Computational Chemistry

reference to experimental quantities (i.e. on parameterization against experiment)
to give actual values of calculated parameters: the SHM gives energy levels in
terms of a parameterbwhich we could try to assign a value by comparison with
experiment (actually the results of SHM calculations are usually left in terms ofb),
while the EHM needs experimental ionization energies to calculate the Fock matrix
elements. The need for parameterization against experiment makes the SHM and
the EHMsemiempirical(“semiexperimental”) theories. In this chapter we deal with
a quantum mechanical approach that does not rely on calibration against measured
chemical parameters and is therefore called ab initio [ 1 , 2 ] meaning “from the first”,
from first principles. It is true that ab initio calculations give results in terms of
fundamental physical constants – Planck’s constant, the speed of light, the charge of
the electron – that must be measured to obtain their actual numerical values, but a
chemical theory could hardly be expected to calculate the fundamental physical
parameters of our universe (for that task we might be content to defer to something
like string theory).

5.2 The Basic Principles of the Ab initio Method

5.2.1 Preliminaries......................................................

InChapter 4we saw that wavefunctions and energy levels could be obtained by
diagonalizing a Fock matrix: the equation

H¼CeC#^1 $ð 5 : 1 Þ

is just another way of saying that diagonalization ofHgives the coefficients or
eigenvectors (the columns ofCthat, combined with the basis functions, yield the
wavefunctions of the molecular orbitals) and the energy levels or eigenvalues (the
diagonal elements ofe). Eq.5.1followed from

HC¼SCe $ð 5 : 2 Þ

which gives Eq.5.1whenSis approximated as a unit matrix (simple H€uckel
method, Section 4.3.4) or when the original Fock matrix is transformed intoH
(intoH^0 in the notation of 4.4.1.2) using an orthogonalizing matrix calculated from
S(extended H€uckel method, Section 4.4.1). To do a simple or an extended H€uckel
calculation the algorithm assembles the Fock matrixHand diagonalizes it. This is
also how an ab initio calculation is done; the essential difference compared to the
H€uckel methods lies in theevaluation of the matrix elements.
In the simple H€uckel method the Fock matrix elementsHijare not calculated, but
are instead set equal to 0 or#1 according to simple rules based on atomic
connectivity (Section 4.3.4); in the extended H€uckel method theHijare calculated

176 5 Ab initio Calculations