Chemistry Reference
In-Depth Information
different infinitesimally later than
t
. Thus, by imposing these boundary condi-
tions, we have shown that the external potential,
v
ext
½
n
;
0
ð
rt
Þ
, is a functional
of the time-dependent density
n
ð
rt
Þ
and the initial wave function
0
.
In this regard please note that:
and
n
0
ð
rt
1. The difference between
n
ð
rt
Þ
Þ
is nonvanishing already in first
order of
v
ext
ð
rt
Þ
, ensuring the invertibility of the linear response operators
described later.
2. Because the density determines the potential up to a time-dependent
constant, the wave function is in turn determined up to a time-dependent
phase, which cancels out of the expectation value of any operator.
3. We write
v
ext
because the potential depends on both the history
of the density and the initial wave function. The functional
v
ext
½
n
;
ð
rt
Þ
0
Þ
is a very complex one, much more so than the ground-state case;
knowledge of it implies solution of all time-dependent Coulomb
interacting problems.
4. If we always begin in a nondegenerate ground state,
181,182
the initial-state
dependence can be subsumed by the Hohenberg-Kohn theorem,
1
½
n
;
ð
rt
0
then
v
ext
ð
rt
Þ
is a functional of
n
ð
rt
Þ
alone:
v
ext
½
n
ð
rt
Þ
.
5. A spin-dependent generalization exists,
183
so that
v
ext
ð
rt
Þ
will be a func-
tional of the spin densities
n
,
n
. This is usually used in practical
a
b
calculations.
6. Since RGI establishes that the external potential is a functional of the
current density, one could choose to use the current density as the basic
variable instead of the density. This is known as time-dependent current
density functional theory (TDCDFT) (See discussion below on solids.)
Kohn-Sham Equations
Once we have a proof that the potential is a functional of the time-
dependent density, it is simple to write the time-dependent Kohn-Sham
(TDKS) equations as
i
qf
j
s
ð
rt
Þ
2
2
þ
t
¼
r
v
S
s
½
n
ð
rt
Þ
f
j
s
ð
rt
Þ
½
31
q
whose potential is uniquely chosen (via the RG theorem) to reproduce the
exact spin densities of the interacting system. For simplicity, the initial-state
dependence of the KS potential is not written explicitly. As noted, if we start
in a nondegenerate ground state, this dependence is subsumed into the density.
X
N
s
2
n
s
ð
rt
Þ¼
1
j
f
j
s
ð
rt
Þj
½
32
j
¼
