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Three- and four-center Coulomb integrals in the solid spherical harmonic Gaussian basis are solved by expansion in terms of two-center integrals. The two-electron Gaussian product rule, coupled with the addition theorem for solid spherical harmonics, reduces four-center Coulomb integrals into a linear combination of two-center Coulomb integrals and one-center overlap integrals. With this approach, three- and four-center Coulomb integrals can be reduced to the same form of two-center integrals. Resulting two-center Coulomb integrals can be further simplified into a simpler form, which can be related to the Boys function. Multi-center Coulomb integrals are solved hierarchically: simple two-center Coulomb integrals are used for calculation of more complicated two-center Coulomb integrals, which are used in the calculation of multicenter integrals. [ABSTRACT FROM AUTHOR]

The Gaussian product rule for two-electron wave functions is introduced. The two-electron Gaussian product rule enables a new way for solving two-electron integrals. The solution is demonstrated with an example of the two-center two-electron integral in solid harmonic Gaussian basis. The solution is obtained by expanding inverse inter-electron separation and integrating in spherical coordinates. The resulting integral separates into four integrals, three of which are straightforward to solve. The remaining integral can be solved with Boys-like functions. It is demonstrated that the solution can deliver results with accuracy comparable with that of the McMurchie–Davidson scheme. [ABSTRACT FROM AUTHOR]