Extraction of the frequency moments of spectral densities from imaginary-time correlation function data

We introduce an exact framework to compute the positive frequency moments $M^{(\alpha)}(\mathbf{q})=\braket{\omega^\alpha}$ of different dynamic properties from imaginary-time quantum Monte Carlo data. As a practical example, we obtain first five structure factor $S(\mathbf{q},\omega)$ uniform electron gas at electronic Fermi temperature based on \emph{ab initio} path integral simulations. find excellent agreement with known sum rules for $\alpha=1,3$, and, our knowledge, present results $\alpha=2,4,5$. Our idea can be straightforwardly generalized other such as single-particle spectral function $A(\mathbf{q},\omega)$, and will useful number applications, including study ultracold atoms, exotic warm dense matter, condensed matter systems.