Knowledge Bank

Four center integrals of a general two electron irregular solid spherical harmonic operator, i.e. $Qlm(r12)=4\pi 2l+1Ylm(r^12)r12-(l+1)$, over the homogeneous solid spherical harmonic Gaussian type functions, i.e. $ri\alpha 2n\alpha +lmYl\alpha m\alpha (r^i\alpha )\exp (-\alpha r1\alpha 2)(i=Ior2;\alpha =a,b,c,ord)$, have been evaluated analytically. When l = 2, 1 or 0 the operator $Q1m(r12)$ is respectively the operator for spin-spin interaction, spin-other-orbit interaction and Coulomb repulsive interaction. Through coincidence of centers, the four-center integral is first transformed into a linear combination of two-center integrals which are then integrated analytically by Fourier transformation convolution theorem. The integral results are in terms of nuclear wave functions of the relative coordinates. All of the nuclear wave functions are in the format of spherical Laguerre Gaussian type function, except one term which is the product of solid spherical harmonic and F-function (error type function). The expressions, which are similar to that obtained by Talmi transformation, are simpler than the previous results obtained by expansion $method.1$ Two-center and three-center overlap, three-center Coulomb repulsion and three-center nuclear attraction integrals needed in the context of density functional formalism have also been integrated explicitly.