Essentials of Computational Chemistry

5.3 CNDO FORMALISM 137
Thus, the only integrals that are non-zero have µ and ν as identical orbitals on the
same atom, and λ and σ also as identical orbitals on the same atom, but the second
atom might be different than the first (the decision to set to zero any integrals involving
overlap of different basis functions gives rise to the model name).
4. For the surviving two-electron integrals,
(µµ|λλ) = γAB
(5.6)
where A and B are the atoms on which basis functions µ and λ reside, respectively. The
term γ can either be computed explicitly from s-type STOs (note that since γ depends
only on the atoms A and B, (sAsA|sBsB) = (p A p A |sBsB) = (p A p A |p B p B ), etc.) or it can
be treated as a parameter. One popular parametric form involves using the so-called
Pariser–Parr approximation for the one-center term (Pariser and Parr 1953).
γAA = IPA − EAA
(5.7)
where IP and EA are the atomic ionization potential and electron affinity, respectively.
For the two-center term, the Mataga–Nishimoto formalism adopts
γAB =
γAA + γBB
2 + rAB(γAA + γBB)
(5.8)
where rAB is the interatomic distance (Mataga and Nishimoto 1957). Note the intuitive
limits on γ in Eq. (5.8). At large distance, it goes to 1/rAB, as expected for widely
separated charge clouds, while at short distances, it approaches the average of the two
one-center parameters.
5. One-electron integrals for diagonal matrix elements are defined by
µ
−1
2 ∇2 −
Zk
µ
=−IPµ −
(Zk − δZAZk )γAk
k
rk
k
(5.9)
where µ is centered on atom A. Equation (5.9) looks a bit opaque at first glance, but
it is actually quite straightforward. Remember that the full Fock matrix element Fµµ is
the sum of the one-electron integral Eq. (5.9) and a series of two-electron integrals. If
the number of valence electrons on each atom is exactly equal to the valence nuclear
charge (i.e., every atom has a partial atomic charge of zero) then the repulsive twoelectron
terms will exactly cancel the attractive nuclear terms appearing at the end of
Eq. (5.9) and we will recapture the expected result, namely that the energy associated
with the diagonal matrix element is the ionization potential of the orbital. The Kronecker
delta affecting the nuclear charge for atom A itself simply avoids correcting for a nonexistent
two-electron repulsion of an electron in basis function µ with itself. (Removing
the attraction to nuclei other than A from the r.h.s. of Eq. (5.9) defines a commonly
tabulated semiempirical parameter that is typically denoted Uµ.)