Kinetic energy from the cubic sum rule of the dynamic structure factor

The third frequency moment sum rule of the dynamic structure factor $S(\mathbf{q},\omega)$ is explored for the first time as an alternative estimator of the kinetic energy $K$ of quantum many-body systems. As a practical example, the uniform electron gas at warm dense matter conditions is considered. First, $K$ is extracted from quasi-exact \emph{ab initio} path integral Monte Carlo results for the imaginary-time density--density correlation function $F(\mathbf{q},\tau)$ and the expected excellent self-consistency with the thermodynamic differentiation route is confirmed. Second, $K$ is extracted from approximate dielectric formalism results for $S(\mathbf{q},\omega)$ and it is observed that common semi-classical approximations lead to a wave-number dependent $K$ with an incorrect short-wavelength limit. Our results are expected to be of broad interest for a great variety of applications, including time-dependent density functional theory, dielectric formalism schemes and warm dense matter models, as well as for the design of dedicated x-ray Thomson scattering experiments with the potential to provide model-free access to the full electronic equation of state.