200
Y.-W. Huang et al.
energies. As can be seen in Table 11.1, the HF method strongly underestimates the
S-T excitation energy as compared to the experimental value of 6.32 eV [73]. After
considering the second order perturbation correction, the error of S-T excitation energy for MP2 is largely reduced to 0.09 eV. For the density functional theories, the
error of S-T excitation energy at LDA (SVWN) is −0.46 eV.
For the GGA and meta-GGA functionals, the error of S-T excitation energies
is increased to ca. −0.65 eV. For the hybrid GGA functionals, the errors of S-T
excitation at functionals with low ratio of HF exchange energy, like HSE03, HSE06,
B3PW91 and PBE0 are only slightly reduced by ca. 0.03 eV. Increasing the ratio of
HF exchange energy, the accuracy of S-T excitation energy can be largely improved.
It can be found that the S-T excitation energy of M06HF agrees excellently well with
the experimental value.
The electronic dipole moment of CO was also collected in Table 11.1. As is
well known [74], the HF method predicts wrong sign of CO dipole moment and
MP2 method overestimates the CO dipole moment comparing with the experimental
value of 0.123 D [75]. It can also be found that the LDA and GGA functionals all
overestimate the CO dipole moment except for B97D. However, the meta-GGA and
hybrid functional all underestimates the CO dipole moment. When adding more
than 40 % HF contribution, it can be found most functionals predict wrong sign of
CO dipole moment except for wB97 and wB97X.
It is well-known that DFT schemes of LDA and GGA levels give a poor description of CO LUMO. As can be seen in Table 11.1, LDA and GGA levels give
a negative energy for CO LUMO. By adding the HF exchange contribution, the
hybrid functional can raise the LUMO energy, lower the HOMO energy, and thus
increase the HOMO-LUMO gap. This is important as the CO LUMO energies become positive after adding more than 40 % HF contribution in exchange-correlation
functionals. It can also be found that the linear relationship exists between HOMOLUMO gap and S-T excitation energy except only for the case of HF which the
HOMO-LUMO gap is high but the S-T excitation energy is far from the experimental value.
11.3.2 The Properties of Pt 7–3 and Pt 9–9–9 Clusters
Performance of DFT on metal clusters has long been an intense subject however
the accurately calculating properties of metal cluster has been shown to be problematic [76]. Here HF, MP2 and 21 DFT schemes were adopted to calculate the
electronic structure of Pt clusters. To understand the effect of functionals to the
electronic structure of Pt clusters, the projected d-band center (DBC) energy is analyzed. According to the adsorption position and symmetry, the projected d-band
center energies for one Pt atom at top site and three Pt atoms at fcc site, and the σ/π -
projected DBC energies at top site are considered. Figure 11.2 shows the projected
DBC as respect to the ratio of HF exchange energy for Pt 7–3 and Pt 10−6 cluster. As
can be seen in Fig. 11.2, the projected DBC energies are strongly dependent on the
Y.-W. Huang et al.
energies. As can be seen in Table 11.1, the HF method strongly underestimates the
S-T excitation energy as compared to the experimental value of 6.32 eV [73]. After
considering the second order perturbation correction, the error of S-T excitation energy for MP2 is largely reduced to 0.09 eV. For the density functional theories, the
error of S-T excitation energy at LDA (SVWN) is −0.46 eV.
For the GGA and meta-GGA functionals, the error of S-T excitation energies
is increased to ca. −0.65 eV. For the hybrid GGA functionals, the errors of S-T
excitation at functionals with low ratio of HF exchange energy, like HSE03, HSE06,
B3PW91 and PBE0 are only slightly reduced by ca. 0.03 eV. Increasing the ratio of
HF exchange energy, the accuracy of S-T excitation energy can be largely improved.
It can be found that the S-T excitation energy of M06HF agrees excellently well with
the experimental value.
The electronic dipole moment of CO was also collected in Table 11.1. As is
well known [74], the HF method predicts wrong sign of CO dipole moment and
MP2 method overestimates the CO dipole moment comparing with the experimental
value of 0.123 D [75]. It can also be found that the LDA and GGA functionals all
overestimate the CO dipole moment except for B97D. However, the meta-GGA and
hybrid functional all underestimates the CO dipole moment. When adding more
than 40 % HF contribution, it can be found most functionals predict wrong sign of
CO dipole moment except for wB97 and wB97X.
It is well-known that DFT schemes of LDA and GGA levels give a poor description of CO LUMO. As can be seen in Table 11.1, LDA and GGA levels give
a negative energy for CO LUMO. By adding the HF exchange contribution, the
hybrid functional can raise the LUMO energy, lower the HOMO energy, and thus
increase the HOMO-LUMO gap. This is important as the CO LUMO energies become positive after adding more than 40 % HF contribution in exchange-correlation
functionals. It can also be found that the linear relationship exists between HOMOLUMO gap and S-T excitation energy except only for the case of HF which the
HOMO-LUMO gap is high but the S-T excitation energy is far from the experimental value.
11.3.2 The Properties of Pt 7–3 and Pt 9–9–9 Clusters
Performance of DFT on metal clusters has long been an intense subject however
the accurately calculating properties of metal cluster has been shown to be problematic [76]. Here HF, MP2 and 21 DFT schemes were adopted to calculate the
electronic structure of Pt clusters. To understand the effect of functionals to the
electronic structure of Pt clusters, the projected d-band center (DBC) energy is analyzed. According to the adsorption position and symmetry, the projected d-band
center energies for one Pt atom at top site and three Pt atoms at fcc site, and the σ/π -
projected DBC energies at top site are considered. Figure 11.2 shows the projected
DBC as respect to the ratio of HF exchange energy for Pt 7–3 and Pt 10−6 cluster. As
can be seen in Fig. 11.2, the projected DBC energies are strongly dependent on the