resulting signals are sensitive to the change in chemical and electronic structure
(Fig. 13a, left). Moreover, the contributions from ρ 00 and ρ 11 populations and ρ 01
and ρ 10 coherences may be separated. Below 29.0 fs the signals mainly come from
the V 1 state and after 29.0 fs from the V 0 state. At the transition point T ¼ 29.0 fs
both V 0 and V 1 populations have comparable contributions to the signals, and
coherence terms have larger contributions than at any other time. We had further
analyzed two major peaks in the total signals (denoted as “o” and “+”, the former is
always lower in energy) and found that they can be tracked by varying timedependent strength: peak “o” is stronger before T ¼ 29.0 fs, when peak “+”
becomes stronger. Both peaks correspond to transitions from the oxygen p orbital
to π* transitions, as indicated by the dominant natural transition orbital (NTO)
[215–217] pairs (Fig. 13c). As time increases, for peak “o” the hole orbital changes
from localized on the C–O bond to delocalized. For peak “+”, the particle orbital
changes from delocalized to localized on the C–O bond.
3.4 Other Core Hole State Simulation Techniques
Unlike most of the practical implementations of DFT, which contain empirical
parameters from numerical fitting to experimental data or results from higher level
theories, the Green’s function method based on many-body perturbation theory
provides a systematic way to achieve higher accuracy. Quasiparticle orbital energies can be obtained by solving a set of coupled Hedin equations [218]. These
orbital energies offer a much better estimation of the ionization potential and the
electron affinity of the system than do Kohn–Sham orbital energies. In Hedin’s
equations, the one-particle Green’s function is solved through a Dyson-like equation with a self-energy, which is complex and energy-dependent. Self-energy plays
a similar role as does exchange-correlation energy in DFT. The most popular
approximation of self-energy is the GW approximation, where the vertex operator
is simplified as product of δ-function and self-energy becomes the product of
single-particle Green’s function (G) and the dynamically screened Coulomb interaction (W) [218, 219]. Moreover, particle-hole interaction, which is neglected in
TDDFT, can be considered and another Dyson-like equation for the four-point
polarization function (two-particle Green’s function) can be derived [220]. This is
the Bethe–Salpeter equation (BSE). GW/BSE equations have to be solved iteratively in a self-consistent way. DFT orbitals and their energies can be used as
initial guesses, but the full self-consistent solutions are independent on the initial
guesses [221]. BSE can be recast into a form similar to the Casida equation in linear
response TDDFT (see (52)), but the kernel in BSE could be frequency-dependent
[222–224], which offers a model for designing non-adiabatic exchange-correlation
kernel in TDDFT. For a thorough comparison of GW/BSE and TDDFT, see
320
Y. Zhang et al.
(Fig. 13a, left). Moreover, the contributions from ρ 00 and ρ 11 populations and ρ 01
and ρ 10 coherences may be separated. Below 29.0 fs the signals mainly come from
the V 1 state and after 29.0 fs from the V 0 state. At the transition point T ¼ 29.0 fs
both V 0 and V 1 populations have comparable contributions to the signals, and
coherence terms have larger contributions than at any other time. We had further
analyzed two major peaks in the total signals (denoted as “o” and “+”, the former is
always lower in energy) and found that they can be tracked by varying timedependent strength: peak “o” is stronger before T ¼ 29.0 fs, when peak “+”
becomes stronger. Both peaks correspond to transitions from the oxygen p orbital
to π* transitions, as indicated by the dominant natural transition orbital (NTO)
[215–217] pairs (Fig. 13c). As time increases, for peak “o” the hole orbital changes
from localized on the C–O bond to delocalized. For peak “+”, the particle orbital
changes from delocalized to localized on the C–O bond.
3.4 Other Core Hole State Simulation Techniques
Unlike most of the practical implementations of DFT, which contain empirical
parameters from numerical fitting to experimental data or results from higher level
theories, the Green’s function method based on many-body perturbation theory
provides a systematic way to achieve higher accuracy. Quasiparticle orbital energies can be obtained by solving a set of coupled Hedin equations [218]. These
orbital energies offer a much better estimation of the ionization potential and the
electron affinity of the system than do Kohn–Sham orbital energies. In Hedin’s
equations, the one-particle Green’s function is solved through a Dyson-like equation with a self-energy, which is complex and energy-dependent. Self-energy plays
a similar role as does exchange-correlation energy in DFT. The most popular
approximation of self-energy is the GW approximation, where the vertex operator
is simplified as product of δ-function and self-energy becomes the product of
single-particle Green’s function (G) and the dynamically screened Coulomb interaction (W) [218, 219]. Moreover, particle-hole interaction, which is neglected in
TDDFT, can be considered and another Dyson-like equation for the four-point
polarization function (two-particle Green’s function) can be derived [220]. This is
the Bethe–Salpeter equation (BSE). GW/BSE equations have to be solved iteratively in a self-consistent way. DFT orbitals and their energies can be used as
initial guesses, but the full self-consistent solutions are independent on the initial
guesses [221]. BSE can be recast into a form similar to the Casida equation in linear
response TDDFT (see (52)), but the kernel in BSE could be frequency-dependent
[222–224], which offers a model for designing non-adiabatic exchange-correlation
kernel in TDDFT. For a thorough comparison of GW/BSE and TDDFT, see
320
Y. Zhang et al.