Study of Electron Correlation in Helium-Like Systems Using an Exactly Soluble Model

Nonuniversality of commonly used correlation-energy density functionals

“…Its separability, in center-of-mass and relative coordinates, was pointed out by Kestner and Sinanoglu [20], and the fact that for certain values of the force constant it is analytically solvable was noted by Kais et al [21] and, more systematically, by Taut [22]. Several studies for Hooke's atom have been performed, aimed at deriving exact density functionals or Kohn-Sham type exchange-correlation potentials [23][24][25], most notably by Burke et al [26][27][28][29][30] and Ludena et al [31].…”

Biconfluent Heun equation in quantum chemistry: Harmonium and related systems

“…On one hand, the existence of simple polynomial solutions of this equation, discovered in connection with studies on electron correlation, motivated numerous works in the community of quantum chemists [18][19][20][21][22][23][24][25][26][27][28]. However, this equation is known in mathematics since more than a century as the biconfluent Heun equation and its properties were studied from both purely mathematical perspective [29,30,[35][36][37][38] and in the context of its applications in different areas of physics [39][40][41].…”

“…(14). The structure of expansion (18) for two electrons in a uniform magnetic field was studied by Kandemir [34]. In particular, he reduced the recurrence relation to a closedform expression.…”

“…Equation (34) may be represented by a diagram which facilitates an easy derivation of a closed-form formula for the coefficients of the expansion (18). It also helps to better understand the structure of this expansion and shows interrelations between specific coefficients.…”

“…and confined in a central harmonic (i.e., quadratic or parabolic) potential is separable [18]. The problem was shown quasi-solvable analytically six years later by Santos [19].…”