Convenient formulae for some integrals in perturbation theory
The free energy and pressure of a fluid, as given by perturbation theory, involve integrals of the hard sphere correlation functions and their density derivatives. In most applications a straightforward procedure would be to obtain these integrals, possibly numerically, using the formulae and comput...
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irk-123456789-320402012-04-07T12:19:37Z Convenient formulae for some integrals in perturbation theory Henderson, D. The free energy and pressure of a fluid, as given by perturbation theory, involve integrals of the hard sphere correlation functions and their density derivatives. In most applications a straightforward procedure would be to obtain these integrals, possibly numerically, using the formulae and computer codes for the hard sphere correlation functions, given previously [Mol. Phys., 2007, 106, 2; Condens. Matter Phys., 2009, 12, 127], followed by numerical differentiation with respect to the density and a possible compounding of errors. More sophisticated methods are given in this paper, which is the second in a planned trilogy, drawn from the author's lecture notes. Three representative model fluids are considered. They are the square-well fluid, the Yukawa fluid, and the Lennard-Jones fluid. Each model fluid is popular for theoretical and engineering calculations and can represent a simple fluid such as argon. With the methods presented here, numerical integration and differentiation are not necessary for the square-well and Yukawa fluids. Numerical integration cannot be easily avoided in the case of the Lennard-Jones fluid. However, numerical differentiation with respect to the density is not required. Зазначено, що вирази для вільної енергії та тиску плину, одержані за допомогою теорії збурень, включають в себе інтеграли як від кореляційних функцій твердих сфер, так і від їх похідних за густиною. В більшості застосувань ці інтеграли можна одержати, також і числово, з застосуванням простої процедури, що використовує формули та комп'ютерні коди для кореляційних функцій твердих сфер, які одержано раніше [Mol. Phys., 2007, 106, 2; Condens. Matter Phys., 2009, 12, 127], а також числове диференціювання за густиною, що може призвести у цьому випадку до можливих похибок. Запропоновано складніші методи. Розглянуто відомі моделі плину, взаємодія в яких представляється за допомогою трьох потенціалів, а саме: потенціалу типу прямокутної ями, потенціалу Юкави та потенціалу Леннарда - Джонса. Кожний із цих модельних плинів широко використовується у теоретичних та інженерних обчисленнях для опису такого простого плину як аргон. Із використанням представлених методів числове інтегрування та диференціювання перестає бути необхідним у випадках потенціалу типу прямокутної ями і потенціалу Юкави. Числового інтегрування не можна легко уникнути у випадку потенціалу Леннарда - Джонса. Проте, числове диференціювання за густиною не вимагається. 2010 Article Convenient formulae for some integrals in perturbation theory / D. Henderson // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13002: 1-10. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 02.30.Qy, 02.30.Rz, 05.20.Jj, 05.70.Ce, 64.30.+t http://dspace.nbuv.gov.ua/handle/123456789/32040 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The free energy and pressure of a fluid, as given by perturbation theory, involve integrals of the hard sphere correlation functions and their density derivatives. In most applications a straightforward procedure would be to obtain these integrals, possibly numerically, using the formulae and computer codes for the hard sphere correlation functions, given previously [Mol. Phys., 2007, 106, 2; Condens. Matter Phys., 2009, 12, 127], followed by numerical differentiation with respect to the density and a possible compounding of errors. More sophisticated methods are given in this paper, which is the second in a planned trilogy, drawn from the author's lecture notes. Three representative model fluids are considered. They are the square-well fluid, the Yukawa fluid, and the Lennard-Jones fluid. Each model fluid is popular for theoretical and engineering calculations and can represent a simple fluid such as argon. With the methods presented here, numerical integration and differentiation are not necessary for the square-well and Yukawa fluids. Numerical integration cannot be easily avoided in the case of the Lennard-Jones fluid. However, numerical differentiation with respect to the density is not required. |
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Henderson, D. Convenient formulae for some integrals in perturbation theory Condensed Matter Physics |
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Henderson, D. |
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Henderson, D. |
| title |
Convenient formulae for some integrals in perturbation theory |
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Convenient formulae for some integrals in perturbation theory |
| title_full |
Convenient formulae for some integrals in perturbation theory |
| title_fullStr |
Convenient formulae for some integrals in perturbation theory |
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Convenient formulae for some integrals in perturbation theory |
| title_sort |
convenient formulae for some integrals in perturbation theory |
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Інститут фізики конденсованих систем НАН України |
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2010 |
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| citation_txt |
Convenient formulae for some integrals in perturbation theory / D. Henderson // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13002: 1-10. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT hendersond convenientformulaeforsomeintegralsinperturbationtheory |
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2025-07-03T12:31:20Z |
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2025-07-03T12:31:20Z |
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1836628973966065664 |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 1, 13002: 1–10
http://www.icmp.lviv.ua/journal
Convenient formulae for some integrals in perturbation
theory
D. Henderson∗
Department of Chemistry and Biochemistry, Brigham Young University, Provo UT 84602
Received March 1, 2010, in final form March 10, 2010
The free energy and pressure of a fluid, as given by perturbation theory, involve integrals of the hard sphere
correlation functions and their density derivatives. In most applications a straightforward procedure would be
to obtain these integrals, possibly numerically, using the formulae and computer codes for the hard sphere
correlation functions, given previously [Mol. Phys., 2007, 106, 2; Condens. Matter Phys., 2009, 12, 127],
followed by numerical differentiation with respect to the density and a possible compounding of errors. More
sophisticated methods are given in this paper, which is the second in a planned trilogy, drawn from the author’s
lecture notes. Three representative model fluids are considered. They are the square-well fluid, the Yukawa
fluid, and the Lennard-Jones fluid. Each model fluid is popular for theoretical and engineering calculations
and can represent a simple fluid such as argon. With the methods presented here, numerical integration and
differentiation are not necessary for the square-well and Yukawa fluids. Numerical integration cannot be easily
avoided in the case of the Lennard-Jones fluid. However, numerical differentiation with respect to the density
is not required.
Key words: perturbation theory, inverse temperature expansion, compressibility approximations, analytic
methods
PACS: 02.30.Qy, 02.30.Rz, 05.20.Jj, 05.70.Ce, 64.30.+t
1. Introduction
Only a venerable scientist, such as the author, can recall the times when the theory of a dense
fluid was considered to be an intractable problem. This unsatisfactory situation changed in the
last half of the previous century first through the availability of computer simulations [1] of the
properties of a fluid that gave unambiguous results for their thermodynamic (and even transport)
properties, followed by the development of perturbation theory [2–4] and some integral equations,
for example, the Percus-Yevick (PY) [5] and the mean spherical (MSA) [6] approximations. Of
course, there has been a great deal of additional progress during the past forty years. However,
this paper will deal with the early work.
The thermodynamic properties and correlation functions of a simple fluid result from the pair
interactions, u(Rij), between pairs of particles (i and j), where Rij = |ri−rj | and ri is the position
of the center of particle i. For simplicity, the further assumption that the forces are central is made.
This discussion is further limited to one component fluids.
In its simplest form, the idea in perturbation theory is to divide the pair potential into two
parts,
u(R) = u0(R) + u1(R), (1)
where u0(R) is the pair interaction of the unperturbed system (usually the hard sphere or HS fluid),
whose thermodynamic properties and correlation functions are well-known, preferably through
accurate analytic functions or accurate tabulations, and u1(R) is the perturbation potential. The
Helmholtz free energy, which is simply related to the partition function, is then expanded in a
series. Less frequently, the pair correlation function is also expanded in a series. For perturbation
theory to be successful, the perturbation must be small, in some sense, so that the series converges
∗E-mail: doug@chem.byu.edu
c© D. Henderson 13002-1
http://www.icmp.lviv.ua/journal
D. Henderson
rapidly and one or two perturbation terms are sufficient to give good results. One reason why the
HS fluid is so useful in perturbation theory is that the thermodynamic properties and correlation
functions of this fluid, that are functions of the density but not the temperature, are given accurately
by the Percus-Yevick (PY) theory. Interestingly, the PY and MSA theories become identical for
the HS fluid. They differ for other fluids. Generally speaking the PY is not very accurate for fluids
other than the HS fluid. Perhaps it would be preferable to refer to the PY HS results as MSA
results. However, for historical reasons the HS results are called PY results.
Thus, the most frequent choice for the unperturbed system is the HS potential, for which
u0(R) =
{
∞, R < σ,
0, R > σ,
(2)
where the quantity σ is the diameter of the (reference) hard sphere system. Often the hard sphere
diameter is represented by the symbol d. The HS fluid is useful in perturbation theory because the
thermodynamic properties and correlation functions are given accurately by analytic expressions
when the PY theory is employed and the HS correlation functions are a good approximation to
the correlation functions of the fluid of interest.
Here we will discuss perturbation theory for three cases, where u1(R) is given firstly by the
square-well fluid, where
u1(R) =
{
−ε, σ < R < λσ,
0, R > λσ,
(3)
secondly, by the Yukawa fluid, where
u1(R) = − εσ
R
exp
[
− z
σ
(R − σ)
]
, (4)
for R > σ, and, thirdly, by the Lennard-Jones (LJ) 12-6 fluid, where
u(R) = 4ε
(
{ σ
R
}12
−
{ σ
R
}6
)
. (5)
The LJ potential gives the properties of a simple fluid, such as argon, with reasonably good
accuracy. The values, λ = 1.5 and z = 1.8 are used conventionally for the square-well and Yukawa
fluids, respectively. With these values, the square-well and Yukawa fluids mimic the LJ fluid, and
thus argon, with similar values of ε and σ.
The division of the LJ potential into u0(R) and u1(R) is less clear because the repulsive part of
this potential is not infinitely steep. However, Barker and Henderson(BH) [3] showed that u1(R)
can be taken to be the portion of the LJ potential for the region, R > σ, for which the LJ potential
is negative. Thus, u0(R) is the portion of the LJ potential for the region, R < σ, for which the LJ
potential is positive. This is not a very convenient reference fluid since its direct use would require a
knowledge of this fluid for every temperature and density. However, BH showed that this reference
fluid could be replaced, with good accuracy, with a HS fluid whose diameter d was given by
d =
σ
∫
0
[1 − exp{−βu(R)}] dR, (6)
where β = 1/kT with k and T being the Boltzmann constant and temperature, respectively. This
effective diameter, d, is slightly smaller than σ and is a weak function of T and independent of
the density. The effective diameter can be obtained at the beginning of a calculation and fit to
some function. Then a value of d can be determined from this fitted function or interpolated from
a table. The BH procedure makes the application of perturbation theory to the LJ fluid almost as
simple as for the other fluids.
13002-2
Convenient formulae for some integrals in perturbation theory
2. Perturbation theory
The idea of perturbation theory is that the Helmholtz free energy, A, the pressure, p, or any
thermodynamic property can be expanded in a power series in ε. Ultimately, this becomes a power
series in βε. Less frequently, the radial distribution function, g(R) is also expanded in a power
series in this same parameter. At high densities, this series is rapidly convergent even though βε is
as large as 1.4 near the triple point. The convergence at high densities is due to the fact that the
higher order terms are fluctuations and these fluctuations are inhibited at high densities because
the hard (or nearly hard) cores of the molecules are pushed up against each other, either by external
pressure or the attractive forces between the molecules (often called an internal pressure).
Thus,
A
NkT
=
A0
NkT
+
∞
∑
n=1
(βε)n An
NkT
, (7)
where N is the number of molecules in the system. The first term is the Helmholtz function for
the hard sphere reference system and is given by
A0
NkT
=
3
2
ln Λ − 1 − ln ρ +
4 − 3η
(1 − η)2
, (8)
with Λ2 = h2/2πmkT , h being Planck’s constant, m the molecular mass, η = πρd3/6, ρ = N/V ,
V is the volume of the system, and d is equal to σ for the square-well and Yukawa potentials
and the value given by equation (6) in the case of the LJ potential. There is no inconsistency.
Equation (6) yields d = σ for the square-well and Yukawa potentials, or any potential with a hard
core. Equation (8) results from, and is consistent with, the well-known Carnahan-Starling equation,
p0V
NkT
=
1 + η + η2 − η3
(1 − η)3
, (9)
where p0 is the pressure of the reference system.
The first order term, A1 is given by
A1
NkT
= 2I1, (10)
where
I1 = πρ
∞
∫
σ
R2u∗
1
(R)g0(R)dR, (11)
with u∗
1
(R) = u1(R)/ε and g0(R) is the radial distribution function, of the reference fluid. The
resulting expression for the first order term, p1, in an analogous expansion for the pressure, p, of
the fluid is
p1V
NkT
= 2η
∂I1
∂η
. (12)
A rigorous, but not very useful, expression for A2 involves three and four particle correlation
functions. BH have calculated A2 from these rigorous expressions but with limited success due
to the fact that the final result is a small difference between much larger values. Because of this
problem, two approximations [2], also due to BH, are of interest. One is the macroscopic com-
pressiblity approximation and the other is the microscopic compressiblity approximation. Both
approximations derive their name and form from the fact that the second order term is, in a gen-
eral sense, a density fluctuation and fluctuations in a bulk system are related to the compressiblity.
The approximations of BH are fairly accurate and appealing since fluctuations are related to the
compressiblity.
13002-3
D. Henderson
Macroscopic approximation
At low densities, one would expect A2 to be given by an expression similar to equation (10) but
involving [u∗(R)]2. It is plausible that the fluctuation term, A2 might be such an integral but with a
prefactor of kT (∂ρ/∂p)0, which is the compressiblity of the reference HS fluid. This approximation
is the macroscopic compressiblity (mac) approximation. Thus,
A2
NkT
= − (1 − η)4
(1 + 2η)2
I2 , (13)
where
I2 = πρ
∞
∫
σ
R2[u∗(R)]2g0(R)dR. (14)
The PY expression for the HS fluid compressiblity has been used. This is a little less accurate
than the Carnahan-Starling expression but, given the fact that the macroscopic approximation is
itself an approximation, the use of a slightly more complex result does not seem warranted. The
corresponding expression for the pressure is
p2V
NkT
= 4η
(1 − η)3(2 + η)
(1 + 2η)3
I2 − η
(1 − η)4
(1 + 2η)2
∂I2
∂η
. (15)
Microscopic approximation
BH suggested further that since A2 was a fluctuation and this was a microscopic fluctuation,
it would be more attractive to use what they called the microscopic compressiblity (mic) approxi-
mation, ∂[ρg0(R)]/∂p, instead of [∂ρ/∂p]g0(R). Using
(
kT
∂
∂p
)
0
=
(
kT
∂ρ
∂p
∂
∂ρ
)
0
=
(1 − η)4
(1 + 2η)2
∂
∂ρ
, (16)
this yields
A2
NkT
= −η
(1 − η)4
(1 + 2η)2
∂I2
∂η
. (17)
Again the PY expression for the HS compressiblity has been used. Differentiating once more yields
p2V
NkT
= η
(1 − η)3(1 − 7η − 6η2)
(1 + 2η)3
∂I2
∂η
− η2
(1 − η)4
(1 + 2η)2
∂2I2
∂η2
. (18)
The mic approximation is mentioned mainly for historical reasons. For those fluids to which they
have been applied, the mac and mic approximations give very similar results. It is probably sat-
isfactory to use the mac approximation and avoid an extra differentiation. This will be done here
and the mac approximation will be referred to simply as the compressiblity approximation (CA).
An extension of the CA to higher order terms has been given by Praestgaard and Toxvaerd [7]
3. Integrals
The task is to calculate I1 and its density derivative and I2 and its first and second derivatives.
The function g0(R) is given quite accurately from the PY theory; its Laplace transform was ob-
tained exactly by Wertheim [8, 9]. An exact expression for the PY g0(R) is not available but an
exact analytic result for the PY g0(R) for d < R < 5d has been found by Smith and Henderson [10]
and for d < R < 6d by Henderson and Smith [11]. A computer code that gives these results can
be found in the supplementary material of Smith et al. [12]. A straightforward procedure would
be to obtain I1 and I2 by numerical integration of the Smith-Henderson results and then obtain
13002-4
Convenient formulae for some integrals in perturbation theory
the derivatives numerically with a probable compounding of numerical errors. However, simpler
methods are available and these will now be considered. They differ somewhat for the square-well,
Yukawa, and LJ potentials and, as a result, these three potentials are considered separately.
Before doing so, it is useful to define the Wertheim polynomials [9],
L(s) = (1 + η/2)s + 1 + 2η (19)
and
S(s) = (1 − η)2s3 + 6η(1 − η)s2 + 18η2s − 12η(1 + 2η). (20)
Also, we will need the density derivatives of these polynomials. They are
∂L(s)
∂η
= L′(s) =
s
2
+ 2, (21)
∂S(s)
∂η
= S′(s) = −2(1− η)s3 + 6(1 − 2η)s2 + 36ηs− 12(1 + 4η), (22)
and
∂2S(s)
∂η2
= S′′(s) = 2s3 − 12s2 + 36s− 48. (23)
The higher order density derivatives are zero. For a simple notation, these derivatives will be
represented by L′(s), etc. It should be kept in mind that L′(s), etc, are derivatives with respect to
η, not s.
Square-well potential
In principle, the integrals In could be obtained for any value of λ by integrating the analytical
expressions for g0(R) that were obtained by Smith and Henderson [10, 11]. In general, this would be
impractical because these expressions become rather complicated if λ > 2. However, for d < R < 2d,
the result is fairly simple,
xg0(x) =
3
∑
i=1
ci exp(mix), (24)
where x = R/d (d = σ for the square-well potential) and the mi are the three roots of the cubic
equation
S(mi) = 0. (25)
The mi are given by analytic expressions but two are complex (i.e, have an imaginary part)
and the calculation is best done using complex arithmetic. The ci and mi can be obtained from
the subroutine GRPRLM of the FORTRAN code of Smith and Henderson that is given in the
supplementary material of the recent paper of Smith et al. [12].
As long as λ < 2, these simple expressions are all that is needed. Using the fact that u∗
1(R) is
either zero or −1, In = (−1)n−1I1 and, hence, only I1 need be determined. The result for I1 is
I1 = −6η
3
∑
i=1
ciJi
m2
i
, (26)
where
Ji = (λmi − 1) exp(λmi) − (mi − 1) exp(mi). (27)
The density derivatives I ′
1
and I ′′
1
can be obtained by straightforward, but tedious, differentiation.
For example,
I ′
1
= −6
3
∑
i=1
ciJi
m2
i
− 6η
3
∑
i=1
Ji
m2
i
∂ci
∂η
+ 12η
3
∑
i=1
ciJi
m3
i
∂mi
∂η
− 6η
3
∑
i=1
ci
m2
i
∂Ji
∂η
(28)
13002-5
D. Henderson
and
∂Ji
∂η
= mi
[
λ2 exp(λmi) − exp(mi)
] ∂mi
∂η
. (29)
The derivative of mi is obtained by differentiating equation (25) to get
∂mi
∂η
=
∂S(mi)
∂η
(
∂S(mi)
∂mi
)
−1
. (30)
An alternative procedure would be to use a result of Barker and Henderson [4] who have fit I1
for the square-well potential (λ = 1.5). Their simple result is
I1 = C
[
1 − exp
(
− 1.5ρ∗√
2 − ρ∗
)
− 1.5ρ∗√
2
]
+ Pρ∗ + Qρ∗2, (31)
where C = −4.230411, P = −2.487096, and Q = −1.213608, and ρ∗ = ρσ3. This fit of I1 was made
using the simulation results for g0(x). However, the difference between the PY and simulation
results for g0(x) is not significant. This result is easily differentiated.
Yukawa potential
The Yukawa potential is the easiest of the three potentials because I1 is directly related to
the Laplace transform of the PY g0(x) that has been obtained as an analytic expression by
Wertheim [9]. The result for I1 is
I1 = −6z exp(z)
ηL(z)
12ηL(z) + S(z) exp(z)
. (32)
The density derivative is obtained easily.
I ′
1
= −6z exp(2z)
L(z)S(z) + η[L′(z)S(z) − L(z)S′(z)]
[12ηL(z) + S(z) exp(z)]2
. (33)
This may be continued to obtain the higher order derivatives.
Additionally, the integral I2 need be calculated. This is not so straightforward. It could be
calculated numerically but this is unappealing. Since the compressiblity approximation is an ap-
proximation, a good estimate to I2 could be obtained by renormalizing I1 so that it is equal to I2
at low densities. The result is
I2 = − z
2(z + 1)
I1. (34)
Since I1 is negative, I2 is positive. This the version of the compressiblity approximation that will be
used here for the Yukawa fluid. Equation (34) is accurate at low densities but is an underestimation
of the magnitude of I2 at high densities. However, this is not so important since the density
dependence of I2 is determined mainly by the compressiblity prefactor that is small at high densities
and suppresses I2.
Lennard-Jones potential
The application of perturbation theory to the LJ potential is complicated by the fact that
d < σ. The value of the first order term, A1, is determined by the integral from σ to ∞. However,
we may write
∞
∫
σ
R2u∗(R)g0(R)dR =
∞
∫
d
R2u∗(R)g0(R)dR −
σ
∫
d
R2u∗(R)g0(R)dR. (35)
13002-6
Convenient formulae for some integrals in perturbation theory
One procedure would be to integrate numerically the integral on the left hand side of the above
equation and differentiate this result numerically or fit the value of the integral to a function
and differentiate this fitted function. Of course, analytic integration, using the Smith-Henderson
formulae, is, in principle, possible but this would involve repeated integrations by parts.
Although numerical integration of the more important first integral cannot be avoided easily,
a method that avoids numerical differentiation with respect to the density will be outlined. Using
equation (52) of the earlier paper in this series [14], it follows that
σ
∫
d
R2u∗(R)g0(R)dR = 4d3
∞
∫
0
[
(σ
d
)12 s10
10!
−
(σ
d
)6 s4
4!
]
G0(s)ds, (36)
where
G0(s) =
sL(s)
12ηL(s) + exp(s)S(s)
(37)
is the Laplace transform of g0(R) that is obtained from the PY theory. Since numerical integration
cannot be avoided easily, equation (36) by itself would seem not to have much advantage over
an integration in direct space. However, it does have the advantage that G0(s) is known for all s
whereas g0(R) is known analytically only for R < 6d. Further, equation (36) can be differentiated
directly without numerical differentiation. All that is needed is the density derivative of G0(s). The
result is
∂G0(s)
∂η
= s
−12L2(s) + [L′(s)S(s) − L(s)S′(s)] exp(s)
[12ηL(s) + exp(s)S(s)]2
. (38)
Equation (38) is perfectly satisfactory. However, the elegant result of Frisch et al. [14] is worth
noting,
∂[ηG0(s)]
∂η
= s4 exp(s)(1 + 2η)
1 + 2η + (1 − η)s
[12ηL(s) + exp(s)S(s)]2
. (39)
Higher order density derivatives can be obtained easily.
In principle, the second integral could be annoying. However, at the temperatures for which
the LJ fluid is a liquid (or a near liquid), d is only slightly less than σ. As a result, the trapizoid
rule may be used. Thus,
σ
∫
d
R2u∗(R)g0(R)dR =
d2
2
u∗(d)g0(d)(d − σ) (40)
with g0(d) = (1 + η/2)/(1 − η)2 from the PY theory. Equation (40) makes use of the fact that
u(σ) = 0. Because d is independent of the density, the density derivatives of equation (40) are
obtained straightforwardly. The integral I2 and its density derivative can be obtained by analogous
methods.
Bear in mind that because d is a function of T , the An/NkT for the LJ potential are (weak)
functions of T . Hence, the values of An/NkT should be computed at each temperature.
4. Some typical results
Since the square-well potential with λ = 1.5, the Yukawa potential with z = 1.8 and the LJ
potential are all meant to yield the thermodynamic properties of a simple fluid such as argon, the
results obtained using these potentials are similar. There is no reason to consider all three fluids;
the Yukawa fluid will be used here.
Results for A1/NkT and A2/NkT are plotted in figures 1 and 2. Two facts emerge from figure 1.
Firstly, A1/NkT is approximately a linear function of ρ. Even though g0(R) is a strong function of
ρ, its integral is not. The assumption that A1/NkT is a linear function of ρ is one of the assumptions
of the van der Waals theory. Secondly, there is no significant difference if the PY or the simulation
results are used to calculate A1/NkT from g0(R).
13002-7
D. Henderson
Figure 1. First order perturbation term for a Yukawa fluid. The points give the simulation
results and the curve is calculated from the PY g0(R) using equation (32).
Figure 2. Second order perturbation term for the Yukawa fluid. The points give the simulation
results, the dashed curve gives the MSA results, and dot dash curve gives the CA results.
Results for A2/NkT are plotted in figure 2, where exact results, obtained from simulations, are
compared with the MSA and the CA results. The second order term, A2/NkT , is small compared
with the first order term, A1/NkT , indicating rapid convergence of the perturbation theory. The
higher order An/NkT would be even smaller. Only in the immediate vicinity of the critical density,
do the higher order terms contribute significantly. We will not concern ourselves with this point
since perturbation theory cannot be expected to be a satisfactory theory of the critical point because
an infinite number of terms in the series would be required. The convergence is particularly rapid
at high densities, where A2/NkT becomes small, and fluctuations are suppressed because the fluid
and reference fluid are nearly incompressible. Further, it is to be noted that there is little difference
13002-8
Convenient formulae for some integrals in perturbation theory
between the MSA and perturbation theory approximations for A2/NkT . They are alike and close to
the exact results. The MSA results are slightly better but the CA results are more easily obtained.
The CA results are a little too large in magnitude (more negative) but this compensates in part
for the neglect of the higher order terms. If the results of the CA, based on a calculation of I2
rather than equation (14), were used, the results would be somewhat larger in magnitude (more
negative) but would still be a good approximation to the simulation results.
Results for the thermodynamics properties of these fluids are not reported since they have been
given for the square-well and LJ fluids by Barker and Henderson [2–4]. Henderson et al. [15] have
reported the thermodynamic properties of the Yukawa fluid using the MSA. The perturbation
theory results are almost identical.
Weeks, Chandler, and Andersen(WCA) [16] have formulated a somewhat different version of
perturbation theory. It is different only for pair potentials with a soft core, such as the LJ potential.
There is no difference between the two versions of perturbation theory for the square-well and
Yukawa fluids. The WCA theory converges somewhat faster for the LJ fluid However, the BH
theory with the CA second order term is equally user friendly and accurate.
5. Summary
The purpose of this paper is to report some methods other than straightforward numerical
integration and differentiation that yield the integrals that arise from perturbation theory and to
draw attention to the compressiblity approximations of Barker and Henderson. After their initial
work Barker and Henderson attempted to compute the integrals that contribute to A2/NkT using
simulations or higher order correlation functions. In retrospect, this was a nobel effort but probably
a mistake. It is difficult to obtain A2/NkT with sufficient accuracy by such methods because this
second order term results from extensive cancellation between much larger quantities. The simpler
CA is recommended. It is easier to use and, for simple liquids, is of comparable accuracy to such
approximations as the MSA.
In closing, it is worth noting that the CA gives some intuitive insight into the singular behavior
of the thermodynamic functions at the critical point. In principle, the compressiblity of the fluid
might be used in the CA instead of the compressiblity of the reference hard sphere fluid. Since
∂p/∂ρ vanishes at the critical point, a singularity would result. It is likely that this argument is too
qualitative to produce the correct behavior of the thermodynamic properties at the critical point
but it does have some appeal.
References
1. Hansen J.-P., McDonald I.R., Theory of Simple Liquids, 3rd. Edition. Academic Press, London, UK,
2006.
2. Barker J.A., Henderson D., J. Chem. Phys., 1967, 47, 2856.
3. Barker J.A., Henderson D., J. Chem. Phys., 1967, 47, 4714.
4. Barker J.A., Henderson D., Rev. Mod. Phys., 1976, 48, 587.
5. Percus J.K., Yevick G.J., Phys. Rev., 1958, 110, 1.
6. Lebowitz J.L., Percus J.K., Phys. Rev., 1966, 144, 251.
7. Praestgaard E., Toxvaerd S., J. Chem. Phys., 1970, 53, 2389.
8. Wertheim M., Phys. Rev. Lett., 1963, 10, 321.
9. Wertheim M., J. Math. Phys., 1964, 5, 643.
10. Smith W.R., Henderson D., Mol. Phys., 1970, 19, 411.
11. Henderson D., Smith W.R., J. Stat. Phys., 1978, 19, 191.
12. Smith W.R., Henderson D., Leonard P.J., Barker J.A., Grundke E.W., Mol. Phys., 2008, 106, 3.
13. Henderson D., Cond. Matter Phys., 2009, 12, 127.
14. Frisch H.L., Katz J.L., Praestgaard E., Lebowitz J.L., J. Phys. Chem., 1966, 70, 2016.
15. Henderson D., Waisman E., Lebowitz J.L., Blum L., Mol. Phys., 1978, 35, 241.
16. Weeks J.D., Chandler D., Andersen H.C., J. Chem. Phys., 1971, 54, 5237.
13002-9
D. Henderson
Зручнi формули для деяких iнтегралiв у теорiї збурень
Д. Гендерсон
Факультет хiмiї i бiологiї, Унiверситет Брайхем Янг, Прово, США
Вирази для вiльної енергiї i тиску плину, отриманi за допомогою теорiї збурень, включають в себе
iнтеграли як вiд кореляцiйних функцiй твердих сфер, так i вiд їхнiх похiдних за густиною. В бiльшостi
застосувань цi iнтеграли можуть бути отриманi, також i чисельно, iз застосуванням простої процеду-
ри. що використовує формули i комп’ютернi коди для кореляцiйних функцiй твердих сфер, якi були
отриманi ранiше [Mol. Phys., 2007, 106, 2; Condens. Matter Phys., 2009, 12, 127 ], а також чисельне
диференцiювання за густиною, що може привести при цьому до можливих похибок. В цiй статтi,
яка є другою iз запланованої серiї трьох статей, що представляють записки лекцiй автора, пропону-
ються складнiшi методи. Розглядаються вiдомi моделi плину, взаємодiя в яких представляється за
допомогою трьох потенцiалiв, а саме: потенцiалу типу прямокутної ями, потенцiалу Юкави i потен-
цiалу Леннарда-Джонса. Кожний з цих модельних плинiв широко використовується у теоретичних
та iнженерних обчисленнях для опису такого простого плину як аргон. З використанням методiв,
представлених у цiй статтi, чисельне iнтегрування i диференцiювання перестає бути необхiдним у
випадках потенцiалу типу прямокутної ями i потенцiалу Юкави. Чисельного iнтегрування не можна
легко уникнути у випадку потенцiалу Леннарда-Джонса. Проте, чисельне диференцiювання за гу-
стиною не вимагається.
Ключовi слова: теорiя збурень, розклад за степенями оберненої температури, наближення
стисливостi, аналiтичнi методи
13002-10
Introduction
Perturbation theory
Integrals
Some typical results
Summary
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