- Email: [email protected]
A new perturbation theory for potentials with a soft core
I_ Introduction
tion to _-1is identically zero and higher order corrections dre of order Ea. where E is the non-lero
In boib the illain perrurb~tion theories of ~inlplc liquids [ 121 tendtile perturbation theory for molecular fluids [3,-C] based on functional espansions in terms oftile Boltzmann factor, it is necessary to generate rtldhl distribution functions&) for 3 reference system corresponding to 3 finite range soft-core interaction potentiid r&). The simplest method of achieving this is the zeroth order espansion of Weeks_ Chandler and Andersen (WCA) [5]_ By considering the functional Taytor series expansion of tile Hehnholtz free energy d ~ in terms of the Boitzmann factor rhey show that rhc softcore correkztion function y(r) = enzflt”g(r) ,
p= IjkT
(I-1)
may be approximated by tile same function for hard spheres of diamerer d.r_l&~; d)_ The diameter J is set by the criterion .r
B(r)d’r = 0 )
(I 2)
where B(r) =YHS(~J)fe.up[-PLf(r)l
- exp kPwsO-~ll(~
-3
is called
the blip function_ Satisfying (12) ensures that the first order correc-
* Resent
address
Department
Melbourne, 30.57 ParkCUe,
274
of Mathematics.
Victoria,Australia_
University
of
B(r). Further it is found th,tt corrections g(r) =_I’t,&-:~)e-~r~(r)
1%idth of
to (I.-O
are of order I’_ For steeply repulsive potentials in which the Boftzmann factor rises sharply from zero to one_ eq_ (I-4) proves to be a \cry useful result [5,6j. The parameter E is smaI1 and eq. (I-4) is very accurate. The availabihty of hard sphere radial distribution functions from either computer simulations or the analytic solution of the Percus-Yevick (PY) equation renders it a very simple and pracrica1 approximation. However
when the potential
sphere reference system becomes
is less harsh, the hard
less reliable. The slower change in the Boltzmann factor from zero to one results in huger ,$ values and larger correction terms to eq_ (l-4) A further problem occurs in that the hard sphere diameter needed to satisfy eq. (l-2) becomes too targe and the use of the PY hard sphere resuits overestimates the structure [?I_ To improve the zeroth order results for softer potentials a more accurate treatment of the reference system is needed_ In this paper we propose a new method of handling the reference system by combining the WCA results and the apparent reliability of the PY approximation for soft-core potentials [S]_
Ctil-wC.\L
3. Pnnmeterization
of the tot31 correlation
function
We consider rhe Ornsrein-Zermke equation [91 defining the direct correkion function C(rj in terms of the total correlation fumtion 110-j = g(r) - I_
where p is the number densit) of particles. For the case that c(r) vanishes outside some finite range R, Baxter IlO] has shown that eq. (3.1) may be transformed into It
(I) They must be even in view of the modulus sign in eq. (3.2). (ii) They should have a simpie addition formula rendering eq. (2.3) of Fredholm type with degenerate kerneI_ (6) They should have sufficient fiexibility to enable comparison with existing correlation functions from either computer simulations_ scattering esperimews or perturbation theories. To proceed we assume that on 0 be represented by a function of the form
The corresponding C(r) = Q(r) - 25;~ j-
’ Q(r)Q!(r + r) dr ,
IOctober 1979
PHYSICS LI--l-rI:RS
H(s) is AI
(2.3)
0
where Q(t-I=O_
H(r)=
t->A!,
jm tft(r) dt ) t-
(3.4) (3.5)
(2.6) The uttlity of eqs. (3.2) and (2.3) is twofold. Firstly. it can be seen that if II(~) and hence U(r) is known only on the range 0 < r < R. then eq. (22) yields Jn mtegrai equation which may be solved ta give the function Q(r)_ A simple numerical algorithm 1111 may then be used to csluclate It(r) sccurately on its entire range. Secondly, from a point of view of applying the WCA theory, knowledge of the function Q(r) enables simpIe evahtation of c(r) from eq. (2.2) and a potenti& suitable as a perturbation reference system potenti,tl may be obtained from thePY approximation which may be written g(r) e@lo W = g(r) - c(r) _
(2.7)
This suggests that progress might be made if we could parameterize It(r) in some way on 0 < r
The Uf parameters Oli, hj as yet dre not specified. if eq_ (3-9) is substituted into (2.3)_ then the resulting integral equation for Q(r) may be readily solved to give
where, from eqs. (22) and (23). Q(x) at x = 1 requires that M
the continuity
Ql=
Xi
-Qz
- Qs - C
I=1
(Qil
COSh
+
of
Q12 sinh hi) _
The remainins 2M + 2 coefficients Q2, Q3, QzI, Qi2 are determined by the ?M f 2 linear equations Q,-12q
/tQ(r)dr=O, 6
LXplicit clpressions for these equ3tions m3y bc witten do\\n 3nd e3siIy proSr3mmed to yrcld the function Q(X)_ The srrrrapondir;S dire:: zorrcl.rtion function now nt3y be obtained 3naIytic3tly using ccl_ (2.3) [ 13 1_The resulting cspression is r.rther lengthy and is omitted at this surge_ The reference system pott2nti.d to be used in the WCX theory in phcc of the usual h3rd sphere reference system can now be obtrtincd from cq_ (2_7)_
3_ Truncated Lcmutrd-Jones potcntki
t-ig_ 1. Itlrp funLtiuns B(r) for ttw truncated Lemxsrd-Jones potentid
The results of section 1 h3\e been applied to the Lenn;lrd-Jones potenthi trunc3ttd 3t its first /cm: “tJ (r) = 4E [(crfr,” =0
- (u/r)6 ] _
r
r>a.
*
(3.1 )
To describe this system we use only one term in the parztmrtcrization of eqs_ (2-S) and (29). That is, WC take BZ = 1 Iewing two pruametcrs Q and X which must be specified to completely detcrntine the reference system_ For convenience we take R = o ensuring that the crtsp in the reference system and resi system occur at the same point_ The first condition used to obtain a- and X is gilen by adopting the WCA blip function criterion eq. (I 2) using our new reference system in place of hard spheres- The second is provided by the finite range transformation by requiring that Izu(_r) be continuous atx= I-/R = I_ From eq_ (22) upon differentiation we have
_xfq)(x) = -Q;(x)
+ 13.) j
(x-
r)fz,(ix-
rI)Qo(f)
dr ,
(32)
0
where q = %irpR=_ A necessary condition that fzo(_r) be continuous is then &(I) 276
=o _
(3.3)
using the p.wxnctric
rctercnce
a> stern P* = O_S6_
The non-Iinertr equations (1.2) and (3.3) \\crc solvcd numcricali_v for Q and X using an irerrttike scheme_ From 3 machine-time point ofbiew the ewluation of the blip function integral presents no problem beczmse of the anal~ tic expressions obttlined for co(~)_ llo(r) 3nd Qo(r)_ The Lennsrd-Jones correlation functions were given by gu (x) = y&)
esp [ -&crJ (x)]
_
(s?)
A qualitative estimate of the usefulness of our approach c3rt be obtained by looking at the blip functions generated- In fiS_ 1 these have been plotted 3t reduced density p’ = OS6 for a range of temperatures_ The non-zero widths E dccrrzrsc 3s the temperature dccreases_ At T* = kT/e = 1 the width is approximately 03. indicating that errors induced by adoptinS cq_ (S-4) shoutd be in the second decimal pIace. For a quantitative appraisal we turn to fig_ 2. Here is a plot ofg&) from eq. (3.4) at the stnte T* = 1, p* = 0235 for which Monte C3rlo results are available [ 12]_ The reference system parameters for this state were found to be cr = 0.0855, h = 34.6. Comparison with the Monte Carlo results shows excellent agreement _ To describe softer potentials it is clear that more terms in the parameterization will need to be considered. For example, the potential obtained by truncation
\
olumc
66.
number
1
CIfII\lIC11
PIIYSICS
LLTTtRS
1 October
19791
AcknowIedgement Supported in part by grants from the Danish Science Research Council and the Australian Research Grants Committee_
References
I ig. 2. Rradts for the abore prrturbarion theor? for the iruncatcd Lenn.~rd-Jones potaria 31 T* = 1.0 and p* = O.S5_ C~cles..c~~~r~
using eq. t3.d).
of the Lennarcl-Jones potential at irs minimum as used b> WCA has Lero slope ;tt r = Iz, = 21’6cr_ To generate a suitable total correiation function to describe this potential, it would be desirable to construct one with a peak in the vicinity of r = R,. This wouId require taking at least two terms in the parsmeterization. At present work is continuing to estend our results to such potentials.
[ 1 j J..\. Barher .md D. Ifenderson. J. Chcm. Phys. 17 t 1967) 1856,471-l. [ 1 i J-D. \\iehs_ D. Ch.md1e-r and KC_ Xndcrsen. J_ Chcm. Ph>c. 5-l (1971) Sl3-‘: 55 t 1972) 5421: D_ ChmdIer and J.D. \\ seks. Ph> s. Rex. Letters 25 (1970) 149. [ 3 J J-N’_ PerrAm .md L.R. \\ hue, Mol. Phks. 28 (1971) 527. [II K.E. Cubbinc and C-G. Grey, \IoL Ph) s. 73 (I 973) 187_ [ 5 1 f LC. Andersen, J.D. \\ ceks and D_ Chandler. Ph) s. Rev. A4 (1971) 1597_ [ 6 I If-C_ Andersen. D. Ch~ndlcr snd J-D. \\ e&s, Xdr an. Chem. Ph> s. 3-I t 1976) 105. [71 J.P. Hansen and J.J. \Yebs. 1101. Ph)s_ 23 (1971) 653. [ Sl I _ Kohlcr and J-N. Perram. unpublished results. [ 9 [ J-S. Ornstein and I_ Zermke. Proc_ XL=d_ Sci Xmstcrd.mi 17 (191-I) 793. [ 101 R-J. Ba.ter. in: Ph? siczl chemistry: .m .tdvrtnced treatise, Vol. Sa. ed. D. Henderson (Academic Press, Se\\ York. 1971) ch_ 4 [ 11 J J.\Y. Perran, MoL Phys. 30 (1975) 1505. [ 121 1; Kohler, J-11 _ Perrrtm .md L.R. Whits. Chem. Ph> s. Letters 32 (1975) 42. [ I31 CC_ \\ right_ Thesis, Aurtralim X.itumI Univerair\ (197s).
277
tion to _-1is identically zero and higher order corrections dre of order Ea. where E is the non-lero
In boib the illain perrurb~tion theories of ~inlplc liquids [ 121 tendtile perturbation theory for molecular fluids [3,-C] based on functional espansions in terms oftile Boltzmann factor, it is necessary to generate rtldhl distribution functions&) for 3 reference system corresponding to 3 finite range soft-core interaction potentiid r&). The simplest method of achieving this is the zeroth order espansion of Weeks_ Chandler and Andersen (WCA) [5]_ By considering the functional Taytor series expansion of tile Hehnholtz free energy d ~ in terms of the Boitzmann factor rhey show that rhc softcore correkztion function y(r) = enzflt”g(r) ,
p= IjkT
(I-1)
may be approximated by tile same function for hard spheres of diamerer d.r_l&~; d)_ The diameter J is set by the criterion .r
B(r)d’r = 0 )
(I 2)
where B(r) =YHS(~J)fe.up[-PLf(r)l
- exp kPwsO-~ll(~
-3
is called
the blip function_ Satisfying (12) ensures that the first order correc-
* Resent
address
Department
Melbourne, 30.57 ParkCUe,
274
of Mathematics.
Victoria,Australia_
University
of
B(r). Further it is found th,tt corrections g(r) =_I’t,&-:~)e-~r~(r)
1%idth of
to (I.-O
are of order I’_ For steeply repulsive potentials in which the Boftzmann factor rises sharply from zero to one_ eq_ (I-4) proves to be a \cry useful result [5,6j. The parameter E is smaI1 and eq. (I-4) is very accurate. The availabihty of hard sphere radial distribution functions from either computer simulations or the analytic solution of the Percus-Yevick (PY) equation renders it a very simple and pracrica1 approximation. However
when the potential
sphere reference system becomes
is less harsh, the hard
less reliable. The slower change in the Boltzmann factor from zero to one results in huger ,$ values and larger correction terms to eq_ (l-4) A further problem occurs in that the hard sphere diameter needed to satisfy eq. (l-2) becomes too targe and the use of the PY hard sphere resuits overestimates the structure [?I_ To improve the zeroth order results for softer potentials a more accurate treatment of the reference system is needed_ In this paper we propose a new method of handling the reference system by combining the WCA results and the apparent reliability of the PY approximation for soft-core potentials [S]_
Ctil-wC.\L
3. Pnnmeterization
of the tot31 correlation
function
We consider rhe Ornsrein-Zermke equation [91 defining the direct correkion function C(rj in terms of the total correlation fumtion 110-j = g(r) - I_
where p is the number densit) of particles. For the case that c(r) vanishes outside some finite range R, Baxter IlO] has shown that eq. (3.1) may be transformed into It
(I) They must be even in view of the modulus sign in eq. (3.2). (ii) They should have a simpie addition formula rendering eq. (2.3) of Fredholm type with degenerate kerneI_ (6) They should have sufficient fiexibility to enable comparison with existing correlation functions from either computer simulations_ scattering esperimews or perturbation theories. To proceed we assume that on 0
The corresponding C(r) = Q(r) - 25;~ j-
’ Q(r)Q!(r + r) dr ,
IOctober 1979
PHYSICS LI--l-rI:RS
H(s) is AI
(2.3)
0
where Q(t-I=O_
H(r)=
t->A!,
jm tft(r) dt ) t-
(3.4) (3.5)
(2.6) The uttlity of eqs. (3.2) and (2.3) is twofold. Firstly. it can be seen that if II(~) and hence U(r) is known only on the range 0 < r < R. then eq. (22) yields Jn mtegrai equation which may be solved ta give the function Q(r)_ A simple numerical algorithm 1111 may then be used to csluclate It(r) sccurately on its entire range. Secondly, from a point of view of applying the WCA theory, knowledge of the function Q(r) enables simpIe evahtation of c(r) from eq. (2.2) and a potenti& suitable as a perturbation reference system potenti,tl may be obtained from thePY approximation which may be written g(r) e@lo W = g(r) - c(r) _
(2.7)
This suggests that progress might be made if we could parameterize It(r) in some way on 0 < r
The Uf parameters Oli, hj as yet dre not specified. if eq_ (3-9) is substituted into (2.3)_ then the resulting integral equation for Q(r) may be readily solved to give
where, from eqs. (22) and (23). Q(x) at x = 1 requires that M
the continuity
Ql=
Xi
-Qz
- Qs - C
I=1
(Qil
COSh
+
of
Q12 sinh hi) _
The remainins 2M + 2 coefficients Q2, Q3, QzI, Qi2 are determined by the ?M f 2 linear equations Q,-12q
/tQ(r)dr=O, 6
LXplicit clpressions for these equ3tions m3y bc witten do\\n 3nd e3siIy proSr3mmed to yrcld the function Q(X)_ The srrrrapondir;S dire:: zorrcl.rtion function now nt3y be obtained 3naIytic3tly using ccl_ (2.3) [ 13 1_The resulting cspression is r.rther lengthy and is omitted at this surge_ The reference system pott2nti.d to be used in the WCX theory in phcc of the usual h3rd sphere reference system can now be obtrtincd from cq_ (2_7)_
3_ Truncated Lcmutrd-Jones potcntki
t-ig_ 1. Itlrp funLtiuns B(r) for ttw truncated Lemxsrd-Jones potentid
The results of section 1 h3\e been applied to the Lenn;lrd-Jones potenthi trunc3ttd 3t its first /cm: “tJ (r) = 4E [(crfr,” =0
- (u/r)6 ] _
r
r>a.
*
(3.1 )
To describe this system we use only one term in the parztmrtcrization of eqs_ (2-S) and (29). That is, WC take BZ = 1 Iewing two pruametcrs Q and X which must be specified to completely detcrntine the reference system_ For convenience we take R = o ensuring that the crtsp in the reference system and resi system occur at the same point_ The first condition used to obtain a- and X is gilen by adopting the WCA blip function criterion eq. (I 2) using our new reference system in place of hard spheres- The second is provided by the finite range transformation by requiring that Izu(_r) be continuous atx= I-/R = I_ From eq_ (22) upon differentiation we have
_xfq)(x) = -Q;(x)
+ 13.) j
(x-
r)fz,(ix-
rI)Qo(f)
dr ,
(32)
0
where q = %irpR=_ A necessary condition that fzo(_r) be continuous is then &(I) 276
=o _
(3.3)
using the p.wxnctric
rctercnce
a> stern P* = O_S6_
The non-Iinertr equations (1.2) and (3.3) \\crc solvcd numcricali_v for Q and X using an irerrttike scheme_ From 3 machine-time point ofbiew the ewluation of the blip function integral presents no problem beczmse of the anal~ tic expressions obttlined for co(~)_ llo(r) 3nd Qo(r)_ The Lennsrd-Jones correlation functions were given by gu (x) = y&)
esp [ -&crJ (x)]
_
(s?)
A qualitative estimate of the usefulness of our approach c3rt be obtained by looking at the blip functions generated- In fiS_ 1 these have been plotted 3t reduced density p’ = OS6 for a range of temperatures_ The non-zero widths E dccrrzrsc 3s the temperature dccreases_ At T* = kT/e = 1 the width is approximately 03. indicating that errors induced by adoptinS cq_ (S-4) shoutd be in the second decimal pIace. For a quantitative appraisal we turn to fig_ 2. Here is a plot ofg&) from eq. (3.4) at the stnte T* = 1, p* = 0235 for which Monte C3rlo results are available [ 12]_ The reference system parameters for this state were found to be cr = 0.0855, h = 34.6. Comparison with the Monte Carlo results shows excellent agreement _ To describe softer potentials it is clear that more terms in the parameterization will need to be considered. For example, the potential obtained by truncation
\
olumc
66.
number
1
CIfII\lIC11
PIIYSICS
LLTTtRS
1 October
19791
AcknowIedgement Supported in part by grants from the Danish Science Research Council and the Australian Research Grants Committee_
References
I ig. 2. Rradts for the abore prrturbarion theor? for the iruncatcd Lenn.~rd-Jones potaria 31 T* = 1.0 and p* = O.S5_ C~cles..c~~~r~
using eq. t3.d).
of the Lennarcl-Jones potential at irs minimum as used b> WCA has Lero slope ;tt r = Iz, = 21’6cr_ To generate a suitable total correiation function to describe this potential, it would be desirable to construct one with a peak in the vicinity of r = R,. This wouId require taking at least two terms in the parsmeterization. At present work is continuing to estend our results to such potentials.
[ 1 j J..\. Barher .md D. Ifenderson. J. Chcm. Phys. 17 t 1967) 1856,471-l. [ 1 i J-D. \\iehs_ D. Ch.md1e-r and KC_ Xndcrsen. J_ Chcm. Ph>c. 5-l (1971) Sl3-‘: 55 t 1972) 5421: D_ ChmdIer and J.D. \\ seks. Ph> s. Rex. Letters 25 (1970) 149. [ 3 J J-N’_ PerrAm .md L.R. \\ hue, Mol. Phks. 28 (1971) 527. [II K.E. Cubbinc and C-G. Grey, \IoL Ph) s. 73 (I 973) 187_ [ 5 1 f LC. Andersen, J.D. \\ ceks and D_ Chandler. Ph) s. Rev. A4 (1971) 1597_ [ 6 I If-C_ Andersen. D. Ch~ndlcr snd J-D. \\ e&s, Xdr an. Chem. Ph> s. 3-I t 1976) 105. [71 J.P. Hansen and J.J. \Yebs. 1101. Ph)s_ 23 (1971) 653. [ Sl I _ Kohlcr and J-N. Perram. unpublished results. [ 9 [ J-S. Ornstein and I_ Zermke. Proc_ XL=d_ Sci Xmstcrd.mi 17 (191-I) 793. [ 101 R-J. Ba.ter. in: Ph? siczl chemistry: .m .tdvrtnced treatise, Vol. Sa. ed. D. Henderson (Academic Press, Se\\ York. 1971) ch_ 4 [ 11 J J.\Y. Perran, MoL Phys. 30 (1975) 1505. [ 121 1; Kohler, J-11 _ Perrrtm .md L.R. Whits. Chem. Ph> s. Letters 32 (1975) 42. [ I31 CC_ \\ right_ Thesis, Aurtralim X.itumI Univerair\ (197s).
277