Analytical Evaluation of Multicenter Integrals needed in Molecular Quantum Mechanical Calculations of some Magnetic Tensors over Slater Type Basis Functions

In the quantum theory of atoms or molecules in magnetic fields of different origin that can be an external homogenous field and that created by the nuclear magnetic moment of a nucleus, some important magnetic tensor appears. The integrals due to magnetic perturbations appeared within coupled Hartree-Fock-Roothaan perturbation Gauge Invariant Atomic Orbitals or in Density Functional Theory (DFT) are analytically evaluated for Slater Type nsnp basis functions. Reduced analytical formulas are obtained using properties of the Levi-Civita tensor of rank 3, properties of rotation matrix and symmetry properties of field-independent atomic orbitals. The condensed formalism is expressed in term of auxiliary functions and integrals. The angular dependence of integrals is also discussed in detail. The multicenter integrals can be monocentric, bicentric or three centric. All monocentric and bicentric integrals with monocentric electron distribution can be calculated rigorously in spherical polar coordinates with the nucleus as origin. Bicentric integrals with bicentric electron distribution were also integrated using elliptic coordinates after an appropriate axis transformation. The three centre integrals with bicentric electron distribution were determined analytically within the London Type approximation. The resulting angular momentum and impulsion integrals whether dipolar or quadrupolar are all computed rigorously in elliptic coordinates.