Hooke’s law correlation in two-electron systems

Abstract

We study the properties of the Hooke's law correlation energy (Ec), defined as the correlation energy when two electrons interact via a harmonic potential in a D-dimensional space. More precisely, we investigate the 1 S ground state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made, which highlights the main differences of the Ec in both the weakly and strongly correlated limits. Moreover, we show that the Schrödinger equation of the spherium model is exactly solvable for two values of the dimension (D = 1 and 3), and that the exact wave function is based on Mathieu functions.

Statistical correlations in the Moshinsky atom

“…This model has been extensively studied in many areas [19,30,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. In particular, it has been studied from the perspective of the correlation as measured by the correlation energy [47].…”

Distribution of r12· p12in quantum systems

“…In (2), u = r 1 − r 2 , u = |u| ≡ r 12 and Ω u is the angular part of u. P (u) gives the probability density for finding two particles separated by a distance u and has been widely studied [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

Correlation energy of anisotropic quantum dots

“…The two-electron problem is one of the fundamental problems of quantum physics [1][2][3][4] and, although it looks simple, it has only been solved in certain very special cases [5][6][7][8][9][10][11][12][13]. Many of the methods that have been developed to provide approximate solutions to the two-electron problem have been central in the development of molecular physics and quantum chemistry [14,15].…”