Vector coupling coefficients for calculations of transition-metal atoms and ions by the SCF coupling operator method

We derived the necessary conditions to which the vector coupling coefficients (VCC ) a and b describing atomic L,S-multiplets of the configurations d^N (1 ≤ N ≤ 9), should satisfy. Special attention is paid to the states of non-Roothaan type for which VCC depend on the choice of degenerate d-orbitals basis set determined within the accuracy up to an orthogonal transformation u. It is shown that for such states the direct sum of matrices ‖a‖ and ‖b‖ must be the non-symmetric matrix. Obtained VCC were used for the ab initio calculations (basis set (14s9p5d)/8s4p2d] from 15]) on first-row transition atoms (from Sc to Cu) to compare to similar calculations 16], in which the Peterson's VCC have been used, and with calculations 15] carried out by the atomic SCF program 4] as well.