Helium atom in an external electric field: Exact diagonalization

Abstract

An exact diagonalization method is applied to solve the quantum-mechanical problem of spinless helium atom in an external electric field of arbitrary magnitude. The basis set for two-electron problem is built from different pair combinations ψ nalama (αr a )ψ nblbmb (αr b ) of orthonormalized single-particle hydrogen-like wave functions ψ nml (r) belonging to any possibly bound states of the individual a-and b-electrons in the Coulomb central field renormalized by the scale parameter α > 0. Within the selected basis the matrix elements of the total Hamiltonian allows an exact analytical representation in the form of finite numerical sums. The diagonalization procedure is performed by Jacobi algorithm for N×N square Hermitian matrix built on the basis of dimension N = 25. The systematics and the numerical values of the low-lying energy levels at zero field are in good agreement with known experimental data. The field dependences of low-lying levels (Stark effect) and polarizability in the ground state of helium atom are presented. It is shown that even extremely high external fields lead only to shifting or splitting of existing low levels, without disturbance of their systematics. Typically, no new lowenergy excitation can be created under external electric field of moderate intensity. Radical reconstruction in spectrum of individual helium atoms can be expected in condensed helium phases where each atom is deeply affected by interaction fields from neighbors. This result should be taken into account at interpretation of electrodynamic experiments on superfluid helium. PACS: 31.15.ac High-precision calculations for few-electron (or few-body) atomic systems.

Low-energy excitations in helium-like dimer within an exact diagonalization approach

“…The Hamiltonian (1) describes an exact non-relativistic dynamics of the system, including internuclear motion. To take into account the dynamics of heavy repulsive nucleus we have to use a basis incorporate the functions of continuous spectrum, whereas the electronic part of the problem should be based on functions of discrete spectrum [12]. However, if we are interested in some specific applications of the problem restricted to interatomic interactions in condensed helium matter then the interatomic distance 0 R has a rather narrow range of variation, and the system can be treated on the qualitative level within Heitler-London approximation (HLA) [10].…”

“…where ( , ) lm Y ϑ ϕ are spherical harmonics in standard determination [17,18], and radial function ( ) nl r α  depends on radial coordinate renormalized by scale parameter [12] > 0 α , and radial function is…”

“…The case = 1 α in single-particle function corresponds to the simple hydrogen atom. For two-electron helium shell we have 0 = α − σ  , where 0 < < 1 σ is the screening parameter [12,18,19].…”

“…We use the basis (7) [12] built as direct composition of four five-component single-particle bases (8)…”

“…In this paper we present the formally exact solution for the non-relativistic quantum mechanical problem of helium-helium interaction treated within Born-Oppenheimer-Heitler-London approach [10,11] (four spinless electrons moving in the Coulomb field of two nuclei with 0 = 2  spaced at fixed distance 0 R ) improved with spin-conditioned relativistic corrections of order 2 1/ c . The oneparametric spectrum 0 ( ) n R  of bound states for the problem has been obtained with exact diagonalization procedure [12,13] at arbitrary values of 0 R (Sec. 4).…”