Numerical evaluation of overlap integrals between atomic orbitals

The algorithm evaluating the overlap integrals for the numerical atomic orbitals is presented. The described algorithm is general and is based on the adaptive integration over the rectangular domain in cylindrical coordinate system. It is designed to apply for Kohn-Sham eigenproblem, which is solved by Linear Combination of Atomic Orbitals with the finite support. Double application of Wigner D-matrix relation, reduces the problem to specific case, where the overlap integral is transformed from R3 to R2. Assuming the finite support of the numerical atomic orbitals, the problem is reformulated in cylindrical coordinate system and simplified to the integration over the rectangular domain. In order to perform the numerical integration over the rectangular domain, the adaptive scheme is applied, based on the seven and five order quadratures. The exemplary results for selected atomic orbitals are presented.