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(usually taken as 1.75). The Hamiltonian neglects
electron repulsion matrix elements but retains the
overlap integrals calculated using Slater-type basis
orbitals. Because the approximated Hamiltonian
(H) does not depend on the MO expansion
coefficient C νi , the matrix form of the EH
equations:
HC = SCE
can be solved without the iterative SCF procedure.
Methods Available in CS
MOPAC
The approximations that MOPAC uses in solving
the matrix equations for a molecular system follow.
Some areas requiring user choices are:
• RHF or UHF methods
• Configuration Interaction (CI)
• Choice of Hamiltonian approximation
(potential energy function)
RHF
The default Hartree-Fock method assumes that the
molecule is a closed shell and imposes spin
restrictions. The spin restrictions allow the Fock
matrix to be simplified. Since alpha (spin up) and
beta (spin down) electrons are always paired, the
basic RHF method is restricted to even electron
closed shell systems.
Further approximations are made to the RHF
method when an open shell system is presented.
This approximation has been termed the 1/2
electron approximation by Dewar. In this method,
unpaired electrons are treated as two 1/2 electrons
of equal charge and opposite spin. This allows the
computation to be performed as a closed shell. A CI
calculation is automatically invoked to correct
errors in energy values inherent to the 1/2 electron
approximation. For more information see
“Configuration Interaction” on page 143.
With the addition of the 1/2 electron
approximation, RHF methods can be run on any
starting configuration.
UHF
The UHF method treats alpha (spin up) and beta
(spin down) electrons separately, allowing them to
occupy different molecular orbitals and thus have
different orbital energies. For many open and
closed shell systems, this treatment of electrons
results in better estimates of the energy in systems
where energy levels are closely spaced, and where
bond breaking is occurring.
UHF can be run on both open and closed shell
systems. The major caveat to this method is the
time involved. Since alpha and beta electrons are
treated separately, twice as many integrals need to
be solved. As your models get large, the time for the
computation may make it a less satisfactory
method.
Configuration Interaction
The effects of electron-electron repulsion are
underestimated by SCF-RHF methods, which
results in the overestimation of energies.
SCF-RHF calculations use a single determinant that
includes only the electron configuration that
describes the occupied orbitals for most molecules
in their ground state. Further, each electron is
assumed to exist in the average field created by all
other electrons in the system, which tends to
overestimate the repulsion between electrons.
Repulsive interactions can be minimized by
allowing the electrons to exist in more places (i.e.
more orbitals, specifically termed virtual orbitals).
The multi-electron configuration interaction
(MECI) method in MOPAC addresses this
problem by allowing multiple sets of electron
assignments (i.e., configurations) to be used in
constructing the molecular wave functions.
ChemOffice 2005/Chem3D Computation Concepts • 143
Quantum Mechanics Theory in Brief