1-RDMs. The formulation of a solid foundation for TD-RDMFT which allows for
non-local potentials is still an open challenge.
The time-dependent 1-RDM can be diagonalized at each time t, allowing one to
consider the time-dependent natural spinorbitals, φ p (xt), and time-dependent occupation numbers, n p (t). The equation of motion for the 1-RDM (94) can be transformed to the time-dependent natural spinorbital basis, which gives the equations of
motion for the natural spinorbitals and occupation numbers separately [118, 119]
i n q t
ð Þ À n p t
ð Þ
À
Á φ p t
ð Þj _
φ q t
ð Þ
¼ n q t
ð Þ À n p t
ð Þ
À
Á
h pq t
ð Þ þ W
{
pq t
ð Þ À W pq t
ð Þ
8 p6 ¼q ,
ð96aÞ
i _
n p t
ð Þ ¼ W
{
pp t
ð Þ À W pp t
ð Þ
;
ð96bÞ
where the dot indicates a time-derivative and we introduced a short-hand notation
for the partial contraction of the 2-RDM with the two-electron integrals
W pq t
ð Þ ¼
X
rst
Γ prst t
ð Þ st
qr
t
ð Þ:
ð97Þ
It should be noted that the two-electron integrals are also time-dependent here,
because they are evaluated with the time-dependent natural spinorbitals.
5.2 Time-Dependent Response Equations
The time-dependent response equations can be derived from the equation of motion
of the 1-RDM (96) by considering a small time-dependent perturbation to a
stationary system, with the stationary 1-RDM γ
0 . The first-order perturbation in
the 1-RDM is directly related to perturbation in the natural spinorbitals and
occupation numbers as [compare with (85)]
δγ pq t
ð Þ ¼ δn p t
ð Þδ pq þ n q À n p
À
Á
δU pq t
ð Þ;
ð98Þ
where the indices refer to the natural spinorbital basis at t ¼ 0 and
δU pq (t) ¼ hφ p |δφ q (t)i. Collecting the perturbations in all the quantities up to first
order, we obtain the first-order time-dependent response equation for the 1-RDM
iδ_ γ pq t
ð Þ ¼
X
r
h pr t
ð Þδγ rq t
ð Þ À δγ pr t
ð Þh rq t
ð Þ
À
Á
þ
X
rs
ð 1
À1
K pq, rs γ
0
 Ã
t À t
0
ð
Þδγ rs t
0
ð Þdt
0
þ n q À n p
À
Á
δv pq t
ð Þ:
ð99Þ
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
159
non-local potentials is still an open challenge.
The time-dependent 1-RDM can be diagonalized at each time t, allowing one to
consider the time-dependent natural spinorbitals, φ p (xt), and time-dependent occupation numbers, n p (t). The equation of motion for the 1-RDM (94) can be transformed to the time-dependent natural spinorbital basis, which gives the equations of
motion for the natural spinorbitals and occupation numbers separately [118, 119]
i n q t
ð Þ À n p t
ð Þ
À
Á φ p t
ð Þj _
φ q t
ð Þ
¼ n q t
ð Þ À n p t
ð Þ
À
Á
h pq t
ð Þ þ W
{
pq t
ð Þ À W pq t
ð Þ
8 p6 ¼q ,
ð96aÞ
i _
n p t
ð Þ ¼ W
{
pp t
ð Þ À W pp t
ð Þ
;
ð96bÞ
where the dot indicates a time-derivative and we introduced a short-hand notation
for the partial contraction of the 2-RDM with the two-electron integrals
W pq t
ð Þ ¼
X
rst
Γ prst t
ð Þ st
qr
t
ð Þ:
ð97Þ
It should be noted that the two-electron integrals are also time-dependent here,
because they are evaluated with the time-dependent natural spinorbitals.
5.2 Time-Dependent Response Equations
The time-dependent response equations can be derived from the equation of motion
of the 1-RDM (96) by considering a small time-dependent perturbation to a
stationary system, with the stationary 1-RDM γ
0 . The first-order perturbation in
the 1-RDM is directly related to perturbation in the natural spinorbitals and
occupation numbers as [compare with (85)]
δγ pq t
ð Þ ¼ δn p t
ð Þδ pq þ n q À n p
À
Á
δU pq t
ð Þ;
ð98Þ
where the indices refer to the natural spinorbital basis at t ¼ 0 and
δU pq (t) ¼ hφ p |δφ q (t)i. Collecting the perturbations in all the quantities up to first
order, we obtain the first-order time-dependent response equation for the 1-RDM
iδ_ γ pq t
ð Þ ¼
X
r
h pr t
ð Þδγ rq t
ð Þ À δγ pr t
ð Þh rq t
ð Þ
À
Á
þ
X
rs
ð 1
À1
K pq, rs γ
0
 Ã
t À t
0
ð
Þδγ rs t
0
ð Þdt
0
þ n q À n p
À
Á
δv pq t
ð Þ:
ð99Þ
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
159