Highly accurate calculation of the hyperpolarizabilities of hydrogenic atoms interacting with screened Coulomb potentials: a variation–perturbation approach

Variational expressions for functionals yielding the second-order and fourth-order corrections to the nondegenerate eigenenergies of a Hamiltonian subject to a given perturbation have been derived. Discrete basis sets consisting of the suitably chosen eigenfunctions of the unperturbed Hamiltonian along with a nonlinear variational parameter can be used to evaluate the stationary functional quite accurately. The method has been applied conveniently to calculate the dipole hyperpolarizabilities of the hydrogenic atoms interacting with screened Coulomb potentials (SCP) quite accurately. Three SCPs are considered, namely the static screened Coulomb potential, the exponential cosine screened Coulomb potential and the generalized exponential cosine screened Coulomb potential. It is found that a basis set consisting of 20 eigenfunctions of the p-states of the hydrogenic atoms yields stationary functional corresponding to the second-order correction to the energy, whereas a basis set consisting of 20 d-state eigenfunctions and 19 s-state eigenfunctions of the hydrogenic atoms produces stationary functional corresponding to the fourth-order correction to the energy. Moreover, the stationary values of the functionals are highly accurate (up to 14th place of decimal) and converge rapidly with the increase in the number of terms in the basis sets.