Mixed charge–spin response functions of an arbitrarily polarized electron gas

In this paper, we demonstrate the equality of the two mixed charge–spin response functions of a spin-polarized electron gas when orbital effects are negligible using different approaches. Within a generalized STLS approximation we show that the two mixed responses are equal. We also present arguments for the equality of the two dynamic responses by considering a symmetry of the effective screened interaction between two opposite spin electrons. Furthermore, using the reflection symmetry of the system and the fact that the Hamiltonian is real we prove rigorously that the two responses coincide identically.