108
3 Fundamentals of the Analysis Tools
the ordinary HF method, they are still sometimes employed for convenience due to
quite short computation time required. On the other hand, the calculation scheme
without any empirical parameters or abbreviation of molecular integrals is called
non-empirical HF method (or simply HF method), which is employed in most cases.
3.1.1.3 Excited State
Electron excitations in the HF picture are represented by using the unoccupied (or
virtual) MO’s mentioned above as the destination of the excited electron. Various
excited-state configurations due to one-electron excitation from the ith to the jth
MO’s are illustrated in Fig. 3.2. Care should be taken that the excitation energy
i→j is not only the energy difference in the ith and the jth MO’s but includes the
adjustment occurring from the above J ij and K ij integrals in Eqs. (3.9) and (3.10).
The one-electron excitation energies in the framework of the HF method is thus
expressed as follows:
1
i→j = ε j − ε i − J ij + 2K ij
(3.16)
for the singlet-state excitation (see Figs. 3.2a–c) and
3
i→j = ε j − ε i − J ij
(3.17)
for the triplet-state excitation (see Figs. 3.2d–f). It is obvious that the triplet-state
excitation energy is smaller than the corresponding singlet-state one due to the lack
of exchange integral K ij having always a positive value. It is noted that, however,
the triplet-state excitation is forbidden since the corresponding oscillator strength for
this excitation (see Sect. 2.5) vanishes due to the integration of different spin states
between the initial and the final configurations.
Formally, there can be excited-state configurations due to two-electron excitation
as also shown in Fig. 3.3. However, one cannot accurately estimate the excitation
i
i
j
j
i
j
i
i
j
j
i
j
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3.2 a–c Various configurations of one-electron singlet excitation (singles) from the ith to
the jth MO. Small circles with the up and down arrows signify the electrons with α and β spins,
respectively. d–f represent the triplet excitation corresponding to (a)–(c)
3 Fundamentals of the Analysis Tools
the ordinary HF method, they are still sometimes employed for convenience due to
quite short computation time required. On the other hand, the calculation scheme
without any empirical parameters or abbreviation of molecular integrals is called
non-empirical HF method (or simply HF method), which is employed in most cases.
3.1.1.3 Excited State
Electron excitations in the HF picture are represented by using the unoccupied (or
virtual) MO’s mentioned above as the destination of the excited electron. Various
excited-state configurations due to one-electron excitation from the ith to the jth
MO’s are illustrated in Fig. 3.2. Care should be taken that the excitation energy
i→j is not only the energy difference in the ith and the jth MO’s but includes the
adjustment occurring from the above J ij and K ij integrals in Eqs. (3.9) and (3.10).
The one-electron excitation energies in the framework of the HF method is thus
expressed as follows:
1
i→j = ε j − ε i − J ij + 2K ij
(3.16)
for the singlet-state excitation (see Figs. 3.2a–c) and
3
i→j = ε j − ε i − J ij
(3.17)
for the triplet-state excitation (see Figs. 3.2d–f). It is obvious that the triplet-state
excitation energy is smaller than the corresponding singlet-state one due to the lack
of exchange integral K ij having always a positive value. It is noted that, however,
the triplet-state excitation is forbidden since the corresponding oscillator strength for
this excitation (see Sect. 2.5) vanishes due to the integration of different spin states
between the initial and the final configurations.
Formally, there can be excited-state configurations due to two-electron excitation
as also shown in Fig. 3.3. However, one cannot accurately estimate the excitation
i
i
j
j
i
j
i
i
j
j
i
j
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3.2 a–c Various configurations of one-electron singlet excitation (singles) from the ith to
the jth MO. Small circles with the up and down arrows signify the electrons with α and β spins,
respectively. d–f represent the triplet excitation corresponding to (a)–(c)