Computational Chemistry

The two-electron integrals are evaluated applying ZDO (Section6.2.1) within
the framework of the NDDO approximation (Section6.2.5). As with the PPP
(Section6.2.2) and CNDO (Section6.2.3) methods, this makes all two-electron
integrals become (rs|tu)¼drsdtu(rr|tt), i.e. only one- and two-center two-electron
integrals are nonzero. The one-center integrals are evaluated from valence-
state ionization energies. The two-center integrals are evaluated from the one-center
integrals and the separation of the nuclei by an involved procedure in which
the integrals are expanded as sums of multipole-multipole interactions [38a, 40 ]
that make the two-center integrals show correct limiting behavior at zero and
infinite separation.
As in CNDO, in MNDO the penetration integrals are neglected (Section6.2.3,
CNDO/2). A consequence of this is that the core–core repulsions (VCCin Eq.6.2)
cannot be realistically calculated simply as the sum of pairs of classical electrostatic
interactions between point charges centered on the nuclei. Instead, Dewar and
coworkers chose [ 38 ] the expression

VCC¼

X

B>A

X

A

½CACBðsAsBjsBsBÞþfðRABފ (6.18)

whereCAandCBare the core charges of atoms A and B andsAandsBare the
valencesorbitals on A and B (the two-electron integral in Eq.6.18is actually
approximately proportional to 1/RAB, so there is some connection with the simple
electrostatic model). Thef(RAB) term is a correction increment to make the result
come out better; it depends on the core charges and the valencesfunctions on A and
B, their separationR, and empirical parametersaAandaB:

fðRABÞ¼CACBðsAsAjsBsBÞðe"aARABþe"aBRABÞ (6.19)

The above mathematical treatment constitutes the creation of theformof the
semiempirical equations. To actually use these equations, they must be parameter-
ized somehow (as stressed above, Dewar used experimental data). This is analogous
to the situation in molecular mechanics (Chapter 3), where a force field, defined by
the form of the functions used (e.g. a quadratic function of the amount by which a
bond is stretched, for the bond-stretch energy term) is constructed, and must then be
parameterized by inserting specific quantities for the parameters (e.g. values for the
stretching force constants of various bonds). For each kind of atom A (a maximum
of) six parameters is needed:

1. The kinetic-energy-containing termUrr of Eq. 6.10 (as explained above,
   this CNDO equation is also used in MNDO to evaluateHcorerArA) wherefrAis a
   valencesAO.


2. The termUrrof Eq.6.10wherefrAis a valencepAO.


3. The parameterzin the exponent of the Slater function (e.g. Section5.3.2,
   Fig.5.12) for the various valence AOs (MNDO uses the samezfor thesand
   pAOs).

6.2 The Basic Principles of SCF Semiempirical Methods 405