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 [ ΔH_f, 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 ψ_μ and ψ_μ are on different centers the NDDO matrix element F_μν^α reduces to

F_μν^α = H_μν^α +Σ_λ^AΣ_σ^BP_λσ^α <μν|λσ>.

Equivalent expressions exist for F_μν^β and P_μν^β. Thus no Coulombic terms are present in the two-center Fock matrix elements.

If ψ_μ and ψ_μ 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:

F_μν^α = H_μν +2P_μν^α+β <μν|μν> - P_μν^α (<μν|μν> + <μμ|νν>).

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

F_μμ^α = H_μμ + Σ_ν^A(P_μμ^{α + β} <μμ|νν> - P_ν_ν^a <μν|μν>) + Σ_BΣ_λ ^BΣ_σ^BP_λσ^{α + β}< μμ|λσ>.