Essentials of Computational Chemistry

138 5 SEMIEMPIRICAL IMPLEMENTATIONS OF MO THEORY
6. The only terms remaining to be defined in the assembly of the HF secular determinant
are the one-electron terms for off-diagonal matrix elements. These are defined as
µ
−1
2 ∇2 −
k
Zk
rk
ν
= (βA + βB)Sµν
2
(5.10)
where µ and ν are centered on atoms A and B, respectively, the β values are semiempirical
parameters, and Sµν is the overlap matrix element computed using the STO basis
set. Note that computation of overlap is carried out for every combination of basis functions,
even though in the secular determinant itself S is defined by Eq. (5.4). There are,
in effect, two different S matrices, one for each purpose. The β parameters are entirely
analogous to the parameter of the same name we saw in Hückel theory – they provide
a measure of the strength of through space interactions between atoms. As they are
intended for completely general use, it is not necessarily obvious how to assign them
a numerical value, unlike the situation that obtains in Hückel theory. Instead, β values
for CNDO were originally adjusted to reproduce certain experimental quantities.
While the CNDO method may appear to be moderately complex, it represents a vast
simplification of HF theory. Equation (5.5) reduces the number of two-electron integrals
having non-zero values from formally N 4 to simply N 2 . Furthermore, those N 2 integrals are
computed by trivial algebraic formulae, not by explicit integration, and between any pair of
atoms all of the integrals have the same value irrespective of the atomic orbitals involved.
Similarly, evaluation of one-electron integrals is also entirely avoided, with numerical values
for those portions of the relevant matrix elements coming from easily evaluated formulae.
Historically, a number of minor modifications to the conventions outlined above were
explored, and the different methods had names like CNDO/1, CNDO/2, CNDO/BW, etc.; as
these methods are all essentially obsolete, we will not itemize their differences. One CNDO
model that does continue to see some use today is the Pariser–Parr–Pople (PPP) model
for conjugated π systems (Pariser and Parr 1953; Pople 1953). It is in essence the CNDO
equivalent of Hückel theory (only π-type orbitals are included in the secular equation), and
improves on the latter theory in the prediction of electronic state energies.
The computational simplifications inherent in the CNDO method are not without chemical
cost, as might be expected. Like EHT, CNDO is quite incapable of accurately predicting good
molecular structures. Furthermore, the simplification inherent in Eq. (5.6) has some fairly
dire consequences; two examples are illustrated in Figure 5.1. Consider the singlet and triplet
states of methylene. Clearly, repulsion between the two highest energy electrons in each
state should be quite different: they are spin-paired in an sp 2 orbital for the singlet, and spinparallel,
one in the sp 2 orbital and one in a p orbital, for the triplet. However, in each case the
interelectronic Coulomb integral, by Eq. (5.6), is simply γCC. And, just as there is no distinguishing
between different types of atomic orbitals, there is also no distinguishing between
the orientation of those orbitals. If we consider the rotational coordinate for hydrazine, it is
clear that one factor influencing the energetics will be the repulsion of the two lone pairs, one