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The solution to the reference hypernetted-chain approximation for fluids of hard spheres with dipoles… Perkyns, John Stephen 1985

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THE SOLUTION TO THE REFERENCE HYPERNETTED-CHAIN FOR FLUIDS OF HARD SPHERES WITH DIPOLES AND  APPROXIMATION QUADRUPOLES  WITH APPLICATION TO LIQUID AMMONIA BY JOHN STEPHEN PERKYNS B.A., DALHOUSIE UNIVERSITY, 1981 B . S c , DALHOUSIE UNIVERSITY, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT  FOR THE DEGREE OF  MASTER OF SCIENCE IN THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF CHEMISTRY) WE ACCEPT THIS THESIS AS CONFORMING 51HE REQUIRED STANDARD  THE UNIVERSITY OF BRITISH OCTOBER,  COLUMBIA  1985  ©JOHN STEPHEN PERKYNS, 1985  In p r e s e n t i n g  this thesis  r e q u i r e m e n t s f o r an of  British  it  freely available  agree t h a t  in partial  advanced degree a t  Columbia,  understood that for  Library  s h a l l make  for reference  and  study.  I  for extensive copying of  h i s or  be  her  g r a n t e d by  s h a l l not  be  Cl^gM  ISTILV  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3  Date  Oct  10 j<85~  of  further this  Columbia  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  permission.  Department o f  the  representatives.  copying or p u b l i c a t i o n  f i n a n c i a l gain  University  the  f o r s c h o l a r l y p u r p o s e s may by  the  the  I agree that  permission  department or  f u l f i l m e n t of  written  ii  ABSTRACT  This reference fluid  thesis  hypernetted-chain  of hard  quadrupoles. constant,  spheres  with  e, and t h e p a i r calculated  from  integral  other  into  (RHNC) a p p r o x i m a t i o n  correlation  In  Monte C a r l o d a t a methods.  Part B a s e l f - c o n s i s t e n t  approximation  used  fluid  i s used,  with ammonia-like  i s calculated  is  t o be q u i t e s e n s i t i v e  found  mean f i e l d  together  with  parameters.  uncertainty,  s e t by t h e q u a d r u p o l e  theory f o r t h e RHNC dipole-linear The d i e l e c t r i c  temperatures  t o the quadrupole  results  and i t  moment.  f o r e a r e shown t o be w e l l w i t h i n t h e  These c a l c u l a t e d the d i e l e c t r i c  non-polarizable  a n d i s shown t o  at three s u b - c r i t i c a l  Experimental  than  results  The RHNC i s f o u n d t o  i n P a r t A, f o r a p o l a r i z a b l e  constant  larger  as w e l l as with  with  methods.  polarizability  quadrupole  i s solved for a  f u n c t i o n a r e compared  a p p r o x i m a t e t h e Monte C a r l o r e s u l t s  molecular  In P a r t A t h e  embedded p o i n t d i p o l e s a n d l i n e a r  equation  i m p r o v e on t h e o t h e r  values.  two p a r t s .  The t h e r m o d y n a m i c p r o p e r t i e s , t h e d i e l e c t r i c  previously  closely  i s divided  system.  moment, i n t h e c a l c u l a t e d e  e v a l u e s a r e shown t o be s i g n i f i c a n t l y constants  f o rthe equivalent  i ii  TABLE OF  CONTENTS  Abstract Table  i i  of Contents  List  of T a b l e s  List  of F i g u r e s  i i i v i i viii  Acknowledgements  x  CHAPTER 1 Introduction  1  PART A CHAPTER 2 Theory 2.1  9 Introduction  9  2.2 E q u i l i b r i u m D i s t r i b u t i o n 2.3 The P o t e n t i a l for  Energy  Functions  Expression  the Model  13  (a) Form o f t h e F u n c t i o n (b) E x p a n s i o n  Equation  Invariants  Theories  (a) The H y p e r n e t t e d Equation (b) R e d u c t i o n  13  of U(12) i n  Rotational 2.4 I n t e g r a l  Chain  16 23  (HNC) I n t e g r a l  Theory  23  of the  Ornstein-Zernike Equation (c) Reduction  10  o f t h e HNC C l o s u r e  25 32  i v  (d)  The R e f e r e n c e HNC C l o s u r e  2.5 E x p r e s s i o n s  37  f o r Thermodynamic Quantities  (a)  Configurational  (b)  Isothermal  (c) S t a t i c  39 Energy  Compressibility  Dielectric  2.6 C o m p u t a t i o n a l  40 Factor  41  Constant  41  Considerations  42  (a) Method o f S o l u t i o n  42  (b)  43  Program E f f i c i e n c y  CHAPTER 3 Results  and D i s c u s s i o n  3.1  Input  45  P a r a m e t e r s and B a s i s  3.2 R e s u l t s  f o r D i f f e r e n t Basis  3.3 C o m p a r i s o n  w i t h Monte C a r l o  (a)  Configurational  (b)  Dielectric  Sets Sets Calculations  Energy  47 ...  52 52  Constant  52  (c) The P a i r C o r r e l a t i o n F u n c t i o n 3.4 Summary  45  and C o n c l u s i o n s  54 93  PART B CHAPTER 4 Polarizability  Theory  96  4.1 The SCMF Method  96  4.2 P o l a r i z a b l e M o l e c u l e s w i t h and  Quadrupoles  4.3 Method o f S o l u t i o n  Dipoles 99 104  V  CHAPTER 5 Results  and D i s c u s s i o n  105  5.1 C a l c u l a t i o n s w i t h  Ammonia-like  Parameters 5.2 S e n s i t i v i t y  to P a r t i c l e and B a s i s  5.3 C o m p a r i s o n  List  105 Diameter  Set  106  t o E x p e r i m e n t a l Data  109  5.4 Summary and C o n c l u s i o n s  110  of References  111  APPENDIX A Identities  Used  i n Subsequent  Appendices  114  APPENDIX B Proofs  o f Symmetry P r o p e r t i e s o f Rotational  Invariants  117  APPENDIX C Calculation  o f U ( 1 2 ) i n Terms o f Multipole  Interactions  119  APPENDIX D Derivation  of E q u a t i o n s  2.4.12 t o 2.4.16  120  APPENDIX E Angular  Integration of  and  Simplification  t h e OZ E q u a t i o n  124  APPENDIX F Chi  Transformation  o f t h e OZ E q u a t i o n  127  vi  APPENDIX G Back T r a n s f o r m a t i o n s  ,  129  APPENDIX H Binary  Products  of R o t a t i o n a l  Invariants  131  vi i  L I S T OF  I  Basis  II  Cut-off  III  Basis  TABLES  Sets  46  Dependence  in c  S e t Dependence  m n l  (r)  48  o f e,  -j3<U>/N and 0p/p Contact Values  49  IV  Projection  V  P r o j e c t i o n V a l u e s a t r=1.1d  51  VI  The C o n f i g u r a t i o n a l  53  VII  The S t a t i c  Energy  Dielectric  Constant  50  55  vi i i  L I S T OF  I.  The  E u l e r angles  2a.  The  short  2b.  The  long  The  short  3b.  The  long  4a.  The  short  The  long  for  s e t (a)  59  s e t (b)  61  g^^(r)  parameter  s e t (b)  63  g^^(r)  parameter  r a n g e p a r t of  57  g^^(r)  parameter  range p a r t of for  4b.  of  r a n g e p a r t of for  s e t (a)  g^^(r)  parameter  range p a r t for  g^^(r)  parameter  range p a r t of for  3a.  18  range p a r t of for  FIGURES  s e t (c)  65  g^^(r)  parameter  s e t (c)  67  5.  h  1 1  ^(r)  f o r parameter  s e t (a)  69  6.  h  1 1  ^(r)  f o r parameter  s e t (b)  71  7.  h  1 1  f o r parameter  s e t (c)  73  f o r parameter  s e t (a)  75  f o r parameter  s e t (b)  77  f o r parameter  s e t (c)  79  II.  ^(r) 1i 2 h (r) 1i 9 h (r) 1i 2 h (r) 1 23 h (r)  f o r parameter  s e t (a)  81  12.  h  f o r parameter  s e t (b)  83  13.  h  1 23 ( r ) f o r parameter  s e t (c)  85  8. 9. 10.  1 2 3  (r)  ix  14.  h  2 2 4  (r)  f o r parameter  s e t (a)  87  15.  h  2 2 4  (r)  f o r parameter  s e t (b)  89  16.  h  2 2 4  (r)  f o r parameter  s e t (c)  91  17.  Dielectric  constant liquid  vs. temperature for ammonia  107  X  ACKNOWLEDGEMENTS  I wish t o thank for  my r e s e a r c h a d v i s o r ,  h i s g u i d a n c e and t h e time and e f f o r t  behalf.  Thanks a r e a l s o  for  'showing  the  University  would  also  initiative  D r . G.N.  Patey,  he h a s s p e n t on my  due t o D r . P.H. F r i e s  a n d P.G. K u s a l i k  me t h e r o p e s ' a n d t o t h e C h e m i s t r y D e p a r t m e n t o f of B r i t i s h Columbia  l i k e t o thank  my w i f e ,  made i t p o s s i b l e  responsibilities  for financial Jane,  whose  support.  u n d e r s t a n d i n g and  f o r me t o i g n o r e many  i n o r d e r t o meet  frightening  I  deadlines.  1 CHAPTER 1  Introduction.  Thirty liquids  years  ago, r e l a t i v e l y  from a t h e o r e t i c a l  knowledge o f s o l i d largely  i s no i d e a l i z e d  comparable  to the i d e a l  understand  liquids  theories high  temperature  solids.  i s clear  for liquids  theory  began.  equilibrium  work o f U r s e l l the c l u s t e r  imperfect  distribution theory,  results  Expressions gas law, s u c h  of l i q u i d s  The accurately  lattice d e s c r i b e d as  which c o r r e c t f o r as t h e v i r i a l  theory  expansion,  b u t do n o t  of l i q u i d s [ 2 ] .  statistical  I t i s not c l e a r  mechanical  when s u c h a  work now u s e d  was d e v e l o p e d  techniques initially  i n other  i n the  contexts.  now u s e d e x t e n s i v e l y .  d e a l i n g with  These  the problem of  Z e r n i k e and P r i n s [ 6 ] i n t r o d u c e d t h e r a d i a l i s a central  d i d so i n o r d e r  general  predict  Simple  [ 3 ] , Y v o n [ 4 , 2 ] , a n d Mayer, e t . a l . [ 5 , 2 ] , l e d  f u n c t i o n , which  but they  Attempts t o of t r y i n g t o  g a s e s a t low d e n s i t i e s ,  was n e e d e d .  expansion  gas.  for liquids  which a r e b e s t  Much o f t h e g r o u n d - b r e a k i n g  theory  T h i s was  been t h o s e  t h a t an 'a p r i o r i '  a d v a n c e s were, however, the  model  and g a s t h e o r i e s .  the basis f o r a s a t i s f a c t o r y  theory  to  solid  i n the i d e a l  It  The  have t r a d i t i o n a l l y  useful f o r imperfect  provide  when compared t o  gas o r t h e p e r f e c t s o l i d .  [1] g e n e r a l l y g i v e  deviations are  between  point-of-view  was known about  or gaseous s t a t e s of m a t t e r .  because t h e r e  interpolate  little  t o e x p l a i n X-ray  g o a l of s t a t i s t i c a l  macroscopic  concept  in liquid scattering.  mechanics  q u a n t i t i e s from  i s to  reasonable  2 microscopic  postulates.  p r o c e s s e s or the  liquid  [7], of  The  configurational  state  hamiltonian  can  only  be  gases or  present  classical  study  description,  accurately  by  inter-particle  the  i s devoted  total  to  energy.  potentials  to  the  described  configurational  l i q u i d s , which g r e a t l y to  s o l i d s where k i n e t i c  a s p e c t s dominate the  w i t h b o t h k i n e t i c and  contribution  our  Unlike  the  contributions  equilibrium  s i m p l i f i e s the It  i s the  total  a  energy  behavior  kinetic  contribution  of  t o w h i c h we  turn  attention. We  expression. symmetric, particles.  define The  and For  a model by  first  specifying  m o d e l s u s e d were s i m p l e ,  described  by  the  example, t h e  square-well  u(r)  =  { -c  between  potential, defined  r>R, the  two  particles, d  R are  constants,  c and  used e x t e n s i v e l y ,  Lennard-Jones  has  u(r)  where  e and  a are  experimentally. sphere  the  = 4e[(f)  constant  1 2  00  r<d,  0  r<d.  {  6-12  is has  the been  potential  (f) ], 6  determined  used p o t e n t i a l  interaction,  =  -  (LJ)  parameters usually  A n o t h e r commonly  u(r)  by  d<r^R,  ( h a r d ) p a r t i c l e d i a m e t e r and as  two  r<d,  0 distance  energy  spherically  i n t e r a c t i o n between  *  where r i s t h e  a potential  i s the  hard  3 Although the L J p o t e n t i a l success  f o r Argon  mathematically systems  [ 2 ] , such i d e a l i z e d  simple  in general.  more r e a l i s t i c  that They  been  so a n g u l a r  added This  t o one  still  real  A,  define an  i t a s an intrinsic  molecular  by  thermodynamic  electrostatic spherical a pair  potential  defined,  quantities,  Computer  by  [8-11].  c o m p r i s e d of These  simple  potential  has  terms  potentials  pair  in general interactions.  with experimental data, of P a r t  we  A, but  to take i n t o account,  terms w h i c h a r i s e  we  computer  have  times  sufficiently  long  [12] t o o b t a i n  static  dielectric  well.  Approximate  on b o u n d a r y  constant, theories  due  to  routes to  and a p p r o x i m a t e essentially  o f many p a r t i c l e s f o r  statistically  even on modern conditions,  cause problems only  basic  r e g a r d e d as  b u t t o keep t r a c k  i s an e x p e n s i v e p r o p o s i t i o n , w h i c h depend  two  simulations  s i m u l a t i o n s c a n be  f o r a g i v e n model,  Quantities  interactions  systems because  results  pair  Much work  interaction  potential  of r e a l  rare  symmetric  terms a r e added.  more c o m p l i c a t e d  exact  results  spherically  of t h e same form as t h a t  'effective' way,  real  polarizability. The  methods.  compare  potential  a r e so  systems, b e s i d e s  and q u a d r u p o l a r t e r m s .  poor d e s c r i p t i o n s  B, where we  use a p a i r  order  real  p o t e n t i a l s c a n n o t be d e s c r i b e d  In P a r t  in  with  of these s i m p l e  hard-sphere, dipolar are  Few  described  study uses, in Part  potentials  some  do, however, form a good b a s i s f o r  dependent  done on p o t e n t i a l s  used w i t h  t h e y c a n n o t be e x p e c t e d t o mimic  potentials.  g a s e s , c a n be a c c u r a t e l y potentials,  has been  computers.  s u c h as t h e  for simulations  take computing  of a few m i n u t e s , so much e f f o r t  sound  has been  as  t i m e s of t h e s p e n t on  finding  4 accurate  theories.  approximate the  results  theoretical  Therefore,  U n f o r t u n a t e l y , d i s c r e p a n c i e s between and e x p e r i m e n t a l  approximations  results  experimental  data.  with  use.  computer  equations  of motion  used. in  Molecular  s i m u l a t i o n s as w e l l as with  the s t a t i s t i c a l  simulated  attention equation  Dynamics c a l c u l a t i o n s and t i m e  solve the averaging i s  (MC) method e v a l u a t e s ensemble  mechanical  u s i n g MC t e c h n i q u e s The  methods o f computer s i m u l a t i o n  f o r the p a r t i c l e s  The Monte C a r l o  itself.  f o l l o w e d i s t o compare  T h e r e a r e two i m p o r t a n t i n common  c o u l d be due t o e i t h e r  o r t o t h e model  the usual procedure  approximate  data  sense.  averages  The model we u s e h a s been  by P a t e y , L e v e s q u e a n d Weis [ 8 ] .  a p p r o x i m a t e t h e o r i e s w h i c h have r e c e i v e d most  recently methods.  a r e p e r t u r b a t i o n a p p r o a c h e s and i n t e g r a l T h e r e a r e many v a r i a t i o n s  in perturbation  a p p r o a c h e s b e c a u s e o f t h e many f u n c t i o n s w h i c h c a n be e x p a n d e d . There are s e v e r a l i n t e g r a l  equation  t h e o r i e s which  have been a p p l i e d t o h a r d  spheres.  agrees  s i m u l a t i o n s a t low d e n s i t i e s  less  w e l l w i t h computer  a c c u r a t e when t h e d e n s i t y i s i n c r e a s e d .  Approximation  (MSA) f o r h a r d  Percus-Yevick  (PY) t h e o r y ,  this  The B o r n - G r e e n  potential  obtained  Together  forms t h e HNC  which  series  with  integral  (HNC) e q u a t i o n  of the d i r e c t  theory.  i t is inferior.  For hard  c a n be  correlation  the Ornstein-Zernike equation  seems t o be s u p e r i o r t o t h e HNC  potentials  t o the  i s solvable analytically for  expansion  equation  but i s  The Mean S p h e r i c a l  becomes i d e n t i c a l  The H y p e r n e t t e d - C h a i n  by a c l u s t e r  function.  solution  [2].  spheres  t h e o r y [2]  [13]  spheres  it  t h e PY  [ 2 ] , but f o r a l l other  5 The Wertheim  MSA h a s been a p p l i e d  [ 1 4 ] and t h e n  a n d Blum  s o l v a b l e even by n u m e r i c a l recently  linearized  solved  [10].  i m p r o v e d by u s i n g  HNC (QHNC)  spheres  so i t e r a t i v e with  [19].  i n t o exact  and p e r t u r b a t i o n  spheres only. In P a r t  equation  A of the present  theory  theoretical experimental  many i n d u s t r i a l intensely  ammonia i s a good c h o i c e  data  studied  purposes, accurate pressure  available.  uses.  i s given  This  In f a c t ,  inorganic liquid  MC  and t h e p o l a r i z a b i l i t y  i s the exact  s t u d y we s o l v e t h e spheres  with  results. with  i s a l a r g e amount o f a c c u r a t e i s probably  solvents.  [20].  are divided  f o r comparison  a f t e r water,  density  by V a n c i n i  part  data  due t o ammonia's i t i s one t h e most  In p a r t i c u l a r f o r o u r at equilibrium  The d i p o l e moment  been e x t e n s i v e l y m e a s u r e d , t h e q u a d r u p o l e moment available,  first  f o r a system of hard  r e s u l t s because there  core i s  i s known a s t h e r e f e r e n c e  d i p o l e s and q u a d r u p o l e s and compare w i t h Liquid  approach  The e x a c t  This  The  t o the hard  The c o r r e l a t i o n f u n c t i o n s parts.  HNC was  c a n n o t be  a r e used.  t h e HNC a p p l i e d  by Lado  (RHNC) t h e o r y .  extensively  [ 1 1 ] and S t o c k m a y e r  techniques  a type of p e r t u r b a t i o n  f o rhard  by t h e  p o t e n t i a l s [8-11] a n d t h e f u l l  introduced  RHNC i n t e g r a l  potentials  The s y s t e m s o f e q u a t i o n s o b t a i n e d  p r o b l e m of i n a c c u r a c y  HNC  The HNC was n o t  means f o r n o n - s p h e r i c a l  f o rd i p o l a r hard  analytically  solution  p o t e n t i a l s by  The LHNC and QHNC have been a p p l i e d  non-spherical  solved  particles  f l u i d s by  [ 1 1 ] a n d s o i t was f u r t h e r a p p r o x i m a t e d  approximations.  recently  [16-18].  HNC (LHNC) o r t h e q u a d r a t i c  to various  to dipolar  to arbitrary multipolar  Blum a n d T o r r u e l l a [ 1 5 ] ,  until  first  [22]  i s a l s o known [ 2 3 ] .  vapor [21] has is Also, the  6 roughly  spherical  shape o f ammonia  lends  itself  w e l l t o simple  models. In liquid  ammonia  Theoretical two  order  i t i s necessary  treatments  c a t e g o r i e s , those  polarizabilities. Patey  of p o l a r i z a b l e  A reveiw  pair  of these  systems  field  (SCMF) t h e o r y ,  system  actual  used  at similar  here,  the s e l f  t h e many-body p o t e n t i a l  permanent m u l t i p o l e moments. d e f i n e s the e f f e c t i v e  properties  of a p o l a r - p o l a r i z a b l e  calculate  these  approximation.  ammonia  i s due  of a p o l a r i z a b l e  The r e s u l t i n g  system. system,  by s o l v i n g  The SCMF t h e o r y  which  permanent such a s  and d e n s i t y a s w e l l a s t h e  properties f o r the related  w h i c h we do f o r l i q u i d  by an  c o n s i s t e n t mean  This effective  temperature  expression  results  p o t e n t i a l s w h i c h depend on an  permanent d i p o l e moment.  polarizability,  into  and f l u c t u a t i n g  depends on v a r i o u s p r o p e r t i e s o f t h e s y s t e m  molecular  roughly  t o previous approaches i n  called  i s w r i t t e n a s a sum o f p a i r  effective dipole  The t h e o r y  theory,  model.  t h e o r i e s i s g i v e n by S t e l l ,  [25], i s similar  In t h i s  fall  r e p l a c e many-body e f f e c t s  potential.  C a r n i e and Patey respect.  both  c o n s t a n t of  to consider a polarizable  w h i c h assume c o n s t a n t  because they  'effective'  this  the d i e l e c t r i c  and H^ye [ 2 4 ] . B o t h a p p r o a c h e s a r r i v e  probably  to  to calculate  In order  potential to find the  i t i s necessary t o effective  system,  t h e RHNC  has r e c e n t l y  been shown t o be  very accurate [26]. This has  a chapter  RHNC i n t e g r a l spheres  thesis  i s divided  on t h e o r y equation  into  and a c h a p t e r theory  two p a r t s , e a c h of w h i c h on r e s u l t s .  In P a r t A t h e  i s a p p l i e d t o a system of hard  embedded w i t h d i p o l e s a n d q u a d r u p o l e s ,  and r e s u l t s a r e  compared w i t h MC derived  Most e q u a t i o n s  i n A p p e n d i c e s B-G.  mathematical the  data.  identities  SCMF t h e o r y  A,  and  is  found  Appendix A  needed  i s s o l v e d u s i n g the  obtained  from  compared w i t h each p a r t a r e  i s devoted  i n the d e r i v a t i o n s . integral  u s i n g ammonia-like parameters. and  s t a t e d i n Chapter  The  experimental given at  the  to  the In P a r t  equations  dielectric  data. end  2 are  of  B  Part  constant  Conclusions of  their sections.  8  PART A  9 CHAPTER 2  Theory.  2.1  This chapter distribution  macroscopic spheres  integral  function  q u a n t i t e s of  with  describing  interest.  approximation  system  f o r most  l i q u i d He  Appendices  is treated  B-H.  [2].  liquids, Most  the t h e o r y  equations.  The  These are  model c o n s i s t s  used, the  of  hard  quadrupoles.  classically. although  equations  of  t h e model, t o g e t  w i t h embedded p o i n t d i p o l e s and The  such as  i s concerned  f u n c t i o n s and  w i t h the p o t e n t i a l  Introduction.  This i s a  there are  stated  good  exceptions  are d e r i v e d i n  10 2.2  Equilibrium Distribution  Consider particles.  The H a m i l t o n i a n  H  3N = I i=1  N  where t h e p^ particle,  i s o f t h e form  2  (2.2.1)  N  t h e 3N momenta, iru i s t h e mass of t h e  i s the c o n f i g u r a t i o n a l e n e r g y . the t h r e e  position  Here  i  f c  ^  X^  and t h r e e o r i e n t a t i o n  of p a r t i c l e i .  The  probability  infinitesimal  volume  dX  of f i n d i n g  every  other  particle  of the p o s i t i o n  1  t h e momentum r e g i o n  finding  identical  ,X ,...,X ),  N  (r^,fi^),  coordinates  of p  N  of N r i g i d  f o r t h e ensemble  2 l i _ + U (X 2m. l  represent  and U  represents  within  a c a n o n i c a l ensemble  Functions.  dp  particle  of p ,  1  1  1 w i t h i n the  and o r i e n t a t i o n X  while  i w i t h i n dX^  1  and  simultaneously  of X^ a n d w i t h i n  dp^  i s [27]  i ?  P N  (X ,X ,...,X ,p ,...,p )dX dX ...dX dp dp ...dp 1  2  N  =  1  N  1  2  N  1  2  ^-exp[-^H (X ,X ,...,X )] N  1  2  N  x dX dX ...dX dp ...dp , 1  where /3-(kT) temperature. defined  by  1  2  N  1  , k i s the Boltzmann constant Q  [28]  N  N  i s the c l a s s i c a l  N  and T t h e  canonical partition  (2.2.2)  absolute function  11 Q  N  NT ~ 3 N * exp[-/3H (X ,X , . . . , X , h  =  N  1  2  N  P l  , . . . ,p ) ] N  x dX ^ d X • • • dX^dp ^ • • • ^Pj^'  (2.2.3)  2  where h i s P l a n c k ' s position,  Q  N  (m^=m  h  =  N  o f t h e form e q n . 2.2.1, e q n . 2.2.3  for identical  particles)  ( ^ r - )  i s called  integrate  coordinates.  a Hamiltonian  f  /  N  1_ (iTnnkTjfN  =  Z  and t h e i n t e g r a t i o n i s over a l l  o r i e n t a t i o n and momentum For  becomes  constant  exp[-/3U (X ,X ,...,X )]dX dX ...dX , N  1  2  N  1  2  N  ^  {  the c l a s s i c a l  configurational integral.  t h e momentum c o o r d i n a t e s  2  >  2  >  4  )  We c a n  i n e q n . 2.2.2 i n j u s t t h e  same way t o o b t a i n  (X ^ ^2'* * * ' ^  ^  r  * * *  = ^-exp[-^U (X ,X ,...,X )]dX dX ...dX . N N  The p r o b a b i l i t y any o t h e r particle  P  (n) N  ix ( y  l f  of f i n d i n g  1  dX  n  of X  „ v \ x ,...,x ; 2  n  (  n  N! _ j  N  n  )  N  any p a r t i c l e  molecule within dX within  2  of X  2  is  1  2  within  dX  (2.2.5)  N  1  of X , and 1  a n d s o on up t o any n*"*  1  2  [27]  1 z  N  x J exp[-/?U (X , X , . . . , X ) ] d X N  1  2  N  n + 1  dX  n +2  ...dX . N  (2.2.6)  12 The  integration  coefficients particles  i s o v e r p a r t i c l e s n+1  are necessary to avoid  more t h a n o n c e .  These  t o N and t h e f a c t o r i a l  counting indistinguishable  p^ ^  a r e known a s n - p a r t i c l e  n  densities. The  n-particle d i s t r i b u t i o n functions,  homogeneous s y s t e m a r e d e f i n e d  g  n  )  N  ( X ,X , . . . , X ) 1  2  n  where p i s t h e number  as [ 2 7 ]  = (87r ) p" p 2  density  for a  n  n  n ) N  (X ,X ,...,X ), 1  2  n  (2.2.7)  of the system. (2)  Only  the p a i r  d i s t r i b u t i o n function  be u s e d h e r e , and i t w i l l convenience. defined  Also  be d e n o t e d  used w i l l  g  N  (X^Xj)  throughout as g(12) f o r  be t h e p a i r c o r r e l a t i o n  function  by h( 12) = g ( 1 2 ) - 1.  Generally  g ( 1 2 ) —>1  1 and 2 i n c r e a s e s , limiting  and h(12) measures t h e d e v i a t i o n  c o n s t a n t c a n be w r i t t e n  procedure u s u a l l y  techniques  followed  i s to define  use e q u i l i b r i u m  distribution properties  of p a r t i c l e s from  this  value.  dielectric  and  (2.2.8)  as the i n t e r p a r t i c l e separation  Thermodynamic q u a n t i t i e s  The  will  i n terms  when u s i n g  mechanics  of g(12) or h ( l 2 ) .  d i s t r i b u t i o n function  the c o n f i g u r a t i o n a l  statistical  functions,  s u c h a s i n t e r n a l e n e r g y and  energy  for a  model,  to c a l c u l a t e the  from which a r e found the m a c r o s c o p i c  of the system.  o  2.3  The  13  Potential  Energy  E x p r e s s i o n f o r the  Model.  (a) Form of t h e F u n c t i o n .  To solve  treat  system  exactly  i t would be  necessary to  t h e many-body S c h r o d i n g e r e q u a t i o n d e s c r i b i n g  of a l l n u c l e i task  any  and  electrons.  so a number o f  T h i s would be a v e r y  simplifying  Born-Oppenheimer a p p r o x i m a t i o n expresses  because  i s the  the n u c l e i  motion  difficult  a p p r o x i m a t i o n s are needed.  t h e many-body p r o b l e m  justifyable  the  first  i n terms  used.  of f i x e d  This  simply  nuclei  a r e much h e a v i e r t h a n  The  and  is  the  electrons. The particles  second  approximation  are r i g i d .  intramolecular  This  vibrations  i g n o r e s any and  i n the  each  t o by  of  i s referred  i t s c e n t e r of mass  treating  the l i q u i d  w i t h a few  as t h e  V  can  that  the  between  f o r c e s and  allows  form U" (X ,X , .. . ,X ) , where N  1  2  N  the o r i e n t a t i o n  (or c h a r g e ) .  classically,  The  third  and  the  i s that  position  of  which  i s a good  approximation  express the r e s u l t i n g  potential  energy  exceptional We  coupling  intermolecular  us t o w r i t e t h e p o t e n t i a l particle  i s the assumption  cases [ 2 ] . function  expansion  X  1 '  X  I i<j  2  u  +  v o ;  U  (x  (2)  i f  (X. ,X . )  Xj,x )  +  two  t h r e e body  k  (2.3.1)  i< j<k  where U  (2)  and  U  (3)  represent  and  interactions.  14 In o r d e r  to describe  a real  probably  be n e e d e d .  This  system d i s c u s s e d  i n Part  terms a r e i n c l u d e d large  amount  s y s t e m e x a c t l y many  i s certainly  true  t e r m s would  of t h e p o l a r i z a b l e  B, where many p a r t i c l e i n t e r a c t i o n  i n an e f f e c t i v e  pair potential.  o f work h a s been done  [2] using  However, a  m o d e l s d e f i n e d by  (2) pair  p o t e n t i a l s o f t h e form U Interactions  strong,  short  interactions the  short  making  ranged  ranged  a distance  longer  ranged  about  i s done by f o r b i d d i n g the hard  sphere  i n the p o t e n t i a l energy  (2.3.2)  the centers  of t h e s p h e r e s .  i n t e r a c t i o n s a r e c a u s e d by a n i s o t r o p i c s i n the e l e c t r o n i c The  instantaneous  i n a l l molecules give  rise  b u t weak compared  t o Van d e r Waals  t o t h e f o r c e s due  anisotropies.  The  permanent  which a r e p r e s e n t  described  term  by  term,  the molecules.  f o r c e s which a r e a t t r a c t i v e  be  d, c a l l e d  between  distributions  t o permanent  I n o u r model  hc  and permanent  present  ranged  °° f o r r<d, = U ,(12) = { . 0 f o r r>d,  instantaneous  anisotropies  or r e p u l s i v e .  This  The f i r s t sphere  longer  c o n s i s t of a  of the p o t e n t i a l i s g i v e n  inpenetrable.  where r i s t h e d i s t a n c e The  u(12).  r e p u l s i o n a n d weaker,  i s then a hard u(12)  =  2  between m o l e c u l e s g e n e r a l l y  of each o t h e r .  anisotropic  ]t  repulsive aspect  them t o come w i t h i n  function  '{X X )  w h i c h c a n be a t t r a c t i v e  the p a r t i c l e s  diameter,  v  non-spherical  electron distributions  i n most m o l e c u l e s c a u s e a c o r r e s p o n d i n g  charge d i s t r i b u t i o n . by a m u l t i p o l e  This  expansion.  .  charge d i s t r i b u t i o n can The e l e c t r o s t a t i c  15  (\,\ potential  f o r k charges q  be c a l c u l a t e d . charge  At a p o i n t  as the o r i g i n )  electrically  R  neutral  at points  r  ' = (r)  v  t = (t^tjft^)  the e l e c t r o s t a t i c molecule w i l l  (]r) (]r) ,*  (\r)  ' 3  '  r  2  G  a  n  (with the c e n t e r of  potential,  approach,  V, f o r an  f o r large t=|t|  [29] 3 n.t. 3 3 Q.,t.t. Z - ^ i + Z Z J 1 + i =1 t i = 1 j=1 t  V(t) =  1  ,  1  (2.3.3)  - 3  where N u - Z Z k=1  Q  (k) r  i j  5 J ,  =  q r  3 (3rj k  (  k  )  k  k  )  r(  k  )  ,  (2.3.4a)  -(r[  k  )  )  2 5  i  j  ),  (2.3.4b)  (k) = |r  | and 5 ^  d i p o l e moment  with elements  tensor  with elements Q ^ j -  taking  the large  by c o n s i d e r i n g mathematical  t limit  and Q i s t h e e l e c t r i c  of the g e n e r a l p o t e n t i a l  t h e m u l t i p o l e moments a s p o i n t  simplification  terms  i n the expansion are very short n + 1  )  these a d d i t i o n a l ignored here.  of the problem  dependence. terms  e x p r e s s i o n or  quantities.  and i s u s e d h e r e .  Although  are n e g l i g i b l e  U n f o r t u n a t e l y , we must  Other  r a n g e d due t o t h e 2 - p o l e n  i t i s n o t known whether for real rely  systems  they a r e  on quantum c h e m i c a l  calculations  f o r v a l u e s o f o c t u p o l e and h i g h e r m u l t i p o l e  which  be v e r i f i e d  cannot  2.3.4 c a n a l s o  The  r e p r e s e n t e d by e q n . 2.3.3 g r e a t l y  the d i f f i c u l t y  (1/t  quadrupole  E q u a t i o n 2.3.3 c a n be d e r i v e d by  eases  having  M i s the e l e c t r i c  i s Kronecker's d e l t a ,  by e x p e r i m e n t s .  be e x p r e s s e d i n terms  moments  E q u a t i o n s 2.3.3 and  of c o n t i n u o u s charge  1 6 distributions. The point  model we  quadrupoles  distributions) The  to  interactions  the  p o t e n t i a l energy will  the  has  (representing  embedded a t  such p a r t i c l e s added due  consider  consist  of  p r i n c i p l e of  [30],  The  hard due  and  pair potential function  u(12)  = u  H S  be  (12) + U  where DD,DQ and  QQ  f o r the  (b)  Expansion  of  12  the  of  i n t e r a c t i o n of  of  quadrupole  U(12)  (12)  + U  two  d q  (12)  + u  Q Q  u(12)  really  depend on  in Rotational  two  the  equations necessary  i n eqn.  of  particle particle  1 will 2.  hard  (12),  sphere  (2.3.5)  three  a total  relative of  to obtain  Explicit  below.  are  p o s i t i o n and These  p o s i t i o n s and  6 variables.  and  Invariants.  2.3.5  f o r each p a r t i c l e .  the  the  given  be  electrostatic  i n c l u d i n g the  terms are  components  coordinates  formidable  centers.  q u a d r u p o l e of  form  v a r i a b l e s , the  particles,  charge  interactions, respectively.  orientation only  and  the  angle dependent  E a c h of functions  d d  the  linear  denote d i p o l e - d i p o l e , d i p o l e - q u a d r u p o l e  quadrupole-quadrupole forms  of  sphere to  and  symmetric  superposition  Thus our  will  dipoles  s e v e r a l components w h i c h can  with both  component  dipole  axially  interact  core  the  dipole  point  Even  macroscopic  written  as  three components orientations  of  so,  of  solution  properties  is a  task. This  difficulty  can  be  easily  overcome  i f a l l pair  functions  a r e expanded  i n a basis  w h i c h span t h e c o m p l e t e The  coefficients  s e t of orthogonal  space of p a r t i c l e  of the b a s i s  interparticle  separation  translational  symmetry U ( 1 2 )  set w i l l  only.  This  polynomials  orientations  be f u n c t i o n s  i sequivalent  must p o s s e s s  [31].  of the t o using the  i n order  to rewrite i t  as  U(12)  = u(r ,n ,r ' n ) 1  i  2  r  = Zu  2  1 (  }  (| r - r | ) f ^ 1  2  a  where r=(0,0,O) a n d 6 a n d <p a r e t h e p o l a r . r 2 r= , — - — , l 1 2' r  of  thevector  third  angles  (0, ,Q , r ) ,  (2.3.6)  2  and a z i m u t h a l  angles  a n d Q. , $2, a r e t h e e u l e r o r i e n t a t i o n  1 a n d 2, r e s p e c t i v e l y ( s e e f i g u r e  angle of f i s set i d e n t i c a l l y enabling  e  r  of p a r t i c l e s  variable  l  r  r  angles  g  J2 , f l 1  are defined  2  with  t o zero  and a c t s  1 ) . The a s a dummy  a n d r t o be t r e a t e d u n i f o r m l y . A l l respect  t o t h e same a r b i t r a r y  (lab.)  frame o f r e f e r e n c e . Using R  m  t h e g e n e r a l i z e d Wigner  • ( f i ) [ 3 2 ] we c a n u s e a s b a s i s  s p h e r i c a l harmonics  functions  the r o t a t i o n a l  invariants  Ci- v 2'*> ( s  n  where f  m  n  possible  - £  m  n  l  s ( ; ;1  >c.<vC-< 2> io *>' n  ^ c a n be any n o n - z e r o c o n s t a n t u,v,\ w h i c h a l l o w  (  m  n  R  (  ( 2  - - » 3  7  a n d t h e sum i s o v e r a l l  > ), the usual  3 - j symbol  [32],  t o be n o n - z e r o . The functions conform  r o t a t i o n a l i n v a r i a n t s a r e chosen as b a s i s  because of t h e i r  t o t h e symmetry  r e m a r k a b l e symmetry p r o p e r t i e s ,  restrictions  required  of u(!2).  which  These  18  Figure  The  three euler  angles  angle  a about  about  the y ' - a x i s  the  z'-axis.  (a,0,7).  the z - a x i s , and  II  1  I i s a r o t a t i o n through  i s a r o t a t i o n through  I I I i s a' r o t a t i o n t h r o u g h  an  an  an  angle  angle  7  0 about  19  20 restrictions are: ( i ) U ( 1 2 ) must be t r a n s l a t i o n a l l y c h o i c e of o r i g i n intermolecular equation (ii)  f o r the lab.  forces.  This  invariant  because our  frame does not a f f e c t the was t a k e n c a r e  of by  writing  2.3.6. u(12) must be r o t a t i o n a l l y  definition  of t h e l a b frame must  invariant.  Again, the  have no e f f e c t  on the  potential. (iii)  U ( 1 2 ) must be unchanged on p e r m u t a t i o n  particles (iv)  of  (identical)  1 and 2. U ( 1 2 ) must remain unchanged when o p e r a t e d  symmetry o p e r a t o r s of t h e i n d i v i d u a l The  on by t h e  particles.  rotational  invariants  have a l l t h e s e  Proof  of ( i ) i s inherent  i " 2.3.6.  P r o o f s of ( i i ) and ( i i i ) a r e  given  i n Appendix B.  Condition  (iv)  properties.  i s dependent on the  symmetry o p e r a t o r s of t h e m o l e c u l e s b e i n g c o n s i d e r e d , simply  has t h e e f f e c t  needed  i n the expansion  that In  any e x p a n s i o n  Chapter  basis that  i n these  We n o t e here t h a t  rotational  invariants  3 dependence of v a r i o u s q u a n t i t i e s  the t h e o r e t i c a l l y  the b a s i s  each other  basis  we must must  assume  converge.  on t h e s i z e  of t h e  [10,11] we see  s e t seems t o c o n v e r g e  with  an example of m o l e c u l a r  symmetry  shrinking  s e t we d e r i v e  function  restrictions for  model b e i n g  dependent  infinite  work  functions  few e l e m e n t s . As  the  [15],  some of t h e b a s i s  s e t i s examined and as i n p r e v i o u s  relatively  of  of removing  and i t  point  used.  the b a s i s  the s i z e  Our model m o l e c u l e has d i r e c t i o n a l l y  d i p o l e s and q u a d r u p o l e s which a r e a l i g n e d  and t h i s d i r e c t i o n  i s a body  fixed  symmetry  with  axis  21 arbitrarily under  defined  as the z - a x i s .  r o t a t i o n about t h i s  rotation and  7  The  only  2  axis  R  m  MM  ,(&,) 1  i n the t h i r d  and fi = ( a , 0 , 7 ) 2  2  which c a n c o n t r i b u t e  those which a r e independent of 7 ^ this are  requirement are those left  be  invariant  f o r both molecules.  c o r r e s p o n d s t o a change  where $^ = ( ^ , ^ , 7 ^  u(12) must  The  2  2  to *  R  euler  angles  7  1  (see f i g u r e 1).  ? , (S2. , 0 , f ) a r e M V 1 2' m i  1  o  ™ '(^)  which  M  f o r w h i c h M ' = 0.  This  Similarly  fulfil J > ' = 0.  We  with  u(12) =  I mnl  u  m n l  (r)$  m n l  (J2  1 1  ,C„r), *  (2.3.8)  , «mnl iiiinl where s> = 9>QQ . It expansion  would  would  seem t h a t  still  an  infinite  be needed  to give  number  of terms  u(12) e x a c t l y ,  but  because the m u l t i p o l e  e x p a n s i o n h a s been  equivalent  truncation  i n the r o t a t i o n a l i n v a r i a n t expansion i s  produced.  F o r our p r e s e n t  that the  the terms  i n eqn. 2.3.5  i n v a r i a n t expansion.  u  (12)  = u  1  D D  U  (12)  = u  1  D Q  = u  2 2 4  U  QQ  model  ( 1 2 )  1  2  2  3  (r)$  identified  an  (Appendix  with  terms  C) from  has  1  1  (r)$  1  (r)*  i t c a n be shown  c a n be  One  truncated,  i n the  2  2 2 4  2  02),  3  02) + u  (12),  (2.3.9a)  2  1  3  (r)$  2  1  3  02),  (2.3.9b)  (2.3.9c)  where  1 12,  2 ^, r  v  ( 2 . 3 .1 Oa)  J  u  1 2 3  (r)  = -u  2 1 3  (r)  = U&j, 2r  (2.3.10b)  4  2 u  and  224  ( r )  _Cj ^ 4r°  =  <*> (l2) = <*> (0 ,0,, r ) , n = u mnl  mnl  (2.3.10c)  and Q = Q  o f eqn.  2.3.3.  23 2.4 I n t e g r a l  Equation Theories.  (a) The H y p e r n e t t e d - C h a i n  This equations.  integral  Integral  e q u a t i o n method  Equation  Theory.  i s defined  by two  These a r e t h e O r n s t e i n - Z e r n i k e (OZ) e q u a t i o n and t h e  Hypernetted-Chain  (HNC) c l o s u r e  a p p r o x i m a t i o n . The OZ e q u a t i o n  is  h(l2)  = c ( 1 2 ) + -£=• / c ( 1 3 ) h ( 3 2 ) d X 8TT  where t h e i n t e g r a t i o n  i s over  variables  3.  1914  of m o l e c u l e  [13],  was o r i g i n a l l y u(12) It The  the  thought  known t h a t  here  that  correlation following  function.  h(12)  critical  i s the g e n e r a l i z a t i o n  the ' d i r e c t  correlation  regarded as a d e f i n i n g  called  expansion  near  for rigid  [33].  It  C ( 1 2 ) was g o i n g t o depend o n l y on  Because of t h e o r i g i n a l i s often  fluctuations  introduced in  function'.  c ( 1 2 ) depends on h ( l 2 ) ( o r g ( 1 2 ) ) a s w e l l .  OZ e q u a t i o n i s now  right  and o r i e n t a t i o n  g i v e n by Workman and Fixman  and so i t was c a l l e d  i s now  c(12).  of d e n s i t y  The f o r m u s e d  non-spherical molecules  the s i x p o s i t i o n  T h i s e q u a t i o n was f i r s t  i n an i n v e s t i g a t i o n  points  (2.4.1)  3'  relationship for  t e r m i n o l o g y , t h e second  the i n d i r e c t  I f one i t e r a t e s  part  t e r m on  of t h e p a i r  e q u a t i o n 2.4.1 t h e  i s obtained  = c ( 1 2 ) + -£=• / c d 3 ) c ( 3 2 ) d X 8TT  +  (-Pj) 87T  2  J cd3)c(34)c(42)dX dX 3  4  +  (2.4.2)  and  one  total  can  see  that  the  c o r r e l a t i o n between m o l e c u l e s  successively  longer  The h(12)  OZ  c h a i n s of  equation  to a s e a r c h  equation  equation,  the as  s y s t e m of  the  C(12)  1953  powers o f  be  density.  divided  these c l a s s e s one  c l a s s , the  The  approximation  that  of  The  applied the  the  c ( i j ) via  an  search  closed.  exact  for  independent  i n t e r m s of  be  i s not  known  The  OZ  result.  known, so  Scoins  i t can  be  [34]  an  approximate  showed t h a t  written  clusters.  according  identified  elementary  ignoring  find  as  c o e f f i c i e n t s in this  into classes be  can  to c(12)  i n t e g r a l s known as  can  2 imposed by  as  used.  transformed  multi-dimensional can  i f we  R u s h b r o o k e and  is Fourier  interpreted  r e d i r e c t s our  e q u a t i o n s can  closure  be  molecules.  r e l a t i o n , i s an  exact  r e l a t i o n s h i p must be In  but  can  1 and  only  relates h(l2)  a defining  Unfortunately  other  really  for c(12),  which a l s o  functions,  in  i n d i r e c t part  with  to  define  elementary  expansion  expansion These  known f u n c t i o n s .  the  clusters  HNC  be  most  of  However,  easily calculated.  equation  [35].  are  clusters  t h e i r form and  c l u s t e r s , cannot to  an  if  The  is HNC  simply equation  is  C(12)  where In basis for  for  d e n o t e s the ignoring  mathematical  integrals  = h(l2)  are  - ln(g(12))  natural  the  logarithm.  Equations  There  (2.4.3)  i s no  physical  elementary c l u s t e r s ; t h e i r omission  simplicity.  only  - 0U(12),  significant 2.4.1  and  However, t h e at 2.4.3  short  elementary  range  cannot  be  [7], solved  is  cluster  25 analytically  so n u m e r i c a l  s y s t e m , h ( 1 2 ) , c d 2 ) and sphere eqn.  diameter,  2.4.1  methods must be  u(12)  used.  are d i s c o n t i n u o u s at  so t o a v o i d c o m p u t a t i o n a l  =  The  OZ  equation  For  spherically  i n F o u r i e r space  ^  where t h e  17( 12)  tilde exists  k  )  =  of  2.4.1  or, a l t e r n a t i v e l y ,  separable  solution  the  hard  rewrite  i s a continuous  t h e OZ  e x a c t l y r e w r i t e s h(12)  eqn.  2.4.4  symmetric with  f u n c t i o n of  the  r e w r i t e s 77 (12)  i n terms i n terms  functions t h i s equation  (  Since  f o r n o n - s p h e r i c a l f u n c t i o n s we with  is  solution  T ^ c T T T '  invariants  r.  Equation.  denotes F o u r i e r transform.  in rotational  (2.4.4)  J  (b) R e d u c t i o n  c(12).  present  J c ( 13) [TJ(32) - c ( 3 2 ) ] d X _ ,  where r){ 1 2 ) =h (1 2 )-c ( 1 2 ) , and  of  r=d,  p r o b l e m s we  8TT  c(12)  the  as  T?(12)  of  For  no  such  - -  2  4  and  the  problem. What [15]  but  recent  with  as  the  n o t a t i o n and  l i t e r a t u r e [10,11].  completely such  f o l l o w s i s e s s e n t i a l l y due  g e n e r a l but  the present  The  is valid  model.  Let  t o Blum and  d e f i n i t i o n s used derivation  given  Torruella  i n t h e more below  f o r a x i a l l y symmetric  )  simple  expand C(12)  a view t o s i m p l i f y i n g  5  i s not particles  L  77(12) =  T7  m  n  l  (r ) *  (12),  (2.4.6a)  m n : L  (12),  (2.4.6b)  m n l  (12),  (2.4.6c)  m  n  l  mnl  c(12)  with  =  I c mnl  m n l  h(12) =  Z h mnl  m n l  $ ^"(12) given  noting  We  / dn dr c(13)[i?(32) 3  8 7T  Then  (r)*  by eqn. 2.3.7.  mn  T?(12) =  (r)$  r  l  2  can w r i t e  e q n . 2.4.4  - c(32)].  3  as  (2.4.7)  1  = r ^  r  +  w 3  2  e  take the F o u r i e r  transform to  obtain  /dr T?( 12)exp(ik-r ) 1 2  1 2  =  / dfl dr dr 3  3  2  c (1 3)  87T  x  If  we  fluid  [T?(32) - c ( 3 2 ) ] e x p ( i k - ( r  l e t molecule  1 be t h e o r i g i n ,  13  +r  3 2  ) ).  then d r = d r 3  (2.4.8)  and s i n c e  3 1  i s i s o t r o p i c , i n t e g r a t i o n on t h e r i g h t o v e r  equivalent  t o i n t e g r a t i o n over d r  /dr r?( 12)exp(ik-r ) l 2  =  l 2  / dJ2  3  1 3  dr  3 2  .  dr  3 1  dr  l 2  our is  Therefore  J d r c (1 3) exp( i k • r . ) 3  8TT  x ; dr [rj(32) 3 2  - c ( 32) ]exp( i k • r  3  2  ) ,  (2.4.9)  or  rj(12) =  / dO8TT  C(13)[T?(32) - c ( 3 2 ) ] , .  (2.4.10)  27 where,  as d e r i v e d  c(12)  = c(n  i n A p p e n d i x D,  i f  O, k)  =  f  1  z  77(12) = ^ ( 0 , ,Q,,k) = 1  z  where k i s a u n i t v e c t o r  we  have  Z mnl  c  (k)$  I mnl  *j (k)* (Q.,Q-,k),  m n l  m n l  (fl  1  ,fl,,k) ,  1  mnl  (2.4.11a)  z  (2.4.11b)  mnl  and  00  c  m n l  (k)  = 4TT J d r r j ( k r ) c 2  m n l  n  0  (r)  f o r 1 even,  (2.4.12a)  f o r 1 odd.  (2.4.12b)  U  00  ?  m n l  (k)  = 47ri / d r r j . ( k r ) c 2  0  The t r a n s f o r m a t i o n first  order  kernels  (r)  in equations  spherical Bessel  j (x)  =  0  j  m n l  1  (  x  functions  S  i  ;  (  x  2.4.12a,b a r e z e r o t h which a r e , e x p l i c i t l y  ,  )  (2.4.13a)  « sinix). _ c o s i x i ^ .  )  and  (2.4.13b)  x The  functions  c  given  by  c  m n l  i (r)  = c  m n l  c  m n l  (r)  = c  m n l  m n l  (r)  -i  (r)  (r)  which a r e t r a n s f o r m e d  o mnl, > - / S — _ i s i r - J c ^  m  n  ( s ) s  e  p  l  (  (  |  )  d  s  f  r o | S I S )  p  where P^(x) and P ° ( x ) a r e p o l y n o m i a l s  (  )  d  o  i n 2.4.12a,b a r e  r  s  given  1  f  e  o  r  by  v  e  1  ,  n  Q  d  (2.4.14a)  d  f  ( 2  . .l4b) 4  28 P®(x)  P  e  = P°(x)  2  L 9( ) X  = 7TT q! .  +  (2.4.15a)  2i a-i a x^(-) (?) i  q 2  2q+2  = 0,  q  = Q  !  X  (  r^—, . ^,  ( p  ?o ^ > 2q 3  = WT q!  x  ++  f o r q>0.  We  2  x  i= Q  the  the hat  note t h a t  m n l  (k)  V (k)  of  Blum  = 47ri  direct  necessary  By  order  Fourier transform  n  ^  m n l  2  given  here  i n terms  (r),  have  m n l  (r) ,  (2.4.16b)  would be  the hat  an  extremely  transforms  functions  (eqns.  can  techniques.  be  functions. time  only  the  2.4.13a,b)  c  mnl, (r). N  c  112,  x  (r), c  f o r n=3,4,5 r e s p e c t i v e l y .  123,  are  calculated using  The  hat  transforms  a d d i t i o n a l f e a t u r e of a c c u r a t e l y i n c l u d i n g t h e of c  of  (2.4.16a)  spherical Bessel  transforms  (FFT)  note  projections could  r j , (kr)T7  order  Bessel  We  i s used.  1  using  w h i c h means t h e  range p a r t r  first  f c  The  are  2  computation  consuming p r o c e d u r e . z e r o t h and  l  Z!=T(Z+1)  similar.  r j,(kr)c  OO J dr  1  (2.4.15c)  step  oo / dr 0  1  2  c^^k)  [16].  single  ^) ',  \ 3)j  are  m n  where j ^ ( k r ) d e n o t e s t h e However, t h i s  q  definition  for 7j ^(k)  = 47ri  mnl  as  general  m n  c  —  i  p r o j e c t i o n s 7 j ^ ( k ) and transforms  have t h e  q  the  been c a l c u l a t e d i n t h e  fast  (-) " (?) i  ( i +  Expressions that  2 i  (2.4.15b)  +  long  \ , 224, , , ( r ) and c ( r ) depend on x  However, c  1 12  (r), c  12 3 (r)  and  224 c  ( r ) have no  such  r d e p e n d e n c e and  are  relatively  short  ranged. Equation  2.4.10 g i v e s a  formal  solution  but  as  yet i t  r  is  not separable  perform  this  because of t h e a n g u l a r  integration e x p l i c i t l y  form o f t h e e x p a n s i o n s . (Appendix  E) u s i n g  spherical  h a r m o n i c s and 3 - j symbols  -mnl,. x * U  )  =  _ ? , nilil  p  (  n  x  _  )  n  i  l  '(k)(7?  n  which c a n c o n v e n i e n t l y  be w r i t t e n  9j  c  m n l  (k)  f  = p I zi'i'i n,l,l2  m  n  i  l  l  m  n  1  1  f  1  n  1  n  l  {  Mk)-c  l  n  n  i  1 1,1 , m n n, 2  f  ImTQ f  i  generalized  we end up w i t h  2  m  amount o f s i m p l i f i c a t i o n  o f t h e Wigner  , .m+n+n,(21+1) (2n,+l)  ( J'J'J)c  We c a n  i f we t a k e a d v a n t a g e o f t h e  With a great  the p r o p e r t i e s  integration.  n  l  2  2  }  (k)),  (2.4.17)  i n t h e form  (k)(??  n  i  n  l  2  (k)-c  n  i  n  l  2  (k)),  (2.4.18a)  1  where  1 1,1 m n n, 2  .m+n+n,(21+1) f (2n,+1)  =  x  Now is the  projections  could  easily  solution value  1  1  f  2l 0  f mnl 1  I  applying  n  l  1 1,1 , m n n,  2 f  2  1  (2.4.18b)  form  space  We c o u l d set c  2.4.18 and t h e n  would be e q u i v a l e n t  n  in Fourier  f o r a given  be p u t i n t o m a t r i x  1  ).  relationship.  rj " '(r) mn  n  l'l  i t i s c l e a r that  j u s t an a l g e b r a i c  transforming,  (  m  m n  at this  *(r)  back  t h e OZ  equation  point  by F o u r i e r  transforming.  such t h a t  find  f i n d i n g the  to i n v e r t i n g a matrix  f o r each  o f k. However, o u r e x p a n s i o n s o f 1 7 ( 1 2 ) and c ( 1 2 ) i n  This  rotational  invariants  terms.  While  expect  fairly  shown  we  need,  i n t h e o r y , an  large basis B that  sets u  m  n  l  must have t h e same symmetry  a l s o have h  of p a r t i c l e s ) m i i i  h ^ " ( r ) used  (r)  projections. m<n<6 we  = (-)  I f we  s u c h a s y s t e m would  in  an  iterative By  matrix  of 84  last  t o be  to  We  and  (r).  in this  retain  our p r o j e c t i o n s  independent the  already  Since  h(12)  under  condition,  can, then, r e s t r i c t  still  we the  a l l independent  t o those f o r which  terms.  inversion  might  The  of an  solution  of  84x84 m a t r i x iteration  identity  f o r 3-j symbols  [16] w h i c h  inverted  This  greatly  (A.11) we  i s Blum's  reduces the s i z e  I n t h e example above t h e  i s now  reduced to  introducing  can  of t h e  largest  6x6.  the x ~ t r a n s f o r m a t i o n i t i s  choose  m  n  l  =  [(2m+1)(2n+1)P.  (2.4.19)  2.4.18b t h e n becomes  z i i i = m n n!  Now  (r) .  n m l  (invariance  resulted  transformation.  f  2  u  m + n  have  a t e a c h v a l u e of k f o r e a c h  inverted.  Before  Equation  , , u , A  require  u s i n g an  t o be  convenient  h  (-)  We  we  procedure.  X-transformation matrices  which  u , T , ,  be c a l c u l a t e d  p e r f o r m one  (r) =  requirement  restrict  get a t o t a l  w h i c h must  t o be n e c e s s a r y .  t o t h o s e w i t h m^n  m n  number of  assume s u c h e x p a n s i o n s a r e t r u n c a t a b l e  i n Appendix  interchange  infinite  1  i f we l e t  (2l+1)(-)  m + n + n i  { 1*1*1 mnn,  }(  n  '  j ! : ). 0 0 0 2  n  (2.4.20)  31 ~ c  m+n mn  ( k )  =  L  ~ mn N  OZ  eqn.  { k )  L  x  i n eqn.  x  m n l  , (k),  (2.4.21a)  , (k),  (2.4.21b)  0  , )r,  (  x  m n l  U  2.4.18 a n d s i m p l i f y  (Appendix F ) , t h e  becomes  N°J (k) = p A  I (-) — i  n  It C  _  a n l=|m-n|  =  2.4.21  equation  x  m+n  x  apply  n  (  l=|m-n|  x  , )c  C™ Mk)[N  X  n  A  n  i s convenient  n i n  A*  (k)-C  t o express  n , n  X  (k)].  (2.4.22)  2.4.22 i n m a t r i x  form  with  ( k ) a n d N ( k ) b e i n g t h e (m+1,n+1) e l e m e n t s o f C (k) a n d A* X N (k) r e s p e c t i v e l y . Our f i n a l form f o r t h e OZ e q u a t i o n i s t h e n m n  m n  A*  a matrix  equation  N (k) x  for  each v a l u e  d e f i n e d by  = p(-)  x + 1  -  smaller  It  2  of x a l l o w e d  Thus t h e x t r a n s f o r m a t i o n 2.4.18 i n t o  C (k)(I + p(-)  Q (k))~ ,  breaks  (2.4.23)  1  x  by t h e 3 - j symbols  independent  i s now o n l y  X + 1  i n eqn.  2.4.21.  up t h e a l g e b r a i c e q u a t i o n s o f s e t s of e q u a t i o n s .  necessary  t o back t r a n s f o r m  C (k) and A*  N  (k).  T h e s e back t r a n s f o r m a t i o n s  ^  m n l  (k)  = (21+1) I ( "J " I ) N ™ ( k ) , n  ^  where x must  a r e g i v e n by  A  A  ^  r a n g e between ± m i n ( m , n ) ,  A  f o l l o w e d by  (2.4.24)  32 f)  m n l  (s)  = - L - / dk 2TT 0  r, (s)  = ^  mnl  T,  m n l  we t a k e  (r)  = 7j  m n l  2  k j (ks)7? 2  = f  m n l 7  the inverse  (k)  (r) " -3  for  1 even,  (2.4.25a)  for  1 odd.  (2.4.25b)  (k)  hat transforms  J ds s P ^ ( f ) f ? 2  m n l  t o get  (s)  f o r 1 odd, (2.4.26b)  ( r ) - ±j J ds s ^ & ^ i s ) r* 0  where P-^(x) and P ^ ( x ) a r e g i v e n equations are obtained transforms are given  f o r 1 even , (2 . 4 . 26a )  0  1  for c  m n  r  by e q n . 2.4.15 and s i m i l a r  ^(r).  i n Appendix  (c) R e d u c t i o n  be  m n l  1  r ^ ( r )  m n l  n  U  JT dk 0  2TT  Lastly  k j (ks)r?  Derivations  o f t h e back  G.  o f t h e HNC  Closure.  Until  recently  t h e HNC c l o s u r e  analytically  expanded  i n r o t a t i o n a l i n v a r i a n t s due t o t h e  logarithmic taylor  term.  This  ( e q n . 2.4.3) c o u l d n o t  p r o b l e m was t r e a t e d  series representation  by t r u n c a t i n g t h e  of the l o g term.  If truncation  occurs a f t e r the l i n e a r  term, t h e r e s u l t i n g e q u a t i o n  linearized  If truncation  HNC  (LHNC).  i s the  occurs a f t e r the quadratic  t e r m t h e QHNC i s p r o d u c e d . The quickly point  LHNC, due t o i t s s i m p l e  i n an i t e r a t i v e  for a full  closure  here.  HNC  Let  form, c a n be  solved  p r o c e s s and so i t makes a good s t a r t i n g  solution.  For t h i s  r e a s o n we g i v e  this  and  separate  2.4.3  non-spherical  c(12)  = h  -  0 0 0  0 0 0  i t s spherical (000 projection)  This  (12)  gives  - ln[g  0 0 0  [h(12) -  we expand  ln[l  h  0  0  0  (r)*  l n [ l + X(12)]  0  0  G  000  after  i n the T a y l o r  the f i r s t  0 0 0  (r ) *  0  (12)]  series  3  t e r m we g e t  + X(12)] - ( h d 2 ) - h°°°(r)*  ( r )  ( r ) c  0 U U  =  h  ooo  mnl  ( r )  ( r )  a l l m,n,l e x c e p t  _  l n [ g  ooo  _ mnl  =  h  G  000  ( r )  ( r ) h  m=n=l=0.  ( r ) ]  _  0 0 0  (l2))  (r)  i n 2.4.28 a n d s e p a r a t i n g  +  for  (r)*  2  2.4.29  ooo  0 0 0  0 0 0  (l2)].  0  g  c  u  - /3u  + X(12)] = X(12) - ^ X ( 1 2 ) + ^ X ( 1 2 ) -  truncate  Using  (r)]  l n [ l + X(12)] - 0[(u(12) -  ln[1  and  parts.  (r)$  +  If  into  p g  mnl  we g e t  _ p ooo u  000  ( r ) u  ( r ) r  mnl  ( r }  ( r ) f  In t e r m s o f T J and c  0  0  (i  34 „  c  (r)  0 0 0  = ln[g  0 0 0  (r)]  0u  +  O O O  (r),  (2.4.31a)  mnl/ x 000/ \r mnl/ \ „ m n l , *-, mnl/ \ (r) = g (r)[tj ( r ) - /3u ( r ) ] - T? (r),  except  f o r m=n=l=0.  Equations  /_ . -,,\ (2.4.31b)  2.4.31a,b c o n s t i t u t e t h e LHNC  closure. The a n a l y t i c a l possible first r  expansion  due t o t h e r e c e n t  take  the p a r t i a l  of the f u l l  work o f F r i e s  HNC was made  and P a t e y [ l l ] .  d e r i v a t i v e o f e q n . 2.4.3 w i t h  We  respect to  to get  9c(12) 3r  Then  =  3h(12) ~97  1 gTiTF  rearranging  h ( 1 2 ) [  For  a &  3u(12) 9r *  . „ v (2.4.32)  using  *?<12> 3r  and  3q(l2) 3r  3£il2i  = 8h(12) 3r '  (  u  2  3  3  )  * * ' " '  we o b t a i n  _ 3aii2i  +  plugii]  -  -  &  ^ 2 l ,  (.. ) 2  4  34  c o n v e n i e n c e we d e f i n e  W(12)  = -77(12)  =  Clearly  +  /3u(l2)  1 - g d 2 ) + c(12) + 0U(12).  3W(12) — ^ — - i s the expression  i n brackets  (2.4.35)  i n 2.3.34, so we  35 can  rewrite  2.4.34 as  l£|i2)  If  we  rearrange  W(12)  of mean f o r c e ' . equation To eqn.  awji2) _ ^auUii^  {  ^  2  3  6  )  find  + W(12)  that  note  and hence  that  this  i s only  get the c l o s u r e  (2.4.37)  W(12) c a n be i d e n t i f i e d  i s the u n i t l e s s ,  We  - 0U(12),  angle  dependent  identification  true  i n t h e HNC  with  'potential  d e p e n d s on t h e  approximation.  i n a more u s a b l e  form we  integrate  2.4.36 o v e r r  1 This  h ( 1 2 )  = h(l2)  compare t o 2.4.3 we  -ln(g(l2)).  HNC  _  2.4.35 t o g e t  C(12)  and  =  dr  3£ii2i  = - J d r h( 1 2 ) ^ 1 2 1  - J  d  r  f U ^ l l i .  ( . . 8) 2  4  3  gives  00  lim [c(12,r)] r—>=>  -  Fortunately,  lim  both  - c ( 1 2 , r ) = - / d r h( 1 2 ) r  9  l i m [/3u(12,r)] - /3c ( 1 2 , r ) . r— limits  [c(12,r)]  are well  W  ^  2  )  (2.4.39)  b e h a v e d and  = l i m [ 0 u ( 1 2 , r ) ] = 0. (2.4.40)  36 Therefore,  we  have t h e  C(12)  It the  be  expandable  a product  multiple  of a s i n g l e  Appendix  H with  m  i  n  i  1  9  W  ^  2  - 0u(l2).  )  only necessary  requires  *  result  = / dr h ( 1 2 ) r  i s now  integral  final  of  the b i n a r y p r o d u c t  in r o t a t i o n a l  two  invariants  invariant.  the  that  (2.4.41)  invariants.  t o be  expressed  This relationship  under  This as  a  i s derived in  result  i ( l 2 ) $  m  2  n  2  1  2 ( 1 2 )  =  P(m,n,l)*  I  m n l  (l2),(2.4.42)  mnl  where  f  P(m,n,l) =  x  and into  1  m,n,l m n,l, - J — (2m+1)(2n+1)(2l+l)(-) 1 f  m i n 111 I m n l jl m n l 2  2  2  { } i s the u s u a l individual  c  m  n  l  (r)  ,  0 0 0  9-j  equations  =  I 2  0 0 0  M  symbol.  m  2  1  m n  ,  m + n m + n  +l 1  ,  0 0 0 ' '  M  Eqn.  f o r the c  P( ,n,l) I d r  111,11,1,  mn  2  2.4.41  (2.4.43)  i s now  separable  ^"(r) coefficients.  m h  '  n  '  1  One  has  M r ) ^ ^  r  2  -  0u  m n l  (r).  (2.4.44)  (d) The R e f e r e n c e HNC  To perturbation divided  improve  accuracy,  [19] i n t r o d u c e d  scheme whereby a l l c o r r e l a t i o n  into reference  convenient  Lado  and p e r t u r b a t i o n  t o choose a s p h e r i c a l l y  s y s t e m , and i n t h e p r e s e n t  spheres.  Writing  where X ( 1 2 ) i s a p r o p e r t y  of  hard  = h  H S  (r)  - 0[u  (2.4.45)  R  2.4.3 becomes  know  c a s e we use t h a t  o f t h e s y s t e m and X ( r ) i s t h e same  Equation  We  system as the  R  of t h e r e l a t e d s p h e r i c a l l y  + Ac(12)  It i s usually  = X ( r ) + AX(12),  property  (r)  parts.  a l l t h e v a r i a b l e s i n t h e form  X(12)  H S  a  functions are  symmetric  reference  c  Closure.  + Ah(l2)  H S  (r)  symmetric  - ln[g  H g  reference  system.  ( r ) + Ag(12)]  + Au(12)].  (2.4.46)  that  c  H S  (r)  = h  H S  (r)  - ln(g  H S  (r)) - 0u  H S  (r),  (2.4.47)  and  Ah(12) = A g ( 1 2 ) .  Therefore,  (2.4.48)  38 Ac (12) = A h d 2 )  - ln[g  H S  (r)  + Ah(l2)] +  -  This  ensures  that  g(l2) = g  H g  (r)  Taking  integrating  the p a r t i a l  with  (r)]  (2.4.49)  /3AU(12) =  in a similar  derivative  H S  0AU(12).  i n the l i m i t  The RHNC c a n be t r e a t e d HNC.  ln[g  manner  respect  0.  t o the  t o r and  gives  Ac(12) = J dr A h ( 1 2 ) ^ f i I 2 i  +  j  d  ^  r  ( ) r  » 31n[g„_(r)] / dr Ah(12) 1|  -  *™±12i  0AU(12),  (2.4.50)  where  AW( 12) = -AT?( 12) + 0Au( 12) .  The b i n a r y  product  Ac  I P ( m , n , l ) J d r m, m, n , 1, r m n 12  m n l  (r)  =  i s expanded  i n t h e same way  Ah  2  +  (2.4.51)  n , 1,  3 A W ^ M r) m  (r }  to get  1  2  ; ar ° ° ° < , l ^ i < r > . J h  r  d r  ^ n l  ( r }  - (JAu  »ln [  m n l  (r).  ( r) 1  (2.4.52)  39  purposes results  2.5  Expressions  The  thermodynamic q u a n t i t i e s  of c o m p a r i s o n can  invariant  be  with other  coefficients  to d e s c r i b e the  p*=pd , M * = ( 0 M / d ) ^ 3  2  for h(l2).  system 2  n  d  t h e more r e c e n t l i t e r a t u r e  f-mnl  11 , m •n- 1 0 0 0 L  v  throughout  this  convention.  f  which  convert  m  n  =  l  [(2m+1)(2n+1)]*  of  the  integral  h ^ " ( r ) from m n  the p r e s e n t  multiplied  by  the  =  section  w i t h Monte C a r l o the  rotational  reduced  parameters  c o n s i s t e n t with  x,  (2.5.1)  ;  i n Chapter 2.4(b)  3 use  this  use  (2.5.2)  f  equation  s i n c e the results  expressions. f  m  n  The  ^ are a r b i t r a r y .  of p r e v i o u s  requires only  To  s e c t i o n s to  t h a t e a c h h "'"(r) mn  be  factor  , m n i 7  the  using for  choose  A l l results  is insignificant  the  be  However, s e c t i o n s 2.3(b) and  simplifies  difference  those  section.  To  5  be  It i s also  i n t e r m s of  [10,11] we  =  will  i n terms of  Q*=(/3Q /d ) *.  3  a  t h a t we  t h e o r i e s and  conveniently expressed  expansion  convenient  f o r Thermodynamic Q u a n t i t i e s .  [ (2m+1  W m n 1 \ ) (2n+1 ) ] _ 0 _ 0 _ 0 _ 1! 2  (2.5.3)  40 (a) C o n f i g u r a t i o n a l E n e r g y .  The a v e r a g e c o n f i g u r a t i o n a l e n e r g y <U>  i s given  by  [31]  Using  <U>  <U>  = 2"  e q n . A.15 we  find  = 2TT V  / r  2  P  u  +  Simplifying  ff<U>  2  d r [  1 2 3  2 J " d X d X u ( 12)g( 12) . 8 7T ( - £  1  (  £  (r)g  ^  2  l 2 3  )  u  2  (r)  2  ( r) g  1 12  1 12  ^ 1  +  (r)  u  2 2 4  (r)g  2 2 4  (r)].(2.5.5)  we g e t  ,  f l  r  2 2 " h  1 1 2  (r)  M  ,  » ,123, > - » , 224, v / * ^ d r + fr - Q / * ill d d r c  + 4 Q  c  6  D  which  (2.5.4)  ) 2  dr],  (2.5.6)  J  i n terms of t h e r e d u c e d p a r a m e t e r s becomes  /3<U> 4TT * *2 " h ^ — = " O-P M / 1  +  1 1 2  I 1 2 £  (r)  p  V  2  , , Q * * * " h d r + 8irp v Q J 1 J 1  dr. r"  3  l 2 3  (r) — r 5  , dr  (2.5.7)  41 (b)  The  EV_ .  ,  Isothermal  isothermal  .  |,^_)2  ;  Compressibility  Factor  c o m p r e s s i b i l i t y factor i s given  d  r  d  !  i  )  d  n  r  3  auii2)  g  (  ]  2  by  )  (  2  [31]  5  8  )  87T  from w h i c h we  NkT  3  1  obtain  7 r p  N  9  3  N  3  0<u > p<u > ft<V > where — r : , — and — are the f i r s t N N N t e r m s of eqn. 2.5.7, r e s p e c t i v e l y .  N  '  ^ • • ' 5  y  DD  V  T  V T  (c) S t a t i c  For be o b t a i n e d  an  using  infinite  Dielectric  third  Constant.  system the d i e l e c t r i c  the Kirkwood  s e c o n d and  constant  e can  r e l a t i o n s h i p [24]  (e-1 ) (2e + 1 ) _ „„ = yg,  fo c m l (2.5.10)  where  y =  The K i r k w o o d  g-factor  4 f f  P/  i s given  .  by  (2.5.11)  42 <M >  ,• . N-1  2  g = —  2 = 1 + —  =  <^'E > 2  1 +  J dr r h 0 2  J  1  l  0  (r),  2 where <M > i s t h e mean s q u a r e o f t h e t o t a l  2.6 C o m p u t a t i o n a l  (2.5.12)  moment.  Considerations.  (a) Method o f S o l u t i o n ,  To for  begin  the i t e r a t i v e  each of the c ^ " ( r ) a t 2 m n  appropriate must  grid  width A r .  be s u f f i c i e n t l y  during  the c y c l e .  choice  such  Iterative  that  This  grid  parameters,  an i n i t i a l  points  The i n i t i a l  separated  values  i n the large  r  be a c h i e v e d  altering  c " ^ " ( r ) t o c a l c u l a t e a new s e t o f c m n l  (r)  the greatest  process  m  a r e compared w i t h  change  solution  I f they  close  fora  a r e then  a r e used w i t h the ^ ( r )  the c  i n each p r o j e c t i o n  are s u f f i c i e n t l y  i s stopped.  n  results  the reduced  ^(r)  mn  mn  t h e new p r o j e c t i o n s  iterative  These  m n  7j ''"(r)  original  If  t h e OZ e q u a t i o n .  The T j  by any  limit.  by u s i n g  r e s u l t s from a p r e v i o u s  s u c h a s t h e RLHNC.  The c  ^(r)  m n  C ( 1 2 ) —> - 0 U ( 1 2 )  using  because  of the c  satisfied  obtained  r=d  by an  i s generally  o r by u s i n g  RHNC c l o s u r e .  i s made  condition  s o l u t i o n and g r a d u a l l y  closure  guess  t o t h e s o l u t i o n so a s n o t t o d i v e r g e  convergence can u s u a l l y  from a p r e v i o u s  different  close  n  cycle  n  m  e  n  w  l  using the ( r )  n  occurs  e  at  w  there..  t o the o l d , the  a r e not s u f f i c i e n t l y  close  the  mnl  C  ( r )  c  =  are  _  ( 1  mixed w i t h  a  mnl  superscript  are  mn  (r)  (i+1)  where t h e a^  m n l  occurs  mnl  ( r )  (i)  c "^ ( r ) m n  +  a  in 0 < a  i f the  a^  m n  are  mn  ^  n  e  mnl mnl c  ( i ) i n d i c a t e s the  scalars varying  divergence  ) c  the  <  i  f c  ^  1.  chosen  according  w  ( f }  new^  (  iteration, It  too  >  6  >  and  i s found l a r g e at  2  to  1  )  the  that first,  so  mnl a given a  m n  ^  a  are  i s increased varied One  the  dimensional  using  i n t e g r a l s are the  standard  (b)  As  Fourier  various  routines  save  would be  of  the  virtually stored,  dramatically  i s minimized  always using  them  by  we  at  rather  using  are  [36].  the  2  faced  space.  In  trade-off to  i t i s necessary  to  points.  amount of without  cpu  The  time  use  FFT  that  the  them.  than c a l c u l a t e d a t  each  iteration,  eqn.  2.4.44.  About  n o n - z e r o but  even  so  number  saving  given  a  order  grid  n  with  by  with  i n the  calculated  derivatives  routines  impossible  are  easily  are  storage  P(m,n,l) c o e f f i c i e n t s  increases  The  Efficiency.  case,  (FFT)  functions  these c o e f f i c i e n t s  space  Program  such a s i g n i f i c a n t  Also are  numerical  t i m e and  transform  tabulate  task  i s approached.  d i f f e r e n c e formulas  i s u s u a l l y the  between c o m p u t a t i o n fast  convergence  independently.  t r a p e z o i d a l r u l e and  calculated  as  the  size  only  of  the  same o r d e r  the  their  basis  set.  20%  Storage  n o n - z e r o P ( m , n , l ) and  to avoid  indexing.  This  by  cuts set  cpu time per of  iteration  35 p r o j e c t i o n s , and  a f a c t o r of about Another taking  advantage  by a f a c t o r of a b o u t  2 for a  f o r an  84  term b a s i s  50, making t h i s  an  u n a v o i d a b l e measure.  decrease of  i n the cpu  the l a r g e  r limit  time can of c  m n  basis  set i t i s cut  be a c h i e v e d by  *(r).  In p r a c t i s e ,  mn 1  f o r m o r n > 2,  the c  zero at r e l a t i v e l y distance  the c  In C h a p t e r c  m n l  (r)  3 we  short  ( r ) can discuss  is valid.  (r) are range. be  insignificantly Thus,  d i f f e r e n t from  for r greater  set to zero for these  a t what d i s t a n c e  this  by  than  some  projections.  truncation  in  45 CHAPTER 3  Results  3.1  The using  results  512 g r i d  reference  Input  P a r a m e t e r s and B a s i s  obtained  points  radial  and D i s c u s s i o n .  with  in this  a grid  distribution  Sets.  chapter  were a l l f o u n d  w i d t h Ar = 0.02d.  function g  u c  (r)  The  was t a k e n  t o be  rib  the  Verlet-Weis Four  f i t [ 3 7 ] t o computer basis  s e t s were o b t a i n e d  s e t s were u s e d t h r o u g h o u t .  by i n c l u d i n g i n d e p e n d e n t  from t h e c o r r e l a t i o n m,n<i The  the  terms a r e denoted  These b a s i s  by t h e i r  s e t so o n l y  the complete  includes  those  data. These  i n Table  I.  s e t i s an e x t e n s i o n  of  (mnl) and o n l y  Each b a s i s  a d d i t i o n a l terms a r e g i v e n .  s e t of independent  terms l i s t e d  terms  the r e s t r i c t i o n  sets are given  indices  basis  (by symmetry)  expansions with  terms a r e l i s t e d .  previous  example, also  function  f o r i=3,4,5 and 6.  independent  simulation  For  terms f o r b a s i s  i n Table  I for basis  set III  sets  I and  II. The reduced are  parameters  used.  previous  s y s t e m c a n be c o m p l e t e l y * * * p ,n  and Q .  T h e s e s e t s were c h o s e n Monte C a r l o  results  characterized  Three  s e t s of these  t o make c o m p a r i s o n s  [8] p o s s i b l e .  parameters with  The p a r a m e t e r  used a r e as f o l l o w s : (a)  p* =  0.8 ,  ix*  (b)  p* =  0.8  ,  M *=  1.5  , Q* = 1.0 ,  (c)  p* =  0.8  ,  M *=  1.0  , Q* = 1.0 .  = 1.5  by t h e  , Q* = 0.5 ,  sets  Table I  B a s i s S e t s U s e d i n RHNC  Basis Set  Number o f Independent 20  m,n<3  II  35  m,n<4  III  m,  56  n<5  calculations,  Terms I n c l u d e d ( v a l u e s of (mnl))  Terms  (011),(110)  (112)  (022) ,  (121 ) ( 1 2 3 ) , ( 2 2 0 )  (222)  (224) ,  (033)  (132),(134)  (231 ) (233) ,  (235)  (330),(332)  (334)  (336)  (044)  (143),(145)  (242)  (244),  (246)  (341),(343)  (345)  (347) ,  (440)  (442),(444)  (446)  (448)  (055)  (154),(156)  (253)  (255),  (257)  (352),(354)  (356)  (358),  (451 ) ( 4 5 3 ) , ( 4 5 5 )  (457)  (459),  (552),(554)  (556)  (558),  (000)  (550)  (5,5,10) IV m, n<6  84  (066),(165),(167),(264),(266), (268),(363),(365),(367),(369), (462),(464),(466),(468),(4,6,10), (561),(563),(565),(567),(569), (5,6,11),(660),(662),(664),(666), (668),(6,6,10),(6,6,12)  As at  discussed  a distance  R.  i n s e c t i o n 2.6 t h e c  Table  I I shows t h e e f f e c t  -0<U>/N when R i s v a r i e d . calculations. m n  ^(r)  and  cut-off.  Also  shown  i n these  f o r parameter  s e t (a)  the  savings  results  achieved  these  by t h e  cut-off  distances  used.  pV/NkT.  Results  values  gives  The b a s i s  of u  acceptable  and ( c ) g i v e  when b a s i s  s e t d e p e n d e n c e t o some e x t e n t  and Q . values  f o r these  reduced. which  -/3<U>/N  d e p e n d s on  s e t (a) b a s i s  set I  q u a n t i t i e s , but parameter  results  for a l l three  IV and V compare c o n t a c t a t r = 1.1d o f t h e f i v e  e, /3<U>/N a n d pV/NkT.  the l a t t e r  s e t than  sets  q u a n t i t i e s only  i s the only  ( r = 1.0d) a s  p r o j e c t i o n s used t o  The c o n t a c t  values  t h e i n t e g r a l s over  This  i sclearly  a s r= 1.1d t h e s e n s i t i v i t y The l e a s t  values  a r e much these  of which a r e used t o f i n d t h e  thermodynamic q u a n t i t i e s . to contact  F o r parameter  converged  more s e n s i t i v e t o b a s i s functions,  s e t dependence o f e,  *  as the values  calculate  Sets.  s e t I I I i s used. Tables  well  for D i f f e r e n t Basis  I I I shows t h e b a s i s  *  (b)  s e t I I was used  i s the time  F o r a l l subsequent  Table  the  Basis  The t a b l e shows t h a t  3.2  and  on t h e d i e l e c t r i c  c a n be c u t o f f s a f e l y a t 5.5d, and a t 3. Od f o r s e t s (b)  (c).  were  truncated  e and t h e average c o n f i g u r a t i o n a i energy p e r p a r t i c l e  constant  c  ( r ) were  b e c a u s e even a s c l o s e  to basis set i s greatly  s e n s i t i v e p r o j e c t i o n at contact  contact  thermodynamic q u a n t i t i e s .  value  used  i s g^^(d)  i n the c a l c u l a t i o n of  Table II  Cut-off  Parameter  Cut-off  Set  in  (a)  (b)  (c)  c  dependence  in c  (r).  Radius  mnl, x (r)  cpu -0<U>/N  e  time per I t e r a t ion  10.00  3.51  20.55  22.0s  8.00  3.51  20.55  19.1s  5.50  3.51  20.54  1 5.6s  3.00  3.51  1 9.52  1 1 ,6s  1 0.00  6.25  10.21  22.3s  8.00  6.25  10.21  1 9.0s  5.50  6.25  10.21  1 5.4s  3.00  6.25  10.17  12.1s  1 0.00  4.21  4.67  22.3s  8.00  4.21  4.67  19.1s  5.50  4.21  4.67  15.3s  3.00  4.21  4.66  1 1 .6s  Table I I I  B a s i s S e t Dependence o f e,0<U>/N and pV/NkT.  Parameter  Basis  Set  Set  (a)  I  (b)  (c)  c  -/3<U>/N  pV/NkT  20.5  3.51  5.22  II  20.5  3.51  5.24  III  20.4  3.52  5.21  IV  20.4  3.52  5.21  I  10.7  6.21  2.80  II  10.2  6.25  2.78  III  10.1  6.58  2.72  IV  10.1  6.58  2.70  I  4.80  3.98  4.11  II  4.66  4.21  4.01  III  4.68  4.37  3.97  IV  4.67  4.38  3.97  T a b l e IV  Projection  Contact  Values  Parameter  Basis  Set  Set  (a)  I  4.799  1 .942  3.617  -.7007  -.06988  II  4.804  2.000  3.695  -.7032  -.05925  III  4.807  1 .979  3.675  -.7133  -.06716  IV  4.806  1 .983  3.680  -.7153  -.06770  I  5.998  .4524  2.885  -1 .426  -.5106  II  6.039  .0654  2.695  -1.518  -.5366  III  6.290  .2253  2.893  -1.726  -.6359  IV  6.287  .2408  2.997  -1.738  -.6255  I  5.330  -.1428  1 .462  -0.954  -.5273  II  5.507  -.4745  1 . 1 08 -1.090  -.6263  III  5.628  -.2964  1 .381  -1.212  -.6703  IV  5.630  -.3532  1 .322  -1.218  -.6783  (b)  (c)  g  0 0 0  (d)  h  1 1  °(d) h  1 l 2  (d)  h  1 2 3  (d)  h  2 2 4  (d)  51 Table V  Projection  V a l u e s a t r=1.1d.  Parameter  Basis  Set  Set  (a)  I  2.254  .5643  1 .096  -.1895  -.01883  II  2.253  .5542  1 .091  -.1891  -.01782  III  2.252  .5575  1 .091  -.1900  -.01879  IV  2.252  .5572  1 .091  -.1900  -.01877  I  2. 1 29  -.0839  .6300  -.2968  -. 1089  II  2. 1 28  -.1328  .5842  -.3043  -.1096  i n  2.091  -.1507  .5693  -.3067  -. 1124  IV  2.090  -.1533  .5668  -.3068  -. 1124  I  2.191  -.1394  .3782  -.2518  -. 1437  II  2.171  -.1824  .2991  -.2610  -.1518  III  2.151  -.1834  .3101  -.2630  -.1527  IV  2. 149  -.1856  .3052  -.2636  -.1531  (b)  (c)  g  0 0 0  (r)  h  1 1 0  (r)  h  l 1 2  (r)  h  l 2 3  (r)  h  2 2 4  (r)  52 3.3 C o m p a r i s o n w i t h Monte C a r l o  The compared  MC c a l c u l a t i o n s w i t h w h i c h o u r r e s u l t s a r e  [ 8 ] were c a l c u l a t e d  condition  Calculations.  at a radius  R  with a s p h e r i c a l  = 3.40d.  cut-off  Therefore,  boundary  i n order to  c compare t h e p a i r the  correlation  RHNC a p p r o x i m a t i o n  3.40d.  This  function  was s o l v e d  and c o n f i g u r a t i o n a l  truncating  was done f o r a l l p a r a m e t e r  (a)  The  Configurational  average c o n f i g u r a t i o n a l  energy  the p o t e n t i a l at  sets.  Energy.  energy per p a r t i c l e i s  compared w i t h MC r e s u l t s  i n Table VI.  calculated  t h e RLHNC, RQHNC and MSA t h e o r i e s a s  well  r e s u l t s using  as those  o f t h e Pade v e r s i o n  theory  [8].  It i s clear  closer  t o t h e MC s i m u l a t i o n  approximations, parameter  dielectric  with  spherical  uncertainty  this  are previously  perturbation  gives  constants  -0<U>/N by ~ 3 % f o r  parameter  sets.  Constant.  from t h e MC c a l c u l a t i o n s  c u t o f f boundary c o n d i t i o n s  using  values  r e s u l t s t h a n any o f t h e o t h e r  Dielectric  i n these values  r e s u l t s were o b t a i n e d In  t h e RHNC c l o s u r e  s e t ( a ) and ~ 5 % f o r t h e o t h e r  The  shown  o f thermodynamic  but tends t o underestimate  (b)  obtained  that  Also  i squite  large  [24].  so t h e  The RHNC  e q n . 2.5.10 f o r an i n f i n i t e  c a l c u l a t i o n the p o t e n t i a l  were  was n o t t r u n c a t e d .  system. Results  T a b l e VI  The C o n f i g u r a t i o n a l  Energy,  -0<U>/N.  Parameter Set  Pade  MSA  RLHNC  (a)  3.93  3.11  4.27  (b)  6.27  5.86  8.31  (c)  4.40  3.44  5.35  RQHNC  RHNC  -  3.62  3.72±0.02  7.58  6.69  6.99±0.02  4.97  4.41  4.64±0.02  MC  from p r e v i o u s a p p r o x i m a t i o n s a s w e l l given  i n Table VII.  consistently For  MC v a l u e s (c)  The RHNC d i e l e c t r i c  smaller  parameter than  sets  uncertainty  than  those  results.  (c) The P a i r  h  1 1 i  ^(r),  h  parameter  1  1  five  1 2 3  (r)  set in figures  equivalent  functions  range  the  than  range sets.  the  Also  = h  u u u  (r)  a r e graphed on t h e s e  to exist.  + 1,  f o r each  figures  a r e the  with  We  s e t ( a ) c a n n o t be  The RHNC r e s u l t s f o r basis  that  s e t I I , and f o r  g^^(r)  t h e RHNC g i v e s  f o r a l l parameter  the other  closures.  sets  We n o t e  T h e same f u n c t i o n  (b)  i s plotted at sets  (a)-(c),  a good  and i s c l o s e r t o that  f o r the  i splotted  a t longer  2b, 3b a n d 4b f o r t h e same r e s p e c t i v e  parameter  H  Good agreement  parameter  (r)  2 a , 3a a n d 4a f o r p a r a m e t e r  = g g (r).  in figures  The  u u u  f o r parameter  I t c a n be s e e n  of g ^ ^ ( r )  RLHNC, g ^ ^ ( r )  within the  Function.  r a d i a l d i s t r i b u t i o n function  description MC d a t a  of parameter s e t  s e t IV.  in figures  • respectively.  g  (r)  s e t (a) were c a l c u l a t e d  (c) with basis  lies  to the  f o r RLHNC, RQHNC and MC c a l c u l a t i o n s .  a n d does n o t a p p e a r  short  still  2 2 4  2-16.  obtained  The  theories.  data.  and h  an RQHNC s o l u t i o n  and  i s not t r u e  Correlation  note that  parameter  This  RHNC p r o j e c t i o n s  (r), h  2  o f t h e RLHNC a n d RQHNC  for e probably  i n t h e e from MC  The  constants are  (a) and ( b ) t h i s p u t s t h e RHNC c l o s e r  other  but t h e value  a s RHNC a n d MC d a t a a r e  between RHNC and MC i s a l s o  projection sets  h  1 1  (a)-(c).  ^(r)  i s plotted  F o r parameter  evident  in figures  here.  5-7 f o r  s e t ( a ) t h e RHNC  Table VII  The  Static  Dielectric  C o n s t a n t , e.  Parameter Set  MSA  RLHNC  RQHNC  RHNC  MC  (a)  20.7  29.9  -  20.4  15.6  (b)  16.3  12.7  10.3  10.1  10.2  (c)  6.27  5.67  4.61  4.38  6.4  projection feature  closely  ressembles that  of f i g u r e s  sources.  'infinite'  constant  leads  would  to the p r e d i c t i o n  approximate  s y s t e m more c l o s e l y  approximations  striking range  The RHNC c u r v e i s c l e a r l y t h e  t o t h e MC c u r v e w h i c h  RHNC d i e l e c t r i c  The most  6 and 7 i s the disagreement a t s h o r t  between t h e v a r i o u s closest  o f t h e MC.  that the  e from a s i m u l a t e d  t h a n t h e RLHNC o r RQHNC  (see eqn. 2.5.12). 1]2  Figures  8-10 show h  ( r ) and f i g u r e s  11-13 show  1 23 h  ( r ) , f o r parameter  t h e g r a p h s have  been  sets  enlarged  t h e MC d a t a i s r e a d i l y follow other  t o the point  though  the s c a l e s of  where t h e ' n o i s e ' i n closely  The RHNC a p p r o x i m a t i o n i m p r o v e s  on t h e  t h e o r i e s as w e l l . 14-16 show h  data  i s an improvement  1.5d  f o r parameter (c)(fig.  results  ( r ) . In a l l c a s e s t h e RHNC  on t h e RLHNC and RQHNC d a t a e x c e p t a r o u n d  set (b)(fig.  15) and a t c o n t a c t  1 6 ) . However, t h e RHNC r e s u l t s  significantly 224  underestimating set  Even  a p p a r e n t , t h e RHNC c u r v e s s t i l l  t h e MC r e s u l t s .  Figures  set  (a)-(c).  h  f o r sets  ( c ) shows much b e t t e r  deviate  (a) and (b) w h i l e  (r) at close  range  agreement.  f o r parameter from t h e MC  seriously  forset (b).  Parameter  F i g u r e 2a  The  short-range part  *  *  u  = 1.5, Q  curves  of the r a d i a l  *  = 0.5 and p  represent  distribution  = 0.8. The s o l i d ,  RHNC, RLHNC and MC r e s u l t s ,  function for  dashed and d o t t e d respectively.  58  59  Figure  The  long-range part  p a r a m e t e r s and  2b  of the r a d i a l  c u r v e s are as  distribution  in figure  2a.  function.  The  61  F i g u r e 3a  The n  short-range = 1.5, Q  dotted  curves  respectively.  p a r t of t h e r a d i a l  = 1 . 0 and p  distribution  = 0.8. The s o l i d ,  function f o r  dash-dot,dashed  r e p r e s e n t RHNC, RQHNC, RLHNC a n d MC  results,  and  F i g u r e 3b  The  u  long-range part = 1.5, Q  curves  of the r a d i a l  = 1 . 0 and p  represent  distribution  = 0.8. The s o l i d ,  RHNC, RLHNC a n d MC r e s u l t s ,  function f o r  dashed  and d o t t e d  respectively.  65  F i g u r e 4a  The  u  *  short-range  = 1.0, Q  dotted  *  curves  respect i v e l y .  part  of t h e r a d i a l  *  = 1 . 0 and p represent  distribution  function for  = 0.8. The s o l i d , d a s h - d o t , d a s h e d and  RHNC, RQHNC, RLHNC a n d MC  results,  66  F i g u r e 4b  The  u  ft  long-range part = 1.0, Q  curves  *  =1.0  represent  of the r a d i a l  ft  and p  distribution  = 0.8. The s o l i d ,  RHNC, RLHNC a n d MC r e s u l t s ,  function for  dashed  and d o t t e d  respectively.  68  69  Figure  The  projection  figure  2a.  h  1 1  ^(r).  The  5  p a r a m e t e r s and  curves are  as  in  71  Figure  The  projection  figure  3a.  h  ( r ) . The  6  p a r a m e t e r s and  curves are  as  in  72  73  Figure  The  projection  figure  4a.  h  1 1 (  ^ ( r ) . The  7  p a r a m e t e r s and  curves are  as  in  74  I  75  Figure  8  112 The  projection  figure  2a.  h  ( r ) . The  p a r a m e t e r s and  curves are  as  in  76  77  Figure  The  projection  figure  3a.  h  ( r ) . The  9  p a r a m e t e r s and  curves are  as  in  79  Figure  The  projection  figure  4a.  h  ( r ) . The  10  p a r a m e t e r s and  c u r v e s are  as  in  81  Figure  11  123 The  projection  figure  2a  h  ( r ) . The  p a r a m e t e r s and  curves are  as  in  83  Figure  The  projection  figure  3a.  h  ( r ) . The  12  p a r a m e t e r s and  c u r v e s are  as  in  85  Figure  The  projection  figure  4a.  h  1 2 3  (r).  The  13  p a r a m e t e r s and  curves are  as  in  87  Figure  14  224 The  projection  figure  2a.  h  ( r ) . The  p a r a m e t e r s and  curves are  as  in  89  Figure  15  224 The  projection  figure  3a.  h  ( r ) . The  p a r a m e t e r s and  c u r v e s are  as  in  91  Figure  16  224 The  projection  figure  4a.  h  ( r ) . The  p a r a m e t e r s and  curves  are  as  in  92  93 3.4  The spheres  RHNC t h e o r y h a s been s o l v e d f o r f l u i d s  w i t h d i p o l e s and q u a d r u p o l e s .  expanding  the p a i r  the d i r e c t computing  correlation  function  easily  explicit  analytical  of e q u a t i o n s  dielectric Three and  properties  i s not found,  convergence  extent  realistic  space  way w h i c h  the solution  of the  methods. but c l o s e  o f t h e thermodynamic a n d relatively  parameters  few t e r m s .  were c o n s i d e r e d ,  t h e number o f terms needed  t o converge  * t h e e x p a n s i o n s d e p e n d s on t h e * * M and Q a r e s i m i l a r i n s i z e g e t a d e q u a t e v a l u e s f o r e and * considered. When n i s large  t h e OZ  A l t h o u g h an  are i n f i n i t e ,  i s achieved with  sets of p h y s i c a l l y  to a certain  i n a simple a l g e b r a i c  the expansions  shows t h a t  both  In F o u r i e r  i s a c h i e v e d by i t e r a t i v e  In p r i n c i p l e examination  which s a t i s f y  by m a t r i x e q u a t i o n s .  solution  f u n c t i o n and  i n v a r i a n t s and  approximation.  are related  be e x p r e s s e d  correlation  in rotational  coefficients  of hard  T h i s was done by  the pair  and t h e RHNC c l o s u r e  these c o e f f i c i e n t s  system  potential,  the expansion  equation  can  Summary and C o n c l u s i o n s .  s i z e s of n  relative  * and Q .  When  56 i n d e p e n d e n t t e r m s a r e needed t o f o r t h e thermodynamic q u a n t i t i e s * compared t o Q o n l y 35 i n d e p e n d e n t  terms a r e r e q u i r e d . The compared average was  RHNC r e s u l t s  f o r thermodynamic q u a n t i t i e s  t o MC r e s u l t s  and d a t a  configurational  energy  found  theories.  t o be a s i g n i f i c a n t The RHNC d i e l e c t r i c  from  were  various other sources.  g i v e n by t h e RHNC improvement constant  over  The  approximation  previous  i sconsistently  lower  * than  that  o b t a i n e d by o t h e r m e t h o d s .  When Q  i s small  relative  to n  this  l e a d s t o an improvement  * When Q other  i s comparable integral  the  to u  equation  in size,  p r o j e c t i o n s of the p a i r  as w e l l as with  e.  approximations,  results  and i n a l m o s t  are closely  e values are similar to  correlation  been compared from  RHNC p r o j e c t i o n s s i g n i f i c a n t l y  results  theories for  results.  t h e RHNC t h e o r y have a l s o  data  other  *  The by  over  graphically  other c l o s u r e s .  w i t h MC  In a l l c a s e s  improve on p r e v i o u s  a l l cases  approximated.  function given  t h e MC s i m u l a t i o n  95  PART B  96 CHAPTER 4  Polarizability  Theory.  4.1 The SCMF  The molecular  self-consistent  polarizibility  C a r n i e and P a t e y that  [25].  pair  approximations physical  proposed  used  (SCMF) t r e a t m e n t of  i s essentially  i n t h e SCMF t h e o r y  due t o  t o p r e v i o u s work  [24]  in  w i t h an e q u i v a l e n t  U n l i k e o t h e r methods, have a  however, t h e  distinctly  basis. a system of N i d e n t i c a l  polar-polarizable particles. = t h e permanent = the l o c a l  Pi  here  It i s similar  potential.  Consider  a  mean f i e l d  i t r e p l a c e s many body p o t e n t i a l s  effective  Hi.  Method.  interacting  Let  d i p o l e moment o f p a r t i c l e i ,  electric  = the p o l a r i z a b i l i t y = the instantaneous  felt  tensor  by p a r t i c l e i ,  f o r each of the N  induced  m. = the instantaneous l We now have  P  field  particles,  d i p o l e moment o f p a r t i c l e i ,  d i p o l e moment of p a r t i c l e i .  i  = g-(E ) , 1  i  (4.1.1)  and  " * i  + p  i '  (4.1.2)  97 Now  l e t <m>  = m' be t h e a v e r a g e t o t a l  (measured w i t h and  l e t <E^>  Then  repect  This  field  = <M>  +  <a«E^>  by a  we now r e s t r i c t  = M + a«<E^>.  ourselves  <E^> w i l l be n o n - z e r o o n l y  implies  felt  moment  particle), particle.  particles  m' = <m>  molecules,  dipole  o f mass f o r e a c h  be t h e a v e r a g e e l e c t r i c  for identical  If  to the center  molecular  (4.1.3)  to a x i a l l y  symmetric  i n the d i r e c t i o n  t h a t m' w i l l be i n t h e d i r e c t i o n  of M.  o f M S O we c a n  write  <E >  = C(m')m',  1  where C(m') i s a If  scalar.  we i n s e r t  e q n . 4.1.4 i n t o  e q n . 4.1.3 we  m' = M + a-C(m')m',  w h i c h we c a n i t e r a t e  m' = M +  =  M  (4.1.4)  with  q>C(m')[u  itself  find  (4.1.5)  to get  + a*C(m')m']  + a-C(m')£ +  g.'q-C  (m')m'.  (4.1.6)  98 This  can  m'  be  repeated  to get  = M + g«C(ni')M  the  If  we  now  define  expansion  (m')v  + g«g«C  + g'g-g'C  infinite  (m')M  +  .  (4.1.7)  a renormalized p o l a r i z a b i l i t y  g'  by  g' = g + C(m')g'-g,  we  can  rewrite  eqn.  4.1.7  total of  exactly  average  the  same f o r m  molecular  t h e permanent d i p o l e  This  means t h a t  polarizability isolated related  a fluid g can  molecule t o g by system  permanent  average  effective  system  they still  do  not  have  be  dipole  as  eqn.  4.1.5, but  moment m'  moment M and surrounded treated  (4.1.9)  as an  t h e new  molecule  in exactly  gives  equation  as  4.1.8. one  dipole  of  with the  can  now  are  a polarizability i s o l a t e d only  same way  can  be  as  g'.  g'.  g' i s  each  with a  g'.  These  i n the  an  our  sense  others p o l a r i z a t i o n properties;  interaction energies.which  i n terms  a  characterize  i s o l a t e d molecules,  £ and  molecules  a f f e c t each  We  the  polarizability  with a renormalized p o l a r i z a b i l i t y  eqn.  effective  as  = u + C(m')a'-M,  m'  w h i c h has  (4.1.8)  that  they  easily calculated.  99 4.2  Polarizable  If is possible pair  we to  potential  effective  M o l e c u l e s w i t h D i p o l e s and  make the  p a r t i c l e s of  derive  the  of  s y s t e m by  the  system which  Chapter  Quadrupoles.  2 polarizable i t  average c o n f i g u r a t i o n a l considering  implicitly  contains  an  energy  and  equivalent  the  polarization  ef f e c t s . The of  the  total  where t h e  =  extra  U  HS  +  term U  U  DD  +  U  DQ  factors are get  to  +  i s the  p  " - HS U  " 1\  m  i'  {  E  U  QQ  V  +  energy  lB i  = E^ due of  D  + E^Q  and  to dipoles ^  avoid  i n the  2  counting  E  and n d  -  ]  + 1 I i  fields  be  of  U  '  2  polarization.  '  1  We  )  can  [38]  U  where  energy w i l l  form  U  write  instantaneous c o n f i g u r a t i o n a l  P  i  irj' iQ E  2 [  1  f T ^ l Q * !  pairs  3  r d  of  +  U  QQ  -(E ) , 1  a  r  e  (4.2.2)  i  fc  ^  local  e  quadrupoles,  and  3  instantaneous  respectively.  t e r m s on  the  r i g h t of  molecules twice.  The eqn.  Simplifying  4.2.2 we  100  " HS  U  U  +  U  QQ " 1  ^i-  Z  ( K  lD>i  "  &i' lQh  Z  {E  l  l  - 1 2 Pi-(E  For  <U>  =  where  <U  <  E  m  fields,  H S  >  >  '  axially  +  <  E  <U  IQ  >  Q Q  >  a  r  symmetric  4NM.<E  e  fc  ^  1 D  >  )..  (4.2.3)  identical  - NM-<E  average  e  1 Q  1 q  >  dipolar  r e s p e c t i v e l y , and < E ^ >  +  D  < E  ^Q>  particles  - lN<  P  I  .E  1  Q  >,  (4.2.4)  and q u a d r u p o l a r = <E^>.  As  i n eqn.  4.1.4 we c a n w r i t e  <E  1 D  >  = C (m')m',  (4.2.5a)  <E  1 Q  >  = C (m')m',  (4.2.5b)  D  Q  and  <  E  1  >  =  <  E  1 D  >  +  <  E  1 Q  >  = C (m')m' + C (m')m' D  Q  = C(m')m'.  The by  last  term  ignoring  (4.2.6)  i n e q n . 4.2.4 i s a p p r o x i m a t e d  fluctuations  i n the f i e l d  such  ( i n t h e SCMF t h e o r y ) that  101 <  P i  .E  1 Q  =  Now c o m b i n i n g  <U> = < U > HS  =  <U  H S  >  >  - < >'<E > P i  (m' - M )  1 Q  , < E  IQ>-  (4.2.7)  e q n s . 4.2.4-4.2.7 we o b t a i n  + <U >  - ^NM-m'C (m')  + <U  - ^NMm'C (m') - ^ N ( M + m')m'C (m'),  QQ  Q Q  - ^ N ( £ + m').m'C (m')  D  >  Q  D  Q  (4.2.8)  where m'= |m'| a n d n = |M|. In  order  <  U  t o e v a l u a t e C (m') we u s e D  DD  >  e  =  <  _  =  I  =  ^ i -  (  E  l Q  )  i  >  -l (m'.<E >) N  l D  = -^Nm' C (m')  (4.2.9a)  2  D  where t h e s u b s c r i p t  e indicates  f  the e f f e c t i v e  system.  Similarly  we have  < u  n n > ~ = -Nm' C (m'), DQ e Q  (4.2.9b)  2  n  and  e q n . 4.2.8 becomes  <U> - < U > HS  +  <U > QQ  +  ^<U  D D  >  e  +  f^<U  D Q  > . e  (4.2.10)  1 02 In o r d e r effective written  to d e r i v e the p a i r  s y s t e m we n o t e  i n the  t h a t the  potential  c o n f i g u r a t i o n a l e n e r g y can be  2 Piy . a  2 I Pix - . a . i  3  XX  a n o  x  a  xx'  a  yy a  n  d a  zz  SCMF t h e o r y  a  P±z  " r  e t  *  a  i e  r  t  XX  et  ^  we i g n o r e  2 p.  n  i r e e  J  eX  a  do  similar  not  diagonal  As with  components o f a . I n in  , s o we have  (4.2.12)  1  and  can  11? ^(r)  u ^ ( r )  2  1  i  z  These c o n s t a n t  (r)  m  =  a given  i s o f the  form  o f eqn. 2.3.8  2.3.10  i -i L _ l m  (4.2.13a)  r  = -4, 2r* =  terms  be i g n o r e d .  2, u ' ( i , j )  1 1  u  P^ -  p r o b a b a b i l i t y of finding  a n a l o g o u s t o eqn.  u  Again,  2  f o r p^  so they  i n Chapter  coefficients  and  2 = <p.> = c o n s t a n t ,  c o n t r i b u t e t o the and  (4.2.11)  d z components o f  n  fluctuations  expressions  configuration,  2 Piz . a zz  yy  2 with  the  form  i<  where P i » P j y  for  (4.2.13b)  42r*  we a p p r o x i m a t e by i g n o r i n g f l u c t u a t i o n s  2 2 m.m.=<m>=m, l ] e'  (4.2.13c)  in  t o get  (4.2.14)  1 03 where  i s the t o t a l  m  2 g  dipole  s y s t e m g i v e n by  2 , 2 ^ , 2 2 = <m > = m + (<p > - <p> ) N  = m'  with  of t h e e f f e c t i v e  a' being  2  + 3a'kT,  (4.2.15)  trg' .  The d i f f e r e n c e  between m  a n d m' t u r n s o u t t o be s m a l l e  compared  t o other  approximation  approximations  made so we make a  final  by s e t t i n g  m Q = m'Q. e  (4.2.16)  Then e q n s . 4.2.13 a n d 4.2.9 become m = - -f, r 2  u  1 1 2  (r)  (4.2.17a)  J  u  1 2 3  (r)  = -u  2 1 3  (r)  =  (4.2.17b) 2r  q  2<U > Y ~ ^ - , m e nn  C (m') = D  (4.2.18a)  <u > n n  C^(m') = 'Q  and  (4.2.18b)  Nm m e  finally  <U> = < U > HS  +  <U  >  +  iBf m  <U e  D D  >  e  +  ^  <u > e  (4.2.! 9)  4.3 Method'of S o l u t i o n .  The  s y s t e m o f P a r t A i s s o l v e d f o r a number  values  which cover  fall.  T h i s data  cubic Then  splines using  the range w i t h i n which the a c t u a l m  i s then  f i t t o an i n t e r p o l a t i n g  [ 3 9 ] and a f u n c t i o n r e l a t i n g  given  values  f o r M,Q and a,  using  e q n s . 4.1.5, 4.1.8 and 4.2.15  m',g'  and m  c o n s i s t e n t with  m  g  curve  t o C(m')  an i t e r a t i v e  i s used t o f i n d  the given  data.  of g  m  g  must using i s found.  process v a l u e s of  105 CHAPTER 5  Results  and  Discussion.  5.1 C a l c u l a t i o n s w i t h A m m o n i a - l i k e  This as  data  chapter  uses  results  f o r t h e SCMF p o l a r i z a b i l i t y  calculations  were c a r r i e d  Parameters.  o f measurements f o r ammonia theory.  out a t three  The SCMF  sub-critical  temperatures,  and t h e l i q u i d d e n s i t i e s used a r e a t  equilibrium.  The v a l u e s a r e :  (i)  T=-35°C,  p=0.6840gcm~ ,  (ii)  T=-5°C,  p=0.6315gcm~ ,  (iii)  T=35°C,  p=0.5875gcm~ .  The Buckingham  which agrees w e l l values  which  3  3  The v a l u e  tensor  should  be u s e d  these are discussed  g used  i s due t o B r i d g e and  o f t h e d i p o l e moment u s e d  with a large  moment a r e n o t a s w e l l and  3  polarizability  [23],  liquid-vapour  number o f e x p e r i m e n t s  for particle  established in sections  diameter  as that  i s 1.47D [21].  The  and q u a d r u p o l e  f o r d i p o l e moment,  5.2 and 5.3.  106 5.2  Sensitivity  The agreement results hard  particle  with  leave  3.1A  d=3.2A, w h i c h a l s o  using Molecular  and  3.3A  shown  with  ammonia  constant  by  particle  SCMF c a l c u l a t i o n s  examined  in chapter  errors  by  the  below.  reasonable with  distribution  McDonald and  Klein  Some r e s u l t s  a t any  [41]  were  for  the  temperature.  found  t o be  The  small for  i n f i g . 17 were done w i t h  1.0%  error  l e d t o a 2.5% 3,  a  be  [25],  i n t r o d u c e d by when compared  was  done  i n -/3<U >/N and DD  i n c r e a s e i n e.  gives essentially  b a s i s set I I I f o r the  insignificant  introduced  radial  A representative calculation  D  are  should  to f i x a  17 were c a l c u l a t e d  by  was  Such  when t r y i n g  i n c r e a s e d the v a l u e  shown  showed a  [40].  d=3.2A b e i n g  t h e N-N  water  calculations  i n -|3<U Q>/N w h i c h  set  with  also  error  The  by N a r t e n  diameter  set I I I which  from  in  4%  basis  to those  t o be  no more t h a n  done on  II of P a r t A.  shown  been c h o s e n  Dynamics c a l c u l a t i o n s .  dielectric  as  Set.  v a l u e chosen  found  f o r d=3.lA, w h i c h  IV,  Basis  in figure  calculated  set  and  data  diameter  agrees  for l i q u i d  All  has  scattering  The  A l l results  dependence on  Diameter  some room f o r i n t e r p r e t a t i o n  somewhere between  function  diameter  neutron  core diameter.  choice.  to P a r t i c l e  energy particle  with a  2.5%  Basis set  identical  results  components and diameter  basis  and  e. basis  t o the p o s s i b l e e r r o r  u n c e r t a i n t y i n the quadrupole  moment.  This is  Figure  Dielectric  constant  experimental  data,  — 26 Q=-3.3 x 10  v s . Temperature. t r i a n g l e s denote  Closed c i r c l e s  denote  SCMF c a l c u l a t i o n s  with  2 e s u cm , s q u a r e s d e n o t e SCMF c a l c u l a t i o n s wi  —  Q=-2.32 x 10 calculated  17  26  2 e s u cm  and open c i r c l e s  indicate  data  from t h e e q u i v a l e n t n o n - p o l a r i z a b l e s y s t e m .  108  40 r  30 h  20  U  10  0l—I -40  1  1— -20  J  l  0  T(°C)  I  L_  20  L  40  109 5.3 C o m p a r i s o n W i t h  The appear  data  t o f i t a smooth c u r v e ,  different by  experimental  sources.  Baldwin  and G i l l  the CRC Handbook different  Experimental  p o i n t s shown on f i g . 17 do n o t  probably  because they  [42], but t h o s e  [22b] w i t h these  results  while  While t h i s  i s a fairly  shown  consistently  these  i n f i g . 17. be v e r y  underestimating  t o t h a t of  e by a b o u t 30%.  i t should  be n o t e d  that  o u t whether t h e SCMF e  to find  that a Q value  within  bounds o f u n c e r t a i n t y c o u l d e x p l a i n t h e d i s c r e p a n c y .  —26 Q=-2.32 x 10  on f i g . 17 a s e t o f p o i n t s  above.  e s u cm .  T h i s erroneous  literature  This result,  value  effect  a l s o given  with  given  h a s been used q u i t e e x t e n s i v e l y i n  [8,44,45] and a l t h o u g h  As c a n be seen  by K u k o l i c h  [22b] t o the v a l u e  p o i n t s i n f i g . 17 g i v e s a good  sensitivity.  To  2  [ 2 2 a ] , was c o r r e c t e d i n an e r r a t u m  large  The shape o f a  similar  enough t o q u a d r u p o l e  end we have a l s o p l o t t e d  these  using a quadrupole  measured e x p e r i m e n t a l l y by  large discrepancy  I t i s necessary  values are s e n s i t i v e  the  from  i s a l a r g e u n c e r t a i n t y i n t h e q u a d r u p o l e moment a s g i v e n  Kukolich.  this  e s u cm  p o i n t s would  experiment  by  from  sources.  through  there  paper  above T=0°C a r e t a k e n  [43], a l l of which a r e o r i g i n a l l y  moment o f Q = ( - 3 . 3 ± 0 . 4 ) x 10  curve  a r e from  P o i n t s below T = 0 ° C a r e from a r e c e n t  SCMF c a l c u l a t i o n s were p e r f o r m e d  Kukolich  Data.  from  on t h e SCMF v a l u e s  i t i s incorrect, indication  of quadrupole  f i g . 17, t h e c h a n g e  f o r e.  overestimates  e, and t h e o v e r e s t i m a t i o n  temperature.  I t i s therefore probable  The t h e o r y i s very  including  i n Q has a  now  l a r g e a t low  that a Q value  w i t h an  1 10 absolute  value  lower  —26 3.3 x 10  2 e s u cm  uncertainty, with  than  , but not o u t s i d e  could give  SCMF v a l u e s  experiments at a l l Calculations  f o r e t h a t would a g r e e w e l l  the values  t h e p o l a r i z a b l e model were c o m p l e t e d  that  bounds o f  temperatures.  using  n o n - p o l a r i z a b l e model.  the given  It i s clear  of  T,P,M,Q  f o r the equivalent, from t h e r e s u l t s  a n o n - p o l a r i z a b l e model d o e s n o t c o r r e c t l y  dielectric  constant  and d used i n  ( f i g . 17)  p r e d i c t the  o f ammonia.  5.4 Summary and C o n c l u s i o n s .  SCMF c a l c u l a t i o n s dielectric  constant  liquid-vapour The  results  o f ammonia a t t h r e e  coexistence were f o u n d  t h e q u a d r u p o l e moment.  line  This  [ 4 6 ] and a g r e e s  reasonably  good v a l u e  discrepancies quadrupole  temperatures  sensitivity  with  data  along the  o f t h e phase d i a g r a m o f ammonia.  earlier  f o r Q i s used  the experimental  out t o f i n d the  t o be q u i t e s e n s i t i v e  Rushbrooke  follow  were c a r r i e d  t o t h e v a l u e of  was p r e d i c t e d by MC work  [47].  the c a l c u l a t e d  quite well,  and any  e values observed  a r e w i t h i n t h e bounds o f e r r o r s e t by t h e  moment.  When a  111  L I S T OF REFERENCES  1.  J.A. B a r k e r , L a t t i c e T h e o r i e s o f t h e L i q u i d S t a t e , T o p i c 1 0 , Volume 1 o f The I n t e r n a t i o n a l E n c y c l o p e d i a o f P h y s i c a l C h e m i s t r y a n d C h e m i c a l P h y s i c s , Pergamon P r e s s , Oxford, 1 9 6 3 .  2.  J.A. B a r k e r ,  4 8 , 587  D. H e n d e r s o n , Rev. Mod. P h y s . ,  (1976).  3.  H.D. U r s e l l ,  4.  J . Yvon, L a T h e o r i e S t a t i s t i q u e d e s F l u i d e s e t 1 ' e q u a t i o n d'Etat, A c t u a l i t e s S c i e n t i f i q u e et I n d u s t r i e l l e s , V o l . 2 0 3 , Hermann, P a r i s , 1 9 3 5 .  5.  J . E . Mayer, M.G. Mayer, S t a t i s t i c a l W i l e y , New Y o r k , 1 9 4 0 .  6.  F. Z e r n i k e ,  7.  C.A. C r o x t o n , I n t r o d u c t i o n t o L i q u i d S t a t e W i l e y and Sons, L o n d o n , 1 9 7 5 .  8.  G.N. P a t e y , D. L e v e s q u e , J . J . W e i s , M o l . P h y s . ,  Proc.  Cambridge  Phil.  J.A. P r i n s , Z. P h y s i k ,  Soc., 2 3 , 685 ( 1 9 2 7 ) .  Mechanics,  John  4J_, 1 8 4 ( 1 9 2 7 ) . Physics,  John  3J3,  1635  5J_,  333  (1979).  9.  D. L e v e s q u e , J . J . W e i s , G.N. P a t e y , M o l . P h y s . , (1984).  10.  L.Y. L e e , P.H. F r i e s ,  11.  P.H. F r i e s ,  12.  F.H. S t i l l i n g e r ,  13.  L . S . O r n s t e i n , F. Z e r n i k e , P r o c . Amsterdam, J_7, 7 9 3 ( 1 91 4 ) .  14.  M.S. W e r t h e i m , J . Chem. P h y s . ,  15.  L . Blum, A . J . T o r r u e l l a , J . Chem. P h y s . ,  16.  L . Blum, J . Chem. P h y s . ,  5 7 , 1862 (1972).  17.  L . Blum, J . Chem. P h y s . ,  5 8 , 3295  18.  L . Blum, Chem. P h y s . L e t t . , 2 6 , 2 0 0 ( 1 9 7 4 ) .  19.  F. Lado, P h y s . Rev. A, J _ 3 5 , 1 0 1 3 ( 1 9 6 4 ) ; 1117  20.  G.N. P a t e y , M o l . P h y s . ,  G.N. P a t e y , J . Chem. Phys., 82, Adv. Chem. P h y s . ,  429 (1985).  3J_, ( 1 9 7 5 ) .  K. Ned. A k a d . Wet.  5 5 , 4291 ( 1 9 7 1 ) . 56,  303 (1972).  (1973).  M o l . P h y s . , 3J_,  (1976).  C.A. V a n c i n i , S y n t h e s i s 1971.  i n press.  of Ammonia, M a c m i l l a n ,  New  York,  11 2 21. A . L . M c C l e l l a n , T a b l e s o f E x p e r i m e n t a l D i p o l e Moments, W.H. Freeman, San F r a n c i s c o , 1963. 22.  (a) S.G. K u k o l i c h , Chem. P h y s . (b) S.G. K u k o l i c h , Chem. P h y s .  Lett., Lett.,  5, 401 ( 1 9 7 0 ) . J_2, 216 ( 1 9 7 1 ) .  23. N . J . B r i d g e , A.D. Buckingham, P r o c . Roy. S o c . A, 205, 135 (1966). 24.  G. S t e l l , G.N. P a t e y , J . S . 183 ( 1 9 8 1 ) .  Adv. Chem. P h y s . ,  25.  S.L. C a r n i e , G.N. P a t e y , M o l . P h y s . ,  26.  G.N. P a t e y , D. L e v e s q u e ,  27.  J . P . Hansen, I.R. M c D o n a l d , T h e o r y A c a d e m i c P r e s s , L o n d o n , 1976.  Htfye,  47, 1129 ( 1 9 8 2 ) .  J . J . Weis, M o l . P h y s . ,  28. D.A. M c Q u a r r i e , S t a t i s t i c a l New Y o r k , 1973.  of Simple  Mechanics,  29. G.R. W a l k e r , Ph.D. T h e s i s , A u s t r a l i a n U n i v e r s i t y , C a n b e r r a , 1983. 30.  in press.  Liquids,  Harper  a n d Row,  National  S t a t e s o f M a t t e r , Volume 2 o f A T r e a t i s e on P h y s i c a l C h e m i s t r y , 3 r d . e d . , e d i t e d by H.S. T a y l o r a n d S. G l a s s t o n e , D. Van N o s t r a n d , New Y o r k , 1951.  31. W.A. 32.  48,  Steele,  J . Chem. P h y s . ,  39, 3197  A. M e s s i a h , Quantum M e c h a n i c s , Sons, New Y o r k , 1958.  (1963).  V o l . I I , John  33. H. Workman, M. F i x m a n , J . Chem. P h y s . ,  W i l e y and  58, 5024  (1973).  34.  G.S. R u s h b r o o k e , H . I . S c o i n s , P r o c . Roy. S o c . A, 216, 203 (1953).  35.  C.A. C r o x t o n , L i q u i d S t a t e P h y s i c s , P r e s s , London, 1974.  Cambridge  University  36. Handbook o f M a t h e m a t i c a l F u n c t i o n s , e d i t e d by M. A b r a m o w i t z and I.A. S t e g u n , D o v e r , New Y o r k , 1970. 37.  L. V e r l e t ,  J . J . W e i s , Phys.  Rev. A, 5, 939  38.  C . J . F . B o t t c h e r , T h e o r y o f E l e c t r i c P o l a r i z a t i o n , 2nd. e d . , E l s e v i e r S c i e n t i f i c , Amsterdam, 1973.  39.  C.F. G e r a l d , A p p l i e d N u m e r i c a l A n a l y s i s , 2nd. e d . , A d d i s o n - W e s l e y , Don M i l l s , O n t a r i o , 1978.  40. A.H. N a r t e n , J . Chem. P h y s . , 41.  I.R. McDonald, M.L. K l e i n , (1976).  (1972).  66, 3117 ( 1 9 7 7 ) .  J . Chem. P h y s . ,  64, 4790  11 3 42.  J . Baldwin, J.B. G i l l , 2, 25 ( 1 9 7 0 ) .  Physics  43. CRC Handbook o f C h e m i s t r y by R.C. Weast, CRC P r e s s , 44.  A. H i n c h l i f f e ,  and C h e m i s t r y  of L i q u i d s ,  and P h y s i c s , 6 0 t h . e d . , e d i t e d C l e v e l a n d , O h i o , 1975.  e t . a l . , J . Chem. Phys.,  7_4, 1211  45. M.L. K l e i n , I.R. McDonald, R. R i g h i n i , J . Chem. 7J_, 3673 ( 1 9 7 3 ) . 46.  G.S. R u s h b r o o k e , M o l . Phys.,  47. G.N. P a t e y , J . P . V a l l e a u ,  37, 761  (1981). Phys.,  (1979).  J . Chem. P h y s . ,  64, 170 ( 1 9 7 6 ) .  48.  A.R. Edmonds, A n g u l a r Momentum i n Quantum M e c h a n i c s , P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , N . J . , 1957.  49.  S.L. C a r n i e , D.Y.C. Chan, G.R. W a l k e r , 1115 ( 1 9 8 1 ) .  M o l . P h y s . , _4_3,  11 4 APPENDIX A  I d e n t i t i e s Used  i n Subsequent A p p e n d i c e s .  Some o f t h e f o l l o w i n g once  i n A p p e n d i c e s B-H.  needed  A.1  for their  R  first  (fl ) = MO 1  Z  A.2  R  io  A.3  R  A.4  Under  (  e  m  MP  )  5  (  are given  i n t h e form  application.  (fijR  =  identities  [48]  m  1  ^  These  a r e a p p l i e d more t h a n  I R™ ( f i , ) R (S2J p pu 3 MP 5  m  n  identities  _  )  L  R  (fl )Rl (fi.H 5 Xs 5 M  m  m  n  R  vq  X 0  (  _  F  V  J ) = (!!!" I ) X p q s  [48]  )  [  , m n 1 % _ , \irt+n+i, n m 1 \ M v X ' v u \ '  I  ;  R  2  ]  odd p e r m u t a t i o n o f columns  K  A.5  3  ^ ^ i  V  )  ;  r p  R  3  (  r^9l  K  i 3  e  )  =  6  1  1,1' XX' 6  [  3  1  2  ]  00  A.6  / dk k  j (kr)j (ks) = ^flMr-s)  A.7  ^  J d O R ; ; ( O 3 ) R ^ ( O 3 ) - 6_„ ^ 6  A.8  ^  ;dk  2  1  3  ~  (  0  R  [17]  1  0  i  n  [48]  i; (k)R^ (k)Ri; u) 0  '  0  1  o  0 0 0 -x,-x x ' M  2  ;  [48]  1 15 A.9  (21+1) Z  (  v fX  '  i ) ( X  n  V  ( I2I1I i _ y m n n, \ \\  in  A  m  jU  1  m  J ) = 1 X  n  u  V  m n , l , M "1X1  (  ]  w  [32]  n,n 1 *, v X  2  W M  2  l i l l x X,X X ' 2  2  M f1 f m n 1 j _ m + n + n + X + X + y + l , + l + l+M J' X +  x  A A  11 >  1  1  1  (  (  1  ( I 2 I 1 I 1/ 1 1 1 2 I \ _ m n n , 0 0 0 " M  x-x  1  1  l 0  1 M  )  x-X  m n 1 \, m n l ' I ( x-x 0 x-x 0 X M  0  2  1  '  +  2  /_\m+n+n,+x# n,n l (  m n l  (  1  v L  ;  m n  ( X  A. 1 2  )  (  0  x~X  2  ]  x  )  ,  f L  °ir (21+1)  ;  4 8  '  '  x  ^  4  8  J  [48]  A. 1 3  • 5B ( 2 M + , ) ( A.14  2  ( inili)(  Mii'iX,  ^l^l^l  m  < S'S2S > R W  m2"2l2)( M2^2X  2  mTiTi m x / n , n n x 2  M!M2"M  2  h ^ " "  M2^2^2 „  x  (  / l i l l i X \ -X X,X X 2  2  ) =  / »m+n+l, m n 1 x r ' ' ? - i „ v \ ^ m n l } m n l m  (  _  )  n  (  2  2  2  _ [32] r  o  l  11 6 A.15  / d^cMg  $  m  i  n  i  l  l  (l2)#  m  2  n  2  l  2  (12)  " (2m +1)(2n,+1)(21,+1)  v  1  if  = 0  All with  symbols  6 i n A.6 b e i n g  A. 12 b e i n g  (m,,n,,1,) =  (m ,n ,1 ), 2  2  2  otherwise.  u s e d a r e as d e f i n e d  Dirac's  the kronecker  '  6.  in previous  chapters,  6 - f u n c t i o n and 6 i n A.5, A.7 and  11 7 APPENDIX B  Proofs  of  Symmetry P r o p e r t i e s of R o t a t i o n a l I n v a r i a n t s .  Proof simplicity general we  we  u(12).  Invariance  Blum  f o r a l l m,n,l  S mnl  u  m l  a t hand,  [15].  with  * 1 \  <r)(  Under R o t a t i o n .  f o r the case  i s o u t l i n e d by  =i  m n l  (ii):  prove only  proof  let f  of  no  For  For  i.e.  A  n'=v'=0.  mathematical  ease  l o s s of g e n e r a l i t y .  Let  ( R ^ X o ' V " ^ ' '  (B.I)  nv\  where 0  l f  8 ,r  a  t o a frame o f U(12)  r  relative  e  2  reference  i s given  u(12).  T,  obtained  (r)(  m n l  «  J  1  mnl The Similarly  °>  2  by  reference  S.  Relative  r o t a t i n g S through  P* , 5  by  u  Z  t o a frame of  r o t a t i o n $2 and  f are  (R^lRynXoU,).  is a result  the  of 0^  (B.2)  f o l l o w e d by  Si^.  Si^  and  r e s u l t s of flg f o l l o w e d  by  r^ ,  respectively. If  U(12)  -  we  use  f o r Si^ ,J2  A.1  2  Z u» (r)( I mnl nv X pqs n l  X  Substituting  A.2  R  i n B.3  J \  and  r i n B.1,  we  obtain  i R ^ ^ l R ^ f ^ l R ^ l r , ) * M  i  M P we  {  v  r  get  V v  R  A V V -  ( B  -  3 )  1 18 u<12>- ^  u  m n l  I  (r)(  I  I  JHjo^jR^CO^R^tep,  (B.4)  pqs which  i s obviously Proof  Particles  t h e same as B.2.  of ( i i i ) :  1 and 2.  substitute  Again  A.3 i n B.1 we Z  u(12)= M  N  ^  u  m n l  I n v a r i a n c e Under I n t e r c h a n g e o f  (r)(  we t r e a t  the  p r o b l e m . I f we  n'=v'=0  obtain " { )R v X  m  M  n n  (n,)R  vO  m  (fi.)(-) Rj (-r). 1 XO 1  n  2  n  nO  (B.5)  MfX Next we use A.4 i n B.5 t o g e t  Z  U(12)= m  n  ^  u  m n l  (r)(  n  m  v  M  \  X  )R  n n  (n )R 2  vO  9  m n  (fi )(-) M0 1 1  m + n  Rl  O r ) . (B.6) XO  vtx\  It  i s now c l e a r  B.I,  f o r u(21),  that  B.6 i s e x a c t l y  i f and o n l y  if  u  n m l  the expression, (r)=(-)  m + n  u  m n l  analogous t o  (r).  119 APPENDIX C  o f u ( 1 2 ) i n Terms o f M u l t i p o l e I n t e r a c t i o n s .  Calculation  The  general  form o f t h e m u l t i p o l e e x p a n s i o n  electrostatic  interaction  o f two n o n - o v e r l a p p i n g  distributions  f o r two a x i a l l y  ,, »<'*>•-  E  U 1  symmetric  |.  ''>"< ( 2  _L  2n>!>'  "  mn =  where 1. If  ^ Z mn  u  (  1  2  )  =  i s truncated  -(30)'M  +  l  (l2) < C >  x  m  l  n  [49]  .Snl 1 „ f r  2 -pole  2  * 1 1 2  (  1  2  )  +  3i70llfi! 224 #  = 0 , Q^=  a t m,n<2,  (lOSPQM  (1 2  K  moment  m  F o r o u r p u r p o s e s , QQ = t h e n e t c h a r g e  u  m  is  mnl/ \»mnl,,_ (r)$ ( 1 2 ) ,  l=m+n a n d Q ( ) i s t h e a x i a l  the expansion  charge  particles  Q°(l)Q°(2)$  of the  direct  1 [  $  2  3  for particle ju and Q = 2  calculation  ( 1 2 ) - $  2  1  3  Q. gives  ( 1 2 ) ]  { c > 2 )  1 20 APPENDIX D  Derivation  We  begin  with  C(13)  and  d e r i v e eqns.  for  1 even.  outlined  2.4.12 t o 2.4.16.  the d e f i n i t i o n  =/dr  l 3  of c(13)  c(13)exp(ik«r  ),  1 3  (D.1)  2.4.12, and 2.4.14 t o 2.4.16 f o r t h e f o r 1 odd and 7 7  Proofs  i n Blum  of E q u a t i o n s  [15,16].  m n l  (k)  E q n . 2.4.13  are s i m i l a r  is a  c  m n l  (k)  and a r e  defining  relat ionship. Using  exp(ik-r)  =  the Rayleigh expansion  Z 1=0  l  j _,(kr)  [15]  L R^, (k)R^, X = 1  ]  {r) ,  Q  (D.2)  Q  —  we o b t a i n  c(13)  x  R  -fdr,-  ^o  (  n  Substitution  i  1  3  )  R  I m,n,li l'X'  ";o  (  f  i  2  i  l  ' c  m  i  n  i  l  l  ( r  - ) f  1  i  n  i  n  i  l  l  (  m  i  i ^ )  n  M  )  R  i;o ^3 (  o f e q n . A.5 g i v e s  )  R  x o ^ ;  (  )  R  x o :  (  f  i3 ^i )  (  k  r  i3 )  ( D  '  3 )  121 c(13)  «Jdr  i > 1  I  1 3  n  ,  1  Mr  1  )f  3  n  ,  ,  n  i  l  ,  (  M " X 1  1  M  MiViX, x R  L  m i n  (J2. ) R  M, 0  1  [4, J d r  01,11,1,  n  ( n ) R ^ ( k ) j , (kr..) 2 X, 0 1 13 9  1 3  n  (D.4)  J  r V ' c ^ ^ M r ^ j  0  (kr  l 3  )]  1  ^'"^Mfl^B^k),  x  from w h i c h , w i t h  n i  J'IO  (D.5)  2.4.11a, we have  CO  c  m n l  (k)  = 47ri  1  ; dr r c 2  m  n  l  j  In o r d e r Hankel t r a n s f o r m  C  m n l  to obtain  of c  (k)  m n  i t s inverse  c  m n l  (2.4.16a)  1  2.4.12 l e t C  m n l  (k)  be t h e 1  T  order  H  ^(r)  = 47ii  1  ; dr r j , ( k r ) c 2  0  with  (kr) .  1  0  m n l  (r),  (D.6)  (k).  (D.7)  1  transform  = Hi  (r)  / dk k j , ( k r ) C  m n l  2  0  2TT  1  Also l e t  ^ n l  (  r  )  =  7 dp p j ( p r ) C  1  2  0  2TT  C  m n l  (p)  D.7  (p),  (  D  >  8  )  U  = 4TT / d r r j ( p r ) c 2  n  0  Then u s i n g  m n l  n  i n 2.4.16 we g e t  U  m n l  (r).  (D.9)  1 22 -mnl,, v 2 °1 , 2. , , v, c ( k ) = ± / dr r * j , ( k r ) ( 0  , , _ 2 . ., \„mnl J dp p ' j , <pr ) C (p)). 0  m u x  Changing  the order  of i n t e g r a t i o n y i e l d s  00  c  for  m n l  00  = Z J dp p C * 0  (k)  (D.10)  u , U A  w  2  m n l  (p)(  w h i c h we c a n i n t e g r a t e  J dr r j ( k r ) j ( p r ) ) , 0  (D.11)  2  1  over  r  1  obtaining  00  c  m n l  ( k ) = ^ ; dp p c 2  m n l  (p)(^-6(k- )) P  ZK.p  7T Q  ^(|)C  m n l  7T Z.  Finally  we  i n s e r t D.9,  (k).  which  (D.12)  gives  oo  c  m n l  (k)  = 4TT J d r r j ( k r ) c 0 2  m n l  n  (r).  (2.4.12a)  u  In simplify,  order  to obtain  2.4.14 we use D.6  c  (r)  and  yielding  co m n l  i n D.8  =  oo  / dk k j ( k r ) ( 4 7 r i ) / ds s j , ( k s ) c 0 0 2  1  2  n  2TT  u  m  n  l  ( s ) (D. 1 3)  1  CO  Co  = ; ds s c 0 2  m  n  l  (s)(|)i  1  / dk k j ( k r ) j , ( k s ) 0 2  n  (D.14)  CO  = / ds s c 2  m n l  (s)0 (s,r),  which d e f i n e s  the t r a n s f o r m a t i o n  We  that  s t a t e here  (D.15)  e  kernel,  © (s,r), 1  f o r 1 even.  123  0^(s,r)  where  8(s-r)  2.4.l5a-c  =  2 —  "  3  i s the D i r a c d e l t a  and 9(s-r)  i s a step  »  (D.16)  f u n c t i o n , P-^(x) i s g i v e n function defined  by  by  1 s>r, O(s-r)  = {  (D.17) 0 s<r.  We  refer  D.15  we  mnl c (r) m  n  l  t o Blum  [17] f o r t h e p r o o f  o f D.16.  Now,  using  D.16 i n  have  ^ mnl °" 0 ( S T ) P* = / ds 6 ( s - r ) c ( s ) - j ds i-S 0 0 m n l  ) a"™" ( S ) ( .18) 1  D  s  m =  c  n  l  , ( r )  °° - / C  mnl, P^(|)ds. x  ( s )  s  (2.4.14a)  124 APPENDIX E  Angular  Integration  We  begin  and S i m p l i f i c a t i o n o f t h e OZ  by w r i t i n g  Equation.  2.4.10 i n t e r m s o f t h e f o l l o w i n g  expansions.  -nl  (  k  )  £  „l  m  . n 1  £  )  »  R  (  )R  ,,  {  )  B  1  (  I  )  mnl P  =JdO. 8TT 2  x R  n i n  " °  3  Z c li  m  (fi_)R^  1  x  [  3  X  l  (k)][  n  ,  l  ,  (k)f  B  ,  i  '  n  (^2n l 2  m n l 2  m  2  1  Z ( Mi^X,  1  2 ) R  2  2  2  2  (  k  )  m  i  M  l  "  l  _~m n l  2  (  )  2  2  n  k  i  J )R 1  X  l  ) f  M  m  2  (n ) 1  °  1  m n l 2  2  2  m 2 (Q n (o 1 2 ( J ) ] M0 3 v20 2 X 0 )  i n  1  R  2  (E.1)  ) R  2  2  2  Applying  A.7 we  - n l  (  k  )  £  obtain  n l  m  • „ 1  (  mnl  )  J  H  <  0  ) R  n  ,„  ) R  1  (  J ,  MfX  = p[  Z min,l, n l 2  x  n  Z  °  Z ( 2n l „ „ \ M ^ X M V 2A 2  ,  i n i  1  2  8  c  m  i  n  i  l  l  (k)(^  (  )  xR  unchanged-.  2  n  2  l  2  (k)-c  m  2  n  2  l  2  (k))(2n +l)"  1  1  2  Z _ ^ min l „ „ % M1 V1 A 1 J>2 2 * 2 X  where -v,  m  f  n 2  1  1  f  n n l 1  2  (  m n l Mi V1X1 1  1  l ) (  n ,n l m, ^iV X M1O 2  2  )R  2  (  2  (n,)R^ (k)R^ (k), 2 X, 0 X 0 1  n  v 2V  2  h a s been  1  (E.2)  2  n  }  n  2  replaced  Multiplying  by v y w h i c h l e a v e s  both sides  o f E.2 by  the  expression  125  n? and  M 0  R  :  ( i 2  1  p 0  ) R  ;  ( n  2 ' }  u s i n g A . 5 we g e t  ~ ~mnl /, x , m n l  I  ,  v  -  ^  p  I  M "  x  (-) ' (2n ,  ,?s  / m n 1 »_1  x  ( k ) f  71  xo  ) R  1)" c 1  l  l+  ( k )  m n i l l  (k)(?J ' n  n l 2  (k)-c  n i n l  Mk))  X,X 12 2  f  mn l 1  l f  n nl 1  m n l (I ^ A ,  2 (  1  1  )  n,n 1 1, ( k t> •, 1^ A 2 X,0  (  2 )  )  R  l  2  (J). X, 0  (  E  .  3  )  1 '* "  We  now  multiply  both  sides  o f E.3 by  Z R . , ( k ) and l'X' dk u s i n g A.7 and A.8 w i t h t h e r e s u l t n  X  integrate  Z ?? X  which  (k)f  m n l  m n l  (  m  Z ( - ) n 1 v , 1, X1 X2 1 2 X f  mn l 1  1 f  i s true  we m u l t i p l y c  over  sides  "  *  X  n nl 1  l  +  2 (  over  J ) (21+1 ) "  n  M  = p  x  both  X  a l l possible  1  X  2  +  y  M 2 n  1  +  m n,l fl V 1 A ) 1  )  (  l ) -  sides  by J  M and v,  (  1  c  m  n,n 1 I* , y  f o r each p o s s i b l e  both  0  n  i  l  2 ) (  A 2  l  ( k ) ( ^  1,1,1 UUU  i  n  ) (  l  2  ( k ) - c  2  i  n  l  ( k ) )  2  (  E  >  4  )  2  and sum b o t h  a n d u s e A . 9 , we g e t  n  1, 1 1 > A , A A.  v a l u e o f m,n,l,ju and v.  m n 1 . )(21+1) X  fJL V  n  Now, i f sides  126 ~mnl/, m n l T? (k) f  a  N £  p  r .  L n  (21+1) 2n, + 1  f  mn l 1  £  n nl,  l f  1  f  X,X 12 M v X 2  n i l i w  n,n  vyv  f,X,  x c  m  n  i  l  \  l i l  l  2  w  n  n  l  2  1  unchanged.  (k)-c  we  X,X X  m  n  i  n  l  n  ^ '  1  JLX v X  (k)),  2  (E.5)  ( ^ ^ ^ ) which l e a v e s the A] A A  2  1  _  Lastly  w  2  ( * } \ ) by A 1 A2 A _  I1I2I  , K  0 0 0  2  Mk)(?j i  l  where we have r e p l a c e d expression  w  2  2  2  use A. 10, 1 + 1 1  + 1 = 0 and  2  UL+U+\=0  to get * n  m n l  (k) [  k  = 0  )  n  x  (  L -)^n+n (2l+D T 1 (2n,+1) 1 J-1 J- 2 (  P n  1,1,1  )  «nn , 1, n , n l mnl m n  1  ^ n  1  l  l  (  k  )  (  - n  l  n l  £  f  a  1  1  f  2  (  k  )  _ ~ n  r  l  n l  2  (  k  )  K  ±  n,  ]  (2.4.17)  127 APPENDIX F  Chi  Applying  Z  m  n n, "  Then,  ^  n  A.11  using  l  (k)  Transformation  t o 2.4.20  (  2  1  +  F.1  1  )  *  Equation.  gives  x~X 0  {  of t h e OZ  M X  "X  0  , ( X  "X  0  H  '  )  (  F  *  1  )  i n 2.4.18a, we have  = P  z  (2i D( +  x  :  x  J*x ™_  1 X  J >(  _  1  x  x  J )(-)  x  11112  n  X x c  Multiplying  both  (k)(??  m n i l l  n i n l 2  (k)-c  n i n l 2  (k)).  (F.2)  s i d e s by  1 2 1 +1  » J  (  I >•  x-x  A.  and  1  x  L  summing b o t h  x  (21+1M  (-) [( x  m  s i d e s over  " J  x x u  ™ ? i A A ^*  )  (  )7?  m n l  (k)  1 from  = p  v-v J ) ] ? A A  w h i c h c a n be r e a r r a n g e d  m n i l l  |m-n|  2  t o m+n  (21 + 1 ) (  yields  2  n l l 1  1  1  x  2  (k)(?J ^ n  n l 2  2  (k)-c  by u s i n g A.9 t o o b t a i n  J )(  x x u  n , n l 2  m  x x u  (k)),  (F.3)  1 28  z  (  1  m n 1 )^nnl X X u  x  (??  ( k )  =  p  L  n  n i n l 2  (k)-c  (  , l , l  n i n l 2  _  )  X  n,n 1 X X u  (  2 )  2  (k))(  m  _" J )c 1  A  Using  1  t h e d e f i n i t i o n 2.4.21 of x ~ t r a n s f o r m s  N  m n  A  (k)  = p  I (-) —.  X  C  m n i  X  (k)[N  n i n  A*  m  n  i  l  l  (k).  (F.4)  A  (k)-C  n i n  A*  we g e t  (k)].  (2.4.22)  1 29 APPENDIX G  Back  In o r d e r 2.4.21b (  m  ~  n  x-x 0  t o get  ) and  sum  over  apply  A.12  m n l  n  U  A.6  A  "  t o get  n  p j (ps)c  Changing  the  x  x  n  A  2 p j 0( p r ) and  2  0  ( ™ " I )N^ (k) = ^  range between  In o r d e r 2.4.12a by  x  /^  (G.1)  Q  \  yielding  ^  where x must  x  x,  (21+1) Z  using  s i d e s of  , m n l \ , m n l ' \ ~mn 1 •, \ ( _ )( _ )r, (k),  z  x  X  J dp 0  m u l t i p l y both  x to obtain  _ rTmn / , \ N (k) =  we  2.4.24 we  by  z  and  Transformations.  of  (k),  (2.4.24)  ±min(m,n).  2.4.25a we  m u l t i p l y both  i n t e g r a t e over  ( p ) = 4TT / ds 0  order  m n l  A  p which  s j (ps)c 2  n  U  integration,  m n l  s i d e s of  gives  ( s j dp 0  p j (pr).(G.2) 2  n  U  w r i t i n g i n t e r m s of  r\ and  yields 00 1  2?r The  case  2  ; dp 0  f o r 1 odd  In o r d e r  p j (ps)?j 2  m n l  n  is  (p) = rj  m n l  (s).  (2.4.25a)  similar.  t o get  2.4.26a we  d e f i n e the  equations .  1 30 r,  m n l  (r)  / dk k jAkr)N (k), 0  = ^  2  2TT  N (k)  = 4TT S ds s j ( k s ) r j 0 2  mnl  (G.3)  mnl  1  m n l  n  (s),  (G.4)  U  w h i c h a r e a n a l o g o u s t o e q n s . D.7 we  insert  e e?(r,s) 1  r,  G.4  i n G.3,  and D.9.  As b e f o r e  (D.13-D.15)  r e a r r a n g e and use t h e d e f i n i t i o n o f  to get  m n l  (r)  = / ds s r 7 0 2  7 He  r n n l  (s)0 (r,s)  -2 6(r-s) = / ds s L j— " O r =  f, {r) mnl  r  - 1 r  (G.5)  e  1  J  J 0  li[2!!i!ll^n.nl, r  s  3"  s P (f)^ 2  d  e  1  J? 7  r  m n l  (s).  , ( ' s  (2.4.26a)  131 APPENDIX H  Binary  We invariants  Products of R o t a t i o n a l  begin  Invariants.  by e x p a n d i n g t h e p r o d u c t  of two  rotational  to get  B = ^ i " '  1  !  ( I 2 ) $  _ j i t ^ n , 1, j m n l 2  2  m  2  n  2  l  2  d 2 )  Z  2  „  i>  ( i ili)( m  n  n  m 2  \  V  M1 f1A1  l  2  2  )  R  m  V 2 ^ 2  2  i  (  R  MlO  }  1  HV\ 2  2  2  x R"' (njRt' V, 0  (r)R  n  U s i n g A. 13 f o r S? , S2 1  B  =  f  and r we  2  n i i n , 1, m n l f  2  2  ^  2  m  2  M2 0  „  v  (fijR" (flJ^ {r). 1 ^ 20 x2o 2  n  2  n  n  (H. 1 )  have ^ m,n,l,^ \ Mi*»iX,  m n l j 2  2  2  tx2v2\2  ^2^2X2  x 1  (2n ,x +  * ^x ( 2 i + , , < Finally  A.14  gives  J;^„><  i;ij-x»  (  n  )R „ (a ) n  < >"  0  n  0  0  0  2  o'o J > xo »!  E  ( f  < - > h  2  1 32 B =  Z f . mnl  m  i  n  i  =  2  2  Z mnl  which d e f i n e s  l  f  m  2  n  2  l  2  -U (2m l)(2n 1)(2l+l)(-) mnl f r  +  m  +  n + l  +  r  x ( n / n V H x i m n i M m n l 2  l  m  ' 0 0 0 m 2 i n  )(  n  P(m,n,l)*  n  i 2 0 0 0  m n l  n  n  )(  M  1  1 zl )* 0 0 0 1  (l2),  the c o e f f i c i e n t s  P(m,n,l).  ,  9  m n l  (i2)  K  1  K  (H 3) ' '  (2.4.42)  

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