Adiabaticity of time-dependent Hartree-Fock solutions

A tractable method is presented for the study of the adiabaticity of a given time-dependent Hartree-Fock solution. In terms of the Baranger-Vénéroni approach, this amounts to extracting the two operators ${\rho }_0$ and $\chi$, for which rather simple expressions are derived. These explicit formulas are exploited to illustrate some implications of the adiabatic approximation itself and several related approximations. In particular, the incompatibility of two current hypotheses for the $\chi$ operator (locality and absence of particle-particle, hole-hole matrix elements) is pointed out. Finally, the whole discussion is exemplified by the simple case of uniform translational motion.