A direct approach to computing the non-interacting kinetic energy functional

The non-interacting kinetic energy functional, $T_{KS}(\rho)$, plays a fundamental role in Density Functional Theory (DFT), but its explicit form remains unknown for arbitrary $N$-representable densities. Although it can, in principle, be evaluated by solving a constrained optimization problem, the associated adjoint problem is not always well-posed; moreover, even when it is, the corresponding adjoint operator may be singular. To the best of our knowledge, none of the existing approaches in the literature precisely determines the non-interacting kinetic energy functional for a given $N$-representable electron density, $\rho$. In this work, we present a variational framework for computing an extension of $T_{KS}(\rho)$ using an exact trigonometric reparametrization of the density that eliminates the need for an adjoint equation. We present a proof-of-concept numerical validation of the variational principle for the special case of one-dimensional Kohn-Sham systems. Our method, however, is general and provides a systematic foundation for computing $T_{KS}(\rho)$ in higher dimensions too, paving the way for improved kinetic energy functionals in DFT.