Integral equations for fluids of linear molecules. I. General formulation

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Integral equations for fluids of linear

molecules

F. Lado a a

Department of Physics, North Carolina State University, Raleigh, North Carolina, 27650, U.S.A.

Available online: 22 Aug 2006

To cite this article: F. Lado (1982): Integral equations for fluids of linear molecules, Molecular Physics, 47:2, 283-298

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MOLECULAR PHYSICS, 1982, VOL. 47, NO. 2, 283-298

I n t e g r a l e q u a t i o n s f o r f l u i d s o f l i n e a r m o l e c u l e s I. G e n e r a l f o r m u l a t i o n

by F. L A D O

Department of Physics, North Carolina State University, Raleigh, North Carolina 27650, U.S.A.

(Received 29 January 1982 ; accepted 21 April 1982)

A general procedure is described that puts the practice of integral equation theory for molecular fluids on a par with that of simple fluids : any integral equation approximation can be solved for any intermolecular potential with no additional approximations beyond those inherent in numerical analysis. The essential elements are expansions in spherical harmonics and numerical evaluation of the spherical harmonic coefficients of the pair distribution function. An explicit formula is derived giving the Helmholtz free energy from the computed coefficients.

1. INTRODUCTION

Integral equations such as the Percus-Yevick (PY) and hypernetted chain (HNC) equations [1] constitute a reliable and by now standard approach to the equilibrium properties of simple fluids, i.e. fluids with spherically symmetric intermolecular potentials. While improved approximations are still to be sought for better quantitative agreement with computer simulations and ulti- mately with experiment, the solution of any new equation will proceed along the familiar lines already established for current versions. The situation is otherwise for molecular fluids with orientation dependent forces. Not only has the number of integral equation solutions lagged behind the vigorous output of computer simulations of these models in recent years [2], but the techniques used thus far have been closely tailored to specific calculations and are not generalizable to arbitrary potentials and integral equation approximations. It is with the development of such a general technique that this paper is con- cerned.

The central task in applying integral equation theory is the iterative solution of the coupled, non-linear equations [1]

h(12)_=g(12)- 1 = C ( 1 2 ) - ~ ~ d3 C(13)h(32) (1)

and

C(12) = h ( 1 2 ) - l n [g(12) exp [fir + B(12). (2)

The first of these is the Ornstein-Zernike equation, which defines the direct correlation function C(12) in terms of the pair distribution function g(12), here written for a general orientation dependent potential so that, e.g.

g(12) =g(rl2 , r r , (3)

0026-8976]82]4702 0283 $04.00 9 1982 Taylor & Francis Ltd

M.P. K

where rl~ is the separation between molecular centres and o~1, ~o 2 the sets of Euler angles needed to specify the orientations of the two molecules. The integral in equation (1) is over r 3 and c%, with

= Saw. (4)

For linear molecules, two Euler angles suffice and ~ =4~r. The notation to be used in the remainder of this work will be specialized to this case, although the method can be readily extended to non-linear molecules, at the expense of greater notational and computational complexity.

Equation (2) is the closure relation, giving C(12) in terms of g(12) and the intermolecular potential 4(12) and containing as well the unknown function B(12), the sum of ' bridge' graphs, approximations to which account for the variety of integral equations currently in use. We shall carry B(12) along in the sequel as if known, so that the results can later be specialized to any arbitrary approximation.

The earliest study of an integral equation with non-spherical potential is by Chen and Steele [3], using the PY closure

Cpy(12) = g ( 1 2 ) - g ( 1 2 ) exp (/34(12)) (5)

for hard dumbell molecules. The replacement of the logarithmic non-linearity in (2) with a quadratic one in (5) simplifies the subsequent analysis in that the spherical harmonic decomposition of equation (5) can be carried out analytically, a procedure not possible with, say, the H N C closure. An even greater simplifi- cation in this sense is effected in the linearized H N C ( L H N C ) equation, separately put forward by Wertheim [4], Patey [5], and Henderson and Gray [6]. This is essentially a perturbation-type equation, constructed by extracting the spherically symmetric part $0(r12) of 4(12) and defining associated functions go, Co, and B 0 satisfying a closure relation like equation (2). The difference of the two closures is then

AC(12) = -/3A~(12) + Ah(12) - In [1 +

Ah(12)/go(r~)]

+

AB(12), (6) with

AC(12)=-C(lZ)-Co(r12 ),

etc. Neglect of AB(12) and linearization of the logarithm in (6) yields the L H N C closure

ACLn~c(12) = - BA~(12) +

ho(ra2)Ah(12)/go(r~2 ),

(7)

which is easily evaluated for its spherical harmonic coefficients. Inclusion of the quadratic term in the expansion (QHNC) [7] raises the difficulty of extracting these coefficients to that of the PY equation. The L H N C and Q H N C equations have been successfully used to study multipolar hard spheres [5, 7, 8], but because of the manner in which A6(12 ) appears in equation (7) and its quadratic extension, they are not appropriate for intermolecular potentials with a non-spherical core, such as the dumbell model. Finally, specializing q~o(rl~) to the hard sphere model and further neglecting

ho(r~ )

in equation (7) gives the closure of the mean spherical approximation (MSA) [9],

c ~ s A ( 1 2 ) = -/3A~(12) (8)

(for rl~ beyond the hard core range), which has been solved analytically for a variety of multipolar potentials [9, 10, 11]. These solutions are generally not

Integral equations for molecular fluids

285

very accurate [12], but the analyses of the MSA by Wertheim [9] and especially Blum [10] have assembled formal machinery useful in a more general approach. Our purpose then is to return to the basic equations (1) and (2) and develop a procedure whereby they can be solved for arbitrary potential 4~(12) and arbitrary approximation B(12). T h e most general approach, implying no preset limitations, is in terms of spherical harmonic expansions; this analysis is developed in w 2, based largely on the work of Blum [10]. A few comments on the numerical version of the procedure are made at the end of this section and the thermodynamic quantities obtainable from the calculationare summarized in w 3 ; in particular an explicit formula for the free energy is derived which generalizes a long-known expression for simple fluids [13]. Solutions of the reference-HNC equation [14] for a hard dumbell fluid, using the procedure described below, are presented in the following paper [15].

2. GENERAL FORMULATION

2.1.

Expansions in spherical harmonics

The usual approach in solving equations (1) and (2) for simple liquids expresses these equations in terms of the ' series ' function defined by

S(12) = h(12) - C(12) (9)

and deconvolutes equation (1) through Fourier transformation. We shall follow the same prescription here.

Thus, in terms of S(12), equation (1) reads

S(12) = P j dr a I

dc~

+

C(32)] (10)

4rr

and its Fourier transform is

~(12)-- I

dr12S(12)

exp (ik. r~2 )

P

=4-~ I d~~ ~ draaC(13) exp ( i k . rta ) j"

dra2[S(32 ) +

C(32)] exp ( i k . raz )

- P I d~ + 0(32)], (11)

4rr

while the closure is now

C(12) = exp [ - / ~ ( 1 2 ) + S(12) + B(12)] - 1 - S(12). (12)

These expressions still contain the full dependence on molecular orientations and so are too detailed to be used directly in a numerical calculation. The simplest way of extracting this dependence is to refer the specification of mole- cular orientations in, say, S(12) and ~'(12), to the directions rl~ and k, respec- tively, so that in a coordinate system with z-axis along rl= we have [2]

S(12) = S(ra2 , 01,

0g, ~12),

(13 a)

with q~12 = $ 1 - $=, while in a rotated frame with z-axis along k we have similarly

•(12) = ~q(k, 0'1, 0'2, ~'12). (13 b)

K2

Both these functions can now respective spaces,

CO /rain

S(12) =4~ Y~ E

la, l z = O m = ~ l m t a

oo lmin

~'(12) = 4rr ~] E

lx, 1 , = 0 m = --Imln

be expanded in spherical harmonics in their

(14 a)

, y ,

S,,t~(k) Y,,m(~o 1) ,2~(w 2), (14 b)

with m = - m, lmi n the smaller of 11, 12, and w = 0, ~. [For a system of identical

molecules, the set of independent coefficients in (14) is greatly reduced by the condition

Sl~l,m = ( - 1) t~+t' St, t ~ (15)

in either direct or Fourier space. (See Appendix 1.)] It is to be noted that because the expansions are made in different frames, S h hm(k) is not the Fourier transform of Sl, l~(r ) ; indeed, the final task below will be to link the expansion coefficients in the two frames of reference.

2.2. The Ornstein-Zernike equation

Writing the angular dependence explicitly in equation (14 b) enables a notable simplification of the Ornstein-Zernike equation. With a similar expansion for C(12), equation (11) becomes

S(12)=47rp Y~ (-1)mC~l,3m(k)[S,3t~(k)+~,3,2m(k)]

l~lfl~m

x Yhm(oJ1) Y,2~(~o'2) , (16)

by virtue of the orthogonality of the spherical harmonics [16], leading finally to a relation between the expansion coefficients themselves [10, 17],

o0

S,,,2m(k)=(-1)~P E ~,,,3~(k)['q,~m(k)+C,~,,~(k)]. (17) l~=m

Defining matrices ~.,(k), Cm(k ) with elements ~l, hm(k), ~ht~m(k), l~, 12>~ m, we may write (17) in matrix form as

~m(k) = ( - 1 )~pe,~(k)[~(k) + ~ ( k ) ] , (18)

with the solution

~ ( k ) = ( - 1 ) ~ p [ I - ( - 1) p ( 1 9 )

as the final version of the Ornstein-Zernike equation for S(12) in terms of C(12).

2.3. The closure equation

Similarly, the expansion coefficients of the closure, equation (12), in their space, satisfy

C,,,2m(r) = ght 2m( r ) -- 8 h ,2m. 000 -- Sh,2m( r ), (20)

with

g,,,2o,(rt2) =---~ j dwx dco2 exp [ - fir + S(12) + B(t2)] Y,.~*(wx) Y,2~*(w2)

= (exp [ - fl~(12) + S(12)+B(12)][ll12m). (21)

Integral equations for molecular fluids

287

Short of new approximations, the integral in (21) cannot be reduced to a simple function of the coefficients of the exponent and must be evaluated as it appears. Using the fact that the coefficients with plus and ,minus m in (21) are identical, or equivalently that g~, ~2m(r) is real, we may wril/6 the integral more explicitly as

1 2 ~ r 1 1

g,,tv~(r)

=

!

dq~12 / d(co8 01) I d(cos 02)g(r, 0~, 02, ~)12)

-1 -1

x #l,~(cos 01)~12.~(cos 02)(- 1) m cos m ~ l 2

1 i

dy

}ldx. i

= 4---~ (1

_y2)ltz

dx~ g(r, xl,

x2, y)

-1 - --1]

x t~h,,~(xl)~,=m(x2) ( - 1)~T,,~(y),

(22)

where we have used the additional symmetry

g(r, 01, 02, "if- ~12) =g(r, 01, 02, ~12) (23)

and have introduced the Chebyshev polynomials

Tin(cos ~ ) = cos mq~ (24)

and the associated Legendre functions

Ptm(x),

with

:~,m(x)

= [(2/+ 1 ) ( l - m)!/(1+ m)!] ~/2

Ptm(x)

(25)

normalized to 2 for convenience. T h e integrand in (22) is to be obtained as in (21),

g(r,

Xl, x~, y) = exp

[ - 19d?(r, x~, x2, y) + S(r, Xl, x2, y) + B(r, Xl, x2,

7)], (26)

with, e.g. the expansion (14 a) now reading

S(r, Xl, x2, y)= ~

~ St, t2m(r)#hm(Xl)~t2m(X2)~m(--

1)~Tm(y), (27)

where

m=O I~,l~=m

c% = 1, m = 0

=2, m > 0 (28)

and the negative m terms of the sum have been absorbed by symmetry. The potential ff(12) may have a similar expansion or may be given in terms of atom- atom interactions. Some approximation must be supplied for the bridge function B(12).

2.4.

Linking the expansions

Equations (19) and (20) constitute the reduced versions of (1) and (2). Containing only functions of a single variable, they are amenable to numerical handling. There remains now the task of linking the spherical harmonic expansions in the two rotated frames.

The spherical harmonics in equation (14 a) refer to a coordinate frame with z-axis along r12 ; their representation in a rotated frame is generated by appli- cation of the rotation operators. Following Edmonds [16], we write for an

arbitrary rotation through the Euler angles a,/3, 7

l

Yl~(~176176

E ~m'~(O(a/37)Ylm'(~163

m ' = - - l

with

(29)

D(~/37)

= exp

(irLJh)

exp

(i/3Lu/h)

exp

(iaLz/h),

(30)

are the quantum mechanical angular m o m e n t u m operators ; the

477"

'~1/2 1,.1.

S(rl2 ; 11lfl)=\21+1]

re=Z-l,.,. (llml2~ll~1210)Shh~(r~2)"

(35)

With S(12) now referred to a new coordinate frame with z-axis aligned along

k,

we may compute its Fourier transform as

oo

~q(12) = 4~" ~] Y,

(llmtl2m 2 l lll21ml + m2) Yhm,(m:l) Ytam2(m'2) $ dr

lxl~l mxm~ 0

2~r

t ~ t

x r 2 S(r ; lll21 ) "~ d/3'

sin/3' exp

(ikr

cos/3')

I d~ Y l,m,+m,(fl ' a')

o o

=47r

,.1..,

~" [ E, (llmlsff~llflflO) (21+1~112\

~

]

x 4-rri* dr r 2 S(r ; l~lfl)j,(kr) Yt,m(o'~) Yh~,(ofs),

(36) o

with

where L~, L u

summation in (29) uses the matrix elements of the rotation operators,

~m.,.ct)(=fiy) =

(Im'lD(o~fiy)llm).

(31)

Substitution of equation (29) in (14 a) yields S(12) in the rotated frame as

S(12)=4~r Z Sh12m(r12) Z "@ra,mClx)(a/3~/)-@m2m(l')(a/3~)

l l l a m mlm~

x Yt,m,(~o'l) Yt2m2(~o's). (32)

But the product of matrix elements in (32) can be written [16]

= Z (llml2mllllflO) ( llm112m2 ]11121ml + ms ) ~ m , +mz 0 (t)(a/3T ) l

= E (l ml#lZAlO)

(t mAm lUdml+

m,)

l

(

y,,

x \2-1-+-11 Y*,.~,+,,~(/3', a')

(33)

(with /3'= -/3, a ' = - y the orientation of rl~ in the rotated frame) where the brackets denote Clebsch-Gordan coefficients, and so (32) becomes [10]

3(12)=4= E

Ud) E

(l mAm llAlm +m )

l d f l m~m2

y

,

,

.

,

•

1,ra,(oJ 1)Yl2mz(O9 2)Y ,,..+,,~.(/3, a')

(34)

Integral equations

for

molecular fluids

289

where we have used [18]

i d ~ s i n l 3 e x p ( i x c ~

I d~

Y'm*(/9,~)=\--~--w ]

47ci'j'(x)Sm, o'

(37)

0 0

with

jr

the spherical Bessel function of order 1. Finally, comparison of equations (14 b) and (36) shows that

,,+,, (21+ 1"~,1 ~,

S,,,~m(k) = E

(l, ml~mllflflO)

_/

S ( k ' l t l f l ) ,

_' (38)

with

S(k ; lflfl)---4~ri I I dr r 2 S(r ; llM)j~(kr).

o

This completes the linkage of the coefficients in (14 a) and (14 b). the steps are

St, t2m(r )

>S(r ; lfl~l)

>S(k;

1,12l )

>~qg~=m(k). equation (35) equation (39) equation (38)

Inversion of the procedure is readily produced using the orthogonality of the spherical Bessel functions and the Clebsch-Gordan coefficients; the transformations are

~(k ; lflfl)

= \ 2 l +

1] m=~--lmin

(l*ml~mllflflO)St't2m(k)'

(40)

S(r" lfl2l ) = ( - i)t ~ dk k S S(k" lfl2l)j~(kr),

(41)

' 2 r r 2 o '

St, t=m(r) =

~

(l, ml2~]l,12lO ~

S(r ; lfl~l).

(42)

1= [l,-1,[

For linear molecules, parity considerations require that 11+ lz+ l be even [12]. (39)

Symbolically,

2.5.

The Fourier-Bessel transforms

Fourier transformation for spherically symmetric potentials involves just the kernel

sin kr

J~

kr

(43)

Here we must evaluate transforms with kernels

jt(kr)

for arbitrary integer l. An accurate numerical algorithm for this purpose, based on a grid scaled to the zeros of

jl(x),

could easily be generated [19]. However, this would require systematic interpolation whenever functions of different l values were mixed, as in equation (38), since the various grid points would not coincide. A neater resolution of this step makes use of the differentiation formulas of the spherical. Bessel functions [10, 18],

d F j ~ _ l ( x ) ] _

j~(x)

(44)

L

j

d

= ( 4 5 )

to reduce all transforms to the

jo(kr)

or

jt(kr)

variety.

Thus, for a function f(~)(r) that vanishes sufficiently rapidly at infinity, we

have from (44) oo

F ( k ) ==- I dr r 2 f(n)(r)j.(kr)

0

1 ~ dr d j~_l(kr)

: k

o [rn+l f ( n ) ( r ) ] r -I ' (46)

after an integration by parts. Define now a new function f(~-2)(r) such that

a [r,~+a f(~)(r)] =r2n_l d Ft("-2)(r)l (47)

j

and use (45) to get

F(k)= Io

L

j

o

oo

= - I dr r ~ i("-2)(r)j,_2(kr), (48)

o

after another integration by parts. Repeated applications as needed of this ' step-down ' operation to the transform of equation (39) finally yields

S ( k ; lflfl) =47r S dr r 2 S(~ ; lfl21)jo(kr ) (49 a) 0

for l even, or

S ( k ; lllfl) =47d ~ dr r ~ S(1)(r ; lflfl)j~(kr) (49 b) o

for l odd, with

S(~-2)(r ; 11121 ) = S(~)(r ; 11121 ) - ( 2 n - 1 )r n-2 ~ dx S('~)(x ; 1112l)

r xn--1 '

n = l, l - 2, ... , 3 or 2, (50)

by integration of equation (47), starting with

S ( ' ) ( r ; l l l 2 l ) - ~ S ( r ; lll2l ).

(51)

This is Blum's [10] generalization of Wertheim's [9] transformation.

An inverse transform proceeds backwards through these steps. By inversion of the transforms in (49), we get, for v = 0 or 1,

S(")(r " lfl2l ) = ( - i ) " ~ dk k s S ( k ; lfl2l)j.(kr), (52) ' 2 ~ 2 o

from which a sequence of ' step-up ' operations using (47) gives

2 n - 1 ~

S('~)(r ; lllfl)= S('~-2)(r ; lflfl) r~+~ S dx xnS(~-2)(x ; lflfl),

n = v + 2 , r ..., l, (53) terminating with

S ( r ; l l l f l ) - S(')(r ; lflfl). (54)

Integral equations for molecular fluids 291

2.6. Summary of the procedure

T h e elements developed above are the necessary ingredients in analysing equations (1) and ( 2 ) ; their final assembly into a procedure to solve these equations is best presented in s u m m a r y form. This is done below. I n the next subsection, we shall examine some aspects of the translation of this s u m m a r y into a numerical version, requiring the usual conversions of infinities and infinitesimals into finite quantities. It is convenient to anticipate one of these conversions in presenting the summary.

Both of the fundamental equations in coefficient form, equations (19) and (20) [which uses equation (27)], suggest the index m as the natural ordering parameter for the finite set of spherical harmonic coefficients to be used in a solution. Letting M be the m a x i m u m value of m at a given stage of the calcula- tion and terminating the ll, 12 sums also at this value, the matrices in (19) are of rank M - m + 1 for m = 0, 1, . . . , M and the s u m in (27) terminates with S ~ M ( r ) .

T h e equations are to be solved iteratively for the Stlt2,,(r). One iteration for a new set of Sl, t2~(r) from the current set consists of the following steps :

(1) C o m p u t e the direct correlation function coefficients

C,,,2m(r ) =g,l,2.,(r)- 8,,, ...

00o-

S,,,=~(r),

(55)

with

gt,~2m(r) = (exp [-fir ) (56)

evaluated as a triple integral using

M M

S(12) = Y.

Z

Sl,12,n(r)#l,m(Xl)~12m(X2)~ 1)mTm(y)"

(57)

m = 0 lx, l~=m

T h e potential r will be given in some form and the bridge function B(12) determined by some approximation.

(2) Convert the Cz,l~,~(r) to the set

{ 4rr ~a/2l ...

C(r

; 111211=\2--~-+--~]

m~=o%,,(l,ml2m[lll2lO)C,,,,m(r).

(58)

( 3 ) ' L o w e r ' the l > 1 members to C(~)(r ; lll2l), v=O or 1, by repeated use of

Cr ; lll2l ) = C(n)(r ; lal2l ) - (2n - 1 )r n-1 ~ dx C(n)(x x~_l lll2l) , (59)

starting from C<t)(r ; lJ21 ) = C(r ; 11131 ).

(4) Compute their Fourier transforms to get

CX3

C ( k ; l~12l ) =4rri ~ j dr r 2 C(")(r ; lll2l)j.(kr). (60)

o

(5) Convert these to the set

t,+,, (2l+1~1/2

Ct, t~,,,(k) = Z (llm12fftl111210) C(k " lJJ). (61)

l = I / , - t o l

(6) F r o m the Ornstein-Zernike equation get

~,.(k) -- ( - 1 )mp[I - ( - 1 )mp(~m(k)]-I e,.(k)r ). (62)

(7) Convert the Shl2,,~(k) to the set

= ~ 41r ~ 112

~q(h;

lllfl)

\ 2 l + 1 ]

~

m

%Alxml~m[lfl2lO>S*'*2m(k)"

(63)

(8) Compute their inverse transforms to get

S(~)(r 9 1,12l ) = ( - i)~ S dk k s ~(k " lfl~l)j~(kr). (64)

' 2 ~r2 o '

(9) ' Raise' the l > 1 members to S ( r ; lflfl)-S(Z)(r ; l, lfl) by repeated use of

S(~)( r" lflfl)=SC~-z)( r ; lllfl) -2n---~l- i dxx~S(~-2)( x ; l, lfl). (65)

' r n + l 0

(10) Convert these to obtain the new set

Sl,l~.,(r) = E _l (l*ml~m]lfl2lO) ( 2 l + _{\ ~ j} ]x~,/2 _{S(r ; lf121 ). (66)}

This completes the iteration.

When convergence is obtained, M can be incremented if desired and a new round of iterations begun. Since convergence is most difficult to reach for the lowest order coefficients, this approach increases the scope of the calculation only as it becomes easier. [In a sophisticated program, truncation of the harmonic series could be done dynamically in I and m separately, as the computed coefficients St,t~m(r) fell below some predetermined measure of significance.] It should be noted that the often-mentioned ' p o o r convergence' of the harmonic expansion does not apply here. Ironically, it is indeed a troublesome feature of the ' exact ' computer simulation methods, where one must perforce work with the coefficients g~z2,,(r) directly [20]. Reconstructing g(12) from these requires that in the region of the potential core, a few coefficients reproduce a complex, rapidly changing or even discontinuous function of four variables, a heroic demand indeed. In contrast, the key expansions here are of S(12) and B(12), b o t h continuous functions, from which g(12) is obtained as

g(12) = exp [ - fi~(12) + S(12) + B(12)], (67)

so that the burden of a strong anisotropy in the vicinity of the core is carried by the exact ~b(12). This is not to claim, of course, that the g(12) computed in this way will necessarily be accurate ; that depends primarily on how well B(12) is approximated, as well as on the host of numerical compromises that must be made to render the procedure computable.

Assuming this more rapid convergence of the expansions for S(12) and B(12) compared to that for g(12), it will likely be possible to reasonably compute more coefficients of g(12) through equation (56) than are included of S(12) and B(12) in the exponent. We note that already the zeroth order approximation of these latter functions, giving

g,,~2,.(r)_~ exp [Sooo(r)+Booo(r)]<exp [-flq~(12)]llfl2m), (68)

studied for hard dumbells by Melnyk and Smith [21] in connection with their perturbation theory for molecular fluids, was found by them to give quite reasonable results for gt~z~.~(r), particularly at small r.

Integral equations [or molecular fluids

293

2.7.

Some numerical details

The full generality of the procedure is maintained in Step (1) of the Summary by numerically evaluating the integral for gh hm(r) in (56), rather than expanding the exponential regardless of the size of the exponent. (To be sure, when the anisotropic part of the exponent becomes small enough as r increases, lineariza- tion will save a considerable amount of computation without sacrificing accuracy, as is illustrated in the following paper.) Since these integrals are over cos 0 and cos ~, gaussian quadrature using the zeroes of the Legendre and Chebyshev polynomials, respectively, seems most natural.

Let x l , . . . , xn be the n zeroes of the Legendre polynomial

P~(x).

Then the integral of a polynomial R 2 n _ l ( X ) of degree 2 n - 1 or less is exactly evaluated as [18]

1

S dxR~n-x(X)=2 ~ wkR2n-l(Xk)'

(69)

- 1 k = l

where the weights w k are given by

w k = {(1 -

Xk2)[P',~(Xk

)]2}-1. (7 O)

Similarly, with the n zeroes Y l , . . . , Yn of the Chebyshev polynomial

Tn(y),

we have [18]

n

1

R2~-I(Y) =~r ~ 1

dy

(1 _y2)1/2 -

R~n_x(yj) ,

(71)

- - 1 j = l n

with equal weights

1/n.

With these rules, the integral in Step (1) (equation (22)) is evaluated as

ks, k~, j = 1

•

(72)

Since this involves a triple sum, the time required for a quadrature increases with n as n a and would very quickly become prohibitive for an iterative scheme that is to repeatedly perform such integrals for several hundred r values. Fortunately, gaussian quadrature is accurate for quite moderate n ; furthermore, different n values can be used in segments of the r-range where the smoothness of the integrand appreciably differs; finally, as noted above, the quadrature can be avoided altogether when the anisotropic part of the exponent in (56) is small. All these devices combine to make this step much less formidable than it appears at first sight, though it is still likely to be the most time consuming step in most cases.

A similar practical simplification can be made in Step (6), where the trans- forms Shh,~(k ) are proportional to the square of the Ct,hm(k ). When the latter become small enough with increasing k, the S h l ~ ( k ) functions can be simply set to zero and the matrix manipulations of this step entirely bypassed. The Jo transforms of Steps (4) and (8) can be done with a Fast Fourier Transform ( F F T ) routine [22] ; one could also adapt the F F T to the Jl trans- forms, or, alternatively, design a specific algorithm for this case, based on the zeroes of

jl(x)

[19]. That th e grid points for the even and odd l components would not coincide using the latter choice is of no consequence, since the two sets are never mixed.

All other integrals, including those of the thermodynamic functions to be discussed below, are adequately evaluated with the trapezoidal rule, assuming a reasonably small interval for the integration grid. Special care may be needed in Step (9), where small errors in quadrature can be magnified near the origin when the integral is divided by r ~+1. A modified trapezoidal rule, wherein only the

S(x)

part of the integrand is fitted by linear interpolation, will circumvent this problem, as noted earlier by Patey [7].

3. THERMODYNAMIC QUANTITIES

T h e generalization to anisotropic potentials of the familiar expressions for internal energy, pressure, and compressibility in terms of the pair distribution function g(12) is straightforward. For completeness, they are listed below. Here we focus first on the extension of a less familiar expression for the Helmholtz free energy A.

T h e artifice of a ' charging ' parameter ~, 0 ~< ~ ~< 1, permits one to define an excess free energy Aex for a partially turned-on system,

fiAex(~ )

= - i n Q(~) (73)

Q(~)=(4~rV)-N I arN do;V exp [--~fl ~j~(ij)],

(74)

so that

d~A ex(~)

_{-89 S dr}

_{(g(12 ; ~)fl~(12)[000). (75)}

d~

Integration of (75) then gives

flAex 1

X -~p j d; Io ae<g(12

; ~)~(12)1000>.

(76)

We shall follow the analysis for simple fluids of Morita and Hiroike and of Green [13] in evaluating equation (76).

Generalizing equation (67), we write

g(12; ~ ) = e x p [ - ~ f i ~ ( 1 2 ) + S ( 1 2 ; ~ ) + B ( 1 2 ; ~)], (77) so that

ag(12 ; se)_

g(12 ; ~:)fi~(12) +g(12 ; ~:) ~ [S(12

or

; ~:)+B(12; ~:)], (78)

g(12; ~:)flq,(12)

-a~: [ - g ( 1 2 ; ~:)+g(12 ; s e) lng(12 ; ~:) exp

{~:5,b(12)}]

-S(12"

~:) ag(12; ~:) B(12. ~:) ag(12 ; ~:)

, a~: , a~:

- - - - [89 ; ~:)+g(12 ; ~:)-g(12 ; ~:)lng(12 ; s r exp {~:fi~(lZ)}]

+ C ( 1 2 " ~:) ~g(12; ~:) B(12" r ag(12; ~:) (79)

, a r , a r ,

where we have used S = h - C .

Integral equations [or molecular fluids

295

Returning to (76) with this last result, we have now

flA.x/N = fiA1/N + fiA2/N + fina/N,

(80)

where

fl A1 1

_{= ~p j" dr ({hZ(12) + h(12) -g(12) In g(12) exp [fl~(12)] [000),}

_{(81 a)}

N = ~o f d,- i df C(12" ~) 0g(12 ; ~) 000 (81 b)

o ' ~ '

3A3

_{N -}

_{I p l d r}

i

_{d~ B(12; ~)}

(

0g(12; ~:)000)

_O~

_"

(81c)

m|0

The integration over ~ has been carried out in A v The second term can also

be integrated, as follows.

From the Ornstein-Zernike equation, (17), we have that

htl,2~(k;

~)=C,lz2,~(k; ~ ) + ( - 1 ) m p Z h l , t3m(k; ~)C~3z~m(k; ~),

(82)

13

or, in matrix form,

[lm(k ; ~)=Cm(k; ~)+(-1)mp~m(k;

~:)~m(k; ~:),

(83)

from which it follows that

[I+(-1)~p~,n(k; ~ ) ] - * = I - ( - 1 ) ~ p ~ , ~ ( k ; ~).

(84)

Now, using the rule, shown in Appendix 2, that for any non-singular, square

matrix A(~),

d In Det [A(~)] = Tr [A(~:)-ad~(J) 1

dse

,

(85)

where Det and Tr stand for the determinant and trace operations, it follows

from (84) that

~--~ In {Det [ I + ( - 1)mp~(k ; ~)]}

= T r { [ I - ( - 1)mp~'m(k '

~)](-1)~PO~(k;8~

~)}'

(86)

whence we get that

p Tr [em(k ; ~) O~m(k;~)]~: =

Pl O~O {lnDet[i+(_l)~p~,~(k ' ~)]

- ( - 1 ) r a p Tr [~m(k ; r

(87)

We are now ready to integrate over ~ in A 2. Transforming to Fourier space

and expanding in spherical harmonics, we have finally from (81 b) and (87),

3A~

dk i d~ lO.(12. ~) ~h(12 ; ~)

)

N =.IP I ~

o

'

O~

000

I n 1

=lpl

ol

y,

e)

~)

[

1

=IP I (-2---~)rr)a !dsr m~ Tr em(k" ~)

O[~m(k;o~ ~)

1

dR

-

2p / ~ - ~ ~ {In Det [ I + ( - 1)mphm(k)]

- ( - 1)rap Tr [~m(k)]},

(88)

where we have used

O*,,,2,,~(k) = O,~,,,Ak),

(89)

shown in Appendix 1.

As with simple fluids, no progress can be made with Aa, save by approxi- mation. Our final expression for the free energy in terms of spherical harmonic coefficients thus reads

= - {P I dr Z g,,,2,,(r)[{-g,,,v,,(r)- Sh,2,,,(r)- Bh,:,,,(r)] - 8 9

lllzm

1 dk

- 2 p I ~ ~ {ln Det [I + ( - 1 y'*ph,,,(k)]- ( - 1)'p Tr [~.,(k)]}

_ l p f d r [ d~ B ( 1 2 ; ~) ag

In the following paper, a perturbative version of (90), leading to the reference- H N C equation, is presented.

Finally, the isothermal compressibility X, internal energy U, and pressure p are given by [12]

pkt~TX= 1 + p j dr ( g ( 1 2 ) - 11000> (91)

f l U ] N = 89 5 dr {g(12)flr (92)

flp/p = 1 - ~p I dr (g(12)rflr (93)

The first of these depends only on the spherically symmetric part of g(12) :

pkuTX = 1 + p I dr [go0o(r) - 1]. (94)

For atom-atom potentials, (92) and (93) are most conveniently evaluated in the forms shown. If the potential is given in terms of a spherical harmonic expan- sion, however, they may be written instead

fiU/N~- 89 j dr ~. ghh,,,(r)flehl:(,f), (95)

l~l,m

flp/p -~ 1 - { I dr E gh,~,,(r)rflr 9 (96)

l~l~m

The usual special handling is required in the pressure integrand for hard core potentials in either form.

APPENDIX 1

Equation (15) follows directly from an expansion in spherical harmonics of the symmetry condition

S(r12, 01, 02, r = 3(~'12, ,N- - 02, 7r- 01, - r (A 1)

Alternatively, it can be obtained from the equivalent expression [12]

S(Y ; 12111 ) = ( - 1)/1+/2 S(r ; 11121 ) (A 2)

using the transformation (58) and the symmetries of the Clebsch-Gordan coefficients [16].

Integral equations for molecular fluids

Similarly, (89) follows from

~*(k,

01, 03, 412)= ~q(-

k,

01, 03, 412)

= S ( k , 77-- 01, 7 r - 02, - 4 1 2 ) ,

or can be obtained from the obvious relation [see equation (39)]

lll l)

using (58).

297

(A 3)

(A 4)

APPENDIX 2

Let ckj be the cofactor of the element akj of square matrix A.

for any j, whence

T h e n

Det A = ~

ckjakj

(A5)

k

Det A

- - =ckj. (A 6)

akj

Now using the chain rule, we have

d

da~j

(A 7)

d--~ Det A = ~ ckj d~ '

k,j

d

CkJ dakj

dakj

~ A 1 d A ~

In [Det A ] = k,y ~ D e t A d~: - k,y ~ b j k - - ~ = T r ~ - ~ - j , ( A 8 )

d-~

since bjk =ckj/Det A defines the matrix elements of the inverse matrix B - - A -1.

REFERENCES

[1] HANSEN, J. P., and McDONALD, I. R., 1976, Theory of Simple Liquids (Academic Press).

[2] STREETT, W. B., and GUBBINS, K. E., 1977, A. Rev. phys. Chem., 28, 373. [3] CHEN, Y.-D., and STEELE, W. A., 1971, ft. chem. Phys., 54, 703.

[4] WERTHEIM, M. S., 1973, Molec. Phys., 26, 1425. [5] PATEY, G. N., 1977, Molee. Phys., 34, 427.

[6] HENDERSON, R. L., and GRAY, C. G., 1978, Can. J. Phys., 56, 571. GRAY, C. G., and HENDERSON, R. L., 1979, Can. J. Phys., 57, 1605.

[7] PATEY, G. N., 1978, Molec. Phys., 35, 1413.

[8] PATEY, G. N., LEVESQUE, D., and WEIS, J. J., 1979, Molec. Phys., 38, 1635. [9] WERTHEIM, M. S., 1971, .7. chem. Phys., 55, 4291.

[t0] BLUM, L., 1972,.7. chem. Phys., 57, 1862. [11] BLUM, L., 1973, .7. chem. Phys., 58, 3295.

[12] EGELSTAFF, P. A., GRAY, C. G., and GUBBINS, K. E., 1975, International Review of Science, Series 2, Volume 2, edited by A. D. Buckingham (Butterworths).

[13] MORITA, T., and HIROIKE, K., 1960, Pros. theor. Phys., 23, 1003. GREEN, M. S., 1960, ft. chem. Phys., 33, 1403.

[14] LADO, F., 1973, Phys. Rev. A, 8, 2548. T h e apt term ' r e f e r e n c e - H N C ' was first used by: CEPERLEY, D. M., and CHESTER, G. V., 1977, Phys. Rev. A, 15, 755. [15] LADO, F., 1982, Molec. Phys., 47, 299.

[16] EDMONDS, A. R., 1960, Angular Momentum in Quantum Mechanics (Princeton Uni- versity Press).

[17] JEPSEN, D. W., and FRIEDMAN, H. L., 1963, J. chem. Phys., 38, 846.

[18] ABRAMOWITZ, M., and STEGUN, I. A., editors, 1964, Handbook of MathematicalFunctions

(U.S. Government Printing Office). [19] LADO, F., 1971, J. comput. Phys., 8, 417.

[20] TILDESLEY, D. J., STREETT, W. B., and WILSON, D. S., 1979, Chem. Phys., 36, 63. [21] MELNYK, T. W., and SMITH, W. R., 1980, Molec. Phys., 40, 317.

[22] BRIGHAM, E. O., 1974, The Fast Fourier Transform (Prentice-Hall), Chap. 10.