Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The statistical mechanical theory of homonuclear diatomic fluids with application to liquid nitrogen Tam, Frank Wai-ming 1982

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1983_A6_7 T35.pdf [ 7.19MB ]
Metadata
JSON: 831-1.0060702.json
JSON-LD: 831-1.0060702-ld.json
RDF/XML (Pretty): 831-1.0060702-rdf.xml
RDF/JSON: 831-1.0060702-rdf.json
Turtle: 831-1.0060702-turtle.txt
N-Triples: 831-1.0060702-rdf-ntriples.txt
Original Record: 831-1.0060702-source.json
Full Text
831-1.0060702-fulltext.txt
Citation
831-1.0060702.ris

Full Text

THE STATISTICAL MECHANICAL THEORY OF HOMONUCLEAR DIATOMIC FLUIDS WITH APPLICATION TO LIQUID NITROGEN BY FRANK WAI-MING TAM B . S c , ANDREWS UNIVERSITY, 1980 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 TO THE REQUIRED STANDARD THE UNIVERSITY OF BRITISH COLUMBIA © AUGUST, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date ^ A 4 Z . SIXT i TABLE OF CONTENTS T a b l e o f C o n t e n t s i L i s t o f F i g u r e s . . i i i L i s t o f T a b l e s v Acknowledgement v i A b s t r a c t ...... . v i i CHAPTER I INTRODUCTION 1 CHAPTER I I THE MODEL POTENTIAL 6 1. I n t r o d u c t i o n 6 2. The I n t e r a c t i o n - S i t e Model 8 CHAPTER I I I THE STATISTICAL MECHANICAL THEORY 25 1. I n t r o d u c t i o n 25 2. D i s t r i b u t i o n F u n c t i o n s 25 3. I n t e g r a l E q u a t i o n Method 28 A The O r n s t e i n - Z e r n i k e E q u a t i o n 28 B C l o s u r e A p p r o x i m a t i o n s 32 4. A p p l i c a t i o n o f I n t e g r a l E q u a t i o n Methods t o Systems o f N o n s p h e r i c a l P a r t i c l e s 34 A R e d u c t i o n o f t h e O r n s t e i n - Z e r n i k e E q u a t i o n 34 B The L i n e a r i z e d H y p e r n e t t e d - C h a i n A p p r o x i m a t i o n 40 C The Method o f S o l u t i o n 44 i i CHAPTER IV RESULTS AND DISCUSSION 51 1. A Model f o r L i q u i d N i t r o g e n 52 A P h y s i c a l P a r a m e t e r s 52 B Convergence o f t h e Expanded P o t e n t i a l 54 2. LHNC R e s u l t s , 7 4 A The P a i r C o r r e l a t i o n F u n c t i o n 74 B Comparison w i t h M o l e c u l a r Dynamic R e s u l t s 83 99 3. Thermodynamic P r o p e r t i e s CHAPTER V LIST OF FIGURES The sphe r i c a l polar coordinate system The intermolecular coordinate system f o r diatomic molecules A comparison of the d i r e c t c o r r e l a t i o n function and and p a i r c o r r e l a t i o n function The convergence of the p a i r p o t e n t i a l f o r the end-end configuration The convergence of the p a i r p o t e n t i a l f o r the T-shape configuration The convergence of the p a i r p o t e n t i a l f o r the p a r a l l e l configuration The convergence of the p a i r p o t e n t i a l f o r a fixed distance , r = 1.0a , but varying orientations ... The convergenve o f the p a i r p o t e n t i a l f o r a fixed distance , r = 1.3 a , but varying orientations ... The c o e f f i c i e n t s of the p a i r p o t e n t i a l The effect of the quadrupolar i n t e r a c t i o n upon the p a i r p o t e n t i a l for the end-end configuration The e f f e c t of the quadrupolar i n t e r a c t i o n upon the p a i r p o t e n t i a l for the T-shape configuration ... The e f f e c t of the quadrupolar i n t e r a c t i o n upon the p a i r p o t e n t i a l f o r the p a r a l l e l configuration ... The c o e f f i c i e n t s of the p a i r c o r r e l a t i o n function The p a i r d i s t r i b u t i o n function for three d i f f e r e n t configurations without quadrupolar i n t e r a c t i o n i v 15. 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 f o r t h e t h r e e d i f f e r e n t c o n f i g u r a t i o n s w i t h q u a d r u p o l a r i n t e r a c t i o n i n c l u d e d 80 16. The a n g l e - a v e r a g e d p a i r 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 n i t r o g e n a t p* = 0.6964 86 17. The a n g l e - a v e r a g e d p a i r 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 n i t r o g e n a t p* = 0.6220 88 18. The a n g l e - a v e r a g e d p a i r 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 n i t r o g e n a t p* = 0.3500 90 19. The c o e f f i c i e n t s g200^ a n d g220^ r-' f o r l i c * u i d n i t r o g e n a t 79°K w i t h o u t q u a d r u p o l a r i n t e r a c t i o n 93 20. The c o e f f i c i e n t s §2oo'- r-' a n t * ^220^ ^ T 1 1 ^ n i t n o g e n a t 79°K w i t h q u a d r u p o l a r i n t e r a c t i o n i n c l u d e d .. . 95 21. Flow d i a g r a m f o r t h e i t e r a t i o n p r o c e d u r e 132 V LIST OF TABLES I The i n v a r i a n t e x p a n s i o n c o e f f i c i e n t s , h m n ^ ( r ) 43 I I M a t r i x £ o f eq. (3.44) 49 I I I P h y s i c a l p a r a m e t e r s f o r n i t r o g e n - l i k e f l u i d s 53 IV Thermodynamic P r o p e r t i e s o f t h e n i t r o g e n - l i k e f l u i d .... 100 V S p h e r i c a l harmonic f u n c t i o n s 110 VI R o t a t i o n a l I n v a r i a n t s ••• 114 V I I I n t e g r a l t r a n s f o r m a t i o n s f o r even I 120 V I I I G e n e r a l i z e d c o n v o l u t i o n c o e f f i c i e n t s 122 v i Acknowledgement I w i s h t o e x p r e s s my s i n c e r e g r a t i t u d e t o Dr. G.N. P a t e y f o r h i s t i r e l e s s e f f o r t s on my b e h a l f , i n t h e d i r e c t i o n o f t h i s r e s e a r c h and i n t h e p r e p a r a t i o n o f t h e t h e s i s . A l s o my t h a n k s t o R a n i T h e e p a r a j a h f o r h e r t i m e and p a t i e n c e i n t h e t y p i n g o f t h i s m a n u s c r i p t . F i n a l l y , I w i s h t o thank t h e C h e m i s t r y Department o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r p r o v i d i n g f i n a n c i a l s u p p o r t i n t h e form o f t e a c h i n g and r e s e a r c h a s s i s t e n t s h i p s . v i i ABSTRACT The p u r p o s e o f t h i s t h e s i s i s t o s e t up a b a s i c t h e o r e t i c a l scheme wh i c h w i l l e n a b l e us t o s t u d y t h e thermodynamic and s t r u c t u r a l p r o p e r t i e s o f homonuclear d i a t o m i c f l u i d s . We c o n s i d e r s o - c a l l e d i n t e r a c t i o n - s i t e models w h i c h assume t h a t t h e p a i r p o t e n t i a l can be w r i t t e n as a sum o f atom-atom i n t e r a c t i o n s o f t h e Lennard-Jones t y p e . The p a i r p o t e n t i a l i s t h e n expanded and e x p r e s s e d as a s e r i e s i n r o t a t i o n a l i n v a r i a n t s . T h i s a l l o w s t h e s h o r t - r a n g e d a n i s o t r o p i c f o r c e s and the l o n g - r a n g e d m u l t i p o l a r i n t e r a c t i o n s t o be t r e a t e d i n a u n i f i e d manner. I t i s found t h a t f o r n i t r o g e n - l i k e p a r t i c l e s t h e expanded p o t e n t i a l g i v e s an a c c u r a t e r e p r e s e n t a t i o n o f t h e f u l l p a i r i n t e r a c t i o n i n t h e p h y s i c a l l y i n t e r e s t i n g r a n g e . The s t a t i s t i c a l m e c h a n i c a l t h e o r y we c o n s i d e r i s based upon d i s t r i b u t i o n f u n c t i o n methods and employs i n t e g r a l e q u a t i o n t e c h n i q u e s . We d e v e l o p a g e n e r a l f o r m u l a t i o n and s o l v e t h e r e l a t i v e l y s i m p l e l i n e a r i z e d h y p e r n e t t e d - c h a i n (LHNC) a p p r o x i m a t i o n f o r a f l u i d o f n i t r o g e n - l i k e p a r t i c l e s . The s t r u c t u r a l p r o p e r t i e s p r e d i c t e d by t h e LHNC t h e o r y a r e i n q u a l i t a t i v e agreement w i t h computer s i m u l a t i o n s , but r a t h e r p o o r r e s u l t s a r e o b t a i n e d f o r some thermodynamic p r o p e r t i e s . F i n a l l y , some s u g g e s t i o n s a r e made as t o how t h e p r e s e n t t h e o r y might be improved and extended i n f u t u r e work. 1 CHAPTER I INTRODUCTION The s t a t i s t i c a l m e c h a n i c a l t h e o r y o f m o l e c u l a r f l u i d s has undergone r a p i d development i n t h e l a s t d e c ade, m a i n l y due t o t h e f o u n d a t i o n s l a i d by t h o r o u g h s t u d i e s o f s i m p l e ( a t o m i c ) f l u i d s . The p r e s e n t s t a t e o f a f f a i r s f o r s i m p l e f l u i d s has been r e v i e w e d by B a r k e r 1 2 3 and Henderson , and by Hansen and McDonald , and S t r e e t t and Gubbins have d i s c u s s e d more c o m p l i c a t e d systems w i t h l i n e a r m o l e c u l e s r e c e i v i n g p a r t i c u l a r a t t e n t i o n . However, o u r p r e s e n t knowledge o f t h e s t a t i c and dynamic p r o p e r t i e s o f m o l e c u l a r f l u i d s i s s t i l l f a r f r o m b e i n g c omprehensive. One i m p o r t a n t r e a s o n f o r t h i s i s t h a t o u r u n d e r s t a n d i n g o f t h e n a t u r e o f i n t e r a c t i o n s between m o l e c u l e s i s i n c o m p l e t e and a n o t h e r r e a s o n i s t h a t t h e t h e o r e t i c a l t r e a t m e n t o f t h e s u b j e c t i s s t i l l i n a d e v e l o p m e n t a l s t a g e . I n t h e p r e s e n t s t u d y we c o n s i d e r f l u i d s o f homonuclear d i a t o m i c m o l e c u l e s w i t h p a r t i c u l a r a t t e n t i o n b e i n g f o c u s s e d upon l i q u i d n i t r o g e n . 2 Computer s i m u l a t i o n s o f t h i s s y s t e m have been f a i r l y e x t e n s i v e . S t r e e t t 4 5 and T i l d e s l e y * have r e p o r t e d Monte C a r l o and m o l e c u l a r dynamics r e s u l t s f o r s e v e r a l systems o f d i a t o m i c m o l e c u l e s . These i n c l u d e h a r d homonuclear d i a t o m i c s as w e l l as more r e a l i s t i c models w i t h atom-atom p o t e n t i a l s o f t h e Lennard-Jones t y p e and q u a d r u p o l e - q u a d r u p o l e f o r c e s . Cheung and Powles ' have c a r r i e d o ut m o l e c u l a r dynamics s t u d i e s on n i t r o g e n - l i k e models i n t h e l i q u i d s t a t e u s i n g Lennard-Jones p o t e n t i a l s and sometimes i n c l u d i n g q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n s . A d d i t i o n a l computer s i m u l a t i o n s t u d i e s u s i n g v a r i o u s model p o t e n t i a l s a r e 3 d e s c r i b e d i n t h e r e v i e w by S t r e e t t and Gubbins . L i q u i d n i t r o g e n i s c o n v e n i e n t t o s t u d y t h e o r e t i c a l l y f o r s e v e r a l r e a s o n s . Many models have been s u g g e s t e d and s t u d i e d u s i n g computer s i m u l a t i o n , and a l t h o u g h no s y s t e m a t i c p r o c e d u r e has been f o l l o w e d i n t h e s e s t u d i e s , a c o n s i d e r a b l e amount o f d a t a has been c o l l e c t e d and i s u s e f u l f o r e v a l u a t i n g a p p r o x i m a t e t h e o r i e s . The q u a d r u p o l e moment o f n i t r o g e n i s known and r e l a t i v e l y s i m p l e models f o r t h e m o l e c u l a r 7 i n t e r a c t i o n s a r e q u i t e a c c u r a t e . M o r e o v e r , t h e n i t r o g e n - n i t r o g e n bond i s n o t e x t r e m e l y l o n g and t h u s a p p r o x i m a t e e x p a n s i o n s o f atom-atom p o t e n t i a l s work w e l l . F i n a l l y , a l a r g e amount o f r e a l e x p e r i m e n t a l d a t a i s a v a i l a b l e f o r t h i s system. There have r e c e n t l y been a number o f t h e o r e t i c a l s t u d i e s o f d i a t o m i c f l u i d s ( s ee r e f e r e n c e 3 ) . E f f o r t s have been made t o a p p l y t h e i n t e g r a l e q u a t i o n methods and t h e thermodynamic p e r t u r b a t i o n a p p r o a c h , w h i c h a r e q u i t e s u c c e s s f u l f o r s p h e r i c a l m o l e c u l e s , t o t h e n o n s p h e r i c a l c a s e . The p r e s e n t s t u d y employs a t h e o r e t i c a l scheme c a l l e d t h e L i n e a r i z e d H y p e r n e t t e d - c h a i n a p p r o x i m a t i o n , w h i c h i s an i n t e g r a l e q u a t i o n method, 3 t o c a l c u l a t e t h e s t r u c t u r a l and thermodynamic p r o p e r t i e s o f d i a t o m i c systems. T h i s a p p r o x i m a t i o n has been found t o work f a i r l y w e l l f o r 9 model f l u i d s i n t e r a c t i n g w i t h e l e c t r i c a l m u l t i p o l e moments . For example, f l u i d s o f s p h e r e s w i t h embedded d i p o l e and q u a d r u p o l e moments. The u s u a l approach i n t h e o r e t i c a l s t u d i e s o f t h e l i q u i d s t a t e i s t h e f o l l o w i n g . F o r a model system w i t h a w e l l d e f i n e d i n t e r a c t i o n p o t e n t i a l , t h e s t r u c t u r a l and thermodynamic p r o p e r t i e s c a l c u l a t e d by a p p r o x i m a t e t h e o r i e s ( i . e . by i n t e g r a l e q u a t i o n methods, o r p e r t u r b a t i o n a p p r o a c h e s , e t c . ) a r e compared w i t h r e s u l t s o b t a i n e d f r o m computer s i m u l a t i o n o f t h e same model. The advantage o f t h i s i s t h a t computer s i m u l a t i o n r e s u l t s can be r e g a r d e d as e s s e n t i a l l y e x a c t f o r t h e model sys t e m and hence any d i s c r e p a n c i e s a r i s i n g from a p p r o x i m a t e t h e o r i e s can be d e t e c t e d i n t h i s way. I f one compares a p p r o x i m a t e t h e o r i e s f o r a p a r t i c u l a r model w i t h r e a l e x p e r i m e n t a l r e s u l t s f o r t h e system o f i n t e r e s t one cannot be c e r t a i n i f o b s e r v e d d i s c r e p a n c i e s o r i g i n a t e w i t h t h e a p p r o x i m a t e t h e o r e t i c a l t r e a t m e n t o r w i t h i n a c c u r a c i e s i n t h e model i t s e l f . 10 S t i l l i n g e r has p o i n t e d out t h a t w i t h p r e s e n t l y a v a i l a b l e g computers and s i m u l a t i o n methods i t would t a k e ~ 3 x 10 y e a r s t o s i m u l a t e one second o f r e a l t i m e i n a m o l e c u l a r dynamics s t u d y o f s e v e r a l hundred model w a t e r m o l e c u l e s . F o r t u n a t e l y , many o f t h e e q u i l i b r i u m and time-dependent p r o p e r t i e s o f i n t e r e s t can be e s t i m a t e d f r o m s i m u l a t e d r e a l t i m e s o f t h e o r d e r o f 10 ^ t o 10 ^ seconds,, which r e q u i r e s computing t i m e s o f t h e o r d e r o f a few m i n u t e s t o a few h o u r s . W i t h t h e h i g h c o s t o f t i m e on modern f a s t c o m p u t e r s , s i m u l a t i o n s 4 continue to be expensive, i n many instances, p r o h i b i t i v e l y so. The application of theoretical approximations to study the properties of molecular f l u i d i s an a t t r a c t i v e compromise. Usually, to calculate numerically the structural properties of a homonuclear diatomic system by approximate theories takes computing times of the order of few seconds. Moreover, comparisons of simulation and theoretical results w i l l often lead to better theories as well as a better understanding of the system of interest. Several integral equation methods have been applied to the study 11 12 of nonspherical f l u i d s . Steele and coworkers ' have studied homonuclear diatomics i n some d e t a i l s using an approximation derived from the Percus-Yevick equation which can be solved for diatomic f l u i d s . We note that the Percus-Yevick equation i t s e l f cannot be solved for this model. The pair correlation functions and thermodynamic 1112 properties of several hard-core models were obtained ' but not 13 compared with computer simulations. Also Morrison has solved a similar approximation for diatomics with Lennard-Jones atom-atom interactions. Chandler and c o w o r k e r s * ^ ' ^ have developed a so-called Reference Interaction S i t e Model (RISM) integral equation theory which has also been applied to the study of diatomics. A l l of these approximate theories work well i n some cases but are rather poor i n others^. Several perturbation schemes have been developed i n the l a s t decade for systems of nonspherical p a r t i c l e s . The most widely applied 17 of these are the BLIP function theory of Anderson, Weeks and Chandler , and the Reference System Average Mayer-function expansion theory (RAM) 5 o f S m i t h and c o w o r k e r s 1 0 A 1 , and Perram and White'" 1'. The BLIP f u n c t i o n 23 t h e o r y was a p p l i e d by C h a n d l e r and c o w o r k e r s t o c a l c u l a t e t h e thermodynamic p r o p e r t i e s o f a model f o r n i t r o g e n and gave r e s u l t s i n 24 good agreement w i t h Monte C a r l o c a l c u l a t i o n s . S t e e l e and S a n d l e r s t u d i e d t h e s t r u c t u r a l p r o p e r t i e s o f d i a t o m i c s u s i n g t h e same method and a t moderate d e n s i t i e s , t h e r e s u l t o b t a i n e d were r o u g h l y comparable w i t h P e r c u s - Y e v i c k t y p e a p p r o x i m a t i o n s . However, t h e BLIP f u n c t i o n 24 t h e o r y i s fo u n d t o g i v e p o o r p r e d i c t i o n s a t low d e n s i t i e s . Nezbeda 25 and S m i t h i n v e s t i g a t e d t h e thermodynamic p r o p e r t i e s o f d i a t o m i c s u s i n g t h e RAM t h e o r y and o b t a i n e d r e s u l t s i n b e t t e r agreement w i t h Monte C a r l o c a l c u l a t i o n t h a n t h a t g i v e n by t h e RISM e q u a t i o n . I n t h e p r e s e n t s t u d y we o b t a i n e d c o n v e n i e n t and a c c u r a t e e x p a n s i o n s f o r d i a t o m i c atom-atom p o t e n t i a l s . These a r e t h e n u s e d i n o r d e r t o s o l v e t h e LHNC i n t e g r a l e q u a t i o n a p p r o x i m a t i o n f o r model f l u i d s w i t h n i t r o g e n - l i k e p a r a m e t e r s and t h e r e s u l t s a r e compared w i t h Monte C a r l o and m o l e c u l a r dynamics c a l c u l a t i o n s . The d e t a i l s o f t h e r i g i d d u mbbell model w i t h Lennard-Jones atom-atom i n t e r a c t i o n s c o n s i d e r e d i n t h i s s t u d y a r e d e s c r i b e d i n C h a p t e r I I . The p a i r c o r r e l a t i o n f u n c t i o n f o r m a l i s m and t h e LHNC t h e o r y a r e d i s c u s s e d i n C h a p t e r I I I . R e s u l t s f o r a n i t r o g e n - l i k e s y s t e m a t d e n s i t i e s i n t h e l i q u i d r a nge a r e g i v e n and d i s c u s s e d i n C h a p t e r IV. C o n c l u s i o n and s u g g e s t i o n s f o r f u r t h e r work a r e d i s c u s s e d i n C h a p t e r V. 6 CHAPTER I I TIE MODEL POTENTIAL 1. I n t r o d u c t i o n I n o r d e r t o s t u d y t h e p h y s i c a l p r o p e r t i e s o f a system o f i n t e r -a c t i n g m o l e c u l e s , i t i s e s s e n t i a l t o u n d e r s t a n d t h e n a t u r e o f t h e i n t e r a c t i o n between i n d i v i d u a l m o l e c u l e s . The i n t e r a c t i o n p o t e n t i a l * o f a system composed o f N m o l e c u l e s , n e g l e c t i n g e l e c t r o n i c e f f e c t s , can be r e p r e s e n t e d by U N = U N < V V V (2-1} where ^ r e p r e s e n t s t h e c o o r d i n a t e s o f m o l e c u l e i ( i n c l u d i n g p o s i t i o n , o r i e n t a t i o n , e t c . ) . I n p r i n c i p l e , t h e N-body i n t e r a c t i o n f o r d i a t o m i c * T h i s i s c a l l e d t h e Born-Oppenheimer a p p r o x i m a t i o n a c c o r d i n g t o which we can s o l v e t h e e l e c t r o n i c p r o b l e m f o r a s t a t i c c o n f i g u r a t i o n o f t h e n u c l e i , t h u s d e r i v i n g a p o t e n t i a l energy f u n c t i o n d e p e n d i n g o n l y on t h e n u c l e a r c o o r d i n a t e s . T h i s a p p r o x i m a t i o n i s j u s t i f i e d s i n c e t h e n u c l e i a r e much h e a v i e r t h a n t h e e l e c t r o n s . 7 systems a t f l u i d d e n s i t i e s i s v e r y c o m p l i c a t e d so t h a t a number o f s i m p l i f i c a t i o n s a r e needed b e f o r e u s e f u l a p p l i c a t i o n s can be endeavoured. The f i r s t s i m p l i f i c a t i o n a r i s e s from t h e f a c t t h a t i n t r a m o l e c u l a r f o r c e s between atoms a r e much s t r o n g e r t h a n i n t e r m o l e c u l a r f o r c e s between m o l e c u l e s . Thus, f o r r e l a t i v e l y r i g i d m o l e c u l e s we can o f t e n i g n o r e 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 and t h e m o t i o n o f t h e m o l e c u l e as a whole. U s i n g t h i s a p p r o x i m a t i o n , we can t r e a t homonuclear d i a t o m i c s as r i g i d m o l e c u l e s and t h e c o o r d i n a t e , X^ , r e p r e s e n t s t h e p o s i t i o n and o r i e n t a t i o n o f m o l e c u l e i . The s econd s i m p l i f i c a t i o n comes from t h e a s s u m p t i o n t h a t i n t e r -m o l e c u l a r p o t e n t i a l e n e r g i e s a r e , t o a f i r s t a p p r o x i m a t i o n , a d d i t i v e . Thus, U may be w r i t t e n as i n w h i c h t h e f i r s t t e r m i s a sum o f p a i r i n t e r a c t i o n s and t h e second a sum o f t r i p l e t i n t e r a c t i o n s . I n t h e case o f d i a t o m i c m o l e c u l e s a t l i q u i d d e n s i t y , i t i s assumed t h a t a l l terms beyond t h e p a i r i n t e r a c t i o n i n (2.2) can be n e g l e c t e d . D e s c r i p t i o n s o f t h e d i f f e r e n t t y p e s o f 26 p a i r i n t e r a c t i o n can be found i n a book e d i t e d by H i r s c h f e l d e r and 27 28 i n r e v i e w a r t i c l e s by L o n g u e t - H i g g i n s , and by Buckingham and U t t i n g T here a r e many models w h i c h have been used t o s i m u l a t e systems o f r i g i d m o l e c u l e s w i t h t h e t o t a l p o t e n t i a l a p p r o x i m a t e d by t h e p a i r 3 i n t e r a c t i o n o f (2.2) . A r e a l i s t i c p a i r p o t e n t i a l w i l l have a weak a t t r a c t i v e f o r c e when t h e m o l e c u l e s a r e f a r a p a r t , and a s t r o n g r e p u l s i v e f o r c e when t h e y a r e v e r y c l o s e ( s i n c e d e n s i t i e s a r e f i n i t e ) . (2.2) To a f i r s t a p p r o x i m a t i o n , t h e p a i r p o t e n t i a l c a n be a p p r o x i m a t e d by a h a r d - c o r e i n t e r a c t i o n * - r > r i 2 < d ' ( o , r 1 2 > d where T ^ = - | ; ^•^'l a r e P o s i t i ° n s °f m o l e c u l e s 2 and 1 , r e s p e c t i v e l y , and d i s t h e e f f e c t i v e h a r d s p h e r e d i a m e t e r o f t h e m o l e c u l e . I t s h o u l d be n o t e d t h a t t h e h a r d - c o r e p o t e n t i a l does n o t have a weak a t t r a c t i v e f o r c e when t h e m o l e c u l e s a r e f a r a p a r t and t h e r e f o r e i s n o t a " r e a l i s t i c " r e p r e s e n t a t i o n o f t h e t r u e p o t e n t i a l . A more " r e a l i s t i c " r e p r e s e n t a t i o n i s g i v e n by so c a l l e d " s o f t - c o r e " p o t e n t i a l s . Examples a r e t h e Lennard-Jones (12-6) p o t e n t i a l ( c f . 29 s e c t i o n 2 ) , and t h e K i h a r a p o t e n t i a l , b o t h o f w h i c h have a r a p i d l y r i s i n g r e p u l s i v e p a r t and a l o n g range w e a k l y a t t a c t i v e t e r m . The i n t e r a c t i o n p o t e n t i a l chosen i n t h i s s t u d y f i t s t h e above d e s c r i p t i o n s o f a r e a l i s t i c model. 2. The I n t e r a c t i o n - S i t e Model For d i a t o m i c m o l e c u l e s i n t h e i r ground e l e c t r o n i c s t a t e , t h e i n t e r m o l e c u l a r p a i r p o t e n t i a l u ( r ^ > ^ J ^ - * depends o n l y on t h e v e c t o r f r o m c e n t r e o f m o l e c u l e 1 t o t h e c e n t r e o f m o l e c u l e 2, and on t h e m o l e c u l a r o r i e n t a t i o n s and 9,^. The o r i e n t a t i o n 0,^ can be d e s c r i b e d i n s p h e r i c a l p o l a r c o o r d i n a t e s as shown i n F i g . 2.1 where 6 and <j) a r e t h e p o l a r and a z i m u t h a l a n g l e s , r e s p e c t i v e l y . I f t h e frame u ( r 1 9 , n . , £ U i s e q u i v a l e n t t o u f X ^ X ^ where X i = r . , ^ , i f one chooses t h e c e n t r e o f mass o f m o l e c u l e 1 as t h e o r i g i n o f t h e c o o r d i n a t e system. 9 Figure 1 The spherical polar coordinate system where 6 is the polar and <f>, the azimuthal angle. A(Ax,Ay,Az) Ay 11 of reference i s chosen such that the Z-axis l i e s along t n e n t n e p a i r i n t e r a c t i o n depends upon r^> 9^ ^2 a n c l ^ ^ = *1 ~ ^ a S s t l 0 W T 1 i n F i g . 2.2. The p o t e n t i a l energy of two diatomic molecules i n t e r a c t i n g v i a an atom-atom p o t e n t i a l described by an i n t e r a c t i o n - s i t e model (Fig. 2.2) i s the sum of the four i n t e r a c t i o n s between pai r s of atoms not on the same molecule. One has u(12) = .1 u (r ) (2.4) v ' s i t e s ss ss where U(12) i s a short-hand notation f o r u f r ^ , ^ , ^ ) and u s g ( r s s ) represents the s i t e - s i t e i n t e r a c t i o n s . One of the atom-atom pot e n t i a l s commonly used i s the Lennard-Jones p o t e n t i a l (u _ ( r ) = u T T ( r__)) S S S S lj J S S which has the form where e i s the depth of the a t t r a c t i v e w e l l , and a i s r e l a t e d to the 30 " e f f e c t i v e atomic diameters". Powles and Gubbins found that the s i t e - s i t e (or i n t e r a c t i o n - s i t e ) Lennard-Jones (12-6) p o t e n t i a l using appropriate parameters can be made to f i t both the second v i r i a l c o e f f i c i e n t and the l i q u i d properties (thermodynamic, s t r u c t u r a l , and dynamic) f o r nitrogen. Writing down e x p l i c i t l y the potentials f o r the s i t e s , we have *W = U a l a 2 ( r a l a 2 ) + U alb2 ( r a l b 2 } U a 2 b l ( r a 2 b l ) + U blb2 C W ( 2 ' 6 ) where the labels r e f e r to Fi g . 2.2. 12 Figure 2 The intermolecular coordinate system for diatomic molecules. The Z-axis lies along r J 2 = ?2~*l* a n d * = $i~$2' £ i s t h e internuclear distance, a + B = 1 and r is the distance betwe the molecular centres of mass. 13 14 The i n t e r a c t i o n - s i t e model i s a u s e f u l way i n w h i c h t o r e p r e s e n t t h e i n t e r a c t i o n s between d i a t o m i c m o l e c u l e s because o f i t s s i m p l e f o r m and b e c a u s e i t can be e xtended t o p o l y a t o m i c systems. However, i t s h o u l d be emphasized t h a t i t may n o t always be p o s s i b l e t o e x p r e s s t h e t r u e p o t e n t i a l i n t h i s form. I f m u l t i p o l a r , i n d u c t i o n , a n i s t r o p i c d i s p e r s i o n o r o v e r l a p f o r c e s e t c . , a r e s i g n i f i c a n t t h e p o t e n t i a l cannot be a d e q u a t e l y r e p r e s e n t e d by a s i m p l e s i t e - s i t e model. T h i s c a n sometimes be c o r r e c t e d by i n c l u d i n g a d d i t i o n a l terms i n ( 2 . 6 ) . F o r example, q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n s a r e o f t e n i n c l u d e d i n t h i s way ( s e e b e l o w ) . The f o u r i n t e r a c t i o n s shown i n F i g . 2.2 can be w r i t t e n as f u n c t i o n s o f a, 8 and I i n a d d i t i o n t o t h e d i s t a n c e a n d t n e a n g l e s , and ft . As a s p e c i a l c a s e , t h e o r i g i n o f t h e m o l e c u l a r f r a m e , G\, can be chosen as t h e c e n t r e o f mass, such t h a t a = B f o r homonuclear m o l e c u l e s . F o r s u c h s y s t e m s , t h e c e n t r e o f mass i s a c o n v e n i e n t c h o i c e f o r o u r p u r p o s e s i n c e e x p a n s i o n s about t h i s c e n t r e a r e s i m p l e r m a t h e m a t i c a l l y and r e l a t i v e l y r a p i d l y c o n v e r g e n t f o r a l l o r i e n t a t i o n s . I f we i d e n t i f y t h e s i t e s w i t h t h e a t o m i c n u c l e i , t h e n £ i s t h e u s u a l bond l e n g t h . I t i s p o s s i b l e t o expand u ( 1 2 ) as a power s e r i e s i n ft/r i n t h e f o l l o w i n g manner. The f o u r s i t e s , a , b.^, and b 2 , shown i n F i g . 2.2, can be w r i t t e n i n terms o f t h e C a r t e s i a n c o o r d i n a t e s , ( x , y, z ) . One o b t a i n s a.^  (cdLSine^ostJ^, a J L S i n S j S i n ^ , a&CosBj) > b (-p^SinOjCosfj^, - B a S i n e ^ i n ^ , - B X X o s e p , 15 a„ (aJtSin62Cos<i>2, a£.Sin92Sin<f>2, r + a£Cos92) , b 2 (-6£Sin62Cos())2, -B£Sin92Sin<|>2, r - B£Cos92) , (2.7) and hence, the site-distances can be found and simplified to rala2 = r { 1 + 2(~f^ ^-WV " 2 f [ T l - T 2 ] } h > ralb2 = r { 1 + («2^fy2 • 2aH^)2 [ T , T ^ ] - 2 i [ a T ^ ] } * , rbla2 = r { 1 + Ca 2 +B 2)(|) 2 * 2aB (f f T ^ T ^ ] + 2 | [BT^aT,]}* , (2.8) rblb2 = T { 1 + 2 C f ) 2 t ^ l W + 2 f , g where, following the notation of St.ell, Patey and Hftye , T = Cos91 , T 2 = Cos9 2 , T 3 = Sine Sine 2Cos<|> , (2.9) and $ = <j>^  - (j>2. If the interaction potential between sites contains terms of the form — , then these terms can be expanded using a McLaurin r series expansion £ CX) = Z *- (0) (2.10) m=0 m! where f(m) = d mf(X) dXm X = 0 (2.11) 16 When #(X) = (1 + X ) 1 ^ 2 , the series can be written e x p l i c i t l y as 2 3 ,.. v,n/2 . n v n.n X n,.n .,n X (1+X) = 1 - 2 X + 2 (2 + 1 ) I T " 2 (2 + 1 ) ( 2 + 2 ) T T * (2.12) This method of expansion i s analogous to,the multipole expansion for coulombic potential ( i . e . n = 1) as described by 31 Buckingham . Notice, however, that (2.10) i s only v a l i d for X < 1; and when X approaches 1, more terms must be retained i n the expansion. Substituting appropriate variables for X, retaining terms to 4th order i n — , and rearranging one obtains the following i—£ = r " n i l + an(T -T2) | + a 2 [-nQ-T^-T^ ala2 + | ( n + 2 ) ( T r T 2 ) 2 ] ( ^ ) 2 + a 3[-n(n +2)(l-T 1T 2-T 3)(T 1-T 2) + | (n+2)(n +4)(T 1-T 2) 3](|) 3 + a 4 ["-(n+2) ( l - T ^ - T ^ 2 - \ (n+2)(n +4)(l-T 1T 2-T 3)(T 1-T 2) 2 + ~ ( n + 2 ) ( n + 4 ) ( n + 6 ) ( T r T 2 ) 4 ] ( i ) 4 + . . . j , (2.13a) T^—-h = { l + n(aT +gT )£ + [3 { a 2 + g 2 + 2 a g ( T T 2 +T )} alb2 1 + § (n +2)(aT 1 + B T 2 ) 2 ] ( ^ ) 2 + [-^(n+2) (aT^T,,) {a 2 +B 2 +2aB(T 1T 2 +T 3)} + £(n+2)(n+4)(aT^gT^3](i) 3+ [ J(rl +2){a 2 +B 2+2a6(T 1T 2 +T 3)} 2 - J(n+2) (n+4) 17 + (n+4) (n+6) ( a T ^ B T . , ) 4 ] ( | ) 4 + ... J. . (2.13b) E x p r e s s i o n s f o r — - — n and — - — n can be o b t a i n e d by s i m p l y r a l b 2 r b l a 2 e x c h a n g i n g a and B and r e p l a c i n g T by -T , and 1^ by - T 2 i n (2.13a) and (2.13b) r e s p e c t i v e l y . We c o n s i d e r p a i r p o t e n t i a l s , u ( 1 2 ) , o f t h e form where t h e C a r e c o n s t a n t s and 1 1 1 1 + + ±—— + ± . (2 15") n n _ n n n ^ - - ^ j r ( I 2 ) r a l a 2 r a l b 2 r a 2 b l r b l b 2 Then s u b s t i t u t i n g (2.13) i n t o ( 2 . 1 5 ) , p e r f o r m i n g some a l g e b r a i c m a n i p u l a t i o n s and r e t a i n i n g terms t o f o u r t h o r d e r i n (—") we f i n d r = r " n {4 + 2 n ( a - B ) ( T 1 - T 2 ) | + [-n{2 ( a 2 + B 2 ) - (a-B) V ^ + T ^ } r ( 1 2 ) n + n ( n + 2 ) { ( T 1 2 + T 2 2 ) ( a 2 + B 2 ) - 2 ( a - B ) ] ( | ) 2 + [ - n ( n + 2 ) { ( 3 a 3 - a 2 B + a B 2 - 3 B 3 ) - 2 ( a 3 - a 2 g + B 2 a - B 3 ) x ( T ^ + ^ J K T j - ^ ) + J ( n + 2 ) ( n + 4 ) { 2 ( a 3 - B 3 ) ( T 1 3 - T 2 3 ) - 3 ( a 3 - a 2 B + a B 2 - B 3 ) ( T 1 2 T 2 - T 1 T 2 2 ) } ] ( | ) 3 + [ J ( n + 2 ) * . . . ( c o n t ' d ) 18 x { ( 3 a 4 + 2 a V + 3 B 4 ) - 4 ( a 4 - a V a 8 3 - 8 4 ) ( T j ^ + T ) + 2 ( a 2 + B 2 ) 2 [ T 1 2 T 2 2 + 2 T 1 T 2 T 3 + T 3 ] } - j (n+2) (n+4) 4 2 2 4 2 2 4 3 3 4 x { ( 3 a +2a g +3g DCTj +T 2 ) - 4 ( a -a S-a3 +3 J T ^ - 2 ( a 4 - a 3 B - B 3 a + B 4 ) ( T 1 T 2 + T 3 ) ( T 1 2 + T 2 2 ) + 4 ( a 2 + g 2 ) 2 x ( T ^ T ^ + T j T ^ ) } + £ ( n + 2 ) ( n + 4 ) ( n + 6 ) { ( a 4 + B 4 ) ( T 1 4 +.T24) - 2 [ ( a - B ) 4 + 3 a B ( a - B ) 2 ] ( T 1 3 T 2 + T 1 T 2 3 ) + 3 (a 2+B 2) * T l 2 T 2 2 ] ( i ) 4 + .... > • C2.16) F o r homonuclear d i a t o m i c s y s t e m s , t h e o r i g i n o f t h e m o l e c u l a r c o o r d i n a t e s y s t e m i s t a k e n t o be t h e c e n t r e o f mass. Then a = B and t h e terms w i t h odd powers o f Jt/r v a n i s h t o y i e l d r ( 1 2 ) " n = r ~ n (4 + ( | ) 2 [ - ^ ( 1 - T ^ T ^ - Y ( 1 + T I T 2 + T 3 ) + f(rD(TrV2 + fcl-+ I)CT X + T 2J 2] - ^ ( r | + i ) r J i + 2 ) { ( l - T 1 T 2 - T 3 ) ( T 1 - T 2 ) 2 + ( l + T ^ + T ^ x ( T 1 + T 2 ) 2 } + ^ + i ) ( n + 2 ) ( H + 3 ) { ( T 1 - T 2 ) 4 + ( V T 2 ) 4 } ] + ( | ) 6 [ ^ L ( | + i ) ( ^ + 2 ) ( l - T ^ - T j ) 3 } + ... (c o n t ' d ) 19 + ( l + T l T 2 + T 3 ) 3 } + * d + T l V T 3 ) 2 ( T 1 + T 2 ) 2 } - £ ( ^ + 1 K ^ 2 ) § + 3 ) ( | + 4){(1-T 1T 2-T 3) x ( T r T 2 ) 4 + ( l + T l V T 3 ) ( T 1 + T 2 ) 4 } + ^ + l ) ( ^ 2 ) ( 5 + 3 ) $ + 4 ) x (^  + 5 ) { : T 1 6 + T 2 6 + 1 5 ( T 1 4 T 2 2 + T 1 2 T 2 4 ) } ] + .... } (2.17) where we have kept terms to order (Jl/r)^. 32 33 Pople and Steele have shown that any macroscopic configuratio-nal property which depends upon the positions and orientations of a pair of molecules, 1 and 2, can be written as a series in a complete set of orthonormal polynomials which span the space of the Eulerian angles denoting the orientations of 1 and 2. Therefore, i t is convenient to express the pair potential of linear molecules as an expansion in rotational invariants to g i v e ^ = mL u <r12? * C V V ^ ( 2 ' 1 8 ) , . . ,.mnJl,,„., , ... , , 34-36 where the rotational invariants, $ (12),are defined by .-v.«.,.;,,) - * f ^ O D l f v % > 2 > ° „ \ ^ - ( 2 - 1 9 ) x z. i. z. y \) A ft and ^ 2 are the orientations of molecules 1 and 2 respectively; r 1 2 is a unit vector defined by r 1 2 / | r 1 2 | , and the Wigner matrix elements D X fn) and the 3-j symbols ( m n * J a r e defined in Edmonds37. The p V A 20 co efficients f™1^ are given by ^ml _ ft! (2.20) * 0 0 0 with the exception of = -2/5, and = 8/35/2. This i s an expression f i r s t used by Blum 3 4 3 ^ for multipolar potentials and has 9 been employed in subsequent calculations . Explicit forms for selected rotational invariants are given in Appendix I. One of the advantages of expanding the pair potential in this way is that any additional interactions involving electrical multipoles can be easily included. For example in the case of homonuclear diatomic systems the quadrupole-quadrupole interaction i s the most important term and i t can be expressed in the form 2 30 224 U (12) = ^ L - $ / Z 4(12) (2.21) 38 where Q is the quadrupole moment of the molecule. Downs et.al. have shown that the site-site model for homonuclear diatomics is inadequate to represent the true pair potential, since i t neglects multipolar and induction interactions. This deficiency can be partially corrected by adding eq. (2.21) to eq. (2.18) thus accounting for the quadrupolar contribution. Using the relationships given in Appendix I, eq. (2.17) can be written in terms of the rotational invariants to give 21 r ( 1 2 ) n s A ^ O O O r A000 n, -,,1,^202 ^022, A000,,J,,2 {4$ + [-n$ + -(n+2) i y ^ +$ )+$ }] (—) rn, _ w l . . . 000 1 ,000 A220,, n, _ w ., , 5 , .202 A022, 4 1)00 1 ,,222 A220,, - j(n+2) (n+4){yg-i[ $ +$ ] + - F + -9 1> + $ ]> n r > n r « w 2 *000 5 r * 2 0 2 * 0 2 2 i + 32^ n + ^ ^ 15 + 4 2 ^  + ^ 1 p-ft.224 o n*222 ...220, 1 r^404 ,044,., .4.4 + 420 C 9 + 2 0 $ " 1 4 $ 1 + 840 [ + W Q r n r ^ r o * n n 0 ^ 2 2 0 , n , „, , „ , , 1 ,137,^422 .242, + [ - "96ot n + 2 ^ n + 4 ^ " $ 3 + 64 Cn+2)(n+4)(n+6)(22o[^-($ +$ ) 10, 424 244, 1 ,426 A246,, 1 ,-,404 „044, ^ 1 r ~7^220 -,.222, 197 r 202 0221 19.000,. + —1-27$ +26$ J + 2YQL* ] + J 4 2 ' , 1 r 1 1 8 8 , . 4 2 2 . 2 4 2 , ^Cn+2 ) ( n +4 ) ( n + 6D ( n +8K T I5 5-[— C» + * ' 58 r . 4 0 4 . , 0 4 4 ^ 21 r 1 r . 6 0 6 . 0 6 6 , (n+2) (n + 4 ) (n + 6 ) (n + 8 ) (n+10){30395 [« 1 + 1 1 5 2 0 . ^Afi (* 4 2 2- 2 4 2) . f c 4 2 4 - 2 4 4 ) ^ ^ 4 2 6 - 2 4 6 ) ] ^ * 4 0 4 -044' (2.22) 22 C o l l e c t i n g terms we obtain -n „-n T A «. Q l ) n f £ . 2 (n-l)n(n+l)(n+2) fft 4 (lln+30)(n-l)n(n+l)(n+2)(n+4) ± 6,*0Q0- pn(n+2) 1 2 + 12.7! V J L 6 V 5n(n+2)(n+4)(9n-16) r£,4 197(n-l)n(n+l) (n+2) (n+4) (n+6) f 6, x ( $ 2 0 2 + $ 0 2 2 ) + r-"(" +2)(n +4)(n-4) £ 4 +  J L 360 r (-79n3+222n2+27n-90)n(n+2) (n+4) ,£,6.^220 rn(n+2) (n+4) (n-1) .ft.4 130(n-l)n(n+l) (n+2) (n+4) (n+6) ,£,6-^222 ,n(n+2) (n+4) (n+6) . U + 2Y78J l r J L 5 6 0 9(n-l)n(n+2) (n+4) (n+6) ,il,6 1 / f i224 rn(n+2) (n+4) (n+6) fft 4 + 28.7! l r J J 1 2.7! V (109n+795) (n-l)n(n+2) (n+4) (n+6)j r^404+$044-| 462.8! + r 3 ( n - l l n 0 i + l X n ^ ( $ 4 2 2 + $ 2 4 2 ) [ 28771 * m r10(5n+55)(n-l)n(n+2)(n+4)(n+6),£ 6 424^ 244) 1 462.8! V J l r(2n 2+42n+187)n(n+2)(n+4)(n+6) £ 6 426 246 L 462.8! r J + rn(n+2) (n+4) (n+6) (n+8) (n+10) ^606} ^066 ) + # i > j (2.23) which allows u(12) to be written as i n (2.17). i n t h i s study we consider models where the s i t e - s i t e i n t e r a c t i o n i s the Lennard-Jones (12-6) p o t e n t i a l defined by equation (2.5). Then 23 the total pair potential is of the form u(12) = 4. - / - 6 1 <2'24) T(12) r(12) I 6 which to order (—) becomes U(12) = u 0 0 % 0 0 0 + u 2 0V 0 2 + u 0 2 2 * 0 2 2 + u 4 0V 0 4 + u044$044 + u 6 0 6 * 6 0 6 + u 0 6V 6 6 + u 2 2 0 * 2 2 0 + u 2 2 2 $ 2 2 2 + u 2 2 4 $ 2 2 4  + u422$422 + u242$242 + ^424^424 + ^44,244 + ^ 4 2 6 + u246 $246 X2.25) where u « " . U 2 ° 2 . 4S ( [ 2 8 £ 2 * * ^ C | ) 6 1 ^ ) 1 2 - K ) 2 500,2.4 985,11.6, .o, 6, u°<* . u 6 0 6 . 4. f [ £ 4 ) 6 K f ) 1 2 - I ^ ) 6 ] ^ 6 ! . .220 _ r r 1001.L4 3718,£.6, .a.12 r 14 1 4 290,£ 6, a. 6, 222 .-,,352,4-4 7436 ,2,,6,'a,12 r200 r£.4 650r«,.6, .a. 6, 24 224 A J r432 ,£,4 4752 ,^6., ra,12 72 1 4 1801 6 u 6 242 422 . , r5148r£,6..,a 12 f30J.6 o 6, u = U = 4 £{[- 3 T-(-) ]{-) - [— (7) )(.-•)}, 244 424 . ,r186,£,6,ra,12 r75,L6 a 6. U = u = 4e.{ [ — ( - ) ] (7) " L77C7) ] (7) } » and 246 426 ^ 89i.6..o 12 r 73 £ 6 0 6 , f 2 2 f t 1 The convergence of this expansion is investigated in Chapter IV. 25 CHAPTER I I I THE STATISTICAL MECHANICAL THEORY 1. I n t r o d u c t i o n The s t a t i s t i c a l m e c h a n i c a l t h e o r y o f f l u i d s can be f o r m u l a t e d 39 40 i n s e v e r a l d i f f e r e n t ways ' . The d i s t r i b u t i o n f u n c t i o n language 41 i s p a r t i c u l a r l y u s e f u l and w i l l be a p p l i e d i n t h e p r e s e n t s t u d y . As i s u s u a l l y done f o r m o l e c u l a r f l u i d s , we s h a l l t r e a t t h e p r o b l e m as b e i n g e n t i r e l y c l a s s i c a l . T h i s i s a v a l i d a p p r o x i m a t i o n f o r most l i q u i d s a t o r d i n a r y t e m p e r a t u r e s but i s i n a d e q u a t e i n a few cases such as f o r l i q u i d H e l i u m o r Hydrogen a t low t e m p e r a t u r e . 2. D i s t r i b u t i o n F u n c t i o n s C o n s i d e r a homogeneous system o f N p a r t i c l e s i n a volume V and a t a t e m p e r a t u r e T, t h e p r o b a b i l i t y t h a t m o l e c u l e 1 i s i n d r ^ d f i ^ a t r ^ , w i t h o r i e n t a t i o n Q^, m o l e c u l e 2 i n ar^dn7 a t r ^ , w i t h o r i e n t a t i o n ft0 e t c . i s g i v e n by 26 P ( r 2 . f i j , r 2 , n 2 , . . »r N,Q N)dr 1. .. d ^ d f ^ . .. 1 N 1 N N where i s t h e c o n f i g u r a t i o n i n t e g r a l ( a l s o a n o r m a l i z a t i o n f a c t o r ) and i s d e f i n e d by r r -8U ( r ... , r , • • • ,° N) - - J I e N 1 N 1 N d r . . . . d r d a . . . d o -N " •*" 1 n 1 N V (3.2) The p r o b a b i l i t y t h a t any m o l e c u l e i s i n d r j d a t r ^ , ^ , . . . , and any m o l e c u l e i s i n d r do a t r ,!1 , i r r e s p e c t i v e o f t h e c o n f i g u r a t i o n o f n n n n t h e r e s t o f t h e m o l e c u l e s i s o b t a i n e d by i n t e g r a t i n g (3.1) o v e r t h e c o o r d i n a t e o f m o l e c u ! e s m + l t h r o u g h N, and m u l t i p l y i n g by a p e r m u t a t i o n f a c t o r : (n) p '^••••rn!V,--'V = (N-n)! ( r -3U M . N! J . . J e N d r n + r - - d r N d n n + l - - - % (3.3) i s o f t e n c a l l e d a g e n e r i c r e d u c e d d i s t r i b u t i o n f u n c t i o n . One d e f i n e s an 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 g by where p = ^  i s t h e number d e n s i t y o f t h e system. 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 e x p r e s s e s t h e p r o b a b i l i t y o f o b s e r v i n g d i f f e r e n t c o n f i g u r a t i o n s o f a s e t o f n m o l e c u l e s o u t o f 27 t h e t o t a l number N. These a r e u s e f u l i n s t a t i s t i c a l m echanics p r i m a r i l y because a l l thermodynamic f u n c t i o n s o f t h e system can be e x p r e s s e d i n terms o f r e l a t i v e l y s i m p 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 n = 1 and n = 2 . Of p a r t i c u l a r i m p o r t a n c e i s t h e 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 ( r , ( F o r c o n v e n i e n c e , s i n c e we s h a l l be c o n c e n t r a t i n g s o l e l y on t h i s f u n c t i o n , we s h a l l d r o p t h e s u p e r s c r i p t and w r i t e s i m p l e g ( 1 2 ) ) d e f i n e d by " B U N /...Je dr . . .dr dn ...dn g(12) = u 2 l ) - - — . • (3.5) p Z N The angle-averaged p a i r 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 s can be 2 42 o b t a i n e d by X - r a y and n e u t r o n s c a t t e r i n g e x p e r i m e n t s ' I t w i l l be u s e f u l t o have e x p l i c i t f o r m u l a e f o r a few thermo-dynamic p r o p e r t i e s o f f l u i d s . 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 , £(12), d e s c r i b i n g a system w i t h m o l e c u l e s i n t e r a c t i n g t h r o u g h t h e p a i r p o t e n t i a l , u ( 1 2 ) , i s r e l a t e d t o t h e a v e r a g e energy E, t h e p r e s s u r e p, 34 43 and t h e 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 K ' by t h e e q u a t i o n s The t y p e o f d i s t r i b u t i o n f u n c t i o n s i n v o l v e d i n t h e c a l c u l a t i o n o f t h e thermodynamic f u n c t i o n s depends v e r y much upon t h e p o t e n t i a l energy o f t h e system. I f t h e p o t e n t i a l o n l y i n v o l v e s p a i r i n t e r -a c t i o n s , t h e n t h e n=l and n=2 d i s t r i b u t i o n f u n c t i o n s a r e r e q u i r e d . I f , however, t h e p o t e n t i a l i n v o l v e s t r i p l e t i n t e r a c t i o n s as w e l l as p a i r i n t e r a c t i o n s , t h e n t h e n=3 d i s t r i b u t i o n f u n c t i o n i s a l s o needed. 28 2 E = " k i n + / u ( 1 2 ) § ( 1 2 ; i d * i d * 2 ' ( 5 - 6 ) (4TT) p ' = p k T - / 9 u H 2 ) . r 1 2 g(12) d X ^ (3.7) 6(4TT) 2 3 r 1 2 and K = - ^ f + 1 6 k T V ^ d ? ) " ^ d * l d h f3'8) where, as b e f o r e , p i s t h e d e n s i t y and V t h e volume o f t h e system. From thermodynamic r e l a t i o n s h i p s i n v o l v i n g E, p and K , a l l o t h e r thermodynamic p r o p e r t i e s o f t h e system can be o b t a i n e d . An a p p r o x i m a t e t h e o r y o f g(12) and hence f o r t h e thermodynamic p r o p e r t i e s i s d e s c r i b e d below. 3. I n t e g r a l E q u a t i o n Methods (A) The O r n s t e i n - Z e r n i c k e E q u a t i o n . I t i s f i r s t n e c e s s a r y t o i n t r o d u c e t h e p a i r c o r r e l a t i o n f u n c t i o n , h ( 1 2 ) , d e f i n e d by h(12) = g,(12) - 1 (3.9) C l e a r l y , h(12) i s a measure o f t h e d e p a r t u r e o f t h e d i s t r i b u t i o n f u n c t i o n f r o m t h e random v a l u e o f u n i t y . I t d e s c r i b e s t h e c o r r e l a t i o n between t h e p o s i t i o n s and o r i e n t a t i o n s o f p a r t i c l e s 1 and 2. O r n s t e i n 44 and Z e r n i k e p r o p o s e d a d i v i s i o n o f t h e p a i r c o r r e l a t i o n f u n c t i o n i n t o two p a r t s : h(12) = c ( 1 2 ) + ^ Jc(13) h(32) d^ (3.10) 29 where = dr^dn^. T h i s i s c a l l e d t h e O r n s t e i n - Z e r n i k e (O-Z) e q u a t i o n , and i t i s one o f t h e b a s i c r e l a t i o n s h i p s i n t h e e q u i l i b r i u m t h e o r y o f l i q u i d s . The 0-Z e q u a t i o n may be r e g a r d e d as an i n t e g r a l e q u a t i o n f o r t h e 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(12) i n terms o f t h e unknown f u n c t i o n c ( 1 2 ) . The o r i g i n a l a u t h o r s c a l l e d c(12) a 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 t h i n k i n g i t t o be dependent o n l y upon t h e p a i r i n t e r a c t i o n u ( 1 2 ) . We now know t h a t t h i s i s i n c o r r e c t and t h a t c(12) depends upon h(12) as w e l l . The second t e r m o f (3.10) i s o f t e n c a l l e d t h e i n d i r e c t p a r t and can be i n t e r p r e t e d as t h e c o r r e l a t i o n imposed on t h e r e l a t i v e c o n f i g u r a t i o n o f p a r t i c l e s 1 and 2 by a t h i r d p a r t i c l e , a f o u r t h p a r t i c l e , e t c . T h i s can be seen by i t e r a t i n g (3.10) t o o b t a i n t h e e x p a n s i o n h(12) = c(12) + J c(13)<c(32)d 3 + (^r)2 / c ( 1 3 ) c ( 3 4 ) c ( 4 2 ) d ^ + .... (3.11) The O-Z eq. (3.10) i s e x a c t and can be d e r i v e d by c l u s t e r 45 d i a g r a m e x p a n s i o n methods . However, as mentioned above, t h e i n t e r p r e t a t i o n o f C ( 1 2 ) as a d i r e c t c o r r e l a t i o n , dependent o n l y upon 46 t h e p a i r i n t e r a c t i o n , t u r n s out t o be i n e x a c t , and t h e 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 does not have a s i m p l e p h y s i c a l i n t e r p r e t a t i o n . The b e h a v i o r o f c ( 1 2 ) and h(12) f o r s p h e r i c a l m o l e c u l e s i s shown i n F i g . 3.1. I f one i s d e a l i n g w i t h a s p h e r i c a l l y s y m m e t r i c p o t e n t i a l , t h e 0-Z e q u a t i o n can be F o u r i e r T r a n s f o r m e d t o g i v e t h e k-space e q u a t i o n 30 Figure 3 A comparison of the direct correlation function, c(r) ( ) and the pair correlation function, h(r) ( ) for spherical particles. Note that the direct correlation function has less structure and i s shorter ranged. h(r) and c(r) -1.0 co H t£ 32 h(k) = c ( k ) + ph(k)c(k). (3.12) (The transformation of the second term i n (3.10) into what appears i n (3.12) i s c a l l e d a convolution). Hence, there i s an e x p l i c i t s o l u t i o n f o r the p a i r c o r r e l a t i o n function i n k-space, namely 1 - P c(k) (B) Closure Approximations The general approach in constructing an i n t e g r a l equation theory i s to look for a second equation which r e l a t e s the p a i r c o r r e l a t i o n function, h ( l 2 ) , and the p a i r p o t e n t i a l , u(12), to the d i r e c t c o r r e l a t i o n function, c(12). Such a r e l a t i o n s h i p i s c a l l e d a closure f o r the 0-Z equation since one can then obtain a so l u t i o n for the two unknown functions c(12) and h(12). The most widely used closures are 9 36 47 the Mean Spherical (MSA) ' ' , the Percus-Yevick (P-Y), and the Hypernetted-chain (HNC) approximations. The MSA i s defined by the equations - • . c(12) = -Bli(12) r 1 2 > d (3.14) and g.(12) = 0 r l 2 ^ d (3.15) where d i s a hard-core diameter. Eq. (3.14) i s the correct asymptotic form becoming exact as r 1 2 ». This approximation, which involves 33 p u t t i n g a h a r d - c o r e i n t h e m o l e c u l e , g i v e s r e a s o n a b l e r e s u l t s f o r t h e e q u i l i b r i u m p r o p e r t i e s o f c l a s s i c a l monoatomic f l u i d s a t low d e n s i t y . However, i t b r e a k s down at h i g h d e n s i t y when, as e x p e c t e d , s h o r t range i n t e r a c t i o n s become i m p o r t a n t . Both t h e P-Y and HNC 2 48 e q u a t i o n s can be d e r i v e d f r o m f u n c t i o n a l T a y l o r s e r i e s e x p a n s i o n ' , o r by n e g l e c t i n g c e r t a i n terms i n c l u s t e r s e r i e s e x p a n s i o n s f o r 7 45 c ( 1 2 ) - ' 4 b . The P-Y e q u a t i o n c(12) = g ( 1 2 ) [1 - e B " ( 1 2 ) ] (3.16) g i v e s good agreement w i t h r e s u l t s from computer s i m u l a t i o n s f o r some monoatomic f l u i d s ^ , and an a p p r o x i m a t e form has been a p p l i e d t o t h e 11 13 d i a t o m i c c a s e ' . The HNC e q u a t i o n , w h i c h i s used i n t h i s s t u d y , i s d e f i n e d by c ( 1 2 ) = -h(l 2) - In [ g ( 1 2 ) e B u C 1 2 ) ] - 1 . (3.17) A l t h o u g h t h e P-Y a p p r o x i m a t i o n g i v e s b e t t e r r e s u l t s f o r h a r d - s p h e r e 1 2 f l u i d s , i n g e n e r a l t h e HNC e q u a t i o n i s s u p e r i o r . I n some c a s e s , i n t e g r a l e q u a t i o n a p p r o x i m a t i o n s can be s o l v e d a n a l y t i c a l l y . F o r example, t h e P-Y e q u a t i o n has been s o l v e d a n a l y t i c a l l y 49 50 51 5 7 53 f o r h a r d s p h e r e s by T h i e l e and by Wertheim ' . B a x t e r has a l s o s o l v e d t h e P-Y e q u a t i o n f o r h a r d s p h e r e s and f o r " s t i c k y " h a r d s p h e r e s , w h i c h a r e h a r d p a r t i c l e s w i t h a s u r f a c e a d h e s i v e p o t e n t i a l . _ F o r most s y s t e m s , however, i n t e g r a l e q u a t i o n a p p r o x i m a t i o n s must be s o l v e d by r e s o r t i n g t o n u m e r i c a l t e c h n i q u e s . These a r e u s u a l l y i t e r a t i v e p r o c e d u r e s 48 w h i c h have been d i s c u s s e d by Watts t o some e x t e n t and t h e a p p l i c a t i o n 34 of these methods to the present problem is described in Appendix IV. 4. Application of Integral Equation Methods to Systems of  Nonspherical Potentials The solution of the 0-Z equation (3.10) becomes significantly more complicated when nonspherical systems are considered. Of the three closure approximations mentioned above, (3.14-15), (3.16) and (3.17), only the MSA can be solved without making additional approximations. Wertheim has obtained a completely analytical solution 47 34-36 of the MSA for dipolar hard spheres , and Blum has shown that formal solutions can be obtained for systems with general electrical multipoles. The HNC approximation, on the other hand, cannot be solved as i t stands and additional approximations must be made. One 54 such closure i s the linearized HNC (LHNC) approximation described 9 54 below (cf. § III.4B). For multipolar fluids, i t has been shown ' that the LHNC theory is much more accurate than the MSA and this is the closure applied in the present study. (A) Reduction of the Ornstein-Zernike Equation The following i s essentially Blum's reduction of the 0-Z equation for linear molecules. Equation (3.10) can be written in the form h(r 2,8 ,82) - a ( r 1 2 , n i , n 2 ) = J L J d r ^ d ^ h C r ^ , ^ , n ^ c ( T ^ , ^ ^ , ^ ^ (3.20) 35 where Jdftf(ft) is the integration over a l l angular orientations of f ( f i ) and Jdfl = / / d<(» Sin 0de = 4TT . (3.21) <(>=0 6=0 Expanding h and c in rotational invariants, we obtain where ( r 1 2 3 = (3.^3j { [ $ m n S ' ( f t 1 , f t 2 , r 1 2 ) ] d f i ^ d r ^ with similar expressions for c(12)*. Substituting these expressions in eq. (3.20) the 0-Z equation becomes t This expression applies only to linear molecules, for more complex molecules , h(12) = „ h ( r 1 0 ) $ (12) J m,n,£,y,v yv v 12' yv We will adopt the notation h(12) and h ( r 1 2 , n , f^) ;$ m n S ' (12) and ^ m ! i (ftj , Q ^ , T ^ interchangeably. Since c (12) can be expanded in rotational invariants in exactly the same way as h(12), unless otherwise stated, the £(12) expressions can be assumed to be equivalent to those for h(12). 36 J- J L I Jdr\do h 1 1 hr ) c 2 2 ( r ) 4TT m.nft. -"3 3 13 4n m i " i ~ i m2n2l2 x » 1 1 1 C a i^ 3 >r 1 3 D $ 2 2 2 C V V r 3 2 ) * C 3 ' 2 4 ) Fourier transforming both sides of eq. (3.24) we obtain m n * J e x p ( i t - r l 2 H h ^ ( r l 2 ) - ^(r^)] t ^ P * V l ^ V V 0 ! ^ m nJt m7n2Z2 JL * ? Jexp[ik.tf 1 3 + r 3 2 ) ] h ( r13 D ° ^ 4TT m 1 n i £ , i m_n.S, _ 2 2 2 * • C^,^,r 1 3) $ ( n 3,^,r 3 2)dr 1dr 2dr 3dQ 1d f2 2dQ 3 . (3.25) We now apply the Rayleigh expansion for exp(ik"-R) exp(it.fo = / = 0 i * j £(kR) J_t D^(k) D^fR) (3.26) where j (kR) is the spherical Bessel function of order I, and the D X» UA are the Wigner matrix elements as defined previously. Substituting eq. (3.26) into eq. (3.25), taking particle 1 as the origin, integrating over d r 1 2 on the l e f t , and dr^dr.^* on the right, and using the The integration oyer d r 2 1 d r 3 1 i s equivalent to integrating over d r ^ d r ^ 37 orthogonality property of the Wigner matrix elements, one obtains ,= - f / d o d o d n E 0 : m l V l r t ~m2V2r, ' 4TT J 1 2 3 n ^ n ^ h (k) c (k) m 2 n 2 £ 2 x m n a a m n £, $ 1 1 1 (n 1,ft 3,k) $ z ^ 'oa3,n2,k) (3.27) where $ m n i l ( f l j , n 2 , i b is defined by (2.18). Writing out $ m n £ in f u l l , and carrying out the integration over 0,^ e x p l i c i t l y , one obtains = fh^w - cmM(k)] yJx O f^ K*>2 W Do>2> Dox« P „ n , fi11^)"12^! U l V l V1 V2 X2 n 2^ V 2 rn J ° " > i > »o2JV D o \ c w Do2^« d n i d n 2 X 2 " l + 1 - - ' - - (3.28) of the orthogonality relations for D X yields Application of tne ortnogunain.; >.^^^.~ ~ y V L [h (k) - c 00] f tyvX3 U0X l K J I ^ - ^ ^ ( k ) ^ 2 ^ ) ^ 2 * ... (cont'd) VlV2 2 n l + 1 X1X2 38 ran £ ^ n ^ o £ i ~ £ ? -X ^ J x ^ v j v x ^ D 0^(k) D ^ ( k ) • (3.29) Finally, multiplying both sides by D (k), integrating over k, and UA using the properties of 3J and 6J symbols3 , we get (after some rearrangements), . . . mn,£, _n,n£,. , _ ~mn£ _ ~mn£ _ p • l 2 l l 4 h 1 1 c 1 2 (3.30) v J K J n1S'i J'2 m n n i 2J 2^  & where Z is defined by m n n^ W _ / ^ V ^ ^ ^ W w m n n, „ ^ m n n , v 0 0 0' 1 f™1 (2n 1 +l) 1 3. b c 5 6 {, -} is a 6-j symbol and the coefficients of the correlation d e f functions in k space are given by the Hankel transforms h^ O O = 4u i 4 f dr r 2 j , (kr) h m n \ r ) • (3.32) The generalized convolution of the 0-Z equation (3.30) is essentially 36 that given by Blum . This constitutes a formal solution of the 0-Z equation in terms of the coefficients h m n 8 , (r) and c m n S , ( r ) . Since Hankel transforms using spherical Bessel functions of second or higher order are very d i f f i c u l t to evaluate numerically, i t is useful to introduce the following transformations 39 n m n i l ; e(R) = d r r 2 6* (r ,R) h ^ f r ) for even I, (3.33a) J ° ( R ) = f d r r 2 8° (r,R) h ^ C D for odd I, (3.33b) 'mn£ uhere the transformation Kernels 9*, 6° are defined by Then i t can by be shown37 that the required Hankel transforms are given OO hm^ ( k ) = 4 l T J dRR 2 j Q(kR) n m n S- ; e(R) for even I , (3.35a) ~mne, = I d R R 2 R ) -mnfc;o(R) f o r o d d % _ (3.35b) o 1 Thus we need only calculate zeroth (i.e. Fourier) and f i r s t order Hankel transforms numerically both of which can be done by Fast Fourier Transform techniques (See Appendix IV). Although the sums i n Eq. (3.30) are in principle i n f i n i t e there are often restrictions upon the number of possible terms. These are imposed both by molecular symmetry and by the closure approximation applied (cf. § III. 4B). 40 6. The LHNC Closure Approximation Writing the hypernetted-chain equation (3.17) i n the form c(12) = h(12) - £n g(12) - Bu(12) (3.36) and separating the spherical (i.e. the (0,0,0) projection) and angular parts, one obtains c(12) = h 0 0 0 ( r ) - in g 0 0°(r) - g u 0 0 0 ( r ) - *n [1+X(12)] - g[u(12) -u° 0 0(r)] + [h(12) - h° 0 0(r)] (3.37) where X(12) = h ( 1 2 ) " h (3.38) A U ^ J ooo, , • g (r) If we expand Jln[l+X(12)] and retain only the term linear in X(12), we arrive at the linearized HNC (LHNC) equations, 000, , ,000, . „ 000, . &.ooo, r^ , Q , c (r) = h (r) - in g (r) - Bu (r) (3.39a) and J f f l i l , . _ hrml . h 0 0° (rD _ ^ ( r ) (3.39b) G- (r) - n i r j 0 0 0 g 00 for a l l other projections. 41 Upon rearrangement, these equations become 000. , , 000, , 0 000, ,, 000, , . ,. , c (r) = exp[n (r) - Bu (r)] - n (r) - 1 , (3.40a) .mnil . , mnft. , r 000. , ., „ mn£, , 000, , ,_ c (r) = n (r) [g (r) - 1] - Bu (r) g (r) (3.40b) where we have defined the function n(12) = h(12) - c(12).* Equation (3.39) i s obtained from eq. (3.37) by matching terms. It i s interesting to note that c°°°(r) and h°°°(r) are related by an expression i d e n t i c a l to the f u l l HNC equation (3.36) f o r spherical molecules. Upon examination of eqs. (3.40a) and (3.40b), i t i s clear that g(12) —*- 0 as ~* 0 which i s the correct l i m i t i n g b e h a v i o r . T h e LHNC equation 9 has been shown to give good results for spherical p a r t i c l e s with embedded point dipole and/or quadrupole moments. m^nft ^ s generally preferred over h™1^ i n numerical calculations because i t i s a well behaved smooth function. This i s unlike h m n 2 , (r) which ri s e s very sharply near r = cr This i s obvious i f one rearranges the two equations (3.40a) and (3.40b) to get 000, , 000, . , 000. , r 000. , . 000, . . „ ^  . c (r) +n (r) + 1 = g (r) = exp[n (r) - Bu (r)] = 0 at r 0 (3.41a) and mnfc,^ „..mnS,,„,-, „000 _ (3.41b) cmn£ ( r ) + T i m n £ ( r ) = g*n4 ( r ) = ^mni ( r ) _ g u ^ ( r ) ] g u ^ ( r ) =0 at r - 0 42 The formal solution of the MSA and LHNC approximation includes only the 16 projections found i n u(12). That is h(12) and c(12) are of the form , „„. , 000, . ,202, .^202,10. ,022, v 022, ,404, . 404,... h(12) = h (r) + h (r)$ (12) + h (r)$ (12) + h (r)$ (12) 044, , 044, 1 0. ,606, . 606,. 0. ^ ,066, .066,... + h (r)$ (12) + h (r)$ (12) + h (r)$ (12) , 220, , 220,^, ,222, , 222,.„, ,224, 224 + h (r)$ (12) + h (r)$ (12) + h (r)$ (12) ,422, .422,... ,242, ,.242,10. ,424, . 424 + h (r)$ (12) + h (r)$ (12) + h (r)$ (12) + h 2 4 2 ( r ) $ 2 4 2 ( 1 2 ) + h 4 2 6 ( r ) $ 4 2 6 ( 1 2 ) + h 2 4 6 ( r ) » 2 4 6 C 1 2 ) (3.42) where the coefficients h m n 4 ( r ) are as defined by eq. (3.23) and explicit expressions are given in Table I. The $ m n 4(12) included in eq. (3.42) form a closed set under the generalized convolution of the 0-Z equation. This means that under the generalized convolution, these functions w i l l generate only themselves and each other. Therefore, eqs. (3.30) and (3.40) constitute a formal solution to the LHNC approximation for a system of homonuclear diatomic molecules interacting * In many cases the terms in the pair potential do not form a closed set and the solution to the LHNC approximation contains terms which do not occur in u(12). For example for dipolar systems h(12) and c(12) contain three terms, (000), (110) and (112), but only (000) and (112) occur in the pair potential. The invatiant expansion coefficients, h- (.rj h ooo ( r ) = i j h ( 1 2 ) $ ooo ( 1 2 ) ^ (4*) h ° 2 2 ( r ) = / h(12) $ 2 0 2C12) d ^ d ^ 4(4TT) h 0 4 4 ( r ) = _ 1 J h ( 1 2 ) $ 4 0 4 ( 1 2 ) d ^ d f t 2 64(4TT) h0 6 6( r ) = — 2 J h(12) * 6 0 6 (12) d f i ^ 518400(4TT) h 2 0 2 ( r ) = __5 | h ( 1 2 5 $ 2 2 0 ( 1 2 ) d ^ d ^ 4(4TT) h 2 2 2 C r ) = 25 J h ( 1 2 ) $222 C 1 2 ; ) 14(4TT) h 2 2 4 ( r ) = _JS / h ( 1 2 ) $ 2 2 4 ( 1 2 ) d ^ d f t 2 224 C4ir) h 2 4 2 ( r ) = 5 J h ( 1 2 ) $ 4 2 2 ( 1 2 D 14(rir) h 2 4 4 ( r ) b 25 J h ( 1 2 ) $ 4 2 4 ( 1 2 ) d ^ d f t 2 1232 (4u) h 2 4 6 ( r ) = 1 / h (12) $ 4 2 6 ( 1 2 ) d « 1 d n 2 25344(4ir) 44 with the potential defined by eq. (2.25). It should be noted that the limitations upon the number of terms occuring in h(12) and c(12) is imposed by the LHNC closure. In an exact result, terms of higher order w i l l occur but these are expected to be small in magnitude. 7. The Method of Solution In the LHNC approximation, eq. (3.30) yields 16 linear simultaneous equations. Evaluating the Z 1 2 these can be written M 6 in n i ij in the form 000 r, 000 000 n 0 ,.022 202 n . ,044 404 5.184><105 ,066 606.. n = P [h c + 0.8 h c + 0.4 h c + ^ h e ] , 202 r i 202 000 . .. 220 202 „ ,,222 202 , ^,224 202 8, 242 .404 n = p[h c - 0.4h c + 0.4h c + 1.6h c + ^ h c A 244 404 246 404, + 32h c + 960h c J, 022 r,000 022 n .,022 220 w .,022 222 , ,,022 224 8, 044 422 n = p[h c - 0.4h c + 0.4h c + 1.6h c + ^ h c _0, 044 424 _,n, 044 426, + 32h c + 960h c ], 220 r . ..202 022 _ .,220 220 n „, 222 222 112 ,224 224 r\ = p[-0.4h c - 0.4h c - 0.28h c 45~ 7,242 422 ,,,244 424 246 426, -r-=h c - 24.64h c - 12672h c ], 45 2 2 2 . p ^ c " 2 2 - 0.4h 2 2°c 2 2 2 - 0.4„ 2 2 2c 2 2° - £ W » * 242 424 44, 244 422 104, 244 424 160 224 224 + _4, 242 422 4^242 424 + ^"J" _ + Tx* c + 63* C 2T 2 1 7 T1 ^ 63 + 3 8 4 h 2 4 4 c 4 2 6 + 3 8 4 h 2 4 6 c 4 2 4 + 11520h 2 4 6c 4 2 6] , 45 p $ h * » c 0 2 2 - O.4h 2 2 0c 2 2 4 . ^ 2 2 2 c 2 2 2 * ^ 2 2 2 c 2 2 4 - 0.4h 2 2 4= 2 2° 242c426 2^224^22 + 24^224c224 + ^ 2 4 2 c 4 2 2 + l h 2 4 2 c 4 2 4 + 20h< 4244 c422 + 1 4 4 h244 c426 + 2 0 h246 c422 + 1 4 4 h246 c424 + ^ 2 4 6 ^ r p [ h404 c000 + ^ 4 2 2 ^ 0 2 + 0. 4 h424 c202 + ^ 4 2 6 ^ 0 2 ^ 30 r,000 044 1,022 242 . ,,022 244 1 0, 022 246, p [h c + c + 0.4h c + 12h c ], 48,404^022 . ,,422 220 8,422 222 8,422 224 264,424 222 p [ — h c _ 0.4h c + c c + T q ^ c 16, 424 224 426 224, — n G + 320h c J , P [ f h 2 0 2 c 0 4 4 _ 0 . 4 h 2 2 ° c 2 4 2 + J l h 2 2 2 c 2 4 2 + ^ h 2 2 4 c 2 4 2 + ^ 2 2 2 ^ 2 4 i ^ h 2 2 4 c 2 4 4 + 3 2 0 h 2 2 4 c 2 4 6 ] , r40,404 022 1.422 222 10,422 224 - . ,,,424 220 13,424 222 p [ 7 7 h ° + 4 2 * C + 693* C " ° - 4 h ° ~ 77*  0 432, 424 224 48, 426 222 160, 426 224, .__h c + c + _ * c ] f r40,202 044 1,222 242 10,224 242 . ,,220 244 13,222 242 = P[y7 h C + 42* C + 693h C " °- 4 h C "77* C 432, 224 244 48, 222 246 160, 224 246, ^ 770* c " i f 1 c + TT^ C ]> T 1,404 022 7 ,422 224 0.28,424 222 2.8^424 224 = ^ 33* ° +1782* C + ~ 3 T ^ C +~99* C _ ,,426 220 2.8,426 222 22.4,426 224, - 0.4h c --JY* c + _ h c ], 48 246 r 1 , 202 044 7 ,224 242 0.28,222 244 2.8,224 244 . ..220 246 2.8, 222 246 22.4, 224 246, - 0.4h c + -jy4i c + -gg-h c ], 606 u606 000 n = ph c , and 066 ,000 066 f, n = ph c LJ.^ JJ , mnil = rmnS, . , where h n (kj. -mn£ The above 16 linear simultaneous equations, with ri""1*'(k) replaced by n^^Ck) + c ^ (k) can be arranged into a matrix equation & + I = 0 C3.44) where A is a 16 x 16 matrix with terms dependent upon p and c m n^(k) as shown in Table II, X is -a 16 dimensional column vector of the form , - 0 0 0 , ~ 0 2 2 , 1 A -044,,, -066,.. ~202„ , ~220„ , ~222„ , ~224,,, & = (n 0 0 , n Ck), n 0 0 , n 0 0 , n 0 0, n (k)» n (k),n 0 0 , ^242,,, -244,., -246,., -404,., -422,., -424,., -426,., ri (k), n (k), n (k), n 00, n (k), n (k), n (k), ~606 ( k ) ) > (3.45) Y l and Y i s a 1 x 16 matrix, Y = '. where Y16 47 , 000 000 n „ 022 202 , A 044 404 5.184xl05 066 606. - {c c + 0.8c c + 64c c + Y% C C *> • • f 000 022 n „ 022 220 . . 022 222 . , 022 224 8 044 422 - {c c = 0.4c c + 0.4c c + 1.6c c + —c c 044 424 n , n 044 426, 32c c + 960c c }, 000 044 J_.022 242 4 c022 c244 + 1 2 c022.246 } j  _ t L 30 000 066 - c c , _ { c202 c000 _ 0. 4 G220 c202 + ,,^222 ^2 + ^224.202 + 8^42^04 3 2 c 2 4 4 c 4 0 4 + 9 6 0 c 2 4 6 c 4 0 4 } , - { - 0 . 4 c 2 0 2 c 0 2 2 - 0.4c 2 2°c 2 2 0 - 0.28c 2 2 2c 2 2 2 - 2 » c 2 2 4 c 2 2 4 Z c 2 4 2 c 4 2 2 - 24.64c 2 4 4c 4 2 4 - 12672c 2 4 6c 4 2 6}, 45 {4.202.022 _ 0 > 4 C220 C222 _ 0. 4 c222.220 _ ^ 222.222 + 32.222.224 32 224 222 160 224 274 ^ 4 242 422 44 244.422 _ 104.244.424 70° ° + ~65 63 2 1 7 3 8 4 c 2 4 4 c 4 2 6 + 3 8 4 c 2 4 6 c 4 2 6 + 11520c 2 4 6c 4 2 6}, _ {18.202.022 _ 0 - 4 c220.224 + _ ^ - c 2 2 2 c 2 2 2 + 2 c 2 2 4 c 2 2 4 - 0 . 4 c 2 2 4 c 2 2 0 2.224.222 + 24.224.224 + 1 4 4 c242.422 + 1..242.424 + 2 0 c242.426 1.244.422 3888.244.424 + 1 4 4 c244.426 + ^246.422 + 1 4 4 c 246.424 7 700 1152c 2 4 6c 4 2 6}, 48 Y = _{i8 c202 c044 _ 0 > 4 c220 c242 + ^ 2 2 2 ^ 4 2 + 8 ^ 2 4 2 + 264 222^44 9 7 70 63 70 + l^ c224 c244 + 3 2 0 c 2 2 4 c 2 4 6 } j Y - - { 4 0 c 2 0 2 c 0 4 4 + J _ c 2 2 2 c 2 4 2 . + 2 2 _ c 2 2 4 c 2 4 2 . o 4 c 2 2 0 c 2 4 4 + M c 2 2 2 c 2 4 4  X10 ~ X70 42 690 17 432 c224 c244 48 £222 c246 160^224^246, + 770 + 11 11 Y r 1 -202 044 + 7 224 242 0.28 222 244 2,8 224 244 Y l l = " {33° ° + 1782 C C + " I T 0 ° 99 C ° - 0 . 4 c 2 2 0 c 2 4 6 + ^ c 2 2 2 ^ 4 6 + i ^ l c 2 2 4 ^ 4 6 } , Y 1 2 = - { c 4 0 4 c 0 0 0 + ^ c 4 2 2 c 2 0 2 + 0 . 4 , 4 2 4 c 2 0 2 + 1 2 c 4 2 6 c 2 0 2 } , ,48 404 022 . .422 220 8 422 222 8 422 224 264 424 222 - {—c c - 0.4c c + — c c + — c c + — c c + ^ c424 c224 + 3 2 0 c 4 2 6 c 2 2 4 } ) Y - - { 40 c404 c022 ^ c422 c222 J0_ c422 c224 _ 4 c 422 c220 _ 13.424^22 14 77 42 693 " 77 Y 1 3 432 424 224 • 48 426 222 160 426 224, + 770 c c + i T c c + T T C c }> , J _ 404 022 7 422 224 O ^ c 4 2 4 ^ 2 2 + 2 - 8 c 4 2 4 c 2 2 4 *15 ' " 33 1782^ 33 99 n „ 426 220 2.8 426 222 22.4 426 224, - 0.4c c + — c c + - — c c }, and _ 606 000 Y16 ~ * (3.46) I Table II. Matrix A of eq.(^.^) u " " = c u ; ) . e o o o _ i 51 ene -O.fc „ i . . 2 " . 0 . » c • 32 c * 2 » . 1.6 c 2 2 » - t *« •* 35 ° . . . . 1 . 1 000. 1 c202 8 zUOk .„„»»- J - 0 . 2 8 c 2 2 2 - V ^ c » » " f t ' k 022 1  c 1^  Oow. 1 owt 73 0 ToTS ' To c * 261. „2W> 1 „2»a 8: 28 2W>. 7500 0 * 28 „2»6 TO e * ' 6 c222 -0.1. c " " » f c 2 2 2 • $ i ,20 c 2 " 6 10 0 2 » 2 . 160 „2»6 I7H5 , 2»2 . 28 2Wt . 590 c * 22"t „2">6 955 c It ,.1>22. S „»2» IT 0 32 c W " £ c " 2 2 -l ^ i c»2». 38» c » 2 « ^888 , « » . 380 c * 2 t . 1)26 11)20 o 20 c " 2 2 . 1W c » 2 » . 1152 e l>26 50 H8 .022 T c 8' 1 .022 75 c - O . l l c 8 75 .220. 222 535 51 c * 10 .224 575 C T7TC 221* f 5 ' -0.1. c 2 2 0 -# , 2 2 2 . 28 , 2 2 2 . 7300 c 28 , 221 5555 c 320 c 22» »B 222. n c 160 ,221 I T c -0 .& c c 2 2 2 221 „22» 555 c 50 The matrix equation (3.44) can be reduced into 4 sets of independent simultaneous equations A i « ! + h = °> (3-47a) £2 h + *2 = °» ( 3 - 4 ? b ) £3 h + % 3 = °' ( 3 , 4 7 C ) and M 4 + * 4 = 0 C 3 ' 4 7 d ) where ,~000„, -022,,, -044,,, -066,.,. £ includes the pro3ections (n (k), n (k), n (k), n (k)), ,~202„, -220,., -222,., -224,,. X„ includes (n (k), n (k), n (k), n (k), -242,., -244,,, "246,,,, n (k), n (k), n (k)), ,-404 „, ~422„, -424,. T, -426,,,, X includes (n (k), n (k), n ( k ) , n (k)), <\,3 ,-606,, ,, and X, includes U l (k)) ^4 (3.48) Solutions can be obtained for the four vectors by solving the four sets of linear equations (see Appendix IV). Equation (3.46) in conjunction with the LHNC closure relation (3.40) are solved numerically by the iteration procedure described in Appendix IV. 51 CHAPTER IV RESULTS AND DISCUSSION In Chapter I I , we have derived expansions for homonuclear diatomic atom-atom potentials retaining terms up to 6 t h order and i n Chapter I I I we have solved the LHNC approximation for systems characterized by t h i s interaction. In the present chapter we discuss the convergence of the potential expansion for a nitrogen-like model and give LHNC results for l i q u i d nitrogen at several densities and temperatures. The structural and therodynamic properties are described and compared with previous computer simulations and with experimental data. In discussing the convergence of the potential and the structure of the system, i t i s convenient to express a l l parameters i n reduced units. Length w i l l be expressed i n units of c, and energies i n units 52 o f kT where T i s t h e a b s o l u t e t e m p e r a t u r e o f t h e s y s t e m and k i s t h e Bodtzmann c o n s t a n t . I n t h e s e u n i t s , w h i c h we d e n o t e by an a s t e r i s k , * SL t h e i n t e r n u c l e a r s e p a r a t i o n o f t h e m o l e c u l e i s H = — , t h e w e l l d e p t h i s e = — , t h e q u a d r u p o l e moment i s Q = ± ( — F — ) ' , and f i n a l l y , kT * 3 a kT t h e r e d u c e d d e n s i t y i s p = pa . Thus i n t h e s e u n i t s t h e p o t e n t i a l (eq. ( 2.24)) becomes Bu(12) = 4 e * ' { [ ~ ] U - [ ] 6 } (4.1) r ( 1 2 ) r ( 1 2 ) * where r ( 1 2 ) i s as p r e v i o u s l y d e f i n e d ( c f . eq. ( 2 . 2 3 ) ) . 1. A Model F o r L i q u i d N i t r o g e n A P h y s i c a l p a r a m e t e r s I n p r a c t i c e , t h e p h y s i c a l p a r a m e t e r s one u s e s i n s i m u l a t i o n o r t h e o r e t i c a l s t u d i e s a r e u s u a l l y o b t a i n e d e x p e r i m e n t a l l y o r by f i t t i n g 6 7 e x p e r i m e n t a l d a t a . Cheung and Powles ' have c a r r i e d o u t m o l e c u l a r dynamics c a l c u l a t i o n s on l i q u i d n i t r o g e n u s i n g a model w i t h Lennard-J o n e s atom-atom p o t e n t i a l s . The p h y s i c a l p a r a m e t e r s u s e d i n t h e i r c a l c u l a t i o n s a r e l i s t e d i n T a b l e I I I . The p a r a m e t e r s , e, a and I, were o b t a i n e d by f i t t i n g computed ppT d a t a t o e x p e r i m e n t a l r e s u l t s f o r n i t r o g e n . We n o t e t h a t t h e i n c l u s i o n o f t h e q u a d r u p o l a r i n t e r -58 a c t i o n s g i v e s a v a l u e c l o s e r t o t h e e x p e r i m e n t a l bond l e n g t h o ( i . e . 1.102 A) t h a n t h a t f o u n d when t h e q u a d r u p o l e i s i g n o r e d . 53 T a b l e I I I . P h y s i c a l p a r a m e t e r s f o r a n i t r o g e n - l i k e f l u i d ( c f . Ref. 6, The t e m p e r a t u r e s and d e n s i t i e s a r e f o r p o i n t s on t h e v a p o r - p r e s s u r e c u r v e . 3 * *2 T(°k) P(g/cm ) e Q -P * * 59 0.8932 0.6452 0.0 0.6964 0.3292 79 0.7978 0.4587 0.0 0.6220 0.3292 1 2 4 a 0.4490 0.2976 0.0 0.3500 0.3292 79 0.7978 0.4587 0.5179 0.6220 0.3323 F o r t h i s model: e = 0.515 x 1 0 " 2 1 J 6, ° fi a = 3.310 A , Q = -1.52 x 1 0 " 2 6 e . s . u . 5 6 = -5.07 x 1 0 _ 4 ° Cm 2 ° 57 H = 1.090 A when Q = 0 o = 1.100 A when Q i s i n c l u d e d T h i s t e m p e r a t u r e and d e n s i t y a r e c l o s e t o t h e c r i t i c a l po 3 T = 126 0K, p = 0.45 g/cm ) f o r l i q u i d n i t r o g e n . 54 B Convergence of the expanded potential Figs. 4,5 and 6 show the convergence of the expanded potential for end-end, T-shape and p a r a l l e l orientations respectively, with I =0.33 and e = 0.4587, which are appropriate for nitrogen at 79°K. The 0th, 2nd, 4th and 6th order expansions are compared with the f u l l atom-atom potential (eq. (4.1)). In the T-shape and p a r a l l e l orienta-t i o n s , both the 4th and the 6th order expansions are indistinguishable from the f u l l potential on the scale used i n the Figures. The convergence i s somewhat poorer for the end-end orientation (Fig. 4) but the 6th order expansion i s much closer to the f u l l potential than those of lower order. One noticeable feature of the potential i s that as the orientation changes from end-end to p a r a l l e l (cf. Figs. 4-6), the potential becomes shorter ranged and the well increases i n depth. To further i l l u s t r a t e how the expanded potential deviates from the f u l l p o t e n t i a l , plots of the f u l l potential together with the expansions are shown i n Figs. 7 and 8 for orientations varying from end-end to T-shape at fixed distances. At a separation of 1.0c, where the potential i s purely repulsive, good agreement i s found near T-shape orientations, but a l l orders of expansion are i n poor agreement with the f u l l potential near end-end orientations (cf. Fig. 7). Nevertheless, we would not expect deviations i n the strongly repulsive region to seriously influence the structural properties or the average configurational energy of the system, since strongly repulsive orientations w i l l be r e l a t i v e l y improbable regardless of the absolute magnitude of the interaction. However, th i s i s not necessarily true of the pressure which depends upon the potential gradient and i s more 55 sensitive to the repulsive part of the potential. This sensitivity-is demonstrated in § 4;3. Fig. 7 is for the separation of 1.3a which is near the beginning of the potential well. Here the agreement is much better but the end-end orientations s t i l l constitute the worst f i t s . We emphasize that in the physically, important regions near the potential minimum the agreement is very good even at small separations. The coefficients of u(12) expanded in rotational invariants 224 are shown in Fig. 9. Included also in this figure i s u (r) when the quadrupole interaction is included (cf. eq. (2.21)). From Fig. 9, one can see that the coefficients corresponding to the projections (066), (246), (244), (242), and (044) are relatively short ranged and that there is 044 l i t t l e or no long range contribution. With the exception of u (r) these terms occur only in the 6th order expansion and we shall see below that they make very l i t t l e contribution to the liquid structure. The coefficient u ^ ^ ( r ) is the most important since i t has the deepest attractive well and the longest range. This is perhaps expected since nitrogen is close to spherical and hence the angle independent part of the potential w i l l make a large contribution. However, the coefficient of two other terms are significant as well. These are 022 224 u (r) and u (r) when the quadrupole interaction is added. The important feature of these projections is that they are relatively long-ranged and tend to dominate the angular part of the intermolecular 224 force. The particular importance of u (r) when the quadrupole inter-action i s included i s illustrated in Figs. 10-12. We see that in this case the end-end (cf. Fig. 10) and parallel (cf. Fig. 12) interactions become purely repulsive but those for the T-shape configuration become 56 Figure 4 The convergence of the pair potential for the end-end configu-ration. The Oth ( ), 2nd (---), 4th ( ), and 6th ( ) order expansions are compared with the f u l l atom-atom potential ( ) for £* = 0.33 and e*=0.4587. 57 9"l 80 — i — 00 ro-(2o,l5'J)nfif 80-—r~ 9 > 03-58 Figure 5 The convergence of the pair potential for the T-shape configuration. The notation and parameters are the same as those in Fig. 4. Note that the 4th and 6th order expansions are unresolvable from the f u l l atom-atom potential on the scale of the plot. 59 Figure 6 The convergence of the pair potential for the parallel configuration. The notation and parameters are the same those in Fig. 4. 61 62 Figure 7 The convergence of the p a i r p o t e n t i a l f o r f i x e d distance but varying o r i e n t a t i o n s . The 2nd ( ), 4th ( ), and 6th ( ) order expansions are compared with the f u l l p o t e n t i a l ( ) at an intermolecular separation of 1.0a with 9^ varying from 0 to it. The physical parameters are as i n F i g . 4. 10.0 29.0 48.0 j3u(r,e1.e2^ ) I r = 1.0,0! = 0 0 , ^ = 0 0 105.0 67.0 124.0 143.0 Q3 r o UJ o. cu" £9 64 Figure 8 The convergence of the pair potential for fixed distance but varying orientations. The various expansions are compared with the f u l l potential as in Fig. 7 for an intermolecular separation of 1.3a. 65 66 Figure 9 The coefficients u m n 4 ( r ) of the pair potential u(12) for Z* = 0.33, e* = 0.4587 and Q*2 = 0.5179. The (mn£) are as .. , rr- • * 606. . . 066, , 246, . indicated. The coefficients u (rj and u ( r j , u (r) . 426, , 244, , , 424, , 242, , , 422, , 404, and u ( r ] , u (r) and u (r) , u (r) and u (r) , u (r) and u ^ 4 4 ( r ) , u 2 (^ 2(r) and u ^ 2 2 ( r ) are equivalent and hence 220 only one of each pair i s shown. Also -u (r) rather than u 2 2 ^ ( r ) i s shown in the plot. L9 68 Figure 10 The effect of the quadrupolar interaction upon the total pair potential, u(12) = u M(12) + U Q Q ( 1 2 ) , for the end-end o o configuration (6j = 0 , &2 = 0 , cf>). The parameters are as in Fig. 9. The solid line i s u M and the dash-dot line i s U M + U Q Q " 70 Figure 11 The effect of the quadrupolar interaction upon the pair potential for the T-shape configuration (6^ = 90°, 6^  = 0°, 40- See the caption of Fig. 10 for details. Note that u ^ increases the depth of the potential well. 72 Figure 12 The effect of the quadrupolar interaction upon the pair potential for the parallel configuration (8^ = 90°, 6^ ~ 90°> (f> = 0°). See the caption of Fig. 10 for details. 74 more attractive (cf. Fig. 11). We shall see below that has a significant influence upon the structure of liquid nitrogen. 2. LHNC Results A The pair correlation function The solution to the 0-Z equation with the LHNC closure approxi-mation gives the pair correlation function h(12) (cf. eq. (3.42)). Fig. mn Q. * 13 shows the h (r) obtained from a LHNC calculation with p = 0.6220, * * e = 0.4587 and I = 0.3292. Calculations were carried out both with and without the quadrupole interaction. As expected the quadrupolar 224 interaction makes a large contribution to h (r) but the other projections are relatively unaltered. With the inclusion of the 224 quadrupole interaction, h (r) becomes more negative and longer 404 ranged. Fig. 13 shows that h (r) is insignificant compared to the other coefficients. Indeed, a l l of the 6th order coefficients (i.e. ,242 , , , 422, , , 244, , , 424, . ,246, . , ,426, . ., h (r), h (r), h (r), h (r), h (r) and h (r)) are either similar 404 in magnitude or smaller than h (r) and are not included in Fig. 13. The dominant projection is h ^ ^ ( r ) which has the highest peak and exhibits more structure in the long range part.(cf. Fig. 13). As we have mentioned before, the coefficients of the (202) and (022) projections are expected to be particularly important. This can be 202 clearly seen in Fig. 13 where near r = 1.1a, h (r) is nearly as large in magnitude as h ^ ^ ( r ) . Clearly for many orientations, the (022) and (202) terms wi l l dominate the g(12) expansion. The pair distribution function, g(12), expressed as a series expansion is useful for analytic purpose but gives l i t t l e direct physical 75 Figure 13 The coefficients h™ 1*(r) of the pair correlation function, h(12), at e* = 0.4587, p* = 0.6220, and I = 0.3292 for Q*2 = 0 and 0.3323 for Q*2 = 0.5179. The (mn£) are as . _. . . . ,606, , , ,066, , ,246, , indicated. The coefficients h (r) and h (rJ, h (r) , , 426, , ,244 , , , ,424, , ,242, , , ,422, . ,404 and h (r), h (r) and h (r), h (r) and h (r), h (r) and h ^ 4 4 ( r ) , h 2 ^ 2 ( r ) and h ^ 2 2 ( r ) are equivalent and hence * only one of each pair i s shown. This value of e corresponds to 79°K which is just below the normal boiling point of liquid nitrogen. 9L 77 i n s i g h t i n t o t h e n a t u r e o f t h e o r i e n t a t i o n a l c o r r e l a t i o n s . The o r i e n t a t i o n a l s t r u c t u r e can be b e s t i l l u s t r a t e d by e x a m i n i n g g(12) f o r s p e c i f i c v a l u e s o f t h e a n g u l a r c o o r d i n a t e . I n F i g s . 14 and 15 we show g(12) f o r s e l e c t e d o r i e n t a t i o n s b o t h w i t h and w i t h o u t q u a d r u p o l a r i n t e r a c t i o n . As m ight be e x p e c t e d , t h e n e a r e s t n e i g h b o u r peaks i n F i g . 14 o c c u r n e a r s e p a r a t i o n s f o r w h i c h t h e p a i r p o t e n t i a l i s a minimum ( c f . F i g s . 4-6). The peaks o c c u r a t 1.31a, 1.20a and 1.14a f o r end-end, T-shape and p a r a l l e l p r i e n t a t i o n s , r e s p e c t i v e l y , and t h e c o r r e s p o n d i n g minima i n t h e p o t e n t i a l s o c c u r a t 1.41a, 1.26a and 1.12a. T h i s i n d i c a t e s a c l o s e c o n n e c t i o n between t h e f l u i d s t r u c t u r e and t h e shape o f t h e p a i r p o t e n t i a l . When U Q Q i s n o t i n c l u d e d t h e h e i g h t s o f t h e f i r s t peak a r e o f s i m i l a r magnitude f o r a l l t h r e e o r i e n t a t i o n s ( c f . F i g . 1 4 ) . W i t h t h e i n c l u s i o n o f u ^ , t h e r e l a t i v e peak h e i g h t s change d r a s t i c a l l y as shown i n F i g . 15. The f i r s t peak f o r t h e T-shape o r i e n t a t i o n becomes much h i g h e r a t t h e expense o f t h e o t h e r two w h i c h d i m i n i s h i n magnitude. T h i s shows t h a t , as e x p e c t e d , a s t r o n g q u a d r u p o l a r i n t e r a c t i o n g r e a t l y i n c r e a s e s t h e p r o b a b i l i t y o f T-shape o r i e n t a t i o n s f o r n e a r e s t n e i g h b o u r p a i r s . The second n e i g h b o u r peaks i n F i g . 14 o c c u r a t 2.20a and 2.48a f o r end-end, 2.36a f o r T-shape, and 2.23a f o r p a r a l l e l o r i e n t a t i o n s . The most p r o b a b l e m o l e c u l a r c o n f i g u r a t i o n g i v i n g r i s e t o t h r e e o f t h e s e peaks a r e i l l u s t r a t e d below. F o r t h e s e c o n f i g u r a t i o n s t h e second n e i g h b o u r d i s t a n c e s can be e s t i m a t e d f r o m t h e p o s i t i o n s o f t h e f i r s t n e i g h b o u r peaks as shown i n t h e i l l u s t r a t i o n . I t i s found t h a t t h e v a l u e s o b t a i n e d a r e i n good agreement w i t h a c t u a l o b s e r v e d peak p o s i t i o n s . 78 Figure 14 The pair distribution function g(12) for parallel, T-shape and end-end configurations as shown. The physical parameters are p* = 0.6220, e* = 0.4587, I* = 0.3292 and Q*2 = 0. 79 (Zc)'lc5'J)6 80 Figure 15 The pair distribution function for par a l l e l , T-shape and end-end configuration as shown. The physical parameters are •k 'fc * ^ 9 e = 0.6220, p = 0.4587, I = 0.3323 and Q = 0.5179. Note that by far the highest f i r s t neighbour peak is found for the T-shape configuration. 81 82 K L20a H I I I I I I O - H O o o h- 1.14a 1.20aH I t I I I I I 2.40a H I— 1.14oH I I 2.34a 2.28a However, the peak at 2.20a in the end-end correlation function i s s t i l l unaccounted for. A possible explanation for the peak at 2.20a in the correlation function for the end-end orientation i s the configuration f— l . l O a H I 6 "T 0.57o A o — o B o o - - L 83 where molecule C and D are perpendicular to the paper and parallel to each other, and molecules A and B are end-end. If we assume this to be the case, we can calculate the expected distance between the neighbouring molecules A and C. The calculated distance turns out to be 1.24a, whereas the pair distribution function for this configuration (i.e. Q = 90°, 6 2 = 27°, o> = 90°) has a f i r s t neighbour peak at 1.17a. These numbers are not close enough to show conclusively that this indeed is the configuration giving rise to the peak at 2.20a. However, the distances for this configuration may well be somewhat greater than these we predict from near neighbour correlations. An accurate determination of these distances would require the four-body correlation function not given by the present theory. The above discussion serves to show that the positions of the second neighbour peaks are strongly influenced by near neighbour correlations. Therefore, by studying the pair distribution, g(r, fi^, fl2), i t is possible to gain significant insights into the f l u i d (Structure. B Comparison with molecular dynamics- results. The physical parameters we have chosen in our LHNC calculations are those of Cheung and Powles^'7 and are listed in Table III. Cheung and Powles study the pair distribution function by calculating several coefficients in the expansion ^ " 4* hI2m \z2n^ \ l 2 m ™ (4.2) 84 where 1 2 (4TT) 1 2 and Y™(fl) represents the spherical harmonics described in Appendix I. We note that this expansion of g(12) i s different from ours in that we use a basis; set of rotational invariants. We can, however, relate some of the coefficients in the two expansions by taking linear combinations of rotational invariants. Consider, for example, S 2 2 g ( r ) ' It is easy to show that * 2 2 0 (12) = | ( 9 T l 2 T 2 2 - ST/ - 3T 2 2 • 1) =4*22V) *l * 2 2 2 (12) + i * 2 2 4 C 1 2 5 where the $ m n 4(12) are rotational invariants and T , iand are defined by eq. (2.9). Substituting eq. (4.6) into eq. (4.2) gives g 2 2 0 ( r ) = — ~ JdJl dn g(12) [-1 $ 2 2 0(12) t.y $ 2 2 2(12) $ 2 2 4(12)] (4ir) (4.7) = - i — {-\ h 2 2 0 ( r ) J > 2 2 ° ( 1 2 ) ] 2 d^dp^ + .... (cont'd) (4ir) (4.3) (4.4) (4.5) (4.6) 85 + | h 2 2 2 ( r ) / [ $ 2 2 2 C 1 2 ) ] 2 d f l ^ + ^  h 2 2 4(4) / [ $ 2 2 4 ( 1 2 ) ] 2 d O i d 0 2 (4-8) 2 , 2 2 0 , , 2 , 222, _ 8 ,224, , = h (r) + j h (r) + ^  h (r) (4.9) where the h m n ^ ( r ) are the coefficients of the pair correlation function in our expansion (cf. eq. (3.42)). Three other coefficients, gQ 0 0( r)> 7 g _ n(r) and g d n n( r)» a*"e reported by Cheung and Powles and these are simply related to our coefficients by the equations g000 (r) - g°0 0(r) = h°°°(r) + 1 , (4-10) 2_ h202 ( 4.ll) and In Figs. 16, 17 and 18 the LHNC approximation for the angle-averaged (radial) pair distribution function is compared with the 6 7 molecular dynamics results of Cheung and Powles ' for the three states given i n Table III. At low density and high temperature (cf. Fig. 18) the long range part of the pair correlation function is very close to that obtained in the molecular dynamics calculations. The agreement is somewhat poorer when comparisons are made at higher densities (cf. Fig. 16 and 17). This is to be expected since integral equation 2 theories are generally less accurate at high density. At a l l three densities the f i r s t peak in the LHNC result i s higher and somewhat 86 Figure 16 The angle-averaged pair distribution function for liquid ic ie it nitrogen at p = 0.6964, e = 0.6451 and £ = 0.3292. The solid line is the LHNC result and the open circles are from the molecular dynamics calculations of Cheung and Powles.^ * Note that this value of e corresponds to 59°K which is a supercooled fl u i d on the vapor-pressure curve for nitrogen. 87 F i g u r e 17 The a n g l e - a v e r a g e d p a i r 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 n i t r o g e n at p = 0.6220, e = 0.4587 and i = 0.3292 f o r Q*2 = 0 and 0.1223 f o r Q*2 = 0.5179. The s o l i d l i n e and t h e open c i r c l e s a r e as i n F i g . 16, and t h e dashed l i n e i s t h e m o l e c u l a r dynamics r e s u l t o f Cheung and P o w l e s 7 when u ^ i s i n c l u d e d . I n t h e LHNC a p p r o x i m a t i o n , t h e e f f e c t o f upon t h e a n g l e - a v e r a g e d p a i r d i s t r i b u t i o n f u n c t i o n i s t o o s m a l l t o be shown i n t h e F i g u r e . 89 90 F i g u r e 18 The a n g l e - a v e r a g e d p a i r 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 n i t r o g e n a t p* = 0.3500, e* = 0.2976 and Z* = 0.3292. The n o t a t i o n i s as i n F i g . 16. 91 92 narrower than that given by the computer simulations. At high density, the discrepancy i n the peak height can be as much as 30%. The positi o n s of the f i r s t peak i n Fig s . 16-18 are 1.20a, 1.18a and 1.23a, r e s p e c t i v e l y , for the molecular dynamics ca l c u l a t i o n s and 1.20a, 1.21a and 1.23a, re s p e c t i v e l y , i n the LHNC approximation. The fact that the positi o n s of the f i r s t peak are i n good agreement suggests that f o r s t r u c t u r a l p r o p e r t i e s , at l e a s t , the rep u l s i v e part of the atom-atom p o t e n t i a l i s adequately represented by the 6th order expansion since i t i s known that the p o s i t i o n of the f i r s t peak i n the p a i r d i s t r i b u t i o n function i s strongly dependent upon the density and the rep u l s i v e part of the p o t e n t i a l . The second peaks i n Figs. 16-18 occur at 2.17a, 2.25a and 2.43a, r e s p e c t i v e l y , i n the molecular dynamics c a l c u l a t i o n s and at 2.28a, 2.30a and 2.43a i n the LHNC theory. The computer simulations give broader second peaks and at the higher d e n s i t i e s the second neighbour separations are somewhat smaller than the LHNC r e s u l t s . F i g . 17 shows the influence o f quadrupolar i n t e r a c t i o n s upon the angle-averaged p a i r d i s t r i b u t i o n function. In the molecular dynamics c a l c u l a t i o n , the i n c l u s i o n of quadrupolar forces s l i g h t l y increases the height o f the f i r s t peak. In the LHNC approximation the increase i n height i s very small and i s not r e a d i l y apparent i n the p l o t . The c o e f f i c i e n t s , g200^ a n d g 2 2 0 ^ a t p = ° - 6 2 2 0 > a r e i l l u s t r a t e d i n Fi g . 19 f o r the p o t e n t i a l without u ^ , and i n F i g . 20 with U Q Q included. I t can be seen from F i g . 19 that the LHNC and molecular dynamics r e s u l t s f o r g2Qo'-r^ a r e q u a l i t a t i v e l y s i m i l a r but that the LHNC approximation gives a much deeper negative well. However, the agreement i s much poorer f o r the c o e f f i c i e n t §220^''* ^ e ^HNC 93 Figure 19 The coefficients g 2 o n ( r ) a n d g 2 2 0 ^ f ° r l i c * u i d n i t r ° g e n * * at 79°K. The physical parameters are p = 0.3292, e = 0.622 I = 0.4587 and Q =0. The LHNC results are compared with 7 the molecular dynamics (MD) results of Cheung and Powles. The curves represent the following: ( ): g 2 0 Q ( r ) ( L H N C ) > ( ): g 2 2 0 ( r ) ( L H N C ) > g 2 0 0 ( r ) ( M D ) ; a n d (° ° ° ) : g 2 2 0 ( r ) (MD). 94 95 Figure 20 The coefficients g 2 0 o^ r ^ a n d g 2 2 0 ^ f ° T l i q u i d n i t r o S e n * * at 79°K. the physical parameters are SL = 0.3323, p = 0.6220, £ = 0.4587 and Q = 0.5179. The notation i s as in Fig. 19. 96 97 t h e o r y n o t o n l y g i v e s a much d e e p e r n e g a t i v e w e l l b u t a l s o m i s s e s e n t i r e l y t h e p o s i t i v e peak o c c u r r i n g n e a r r = 1.1a i n t h e m o l e c u l a r dynamics r e s u l t . We n o t e t h a t t h e m o l e c u l a r dynamics r e s u l t f o r g^Q^Cr) has a p o s i t i v e peak a t t h e same p o s i t i o n w h i c h i s a l s o m i s s i n g i n t h e LHNC a p p r o x i m a t i o n . However, t h e c o e f f i c i e n t , S ^ Q Q CT) i s r e l a t i v e l y v e r y s m a l l i n magnitude and t h e (400) t e r m does n o t make an i m p o r t a n t c o n t r i b u t i o n t o g ( 1 2 ) . F i g . 20 i s v e r y s i m i l a r t o F i g . 19 and t h e comments made above a p p l y e q u a l l y h e r e . When u ^ i s i n c l u d e d t h e n e g a t i v e w e l l i n g 2 2 Q ( r ) b e c o m e s d e e p e r (-20%) and g 2 0 0 ( r ) s n r i n k s (-10%) i n b o t h t h e m o l e c u l a r dynamics and LHNC c a l c u l a t i o n s . W i t h t h e e x c e p t i o n o f t h e p o s i t i v e peaks i n g 2 0 n ( r ) and g 4 0 0 ( r ) t n e L H N C t h e o r y p r e d i c t i o n s a r e i n q u a l i t a t i v e agreement w i t h t h e m o l e c u l a r dynamics r e s u l t s . We n o t e t h a t t h i s i s a l s o t r u e o f g ( r , si , Q^) f ° r t n e o r i e n t a t i o n s d i s c u s s e d above ( c f . F i g . 14) and t h a t t h e LHNC t h e o r y and m o l e c u l a r dynamics c a l c u l a t i o n s a r e i n agreement c o n c e r n i n g t h e dominant p a i r o r i e n t a t i o n s and t h r e e - b o d y c l u s t e r s one i s l i k e l y t o f i n d i n l i q u i d n i t r o g e n . I n summary, t h e most s e r i o u s d i s c r e p a n c i e s between t h e LHNC and m o l e c u l a r dynamics r e s u l t s o c c u r i n t h e r e g i o n r = 1.0a t o 1.2a. Two p o s s i b l e f a c t o r s may c o n t r i b u t e t o t h e s e d i s c r e p a n c i e s . F i r s t l y , t h e 6 t h o r d e r e x p a n s i o n may n o t be s u f f i c i e n t l y c o n v e r g e d t o r e p r e s e n t t h e f u l l atom-atom p o t e n t i a l and, s e c o n d l y , d i s c r e p a n c i e s may a r i s e from t h e LHNC c l o s u r e a p p r o x i m a t i o n . As n o t e d e a r l i e r ( c f . § 4.1 B) t h e r e i s some i n a c c u r a c y i n t h e p o t e n t i a l e x p a n s i o n p a r t i c u l a r l y f o r t h e end-end o r i e n t a t i o n s . However, we do n o t b e l i e v e t h i s t o be t h e maj o r p r o b l e m s i n c e t h e a t t r a c t i v e p a r t s o f t h e atom-atom p o t e n t i a l 98 are well represented by the 6 th order expansion. This question could only be completely resolved by computer simulations using the expanded po t e n t i a l , but i n a l l l i k e l i h o o d the problem l i e s i n the LHNC theory. In chapter I I I , we mentioned that the HNC theory i t s e l f i s r e l a t i v e l y poor for short ranged interactions and we cannot expect the linearized version to be any better. The fact that the LHNC theory works very well for systems with e l e c t r i c a l multipolar forces i s perhaps due to the long range nature of these interactions. The atom-atom potential i s very short-ranged and hence i t i s not surprising that the LHNC theory gives poorer results for t h i s system. Moreover, i t i s l i k e l y that the LHNC closure does not contain enough coupling between the spherical and angular parts of g (12 ) . In the LHNC approxi-mation, the coupling occurs only through the 0-Z equation and not i n the closure relations themselves. This i s perhaps why i n the LHNC theory the effect of  UQQ upon g ^ ^ ( r ) i s small compared with the molecular dynamics r e s u l t . Therefore,' i t i s probable that i n order to obtain better results for the short range behavior of homonuclear diatomic systems one must derive better closure approximations. Some suggestions as to how t h i s might be done within the present theoretical framework are discussed i n chapter V. 99 3. Thermodynamic Properties Eqs. (3.6 - 3.8) give expressions for the configurational energy, pressure and isothermal compressibility, respectively, in terms of g(12) and u(12). Expanding the pair distribution function and the pair potential in rotational invariants, these equations become J L = E Z J u ^ C r ) g m n l M 41rr2drH$mn* ( 1 2 ) ] 2 dndn 2 . NkT 2 k T ( 4 i r ) 2 ^ (4.13) _JL_ = i _ P = ( I i g ^ f r ) 4 l r r 3 d r / [ ^ ( 1 2 ) 1 ^ ^ 2 , pkT 6 k T ( 4 7 T ) 2 m n £ 9r (4.14) and p<kT = 1 + 4irp j [g0°°(r) - 1] r 2 dr (4.15) where the u m i ( r ) are defined by eq. ( 2 . 2 5 ) . The g"1"2, (r) are given by 000, , ,000. . . (A lfi<, g (r) = h (r) + 1, (4.lb) and g m n £ ( r ) = h ^ C r ) (4.17) for a l l other projections, where the h™1*-(r) are as in eq. (3.42) configurational energy, pressure, and isothermal compressibility for the three states given in Table I I I were obtained from eqs. (4.13 - 15) and the results are summarized in Table IV. The three state points The 100 Table IV. Thermodynamic Properties of the nitrogen-like fl u i d . e* p* Q*2 ^(LHNC) p<kT(LHNC) (LHNC) (MD)6'7 (Experiment) 0.6451 0.6964 0 -20.5 -1.71 -14.9 -12.2 -11.9 0.4587 0.6220 0 -17.5 -0.83 -10.5 - 7.5 - 7.2 0.2976 0.3500 0 - 4.6 0.81 - 3.3 0.4587 0.6220 0.5179 -19.2 -0.84 -11.6 - 7.2 101 (cf. Table III) used in our calculations cover the liquid range of nitrogen from a supercooled fl u i d at 59°K to a state near the c r i t i c a l point at 124°K.* The configurational energies obtained from the LHNC theory are given in Table IV and at the high densities a comparison is made with 6 7 the molecular dynamics results of Cheung and Powles. ' The LHNC energies are much more negative than those given by the computer * simulations with the discrepancy being 25% at p = 0.6964 and 40% * at p = 0.6220. These large discrepancies are not unexpected for several reasons. As mentioned earlier the expanded potential i s more negative than the f u l l potential for several configurations (cf. Figs. 4-6) and this w i l l make some contribution to the discrepancy. Also the LHNC theory does not accurately predict the short range behavior of g(12). In the LHNC theory the contributions made to the total configurational energy by the various terms in eq. (4.13) are as follows: (000), 60%; (022), 13%; (202), 13%; (224), 7%; and the remaining 7% is the combined contribution of a l l other terms. We have previously seen that the LHNC approximation overestimates the f i r s t neighbour peak in g^^(r) (cf. Figs. 16-18) and that the coefficients , , 2 ,202, , , , 2 ,220, , 2 ,222, , 8 ,224, . W R ) = 7 = H W a n d g 2 2 0 ( r ) = "5 h ( r ) + 5 h ( r ) + 5 h ( r ) have negative wells which are much deeper than the molecular dynamics results (cf. Figs. 19 and 20). Thus a l l of these terms make important contributions to the total energy which are negative and too large in The normal freezing point and c r i t i c a l point are 64°K and 126°K, respectively. 102 magnitude. In a l l likelihood these inaccuracies in g(12) are the principal source of error in the LHNC energies. The negative compressibility for the high density region probably results from the fact that the height of the f i r s t peak in g^^(r) is overestimated by the LHNC theory. The huge negative pressures given by the LHNC theory are non-physical. The problem is that the pressure depends upon the gradient of the potential and i s , therefore, much more sensitive to inaccuracies in the potential expansion. Also the pressure is very sensitive to the short range behaviour of g(12) and the discrepancies described above will make a large contribution to the error in the pressure. At the densities we consider, the (000) term makes the only positive contribution to - 1. The negative part i s dominated by (202) and (022) and i t s magnitude is overestimated by the LHNC theory. We note that pressures are notoriously d i f f i c u l t to calculate theoretically and that often even computer simulations are not sufficiently accurate to give good values. In their molecular dynamics studies Cheung and Powles adjust parameters in their potential in order to f i t the experimental pressure data. It is lik e l y that one could carry out a similar procedure for the expanded potential and readjust £, a and e in order to obtain physically r e a l i s t i c pressures. However, in the absence of an exact theory, this would require computer simulations and we are presently more interested in setting up a general theoretical framework than in obtaining accurate potentials for nitrogen. Both the internal energy and the pressure drop significantly when u ^ is included. This is expected in the LHNC result since the 103 c o e f f i c i e n t J^O^1"'' 2^" v e s a m u c n d e e p e r w e l l when i s p r e s e n t . T h i s i s , however, i n c o n t r a d i c t i o n t o t h e computer s i m u l a t i o n r e s u l t s , w h i c h shows a s l i g h t i n c r e a s e i n t h e i n t e r n a l energy. T h i s may 202 perhaps be due t o t h e f a c t t h a t i n t h e LHNC t h e o r y , h ( r ) i s not much i n f l u e n c e d by u ^ whereas i n t h e m o l e c u l a r dynamics c a l c u l a t i o n s t h e w e l l d e p t h i s s i g n i f i c a n t l y r e d u c e d ( c f . F i g s . 19 and 20). A g a i n , t h i s i s p r o b a b l y t h e r e s u l t o f i n s u f f i c i e n t c o u p l i n g between t h e v a r i o u s p r o j e c t i o n s i n t h e LHNC a p p r o x i m a t i o n . I n summary t h e LHNC t h e o r y g i v e s v e r y p o o r r e s u l t s f o r t h e thermodynamic p r o p e r t i e s . The c o n f i g u r a t i o n a l e n e r g i e s a r e s u b s t a n t i a l l y u n d e r e s t i m a t e d and t h e p r e s s u r e s a r e n e g a t i v e and n o n - p h y s i c a l . The d i s c r e p a n c y i n t h e energy i s most l i k e l y due t o i n a c c u r a c i e s i n t h e LHNC t h e o r y and c o u l d be c o r r e c t e d w i t h a b e t t e r c l o s u r e a p p r o x i m a t i o n . However, i n t h e case o f t h e p r e s s u r e i t i s p r o b a b l e t h a t t h e 6 t h o r d e r p o t e n t i a l e x p a n s i o n i s not s u f f i c i e n t l y converged and i n o r d e r t o o b t a i n p h y s i c a l l y r e a l i s t i c v a l u e s one must e i t h e r r e t a i n more terms i n t h e e x p a n s i o n o r r e a d j u s t t h e p a r a m e t e r s i n t h e p o t e n t i a l . 103 CHAPTER V CONCLUSION I n t h e p r e s e n t s t u d y , two b a s i c p roblems have been c o n s i d e r e d . I n t h e f i r s t p l a c e , we have a t t e m p t e d t o o b t a i n a r e p r e s e n t a t i o n f o r t h e p a i r p o t e n t i a l o f homonuclear d i a t o m i c m o l e c u l e s w h i c h a l l o w s t h e s h o r t r a n g e a n i s o t r o p i c f o r c e s and r e l a t i v e l y l o n g r ange m u l t i p o l a r i n t e r a c t i o n s t o be t r e a t e d s i m u l t a n e o u s l y i n a u n i f o r m manner. We choose t o examine t h e so c a l l e d i n t e r a c t i o n - s i t e model because such systems have r e c e i v e d a good d e a l o f a t t e n t i o n i n t h e l i t e r a t u r e . S e c o n d l y , we have s e t up a t h e o r e t i c a l framework w h i c h can be u s e d t o s t u d y t h e s t r u c t u r a l and thermodynamic p r o p e r t i e s o f homonuclear d i a t o m i c f l u i d s and have a p p l i e d t h e r e l a t i v e l y s i m p l e l i n e a r i z e d h y p e r n e t t e d - c h a i n a p p r o x i m a t i o n t o a n i t r o g e n - l i k e f l u i d a t l i q u i d d e n s i t i e s . The p r i m a r y p u r p o s e o f t h i s work has n o t been so much 105 t o o b t a i n a c c u r a t e n u m e r i c a l r e s u l t s f o r d i a t o m i c systems but t o l a y a t h e o r e t i c a l groundwork w h i c h can be a p p l i e d i n f u t u r e s t u d i e s . The f o l l o w i n g i s a b r i e f summary o f o u r p r i n c i p a l r e s u l t s and c o n c l u s i o n s . Our b a s i c p r o c e d u r e has been t o expand atom-atom p o t e n t i a l s o f t h e Lennard-Jones (12-6) t y p e i n a s e r i e s o f r o t a t i o n a l i n v a r i a n t s w h i c h s e p a r a t e t h e a n g l e i n d e p e n d e n t and a n g l e dependent p a r t s o f t h e p o t e n t i a l . We r e t a i n terms t o 6 t h o r d e r i n l/r and i n v e s t i g a t e t h e con v e r g e n c e o f t h e expanded p o t e n t i a l . I t i s fo u n d t h a t t h e s e e x p a n s i o n s g i v e an a c c u r a t e r e p r e s e n t a t i o n o f t h e f u l l atom-atom p o t e n t i a l e x c e p t i n t h e n e i g h b o u r h o o d o f end-end o r i e n t a t i o n s . Of t h e 16 c o e f f i c i e n t s w h i c h o c c u r i n t h e 6 t h o r d e r e x p a n s i o n o f t h e p a i r p o t e n t i a l , u ^ ^ ( r ) , 202 022 224 u ( r ) , u ( r ) and u ( r ) a r e t h e l o n g e s t ranged and make t h e most s i g n i f i c a n t c o n t r i b u t i o n s . I t i s a l s o found t h a t t h e i n c l u s i o n o f t h e q u a d r u p o l a r i n t e r a c t i o n , u Q Q » n a s a s i g n i f i c a n t e f f e c t , r e n d e r i n g t h e p a i r p o t e n t i a l more r e p u l s i v e f o r end-end and p a r a l l e l c o n f i g u r a t i o n s , and more a t t r a c t i v e f o r t h e T-shape c o n f i g u r a t i o n . Our t h e o r e t i c a l t r e a t m e n t i s an i n t e g r a l e q u a t i o n method based upon t h e O r n s t e i n - Z e r n i k e e q u a t i o n . By ex p a n d i n g t h e c o r r e l a t i o n f u n c t i o n s i n r o t a t i o n a l i n v a r i a n t s t h e O r n s t e i n - Z e r n i k e e q u a t i o n i s r e d u c e d t o a s e t o f l i n e a r e q u a t i o n s i n F o u r i e r s p a c e . These e x p r e s s i o n s c o u p l e d w i t h t h e LHNC c l o s u r e r e l a t i o n s c o n s t i t u t e a c l o s e d s e t o f e q u a t i o n s w h i c h can be s o l v e d i t e r a t i v e l y t o g i v e n u m e r i c a l r e s u l t s f o r t h e p a i r c o r r e l a t i o n f u n c t i o n . I t i s found t h a t t h e c o e f f i c i e n t s o f t h e p a i r c o r r e l a t i o n f u n c t i o n a r i s i n g from t h e 6 t h o r d e r t e r m a r e 202 022 r e l a t i v e l y i n s i g n i f i c a n t . The c o e f f i c i e n t s , h ( r ) , h ( r ) and 1 0 6 224 h (r) are very important and tend to dominate the angular part of the pair correlation function. In l i g h t of the fact that the 6th order coef f i c i e n t s are r e l a t i v e l y small, i t i s perhaps not necessary to retain terms beyond 4th order i f one wishes only to study the structural properties. I t i s also found that the quadrupolar i n t e r -action has a strong influence upon the structure greatly increasing the r e l a t i v e p robability of T-shape orientations. Comparisons with molecular dynamics calculations show that the LHNC theory gives an accurate description of the long-ranged behaviour of the p a i r correlation function but i s rather poor for the short ranged part. Nevertheless, the LHNC theory does give e s s e n t i a l l y correct results for the structural properties of the nitrogen-like f l u i d . For example common pair orientations and the nature of some three-body clusters are correctly predicted by the LHNC theory. However, due to the inaccuracies i n the short-ranged behaviour of g(12) the thermodynamic properties given by LHNC theory are i n very poor agreement with the computer simulations. Although the LHNC theory i s not as successful as one would l i k e , we have, nevertheless, l a i d the foundation for a better theory. I t i s possible to find better closures which f i t into our theoretical framework. One such closure which has been applied to multipolar f l u i d s i s the so-called quadratic HNC (QHNC) equation^, which retains both l i n e a r and quadratic terms i n the logarithmic expansion of eq. (3.37), This approximation gives more coupling between the various coefficients of h(12) but i s generally more d i f f i c u l t to solve numerically than the LHNC equation. One might also consider expanding the Percus-Yevick 107 approximation which i s known to give good results for systems with spherically symmetric short-ranged interactions. Therefore, one could perhaps use a Percus-Yevick type closure to calculate the short-ranged behaviour and the LHNC equation for the long-ranged part. I t i s l i k e l y that such mixed closure approximation would give accurate predictions for the structural properties of homonuclear diatomic f l u i d s at a l l densities. Another extension of the present work would be to investigate the properties of dipolar diatomic f l u i d s . This could be done by applying present methods since the potential expansion and integral equation theory can both be easily adapted to heteronuclear molecules. Some projections which do not appear i n the homonuclear case, such as (110), (112) etc., would occur i n t h i s extension of the theory. It would be interesting to apply the LHNC approximation to study the d i e l e c t r i c constant of polar diatomic f l u i d s since t h i s property should be p a r t i c u l a r l y sensitive to the coupling between long and short-ranged anisotropic forces. 108 APPENDIX I ROTATIONAL INVARIANTS AND SPHERICAL HARMONICS The rotational invariants for linear molecules are defined by •""^12) - V J X ^  ( y " J> W D0>2> D i C*12> ( 2 - 1 9 ) where ( 1 2 ) is the 3-j symbol. D (0) is the Wigner matrix m2 m uy-element defined by D > ) - C - ^ ) * C A i . D °^ 2m + 1 m where Y V(Q) is the spherical harmonic 6 1 for Euler angle Q = 6, <>, m v^ ra *1 - f nw / 2 n i l CntjOl P^ (Cos 0) elw* CM. 2) Ym(0,<t>) " C-l) / 4 l T ( m +y)! m V and v , 2 y/2 dm+y 2 .,m " ^ ( 1 - X 1 ( X • -m < y < m- (AI.3) fmn£ i s d e f i n e d b y finnX, = (2.19) .m n l0 0 0J o 109 with exception of f 2 2° = -2/5 and f 2 2 4 = sTIsJl. If we align r J 2 with the z-axis, we have D * ^ ) = 6 ^ a n d therefore 9J - % fx ^  The 3-j symbol ( J ; ^  m ) is zero unless a + i 2 > 4 > I' - *2 I a n d ( A I * 5 ) ^ m - m (AI.5) m1 + 2 - m. J Therefore we must have y = -v in (AI.4). Then using the definitions for f m n Z and DQ™ (fl), we obtain ,m n 1 J V (m n S, / ( 2 m + l ) ( 2 n + n " ( ° 0 ° (AI.6) Explicit expressions for the relevant spherical harmonics and rotational invariants are given in Table V and VI respectively. Table V. Spherical Harmonics Oth Order Y°(e,<|0 = 1st Order rl(e,<t>) = -H- S i 1 V 8TT Sine e i<f» cose 4TT Y"J(e,<|.) = + / L sine e - 1 * 07T 2nd Order 5 3 Sin 26 e 2 1* 5 3 Sine Cose e 1* Y°(e,»>D S ,3 „ 2. K 4T (2 C ° S 9 " 25 Y~2 (9 »<f>) = i/^r- 3 Sine Cos6 e _ 1 < t > Y~2(e,<|>) = 5 „ 0. 2n -2i<() 3 Sin 6 e 3rd Order Y^e,*) X 15 S i n 3 e e 3 i* 4 7 r /6T Y2(e,<to = A 7 - — s i n 2 e C o s e e 2 1  3 • 1 / 4 7 1 /5T Y ^ e ^ = ywi? s i n e ( 5 C o s 2 e I l l Y3 (e,40 = fj- (| Cos 3 6 - \ Cos6) 4TT 2 2 Y^CM) = /k/rh- sin0 (5 Cos2e - ^ e_1* ' 7 1 5- S i n 2 6 cose e " 2 i * 4ir 3 / 4 7 T /oT 4 t h O r d e r Y (6,<|>) '9 /35 _. 4Q 4i<}) 47/T28 S i n 9 6 4TT/ 16 S i n3 9 Cos6 e 3 i * 4TT/32 2 2 i2<t> S i n 6 (7 Cos 6 - 1) e f—/4r S i n e (7 C o s 3 e - 3 Cos6) e 4TTV 16 i<J> /* r f c o s 4 e - *° Cos 2 e + 3 ] 4TT L 8 8 o Y - 1 fe 6") = / V - /S S i n e (7 C o s 3 6 - 3 Cos6) e 1 < J > 4 y 4ir v 16 y ~ ; ^ ) = s i n 2 e ( ? c o s 2 e - l D e _ 2 i * Y~>'«  = Jhf%  s i n 5 e C o s e e _ 3 i * 'l/I- 4-- 4 i* 112 5th Order Y ^ M ) -r— Sm 0 e 4ir Y4(e,<l>) H S i n 4 e cose e i 4 * 4TT 6 Y 3 ( M ) 5-T^  Sin 3e (5 Cos 2e - 1) e 3 i * 4TT 8 YgCe,*D 'ik-4H> S m2e (3 Cos3e - cose) e 2 i * 4if 8 Y J C M ) H /l£ Sine (21 Cos46 - 14 Cos26 + l ) e 1 ( | 4TT 16 Y°(e,40 5l I (43 Cos56 - 50 Cos 3e + 15 Cos6) 4TT 8 Y_1(e,<t>) 5 1 ^ Sine (21 Cos46 - 14 Cos2e + 1) e"1^ 4TT 16 Y"2(e,4») 51 vfio 4TT 8 •2 2 -2i<f> (3 Cos 6 - Cos6) Sin 6 e Y"3(e,<iO 51^1 Sin 3e (5 Cos 2e - 1) e " 3 i * 4TT 8 Y " 4 ( 6 , * ) 51 vff sin 4e cose e- i 4* 4TT 16 Y_3(e,<t>) 51 v f S i n 5 e e"** 4TT 16 6th Order Y6(8,<(>) '51 <§T Sin 6e e 6 1 * 4TT 32 U 1^1 s i n 5 e cose e 5 i * 4TT 16 113 51 S i n 4 Q ( n C o s 2 e _ ^ e 4ir 32 4i<f> Ai'-4o| s i n 3 6 (11 Cos36 - 3 Cos9)e 4TT 16 31* Yg(e,40 Sin20 ( 33 Cos46 - 18 Cos29 + 1) e 2i<t> 4TT 8 Yj.(e,<|>) 51 1^ Sine (33 Cos56 - 30 Cos 3e + 5 Cos6) 4ir 16 '11 _L (231 Cos66 - 315 Cos46 + 105 Cos26 - 5) 4TT 16 ^ Y"g(6,<|.) = f}.± V-H sine (33 Cosb6 - 30 CosJ6 + 5 Cos6) e 4TT 16 •i<f> 51^ 5°! Sin 26 (33 Cos46 - 18 Cos26 + 1) e' 4TT 8 •2i<t> Y"gCe,+) 5I'v5°i S i n 3 e (ii cos 3e - 3 cose) e 4ir 16 -3i+ YgCe,*) = 51 s i n 4 e (11 Cos 2e - i) e" 4 i* 4ir 32 Y"^ (6,4>) = 'II 3/77 c. 5 4ir 1"6: Sin e Cos6 e •5i<f> Y"^(e,(t>) 51 Sin 6e e" 6 i < t > 4-rr 32 Table VI. Rotation Invariants $ =1 $110 = T T +T 112 $ = 21^1 -T 3 * 2 0 2 - 3T/-1 » 0 2 2 = 3T 2 2-1 $ 2 U = | [T 2 (31^-1) + 3T XT 3] $ 1 2 1 = I [ T ^ T ^ - l ) + 3T 2T 3 $ 2 1 3 = 3T 2(3T 1 2-1) - 6T XT 3 * 1 2 3 = 3T l (3T 2 2-l) - T 2T 3 $ 3 0 3 = 3T (51^-3) $033 = 5T^ST2_3>j 115 $220 = 1-3T 1 2T 2 2 - 6T 1T 2T 3 - 3T 3 2 $ 2 2 2 = 2 - 3T 2 - 3T 2 2 + 6 T 1 2 T 2 2 + S T j T ^ -O 2 2 4 = 1 - 5Tj 2 - 5T 2 2 + 171*1 * + 2T 3 2 -? 2 2 = T T 2 (5T -3) + T 3(5T 2 -1) $ 1 3 2 = T1 T2 ( 5 T 2 2 " 3 ) + T3 ( 5 T i 2 - ^ $ 3 1 4 = 3[4(5T 1 2-3) T T 2 - 3(51^-1) $ 1 3 4 = 3[4(5T 2 2-3) T T 2 - 3(5T 2 2-1) Tj] $ 4 0 4 = 3[35T 4 - 30TJ 2 + 3] $ 0 4 4 = 3[35T 2 4 - 30T 2 2 + 3] $321 = ^ [ 1 0 T ^ 2 ^ 2 _ 5 T^2 _ 7 T 2 + 4 ] + T ^ _ 5 1 ^ 2 $ 2 3 1 = ^ T 2 [ 1 0 T l 2 T 2 2 - 5T 2 2 - 77* + 4] + (5T 2 2-1) - § T ^ 2 * 3 2 3 = | T 1[45T 1 2T 2 2 - 2 5 T l 2 - 321^21] + f - f * 2 3 3 = | T 2 [ 4 5 T l 2 T 2 2 - 25T 2 2 - 327^ * 21] * f CST^-l) -f $ 3 2 5 = 15 T 1(27T 1 2T 2 2 - 7T 1 2-3T 2 2+3) - 90(51^-1) + g O T ^ 2 $ 2 3 5 = 15 T 2(27T 1 2T 2 2 - 7T 2 2 - 31^ 2 + 3) -90 (ST^-l) + g O T ^ 2 116 422 1 . 4 2 2 2 3 2 2 2 2 2 $*z^ = j[35T T 2 -25T T +70T T T -30T T2T3+70T T 3 -5T3 -5T +2T2 +1] 242 1 4 2 2 2 3 2 2 2 = j[35T T -25T, T» +70T T T -30T.T T -5T -5T„ +2T. +1] 4 2 1 1 2 2 1 3 1 2 3 3 2 1 J $ 4 2 4 = 3[84T 1 4T 2 2-81T 1 2T 2 2+21T 1 3T 2T 3-9T 1T 2T 3-63T 1 2T 3 2-49T 1 4+9T 3 2+51T 1 2+9T 2 2-6] 244 2 4 2 2 3 2 2 4 2 2 2 ^ = 3[ 8 4 T T 2 - 8 1 T % T 2 + 2 1 T 2 T ^ - g ^ T ^ - e S ^ T 3-49T 2*+9T 3+51T 2 + 91^ -6] $ 4 2 6 = 45[91T 1 4 2 2-74T 1 2T 2 2-112T 1 3T 2T 3+48T 1T 2T 3+28T 1 2T 3 2-21T 1 4-4T 3 2+14T 1 2 -+ 7T 2 2-1] $ 2 4 6 = 45[91T 2 4T 1 2-74T 1 2T 2 2-112T 2 3T 1T 3+48T 1T 2T 3+28T 2 2T 3 2-21T 2 4-4T 3 2+14T 2 2 + 7T 2 - l ] * 6 0 6 = 45[231T 1 6-315T 1 4+105T 1 2-5] $ 0 6 6 = 45[231T 2 6-315T 2 4+105T 2 2-5] where T = Cos61 T 2 = C o s e 2 T, ••••= Sin8 1 S i n e 0Cosi() 117 APPENDIX II INTEGRAL TRANSFORMS Eq. (3.33) defines the following transformations for a function h m n £(R) emn*;e C r ) = J°° d R R 2 0 e ( R > r ) h ™ ^ ( R ) £ ( j r e y e n 4 (3.33a) h and h m n £ ; o ( r ) = f dRR2 0° (R,r) h m n i l(R) for odd 4 (3.33b) where the Kernels are defined by Q e r R r^ = i * I f d k k 2 . _ ( m ) • ( k r ) (3.34a) (R r) = . i * - f dkk^ j (kR) j (kr) and 0° (R,r) = i £ _ 1 | /~ dkk 2 J a(kR) jj'Ckr). (3.34b) Evaluation of (3.34) leads to the simplified expressions Q°0 (R,r) = 4 6 (R-r) - A H(R"r) (All. la) R R 16 K and 0? (R.r) = R R ej C , ) - ^  « (R-r) - -\ H(R-r) P°(£) (All. lb) 62 where 6(R-r) is the Dirac delta function defined to be o i f X ¥ X « C * - V - l ~ i f x = x° ( A I I - 2 ) and / 6(X-XQ)dX = 1 . (All.3) 118 From this definition, we know that / f(X) 6(X-X0)dX = f(X Q) . (All.4) H(R-r) is the heavyside step function: 0 R < r H(R-r) =' { (All. 5) 1 R > r and P £(X) is related to Gegenbauer polynomials and hypergeometric functions. 6 3 The f i r s t few members of the even and odd series are listed below p e ( x ) = 3 } (All.6) PjCX) = \ (35X2 - 15), (All.7) Pf(X) = i (105 - 630X2 + 693X4), (All.8) 6 8 P 0 ( X ) = S ) (All.9) P°(X) =1 (63X2 - 35), (All.10) pOj-X) = I ( 3 1 5 _ 1386X2 + 1287X4). (All. 11) 7 8 From these definitions we can write 119 oo .2 „e,r, ^ mnX-; r f o r even «. ( A l l . 12a) , Tmnil;o. , , mn£, , 1 f°°Jr>r)2 _o r , mn£, D, and h ( r ) = h ( r ) - — J r d R R P £ (^ } h (R) r f o r odd I . ( A l l , 1 2 b ) The back i n t e g r a l t r a n s f o r m o f (3.33) i s a c c o m p l i s h e d by 2 m u l t i p l y i n g b o t h s i d e s by H ( R , r ) r d r and i n t e g r a t i n g o v e r a l l p o s i t i v e space t o o b t a i n h ^ O O - (-1)* / 0 d r r 2 9£(R,rD h™1 ( r ) ( A l l . 1 3 ) Then a p p l y i n g t h e d e f i n i t i o n s ( A I I . l ) f o r G^CR.r) we f i n d R h m nV) = h ^ ^ C R ) - -\ J0 d r r 2 p j ( | ) h ^ ^ r ) f o r even I R ( A l l . 1 4 a ) and h^V) = h m n*'°(R) - ± jQ d r r 2 P°(£) h m n ^ ° ( r ) f o r odd £ R ( A l l . 1 4 b ) E x p l i c i t e x p r e s s i o n s f o r t h e t r a n s f o r m a t i o n s a p p l i e d i n t h e p r e s e n t work a r e l i s t e d i n T a b l e V I I . 120 Table VII. Integral Transformations for Even I. Forward transforms "mnO. , K m n 0 r ^ ^ h (r) = n (r) oo mn2 hmn2 ( r ) = hmn2 ( r ) _ 3 /.^  - — R - ^ - dR lmn4_ .mn4_ . 15 " h " " ^ „„ _ 35 2 " h ^ j R ) d R Tmn4r , Kmn4. 15 r n (RJ . p ^ f h (r) = h (r) + -5- J R dR - 2 r J r R 3 ir-« C r ) . *»«(r) - i f * J r ^ 1 dR * f l r 2 J r dR 8 Jr,- R5 Backward transforms mnO, . AmnO, , h Cr) = h (r) hmn2 ( r ) = ^mn2(r) 3 f d R R 2 ^mn2(R) 3 •'.o r h- 4(r) - h m n 4(r) + » / dRR2 h™4 (R) - ^ fQ dRR4 ^V) 2r ° 2r h m n 6 ( r ) = ;mn6(r) _ 105 ; * d R R 2 ^mn6(R) + 630 f dRR4 j^V) 8r ° 8r .693 j r d R R 6 h n » n 6 ( R ) 8r 7 ° 121 APPENDIX III. GENERALIZED CONVOLUTION COEFFICIENTS Eq. (3.30) gives the generalized convolution of the Ornstein-Zernike equation ~nm£ _ ~ m no _ E V l * - " " i V - n . n l , C 3 ~ V l £ 2 m n n i 00 c 1 2 C k ) C3-3°) where o o o mn£ n no Z 2 1 - f 1 f 1 2 T2£ + l) m + n + n r W r m n n l f™* (2n i +i) ( 3 f mnn^ C0 0 0 5 <3-31) £ £, 5, Table VIII l i s t s a l l coefficients, Z 2 1 m n , up to 6th order. Table VIII. Generalized convolution coefficients Z ^ . The m n n^ three integers in the upper lef t corner are (m, n, £) and that underneath are (m, n ^ 2^) and , n, l^). The ^ m n n are l i s t on the right. 0 0 0 O 0 0 1 OOOOOOOOOOOOOOE +00 0 1 1 1 0 1 3 33333333333333E-01 o 2 2 2 0 2 8 OOOOOOOOOOOOOOE-01 0 3 3 3 0 3 5 142857 14285714E+00 0 4 4 4 0 4 6 4OOOO0OO0O000OE +01 o 5 5 5 0 5 1 30909090909091E +03 0 6 6 6 0 6 3 98769230769231E+04 1 O 1 1 1 1 1 1 o 1 O 0 0 1 . OOOOOOOOOOOOOO E + OO 1 1 0 1 0 1 3 . 33333333333333E-01 1 1 2 1 0 1 6. 66666666666666E-01 1 2 1 2 0 2 4 . OOOOOOOOOOOOOOE-01 1 2 3 2 0 2 2. 400O000OO000OOE +00 1 3 2 3 0 3 1 . 71428571428571E+00 1 3 4 3 0 3 2. 05714285714286E+01 1 4 3 4 0 4 1 60000000000OOOE+01 1 4 5 4 0 4 3 20000000000000E+02 1 S 4 5 0 5 2 61818181818182E+02 1 5 6 5 0 5 7 85454545454545E+03 1 6 5 6 0 6 6 64615384615384E+03 1 0 0 1 0 1 1 3 33333333333333E-01 1 1 0 1 1 0 3 33333333333333E-01 1 1 2 1 1 2 6 66666666666666E-01 1 2 1 2 1 1 1 66666666666667E-01 1 2 • 3 2 1 3 4 OOOOOOOOOOOOOOE+oo 1 3 2 3 1 2 4 44444444444444E-01 1 3 4 3 1 4 4 80OO0OOO0O0000E +01 1 4 3 4 1 3 3 OOOOOOOOOOOOOOE +oo 1 4 5 4 1 5 9 60000000000000E +02 1 5 4 5 1 4 3 84000000000000E+01 1 5 6 5 1 6 2 88000000000000E+04 1 6 5 6 1 5 7 99999999999999E+02 1 2 0 1 0 1 1 3 33333333333333E-01 1 1 O 1 1 2 3 33333333333333E-01 1 1 2 1 1 0 3 .33333333333333E-01 1 1 2 1 1 2 3 .33333333333333E-01 1 2 1 2 1 1 1 66666666G66667E-02 1 2 1 2 1 3 5 .99999999999999E-01 1 2 3 2 1 1 5 .99999999999999E-01 1 2 3 2 1 3 1 .60OOOOOOO0O00OE+00 1 3 2 3 1 2 6 .34920634920635E-02 1 3 2 3 1 4 3 42857142857143E+00 1 3 4 3 1 2 3 .42857142857143E+00 1 3 4 3 1 4 1 71428571428571E+01 1 4 3 4 1 3 5 .OOOOOOOOOOOOOOE-01 1 4 3 4 1 5 4 .OOOOOOOOOOOOOOE +01 1 4 5 4 1 3 4 .OOOOOOOOOOOOOOE +01 1 4 5 4 1 5 3 .200000OOOO0000E +02 1 5 4 5 1 4 6 98181818181818E+00 1 5 4 5 1 6 7 .85454545454545E+02 1 5 6 5 1 4 7 . 85454545454545E+02 1 5 6 5 1 6 9 . 16363636363636E+03 2 0 1 6 5 6 1 5 2 2 O 2 0 0 O 2 1 1 1 0 1 2 1 3 1 0 1 2 2 O 2 0 2 2 2 2 2 0 2 2 2 4 2 0 2 2 3 1 3 0 3 2 3 3 3 O 3 2 3 5 3 0 3 2 4 2 4 O 4 2 4 4 4 0 4 2 4 6 4 0 4 2 5 3 5 0 5 2 5 5 5 0 5 2 6 4 6 0 6 2 6 6 6 O 6 1 2 0 2 0 1 1 2 1 1 1 1 0 2 1 1 1 1 2 2 1 3 1 1 2 2 2 0 2 1 1 2 2 2 2 1 1 2 2 2 2 1 3 2 2 4 2 1 3 2 3 1 3 1 2 2 3 3 3 1 2 2 3 3 3 1 4 2 3 5 3 1 4 2 4 2 4 1 3 2 4 4 4 1 3 2 4 4 4 1 5 2 4 6 4 1 5 2 5 3 5 1 4 2 5 5 5 1 4 2 5 S 5 1 6 2 6 4 6 1 5 2 6 6 6 1 5 3 2 0 2 O 1 1 2 1 1 1 1 2 2 1 3 1 1 0 2 1 3 1 1 2 2 2 0 2 1 3 2 2 2 2 1 1 2 2 2 2 1 3 2 2 4 2 1 1 2 2 4 2 1 3 2 3 1 3 1 2 2 3 1 3 1 4 2 3 3 3 1 2 2 3 3 3 1 4 2 3 5 3 1 2 2 3 5 3 1 4 2 4 2 4 1 3 1.53846153846154E+02 1.OOOOOOOOOOOOOOE+00 1 66666666666667E-01 9.99999999999999E-01 -4.OOOOOOOOOOOOOOE-01 4.OOOOOOOOOOOOOOE-01 1.60000000000000E+00 4.28571428571428E-01 2.57142857 14 2857E+0O 5. 14285714285714E+01 2.66666666666667E+00 3.20OOOOOOOO0O00E+01 9.600O0O000O0000E +02 3.27272727272727E+01 6.54545454545454E+02 6.64615384615384E+02 1 99384615384615E+04 7.99999999999999E-01 3.33333333333333E-01 6 66666666666666E-02 2.4 OOOOOOOOOOOOOE+00 -4.OOOOOOOOOOOOOOE-01 2.80OOO0O0O0000OE-01 4.80000000000000E-01 6.39999999999999E+0O 2.66666666666667E-01 1.02857142857143E+00 5. 14285714285714E+00 2.880OOOOOOO00OOE+02 1.20000OOO0000O0E+00 8.800OOOO0OO000OE+0O 8.960O0OOOO0OOO0E+01 6.912O0O0OOOOOO0E +03 1.152OO00O0O000OE+01 1.36145454545454E+02 2.35636363636364E+03 1.92000000000000E+02 3.32307692307692E+03 2.OOOOOOOOOOOOOOE-O1 9.99999999999999E-02 3.33333333333333E-01 2 66666666666666E-01 -4.OOOOOOOOOOOOOOE-01 2 . OOOOOCKXKXKXXJOE -02 3.20000000000000E-O1 2.66666666666666E-01 5.33333333333333E-01 3.17460317460317E-03 5.7 1428571428571E-01 1.1428S714285714E-01 2.57142857142857E+00 5.7142857142857lE+OO 2.05714285714286E+01 2.22222222222222E-02 3 O 2 4 2 4 1 5 2 4 4 4 1 3 2 4 4 4 1 5 2 4 6 4 1 3 2 4 6 4 1 5 2 5 3 5 1 4 2 5 3 5 1 6 2 5 5 5 1 4 2 5 5 5 1 e 2 6 4 6 1 5 2 6 6 6 1 5 3 3 0 3 0 0 0 3 1 2 1 0 1 3 1 4 1 0 1 3 2 1 2 0 2 3 2 3 2 0 2 3 2 5 2 0 2 3 3 0 3 0 3 3 3 2 3 0 3 3 3 4 3 0 3 3 3 6 3 0 3 3 4 1 4 0 4 3 4 3 4 0 4 3 4 5 4 0 4 3 5 2 5 0 5 3 5 4 5 0 5 3 5 6 5 0 5 3 6 3 6 0 6 3 6 5 6 0 6 0 2 0 2 O 2 2 2 1 1 1 2 1 2 1 3 1 2 3 2 2 0 2 2 0 2 2 2 2 2 2 2 2 4 2 2 4 2 3 1 3 2 1 2 3 3 3 2 3 2 3 5 3 2 5 2 4 2 4 2 2 2 4 4 4 2 4 2 4 6 4 2 6 2 5 3 5 2 3 2 5 5 5 2 5 2 6 4 6 2 4 2 6 6 6 2 6 2 2 0 2 0 2 2 2 1 1 1 2 1 2 1 1 1 2 3 2 1 3 1 2 1 2 1 3 1 2 3 2 2 0 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 2 2 2 4 4.44444444444444E+00 1 .20O0OO000OO00OE+00 3.84O0O0OO0OOOO0E+01 8.OOOOOOOOOOOOOOE +01 4 .480OOO0000O0OOE+02 2.61818181818182E-01 6.54545454545454E+01 2.09454545454545E+01 9.16363636363636E+02 4.92307692307692E+00 5.53846153846154E+02 1 . OOOOOOOOOOOOOOE +00 1 .11111111111111E-01 1 .33333333333333E+00 6.66666666666666E-02 4.OOOOOOOOOOOOOOE-01 8.OOOOOOOOOOOOOOE +00 1 .42857142857143E-01 2.85714285714286E-01 3.42857142857143E+00 1 .02857142857143E+02 4 44444444444444E-01 2.66666666666666E+00 5.33333333333333E+01 3.63636363636364E+00 4.36363636363636E+01 1.3090909090909 IE+03 5.53846153846154E+01 1.10769230769231E+03 -4.OOOOOOOOOOOOOOE-01 -8.33333333333333E-02 -2.OOOOOOOOOOOOOOE+00 -4.OOOOOOOOOOOOOOE-01 -2.8O0OOOOO0O00OOE-01 -2.48888B88888889E+00 -5.55555555555555E-02 -1.92857142857143E+00 -4.3200OO0OO000O0E+02 -1.55555555555556E-01 -2.46400000000000E+01 -1.26720OOOOOOOOOE+04 - 1.080OOOOO0OOO00E+O0 -5.10545454545454E+02 - 1.40800OOO0000O0E+01 - 1.56659340659341E+04 5.7142857142B571E-01 8.33333333333333E-02 1.42857142857143E-01 1 42857142857143E-01 2.28571428571428E+00 -4.OOOOOOOOOOOOOOE-O1 -4.OOOOOOOOOOOOOOE-O1 -8.57142857142857E-02 4.57142857142857E-01 2 2 4 2 2 2 2 2 4 2 2 4 2 3 1 3 2 1 2 3 1 3 2 3 2 3 3 3 2 1 2 3 3 3 2 3 2 3 3 3 2 5 2 3 5 3 2 3 2 3 5 3 2 5 2 4 2 4 2 2 2 4 2 4 2 4 2 4 4 4 2 2 2 4 4 4 2 4 2 4 4 4 2 6 2 4 6 4 2 4 2 4 6 4 2 6 2 5 3 5 2 3 2 5 3 5 2 5 2 5 5 5 2 3 2 5 5 5 2 5 2 6 4 6 2 4 2 6 4 6 2 6 2 6 6 6 2 4 2 6 6 6 2 6 2 4 2 0 2 O 2 2 2 1 1 1 2 3 2 1 3 1 2 1 2 1 3 1 2 3 2 2 0 2 2 4 2 2 2 2 2 2 2 2 2 2 2 4 2 2 4 2 2 0 2 2 4 2 2 2 2 2 4 2 2 4 2 3 1 3 2 3 2 3 1 3 2 5 2 3 3 3 2 1 2 3 3 3 2 3 2 3 3 3 2 5 2 3 5 3 2 1 2 3 5 3 2 3 2 3 5 3 2 5 2 4 2 4 2 2 2 4 2 4 2 4 2 4 2 4 2 6 2 4 4 4 2 2 2 4 4 4 2 4 2 4 4 4 2 6 2 4 6 4 2 2 2 4 6 4 2 4 2 4 6 4 2 6 2 5 3 5 2 3 2 5 3 5 2 5 2 5 5 5 2 3 2 5 5 5 2 5 2 6 4 6 2 4 2 6 4 6 2 6 2 6 6 6 2 4 4.57142857142857E-01 2.53968253968254E+00 1 58730158730159E-02 3.67346938775510E-01 3.67346938775510E-01 - 1.01020408163265E+00 1.83673469387755E+01 1.83673469387755E+01 4.11428571428571E+02 6.34920634920635E-02 2.09523809523809E+00 2.O95238095238O9E+0O -1.4857 1428571428E+01 3.84O0O0OOOOOO0OE+02 3.840000O0O00O00E+02 1.1520000O0O00O0E+04 5.14285714285714E-01 2.431 168831 16883E+01 2.43116883116883E+01 -3.27272727272727E+02 7.31428571428571E+00 4.74725274725274E+02 4.74725274725274E+02 -1.03761381475667E+04 2.57142857142857E-01 2.14285714285714E-01 2.14285714285714E-01 4.28571428571428E-01 -4.OOOOOOOOOOOOOOE-01 5.14285714285714E-02 2.85714285714286E-01 -4 . CKDOOOOOOOOOOOOE-01 2.85714285714286E-01 3.42857142857143E-01 1.53061224489796E-02 2.14285714285714E+00 1.53061224489796E-02 4.13265306122449E-01 8.26530612244897E+00 2.14285714285714E+00 8.26530612244898E+00 4.62857142857143E+01 7.93650793650794E-04 1.42857142857143E-01 2.OOOOOOOOOOOOOOE+O1 1.42857142857143E-01 5.55428571428571E+00 1.440O00O0O00OO0E+02 2.OOOOOOOOOOOOOOE +01 1 .44C)OOOOOOOOOOOE+02 1.1520OO0OO0OOO0E+O3 1 .05194805194805E-02 2.10389610389610E+00 £ 2.10389610389610E+00 1.17818181818182E+02 1.89390109890110E-01 4. 74725274725274E+01 4.74725274725274E+01 3 1 3 1 2 6 6 6 2 6 2 3 0 3 0 1 1 3 1 2 1 1 0 3 1 2 1 1 2 3 1 4 1 1 2 3 2 1 2 1 1 3 2 1 2 1 3 3 2 3 2 1 1 3 2 3 2 1 3 3 2 5 2 1 3 3 3 0 3 1 2 3 3 2 3 1 2 3 3 2 3 1 4 3 3 4 3 1 2 3 3 4 3 1 4 3 3 6 3 1 4 3 4 1 4 1 3 3 4 3 4 1 3 3 4 3 4 1 5 3 4 5 4 1 3 3 4 5 4 1 5 3 5 2 5 1 4 3 5 4 5 1 4 3 5 4 5 1 6 3 5 6 5 1 4 3 5 6 5 1 6 3 6 3 6 1 5 3 6 5 6 1 5 4 3 0 3 0 1 1 3 1 2 1 1 2 3 1 4 1 1 0 3 1 4 1 1 2 3 2 1 2 1 3 3 2 3 2 1 1 3 2 3 2 1 3 3 2 5 2 1 1 3 2 5 2 1 3 3 3 0 3 1 4 3 3 2 3 1 2 3 3 2 3 1 4 3 3 4 3 1 2 3 3 4 3 1 4 3 3 6 3 1 2 3 3 6 3 1 4 3 4 1 4 1 3 3 4 1 4 1 5 3 4 3 4 1 3 3 4 3 4 1 5 3 4 5 4 1 3 3 4 5 4 1 5 3 5 2 5 1 4 3 5 2 5 1 6 3 5 4 5 1 4 3 5 4 5 1 6 3 5 6 5 1 4 3 5 6 5 1 6 3.66216640502355E+03 1 28571428571428E+00 3.33333333333333E-01 9.52380952380952E-02 5. 14285714285714E+00 9.99999999999999E-02 2.85714285714286E-02 3.85714285714285E-01 1 .02857142857143E+00 5. 14285714285714E+01 1 .42857142857143E-01 2 .44897959183673E-01 3.67346938775510E-01 1 . 79591836734694E+00 1 .46938775510204E+01 9. 257 14285714285E+02 3.21428571428571E-01 1 .57142857142857E+00 5. 71428571428571E+00 1.85714285714286E+01 3.20OOOOOO00OO0OE+02 2.05714285714286E+00 1 .94493506493506E+01 1 .30909090909091E +02 3.36623376623376E+02 1 O0987012987013E+O4 2.57142857142857E+01 3.95604395604395E+02 1 .42857142857143E-01 4.76190476190476E-02 3.33333333333333E-01 2.38095238095238E-01 4.76190476190476E-02 1 .78571428571428E-02 2. 14285714285714E-01 9 99999999999999E-01 1 .71428571428571E+00 1.42857142857143E-01 3.40136054421769E-03 2.55102040816326E-01 1 .36054421768707E-01 2.20408163265306E+0O 8.57142857142857E+00 2.57142857142857E+01 9.92063492063491E-04 5.55555555555555E-01 3.57142857142857E-02 2 . 85714285714285E+00 1 .78571428571428E+00 4.OOOOOOOOOOOOOOE +01 1 .03896103896104E-02 5 . 45454545454545E+00 5.6 103896103896 IE-01 5. 45454545454545E+01 3.74025974025974E+01 1 . 12207792207792E+03 3 2 3 3 6 3 6 1 5 3 6 5 6 1 5 4 4 0 4 O 0 0 4 1 3 1 0 1 4 1 5 1 0 1 4 2 2 2 0 2 4 2 4 2 0 2 4 2 6 2 0 2 4 3 1 3 0 3 4 3 3 3 0 3 4 3 5 3 0 3 4 4 0 4 0 4 4 4 2 4 0 4 4 4 4 4 0 4 4 4 6 4 0 4 4 5 1 5 o 5 4 5 3 5 o 5 4 5 5 5 0 5 4 6 2 6 0 6 4 6 4 6 0 6 4 6 6 6 0 6 » 1 3 0 3 0 2 2 3 1 2 1 2 1 3 1 2 1 2 3 3 1 4 1 2 3 3 2 1 2 2 0 3 2 1 2 2 2 3 2 3 2 2 2 3 2 3 2 2 4 3 2 5 2 2 4 3 3 0 3 2 1 3 3 2 3 2 1 3 3 2 3 2 3 3 3 4 3 2 3 3 3 4 3 2 5 3 3 6 3 2 5 3 4 1 4 2 2 3 4 3 4 2 2 3 4 3 4 2 4 3 4 5 4 2 4 3 4 5 4 2 6 3 5 2 5 2 3 3 5 4 5 2 3 3 5 4 5 2 5 3 5 6 5 2 5 3 6 3 6 2 4 3 6 5 6 2 4 3 6 5 6 2 6 2 3 3 0 3 0 2 2 3 1 2 1 2 1 3 1 2 1 2 3 3 1 4 1 2 1 3 1 4 1 2 3 3 2 1 2 2 2 1.64835164835165E-01 1.31868131868132E+01 0.99999999999999E+00 8.33333333333333E-02 1 .66666666666667E+0O 3.33333333333333E-02 4.OOOOOOOOOOOOOOE-01 1.2 OOOOOOOOOOOOOE+01 3.57142857142857E-02 2. 142857 14285714E-01 4 . 28571428571428E+00 1 . 1 1 1 1 1 1 1 1 1 1 1111E-01 2 . 22222222222222E-01 2.66666666666667E+00 8.OOOOOOOOOOOOOOE +01 4 54545454545454E-01 2 . 72727272727273E+00 5.45454545454545E+01 4.6153846 1538461E+00 5.53846153846154E+01 1 .66153846153846E+03 3.08571428571428E+00 4.OOOOOOOOOOOOOOE-O1 1 . 14285714285714E-01 2.057142857142B6E+01 -4.OOOOOOOOOOOOOOE-O1 8.OOOOOOOOOOOOOOE-02 1 B5142B57142857E+00 6.85714285714285E-01 9.6OOOOOOOOOOOOOE+01 1 .42857142857143E-01 1 . 71428571428571E-01 6.61224489795918E-01 1 61632653061224E+01 4. 1 1428571428571E+01 6.66514285714286E+03 2.OOOOOOOOOOOOOOE-O1 6.28571428571428E-01 7.54285714285714E+00 2.496OOOOOOOOO0OE+O2 1.1520OOO0OO00O0E+03 9 . 257 14285714285E-01 5. 34857142857143E+00 1 . 36145454545455E+02 6.05922077922078E+03 9.05142857142857E+00 8.22857142857143E+01 3.66216640502355E+03 5.33333333333333E-01 4.44444444444444E-02 1.18518518518518E-01 2.22222222222222E-01 2.66666666666666E+00 5.33333333333333E-02 3 2 3 2 1 2 2 4 3 2 3 2 2 0 3 2 3 2 2 2 3 2 3 2 2 4 3 2 5 2 2 2 3 2 5 2 2 4 3 3 0 3 2 3 3 3 2 3 2 1 3 3 2 3 2 3 3 3 2 3 2 5 3 3 4 3 2 1 3 3 4 3 2 3 3 3 4 3 2 5 3 3 6 3 2 3 3 3 6 3 2 5 3 4 1 4 2 2 3 4 1 4 2 4 3 4 3 4 2 2 3 4 3 4 2 4 3 4 3 4 2 6 3 4 5 4 2 2 3 4 5 4 2 4 3 4 5 4 2 6 3 5 2 5 2 3 3 5 2 5 2 5 3 5 4 5 2 3 3 5 4 5 2 5 3 5 6 5 2 3 3 5 6 5 2 5 3 6 3 6 2 4 3 6 3 6 2 6 3 6 5 6 2 4 3 6 5 6 2 6  5 3 0 3 0 2 2 3 1 2 1 2 3 3 1 4 1 2 1 3 1 4 1 2 3 3 2 1 2 2 4 3 2 3 2 2 2 3 2 3 2 2 4 3 2 5 2 2 0 3 2 5 2 2 2 3 2 5 2 2 4 3 3 0 3 2 5 3 3 2 3 2 3 3 3 2 3 2 5 3 3 4 3 2 1 3 3 4 3 2 3 3 3 4 3 2 5 3 3 6 3 2 1 3 3 6 3 2 3 3 6 3 2 5 1 4 1 4 2 4 1 4 1 4 2 6 1 4 3 4 2 2 1 4 3 4 2 4 9 4 3 4 2 6 9 4 5 4 2 2 1.97530864197531E-02 -4.OOOOOOOOOOOOOOE-O1 - 1.46666666666666E-01 5.33333333333333E-01 2.6666G666666667E+00 1.06666666666667E+01 1.42857142857143E-01 1.90476190476190E-02 9.04761904761904E-02 9.52380952380952E-01 4.65608465608465E-01 -1.7142857142857 1E+OO 2.74285714285714E+01 4.28571428571428E+01 6.71999999999999E+02 3.7037037O37O37OE-O3 4 074O74O74O7407E-O1 1.08641975308642E-01 2.66666666666666E-O1 1.77777777777778E+01 3.20987654320987E+00 -2.82666666666666E+01 6.96888888888889E+02 2.66666666666666E-02 3.15151515151515E+00 1.13454545454545E+00 -8.72727272727272E-01 4.36363636363636E+01 -6.98181818181818E+02 3.20000000000000E-O1 4.61538461538461E+01 1 96923076923077E+01 -1.05494505494505E+02 4.76190476190476E-02 2 64550264550264E-02 5.55555555555555E-02 9 .52380952380951E-02 1 .23456790123457E-02 1 . 19047619047619E-02 4.76190476190476E-02 -4.OOOOOOOOOOOOOOE-O1 2.66666666666666E-01 2 66666666666666E-01 1 .42857142857143E-01 4.25170068027211E-03 2.38095238095238E-Ol 5 . 2910O529100529E-03 1 .22448979591837E-01 1 .71428571428571E+OO 8.57142857142857E-01 3.OOOOOOOOOOOOOOE +00 1 .37142857142857E+01 1 .85185185185185E-03 6.66666666666666E-01 4.409171075B3774E-04 5.71428571428571E-02 3.11111111111111E+00 6. 17283950617284E-02 3 4 5 4 2 4 3 4 5 4 2 6 3 5 2 5 2 3 3 5 2 5 2 5 3 5 4 5 2 3 3 5 4 5 2 5 3 5 6 5 2 3 3 5 6 5 2 5 3 6 3 6 2 4 3 6 3 6 2 6 3 6 5 6 2 4 3 6 5 6 2 6 3 4 0 4 0 1 1 4 1 3 1 1 0 4 1 3 1 1 2 4 1 5 1 1 2 4 2 2 2 1 1 4 2 2 2 1 3 4 2 4 2 1 1 4 2 4 2 1 3 4 2 6 2 1 3 4 3 1 3 1 2 4 3 1 3 1 4 4 3 3 3 1 2 4 3 3 3 1 4 4 3 5 3 1 2 4 3 5 3 1 4 4 4 0 4 1 3 4 4 2 4 1 3 4 4 2 4 1 5 4 4 4 4 1 3 4 4 4 4 1 5 4 4 6 4 1 3 4 4 6 4 1 5 4 5 1 5 1 4 4 5 3 5 1 4 4 5 3 5 1 6 4 5 5 5 1 4 4 5 5 5 1 6 4 6 2 6 1 5 4 6 4 6 1 5 4 6 6 6 1 5 1 5 4 0 4 0 • 1 1 4 1 3 1 1 2 4 1 5 1 1 0 4 1 5 1 1 2 4 2 2 2 1 3 4 2 4 2 1 1 4 2 4 2 1 3 4 2 6 2 1 1 4 2 6 2 1 3 4 3 1 3 1 4 4 3 3 3 1 2 4 3 3 3 1 4 4 3 5 3 1 2 4 3 5 3 1 4 2.OOOOOOOOOOOOOOE +00 3.55555555555556E+01 4.32900432900433E-05 2 .42424242424242E-02 7.79220779220779E-03 1.09090909090909E+00 1.09090909090909E+00 4 98701298701298E+01 8.79120879120878E-04 4.61538461538461E-01 1 .75824175824176E-01 3.01412872841444E+01 1.77777777777778E+00 3.33333333333333E-01 1.111111111111 11E-01 8.8888888888B888E+00 6.66666666666666E-02 2.96296296296296E-02 4.88888888888888E-01 1.600OOOO0OOO0O0E+00 1.06666666666667E+02 4.76190476190476E-02 1.58730158730158E-02 2.32804232804233E-01 5.71428571428571E-01 2.75132275 132275E+00 2.857 14285714286E+01 1.11111111111111E-01 2.03703703703704E-01 2.96296296296296E-01 1 .92592592592592E+00 1 . 1851S518518518E+01 3.33333333333333E+01 7.46666666666666E+02 3.55555555555555E-01 1 .89090909090909E+OO 6.06060606060606E+OO 2.90909090909091E +01 3.39393939393939E+02 2.96296296296296E+00 3.07692307692308E+01 6.97435897435897E+02 1.111111111111 11E-01 2 . 77777777777778E-02 3.33333333333333E-01 2.22222222222222E-01 1 .85185185185185E-02 1.55555555555555E-02 1.60OOO000OO0000E-01 1.20000000000000E+00 ! .86G66666666667E+00 2.77777777777778E-02 2.64550264550264E-03 1.42857142857143E-01 1.48148148148148E-01 2.OOOOOOOOOOOOOOE+00 4 4 0 4 1 5 4 4 2 4 1 3 4 4 2 4 1 5 4 4 4 4 1 3 4 4 4 4 1 5 4 4 6 4 1 3 4 4 6 4 1 5 4 5 1 5 1 4 4 5 1 5 1 6 4 5 3 5 1 4 4 5 3 5 1 6 4 5 5 5 1 4 4 5 5 5 1 6 4 6 2 6 1 5 4 6 4 6 1 5 4 6 6 6 1 5 0 5 5 0 5 0 0 0 5 1 4 1 0 1 5 1 6 1 0 1 5 2 3 2 0 2 5 2 5 2 0 2 5 3 2 3 0 3 5 3 4 3 0 3 5 3 6 3 0 3 5 4 1 4 0 4 5 4 3 4 0 4 5 4 5 4 0 4 5 5 0 5 0 5 5 5 2 5 0 5 S 5 4 5 0 5 5 5 6 5 0 5 5 6 1 6 0 6 5 6 3 6 0 6 5 6 5 6 0 6 3 0 3 O 3 0 3 3 3 1 2 1 3 2 3 1 4 1 3 4 3 2 1 2 3 1 3 2 3 2 3 3 3 i 2 5 2 3 5 3 1 3 0 3 3 0 3 1 3 2 3 3 2 3 1 3 4 3 3 4 3 1 3 6 3 3 6 3 1 4 1 4 3 1 1 4 3 4 3 3 1 4 5 4 3 5 1 5 2 5 3 2 ) 5 4 5 3 4 ) 5 6 5 3 6 3 6 3 6 3 3 3 6 5 6 3 5 3 2 3 0 3 0 3 3 3 1 2 1 3 2 1.11111111111111E-01 9.25925925925926E-04 2.07407407407407E-01 3 . 70370370370370E-02 2.074074074O7407E+OO 2 . 33333333333333E+00 4 . 26666666666667E+01 4.04O4O4O4O4O4O4E-04 5.45454545454545E-01 1 .454S4545454545E-02 2.96969696969697E+00 7 . 27272727272727E-01 4 84B48484848485E+01 5 .698005G9800569E-03 3.07692307692308E-01 2.05128205128205E+01 9.99999999999999E-01 6.666666666G6666E-02 2.OOOOOOOOOOOOOOE+00 2.OOOOOOOOOOOOOOE-02 4.OOOOOOOOOOOOOOE-O1 1 .42857142857143E-02 1 .71428571428571E-01 5. 14285714285714E+00 2 .22222222222222E-02 1 .33333333333333E-01 2.66666666666666E+00 9.09090909090908 E-02 1 81818181818182E-01 2.18181818181818E+00 6.54545454545454E+01 4 .615384615384G1E-01 2.7G923076923077E+00 5 .53846153846154E+01 5. 14285714285714E+00 4 .44444444444444E-01 4 . 8OOOOOOOOOOOOOE+01 1 . 1 1 1 1 1 1 1 11 1111 IE-01 3.857 14285714286E+00 8.640000O0000O00E+02 1 42857142857143E-01 4.28571428571429E-01 5.02857142857143E+01 2 . 44388571428571E+04 8 33333333333333E-02 3. 14285714285714E+00 1 .04OO0OOOO000O0E+03 2.40OO0O0OOO000OE-O1 4.160OO00OOOOO00E +01 3. 17387755102041E+04 1 69714285714286E+00 8. 8 1632653061224E+02 3 . 42857142857143E+00 2 .53968253968254E-01 3 1 2 1 3 4 3 1 4 1 3 2 3 1 4 1 3 4 3 2 1 2 3 1 3 2 1 2 3 3 3 2 3 2 3 1 3 2 3 2 3 3 3 2 3 2 3 5 3 2 5 2 3 3 3 2 5 2 3 5 3 3 0 3 3 2 3 3 2 3 3 O 3 3 2 3 3 2 3 3 2 3 3 4 3 3 4 3 3 2 3 3 4 3 3 4 3 3 4 3 3 6 3 3 6 3 3 4 3 3 6 3 3 6 3 4 1 4 3 1 3 4 1 4 3 3 3 4 3 4 3 1 3 4 3 4 3 3 3 4 3 4 3 5 3 4 5 4 3 3 3 4 5 4 3 5 3 5 2 5 3 2 3 5 2 5 3 4 3 5 4 5 3 2 3 5 4 5 3 4 3 5 4 5 3 6 3 5 6 5 3 4 3 5 6 5 3 6 3 6 3 6 3 3 3 6 3 6 3 5 3 6 5 6 3 3 3 6 5 6 3 5 3.80952380952381E-01 3.80952380952381E-01 2.85714285714286E+01 4.44444444444444E-02 1 .71428571428571E-01 1 .71428571428571E-01 8. 14285714285713E-01 8.57142857142857E+00 8.57 14285714 2857E+00 4.80000000000000E +02 > 1 .42857142857143E-01 1 .42857142857143E-01 -1 . 12244897959184E-01 1 .79591836734694E+00 1 .79591836734694E+00 3.26530612244898E+00 2.05714 285714286E+02 2.05714285714286E+02 1.29600O0OOOO000E+04 1.38888888888889E-02 2.619O4761904762E-O1 2.61904761904762E-Ol -1.04761904761905E+00 2.47619047619047E+01 2.47619047619047E+01 - 1 . 33333333333333E+01 5.71428571428571E-02 1.48571428571429E+00 1.48571428571429E+0O -1.43168831168831E+01 5 . 23636363636364E+02 5.2363G363636364E+02 -1 92356215213358E+03 4.71428571428571E-01 1 71428571428571E+01 1 .71428571428571E+01 -3.01412872841444E+02 3 0 3 0 3 3 3 1 2 1 3 2 3 1 2 1 3 4 3 1 4 1 3 2 3 1 4 1 3 4 3 2 1 2 3 3 3 2 1 2 3 5 3 2 3 2 3 1 3 2 3 2 3 3 3 2 3 2 3 5 3 2 5 2 3 1 3 2 5 2 3 3 3 2 5 2 3 5 3 3 O 3 3 4 3 3 2 3 3 2 3 3 2 3 3 4 3 3 2 3 3 6 3 3 4 3 3 0 3 3 4 3 3 2 3 3 4 3 3 4 3 3 4 3 3 6 3.50649350649350E-01 1.58730158730159E-02 1 .29870129870130E-01 1.29870129870130E-01 2.10389610389610E+00 3.57142857142857E-02 9.09090909090909E-02 3.57142857142857E-02 - 1.31493506493506E-01 2. 103896103896lOE+OO 9.09090909090909E-02 2. 103896 103896 lOE+OO 2.94545454545454E+01 1 .42857142857143E-01 1 53061224489796E-02 ^ 2 . 78293135435992E-02 NJ t.75324675324675E+00 ^ 1.42857142857143E-01 2 . 78293135435992E-02 - 1.53617810760668E+00 4.20779220779220E+01 3 3 3 3 6 3 3 2 3 3 6 3 3 4 3 3 6 3 3 6 3 4 1 4 3 3 3 4 1 4 3 5 3 4 3 4 3 1 3 4 3 4 3 3 3 4 3 4 3 5 3 4 5 4 3 1 3 4 5 4 3 3 3 4 5 4 3 5 3 5 2 5 3 2 3 5 2 5 3 4 3 5 2 5 3 6 3 5 4 5 3 2 3 5 4 5 3 4 3 5 4 5 3 6 3 5 6 5 3 2 3 5 6 5 3 4 3 5 6 5 3 6 3 6 3 6 3 3 3 6 3 6 3 5 3 6 5 6 3 3 3 6 5 6 3 5  6 3 0 3 0 3 3 3 1 2 1 3 4 3 1 4 1 3 2 3 1 4 1 3 4 3 2 1 2 3 5 ' 3 2 3 2 3 3 3 2 3 2 3 5 3 2 5 2 3 1 3 2 5 2 3 3 3 2 5 2 3 5 3 3 0 3 3 6 3 3 2 3 3 4 3 3 2 3 3 6 3 3 4 3 3 2 3 3 4 3 3 4 3 t 3 4 3 3 6 3 l 3 6 3 3 0 3 1 3 6 3 3 2 3 1 3 6 3 3 4 3 1 3 6 3 3 6 3 1 4 1 4 3 5 3 1 4 3 4 3 3 1 4 3 4 3 5 ) 4 5 4 3 1 ) 4 5 4 3 3 i 4 5 4 3 5 1 5 2 5 3 4 3 5 2 5 3 6 3 5 4 5 3 2 3 5 4 5 3 4 3 5 4 5 3 6 3 5 6 5 3 2 3 5 6 5 3 4 3 5 6 5 3 6 t.75324675324675E+OD 4.20779220779220E+01 7.069090909O9090E+O2 5.95238095238095E-03 4.92424242424242E-01 5.95238095238095E-03 1.04978354978355E-01 -4.97835497835498E-01 4.92424242424242E-01 -4.97835497835498E-01 -2.60606060606061E+01 3.89610389610389E-04 5.52538370720189E-02 4.46280991735537E+00 5.52538370720189E-02 1.228335301O6257E+OO - 1.78512396694215E+01 4.46280991735537E+00 -1.78512396694215E+01 -6.55759824590993E+02 5.25974025974026E-03 8.09190809190809E-01 8.09190809190809 E-01 2.31197374054517E+01 2.16450216450216E-02 1 .68350168350168E-02 1.68350168350168E-02 5.05050505050504 E-02 3.0303O3O30303O3E-02 6 .76406926406926E-03 1.06060606060606E-01 3.03030303030303E-02 1 .06060606060606E-01 4 .84848484848485E-01 1 .42857142857143E-01 3.60750360750361E-03 2.27272727272727E-01 3.60750360750361E-03 8 65800865800864E-02 1 .45454545454545E+00 1 .42857142857143E-01 2.27272727272727E-01 1 .45454545454545E+00 9.35064935064934E+00 2.10437710437710E-03 6.0125060125060IE-04 5.89225589225589E-02 2.10437710437710E-03 5.89225589225589E-02 1.68350168350168E+00 9. 18273645546373E-05 3.30578512396694E-02 9.18273645546373E-05 1 .37741046831956E-02 1 .32231404958678E+00 3.30578512396694E-02 1.32231404958678E+00 4.85748018215550E+01 3 6 3 6 3 3 3 6 3 6 3 5 3 6 5 6 3 3 3 6 5 6 3 5 4.16250416250416E-06 2.33100233100233E-03 2.33100233100233E-03 3.80571809143238E-01 4 0 4 0 2 2 4 1 3 1 2 1 4 1 3 1 2 3 4 1 5 1 2 3 4 2 2 2 2 0 4 2 2 2 2 2 4 2 2 2 2 4 4 2 4 2 2 2 4 2 4 2 2 4 4 2 6 2 2 4 4 3 1 3 2 1 4 3 1 3 2 3 4 3 3 3 2 1 4 3 3 3 2 3 4 3 3 3 2 5 4 3 5 3 2 3 4 3 5 3 2 5 4 4 0 4 2 2 4 4 2 4 2 2 4 4 2 4 2 4 4 4 4 4 2 2 4 4 4 4 2 4 4 4 4 4 2 6 4 4 6 4 2 4 4 4 6 4 2 6 4 5 1 5 2 3 4 5 3 5 2 3 4 5 3 5 2 5 4 5 5 5 2 3 4 5 5 5 2 5 4 6 2 6 2 4 4 6 4 6 2 4 4 6 4 6 2 6 4 6 6 6 2 4 4 6 6 6 2 6 6 . 85714285714285E+00 6.42857142857142E-01 2.85714285714286E-01 5.71428571428571E+01 -4.OOOOOOOOOOOOOOE-01 1 . 14285714285714E-01 1.26984126984127E-02 3.77142857142857E+00 2.285714 2857 14 28E+00 3.20OO00OO0OO0O0E+02 7. 14285714285714E-02 4.59183673469388E-02 2.24489795918367E-01 1 .34693877551020E+0O 1 . 22448979591837E+00 3.97959183673469E+01 1 71428571428571E+02 1.11111111111111E-01 1 .74603174603175E-01 6.28571428571428E-01 1.0O881834215167E+00 1.98095238095238E+01 3.55555555555555E+01 7.20O0OOOOOO0OO0E +02 5.76OO0OOOO0OOOOE+03 2.57142857142857E-01 1. 1 1428571428571E+00 1 .01298701298701E+01 1.O1298701298701E+01 4.36363636363636E+02 1 .67619047619048E+00 1 37142857142857E+01 2 .37362637362637E+02 1.79340659340659E+02 1.38348508634223E+04 4 0 4 0 2 2 4 1 3 1 2 1 4 1 3 1 2 3 4 1 5 1 2 1 4 1 5 1 2 3 4 2 2 2 2 2 4 2 2 2 2 4 4 2 4 2 2 0 4 2 4 2 2 2 4 2 4 2 2 4 4 2 6 2 2 2 4 2 6 2 2 4 4 3 1 3 2 3 4 3 1 3 2 5 4 3 3 3 2 1 4 3 3 3 2 3 4 3 3 3 2 5 5.19480519480519E-01 2.97619047619047E-02 9.74025974025973E-02 3.03030303030303E-01 3.11688311688311E+OO 2.38095238095238E-02 1.44300144300144E-02 -4.OOOOOOOOOOOOOOE-01 -1 .68831 168831169E-01 5.6103896103896 1E-01 4.36363636363636E+00 £ 1.45454545454545E+01 Oo 3.18877551O20408E-02 3. 24675324675325E-02 1.70068027210884E-02 2.08719851576994E-02 1.0O185528756957E+00 4 2 4 3 5 3 2 1 4 3 5 3 2 3 4 3 5 3 2 5 4 4 0 4 2 4 4 4 2 4 2 2 4 4 2 4 2 4 4 4 2 4 2 6 4 4 4 4 2 2 4 4 4 4 2 4 4 4 4 4 2 6 4 4 6 4 2 2 4 4 6 4 2 4 4 4 6 4 2 6 4 5 1 5 2 3 4 5 1 5 2 5 4 5 3 5 2 3 4 5 3 5 2 5 4 5 5 5 2 3 4 5 5 • 5 2 5 4 6 2 6 2 4 4 6 2 6 2 6 4 6 4 6 2 4 4 6 4 6 2 6 4 6 6 6 2 4 4 6 6 6 2 6  6 4 0 4 0 2 2 4 1 3 1 2 3 4 1 5 1 2 1 4 1 5 1 2 3 4 2 2 2 2 4 4 2 4 2 2 2 4 2 4 2 2 4 4 2 6 2 2 0 4 2 6 2 2 2 4 2 6 2 2 4 4 3 1 3 2 5 4 3 3 3 2 3 4 3 3 3 2 5 4 3 5 3 2 1 4 3 5 3 2 3 4 3 5 3 2 5 4 4 0 4 2 6 4 4 2 4 2 4 4 4 2 4 2 6 4 4 4 4 2 2 4 4 4 4 2 4 4 4 4 4 2 6 4 4 6 4 2 2 4 4 6 4 2 4 4 4 6 4 2 6 4 5 1 5 2 5 4 5 3 5 2 3 4 5 3 5 2 5 4 5 5 5 2 3 4 5 5 5 2 5 4 6 2 6 2 4 4 6 2 6 2 6 4 6 4 6 2 4 5.62770562770563E-01 -2 . 21243042671614E+00 3. 50649350649351E+01 1 . 11 1 1 1 1 1 1111111E-01 3.96825396825397E-03 1.27417027417027E-01 7.27272727272727E-01 1 . 25060125060125E-01 -3.98268398268398E-01 2 .42424242424242E+01 4 .54545454545455E+0O -4. 10909090909091E+01 1 .04727272727273E+O3 1 . 29870129870130E-03 4 . 29752066115702E-01 4 . 14403778040142E-02 1 .07319952774498E+0O 1.5938606B476977E+00 -1 .48760330578512E+01 1 .38528138528138E-02 4 . 19580419580419E+00 6 .47352647352648E-01 1 .54645354645355E+01 3 .26073926073926E+01 -5.70286856001141E+02 3 .03O3O3O3O3O3O3E-O2 1 .26262626262626E-02 4 .54545454545454E-02 7 .07070707070706E-02 3 92817059483726E-03 8.48484848484848E-03 2 .82828282828282E-02 -4.OOOOOOOOOOOOOOE-01 2 .54545454545454E-01 2.26262626262626E-01 2 .27272727272727E-02 2.70562770562770E-03 1 .06O6O6O6O606O6E-O1 5.05050505050505E-03 1 .06060606060606E-01 1 .21212121212121E+00 1 . 1 1 1 1 1 1 1 1 1 11111E-01 1 414 1414 1414 141E-03 1 97979797979798E-01 4 36463399426362E-04 4.71380471380471E-02 1 .75982042648709E+00 7 .07070707070707E-02 2 03636363636364E+00 2.90909090909091E +01 8.26446280991735E-04 6.88705234159779E-05 2 .69972451790633E-02 9. 64 187327823691E-03 1 . 10192837465565E+00 1 0360O103600104E-05 1 .39860139860140E-02 1 86480186480186E-03 5 1 5 1 4 6 4 6 2 6 4 6 6 6 2 4 4 6 6 6 2 6 4 5 0 5 0 1 1 5 1 4 1 1 0 5 1 4 1 1 2 5 1 6 1 1 2 5 2 3 2 1 1 5 2 3 2 1 3 5 2 5 2 1 1 5 2 5 2 1 3 5 3 2 3 1 2 5 3 2 3 1 4 5 3 4 3 1 2 5 3 4 3 1 4 5 3 6 3 1 2 5 3 6 3 1 4 5 4 1 4 1 3 5 4 1 4 1 5 5 4 3 4 1 3 5 4 3 4 1 5 5 4 5 4 1 3 5 4 5 4 1 5 5 5 0 5 1 4 5 5 2 5 1 4 5 5 2 5 1 6 5 5 4 5 1 4 5 5 4 S 1 6 5 5 6 5 1 4 5 5 6 5 1 6 5 6 1 6 1 5 5 6 3 6 1 5 5 6 5 6 1 5 6 5 0 5 O 1 1 5 1 4 1 1 2 5 1 6 1 1 0 5 1 6 1 1 2 5 2 3 2 1 3 5 2 5 2 1 1 5 2 5 2 1 3 5 3 2 3 1 4 5 3 4 3 1 2 5 3 4 3 1 4 5 3 6 3 1 2 5 3 6 3 1 4 5 4 1 4 1 5 5 4 3 4 1 3 5 4 3 4 1 5 5 4 5 4 1 3 5 4 5 4 1 5 5 5 0 5 1 6 5 5 2 5 1 4 5 5 2 5 1 6 5 5 4 5 1 4 5 5 4 5 1 6 5 5 6 5 1 4 6.71328671328671E-01 2.61072261072261E-01 3.59640359640359E+01 2.27272727272727E+00 3 . 33333333333333E-OI 1 .21212121212121E-01 1 .36363636363636E+01 5.OOOOOOOOOOOOOOE-02 2 .72727272727273E-02 5.90909090909091E-01 2. 18181818181818E+00 2.38095238095238E-02 1.29870129870130E-02 2.25108225108225E-01 7.01298701298701E-01 3.89610389610390E+00 4.67532467532467E+01 2.77777777777778E-02 1.01010101010101E-02 1 .47727272727273E-01 3.63636363636363E-01 2.27272727272727E+00 1 .81818181818182E+01 9.09090909090909E-02 1.719008264462B1E-01 2.47933884297520E-01 1 .785 12396694215E+00 9.91735537190082E+00 4.04628099173553E+01 6.24793388429752E+02 3.78787878787879E-01 2.09790209790210E+00 3.56643356643356E+01 9 09090909090909E -02 1 B1818181818182E -02 3 33333333333333E -01 2 121212 12121212E -01 9 09090909090908E -03 1 36363636363636E -02 1 27272727272727E -01 9 09090909090909 E -03 2 02020202020202E -03 9 09090909090908 E -02 1 55844155844156E -01 1 87012987012987E+00 1 81818181818182E -02 6 31313131313131E -04 f) 89398989898989E -02 3 53535353535353E -02 1 61616161616162E+00 9 09090909090909E -02 3 30578512396694E -04 1 73553719008264E -01 1 32231404958678E -02 1 85123966942149E+00 8 33057851239669E -01 5 5 6 5 1 6 5 6 1 € 1 5 5 6 3 6 1 5 5 6 5 6 1 5 > 6 6 0 6 0 0 0 6 1 5 1 O 1 6 2 4 2 0 2 6 2 6 2 0 2 6 3 3 3 0 3 6 3 5 3 0 3 6 4 2 4 0 4 6 4 4 4 0 4 6 4 6 4 0 4 6 5 1 5 0 5 6 5 3 5 0 5 6 5 5 5 0 5 6 6 0 6 0 6 6 6 2 6 0 6 6 6 4 6 0 6 6 6 6 6 0 6 4.46280991735537E+01 1.94250194250194E-04 6.99300699300699E-03 3.49G50349650350E-01 9. 99999999999999E-01 5. 55555555555555E-02 1.33333333333333E-02 4.OOOOOOOOOOOOOOE-O1 7. 14285714285714E-03 1 .42857142857143E-01 7.40740740740740E-03 8.88888888888888E-02 2 . 66666666666666E+OO 1 51515151515151E-02 9.09090909090908E-02 t .81818181818182E+00 7 692307692307G8E-02 1 .5384G15384G154E-01 1 .84G15384615384E+00 5.53846153846153E+01 o 131 APPENDIX IV. THE ITERATION PROCEDURE The f l o w d i a g r a m , F i g . 2 1 , summarizes t h e p r o c e d u r e used i n t h e i t e r a t i v e s o l u t i o n o f eqs. (3.47) s u b j e c t t o t h e LHNC c l o s u r e r e l a t i o n s (eqs. ( 3 . 4 0 ) ) . The f o l l o w i n g i s a s t e p - b y - s t e p e x p l a n a t i o n o f each p r o c e d u r e i n t h e f i g u r e . A l l n u m e r i c a l i n t e g r a t i o n s i n r were p e r f o r m e d u s i n g t h e g r i d w i d t h A r = 0.02a. P r o c e d u r e 0: INPUT. A f i l e w h i c h s t o r e s t h e p r e v i o u s l y c a l c u l a t e d s e t o f ( r ) ( o r a c o m p l e t e l y new s e t o f n m n ^ ( r ) ) i s r e t r i e v e d and used as i n p u t . A l l c o e f f i c i e n t s , u m n A ( r ) o f t h e p a i r p o t e n t i a l a r e c a l c u l a t e d a t t h i s t i m e . P r o c e d u r e 1: A p p l y i n g t h e LHNC C l o s u r e . c°°°(r) i s c a l c u l a t e d f r o m eq. (3.40a) and a l l o t h e r c™"^ ( r ) a r e o b t a i n e d from eq. (3.40b). g ( r ) i s g i v e n by g ( r ) = n ( r ) 000, , , , 000, , , + c ( r ) + 1 where n ( r ) comes from t h e p r e v i o u s p r o c e d u r e . The l a r g e v a l u e s o f t h e c o e f f i c i e n t s , u m n 2 , ( r ) , n e a r r = 0 c o u l d l e a d t o n u m e r i c a l p r o b l e m s . To a v o i d t h i s t h e m o l e c u l e s a r e assumed t o have a h a r d c o r e o f d i a m e t e r , d, such t h a t d << a. Then g ^ ^ ( r ) = 0 f o r r < d and t h e c l o s u r e r e l a t i o n (eq. (3.40b)) becomes c ( r ) = -n ( r ) r < d ( A I V . l ) S i n c e d « a and, t h e r e f o r e , g0 0°Cd) = 0, i n s e r t i n g t h e h a r d c o r e 132 Figure 21 Flow diagram for the iteration procedure. The coefficients above the arrows indicate the functions resulting from each step. INT. T R A N S F O R M T?mn,(r) 6 B A C K INT. T R A N S F O R M cmnl (r) FOURIER T R A N S F O R M cmnl ( k ) G E N . C O N V O L U T I O N ^mnl (r) 5 B A C K FOURIER T R A N S F O R M 134 condition should have no influence upon the physical properties of the flu i d . In fact i t is found that for the present model varying d from 0.2a to 0.9a had no effect upon the results. Procedure 2. Integral Transformations. The c™1^ (r) are calculated from the c ^ W using the f i r s t set of equations in Table VII. The integrations are performed using the •A n 1 6 4 trapezoidal rule. Procedure 3. Fourier Transforms. ~c m nJl(k) i s c a l c u l a t e d f r o m c™1^!,) using a Fast Fourier Transform (FFT) package 6 5 which can transform two real functions simultaneously. Procedure 4. Generalized Convolution. The two sets of linear equations (4.7a,b) are solved by usinj the subroutine DSLIMP available at the UBC Computer Centre. 6 7 The iterative solution i s converged to an accuracy in the neighbourhood of 1 x IO - 1 6. Procedure 5. Back Fourier Transforms T i m n S , ( r ) is calculated from n™^ 00 using the same procedure as 3. Instead of using the r vector as variable for integration, the k 135 r e c t o r i s us e d where k. = j A k , CAIV.2) and Ak = - 5 — (AIV.3) NAr where N i s t h e number o f d a t a p o i n t s i n t h e f u n c t i o n T 1 m n 2 ' ( r ) , T h i s 66 . „ r e s t r i c t i o n on Ak i s r e q u i r e d by t h e FFT t e c h n i q u e . F o r more i n f o r m a t i o n on t h e n u m e r i c a l c o m p u t a t i o n o f F o u r i e r T r a n s f o r m s t h e r e a d e r i s 68 r e f e r r e d t o t h e a r t i c l e by Lado. P r o c e d u r e 6. Back I n t e g r a l T r a n s f o r m s . An i n t e g r a t i o n t e c h n i q u e s i m i l a r t o t h a t a p p l i e d i n p r o c e d u r e 2 i s used t o t r a n s f o r m n f r ) t o n ( r ) a c c o r d i n g t o t h e second ^ 1 new' to s e t o f e q u a t i o n s i n T a b l e V I I . I n s p e c t i o n o f t h e s e e q u a t i o n s s u g g e s t s t h a t c o m p u t a t i o n a l e r r o r s may a r i s e when r ->• 0 s i n c e t h e i n t e g r a l s f o r £ = 2 , 4 and 6 a r e t o be d i v i d e d by powers o f r . To c i r c u m v e n t t h i s R p r o b l e m , f o r t h e i n t e g r a l , / f ( r ) d r , we use n t h o r d e r p o l y n o m i a l s i t o f i t a s m a l l segment o f f ( r ) between 0 and R<<a. The s u b r o u t i n e 59 DOLSF fr o m t h e UBC Computer C e n t r e i s employed t o a c c o m p l i s h t h i s t a s k . Then t h e r e s u l t i n g p o l y n o m i a l i s i n t e g r a t e d a n a l y t i c a l l y up t o R'. I t i s found t h a t v a r y i n g n from 4 t o 7 and R' from 0.2 a t o 0.4 a has no e f f e c t upon t h e r e s u l t o b t a i n e d . 136 Procedure 7. Mixing 48 It is found that solutions obtained by direct iteration using the above procedures do not converge and hence another method must be 69 used. Broyles found that the solution could be forced to converge by mixing successive iterates. For our computations, the following formula is used mn£,..% „ ... _mn*r^ ^ „ „ ^ f r 1 (AIV.4) where a i s a mixing parameter (ranging from 0.7 to 0.95) that must be increased steadily with increasing density. Proceeding in this way several hundred iterations (the exact number depends upon the temperature and density) are required in orde to obtain a converged result. It takes approximately 60 sec. per 100 iterations in the Amdahl 470 V/6, Model II computer. 137* REFERENCES 1. Barker, J.A., Henderson, D., Rev. Mod. Phys. 48, 587 (1976). 2. Hansen, J.P., McDonald, I.R., Theory of Simple Liquid, Academic Press, London, (1976). 3. Streett, W.B., Gubbins, K.E., Ann. Rev. Phys. Chem. 28, 373 (1977). 4. Streett, W.B., Tildesley, D.J., Proc. R. Soc. Lond. A. 348, 485 (1976). 5. Streett, W.B., Tildesley, D.J., Proc. R. Soc. Lond. A. 355, 239 (1977). 6. Cheung, P.S.Y., Powles, J.G., Mole. Phys. 30, 921 (1975). 7. Cheung, P.S.Y., Powles, J.G., Mole. Phys. 32, 1383 (1976). 8. Buckingham, A.D., Disch, R.L., Dunmur, D.A., J.A.C.S. 90, 3104 (1968). 9. S t e l l , G., Patey, G.N., Htye, J.S., Adv. Chem. Phys. 48, 183 (1981). 10. St i l l i n g e r , F.H., Adv. Chem. Phys. 31, 1 (1975). 11. Chen, Y., Steele, W.A., J. Chem. Phys. 50, 1428 (1969). 12. Chen, Y., Steele, W.A., J. Chem. Phys. 54, 703 (1971). 13. Morrison, P.F., Ph.D. Thesis, California Institute of Technology (1972). 14. Lowden, L.J., Chandler, D., J. Chem. Phys. 59, 6587 (1973). 15. Lowden, L.J., Chandler, D., J. Chem. Phys. 62, 4246 (1975). (Erratum and Addenda of Ref. 14). 16. Chandler, D., Anderson, H.C., J. Chem. Phys. 57, 1930 (1972). 17. Anderson, H.C., Weeks, J.D. , Chandler, D., Phys. REv. A. 4_, 1597 (1971). 18. Smith, W.R., Can. J. Phys. 52, 2022 (1974). 19. Smith, W.R., Madden, W.G., F i t t s , D.C. Chem. Phys. Let. 36, 195 (1975). 138 20. S m i t h , W.R. , Chem. Phys. L e t . 40_, 313 (1976). 21. Madden, W.G., F i t t s , D.D., S m i t h , W.R., Mole. Phys. 35, 1017 (1978). 22. Perram, J.W. , W h i t e , L.R., Mole. Phys. 27_, 527 (1 9 7 4 ) . 23. Sung, S., C h a n d l e r , D., J . Chem. Phys. 56, 4989 (1972). 24. S t e e l e , W.A., S a n d l e r , S . I . , J . Chem. Phys. 61, 1315 (1974). 25. Nezbeda, I . , S m i t h , W.R., Chem. Phys. L e t . 64, 146 (1979). 26. H i r s c h f e l d e r , J.O., ed. I n t e r m o l e c u l a r F o r c e s (Adv. Chem. Phys. 12) (1967). 27. L o n g u e t - H i g g i n s , H.C., D i s s . F a r a . Soc. £0, 7 (1965). 28. Buckingham, A.D., U t t i n g , B.D. , Ann. Rev. Phy. Chem. 21_, 287 (1970). 29. Sweet, J.R., S t e e l e , W.A., J . Chem. Phys. 47, 3022 (1967). 30. P o w l e s , J.G., Gu b b i n s , K.E., Chem. Phys. L e t . 38, 405 (1 9 7 6 ) . 31. Buckingham, A.D., Q u a r t . Rev. 8_, 183 (1959). 32. P o p l e , J.A. , P r o c . Roy. Soc. Lon. A. _221, 498 (1954). 33. S t e e l e , W.A., J . Chem. Phys. 39, 3197 (1963). 34. Blum, L., T o r r u e l l a , A . J . , J . Chem. Phys. 56, 303 (1972). 35. Blum, L. , J . Chem. Phys. _57, 1862 (1972). 36. Blum, L., J . Chem. Phys. 58, 3295 (1973). 37. Edmonds, A.R., A n g u l a r Momentum i n Quantum M e c h a n i c s , P r i n c e t o n U. P r e s s , P r i n c e t o n ( 1 9 5 7 ) . 38. Downs, J . , G u b b i n s , K.E., Murad, S., Mole. Phys. 37, 129 (1979). 39. R i c e , S.A., Gray , P., The S t a t i s t i c a l M e c h anics o f S i m p l e L i q u i d s , I n t e r s c i e n c e , N.Y., (1965). 40. Temperley, H.N.V., R o w l i n s o n , J . S . , Rushbrooke, G.S., ed., P h y s i c s o f S i m p l e L i q u i d , North-Ho 11and, Amsterdam, (1968) . 41. H i l l , T.L., S t a t i s t i c a l M e c h a n i c s , M c G r a w - H i l l , N.Y., (1956). 139 42. Egelstaff, P.A., An Introduction to the Liquid State, Acad. Press, N.Y., (1967). 43. McQuarrie, D.A., Statistical Mechanics, Haper § Row, N.Y., (1976). 44. Ornstein, L.S., Zernike, F., Proc. Acad. Sci., Amsterdam 17, 793 (1914). 45. Anderson, H.C, Statistical Mechanics A, Nerne, B.J. Ed., Plenum Press, N.Y., (1977). 46. Baer, S., J. Chem. Phys., 60, 435 (1974). 47. Wertheim, M.S., J. Chem. Phys. 55, 4291 (1971). 48. Watts, R.O., A. Specialist Periodical Report: Statistical Mechanics, Vol. 1, Singer, K. ed., The Chemical Soc., London, (1973). 49. Thiele, E., J. Chem. Phys. 39, 474 (1963). 50. Wertheim, M.S., Phys. Rev. Let. 1_0, 321 (1963). 51. Wertheim, M.S., J. Math, and Phys. S, 643 (1964). 52. Baxter, R.J., Phys. Rev. 154, 170 (1967). 53. Baxter, R.J., J. Chem. Phys. 4_9, 2770 (1968). 54. Patey, G.N., Mole. Phys. 34, 4 2 ? (1977). 55. Rotenberg, M., Bivins, R., Metropolis, N., Wooten, J.K. Jr., The 3-j and 6-j Symbols, The Technology Press, M.I.T., Cambridge, (1959). 56. Stogryn, D.E., Stogryn, A.P., Mole. Phys. L l , 371 (1966). 57. Barojas, J. , Levesque, D. , Quentrec, B. , Phys. Rev. A, 1_> 1092 (1973). 58. Bartell, L.S., Kuchitsu, K., J. Phys. Soc. Japan 17_, 20 (1962). 59. Rowlinson, J.S., Liquid and Liquid Mixtures, 2nd Ed., Butterworth, London, (1969). 6.0. Patey, G.N., Mole. Phys. 35, 1413 (1978). 140 61. Arfken, G., Mathematical Methods for Physicists 2nd Ed. , Academic Press, N.Y. (1970). 62. Messiah, A., Quantum Mechanics, Vol. 1, John Wiley § Sons, N.Y. (1958). 63. Abramowitz, M., Stegun, I. A., ed., Appl. Math. Ser. 55_, Natl. Bur. Std., D.C. (1965). 64. Scarborough, J.B., Numerical Mathematical Analysis 6th Ed., The John Hopkins Press, Baltimore (1966). 65. Westwell, A., NRC TSS User's Manuel, Computational Centre, N.R.C. 335.010.01. (1974). 66. Brigham, E.O., Morrow, R.E., The Fast Fourier Transform, IEEE Spectrum (1967). 67. Bird, C. , ed., UBC. Matrix, Computer Centre, U.B.C, (1972). 68. Lado, F. , J. Comp. Phys. 8_, 417 (1971). 69. Moore, C., UBC Curve, Computer Centre, U.B.C, (1981). 70. Broyles, A.A., J. Chem. Phys. 33, 456 (1960). 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0060702/manifest

Comment

Related Items