Compact wave functions for four-electron atomic systems

Abstract

A procedure is developed to construct improved wave functions for the bound states in various four-electron atoms and ions. The bound state wave functions are improved in the sense that our trial wave functions are compact and accurate. We employ two independent four-electron spin functions ${\ensuremath{\chi}}_{1}=\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}+\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}$ and ${\ensuremath{\chi}}_{2}=2\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}+2\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}$ in each trial wave function. Optimization of the nonlinear parameters is performed for each radial function. Our procedure is applied to obtain the total energies of the ground $^{1}S^{e}$ states in the following four-electron berylliumlike systems: ${\mathrm{Li}}^{\ensuremath{-}}$, Be, ${\mathrm{B}}^{+}$, ${\mathrm{C}}^{2+}$, ${\mathrm{N}}^{3+}$, ${\mathrm{O}}^{4+}$, ${\mathrm{F}}^{5+}$, ${\mathrm{Ne}}^{6+}$, ${\mathrm{Na}}^{7+}$, and ${\mathrm{Mg}}^{8+}$. All our trial wave functions used in this study are very compact. In fact, each of these functions contains only 300 basis (radial) functions which are constructed from ten-dimensional gaussoids of the five-body relative coordinates ${r}_{12}$, ${r}_{13}$, ${r}_{14}$, ${r}_{15}$, ${r}_{23}$, ${r}_{24}$, ${r}_{25}$, ${r}_{34}$, ${r}_{35}$, and ${r}_{45}$.