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For the symmetric spin-saturated homogeneous nuclear matter, the EDF coupling constants expressed in terms of the pseudopotential parameters can easily be obtained from those derived for finite nuclei. Indeed, in this former system, isovector densities are zero because of the equality of proton and neutron densities, gradients of local densities are zero because of its homogeneity, and all tensors of rank greater than zero vanish because of its isotropy.
At zero-order, expressions relating the EDF non-vanishing coupling constants
to pseudopotential parameters can be read from the first and fifth lines
of Eq. (17). They explicitly read,
For higher-order terms of the EDF,
we note that terms depending on gradients 
of nonlocal densities
do contribute provided that they are coupled to a scalar. There are 1, 4, and 8
such a terms at second, fourth, and sixth order, respectively;
below they are
shown in bold face. For these
densities, the local limit does not give rise to the cancellation of
the corresponding term of the functional.
At second-order, we obtain three nonvanishing coupling constants,
which are given by,
At fourth (sixth) order, there are 8 (12) non-vanishing
coupling constants, which are linear
combinations of the 16 (24) parameters corresponding to the central terms of the pseudopotential. These lengthy
expressions are collected in the supplemental material. Here we only list
these non-vanishing coupling constants, which at fourth and sixth order are
,
,
,
,
,
,
,
and
,
,
,
,
,
,
,
,
,
,
,
, respectively.
