3.1 Molecular Orbital Calculations
107
(a)
(b)
(c)
Fig. 3.1 Examples of MO energy levels obtained by each HF scheme. a Closed-shell structure,
b open-shell structure (restricted open HF (ROHF) picture), and c open-shell structure (unrestricted
open HF (UHF) picture). Small circles with the up and down arrows signify the electrons with α
and β spins, respectively
mechanical points of view, it gives lower energy than ROHF due to larger freedom
of variation. The spin contamination in the UHF method can be annihilated by a
certain numerical procedure. On the other hand, there is no spin contamination in the
ROHF description from the beginning. To the author’s knowledge, the UHF tends
to be more frequently employed for the open-shell structure calculation. The spin
density values resulting from the unrestricted calculations are often used and has
already been explained in Sect. 2.3.
Since the HF equation is for an electron moving in the averaged potential of
the other electrons as mentioned above, it does not include the direct interelectron
interactions. That is, the electron correlation is incompletely incorporated into the
HF scheme and correlation energy E corr virtually defined as follows:
E corr = E exact − E HF
(3.15)
is inevitably shown up, where E exact stands for the electronic energy obtained from
the exact Schrödinger equation and E HF being ideally the best value of the electronic
energy from the HF framework (Löwdin 1955; Sinano˘ glu 1964). This best value is
often called HF limit. The existence of correlation energy is a typical drawback in
the HF method and inclusion of the electron correlation effect is desirable for more
quantitative estimations of bond lengths, binding energy, excitation energy, and so
on.
There are several simplified versions of the HF method being often called semiempirical HF ones that deal with only the valence electrons and use some effective
parametrizations to build up molecular integrals shown in Glossaries 3 (Pople and
Segal 1966; Del Bene and Jaffé 1969; Bingham et al. 1975). Although the semiempirical HF methods generally afford less quantitative results compared with that of
107
(a)
(b)
(c)
Fig. 3.1 Examples of MO energy levels obtained by each HF scheme. a Closed-shell structure,
b open-shell structure (restricted open HF (ROHF) picture), and c open-shell structure (unrestricted
open HF (UHF) picture). Small circles with the up and down arrows signify the electrons with α
and β spins, respectively
mechanical points of view, it gives lower energy than ROHF due to larger freedom
of variation. The spin contamination in the UHF method can be annihilated by a
certain numerical procedure. On the other hand, there is no spin contamination in the
ROHF description from the beginning. To the author’s knowledge, the UHF tends
to be more frequently employed for the open-shell structure calculation. The spin
density values resulting from the unrestricted calculations are often used and has
already been explained in Sect. 2.3.
Since the HF equation is for an electron moving in the averaged potential of
the other electrons as mentioned above, it does not include the direct interelectron
interactions. That is, the electron correlation is incompletely incorporated into the
HF scheme and correlation energy E corr virtually defined as follows:
E corr = E exact − E HF
(3.15)
is inevitably shown up, where E exact stands for the electronic energy obtained from
the exact Schrödinger equation and E HF being ideally the best value of the electronic
energy from the HF framework (Löwdin 1955; Sinano˘ glu 1964). This best value is
often called HF limit. The existence of correlation energy is a typical drawback in
the HF method and inclusion of the electron correlation effect is desirable for more
quantitative estimations of bond lengths, binding energy, excitation energy, and so
on.
There are several simplified versions of the HF method being often called semiempirical HF ones that deal with only the valence electrons and use some effective
parametrizations to build up molecular integrals shown in Glossaries 3 (Pople and
Segal 1966; Del Bene and Jaffé 1969; Bingham et al. 1975). Although the semiempirical HF methods generally afford less quantitative results compared with that of