Curvature of the total electron density at critical coupling

A structural property of the spherically averaged total charge density ${\rho }_{\lambda }\left(r\right)$ of a degenerate electron gas distorted by the action of an attractive, screened potential $\lambda V\left(r\right)$ is the subject of this study. The total charge density is described as a sum over occupied bound and scattering wave functions of independent electrons moving in the effective central field. It is shown that the second radial derivative ${\rho }_{\lambda }^{″}\left(0\right)$ is a differentiable function of $\lambda$ for any $\lambda ⩾0$. The smooth behavior implies that this curvature ${\rho }_{\lambda }^{″}\left(0\right)$ does not provide a criterion for the bound-to-free transition in metallic environments.

Figure 1

(Color online) Partial components of the electronic density at the origin ${\rho }_{\lambda ,\mathrm{bd}}\left(0\right)$ (upper panel) and ${\rho }_{\lambda ,\mathrm{sc}}\left(0\right)$ (middle panel) as a function of the coupling constant $\lambda$ for a Hulthén potential $\lambda V\left(r\right)=-\lambda ∕({\mathrm{e}}^r-1)$. The total density ${\rho }_{\lambda }\left(0\right)≔{\rho }_{\lambda ,\mathrm{bd}}\left(0\right)+{\rho }_{\lambda ,\mathrm{sc}}\left(0\right)$ is shown as well in the lower panel. The insets show the derivatives of these functions with respect to $\lambda$ [$\delta {\rho }_{\lambda ,\mathrm{bd}}\left(0\right)∕\delta \lambda$, $\delta {\rho }_{\lambda ,\mathrm{sc}}\left(0\right)∕\delta \lambda$, and $\delta {\rho }_{\lambda }\left(0\right)∕\delta \lambda$ in the upper, middle, and lower panels, respectively] in the vicinity of the threshold $\lambda ={\lambda }_{0,1}$ for the appearance of the first $(L=0)$-bound state.

Figure 2

(Color online) Partial components of the curvature at the origin ${\rho }_{\lambda ,\mathrm{bd}}^{″}\left(0\right)$ (upper panel) and ${\rho }_{\lambda ,\mathrm{sc}}^{″}\left(0\right)$ (middle panel) as a function of the coupling constant $\lambda$ for a Hulthén potential $\lambda V\left(r\right)=-\lambda ∕({\mathrm{e}}^r-1)$. The total curvature ${\rho }_{\lambda }^{″}\left(0\right)≔{\rho }_{\lambda ,\mathrm{bd}}^{″}\left(0\right)+{\rho }_{\lambda ,\mathrm{sc}}^{″}\left(0\right)$ is shown as well in the lower panel. The insets show the same functions in the vicinity of the threshold $\lambda ={\lambda }_{1,1}$ for the appearance of the first $(L=1)$-bound state.