A new variational wave-function ansatz for confined two-electron atomic systems

Abstract

In this paper, we propose a simple variational ansatz, Ψ(r⃗<inf>1</inf>,r⃗<inf>2</inf>) = C sin (π rc r1) sin( π rc r2) exp(−Z(r<inf>1</inf> + r<inf>2</inf>))[cosh(ar<inf>1</inf>) + cosh(ar<inf>2</inf>)][1 + 0.5r<inf>12</inf> exp(−br<inf>12</inf>)], r<inf>1</inf> r<inf>2</inf> to study confined two-electron atomic systems. Here, r<inf>12</inf> = |r-<inf>1</inf> −-<inf>2</inf>| is the inter-electronic distance with the electron coordinatesr-1 andr-<inf>2</inf>, r<inf>c</inf> is the radius of the impenetrable well in which the two-electron atoms are confined, and C is the normalization constant. The function sin(πr/r<inf>c</inf>)/r incorporates the Dirichlet boundary conditions at r = r<inf>c</inf> needed for the wave function of two-electron systems, and a and b are the variational parameters evaluated by minimizing the total energy functional of confined two-electron atoms. We also calculate the pressure and check the satisfaction of the virial relation for such systems. Our results for the ground-state energy and its components, radial distance moments, and pressure show agreement with the existing literature. © 2025 IOP Publishing Ltd. All rights, including for text and data mining, AI training, and similar technologies, are reserved.