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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 APPROXIMATION FOR FLUIDS OF HARD SPHERES WITH DIPOLES AND 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 COLUMBIA OCTOBER, 1985 ©JOHN STEPHEN PERKYNS, 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Cl^gM ISTILV The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Oct 10 j<85~ i i ABSTRACT T h i s t h e s i s i s d i v i d e d i n t o two p a r t s . In Part A the re f e r e n c e hypernetted-chain (RHNC) approximation i s solved f o r a f l u i d of hard spheres with embedded p o i n t d i p o l e s and l i n e a r quadrupoles. The thermodynamic p r o p e r t i e s , the d i e l e c t r i c c o n s t a n t , e, and the p a i r c o r r e l a t i o n f u n c t i o n are compared with p r e v i o u s l y c a l c u l a t e d Monte C a r l o data as w e l l as with r e s u l t s from other i n t e g r a l equation methods. The RHNC i s found to c l o s e l y approximate the Monte C a r l o r e s u l t s and i s shown to improve on the other methods. In Part B a s e l f - c o n s i s t e n t mean f i e l d theory f o r molecular p o l a r i z a b i l i t y i s used, together with the RHNC approximation used i n Part A, f o r a p o l a r i z a b l e d i p o l e - l i n e a r quadrupole f l u i d with ammonia-like parameters. The d i e l e c t r i c c onstant i s c a l c u l a t e d at three s u b - c r i t i c a l temperatures and i t i s found to be q u i t e s e n s i t i v e to the quadrupole moment. Experimental r e s u l t s f o r e are shown to be w e l l w i t h i n the u n c e r t a i n t y , set by the quadrupole moment, i n the c a l c u l a t e d e v a l u e s . These c a l c u l a t e d e valu e s are shown to be s i g n i f i c a n t l y l a r g e r than the d i e l e c t r i c c o n s t a n t s f o r the e q u i v a l e n t n o n - p o l a r i z a b l e system. i i i TABLE OF CONTENTS A b s t r a c t i i Table of Contents i i i L i s t of Tables v i i L i s t of F i g u r e s v i i i Acknowledgements x CHAPTER 1 I n t r o d u c t i o n 1 PART A CHAPTER 2 Theory 9 2.1 I n t r o d u c t i o n 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 F u n c t i o n s 10 2.3 The P o t e n t i a l Energy E x p r e s s i o n for the Model 13 (a) Form of the Fu n c t i o n 13 (b) Expansion of U(12) i n R o t a t i o n a l I n v a r i a n t s 16 2.4 I n t e g r a l Equation T h e o r i e s 23 (a) The Hypernetted Chain (HNC) I n t e g r a l Equation Theory 23 (b) Reduction of the O r n s t e i n - Z e r n i k e Equation 25 (c) Reduction of the HNC C l o s u r e 32 i v (d) The Reference HNC Closure 37 2.5 Expressions f o r Thermodynamic Q u a n t i t i e s 39 (a) C o n f i g u r a t i o n a l Energy 40 (b) Isothermal C o m p r e s s i b i l i t y F a c t o r 41 (c) S t a t i c D i e l e c t r i c Constant 41 2.6 Computational C o n s i d e r a t i o n s 42 (a) Method of S o l u t i o n 42 (b) Program E f f i c i e n c y 43 CHAPTER 3 R e s u l t s and D i s c u s s i o n 45 3.1 Input Parameters and B a s i s Sets 45 3.2 R e s u l t s f o r D i f f e r e n t B a s i s Sets 47 3.3 Comparison with Monte C a r l o C a l c u l a t i o n s ... 52 (a) C o n f i g u r a t i o n a l Energy 52 (b) D i e l e c t r i c Constant 52 (c) The P a i r C o r r e l a t i o n Function 54 3.4 Summary and C o n c l u s i o n s 93 PART B CHAPTER 4 P o l a r i z a b i l i t y Theory 96 4.1 The SCMF Method 96 4.2 P o l a r i z a b l e Molecules with D i p o l e s and Quadrupoles 99 4.3 Method of S o l u t i o n 104 V CHAPTER 5 R e s u l t s 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 with Ammonia-like Parameters 105 5.2 S e n s i t i v i t y to P a r t i c l e Diameter and B a s i s Set 106 5.3 Comparison to Experimental Data 109 5.4 Summary and C o n c l u s i o n s 110 L i s t of References 111 APPENDIX A I d e n t i t i e s Used i n Subsequent Appendices 114 APPENDIX B P r o o f s 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 117 APPENDIX C C a l c u l a t i o n of U(12) i n Terms of M u l t i p o l e I n t e r a c t i o n s 119 APPENDIX D D e r i v a t i o n of Equations 2.4.12 to 2.4.16 120 APPENDIX E Angular I n t e g r a t i o n and S i m p l i f i c a t i o n of the OZ Equation 124 APPENDIX F Chi Transformation of the OZ Equation 127 v i APPENDIX G Back Transformations , 129 APPENDIX H Binary Products of R o t a t i o n a l I n v a r i a n t s 131 v i i LIST OF TABLES I B a s i s Sets 46 II C u t - o f f Dependence i n c m n l ( r ) 48 III B a s i s Set Dependence of e, -j3<U>/N and 0p/p 49 IV P r o j e c t i o n Contact Values 50 V P r o j e c t i o n Values at r=1.1d 51 VI The C o n f i g u r a t i o n a l Energy 53 VII The S t a t i c D i e l e c t r i c Constant 55 v i i i LIST OF FIGURES I. The E u l e r angles 18 2a. The sho r t range part of g ^ ^ ( r ) for parameter set (a) 57 2b. The long range part of g ^ ^ ( r ) for parameter set (a) 59 3a. The short range part of g ^ ^ ( r ) for parameter set (b) 61 3b. The long range p a r t of g ^ ^ ( r ) for parameter set (b) 63 4a. The sho r t range part of g ^ ^ ( r ) for parameter set (c) 65 4b. The long range p a r t of g ^ ^ ( r ) for parameter set (c) 67 5. h 1 1 ^ ( r ) f o r parameter set (a) 69 6. h 1 1 ^ ( r ) f o r parameter set (b) 71 7. h 1 1 ^ ( r ) f o r parameter set (c) 73 1 i 2 8. h (r) f o r parameter set (a) 75 1 i 9 9. h (r) f o r parameter set (b) 77 1 i 2 10. h (r) f o r parameter set (c) 79 1 23 I I . h (r) f o r parameter set (a) 81 12. h 1 2 3 ( r ) f o r parameter set (b) 83 1 23 13. h (r) f o r parameter set (c) 85 i x 14. h 2 2 4 ( r ) f o r parameter set (a) 87 15. h 2 2 4 ( r ) f o r parameter set (b) 89 16. h 2 2 4 ( r ) f o r parameter set (c) 91 17. D i e l e c t r i c constant vs. temperature f o r l i q u i d ammonia 107 X ACKNOWLEDGEMENTS I wish t o thank my r e s e a r c h a d v i s o r , Dr. G.N. Patey, f o r h i s guidance and the time and e f f o r t he has spent on my b e h a l f . Thanks are a l s o due to Dr. P.H. F r i e s and P.G. K u s a l i k f o r 'showing me the ropes' and to the Chemistry Department of the U n i v e r s i t y of B r i t i s h Columbia f o r f i n a n c i a l support. I would a l s o l i k e to thank my wi f e , Jane, whose understanding and i n i t i a t i v e made i t p o s s i b l e f o r me to ignore many r e s p o n s i b i l i t i e s i n order to meet f r i g h t e n i n g d e a d l i n e s . 1 CHAPTER 1 I n t r o d u c t i o n . T h i r t y years ago, r e l a t i v e l y l i t t l e was known about l i q u i d s from a t h e o r e t i c a l p o i n t - o f - v i e w when compared to knowledge of s o l i d or gaseous s t a t e s of matter. T h i s was l a r g e l y because there i s no i d e a l i z e d model f o r l i q u i d s comparable to the i d e a l gas or the p e r f e c t s o l i d . Attempts to understand l i q u i d s have t r a d i t i o n a l l y been those of t r y i n g to i n t e r p o l a t e between s o l i d and gas t h e o r i e s . Simple l a t t i c e t h e o r i e s [1] g e n e r a l l y give r e s u l t s which are best d e s c r i b e d as high temperature s o l i d s . E x p r e s s i o n s which c o r r e c t f o r d e v i a t i o n s i n the i d e a l gas law, such as the v i r i a l expansion, are u s e f u l f o r imperfect gases at low d e n s i t i e s , but do not provide the b a s i s f o r a s a t i s f a c t o r y theory of l i q u i d s [ 2 ] . It i s c l e a r that an 'a p r i o r i ' s t a t i s t i c a l mechanical theory f o r l i q u i d s was needed. I t i s not c l e a r when such a theory began. Much of the ground-breaking work now used i n the e q u i l i b r i u m theory of l i q u i d s was developed i n other c o n t e x t s . The work of U r s e l l [ 3 ] , Yvon[4,2], and Mayer, e t . a l . [5,2], l e d to the c l u s t e r expansion techniques now used e x t e n s i v e l y . These advances were, however, i n i t i a l l y d e a l i n g with the problem of the imperfect gas. Zernike and P r i n s [6] in t r o d u c e d the r a d i a l d i s t r i b u t i o n f u n c t i o n , which i s a c e n t r a l concept i n l i q u i d theory, but they d i d so i n order to e x p l a i n X-ray s c a t t e r i n g . The g e n e r a l goal of s t a t i s t i c a l mechanics i s to a c c u r a t e l y p r e d i c t macroscopic q u a n t i t i e s from reasonable 2 m icroscopic p o s t u l a t e s . U n l i k e gases or s o l i d s where k i n e t i c processes or c o n f i g u r a t i o n a l a s p e c t s dominate the d e s c r i p t i o n , the l i q u i d s t a t e can only be a c c u r a t e l y d e s c r i b e d by a h a miltonian with both k i n e t i c and c o n f i g u r a t i o n a l c o n t r i b u t i o n s [ 7 ] , The present study i s devoted to the e q u i l i b r i u m behavior of c l a s s i c a l l i q u i d s , which g r e a t l y s i m p l i f i e s the k i n e t i c c o n t r i b u t i o n to the t o t a l energy. I t i s the c o n t r i b u t i o n of i n t e r - p a r t i c l e p o t e n t i a l s to the t o t a l energy to which we turn our a t t e n t i o n . We d e f i n e a model by s p e c i f y i n g a p o t e n t i a l energy e x p r e s s i o n . The f i r s t models used were simple, s p h e r i c a l l y symmetric, and d e s c r i b e d by the i n t e r a c t i o n between two p a r t i c l e s . For example, the s q u a r e - w e l l p o t e n t i a l , d e f i n e d by * r<d, u(r) = { -c d<r^R, 0 r>R, where r i s the d i s t a n c e between the two p a r t i c l e s , d i s the (hard) p a r t i c l e diameter and c and R are c o n s t a n t s , has been used e x t e n s i v e l y , as has the Lennard-Jones (LJ) 6-12 p o t e n t i a l u(r) = 4 e [ ( f ) 1 2 - ( f ) 6 ] , where e and a are constant parameters u s u a l l y determined e x p e r i m e n t a l l y . Another commonly used p o t e n t i a l i s the hard sphere i n t e r a c t i o n , 0 0 r<d, u(r) = { 0 r<d. 3 Although the L J p o t e n t i a l has been used with some success f o r Argon [ 2 ] , such i d e a l i z e d p o t e n t i a l s are so mathematically simple that they cannot be expected to mimic r e a l systems i n g e n e r a l . They do, however, form a good b a s i s for more r e a l i s t i c p o t e n t i a l s . Few r e a l systems, besides rare gases, can be a c c u r a t e l y d e s c r i b e d by s p h e r i c a l l y symmetric p o t e n t i a l s , so angular dependent terms are added. Much work has been done on p o t e n t i a l s with e l e c t r o s t a t i c i n t e r a c t i o n terms added to one of these simple s p h e r i c a l i n t e r a c t i o n s [8-11]. T h i s study uses, in Part A, a p a i r p o t e n t i a l comprised of hard-sphere, d i p o l a r and quadrupolar terms. These p o t e n t i a l s are s t i l l poor d e s c r i p t i o n s of r e a l systems because i n general r e a l p o t e n t i a l s cannot be d e s c r i b e d by simple p a i r i n t e r a c t i o n s . In Part B, where we compare r e s u l t s with experimental data, we use a p a i r p o t e n t i a l of the same form as that of Part A, but d e f i n e i t as an ' e f f e c t i v e ' p a i r p o t e n t i a l to take i n t o account, in an i n t r i n s i c way, more complicated terms which a r i s e due to molecular p o l a r i z a b i l i t y . The p o t e n t i a l d e f i n e d , we have two b a s i c routes to thermodynamic q u a n t i t i e s , computer s i m u l a t i o n s and approximate methods. Computer s i m u l a t i o n s can be regarded as e s s e n t i a l l y exact f o r a given model, but to keep tr a c k of many p a r t i c l e s f o r times s u f f i c i e n t l y long [12] to o b t a i n s t a t i s t i c a l l y sound r e s u l t s i s an expensive p r o p o s i t i o n , even on modern computers. Q u a n t i t i e s which depend on boundary c o n d i t i o n s , such as the s t a t i c d i e l e c t r i c c onstant, cause problems f o r s i m u l a t i o n s as w e l l . Approximate t h e o r i e s only take computing times of the order of a few minutes, so much e f f o r t has been spent on f i n d i n g 4 ac c u r a t e t h e o r i e s . 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 approximate r e s u l t s and experimental data c o u l d be due to e i t h e r the t h e o r e t i c a l approximations or to the model i t s e l f . T h e r e f o r e , the usual procedure f o l l o w e d i s to compare approximate r e s u l t s with computer s i m u l a t i o n s as w e l l as with experimental data. There are two important methods of computer s i m u l a t i o n i n common use. Molecular Dynamics c a l c u l a t i o n s s o l v e the equations of motion f o r the p a r t i c l e s and time averaging i s used. The Monte C a r l o (MC) method e v a l u a t e s ensemble averages i n the s t a t i s t i c a l mechanical sense. The model we use has been simulated using MC techniques by Patey,Levesque and Weis [ 8 ] . The approximate t h e o r i e s which have r e c e i v e d most a t t e n t i o n r e c e n t l y are p e r t u r b a t i o n approaches and i n t e g r a l equation methods. There are many v a r i a t i o n s i n p e r t u r b a t i o n approaches because of the many f u n c t i o n s which can be expanded. 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 to hard spheres. The Born-Green theory [2] agrees w e l l with computer s i m u l a t i o n s at low d e n s i t i e s but i s l e s s a c c u r a t e when the d e n s i t y i s i n c r e a s e d . The Mean S p h e r i c a l Approximation (MSA) f o r hard spheres becomes i d e n t i c a l to the Percus-Yevick (PY) theory, which i s s o l v a b l e a n a l y t i c a l l y f o r t h i s p o t e n t i a l [ 2 ] . The Hypernetted-Chain (HNC) equation can be obtained by a c l u s t e r s e r i e s expansion of the d i r e c t c o r r e l a t i o n f u n c t i o n . Together with the O r n s t e i n - Z e r n i k e equation [13] i t forms the HNC i n t e g r a l equation t h e o r y . For hard spheres the PY s o l u t i o n seems to be s u p e r i o r to the HNC [ 2 ] , but f o r a l l other p o t e n t i a l s i t i s i n f e r i o r . 5 The MSA has been a p p l i e d f i r s t to d i p o l a r f l u i d s by Wertheim [14] and then to a r b i t r a r y m u l t i p o l a r p o t e n t i a l s by Blum and T o r r u e l l a [15], and Blum [16-18]. The HNC was not s o l v a b l e even by numerical means f o r n o n - s p h e r i c a l p o t e n t i a l s u n t i l r e c e n t l y [11] and so i t was f u r t h e r approximated by the l i n e a r i z e d HNC (LHNC) or the q u a d r a t i c HNC (QHNC) approximations. The LHNC and QHNC have been a p p l i e d e x t e n s i v e l y to v a r i o u s n o n - s p h e r i c a l p o t e n t i a l s [8-11] and the f u l l HNC was r e c e n t l y s o l v e d f o r d i p o l a r hard spheres [11] and Stockmayer p a r t i c l e s [10]. The systems of equations obtained cannot be sol v e d a n a l y t i c a l l y so i t e r a t i v e techniques are used. The problem of inaccuracy with the HNC a p p l i e d to the hard core i s improved by using a type of p e r t u r b a t i o n approach f i r s t i n t r o d u c e d by Lado [19]. The c o r r e l a t i o n f u n c t i o n s are d i v i d e d i n t o exact and p e r t u r b a t i o n p a r t s . The exact p a r t i s the exact s o l u t i o n f o r hard spheres o n l y . T h i s i s known as the ref e r e n c e HNC (RHNC) theory. In Part A of the present study we solve the RHNC i n t e g r a l equation theory f o r a system of hard spheres with d i p o l e s and quadrupoles and compare with MC r e s u l t s . L i q u i d ammonia i s a good c h o i c e f o r comparison with t h e o r e t i c a l r e s u l t s because there i s a l a r g e amount of accurate experimental data a v a i l a b l e . T h i s i s probably due to ammonia's many i n d u s t r i a l uses. In f a c t , a f t e r water, i t i s one the most i n t e n s e l y s t u d i e d i n o r g a n i c s o l v e n t s . In p a r t i c u l a r f o r our purposes, accurate l i q u i d d e n s i t y data at e q u i l i b r i u m vapor pressure i s given by V a n c i n i [20]. The d i p o l e moment [21] has been e x t e n s i v e l y measured, the quadrupole moment [22] i s a v a i l a b l e , and the p o l a r i z a b i l i t y i s a l s o known [23]. A l s o , the 6 roughly s p h e r i c a l shape of ammonia lends i t s e l f w e l l to simple models. In order to c a l c u l a t e the d i e l e c t r i c constant of l i q u i d ammonia i t i s necessary to conside r a p o l a r i z a b l e model. T h e o r e t i c a l treatments of p o l a r i z a b l e systems f a l l roughly i n t o two c a t e g o r i e s , those which assume constant and f l u c t u a t i n g p o l a r i z a b i l i t i e s . A reveiw of these t h e o r i e s i s given by S t e l l , Patey and H^ye [24]. Both approaches a r r i v e at s i m i l a r r e s u l t s probably because they both r e p l a c e many-body e f f e c t s by an ' e f f e c t i v e ' p a i r p o t e n t i a l . The theory used here, which i s due to Carnie and Patey [25], i s s i m i l a r to previous approaches i n t h i s r e s p e c t . In t h i s theory, c a l l e d the s e l f c o n s i s t e n t mean f i e l d (SCMF) theory, the many-body p o t e n t i a l of a p o l a r i z a b l e system i s w r i t t e n as a sum of p a i r p o t e n t i a l s which depend on an e f f e c t i v e permanent d i p o l e moment. T h i s e f f e c t i v e permanent d i p o l e depends on v a r i o u s p r o p e r t i e s of the system such as molecular p o l a r i z a b i l i t y , temperature and d e n s i t y as well as the a c t u a l permanent m u l t i p o l e moments. The r e s u l t i n g p o t e n t i a l e x p r e s s i o n d e f i n e s the e f f e c t i v e system. In order to f i n d the p r o p e r t i e s of a p o l a r - p o l a r i z a b l e system, i t i s necessary to c a l c u l a t e these p r o p e r t i e s f o r the r e l a t e d e f f e c t i v e system, which we do f o r l i q u i d ammonia by s o l v i n g the RHNC approximation. The SCMF theory has r e c e n t l y been shown to be very a c c u r a t e [26]. T h i s t h e s i s i s d i v i d e d i n t o two p a r t s , each of which has a chapter on theory and a chapter on r e s u l t s . In Part A the RHNC i n t e g r a l equation theory i s a p p l i e d to a system of hard spheres embedded with d i p o l e s and quadrupoles, and r e s u l t s are compared with MC data. Most equations s t a t e d i n Chapter 2 are d e r i v e d i n Appendices B-G. Appendix A i s devoted to the mathematical i d e n t i t i e s needed i n the d e r i v a t i o n s . In Part B the SCMF theory i s solved using the i n t e g r a l e q u a t i o n s of Part A, and using ammonia-like parameters. The d i e l e c t r i c constant i s found and compared with experimental data. C o n c l u s i o n s obtained from each part are given at the end of t h e i r s e c t i o n s . 8 PART A CHAPTER 2 9 Theory. 2.1 I n t r o d u c t i o n . T h i s chapter i s concerned with the theory of d i s t r i b u t i o n f u n c t i o n s and i n t e g r a l equations. These are used, with the p o t e n t i a l f u n c t i o n d e s c r i b i n g the model, to get the macroscopic q u a n t i t e s of i n t e r e s t . The model c o n s i s t s of hard spheres with embedded p o i n t d i p o l e s and quadrupoles. The system i s t r e a t e d c l a s s i c a l l y . T h i s i s a good approximation f o r most l i q u i d s , although there are e x c e p t i o n s such as l i q u i d He [ 2 ] . Most equations s t a t e d are d e r i v e d i n Appendices B-H. 10 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 F u n c t i o n s . Consider a c a n o n i c a l ensemble of N r i g i d i d e n t i c a l p a r t i c l e s . The Hamiltonian f o r the ensemble i s of the form 3N 2 H N = I l i _ + U N(X ,X 2,...,X N), (2.2.1) i=1 2m. l where the p^ represent the 3N momenta, iru i s the mass of the i f c ^ p a r t i c l e , and U N i s the c o n f i g u r a t i o n a l energy. Here X^ repr e s e n t s ( r ^ , f i ^ ) , the three p o s i t i o n and three o r i e n t a t i o n c o o r d i n a t e s of p a r t i c l e i . The p r o b a b i l i t y of f i n d i n g p a r t i c l e 1 w i t h i n the i n f i n i t e s i m a l volume dX 1 of the p o s i t i o n and o r i e n t a t i o n X 1 and w i t h i n the momentum region dp 1 of p 1 , while simultaneously f i n d i n g every other p a r t i c l e i w i t h i n dX^ of X^ and w i t h i n dp^ of p i ? i s [27] PN ( X 1 , X 2 , . . . , X N , p 1 , . . . , p N ) d X 1 d X 2 . . . d X N d p 1 d p 2 . . . d p N = ^ - e x p [ - ^ H N ( X 1 , X 2 , . . . , X N ) ] x dX 1dX 2...dX Ndp 1...dp N, (2.2.2) where /3-(kT) 1 , k i s the Boltzmann constant and T the a b s o l u t e temperature. Q N i s the c l a s s i c a l c a n o n i c a l p a r t i t i o n f u n c t i o n d e f i n e d by [28] 11 Q N = NT ~3N * exp[-/3H N(X 1 ,X 2, . . . ,X N, P l , . . . ,p N) ] h x dX ^  dX 2 • • • dX^dp ^  • • • ^ Pj^' (2.2.3) where h i s Planck's constant and the i n t e g r a t i o n i s over a l l p o s i t i o n , o r i e n t a t i o n and momentum c o o r d i n a t e s . For a Hamiltonian of the form eqn. 2.2.1, eqn. 2.2.3 becomes (m^=m f o r i d e n t i c a l p a r t i c l e s ) QN = h ( ^ r - ) f N / exp[-/3U N(X 1,X 2,...,X N)]dX 1dX 2...dX N, = 1_ (iTnnkTjfN ^ { 2 > 2 > 4 ) Z N i s c a l l e d the c l a s s i c a l c o n f i g u r a t i o n a l i n t e g r a l . We can i n t e g r a t e the momentum c o o r d i n a t e s i n eqn. 2.2.2 i n j u s t the same way to o b t a i n (X ^  r^2'* * * ' ^ ^ * * * = ^-e x p [ - ^ U N(X 1,X 2,...,X N)]dX 1dX 2...dX N. (2.2.5) N The p r o b a b i l i t y of f i n d i n g any p a r t i c l e w i t h i n dX 1 of X 1, and any other molecule w i t h i n dX 2 of X 2 and so on up to any n*"*1 p a r t i c l e w i t h i n dX n of X n i s [27] ( n ) ( y „ v \ - N! 1 P N i x l f x 2 , . . . , x n ; - ( N _ n ) j z N x J exp[-/?U N(X 1 , X 2 , . . . , X N ) ] d X n + 1 d X n + 2...dX N. (2.2.6) 12 The i n t e g r a t i o n i s over p a r t i c l e s n+1 to N and the f a c t o r i a l c o e f f i c i e n t s are necessary to a v o i d c o u n t i n g i n d i s t i n g u i s h a b l e p a r t i c l e s more than once. These p^n^ are known as n - p a r t i c l e d e n s i t i e s . The n - p a r t i c l e d i s t r i b u t i o n f u n c t i o n s , f o r a homogeneous system are d e f i n e d as [27] g N n ) (X 1 ,X2, . . . ,X n) = ( 8 7 r 2 ) n p " n p N n ) ( X 1 , X 2 , . . . , X n ) , (2.2.7) where p i s the number d e n s i t y of the system. (2) Only the p a i r d i s t r i b u t i o n f u n c t i o n g N ( X ^ X j ) w i l l be used here, and i t w i l l be denoted throughout as g(12) f o r convenience. A l s o used w i l l be the p a i r c o r r e l a t i o n f u n c t i o n d e f i n e d by h( 12) = g(12) - 1. (2.2.8) G e n e r a l l y g(12) —>1 as the i n t e r p a r t i c l e s e p a r a t i o n of p a r t i c l e s 1 and 2 i n c r e a s e s , and h(12) measures the d e v i a t i o n from t h i s l i m i t i n g v a l u e . Thermodynamic q u a n t i t i e s such as i n t e r n a l energy and d i e l e c t r i c constant can be w r i t t e n i n terms of g(12) or h ( l 2 ) . The procedure u s u a l l y f o l l o w e d when us i n g d i s t r i b u t i o n f u n c t i o n techniques i s to d e f i n e the c o n f i g u r a t i o n a l energy f o r a model, and use e q u i l i b r i u m s t a t i s t i c a l mechanics to c a l c u l a t e the d i s t r i b u t i o n f u n c t i o n s , from which are found the macroscopic p r o p e r t i e s of the system. o 13 2.3 The P o t e n t i a l Energy E x p r e s s i o n for the Model. (a) Form of the F u n c t i o n . To t r e a t any system e x a c t l y i t would be necessary to so l v e the many-body Schrodinger equation d e s c r i b i n g the motion of a l l n u c l e i and e l e c t r o n s . T h i s would be a very d i f f i c u l t task so a number of s i m p l i f y i n g approximations are needed. The Born-Oppenheimer approximation i s the f i r s t used. T h i s simply expresses the many-body problem i n terms of f i x e d n u c l e i and i s j u s t i f y a b l e because the n u c l e i are much heavier than the e l e c t r o n s . p a r t i c l e s are r i g i d . T h i s ignores any c o u p l i n g between i n t r a m o l e c u l a r v i b r a t i o n s and i n t e r m o l e c u l a r f o r c e s and allows us to w r i t e the p o t e n t i a l i n the form U"N(X1 ,X2, .. . ,X N) , where each p a r t i c l e i s r e f e r r e d to by the o r i e n t a t i o n and the p o s i t i o n of i t s center of mass (or charge). The t h i r d i s that of t r e a t i n g the l i q u i d c l a s s i c a l l y , which i s a good approximation with a few e x c e p t i o n a l cases [ 2 ] . We can express the r e s u l t i n g p o t e n t i a l energy f u n c t i o n as the expansion The second approximation i s the assumption that the V X 1 ' X 2 I U i<j (2) (X. ,X . ) + i< j<k u v o ; ( x i f X j , x k ) + (2.3.1) where U (2) and U (3) represent two and three body i n t e r a c t i o n s . 14 In order to d e s c r i b e a r e a l system e x a c t l y many terms would probably be needed. T h i s i s c e r t a i n l y true of the p o l a r i z a b l e system d i s c u s s e d i n Part B, where many p a r t i c l e i n t e r a c t i o n terms are i n c l u d e d i n an e f f e c t i v e p a i r p o t e n t i a l . However, a l a r g e amount of work has been done [2] using models d e f i n e d by (2) p a i r p o t e n t i a l s of the form U v '{X]tX2) = u(12). I n t e r a c t i o n s between molecules g e n e r a l l y c o n s i s t of a strong, short ranged r e p u l s i o n and weaker, longer ranged i n t e r a c t i o n s which can be a t t r a c t i v e or r e p u l s i v e . In our model the short ranged r e p u l s i v e aspect of the p o t e n t i a l i s given by making the p a r t i c l e s i n p e n e t r a b l e . T h i s i s done by f o r b i d d i n g them to come w i t h i n a d i s t a n c e d, c a l l e d the hard sphere diameter, of each other. The f i r s t term i n the p o t e n t i a l energy f u n c t i o n i s then a hard sphere term, °° f o r r<d, u(12) = U h c,(12) = { (2.3.2) . 0 f o r r>d, where r i s the d i s t a n c e between the ce n t e r s of the spheres. The longer ranged i n t e r a c t i o n s are caused by instantaneous and permanent 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 d i s t r i b u t i o n s about the molecules. The instantaneous a n i s o t r o p i e s present i n a l l molecules give r i s e to Van der Waals f o r c e s which are a t t r a c t i v e but weak compared to the f o r c e s due to permanent a n i s o t r o p i e s . The permanent n o n - s p h e r i c a l e l e c t r o n d i s t r i b u t i o n s which are present i n most molecules cause a corresponding . a n i s o t r o p i c charge d i s t r i b u t i o n . T h i s charge d i s t r i b u t i o n can be d e s c r i b e d by a m u l t i p o l e expansion. The e l e c t r o s t a t i c 15 (\,\ (]r) (]r) (\r) p o t e n t i a l f o r k charges q R at p o i n t s r v ' = (r) ,* 2 ' r3 ' G a n be c a l c u l a t e d . At a p o i n t t = ( t ^ t j f t ^ ) (with the center of charge as the o r i g i n ) the e l e c t r o s t a t i c p o t e n t i a l , V, f o r an e l e c t r i c a l l y n e u t r a l molecule w i l l approach, f o r l a r g e t=|t| [29] 3 n.t. 3 3 Q . , t . t . V ( t ) = Z - ^ i + Z Z 1J 1 1 + , (2.3.3) i = 1 t - 3 i = 1 j=1 t where N Z k=1 u - q k r ( k ) , (2.3.4a) Q i j = 5 J , 3 k ( 3 r j k ) r ( k ) - ( r [ k ) ) 2 5 i j ) , (2.3.4b) ( k ) ( k ) r = |r | and 5 ^ i s Kronecker's d e l t a , M i s the e l e c t r i c d i p o l e moment with elements and Q i s the e l e c t r i c quadrupole tensor with elements Q^j- Equation 2.3.3 can be d e r i v e d by t a k i n g the l a r g e t l i m i t of the gen e r a l p o t e n t i a l expression or by c o n s i d e r i n g the m u l t i p o l e moments as p o i n t q u a n t i t i e s . The mathematical s i m p l i f i c a t i o n represented by eqn. 2.3.3 g r e a t l y eases the d i f f i c u l t y of the problem and i s used here. Other terms i n the expansion are very short ranged due to the 2 n - p o l e having ( 1 / t n + 1 ) dependence. Although i t i s not known whether these a d d i t i o n a l terms are n e g l i g i b l e f o r r e a l systems they are ignored here. U n f o r t u n a t e l y , we must r e l y on quantum chemical c a l c u l a t i o n s f o r values of oct u p o l e and higher m u l t i p o l e moments which cannot be v e r i f i e d by experiments. Equations 2.3.3 and 2.3.4 can a l s o be expressed i n terms of continuous charge 1 6 d i s t r i b u t i o n s . The model we c o n s i d e r has p o i n t d i p o l e s and l i n e a r p o i n t quadrupoles ( r e p r e s e n t i n g a x i a l l y symmetric charge d i s t r i b u t i o n s ) embedded at the hard sphere c e n t e r s . The p o t e n t i a l energy due to the i n t e r a c t i o n of two such p a r t i c l e s w i l l c o n s i s t of s e v e r a l components which can be added due to the p r i n c i p l e of s u p e r p o s i t i o n of e l e c t r o s t a t i c i n t e r a c t i o n s [30], The d i p o l e and quadrupole of p a r t i c l e 1 w i l l i n t e r a c t with both the d i p o l e and quadrupole of p a r t i c l e 2. Thus our p a i r p o t e n t i a l f u n c t i o n U(12) i n c l u d i n g the hard sphere core component w i l l be of the form u(12) = u H S ( 1 2 ) + U d d ( 1 2 ) + U d q ( 1 2 ) + u Q Q ( 1 2 ) , (2.3.5) where DD,DQ and QQ denote d i p o l e - d i p o l e , dipole-quadrupole and quadrupole-quadrupole i n t e r a c t i o n s , r e s p e c t i v e l y . E x p l i c i t forms f o r the angle dependent terms are given below. (b) Expansion of u(12) i n R o t a t i o n a l I n v a r i a n t s . Each of the components in eqn. 2.3.5 are w r i t t e n as f u n c t i o n s of 12 v a r i a b l e s , the three p o s i t i o n and three o r i e n t a t i o n c o o r d i n a t e s f o r each p a r t i c l e . These components r e a l l y only depend on the r e l a t i v e p o s i t i o n s and o r i e n t a t i o n s of the two p a r t i c l e s , a t o t a l of 6 v a r i a b l e s . Even so, s o l u t i o n of the equations necessary to o b t a i n macroscopic p r o p e r t i e s i s a formidable task. T h i s d i f f i c u l t y can be e a s i l y overcome i f a l l p a i r f u n c t i o n s are expanded i n a b a s i s set of orthogonal polynomials which span the complete space of p a r t i c l e o r i e n t a t i o n s [31]. The c o e f f i c i e n t s of the b a s i s set w i l l be f u n c t i o n s of the i n t e r p a r t i c l e s e p a r a t i o n o n l y . T h i s i s e q u i v a l e n t to using the t r a n s l a t i o n a l symmetry U(12) must possess i n order to r e w r i t e i t as U(12) = u ( r 1 , n i , r 2 ' r n 2 ) = Z u ( 1 } ( | r 1 - r 2 | ) f a ^ g l e (0, ,Q2 , r) , (2.3.6) where r=(0,0,O) and 6 and <p are the p o l a r and azimuthal angles . r r r 2 of the ve c t o r r= , — - — , and Q. , $2, are the e u l e r o r i e n t a t i o n l r 1 r 2 ' angles of p a r t i c l e s 1 and 2, r e s p e c t i v e l y (see f i g u r e 1). The t h i r d angle of f i s set i d e n t i c a l l y to zero and a c t s as a dummy v a r i a b l e e n a b l i n g J21 , f l 2 and r to be t r e a t e d u n i f o r m l y . A l l angles are d e f i n e d with r e s p e c t to the same a r b i t r a r y (lab.) frame of r e f e r e n c e . Using the g e n e r a l i z e d Wigner s p h e r i c a l harmonics R m •(fi) [32] we can use as b a s i s f u n c t i o n s the r o t a t i o n a l i n v a r i a n t s C i- ( sv n2'*> - £ m n l s ( ; ;1 > c.<vC-< n2> Rio (*>' ( 2 - 3 - 7 » where f m n ^ can be any non-zero constant and the sum i s over a l l p o s s i b l e u,v,\ which allow ( m n > ), the usual 3-j symbol [32], to be non-zero. The r o t a t i o n a l i n v a r i a n t s are chosen as b a s i s f u n c t i o n s because of t h e i r remarkable symmetry p r o p e r t i e s , which conform to the symmetry r e s t r i c t i o n s r e q u i r e d of u ( ! 2 ) . These 18 F i g u r e 1 The three e u l e r angles ( a , 0 , 7 ) . I i s a r o t a t i o n through an angle a about the z - a x i s , II i s a r o t a t i o n through an angle 0 about the y ' - a x i s and I I I i s a' r o t a t i o n through an angle 7 about the z ' - a x i s . 19 20 r e s t r i c t i o n s are: ( i ) U(12) must be t r a n s l a t i o n a l l y i n v a r i a n t because our choice of o r i g i n for the la b . frame does not a f f e c t the in t e r m o l e c u l a r f o r c e s . This was taken care of by w r i t i n g equation 2.3.6. ( i i ) u(12) must be r o t a t i o n a l l y i n v a r i a n t . Again, the d e f i n i t i o n of the lab frame must have no e f f e c t on the p o t e n t i a l . ( i i i ) U(12) must be unchanged on permutation of ( i d e n t i c a l ) p a r t i c l e s 1 and 2. (iv) U(12) must remain unchanged when operated on by the symmetry operators of the i n d i v i d u a l p a r t i c l e s . The r o t a t i o n a l i n v a r i a n t s have a l l these p r o p e r t i e s . Proof of ( i ) i s inherent i " 2.3.6. Proofs of ( i i ) and ( i i i ) are given in Appendix B. Co n d i t i o n ( i v ) i s dependent on the symmetry operators of the molecules being considered, and i t simply has the e f f e c t of removing some of the basi s functions needed in the expansion [15], We note here that we must assume that any expansion in these r o t a t i o n a l i n v a r i a n t s must converge. In Chapter 3 dependence of va r i o u s q u a n t i t i e s on the s i z e of the ba s i s set i s examined and as i n previous work [10,11] we see that the t h e o r e t i c a l l y i n f i n i t e b a s i s set seems to converge with r e l a t i v e l y few elements. As an example of molecular symmetry s h r i n k i n g the s i z e of the b a s i s set we d e r i v e the b a s i s f u n c t i o n r e s t r i c t i o n s for the model being used. Our model molecule has d i r e c t i o n a l l y dependent point d i p o l e s and quadrupoles which are a l i g n e d with each other and t h i s d i r e c t i o n i s a body f i x e d symmetry axis 21 a r b i t r a r i l y d e f i n e d as the z - a x i s . u(12) must be i n v a r i a n t under r o t a t i o n about t h i s a x i s f o r both molecules. T h i s r o t a t i o n corresponds to a change i n the t h i r d e u l e r angles 7 1 and 7 2 where $^ = ( ^ , ^ , 7 ^ and fi2 = ( a 2 , 0 2 , 7 2 ) (see f i g u r e 1). The only R m ,(&,) which can c o n t r i b u t e to * m i ? 1 , (S2. ,0 o, f) are M M 1 M V 1 2' those which are independent of 7 ^ The R ™ M ' ( ^ ) which f u l f i l t h i s requirement are those f o r which M ' = 0. S i m i l a r l y J > ' = 0. We are l e f t with u(12) = I u m n l ( r ) $ m n l ( J 2 1 , C „ r ) , (2.3.8) mnl 1 * , «mnl iiiinl where s> = 9>QQ . It would seem that an i n f i n i t e number of terms i n the expansion would s t i l l be needed to give u(12) e x a c t l y , but because the m u l t i p o l e expansion has been t r u n c a t e d , an e q u i v a l e n t t r u n c a t i o n 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. For our present model i t can be shown (Appendix C) that the terms i n eqn. 2.3.5 can be i d e n t i f i e d w i t h terms from the i n v a r i a n t expansion. One has u D D ( 1 2 ) = u 1 1 2 ( r ) $ 1 1 2 0 2 ) , (2.3.9a) U D Q ( 1 2 ) = u 1 2 3 ( r ) $ 1 2 3 0 2 ) + u 2 1 3 ( r ) $ 2 1 3 0 2 ) , (2.3.9b) U Q Q ( 1 2 ) = u2 2 4 ( r ) * 2 2 4 ( 1 2 ) , (2.3.9c) where 1 12, v 2 ^ , ( 2 . 3 .1 Oa) r J u 1 2 3 ( r ) = - u 2 1 3 ( r ) = U&j, (2.3.10b) 2 r 4 2 u 2 2 4 ( r ) = _Cj ^ (2.3.10c) 4r° and <*> m n l(l2) = <*>mnl (0 ,0,, r) , n = u and Q = Q of eqn. 2.3.3. 2.4 I n t e g r a l Equation T h e o r i e s . 23 (a) The Hypernetted-Chain I n t e g r a l Equation Theory. T h i s i n t e g r a l equation method i s d e f i n e d by two eq u a t i o n s . These are the O r n s t e i n - Z e r n i k e (OZ) equation and the Hypernetted-Chain (HNC) c l o s u r e approximation. The OZ equation i s where the i n t e g r a t i o n i s over the s i x p o s i t i o n and o r i e n t a t i o n v a r i a b l e s of molecule 3. T h i s equation was f i r s t i n t r o d u c e d i n 1914 i n an i n v e s t i g a t i o n of d e n s i t y f l u c t u a t i o n s near c r i t i c a l p o i n t s [13], The form used here i s the g e n e r a l i z a t i o n f o r r i g i d n o n - s p h e r i c a l molecules given by Workman and Fixman [33]. I t was o r i g i n a l l y thought that C(12) was going to depend only on u(12) and so i t was c a l l e d the ' d i r e c t c o r r e l a t i o n f u n c t i o n ' . I t i s now known that c(12) depends on h ( l 2 ) (or g(12)) as w e l l . The OZ equation i s now regarded as a d e f i n i n g r e l a t i o n s h i p f o r c ( 1 2 ) . Because of the o r i g i n a l terminology, the second term on the r i g h t i s of t e n c a l l e d the i n d i r e c t part of the p a i r c o r r e l a t i o n f u n c t i o n . I f one i t e r a t e s equation 2.4.1 the f o l l o w i n g expansion i s obtained h(l 2 ) = c(12) + -£=• / c(13)h(32)dX 8TT 3' (2.4.1) h(12) = c(12) + -£=• / c d 3 ) c ( 3 2 ) d X 8TT + (-Pj)2 J c d 3 ) c ( 3 4 ) c ( 4 2 ) d X 3 d X 4 + 87T (2.4.2) and one can see that the i n d i r e c t part can be i n t e r p r e t e d as the t o t a l c o r r e l a t i o n between molecules 1 and 2 imposed by c ( i j ) v i a s u c c e s s i v e l y longer chains of other molecules. The OZ equation r e a l l y only r e d i r e c t s our search f o r h(12) to a search f o r c ( 1 2 ) , but i f we can f i n d an independent equation which a l s o r e l a t e s h ( l 2 ) to c(12) i n terms of known f u n c t i o n s , the system of equations can be c l o s e d . The OZ equation, as a d e f i n i n g r e l a t i o n , i s an exact r e s u l t . U n f o r t u n a t e l y the exact c l o s u r e i s not known, so an approximate r e l a t i o n s h i p must be used. In 1953 Rushbrooke and Scoins [34] showed that i f C(12) i s F o u r i e r transformed i t can be w r i t t e n as an expansion in powers of d e n s i t y . The c o e f f i c i e n t s i n t h i s expansion are m u l t i - d i m e n s i o n a l i n t e g r a l s known as c l u s t e r s . These c l u s t e r s can be d i v i d e d i n t o c l a s s e s a c c o r d i n g to t h e i r form and most of these c l a s s e s can be i d e n t i f i e d with known f u n c t i o n s . However, one c l a s s , the elementary c l u s t e r s , cannot be e a s i l y c a l c u l a t e d . The approximation a p p l i e d to d e f i n e the HNC equation i s simply that of i g n o r i n g the elementary c l u s t e r s [35]. The HNC equation i s C(12) = h ( l 2 ) - l n ( g ( 1 2 ) ) - 0U(12), (2.4.3) where In denotes the n a t u r a l l o g a r i t h m . There i s no p h y s i c a l b a s i s f o r i g n o r i n g the elementary c l u s t e r s ; t h e i r omission i s f o r mathematical s i m p l i c i t y . However, the elementary c l u s t e r i n t e g r a l s are only s i g n i f i c a n t at short range [ 7 ] , Equations 2.4.1 and 2.4.3 cannot be s o l v e d 25 a n a l y t i c a l l y so numerical methods must be used. For the present system, h ( 1 2 ) , c d 2 ) and u(12) are d i s c o n t i n u o u s at r=d, the hard sphere diameter, so to a v o i d computational problems we r e w r i t e eqn. 2.4.1 as T?(12) = J c( 13) [TJ(32) - c(32)]dX_, (2.4.4) 8TT J where r){ 1 2 ) =h (1 2 )-c ( 1 2 ) , and i s a continuous f u n c t i o n of r . (b) Reduction of the OZ Equation. The OZ equation 2.4.1 e x a c t l y r e w r i t e s h(12) in terms of c(12) or, a l t e r n a t i v e l y , eqn. 2.4.4 r e w r i t e s 77 (12) i n terms of c ( 1 2 ) . For s p h e r i c a l l y symmetric f u n c t i o n s t h i s equation i s separable i n F o u r i e r space with the s o l u t i o n ^ k ) = T ^ c T T T ' ( 2 - 4 - 5 ) where the t i l d e denotes F o u r i e r transform. Since no such simple s o l u t i o n e x i s t s 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 expand C(12) and 17( 12) i n r o t a t i o n a l i n v a r i a n t s with a view to s i m p l i f y i n g the problem. What f o l l o w s i s e s s e n t i a l l y due to Blum and T o r r u e l l a [15] but with the n o t a t i o n and d e f i n i t i o n s used i n the more recent l i t e r a t u r e [10,11]. The d e r i v a t i o n given below i s not completely general but i s v a l i d f o r a x i a l l y symmetric p a r t i c l e s such as the present model. Let 77(12) = L T 7 m n l (r ) * m n l (12), (2.4.6a) mnl c(12) = I c m n l ( r ) $ m n : L ( 1 2 ) , (2.4.6b) mnl h(12) = Z h m n l ( r ) * m n l ( 1 2 ) , (2.4.6c) mnl with $ m n^"(12) given by eqn. 2.3.7. We can w r i t e eqn. 2.4.4 as T?(12) = / d n 3 d r 3 c ( 1 3 ) [ i ? ( 3 2 ) - c ( 3 2 ) ] . (2.4.7) 8 7T 1 Then n o t i n g r l 2 = r ^ + r 3 2 w e take the F o u r i e r transform to o b t a i n / d r 1 2 T ? ( 1 2 ) e x p ( i k - r 1 2 ) = / d f l 3 d r 3 d r 2 c (1 3) 8 7 T x [T?(32) - c(32) ] e x p ( i k - ( r 1 3 + r 3 2 ) ) . (2.4.8) If we l e t molecule 1 be the o r i g i n , then d r 3 = d r 3 1 and s i n c e our f l u i d i s i s o t r o p i c , i n t e g r a t i o n on the r i g h t over d r 3 1 d r l 2 i s e q u i v a l e n t to i n t e g r a t i o n over d r 1 3 d r 3 2 . T h e r e f o r e / d r l 2 r ? ( 1 2 ) e x p ( i k - r l 2 ) = / dJ23 J dr 3 c (1 3) exp( ik • r . ) 8TT x ; d r 3 2 [ r j ( 3 2 ) - c ( 32) ]exp( ik • r 3 2 ) , (2.4.9) or rj(12) = / dO- C(13)[T?(32) - c ( 3 2 ) ] , (2.4.10) 8TT . 27 where, as d e r i v e d i n Appendix D, we have c(12) = c ( n i f O , f k ) = Z c m n l ( k ) $ m n l ( f l 1 ,fl,,k) , (2.4.11a) 1 z mnl 1 z 77(12) = ^ ( 0 , ,Q,,k) = I *jmnl(k)*mnl(Q.,Q-,k), (2.4.11b) 1 z mnl where k i s a u n i t v e c t o r and 00 c m n l ( k ) = 4TT J dr r 2 j n ( k r ) c m n l ( r ) f o r 1 even, (2.4.12a) 0 U 00 ? m n l ( k ) = 47ri / dr r 2 j . ( k r ) c m n l ( r ) f o r 1 odd. (2.4.12b) 0 1 The t r a n s f o r m a t i o n k e r n e l s i n e q u a t i o n s 2.4.12a,b are z e r o t h and f i r s t order s p h e r i c a l B e s s e l f u n c t i o n s which are, e x p l i c i t l y j 0 ( x ) = S i ; ( x ) , (2.4.13a) j ( x ) « s i n i x ) . _ c o s i x i ^ . (2.4.13b) x The f u n c t i o n s c m n l ( r ) which are transformed i n 2.4.12a,b are given by i -i o mnl, > c m n l ( r ) = c m n l ( r ) - / S — _ i s i p e ( | ) d s f o r 1 e v e n , (2.4.14a) r c m n l ( r ) = c m n l ( r ) - J c m n l ( s ) ( r ) p o ( | ) d s f o r 1 Q d d f ( 2 . 4 . l 4 b ) ^ s S I S where P^(x) and P°(x) are p o l y n o m i a l s given by 28 P®(x) = P°(x) = 0, (2.4.15a) e 2 q 2i a - i a ! P L +9( X) = 7TT 2 x ^ ( - ) q X ( ? ) r ^ — , (2.4.15b) 2q+2 q! . = Q i ( . + ^ , ( ^) p ? o + ^ x > = WT 2 x 2 i ( - ) q " i ( ? ) q \ 2 ', (2.4.15c) 2q +3 q! i = Q i ( i + 3)j f o r q>0. We note that the g e n e r a l d e f i n i t i o n Z!=T(Z+1) i s used. E x p r e s s i o n s f o r 7 j m n ^ ( k ) are s i m i l a r . We note here that the p r o j e c t i o n s 7 j m n ^ ( k ) and c ^ ^ k ) are given i n terms of the hat transforms of Blum [16]. The p r o j e c t i o n s c o u l d have been c a l c u l a t e d i n the s i n g l e s t e p oo c m n l ( k ) = 47ri 1 / dr r 2 j , ( k r ) c m n l ( r ) , (2.4.16a) 0 1 OO Vmnl(k) = 47ri 1 J dr r 2 j , ( k r ) T 7 m n l ( r ) , (2.4.16b) where j ^ ( k r ) denotes the l f c ^ order s p h e r i c a l B e s s e l f u n c t i o n s . However, t h i s d i r e c t computation would be an extremely time consuming procedure. By u s i n g the hat transforms only the z e r o t h and f i r s t order B e s s e l f u n c t i o n s (eqns. 2.4.13a,b) are necessary which means the transforms can be c a l c u l a t e d u s i n g f a s t F o u r i e r transform (FFT) techniques. The hat transforms have the 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 the long c mnl, N 112, x 123, \ , 224, x , , range p a r t of c ( r ) . c ( r ) , c (r) and c (r) depend on r — n 1 12 12 3 as r f o r n=3,4,5 r e s p e c t i v e l y . However, c ( r ) , c (r) and 224 c (r) have no such r dependence and are r e l a t i v e l y short ranged. Equation 2.4.10 g i v e s a formal s o l u t i o n but as yet i t i s not separable because of the angular i n t e g r a t i o n . We can perform t h i s i n t e g r a t i o n e x p l i c i t l y i f we take advantage of the form of the expansions. With a great amount of s i m p l i f i c a t i o n (Appendix E) using the p r o p e r t i e s of the Wigner g e n e r a l i z e d s p h e r i c a l harmonics and 3-j symbols we end up with -mnl,. x _ , .m+n+n,(21+1) f m n 1 1 1 f n 1 n l 2 f 1 21,1 , * U ) = p n ? , ( _ ) (2n,+l) ImTQ { m n n, } n i l i l 2 f x ( J ' J ' J ) c m n i l ' ( k ) ( 7 ? n i n l M k ) - c n i n l 2 ( k ) ) , (2.4.17) which can c o n v e n i e n t l y be w r i t t e n i n the form 9 j m n l ( k ) = p I z i ' i ' i c m n i l l ( k ) ( ? ? n i n l 2 ( k ) - c n i n l 2 ( k ) ) , (2.4.18a) n , l , l 2 1 where 1 21,1 = .m+n+n,(21+1) f m n 1 1 1 f n 1 n l 2 f 1 21,1 , m n n, (2n,+1) fmnl 1 m n n, x ( l'l2l0 ). (2.4.18b) Now i t i s c l e a r that i n F o u r i e r space the OZ equation i s j u s t an a l g e b r a i c r e l a t i o n s h i p . We c o u l d at t h i s p o i n t f i n d the p r o j e c t i o n s rj m n" I'(r) f o r a given set c m n * ( r ) by F o u r i e r t r a n s f o r m i n g , a p p l y i n g 2.4.18 and then back t r a n s f o r m i n g . T h i s could e a s i l y be put i n t o matrix form such that f i n d i n g the s o l u t i o n would be e q u i v a l e n t to i n v e r t i n g a matrix f o r each value of k. However, our expansions of 17(12) and c(12) in r o t a t i o n a l i n v a r i a n t s need, i n theory, an i n f i n i t e number of terms. While we assume such expansions are t r u n c a t a b l e we might expect f a i r l y l a r g e b a s i s s e t s to be necessary. We have already shown in Appendix B that u m n l ( r ) = ( - ) m + n u n m l ( r ) . Since h(12) must have the same symmetry requirement ( i n v a r i a n c e under interchange of p a r t i c l e s ) which r e s u l t e d i n t h i s c o n d i t i o n , we a l s o have h m i i i ( r ) = ( - ) u , T , , h , , u , A ( r ) . We can, then, r e s t r i c t the h m n ^ " ( r ) used to those with m^n and s t i l l r e t a i n a l l independent p r o j e c t i o n s . I f we r e s t r i c t our p r o j e c t i o n s t o those f o r which m<n<6 we get a t o t a l of 84 independent terms. The s o l u t i o n of such a system would r e q u i r e the i n v e r s i o n of an 84x84 matrix which must be c a l c u l a t e d a t each value of k f o r each i t e r a t i o n in an i t e r a t i v e procedure. By using an i d e n t i t y f o r 3-j symbols (A.11) we can perform one l a s t t r a n s f o r m a t i o n . T h i s i s Blum's X-transformation [16] which g r e a t l y reduces the s i z e of the matrices to be i n v e r t e d . In the example above the l a r g e s t matrix to be i n v e r t e d i s now reduced to 6x6. Before i n t r o d u c i n g the x ~ t r a n s f o r m a t i o n i t i s convenient t o choose f m n l = [(2m+1)(2n+1)P. (2.4.19) Equation 2.4.18b then becomes z i 2 i 1 i = ( 2 l + 1 ) ( - ) m + n + n i { 1*1*1 }( n ' n 2 j ! : ). (2.4.20) m n n! m n n , 0 0 0 Now i f we l e t 31 ~ m+n , , c m n ( k ) = L ( n ) c m n l ( k ) , (2.4.21a) x l=|m-n| x _ x 0 ~ m+n , , N m n { k ) = L ( a n ) r , m n l ( k ) , (2.4.21b) x l=|m-n| x x U apply eqn. 2.4.21 i n eqn. 2.4.18 and s i m p l i f y (Appendix F ) , the OZ equation becomes N°J n(k) = p I ( - ) X C ™ n M k ) [ N n i n ( k ) - C n , n ( k ) ] . (2.4.22) A — A A* X n i I t i s convenient to express 2.4.22 i n matrix form with C m n ( k ) and N m n ( k ) being the (m+1,n+1) elements of C (k) and 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 the OZ equation i s then A* a matrix equation d e f i n e d by N x(k) = p ( - ) x + 1 C 2 ( k ) ( I + p ( - ) X + 1 Q x ( k ) ) ~ 1 , (2.4.23) for each value of x allowed by the 3-j symbols i n eqn. 2.4.21. Thus the x - t r a n s f o r m a t i o n breaks up the a l g e b r a i c equations of 2.4.18 i n t o smaller independent s e t s of equations. I t i s now only necessary to back transform C (k) and A* N ( k ) . These back t r a n s f o r m a t i o n s are given by ^ m n l ( k ) = (21+1) I ( "J " I )N™ n(k), (2.4.24) ^ A A ^ A where x must range between ±min(m,n), followed by 32 fo r 1 even, (2.4.25a) fo r 1 odd. (2.4.25b) L a s t l y we take the in v e r s e hat transforms to get T , m n l ( r ) = 7 j m n l ( r ) " - 3 J ds s 2 P ^ ( f ) f ? m n l ( s ) f o r 1 even , (2 . 4 . 26a ) r 0 ^ ( r ) = f 7 m n l ( r ) - ±j J ds s ^ & ^ i s ) f o r 1 odd, (2.4.26b) r* 0 1 r where P-^(x) and P^(x) are given by eqn. 2.4.15 and s i m i l a r equations are obtained f o r c m n ^ ( r ) . D e r i v a t i o n s of the back transforms are given i n Appendix G. (c) Reduction of the HNC C l o s u r e . U n t i l r e c e n t l y the HNC c l o s u r e (eqn. 2.4.3) c o u l d not be a n a l y t i c a l l y expanded i n r o t a t i o n a l i n v a r i a n t s due to the l o g a r i t h m i c term. T h i s problem was t r e a t e d by t r u n c a t i n g the t a y l o r s e r i e s r e p r e s e n t a t i o n of the l o g term. I f t r u n c a t i o n occurs a f t e r the l i n e a r term, the r e s u l t i n g equation i s the l i n e a r i z e d HNC (LHNC). I f t r u n c a t i o n occurs a f t e r the q u a d r a t i c term the QHNC i s produced. The LHNC, due to i t s simple form, can be sol v e d q u i c k l y i n an i t e r a t i v e process and so i t makes a good s t a r t i n g p o i n t f o r a f u l l HNC s o l u t i o n . For t h i s reason we give t h i s c l o s u r e here. Let f ) m n l ( s ) = - L - / dk k 2 j n ( k s ) r ? m n l ( k ) 2TT 0 U r,mnl(s) = ^ JT dk k 2 j 1 ( k s ) 7 ? m n l ( k ) 2TT 0 -and separate 2.4.3 i n t o i t s s p h e r i c a l ( 0 0 0 p r o j e c t i o n ) n o n - s p h e r i c a l p a r t s . T h i s gives c(12) = h 0 0 0 ( r ) $ 0 0 0 ( 1 2 ) - l n [ g 0 0 0 ( r ) ] - /3u 0 0 0 (r ) * 0 0 0 ( i - l n [ l + X(12)] - 0[(u(12) - u 0 0 0 ( r ) * 0 0 0 ( 1 2 ) ] + [h(12) - h 0 0 0 ( r ) * 0 0 0 ( l 2 ) ] . I f we expand l n [ l + X(12)] in the T a y l o r s e r i e s ln[1 + X(12)] = X(12) - ^X 2(12) + ^X 3(12) -and t r u n c a t e a f t e r the f i r s t term we get l n [ l + X(12)] - ( h d 2 ) - h ° ° ° ( r ) * 0 0 0 ( l 2 ) ) g 0 U U ( r ) Using 2.4.29 i n 2.4.28 and s e p a r a t i n g we get c o o o ( r ) = h o o o ( r ) _ l n [ g o o o ( r ) ] _ p u o o o ( r ) r G 0 0 0 ( r ) c m n l ( r ) = _ h m n l ( r ) _ p g 0 0 0 ( r ) u m n l ( r } + G 0 0 0 ( r ) h m n l ( r ) f f o r a l l m,n,l except m=n=l=0. In terms of T J and c „ 0 0 0 ( r ) = l n [ g 0 0 0 ( r ) ] + 0 u O O O ( r ) , 34 (2.4.31a) mnl/ x 000/ \r mnl/ \ „ mnl, *-, mnl/ \ /_ . -,,\ c (r) = g (r)[tj (r) - /3u ( r ) ] - T? ( r ) , (2.4.31b) except f o r m=n=l=0. Equations 2.4.31a,b c o n s t i t u t e the LHNC c l o s u r e . The a n a l y t i c a l expansion of the f u l l HNC was made p o s s i b l e due to the recent work of F r i e s and P a t e y [ l l ] . We f i r s t take the p a r t i a l d e r i v a t i v e of eqn. 2.4.3 with respect to r to get 9c(12) 3h(12) 1 3q(l2) a3u(12) . „ v 3r = ~ 9 7 g T i T F 3r & 9r * (2.4.32) Then u s i n g *?<12> = 8h(12) ( 2 3 3 ) 3r 3r ' u * * ' " ' and r e a r r a n g i n g we o b t a i n h ( 1 2 ) [ 3 £ i l 2 i _ 3 a i i 2 i + p l u g i i ] - - & ^ 2 l , (2.4.34) For convenience we d e f i n e W ( 1 2 ) = - 7 7 ( 1 2 ) + /3u ( l 2 ) = 1 - g d 2 ) + c(12) + 0U(12). (2.4.35) 3W(12) C l e a r l y — ^ — - i s the e x p r e s s i o n i n bracke t s i n 2.3.34, so we 35 can r e w r i t e 2.4.34 as l£|i2) = _ h ( 1 2 ) a w j i 2 ) _ ^ a u U i i ^ { 2 ^ 3 6 ) If we rearrange 2.4.35 to get C(12) = h ( l 2 ) + W(12) - 0U(12), (2.4.37) and compare to 2.4.3 we f i n d that W(12) can be i d e n t i f i e d with - l n ( g ( l 2 ) ) . W(12) i s the u n i t l e s s , angle dependent ' p o t e n t i a l of mean f o r c e ' . We note that t h i s i d e n t i f i c a t i o n depends on the HNC equation and hence i s only t r u e i n the HNC approximation. To get the c l o s u r e i n a more usable form we i n t e g r a t e eqn. 2.4.36 over r 1 dr 3£ii2i = - J dr h( 1 2 ) ^ 1 2 1 - J d r f U ^ l l i . ( 2. 4. 38) Th i s g i v e s 00 l i m [ c ( 1 2 , r ) ] - c(12,r) = - / dr h( 1 2 ) 9 W ^ 2 ) r—>=> r - l i m [/3u(12,r)] - /3c ( 1 2 , r ) . (2.4.39) r — F o r t u n a t e l y , both l i m i t s are w e l l behaved and l i m [ c ( 1 2 , r ) ] = l i m [0u(12,r)] = 0. (2.4.40) 36 T h e r e f o r e , we have the f i n a l r e s u l t C ( 1 2 ) = / dr h ( 1 2 ) 9 W ^ 2 ) - 0 u ( l 2 ) . (2.4.41) r I t i s now only necessary that the b i n a r y product under the i n t e g r a l be expandable in r o t a t i o n a l i n v a r i a n t s . T h i s r e q u i r e s a product of two i n v a r i a n t s to be expressed as a m u l t i p l e of a s i n g l e i n v a r i a n t . T h i s r e l a t i o n s h i p i s d e r i v e d in Appendix H with the r e s u l t * m i n i 1 i ( l 2 ) $ m 2 n 2 1 2 ( 1 2 ) = I P ( m , n , l ) * m n l ( l 2 ) , ( 2 . 4 . 4 2 ) mnl where f m , n , l 1 f m 2 n , l , m + n + l P(m,n,l) = 1 - J — ( 2 m + 1 ) ( 2 n + 1 ) ( 2 l + l ) ( - ) m + n 1 m i n 1 1 1 , , , x I m 2 n 2 l 2 j l 0 0 0 M 0 0 0 M 0 0 0 ' ' (2.4.43) m n l and { } i s the usual 9-j symbol. Eqn. 2.4.41 i s now separable i n t o i n d i v i d u a l equations f o r the c m n ^ " ( r ) c o e f f i c i e n t s . One has c m n l ( r ) = I P ( m , n , l ) I d r h m ' n ' 1 M r ) ^ ^ 1 1 1 , 1 1 , 1 , r m2n 2 1 2 - 0 u m n l ( r ) . (2.4.44) (d) The Reference HNC C l o s u r e . To improve accuracy, Lado [19] int r o d u c e d a p e r t u r b a t i o n scheme whereby a l l c o r r e l a t i o n f u n c t i o n s are d i v i d e d i n t o r e f e r e n c e and p e r t u r b a t i o n p a r t s . I t i s u s u a l l y convenient to choose a s p h e r i c a l l y symmetric system as the ref e r e n c e system, and in the present case we use t h a t of hard spheres. W r i t i n g a l l the v a r i a b l e s i n the form X(12) = X R ( r ) + AX(12), (2.4.45) where X(12) i s a property of the system and X R ( r ) i s the same property of the r e l a t e d s p h e r i c a l l y symmetric r e f e r e n c e system. Equation 2.4.3 becomes c H S ( r ) + Ac(12) = h H S ( r ) + Ah(l2) - l n [ g H g ( r ) + Ag(12)] - 0 [ u H S ( r ) + Au(12)]. (2.4.46) We know that c H S ( r ) = h H S ( r ) - l n ( g H S ( r ) ) - 0 u H S ( r ) , (2.4.47) and Ah(12) = Ag(12). (2.4.48) Th e r e f o r e , 38 Ac (12) = A h d 2 ) - l n [ g H S ( r ) + A h ( l 2 ) ] + l n [ g H S ( r ) ] - 0AU(12). (2.4.49) T h i s ensures that g ( l 2 ) = g H g ( r ) i n the l i m i t /3AU(12) = 0. The RHNC can be t r e a t e d i n a s i m i l a r manner to the HNC. Taking the p a r t i a l d e r i v a t i v e with respect to r and i n t e g r a t i n g g i v e s Ac(12) = J dr A h ( 1 2 ) ^ f i I 2 i + j d r ^ ( r) *™±12i » 31n[g„_(r)] - / dr Ah(12) 1| 0AU(12), (2.4.50) where AW( 12) = -AT?( 12) + 0Au( 12) . (2.4.51) The b i n a r y product i s expanded i n the same way to get A c m n l ( r ) = I P(m,n,l) J dr Ahm, n , 1, ( r } 3 A W m ^ 1 M r) m, n , 1, r m 2n 2 1 2 + ; ar h°°°< r,l^ i<r> . J d r ^ n l ( r } »ln [ ( r) 1 - ( J A u m n l ( r ) . (2.4.52) 39 2.5 E x p r e s s i o n s f o r Thermodynamic Q u a n t i t i e s . The thermodynamic q u a n t i t i e s that we w i l l be using f o r purposes of comparison with other t h e o r i e s and with Monte C a r l o r e s u l t s can be c o n v e n i e n t l y expressed i n terms of the r o t a t i o n a l i n v a r i a n t expansion c o e f f i c i e n t s f o r h ( l 2 ) . I t i s a l s o convenient to d e s c r i b e the system i n terms of reduced parameters p*=pd 3, M*=(0M 2/d 3)^ a n d Q*=(/3Q2/d5) *. To be c o n s i s t e n t with the more recent l i t e r a t u r e [10,11] we choose 11 -mnl •L-f = , m n 1 x , (2.5.1) v 0 0 0 ; throughout t h i s s e c t i o n . A l l r e s u l t s i n Chapter 3 use t h i s c onvention. However, s e c t i o n s 2.3(b) and 2.4(b) use f m n l = [(2m+1)(2n+1)]* f (2.5.2) which s i m p l i f i e s the i n t e g r a l equation e x p r e s s i o n s . The d i f f e r e n c e i s i n s i g n i f i c a n t s i n c e the f m n ^ are a r b i t r a r y . To convert the h m n ^ " ( r ) from the r e s u l t s of p r e v i o u s s e c t i o n s to those of the present s e c t i o n r e q u i r e s only that each h m n"'"(r) be m u l t i p l i e d by the f a c t o r , W m n 1 \ 7m n i = [ (2m+1 ) (2n+1 ) ] 2 _ 0 _ 0 _ 0 _ - (2.5.3) 1! 40 (a) C o n f i g u r a t i o n a l Energy. The average c o n f i g u r a t i o n a l energy <U> i s given by [31] <U> = 2" ( - £2 ) 2J" dX 1dX 2u( 12)g( 12) . (2.5.4) 8 7T Using eqn. A.15 we f i n d <U> = 2TT P 2V / r 2 d r [ ( £ ^ 2 ) 2 u 1 1 2 ( r) g 1 1 2 ( r ) + u 1 2 3 ( r ) g l 2 3 ( r ) + ^ 1 u 2 2 4 ( r ) g 2 2 4 ( r ) ] . ( 2 . 5 . 5 ) S i m p l i f y i n g we get ff<U> , f l r 2 2 " h 1 1 2 ( r ) , » ,123, > c c - » , 224, v + 4 MQ / * ^ dr + fr6- Q / * i l l d r ] , (2.5.6) d D d r J which i n terms of the reduced parameters becomes /3<U> 4TT * *2 " h 1 1 2 ( r ) , , Q * * * " h l 2 3 ( r ) , ^ — = " O - P M / dr + 8irp v Q J 5 — dr 1 1 r + I 1 2 £ p V 2 J dr. (2.5.7) 1 r"3 41 (b) Isothermal C o m p r e s s i b i l i t y F a c t o r The i s o t h e r m a l c o m p r e s s i b i l i t y f a c t o r i s given by [31] EV_ . , . | , ^ _ ) 2 ; d r d ! i ) d n r 3 a u i i 2 ) g ( ] 2 ) ( 2 5 8 ) 87T from which we o b t a i n NkT 1 3 7 r p 9 N 3 N 3 N ' ^ • 5 • y ' 0<u D D> p<u > ft<V > where — r : , — V T and — V T are the f i r s t second and t h i r d N N N terms of eqn. 2.5.7, r e s p e c t i v e l y . (c) S t a t i c D i e l e c t r i c Constant. For an i n f i n i t e system the d i e l e c t r i c constant e can be obtained u s i n g the Kirkwood r e l a t i o n s h i p [24] (e-1 ) (2e + 1 ) _ „„ fo c m l = yg, (2.5.10) where y = 4 f f P / . (2.5.11) The Kirkwood g - f a c t o r i s given by 42 <M2> ,• . N-1 g = — 2 = 1 + — <^'E2> = 1 + J dr r 2 h 1 l 0 ( r ) , (2.5.12) J 0 2 where <M > i s the mean square of the t o t a l moment. 2.6 Computational C o n s i d e r a t i o n s . (a) Method of S o l u t i o n , To begin the i t e r a t i v e c y c l e an i n i t i a l guess i s made for each of the c m n ^ " ( r ) at 2 n g r i d p o i n t s separated by an a p p r o p r i a t e g r i d width Ar. The i n i t i a l values of the c m n ^ ( r ) must be s u f f i c i e n t l y c l o s e to the s o l u t i o n so as not to d i v e r g e d u r i n g the c y c l e . T h i s c o n d i t i o n i s g e n e r a l l y s a t i s f i e d by any cho i c e such that C(12) —> -0U(12) i n the l a r g e r l i m i t . I t e r a t i v e convergence can u s u a l l y be achieved by using r e s u l t s from a p r e v i o u s s o l u t i o n and g r a d u a l l y a l t e r i n g the reduced parameters, or by using r e s u l t s from a previous s o l u t i o n f o r a d i f f e r e n t c l o s u r e such as the RLHNC. The T j m n ^ ( r ) are then obtained u s i n g the OZ equation. These 7 j m n''"(r) are used with the o r i g i n a l c m n"^"(r) to c a l c u l a t e a new set of c m n ^ ( r ) n e w u s i n g the RHNC c l o s u r e . The c m n l ( r ) are compared with the c m n l ( r ) n e w at r=d because the g r e a t e s t change i n each p r o j e c t i o n occurs there.. If the new p r o j e c t i o n s are s u f f i c i e n t l y c l o s e to the o l d , the i t e r a t i v e process i s stopped. If they are not s u f f i c i e n t l y c l o s e the c m n l ( r ) are mixed with the c m n"^ ( r ) n e w a c c o r d i n g to C m n l ( r ) ( i + 1 ) = ( 1 _ a m n l ) c m n l ( r ) ( i ) + amnl cmnl ( f }new^ ( 2 > 6 > 1 ) where the s u p e r s c r i p t ( i ) i n d i c a t e s the i f c ^ i t e r a t i o n , and the amn^ are s c a l a r s v a r y i n g i n 0 < a m n ^ < 1 . I t i s found that divergence occurs i f the amn^ are chosen too l a r g e at f i r s t , so mnl a given a i s i n c r e a s e d as convergence i s approached. The a m n ^ are v a r i e d independently. One dimensional i n t e g r a l s are e a s i l y c a l c u l a t e d using the t r a p e z o i d a l r u l e and the numerical d e r i v a t i v e s are c a l c u l a t e d using standard d i f f e r e n c e formulas [36]. (b) Program E f f i c i e n c y . As i s u s u a l l y the case, we are faced with a t r a d e - o f f between computation time and storage space. In order to use f a s t F o u r i e r transform (FFT) r o u t i n e s i t i s necessary to t a b u l a t e v a r i o u s f u n c t i o n s at the 2 n g r i d p o i n t s . The FFT r o u t i n e s save such a s i g n i f i c a n t amount of cpu time that the task would be v i r t u a l l y i m possible without them. A l s o s t o r e d , r a t h e r than c a l c u l a t e d at each i t e r a t i o n , are the P(m,n,l) c o e f f i c i e n t s given by eqn. 2.4.44. About 20% of these c o e f f i c i e n t s are non-zero but even so t h e i r number i n c r e a s e s d r a m a t i c a l l y with the s i z e of the b a s i s s e t . Storage space i s minimized by saving o n l y the non-zero P(m,n,l) and by always using them i n the same order to a v o i d i n d e x i n g . T h i s cuts cpu time per i t e r a t i o n by a f a c t o r of about 2 f o r a b a s i s set of 35 p r o j e c t i o n s , and f o r an 84 term b a s i s set i t i s cut by a f a c t o r of about 50, making t h i s an unavoidable measure. Another decrease in the cpu time can be achieved by t a k i n g advantage of the l a r g e r l i m i t of c m n * ( r ) . In p r a c t i s e , mn 1 f o r m or n > 2, the c (r) are i n s i g n i f i c a n t l y d i f f e r e n t from zero at r e l a t i v e l y short range. Thus, for r g r e a t e r than some d i s t a n c e the c (r) can be set to zero for these p r o j e c t i o n s . In Chapter 3 we d i s c u s s at what d i s t a n c e t h i s t r u n c a t i o n i n c m n l ( r ) i s v a l i d . 45 CHAPTER 3 R e s u l t s and D i s c u s s i o n . 3.1 Input Parameters and B a s i s Sets. The r e s u l t s o btained i n t h i s chapter were a l l found using 512 g r i d p o i n t s with a g r i d width Ar = 0.02d. The ref e r e n c e r a d i a l d i s t r i b u t i o n f u n c t i o n g u c ( r ) was taken to be r i b the V e r l e t - W e i s f i t [37] to computer s i m u l a t i o n data. Four b a s i s s e t s were used throughout. These b a s i s s e t s were obtained by i n c l u d i n g independent (by symmetry) terms from the c o r r e l a t i o n f u n c t i o n expansions with the r e s t r i c t i o n m,n<i f o r i=3,4,5 and 6. These b a s i s s e t s are given i n Table I. The terms are denoted by t h e i r i n d i c e s (mnl) and only independent terms are l i s t e d . Each b a s i s set i s an exte n s i o n of the p r e v i o u s set so only a d d i t i o n a l terms are given. For example, the complete set of independent terms f o r b a s i s s et III a l s o i n c l u d e s those terms l i s t e d i n Table I f o r b a s i s s e t s I and I I . The system can be completely c h a r a c t e r i z e d by the * * * reduced parameters p ,n and Q . Three s e t s of these parameters are used. These s e t s were chosen to make comparisons with p r e v i o u s Monte C a r l o r e s u l t s [8] p o s s i b l e . The parameter s e t s used are as f o l l o w s : (a) p* = 0.8 , ix* = 1.5 , Q* = 0.5 , (b) p* = 0.8 , M * = 1.5 , Q* = 1.0 , (c) p* = 0.8 , M * = 1.0 , Q* = 1.0 . T a b l e I Basis Sets Used i n RHNC c a l c u l a t i o n s , B a s i s Set Number of Independent Terms Terms Included (values of (mnl)) m,n<3 20 (000) (121 ) (033) (235) (011),(110) (123),(220) (132),(134) (330),(332) (112) (222) (231 ) (334) (022) , (224) , (233) , (336) II m,n<4 35 (044) (246) (440) (143),(145) (341),(343) (442),(444) (242) (345) (446) (244), (347) , (448) III m, n<5 56 (055) (257) (451 ) (550) (154),(156) (352),(354) (453),(455) (552),(554) (253) (356) (457) (556) (255), (358), (459), (558), (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 d i s c u s s e d i n s e c t i o n 2.6 the c (r) were t r u n c a t e d at a d i s t a n c e R. Table II shows the e f f e c t on the d i e l e c t r i c constant e and the average c o n f i g u r a t i o n a i energy per p a r t i c l e -0<U>/N when R i s v a r i e d . B a s i s set II was used i n these c a l c u l a t i o n s . The t a b l e shows that f o r parameter set (a) the c m n ^ ( r ) can be cut o f f s a f e l y at 5.5d, and at 3. Od f o r set s (b) and ( c ) . A l s o shown i s the time savings achieved by the c u t - o f f . For a l l subsequent r e s u l t s these c u t - o f f d i s t a n c e s were used. 3.2 R e s u l t s f o r D i f f e r e n t B a s i s Sets. Table III shows the b a s i s set dependence of e, -/3<U>/N and pV/NkT. The b a s i s set dependence to some extent depends on * * the v a l u e s of u and Q . For parameter set (a) b a s i s set I gi v e s a c c e p t a b l e values f o r these q u a n t i t i e s , but parameter s e t s (b) and (c) give converged r e s u l t s f o r a l l three q u a n t i t i e s only when b a s i s set III i s used. Tables IV and V compare con t a c t values (r = 1.0d) as w e l l as the value s at r = 1.1d of the f i v e p r o j e c t i o n s used to c a l c u l a t e e, /3<U>/N and pV/NkT. The contact v a l u e s are much more s e n s i t i v e to b a s i s set than the i n t e g r a l s over these f u n c t i o n s , the l a t t e r of which are used to f i n d the thermodynamic q u a n t i t i e s . T h i s i s c l e a r l y because even as c l o s e to c o n t a c t as r= 1.1d the s e n s i t i v i t y to b a s i s set i s g r e a t l y reduced. The l e a s t s e n s i t i v e p r o j e c t i o n at contact i s g ^ ^ ( d ) which i s the only contact value used i n the c a l c u l a t i o n of thermodynamic q u a n t i t i e s . Table II Cu t - o f f dependence i n c ( r ) . Parameter Set C u t - o f f Radius mnl, x in c (r) -0<U>/N e cpu time per I t e r a t ion (a) 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 (b) 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 (c) 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 III Bas i s Set Dependence of e,0<U>/N and pV/NkT. Parameter B a s i s Set Set c -/3<U>/N pV/NkT (a) I 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 (b) 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 (c) 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 Table IV P r o j e c t i o n Contact Values Parameter Bas i s Set Set g 0 0 0 ( d ) h 1 1 ° ( d ) h 1 l 2 ( d ) h 1 2 3 ( d ) h 2 2 4 ( d ) (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 (b) 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 (c) 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 51 Table V P r o j e c t i o n Values at r=1.1d. Parameter B a s i s Set Set 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 ) (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 (b) 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 (c) 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 3.3 Comparison with Monte C a r l o C a l c u l a t i o n s . 52 The MC c a l c u l a t i o n s with which our r e s u l t s are compared [8] were c a l c u l a t e d with a s p h e r i c a l c u t - o f f boundary c o n d i t i o n at a ra d i u s R = 3.40d. The r e f o r e , i n order to c compare the p a i r c o r r e l a t i o n f u n c t i o n and c o n f i g u r a t i o n a l energy the RHNC approximation was solv e d t r u n c a t i n g the p o t e n t i a l at 3.40d. T h i s was done f o r a l l parameter s e t s . (a) C o n f i g u r a t i o n a l Energy. The average c o n f i g u r a t i o n a l energy per p a r t i c l e i s compared with MC r e s u l t s i n Table VI. A l s o shown are p r e v i o u s l y c a l c u l a t e d r e s u l t s using the RLHNC, RQHNC and MSA t h e o r i e s as wel l as those of the Pade v e r s i o n of thermodynamic p e r t u r b a t i o n theory [ 8 ] . I t i s c l e a r that the RHNC c l o s u r e g i v e s values c l o s e r to the MC s i m u l a t i o n r e s u l t s than any of the other approximations, but tends to underestimate -0<U>/N by ~3% f o r parameter set (a) and ~5% for the other parameter s e t s . (b) D i e l e c t r i c Constant. The d i e l e c t r i c c o nstants from the MC c a l c u l a t i o n s were obtained with s p h e r i c a l cut o f f boundary c o n d i t i o n s so the u n c e r t a i n t y i n these values i s q u i t e l a r g e [24]. The RHNC r e s u l t s were obtained using eqn. 2.5.10 f o r an i n f i n i t e system. In t h i s c a l c u l a t i o n the p o t e n t i a l was not t r u n c a t e d . R e s u l t s Table 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 RQHNC RHNC MC (a) 3.93 3.11 4.27 - 3.62 3.72±0.02 (b) 6.27 5.86 8.31 7.58 6.69 6.99±0.02 (c) 4.40 3.44 5.35 4.97 4.41 4.64±0.02 from p r e v i o u s approximations as w e l l as RHNC and MC data are given i n Table V I I . The RHNC d i e l e c t r i c c o n s t a n t s are c o n s i s t e n t l y smaller than those of the RLHNC and RQHNC t h e o r i e s . For parameter sets (a) and (b) t h i s puts the RHNC c l o s e r to the MC value s than other r e s u l t s . T h i s i s not t r u e of parameter set (c) but the value f o r e p r o b a b l y s t i l l l i e s w i t h i n the u n c e r t a i n t y i n the e from MC data. (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 . The f i v e RHNC p r o j e c t i o n s g u u u ( r ) = h u u u ( r ) + 1, h 1 1 i ^ ( r ) , h 1 1 2 ( r ) , h 1 2 3 ( r ) and h 2 2 4 ( r ) are graphed f o r each parameter set i n f i g u r e s 2-16. A l s o on these f i g u r e s are the eq u i v a l e n t f u n c t i o n s f o r RLHNC, RQHNC and MC c a l c u l a t i o n s . We note that an RQHNC s o l u t i o n f o r parameter set (a) cannot be obtained and does not appear to e x i s t . The RHNC r e s u l t s f o r parameter set (a) were c a l c u l a t e d with b a s i s set I I , and f o r (b) and (c) with b a s i s set IV. The r a d i a l d i s t r i b u t i o n f u n c t i o n g ^ ^ ( r ) i s p l o t t e d at short range i n f i g u r e s 2a, 3a and 4a fo r parameter s e t s ( a ) - ( c ) , • r e s p e c t i v e l y . I t can be seen that the RHNC g i v e s a good d e s c r i p t i o n of g ^ ^ ( r ) f o r a l l parameter s e t s and i s c l o s e r to the MC data than the other c l o s u r e s . We note that f o r the RLHNC, g ^ ^ ( r ) = g H g ( r ) . The same f u n c t i o n i s p l o t t e d at longer range i n f i g u r e s 2b, 3b and 4b f o r the same r e s p e c t i v e parameter s e t s . Good agreement between RHNC and MC i s a l s o evident here. The p r o j e c t i o n h 1 1 ^ ( r ) i s p l o t t e d i n f i g u r e s 5-7 f o r the parameter sets ( a ) - ( c ) . For parameter set (a) the RHNC Table VII The S t a t i c D i e l e c t r i c Constant, 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 p r o j e c t i o n c l o s e l y ressembles that of the MC. The most s t r i k i n g f e a t u r e of f i g u r e s 6 and 7 i s the disagreement at short range between the v a r i o u s sources. The RHNC curve i s c l e a r l y the c l o s e s t to the MC curve which leads to the p r e d i c t i o n that the RHNC d i e l e c t r i c constant would approximate e from a simulated ' i n f i n i t e ' system more c l o s e l y than the RLHNC or RQHNC approximations (see eqn. 2.5.12). 1 ] 2 F i g u r e s 8-10 show h (r) and f i g u r e s 11-13 show 1 2 3 h ( r ) , f o r parameter s e t s ( a ) - ( c ) . Even though the s c a l e s of the graphs have been e n l a r g e d to the p o i n t where the 'noise' i n the MC data i s r e a d i l y apparent, the RHNC curves s t i l l c l o s e l y f o l l o w the MC r e s u l t s . The RHNC approximation improves on the other t h e o r i e s as w e l l . F i g u r e s 14-16 show h ( r ) . In a l l cases the RHNC data i s an improvement on the RLHNC and RQHNC data except around 1.5d f o r parameter set ( b ) ( f i g . 15) and at cont a c t f o r parameter set ( c ) ( f i g . 16). However, the RHNC r e s u l t s d e v i a t e from the MC r e s u l t s s i g n i f i c a n t l y f o r set s (a) and (b) while s e r i o u s l y 224 underestimating h (r) at c l o s e range f o r set ( b ) . Parameter set (c) shows much b e t t e r agreement. F i g u r e 2a The short-range p a r t of the r a d i a l d i s t r i b u t i o n f u n c t i o n f o r * * * u = 1.5, Q = 0.5 and p = 0.8. The s o l i d , dashed and dotted curves represent RHNC, RLHNC and MC r e s u l t s , r e s p e c t i v e l y . 58 59 F i g u r e 2b The long-range part of the r a d i a l d i s t r i b u t i o n f u n c t i o n . The parameters and curves are as i n f i g u r e 2a. 61 F i g u r e 3a The short-range p a r t of the r a d i a l d i s t r i b u t i o n f u n c t i o n f o r n = 1.5, Q =1.0 and p = 0.8. The s o l i d , dash-dot,dashed and dot t e d curves represent RHNC, RQHNC, RLHNC and MC r e s u l t s , r e s p e c t i v e l y . F i g u r e 3b The long-range part of the r a d i a l d i s t r i b u t i o n f u n c t i o n f o r u = 1.5, Q =1.0 and p = 0.8. The s o l i d , dashed and d o t t e d c u rves represent RHNC, RLHNC and MC r e s u l t s , r e s p e c t i v e l y . 65 F i g u r e 4a The short-range part of the r a d i a l d i s t r i b u t i o n f u n c t i o n f o r * * * u = 1.0, Q =1.0 and p = 0.8. The s o l i d , dash-dot,dashed and dott e d curves represent RHNC, RQHNC, RLHNC and MC r e s u l t s , respect i v e l y . 66 F i g u r e 4b The long-range p a r t of the r a d i a l d i s t r i b u t i o n f u n c t i o n f o r ft * ft u = 1.0, Q =1.0 and p = 0.8. The s o l i d , dashed and dotted curves represent RHNC, RLHNC and MC r e s u l t s , r e s p e c t i v e l y . 68 69 F i g u r e 5 The p r o j e c t i o n h 1 1 ^ ( r ) . The parameters and curves are as i n f i g u r e 2a. 71 F i g u r e 6 The p r o j e c t i o n h ( r ) . The parameters and curves are as i n f i g u r e 3a. 72 73 F i g u r e 7 The p r o j e c t i o n h 1 1 ( ^ ( r ) . The parameters and curves are as i n f i g u r e 4a. 74 I 75 F i g u r e 8 112 The p r o j e c t i o n h ( r ) . The parameters and curves are as i n f i g u r e 2a. 76 77 F i g u r e 9 The p r o j e c t i o n h ( r ) . The parameters and curves are as i n f i g u r e 3a. 79 F i g u r e 10 The p r o j e c t i o n h ( r ) . The parameters and curves are as i n f i g u r e 4a. 81 F i g u r e 11 123 The p r o j e c t i o n h ( r ) . The parameters and curves are as i n f i g u r e 2a 83 F i g u r e 12 The p r o j e c t i o n h ( r ) . The parameters and curves are as i n f i g u r e 3a. 85 F i g u r e 13 The p r o j e c t i o n h 1 2 3 ( r ) . The parameters and curves are as i n f i g u r e 4a. 87 F i g u r e 14 224 The p r o j e c t i o n h ( r ) . The parameters and curves are as i n f i g u r e 2a. 89 F i g u r e 15 224 The p r o j e c t i o n h ( r ) . The parameters and curves are as i n f i g u r e 3a. 91 F i g u r e 16 224 The p r o j e c t i o n h ( r ) . The parameters and curves are as i n f i g u r e 4a. 92 93 3.4 Summary and Co n c l u s i o n s . The RHNC theory has been so l v e d f o r f l u i d s of hard spheres with d i p o l e s and quadrupoles. T h i s was done by expanding the p a i r p o t e n t i a l , the p a i r c o r r e l a t i o n f u n c t i o n and the d i r e c t c o r r e l a t i o n f u n c t i o n i n r o t a t i o n a l i n v a r i a n t s and computing the expansion c o e f f i c i e n t s which s a t i s f y both the OZ equation and the RHNC c l o s u r e approximation. In F o u r i e r space these c o e f f i c i e n t s are r e l a t e d i n a simple a l g e b r a i c way which can e a s i l y be expressed by matrix e q u a t i o n s . Although an e x p l i c i t a n a l y t i c a l s o l u t i o n i s not found, the s o l u t i o n of the system of equations i s achieved by i t e r a t i v e methods. In p r i n c i p l e the expansions are i n f i n i t e , but c l o s e examination shows that convergence of the thermodynamic and d i e l e c t r i c p r o p e r t i e s i s achieved with r e l a t i v e l y few terms. Three s e t s of p h y s i c a l l y r e a l i s t i c parameters were c o n s i d e r e d , and to a c e r t a i n extent the number of terms needed to converge * * the expansions depends on the r e l a t i v e s i z e s of n and Q . When * * M and Q are s i m i l a r i n s i z e 56 independent terms are needed to get adequate v a l u e s f o r e and f o r the thermodynamic q u a n t i t i e s * * co n s i d e r e d . When n i s l a r g e compared to Q only 35 independent terms are r e q u i r e d . The RHNC r e s u l t s f o r thermodynamic q u a n t i t i e s were compared to MC r e s u l t s and data from v a r i o u s other sources. The average c o n f i g u r a t i o n a l energy given by the RHNC approximation was found to be a s i g n i f i c a n t improvement over p r e v i o u s t h e o r i e s . The RHNC d i e l e c t r i c constant i s c o n s i s t e n t l y lower * than that obtained by other methods. When Q i s small r e l a t i v e to n t h i s leads to an improvement over other t h e o r i e s f o r e. * * When Q i s comparable to u i n s i z e , e values are s i m i l a r to other i n t e g r a l equation r e s u l t s . The p r o j e c t i o n s of the p a i r c o r r e l a t i o n f u n c t i o n given by the RHNC theory have a l s o been compared g r a p h i c a l l y with MC data as w e l l as with r e s u l t s from other c l o s u r e s . In a l l cases the 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 improve on pre v i o u s approximations, and i n almost a l l cases the MC s i m u l a t i o n r e s u l t s are c l o s e l y approximated. 95 PART B 96 CHAPTER 4 P o l a r i z a b i l i t y Theory. 4.1 The SCMF Method. The s e l f - c o n s i s t e n t mean f i e l d (SCMF) treatment of molecular p o l a r i z i b i l i t y proposed here i s e s s e n t i a l l y due to Carnie and Patey [25]. I t i s s i m i l a r to previous work [24] i n that i t r e p l a c e s many body p o t e n t i a l s with an e q u i v a l e n t e f f e c t i v e p a i r p o t e n t i a l . U n l i k e other methods, however, the approximations used i n the SCMF theory have a d i s t i n c t l y p h y s i c a l b a s i s . Consider a system of N i d e n t i c a l i n t e r a c t i n g p o l a r - p o l a r i z a b l e p a r t i c l e s . Let Hi. = the permanent d i p o l e moment of p a r t i c l e i , = the l o c a l e l e c t r i c f i e l d f e l t by p a r t i c l e i , a = the p o l a r i z a b i l i t y tensor f o r each of the N p a r t i c l e s , P i = the instantaneous induced d i p o l e moment of p a r t i c l e i , m. l = the instantaneous d i p o l e moment of p a r t i c l e i . We now have P i = g - ( E 1 ) i , (4.1.1) and " * i + p i ' (4.1.2) 97 Now l e t <m> = m' be the average t o t a l m olecular d i p o l e moment (measured with repect to the center of mass f o r each p a r t i c l e ) , and l e t <E^> be the average e l e c t r i c f i e l d f e l t by a p a r t i c l e . Then f o r i d e n t i c a l p a r t i c l e s m' = <m> = <M> + <a«E^> = M + a«<E^>. (4.1.3) If we now r e s t r i c t o u r s e l v e s to a x i a l l y symmetric molecules, <E^> w i l l be non-zero only i n the d i r e c t i o n of M. T h i s i m p l i e s that m' w i l l be i n the d i r e c t i o n of M SO we can wr i t e <E1> = C(m')m', (4.1.4) where C(m') i s a s c a l a r . If we i n s e r t eqn. 4.1.4 i n t o eqn. 4.1.3 we f i n d m' = M + a-C(m')m', (4.1.5) which we can i t e r a t e with i t s e l f to get m' = M + q>C(m')[u + a*C(m')m'] = M + a-C(m')£ + g.'q-C (m')m'. (4.1.6) 98 T h i s can be repeated to get the i n f i n i t e expansion m' = M + g«C(ni')M + g«g«C (m')v + g'g-g'C (m')M + . (4.1.7) If we now d e f i n e a renormalized p o l a r i z a b i l i t y g' by g' = g + C(m')g'-g, (4.1.8) we can r e w r i t e eqn. 4.1.7 as m' = u + C(m')a'-M, (4.1.9) which has e x a c t l y the same form as eqn. 4.1.5, but gi v e s the t o t a l average molecular d i p o l e moment m' as an equation i n terms of the permanent d i p o l e moment M and the new p o l a r i z a b i l i t y g'. T h i s means that a f l u i d surrounded molecule with a p o l a r i z a b i l i t y g can be t r e a t e d i n e x a c t l y the same way as an i s o l a t e d molecule with a r e n o r m a l i z e d p o l a r i z a b i l i t y g'. g' i s r e l a t e d to g by eqn. 4.1.8. We can now c h a r a c t e r i z e our e f f e c t i v e system as one of i s o l a t e d molecules, each with a permanent average d i p o l e £ and a p o l a r i z a b i l i t y g'. These e f f e c t i v e system molecules are i s o l a t e d only i n the sense that they do not a f f e c t each others p o l a r i z a t i o n p r o p e r t i e s ; they s t i l l have i n t e r a c t i o n energies.which can be e a s i l y c a l c u l a t e d . 99 4.2 P o l a r i z a b l e Molecules with D i p o l e s and Quadrupoles. If we make the p a r t i c l e s of Chapter 2 p o l a r i z a b l e i t i s p o s s i b l e to d e r i v e the average c o n f i g u r a t i o n a l energy and p a i r p o t e n t i a l of the system by c o n s i d e r i n g an e q u i v a l e n t e f f e c t i v e system which i m p l i c i t l y c o n t a i n s the p o l a r i z a t i o n ef f e c t s . The t o t a l instantaneous c o n f i g u r a t i o n a l energy w i l l be of the form U = UHS + UDD + UDQ + UQQ + V U ' 2 ' 1 ) where the e x t r a term U p i s the energy of p o l a r i z a t i o n . We can w r i t e [38] U" - UHS " 1\ m i ' { E l B ] i - 2 [ 1 f T ^ l Q * ! 3 + UQQ + 1 I P i - ( E 1 ) i , (4.2.2) i where = E ^ D + E^Q and E i r j ' E i Q a r e fc^e l o c a l instantaneous f i e l d s due to d i p o l e s and quadrupoles, r e s p e c t i v e l y . The f a c t o r s of ^ i n the 2 n d and 3 r d terms on the r i g h t of eqn. 4.2.2 are to a v o i d counting p a i r s of molecules twice. S i m p l i f y i n g we get 100 U " UHS + U Q Q " 1 Z ^ i - ( K l D > i " Z &i'{ElQh l l - 1 2 P i - ( E 1 Q ) . . (4.2.3) For a x i a l l y symmetric i d e n t i c a l p a r t i c l e s <U> = < U H S > + < U Q Q > 4NM.<E 1 D> - NM-<E 1 q> - l N < P I . E 1 Q > , (4.2.4) where < E m > ' < E I Q > a r e fc^e average d i p o l a r and quadrupolar f i e l d s , r e s p e c t i v e l y , and < E ^ D > + < E ^ Q > = <E^>. As i n eqn. 4.1.4 we can write < E 1 D > = CD(m')m', (4.2.5a) < E 1 Q > = CQ(m')m', (4.2.5b) and < E 1 > = < E 1 D > + < E 1 Q > = C D(m')m' + C Q(m')m' = C(m')m'. (4.2.6) The l a s t term i n eqn. 4.2.4 i s approximated ( i n the SCMF theory) by i g n o r i n g f l u c t u a t i o n s i n the f i e l d such that 101 < P i.E 1 Q> - < P i>'<E 1 Q> = (m' - M ) , < E I Q > - (4.2.7) Now combining eqns. 4.2.4-4.2.7 we obta i n <U> = <U H S> + <U Q Q> - ^ NM-m'CD(m') - ^ N(£ + m').m'CQ(m') = < U H S > + < U Q Q > - ^ NMm'CD(m') - ^ N(M + m')m'CQ(m'), (4.2.8) where m'= |m'| and n = |M|. In order to eva l u a t e C D(m') we use < U D D > e = < _ I = ^ i - ( E l Q ) i > = -l N(m'.<E l D>) = -^Nm' 2C D(m') f (4.2.9a) where the s u b s c r i p t e i n d i c a t e s the e f f e c t i v e system. S i m i l a r l y we have < unn>~ = -Nm' 2C n(m'), (4.2.9b) DQ e Q and eqn. 4.2.8 becomes <U> - <U H S> + <U Q Q> + ^ < U D D > e + f^<U D Q> e. (4.2.10) 1 02 In order to d e r i v e the p a i r p o t e n t i a l f o r the e f f e c t i v e system we note that the c o n f i g u r a t i o n a l energy can be w r i t t e n i n the form 2 2 2 I Pix P i y P i z i a a a X X y y zz i < 3 - . . . . (4.2.11) where Pi x»Pjy a n o " P±z a r e t n e X J a n d z components of and a x x ' a y y a n d a z z a r e t * i e t ^ i r e e d i a g o n a l components of a. In SCMF theory we ignore f l u c t u a t i o n s i n , so we have 2 2 p. = <p.> = constant, (4.2.12) X X 1 2 2 with s i m i l a r e xpressions f o r p^ and P^ z- These constant terms do not c o n t r i b u t e to the p r o b a b a b i l i t y of f i n d i n g a given c o n f i g u r a t i o n , and so they can be ignored. As i n Chapter 2, u ' ( i , j ) i s of the form of eqn. 2.3.8 with c o e f f i c i e n t s analogous to eqn. 2.3.10 1 1 ? m i m - i u1 1 ^ ( r ) = L _ l r (4.2.13a) u ^ ( r ) = -4, (4.2.13b) 2r* u2 1 i ( r ) = 4- (4.2.13c) 2r* Again, we approximate by i g n o r i n g f l u c t u a t i o n s in to get 2 2 m.m.=<m>=m, (4.2.14) l ] e' 1 03 where i s the t o t a l d i p o l e of the e f f e c t i v e system given by 2 2 , 2 ^ , 2 2 N mg = <m > = m + (<p > - <p> ) = m'2 + 3a'kT, (4.2.15) with a' being t r g ' . The d i f f e r e n c e between m and m' turns out to be small e compared to other approximations made so we make a f i n a l approximation by s e t t i n g m Q = m'Q. (4.2.16) e Then eqns. 4.2.13 and 4.2.9 become m2 u 1 1 2 ( r ) = - - f , (4.2.17a) r J u 1 2 3 ( r ) = - u 2 1 3 ( r ) = (4.2.17b) 2rq 2<U n n> C D(m') = Y ~ ^ - , (4.2.18a) m e < u n n > C^(m') = (4.2.18b) Nm m e 'Q and f i n a l l y <U> = <U H S> + <U > + iBf <U D D> e + ^ < u > (4.2.! 9) m e e 4.3 Method'of S o l u t i o n . The system of Part A i s sol v e d f o r a number of mg v a l u e s which cover the range w i t h i n which the a c t u a l mg must f a l l . T h i s data i s then f i t to an i n t e r p o l a t i n g curve using c u b i c s p l i n e s [39] and a f u n c t i o n r e l a t i n g mg to C(m') i s found. Then using given values f o r M,Q and a, an i t e r a t i v e process using eqns. 4.1.5, 4.1.8 and 4.2.15 i s used to f i n d values of m',g' and m c o n s i s t e n t with the given data. 105 CHAPTER 5 R e s u l t s and D i s c u s s i o n . 5.1 C a l c u l a t i o n s with Ammonia-like Parameters. T h i s chapter uses r e s u l t s of measurements f o r ammonia as data f o r the SCMF p o l a r i z a b i l i t y theory. The SCMF c a l c u l a t i o n s were c a r r i e d out at three s u b - c r i t i c a l temperatures, and the l i q u i d d e n s i t i e s used are at l i q u i d - v a p o u r e q u i l i b r i u m . The values a r e : ( i ) T=-35°C, p=0.6840gcm~ 3, ( i i ) T=-5°C, p=0.6315gcm~ 3, ( i i i ) T=35°C, p=0.5875gcm~ 3. The p o l a r i z a b i l i t y tensor g used i s due to Bridge and Buckingham [23], The value of the d i p o l e moment used i s 1.47D which agrees w e l l with a l a r g e number of experiments [21]. The values which should be used f o r p a r t i c l e diameter and quadrupole moment are not as w e l l e s t a b l i s h e d as that f o r d i p o l e moment, and these are d i s c u s s e d i n s e c t i o n s 5.2 and 5.3. 106 5.2 S e n s i t i v i t y to P a r t i c l e Diameter and B a s i s Set. The p a r t i c l e diameter has been chosen to be in agreement with neutron s c a t t e r i n g data by Narten [40]. Such r e s u l t s leave some room for i n t e r p r e t a t i o n when t r y i n g to f i x a hard core diameter. The diameter value chosen should be somewhere between 3.1A and 3.3A with d=3.2A being a reasonable c h o i c e . A l l r e s u l t s shown in f i g u r e 17 were c a l c u l a t e d with d=3.2A, which a l s o agrees with the N-N r a d i a l d i s t r i b u t i o n f u n c t i o n f o r l i q u i d ammonia found by McDonald and K l e i n [41] using Molecular Dynamics c a l c u l a t i o n s . Some r e s u l t s were c a l c u l a t e d f o r d=3.lA, which i n c r e a s e d the value f o r the d i e l e c t r i c constant by no more than 4% at any temperature. The dependence on p a r t i c l e diameter was a l s o found to be small f o r SCMF c a l c u l a t i o n s done on water [25], A l l c a l c u l a t i o n s shown in f i g . 17 were done with b a s i s set II of Part A. A r e p r e s e n t a t i v e c a l c u l a t i o n was done with b a s i s set III which showed a 1.0% e r r o r i n -/3<UDD>/N and a 2.5% e r r o r in -|3<UDQ>/N which l e d to a 2.5% increase i n e. B a s i s set IV, as shown in chapter 3, g i v e s e s s e n t i a l l y i d e n t i c a l r e s u l t s to those from b a s i s set III f o r the energy components and e. The e r r o r s i ntroduced by p a r t i c l e diameter and b a s i s set are i n s i g n i f i c a n t when compared to the p o s s i b l e e r r o r introduced by the u n c e r t a i n t y i n the quadrupole moment. T h i s i s examined below. F i g u r e 17 D i e l e c t r i c constant vs. Temperature. Closed c i r c l e s denote experimental data, t r i a n g l e s denote SCMF c a l c u l a t i o n s with — 26 2 Q=-3.3 x 10 esu cm , squares denote SCMF c a l c u l a t i o n s wi — 2 6 2 Q=-2.32 x 10 esu cm and open c i r c l e s i n d i c a t e data c a l c u l a t e d from the e q u i v a l e n t n o n - p o l a r i z a b l e system. 108 40 r 30 h 20 U 10 0 l — I 1 1 — -40 -20 J l I L_ L 0 T(°C) 20 40 109 5.3 Comparison With Experimental Data. The experimental data p o i n t s shown on f i g . 17 do not appear to f i t a smooth curve, p r o b a b l y because they are from d i f f e r e n t sources. P o i n t s below T=0°C are from a recent paper by Baldwin and G i l l [42], but those above T=0°C are taken from the CRC Handbook [43], a l l of which are o r i g i n a l l y from d i f f e r e n t sources. SCMF c a l c u l a t i o n s were performed using a quadrupole moment of Q=(-3.3±0.4) x 10 esu cm measured e x p e r i m e n t a l l y by Kuko l i c h [22b] with r e s u l t s shown i n f i g . 17. The shape of a curve through these p o i n t s would be very s i m i l a r to that of experiment while c o n s i s t e n t l y u n d e r e s t i m a t i n g e by about 30%. While t h i s i s a f a i r l y l a r g e d i s c r e p a n c y i t should be noted that there i s a l a r g e u n c e r t a i n t y i n the quadrupole moment as given by K u k o l i c h . I t i s necessary t o f i n d out whether the SCMF e values are s e n s i t i v e enough to quadrupole that a Q value w i t h i n these bounds of u n c e r t a i n t y c o u l d e x p l a i n the d i s c r e p a n c y . To t h i s end we have a l s o p l o t t e d on f i g . 17 a set of p o i n t s with — 2 6 2 Q=-2.32 x 10 esu cm . T h i s r e s u l t , a l s o given by K u k o l i c h [22a], was c o r r e c t e d i n an erratum [22b] to the value given above. T h i s erroneous value has been used q u i t e e x t e n s i v e l y i n the l i t e r a t u r e [8,44,45] and al t h o u g h i t i s i n c o r r e c t , i n c l u d i n g these p o i n t s i n f i g . 17 g i v e s a good i n d i c a t i o n of quadrupole s e n s i t i v i t y . As can be seen from f i g . 17, the change i n Q has a lar g e e f f e c t on the SCMF values f o r e. The theory now overestimates e, and the o v e r e s t i m a t i o n i s very l a r g e at low temperature. I t i s t h e r e f o r e p r o b a b l e that a Q value with an 1 10 ab s o l u t e value lower than — 2 6 2 3.3 x 10 esu cm , but not o u t s i d e the given bounds of u n c e r t a i n t y , c o u l d give SCMF value s for e that would agree w e l l with experiments at a l l temperatures. C a l c u l a t i o n s u s i n g the values of T,P,M,Q and d used i n the p o l a r i z a b l e model were completed f o r the equivalent, n o n - p o l a r i z a b l e model. I t i s c l e a r from the r e s u l t s ( f i g . 17) that a n o n - p o l a r i z a b l e model does not c o r r e c t l y p r e d i c t the d i e l e c t r i c constant of 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 were c a r r i e d out to f i n d the d i e l e c t r i c constant of ammonia at three temperatures along the l i q u i d - v a p o u r c o e x i s t e n c e l i n e of the phase diagram of ammonia. The r e s u l t s were found to be q u i t e s e n s i t i v e to the value of the quadrupole moment. T h i s s e n s i t i v i t y was p r e d i c t e d by Rushbrooke [46] and agrees with e a r l i e r MC work [47]. When a reasonably good value f o r Q i s used the c a l c u l a t e d e values f o l l o w the experimental data q u i t e w e l l , and any observed d i s c r e p a n c i e s are w i t h i n the bounds of e r r o r set by the quadrupole moment. 111 LIST OF REFERENCES 1 . J.A. Barker, L a t t i c e T h e o r i e s of the L i q u i d S t a t e , Topic 1 0 , Volume 1 of 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 of  P h y s i c a l Chemistry and Chemical P h y s i c s , Pergamon Press, Oxford, 1 9 6 3 . 2 . J.A. Barker, D. Henderson, Rev. Mod. Phys., 4 8 , 5 8 7 ( 1 9 7 6 ) . 3 . H.D. U r s e l l , Proc. Cambridge P h i l . Soc. , 2 3 , 6 8 5 ( 1 9 2 7 ) . 4 . J . Yvon, La Theor i e S t a t i s t i q u e des F l u i d e s et 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 Mechanics, John Wiley, New York, 1 9 4 0 . 6 . F. 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Walker, Ph.D. T h e s i s , A u s t r a l i a n N a t i o n a l U n i v e r s i t y , Canberra, 1983. 30. S t a t e s of Matter, Volume 2 of A T r e a t i s e on P h y s i c a l  Chemistry, 3rd. ed., e d i t e d by H.S. T a y l o r and S. Glasstone, D. Van Nostrand, New York, 1951. 31. W.A. S t e e l e , J . Chem. Phys., 39, 3197 (1963). 32. A. Messiah, Quantum Mechanics, V o l . I I , John Wiley and Sons, New York, 1958. 33. H. Workman, M. Fixman, J . Chem. Phys., 58, 5024 (1973). 34. G.S. Rushbrooke, H.I. Sc o i n s , Proc. Roy. Soc. A, 216, 203 (1953). 35. C.A. Croxton, L i q u i d S tate P h y s i c s , Cambridge U n i v e r s i t y P ress, London, 1974. 36. Handbook of Mathematical F u n c t i o n s , e d i t e d by M. Abramowitz and I.A. Stegun, Dover, New York, 1970. 37. L. V e r l e t , J . J . Weis, Phys. Rev. A, 5, 939 (1972). 38. C.J.F. Bottcher, Theory of E l e c t r i c P o l a r i z a t i o n , 2nd. ed., E l s e v i e r S c i e n t i f i c , Amsterdam, 1973. 39. C.F. 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Phys., _4_3, 1115 (1981). 1 1 4 APPENDIX A I d e n t i t i e s Used i n Subsequent Appendices. Some of the f o l l o w i n g i d e n t i t i e s are a p p l i e d more than once i n Appendices B-H. These i d e n t i t i e s are given i n the form needed f o r t h e i r f i r s t a p p l i c a t i o n . A.1 R m n ( f l 1 ) = I R™ ( f i , ) R m (S2J [48] MO 1 p pu 3 MP 5 A.2 Z R m ( f i j R m ( f l R ) R l ( f i . H m n J ) = (!!!" I ) [48] ^ MP 5 vq 5 Xs 5 M V X p q s A.3 R i o ( e ) = ( _ ) L R X 0 ( _ F ) [ 3 2 ] A.4 Under odd permutation of columns , m n 1 % _ , \irt+n+i, n m 1 \ r ^ 9 l K M v X ; ' V ;  K v u \ '  1 1 A.5 I R ^ ^ i 3 ) R r p ( e i 3 ) = 61,1' 6XX' [ 3 2 ] 00 A.6 / dk k 2 j 1 ( k r ) j 1 ( k s ) = ^ f l M r - s ) [17] A.7 ^ JdO 3R;; 0(O3)R^ 0(O3) - 6_„ i ^ 6 n [48] A.8 ^ ;dk R i ; 0 ( k ) R ^ 0 ( k ) R i ; o u ) ~ ( ' 1 0 0 0 M - x , - x 2 x ' ; [48] 1 15 A.9 (21+1) Z ( m n i ) ( m n J ) = 1 [32] jU V X u V X v fX A in ( I2I1I i _ y ( m n , l , w n,n 1 2 W l i l 2 l x ' 1 m n n, ] \ \\ M "1X1 *, v X 2 M X,X 2X ' M f 1 f x ( m n 1 j (_ )m+n+n 1+X 1+X 2 + y 1 + l , + l 2 + l+M +J' +X ^ 4 8 ] A 11 ( I2I1I 1/ 1112I \ _ v /_\m+n+n,+x# n,n l 2 x A > 1 1 1 m n n , M 0 0 0 ; "  L ( '  ( x~X 0 ' ( m n 1 l 1 ) ( m n l ) f , X 1 x-x 0 M x-X 0 ' L 4 8 J A. 1 2 I ( X m n 1 \, m n l ' x x-x 0 M x-x 0 ; ° i r (21+1) [48] A. 1 3 • 5B ( 2 M + , ) ( < S'S2S > R W A.14 2 ( m i n i l i ) ( m2"2l2)( mTiTi2m x / n,n 2n x Mii'iX, ^ l ^ l ^ l M2^2X2 M!M2"M h ^ " " M2^2^2 „ / l i l 2 l i / »m+n+l, m n 1 x r m ' n ' ? - i r _ o l x ( X \ -X ) = ( _ ) ( „ v \ ^ m 2 n 2 l 2 } [32] X,X 2 X m n l 1 1 6 A.15 / d^cMg $ m i n i l l ( l 2 ) # m 2 n 2 l 2 ( 1 2 ) " (2m1+1)(2n,+1)(21,+1) v ' i f (m,,n,,1,) = ( m 2 , n 2 , 1 2 ) , = 0 otherwise. A l l symbols used are as d e f i n e d in pr e v i o u s chapters, with 6 i n A.6 being D i r a c ' s 6-function and 6 i n A.5, A.7 and A. 12 being the kronecker 6. 1 1 7 APPENDIX B P r o o f s 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 of ( i i ) : I n variance Under R o t a t i o n . For s i m p l i c i t y we prove only f o r the case at hand, i . e . n ' = v ' = 0 . A general proof i s o u t l i n e d by Blum [15]. For mathematical ease we l e t f m n l = i f o r a l l m,n,l with no l o s s of g e n e r a l i t y . Let u ( 1 2 ) . S u m l < r ) ( * 1 \ ( R ^ X o ' V " ^ ' ' (B.I) mnl n v \ where 0 l f 8 2 , r a r e r e l a t i v e to a frame of r e f e r e n c e S. R e l a t i v e to a frame of r e f e r e n c e T, obtained by r o t a t i n g S through P*5, U(12) i s given by u ( 1 2 ) . Z u m n l ( r ) ( « J 1 ( R ^ l R y n X o U , ) . (B.2) mnl The r o t a t i o n $2 i s a r e s u l t of 0^ f o l l o w e d by Si^. S i m i l a r l y °>2 and f are the r e s u l t s of flg followed by Si^ and r^ , r e s p e c t i v e l y . If we use A.1 f o r Si^ ,J22 and r i n B.1, we o b t a i n U(12) - Z u » n l ( r ) ( I J \ i R ^ ^ l R ^ f ^ l R ^ l r , ) mnl * M nv X pqs i X R M P { v r V v R A V V - ( B - 3 ) S u b s t i t u t i n g A.2 i n B.3 we get 1 18 u<12>- ^ u m n l ( r ) ( I I I J H j o ^ j R ^ C O ^ R ^ t e p , (B.4) pqs which i s o b v i o u s l y the same as B.2. Proof of ( i i i ) : I n v a r i a n c e Under Interchange of P a r t i c l e s 1 and 2. Again we t r e a t the n'=v'=0 problem. If we s u b s t i t u t e A.3 i n B.1 we o b t a i n u(12)= Z u m n l ( r ) ( m " { ) R n n ( n , ) R m n ( f i . ) ( - ) 1 R j n ( - r ) . (B.5) M N ^ M v X vO 2 nO 1 XO MfX Next we use A.4 i n B.5 to get U(12)= Z u m n l ( r ) ( n m \ ) R n n ( n 9 ) R m n ( f i 1 ) ( - ) m + n R l O r ) . (B.6) m n ^ v M X vO 2 M0 1 XO vtx\ I t i s now c l e a r that B.6 i s e x a c t l y the e x p r e s s i o n , analogous to B.I, f o r u(21), i f and only i f u n m l ( r ) = ( - ) m + n u m n l ( r ) . 1 1 9 APPENDIX C C a l c u l a t i o n of u ( 1 2 ) i n Terms of M u l t i p o l e I n t e r a c t i o n s . The general form of the m u l t i p o l e expansion of the e l e c t r o s t a t i c i n t e r a c t i o n of two non-overlapping charge d i s t r i b u t i o n s f o r two a x i a l l y symmetric p a r t i c l e s i s [ 4 9 ] , , U 1 | . _L Q ° ( l ) Q ° ( 2 ) $ m n l ( l 2 ) » < ' * > • - E ' ' > " < ( 2 2 n > ! > ' " . S n l 1 „ < C > mn f r ^ mnl/ \»mnl,,_ x = Z u ( r ) $ ( 1 2 ) , mn where l=m+n and Q m ( 1 ) i s the a x i a l 2 m - p o l e moment f o r p a r t i c l e 1 . For our purposes, QQ = the net charge = 0 , Q^= ju and Q 2= Q. If the expansion i s t r u n c a t e d at m,n<2, d i r e c t c a l c u l a t i o n g i v e s u ( l 2 ) = - ( 3 0 ) ' M 2 * 1 1 2 ( 1 2 ) + (lOSPQM [ $ 1 2 3 ( 1 2 ) - $ 2 1 3 ( 1 2 ) ] + 3 i 7 0 l l f i ! # 2 2 4 ( 1 2 K { c > 2 ) 1 20 APPENDIX D D e r i v a t i o n of Equations 2.4.12 to 2.4.16. We begin with the d e f i n i t i o n of c(13) C(13) = / d r l 3 c ( 1 3 ) e x p ( i k « r 1 3 ) , (D.1) and d e r i v e eqns. 2.4.12, and 2.4.14 to 2.4.16 f o r the c m n l ( k ) fo r 1 even. P r o o f s f o r 1 odd and 7 7 m n l ( k ) are s i m i l a r and are o u t l i n e d i n Blum [15,16]. Eqn. 2.4.13 i s a d e f i n i n g r e l a t i o n s h i p . Using the Ray l e i g h expansion [15] e x p ( i k - r ) = Z l j ]_,(kr) L R^,Q(k)R^, Q{r) , (D.2) 1 = 0 X = — 1 we obtain c(13) - f d r , - I i l ' c m i n i l l ( r 1 - ) f i n i n i l l ( m i n i ^ ) 1 3 m,n,li M l ' X ' x R ^ o ( n i ) R " ; o ( f i 2 ) R i ; o ( ^ 3 ) R x ; o ( ^ ) R x : o ( f i 3 ) ^ i ( k r i 3 ) - ( D ' 3 ) S u b s t i t u t i o n of eqn. A.5 g i v e s 121 c(13) « J d r 1 3 I i 1 > n , 1 M r 1 3 ) f n , , n i l , ( M 1 " 1 X M MiViX, x R m i n ( J 2 . ) R n i n ( n 9 ) R ^ n ( k ) j , (kr..) (D.4) M, 0 1 J'IO 2 X, 0 J 1 1 3 L [4, J d r 1 3 r V ' c ^ ^ M r ^ j ( k r l 3 ) ] 01,11,1, 0 1 x ^ ' " ^ M f l ^ B ^ k ) , (D.5) from which, with 2.4.11a, we have CO c m n l ( k ) = 47ri 1 ; dr r 2 c m n l j 1 (kr) . (2.4.16a) 0 1 In order to o b t a i n 2.4.12 l e t C m n l ( k ) be the 1 T H order Hankel transform of c m n ^ ( r ) C m n l ( k ) = 47ii 1 ; dr r 2 j , ( k r ) c m n l ( r ) , (D.6) 0 1 with i t s i n v e r s e transform c m n l ( r ) = H i / dk k 2 j , ( k r ) C m n l ( k ) . (D.7) 2TT 0 1 A l s o l e t ^ n l ( r ) = 1 7 dp p 2 j n ( p r ) C m n l ( p ) , ( D > 8 ) 2TT 0 U C m n l ( p ) = 4TT / dr r 2 j n ( p r ) c m n l ( r ) . ( D . 9 ) 0 U Then using D.7 i n 2.4.16 we get 1 22 -mnl,, v 2 °1 , 2. , , v , , , _2. ., \„mnl c m u x ( k ) = ± / dr r * j , ( k r ) ( J dp p ' j , <pr ) C u , U A ( p ) ) . (D.10) w 0 0 Changing the order of i n t e g r a t i o n y i e l d s 00 00 c m n l ( k ) = Z J dp p 2 C m n l ( p ) ( J dr r 2 j 1 ( k r ) j 1 ( p r ) ) , (D.11) * 0 0 for which we can i n t e g r a t e over r o b t a i n i n g 00 c m n l ( k ) = ^  ; dp p 2 c m n l ( p ) ( ^ - 6 ( k - P ) ) 7T Q ZK.p ^ ( | ) C m n l ( k ) . (D.12) 7T Z. F i n a l l y we i n s e r t D.9, which g i v e s oo c m n l ( k ) = 4TT J dr r 2 j n ( k r ) c m n l ( r ) . (2.4.12a) 0 u In order to o b t a i n 2.4.14 we use D.6 i n D.8 and s i m p l i f y , y i e l d i n g co oo c m n l ( r ) = / dk k 2 j n ( k r ) (47ri 1) / ds s 2 j , (ks) c m n l (s) (D. 1 3) 2TT 0 u 0 1 CO Co = ; ds s 2 c m n l ( s ) ( | ) i 1 / dk k 2 j n ( k r ) j , ( k s ) (D.14) 0 0 CO = / ds s 2 c m n l ( s ) 0 e ( s , r ) , (D.15) which d e f i n e s the t r a n s f o r m a t i o n k e r n e l , © 1 ( s , r ) , f o r 1 even. We s t a t e here that 123 0 ^ ( s , r ) = 2 — " 3 » (D.16) where 8(s-r) i s the D i r a c d e l t a f u n c t i o n , P-^(x) i s given by 2.4.l5a-c and 9(s-r) i s a step f u n c t i o n d e f i n e d by 1 s>r, O(s-r) = { (D.17) 0 s<r. We r e f e r to Blum [17] f o r the proof of D.16. Now, using D.16 in D.15 we have mnl ^ mnl °" 0 ( S T ) P* ) a"™" 1 ( S ) c m n l ( r ) = / ds 6 ( s - r ) c m n l ( s ) - j ds i - S ( D.18) 0 0 s , °° mnl, x = cm n l ( r ) - / C s ( s ) P ^ ( | ) d s . (2.4.14a) 124 APPENDIX E Angular I n t e g r a t i o n and S i m p l i f i c a t i o n of the OZ Equation. We begin by w r i t i n g 2.4.10 i n terms of the f o l l o w i n g expansions. - n l ( k ) £ m „ l £ . n 1 ) R » ( ) R,, { ) B 1 ( I ) mnl P = J d O . [ Z c 8 , n , l , ( k ) f B , i n ' 1 1 Z ( m i n i J 1 ) R m i n ( n 1 ) 8TT2 3 m i n i l i M i ^ X , M l " l X l M 1 ° 1 x R n i n ( f i _ ) R ^ 1 n ( k ) ] [ Z ( ^ 2 n 2 l 2 ( k ) _ ~ m 2 n 2 l 2 ( k ) ) f m 2 n 2 l 2 " 1 ° 3 X l ° m 2 n 2 l 2 x Z ( m 2 n 2 l 2 ) R m 2 (Q ) R n 2 (o ) R12 ( J ) ] (E.1) „ „ \ M 2^ 2X 2 M20 3 v20 2 X 20 M2 V 2A2 A p p l y i n g A.7 we ob t a i n - n l ( k ) £ m n l ( • „ 1 ) H J < 0 ) R n ,„ ) R 1 ( J , mnl MfX = p[ Z c m i n i l l ( k ) ( ^ m 2 n 2 l 2 ( k ) - c m 2 n 2 l 2 ( k ) ) ( 2 n 1 + l ) " 1 m i n , l , n 2 l 2 x Z ( _ ) ^ f m i n 1 l 1 f n 1 n 2 l 2 ( m 1 n 1 l l ) ( n ,n2l2 )Rm, ( } „ „ % Mi V1X1 ^ i V 2 X 2 M1O 1 M 1 V 1 A 1 J> 2 X 2 * 2 x R n 2 n ( n , ) R ^ 1 n ( k ) R ^ 2 n ( k ) , (E.2) v 2V 2 X, 0 X 20 where -v, has been rep l a c e d by v y which leaves the expression unchanged-. M u l t i p l y i n g both s i d e s of E.2 by 125 ? , R M : 0 ( i 2 1 ) R p ; 0 ( n 2 } ' n v and using A.5 we get ~ ~mnl /, x,mnl ^ / m n 1 »_1 , ? s I 71 ( k ) f x M " x ) R x o ( k ) - p I ( - ) , ' l ( 2 n l + 1 ) " 1 c m n i l l ( k ) ( ? J n ' n l 2 ( k ) - c n i n l M k ) ) X,X 212 f m n 1 l l f n 1 n l 2 ( m n 1 l 1 ) ( n,n 1 2 ) 1, ( k ) R l 2 ( J ) . ( E . 3 ) (I ^ A, t> •, 1^ A 2 X,0 X , 0 1 ' * " We now m u l t i p l y both s i d e s of E.3 by Z R., n(k) and l' X ' X 0 i n t e g r a t e both s i d e s over dk using A.7 and A.8 with the r e s u l t Z ? ? m n l ( k ) f m n l ( m n J ) (21+1 ) " 1 X M " X = p Z ( - ) X l + X 2 + y M 2 n 1 + l ) - 1 c m n i l l ( k ) ( ^ n i n l 2 ( k ) - c n i n l 2 ( k ) ) n 1 v , 1, X 1 X 2 1 2 X x f m n 1 l 1 f n 1 n l 2 ( m n , l 1 ) ( n,n 1 2 ) ( 1,1,1 ) ( 1, 1 2 1 > ( E > 4 ) fl V 1 A ) I* , y A 2 U U U A , A 2 A. which i s tr u e f o r each p o s s i b l e value of m,n,l,ju and v. Now, i f m n 1 we m u l t i p l y both s i d e s by ( . ) ( 2 1 + 1 ) and sum both s i d e s c * J fJL V X over a l l p o s s i b l e M and v, and use A.9, we get 126 ~mnl/, N £mnl T? (k) f a r (21+1) f m n 1 l l f n 1 n l , p n L . 2n, + 1 £ f X,X 212 M v X n i l i w n,n l 2 w l i l 2 l w I 1 I 2 I w m n 1 ^ ' f,X, vyv \ 2 0 0 0  , K X,X 2X JLX v X x c m n i l M k ) ( ? j n i n l 2 ( k ) - c n i n l 2 ( k ) ) , (E.5) where we have r e p l a c e d ( * 1 } 2 \ ) by ( ^ 1 ^ 2 ^ ) which leaves the _ A 1 _ A 2 A A] A 2 A expre s s i o n unchanged. L a s t l y we use A. 10, 1 1+1 2 + 1 = 0 and UL+U+\=0 to get * m n l ( k ) = 0 L ( - ) ^ n + n 1 ( 2 l + D £«nn , 1, f n , n l a 1 ± n [ k ) P n T 1 (2n,+1) fmnl 1 m n n, ] n 1 J-1 J- 2 r x ( 1,1,1 ) ^ n 1 l l ( k ) ( - n l n l 2 ( k ) _ ~ n l n l 2 ( k ) K (2.4.17) 127 APPENDIX F Chi Transformation of the OZ Equ a t i o n . A p p l y i n g A.11 to 2.4.20 g i v e s Zm n n, " ( 2 1 + 1 ) * { x~X 0 M X"X 0 , ( X"X 0 H ) ' ( F * 1 ) Then, using F.1 i n 2.4.18a, we have ^ n l ( k ) = P z ( 2 i + D ( x : x J * x ™ _ X 1 J 1 > ( x _ x J ) ( - ) x  n 11112 X x c m n i l l ( k ) ( ? ? n i n l 2 ( k ) - c n i n l 2 ( k ) ) . (F.2) M u l t i p l y i n g both s i d e s by 1 2 1 + 1» J ( x-x I >• A. and summing both s i d e s over 1 from |m-n| to m+n y i e l d s L ( 2 1 + 1 M m " J ) 7 ? m n l ( k ) = p 2 ( 2 1 + 1 ) 2 ( J 2)( m 1 x x x u n 1 l 1 l 2 x x u x x u 1 x x ( - ) x [ ( ™ ? i ) ( v-v J ) ] ? m n i l l ( k ) ( ? J n ^ n l 2 ( k ) - c n , n l 2 ( k ) ) , (F.3) A A *^ A A which can be rearranged by using A.9 to o b t a i n 1 28 z ( m n 1 ) ^ n n l ( k ) = p L ( _ ) X ( n,n 1 2 ) 1 X X u n , l , l 2 X X u x ( ? ? n i n l 2 ( k ) - c n i n l 2 ( k ) ) ( m _ " 1 J 1 ) c m n i l l ( k ) . (F.4) A A Using the d e f i n i t i o n 2.4.21 of x~transforms we get N m n ( k ) = p I ( - ) X C m n i ( k ) [ N n i n ( k ) - C n i n ( k ) ] . (2.4.22) A —. X A* A* 1 29 APPENDIX G Back Tran s f o r m a t i o n s . In order to get 2.4.24 we m u l t i p l y both s i d e s of 2.4.21b by ( m n ~ ) and sum over x to o b t a i n x-x 0 _ rTmn / , \ , m n l \ , m n l ' \ ~mn 1 •, \ / ^  \ z N x (k) = z ( x _ x 0 )( x _ x Q )r, ( k ) , (G.1) X x, and we apply A.12 y i e l d i n g (21+1) Z ( ™ " I )N^ n(k) = ^ m n l ( k ) , (2.4.24) ^ A A " A where x must range between ±min(m,n). In order to get 2.4.25a we m u l t i p l y both s i d e s of 0 2 2.4.12a by p j n ( p r ) and i n t e g r a t e over p which g i v e s J dp p 2 j n ( p s ) c m n l ( p ) = 4TT / ds s 2 j n ( p s ) c m n l ( s j dp p 2 j n ( p r ) . ( G . 2 ) 0 U 0 U 0 U Changing the order of i n t e g r a t i o n , w r i t i n g i n terms of r\ and using A.6 y i e l d s 00 1 ; dp p 2 j n ( p s ) ? j m n l ( p ) = r j m n l ( s ) . (2.4.25a) 2?r2 0 The case f o r 1 odd i s s i m i l a r . In order to get 2.4.26a we d e f i n e the equations . 1 30 r , m n l ( r ) = ^ / dk k2jAkr)Nmnl(k), (G.3) 2TT 0 1 Nmnl(k) = 4TT S ds s 2 j n ( k s ) r j m n l ( s ) , (G.4) 0 U which are analogous to eqns. D.7 and D.9. As before (D.13-D.15) we i n s e r t G.4 i n G.3, rearrange and use the d e f i n i t i o n of e 1 e ? ( r , s ) to get r , m n l ( r ) = / ds s 2 r 7 r n n l ( s ) 0 e ( r , s ) (G.5) 0 1 7 He - 2 r 6 ( r - s ) li[2!!i!ll^n.nl, , = / ds s L j— " 3" J7? ( s' O r r = f,mnl{r) - 1 J d s s 2 P e ( f ) ^ m n l ( s ) . (2.4.26a) r J 0 1 r 131 APPENDIX H Binary Products of R o t a t i o n a l I n v a r i a n t s . We begin by expanding the product of two r o t a t i o n a l i n v a r i a n t s to get B = ^ i " ' 1 ! ( I 2 ) $ m 2 n 2 l 2 d 2 ) _ jit^n , 1, j m 2 n 2 l 2 Z ( m i n i l i ) ( m 2 n 2 l 2 ) R m i ( R } „ i> \ V 2 V 2 ^ 2 MlO 1 M 1 f 1 A 1 H2V2\2 x R " ' n ( n j R t ' ( r ) R m 2 n ( f i j R " 2 n ( f l J ^ 2 n { r ) . V, 0 M2 0 1 ^ 2 0 x 2 o (H. 1 ) Using A. 13 f o r S?1 , S22 and r we have B = fniin , 1, f m 2 n 2 l 2 ^ ^ m , n , l , ^ m 2 n 2 l 2 j „ v \ Mi*»iX, tx2v2\2 ^ 2 ^ 2 X 2 x 1 ( 2 n + , x J ; ^ „ > < n0<n0>"0 )R n„ 0(a 2) * x^ ( 2 i + , , < i ; i j - x » ( o ' o ! J > E x o ( f » - < h - 2 > F i n a l l y A.14 g i v e s 1 32 B = Z f m i n i l l f m 2 n 2 l 2 - U r ( 2 m + l ) ( 2 n + 1 ) ( 2 l + l ) ( - ) m + n + l . rmnl mnl f x ( n / n V H m ' m 2 i n )( n i n 2 n )( 1 1 1 z l ) * m n l ( i 2 ) (H 3) x i m 2 n 2 i 2 M 0 0 0 n 0 0 0 M 0 0 0 , 9 K 1 K ' ' m n l = Z P ( m , n , l ) * m n l ( l 2 ) , (2.4.42) mnl which d e f i n e s the c o e f f i c i e n t s P(m,n,l). 

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