Series expansions for an exact two-electron wave function in terms of Löwdin's renormalized natural orbitals

In recent developments on the pair density needed to treat the non-Hartree-Fock-like part of interparticle repulsion, the natural orbitals and sign-correct expansion coefficients play a central role. Since, in principle, an infinite number of natural orbitals must be included, the convergence of expectation values due to finite-term approximations is an important issue. Here we discuss quantitatively this convergence problem based on an exactly solvable two-electron model atom, where the Schrödinger wave function for the ground state is expressible in terms of Löwdin's natural orbitals and sign-correct expansion coefficients. Using properly renormalized truncated series expansions for such an exact decomposition, the corresponding expectation values of the Schrödinger Hamiltonian are calculated analytically. A rapid and uniform convergence is found in these expectation values at given values of the coupling in the interparticle repulsion.

Figure 1

Different dimensionless ratios, defined by ${\overline{K}}^{\left(\text{tr}\right)}/{\overline{K}}_{\text{ex}}$ (dotted curve), $[{\overline{C}}^{\left(\text{tr}\right)}+{\overline{R}}^{\left(\text{tr}\right)}]/{\overline{K}}_{\text{ex}}$ (dashed curve), and $[{\overline{K}}^{\left(\text{tr}\right)}+{\overline{C}}^{\left(\text{tr}\right)}+{\overline{R}}^{\left(\text{tr}\right)}]/\left(2{\overline{K}}^{\left(\text{tr}\right)}\right)$ (solid curve), as a function of $N$ at $\Lambda =0.499$. The curve representation is used only for visualization. The data are at integer $N$ values.