A new Green's function formalism for kinetic energy density functional for atomic and molecular system: Emergence of $N-$dependence using model potentials

An accurate expression of the kinetic energy density of an electronic distribution in terms of the single particle reduced density matrix for atomic and molecular systems is a long-standing problem in electron structure theory. Existing kinetic energy density functionals are generally expressed as modifications over kinetic energy of homogeneous electron gas and/or von Weizs\"acker kinetic energy. A large class of these functionals also require empirical parametrizations to make accurate predictions of the kinetic energy for atomic and molecular systems restricting their transferability. Moreover, the correct kinetic energy density which produces accurate local properties such as atomic shell structure is still an unsolved problem. In this work, we have developed an exact methodology that can be used to derive the kinetic energy of an electronic system of arbitrary spin multiplicity. One of the attractive features of this present analytical formalism is the possibility of systematic improvement of the kinetic energy by virtue of a novel perturbation series. Applying this methodology to simple model systems such as one-dimensional quantum harmonic oscillator and homogeneous electron gas produces a qualitatively correct $N$-dependence of kinetic energy as a result. A one-to-one correspondence between our formalism to the traditional Green's function formalism is also demonstrated.