Uncoupled Correlated Method for Helium Bound States

Abstract

An uncoupled correlated variational method for calculating helium bound states is proposed. The projective coordinates $\mathit{s}={\mathit{r}}_1+{\mathit{r}}_2$ , $\mathit{v}=\frac{{\mathit{r}}_{12}}{{\mathit{r}}_1+{\mathit{r}}_2}$ , $\mathit{w}=\frac{{\mathit{r}}_1-{\mathit{r}}_2}{{\mathit{r}}_{12}}$ are introduced instead of the conventional ones $\mathit{s}={\mathit{r}}_1+{\mathit{r}}_2$ , $\mathit{t}={\mathit{r}}_1-{\mathit{r}}_2$ , $\mathit{u}={\mathit{r}}_{12}$ . All matrix elements of the total Hamiltonian and the weight function are expressed by simple products of three one-dimensional integrals. The variational basis is formed by a set of Laguerre polynomials with a single nonlinear parameter and two sets of Jacobi polynomials for the projective coordinates $\mathit{s}\mathrm{,}\mathit{v}\mathrm{,}\mathit{w}$ , correspondingly. It provides a reasonable degree of convergence of the energy, $\mathit{E}=\mathit{E}\mathrm{(}\mathit{N}\mathrm{)}$ , with respect to the number N of expansion terms over the basis of the eigenvectors. In the case of infinite and finite nuclear mass, calculations give the energy of the helium ground state.