Exact hierarchy of equations for the kinetic energy functional

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Abstract
The virial theorem is used to derive an exact hierarchy of coupled equations for the kinetic energy functional T[ρ] of an inhomogeneous system of noninteracting fermions. The hierarchy links the nth functional derivative of T[ρ] with respect to the particle density ρ(r) to the (n+1)th functional derivative and to ρ(r) for n=1,2,. . . . These functional derivatives are the analogs of the so-called ‘‘direct correlation functions’’ in the theory of classical fluids, and the hierarchy obtained resembles the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for the distribution functions associated with such fluids. Some general properties of the two- and three-point correlation functions are described, in particular, their relations to the linear and quadratic response functions that describe the response of the inhomogeneous system to a change in the potential. The hierarchy is then used to obtain several equivalent expressions for T[ρ], one of which is an infinite series of successively higher-order terms, of which the nth term represents the contribution from the n-point correlation function and the densities associated with those points. The equations are then examined in the homogeneous (uniform) gas limit and the first and second equations are verified explicitly with the aid of the associated response functions. The analysis indicates that the infinite series expression converges rapidly for the three-dimensional gas; it also reveals that for both the one-dimensional and two-dimensional gas, the series terminates after the first few terms, suggesting that the structure of T[ρ] is much simpler in one and two dimensions than in three dimensions.