Computational Chemistry

and

c 21 ¼

ffiffiffi

2

p

c 11 ¼

1

ffiffiffi

2

p

and

c 31 ¼c 11 ¼ 1

By substituting into the secular equations4.83theEvalues forc 2 andc 3 we
could find the ratios of thec’s forc 2 andc 3 and with the aid of the orthonormaliza-
tion equation analogous to Eq.4.88we could get the actual values ofc 12 ,c 22 ,c 32
andc 13 ,c 23 , andc 33. Although this somewhat clumsy way of finding thec’s from
the energy levels was streamlined (see e.g. [ 21 d]), the determinant method has been
replaced by matrix diagonalization implemented in a computer program.

4.4 The Extended Huckel Method€

4.4.1 Theory............................................................

In thesimpleH€uckel method, as in all modern molecular orbital methods, a Fock
matrix is diagonalized to give coefficients (which with the basis set give the wave-
functions of the molecular orbitals) and energy levels (i.e. molecular orbital energies).
The SHM and the extended H€uckel method (EHM, extended H€uckel theory, EHT)
differ in how the elements of the Fock matrix are obtained and how the overlap matrix
is treated. The EHM was popularized and widely applied by Hoffmann^28 [ 56 ],
although earlier work using the approach had been done by Wolfsberg and Helmholz
[ 57 ]. We now compare point by point the SHM and the EHM.

4.4.1.1 Simple Huckel Method€

1.Basis set is limited to p orbitals. Each element of the Fock matrixHis an integral
that represents an interaction between two orbitals. The orbitals are in almost all
cases a set ofporbitals (usually carbon 2p) supplied by ansp^2 framework, with
theporbital axes parallel to one another and perpendicular to the plane of the
framework. In other words, the set of basis orbitals – thebasis set– is limited (in

(^28) Roald Hoffmann, born Zloczow, Poland, 1937. Ph.D. Harvard, 1962, Professor, Cornell. Nobel
Prize 1981(shared with Kenichi Fukui; Section 7.3.5) for work with organic chemist Robert B.
Woodward, showing how the symmetry of molecular orbitals influences the course of chemical
reactions (the Woodward–Hoffmann rules or the conservation of orbital symmetry). Main expo-
nent of the extended H€uckel method. He has written poetry, and several popular books on
chemistry.
152 4 Introduction to Quantum Mechanics in Computational Chemistry