3.1 Molecular Orbital Calculations
103
4πε 0 = 1 1.1127 × 10
−10 C
2 J
−1 m
−1
The first and the second terms in the Hamiltonian H
in Eq. (3.2) are called,
respectively, kinetic operator and nuclear attraction potential, and the third term is
interelectron repulsion potential. It is the presence of this third term that makes
it impossible to analytically solve the Schrödinger equation in case of molecules
having multielectron as is well known. In this sense, one has to introduce a certain
approximation to solve the Schrödinger equation. Theoretical chemists working for
the description of atoms and molecules based on quantum mechanics have long been
making efforts to improve this approximation as well as solid-state physicists dealing
with solids.
The most popular way of approximation is to use the molecular orbitals (MO’s) for
a description of the electronic structure of molecules. Earlier MO methods include
the Hückel (Hückel 1930) and the extended Hückel (Hoffmann 1963) schemes. The
former is the simplest one dealing with only π electrons in π-conjugated organic
molecules employing simplified parameters called Coulomb and resonance integrals.
The latter can handle all the valence electrons including both σ and π electrons.
Both of these methods are considered to be too simple to afford quantitative results
but nonetheless provide certain insight particularly into the discussion of chemical
reactions.
Currently, there are two major tools for the MO calculations: one is the HF method
and another is based on the DFT method. The former is making further progress to
the various post-HF methods, which give more accurate results compared with those
by the ordinary HF method and nowadays are being selected toward obtaining more
quantitative results particularly to the theoreticians. On the other hand, the latter has
recently joined the theoretical chemistry method and seems to be more tractable due
to less time-consuming and providing a certain accuracy. In the next few sections,
these methods are to be elucidated.
3.1.1 Hartree-Fock (HF) Method
3.1.1.1 Basic Idea
The ordinary MO calculations in quantum chemistry have traditionally been
performed with the use of the HF method described above. This has been and is still
now considered to give reference to quantum chemical calculations for molecules
and polymers. For instance, the HF scheme also gives the starting point to launch
into the post-HF calculation which can more quantitatively describe the electronic
structures. In this sense, comprehension of the basic concept of the HF method would
be useful to the users of theoretical chemistry as well.
The HF framework lies in the usage of the wavefunction called “orbital” based on
the one-electron approximation, i.e., an orbital is a function of the coordinate of one
103
4πε 0 = 1 1.1127 × 10
−10 C
2 J
−1 m
−1
The first and the second terms in the Hamiltonian H
in Eq. (3.2) are called,
respectively, kinetic operator and nuclear attraction potential, and the third term is
interelectron repulsion potential. It is the presence of this third term that makes
it impossible to analytically solve the Schrödinger equation in case of molecules
having multielectron as is well known. In this sense, one has to introduce a certain
approximation to solve the Schrödinger equation. Theoretical chemists working for
the description of atoms and molecules based on quantum mechanics have long been
making efforts to improve this approximation as well as solid-state physicists dealing
with solids.
The most popular way of approximation is to use the molecular orbitals (MO’s) for
a description of the electronic structure of molecules. Earlier MO methods include
the Hückel (Hückel 1930) and the extended Hückel (Hoffmann 1963) schemes. The
former is the simplest one dealing with only π electrons in π-conjugated organic
molecules employing simplified parameters called Coulomb and resonance integrals.
The latter can handle all the valence electrons including both σ and π electrons.
Both of these methods are considered to be too simple to afford quantitative results
but nonetheless provide certain insight particularly into the discussion of chemical
reactions.
Currently, there are two major tools for the MO calculations: one is the HF method
and another is based on the DFT method. The former is making further progress to
the various post-HF methods, which give more accurate results compared with those
by the ordinary HF method and nowadays are being selected toward obtaining more
quantitative results particularly to the theoreticians. On the other hand, the latter has
recently joined the theoretical chemistry method and seems to be more tractable due
to less time-consuming and providing a certain accuracy. In the next few sections,
these methods are to be elucidated.
3.1.1 Hartree-Fock (HF) Method
3.1.1.1 Basic Idea
The ordinary MO calculations in quantum chemistry have traditionally been
performed with the use of the HF method described above. This has been and is still
now considered to give reference to quantum chemical calculations for molecules
and polymers. For instance, the HF scheme also gives the starting point to launch
into the post-HF calculation which can more quantitatively describe the electronic
structures. In this sense, comprehension of the basic concept of the HF method would
be useful to the users of theoretical chemistry as well.
The HF framework lies in the usage of the wavefunction called “orbital” based on
the one-electron approximation, i.e., an orbital is a function of the coordinate of one