Exact properties of the Pauli potential for the square root of the electron density and the kinetic energy functional

It is known that the square root of the electron density satisfies {-(1/2${\nabla }^2$+$v_{\theta }$([n];r) +$v_s$([n];r)}$n^{1/2}$(r) =${\varepsilon }_M$ $n^{1/2}$(r), where $v_s$ is the Kohn-Sham potential and ${\varepsilon }_M$ is its highest-occupied orbital energy. The Pauli potential $v_{\theta }$ is defined as the functional derivative of the difference between the noninteracting kinetic energy $T_s$[n] and the full von Weizsäcker kinetic energy. It has already been proven that $v_{\theta }$([n];r)≥0 for all r. By starting primarily with a slightly modified version of an equation of Bartolotti and Acharya, new exact properties of $v_{\theta }$([n];r) are derived for the purpose of approximating it. The gradient expansion for $T_s$[n] gives a $v_{\theta }$([n];r) that is found to violate several of the exact conditions. For instance, $v_{\theta }$≥0 is violated unless the full von Weizsäcker term is employed. A new approximate form for $v_{\theta }$([n];r) is proposed.