226
A. V. Pomogaeva and A. Y. Timoshkin
oligomerization degree. In absolute value the difference is 10.2 kcal mol −1 for
n = 10. The HOMO-LUMO gaps of H 3 [HGaNH] 3n H 3 obtained using the TZVP
basis set are by 0.27 eV (n = 3) and by 0.20 eV (n = 10) smaller than the values
obtained using the SVP basis set. In general, HOMO-LUMO energies are sensitive
to the particular choice of DFT method. It was found that a choice of PBE0 instead
of B3LYP causes an increase of the HOMO-LUMO gap value of about 0.55 eV,
while the overall tendencies in the changes in the HOMO-LUMO gap upon the rod
elongation are qualitatively similar.
Thus, in the following discussion, geometries of all compounds were fully
optimized at the B3LYP/SVP level of theory. All electron basis sets were applied
for all atoms except for In, where effective core potential (ECP-28 core AOs 1s
to 3d) [157] is used to reflect the varying degree of separation between core and
outer shells. The vibrational frequency computations were performed to verify that
obtained structures are true minima on their respective potential energy surfaces.
The same level of theory was used to evaluate the characteristic features of the
ground state electronic structure.
Computations of excitation spectra required additional testing on the reliability
of computational methods. TDDFT method is highly popular tool to estimate
excitation energies of rather complicated systems [158]. Time-dependent response
theory is used to reduce electronic excitations to ground state properties. It considers
a molecule in its ground state to be a subject of a periodic perturbation by
uniform electric field. The perturbation theory in this case operates with frequencydependent one-particle density matrix. The excitation energies are obtained as
eigenvalues of an electronic Hessian which may be imagined as the matrix of second
derivatives of the electron energy with respect to the electronic degrees of freedom.
Efficiency of TDDFT calculations with B3LYP, PBE0, M06 [159], and HSSE
functionals was tested against the benchmark EOM-CCSD [160, 161] calculations
on the example of [HGaNH] 4 . Effect of diffusion functions extension to TZVP and
QZVP basis sets was studied by considering augmented TZVPD and QZVPD basis
sets developed by Rappoport and Furche [162]. Energies of several low-lying singlet
states calculated at different levels of theory are provided in Table 6.2.
The results show that all TDDFT methods underestimate excited state values
with respect to the values obtained at EOM-CCSD level. An underestimation of
valence excitation energies by TDDFT methods is often attributed to so-called selfinteraction problem of semi-local DFT functional [163]. The larger the basis set,
the greater is the underestimation in the obtained values of energy for the tested
compound. Thus, the smallest tested basis set, TZVP, provides the best agreement
with the benchmark EOM-CCSD/TZVPD computations for any of the considered
DFT functional.
It was found that excitation energies computed with PBE0 functional are in better
agreement with EOM-CCSD/TZVPD results. In case of PBE0/TZVP level, the
correct order of three- and twofold degenerate levels is preserved (see Table 6.2).
The error is less than 0.01 eV for the lowest triply degenerate spin-singlet state and
is smaller than 0.06 eV for higher singlet states.
A. V. Pomogaeva and A. Y. Timoshkin
oligomerization degree. In absolute value the difference is 10.2 kcal mol −1 for
n = 10. The HOMO-LUMO gaps of H 3 [HGaNH] 3n H 3 obtained using the TZVP
basis set are by 0.27 eV (n = 3) and by 0.20 eV (n = 10) smaller than the values
obtained using the SVP basis set. In general, HOMO-LUMO energies are sensitive
to the particular choice of DFT method. It was found that a choice of PBE0 instead
of B3LYP causes an increase of the HOMO-LUMO gap value of about 0.55 eV,
while the overall tendencies in the changes in the HOMO-LUMO gap upon the rod
elongation are qualitatively similar.
Thus, in the following discussion, geometries of all compounds were fully
optimized at the B3LYP/SVP level of theory. All electron basis sets were applied
for all atoms except for In, where effective core potential (ECP-28 core AOs 1s
to 3d) [157] is used to reflect the varying degree of separation between core and
outer shells. The vibrational frequency computations were performed to verify that
obtained structures are true minima on their respective potential energy surfaces.
The same level of theory was used to evaluate the characteristic features of the
ground state electronic structure.
Computations of excitation spectra required additional testing on the reliability
of computational methods. TDDFT method is highly popular tool to estimate
excitation energies of rather complicated systems [158]. Time-dependent response
theory is used to reduce electronic excitations to ground state properties. It considers
a molecule in its ground state to be a subject of a periodic perturbation by
uniform electric field. The perturbation theory in this case operates with frequencydependent one-particle density matrix. The excitation energies are obtained as
eigenvalues of an electronic Hessian which may be imagined as the matrix of second
derivatives of the electron energy with respect to the electronic degrees of freedom.
Efficiency of TDDFT calculations with B3LYP, PBE0, M06 [159], and HSSE
functionals was tested against the benchmark EOM-CCSD [160, 161] calculations
on the example of [HGaNH] 4 . Effect of diffusion functions extension to TZVP and
QZVP basis sets was studied by considering augmented TZVPD and QZVPD basis
sets developed by Rappoport and Furche [162]. Energies of several low-lying singlet
states calculated at different levels of theory are provided in Table 6.2.
The results show that all TDDFT methods underestimate excited state values
with respect to the values obtained at EOM-CCSD level. An underestimation of
valence excitation energies by TDDFT methods is often attributed to so-called selfinteraction problem of semi-local DFT functional [163]. The larger the basis set,
the greater is the underestimation in the obtained values of energy for the tested
compound. Thus, the smallest tested basis set, TZVP, provides the best agreement
with the benchmark EOM-CCSD/TZVPD computations for any of the considered
DFT functional.
It was found that excitation energies computed with PBE0 functional are in better
agreement with EOM-CCSD/TZVPD results. In case of PBE0/TZVP level, the
correct order of three- and twofold degenerate levels is preserved (see Table 6.2).
The error is less than 0.01 eV for the lowest triply degenerate spin-singlet state and
is smaller than 0.06 eV for higher singlet states.