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We present a theory for the rate of energy exchange between electrons andions -- also known as the electron-ion coupling factor -- in physical systemsranging from hot solid metals to plasmas, including liquid metals and warmdense matter. The paper provides the theoretical foundations of a recent work[J. Simoni and J. Daligault, Phys. Rev. Lett. {\bf 122}, 205001 (2019)], wherefirst-principles quantum molecular dynamics calculations based on this theorywere presented for representative materials and conditions. We first derive ageneral expression for the electron-ion coupling factor that includesself-consistently the quantum mechanical and statistical nature of electrons,the thermal and disorder effects, and the correlations between particles. Weshow that our theory reduces to well-known models in limiting cases. Inparticular, we show that it simplifies to the standard electron-phonon couplingformula in the limit of hot solids with lattice and electronic temperaturesmuch greater than the Debye temperature, and that it extends theelectron-phonon coupling formula beyond the harmonic phonon approximation. Forplasmas, we show that the theory readily reduces to well-know Spitzer formulain the hot plasma limit, to the Fermi golden rule formula in the limit of weakelectron-ion interactions, and to other models proposed to go beyond the latterapproximation. We explain that the electron-ion coupling is particularly welladapted to averaged atom models, which offer an effective way to includenon-ideal interaction effects to the standard models and at a much reducedcomputational cost in comparison to first-principles quantum molecular dynamicssimulations.

Theory of the electron-ion temperature relaxation rate spanning the hot solid metals and plasma phases