Theory and computation have made huge advances over the last 30-40 years in meeting the challenges of ground state quantum mechanics for solids, clusters, molecules, and atoms. Warm dense matter confronts that success with the reality of complicated quantum statistical mechanics. WDM forces treatment of thermal occupancy of excited electronic states in the context of multiple ionization states and transient (sometimes exotic) molecular species, all in a pressure-temperature-density regime far removed from the ground state. Moreover, a WDM system may include locally crystalline, liquid, and gaseous regions concurrently. Differences in vocabulary and perspective of the two main communities, solid state physics
and quantum chemistry, can get in the way of capitalizing on ground-state QM successes. About 65-70 % of this talk will survey the landscape (or toolbox) of QM methods which have been and are being used for WDM. The goal is to link relevant ground-state approximations and implementations, both for solids and molecules, with nite-temperature uses and extensions. This will include wave-function methods (Hartree-Fock, CI, coupled cluster), their extension to average atom schemes and their relationship with Quantum Monte Carlo calculations, and with density functional methods. The survey will include some basic information on computational scalings and on known and suspected limitations of approximations. The remainder of the talk will summarize progress and problems in recent development (with collaborators at Univ. Florida and in Mexico) of constraint-based, non-empirical free-
energy density functionals for use in the so-called orbital-free approach. Supported by US DoE Grant DE-SC0002139.