Mass parameters in the adiabatic time-dependent Hartree-Fock approximation. I. Theoretical aspects; the case of a single collective variable

A self-consistent method for evaluation of mass parameters is presented in the framework of the adiabatic limit of the time-dependent Hartree-Fock approximation, reduced to a single collective variable. The corresponding collective path is assumed to be given either by solving a constrained Hartree-Fock problem with a given time-even constraining operator $Q$, or by scaling a static Hartree-Fock equilibrium solution. In the former case, once the path is given, a method for solving the equation of motion (of the Hamilton type) is provided, which reduces to a double-constrained Hartree-Fock problem with the time-even constraint $Q$ and with a time-odd constraining operator $P$. In the case of the scaling path, an analytical solution of the Hamilton equation is discussed and the adiabatic mass for the particular case of an isoscalar quadrupole $Q_{20}$ mode is given. The operator $P$, which is uniquely determined from the knowledge of $Q$, has the physical meaning of a momentum operator; it satisfies, together with $Q$, a weak quantal conjugation relation. Finally, the connection between the two paths is discussed in terms of generalized random-phase approximation sum rules.