Chemistry Reference
In-Depth Information
where in contrast to the first definition of the electron density the integration is
now carried out over all dynamical variables. The charge density
r
c
is directly
related to the electron density via the negative elementary charge:
r
c
(
).
The propagation of this expectation value in time can then be described by the
Ehrenfest theorem, which allows us to express the total time derivative of an expecta-
tion value as an expectation value of the partial time derivative of the operator and an
expectation value of the commutator of the operator with the Hamiltonian:
d
r r;
r
)
¼
er
(
r
ðÞ
d
t
¼
t
d
C
h
ð
rfg;
t
Þ rj C
ð
rfg;
t
Þ
i
d
t
Þ
@
r
r
@
i
¼
C
ð
rfg;
t
C
ð
rfg;
t
Þ
þ
h
C
ð h i
|
{z
}
r j
ð
rfg;
t
Þ
j
½
H
; r
r
jC
rfg;
t
:
(5)
t
|
{z
}
¼
0
The partial derivative of the density operator with respect to time vanishes
in this equation, because this operator does not depend on time. If one chooses
a Hamiltonian and a wave function, the second term can be evaluated and yields the
negative divergence of the current density. Because the position
does not depend
on the time
t
, the total derivative in (
5
) is equal to the partial derivative d
r
(
r
,
t
)/
d
t
r
¼
∂
r
(
r
,
t
)/
∂
t
, and one therefore arrives at the continuity equation
@r r;
ðÞ
@
t
t
þrj ¼
0
;
(6)
which is the fundamental equation defining both the electron density and the
current density
j
[
22
]. The deduction of the continuity equation from the expecta-
tion value of the density operator
C
h
ð
rfg;
t
Þ rj C
ð
rfg;
t
Þ
i
uniquely defines the
density distribution of an
N
-particle system. The continuity equation can alterna-
tively be deduced from the Heisenberg equation of motion written for the density
operator [
22
], which is omitted here for the sake of brevity.
3 Dependence on Wave Function and Hamiltonian
From the continuity equation, it is clear that one has to choose a wave function and
a Hamiltonian operator [see (
5
)] to resolve the electron density and the current
density. We will therefore give a brief overview on approximations to the electronic
wave function and on the different Hamiltonians relevant to chemistry before
proceeding with the analysis of the electron density.
3.1 The Wave Function
The most general ansatz for the total wave function of a molecule consisting of
N
electrons and
M
nuclei,
