Computational Chemistry

TheSmatrix is theoverlap matrix, whose elements areoverlap integrals Sij
which are a measure of how well pairs of basis functions (roughly, atomic orbitals)
overlap. Perfect overlap, between identical functions on the same atom, corre-
sponds toSii¼1, while zero overlap, between different functions on the same
atom or well-separated functions on different atoms, corresponds toSij¼0.
The diagonalematrix is an energy-levels matrix, whose diagonal elements are
MO energy levelsei, corresponding to the MOsci. Eacheiis ideally the negative of
the energy needed to remove an electron from that orbital, i.e. the negative of the
ionization energy for that orbital. Thus it is ideally the energy of an electron
attracted to the nuclei and repelled by the other electrons, relative to the energy
of that electron and the corresponding ionized molecule, infinitely separated from
one another. This is seen by the fact that photoelectron spectra correlate well with
the energies of the occupied orbitals, in more elaborate (ab initio) calculations [ 26 ].
In simple H€uckel calculations, however, the quantitative correlation is largely lost.
Now suppose that the basis functionsfhad these properties (theHand S
integrals, involvingf, are defined in Eqs.4.44):

S 11 ¼ 1

S 12 ¼S 21 ¼ 0

S 22 ¼ 1

(4.56)

More succinctly, suppose that

Sij¼dij (4.57)

wheredijis the Kronecker delta (Leopold Kronecker, German mathematician, ca.
1860) which has the property of being 1 or 0 depending on whetheriandjare the
same or different. Then theSmatrix (Eqs.4.55) would be

S¼

10

01



(4.58)

Since this is a unit matrix Eq.4.54would become

HC¼Ce (4.59)

and by multiplying on the right by the inverse ofCwe get

H¼CeC"^1 '(4:60)

So from the definition of matrix diagonalization, diagonalization of theHmatrix
will give theCand the«matrices, i.e. will give the coefficientscand the MO

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 125