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5.2 PFC Model Basics
5.2.1 Classic Density Functional Theory
In this section, a brief review of CDFT will be given. More detailed derivation of
the CDFT were proposed by Ramakrishnan and Yussouff [1] and reviewed by Singh
[2]. In addition, the lecture note of Oxtoby [3] provided another good introduction
on this subject.
In general, the free energy F of a material system depending on its density ρ can
be written as a function F(ρ). Because the density can vary with space and itself a
function of the position vector r, i.e., ρ(r), the free energy of such density becomes a
functional, represented by F[ρ(r)]. Note that the dependent variable r is frequently
omitted and the free energy functional is written as F[ρ] in this chapter.
Usually, a function f (x) can be expressed as a Taylor expansion about a point x 0
in the form of:
f (x) = f (x 0 ) +
df
dx
x 0
(x − x 0 ) +
1
2
d
2 f
dx 2
x 0
(x − x 0 )
2
+ · · ·
(5.1)
A function of multivariables can be expressed as:
f (x 1 , x 2 , . . . , x n ) ≡ f (x) = f (x 0 ) +
n
i=1
df
dx i
x 0
(x i − x i0 )
+
1
2
n
i=1
n
j=1
d
2 f
dx i dx j
x 0
(x i − x i0 )(x j − x j0 ) + · · ·
(5.2)
Now, consider a functional F[ρ(r)], which can be thought of a continuous version
of Eq. (5.2) with infinite variables. Therefore, the sums should be replaced by integrals
over r. Analogously, the Taylor expansion of F[ρ] for the number density ρ(r), i.e.,
atom number per unit volume, near the density ρ 0 (r) is written:
F[ρ(r)] = F[ρ 0 (r)] +
dr
δF
δρ(r)
ρ(r) − ρ 0 (r)
+
1
2
drdr
δ
2 F
δρ(r)δρ(r
)
ρ(r) − ρ 0 (r)
ρ(r
) − ρ 0 (r
)
+ · · ·
(5.3)
in which δF/δρ(r) is the first order functional derivative, or the first variation.
According to Ramakrishan and Yussouff [1], the second term of Eq. (5.3) is the
“ideal gas” contribution, driving the system to a stable uniform field. This ideal free
energy can be obtained from an non-interacting system, i.e., without accounting for
interaction potential energy, and therefore the integrand in this term can be written