Nonlocal exchange effects on the bulk plasmon dispersion relation

Abstract

Starting from the action principle, a bulk plasmon dispersion relation is obtained. We work with Slater determinants built of plane waves and consider a Hermitian generator of plasma oscillations (with well-defined q momentum) S=αQ-βP, where Q= tsum j cos(q⋅ x j ) is a time-even Hermitian generator, P=scrB tsum j [ p j ⋅q sin(q⋅ x j )+sin(q⋅ x j ) p j ⋅q] is a time-odd Hermitian generator, α(t) and β(t) are real time-dependent functions, and scrB is a real normalization constant. If the parameters α and β are small, the amplitude of the plasma oscillations generated by S is small. The quantum-mechanical action principle leads, in the harmonic approximation, to a quadratic Lagrangian L ( 2 ) (α,β) from which the dispersion relation is obtained. The nonlocal expression of the exchange contribution is explicitly obtained. The resulting bulk plasmon dispersion relation is related to the energy-weighted and cubic-energy-weighted sum rules. Finally, we compare our results with the experimental data.