Dynamic coordinate separation and state-dependent potentials. An analysis of localization in helium

Abstract

We have examined in detail the construction of state-dependent dynamic potentials (SDPs) based on an exact separation scheme of the wavefunction as a product of coupled conditional and marginal probability amplitudes. For particles of equal mass and for rather general forms of the kinetic energy operator in curvilinear coordinates, it is proven that the SDP becomes the expectation value of the total hamiltonian with respect to the conditional amplitude obtained from an assumed real wavefunction. The use of these SDPs for the description of localized wavefunctions is also discussed. As an example we have undertaken the analysis of SDPs arising from various separation schemes of Hylleraas coordinates for two simple wavefunctions approximating the ground state of helium. The separation of s  r 1 + r 2, t  r 2  r 1 and r 12 leads to SDPs exhibiting minima in the regions where localization of the wavefunction takes place. The possibility of an adiabatic breakup, Born—Oppenheimer type, of the problem is also considered. Our results show that strong “non-adiabaticity” is a distinct feature of all the separations we considered. The separation of r 1 is treated too as an example of a non-collective variable and the corresponding SDP resembles an effective Hartree-like one-electron potential. We finally sketch some possible applications of the present approach for the analysis of localization and non-adiabaticity in more complex situations in atomic and molecular cases.