Calculation of the density $\Delta\mbox{{\boldmath {$s$ }}}$

There are two misprints in formulas (I-46)³ for the Cartesian components of the vector density $\Delta\mbox{{\boldmath {$s$ }}}$. The correct expressions are as follows:

   

The misprints were present only in the text of I, and did not affect the code HFODD (v1.60r). Unfortunately, the factors of 2, which should multiply densities T₁ and T₂in Eqs. (1a) and (1b), respectively, where missing in version (v1.60r) of the code. As far as the numerical values are concerned, this error has been fairly unimportant for the final results, however, it may have created a weak spurious dependence of the results on the orientation of the nucleus with respect to the Cartesian reference frame, because only the x and y components of $\Delta\mbox{{\boldmath {$s$ }}}$ were affected.

Incorrect expressions for the densities $\Delta s_1$ and $\Delta s_2$amounted to adding the erroneous term $-C_t^{\Delta s}\left({s}_{xt}{T}_{xt}+{s}_{yt}{T}_{yt}\right)$to the time-odd energy density ${\cal H}^{\mbox{\scriptsize {odd}}}_t(\mbox{{\boldmath {$r$ }}})$of Eq. (I-12a), and simultaneously adding the erroneous terms $-2C_t^{\Delta s}{T}_{xt}$ and $-2C_t^{\Delta s}{T}_{yt}$to the time-odd spin potentials $\Sigma_{xt}$ and $\Sigma_{yt}$, respectively, Eq. (I-29b). Therefore, up to a very small difference between the matrix elements of spin and kinetic-spin potentials, the incorrect densities $\Delta s_1$ and $\Delta s_2$ where equivalent to a (direction-dependent) modification of the coupling constant C_t^T. This is the main reason why the error went undetected for a relatively long time. Needless to say, calculations performed with $C_t^{\Delta s}$=0, and in particular those with all time-odd terms neglected, are unaffected.

Since the term $\mbox{{\boldmath {$s$ }}}_t\cdot\Delta\mbox{{\boldmath {$s$ }}}_t$ gives anyhow fairly small contribution to the rotational properties of nuclei (compare curves denoted by open circles with those denoted by full squares in Figs. 3 and 4 of Ref. [3]), the incorrect expressions for densities $\Delta s_1$ and $\Delta s_2$ had numerically relatively small importance. The total energies could have been affected at the level of about 0.3MeV and the total spins at the level of about 0.3$\hbar$ (for details compare the output file reproduced in the section 8 TEST RUN OUTPUT below with that given in II).