Evaluation of the CPY and PYX approximations for ... - Semantic Scholar

MOLECULAR PHYSICS,

1983, VOL. 50, NO. 5, 1 1 3 3 - 1 1 4 0

Evaluation of the CPY and PYX approximations for short ranged anisotropie potentials by P. T. C U M M I N G S t Departments of Mechanical Engineering and Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794, U.S.A. J. RAMS and C. G. GRAY Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1, Canada and M. S. W E R T H E I M Department of Mathematics, Hill Centre for the Mathematical Sciences, Rutgers University, Busch Campus, New Brunswick, New Jersey 08903, U.S.A. (Received 20June 1983 ; accepted 18 July 1983) In the first paper in this series, a new integral equation approximation - - t h e Percus-Yevick excess (PYX) approximation--was introduced and contrasted with the corrected Percus-Yevick (CPY) approximation. In the present paper, these two approximations are evaluated for a molecular fluid, the molecules of which interact via a short ranged anisotropic potential prototypical of the overlap interactions encountered between homonuclear diatomic molecules. We conclude that the CPY and PYX approximations are quite accurate; interestingly, however, they unexpectedly yield very similar results leading us to conclude that, on balance, both approximations are equally good.

1. INTRODUCTION

In the previous paper in this series [1, henceforth referred to as I ; equation (m.n) of I is denoted by (I.m.n)], the single super chain/linearized hypernetted chain (SSC/LHNC), single super chain f-expansion (SSCF) and PercusYevick (PY) approximations were evaluated for short ranged anisotropic potentials. This was done by solving these approximations numerically for the delta model (DM) fluid, the molecules of which interact via the DM potential given by (I.1.14). The molecular total correlation function h(12) (which depends on both orientations and positions of molecules 1 and 2 and is defined by (I.1.2)) predicted by the various approximations is resolved into a set of spherical harmonic coefficients h(11121 ; r) (see § 3 and Appendix B of I) and then compared with Monte Carlo (MC) simulation results for the same quantities. The DM potential is prototypical of the short ranged anisotropic overlap t Present address : Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, Virginia 22901, U.S.A. $ On leave from : Physics Department, Banaras Hindu University, Varanasi 221005, India.

1134

P . T . Cummings et al.

forces that characterize homonuclear diatomic molecules. On the basis of the results presented in I, it was concluded that SSC/LHNC is not a good approximation for short ranged anisotropic interactions. The SSCF approximation was found to be superior to SSC/LHNC, but nevertheless is unable to predict certain features in the MC results that are rendered qualitatively correctly by the PY approximation. In I, an additional new approximation--the Percus-Yevick excess (PYX) approximation--was introduced and contrasted to the corrected Percus-Yevick (CPY) approximation of Smith and Henderson [2]. The CPY and PYX approximation are expressed by approximate closure relations between h(12) and the molecular direct correlation function c(12) defined by the OrnsteinZernike equation (I.1.3). If the molecular pair potential u(12) is divided into a reference part (ura(12)) and a perturbation part (Upert(12)), then the Mayer f-function f(12)=exp ( - u ( 1 2 ) / k B T - 1 (k B is Bohzmann's constant and T the absolute temperature) can be written as f(12) = fre~(12) + fp~rt(12) + f~t(12)fp~t(12).

(1.1)

Dividing h(12) and c(12) into reference and perturbation parts as well (i.e. similarly for c(12), where hre~(12) and Cref(12) are the total and direct correlation functions for a fluid interacting via the potential u~a(12)), the CPY and PYX approximations are respectively given by (see (I.A.17) and I.A.18))

h(12)=hret(12)+hpert(12) and

Cpert(12 ) = fpert(12)[gref(12) + [1 + fre~(12)J[hpert(12) -- Cpert(12)] ] + f~(12)[hve,t(12)- %~t(12)] ,

(1.2)

%,n(12) = fp,rt(12)[gre,(12) +gra(12)[hpe~t(12)- %~t(12)]]

+h,a(12)[hvc~t(12)-%¢rt(12)],

(1.3)

where g ~ e f ( 1 2 ) = h ~ ( 1 2 ) + l . In I it was suggested that renormalized PY approximations such as PYC and PYX (i.e. approximations that yield exact reference system quantities in the absence of the anisotropic perturbation) should yield superior results to PY by bringing the qualitative predictions of PY into better quantitative agreement with MC. In this paper, we report results for PYX and CPY that bear out this expectation. Perhaps a little surprisingly, the difference between CPY and PYX is not significant enough to definitively choose one as more accurate than the other; however, it will be clear from the results presented that both are superior overall to PY and SSCF. For the computations reported in § 2, we choose a spherically symmetric reference system (so that qra(12)=qret(ra2) for q--u, f, h and c, rl~ being the distance between the centres of molecule 1 and 2). This simplifies the application of the closure (1.2) and (1.3) considerably. The spherically symmetric reference system chosen is the same as that used in § 2 of I for SSCF : that is, the reference system Mayer f-function f~,,(rle ) is defined to be the unweighted angular average of f(12) as described by (I.2.10) and (I.2.11). Thus, in all the comparisons that follow, SSCF, PYX and CPY all share the same reference system, and the ability of each approximation to incorporate correctly the detailed effects of the anisotropic perturbation in the Mayer f-function is being evaluated.

CP Y and P Y X approximations

1135

All the computational details for producing the results in the following section are to be found explicitly in I, or can be inferred from it. (The one noteworthy point is that products of quantities having orientational dependence in the CPY and PYX closures (1.2) and (1.3) are handled using the method applied in I to the similar product that arises in the PY approximation.) Thus, we do not discuss technical aspects of the computation here, and refer the interested reader to I for details. T h e state conditions employed in the paper are the same as those of I ; hence, the M C results used here are those described in detail in I. 2. RESULTS We shall consider g(r), the unweighted angular average of the molecular correlation function g ( 1 2 ) = h ( 1 2 ) + l , at the reduced density p * = p o a = 0 . 8 , reduced temperature T*=kBT/¢=0.719 and three values of 8, the strength of the anisotropic perturbation. (The parameters or, E and 8 are defined in (I.1.14).) In figure 1, g(r) for 8=0.1 is shown. At this small value of 8, C P ¥ , PYX and S S C F are all equally good, being indistinguishable on the scale of this figure ; the PY approximation is not good due to overestimation of the first peak, a point discussed in greater detail in I. For ~=0"25 (figure 2), we find that PYX and CPY are again indistinguishable and are very accurate, differing by at most 2 per cent from the M C results. The position of the principal peak in the S S C F g(r) is a little to the left of the M C results, as is the PY result. Thus, PYX and CPY are clearly superior at this value of 8. For 8 = 0.3 (figure 3), both S S C F and PY have incorrect peak positions and heights, while PYX and CPY are nearly indistinguishable and in good agreement with

]

g(r} 2

i

I

i

I 2

=

t

i

Z! I

l r/o"

3

Figure 1. The angle average pair co-_relation function g(r) for the DM potential calculated at p*=0-8, T*=0.719 and 3=0.1 using the PY ( ), CPY (. . . . . . . ), PYX( . . . . ) and SSCF( ......... ) approximations. The circles are MC results. Note that on the scale of this graph CPY, PYX and SSCF are indistinguishable.

P . T . C u m m i n g s et al.

1136 [

i

I

i

3

g(r) 2

0

~

I

1

,

rio"

I

2

Figure 2. The angle average pair correlation function g(r) for the DM potential calculated at p*=0.8, T*=0.719 and 8--0-25. The legend used is the same as that for figure 1. Note that on the scale of this graph CPY and PYX are indistinguishable.

[

i

]

i

2 -

g(r)

,

I

r/o.

2

Figure 3. The angle average pair correlation function g(r) for the DM potential calculated at p*=0.8, T~=0.719 and ~=0-3. The legend used is the same as that for figure 1.

M C with an error of at most 4 per cent. Clearly, however, even P Y X and C P Y are showing the initial signs of breakdown in accuracy. T u r n i n g now to the principal harmonic h(202; r) of h(12), we find in figure 4 ( 8 - - 0 . 1 ) a mixed set of results. T h e CPY result obtains the first t r o u g h correctly, but overestimates the height of the first peak ; P Y X performs

CP Y and P Y X approximations

1

r/d

2

1137

3

Figure 4. The spherical harmonic coefficient h(202 ; r) for the DM potential calculated at p*=0-8, T*=0.719 and 8=0-1. The legend used is the same as that for figure 1.

10

0 h(202;r) -I0

-20

-30 I 1

, r/O"

I 2

Figure 5. The spherical harmonic coefficient h(202 ; r) for the DM potential calculated at p*=0.8, T*=0.719 and 8=0.25. The legend used is the same as that for figure 1.

better overall since it agrees with the first peak quite well and makes only a small underestimation of the depth of the first trough. T h u s we would rate the approximations in descending order of accuracy as (PYX, CPY, S S C F , PY). For ~ =0-25, figure 5 shows that PYX is again superior, since P Y X and CPY are equally good in predicting the first trough (both overestimating by about

1138

P . T . Cummings et al. i

loi

I

i

0 I

h(202;r) I -10 :

°q

2

-20

-30

-40

-5C

I

z

I

1

r/o"

2

p

3

Figure 6. The spherical harmonic coefficient h(202 ; r) for the DM potential calculated at 0*=0-8, T*=0.719 and 8=0.3. The legend used is the same as that for figure 1. 30

I

!

20 h(220;r)

'

I

'

10

J

o

•'o° °•

-

. . . . . . . . . . . . . . .

Lo

I

-10 a

I

rio"

2

i

3

Figure 7. The spherical harmonic coefficient h(220 ; r) for the DM potential calculated at 0*=0.8, T*=0-719 and 8=0.25. The legend used is the same as that for figure |.

15 per cent) while PYX is better for the first peak and beyond. T h u s , the accuracy ordering is again (PYX, CPY, S S C F , PY). In figure 6, we display h(202 ; r) for 5 = 0 . 3 , and here we find that CPY and P Y X are overestimating the true result (although overall PYX is better)• For this value of 5, S S C F appears to be best, so that the accuracy ordering is ( S S C F , PYX, CPY, PY).

CP Y and P Y X approximations

1139

We now consider two other harmonics (h(220 ; r) and h(224 ; r)) in detail for 3=0.25 only. In figure 7, we show the results for h(220 ; r). The SSCF result has the best magnitude for the first peak; however the position is not rendered well by SSCF and additionally the first trough is missed. CPY and PYX are roughly equivalent here, since they both have the peak and trough in their correct positions; in magnitudes, CPY is better than PYX for the first peak, while the reverse is true for the first trough. The PY approximation is qualitatively correct but the magnitudes are excessive. Thus we conclude that the accuracy ordering is (PYX = CPY, PY, SSCF). Although not shown, the 8=0-3 results yield the same accuracy ordering while at 3=0.1 we find (CPY, PYX, PY, SSCF).

lfi

I

I

'

,

12 h(224;r) 8

4

0 I

,

I

1

r/d

2

Figure 8. The spherical harmonic coefficient h(224 ; r) for the DM potential calculated at p*=0-8, T*=0.719 and 8=0.25. The legend used is the same as that for figure 1.

The h(224 ; r) harmonie for 3 = 0-25 is shown in figure 8. No approximation is very satisfactory for this harmonic coefficient and it is difficult to draw up an accuracy ordering for this harmonic coefficient; the same is true for 3=0-1 and 3=0.3, although we have not exhibited those results here. Also, we have not shown the h(222; r) harmonic coefficient. For 8=0.3, all the approximations are qualitative only and no accuracy ordering can be deduced. For 3=0.1, we find (CPY, PYX, PY, SSCF) and for 3=0-25 (CPY=PYX, PY, SSCF). 3. CONCLUSION As the results reported in § 2 make clear, the CPY and PYX approximations represent definite improvements over PY and SSCF. For the centres distribution and the principal harmonic coefficient h(202 ; r) they are especially

1140

P . T . Cummings et al.

good, while the accuracy deteriorates for the higher order harmonic coefficients considered. In choosing between CPY and PYX, we note that the differences between the approximations are small and neither one is consistently more accurate than the other when results are compared with MC. Consequently, we conclude that CPY and PYX are equally good. This is a little surprising in view of the somewhat richer structure of PYX that was noted in I ; however, as the example of the hypernetted chain and Percus-Yevick approximations for hard spheres illustrates [3], in liquid state physics approximations which in principle have a richer structure do not necessarily yield better results for some hamiltonian models. Overall, we have been encouraged by the success of the relatively simple CPY and PYX approximations; however, their utility for long-ranged interactions has yet to be determined and is presently under study. The authors gratefully acknowledge N S E R C (Canada) for financial support, and the Computer Services Council of the University of Guelph for the allocation of computer resources. P.T.C. acknowledges the support of the National Science Foundation, under grant CHE-8119143 for this research; M.S.W. acknowledges the support of the National Science Foundation under grants CHE-7808730 and CHE-8114968. REFERENCES [1] CUMMINCS,P. T., RAM, J., BARKER,R., GRAY, C. G., and WERTHEXM,M. S., 1983, Molec. Phys., 48, 1177. [2] SMITH,W. R., and HENDERSON,D., 1978, ~. chem. Phys., 69, 319. [3] See, for example, HANSEN,J.-P., and McDoNALD, I. R., 1976, Physics of Simple Fluids (Academic Press).