middle of a conjugated chain should have differentHiiandHijparameters than ones
at the end of the chain. Of course, this approximation simplifies the Fock matrix
(or the determinant in the old determinant method,Section 4.3.7).
The neglect of electron spin and the deficient treatment of interelectronic
repulsion is obvious. In the usual derivation (Section 4.3.4): in Eq.4.40the
integration is carried out with respect to only spatial coordinates (ignoring spin
coordinates; contrast ab initio theory, Section 5.2), and in calculatingpenergies
(Section 4.3.5.3) we simply took the sum of the number of electrons in each
occupied MO times the energy level of the MO. However, the energy of an MO
is the energy of an electron in the MO moving in the force field of the nuclei and
all the other electrons (as pointed out inSection 4.3.4, in explaining the matrices
of Eqs.4.55). If we calculate the total electronic energy by simply summing MO
energies times occupancy numbers, we are assuming, wrongly, that the electron
energies are independent of one another, i.e. that the electrons do not interact. An
energy calculated in this way is said to be a sum of one-electron energies. The
resonance energies calculated by the SHM can thus be only very rough, unless
the errors tend to cancel in the subtraction step, which in fact probably occurs to
some extent (this is presumably why the method of Hess and Schaad for
calculating resonance energies works so well [ 53 ]). The neglect of electron
repulsion and spin in the usual derivation of the SHM is discussed in reference
[ 30 a].
4.3.7 The Determinant Method of Calculating the Huckel c’s€
and Energy Levels................................................
An older method of obtaining the coefficients and energy levels from the secular
equations (Eqs.4.49for a two-basis-function system) utilizes determinants rather
than matrices. The method is much more cumbersome than the matrix diagonaliza-
tion approach ofSection 4.3.4, but in the absence of cheap, readily-available
computers (matrix diagonalization is easily handled by a personal computer) its
erstwhile employment may be forgiven. It is outlined here because traditional
presentations of the SHM [ 21 ] use it.
Consider again the secular equations4.49:
ðH 11 "HS 11 Þc 1 þðH 12 "ES 12 Þc 2 ¼ 0
ðH 21 "HS 21 Þc 1 þðH 22 "ES 22 Þc 2 ¼ 0By considering the requirements for nonzero values ofc 1 andc 2 we can find
how to calculate thec’s and the molecular orbital energies (since the coefficients are
weighting factors that determine how much each basis function contributes to the
MO, zeroc’s would mean no contributions from the basis functions and hence no
MOs; that would not be much of a molecule). Consider the system of linear
equations
146 4 Introduction to Quantum Mechanics in Computational Chemistry