Essentials of Computational Chemistry

136 5 SEMIEMPIRICAL IMPLEMENTATIONS OF MO THEORY
the relevant filled and empty orbitals in spectroscopic transitions. In the 21st century, such
a molecular problem has become amenable to more accurate treatments, so the province of
EHT is now primarily very large systems, like extended solids, where its speed makes it
a practical option for understanding band structure (a ‘band’ is a set of MOs so densely
spread over a range of energy that for practical purposes it may be regarded as a continuum;
bands derive from combinations of molecular orbitals in a solid much as MOs derive from
combinations of AOs in a molecule).
Thus, for example, EHT has been used by Genin and Hoffmann (1998) to characterize the
band structure of a series of organic polymers with the intent of suggesting likely candidates
for materials exhibiting organic ferromagnetism. Certain polymers formed from repeating
heterocycle units having seven π electrons were identified as having narrow, half-filled
valence bands, such bands being proposed as a necessary, albeit not sufficient, condition for
ferromagnetism.
Note that one drawback of EHT is a failure to take into account electron spin. There is
no mechanism for distinguishing between different multiplets, except that a chemist can, by
hand, decide which orbitals are occupied, and thus enforce the Pauli exclusion principle.
However, the energy computed for a triplet state is exactly the same as the energy for the
corresponding ‘open-shell’ singlet (i.e., the state that results from spin-flip of one of the
unpaired electrons in the triplet) – the electronic energy is the sum of the occupied orbital
energies irrespective of spin – such an equality occurs experimentally only when the partially
occupied orbitals fail to interact with each other either for symmetry reasons or because they
are infinitely separated.
5.3 CNDO Formalism
Returning to the SCF formalism of HF theory, one can proceed in the spirit of an effective
Hamiltonian method by developing a recipe for the replacement of matrix elements in the
HF secular equation, Eq. (4.53). One of the first efforts along these lines was described by
Pople and co-workers in 1965 (Pople, Santry, and Segal 1965; Pople and Segal 1965). The
complete neglect of differential overlap (CNDO) method adopted the following conventions:
1. Just as in EHT, the basis set is formed from valence STOs, one STO per valence orbital.
In the original CNDO implementation, only atoms having s and p valence orbitals were
addressed.
2. In the secular determinant, overlap matrix elements are defined by
where δ is the Kronecker delta.
Sµν = δµν
(5.4)
3. All two-electron integrals are parameterized according to the following scheme. First,
define
(µν|λσ ) = δµνδλσ (µµ|λλ) (5.5)