A prerequisite for the calculation of the integrals in Eqs. (2), (10), and (13) is the isospin rotation of the Slater determinant. To this end, one has to perform an independent rotation of s.p. neutron,
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(19) |
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(20) |
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(21) |
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(22) |
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(23) |
To calculate the kernel
of an arbitrary
operator
between two non-orthogonal Slater determinants,
we apply the generalized Wick's theorem,
see, e.g., Ref. [39]. In particular, the norm kernel
can be written in a compact form:
To calculate kernels appearing in the projection formalism, one needs to
invert the
overlap matrix
. This can cause serious problems due to the presence of singularities [22,40].
The regularization of kernel singularities is a difficult problem [23]. Thus far, a
regularization scheme has been worked out only for a very specific class
of functionals (or effective density-dependent
interactions) solely involving integer powers of local densities [23].
Unfortunately, almost all commonly used Skyrme and Gogny parameterizations, except for SIII [41],
involve fractional powers of the density. The appearance of singularities prevents us
from using the local Slater approximation for the
Coulomb exchange. In the present work we treat it exactly using the method of the Gaussian
decomposition of the Coulomb interaction, as described in
Refs. [42,43].
Compared to the particle number
or the angular momentum projection schemes, isospin projection is a relatively simple procedure. In particular,
the dependence of the inverse matrix
on the isorotation angle
can be determined
analytically and this enables us to demonstrate that
the isospin projection is free from kernel singularities. To this end, we write
the overlap matrix (25)
in the form:
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(28a) |
The SVD decomposition allows us to analytically diagonalize the overlap matrix
. Without loss of generality we assume
that
. Hence, the product
can be written as:
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(30) |
The
matrix, on the other hand, is the
matrix
completed to the dimension
by zeros:
Since the first and third matrices on the right hand side of Eq. (29) are unitary and the second matrix is diagonal, the inverse of the overlap matrix reads:
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(32) |
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(33) |