Essentials of Computational Chemistry

5.5 BASIC NDDO FORMALISM 143
error for ionization potentials was 0.7 eV (46 molecules), for heavy-atom bond lengths
0.022 ˚A (81 molecules), and for dipole moments 0.45 D (31 molecules). While mean errors
of this size exceed what would be tolerated today, they were unprecedentedly small in 1975.
Dewar’s subsequent work on other semiempirical models (see below) rendered MINDO/3
effectively obsolete, but its historical importance remains unchanged.
A modified INDO model that is not entirely obsolete is the symmetric orthogonalized
INDO (SINDO1) model of Jug and co-workers, first described in 1980 (Nanda and
Jug 1980). The various conventions employed by SINDO1 represent slightly different modifications
to INDO theory than those adopted in the MINDO/3 model, but the more fundamental
difference is the inclusion of d functions for atoms of the second row in the periodic table.
Inclusion of such functions in the atomic valence basis set proves critical for handling hypervalent
molecules containing these atoms, and thus SINDO1 performs considerably better for
phosphorus-containing compounds, for instance, than do other semiempirical models that
lack d functions (Jug and Schulz 1988).
5.5 Basic NDDO Formalism
The INDO model extends the CNDO model by adding flexibility to the description of the
one-center two-electron integrals. In INDO, however, there continues to be only a single
two-center two-electron integral, which takes on the value γAB irrespective of which orbitals
on atoms A and B are considered. As already noted, this can play havoc with the accurate
representation of lone pair interactions.
The neglect of diatomic differential overlap (NDDO) method relaxes the constraints on
two-center two-electron integrals in a fashion analogous to that for one-center integrals in
the INDO method. Thus, all integrals (µν|λσ ) are retained provided µ and ν are on the
same atomic center and λ and σ are on the same atomic center, but not necessarily the
center hosting µ and ν. How many different integrals are permitted? The order of µ and
ν does not affect the value of the integral, so we need only worry about combinations, not
permutations, in which case there are 10 unique combinations of s, p x ,p y ,andp z . With
10 unique combinations on each atom, there are 100 possible combinations of combinations
for the integrals. If we include d functions, the number of unique integrals increases to 2025.
Although these numbers seem large, this is still a considerable improvement over evaluating
every possible integral, as would be undertaken in ab initio HF theory. Most modern
semiempirical models are NDDO models. After examining the differences in their formulation,
we will examine their performance characteristics in some detail in Section 5.6.
5.5.1 MNDO
Dewar and Thiel (1977) reported a modified neglect of differential overlap (MNDO) method
based on the NDDO formalism for the elements C, H, O, and N. With the conventions
specified by NDDO for which integrals to keep, which to discard, and how to model oneelectron
integrals, it is possible to write the NDDO Fock matrix elements individually for