associated with a wavelengthl¼h/p, wherepis the momentum. The four quantum
numbers follow naturally from the wave mechanical treatment and the model does
not break down for atoms beyond hydrogen.
H€uckel was the first to apply QM to species significantly more complex than the
hydrogen atom. The H€uckel approach is treated nowadays within the framework of
the concept of hybridization: thepelectrons inporbitals are taken into account and
theselectrons in ansp^2 framework are ignored. Hybridization is a purely mathe-
matical convenience, a procedure in which atomic (or molecular) orbitals are
combined to give new orbitals; it is analogous to the combination of simple vectors
to give new vectors (an orbital is actually a kind of vector).
The simple H€uckel method (SHM, simple H€uckel theory, SHT, H€uckel molecu-
lar orbital method, HMO method) starts with the Schr€odinger equation in the form
Hˆc¼EcwhereHˆis a Hamiltonian operator,cis a MO wavefunction andEis the
energy of the system (atom or molecule). By expressingcas a linear combination
of atomic orbitals (LCAO) and minimizingEwith respect to the LCAO coefficients
one obtains a set of simultaneous equations, thesecular equations. These are
equivalent to a single matrix equation,HC¼SC«;His an energy matrix, the
Fock matrix,Cis the matrix of the LCAO coefficients,Sis the overlap matrix and
«is a diagonal matrix whose nonzero, i.e. diagonal, elements are the MO energy
levels. The columns ofCare called eigenvectors and the diagonal elements of«are
called eigenvalues. By the drastic approximationS¼ 1 ( 1 is the unit matrix), the
matrix equation becomesHC¼C«, i.e.H¼C«C"^1 which is the same as saying
that diagonalization ofHgivesCand«, i.e. gives the MO coefficients in the LCAO,
and the MO energies. To get numbers forHthe SHM reduces all the Fock matrix
elements toa(the coulomb integral, for AOs on the same atom) andb(the bond
integral or resonance integral, for AOs not on the same atom; for nonadjacent atoms
bis set¼0). To get actual numbers for the Fock elements,aandbare defined as
energies relative toa, in units of |b|; this makes the Fock matrix consist of just 0’s
and"1’s, where the 0’s represent same-atom interactions and nonadjacent-atom
interactions, and the"1’s represent adjacent-atom interactions. The use of just two
Fock elements is a big approximation. The SHM Fock matrix is easily written down
just by looking at the way the atoms in the molecule are connected.
Applications of the SHM include predicting:
The nodal properties of the MOs, very useful in applying the Woodward–Hoffmann
rules.
The stability of a molecule based on its filled and empty MOs, and its delocalization
energy or resonance energy, based on a comparison of its totalp-energy with
that of a reference system. The pattern of filled and empty MOs led to H€uckel’s
rule (the 4n þ2 rule) which says that planar molecules with completely
conjugatedporbitals containing 4nþ2 electrons should be aromatic.
Bond orders and atom charges, which are calculated from the AO coefficients of the
occupiedpMOs (in the SHM LCAO treatment,pAOs are basis functions that
make up the MOs).
166 4 Introduction to Quantum Mechanics in Computational Chemistry