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Table 1 displays the results of PNP calculations performed
for O by using circular integration contours (53)
of different radii. The precision of numerical integrations was
confirmed by calculating contributions from individual poles.
This was done by
carrying out contour integrals
over small circles surrounding the poles. In this way, we determined
residues from the individual poles
and checked
that their sums,
,
agree very well
with the results of contour integrals along circular contours
, as
required by the Cauchy theorem (44).
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As seen in Table 1, contributions of the poles at
are huge. Therefore, the DFT residues at
cannot at
all be interpreted as the projected energies, as was the case for the
PNP HFB theory, Eq. (15). Residues at
are cancelled, to a
large extent, by contributions from the 1s
deep-hole
states, which are
large because they contain large factors of the type
for
[see
Eq. (47]. Contributions from other poles are also quite
large, and apart from the integration contour at
,
none of the other contours reproduce the correct
projected energy shown by a boxed number.
For the SIII parametrization, one can see that contributions
from poles associated with
spherical states with
(
)
are indeed equal to zero, cf.
discussion in Sec. 3.3. This property does not hold for
SLy4, for which the projected DFT energies have jumps
also when the integration contours cross the
poles. In
this case, the jumps are not related to non-zero residues, but, as
discussed in Sec. 3.6, they are caused by the fact that
the integration contours are not closed for the fractional-power terms.
Figure 6 shows the projected DFT-SLy4 energies obtained by
using circular integration contours of different radii .
These calculations illustrate
properties of poles listed in Table 1. The
contributions originating from the
density-independent and density-dependent terms of the
Skyrme force are separated.
The latter terms yield the
fractional-power terms in the DFT energy density discussed in
Sec. 3.6. As in the SIII case,
the density-independent terms exhibit jumps only at the two
poles. On the other
hand, the density-dependent terms show jumps at all poles, and
these jumps carry over to the total projected DFT energies shown in
Table 1.
[The small jump at the 1d
pole, 120keV, is practically
invisible in the scale of Fig. 6.]
Moreover, contributions of the density-dependent terms are not constant
between the poles, as would be required by the Cauchy theorem. This is
caused by the prescription (52) to step over the cuts in the
complex plane, and illustrates spurious contributions to the projected
DFT energies discussed in Sec. 3.6. As shown in the blown-up
inset in Fig. 6(b), these spurious contributions appear just
below the pole thresholds (i.e., for small negative values of
), and they can be quite large - of the order of several
tens of MeV. The gradual development of spurious contributions below
threshold has been explained in Sec. 3.6. Namely, if the contour
radius is only slightly greater than
, the branching point
associated with the pole
is always outside for all values of
. With increasing
, more and more branching points corresponding
to different regions of space fall inside the contour, leading to the
spurious behavior. As discussed earlier, one can eliminate this
subthreshold effect by taking equivalent contours discussed in the
context of Fig. 5. Such a procedure is illustrated by
a dotted line in the inset of Fig. 6(b).
The spurious contributions may result in large errors in the
projected PNP energies, making results of the standard PNP
calculations meaningless. Unfortunately, this is true not only for
Skyrme forces that use density-dependent terms of
fractional orders but also for the Gogny
force, which contains a density-dependent term of order .