —;o";ori944B3o  0,000000000 0,000000000 0,000000000•0,002828014 •0,002798729 "•0,002769741" •0,003116601 •0,003084106 —.07003051942" •0,003439539 •0,003403457 "•0,003367747" -0,004216173 •0,004171496 "~0—J04T272BJ— •0,005234788 •0,005176900 "-0,00512359;•0,006626588 •0,006557460  0,000000000 0,000000000 0,000000000 •0,002803989 •0,002778761 •0,002753729 •0,002970325 •0,002943368 "SO;002916625 "0,003143525 "0,003(14769 "•0,003086245 •0.003S1266S •0,003480077 -~)700J4TI7757 •0,003916569 •0,003879786 "•0,003643313 •0,004162456 •0.004121039  "V0"7o-Tj6TJB7i20" "TTOrO—>T9 975  t.00  f-575-  "OTBfl  "T770 "O7B0 * •  T700  5,490 5,500 5,510  -J,221515781  -1.223851055 -3,223188822  6 T 0 7 0 - "•1,045578966"  6,500 6,530  •3.044722559 •3,043874046 6,470- -"•3,076492286" 6,500 •3,075493119 6,530 •3,074503166 6,470" "•3,1074059566,500 •3,106264009 6,510_ •3,105132586 "•I,1183201266)500 -1,137015359 6,510 •3,135762446 5747O— --37I59235096 6,500 •3,167807459 6.S30 -3,166393006  •0,967210562 -0,967211631 -0,967236637  0,09751914* 0,097522842 0,097526466  «0,151710180 -0,153719507 -0,153728652  "•0,967373228•0,967373676 •0,967374097 —•0,96737146J" •0,967372021 •0,967372545 -•0,967369168" •0.967369869 •0,967370528 "50,967366043•0,967166940 •0,967367783 -•"079 6 736146^•0,967362651 •0,967363765  -0709T6857650,097686629 0,097667441 -07097680250 0,097685214 0,097686119 -0,097682453" 0.097681S33 0,097684546 "0,0976802060,097681430 0,097682579 "070976771730,097678591 0,097679921  -STJ7154137750" •0,154139976 •0,154142067 -»0,154133602•0,154136091 •0,154138433 "•0,154128654" •0,154131460 •0,154134096 "•0,154122450" •0,154125649 •0,154128654  ~"-07154nT067" •0,154117796 •0,154121297  •0.0086S0121 -0.008557195 -0,008465244  .0.004861049 -0,004814823 -0,004768601  -*07O01212182 •0,001173789 •0,001136596 "50;00|471159 •0,001424159 •0,001378644 -•0,001812446 •0,001754142 •0,001697694 ~tfO", 002282517 •0,002206713 _»0,002137273 • 070 0297081. •0,002874535 •0,002781281  5"<M>01?2Tff39 -0,001185485 -0,001150140 SO",OOI3S0661 •0,001310121 •0,001270715 -0,001491612 .0,001446486 •0,001402637 •0,001647025 •0,001596639 •0,001548089 5TJ70"OT8T03T2 •0,001764539 •0,001710109  Appendix F, O.J  0,00 0,20 0,30 0,40 0.4S  0,50 J>.S5_ 0,b0 0.70 0.5  0,00 0.25 0,30 6,35" 0,40 0.45 0.50 0.55 o,*o_  "6,75 1,00 Oo»  0,9  1.1  0,00 0,40 _0j45 0,56" 0,55 0,60_ 0,00 0,40_ 0,45 0,50 0j55_ 0,60 0,65 0,00 0,45 0.S0 0,55 0,60 J.,65 1,00 0,00 0,45 0,50 0.55 0,60 0,65 • t.Og-  1.5 i f~  0,00 _0.45_ 0,50 0.55 0.60  1,4948 1.623S 1,6859 1.7480 1,7790 1.8100 1,8410 "1,8722 1,9356  _  3,9304 4,6474 5,0044 5,3736_ 5.5632 5,7565 5,9542 6,157t' 6,5836  Matrix Elements between Molecular Orbital §  0,0000 •0,0303 •0.0654 _»0jU>93_ •6,1381 • 0,1714 •0.2101 • 0,2514 " •0,3654  2,7134 1,2995 1,4059 3,2211 _ 3 ,3268_ J . ^ ' l 1,4483" 3,4347 1,4695 3,5445 _1_,4 90B_ 3,6563 1.5120 3,7699 1.5332 3,8852 J . 5 5 4 4 _4,0022_ 1.6174" 4,3615 1.7196 4,9805  0,0000 0,0000 •1,2274 0,7705 •1.8819 1,5991 =2 ._6 7 0 3^ 2,9249 •3,1480 3,9295 •3,7013 5,3436 •4,3541 •5,1258" 10,6111 •7,1703 23,0488  0,0000 0,"66"00 •0,0327 •0,8161 n o u t •0,0465 ^1 .0039 •6,0624" "•I ,"2024" •0,0822 • 1,4148 ^0, 1028_ ^ U 6 3 6 6 •0.1272 •1,8 7 "20 •0,1532 •2,1182 •0.1833 •2,3783 • 0,2656 •3,2257 •0,5097 •4,8730  0,0000 0,0138 0,0305 0,0537 0,0678 0,0835 _0,I012 6.1211 0,1713  0,0000 0,0000 0,0074 0.0074 0,0161 0,0161 0.02822 0,0282 0.0358 6,0356 0,0444 0,0444 _0,0542 _0,0542 0,0653" 6,0653 0,0924 0,0924  0,0000 0,0000 • 0,0000 0,0000 gCVCVA VA V 0,0159 A 0,0080 0,0080 0,5590 _0.0226 0,0115 _0,0115 0,7933_ 6,0305 0,0i56 0.01S6 1,0748 0,0396 0,0204 0,0204 1.4068 0.0499 0 0259_ 0,0259 1.7914 0,0613 0,0322 0,0 322— 2.2310 0,0739 0,0392 0,0392 2.7282 0,0875 0,0469 0,0469 3,2841 0.1347 " 0,0746" "0,0746"'"' 5.3286 0,2330 0,1361 0,1361 10,0503 1,1190 1,8797 0,0000 0,0000 0,0000 O.OOOO 0,0000 0,0000 1,2359 2,3844 •0,0546 •0,7606 0,7148 0,0256 0,0141 0,0141 2,4556_ •0,0688 _=0,9093_ _0.919t Oj.0324 " Jj* 12_ 1.2667 "2,5286 •6,0858" •1,0465 1,1532" "6 "" 1.2822 2,6033 -0,1038 •1,1903 1,4189 2,6797 •0.1241 -1.3422 1.7172 -L.A?L8_ V  t  s  0,9649 1.3217 0,0000 0,0000 0,0000 0,0000 l_t 0465_ _lj6458_ •0,015S_ •0,4620 0.9028 0,0160 1,0599 1,6937 -6,0456 "•"0.5424 0,5217 ""0,020 4 1,0716 1,7433 •0,0570 •0,6278 0,6585 0,0253 I,0635 JL»7."»£_ •J>,0692_ _«0j,7l82_ _0,8139 _0 0308 1,6956 1,8471 •0,0836 -6,8146 0,9889 0,6369 •0,0990 • 0,9162 1,1078 1,1836 0,0437 1,9015 0,8375 0,94 80 6 , 6 o"6 6" 0,0000 0,0000 ""676566 0,9079 1,1983 •0,0306 -0,3382 0.3177 0,0130 0.9169 L 23J2_ •0,0386 _-0.1944 0.4030 0.0163 •0,6469 0,9263 1,2695 •0.4540" 0,5007 0,0200 . 0,9359 1,3074 •0,0568 •0,5160 0,6107 0,0242 0.9457 1.3469 j.0,0682 _^0,5868_ __0,7342 0,0288 - |vfcvw 1,0215 1,6672" •0.1797 •1,1916 2,6613"" 0,0759 t  1  L  _0,7334 0,7863 0,7935 _0,8009 0,8066 0,8166 0.8814  0,6953_ 0,8678 0,8929 0,9J9J_ 0,9472 0,9765 1.2264  _0,0000_ •0,0209 •0,0261 ^0,033l_ _•0.0396 A A "f A A. •0,0472 •0.1325  0,6465 _0,6B85_ 6,6941 0,7000 0.7062  0,5221 0,0000 0,0000 0.6424 _»0j,0146_ _i0j14 0 2 0,6605 •0,0180 •0,1655 0,6798 0,0233 -0,1944 0.7004 •0.0285 •0.2255  _0,0000_ -0,2161 •0,2537 _M>_,2952_ •0,3390 •0,3664 •0.8297  0,0000 0,0420 0,0923 _0..1S66_ 0,"1969 0,2405 0,2929 0,3618 0,6314  1,0000 0,0000 1,0000 •0,0676 1 ,0000 •0,0974 1 ,0000 •0.1266 •0,1417 i .0060"•0,1577 1,0000 •0,1753 "•6,1957 " J.O00O 1,0006" •0,2531 -  1,0000  0,0000 0,0000 — I T O J o o " 0,0254 •0,0566 1,0000 0,0349 •0,0665 1,0000 "0,6456 ""•6, 0762" " 1,0000 0.0573 •0,0857 1,0000 •0,0949 1,0000 0,6S3f •6,1040 t,6000 0,0966 •0,|127 1,0000 0,1104 -0,1212 _ 1 , 0 0 0 0 "0.1S16 •0,1446 1,0000 0,204V •0,1759 1,0000 0,0000 0,0199  0,0000 1,0000 •0,0543 1,0000 =0,0601 1,0000 •0,0657" "1 .0000" •0,0712 1,0000 •0.0765 1,0000  0,0000 0,0000 0,0000 0,0000 0,0096 0,0096 0,0080 •0.0359 6,0123" 0.0123" ~ 6 , 0 0 9 r •0,0198" 0,0155 0,0155 0,0114 '0,0435 0,0190 J>j0190 0.0133 0,0471 0.0229 0,0152 0,0506 0,0229 0,0272 0,0171 -0,0540 0,0272 0.606T"  0,0006""^  6,6606" " 0,0660  0,0000 • 0,0000 •0,0000 •0,0000_ •6,60oi •0,0001 •0,000l_ •0,0001 •0,0001 "0,6000 •0,0001 •0,0001 •0,0001 •0,0001 •0 ,0001 • 0,0 O i l -0,0001 • 0,0002 •0,0002" •0,0002 0,00"06~ •0,0001 •0,0001 ••0,0001" •0,0001 • 0,0001  1,0000 1,0000 "l.OuOO 1,0000 _ I 0 0 00 1,0006 1,0000  0,0000 •0,0001 •0,0001 • 0,0001 • 0,0001  TToTioo  "T,"6So"S"  A  •o.oOoT" • 0,0001  0,0085 0.0107 0,0132 0,0161 0.0192 vj^v * TI 0,0514  0,0085 0,0107 0,0132 0,0161 _0,0192 v ,ju, T C 0,0514  0,0051 0^00 61 6,0674 0,0086 *0 ^0098 0,0189  •0,0282 1,0000 •0,0001 •0.0109 1,0000 •0,0001 •0,0116 1,0000 • O.OOTf" 0,0162 1,0000 • 0 , 0 0 0 1 • 0«0 5 S 6 _1,0000_ 1, uuoo • 0 , 0 0 0 1 "•0,0511 1,6666" 0,0000 •0,0211 •0,0233 •0,0255 •0,0276 -0,0296 -0,0396  0.0000 0.2017 0,2574 0.3216 0,3948 0. 4771 1. » 3 4  0,0000 0,0067 0,0110 0.0135 0,0165 0,0198 0.0548  0,0000 0,0060 0.0075 0.0093 0,0114 w • v • « -1 " 0.0137 0.0384  0,0000 0,0060 0,0075 0,0093 — ' 0,0114 0,0137 0.0384  ... 0,0000 O.OOle" 0,0047 0.0056 0,0066 0,0077 0.0156  0,0000 p ,1318 0,1690 0,2123 0.2625  0,0000 0.0061 0,0077 0,0096 0.0118  0.0042 JJL? P42 * -'---""  0,0000  0,0000  A AAAfl 0,0000  0,0053 0,0066 0,0081  0,0053 0,0066 0,0081  0,0018 0,0047 0.0056  0,0 042  •6,666?"  AA AA 0,0000  A  0.0 0 30 - 0 . 0162 -O.OIBl -0,0200 -0.0219  1.0000 1,0000 1,0000 1,0000 1,0000 1,0000 1,0000 A A A rt 1,0000  «  0», 0I0 V0 V0W ft •» o , T 6 o l •0,0001 •0,0001 •0,6061 •0,0001 •0,0002  0,0000  1.000 0 •0,0001  1,0000" •0,0001 1,0000 • 0,0001 1.0000 • 0,0001  0  I"  0,65" 1,00 1.7  0,00 0,45 _0j50_ 0,55 0. 60 _0,65_ 1, bo - 0 , 0 0_ 0,<15  0,50 -0.55.... 0,60 0,65 _K00_  2.1  0,00 o««s_  0,50 0.55  _o.*o_ 2,7  6,65 1.00 0,00 0,a5  _o_.so_ 0,55 0,60 _0..65_ 1,00  _o.oo_ 0,45 0,50 _0-.55_ 0,60 0,65  J.J  J.7  0","7l27~ ^ 7 2 2 4 " 0,7684 0,9207  •0,0144" "V0"~!51»S- "673T95 •0,0997 •0,5934 0,9544  0,5789 0,4014 0,0000 0,6091 0,4655 •0,0106 _0,61 J4_ 0,4965 _?0j0133_ 0,6161 0.5126 " •6,0181 0,6230 0,5279 •0,0202 _0,6283_ -°-o5444_ • 0,0246 6,6759 0,7023 •6,0766  0,0000 0,0044 0j,0057  0,6671  0,0086 0,0107 0,0321  0,0290 0,0000 0,0029 0,0037 0,0047 0,0058 0,0070 0,0222  6,006r 0,0144  1,0000 "VOTiTO^r 1,0000 •0,0001 •0,0329 0,0000 0,0000 0,0000 1,0000 0,0000 0,0029 0,0024 •0,0126 1,0000 •0,0001 0,0037 _ 0 , 0 0 3 l _ •0,0143 J^0,0001_ o.ooiir 0.66J9 •0,0159 1,0000 0,0058 0,0047 "0,0176 1,0000 •6,0001 0,0070 0.0056 _"_0,0192 _1,0000_ •0,0001 0,0222 0,6134 •6,0284" 1,6666 - 0 , 0 0 0 l _ 0,0290  ^676«r  •6,6ooi _ 0,0000 0.0000 _O 0OOO 0,0000 J>.,0000_ _0 , 000 p_ _0,0000_ X.oooo 0,0000 •0,0601 0,0590 0,0033 0.0021 0,0021 o",ooi9 •6,6698 "1,666o" •0,0001 " •0,0721 0.0763 0,0043 0,0027 0,0027 •0,0112 1,0000 •0,0001 _^0,0858_ _0,0969 0,0054 _0,0033_ 0,0033_ 0,0024 •0,0126 1,0000 •0,0002 0,0031 •0,0141 •6.1014" " 6.1212" ""6,0067" 0.0041 6,0041 1,0000 •0,0002 6,0038" •O.OtSS •0,1186 0.1497 0,0082 0,0051 0,0051 w ** ~ » • •» » •» a Wffvw^t 1,0000 •0,0002 0,0046 •.0j3171_ _0j5096_ 1,0000 -_0,0003_ JL«0?_57 0,0170 0,01701 - J L n O l f L . •0,0248 0,4737 0,2539 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,0000 0,0000 0,4905 0,2944 •0,0057 •0.039S _0,0402 0,0026 0,0015 0,0014 •0,0076 1,0000 •0,0001 0,4930 0,3010 " • 0 , 0 0 7 4 " •0,0477 " 0,0520" 6,0033" 0,0019" _0,0015 6,0019" 0,0018 " •0,0087' "1,0000 •0,0001 0,4957 0,3063 •0,0092 •0,0571 0,0663 0,0042 0,0024 0,0023 •0,0099 1,0000 •0,0001 0,4967 _0,1164_ •0,0115 =0,0680_ _0,0833_ _0,00S3_ 0,0024 0,0030_ _0,0030 _0,0029_ J ; O , O I U _ _ _1,0000_ •0,0001_ "6,5026" 0,3254 •6,0140 •0,0604 6,0065 "0,6036 6,1036 0.0036" 6,6636 •6,0124 1,00 00 •0,0001 0,5355 0,4242 •0,0477 •0,2337 0.02H 0,3795 0,0131 0,0108 •0,0216 0,0131 1,0000 •0,0002 0,3700 0,1469 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 "i76666~ 0,0000 0,3766 0,1599 .0,0024 •0,0112 0,0134 0,0013 0,0005 0,0005 "0,0033 1,0000 •0,0001 -0,3775.. _0,J62J • 0,0 0 31 _ ; 0 , 0 J 3 7 _ -OiPJIL. 0 ' 0 ° l * _ 0.0007 _0j0007_ 0,0005 0.0006 _^0_,J0 038_ _l OOjOO_ •0,0001 0,3786 0,1646 •0.0040 •6,0166 0,0021 0,0219 0,0009" 0,0009 0,0008 •0,0044 1,0000 •0,0001" 0,3799 0,1674 .0,0049 •0,0201 0,0276 0,0026 0,0011 0,0011 0,0010 •0,0051 1,0000 •0,0001 _0,3812_ . P . J 7 0 7 .0,0061 _-0,0243 0,0346 _0,0033 _0,0013_ _0,0013_ J . 0 0 1 3 »0,0058__ _1,0000 •0,0001_ 0.3983 0,2148 •0.0240" •6,0910 6,i547 6,6i23" 0,0057 6,0057 1,0000 -0,0002 6,66«3" •6.0131 _0,3446_ _0,1260 0,0000 0,0000_ _0,0000_ __0,0000 0,0000 o . o o o o 0„0000 0.0000 1.0000 0,0000 '6,3494 0,1349 '0,0019 •6,6674 0.0095 0,0010 0,0004 0,0003 •0,0024 0,0004 1,0000 • 0,0000 0,3501 0,1363 0,0023 •0,0090 0,0121 0,0013 0,0005 0,0004 •0,0028 0,0005 1,0000 • 0,0001 _P,3508_ J>j 1 J80 0,0030 _«0 0110 _?.0153_ _0,0016 . 0.0006 0.0005 JJ0,0033_ __1,0000 ;0,0001_ 0,0006 0,1400 0,3517 6,6637" •0,6)33 0,0193 0,0008 0,0621 0,0008 0,00 06 1,0000" • 6,6001 •0,0036 0,1422 0,3527 >0,0045 •0,0161 0,0241 0,0009 0,0026 0,0009 0,0008 •0,0043 1 ,0000 •0,0001 0_,3657 JbJ 74 7 -0.0189 _-0,0649 0,1124 0.0042 0,0102 0,0042 0.0047 •0.0107 1.0000 • 0,0001 X  t  x  _  L  0,3030 0,0958 _P,3054 _0,0998 0,3057" 6,1005" 0,3061 0,1013 0,3065 0,l02l_ " 6 , 3 0 7 0 - 6,1032 0,3141 0,1199  0,0000 •0,0011 •0,0013 •0,0017 •0,0020 •0,0026 •0,0113  0,00 0,45 _0,50_ 0.55 0,60 _0,65_ 1,00  "T,"2 7 03"  "o"V6666~  0.00  0,0000 0,0676 0,1129 0,1426 0,1774 0,2176 0,6914  0.5215 0.3158 0 000 0_ 0,5441 0,3744 •0,0078 0,5475 0,3837 •0.0098 0,5510_ _0,39J9 •0,0124 0,5549 6,4051 •0.0155 " 0.5591 0,4173 •0,0188 _0,5994_ - J . . 5 » 2 8 _ _TP-t-?.*04_  0,00 _0 «5_ 0,50 0.55 0,60_ 6,65 1,00 J  0,0000 •0.0917 j^0,1091_ •0,1297 •0,1506 "0,1752_ •0,4314  V~,WH~ 0,0412  0,0755 0,0773 0,0775 0,2718"" 0,0779 0,2720 0,0783 _0 2722_ 0.0787 6,2758 0.0867 0,2714  J  0.2439  0.0610  0,0000 •0,0032 •0,0039 •0,0047 •0,0057 •0,0070 •0,0315  0,0000 0,0049 6,0061 0,0076 0.0095 0,0118 0,0570  0,0000 •0,0014 •0,--. •0,0009 •0,6020 •0,0010 •0,0024 _ » O , O O I J L _?0 0030_ •0,0062 • 6,6143  0,0000 0,0025  •0,0006  t  0,0000  0,0000  v j v v  jw  0,0276" 0,0000  0,0000 0,0000 0,0000 0,0000 0,0000 _0,0006_ _0,0002_ _0j0002_ _0.0O0J_ •0,0013 0,00 08 6,0002 0,0662 0,666| •o'.ooTS" 0,0010 0,0003 0,0003 0,0002 •0,0018 0.0004 0.0004 _it?0J2 0,0002 •0,0020 o.ooiT" O.OOOS 0,0065" ~6"75OTJ- •0,0023" 0,0022 0,0066 0,0022 0,0020 •0,0065 0.0000 0,0004 O.OOOS 0,0006" 0,0007 0.0009  0.0000 0,0001 0,0001 0,0002 0,0002 0.0002  0,0000 0,0001 0.0001 0,0002" 0,0002 0.0002  1.0000 J_,0000_ 1,0066 1,0000 _1.0000  0.0000 •j>,0000_ •6,6600 "0,0000 •0,0000  1,0000  I TSooo •o,6"0oo  0,0000 0,0000 OjOOOO 0,0000 0,0000 0.0000 0,0005  0,0000 •0,0007 •0,0008 •0,0009 •0,0010 •0.0012 •0,0036  1,0000 1 .0000 1,0000 1,0000 1 .0000 1,0000  -0,0001 0,0000 •0,0000 •0,0000_ •0,0000 •0,0000 0,0000 • O.O'OOl  0.0000  0.0000  1,0000  0.0000  1,0666"  I  1  O O O O O O O O O O O O O O O O p o O O O O O O j O O O O O O O O  I ?  0  0  0  0  !  0  0  o  o  *» m m  <t  o  o  O O O O O O O * J O O O O O O O O o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  ;o o  o  !o o  o O O O O O O o o ru  •o o  o  <  o  o  o  o  jo  o  t:  o  o  rt9 i n 411  p urt  O O O O O O JOAI O O O O O O O O  f o o o o o o p o o o jo O  > o  o p  o  Ioo  0.0  o  o  ;o  **  000—•«-•>** j . * , b o O O O O o o 0000000 000000 ;oo  —!  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C  3  O-  Appendix G,  V (r , lj) and £ ( Vp. @, U) p  \  - §*)  f  A AH  0 1  V t UO  •2,9b96 3,6771 •3.1935 3,4732 •3.2532 _ 3 , 4 1 J5 •3 2664 •J , 36 01 •3J2928 3,3738 •3.2924 3,3743 ™* j1eo*» / 3,3620 p H li •3,2691 3,3976 •3,2090 3,4577  3,3920 1,0708 0.43U 0,1566 0,2231 0,5188 1,1774 2,4219 8.1295  •2,599| •2,7538  2,0492 0.5224 0,1461 0,2199 0.1419 0,1286 0,1734 0.2661 0,4680 1,4683 4.9869  •2,2379 0,6192 -2,3707 _ 0,4864 0,4813 tt •2,3736 I 7 t0,4835 •2,3701 _ P.4871 .a TK la 0,4940 C|3DJc  0,20 0,30 , " 0.40 0,45 0.50 0.55 •  0,0000 0.0317 0,0452 0,0611 0,0793 0,0998 0.1227 0,1477 0,1750 0,2694 0,4659  0,0000 0.0160 0,0229 0,0312 0,0406 0,0519 0,0644 0,0783 0,0937 0,1491 0.2721  0,0000 0,0160 0.0229 0.0312 0,0408 0,0519 • 0,0644 0,0783 0.091/ 0,1491 0.2721  1.2553 0.1169 0,0911 0,0984 0,1440 0,2290  0,0000 •0,0905 •0,0819 •0,0744 •0,0620 •0,0467  0,0000 0,0512 0,0647 0,0799 0,0966 0,1150  0,0000 0.0281 0,0159 0,0447 0.0546 0,0656  0.0000 0.0281 0,0159 0,0447 0.0546 0,0656  0,2925 0,1IS I 0,1917 0.1905 0,1918 0,1955 0,2017  0.781S 0,0647 0,0615 0,0566 0,0762 0.1212 0,1903  0,0000 •0,6702 •0,0682 •0,0642 •0,0583 •0,0507 •0,0413  0,0000 0,6326 0,0408 0,0506 0,0616 0,0739 0,0674  0,4931 0,0410 0,0176 0,0469 0,0707 0,1110 0,9957  0,0000 •0,0515 •0,0492 •0,0457 •0,0406 -0,0344 0,0176  0,0000 0,0261 6,0326 0,0400 0,0484 0,0576 0,1519  0,0000 0,0171 0,0215 0,0265 0,0321 0,0184 0,1029  A AA U f uu 0,45 0.50 "*. a, 5 0,60 0.65 11 | 0V 0 U  0,1431 _0,0658 • 1 ,7543 0,0639 0,0637 •1,7544 0,0654 •1,7528 9mi 1 UQ I0,0669 1 ' *• » J 0,1547 •1,6635 •1,4669 0,0716 •1.S270 0,0115 •1.5291 0,0094 • 1 | DCT*J 0,0085 0,0090 •1.5295 0.0109 •1.5276 •I,4648 0,0736  0,0000 0,0171 0,0215 0,0265 0,0321 0,03 6* 0,1029  0,1146 0,0318 0,0271 0,0107 0,0452 0.070S 0,6740  0,0000 •0,0187 •0,0173 -0,0147 •0,0309 •0,0261  0,0000 0,0174 0,0219 0,0271 0,0330 0,0395 6,169}  0,0000 0,0119 0.0150 0,0186 0,0227 0,0273 0,076?  0,0066 0.0119 0,0150 0,0186 0.0227 0.027) 6.6767  0.00  •1.2970  0.2030  0,0000  0,0000  0.0000  •2.7632 •2,7910  0.55  •2,7925 •2,7662  0,60 0.75 1.00  •e.»/31 •2.7135 •2.5180  0.00 0.40 0 n't V , M3 0,50 _.0.S5 0,60 0,00 0,40 0,45 0,50 0 S5. »i ' 0,60 0.65 3  1.1  • I *  a  r  0,00 0,45 0 50 0^55 0,60 0.65 1,00  u  •  ,  1.5  o T o ^ ^ m ^  0,0000 •0,1191 •0,1163 -0,1069 •0,0909 •0,0690 •0,0418 •0,0094 0,0269 0.1586 0,4089  i  f  0,60 0,70 0,00 0,25 0 iO 0J35 0,4 0 0 45  o.so  III  o.oooo i•CIP.J il rill. •0,1324 ' •0,0880 0,0131 0,0820 0,1655 0,2703 0,4105 0,9591  0,0149 0,0321 0,056$ 0,0716 0,0668 0,1084 0,1305 0.1848  0,5  0,9  *2<st-ec) 0,0277 0,0609 0,1074 0,1355 0.1671 0,2020 0,2421 0,3427  w  0.T  •  .3 lino • c | (fUO  _ 3  jnii i "  1,4009 1,2462' 1.2292 1,2168 lj2090 1,2059 1,2075 1,2138 1,2249 1,2865 1,4820  m"3  w  •1,9298 .c) A 3 T ae •2,0306 -2.0317 a > , UhlA.l •C i U It •2.0267 •2.0205 m  f  •1,6750 •Ij7524  l  0.0363  o.oiu  '  0,0000  "  "  0,0000 0,0193 0,0247 0,0309 0,0379 0,0457 0,0544  i _' :  0,0149 0.0321 0,0565 0,0716 0,0888 0.1084 0,1305 0 >1848  1  >  0  f  O  !  -  " " i  0,0000  0.01n •"  0,0247 0,0309 6 ,6179— . 0,0457 0,0544  1  *•>  : 1  " i  -0 —  .-• i  1  ""o7*r  -B.oior •0,0122 •0,0134 •0,0135 •0,0125 0.0345  " 0,0208" 0,0210 0,0224 0,0110 0,0471 0,4756  "•TOW •0,0266 •0,0267 •0,0237 -0,0196 0.0269  0,00 0,50 0.5$ 0,60_ 0,65 1,00  •1,1570 0,0166 .•1,1936 ^OfOITl •1,195 3" •6,016*"" •1,1964 •0.0200 _?1,I966 -0,0203 •1,1961 -0,0199" •1,1601 0,0164  "6,01210.0154 0,0192 0,021* 0,02*4 .6,6624  "B.00B1" 6,616* 6,6112 6,61*2 6,019* 6,6566  "07W61 0,010* 0,01)2 0,01*2 0,019* 0,0560  0,1125 0,0192 "5,01*5" 0,0166 _0,02)0 0,6337""  0,0000 0,0069  •1.041T 0,0096 •1,0702 •0,0175 _-l,671» r0,0169 •1.0725 .676.9T" •1,0729 -0.0202 _-l,0726_-0,0200 •1,0441 "0,0085" •0,9«7«  6,6006 6,6059 "6.607r 0.0094 _6,0115_ 6,0141 0,0443  0,0006 0,005* "070075 0,009* . 0.0115  0,00 0,45 _0,50 0,55" 0,60 0,65_ 1,00  0,0000 J?0j02)0 •6,622*" •0,0209 •0,0166 •0,0156 0,0266  0,0677" 0,0151 0,013) "T,onj0.0172 0,0245 "0,25*1"  0,0000 •0,0179 •0,0176  •0,9606 •0,0164 _"0,969S _iO 017l -6.9696" •6,6l7S~ •0,9697 •0,0171 _-0,9466 _0,0056  "o.otar  0,50 0,55 0,«0 0,65 1.00  OjOO 0,50 •1**.  0,60 0.65 J,M. 0,00 0,«S_ 0,50 0,55 0,*0_ 0,65 1,00 0,00 0,45 0,50 0.55 0,60 •»-•*_ 1,00 OjOO. 6,45 0,50 0,60 0,65 1.00..  0.00 0,50 0,55  ».*9.  *T;J*J* •1,345* -1.3467 •1,3466 -I.3450 •1.2960  ~=6,9i7r  t  •0,7400 _rO,74«l »0,7*«*" •0,7467 •0,7467 •0,7466 •0,7156  0,0007 -0,0071 -0,0077 •0,0079 •0,0060 •0,0076 0,0049  •0,669"r 0,0004 •0,6951 •0,0050 _-0,6953 .•OjOOS* •0,6954 •0,0056" •0,695* •0,0058 j-0,695i •0,0057 •0,660$" 0,0046 _£P,*060_ 0,0001 •0,6084 •0,0028 •0,6090 •0,0029 —6-» JO_iO, 002* •0,60*0 -0,002*" •0,6069 -0,0028 •0.6022 0.0019 60,  •0,5405 0,0006 _S0,S420_ -0.0014 •0,5420 •0,0014 •0,5420 •0,0014  •0.5410. .0 flfl|  fl  0,3446  .0589  0,0106 0,0111 0,0134"" 0,018) 0,1894 0,01*9 _0,0063 6,0059" 0,0059 0.0066 0,0081 0,077) 0,0145 0,0050 0.0047 "67664 r  «.OT*T"  •0,0149 •0,0125 "0,02*3" 0,0000 •0,01)9" •0.01)6 •0,01)2 •6,0126" •0,0102 0.0217  l.oiir  0,0142 0.017* 0.62TJ" 0,0**2  oToOoi" 0,00*7  0,00a*  "ovoio*"  0,0115 0.0165 "0,0514" JljOOOO 0.0052" 0,00*7 0,0085 "6,616*0,0110 0.0422  TYS666"  0,0042 6,005) "670^*7" 0,0063 _6,610t 0,0)46'  _e_,oooo  6,60)9" 0,60)6 6,6046  TTeiil  0,0*43  "9,0066 0,00*2 0,0053 0.0067  0,0063 0,0101 0,03*0  0,0006 "6,0636 8,0036 0,00*6  ~5.6ovr-  T.60S* 0,0073 0,0261  0,6666 _0j00l| 0,0014 0,0017 0.0021 o.oosr 0,0114 O.OOoT" 0,0006 0,00|0  0,0006 0.0011 0,0014 6.0017 0,0021  6,007) 0.02*1  0,0000 -0,0062 •0,00*4" • 0,0061 .•0,00*1 •o.oorr 0,0125  0,0000 _0,002S 0,0031" 0,0042 0.0051 0,66*1" 0,024*  •0*0047 •0.00*9 •0,0049 •0,0048 •0,004*  0,0020 0.0026 0,0013" 0,0041 0.0051 0,020*  "T.ooir  0,0000  0,0000 "6,0012"" 0,001* 0,0020  6,0066 "4,0004" 6,6005 _6.000*  O.OOO* 0,0005 JjOOO*  6.0031 6.01))  0,6669 6.6645  6,600* 6,0045  6,6060 0,0002 O.OOOl" 6,6063 6.666*  0.0000 6,0002 O.OOOl 6.0003 6.6604  0.0052 0,0062 "olossr __0.0081 0,0032" 0,0031 .0.00)1 0,003r 0.00)7 0.0265  •0,5026" •0,0026 0.00*0  0,00*8 0.0021 4,0020 0,0019 0.0020  0,0000 •0.0015 •0.0015 •0,001* •0.0016  0,009T"  •6,0627"  •0,0028 •0,0026  o.oonr  6,0006 _6j6067 6,6609 6.6611 6.6614  0.0015 6,001* "676085"  O^SOT"  6,662f 0,011*  "676056 6,0006 6.0016 "T760I2 6.0015 _6.00l» "o7008"5 0,0000  0,0 008  Appendix H.  oOp,p) as a function of positron energy ep  r  \  IL  8,5  _fi_ 0j_0 0 0,20 0.J0 0,10 0,45 0,50 0.55 0,60 0,70  3.391984 0,914897 0,514798 0.656665 0,991508 1.601421 2.635296 4,348691 11,649500  e^3«V  e *7«y  ep»i?«v  3.391984 0.915113 0,512627 0,650653 0,963069 1,590178 2.620768 4,330179 11,618390  3.191964 0,915015 0,509762 0,602690 0.971883 1.575269 2.601496_ 4.305612 11,577060  3,391984 0,915817 0,506226 0,632612 0,957995 1,556746 2_,5_7750_1 4,275063 11,525620  S,542026 2,558496 4,250764" 11,080660  P  3.191964 0.916158""0,503439 0,624975_ 0,946966  0,00 0,25 0.J0 0,35 0,40 0.45 0,50 0,55 0.60 0,75 1.00  2,049150 0,423108 0.271605 0,181668 0.154155 . 0.198309 0,312794 0,505278 0.775121 2,107690 6,228328  2,049150 0,024054 0.271706 0,180632 0,151692 0.194147 0,306676 0,496953 0.760372 2,088003 6,191834  2,049150 0,425374 _0^271?18_ 0,179350 0,106533 0„18875!_ 6,298703 0,086071 _0 7S0294_ 2,063067 6,103821  2,049150 0,427118 0,272309 0,177910 0,144786 0,182255 6,289036 0,472625 0,733112 2,032621" 6,064859  27609150 0,426539 0,272723 0,176890 0,101957 __0,177260 6.2BiS05 0,462514 _ 0 , 7 19702 2,007680 6,038539  0,00 0.40 0.45 0.50 0,55 0.60  1,255282 0,077032 0.066153 0,091914 0,159453 0.270062  1,255282 0,077271 0.065001 0,069978 0,156142  1.255282 0,078749 0,064066 0,065268 0,147502 0.252047  ~"77255282  0.?h5181  1,255262 0,077785 _0_,064635_ 0,087678 0,152059 _0,259063_  0,00 0.40 0,45 0.50 0.55 0,60 0,65  0,781509 0.054789 0,035686 0,038435 0.065139 0,116629 0,|9S7|8  0,761509 0,056389 0,036746 0,038801 0,064661 0,115168 0,193126  0,781509 _0 058 770_ 0.038456 0,039641 _0j064437 0,113699 0,190219  0,781509 0 062165 0,041094 0,041285 0,060851 0,112668 0.I675U  0,781509 0.065232 0,041627" 0,003100 0.065768 0,112521 0.186126  111  0,00 0,45 0,50 0,55 0,60 0,65 1,00  0,493092 0,026258 0.023047 0,034875 0,062193 0.106602 1,036950  0,093092 0,028114 0,024077 0.035763 0,062011' 0.106039 1,027897  0,493092 0,030936 _0.026791 0", 637422" 0,063209 0,105914 •,017145"  0,093092 0,035065 0,030391 0,000317 0,065245 0.106606  1.1  0.00 0,45 0,50 0.55 0,60 0,65 1.00  0.314643 0,021832 0,018040 0.022815 0,038905 0,066212 0,692765  0.31064] 0,023980 0,019910 0,024302 0,039903 0,066615 0.686011  „ 0.314643 0,027238 0,022864 0,026818 0,041844 0,067847 0.679521  0,00 0.45  0,202972 0.018612  0,202972 0,020814  0.202972 0,024202  0,7  0,9  1.5  1  A  X  17506417  0,314643 T,?3'2o2J 0,027385 0.030921  —67015171 0.070644 0.673793 0,202972 0.029282  0,079784 0.063931 0,083721 0,100305 0.206958  0,491092 0,018690 0,011885 - 0 , 0 Oil09 0,067669 0.106500 ~0799992T"  0j3l0643 "0,6 3 6492"0,031754 0,035075 07649194 0,074027 0,671861 0,202972 0,014152  ~0  0,"5T" 0,55 0,60 0,65 1,00  TToimr  0,017575 0,027026 0,044320 0,485608 0,132490 0,015777 _0,0l323l_ 0,013690 0,020479 -P-ll1'!*!. 0,350537  0,617549" 0,019310 0,028381 0,045193 0,480677  0,026760" 0,022220 0.030B68 0,047128 0,475842  D70"2"S755 0,026987 0,035267 0,051012 0,473149  "6713 FSW" 0,017919 _0,015257_  0,132"490 0,021288 _0,016548 0,018606 0,024776 _0,J>3S245 0,342864  ~o",Tsr49^ 0,026518 _0.023842 0,023831 0,029794 _0,039936 0,342386  om"2'4~j 0,031741 0,029287 0,0294 05 0,035402 0.045504 0,346164  _0j087660_ 0,013310 0,011418 _0j0_12007_ 0,015674 0,023235 _0.,257767_.  _0,087660_ 0,0153.24 0,013371 _0j0i3816_ 0,017249 0,024488 _0,.25422?_  .0,087660 0,016592 0,016643 —Oj_OL6990 0.020213 0,027128 _0,251615  _0, 087660 0,023913 0,022167 _0j 022602 0,025789 0,032547  _ 0 , 087660 6,029549 0,028194 0 028949 0,632179 0,039306  0,058903  0,058903 _0j_pl2943_ 0,011527" 0.0U797 _0,014025_ 0,018660 0,188474  0,058903 0.016046 0,01470! 0,014956 0,017068 "0,621663" 0,186638  _9,251095  0. .25990 J  0,077746  0,019942 0,007299 _0j006906 6,666938"0,007611 0.009Q7S_ 0,075945  0,019942 0,009704 _0,009497 0,669686 0,010477 0.012009 0,076715  0j00_ 0,45 0,50 0.55_ 0,60 0,65  0,058901 0,027644 0;o27210 0,026287 0,031047 0|6161 52 0,201122 676T99 42 0,025256 0,027419 6,010214 0,011811 0jJ>38171 0,11220?"  _0,01446|_ 0,004875 0,004603 _0,0046M_ 0,005089 0,006125 .0,056443  0,056901 _PJ021438 0,020428 0,020929 Jb 021184 0.027 8 31 0,190225 0.019942 0,015290 0,01578? 0,01670| 0,018236 JLLP 20 519 0,090082  _0,014461 0,005916 0,005694 _0,005729_ 0,06626"5 0,007203 _0|055l5a_  0.014461  0,009024 0,010167 _0. 057121  Jj.0|446l 0,014474 0,015182 0,016610 0,018142 0,020651 078396  0,014461 0,011020 0,015154 OjJ 4 00 80 0,046054 0,051226 0,171864  0,00 0,45 0,50 0,55 0 60_ 0,65 1,00  0,008067 _0,003182 6,003027~ 0,003011 _0|0_03213 0,003649 0,028554  0,008067 _0,003919_ 6,003812 0,003835 0 004065 0,664513 0,028270  0,008067 _O,005893_ 0.006018 0,006281 A l i o 67 8 3_ u.gumfa 0,007475 0,032906  0,008067 -?j0l6793_^ 0,018913 0,021470 0j_0246 07 0,028408 0,096747  0,00 0,45 0 , 5 0_  0,004835 0,002081 0.001962 0,001921 0,002018 0.002177  0,004835 0,002569 _0,002509_ 0,002502 0,002634 0.002822  0,004835 0,004465 0.0046S6_ 0,004928 0,005377 0.005919  0,004635 0,226117 0.278420 0,341455 0.417601 0.510325  0,00 0,45 0.S5 0,60 0.65 1.00 OjO0_ 0,45 0,50 0,60 0,65 >i-°.P_ 0,00 0.45 0,50 0.55 0,60_ 0,65 1.00 0,00 0,45 0,50_ 0.5S 0,60 1,00  t  0,55 0,60 0.«>5_  _P,011105_ 0,009710 0,010072 _0,012470_ •0,017561 i>w«i 0,191599  0,019942 0,006063 _0,005646_ 0,005667 0,006369  _0,0079LL  0,615508  0,021992 0.032886 0,346438  L  0,008158 0,008143 _°-i00|373_  :  6703058^ 0,031683 0,040027 0,055534 0,474272  L  0,008067 1,288423 1.62626T 2,035139 2^529 9 21 3,128915 13,043730 0,004815 0,002944 0j_0 03798 0,005026" 0,006740 0,008971  i.  0  I, O  0  J  ''•>  ••> '  l'  O  0  I  I  i 1  -j o 1  77  •- P 5  rvj ru  r^. to o ru  0 0 > 0 0 j 0 « « 9 ' 0 > 9  |LA  — -• . * o o o o o o o o o o K» O O O O O O O ' O O D  | « o t < i N jti/t;^oiM •o to r » ; ^ > -O 4> r*> i _ _  kuooo  O O I O O K I :  o o o o o o io o o; o o o o o o-o o oj  O O O O O O  O rt O* O f - O O O ?  rt A I A I rt , 9 4 > r »  o  o o o o o o o ! « * r t  o  •«IVO!««<ftf>>99 l\J 4>o»9rt41rt'*>»c04) O A4)l4>«MSON ruoooooo-«9 o o o o o o o o o o o o o o o o o o  0  rt r» Al O  o o io o o !o o o  O o o o  IA U V * At * * - | o O - * | <OcOiAAtcOtQ>IA-* « ru AI Al »  O* 9<AI (A < 0> r«. ; 4) 41 C o oio o <  o o o o o o o o  O O O O <  LA rt CO 9 IMIA Al o o -< » r\j j cortrtrtAI Irt 9 4k oo ooojooo o o o o o o o o ooooolooo  00000:00 o o o o o o o o o  O OJO  4> i 4> i 9 . 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