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{^2+v_ (n;r) +vₒ (n;r) }n^1/2 (r) =₌n^1/2 (r), where vₒ is the Kohn-Sham potential and ₌ is its highest-occupied orbital energy. The Pauli potential v_ is defined as the functional derivative of the difference between the noninteracting kinetic energy Tₒn and the full von Weizs\"acker kinetic energy. It has already been proven that v_ (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_ (n;r) are derived for the purpose of approximating it. The gradient expansion for Tₒn gives a v_ (n;r) that is found to violate several of the exact conditions. For instance, v_0 is violated unless the full von Weizs\"acker term is employed. A new approximate form for v_ (n;r) is proposed.