v r; r
0
; t; t
0
ð
Þ¼
dq V ext
½
r; t
ð Þ
dV ext r 0 ; t 0
ð
Þ
V 0
ð1:90Þ
Analogous Kohn–Sham electron density first-order response function for the
reference system of non-interacting particles with unperturbed density q
0 r; t
ð Þ,
moving in external potential V KS r; t
ð Þ, is given by:
v KS r; r
0
; t; t
0
ð
Þ¼
dq V KS
½
r; t
ð Þ
dV KS r 0 ; t 0
ð
Þ
V KS q 0 r;t
ð Þ
½
ð1:91Þ
while the potential V KS r; t
ð Þ of a system of non-interacting particles corresponding
to the potential V ext r; t
ð Þ of the actual system is equal:
V KS r; t
ð Þ ¼ V ext r; t
ð Þþ
Z q r; t
ð Þ
r À r 0
j
j
dr
0
þ V xc q r; t
ð Þ
½
ð1:92Þ
Using the functional chain rule and calculating the functional derivative of
V KS r; t
ð Þ with respect to V ext r; t
ð Þ, we get the necessary link between the electron
density linear response function of the real system of interacting particles and the
reference system of the Kohn–Sham system of non-interacting particles and hence
the expression for the density linear response function of the non-interacting particles system:
q
1 r; t
ð Þ ¼
Z Z
dt
0 dr
0 v KS r; r
0
; t; t
0
ð
Þ V KS;1 r
0
; t
0
ð
Þ
ð1:93Þ
with effective potential V KS;1 r
0
; t
0
ð
Þ equal to:
V KS;1 r; t
ð Þ ¼ V 1 r; t
ð Þþ
Z q 1 r; t
ð Þ
r À r 0
j
j
dr
0
þ
Z Z
dr
0 dt
0 f xc q
0
 Ã
r; r
0
; t; t
0
ð
Þ q
1 r
0
; t
0
ð
Þ
ð1:94Þ
where time-independent kernel:
f xc q
0
 Ã
r; r
0
; t; t
0
ð
Þ¼
dV KS q
½ r; t
ð Þ
dq r 0 ; t 0
ð
Þ
q 0 r;t
ð Þ
ð1:95Þ
is a functional of the initial ground state electron density. The obtained expression
for the electron density linear response function of the system of non-interacting
particles allows practical application of the linear response function approximation
within the TD-DFT formalism for the calculation of time-dependent properties of
multi-electron systems (analogous derivation can be carried out for higher-order
corrections). If the system is subjected to a small time-dependent external potential
V 1 r; t
ð Þ, then there is no need to solve the full time-dependent KS equation, and
30
A. Koleżyński
0
; t; t
0
ð
Þ¼
dq V ext
½
r; t
ð Þ
dV ext r 0 ; t 0
ð
Þ
V 0
ð1:90Þ
Analogous Kohn–Sham electron density first-order response function for the
reference system of non-interacting particles with unperturbed density q
0 r; t
ð Þ,
moving in external potential V KS r; t
ð Þ, is given by:
v KS r; r
0
; t; t
0
ð
Þ¼
dq V KS
½
r; t
ð Þ
dV KS r 0 ; t 0
ð
Þ
V KS q 0 r;t
ð Þ
½
ð1:91Þ
while the potential V KS r; t
ð Þ of a system of non-interacting particles corresponding
to the potential V ext r; t
ð Þ of the actual system is equal:
V KS r; t
ð Þ ¼ V ext r; t
ð Þþ
Z q r; t
ð Þ
r À r 0
j
j
dr
0
þ V xc q r; t
ð Þ
½
ð1:92Þ
Using the functional chain rule and calculating the functional derivative of
V KS r; t
ð Þ with respect to V ext r; t
ð Þ, we get the necessary link between the electron
density linear response function of the real system of interacting particles and the
reference system of the Kohn–Sham system of non-interacting particles and hence
the expression for the density linear response function of the non-interacting particles system:
q
1 r; t
ð Þ ¼
Z Z
dt
0 dr
0 v KS r; r
0
; t; t
0
ð
Þ V KS;1 r
0
; t
0
ð
Þ
ð1:93Þ
with effective potential V KS;1 r
0
; t
0
ð
Þ equal to:
V KS;1 r; t
ð Þ ¼ V 1 r; t
ð Þþ
Z q 1 r; t
ð Þ
r À r 0
j
j
dr
0
þ
Z Z
dr
0 dt
0 f xc q
0
 Ã
r; r
0
; t; t
0
ð
Þ q
1 r
0
; t
0
ð
Þ
ð1:94Þ
where time-independent kernel:
f xc q
0
 Ã
r; r
0
; t; t
0
ð
Þ¼
dV KS q
½ r; t
ð Þ
dq r 0 ; t 0
ð
Þ
q 0 r;t
ð Þ
ð1:95Þ
is a functional of the initial ground state electron density. The obtained expression
for the electron density linear response function of the system of non-interacting
particles allows practical application of the linear response function approximation
within the TD-DFT formalism for the calculation of time-dependent properties of
multi-electron systems (analogous derivation can be carried out for higher-order
corrections). If the system is subjected to a small time-dependent external potential
V 1 r; t
ð Þ, then there is no need to solve the full time-dependent KS equation, and
30
A. Koleżyński