An increasing quality of spectroscopic information demands more and more accurate theory, pushing the efforts beyond the mean-field approximation towards, in particular, symmetry-conserving formalisms and symmetry restoration.[1,2,3] Hereby, we report on development of a new theoretical tool allowing for the angular-momentum projection (AMP) of cranked symmetry-unrestricted Slater determinants.

The standard method used to develop symmetry-conserving theory,
starting from *intrinsic* (deformed) wave function
, obtained within the mean-field approach, is provided
by the *projection techniques* onto eigenspaces of symmetry
operators. There are two practical realizations of the projection
methods: more fundamental and elaborate *variation after
projection* (VAP) and slightly less advanced *projection after
variation* (PAV). In the past, many calculations have been performed
within the angular-momentum PAV method, where non-rotating states
have been projected, see, e.g., Refs.[4,5,6]
and the reviews in Refs.[2,3]. The cranking method provides
the first-order approximation to the angular-momentum VAP
method.[1] However, after the ground-breaking studies in
Refs.[7,8], calculations based on the AMP of
cranked states have not been performed. Here, we present the first
results of our recently developed AMP method of cranked
symmetry-unrestricted Hartree-Fock states. The procedure we use has been
implemented within the code HFODD
(v2.25b).[9,10,11,12]

We determine the optimal product wave function by using the
cranked self-consistent Skyrme-Hartree-Fock (SHF) method. The Ritz variational
principle:

where is expressed as a bilinear form of the time-even (TE) and time-odd (TO) , , local densities and currents, and by their derivatives,[13]

By taking an expectation value of the Skyrme force over the Slater determinant, one obtains the LEDF (3)-(4) with coupling constants that are expressed uniquely through the 10 parameters and of the standard Skyrme force. Because of the local gauge invariance of the Skyrme force, only 14 coupling constants are independent quantities. The local gauge invariance links three pairs of time-even and time-odd constants in the following way: