Essentials of Computational Chemistry

144 5 SEMIEMPIRICAL IMPLEMENTATIONS OF MO THEORY
inspection. For the most complex element, a diagonal element, we have
Fµµ = Uµ −
ZB(µµ|sBsB) +
Pνν (µµ|νν) − 1
2 (µν|µν)
B=A
ν∈A
+
Pλσ (µµ|λσ ) (5.12)
B
λ∈B σ ∈B
where µ is located on atom A. The first term on the r.h.s. is the atomic orbital ionization
potential, the second term the attraction to the other nuclei where each nuclear term is
proportional to the repulsion with the valence s electron on that nucleus, the third term
reflects the Coulomb and exchange interactions with the other electrons on atom A, and the
final term reflects Coulomb repulsion with electrons on other atoms B.
An off-diagonal Fock matrix element for two basis functions µ and ν on the same atom A
is written as
Fµν =−
3 1
ZB(µν|sBsB) + Pµν (µν|µν) −
2 2
B=A
(µµ|νν)
+
Pλσ (µν|λσ )
B λ∈B σ ∈B
(5.13)
where each term on the r.h.s. has its analogy in Eq. (5.12). When µ is on atom A and ν on
atom B, this matrix element is written instead as
Fµν = 1
2 (βµ + βν)Sµν − 1
2
Pλσ (µλ|νσ) (5.14)
λ∈A σ ∈B
where the first term on the r.h.s. is the resonance integral that encompasses the one-electron
kinetic energy and nuclear attraction terms; it is an average of atomic resonance integrals
‘β’ times the overlap of the orbitals involved. The second term on the r.h.s. captures favorable
exchange interactions. Note that the MNDO model did not follow Dewar’s MINDO/3
approach of having β parameters specific to pairs of atoms. While the latter approach allowed
for some improved accuracy, it made it quite difficult to add new elements, since to be
complete all possible pairwise β combinations with already existing elements would require
parameterization.
The only point not addressed in Eqs. (5.12) to (5.14) is how to go about evaluating all
of the necessary two-electron integrals. Unlike one-center two-electron integrals, it is not
easy to analyze spectroscopic data to determine universal values, particularly given the large
number of integrals not taken to be zero. The approach taken by Dewar and co-workers was to
evaluate these integrals by replacing the continuous charge clouds with classical multipoles.
Thus, an ss product was replaced with a point charge, an sp product was replaced with
a classical dipole (represented by two point charges slightly displaced from the nucleus
along the p orbital axis), and a pp product was replaced with a classical quadrupole (again
represented by point charges). The magnitudes of the moments, being one-center in nature,
are related to the parameterized integrals in Eq. (5.11). By adopting such a form for the
integrals, their evaluation is made quite simple, and so too is evaluation of their analytic
derivatives with respect to nuclear motion.