Let us now analyze the terms in the DFT energy density that
depend on fractional powers of the local density. In many
functionals related to the Skyrme interaction, and for the Gogny
force, such terms are quite often postulated, both in the
particle-hole and pairing channels (see Ref. [12]
for a review). In particular, the familiar density-dependent term of the
Skyrme force, which is proportional to
, produces
a contribution of the order of
to the DFT energy density.
Similarly, the density-dependent, zero-range term of the Gogny force
yields a contribution to the DFT energy density that is of
order,
provided the particle-hole and pairing terms are consistently added.
By taking into account the degeneracy factors
discussed
in Sec. 3.3, the resulting poles are of the order of
and
, respectively. Since typical values of
are between 0 and 1, the DFT transition energy density always has poles
at
for non-degenerate
states (
). But more importantly, the fractional
powers lead to the multivalued DFT transition energy density on
the complex-
plane.
In the standard treatment of fractional powers of a complex function,
cuts along the negative real axis must be introduced. In order to
apply this procedure to fractional powers of the local transition
density (46), we must identify on the complex
plane the
lines along which
is real negative.
Obviously,
is real positive along the real
axis
and real along the imaginary
axis. In order to simplify the
discussion, let us assume that the sum in Eq. (46) is finite,
which is always the case in any practical calculation. In
such a case,
has a finite number, say
, of
different first-order poles along the positive imaginary axis, and
the same number
of poles along the negative imaginary axis.
Moreover, since all coefficients in Eq. (46) are positive,
must have a first-order zero between each
pair of poles on the positive imaginary axis, and similarly on the
negative imaginary axis. Since
also has a
second-order zero at
, we conclude that it has altogether
zeros on the imaginary axis. It is also obvious that
is a rational function with a
-order polynomial in the
numerator, and thus we conclude that all the zeros of
are located on the imaginary axis. Therefore, the cuts for possible
fractional powers
must be located along the imaginary
axis, and connect zeros of
with its adjacent poles.
The above discussion is visualized in Fig. 2. The left
portion shows schematically the transition density
along the imaginary
axis
oriented
vertically.
The plot illustrates the transition density (46)
in one selected point of space
, i.e., values of wave
functions at
enter only as numerical coefficients.
There appear four poles
and three zeros of
on the positive imaginary
axis, the same number of poles and zeros on the negative imaginary
axis, and the second-order zero at the origin. Sections of the imaginary
axis where the density is negative are shaded. In the
right portion of Fig. 2 we show poles (crossed circles) and zeros
(full dots) of the transition density on the complex
plane, along
with the three integration contours
,
, and
discussed above.
The cuts in the complex
-plane connecting zeros and poles,
corresponding to
real negative values of
, are indicated by vertical
segments.
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We are now ready to discuss contour integration of terms
depending on fractional powers of the transition density.
Contour shown in Fig. 2 crosses the imaginary axis in
sections where there is no cut, and thus it always stays on the same Riemann
sheet. On the other hand, contours
and
of Fig. 1, or contours
and
of Fig. 2,
cross the imaginary axis by passing through cuts onto another
Riemann sheet.
Since the transition density (46) is
an even function of
, the phase of the fractional power
of the transition density increases or decreases by
when
going across each of the two cuts. Therefore, after returning
to
,
is multiplied by
,
and thus it is not a continuous function at
, unless
. This is quite unacceptable as the presence of the
phase creates
serious problems in interpreting the projected DFT energies
(34) (see, e.g., the sum rule condition discussed in
Sec. 3.5).
Formally, by using powers of square roots in density-dependent terms,
i.e., , one can guarantee that the integration contours
return onto the original Riemann sheet, and that the transition energy
density is a continuous function of
. However, even in such a case,
one important property of the DFT transition energy density (33)
is lost, namely, the density-dependent term in the energy density
becomes an odd function of
, and the corresponding term in the
integrand (35) becomes an even function of
. This is
so, because the square root has opposite signs on the two Riemann sheets
in question. Consequently, contour integrals of such terms would vanish
and the density-dependent terms would yield zero contribution to the
PN-projected energy. This is a rather disastrous result. Hence, we are
forced to conclude that the use of continuous contours for fractional
powers is not a viable prescription for constructing the projected DFT
energies.
Let us now discuss the way of
evaluating contour integrals in all
practical PNP calculations
up to now. Unfortunately,
such calculations have always disregarded
the analytic structure of the underlying integrands.
In fact, the
fractional powers of transition density,
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The contour can be closed by adding a piece that goes around the zero
of the transition density
. This is illustrated in
Fig. 5 that shows a modification
of contour
of Fig. 2
near the positive imaginary
-axis.
[An analogous
mirror-like detour is made near the negative imaginary axis.]
The resulting contour
always stays on the same Riemann sheet and, therefore, the integration
result does not depend on the radius.
On the other hand, the contribution due to the
additional path surrounding the zero is affected by
the discontinuity of
the integrand along the cut. Such a discontinuity in the transition
density
is shown in the top
panel of Fig. 4 for
=1/6 (52). Since the
ordinary density (
) is real and positive, its fractional power
is also real and positive. On the other hand, transition densities
(46) corresponding to
with
, i.e., near the positive imaginary axis, are
complex. While their real parts are practically identical
on both sides of the cut, their
imaginary parts have opposite signs; hence, a
discontinuity is encountered. [Of course, since the directions
of integration are opposite, the contributions
to the PNP energy from both segments
of the additional path
are identical.]
When transforming the contour integration in Eq. (34) into the
integral along the imaginary axis , one must do a change of
variables from
to
. This introduces the additional factor
in the integrand. For that reason, for even
, the
discontinuity in the imaginary part of the density-dependent term
of fractional order contributes to the real part of the projected
DFT energy. As the discussion in Sec. 3.4 proves, the same
holds for odd values of
.
Figure 5 also shows
the circular contour lying between
the zero of
and the previous pole
.
This contour is formally
equivalent to the deformed contour
but it is easier to handle
in practical applications. The radius of
must be slightly greater
than
and smaller than the lowest zero of
,
minimized over the whole space
,
associated with the branching point corresponding to
. The use
of contour
guarantees that the integration of fractional-order
terms is done properly.
Altogether, blind application of prescription (52) can lead to spurious and entirely uncontrolled contributions to the projected DFT energies. Excepting Ref. [23], this fact has been entirely overlooked in all practical applications of the PNP method to date, and casts serious doubts on the reliability of the obtained results. Largest contributions are, of course, obtained when the integration contour passes slightly below a pole of the DFT transition energy density. For the Skyrme functionals in Sec. 4.2, we present specific examples of such situations.
The appearance of spurious contributions is, in fact, independent of
the order of divergence at the pole. Therefore, it also shows up for
``integrable" poles, diverging with powers of
,
discussed for the Gogny force in Ref. [37].