Level crossings and the HF convergence

In all calculations, we diabatically follow configurations, i.e., we always occupy orbitals with given single-particle characteristics, irrespectively of whether they cross with the other orbitals or not. This is technically easy if crossings involve orbitals of different parity/signature blocks, but special techniques [36] must be used in self-consistent methods, to diabatically follow configurations which cross within the common parity/signature blocks. Those crossings are particularly interesting because they usually give rise to an up-bending or back-bending structures, and are thus important for the experimental identification of the underlying structures.

In the present study we separate the diabatic configurations
by proceeding as follows.
If, in the positive-parity orbitals, a particle ``switches on and off''
from the occupation of one orbital to another, we force an occupation of
the state that has a *larger* single-particle alignment, independently
of whether it is slightly higher or lower in energy.
In the present case this implies that
we always force the particle into the
[211]1/2 orbital and leave the extruder orbital [202]5/2 empty.
Incidentally, by occupying the state that has a *smaller* alignment,
at the end of a successful iteration sequence
we obtain a markedly different solution, with much smaller
deformations, i.e., the fact that two configurations mix does not
imply that both manifest all the same physical properties in this case.

A rich collection of the experimental data on the back-bending and up-bending phenomena that exist today in the literature has been interpreted in terms of the theoretical single-particle configurations followed according to the diabatic scheme. Whether the experimental bands exist that follow an adiabatic scheme is an open question, and an unambiguous (while anyway model depended) demonstrations are very difficult. On the theory grounds, this question cannot be settled within a mean-field approach. Therefore, our approach to follow diabatic configurations is dictated by the fact that a great majority of the high spin data has been interpreted accordingly. In case of need demonstrated by future experiments, our present results could immediately be used as a first step in the band-band mixing calculations.

By examining the routhian diagrams obtained self-consistently at a fixed
particle-hole configuration, as, e.g., those shown in Figs. 1 and
2, one cannot predict crossings which may happen in
some other configuration. This is especially true in ^{32}S,
where different configurations correspond to fairly different
deformations, see Sec. 4.2, and therefore, may involve
significantly different ordering of orbitals. As an example, in
Figs. 4 and 5 we present neutron routhians
corresponding to the self-consistently calculated
3^{1}_{-}3^{1}_{-} and
3^{1}_{+}3^{1}_{+}bands, respectively. Since these configurations
produce deformations significantly smaller than that of the
magic SD configuration, the (empty) extruder orbitals [202]5/2(*r*=)
are here much lower in energy, and strongly mix with the
(occupied) [211]1/2(*r*=)
orbitals. In these configurations,
the signature splitting is very large, and strongly depends on the actual
configuration, see discussion in Sec. 4.4. Therefore the order of
routhians in the
3^{1}_{+}3^{1}_{+} configuration (Fig. 5)
is entirely different from that in the
3^{1}_{-}3^{1}_{-} configuration
(Fig. 4) and leads to very strong mixing and level repulsion
at and near the crossing frequency in the latter case.

We can easily identify the crossing regions by the simple fact that the HF iterations are poorly convergent, or non-convergent there [36]. This concerns only those methods of solving the HF equations, which are based on successive diagonalizations of the mean-field Hamiltonian. The gradient method and the imaginary-time method [39], always arrive at the smallest-energy solution, and do converge. However, the obtained solutions simply correspond to one of the infinitely many possible mixed-orbital solutions, with the same or very close energy, and consequently, those methods do not cure the problem, but allow for not seeing it.

We have made every possible effort to
achieve convergence of all configurations at all angular frequencies, however,
in several cases it turned out not to be possible. This is the
case, for example, for the
3^{1}_{+}3^{1}_{+} band at
=1.0-1.4MeV; the non-convergence here results in an irregular
behavior of routhians in Fig. 2.
In the following, we show energies
corresponding to the non-converged points along with the
well-converged ones, however, we remove points corresponding to
non-converged solutions from plots of other observables.