Moment of Inertia in Hartree-Fock Theory

The conditions under which a Hartree-Fock wave function may be a good description of the intrinsic state of a rotational nucleus are discussed from two points of view; first by studying the fluctuations of the modified Hamiltonian $H-\alpha {\mathrm{J}}^2$ as a function of $\alpha$, and second by studying the response of the wave function to a small external perturbation $-\omega J_x$. Following Thouless and Valatin, this response is given in terms of an anti-Hermitian cranking operator $S$. We demonstrate that, when the Hartree-Fock wave function is adequate, $S$, is of the form $\frac{[J_x, \rho ]}{\epsilon }$, where $\rho$ is the single-particle density operator and $\epsilon$ is twice the rotational energy content of the intrinsic state. These considerations lead to simple tests of the adequacy of the Hartree-Fock wave function as a rotational intrinsic wave function. A comparative study of various formulas for the moment of inertia, utilizing the aforementioned result for $S$, is presented.