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MOLECULAR ORBITAL THEORY CALCULATIONS 203

where H is the Hamiltonian operator, E is the energy of the particle (an electron
or nucleus), and ψ is the wave function. The product of ψ with its complex con-
2
jugate (ψ ∗ ψ, often written as |ψ| ) is interpreted as the probability distribution
of the particle (or electron).
Electronic structure methods are characterized by their various mathematical
approximations to its solution, since exact solutions to the Schr¨ odinger equation
are not computationally practical. There are three classes of electronic structure
methods: semi-empirical methods, density functional theory (DFT) methods, and
ab initio methods.

8.2.2. Semi-Empirical Methods
Semi-empirical methods are parameters derived from experimental data to sim-
plify the computation. They are used to solve an approximate form of the
Schr¨ odinger equation that depends on having appropriate parameters available
for the type of chemical system under investigation. Different semi-empirical
methods are largely characterized by their differing parameter sets. One of the
most commonly used semi-empirical computer programs is MOPAC, developed
by Stewart (1990). A commercial package based on MOPAC is available from
Fujitsu Company of Japan. MOPAC includes the semi-empirical Hamiltonians
MNDO (modified neglect of diatomic overlap), MINDO/3 (modified intermediate
neglect of differential overlap), AM1 (Austin model 1), PM3 (parametric method
3), MNDO-d, and PM5. These methods have been calibrated by using experimen-
tal data for thermodynamic properties such as heats of formation (Pople et al.,
1965; Baird and Dewar, 1969; Bodor et al., 1970; Murrell and Harget, 1972).
The advantage of semi-empirical methods is in the economy of computation.
These programs can be performed with a personal computer. However, the bond
energies calculated from these programs are substantially higher than the actual
values. In many instances, the bond energies are divided by an empirical factor
of 5 to give an indication of the real values. MOPAC provides the most accurate
energy calculations among these methods, while the results in bond energies are
still substantially over-estimated (e.g., Chen and Yang, 1997).

8.2.3. Density Functional Theory Methods
The approach of density functional theory (DFT) was developed in the 1960s by
using mathematical functions, called functionals, to describe the electron density
(Hohenberg and Kohn, 1964; Kohn and Sham, 1965). A summary of the theory is
given by Foresman and Frisch (1996), and a detailed description is available from
Parr and Yang (1989). DFT methods are attractive because they include the effects
of electron correlation (i.e., electron–electron interaction) in the energy. Devel-
opments in DFT have led to nonlocal (gradient-corrected) functionals — BLYP
(Becke-Lee-Yang-Parr), and to hybrid functionals — B3LYP. The nonlocal func-
tionals account for the non-uniformity of the overall electron distribution. The
hybrid functionals use a linear combination of the Hartree/Fock (HF) and the