Chemistry Reference
In-Depth Information
charges, and the time-independent Schr ¨ dinger equation produces the wave
functions and energy levels for stationary quantum systems. The Poisson equa-
tion in one dimension is
d
2
Þ
dx
2
¼
f
ð
x
4
pr
ð
x
Þ
½
1
where
f
ð
x
Þ
is the electrostatic potential, and
r
ð
x
Þ
is the charge density. The
Schr ¨ dinger eigenvalue equation is
d
2
1
2
Þ
dx
2
þ
c
ð
x
V
ð
x
Þ
c
ð
x
Þ¼
E
c
ð
x
Þ
½
2
where
is the potential, and
E
is the energy
eigenvalue. Atomic units are assumed throughout this chapter. In typical elec-
tronic structure theory, such as Hartree-Fock or density functional calcula-
tions, the potential
V
c
ð
x
Þ
is the wave function,
V
ð
x
Þ
depends on the charge density produced by all the
electrons and nuclei; thus, the calculation is termed
self-consistent
since, at
convergence, the effective potential ceases to change. Prior to convergence, a
solver must go back and forth between updates of the wave functions and the
effective potential that depends on those wave functions.
We begin with a word on the general properties of these equations and
how they can be ''derived.'' Even though the FD representation is not varia-
tional in the sense of bounds on the ground-state energy, the iterative process
by which we obtain the solution to the partial differential equations (PDEs)
can be viewed variationally, i.e., we minimize some functional (see below)
with respect to variations of the desired function until we get to the lowest
''action'' or ''energy.'' This may seem rather abstract, but it turns out to be
practical since it leads directly to the iterative relaxation methods to be dis-
cussed below. (See Ref. 98 for a more complete mathematical description of
minimization and variational methods in relation to the FE method.)
For the Poisson problem, we make up an action functional
S
ð
x
Þ
½
f
that,
when minimized, yields the Poisson equation:
ð
ð
d
2
1
2
f
dx
2
dx
S
½
f
¼
f
4
p
rf
dx
½
3
A
functional
yields a number when the function over the whole domain
(
here) is specified. We then write a pseudodynamical equation (actually a
steepest-descent equation) for the updates of
f
f
:
qf
qt
¼
d
S
½
f
df
½
4
