Two-electron integrals contributions

So only the two-electron integrals with p > v, and X > a and [pv] > [Xa] need to be computed and stored. Dpv,Xa only appears in Gpv, and Gvp, whereas the original two-electron integrals contribute to other matrix elements as well. So it is much easier to form the Fock matrix by using the supermatrix D and modified density matrix P than the regular format of the two-electron integrals and standard density matrix. 

The overlap integrals are first calculated, and the matrix inverted. The one- and two-electron integral contributions to the electronic energy are summed as they... 

The two electron integrals contribute Jy for each pair of electrons, and (—K ) for each pair of parallel spin. 

Only four-index two-electron integrals contribute to the spin-orbit coupling matrix element ... 

The derivatives of SCF and MCSCF energies with respect to nuclear coordinates present no new tradeoffs. In some ways, calculation of derivatives is simpler than the computation of the energy because the integral derivatives need not be stored since they can be contracted immediately with the density to form the contribution to the gradient. Since each two electron integral contributes to at most 12 nuclear coordinate derivatives, the cpu time grows only as the number of integrals 0(N ), not 0(N Na) where Na is the number of derivatives with respect to nuclear coordinates. MCSCF derivatives are similar, except that... 

If the five independent one-center two-electron integrals are expressed by symbols such as Gss, Gsp, defined above, then the Fock matrix element contributions from the one-center two-electron integrals are ... 

Thus, we can alternatively express the Coulomb contribution in equation (7-12) solely in terms of the basis functions as the following four-center-two-electron integrals (since the four basis functions qfl, r v, q/ r a can be attached to a maximum of four different atoms)... 

The right-hand side can be evaluated with mere 0(nWRl) arithmetic operations (nocc is the number of occupied spin orbitals) using just computationally tractable two-electron integrals. For atoms, nonzero contributions to this sum occur only from the RI basis functions with angular quantum numbers up to 3Locc, where Locc is the maximum angular quantum number of occupied spin orbitals. 

The SAMI parametrization [74] further extends the number of two-electron integrals included in the treatment. They are calculated first for the AOs taken as in the STO-3G Gaussian basis set, but then scaled using the distance dependent functions containing adjustable parameters. The SAMI method has been parametrized for the elements H, Li, C, N, O, F, Si, P, S, Cl, Fe, Cu, Br, and I. Unfortunately, this parametrization was never thoroughly published and studied. The same applies to the PM5 method [75] which is implemented only by a commercial software, without adequate explanation.2 Further refinement of the system of correcting Gaussian contributions to the interatomic interaction functions has been proposed in [71],... 

Single excitations and I ] differ by one pair of spin orbitals i,j. Contributions come from the one-electron spin-orbit integral and three-index two-electron integrals... 

Figure 1-2. Schematic diagrammatic representation of the E correction (Brandow skeletons). The horizontal lines represent the denominators, while the vertical bar separates the monomers A and B. The two-electron integral corresponding to the dotted interaction line is a Coulomb integral. The dashed interaction lines represent antisymmetric two-electron integrals of the monomers. Diagram (a) is the intermolecular perturbation theory form of the MP5 contribution s, diagram (d) of qQ(/7), while (b) and (c) are combinations of 7s T and E (I)...

---