Computational Chemistry

energiese(Eqs.4.55),if Sij¼dij(Eq.4.57). This is a big if, and in fact it is not true.
Sij¼dijwould mean that the basis functions are both orthogonal and normalized,
i.e. orthonormal.Orthogonalatomic (or molecular) orbitals or functionsfhave
zero net overlap (Fig.4.12), corresponding to

R

fifjdv¼0. Anormalizedorbital
or functionfhas the property

R

ffdv¼1. We can indeed use a set of normalized
basis functions: a suitable normalization constantkapplied to an unnormalized
basis functionf^0 will ensure normalization (f¼kf^0 ). However, we cannot
choose a set oforthogonalatom-centered basis functions, because orthogonality
implies zero overlap between the two functions in question, and in a molecule
the overlap between pairs of basis functions will depend on the geometry of
the molecule (Fig.4.12). (However, as we will see later, the basis functions can
be manipulated mathematically to givecombinationsof the original functions
whichareorthonormal).
The assumption of basis function orthonormality is a drastic approximation, but
it greatly simplifies the H€uckel method, and in the present context it enables us to
reduce Eq.4.54to Eq.4.59, and thus to obtain the coefficients and energy levels by
diagonalizing the Fock matrix. Later we will see that in the absence of the
orthogonality assumption the set of basis functions can be mathematically trans-
formed so that a modified Fock matrix can be diagonalized; in the simple H€uckel
method we are spared this transformation. In the matrix approach to the H€uckel
method, then, we must diagonalize the Fock matrixH; to do this we have to assign
numbers to the matrix elementsHij, and this brings us to other simplifying assump-
tions of the SHM, concerning theHij.
In the SHM the energy integralsHijare approximated as just three quantities (the
units are, e.g., kJ mol"^1 ):

C 1

C 2

C 3

* = +, .or –

C

C

C

H

H

H

H

H

a twisted allyl species *

+

+ +

+

Fig. 4.12 We cannot simply choose a set of orthonormal basis functions, because in a typical
molecule many pairs of basis functions will not be orthogonal, i.e. will not have zero overlap. In
the allyl species shown, the 2sand the 2pfunctions (i.e. AOs) on C 1 are orthogonal (theþpart of
theporbital cancels the"part in overlap with thesorbital; in general AOs on the same atom are
orthogonal), and the 2pfunctions on C 2 and C 3 are also orthogonal, if their axes are at right angles.
However, the C 1 (2s)/C 2 (2p) and the C 1 (2p)/C 2 (2p) pairs are not orthogonal

126 4 Introduction to Quantum Mechanics in Computational Chemistry