Exact and Approximate Values of the DistributionFunctions of a Simple Fluid*BY DOUGLAS HENDERSON tDivision of Applied Chemistry, C.S.I.R.O. Chemical Research Laboratories,Melbourne, Victoria, AustraliaSUNGWOON KIM AND LYNN ODENfDepartment of Physics, University of Waterloo, Waterloo, Ontario, CanadaReceiued 29th December, 1966The terms which are proportional to the cube of the density in the expansions of the pair distribu-tion function g(r), and the direct correlation fuiiction c(r), have been evaluated for the 6 : 12 potential.In addition, these terms have been evaluated using the Percus-Yevick (PY), hyper-netted chain (HNC),PY2, and HNC2 theories. Comparison of these approximate and exact values for g(r) and c(r) andof the resulting approximate and exact values of the fifth virial coefficient shows that, to this order inthe density, the PY2 and HNC2 theories are very reliable.Extensions of the PY and HNC theorieswhich are simpler thean the PY2 and HNC2 theories are also considered.If only pair interactions are considered,l the equation of state of a fluid of A7molecules in a volume V can be calculated from the pair distribution function g(r)by means of the pressure equation, or from the direct correlation function c(r) bymeans of the compressibility equation. These two functions are related by theequation :where c12 = c(r12), etc., p = N/V, and h(r) = g ( r ) - 1.The problem is to calculate g(r) and c(r). An approximate theory may be obtainedby postulating an equation, in addition to (l), relating g(r) and c(r).If such a theoryis tested by comparing the resulting approximate equation of state with experiment it isdifficult to make conclusions because of inexact knowledge of the intermolecularpotential u(r) and because of the perturbations resulting from the presence of three-body forces in real systems.An alternate procedure is to expand g ( r ) and c(r) in a power series inp and calculatethe coefficients both exactly and on the basis of the approximate theory. A conipari-son of these coefficients and the resulting virial coefficient Bh involves no assumptionsother than those inherent in the theory itself. The conclusions reached from such acomparison are only valid at low densities but they do not appear to be sensitive to thenumber of terms included in the expansion and are of interest in a discussion of theliquid state.hl2 = c12 +P.fh13C23dr3, (1)* This work has been supported by grants from the National Research Council of Canada and theOffice of Saline Water, United States Department of the Interior.t Alfred P. Sloan Foundation Fellow and Ian Potter Foundation Fellow ; permanent address :Department of Physics, University of Waterloo, Waterloo, Ontario, Canada.$ present address : Fresno State College, Fresno, California, U S A .2I>. HENDERSON, S . KIM A N D L . O D E N 27EXACT RESULTSWe denote the coefficients of pla in the expansions of y ( r ) = exp (pu(r))g(r) andc(r) by gn(r) and cn(r), respectively. The lowest-order coefficients are go = 1 andcg(r) = f(r) = exp ( -pu(r)) - 1.In previous publications 2, 3 we have presentedresults for gl, 9 2 , c1, and c2 using the 6 : 12 potential. In this paper we present our3 -- 1r*and 4 terms in the expansion for g(r).FIG. 1 .-g(r) for p* = 1 and T* = 2.5. The curves labelled 1, 2, 3, and 4 are obtained using 1 , 2, 3results for 93 and c3. Values for g(r), obtained using a four-term expansion, areplotted in fig. 1 for T* = kT/& = 2.5 and p* = pa3 = 1. This high density waschosen so as to magnify the contributions of the higher-order terms. In fig. 2 asimilar plot of c(r) is given. Unfortunately, our calculations for 93 and c3 are notcomplete so that it is not yet possible to show results for lower temperatures.PY A N D HNC THEORIESand 5 that c(r) can be written in the form,Eqn.(2) and (3) are little more than definitions of d(r) and E(r), and are useful only ifd(r) or E(r) can be approximated.The Percus-Yevick (PY) approximation is d(r) = 0 and the hyper-netted chain(HNC) approximation is E(r) = 0. Both approximations give correct results for goand g1 and thus give exact B2 and B3. However, they give approximate results forc(r) = f(r)y(r) +y(r) - 1 -In y(r) + E(r). (328 FUNCTIONS OF A SIMPLE FLUIDthe higher gn. In fig. 3 and 4 the errors in the PY and HNC values for g; = g2/a3 andg: = 93/06 at T* = 2.5 are shown. At this temperature the HNC theory gives theleast error for r* = r/a small but the PY theory gives the least error in the importantregion r * - 1.r*and 4 terms in the expansion for c(r).FIG. 2.-c(r) for p* = 1 and T* = 2.5.The curves labelled I, 2, 3, and 4 are obtained using 1,2, 3,I I 9 I O11 5 2.0 2.5 1.0r*FIG. 3.-A& = gz (approximate)-gg for T* = 2-5.In tables 1 and 2, B4 and B4, obtained from these theories, are compared with theexact values. The parameter b = 2na3/3. The necessary cluster integrals have beentabulated previously.39 6, 7 The compressibility equation yields the best B4 and B5 aD. HENDERSON, S . KIM AND L . ODEN 290.402*m 20- 0.2!!!!\.\ T * : 2.5\i;HNCZI /I1 I I1.0 15 2.0 2.5r*FIG. 4.-Ag4 = gf (approximate)-& for T* = 2-5.TABLE 1 .-FOURTH VIRIAL COEFFICIENTS FOR 6 : 12 POTENTIAL(in units of b3)0-6 -0.81.01.21.41.62.02-53.05.010.0182.17 -- 9.327- 0.2650,33930.27050.1 8980.12300.11310.1 1980.13420.1 156130-89- 7.179-0,31840.21700.20280-15910.12050.11600.12100.12990.1 120- 194.87- 11.699- 0.721 30.27890.30870.25290-18720.16580.16210.15340.1 177- 193.40- 12.597- 1.2473- 0.05020.07770.07650.06660.07750.09 120.1 1090.0937- 196.33- 10.768- 0.47760.33470.30430.23870- 18070-17390.18150- 19490.1681TABLE 2.-FIFTH COMPRESSIBILITY VIRIAL COEFFICIENTS FOR 6 12 POTENTIAL(in units of b4)T* exact PY PY2 HNC HNC2 v 0,0.60.81.01.21.41.62.02.53.05.010.0-- 76.6- 2.770-023- 0.0021- 0.041 5- 0.00990.03650.05790.06290.03901845.4 -- 44.8- 1.930.0020.03410.01 190-021 80.04410.05470.05500.03653818.1-81.6- 3.090.0090.0062- 0.0348- 0.00670,03870.05980.06420.0393-3689.8 -- 105.0- 6.61- 0.502- 0.01 300.02220.03700.05490-06090.04720.02 143836.8 -- 80.8- 3.020.00 1- 0*0040- 0.041 3- 0.00860.03870.06070.06650.04223614.0 -- 85.9-4.14-0.191- 0.0227- 0.02900.00330-04360-061 80.06450.041 8'3580.6- 99.2- 6.08-0.478- 0.0370-040030.02540.051 10-06140.055 10.03330 FUNCTIONS OF A SIMPLE FLUIDhigh temperatures while the pressure equation yields the best B4 and B5 at lowtemperatures. This is true of all theories considered in this paper.The PY theorygives ihe best overall results while the HNC theory gives the best results at lowtemperatures.PY2 AND HNC2 THEORIESVerlet S has proposed extensions of the PY and HNC theories.His expressionsfor d(r) and E(r) are too complicated to be given here. Oden el aZ.9 have given thefirst two terms in the density expansions of the PY2 ar,d HNC2 approximations tod(r) and E(r). Both theories give go, 91, and 92 exactly. In fig. 4 the errors in the PY2and HNC2 values for g; at T* = 2-5 are shown. The HNC2 theory gives the bestoverall values but the PY2 values are best in the most important region r* - 1. Valuesfor B5 which result from these theories are listed in table 2 ; they are very good.However, in the form given byVerlet, they involve an asymmetric approximation for the triplet distribution function,g(3)(rl,~,r3).This asymmetry does not affect 9 2 or 9 3 but will affect the higher-ordergn. A symmetric approximation which could be used is the superposition approxima-tion.10 However, if this approximation is used, the results for 93 are not as good asthose obtained from Verlet's asymmetric approximation. An alternate symmetricexpression isThe PY2 and HNC2 theories are promising.Eqn. (4) yields the same 9 3 as does Verlet's approximation to g(3) and should be moresatisfactory for the higher-order gn. We have not yet tested this approximation for the6 : 12 potential. However, our calculations 11 for Gaussian molecules and hardspheres indicate that eqn. (4) is a considerable improvement over the superpositionapproximation.SIMPLER EXTENSIONSThe PY2 and HNC2 theories are complicated. It would be useful to have somesimpler extensions of the PY and HNC theories.Each of these extensions yields anexact 9 2 and B4. Verlet 12 and Rowlinson 13 have proposed the approximation :We refer to this as the V approximation. In table 2 values for B5 which result fromthis approximation are listed. In general, the results are worse than PY values.However, at low temperatures the results are good and eqn. (5) may be of some valueat low temperatures. Green 14 has proposed an approximation which is similar to (5)but which yields poor values for Bg.An approximation would be to truncate the series expansion for E after one term.ThusEl2 = +P2J f13f14f23f24f34dr&4*This approximation, which we call the El approximation, corrects the HNC expres-sion for g2 but does not greatly affect the higher-order gn.This can be seen from table2. The El values for B5 are slightly improved over the HNC results. The advantageof this approximation is that it can be continued. When our calculations of 9 3 arecomplete it will be possible to add a second term to (6) and obtain the E2 approxirna-tion. The E2 approximation will yield g3 exactly and should give reasonably goodvalues for the higher-order gnD . HENDERSON, S. KIM AND L. ODEN 31A similar series of approximations could be based on the PY theory and could bereferred to as d l , d2, etc. approximations. However, in view of the superiority of theHNC results at low temperatures it would probably be better to use the HNC theoryas a basis.Van Leeuwen et aZ.5 have proposed an approximation where E(r) is givenby (6) with eachf(r) replaced by h(r). However, the 93 and Bs which result from thisapproximation are disappointing.The PY2 and HNC2 theories are promising theories of the liquid state. However,they are complicated and it is harder to obtain numerical values for g(r) from thesetheories than from the simpler PY and HNC theories. On the other hand, the E2and d2 theories, although probably less satisfactory than the PY2 and HNC2 theories,are no more difficult to use than are the PY and HNC theories.One of us (D. H.) is grateful to the staff of the C.S.I.R.O. Chemical ResearchLaboratories for their hospitality during his visit.1 J. S. Rowlinson, Reports Prog. Physics, 1965, 28, 169.2 D. Henderson, Mol. Physics, 1966, 10, 73.3 D. Henderson and L. Oden, Mol. Physics, 1966, 10,405.4 G. Stell, Physica, 1963, 29, 517.5 J. M. J. van Leeuwen, J. Groeneveld and J. de Boer, Physica, 1959, 25, 792.6 S. Kim, D. Henderson and L. Oden, J. Chem. Physics, 1966,45.7 J. A. Barker, P. J. Leonard and A. Pompe, J. Chem. Physics, 1966, 44, 4206.8 L. Verlet, Physica, 1964, 30, 95.9 L. Oden, D. Henderson and R. Chen, Physics Letters, 1966, 21, 420.10 J. G. Kirkwood, J. Chem. Physics, 1935, 3, 300.11 D. Henderson, J. Chem. Physics, 1967, (in press).12 L. Verlet, Physica, 1965, 31, 959.13 J. S. Rowlinson, Mol. Physics, 1966, 10, 533.14 H. S. Green, Physics Fluids, 1965, 8, 1