The NLO quasilocal functional

We are now in a position to discuss the CE for the NLO
quasilocal functional introduced by Carlsson
*et al.* [Carlsson et al.(2008)Carlsson,
Dobaczewski, and Kortelainen]. By imposing on the functional the
gauge-invariance conditions, we can then confirm and explicitly
rederive the results of Sec. 2.2. The explicit
derivation will also allow us to discuss the CEs for densities in
other spin-isospin channels
analyzed in Sec. 2.2.2.

Below we consider the EDF given in terms of a local integral
of the energy density
,

To lighten the notation and avoid confusion with the isospin index , in this section we do not explicitly show the time argument of densities, which within the TDDFT all depend on time.

The quasilocal NLO EDF was constructed [Carlsson et al.(2008)Carlsson, Dobaczewski, and Kortelainen] by building the and potential-energy densities from isoscalar and isovector densities, respectively, and their derivatives up to sixth order. For clarity, we give here a brief summary of definitions and notations used in this construction.

The local higher-order primary densities are defined by the coupling
of relative-momentum tensors [Carlsson et al.(2008)Carlsson,
Dobaczewski, and Kortelainen] with nonlocal
densities (29) to total angular momentum , that is,

where the local secondary densities, , are obtained by acting with derivatives on primary densities and coupling them to total angular momentum . Each term (35) is multiplied by the corresponding coupling constant that is denoted by the same set of indices as those in the term itself.

We note here that the definition of the isovector terms depends on
whether one uses Cartesian or spherical
representation of tensors in isospace. On the one hand, the use of the standard
Cartesian representation, see, e.g., Refs. [16,Carlsson et al.(2008)Carlsson,
Dobaczewski, and Kortelainen],
implies that the isovector terms depend on products of differences of
neutron and proton densities. On the other hand, the use of the
spherical representation, which was assumed in Ref. [Raimondi et al.(2011)Raimondi, Carlsson,
and Dobaczewski]
and is also used in the present study, involves the coupling of two
isovectors to a scalar, whereby there appears a Clebsch-Gordan
coefficient of
. Therefore, for the
isospace spherical representation, the isovector coupling constants
are by the factor of
*larger* than those
for the Cartesian representation.

In the remaining part of this section, we employ the compact notation
introduced in Ref. [11], whereby the grouped indices, such
as the Greek indices
and the Roman
indices
, denote all the quantum numbers of
the local densities
and derivative operators
, respectively. In this notation, the NLO
potential-energy density of Eq. (33) reads

Our following discussion of the CE is mainly focused on the
one-body potential-energy term, defined in Eq. (16) as
the variation of the potential energy with respect to the density
matrix. For the NLO functional, this term was derived
in Ref. [11], where it was shown that in space coordinates
it has the form of a one-body pseudopotential,

In turn, potentials
were derived as linear combinations
of the secondary densities,

For the one-body pseudopotential (37),
the Schrödinger equation that gives the time evolution of
single-particle Kohn-Sham wave functions in space coordinates reads,

By multiplying the Schrödinger equation with the complex-conjugated wave function, and summing over the single-particle index we obtain the time-evolution equation of the density matrix (14), that is,

Before we proceed, we must first consider the complex-conjugated
pseudopotential
. To this
end, we use the property of the Biedenharn-Rose phase convention employed in
Refs. [Carlsson et al.(2008)Carlsson,
Dobaczewski, and Kortelainen,11], by which all scalars are always real.
Note that for the spherical representation of Pauli matrices,
the Biedenharn-Rose phase convention implies the transposition
of spin indices, that is,

Finally, in Eqs. (38) and (39), the complex
conjugation only affects coefficients
[11],
which gives,

and

It means that in all further derivations we must use the second set of potentials with signs of terms modified according to the phase . It is now obvious that the CEs will hold independently of the spin-isospin coordinates if, and only if, the pseudopotentials fulfill the condition

We are now in a position to separate the four
spin-isospin channels in Eq. (41). We do so by multiplying both sides of the
equation with
and
summing over
. From Eq. (29)
it is then obvious that, in close analogy to Sec. 2.1,
after setting
, we obtain the CEs (30)
in the four spin-isospin channels,
provided terms coming from one-body pseudopotentials do
not contribute, as in Eq. (46). When evaluating this
condition for the four spin-isospin channels, we use the
expression for the trace of three Paul matrices in spherical
representation, which reads [17],

For our practical implementation of the CE condition (48), we proceed by transforming the two differential operators, and , which act on two different variables and , respectively, with the recoupling methods developed in Ref. [11], and we obtain,

On the right-hand sides, operators are the higher-order spherical tensor derivatives [Raimondi et al.(2011)Raimondi, Carlsson, and Dobaczewski] built of the relative momenta, , and operators act on variable