On the Atomic Energy Asymptotics

Consider an atom consisting of N quantized electrons at positions x_i and a nucleus fixed at the origin. The Schrödinger Hamiltonian of such a system is given by $${H_{Z,N}} = \sum\limits_{i = 1}^N {\left( { - {\Delta _{{x_i}}} - \frac{Z} {{\left| {{x_i}} \right|}}} \right)} + \frac{1} {2}\sum\limits_{i \ne j} {\frac{1} {{\left| {{x_i} - {x_i}} \right|}}} $$ acting on $$ = \wedge _{i = 1}^N{L^2} $$ (R3) (in this exposition, in order to simplify notation, we neglect spin.) Define the ground state of an atom of charge Z by $$ E\left( Z \right) = {\kern 1pt} \mathop {\inf }\limits_N \mathop {\inf }\limits_{\mathop {\left\| \Psi \right\| = 1}\limits_{\Psi \in } } \;\left\langle {{H_{Z,N\Psi ,\Psi }}} \right\rangle $$.