Effective pseudopotential for energy density functionals with higher order derivatives

Abstract

We derive a zero-range pseudopotential that includes all possible terms up to
sixth order in derivatives. Within the Hartree-Fock approximation, it gives the
average energy that corresponds to a quasi-local nuclear Energy Density
Functional (EDF) built of derivatives of the one-body density matrix up to
sixth order. The direct reference of the EDF to the pseudopotential acts as a
constraint that divides the number of independent coupling constants of the EDF
by two. This allows, e.g., for expressing the isovector part of the functional
in terms of the isoscalar part, or vice versa. We also derive the analogous set
of constraints for the coupling constants of the EDF that is restricted by
spherical, space-inversion, and time-reversal symmetries.

Full text

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arXiv:1103.0682v2 [nucl-th] 17 Apr 2011
Effective pseudopotential for energy density functionals with higher order derivatives
F. Raimondi,1, ∗B. G. Carlsson,1,2and J. Dobaczewski1,3
1Department of Physics, P.O. Box 35 (YFL) FI-40014 University of Jyv¨ askyl¨ a, Finland.
2Department of Physics, Lund University, P.O. Box 118 Lund 22100, Sweden.
3Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, ul. Ho˙ za 69, PL-00-681 Warsaw, Poland.
(Dated: April 19, 2011)
We derive a zero-range pseudopotential that includes all possible terms up to sixth order in
derivatives. Within the Hartree-Fock approximation, it gives the average energy that corresponds
to a quasi-local nuclear Energy Density Functional (EDF) built of derivatives of the one-body density
matrix up to sixth order. The direct reference of the EDF to the pseudopotential acts as a constraint
that divides the number of independent coupling constants of the EDF by two. This allows, e.g.,
for expressing the isovector part of the functional in terms of the isoscalar part, or vice versa. We
also derive the analogous set of constraints for the coupling constants of the EDF that is restricted
by spherical, space-inversion, and time-reversal symmetries.
PACS numbers: 21.60.Jz, 21.30.Fe, 71.15.Mb
I. INTRODUCTION
One of the big challenges of the current research in nu-
clear structure physics is the search for a universal energy
density functional (EDF) [1]. Among different possible
approaches to this search, the consideration of a local or
quasi-local EDF based on the density-matrix expansion
(DME) is in recent years the object of intense studies
[2–7]. These aim at improving the classic work of Negele
and Vautherin [8, 9] and better theoretical understanding
based on the effective theory [10, 11] and on the frame-
work of the density functional theory [12].
In the recent work [2], we proposed a new expansion of
the nuclear energy density in higher-order derivatives of
densities. There, following the effective-theory approach,
a Skyrme-like quasi-local next-to-next-to-next-to-leading
order (N3LO) EDF was derived with terms of the EDF
constrained only by symmetry principles. In the present
derivation, we took the route in opposite direction as
compared to what has been done for the standard Skyrme
next-to-leading order (NLO) EDF. Namely, historically,
the Skyrme force has been initially proposed first as an
expansion of the effective interaction in relative momenta
up to second order [13, 14]. Next, for this force the aver-
age Hartree-Fock (HF) energy was evaluated, giving the
Skyrme EDF with half of the coupling constants con-
straint to the other half, see Ref. [15] for the modern
complete analysis. Only later, a possibility of releasing
these constraints was considered and studied, see, e.g.,
Ref. [16] for the analysis of the spin-orbit term.
In the present work we complete the results of Ref. [2]
by deriving the expansion of the effective interaction
in relative momenta up to N3LO. This generalizes the
Skyrme force up to sixth order and allows us to make
a link with the general N3LO EDF derived in [2]. One
∗Electronic address: francesco.raimondi@jyu.fi
should stress that the present analysis is not at all an
independent repetitive derivation of the same functional.
Indeed, the constraints on the EDF coupling constants,
which are induced by the HF averaging of this general-
ized force, cannot be obtained without following the path
presented in this study.
The complete higher-order EDFs or pseudopotentials
have never yet been applied in practical calculations. The
work towards this goal is now in progress, with basic
derivations like the ones of Ref. [2] and in the present
work coming first, the construction of numerical codes
like the one in Ref. [17] coming next, and the full ad-
justments of coupling constants that will follow. In this
respect, at present, we are in a similar phase of stud-
ies as before the chiral N3LO potentials for two-nucleon
systems were adjusted, see, e.g., Ref. [18], and after the
tools for calculating the corresponding N3LO diagrams
were developed, see, e.g., Ref. [19]. Nevertheless, studies
of particular higher-order EDF terms have already been
performed [20, 21].
In Ref. [22], the question of convergence of the series
in higher-order derivatives was recently addressed within
the DME applied to the Gogny non-local functional, and
it was shown that every next order up to sixth gives con-
tributions smaller by large factors. This gives us confi-
dence that fits of higher-order EDFs have a fair chance
of converging. A rigorous power counting scheme, anal-
ogous to what has been introduced in the chiral pertur-
bation theory [23], would have to use derivatives of reg-
ularized zero-range interactions, see, e.g. Ref. [10]. Such
a regularization would provide a proper cut-off scale,
against which the powers of derivatives could be esti-
mated. A good model of the regularized delta force is
the Gaussian interaction, which, however, leads (through
the exchange term) to non-local functionals. Within the
EDF methodology, an effective theory based on deriva-
tives of finite-range force is in principle possible, but has
not yet been tried because of the degree of numerical com-
plications involved. In the language of the effective field

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theory, the power counting scheme allows us to properly
classify diagrams of the perturbation series, however, the
ideas of an effective theory are much more general than
their applications in the field theory – here we use them
within the framework of standard quantum mechanics of
many-body systems.
The EDF description of nuclear states is phenomeno-
logical in the sense that it depends on the coupling con-
stants, which are usually fitted to available experimen-
tal data, see recent Refs. [24, 25] on fitting the second-
order (NLO) Skyrme functionals. Fits of the full N3LO
EDF are much more complicated because of strong inter-
dependencies of the coupling constants and instabili-
ties [26] occurring in certain regions of the parameter
space. Our main motivation to carry out the present
work was to find constraints on parameters of the general
EDF, which result from its relation to a pseudopotential.
Such a relation reduces the number of parameters that
have to be fit to data, and by this virtue is a positive
change, at least at the preliminary stage of adjustments.
Instead of fitting the coupling constants of the EDF,
it is also possible to derive them directly using the
DME [22]. The DME gives an EDF which approxi-
mates more complicated and time consuming HF cal-
culations based on finite-range forces. When applying
the DME, the relations to pseudopotentials are however
usually broken [4]. By enforcing these relations, as done
here, one ensures that the generated EDF is free from
unphysical self-interaction [27, 28] and can be applied
in beyond-mean-field applications without problems, see,
e.g., Refs. [29, 30].
By following the standard convention, here we call the
generalized Skyrme force pseudopotential, which is the
name denoting a quasi-local operator depending on spa-
tial derivatives. We also consequently use the names
’parameters’ to denote numerical coefficients of different
terms of the pseudopotential, and we use the names ’cou-
pling constants’ to denote numerical coefficients of terms
in the EDF.
The paper is organized as follows. In Sec. II we con-
struct the pseudopotential in two alternative forms and
list all its terms up to N3LO. We also evaluate the con-
straints imposed by the gauge symmetry. In Sec. III we
discuss the procedure of HF averaging to obtain the EDF
from the pseudopotential. In particular, in Sec. IIIA we
derive the general relations connecting the parameters
of the Galilean-invariant pseudopotential to the coupling
constants of the EDF, whereas in Sec. IIIB we derive the
constraints for the case of conserved gauge symmetry. In
Sec. IV we reduce our results to the case of the conserved
spherical, space-inversion, and time-reversal symmetries.
After formulating the conclusions of the present study
in Sec. V, in Appendices A–C we present derivations
related to the time-reversal invariance and hermiticity
of the pseudopotential, we list results pertaining to the
gauge-invariant pseudopotentials, and we give relations
between the two alternative forms of pseudopotentials.
Results obtained in the present work that are too volu-
minous to be published in the printed form are collected
in the supplemental material [31].
II.
PSEUDOPOTENTIAL IN THE
SPHERICAL-TENSOR FORMALISM
GENERAL FORM OF THE
A. Central-like form of the pseudopotential
The Skyrme interaction is one of the most impor-
tant phenomenological effective interaction used in mi-
croscopic nuclear structure calculations: such two-body
interaction is a short-range expansion up to the second
order in derivatives, which contains a certain number of
fit parameters adjusted to reproduce the experimental
data. In the literature the Skyrme interaction is usually
written in cartesian representation, but for our extended
pseudopotential we adopt the spherical-tensor represen-
tation of operators [32], whose building blocks can be
found in [2].
Depending on the specific form of the coupling of the
derivative operators with the spin operators, different
ways to construct the pseudopotential are possible. A
particular form of the pseudopotential, which we call
central-like or LS-like, is constructed in the present Sec-
tion. It is based on coupling together the derivative oper-
ators and spin operators, which are then coupled to rota-
tional scalars. An alternative form, called tensor-like or
JJ-like, is presented in Section IID. There, each deriva-
tive operator is coupled with one spin operator, and then
they are coupled together to rotational scalars.
In the central-like form, the pseudopotential is a sum
of terms,
ˆV =
?
˜ n′˜L′,
˜ n˜L,v12S
C˜ n′˜L′
˜ n˜L,v12SˆV˜ n′˜L′
˜ n˜L,v12S, (1)
where the sum runs over the allowed indices of the tensors
according to the symmetries discussed below. Each term
in the sum is accompanied by the corresponding strength
parameter C˜ n′˜L′
˜ n˜L,v12S, and explicitly reads,
ˆV˜ n′˜L′
˜ n˜L,v12S=
1
2iv12???K′
+(−1)v12+S??K′
×
˜ n′ ˜L′K˜ n˜L
?
SˆSv12S
?
0
˜ n˜LK˜ n′ ˜L′
?
SˆSv12S
?
0
?
?
1 −ˆPMˆPσˆPτ?ˆδ12(r′
1r′
2;r1r2).(2)
In Eq. (2), K˜ n˜Lare the spherical tensor derivatives of
order ˜ n and rank˜L built of the spherical representations
of the relative momenta k = (∇1− ∇2)/2i,
k1,µ={−1,0,1} = −i
?
1
√2(kx− iky),
kz,−1
√2(kx+ iky)
?
; (3)

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TABLE I: Derivative operators KnL up to N3LO as expressed
through spherical tensor representation of relative momenta
k defined in Eq. (3).
No. tensor KnL
order n rank L
1100
2
k
11
3[kk]0
[kk]2
[kk]0k
[k[kk]2]3
[kk]2
20
422
531
633
7
0
40
8[kk]0[kk]2
[k[k[kk]2]3]4
[kk]2
42
944
10
0k
51
11[kk]0[k[kk]2]3
[k[k[k[kk]2]3]4]5
[kk]3
[kk]2
53
1255
13
0
60
14
0[kk]2
62
15[kk]0[k[k[kk]2]3]4
16 [k[k[k[k[kk]2]3]4]5]6
64
66
up to sixth order they are listed in Table I. Similarly,
operators K′
(∇′
The symmetrized two-body spin operatorsˆSv12S are
defined as,
˜ n˜Lare built of the relative momenta k′=
2)/2i.
1− ∇′
ˆSv12S=?1 −1
where v12 = v1+ v2 and σ(i)
components of the rank-v Pauli matrices acting on spin
coordinates of particles i = 1 or 2. They are expressed
as
2δv1,v2
??
[σ(1)
v1σ(2)
v2]S+ [σ(1)
v2σ(2)
v1]S
?
, (4)
vµ are the spherical-tensor
σ(i)
00=ˆ1,(5)
σ(i)
1,µ={−1,0,1}= −i
?
1
√2
?
?
σ(i)
x− iσ(i)
σ(i)
y
?
??
,
σ(i)
z,−1
√2
x + iσ(i)
y
(6)
through the spin unity matrixˆ1 and the standard Carte-
sian components of the Pauli matrices σ(i)
The Dirac delta function,
x,y,z.
ˆδ12(r′
1r′
2,r1r2) = δ(r′
1−r1)δ(r′
1−r2)δ(r′
2−r2)δ(r1−r2)
2−r1)δ(r2−r1).
= δ(r′
(7)
ensures the locality and zero-range character of the pseu-
dopotential. The action of derivatives K˜ n˜Land K′
onˆδ12(r′
dard sense of derivatives of distributions. Whenever the
˜ n˜L
1r′
2,r1r2) has to be understood in the stan-
pseudopotential (1) is inserted into integrals to calculate
the two-body matrix elements, the integration by parts
transfers the derivatives onto appropriate wave functions
in the remaining parts of integrands.
The exchange term is explicitly embedded in the pseu-
dopotential through the operator
ˆPMˆPσˆPτ= (−1)˜ n′ 1
+√3
?
4
?
1 +
√3
?
σ(1)
1σ(2)
1
?
0
τ(1)
1τ(2)
1
?0
+ 3
?
σ(1)
1σ(2)
1
?
0
?
τ(1)
1τ(2)
1
?0?
,(8)
where τ(i)
matrices defined analogously as in Eq. (6). The square
brackets with superscripts and subscripts denote the cou-
pling of spherical tensors in the isospin space and co-
ordinate space, respectively. The above definitions and
conventions exactly correspond to those introduced in
Ref. [2].
1
are the standard spherical-tensor isospin Pauli
The zero range of the pseudopotential has an impor-
tant bearing on the structure of terms in Eq. (2). Indeed,
only for the zero-range force, the space-exchange (Majo-
rana) operatorˆPMcan be replaced, in any individual
term, by the phase (−1)˜ n′appearing in Eq. (8). More-
over, apart from the isospin-exchange operatorˆPτ, terms
of the pseudopotential cannot then depend on isospin.
This fact, effectively reduces by half the number of al-
lowed terms of the pseudopotential, as compared to what
would have been possible for a finite-range potential.
This is at the origin of the numbers of allowed terms
of the pseudopotential being equal one half of the num-
bers of the allowed terms of the EDF, which we discuss
below.
The full antisymmetrization of the pseudopotential in-
cludes the exchange operator in the isospin space; there-
fore, in the following we consider the EDF with the
isospin degree of freedom included, that is, we discuss
both the isoscalar and isovector terms of the N3LO [2],
which allows us to fully incorporate the proton-neutron
mixing at the level of the energy density [15].
The general form of the pseudopotential and the al-
lowed terms listed below reflect the fact that the funda-
mental symmetries of the two-body interaction must be
respected, see Appendix A. In particular, (i) all terms are
scalar operators, that is, they are coupled to the total an-
gular momentum 0, which ensures the rotational invari-
ance, (ii) the total number of derivative operators must
be even, namely, ˜ n + ˜ n′= 0,2,4,6, which ensures the
time-reversal and parity invariances, (iii) the parameters
C˜ n′˜L′
˜ n˜L,v12Sof the pseudopotential must be real, to guar-
antee both the time-reversal invariance and hermiticity,
and (iv) the invariance under exchange of the coordinates
of particle 1 and 2 is respected by expression (2).

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TABLE II: Zero-order terms of the pseudopotential (2).
No.˜ n′
˜L′
˜ n
˜Lv12
S
gauge
1000000Y
2000020Y
TABLE III: Same as in Table II but for the second order
terms.
No.˜ n′
˜L′
˜ n
˜Lv12
S
gauge
1200000Y
2200020Y
3220022Y
4111100Y
5111120Y
6111111Y
7111122Y
B. Lists of terms of the pseudopotentialˆV order by
order
In Tables II-V are listed, respectively, all possible terms
of the pseudopotential (1) in zero, second, fourth, and
sixth order. In each order, the numbers of terms equal 2,
7, 15, and 26, giving the total number of 50 terms up to
N3LO. We see that these numbers of terms are exactly
equal to those corresponding to the EDF in each isospin
channel with the Galilean invariance imposed, cf. Ta-
ble VI of Ref. [2]. One should note that each term of
the pseudopotential (2) is Galilean-invariant by construc-
tion, because it is built with relative-momentum opera-
tors K˜ n˜L; therefore, the pseudopotential is not changed
by a transformation to a system moving with a constant
velocity. When both isoscalar and isovector channels are
considered in the EDF, the number of EDF terms be-
comes in each order twice larger than the number of terms
of the pseudopotential.
This means that the EDF obtained by averaging the
pseudopotential is constrained by as many conditions as
there are terms in each isospin channel. One possible
solution is than to find a one-to-one correspondence be-
tween the EDF and the pseudopotential by relating the
isoscalar part of the EDF to its isovector part, in a way
that will be showed explicitly in the following Sections of
this work.
To make the connection between the pseudopotential
and the standard form of the Skyrme interaction more
transparent, we give here the relations of conversion be-
tween the parameters of the zero- and second-order pseu-
dopotential and those of the Skyrme interaction, see
TABLE IV: Same as in Table II but for the fourth order terms.
No.˜ n′
˜L′
˜ n
˜Lv12
S
gauge
1400000D
2400020D
3420022D
4311100Y
5311120Y
6311111N
7311122D
8331122I
9202000D
10202020D
11222022D
12222200I
13222220I
14222211N
15222222I
Ref. [15] for the definitions used. They read,
t0 = C00
00,00+
1
√3C00
00,20, (9a)
t0x0 = −2
√3C00
1
√3C20
00,20, (9b)
t1 =
00,00+1
3C20
00,20, (9c)
t1x1 = −2
3C20
1
√3C11
00,20, (9d)
t2 =
11,00+1
3C11
11,20, (9e)
t2x2 = −2
3C11
1
√6C11
1
3√5C11
1
3√5C22
11,20, (9f)
W0 =
11,11, (9g)
to = −
11,22, (9h)
te = −
00,22. (9i)
In relations of Eqs. (9), parameters t3 and t3x3 are
missing: they are related to the terms of the Skyrme
interaction depending on density, which have been in-
troduced to mimic the effects of the three-body force in
the phenomenological interaction and to get the satura-
tion feature of the nuclear force. In the same way, the
zero-order parameters C00
tential, see Eqs. (9a) and (9b), should become density-
dependent.
In his effective nuclear potential, Skyrme also intro-
duced [14] one additional term of the fourth order, which
he justified through the presence of considerable D-
waves in the nucleon-nucleon interaction energies around
00,00and C00
00,20of the pseudopo-

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TABLE V: Same as in Table II but for the sixth order terms.
No.˜ n′
˜L′
˜ n
˜Lv12
S
gauge
1600000D
2600020D
3620022D
4511100D
5511120D
6511111N
7511122D
8531122I
9402000D
10402020D
11422022D
12402222D
13422200I
14422220I
15422211N
16422222D
17442222I
18313100D
19313120D
20313111N
21313122D
22333122D
23333300I
24333320I
25333311N
26333322D
100MeV. Also in this case, we give the relation between
the corresponding parameter tD and the parameter of
our full pseudopotential,
tD=1
2C00
20,20. (10)
C.Gauge invariance of the pseudopotential
Besides the Galilean invariance mentioned above, the
standard Skyrme force has been also proved to be invari-
ant with respect to a more general local gauge invariance,
and to give rise to the energy density that is invariant un-
der the same symmetry when specific relations between
the coupling constants are set [33, 34].
The gauge transformation acts on a many-body wave
function by multiplying it with a position-dependent
phase factor, that is,
|Ψ′? = exp

i
A
?
j=1
φ(rj)

|Ψ?, (11)
and its action transferred onto the pseudopotential is,
ˆ V′= e−iφ(r′
2)e−iφ(r′
1)ˆV eiφ(r1)eiφ(r2). (12)
Apart from zero order, the terms of the pseudopoten-
tial are not trivially invariant with respect to the trans-
formation of the Eq. (12) and, in general, the transformed
pseudopotentialˆV′is different than the original pseu-
dopotentialˆV . To impose the gauge invariance on the
pseudopotential, one has to derive a list of constraints
among the parameters, which can be done using the con-
dition
[φ(r1),ˆV ] + [φ(r2),ˆV ] = 0. (13)
As expected, at second order, all the 7 terms of the pseu-
dopotential listed in Table III fulfill condition (13). Then
they all are the stand-alone gauge invariant terms of the
pseudopotential, which in the last column of the Table
is marked by the letter Y. On the other hand, at fourth
order, only two of the terms of the pseudopotential listed
in Table IV, those that correspond to parameters C31
and C31
11,20, fulfill condition (13). At sixth order, none of
the terms are stand-alone gauge invariant.
At fourth order, the gauge invariance forces seven pa-
rameters of the pseudopotential to be specific linear com-
binations of four independent ones. In Table IV, they are
marked by letters D and I, respectively. In Appendix B,
we list such relations between the dependent and inde-
pendent parameters. One should note that other choices
of the four independent parameters are also possible, that
is, at fourth order, there are simply four different gauge-
invariant linear combinations of terms of the pseudopo-
tential (1). Moreover, at this order, there are also two
terms that alone are gauge non-invariant – those that
correspond to parameters C31
they are marked by letters N. Similarly, at sixth order,
there are six gauge-invariant linear combinations of terms
of the pseudopotential, that is, sixteen dependent param-
eters are related to six independent ones, see Appendix B,
and there are also four alone gauge non-invariant terms
corresponding to parameters C51
C33
33,11.
A comparison between the numbers of terms of
the Galilean-invariant pseudopotential and the gauge-
invariant pseudopotential is plotted in Fig. 1. Again we
note that at each order, the numbers of gauge-invariant
parameters (2 for the zero order, 7 for the second order,
6 for the fourth order, and 6 for the sixth order) are ex-
actly the same as the numbers of independent coupling
constants of the EDF in each isospin channel with the
gauge invariance imposed, cf. Table VI of Ref. [2]. Again,
this observation will be crucial when we proceed to derive
11,00
11,11and C22
22,11; in Table IV,
11,11, C42
22,11, C31
31,11, and