7 First-Principles Investigations of Electronically …
171
Finally, we add a couple of comments to the benchmark results. First, the TDA was
employed in all the TDDFT calculations. As explained, we used the TDA to contrast
the CIS, TDDFT, and GW/BSE. For the S 1 excitation energy of the PEN molecule,
the full diagonalization mixing resonant and anti-resonant transitions decreases excitation energies by 0.2–0.3 eV than those with the TDA. However, the inclusion of
anti-resonant transitions has minor effects on the long-range CT states, and thus, the
discussion regarding the long-range asymptotic behavior is valid without the TDA.
Second, although the range-separated functionals provide the reasonable results for
the isolated PEN molecule, they do not necessarily provide accurate results for a
crystal or a molecule in a condensed media. This is because parameters in rangeseparated functionals are generally determined so as to reproduce the properties of
gas-phase molecules. A recent development of the optimally tuned range-separation
scheme [90] allows for the accurate prediction of exciton binding energies for solidstate systems. By contrast, the GW/BSE method can yield reasonable exciton binding
energies for both gas- and condensed-phase systems.
7.3.3 Fragment Molecular Orbital Method
In this section, we summarize our recent attempt to extend the applicability of the
GW to disordered molecular aggregates. The GW and GW/BSE implementation have
been established for isolated molecular systems [10, 92, 99] and periodic systems [20,
100, 109]. Although recent developments of the efficient algorithms and implementations have enabled GW calculations for systems containing more than 100 atoms
[47, 51, 81, 82], it is still computationally formidable to apply the GW to disordered
molecular aggregates of more than 1,000 atoms. We briefly summarize our recent
developments of a large-scale GW based on the fragment molecular orbital method
[39, 45].
Our implementation adopts a fragment-based electronic structure method [50].
In a fragment-based method, an entire system is first divided into numerous small
subsystems, and the physical properties and the wave function of the entire system
are approximated from the multiple quantum-chemistry calculations for the subsystems. In the FMO method proposed by Kitaura and coworkers [61], an entire system
is divided into many small parts refereed as fragments. The self-consistent field
(SCF) calculations for fragment monomers, dimers, and optionally trimers are then
performed to approximate physical properties of the entire system. Theoretical
formulations of the FMO method and its applications have been widely reviewed
[30, 104, 107].
If a molecule is assigned as an independent fragment in an FMO calculation, the
SCF calculation for a fragment monomer provides the one-electron orbitals localized
within the molecule. The orbital energies
I
P and MOs (
ψ
I
p ) of an Ith fragment
(molecule) are given by
171
Finally, we add a couple of comments to the benchmark results. First, the TDA was
employed in all the TDDFT calculations. As explained, we used the TDA to contrast
the CIS, TDDFT, and GW/BSE. For the S 1 excitation energy of the PEN molecule,
the full diagonalization mixing resonant and anti-resonant transitions decreases excitation energies by 0.2–0.3 eV than those with the TDA. However, the inclusion of
anti-resonant transitions has minor effects on the long-range CT states, and thus, the
discussion regarding the long-range asymptotic behavior is valid without the TDA.
Second, although the range-separated functionals provide the reasonable results for
the isolated PEN molecule, they do not necessarily provide accurate results for a
crystal or a molecule in a condensed media. This is because parameters in rangeseparated functionals are generally determined so as to reproduce the properties of
gas-phase molecules. A recent development of the optimally tuned range-separation
scheme [90] allows for the accurate prediction of exciton binding energies for solidstate systems. By contrast, the GW/BSE method can yield reasonable exciton binding
energies for both gas- and condensed-phase systems.
7.3.3 Fragment Molecular Orbital Method
In this section, we summarize our recent attempt to extend the applicability of the
GW to disordered molecular aggregates. The GW and GW/BSE implementation have
been established for isolated molecular systems [10, 92, 99] and periodic systems [20,
100, 109]. Although recent developments of the efficient algorithms and implementations have enabled GW calculations for systems containing more than 100 atoms
[47, 51, 81, 82], it is still computationally formidable to apply the GW to disordered
molecular aggregates of more than 1,000 atoms. We briefly summarize our recent
developments of a large-scale GW based on the fragment molecular orbital method
[39, 45].
Our implementation adopts a fragment-based electronic structure method [50].
In a fragment-based method, an entire system is first divided into numerous small
subsystems, and the physical properties and the wave function of the entire system
are approximated from the multiple quantum-chemistry calculations for the subsystems. In the FMO method proposed by Kitaura and coworkers [61], an entire system
is divided into many small parts refereed as fragments. The self-consistent field
(SCF) calculations for fragment monomers, dimers, and optionally trimers are then
performed to approximate physical properties of the entire system. Theoretical
formulations of the FMO method and its applications have been widely reviewed
[30, 104, 107].
If a molecule is assigned as an independent fragment in an FMO calculation, the
SCF calculation for a fragment monomer provides the one-electron orbitals localized
within the molecule. The orbital energies
I
P and MOs (
ψ
I
p ) of an Ith fragment
(molecule) are given by