A solvation-free-energy functional: A reference-modified density
functional formulation
Tomonari Sumi1, Ayori Mitsutake2,3, and Yutaka Maruyama2
1
Department of Chemistry, Faculty of Science, Okayama University, 3-1-1 Tsushima-Naka, Kita-ku,
Okayama 700-8530, Japan.
2
Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223–
8522, Japan.
3
JST, PREST, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223–8522, Japan.
Supporting Information
An approximation of the intrinsic free-energy functional for an ideal polyatomic molecular gas
The site-density distribution functions for an ideal polyatomic molecular gas under arbitrary external
fields
(
{U ( r )}
IM
l
are given by[1,2]
)
naIM r {U lIM } = n0 exp ( - bUaIM ( r ))
(
)
(
)
ì P
ü
ì P
üì P
ü
´ ò íÕ dr1a ýd ( r - r1a ) íÕ sab r1a - r1b ý íÕ exp - bU aIM ( r1a ) ý
î a=1
þ
î a<b
þ î a=1
þ
(A1)
where P is the number of sites, n0 is the number density of the ideal polyatomic molecular gas
molecules, and sab ( r ) = d ( r - Lab ) 4p Lab is the intramolecular bonding function between sites a
and b. As pointed out by Chandler et al.[3], the external fields
{U ( r )}
IM
l
cannot be expressed as a
simple functional of the site-density distributions, because of the nonlinear inverse nature of
{U ( r )}
IM
l
in Eq. (A1). As a result, we cannot obtain the exact intrinsic free-energy functional for
{ }
IM
the ideal polyatomic molecular gas, FIM éë nl ùû , since we cannot perform the functional integral of
the Euler-Lagrange equation, which is defined as
d FIM éë{ nlIM } ùû
(
d na r {U l
IM
IM
})
= maIM - UaIM ( r )
(A2)
T ,V
where maIM is the chemical potential for site
a of the ideal polyatomic molecular gas. To avoid
having to calculate the functional integral of Eq. (A2) with respect to the density functional of
{U ( r )} , we can use the reference-modified density functional theory (RMDFT) to derive an
IM
l
{ }
IM
approximate functional for FIM éë nl ùû , where the ideal multicomponent gas is employed as the
reference system for RMDFT. Then, the Euler-Lagrange equation for the ideal multicomponent gas
is given by
d FIG éë{nlIG } ùû
(
d na r {U l
IG
IG
})
= maIG - UaIG ( r )
(A3)
T ,V
{ }
IG
where FIG éë nl ùû is the intrinsic free-energy functional of the ideal multicomponent gas, and
UaIG ( r ) and maIG are the external field and chemical potential for component a of the ideal
multicomponent gas, respectively. The external fields for the ideal multicomponent gas,
{U ( r )}
IG
l
are defined so that
(
)
(
naIG r {UlIG } = naIM r {UlIM }
)
(A4)
{ }
IM
Here, we introduce the excess intrinsic free-energy functional for FIM éë nl ùû and the excess
chemical potential for maIM . These can be defined as the differences with respect to the reference
ideal multicomponent gas as follows:
FIMex éë{nlIM } ùû = FIM éë{nlIM } ùû - FIG éë{nlIM } ùû
(A5)
maexIM = maIM - maIG
(A6)
From Eqs. (A2)-(A6), we obtain
Ua ( r ) = Ua
IG
IM
(r) +
d FIMex éë{ nlIM } ùû
(
d na r {Ul
IM
IM
})
- maexIM
(A7)
T ,V
FIMex éë{nlIM } ùû can be approximated using the second-order density-functional Taylor expansion as
follows:
(
P
)
FIMex éë{nlIM } ùû = FIMex éë{ n0 } ùû + å ò dr1a maexIM é naIM r1a {U lIM } - n0 ù
ë
û
a=1
(
)
(
)
(
)
1 P P
dr1a dr1bCabIM r1a - r1b é naIM r1a {U lIM } - n0 ù é nbIM r1b {U lIM } - n0 ù
å
å
ò
ë
ûë
û
2 b a=1 b=1
(A8)
In this equation, we use
ma
exIM
º
d FIMex éë{nlIM } ùû
(
d naIM r {U lIM }
)
(A9)
nlIM =n0
which is given by Eq. (A7) and the definition of the intramolecular direct correlation function[4],
Cab ( r - r ¢ ) º - b
IM
(
d 2 FIMex éë{nlIM } ùû
)
(
d naIM r {U lIM } d nbIM r ¢ {U lIM }
d éë - bUa ( r ) ùû
d nbIG r ¢ U bIG
IG
=
(
)
)
nlIM =n0
d éë - bUa ( r ) ùû
d nbIM r ¢ {U lIG }
(A10)
IM
nlIG =n0
(
)
nlIM =n0
Equation (A10) is equal to Eq. (31) in the main text. Equation (A7) can be approximated using Eq.
(A8) as follows:
UaIG ( r ) = UaIM ( r ) -
å dr Ca ( r - r ) éën
b ò
1
P
a
1
IM
a
a
1
IM
a
a=1
( r {U }) - n ùû
a
1
IM
l
(A11)
0
{ }
IM
Using Eqs. (A8) and (A11) and FIG éë nl ùû , which is given by Eq. (19) in the main text,
FIM éë{nlIM } ùû in Eq. (A5) can be expressed as follows:
FIM éë{ nlIM } ùû = FIM éë{ n0 } ùû -
1
P
å dr
b
a
1
a=1
(
P
(
)
)
P
é n IM r a {U IM } - n ù
l
0
ë a 1
û
(
+ å maexIM dr1a é naIM r1a {U lIM } - n0 ù - å dr1aU aIM ( r1a ) naIM r1a {U lIM }
ë
û a=1
a=1
+
n0
b
å å ò dr dr C ( r
P
P
a
1
b
1
IM
ab
a
1
a=1 b=1
(
-r
b
1
)
,(A12)
) éën ( r {U }) - n ùû
IM
b
)
b
1
IM
l
(
0
)
(
)
1 P P
+
dr1a dr1bCabIM r1a - r1b é naIM r1a {U lIM } - n0 ù é nbIM r1b {U lIM } - n0 ù
å
å
ò
ë
ûë
û
2 b a=1 b=1
where
FIM éë{n0 } ùû = -
PN
b
P
+ å maexIM N
(A13)
a=1
This equation is regarded as the hypernetted-chain-type approximation for the intrinsic free-energy
functional of the ideal polyatomic molecular gas.
3D-RISM calculation
( { })
We require the site-density distribution functions na r UlPR
to calculate the SFE using Eq. (40).
In the present study, we employed the three-dimensional reference-interaction-site-model
( { }) .
(3D-RISM) theory[5,6] to calculate na r UlPR
The 3D-RISM theory is an integral-equation theory that is used to obtain the solvent
distribution functions from the intermolecular potential functions and the thermodynamic conditions
(i.e., temperature and density). It produces the distribution functions of the solvent molecules around
the solute molecule. The theoretical calculations involved in the method have two steps. The first
step is to calculate the pair correlation functions in the aqueous solution on the basis of the
solvent-solvent 1D-RISM theory,[7,8] which represents the microscopic structure of the distribution
of the water molecules. In the second step, a solute molecule is immersed into the solvent and the 3D
distribution functions of the solvent molecule are calculated using the 3D-RISM integral equations.
For an infinitely dilute solution, the solute-solvent 3D-RISM integral equations can be
written as
p
p
ĥg (k) = å ĉa (k)* ŝ ag (k) + n0 å ĉa (k)* Ĥ ag (k)
a=1
(A14)
a=1
Here, ĥg (k) and ĉg (k) are the Fourier transforms of hg (r) and cg (r) , which are the
solute-solvent total and direct correlation functions between the solute molecule and solvent site g .
We should mention that Eq. (A14) contains the solvent-solvent total correlation function Hag (r) ,
which is defined by Eq. (34) in the main text. Thus, one needs to solve Eq. (34) before solving Eq.
(A14).
(
We
)
can
calculate
na r {UlPR } = n0 (ha (r) +1) .[1,2]
(
na r {UlPR }
)
from
hg (r)
using
the
Percus’
relation,
Equations (34) and (A14) both have two unknown functions, namely, the total and direct
correlation functions. Therefore, more than one equation is required to solve them. We employed
partially linearized HNC (PLHNC) equations,[5] which are called Kovalenko-Hirata (KH) equations,
[6]
as the closure equations:
ì exp( c (r)) c (r) < 1
ï
ag
ag
Hag (r) = í
ïî c ag (r) -1 c ag (r) ³ 1
c ag (r) = -b vag (r) + Hag (r) - Cag (r)
(A15)
and
ì exp( c g (r)) cg (r) < 1
ï
hg (r) = í
ïî cg (r) - 1 cg (r) ³ 1
(
(A16)
)
cg (r) = - bUgPR r {r1l } + hg (r) - cg (r)
where vag (r) is the intermolecular interaction potential between sites
a and g on the solvent
( { }) is given by Eq. (35) in the main text. We used Eqs. (34) and (A15) for
molecule, and UgPR r r1l
the first step and Eqs. (A14) and (A16) for the second step. To solve the 1D-RISM and 3D-RISM
integral equations, we respectively introduce the following difference functions:
Tag (r) = Hag (r)- Cag (r)
(A17)
and
tg (r) = hg (r) - cg (r)
(A18)
The numerical procedure for the first/second step can be briefly summarized as follows.
( { }) at each 1D/3D grid point.
1. Calculate uag (r)/ UgPR r r1l
2. Initialize Tag (r) / tg (r) to zero.
3. Calculate Hag (r)/ hg (r) using Eq. A14/A15.
4. Calculate Cag (r) / cg (r) from Cag (r) = Hag (r)-Tag (r) / cg (r) = hg (r) - tg (r) .
5. Transform Cag (r) / cg (r) to Ĉag (k) / ĉg (k) using the 1D/3D fast Fourier transform
(1D-FFT/3D-FFT).
6. Calculate Ĥag (k) / ĥg (k) using Eq. 34/A14.
7. Invert Ĥag (k) / ĥg (k) to Hag (r)/ hg (r) using the 1D-FFT/3D-FFT.
8. Calculate Tag (r) / tg (r) using Eq. A17/A18.
9. Calculate the new Tag (r) / tg (r) from Tag (r) / tg (r) and its history using an acceleration method.
10. Repeat steps 3–9 until the input and output functions become identical within the convergence
tolerance.
Here, we have proposed a few acceleration methods for solving the integral equations, in which a
modified version of the Anderson method is employed.[9]
References
[1]
T. Sumi, T. Imai, F. Hirata, J. Chem. Phys., 2001, DOI:10.1063/1.1401824.
[2]
T. Sumi, H. Sekino, J. Chem. Phys., 2006, DOI:10.1063/1.2215603.
[3]
D. Chandler, J. D. McCoy, S. J. Singer, J. Chem. Phys., 1986, DOI:10.1063/1.451511.
[4]
D. Chandler, J. D. McCoy, S. J. Singer, J. Chem. Phys., 1986, DOI:10.1063/1.451510.
[5]
A. Kovalenko, F. Hirata, J. Chem. Phys., 1999, DOI:10.1063/1.478883.
[6]
F. Hirata, in Molecular Theory of Solvation; Kluwer Academic Publishers, 2003.
[7]
F. Hirata, P. J. Rossky, Chem. Phys. Lett., 1981, DOI:10.1016/0009-2614(81)85474-7.
[8]
F. Hirata, J. Chem. Phys., 1982, DOI:10.1063/1.443606.
[9]
Y. Maruyama, F. Hirata, J. Chem. Theo. Comput., 2012, DOI:10.1021/ct300355r.