Gogny energy density functionals

Odd mass nuclei were described using the HFB method with full blocking and taking into account all possible time-odd fields coming from the potential part of the Gogny interaction. The original formulation of the density-dependent part of the interaction is preserved and therefore no additional time-odd density-dependent terms are added to it. This assumption has to be verified in the future as there are indications that such kind of terms could be required to recover basic properties of odd-odd systems [49].

The computer code ATB was used in the calculations [50]. Axial symmetry was preserved in the calculations but reflection symmetry was not. Therefore, the projection of the angular momentum along the intrinsic axial-symmetry axis, $\Omega=\langle \hat{j}_\parallel \rangle$, is a good quantum number, but this is not necessarily the case for parity. However, many of the states analyzed show a mean value of the parity operator, which is rather close to either plus or minus one, allowing the labelling of those states with a definite parity. As $\Omega$ is a good quantum number, states with different values of $\Omega$ are automatically orthogonal. States with the same $\Omega$ value, same parity and similar deformation parameters are imposed to be orthogonal to each other by using the traditional technique of constraints. The way the constraints are handled does not depend on the nature of the constrained operator as only its mean value and gradient are required [51]. The code expands the quasiparticle operators in a harmonic oscillator basis with 15 shells ($680\times2$ states) although axial symmetry reduces the maximum size of matrices to $120\times2$, corresponding to the $\Omega=1/2$ block.

In the odd-mass (odd number parity) case, the HFB iterative minimization process requires a starting mean-field state with given characteristics. An HFB mean-field state with even number parity (obtained with a time-even code) was used to generate this starting configuration. The mean values of neutron $N$ and proton $Z$ number operators were constrained to the required values for the odd nucleus under consideration. To avoid spurious good-parity solutions, which could result from the propagation of self-consistent symmetries in the HFB method, a small octupole moment was also constrained.

Ten quasiparticle states with the lowest one-quasiparticle energies coming out of this calculation were considered as starting configurations for the time-odd code ATB. They were obtained with the standard ``blocking procedure" consisting of swapping the appropriate columns in the $U$ and $V$ matrices of the Bogoliubov transformation. Each of the starting configurations was labelled with the corresponding $\Omega$ quantum number, which was preserved all along the minimization process. The first starting configuration with a given value of $\Omega$ usually converges to the lowest-energy band head with angular momentum $I=\Omega$. For the same value of $\Omega$, the second starting configuration usually converges to the same first solution, unless an orthogonality constraint to that state is imposed (including both the $\Omega$ and $-\Omega$ states). For the third starting configuration for a given $\Omega$ value, the orthogonality constraints with respect to the first and second states are required. In principle, for the $n$-th starting configuration $2(n-1)$ orthogonality constraints to the states previously obtained are required. Although the number of constraints can be relatively high, this is not a formal problem for the gradient method used in the code, as handling of constraints is very simple to implement [51]. The orthogonality constraint is also important to ensure the orthogonality of the physical states. If such orthogonality is not imposed, even if the states are different, the relative excitation energies can be shifted. In the present calculations only two starting configurations for each value of $\Omega$ were considered.

For the calculation of the properties of even-even nuclei required in the evaluation of the pairing gap parameters, the code HFBAXIAL was used. It has the capability to break reflection symmetry although the phenomenon was not relevant in the present calculation. Moments of inertia were determined using cranked HFB wave functions obtained with the code HFBTRI. In this code, the axial and time-reversal symmetries were allowed to be broken in the minimization process.