Excited States from Time-Dependent Density Functional Theory - Reviews in Computational Chemistry - page 105

where

2 i X

N

1

j ð r Þ¼

1 ½r j d ð r r j Þþ d ð r r j Þr j

½ 26

¼

j

is the current density operator. The equation of motion for the difference of
the two current densities gives 16

t ¼0 ¼

q j ð rt Þ

n 0 ð r Þr v ext ð r ; 0 Þ ½ 27

q t

If the Taylor expansion of the difference of the two potentials about t 0is

not spatially uniform for some order, the Taylor expansion of the current
density difference will then be nonzero at a finite order. Thus, RGI establishes
the fact that the external potential is a functional of the current density,
v ext

¼

½ j ; .

In the second part of the theorem (RGII), continuity is used:

ð r ; t Þ

0

q ð rt Þ

n

¼ r j ð rt Þ ½ 28

q t

which leads to

t ¼0 ¼r½

2

q n ð rt Þ

n 0 ð r Þr v ext ð r ; 0 Þ ½ 29

q t 2

Suppose now that is not uniform everywhere. Might not the left-

hand side of Eq. [29] still vanish? Apparently not, for real systems, because
it is easy to show: 180
ð d 3 r

v ext ð r ; 0 Þ

v ext ð r ; 0 Þr ½ n 0 ð r Þr v ext ð r ; 0 Þ

ð d 3 r

½ 30

¼ ½r ð v ext ð r ; 0 Þ n 0 ð r Þr v ext ð r ; 0 ÞÞ

2

n 0 jr v ext ð r ; 0 Þj

Using Green's theorem, the first term on the right vanishes for physically
realistic potentials (i.e., potentials arising from normalizable external charge
densities) because for such potentials, v ext ð r Þ falls off at least as 1 = r . But,

the second term is definitely negative, so, if is nonuniform, the inte-

gral must be finite, causing the densities to differ in second order in t . This
argument applies to each order and the densities n

v ext ð r ; 0 Þ

and n 0 ð r ;

ð r ; t Þ t Þ will become