Neglect of three and four center integrals

Continuing with the neglect of differential overlap, all two-electron integrals involving charge clouds arising from the overlap of two atomic orbitals on different centers are ignored. Since no rotation can convert a two center two-electron integral into a set of integrals involving three and four center terms, rotational invariance is not compromised by this approximation. Rotational invariance is present if the calculated observables [ ΔHf, Dipole, I.P., etc] are not dependent on the orientation of the system. The effects of this approximation on the Roothaan equations are as follows:

In the Fock matrix, if ψμ and ψμ are on different centers the NDDO matrix element Fμνα reduces to

Fμνα = Hμνα +ΣλAΣσBPλσα  <μν|λσ>.


Equivalent expressions exist for Fμνβ and Pμνβ. Thus no Coulombic terms are present in the two-center Fock matrix elements.

If ψμ and ψμ are different but on the same center, then, since a minimal basis set is being used, all integrals of the type <μν|λσ> are zero by the orthogonality of the atomic orbitals unless μ = ν and λ = σ, or μ = λ and ν = σ. The off-diagonal one-center NDDO Fock matrix elements become:

Fμνα = Hμν +2Pμνα+β <μν|μν> - Pμνα (<μν|μν> + <μμ|νν>).

If ψμ is the same as ψμ, then, because of the symmetry of the two-electron integrals, the diagonal NDDO Fock matrix elements reduce to:

 Fμμα = Hμμ + ΣνA(Pμμα + β <μμ|νν> - Pννa <μν|μν>) +  ΣBΣλ BΣσBPλσα + β< μμ|λσ>.