orbital). The postulate behind the LCAO approach is that an MO can be “synthe-
sized” by combining simpler functions, now calledbasis functions; these functions
comprise a basis set. This way of calculating MOs is based on suggestions of
Pauling (1928) [ 32 ] and Lennard–Jones^25 (1929) [ 33 ]. Perhaps the most important
early applications of the LCAO method were the simple H€uckel method (1931)
[ 19 ], in whichpAOs orbitals are combined to givepAOs (probably the first time
that the MOs of relatively big molecules were represented as a weighted sum of
AOs with optimized coefficients), and the treatment of all the lower electronic
states of the hydrogen molecule by Coulson^26 and Fischer (1949) [ 34 ]. The basis
functions are usually located on the atoms of the molecule, and may or may not (see
the discussion of basis functions in Section 5.3) be conventional atomic orbitals.
The wavefunction can in principle be approximated as accurately as desired by
using enough suitable basis functions. In this simplified derivation of the H€uckel
method we at first consider a molecule with just two atoms, with each atom
contributing one basis function to the MO. Combining basis functions on different
atoms to give MOs spread over the molecule is somewhat analogous to combining
atomic orbitals on the same atom to give hybrid atomic orbitals (Section 4.3.2)[ 27 ].
The combination ofnbasis functions always givesnMOs, as indicated in Fig.4.11,
and we expect two MOs for the two-atomic-orbital diatomic molecule we are
using here.
Using the LCAO approximation
c¼c 1 f 1 þc 2 f 2 (4.41)wheref 1 andf 2 are basis functions on atoms 1 and 2, andc 1 andc 2 are weight-
ing coefficients to be adjusted to get the bestc, and substituting into Eq.4.40
we get
E¼
R
ðc 1 f 1 þc 2 f 2 ÞH^ðc 1 f 1 þc 2 f 2 Þdv
R
ðc 1 f 1 þc 2 f 2 Þ^2 dv(4.42)
If we multiply out the terms in Eq.4.42we getE¼
c^21 H 11 þ 2 c 1 c 2 H 12 þc^22 H 22
c^21 S 11 þ 2 c 1 c 2 S 12 þc^22 S 22(4.43)
(^25) John Edward Lennard-Jones, born Leigh, Lancaster, England, 1894. Ph.D. Cambridge, 1924.
Professor Bristol. Best known for the Lennard–Jones potential function for nonbonded atoms.
Died Stoke-on-Trent, England, 1954.
(^26) Charles A. Coulson, born Worcestershire, England, 1910. Ph.D. Cambridge, 1935. Professor of
theoretical physics, King’s College, London; professor of mathematics, Oxford; professor of
theoretical chemistry, Oxford. Died Oxford, 1974. Best known for his book "Valence" (the 1st
Ed., 1952).
120 4 Introduction to Quantum Mechanics in Computational Chemistry