Computational Chemistry

attempted transition metal parameterization of PM3, PM3(tm), and the successors
to PM3, namely PM5 and PM6.

6.2.5.4 MNDO

MNDO [ 37 ], a modified NDDO (Section6.2.5) method, was reported in 1977 [ 38 ].
MNDO is conveniently explained by reference to CNDO (Section6.2.3). MNDO is
a general geometry method with a minimal valence basis set of Slater-type orbitals.
The Fock matrix elements are calculated using Eq.6.1¼5.82. We discuss the core
and two-electron integrals in the same order as for CNDO.
The core integralsHcorerArA, with the same orbitalfrtwice on the same atom A are
calculated using Eq.6.10. Unlike the case in CNDO, whereUrris found from ioni-
zation energies (CNDO/1) or ionization energies and electron affinities (CNDO/2),
in MNDOUrris one of the parameters to be adjusted. The integrals in the
summation termVABare evaluated similarly to the CNDO/2 method from a two-
electron integral (see below) involvingfrAand the valencesorbital on atom B:

frAð 1 ÞjV^BjfrAð 1 Þ

¼"CBðfrfrjsBsBÞ (6.14)

The core integralsHcorerAsAwith different orbitalsfrandfs, on the same atom A
are not simply taken as being proportional to the overlap integral, as in CNDO
(Eq.6.12), but rather are also (like the case of both orbitals on the same atom)
evaluated from Eq.6.10, which in this case becomes

HrcoreAsA¼ frAð 1 ÞjT^þV^AjfsAð 1 Þ

þ frAð 1 ÞjV^BjfsAð 1 Þ

þ frAð 1 ÞjV^CjfsAð 1 Þ

þ)))

¼Ursþ

X

B 6 ¼A

frAð 1 ÞjV^BjfsAð 1 Þ

ð 6 : 15 Þ

The first term is zero from symmetry [ 39 ]. Each integral of the summation term
is again evaluated, as in CNDO/2, from a two-electron integral:

frAð 1 ÞjV^BjfsAð 1 Þ

¼"CBðfrAfsAjsBsBÞ (6.16)

The core integralsHcorerAsBwith different orbitalsfrandfs, on different atoms A
and B are taken, as in CNDO (cf. Eq.6.12), to be proportional to the overlap
integral betweenfrandfs, where again the proportionality constant is the arith-
metic mean of parameters for atoms A and B:

HrcoreAsB¼

1

2

ðbrAþbsBÞhifrð 1 Þjfsð 1 Þ r 6 ¼s (6.17)

The overlap integral is calculated from the basis functions although the over-
lap matrix is taken as a unit matrix as far as the Roothaan-Hall equations go
(Section6.2.2). These core integrals are sometimes called core resonance integrals.

404 6 Semiempirical Calculations