Chapter 3
Fundamentals of the Analysis Tools
Abstract The concept of molecular orbitals (MO’s) is of central importance in theoretical chemistry and is widely used by not only theoreticians but by experimentalists
for understanding the electronic structures of molecules. In this chapter, simple bases
or essential ingredients of the MO methods including the Hartree-Fock (HF), postHF, density functional theory (DFT), and other useful calculation methods are to be
described for convenience to effectively employ them. Although a few formulations
are to appear in this chapter just for the assistance of understanding the outlook of
the theories, those will be simplified as much as possible.
Keywords Molecular orbital · Hartree-Fock · Post-Hartree-Fock · Density
functional · Crystal orbital
3.1 Molecular Orbital Calculations
The electronic structure of a molecule is best described by the Schrödinger equation based on quantum mechanics. Within the adiabatic (also often called BornOppenheimer, although there is a minute difference which can be neglected here)
approximation and the non-relativistic scheme for simplicity, this equation is written
as follows:
H
Φ I = E I Φ I
(3.1)
H
= −
μ
∇
2
μ
2
−
μ,A
Z A
r μA
+
μ>ν
1
r μν
(3.2)
where operator H
denotes the Hamiltonian and Φ I and E I are the exact wavefunction
and energy, respectively, of the electrons in a molecule in the Ith state (see the
Glossaries 1).
© Springer Nature Singapore Pte Ltd. 2020
K. Tanaka, Theoretical Chemistry for Experimental Chemists,
https://doi.org/10.1007/978-981-15-7195-4_3
101
Fundamentals of the Analysis Tools
Abstract The concept of molecular orbitals (MO’s) is of central importance in theoretical chemistry and is widely used by not only theoreticians but by experimentalists
for understanding the electronic structures of molecules. In this chapter, simple bases
or essential ingredients of the MO methods including the Hartree-Fock (HF), postHF, density functional theory (DFT), and other useful calculation methods are to be
described for convenience to effectively employ them. Although a few formulations
are to appear in this chapter just for the assistance of understanding the outlook of
the theories, those will be simplified as much as possible.
Keywords Molecular orbital · Hartree-Fock · Post-Hartree-Fock · Density
functional · Crystal orbital
3.1 Molecular Orbital Calculations
The electronic structure of a molecule is best described by the Schrödinger equation based on quantum mechanics. Within the adiabatic (also often called BornOppenheimer, although there is a minute difference which can be neglected here)
approximation and the non-relativistic scheme for simplicity, this equation is written
as follows:
H
Φ I = E I Φ I
(3.1)
H
= −
μ
∇
2
μ
2
−
μ,A
Z A
r μA
+
μ>ν
1
r μν
(3.2)
where operator H
denotes the Hamiltonian and Φ I and E I are the exact wavefunction
and energy, respectively, of the electrons in a molecule in the Ith state (see the
Glossaries 1).
© Springer Nature Singapore Pte Ltd. 2020
K. Tanaka, Theoretical Chemistry for Experimental Chemists,
https://doi.org/10.1007/978-981-15-7195-4_3
101