A theory for site–site pair distribution functions of molecular fluidsElijah Johnson

Citation: The Journal of Chemical Physics 67, 3194 (1977); doi: 10.1063/1.435234 View online: http://dx.doi.org/10.1063/1.435234 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/67/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Evaluation of site-site bridge diagrams for molecular fluids J. Chem. Phys. 121, 6922 (2004); 10.1063/1.1789131 Experimental determination of the site–site radial distribution functions of supercooled ultrapure bulk water J. Chem. Phys. 117, 6196 (2002); 10.1063/1.1503337 On the relation between the Wertheim’s twodensity integral equation theory for associating fluids andChandler–Silbey–Ladanyi integral equation theory for site–site molecular fluids J. Chem. Phys. 104, 3325 (1996); 10.1063/1.471094 Site–site correlations in short chain fluids J. Chem. Phys. 93, 4453 (1990); 10.1063/1.458728 Equation of state of siteinteraction fluids from the site–site correlation function J. Chem. Phys. 75, 4060 (1981); 10.1063/1.442564

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A theory for site-site pair distribution functions of molecular fluidsa)

Elijah Johnson

Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 (Received 17 May 1977)

A theory for site-site pair distribution functions of molecular fluids is presented. This theory is analogous to the RISM theory, but it is not restricted to interaction site pair potentials. It was derived directly from the Omstein-Zemike equation using Fourier-Wigner series. Results of the theory for homonuclear diatomic pair potentials are compared with corresponding results obtained from the RISM theory and from computer simulation studies. The relationship between the RISM equation for homonuclear hard sphere diatomic pair potentials and the theory presented is given.

I. INTRODUCTION

The RISM theory is a theory of equilibrium statistical mechanics for site-site pair distribution functions of molecular fluids. 1 A rigorous derivation of the RISM equation has not yet been presented. 2 A set of equations which is analogous to the RISM equation is derived in this paper in a rigorous manner from the Ornstein-Zernike equation. This derivation suggests new theories for site- site pair distribution functions of molecular fluids. One of these new theories is used to treat model pair po­tentials for diatomic molecules.

In Sec. II some of the notation of this paper is dis­cussed. In Sec. III the theory for site-site pair distri­bution functions of molecular fluids is derived. In this section explicit use is made of the fact that the position vector of any site of a molecule is suitable as a position vector argument of the total correlation function. In Sec. IV the theory is applied to a number of homonuclear diatomic pair potential systems. Section V is the dis­cussion section. This paper has four appendices. In Appendixes A, B, and C, some equations presented in this paper are derived. Appendix D is a discussion of the numerical methods which were used to obtain some of the atom-atom pair distribution functions presented in Sec. IV.

II. DEFINITIONS

The systems of interest here are many-component uniform fluids. Species of molecules will be denoted by lower case italic Roman letters. Lower case Greek letters will be used to denote specific sites in a mole­cule. The sites of most interest are the locations of the nuclei of molecules The vector 11(a, a) denotes the po­sition of site a of molecule 1 of species a in a specific molecule-fixed coordinate system. The vectors ql(a, a(3) and q2(a, YIl) are defined by

(2. 1)

and

a) Research sponsored by the U. S. Energy Research and Devel­opment Administration under contract with the Union Carbide Corporation.

(2.2)

The vector q1 (a, a(3) is the vector displacement from site (3 of molecule 1 to site a of molecule 10 The magnitude of a vector x will be denoted by x. The quantity q1 (a, a (3) is the same in each molecule of species a, so q1 (a, a (3) will be denoted by q(a, a(3),

The vector R denotes a position vector in a Cartesian coordinate system. Let R 1(a, (3) be the position vector of site (3 of molecule 1 of molecular species a and let R2(b, Il) be the position vector of site Il of molecule 2 of molecular species b. The vectors R 1(a, (3) and R2 (b, Il) are defined with respect to a space-fixed coordinate system. The quantity Q denotes the set (1), e, X). The quantities 1>, e, and X are Euler angles. Let Q1 (a, (3) be the Euler angles of the chosen molecule-fixed coordi­nate system at site (3 of molecule 1 of molecular species a with respect to a space-fixed coordinate system.

The functions P(R1, R2 , Q1, Q2) and Q(R1, R2 , Q1 , Q2) are defined by

P(R1, R2 , Q1, Q2)

= L P[R1 , R2 ; L 1Z1m 1 , L 2Z2m 2] L 1 l 1 rn1

L212m2

= L Q [R1, R2 ; L 1Z1m 1, L 2Z2m 2] L111m1

L2'2 rn2

(2.3)

(2.4)

The quantities P[R1 , R2; L1Z1m1, L2Z2m2] are the Fou­rier-Wigner coefficients of p( R1, R2 , Q1, Q2)' The quantities D(Q1 ; L1Z1m1) are Wigner functions. A dis­cussion of Fourier-Wigner series and a definition of the Wigner functions are given in Ref. 3. The quanti­ties L 1 , Z1, and m1 denote integers. The ranges of L 1 ,

11 , and m1 follow: OO'S L 10'S 00, - L 10'S 110'S L 1 , and - L1

O'S m10'S L 1 • The function Df11m1 (Q1) of Ref. 3 corresponds to D(Q1 ; L111m1)' The convolution of P(Rh R2, Qh Q2) and Q(Rl> R2, Q1, Q2), P* Q(Rh R2, Qh Q2), is defined by

3194 The Journal of Chemical Physics, Vol. 67, No.7, 1 October 1977 Copyright © 1977 American Institute of Physics

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Elijah Johnson: Pair distribution functions of molecular fluids 3195

P* Q(Rh R2, S1i1 S12)

= f dR3 f d~3 P(Rl, R 3, S1 l, S13)Q(R3, R2, S13, S12) .

(2.5) The quantity w is equal to 8.,f, and f dS1 denotes

fo2r de!> So2r dX So' de sine.

It follows that

P* Q(RiI R2, S1i1 S12)

= L [p* Q(Rh R2; Lll1mh L 2l2m2)]D(S1 l ; Lll1ml) L1'lml L2'2~

(2.6)

The quantities [p* Q(Rh ~; L1l1mh L 2l2m2)] are defined by

[P* Q(Rh R2; L1l1mh L 2l2m2)]

= L J ~ P[Rh ~; Lll1mh L3 l3m 3] L3'3~

Equation (2.6) is derived in Appendix A.

h",y{r1(a, a), r2(b, y); ab; i3/l; Llltmh L 2l2m2}

(2.7)

III. THEORY

The total correlation function, h.b(Rl(a, (3), ~(b, /l), S11(a, (3), S12(b, /l); i3/l), of a fluid system can be written as follows:

hab(Rl(a, (3), ~(b, /l), S1 1(a, (3), S12(b, /l); i3/l)

= L h[Rl(a, (3), ~(b, /l); ab; i3/l; Llllmh L 2l2m2] Ll'l m1 L2'2~

The function Pa Pb(l+h.b(Rl(a, (3), ~(b, /l), S11(a, (3), S12 (b, /l): i3/l» is the probability distribution thatthe mole­cule-fixed coordinate system at site 13 of a molecule of species a has orientation S11(a, (3) and site (3 of this mol­ecule is at R1(a, (3) and that the molecule-fixed coordinate system at site /l of a molecule of species b has orienta­tion S12(b, /l) and site /l of this molecule is at ~(b, /l). The quantity Pa is the number density of molecules of species a. The quantity Pa is equal to the number of molecules of species a in the system divided by the vol­ume of the system.

For a given 13 and /l the element h",y of the site-site total correlation function matrix of order (Llllmh L 2 l 2m2) is defined by

= f dR1(a, (3) f dR2(b, f.l) S dS1l~' (3) J dS12~' /l) hOb (Rl(a, (3), ~(b, /l), S11(a, (3), S12(b, /l); i3/l)

XD(S11(a, (3); L 1l 1ml)D(S12(b, /l); L 2l2m2)Ii(Rl(a, (3)+ql(a, a(3) - rl(a, a» Ii~(b, /l)+~(b, Y/l) - r 2 (b, y» . (3.2)

The quantity Ii(x) is the Dirac delta function. Elements of the site-site total correlation function matrix of order (000, 000) are the site-site total correlation functions of the RISM theory. Only the site-site total correlation func­tion matrix of order (000, 000) will be used in this paper. The matrix elements of this matrix are given by

h",y{rl(a, a), r 2(b, y); ab; i3f.l; 000, OOO}

= S dRl(a, (3) S ~(b, /l) f dS11~' (3) f dS12(~ /l) hab(Rl(a, (3), ~(b, /l), S11(a, (3), S12(b, /l); i3fJ.)

X Ii (Rl (a, (3)+ql(a, a(3)-rl(a, a»Ii(R2(b, /l)+q2(b, Y/l)-r2(b, y». (3.3)

Here the function h",y{rl(a, a), r 2 (b, y); ab; i3/l; 000, OOO} will be denoted by h",y{rl(a, a), r2(b, y); ab; (3/l}. The quanti­ty P. Pb(l + hay {rl (a, a), r2(b, y); ab; .B/l}) is the probability distribution that site (JI of a molecule of species a is at rl(a, a) and that site Y of a molecule of species b is at r 2(b, y).

For a uniform fluid the function hab is a function of the vector (Rl (a,.B) - R2 (b, /l ». This implies that

(3.4)

It follows from Eq. (3.1), Eq. (3.3), and Eq. (3.4) that

hay {r1 (a, a) - r 2(b, y); ab; (3p.} = f (2~3 exp[ik' (rl (a, a) - r 2 (b, y»]jo(kq(a, a.B»

xh[k; ab; (3/l; 000, OOOJjo(kq(b, Y/l»+ V",y{rl(a, a)-r2(b, y); ab; {3/l}. (3.5)

The function V",y{r1(a, a) - r 2 (b, y); ab; (3/l} is defined by

Vay {rl(a, a) - r 2 (b, y); ab; {3/l} '" L (1- IiL1+La,O) f (2rt;)3 exp[t"k' (rl(a, a) - ra(b, y»))(- i)Ll Ll'lml L2'2~

J. Chern. Phys., Vol. 67, No.7, 1 October 1977

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3196 Elijah Johnson: Pair distribution functions of molecular fluids

XjL 1(kq(a, ()I (3»D (0, - ()k' - rPk; L 10-l1)(-1)'I-mlD(n(ql); L1-11110)h[k; ab; {3/lj L1l11111> L2121112]iL2

Xj L2 (kq(b, Y/l »D(O, - ()k' - rPk j LzO - l2)(- 1)'2-mZD(n(Q2); L z - 1112 0).

The quantity oiJ is the Kronecker delta function,

15 0 =1, ifi=j,

= 0, if i* j .

(3.6)

(3.7)

The derivation of Eq. (3.5) is given in Appendix B. The function h[k; abj {31l j Llllm[, L2l2111z] if the Fourier trans­form of h[r1 (a, ()I) - rz(b, y); abj {31l j Lll1111[, L zlz111Z]. The function h[kj ab; {31l; L 1it1111, L zlz111Z] is defined by

h[kj abj {31l; L 1l11111 , L zlz111Z] = f dR exp(- ik· R)h[R; abj (31l j Llll1111> Lzlzmz] • (3.8)

The quantity (rPk' ()k' 0) denotes the Euler angles of the vector k in the space-fixed coordinate system. The quantity (0, - ()k' - rPk) denotes the Euler angles of the rotation which is the inverse rotation of the rotation which corresponds to (rPk' ()k' 0). These rotations are rotations relative to the space-fixed coordinate system. The quantity n1(Q1) de­notes the Euler angles of the vector Q1 (a, ()I{3) in the chosen molecule-fixed coordinate system at site (3. The quantity jL(x) is the spherical Bessel function of order L. The function jo(x) is given by

jo(x) = sinx/x •

When 11 =Y the vector ~(b, YIl) is equal to the zero vector. So,

h",r{rl{a, ()I)-rz{b, Y)j ab; {3y}

= f ([:)3 exp[z"k. (rl(a, ()I) - rz(b, y»]jo(kq(a, ()I{3»h[kj abj (3y; 000, 000] + V"y{rl(a, ()I) - rz(b, Y)j ab; {3y} .

The function V",y{rl(a, ()I) - rz(b, y); ab; {3y} is given by

For {3 = ()I and 11 = Y,

h",y{r1(a, ()I)-rz{b, y); ab; O'y}=h[r1(a, O')-rz(b, y); ab; 000, 000].

For a uniform fluid the Ornstein- Zernike equation 4 is

hab (R1(a, (3) - Rz{b, 11), n 1(a, (3), nz(b, 11); (31l)= Cab (Rl(a, (3) - Rz(b, 11), n 1(a, (3), nz(b, 11); (31l)

+ ~ Pe f dR3(e, Ii) f dn3~' Ii) C.e (R1(a, (3) - Ra(e, 15), n 1(a, (3), n 3(e, Ii); (3o)

xheb<Ra(e, 0)- Rz(b, 11), n 3(e, Ii), nz(b, Il)j lill) .

The function Cab(Rl(a, (3) - Rz(b, 11), n 1(a, (3), nz(b, Il)j (31l) is the direct correlation function. The Ornstein­Zernike equation implies that

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

h,., {rl(a, ()I) - rz{b, Y)j ab; {31l} = Cay {r 1(a, ()I) - rz(b, y); ab; {31l}+ L Pe {c{aej (3lih h(eb; Ii 11 ; rl(a, 0') - rz(b, Y»}"y •

e (3.14) The function {C(ae; (315)* h(ebj lill; rl(a, ()I) - rz(b, Y»}"y is defined by

{C(ae; (3o)*h(ebj 011; rl(a, ()I) - rz(b, Y»},.y= f dR1(a, (3) f dRz(b, 11) f dnl~a, (3) f dnz~, 11) J dRaCe, 0) f dn3~' 0)

x Cae(R1{a, (3) - Ra(e, 0), 0l(a, (3), 03{e, o)j (31i)h"b<Ra(e, 0) - Rz(b, 11), 03(e, 15), Oz{b, 11); lill)

XI5(R1(a, (3)+Ql(a, ()I{3)- rl(a, ()I» Ii (R2 (b, Il)+~{b, YIl)-r2(b, y». (3.15)

The following equations hold for any allowed values of {3, 11, E, and 1]:

R1(a, (3)+Ql(a, 1](3)=R1{a, 1]) ,

Rz(b, Il)+~(b, EIl)=Rz(b, E) .

(3.16)

(3.17)

Let the molecule-fixed coordinate systems at sites ()I, (3, Y, etc., of a molecule be chosen such that the Euler angles, (rP, (), X), between any two of them are equal to CO, 0, 0). That is, the molecule-fixed coordinate system at one site in a molecule can be brought into coincidence

J. Chern. Phys., Vol. 67, No.7, 1 October 1977

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Elijah Johnson: Pair distribution functions of molecular fluids 3197

C",y{fl(a, a) - f2(b, 1'); ab; f3fl} with the molecule-fixed coordinate system at any other site in the molecule by a translation. This means that

=C",y{rl(a, ad-rz(b, Y)j abj T}E} j (3.23) (3.18)

{C(aej f3o)*h(eb; 0flj flea, a)-f2(b, Y»}"y for any allowed values of ° and X. Since 5"2 1 (a, (3) = 5"2 1 (a, T}) and 5"2a(b, fl)=5"22(b, d, the following equation holds: ={C(ae; T}X)*h(ebj XE; flea, a)-f2(b, Y»}"'Y' (3.24)

hob(RI(a, (3) - R2(b, fl), 5"2 1(a, (3), 5"22(b, fl); f3fl)

=hab(RI(a, T}) - R2(b, E), 5"2 1(a, T}), 5"22(b, E); T}E) • (3.19)

Equation (3.24) implies, for example, that

{C(aej (30) *h(ebj 0fl j fl (a, a) - fz(b, y»ty Equation (3.19) is true because arguments of hOb on both sides of Eq. (3.19) correspond to the same positions for molecules 1 and 2 when Eqs. (3.16) and (3.17) hold. Eqs. (3.3) and (3.19) and the following equations

={C(aej (30) *h(ebj oYj rl (a, a) - fz(b, y»} ",y' (3.25)

Equation (3.14) and Eq. (3.25) imply that

h"y{rl(a, a) - rz(b, Y)j abj f3fl}

(3.20) = Cal' {flea, a) - fa(b, Y)j abj f3fl}

and + L Pe {C(aej f3o)*h(eb; 0Yj fl(a, a) - fa(b, Y»}"y • (3.21) e

(3.26) imply Eqs. (3.22), (3.23), and (3.24). Equations (3.20) and (3.21) are equivalent to Eqs. (3.16) and (3.17).

h",y{rl(a, a) - r 2(b, 1'); ab; f3fl}

(3.22)

I

Let each molecule of the fluid have m labeled sites. To obtain an equation which is analogous to the RlSM equa­tion a summation over sites is performed on equation (3.26). This summation is defined by the following equation:

m

L h",r{fl(a, a)- f 2(b, 1'); ab; f3f.L}'" 8"

m m

L C",y{fl(a, a)-fz(b, Y)j abj i3fl} + LL: p,,{c(aej f3fl)*h(ebj flYj flea, a)-fz(b, Y»}"'Y' 6" 8" e

(3.27)

In Eq. (3.27) the range of values of the Greek letters is from 1 to m.

By analogy with the case for h",r {fl (a, a) - fz(b, Y)j abj f3fl} the following relations for C",y {fl (a, a) - fz(b, Y)j abj f3fl} and {C(aej f3fl)*h(ebj flYj flea, a)-fz(b, Y»}",y hold:

C",r{fl(a, a)-fz(b, Y)j ab; f3fl}= f (::'3 exp[t'k'(fl(a, a)-ra(b, y»]jo(kq(a, a(3»C[k; abj f3fl; 000, 000]

and

{C(aej f3fl) *h(ebj fl Yj rl (a, a) - fa(b, y»} ",y = f (::)3 exp[ik' (fl (a, a) - ra(b, y»]jo(kq(a, a(3»

XC[k; aej f3fl; 000, 000] h[kj ebj flYj 000, 000]

+X",y{aej f3flj ebj flYj fl(a, a)-fa(b, y)}+Z",y{aej f3f.Lj ebj f.LYjfl(a, a)-fa(b, y)}.

The functions W",y, X",y, and Z",y are given by

W",y{fl(a, a)-ra(b, Y)j abj f3fl}= L (1-0LI+LZ'O) f (2a:)3 exp[ik'(fl(a, a)-fz(b, 1'»] Ll'lml Lz'zmz

x (- i)Ll hI (kq(a, a(3»D(O, - Bk , - CPk j LID -ll){- 1)'I-m1 D(5"2(q1)j L1 - ml0)C[kj abj f3fl; Lll1ml, LzlzmzJ iLa

(3.28)

(3.29)

x ha (kq(b, Yfl» D(O, - ek , - CPk; LaO - la) (- l)/z-mz D(5"2(qz); Lz - maO) , (3.30)

X",y {ae; f3f.L; eb; fl 1'; r1 (a, a) - rz(b, y)}= ~ (1 - 0L3.O) f (:!-)3 exp [ik· (r1 (a, a) - rz (b, Y))]jo(kq(a, a(3» L3'3ms

XC[kj aej (3)J.j 000,000, L3l3m3]h[k; ebj flY; L3-l3-m3j OOOJ (-21t/3~ms (3.31) L3+

Z",y {ae; f3fJ.; ebj fJ. 1'; r l (a, a) - fz(b, y)} = ~ (1- 0LI'O) L: (::)3 exp [ik . (fl (a, a) - fz(b, 1'))](_ i)LI LI/lml L3 /3ms

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3198 Elijah Johnson: Pair distribution functions of molecular fluids

xh1(kq(a, a(3»D(O, - 11k , - ¢k; L 10 -Zl)(- 1)l1-m1D(I1(q1); L1- m10) C[k; ae; {3/l; L1Zi'm1, L3Z3m3]

A (_ 1t13+ma

x h [k; eb; /l Y; L3 - Z3 - m3 ; 000] 2L3 + 1 (3.32)

Equations (3.5), (3.28), and (3.29) imply that the Fourier transform of Eq. (3.27) is m m

L jo(kq(a, a(3»h[k; ab; /3/l; 000, OOO]jo(kq(b, Y/l»= Ljo(kq(a, a/3»C[k; ab; (3/l; 000, OOO]jo(kq(b, Y/l» el' el'

m m

+ L W",y{k; ab; /3/l}+ L L pejo (kq(a, a(3»C[k; ae; (3/l; 000, 000] h[k; eb; Wr; 000, 000] (3.33) al' el' e

m m

+ L L pJX",y{ae; /3/l; eb; /lY; k}+Z",y{ae; {3/l; eb; /lY; k}]- L V",y{k; ab; /3/l}. el' e el'

Equation (3.22) implies that h",y{k; '1b; {3/l} is independent of the values of /3 and /l, so h",y{k; ab; {3/l} can be written as h",y{k; ab}. Equation (3.23) implies that C"y{k; ab; {3/l} can be written as C:\y{k; ab}. These facts and Eq. (3.12) imply that Eq. (3.33) is equivalent to the following equation:

m m

L jo(kq(a, a (3»he I' {k; ab} jo(kq(b, Y/l» = L jo(kq(a, a(3» Cel' {k; ab}jo(kq(b, Y/l» el' el'

m m

+ L Pe L jo(kq(a, a(3»Cel'{k; ae}hl'y{k; eb}+ L [W",y{k; ab; /3/l} e el' el'

e

When the right-hand side of Eq. (3.34) is truncated at the third term, the following equation is obtained:

t jo(kq(a, a(3» he I' {k ; ab} jo(kq(b, Y/l» el'

m

= Ljo(kq(a, a/3»Cel'{k; ab}jo(kq(b, Y/l» el'

m

+ L Pe Ljo(kq(a, a/3» CIll' {k; ae}hl'y{k; eb} • e el'

(3.35) Only one-component systems which have a pairwise ad­ditive potential energy function will be treated in this paper. For a one-component system Eq. (3.35) becomes

t jo(kq(a{3» hilI' {k} joCk q(Y/l» el'

m

+P Ljo(kq(a/3»Cel'{k}hl'y{k} 0 (3.36) -gl'

Equation (3.36) is analogous to the RISM equation. If the left-hand side of Eq. (3.36) were replaced by h"'l'{k}, Eq. (3.36) would become an equation of the RISM theory. Each theory for the direct correlation function when used in conjunction with Eq. (3.36) yields a different theory for site-site pair distribution functions of molecular fluids. The Percus-Yevick approximation will be used here. The Percus-Yevick approximation5 for the direct correlation function of one-component systems which have a pairwise additive potential energy function is

C(Rl(13) - ~(/l), 111(J3), 112(/ . .t}; (3/l)

= /(R1(/3) - ~(/l), 111(/3), n2(/l); (3/l)

x y{R1 ((3) - ~ (/l), 111 ((3), 112 (/l); (3/l) • (3.37)

(3.34)

I The function/(R1({3) -~(/l), 111(13), 112(/l); (3/l) is the Mayer / function of the system and y(R1({3) - R2 (J.L), 111(/3), 112 (/l): (3J.L) is the y function of the system. The Mayer / function is related to the pair potential of the system, U(R1 ((3) - R2 (/l), 111 ((3), 112 (/l); /3/l), as follows:

/(R1({3) - R2 (/l) , 111(J3), 112 (/l); (3/l)

=exp[- U(R1({3)- ~(/l), 111({3), 112 (/l); (3/l)/k B T]-1 •

(3.38) The quantity k B is the Boltzmann constant, k B = 1. 38044 X 10-16 erg/OK, and T is the temperature of the system. The y function is defined by

(1 + h(R1({3) - R2(/l), 111({3), n2 (/l); (3/l»

= exp( - U(R1({3) - ~(/l), 11 1({3), 112 (/l); (3/l)/k BTl

x y(R1({3) - R2 (/l), 111({3), 112(/l); /3/l) . (3.39)

IV. APPLICATIONS

In this section the theory derived above using the Percus-Yevick approximation will be applied to a num­ber of pair potential systems. In the applications below each molecule has two equivalent labeled sites. A rela­tion between site-site total correlation functions, site­site direct correlation functions, and site- site y functions will now be given. Equations (3.38) and (3.39) imply that the total correlation function is given by

h(Rl(J3)-R2 (J.L), 111({3), 112 (/l); (3/l)

= - 1+ Y (R1 (/3) - R2 (J.L), n1 (J3), n2 (J.L); (3/l)

+ /(R1 ({3) - R2 (/l), 111({3), 112 (/l): /3/l)

x y(Rl(/3) - ~(J.L), 111({3), 112 (J.L); /3J.L) • (4.1)

Equations (3.37) and (4.1) imply that the Percus-Yevick apprOximation total correlation function is given by

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Elijah Johnson: Pair distribution functions of molecular fluids 3199

h(RtUl) - R2(/l), S1 t (l'3), S1 2 (/l): f3/l)

= y'(RtUl) - R2 (/l), S1 t (f3), S12 (/l); f3/l)

+ c(Rt(f3) - ~(/l), S1 t (f3), S1 2 (/l); f3/l) . (4.2)

The function y' (Rt (13) - ~ (/l), S1 t (1'3), S12 (/l); (3/l) is defined by

y'(R t(f3)- R2 (/l), S1 t (f3), S1 2(/l); f3/l)

= Y (Rt (13) - R2 (/l), S1 t (f3), S1 2(/l); f3/l) - 1 •

Equation (4.2) implies that

h"r{r}= y~r{rh C",r{r} .

Since q(aa) = 0, the following equation holds:

joe q(aa»= 1 .

(4.3)

(4.4)

(4.5)

When the labeled sites of each molecule are equivalent, all site-site total correlation functions are equal and all site-site direct correlation functions are equal. The last statement is expressed by the following equations:

h"r {r} = he~ {r} , (4.6)

(4.7)

The following equation is Eq. (3.36) for two labeled sites per molecule:

2

Ljo( kq(af3» h61l {k}jo( kq(Y/l» 61l

2

= L joCk q(af3» C61l {k}jo(kq(Y/l» 61l

2

+ p L jo(kq(af3» C61l {k}hllr{k} • (4.8) 61l

Equations (4.5)- (4.8) give the following equation for yft {k}:

yft{k}= - Cll{k}

+ Cll {k} 1 {I - 2pCll {k} 1[1 + jo(kq(12»]} • (4.9)

Equation (4.9) for Ylt {k} was solved by iteration. To discuss the method of solution further it is necessary to give the relation between Car {r} and Yar {r} which is used here. Equation (3.37) implies that the Percus- Yevick approximation site-site direct correlation functions are given by

C",r{r}=f[r;ay;OOO,OOO]Y",r{r}+ L (1-6 L +L • o) L 3'3"'3 3 4

L 4'4 m4

xf[r; ay; L 3-l3-m3' L 4-l4-m4]

(- It'3-"'3 (- 1)-'4-m4 X y[r; ay; L3 l3m 3' L 4l 4m4J 2L3 + 1 2L4 + 1

(4.10)

Equation (4.10) is dedved in Appendix C. Here it is assumed that the second term on the right-hand side of Eq. (4.10) is negligible. In this case C",.,.{r} is given by

Car{r} = f[r; ay: 000, 000] Y",r{r} . (4.11)

In Ref. 6 it is assumed that the second term on the right­hand side of Eq. (4.10) is equal to zero because of the nature of the Y function. In the iteration to find Ylt {k}

the initial Cll{r} was equal to f[r; 11; 000, 000].

The pair potentials of the systems treated here will now be presented. The vector St(a, i, a) is the posi­tion of the ith interaction site of molecule 1 of species a relative to site a of this molecule. Here only one­component systems will be treated, so St(a, i, a) will be written as St(i, a). The pair potentials of the fluids treated here are of the form

U(rt(a) - r 2 (y), S1 t (a), S12(y); ay) 2 2

= L L ujJ(\rt(a)+St(i, a)-r2(y)-S2(j, y)\). i=l ;=1

(4.12) The function U(rt(a) - r 2 (y), S1 t (a), S12(y); ay) is the pair interaction energy between molecule 1 and molecule 2. The function U(rt(a) - r 2 (y), S1 t(a), S1 2(y); ay) is indepen­dent of the values of a and y for a particular position and orientation of both molecule 1 and molecule 2 be­cause (rt(a)+St(i, a» is independent of a and (r2 (y) +S2(j, y» is independent of y. The quantity S is defined by

(4.13)

The quantity S is the magnitude of the vector (St(1, a) - 8 t (2, a».

For one pair potential treated here, S= 1.1 A and the function u/J(r) is equal to the Lennard-Jones pair po­tential,

(4.14)

with E =0.6067x 10-14 erg and 0'=3.341 A. This pair po­tential will be referred to as pair potential I. Pair po­tential I is a model pair potential for liquid nitrogen. Molecular dynamics results for pair potential I are presented in Refs. 7 and 8. The bond length of a mole­cule of liquid nitrogen is 1. 1 A.

For pair potential II, S = 1. 1 A and the function ujJ(r) is equal to the repulsive part of the Lennard-Jones pair potential of pair potential I. That is,

u lJ (r)=4E[(O'lr)t2-(O'!r)6]+E, r~2t/6(1,

=0, (4.15)

Pair potential II is an approximation of the repulsive part of pair potential I. Results presented in Ref. 9 im­ply that at high denSities the structure of a liquid such as liquid nitrogen is determined primarily by the repul­sive part of the pair potential. 9

For two pair potentials treated here the function ujJ(r) is equal to the hard sphere pair potential,

ufJ(r) = co, r < (1 ,

= 0, r> (1 • (4.16)

Here (1 is the diameter of the hard spheres. The pair potentials presented above are referred to as homonu­clear diatomic pair potentials. Pair potential I is a Lennard-Jones diatomic pair poUmtiai. Hard sphere diatomic pair potentials are treated in Refs. 1 and 10 by the RISM theory.

When the labeled sites of a molecule are located at

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3200 Elijah Johnson: Pair distribution functions of molecular fluids

the nuclei, the site-site pair distribution functions are atom-atom pair distribution functions. Let gll {r} be the atom-atom pair distribution function which is obtained by solving Eq. (4.9) for Yf1{k}. The atom-atom pair dis­tribution function g "y {r} is related to the atom-atom total correlation function h"y {r} as follows:

g"y{r}=1+h"y{r}. (4.17)

The function gll{r} for pair potential I and pair potential II at kBTIE '" 4. 03 and p0'3 = O. 696 and the molecular dy­namics pair distribution function 7 for pair potential I at kB TIE'" 4. 03 and p0'3 = O. 696 are shown in Fig. 1. The maximum difference between gll {r} of pair potential I and the molecular dynamics pair distribution function of pair potential I in the vicinity of the first peak is O. 3 or about 15% of the maximum value of the molecular dy­namics pair distribution function in the vicinity of the first peak. For a Lennard-Jones atomic fluid the Percus­Yevick approximation pair distribution function is also larger than the corresponding computer simulation pair distribution function in the vicinity of the first peak. 11,12

For a Lennard-Jones atomic fluid the positions of the maximums and minimums in the Percus-Yevick approxi­mation pair distribution function are shifted relative to the positions of the corresponding maximums and mini­mums in the computer simulation pair distribution func­tion. ll This is also the case for gl1 {r} of pair potential I.

The function gu{r} of pair potential II shown in Fig. 1 is in better agreement with the molecular dynamics atom-atom pair distribution function for pair potential I than is gll{r} for pair potential I. This suggests that atom-atom pair distribution functions which are obtained using Eq. (4.9) are more accurate for repulsive pair po­tentials than for pair potentials with attractive regions. Percus-Yevick approximation pair distribution functions also seem to be more accurate for repulsive atomic pair potentials than they are for atomic pair potentials with attractive regions. 5

The function g 11{r} for pair potential II at k BTl E = 1. 61 and pa3 = O. 6964 and the molecular dynamics pair dis­tribution functions for pair potential I at kB TIE = 1. 61

- g,l{r} • Pair Potential II

- - gll{r} • Pair Potential I 2

000 MD. Pair Potential I

o j

2 3 rllT

FIG. 1. Atom-atom pair distribution functions for pair poten­tial I and pair potential II at KBT/E =4. 03 and P0'3=:0. 696. The solid curve is the atom-atom pair distribution function of Eq. (4. 17) for pair potential II. The curve which conSist of dashes is the atom-atom pair distribution function of Eq. (4.17) for pair potential I. The circles are molecular dynamics results for pair potential I.

I--~T-- ----------- -. -----~---r-------I

gll{r}, Pair Potential IT

0.00 MD. Pair Potential I

2 3 r / IT

FIG. 2. Atom-atom pair distribution functions for pair poten­tial I and pair potential II at KBT/E = 1. 61 and p0'3 =0.6964. The solid curve is the atom-atom pair distribution function of Eq. (4. 17) for pair potential II. The circles are molecular dynamics results for pair potential I.

and pa3 = O. 6964 are shown in Fig. 2. The maximum dif­ference between gll{r} of Fig. 2 and the molecular dy­namics pair distribution function of Fig. 2 in the vicinity of the first peak is 0.2 or about 10% of the maximum value of the molecular dynamics pair distribution func­tion in the vicinity of the first peak. The function gl1{r} of Fig. 2 does not have the shoulder which occurs in the molecular dynamics pair distribution function at about r/a=1.3.

There are at least two reasons for the differences between gl1{r} for pair potential I and the molecular dy­namics pair distribution function for pair potential I. The first reason is that the right-hand side of Eq. (3.34) was truncated to obtain Eq. (3.36). The second reason is that the Percus-Yevick approximation was used to ob­tain Eq. (4.11).

Let gn{r} be the atom-atom pair distribution function which is obtained from the RISM theory. 10 The gu{r}'s presented here for hard sphere diatomic pair potentials are site-site pair distribution functions for the interac­tion sites. The functions gll{r} and gR{r} for a hard sphere diatomic pair potential system with Sla = O. 6 at pa3 = O. 3 are shown in Fig. 3. The function gn{r} of Fig.

2

?-- ~-------------

OL----2 3

r / IT

FIG. 3. Atom-atom pair distribution functions for the homo­nuclear hard sphere diatomic pair system with S/0'=0.6 and P0'3=0.3. The solid curve is the atom-atom pair distribution function of Eq. (4.17). The curve which consists of dashes is the atom-atom pair distribution function of the RISM theory.

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Elijah Johnson: Pair distribution functions of molecular fluids 3201

2

! o~

\ \

\ , '-

2 r / <7

FIG. 4. Atom-atom pair distribution functions for the homo­nuclear hard sphere diatomic pair potential system with (j =3.3 A, S = 1. 1 A, and p = 1. 729 X 10-2 IA3. The solid curve is the atom-atom pair distribution function of Eq. (4. 17). The curve which consists of dashes is the atom-atom pair distribu­tion function of the RISM theory.

3 is in good agreement with Monte Carlo results, 13 so the function gl1{r} is less accurate than gR{r} in the vi­cinty of the first peak. Note that the function gl1{r} of Fig. 3 does not have the maximum which occurs in gR{r} at r /a = 1. 6. This maximum also occurs in the Monte Carlo pair distribution function 13 which corresponds to gR{r} of Fig. 3. At high densities a Percus- Yevick ap­proximation pair distribution function for a hard sphere system is also significantly different from the corre­sponding computer simulation pair distribution function in the vicinity of the first peak. 5.14

The functions gl1{r} and gR{r} for the hard sphere diatom­ic pair potential system with a = 3. 3 A and S = 1. 1 A at p = 1. 729 x 1O-2/A3 are shown in Fig. 4. The functions gl1{r} and gR{r} of Fig. 4 are in good agreement except in the vicinity of the first peak.

The function y",y(r) of this paper and y",y{r) of Ref. 15 are not equivalent. The function Yl1{r} is given by

(4. 18)

Note that gl1{r} is given by

gll{r}={l + f[r; 11; 000, 000] }Yll{r} . (4.19)

The functionf[r; 11; 000, 000] for the homonuclear hard sphere diatomic pair potential with a=3.3 A and S= 1. 1 A is shown in Fig. 5.

8" 0,0 o· o o :;::

-1 L.. ____ __

, , , __________ L-________ L-______ ~

,/<7 2

FIG. 5. The function! Ir; 11; 000, 000] for the homonuclear hard sphere diatomic pair potential with (j=3.3 A and S=1.1 A.

o

-10

-20

2 , / (J

FIG. 6. Atom-atom direct correlation functions for the homo­nuclear hard sphere diatomic pair potential system with (j=3.3 A, S = 1.1 A, and p = 1. 729 X 10.2 I A 3• The solid curve is the atom-atom direct correlation function of Eq. (4.11). The curve which consists of dashes is the function C~{r}.

The RISM equation for a homonuclear hard sphere diatomic pair potential system consists of the following three equations 10:

for r<a ,

CR{r} = 0, for r> a .

(4.20)

(4.21)

(4.22)

Here hR{k} is the Fourier transform of hR{r}, and hR{r} is equal to hll{r} of the RISM theory. The function CMr} is defined by

(4.23)

The functions C ll{r} and C~{r} for the hard sphere diatomic pair potential system with a = 3.3 A and S = 1. 1 A at p = 1. 729 X 10-2/A3 are shown in Figo 6. It is apparent from Fig. 6 that CMr} is approximately equal to Cll{r}. Similar results for f[r; 11; 000, 000] andfR{r} are dis­cussed in Ref. 15. The function fR{r} is the atom-atom Mayer f function of the RISM theory. This result sug­gests that the atom-atom direct correlation function of the RISM theory for homonuclear hard sphere diatomic pair potential systems is C~{r}.

V. DISCUSSION

The theory presented here is applicable to any pair potential. Some results of the theory for homonuclear diatomic pair potentials are given in this paper. The next two statements refer to the atom-atom pair dis­tribution functions which are presented in this paper and which were obtained by using Eq. (4.19). These atom-atom pair distribution functions are as accurate as the Percus-Yevick approximation pair distribution functions for the corresponding atomic pair potentials. The major cause of errors in these atom-atom pair distribution functions seems to be the use of the Percus­Yevick approximation to obtain Eq. (4.11).

Equation (3.36) is not the only analog to the RISM

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3202 Elijah Johnson: Pair distribution functions of molecular fluids

ACKNOWLEDGMENTS equation which can be derived using the methods pre­sented here. A A different analog might be more appro­priate. Let hlk} be the site-site total correlation func­tion matrix ofAorder (000, 000). The exact h{k} is sym­metric. The h{k} obtained by solving Eq. (3.36) is not necessarily symmetric. This problem is discussed in Ref. 21.

I am grateful to Alfred H 0 Narten, David Chandler, Roger P. Hazoume, and Lesser Blum for their com­ments on the derivational method which was used in this paper. I greatly appreciate the many discussions which I had with Dr. Narten and Roberto Triolo while this work was in progress.

-----------------------------------~

APPENDIX A

The derivation of the Fourier-Wigner series for P* Q(Rl, Rz, S11 , S1z) follows. Equations (2.3) and (2.4) imply the following equation:

P*Q(RI, Rz, S1 1 , S1z) = J ~ f dS1 s L P[RI , Ra; LI1Iml, Ls1smsJ D(S1 I ; LI1Iml) D(S1s j Ls13m 3) w Lilirttl

x L: Q[Ra, Rz; L 414m4, Lz1zmzJ D(S13 ; L414m4) D(S1z ; L212m2) . L 4 14"'4

L212mz

When the right-hand side of Eq. (AI) is rearranged the following equation is obtained:

L: L P[RI , Rs; LI1Iml, Ls1sm3J LIII"'I L414"'4 L3 1S m3 L2 12"'Z

(AI)

(A2)

The function D * (ns j L4 - 14 - m4) is the complex conjugate of D(S13 ; L4 - 14 - m4)' The function D(S13 j L 414m4) is re­lated to D*(S1s; L4 -14 - m4) as followS l6

:

D(S13; L 414m4) = (- 1)14-"'4 D* (S13; L4 -14 - m4) • (A3)

The following reiation I6 holds:

J d~3 D(S13 ; Lsl3m 3) D* (S13 ; L4 - 14 - m4) = 2L: + 1 15 LS' L4 15 ls , -1415 "'3' -"'4 •

The function 15 11 is defined by Eq. (3.7). Equations (A2) and (A4) imply that

P* Q(Rlo Rz, S1 1 , S12) = L L: f dRs P[RI , Ra; LI1Imh Ls1smsJ Q [Ra, Rz; L3 - ls - ms , L 212m2J LIII"'I La l3 1tts

L212"'2

(- 1tl s+ms x D(S1 l j LI!tml) D(S12; L2l2m2) .

2L3 + 1

Equation (A5) is equivalent to Eq. (2.6).

APPENDIX B

The derivation of Eq. (3.5) follows. Equations (3.1) and (3.3) imply that

ha,,{rl(a, 0') - r2(b, Y)j a bj i3J.L}= J dRt(a, f3) S dRz(b, J.L) J dS1t~a, (3) S dS12~' J.L)

xL: h[Rt(a, f3)-R2 (b, J.L); ab; i3J.L; LtlIml, L212m2]D(S1 I(a, f3); L ll Iml)D(n2(b, J.L); LIlt"'}

L2 12"'2

L 2l2m2) 15 (Rt(a, f3)+ql(a, af3) - rl(a, 0') 15(Rz(b, J.L)+qz(b, YfJ.) - r2(b, y» 0

After integration over RI(a, f3) and Rz(b, J.L), Eq. (B1) becomes

{ S dS1lw(a, f3) S dS1Z (wb, J.L) '" hOI" rt(a, a)-r2(b, Y)j abj {3fJ.}= L.J h[rt(a, a)-ql(a, af3)-r2(b, 1') Lilimi

L212"'2

+ qz(b, I'll); ab; {3J.L; LI1Imh L21zm2] D (ni (a, f3); Llllml) D(S1z(b, p.); L212m2) .

Equation (B2) implies that the Fourier representation of hOI" {r i (a, 0') - r2 (b, y); abj {3fJ.} is given by

J. Chern. Phys., Vol. 67, No.7, 1 October 1977

(A4)

(A5)

(B1)

(B2)

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Elijah Johnson: Pair distribution functions of molecular fluids 3203

{ } f d~1(a, (3) f d~z(b, /1) '" S dk hay r 1(a, a) - r2(b, Y)j abj 13/1 = w w ~ (21T)3

L111'"1 LZIZm2

X exp[ik· (r1 (a, a) - r2(b, 1') + ~(b, 1'/1) - q1 (a, a(3))] h[kj abj f3/1j L1l1mb Lzlzmz] D(~1 (a, (3)j L1hm1)D(~a(b, /1); Lalama)'

The function h[k; ab; 13/1; L1hmb Lzlzmz] is the Fourier transform of h[Rj abj 13/1; L1l1mb Lalama]. (B3)

The function exp[ - ik . q1 (a, a (3) ] may be represented as follows16:

exp[ - z'k • q1 (a, a(3)] = L (2L3 + 1)(- i)L3 iL3 (kq(a, a(3» P L3 (cosO (k, q1» LS~O

(B4)

The function h (x) is the spherical Bessel function of order L and the function PL(x) is the Legendre polynomial of order L. The quantity O(k, q1) denotes the angle of the vector q1(a, a(3) relative to the vector k. The quantity (m k ,

Ok' 0) denotes the Euler angles of the vector k in the space-fixed coordinate system. The quantity (0, - Ok' - cf>k) denotes the Euler angles of the rotation which is the inverse rotation of the rotation which corresponds to (cf>k' Ok' 0). These rotations are rotations relative to the space-fixed coordinate system.

Relationships between (0, - Ok' - cf>k)' O(k, q1), and (cf>(q1), O(q1), 0), the Euler angles of the vector q1(a, a(3) rela­tive to the space-fixed coordinate system, are illustrated in Fig. 7. In Fig. 7, a dot represents a vector or co­ordinate system and an arrow between two dots represents the rotation of the vector or coordinate system at the tail of the arrow into the vector or coordinate system at the head of the arrow. The quantity above or below each arrow denotes the Euler angles of the rotation represented by the arrow.

The following relation holds16:

The function P LS (cosO (k, q1» is equal to D(O, 0 (k, q1), 0; L 300), so

PL3(COSO(k, q1» = L D(O, - Ok' - cf>k; LSOm3)D(cf> (q1), O(q1), OJ L3m 30) . '"3

Equations (B4) and (B6) imply the following equations:

(B5)

(B6)

exp[- z'k 'q1(a, a(3)]= L (2L3+ 1)(- i)L3 iL3(kq(a, a(3»D(O, - Ok' - cf>kj LSOm3) D(cf> (q1), O(q1), OJ LSm30) , (B7) L 3 '"3

When the exnressions above for exp [ - ik . q1 (a, a (3) ] and exp [ik • qz (b, 1'/1)] are substituted into Eq. (B3), the following equation is obtained:

hay {r1 (a, a) - r2(b, Y)j abj f3/1} = L L f (2~3 exp [z'k • (r1 (a, a) - ra(b, y»](2L3 + 1) (- i}L3 iL3 (kq(a, a(3» L111 '"1 Ls'"3 L zlz'"2 L 4'"4

XD(O, -Ok' -cf>kj LsOm3)(2L4+1)iL4iL4(kq(b, Y/1»D(O, -Ok' -cf>kj L40m4)h[kj ab; f3/1j L 1l1m1, Lal2ma] f d~1~' (3)

XD(cf>(ql), O(q1), 0; L3m30)D(~1(a, (3); L 1l1m1) f d~2(~ /1) D(cf>(qz), O(~), 0; L4m40)D(~z(b, /1); Lalama) . (B9)

Let the quantity ~(ql) be the Euler angles of the vector Q1(a, a(3) in the molecule-fixed coordinate system at site 13. Relationships between ~1(a, (3), ~(ql)' and (cf>(q1), O(q1), 0) are illustrated in Fig. 8. The general form of equation (B5) implie s that

D(</>(ql), O(Ql), OJ L 3m 30 )= L D(~1(a, (3); L3m3mS)D(~1(Ql)j L3mSO) . ms

The same relationships exist between ~l(b, /1), ~(qz), and (</>(qz), O(qz), 0) which exists between ~1(a, (3), ~(Q1)' and (</> (Ql), 9 (Ql), 0), so,

D(</>(qz), 9(~), OJ L4m40)= L D(~2(b, /1)j L4m4me)D(~(qz); L4meO) . m6

(BI0)

(B11)

When the expressions above for D(</>(Ql), O(Ql), 0; LSm30) and D(<fJ(qz), O(~), 0; L4m40) are substituted into EQ. (B9), the following equation is obtained:

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3204

k

0' \...

Elijah Johnson: Pair distribution functions of molecular fluids

Space-Fixed Coordinate System

Molecule- Fixed Coordinate System

ot Point f3

Space-Fixed Coordinate System

FIG. 7. An illustration of the relationships between a specific set of vectors, coordinate systems, and Euler angles.

FIG. 8. An illustration of the relationships between a specific set of vectors, coordinate systems, and Euler angles.

h .. y {rl(a, O!) - r 2(b, y); abj !31l} = L: L f (::)3 exp [z'k . (r i (a, O!) - rz(b, Y»}(2Ls + 1)(- ilLs hs (kq(a, 0!!3) LIII"'I LS"'S

Lz12"'2 L4"'4

xL D(n2(b, J.-L)j L 4m 4m 6)(-1)IZ-"'ZD*(nz(b, J.-L); L2 -l2-m2)D(n(qz); L4m aO). "'6

Here use was made of Eq. (A3).

Equations (A4) and (B12) imply the following equation:

h .. y{rl(a, O!) - r 2(b, I'); ab; IW}= L: J (~3 exp[ik. (rl(a, O!) - r2(b, 1'»](- i}L1hl(kq(a, 0!!3) LIII"'I

LZIZ"'Z

x D(O, - 11 k, - ¢k; L 10 -ll)iLZ hz(kq(b, YJ.-L» nCo, - 11 k, - ¢k; LzO -l2)

(B12)

x Ii [k; ab; !3J.Lj L 1llml> Lz12m2](- 1)11-"'1 D(~~(ql); Ll - ml0)(- 1)12-"'2 D(n(CIz); L2 - m20 ) .

(B13) Equation (B13) is equivalent to Eq. (3.5).

APPENDIX C

The functions f(R1 (O!) - R2(Y), n l (O!), n 2 (1'); O!Y) and y(Rl (O!) - R2(y), n l ( O!), n 2(y); O!Y) may be represented as follows:

f(RI(O!)-Rz(Y), nl(O!), ~lz(Y); O!Y)= L f[R1(0!)-Rz(Y); O!Yj Lll1ml, L2l2m2]D(·~1(0!); LllIml)D(·~2(Y); L2l2m2) ' (cl) Lllll"l L2IZ"'2

y(RI(O!)-Rz(y), nl(O!), n 2(y); O!Y)= L y[Rl(O!)-Rz(Y); O!y; LllIml, L2l2mz] DPl(O!); Llllml)D(~12(Y); L 2l2m2) . Llll"'l L Z12"'2

Equations (3.37), (C1), and (C2) imply that

(C2)

C(Rl (O!) - Rz(Y), n l (O!), n 2(y); O!Y) = L LIll"'l L2IZ"'2

L: f[RI(O!)-Rz(Y); O!Y; Llltml. Lzlzmz]y[Rl(O!)-Rz(Y)j O!Y; Ls1sms, L,l4m .1 LSIS"'3

L4 14""

(C3)

The following equation holds16:

D(nl; L 1llm1)D(nl ; Lzlzm2) = L V(Ll11Lz12\ LlL2LS1l + 12) V+(L 1m lL2m 2\ LlL2L Sml + m2)D(nl; L s , 11 + 12, ml + m2) . LS (C4)

J. Chem. Phys., Vol. 67, No.7, 1 October 1977

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Elijah Johnson: Pair distribution functions of molecular fluids 3205

The quantity V(LlllL2l21 L1L2L3l3) is a vector coupling or V -C coefficient. The quantity V(LlllLz121 L1L2LSl3) is the complex conjugate transpose of V(LlllL2l21 L1L2L3l3)' Equation (C4) was used to obtain the following equation from Eq. (C3):

e(Rl(a)-R2 (y), a l (a), az(y); 0'1')= L e[Rl(a)-Rz(y); 0'1'; Lslsms. L6l6mS]D(~11(a); Lslsms)D(~lz(Y); L6l6m6) . Ls'sms

(C5) L 6'6m6

The function e[Rl(a) - Rz(y); aYi Lslsms, Lsl6m6] is given by

e[Rl (a)-Rz(y); 0'1'; Ls1sms, Lel6m s] = L L j[Rl(a)-Rz(y); 0'1'; LlJ ls-l3' ms-ms, Lz, ls-l4, m6- m 4] LILa LS'Sm3

L 4'4 m4

X y[Rl (a) - Ra(Y); 0'1'; Lslsm3' L 4l4m4] V(LllS -lsL3l31 L1L3LSls)V+(LlmS - msL3m31 L1L 3L Sm S)

x V(L2l6 - l4L4l41 L2L4Lele) V+(L2mS - m4L4m41 LzL4Leme) .

Equation (C6) implies equation (4.10).

APPENDIX D: NUMERICAL METHODS

Fourier transforms which were obtained numerically were done using Filon's rule. 17,18 The three dimension­al integrals which were evaluated to find the j[r; 11; 000, OOO]'s were done using the Haselgrove integration meth­Od. 19 Except for hard sphere diatomic pair potentials, the number of points used in the Haselgrove integrations was 1001, or the value of the Haselgrove integration variable N of the function SI (N) of Ref. 19 was equal to 500. For hard sphere diatomic pair potentials N' was equal to 1000. Other numerical integrations were per­formed using Simpson's rule. The uncertainty in the j[r; 11; 000, OOO]'s used in this paper is about 1 part in 200.

The quantity [Yl1{r}]n+l denotes yMr} after n iterations of Eq. (4.9). The quantity [Yfl{r}]n denotes the inverse Fourier transform of the right-hand side of Eq. (4.9) after n iterations, The relationship between the [Yl1{r}]n' s and the [Y 11{r}]/s followszo,zl:

[Yl1{r}]n+l=[Y'l1{r}]n+>I' ([yfl{r}]n-[Y lt{r}]n)' (01)

For pair potential II and hard sphere diatomic pair po­tentials, >I' was equal to 0.4 for the results presented here. For pair potential I, >I' was equal to zero at KB T/E = 4. 03 and pu3 = O. 696. A condition on the solu­tion of Eq. (4.9) is that the denominator of the second term on the right-hand side of Eq. (4.9), {1- 2pCu{k}/ [1 + jo(kq(12))]}, is a positive function. 2o No solution of Eq. (4.9) for pair potential I at KBT/E = 1. 61 and P(,s = O. 6964 could be found by iteration. The largest value of >I' used for pair potential I was 0.95.

It was assumed that the solution of Eq. (4.9) had been found when the absolute value of

([Yl1{r}]n+l- [Yl1{r}]n)/{(1- >1')(1 + [Ylt{r}]n+l)}

was less than 1. Ox 10-5 at a set of points on the interval (0, RMAX). The points were at intervals of 0.020'. The first point was at zero. For pair potential I, RMAX

(C6)

was equal to (25 + 3. 00'). For pair potential II and hard sphere diatomic pair potentials, RMAX was the mini­mum value of r for whichj[r; 11; 000, 000] was equal to zero. For hard sphere diatomic pair potentials RMAX was equal to (25+0'). For pair potential II, RMAX was equal to (2S+ 21/60').

lC. S. Hsu, D. Chandler, and L. J. Lowden, Chern. Phys. 14, 213 (1976).

2D. Chandler, Mol. Phys. 31, 1213 (1976). sw. A. Steele, J. Chern. Phys. 39, 3197 (1963). 4J. K. Percus, "The Pair Distribution Function in Equilibrium

statistical MechaniCS, " in The Equilibrium Theory of Classi­cal Fluids, edited by H. L. Frisch and J. L. Lebowitz, (Benjamin, New York, 1964).

5J. A. Barker and D. Henderson, Rev. Mod. Phys. 48, 587 (1976).

GA. Ben-Nairn, J. Chern. Phys. 52, 5531 (1970). 7J. Barojas, D. Levesque, and B. Quentrec, Phys. Rev. A 7,

1092 (1973). BJ. J. Weis and D. Levesque, Phys. Rev. A 13, 450 (1976). 9H• C. Andersen, D. Chandler, and J. D. Weeks, Adv. Chern.

Phys. 34, 105 (1976). IOL. J. Lowden and D. Chandler, J. Chern. Phys. 59, 6587

(1973). l1F. Mandel, R. J. Bearman, and M. Y. Bearman, J. Chern.

Phys. 52, 3315 (1970). 12A. A. Broyles, J. Chern. Phys. 35, 493 (1961). 13W. B. Streett and D. J. Tildesley, Proc. R. Soc. Lond. A

348, 485 (1976). 14L. Verlet and J. J. Weis, Phys. Rev. A 5, 939 (1972). 15B. M. Ladanyi and D. Chandler, J. Chern. Phys. 62, 4308

(1975). 16A. R. Edmonds, Angular Momentum in Quantum Mechanics

(Princeton Univ., Princeton, 1960), 2nd ed. 17L. N. G. Filon, Proc. Roy. Soc. Edinburg A 49, 38 (1928). IBC. E. Froberg, Introduction to Numerical Analysis (Addison­

Wesley, Reading, MA, 1965). 19C. B. Haselgrove, Math. Computation 15, 323 (1961), 20y. -D. Chen and W. A. Steele, J. Chern. Phys. 54, 703

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J. Chern. Phys., Vol. 67, No.7, 1 October 1977

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