Potential-harmonic expansion for atomic wave functions

Abstract

One way to reduce the large degeneracy of the hyperspherical-harmonic basis for solving fewand many-body bound-state problems is to introduce an optimal basis truncation called the potentialharmonic (PH) basis. In this paper we introduce various potential-harmonic truncation schemes and assess their accuracies in predicting the energies of the helium and H ground states and the excited 2 'S level of the helium atom. We first find that the part of the PH basis that accounts for one-body correlations gives a better ground-state energy for He than Hartree-Fock (2.8790 a.u. versus 2.8617 for Hartree-Fock and 2.9037 exact). When an orthogonal complement is introduced to the basis to account for e-e correlations, we find that the error in the binding energy is 0.00025 a.u. , and 0.00015 a.u. for ground-state and excited helium, and 0.00035 a.u. for H . Furthermore, the PH truncation is about 99.9% accurate in accounting for contributions coming from large values of the global angular momentum. This PH scheme is also much more accurate than previous versions based on the Faddeev equations. The present results indicate that the PH truncation can render the hyperspherical-harmonic method useful for systems with X)3. PACS number(s): 31.15.+q

Stability of few-charge systems in quantum mechanics

“…In practice, the number of equations grows very rapidly as L is increased. Several approximation schemes have been proposed, such as "potential harmonics" [257] and "adiabatic approximation" [255,259]. An interesting variant is proposed in [260].…”

“…See, e.g., [72,255,256,257,258], and the detailed review by Lin [259] for details and references. In the accurate calculation by Mandelzweig et al, the method is compared with other variational methods [72].…”

Green function and scattering amplitudes in many-dimensional space

“…We therefore expect that the calculation with the HA basis using PH, what reduces drastically the number of coupled hyperspherical equations, would be of a similar accuracy as the one of Lin using the ZB basis or the one of Klar using the S basis. On the other hand, since the PH basis is complete for describing the asymptotic behaviour of the two-body systems and in particular the bound states, we expect that the eigenpotential would be obtained with a very good accuracy for r --* ov and also at short distances, as the binding energy, closely related to the shape of the potential for r rather small, is well reproduced; furthermore, the accuracy improves for excited states [35].…”

“…In collaboration with Haftel and Larsen we investigated the case of atoms with two electrons in an S-state [35]. We first assumed that the two electrons do not interact with each other and found that the potential basis is complete.…”

“…[35] to be compared with the exact value E0 = -0.5275 a.u. We therefore expect that the calculation with the HA basis using PH, what reduces drastically the number of coupled hyperspherical equations, would be of a similar accuracy as the one of Lin using the ZB basis or the one of Klar using the S basis.…”