Excited States from Time-Dependent Density Functional Theory - Reviews in Computational Chemistry - page 107

We define the exchange-correlation potential via

ð d 3 r 0

ð r 0 t Þ

j r r 0 j þ

n

v S s ð rt Þ¼ v ext s ð rt Þþ v XC s ð rt Þ ½ 33

where the second term is the familiar Hartree potential.
Here the following should be noted:

1. The exchange-correlation potential, v XC s ð rt Þ is in general a functional of

the entire history of the densities, n s ð rt Þ , the initial interacting wave

. 182 But if
both the KS and interacting initial wave functions are nondegenerate
ground states, it becomes a simple functional of n

function ð 0 Þ , and the initial Kohn-Sham wave function, ð 0 Þ

alone.
2. By inverting the single doubly occupied KS equation for a spin-unpolarized
two-electron system, it is straightforward (but technically demanding) to
find the TDKS potential from an exact time-dependent density, a task that
has been done several times 184-186 for simple model systems.
3. Some approximation is used for v XC ð rt

s ð rt

Þ

as a functional of the density in
practical calculations, so that modifications of traditional TDSE schemes
are needed for the propagation. 187
4. Unlike the ground-state case, there is no self-consistency, merely forward
propagation in time, of a density-dependent Hamiltonian.
5. Again, unlike in the ground state, there is no central role played by a single-
number functional, such as the ground-state energy. In fact, while an action
was provided in the original RG study, extremizing it was later shown to
not yield the TDKS equations. 188

Þ

Linear Response
The most common application of TDDFT is the response of a system to a
weak, long-wavelength, optical field:

d v ext ð rt Þ¼ x exp ð i o t Þ z ½ 34

In the general case of the response of the ground state to an arbitrary weak
external field, the system's first-order response is characterized by the nonlocal
susceptibility:

ð dt 0 ð d 3 r 0 w ss 0 ½

X

n 0 ð r ; r 0 ; t 0 Þ d v ext s 0 ð r 0 t 0 Þ

d n

s ð rt

Þ¼ t ½ 35

s 0

This susceptibility w is a functional of the ground-state density, n 0 ð r Þ . A similar

equation describes the density response in the KS system:

ð dt 0 ð d 3 r 0 w S ss 0 ½

X

n 0 ð r ; r 0 ; t 0 Þ d v S s 0 ð r 0 t 0 Þ d n s ð rt Þ¼ t ½ 36

s 0