ChemOffice.Com - CambridgeSoft

Administrator
A variety of other basis sets, such as diffuse
function basis sets and high angular momentum
basis sets, are tailored to the properties of particular
of models under investigation.
The coefficients (C νi ) used for a given AO basis set
(φ ν ) are derived from the solution of the Roothaan-
Hall matrix equation with a diagonalized matrix of
orbital energies, E.
The Roothaan-Hall Matrix Equation
This equation, shown below, includes the Fock
matrix (F), the matrix of molecular orbital
coefficients (C) from the LCAO approximation,
the overlap matrix (S), and the diagonalized
molecular orbital energies matrix (E).
FC = SCE
Since the Fock equations are a function of the
molecular orbitals, they are not linearly
independent. As such the equations must be solved
using iterative, self-consistent field (SCF) methods.
The initial elements in the Fock matrix are guessed.
The molecular coefficients are calculated and the
energy determined. Each subsequent iteration uses
the results of the previous iteration until no further
variation in the energy occurs (a self-consistent field
is reached).
Ab Initio vs. Semiempirical
Ab initio (meaning literally “from first principles”)
methods use the complete form of the Fock
operator to construct the wave equation. The
semiempirical methods use simplified Fock
operators, in which 1-electron matrix elements and
some of the two electron integral terms are replaced
by empirically determined parameters.
Both the SCF RHF and UHF methods
underestimate the electron-electron repulsion and
lead to electron correlation errors, which tend to
overestimate the energy of a model. The use of
configuration interaction (CI) is one method
available to correct for this overestimation. For
more information see “Configuration Interaction”
on page 143.
The Semi-empirical Methods
Semiempirical methods can be divided into two
categories: one-electron types and two-electron
types. One-electron semiempirical methods use
only a one-electron Hamiltonian, while twoelectron
methods use a Hamiltonian which includes
a two-electron repulsion term. Authors differ
concerning the classification of methods with oneelectron
Hamiltonians; some prefer to classify these
as empirical.
The method descriptions that follow represent a
very simplified view of the semiempirical methods
available in Chem3D and CS MOPAC. For more
information see the online MOPAC manual.
Extended Hückel Method
Developed from the qualitative Hückel MO
method, the Extended Hückel Method (EH)
represents the earliest one-electron semiempirical
method to incorporate both σ and p valence
systems. It is still widely used, owing to its versatility
and success in analyzing and interpreting groundstate
properties of organic, organometallic, and
inorganic compounds of biological interest. Built
into Chem3D, EH is the default semiempirical
method used to calculate data required for
displaying molecular surfaces.
The EH method uses a one-electron Hamiltonian
with matrix elements defined as follows:
H µµ = – I µ
H µν = 0.5K( H µµ + H νν )S µν µ ≠ ν
where I µ is the valence state ionization energy
(VSIE) of orbital µ as deduced from spectroscopic
data, and K is the Wolfsberg-Helmholtz constant
142•Computation Concepts
CambridgeSoft
Quantum Mechanics Theory in Brief