on different atoms are taken as being proportional to the overlap integral of the
relevant orbitals:
HrcoreAsB¼bABhifrð 1 Þjfsð 1 Þ r 6 ¼s (6.12)The overlap integral hereis calculated from the basisfunctions,although(asfor
the PPP method, above) the overlap matrixis simply taken as a unit matrix as far as
the matrix Roothaan-Hall equations are concerned. The proportionality constant
bABis taken as the arithmetic mean of parameters for atoms A and B, these
parameters being those that give the best fit of CNDO MO coefficients to those
of minimal-basis-set ab initio calculations. Since different AOs on the same atom
are orthogonal, when A¼Btheseintegralsarezero.NotethatcalculatingbAB
from a best-fit to minimal-basis-set abinitiocalculationsmeansthatCNDO
parameterization isnot purely empirical, but rather, to some extent attempts to
match (low-level) ab initio results. This is a weakness of CNDO and a potential
weakness of its successors INDO and NDDO (below). As repeatedly emphasized
by Dewar, this deficiency was avoided in his methods (Section6.2.5.1)byconsis-
tently parameterizing to matchexperiment.
As with the PPP method, the two-electron repulsion integrals are evaluated by
applying the ZDO approximation to all different orbitalsrands(Eq.6.6). Thus the
two-electron integrals reduce to (rs|tu)¼drsdtu(rr|tt) (Eq.6.8), i.e. only one- and
two-center two-electron integrals are considered. All one-center integrals on the
same atom A are given the same value,gAA, and all two-center integrals between
atoms A and B are given the same value,gAB. These integrals are calculated from
valencesSlater functions on A and B.
CNDO/2 differs from CNDO/1 in two modifications to theHcorerArAmatrix ele-
ments (Eq. 6.1): (1) to account better for both ionization energy and electron affinity,
Urris evaluated not just from ionization energy but as a kind of average of ionization
energy and electron affinity, and (2) the integrals in theVABterm are calculated from
the two-electron integralsgAB, asVAB¼"CBgAB. This latter evaluation amounts to
neglecting so-calledpenetration integrals; these integrals make nonbonded atoms
attract one another, and cause bond lengths to be too short and bond energies to be
too large [ 15 – 18 ]. CNDO energies are valence electron electronic energiesESE, or
electronic energies plus core–core repulsions,EtotalSE, ifVCCis added (Eq.6.2).
CNDO is now obsolete, having served its purpose as a precursor to the more
effective general-geometry methods INDO and NDDO (below).
6.2.4 The Intermediate Neglect of Differential Overlap (INDO)
Method
INDO [ 19 ] goes beyond CNDO by curtailing the application of the ZDO approxi-
mation. Instead of applying it to all different (r 6 ¼s) atomic orbitals in the two-
electron integrals (Eq.6.6), as in the PPP and CNDO methods, in INDO ZDO is not
6.2 The Basic Principles of SCF Semiempirical Methods 399