Ω ia, jb ¼ δ ij δ ab ω
2
jb þ 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n i À n a
ð
Þω ia
p
K ia, jb
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
n j À n b
À
Á ω jb
q
:
(4.2)
The first term contains the Kohn–Sham eigenvalue differences ω ia ¼ ε a À ε i .
The second term contains the coupling matrix K, which gives a correction to the
diagonal terms that causes a singlet–triplet splitting and off-diagonal elements that
are responsible for the mixing of the single-particle transitions. For the singlet case
it reads
K ia, jb ¼
Z Z
ϕ i r
ð Þϕ a r
ð Þ
1
r À r
0
j
j
þ
δ
2 E xc
δρ r
ð Þδρ r
0
ð Þ
ϕ j r
0
ϕ b r
0
drdr
0
(4.3)
The approach also grants access to transition properties, analytical derivatives of
the excitation energy, the excited-state density matrix, and derived observables, like
the excited-state charge distribution and dipole moment [6, 7]. Although timedependent DFT (TDDFT), is an exact method in principle, the unknown exchangecorrelation (XC) potential, a unique functional of the density, must be
approximated in practice, which determines the accuracy of the method. Approximate functionals can be grouped into classes that use similar analytical forms and
behave in a similar way when applied to situations where large errors are expected.
Functionals that employ the local density approximation (LDA), generalised gradient approximation (GGA), or higher-order gradient corrections to LDA (meta
GGA), are usually applied to TDDFT using the adiabatic approximation, i.e.,
approximating the XC potential by a local functional in time v xc
adia (t) À v xc
LDA [ρ(t)],
i.e., simply evaluate the analytical density functional with the time-dependent
density ρ(t). This approximation has well known shortcomings, which must be
kept in mind when applying this method. They shall be discussed in the following.
DFT with local density functionals (LDA, GGA, meta-GGA) breaks down if the
ground state involves strong static correlation, i.e., when wave-function-based
methods describe the ground-state as multi-configurational, with significant
contributions from excited configurations [8–11]. This case is associated with the
triplet instability [9, 12, 13], i.e., the total energy can be lowered by breaking spin
symmetry. In regions of the nuclear configuration space close to triplet instabilities
or in the vicinity of conical intersections between ground and excited state, the
lowest excitation energy can be drastically underestimated by TDLDA. Moreover,
the shape of the PES close to conical intersections between the ground and first
excited state can be qualitatively wrong. Within the two-dimensional branching
space, which is created by the nonadiabatic coupling vector and the gradient
difference vector, the latter is often described artificially large or reaches zero, in
which case the dimensionality of the branching space appears to be reduced to one
[14].
Another problem is the general underestimation of the ionisation energy, which
originates from the wrong asymptotic shape of the XC potential of LDA, which
decays exponentially, whereas the exact Kohn–Sham potential decays like À 1/r
48
M. Wanko and A. Rubio
2
jb þ 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n i À n a
ð
Þω ia
p
K ia, jb
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
n j À n b
À
Á ω jb
q
:
(4.2)
The first term contains the Kohn–Sham eigenvalue differences ω ia ¼ ε a À ε i .
The second term contains the coupling matrix K, which gives a correction to the
diagonal terms that causes a singlet–triplet splitting and off-diagonal elements that
are responsible for the mixing of the single-particle transitions. For the singlet case
it reads
K ia, jb ¼
Z Z
ϕ i r
ð Þϕ a r
ð Þ
1
r À r
0
j
j
þ
δ
2 E xc
δρ r
ð Þδρ r
0
ð Þ
ϕ j r
0
ϕ b r
0
drdr
0
(4.3)
The approach also grants access to transition properties, analytical derivatives of
the excitation energy, the excited-state density matrix, and derived observables, like
the excited-state charge distribution and dipole moment [6, 7]. Although timedependent DFT (TDDFT), is an exact method in principle, the unknown exchangecorrelation (XC) potential, a unique functional of the density, must be
approximated in practice, which determines the accuracy of the method. Approximate functionals can be grouped into classes that use similar analytical forms and
behave in a similar way when applied to situations where large errors are expected.
Functionals that employ the local density approximation (LDA), generalised gradient approximation (GGA), or higher-order gradient corrections to LDA (meta
GGA), are usually applied to TDDFT using the adiabatic approximation, i.e.,
approximating the XC potential by a local functional in time v xc
adia (t) À v xc
LDA [ρ(t)],
i.e., simply evaluate the analytical density functional with the time-dependent
density ρ(t). This approximation has well known shortcomings, which must be
kept in mind when applying this method. They shall be discussed in the following.
DFT with local density functionals (LDA, GGA, meta-GGA) breaks down if the
ground state involves strong static correlation, i.e., when wave-function-based
methods describe the ground-state as multi-configurational, with significant
contributions from excited configurations [8–11]. This case is associated with the
triplet instability [9, 12, 13], i.e., the total energy can be lowered by breaking spin
symmetry. In regions of the nuclear configuration space close to triplet instabilities
or in the vicinity of conical intersections between ground and excited state, the
lowest excitation energy can be drastically underestimated by TDLDA. Moreover,
the shape of the PES close to conical intersections between the ground and first
excited state can be qualitatively wrong. Within the two-dimensional branching
space, which is created by the nonadiabatic coupling vector and the gradient
difference vector, the latter is often described artificially large or reaches zero, in
which case the dimensionality of the branching space appears to be reduced to one
[14].
Another problem is the general underestimation of the ionisation energy, which
originates from the wrong asymptotic shape of the XC potential of LDA, which
decays exponentially, whereas the exact Kohn–Sham potential decays like À 1/r
48
M. Wanko and A. Rubio