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1.2 Functional Differentiation
It may be shown that the equilibrium density, as a function of
, is given by
|
(1.5) |
Proof? 5
Similarly, for the density correlations we have
|
(1.6) |
etc. (subscript
omitted).
Reminder:
is proportional to the
pair correlation function
.
Helmholtz Free Energy as a Density Functional:
Defining
|
(1.7) |
(functional Legendre transform) and
|
(1.8) |
we find a complementary hierarchy of correlation functions:
|
(1.9) |
|
(1.10) |
etc., where is the -th direct correlation function.
Reminder: is the direct pair correlation function which is
related to the net pcf via the Ornstein-Zernike equation.
For some homogeneous systems we know as a function
of density. It is reasonable to take advantage of this knowledge in some
manner (see below).
The direct one-particle pcf vanishes for the homogeneous ideal
gas; for interacting fluids it defines the chemical excess potential, both for
homogeneous and inhomogeneous systems.
Theorems of Hohenberg-Mermin-Kohn:
The following theorems are at the basis of density functional method:
Next: 1.3 Basic Strategy of
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Franz J. Vesely Oct 2001
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001