Extended Thomas–Fermi approximation to the one-body density matrix

Nuclear Physics A 665 Ž2000. 291–317

Extended Thomas–Fermi approximation to the one-body density matrix V.B. Soubbotin a , X. Vinas ˜

b

a

Nuclear Physics Department, Physical Research Institute, St. Petersburg UniÕersity, St. Petersburg, Russian Federation b Departament d’Estructura i Constituents de la Materia, UniÕersitat de Barcelona, ` Facultat de Fısica, ´ Diagonal 645 E-08028 Barcelona, Spain Received 18 February 1999; received in revised form 10 September 1999; accepted 21 October 1999

Abstract The one-body density matrix is derived within the Extended Thomas–Fermi approximation. This has been done starting from the Wigner–Kirkwood distribution function for a non-local single-particle potential. The links between this new approximation to the density matrix with former approaches available in the literature are widely discussed. The semiclassical counterpart of the Hartree–Fock energy at the Extended Thomas–Fermi level is also obtained in the case of a non-local one-body Hamiltonian. The semiclassical binding energies and root mean square radii are compared with those obtained using the Strutinsky averaged method. The full Hartree–Fock values are compared with those obtained using the Kohn–Sham scheme based on the different approximations to the density matrix considered in the text. Numerical applications are performed using the Gogny, Brink–Boeker and BDM3Y1) effective interactions. q 2000 Elsevier Science B.V. All rights reserved. PACS: 21.60.yn; 21.60.Jz; 21.90.qf

Keywords: Nuclear structure; Semiclassical methods in Nuclear Physics; Density Functional Theory; Extended Thomas–Fermi approach

1. Introduction The one-body density matrix ŽDM. r Žr,r’ . s Ý a fa) Žr. fa Žr’ . or equivalently its Wigner transform, the distribution function f ŽR,p. Ždefined below., plays a crucial role in Hartree–Fock ŽHF. calculations. If zero-range Skyrme forces w1x are used, only the diagonal part of the DM is needed. However, full knowledge of r Žr,r’ . Žor f ŽR,p.. is necessary if one considers finite-range effective nuclear forces which are derived from G-matrix calculations in nuclear matter through the local density approximation w2–5x or postulated empirically with parameters fitted to reproduce various properties of nuclear matter and finite nuclei w6,7x. 0375-9474r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 5 5 8 - 8

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For a finite nucleus the full calculation of the density matrix Žor the distribution function. is not an easy task and requires some computational effort w7,8x. Although the HF technique is well developed for effective forces with Gaussian formfactor w7x, the use of effective forces of a different type is rather limited. Consequently, approximations which simplify the calculation and, at the same time, show more clearly the physical content of the DM are in order. The simplest approach is to replace locally the non-diagonal part of the DM by its expression in nuclear matter ŽSlater or Thomas–Fermi approach.. A more elaborate approach developed by Negele and Vautherin ŽNV. w9–11x consists of expanding the DM in such a way that the leading contribution corresponds to the Slater approximation and the corrective terms take into account finite size effects. Another different approach is due to Campi and Bouyssy ŽCB. w12,13x. In this case the DM is taken in the Slater form, but with an effective Fermi momentum which partially resumates the corrective terms to the density matrix expansion ŽDME. of NV. Very recently, the CB approach has been applied to HF calculations of finite nuclei w14x using a density-dependent version of the M3Y interaction w4,5x. On the other hand, semiclassical methods w15x are very useful for describing nuclear properties of a global character such as binding energies or nuclear densities and their moments. Concerning the nuclear ground state properties at HF level, semiclassical approaches are based on the Wigner–Kirkwood ŽWK. "-expansion of the distribution function, which for a set of nucleons moving in a local external potential V up to second order is given by w15x "2 f WK Ž R,p . s Q Ž l y HW . y 1 q m

8m

DVd X Ž l y HW . q

"2 24 m

2 Ž p P = . V d XX Ž l y HW . q O Ž " 4 . ,

Ž =V .

2

Ž 1.

where l is the chemical potential and HW is the Wigner transform w15x of the one-body Hamiltonian, which reads

HW s

p2 2m

q V Ž R. .

Ž 2.

The semiclassical WK distribution function is a representation of the true phase space in terms of distributions. Notice that Eq. Ž1. does not contain any shell structure because it only includes the semiclassical Hamiltonian HW . Due to the fact that Eq. Ž1. is an asymptotic expansion in ", the quantal result can never be recovered even by adding higher order contributions. Thus, f ŽR,p. given by Eq. Ž1. has to be considered as a representation, in terms of distributions of the Strutinsky distribution function w16–18x, that consists in a well-defined mathematical procedure to average out the shell effects. This can be seen as follows: if the ground-state energy of a set of nucleons moving in an external potential, for instance Woods–Saxon or harmonic oscillator ŽHO. type, is obtained starting from Eq. Ž1. one recovers almost the same value as if the Strutinsky averaged method w19–21x is used. In spite of its distribution character, f ŽR,p. is very

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efficient in order to obtain semiclassical expectation values by integrals over the whole phase-space w22,23x. Starting from the WK distribution function one can derive the so-called Extended Thomas–Fermi approximation ŽETF. Žw24–26x and references therein. to the kinetic energy density. The ETF approach together with Skyrme forces have been widely used in nuclear structure calculations w24,27x as well as in Heavy-Ion physics w27x. These ETF calculations are especially well suited in situations where the full HF calculations are rather cumbersome or if one is interested in separating shell effects from the average part of the energy. For instance, in the derivation of the Liquid Droplet Mass Formula w28–30x associated with a given effective nuclear force. Although in this case the surface energy can be obtained quantally, the full curvature energy can only be obtained in the semiclassical framework w31,32x. Another typical example where semiclassical techniques can still be competitive are finite temperature calculations. Of course, they can be performed at the full HF level if the subtraction procedure w33,34x is taken into account. Due to the fact that temperature washes out shell effects, semiclassical calculations using a trial density can be sufficient for describing a hot nucleus at high temperature where the subtraction procedure in the HF approach is unavoidable. Also, the ETF approach can be used for obtaining the Liquid Drop Model fission barrier where shell contributions are suppressed w24,35x at zero and finite temperature. In the astrophysical context, the ETF approximation is again very useful because it allows drop and bubble configurations to be easily obtained and consequently to study phase transitions in hot dense nuclear matter w36x. In Heavy-Ion physics, the ETF approach is also very helpful for obtaining the real part of the nucleus–nucleus optical potential at zero and finite temperature within the energy density formalism. In this case the energy density of the overlapping system is easily written using the ETF approach. The ETF approach can also be applied to obtain the Heavy-Ion potential using the Double Folding Model w37x. For instance, starting from a local proton density obtained from the electron scattering data, which is considered as an independent input, one can restore the full DM using the ETF approach to calculate the exchange contribution. This wide scenario of applicability of the ETF approximation is actually almost restricted to the use of zero-range forces of the Skyrme type. We think that to generalize the ETF approach to the case of finite-range forces is still an interesting and open problem. To establish the grounds and ability of the ETF approach in the case of non-local potentials is the main purpose of this paper. To do this we first derive the ETF DM for non-local potentials starting from the very recently presented WK expansion up to " 2 order of the distribution function for non-local potentials w23x. It is important to note that the lowest order of the distribution function expansion Eq. Ž1. corresponds in coordinate space to the DM in the Thomas–Fermi ŽSlater. approach w15x, as happens in the DME of NV and CB. This fact strongly motivates us to study the " 2-contribution of the ETF DM and point out the interrelation between this new semiclassical expansion and the former expansions of NV and CB. The next step is to obtain the semiclassical counterpart of the ground-state HF binding energy in the case of finite range forces. It should be pointed out that, as happens for Skyrme forces, the ETF HF energy density can also be written as a functional of the local particle density r . This leads to the so-called density functional theory ŽDFT., the basic idea of which is that the ground-state energy of a system of

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interacting fermions can be expressed by an integral over the whole space of a energy density ´ w r x, which depends only on the ground-state local density. The theoretical justification of DFT is provided by the Hohenberg–Kohn theorem w38x which states that the exact non-degenerate ground-state energy of a correlated electron system is a functional of the local density r and that this functional has its variational minimum when evaluated with the exact ground-state density,

d dr

H´ w r x dr y lHr Ž r. dr

s0 ,

Ž 3.

where l is a Lagrange multiplier to ensure the right number of particles. Unfortunately, the energy density ´ w r x is not exactly known and approximations are in order. One possibility is to replace the exact energy density ´ w r x by the ETF energy density ´ ETF w r x and then to solve the Euler–Lagrange equation Ž3. for the density r . Due to its semiclassical character the ETF binding energy is free from shell effects and, consequently, does not coincide with the quantal HF energy. According to the Strutinsky energy theorem w16x, the HF binding energy can be split into two parts. One of them varies smoothly with the number of nucleons and is similar to that obtained with the Liquid Drop Model w28–30x. The other contribution is the shell correction, which has a purely quantal origin, is small and its behaviour is not smooth. The ETF energy corresponds to the smooth part and has to be compared with the smooth HF energy obtained from the DM calculated with the Strutinsky occupation numbers. To carry out the comparison between the semiclassical ETF and Strutinsky energy is another point that we will discuss in this paper. On the other hand, the shell corrections not contained in the ETF approach can be recovered using the Kohn–Sham ŽKS. scheme w39x widely used in atomic physics w40,41x and discussed for the nuclear problem in Ref. w42x. Basically, the KS approach consists of solving a Schrodinger equation using a local effective one-body potential ¨ obtained as the functional derivative of the potential part of the energy density ´ w r x for each nucleon. It should be pointed out that the KS approximation coincides with HF in the case of Skyrme forces w43x. However, this is not the situation for finite-range interactions because in this case we replace the exact exchange energy by its semiclassical counterpart. Comparison of these KS energies with the HF energies will close our analysis about the extension of the ETF approximation for non-local potentials. The paper is organized as follows: In the second section we compare the semiclassical ETF density matrix with the former approximations of NV and CB in the case of a local potential. In the third section we derive the density matrix and the binding energy in the ETF approximation for a non-local potential. The fourth section is devoted to the study of the semiclassical ground-state binding energies and particle densities comparing the results obtained with the ETF approach and Strutinsky averaged method. In the fifth section we discuss the KS approach for finite range-forces and compare the binding energies and root mean square Žr.m.s.. radii obtained in this way with the corresponding full HF results. In the last section we give our conclusions. Technical details concerning the calculation of the DM and HF ground-state binding energy in the ETF approach for a non-local potential are given in the appendix. For numerical applications Gogny w7x, Brink–Boeker w6x and M3Y w4,5x forces are used.

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2. Extended Thomas–Fermi density matrix The first step is to perform the inverse Wigner transform of Ž1. to obtain the semiclassical WK density matrix in coordinate space. The definition used here for the Wigner transform of the one-body density matrix is w15x f Ž R,p . s ds eyi p s r "r R q

ž

H

s 2

,R y

s 2

/

,

Ž 4.

where R s Žr 1 q r 2 .r2, s s r 1 y r 2 and p are the centre-of-mass, the relative coordinates and the phase-space momentum, respectively. Performing the inverse Wigner transform of the distribution function Eq. Ž1., after some lengthy but straightforward algebra one obtains the semiclassical DM in terms of R and s at WK level,

r Ž R,s . s

gk F3 3 j1 Ž k F s . 6p

q

2

g q

kF s g

Ž =k F .

48p 2

kF

g

1

y 48p

2

kF

24p 2

D k F j0 Ž k F s . y k F sj1 Ž k F s .

2

j0 Ž k F s . y 4 k F sj1 Ž k F s . q k F2 s 2 j2 Ž k F s .

= k F =k F

s s s s

y3k F sj1 Ž k F s . q k F2 s 2 j2 Ž k F s .

q O Ž "4 . ,

Ž 5.

where k F s 2"m2 Ž l y V Ž R . . is the local Fermi momentum, jl Ž k F s . are the spherical Bessel functions and g stands for the degeneracy. This expression, although written in a slightly different way, coincides with those obtained previously by Dreizler and Gross w40,41x and Jennings w44x. The first term of the expansion Ž5. corresponds to the Slater approach, whereas the " 2 terms are the part that take into account quantal finite-size effects. The WK density matrix in coordinate space depends on the angle between R and s, however, for practical purposes and following previous literature w9–13x we perform the angular average of Eq. Ž5. obtaining

(

r˜ Ž R, s . s r 0˜ Ž R, s . q r 2˜ Ž R, s . s g q

q

144p 2

gk F3 3 j1 Ž k F s . 6p 2

kF s

D k F 6 j0 Ž k F s . y 3k F sj1 Ž k F s . y k F2 s 2 j2 Ž k F s .

g

Ž =k F .

144p 2

kF

2

3 j0 Ž k F s . y 9k F sj1 Ž k F s . q 2 k F2 s 2 j2 Ž k F s . . Ž 6 .

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The diagonal part Ž s s 0. of Eq. Ž6. is the well-known WK expression for the local density Žwith degeneracy g . w15,23x,

r Ž R. s

gk F3

q

6p 2

g

Ž =k F .

48p 2

kF

2

q 2D k F .

Ž 7.

To obtain the DM in the ETF approach we shall express the Fermi momentum and its derivatives in terms of the local density and its gradients. First, the local Fermi momentum is obtained by inverting Eq. Ž7.,

Ž =k 0 .

k F s k 0 y 241

2

D k0

q2

k 03

k 02

,

Ž 8.

where k 0 s Ž6p 2rrg .1r3. Notice that for inverting the gradient terms in Eq. Ž7. it is enough to replace k F by k 0 to be consistent with the "-order in the expansion of the Fermi momentum Eq. Ž8.. Writing the gradients of k 0 in terms of the spatial derivatives of the local density, 2

k 0 Ž =k 0 . s k 02 D k 0 s

2p 2 Ž =r .

D r y 23

g

,

r

3g

2p 2

2

Ž 9.

Ž =r .

2

,

r

Ž 10 .

one finally obtains the Fermi momentum as kF s

6p 2r

1r3

ž /

1 q

g

72

6p 2r

y1r3

ž /

Ž =r . r

g

2

2

Dr y2

,

r

Ž 11 .

where the first term of the right-hand side is the pure Thomas–Fermi part and the second term, which contains derivatives of the local density, is the " 2 contribution. The semiclassical density matrix for a local potential in the ETF approach is obtained from Eq. Ž6. by expanding consistently the Fermi momentum k F up to " 2-order with the help of Eqs. Ž9. – Ž11.,

r˜ ETF Ž R, s . s r

3 j1 Ž k F s .

y

s2 q

kF s

72

s 2 Ž =r .

r

216

D r j0 Ž k F s . y 6

2

4 j0 Ž k F s . y 9

j1 Ž k F s . kF s

j1 Ž k F s .

,

kF s

Ž 12 .

where now k F s k 0 s Ž6p 2rrg .1r3. Let us now analyze the main properties of this semiclassical approach as compared with the quantal case. Following Refs. w8–10x, the quantal DM averaged over the s direction can be approximated by

r Ž R, s . s

1

Hd V 4p

e is ˆ r "r Ž r,r ’ . p

pˆ

ž /

s j0 s rsr ’sR

"

r Ž r,r ’ .

, rs r ’sR

Ž 13 .

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where pˆ s yi" Ž =r y =r X .r2 is the relative momentum operator. Expanding the Bessel function in a Taylor series, one gets n

Ž y1. s 2 n pˆ Ž 2 n q 1. ! "

`

r Ž R, s . s

ž /

Ý ns0

2n

n

`

r Ž r,r ’ . < rs r ’sR s

Ý ns0

Ž y1. s 2 n M , Ž 2 n q 1. ! 2 n

Ž 14 .

where M2 n are the momentum-weighted integrals defined as w8x M2 n s

1

Ž 2p " .

p

3

2n

Hdp ž " /

f Ž R,p . s

pˆ

2n

ž / "

r Ž r,r ’ .

.

Ž 15 .

rs r ’sR

It should be pointed out that series equation Ž14. as it stands is not useful because it converges very slowly and cannot be truncated for large s-values if the even moments are different from zero. At this point there are two possibilities for approximating the exact DM. One is to sum the series equation Ž14. using some approach for evaluating the momentum weighted-integrals and the other is to rearrange the series equation Ž14. in such a way that truncation is possible. First of all, we will show that the semiclassical ETF approach to the DM equation Ž12. corresponds to the whole sum of the series equation Ž14. if the moments M2 n are calculated in the same ETF approximation. Starting from the semiclassical distribution function, Eq. Ž1., the WK momentum weighted integrals up to " 2-order are easily derived. Expanding consistently the Fermi momentum with the help of Eq. Ž11. and using Eqs. Ž9. and Ž10. one finally obtains the ETF weighted integrals in terms of the local density and its gradients, M2ETF n s

3 2nq3

r k F2 n q nk F2 ny2

8 n y 5 Ž =r . 108

r

2

2ny5 y 36

Dr ,

Ž 16 .

where again k F s k 0 s Ž6p 2rrg .1r3. The first term of the right-hand side of Ž16. is the Thomas–Fermi weighted integral while the second term is just the " 2 correction in the ETF approach. For n s 0 and n s 2, one obtains M0ETF s r

Ž 17 .

and M2ETF s 35 r k F2 q 361

Ž =r . r

2

q 121 D r ,

Ž 18 .

which are just the semiclassical counterparts Žat ETF-" 2 level. of the zeroth and second-order quantal momentum weighted integrals: M0 s r and M2 s t y D rr4, where t is the kinetic energy density. Notice that in the ETF-" 2 approach only second-order gradients of the local density appear in M2ETF for any value of n. n However, higher order derivatives will appear in the moments if the ETF expansion is pushed to higher powers in ".

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Taking into account the Taylor expansion of the Bessel functions in Eq. Ž12. and after some algebra, one finds

r˜ ETF Ž R, s . s r

3 j1 Ž k F s .

y

kF s

=

`

Ý ns0

=

Ý ns0

8 n q 3 Ž =r . 108

s3r

n

`

Ž y1. s 2 nq2 Ž n q 1 . k F2 n Ž 2 n q 3. !

2

2ny3 y

r

36

n

Dr n

2n

` Ž y1. Ž k F s . Ž y1. s 2 n q Ý ns 1 Ž 2 n q 3. Ž 2 n q 1. ! Ž 2 n q 1. !

M2ETF n y

n

`

3 2nq3

r k F2 n

s Ý ns0

Ž y1. s 2 n ETF M . Ž 2 n q 1. ! 2 n

Ž 19 .

From this result it is clear that in the ETF approximation to the DM all the momentumweighted integrals appearing in Eq. Ž14., evaluated within the same semiclassical approach, are consistently summed. Another possibility for approximating the quantal DM is to rearrange the terms in Eq. Ž14. in such a way that the leading term is the Slater term. This is, actually, the way in which the NV and CB approaches to the DM are done. For the sake of completeness we shall once again briefly derive the NV and CB approaches to the DM following the method outlined in Ref. w8x. The starting point is the identity ` n Ý Ž y1. Ž 4 n q 3.

j0 Ž ab . s

ns0

j2 nq1 Ž ak . P2 nq1 Ž brk . ak

Ž 20 .

brk

valid for any k such that y1 F brk F 1 and where P2 nq1Ž x . are the Legendre polynomials P2 nq1Ž x . s Ý nms0 a mn x 2Ž nym.q1 with a mn s

Ž y1. 2

m

2 nq1

Ž 4 n y 2 m q 2. ! . m! Ž 2 n y m q 1 . ! Ž 2 n y 2 m q 1 . !

Ž 21 .

Using Eq. Ž20. in Eq. Ž13., the angular averaged DM equation Ž14. is also written as `

r Ž R, s . s

n Ý Ž y1. Ž 4 n q 3.

ns0 `

s

Ý ns0

j2 nq1 Ž ks . P2 nq1 Ž pr"k . ˆ pr"k ˆ

ks

n Ž y1. jˆ2 nq1 Ž ks . s 2 n Ž 2 n q 1. !

n

a mn

r Ž r,r ’ . < rs r ’sR

Ý ms0 a n M2Ž nym. k 2 m ,

Ž 22 .

0

where jˆ2 nq1Ž ks . s Ž4 n q 3.!! j2 nq1Ž ks .rŽ ks . 2 nq1 are the spherical Bessel functions normalized to unity at s s 0. Of course, the semiclassical r˜ ETF ŽR, s . Eq. Ž19. also fulfills Eq. Ž22.. Starting from this equation if the moments are obtained in the ETF approximation Ž16., the Bessel functions jˆ2 nq1Ž ks . are expanded in a Taylor series and the even powers of s are properly sorted out, one recovers Eq. Ž19..

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The NV approach consists of keeping the first two terms of the expansion equation Ž22. and taking k s k F . In this case the DM reads

r˜ N V Ž R, s . s r

3 j1 Ž k F s .

q

kF s

35 j3 Ž k F s . 2

k F3 s

3 5

k F2 M0 y M2 .

Ž 23 .

The CB approximation keeps only the first term of Eq. Ž23., but with an effective k˜ F fixed in such a way that the second term of this equation identically vanishes,

r˜ CB Ž R, s . s r

3 j1 Ž k˜ F s . k˜ F s

,

Ž 24 .

where k˜ F s 5M2r3 r . The NV and CB approaches are truncations of the true expansion of the quantal DM equation Ž14.. As is discussed in Ref. w8x, in these approximations only the M0 and M2 momentum weighted-integrals correspond to those obtained with the exact DM, whereas any higher momentum-weighted integral in these approaches M2 lŽ l ) 1. has little to do with its exact quantal value. Let us now discuss the results obtained using the different approaches to the DM analyzed previously. To do this we consider a 40 Ca nucleus described using HO wavefunctions with an oscillator parameter a s m vr" s 0.516 fmy1 . Fig. 1 displays the ratio of the off-diagonal to diagonal DM r ŽR, s .rr ŽR. as a function of the interparticle distance s for selected values of the distance from the centre-of-mass R. The different curves shown in this figure correspond to the quantal DM Žblack dots., the NV Ždashed-dotted line. and CB Ždashed line. approximations and the semiclassical ETF approach calculated using the quantal local density Žsolid line.. In general, the approximations to the DM considered here reproduce reasonably well the quantal values in the range of R s 1–3 fm, they show some deficiencies at R s 0 fm and clearly start to separate from the quantal values for R ) 4 fm. However, in the whole range of R-values analyzed, the quantal approaches NV and CB reproduce the quantal DM better than the semiclassical ETF calculation. At this point two comments are in order. First of all, it should be pointed out that all the approaches to the DM considered in this paper are, in fact, distributions Žsee Eq. Ž1. for the ETF DM and Ref. w8x for the discussion of the NV and CB cases. and consequently, the only meaningful comparison should be done through the moments in k and R spaces. On the other hand, the semiclassical approaches to the DM are obtained by switching off shell effects. Consequently, the Slater and ETF approximations to the DM should be compared with the smoothed DM obtained using the Strutinsky averaged occupation numbers w16x rather than with the quantal DM. To do this, one starts from the smooth distribution function, which for closed HO shells reads w17,18x

(

'

` K

f˜S t Ž R,p . s 8 Ý K Ž y1 . n˜ K ey ´ L2K Ž 2 ´ . ,

Ž 25 .

where ´ s p 2rm q m v 2 R 2 , LaK are the generalized Laguerre polynomials and n˜ K the Strutinsky occupation numbers w16x. Performing the inverse Wigner transform of Eq.

300

V.B. Soubbotin, X. Vinas ˜ r Nuclear Physics A 665 (2000) 291–317

Fig. 1. Ratio of off-diagonal to diagonal quantal density matrix r ŽRqsr2,Rysr2.r r ŽR. for 40 Ca as a function of the interparticle distance s for some values of the centre-of-masss distance R Žin fm.. The different curves appearing in the figure are explained in the text.

Ž25. and averaging over the angles, one obtains the Strutinsky DM in coordinate space to which the semiclassical approximations ŽSlater and ETF. should be compared,

r˜S t Ž R, s . s

dp

H Ž 2p " . a3

s

p

3r2

˜ Ž R,p . e ip s r "

f 3 St

`

Ý Ks 0 n˜ K ey a

=L1r2 Ky K 1

a 2s2

ž / 2

.

2

Ž R 2 qs 2 r4.

K

Ý K s0 Ž y1. K 1

1

2 2 L1r2 K1 Ž2 a R .

Ž 26 .

The NV and CB expansions of the DM can also be considered within the semiclassical framework if the M2 moment is calculated in the ETF approach. From Eq. Ž19., it is clear that in this case NV and CB become truncations of the ETF density matrix. It is

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interesting to look at the quality of these approximations because they have been used for calculating the exchange part of the nucleus–nucleus potential Žsee w45x and references quoted therein.. The Strutinsky DM equation Ž26. for 40 Ca is obtained with a HO parameter a s 0.516 fmy1 and with a smoothing parameter w15,16x g s 1.25" v . Fig. 2 collects the semiclassical results where the ratio r˜ ŽR, s .rr ŽR. is displayed for the Strutinsky smoothed DM equation Ž26. Žblack dots. and for the ETF DM Žsolid line. as well as for the semiclassical NV Ždashed-dotted line. and CB Ždashed line. truncations of the ETF DM. Notice that in order to completely remove the shell effects, the ETF, NV and CB density matrices have to be obtained using the Strutinsky local density w46,47x. From this figure it can be seen that the ETF ratio reproduces reasonably well the Strutinsky result. The ETF quotient is better than the NV result in the whole range of R and s distances analyzed and better than the CB for small values of R. The difference between ETF and Strutinsky ratios indicates that the "- expansion in ETF is not fully converged. Consequently, it would be necessary to add " 4 contributions to the ETF DM to obtain ETF expectation values in better agreement with the corresponding Strutinsky results.

Fig. 2. Ratio of off-diagonal to diagonal semiclassical density matrix r ŽRqsr2,Rysr2.r r ŽR. for 40 Ca as a function of the interparticle distance s for some values of the centre-of-mass distance R Žin fm.. The different curves appearing in the figure are explained in the text.

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3. Extended Thomas–Fermi binding energy In this section we will derive the ETF approximation to the HF energy for finite-range forces. As has been pointed out at the introduction, this semiclassical approach represents the part of the energy which varies smoothly with the number of nucleons and is free of shell effects. To obtain the ETF binding energy we shall first derive the DM starting from the non-local HF potential, V HF Ž r,r ’ . s V H Ž r,r ’ . d Ž r y r ’ . q V F Ž r,r ’ . ,

Ž 27 .

where V H and V F are the direct and exchange parts of the HF potential. In the Wigner representation Eq. Ž27. becomes V Ž R,k . s Vdir Ž R . q Vex Ž R,k . ,

Ž 28 .

where Vdir Ž R . s dR’ Õ Ž R,R’ . r Ž R’ .

Ž 29 .

H

and dk ’

Vex Ž R,k . s yg

H Ž 2p .

3

w Ž k,k ’ . f Ž R,k ’ . .

Ž 30 .

In these equations Õ ŽR,R’ . is the two-body effective interaction, w Žk,k’ . is its Fourier transform and g stands for the degeneracy Žfor the sake of simplicity we consider a simple Wigner force in Eqs. Ž29. and Ž30... Consequently, the Wigner transform of the one-body Hamiltonian will be HW Ž R,k . s

"2 k 2 2m

q V Ž R,k . .

Ž 31 .

If the HF potential is spherically symmetric in k, i.e. V ŽR,k ., the WK distribution function required for semiclassical calculations is w23x f˜WK Ž R,k . s

1 4p

Hf "2

y 8m

WK

Ž R,k . d V ksQ Ž l y HW .

d X Ž l y HW . F1 Ž R,k . q

"2 24 m

d XX Ž l y HW . F2 Ž R,k . ,

Ž 32 .

where the functions F1ŽR,k . and F2 ŽR,k . are given by "2 F1 Ž R,k . s

3m " 2 "2

F2 Ž R,k . s

m

m

3m " 2

2

DV Ž 3 f q kf k . y k 2 Ž =f . ,

Ž 33 .

2 Ž = V . Ž 3 f q kf k . q k 2 f 2DV y 2 k 2 f = V =f .

Ž 34 .

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In Eqs. Ž33., Ž34., f is the inverse of the position and momentum-dependent effective mass defined as f Ž R,k . s

m )

m Ž R,k .

s1q

m "2 k

Vex , k Ž R,k . ,

Ž 35 .

where the subscript k indicates a partial derivative with respect to k. Due to the fact that the effective mass corrections are included in the " 2 part of the distribution function Ž32., they are calculated using the " 0 order exchange potential in Eq. Ž35. to be consistent with the "-order in the expansion of the WK distribution function. Following the steps indicated in the appendix, the ETF density matrix for each kind of nucleon in the case of a non-local potential can be written as

r˜ Ž R, s . s r

s2

3 j1 Ž k F s .

q

kF s

216

Ž =r .

y4 j0 Ž k F s .

y 18 r

=

Df f

r

ž

½ž

9y2 kF

kF s

5

f

y 2 k F2

2

y

q 18 y 6 k F

j1 Ž k F s .

fk

ž

18 q 6 k F

fk f

/

fk

f

kF s

=r P =f f

f k2

q k F2

j1 Ž k F s .

/

f

fk k

q 12 k F

f

2

/

j1 Ž k F s . kF s

y 3 j0 Ž k F s . D r

=r P =f k f

,

y 9r

Ž =f .

2

f2

Ž 36 .

where now k F s Ž3p 2r .1r3 and the inverse effective mass f, Eq. Ž35., and its derivatives are computed at k s k F . We use here g s 2 because in this way the ETF DM equation Ž36. can be directly applied to each type of nucleon in non-symmetric nuclei. If all the spatial and momentum derivatives of the inverse effective mass are dropped in Eq. Ž36., one recovers the ETF DM for the local case Ž12.. If only the momentum derivatives of f are eliminated, one obtains the ETF DM corresponding to the case of a position-dependent effective mass. This latter case is just the situation that appears when one uses zero-range forces such as the Skyrme interactions. The next step is to obtain the ETF approach to the HF energy, which for an uncharged and spin-saturated nucleus can be written as E HF s Ý dR

H

q

" 2t Ž R . 2m

q 12 r Ž R . Vdir Ž R . q 12 ds Vex Ž R, s . r R q

H

ž

where the subindex q stands for each type of nucleon.

s 2

,R y

s 2

/

, q

Ž 37 .

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The HF energy in the ETF approximation is obtained as explained in the appendix and reads E˜ETF s Ý dR

H

q

"2 2m

t ETF Ž R . q 12 r Ž R . Vdir Ž R . q ´exETF Ž R .

.

Ž 38 .

q

In this equation t ETF ŽR. is the kinetic energy up to " 2 order for each type of nucleon in the ETF approximation and reads

t ETF Ž R . s t ETF ,0 Ž R . q t ETF ,2 Ž R . s

3 5

k F2 r q

1 Ž =r .

1 =r P =f k 9

1 q 23 k F

r

36

q 121 D r 4 q 32 k F

q

2

fk f 1

f

12

r

f

q 23 k F2

fk k f

y 13 k F2

f k2 f2

1 Df 1 =r P =f fk r q 1 y 31 k F 6 f 6 f f

q

y

fk

Ž =f .

2

Ž 39 .

f2

and ´exETF ŽR. is the exchange energy density for each kind of nucleon in the same approximation given by ETF ´exETF Ž R . s ´exETF ,0 Ž R . q ´ex ,2 Ž R .

s y 12 r 2 Ž R . ds Õ Ž s .

H

qk F f k

ž

1 Ž =r . 27

9 j12 Ž k F s .

2

k F2 s 2 1

y

r

36

Dr

"2 q 2m

/

Ž f y 1 . Ž t ETF y 35 k F2 r y 14 D r .

,

Ž 40 .

where Õ Ž s . is the form factor of the central nucleon–nucleon interaction Žhere again we use a simple Wigner force., k F s Ž3p 2r .1r3 and f and its k derivatives are calculated at k s k F . For Gaussian type forces such as the Gogny or Brink–Boeker interactions used Ž . in this paper, the explicit form of the lowest order exchange energy ´exETF ,0 R can be w x found in Ref. 48 . In the special case of a local potential the " 2 part of the kinetic energy density Eq. Ž39. reduces to the well-known result

t ETF ,2 s

1 Ž =r . 36

2

q 13 D r .

r

Ž 41 .

If in Wigner space the HF potential is quadratic in k, as happens for Skyrme forces, one recovers the result of Ref. w24x,

t ETF ,2 s

1 Ž =r . 36

r

2 1 3

q D rq

1 rD f

1 =r =f q

6

f

y 6

f

1 r Ž =f . 12

f2

2

.

Ž 42 .

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Table 1 Extended Thomas–Fermi ŽETF. Coulomb exchange energies for 4 He, 16 O and 40 Ca compared with the quantal ŽQM., Slater ŽSL., Negele–Vautherin ŽNV. and Campi–Bouyssy ŽCB. values reported in Ref. w12,13x 4

HF NV CB SL ET

16

He

y0.86 y0.47 y0.78 y0.74 y0.82

40

O

y2.98 y2.31 y2.75 y2.75 y2.89

Ca

y7.46 y6.42 y7.03 y7.05 y7.31

For the particular case of the Coulomb potential, the direct calculation of the exchange energy density up to " 2 order in the ETF approach Žlocal case. leads to ETF ´ Coul ,ex s y

3

1r3

3

ž /

r

4r3

Ž =r .

7

2

, Ž 43 . 1r3 r 4r3 432p Ž 3p 2 . where r is the proton density. Eq. Ž43. agrees with the result reported previously in Ref. w40,41x. As a first test of our ETF approach, let us compare the exchange Coulomb energy obtained using Eq. Ž43. with the quantal result as well as the same energy derived through the NV, CB and Slater approximations to the DM. To this end and following Ref. w12,13x, we use HO wavefunctions with fixed parameters a s 0.752 fmy1 for 4 He, 0.546 fmy1 for 16 O and 0.481 fmy1 for 40 Ca. Table 1 shows the HF, NV, CB, Slater ŽSL label. and ETF results for the exchange Coulomb energy. From this table it can be seen that the ETF results almost reproduce the HF values and improve those obtained using the NV, CB and Slater approximations. 4 p

y

4. Semiclassical results Let us now discuss the quality of the Slater and ETF energies within the semiclassical framework. In this case we have to compare them with the energies obtained using the Strutinsky averaged method w16x. The starting point for a Strutinsky calculation of the energy using trial HO wavefunctions is the smooth density matrix Eq. Ž26. from which the particle and kinetic energy can also be derived w17,18x. For an effective nuclear interaction with two Gaussian type form factors Žas in the case of the forces studied in this paper. and HO closed shells, the direct and exchange energies can be obtained analytically, 2

Edir s

16 X d ,i

Ý 'p

is1

K

=

a2

3r2 K max K max

gi

Ks0 Ms0

ž /

Ý Ý n˜ K n˜ M

M

Ý Ý

1r2 L1r2 Ky K 1Ž 0 . L MyM 1Ž 0 .

G Ž K 1 q M1 q 32 .

K 1s0 M 1s0 1 2

=F yM1 ,y K 1 ,y M1 y K 1 y ,1 y

a4

ž

a2 1y

K 1 qM 1

ž / gi

K 1 !M1 !

g i2

a2

gi

2

/

,

Ž 44 .

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306

2

Eexc s y

Ý is1

=

32 X e ,i

p

a2

3r2 K max K maxyK K maxyK

ž / gi

Ý K 1s0

G Ž K q 32 . G Ž K 1 q M1 q 32 . K!

Ý n˜ KqK n˜ KqM

Ý Ks0

K 1 !M1 ! 1 2

ž

1

1

M 1s0

a2 1y

gi a4

=F yM1 ,y K 1 ,y M1 y K 1 y ,1 y

g i2

K 1 qM 1

/ ž

a2 1y

gi

y2

/

,

Ž 45 .

where g i s 2rm2i q a 2 and F are the Gauss hypergeometric functions. In Eqs. Ž44. and Ž45., X d,i s wi q bir2 y h ir2 y m ir4 and X e,i s wir4 q bir2 y h ir2 y m i are the usual combination of the direct and exchange parameters of the central effective interaction and m i is the range of each Gaussian form factor. The Strutinsky occupation numbers that come into the energy calculation are obtained from a HO spectrum. In this way the smooth energy becomes a function of the HO parameter a . The Strutinsky energy Žcorrected from the centre-of-mass motion. is obtained by minimizing with respect to a to simulate the self-consistency w52x with the additional constraint that the minimization procedure is performed in the plateau region w15,16x. With a smoothing parameter g s 1.25" v , the HO parameter which minimize the Strutinsky energies of the 4 He, 16 O and 40 Ca nuclei are a s 0.647, 0.550 and 0.509 fmy1 respectively using the Brink–Boeker force and a s 0.643, 0.567 and 0.516 fmy1 in the case of the Gogny interaction. The binding energies and r.m.s. radii ² r 2 :1r2 obtained in this way for uncharged 4 He, 16 O and 40 Ca nuclei are collected in Table 2 with the label ST. The semiclassical binding energies at Slater and ETF levels are computed using the Strutinsky particle density obtained previously in order to drop the shell effects completely w46,47x. These results are shown in Table 2 labelled SL and ETFŽa., respectively. The Strutinsky value represents the energy which varies smoothly with the number of nucleons A. For each nucleus the difference between the Strutinsky and HF energies Table 2 Total binding energies and root mean square radii of the 4 He, 16 O and 40 Ca nuclei obtained with the Strutinsky averaged method ŽST. and with the different Extended Thomas–Fermi approaches described in the text for the Brink–Boeker Žtop. and Gogny Žbottom. forces 4

16

He

40

O

Ca

E ŽMeV.

² r 2 :1r 2 Žfm.

ST SL ETFŽa. ETFŽb. ETFŽ " 4 .

y16.71 y22.77 y22.51 y21.71 y17.64

1.94 1.94 1.94 1.94 1.94

y89.15 y97.73 y98.06 y96.20 y90.19

2.75 2.75 2.75 2.75 2.75

y296.79 y307.94 y310.51 y307.03 y298.71

3.42 3.42 3.42 3.42 3.42

ST SL ETFŽa. ETFŽb. ETFŽ " 4 .

y22.95 y31.85 y27.78 y27.60 y23.67

1.95 1.95 1.95 1.95 1.95

y125.14 y140.59 y133.39 y132.85 y125.30

2.67 2.67 2.67 2.67 2.67

y386.44 y409.45 y398.18 y397.22 y386.07

3.37 3.37 3.37 3.37 3.37

E ŽMeV.

² r 2 :1r2 Žfm.

E ŽMeV.

² r 2 :1r2 Žfm.

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307

Table 3 Total binding energies and root mean square radius of the 4 He, 16 O and 40 Ca nuclei obtained quantally ŽHF. and within the KS scheme using the exchange-correlation energy densities described in the text for the Brink–Boeker Žtop. and Gogny forces Žbottom. 4

16

He 2 :1r 2

40

O 2 :1r2

Ca

E ŽMeV.

²r Žfm.

E ŽMeV.

²r Žfm.

E ŽMeV.

² r 2 :1r2 Žfm.

HF NV CB SLŽKS. ETFŽKS.

y28.24 y23.69 y25.61 y17.51 y25.38

1.72 1.86 1.80 2.00 1.80

y106.62 y100.32 y101.96 y87.85 y103.87

2.65 2.71 2.70 2.79 2.68

y323.98 y313.49 y315.22 y291.78 y320.45

3.36 3.40 3.39 3.45 3.37

HF NV CB SLŽKS. ETFŽKS.

y29.66 y28.53 y29.04 y25.80 y28.97

1.88 1.91 1.90 1.96 1.90

y138.07 y135.72 y136.31 y129.49 y137.06

2.63 2.65 2.64 2.67 2.64

y406.47 y402.68 y403.22 y392.33 y405.22

3.34 3.36 3.36 3.37 3.35

Žalso obtained with trial HO wavefunctions., and reported in Table 3 Žlabelled HF., is the so-called shell-energy. This is a subtle quantity that is not well reproduced by SL or ETF approaches up to " 2 order. As has been pointed out in previous literature w46,47x, if the ETF kinetic energy density functional only contains the " 0 and " 2 contributions, its integral is not able to reproduce the Strutinsky kinetic energy at least in the case of a set of nucleons moving in a HO or Woods–Saxon type external potential. However, if the " 4 contributions are included in the functional, the ETF kinetic energies are in much better agreement with the Strutinsky values w46,47,51x. In our non-local calculations the differences found between Strutinsky and ETF Žup to " 2 order. total energies are roughly similar to those found for the kinetic energy in the case of an external HO potential w51x. This fact suggests including approximately the " 4 corrections to ETF by adding to t ETF which enters Eqs. Ž39. and Ž40. the " 4 contribution in the local potential case, 1 t4 s 6480 Ž 3p 2 .

y2 r3

r 1r3 24

Dr

ž / r

2

Dr y 27

r

=r

2

=r

ž / ž / r

q8

r

4

.

Ž 46 .

From this approximated calculation we find that almost all the correction comes from the kinetic energy term, whereas the exchange part gives only a minor contribution. The total energy when this " 4 correction is included perturbatively is shown in Table 2 labelled ETFŽ " 4 .. It can be seen that the Strutinsky binding energies are very well reproduced by this approximate ETFŽ " 4 . calculation showing again the importance of including " 4 corrections in ETF in order to obtain a better description of the shell energies w46,47,51x as happens in the case of local potentials. For finite-range forces the non-local effects contribute to the DM equation Ž36. through the gradients and the derivative with respect to k of the inverse effective mass equation Ž35. calculated at k s k F s Ž6p 2rrg .1r3. To investigate the influence of these non-local corrections in the ETF energy, we have again obtained it starting from the DM corresponding to a local potential equation Ž12. instead of the full DM equation Ž36.. In

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308

this case, the kinetic energy density reduces to Eq. Ž41. and the " 2 exchange energy is calculated using Eq. Ž40. but with t given by Eq. Ž41.. The ETF energies calculated in this way are also displayed in Table 2 with the label ETFŽb.. From these results it can be seen that, in fact, the " 2 effective mass corrections to the DM Eq. Ž36. are almost negligible for the Gogny force but they become more important for the Brink–Boeker interaction where the non-local effects are larger.

5. Kohn–Sham approximation to the binding energy for finite-range effective forces Although the ETF energy represents the part of the HF binding energy which varies smoothly with the number of particles A, a direct comparison with the quantal HF results is still possible if shell effects are added to the semiclassical results Žat Slater or ETF levels. according to the Strutinsky energy theorem. As mentioned in the introduction, one way of recovering the shell effects consists of using the KS scheme within DFT. Due to the fact that the exact energy density is, in general, unknown, the KS approximation starts separating ´ w r x into several parts, " 2t w r x

´w rxs

2m

KS w rx , q 12 VH Ž R . r q ´exch

Ž 47 .

where VH ŽR. is the direct ŽHartree. potential and t w r x the non-interacting kinetic energy density of a set of fermions of density r . The last term in Eq. Ž47. is the so-called exchange-correlation energy that contains the exchange energy as well as the correlation energy due to the fact that the exact wave function is not the Slater determinant. The basic idea of the KS ansatz consists of writing the local density in terms of trial single particle wavefunctions f i ŽR. as A

r Ž R. s

Ý < fi Ž R. < 2

Ž 48 .

is1

and also to assume that the kinetic energy density in Eq. Ž47. can be expressed through the same trial single-particle wavefunctions as A

t Ž R. s

Ý < =fi Ž R. < 2 .

Ž 49 .

is1

Using Eqs. Ž48. and Ž49., the variation of Eq. Ž47. with respect to f i) ŽR. enables the following Schrodinger equation for each nucleon to be written: ¨ "2 y 2m

ž

D f i q VH Ž R . q

KS d´exch

dr

/

f i s e KS f i ,

Ž 50 .

where now each nucleon moves in a local single-particle potential which is the sum of the Hartree part plus the functional derivative of the KS exchange-correlation energy density.

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In our case of finite-range forces we use as the exchange-correlation energy density for each kind of nucleon the ETF exchange energy density equation Ž40., but with t given by its KS value Ž49.,

´˜exKS w r x ss y 12 r 2 Ž R . ds Õ Ž s .

H

qk F f k

ž

1 Ž =r . 27

r

9 j12 Ž k F s .

"2 q

k F2 s 2

2m

Ž f y 1 . Ž t y 35 k F2 r y 14 D r .

2

y 361 D r

/

.

Ž 51 .

In this way the KS equation for each type of nucleon, Eq. Ž50., becomes a local Schrodinger equation similar to those found when one uses Skyrme forces w1x. ¨ To check the quality of this approach in a restricted variational approach, instead of solving Eqs. Ž50. we simply minimize the KS energy using the exchange-correlation energy density given by Eq. Ž51.. In this calculation the KS particle and kinetic energy densities ŽEqs. Ž48. and Ž49.. are obtained using trial wavefunctions of HO type where a s m vr" is the variational parameter. At the Slater level only the " 0 part of the exchange energy density is considered in the calculation. Again the centre-of-mass correction to the energy has been taken into account. At the HF level the binding energies are also obtained in a restricted variational approach using HO wavefunctions too. In this case the HF DM can be evaluated analytically for finite-range forces with Gaussian or Yukawa form factors w49x Žsee also Eqs. Ž44. and Ž45. for the Gaussian case.. At this point it is important to note that using the former NV or CB approaches to the DM, the quantal calculation of the binding energies reduces to a KS calculation using different ŽNV or CB. approximations to the exchange-correlation energy. Actually, the HF results calculated with a M3Y force and reported in Ref. w14x are obtained within the KS scheme using the CB approach to the DM for obtaining the exchange-correlation energy. The variational HF energies and r.m.s. radii of the ground-state densities of the 4 He, 16 O and 40 Ca nuclei calculated with the Brink–Boeker and Gogny forces are collected in Table 3. The same results obtained within the KS scheme with the Slater, NV, CB and ETF ŽEq. Ž51.. exchange energy densities are also given in Table 3 with the SLŽKS., NV, CB and ETFŽKS. labels. The differences between the SLŽKS. and ETFŽKS. results and the purely HF results show the quality of the semiclassical approach to the exchange energy. From this comparison one can see that the Slater approach is very poor in the case of the Brink–Boeker force, underbinding all of the considered nuclei and giving r.m.s. radii larger than the quantal values. However, the result is more satisfactory for the Gogny force. This difference is due to the fact that the non-local effects are larger in the Brink–Boeker force than in the Gogny case. The non-local effects are better accounted for in the ETFŽKS. approximation for which the agreement with the HF results is very good for these two effective forces. The binding energies and r.m.s. radii calculated with the ETFŽKS. approach Eq. Ž51. proposed in this paper are in better agreement with the full HF results that those obtained from the NV and CB approximations to the DM for both Brink–Boeker and Gogny interactions. Our KS approach can be easily applied to other kinds of finite-range effective forces. As an example, we have also calculated the binding energies and r.m.s. radii of the 4 He,

'

V.B. Soubbotin, X. Vinas ˜ r Nuclear Physics A 665 (2000) 291–317

310

and 40 Ca nuclei using the BDM3Y1) force w54x, which has a Yukawa-type finite-range form factor. The structure of this density-dependent force is 16 O

Õ Ž R . s C Ž 1 y br g . Õ M3 Y Ž R . ,

Ž 52 .

where Õ M3 Y ŽR. is the finite range of the M3YŽParis. force w5x and C, b and g constants are fitted to reproduce the nuclear matter data w54x. For BDM3Y1) force C s 1.2521, b s 1.7452 fm3 and g s 1. The HF energies and the r.m.s. radii of the same nuclei calculated with this force as well as the corresponding KS values obtained using the Slater, CB and ETF exchange energy are collected in Table 4. Again the results obtained with the ETF approach to the KS exchange-correlation energy density ŽEq. Ž51.. reproduce very well the HF results and improve the values obtained with the Slater and CB approaches. From Table 3 and 4 one obtains the following conclusion: the agreement between ETFŽKS. and HF is better, compared with the other approximations to the DM considered in this paper, when the nucleus is heavier. As we have pointed out in Section 2, the NV and CB truncations of the quantal DM become truncations of the ETF DM if the M2 momentum-weighted integrals are also computed using the same ETF approach Ži.e. obtained through Eqs. Ž23. and Ž24. using t given by Eq. Ž39... To check the quality of the NV and CB approximations to the ETF DM, we have computed variationally the binding energies and r.m.s. radii of our selected nuclei within the KS scheme using the exchange-correlation energies obtained from these semiclassical NV and CB DM. These values have to be compared with the KS results calculated using the full ETF exchange energy density Ž40. instead of the ansatz Ž51.. The corresponding results are collected in Table 5 for the Gogny and Brink–Boeker forces. From this table we can see that the agreement of the semiclassical NV and CB with ETF is similar to that found comparing the NV and CB results with the HF values in Table 3. On the other hand, this agreement improves when the non-locality of the effective force is smaller ŽGogny force.. From the comparison of the KS ETF results in Tables 3 and 5 with the HF values, one can see that to use the ansatz Ž51. as the exchange-correlation energy density better reproduces the HF than to use to this end the ETF exchange energy density equation Ž40.. Some time ago another different approximation to the DM was presented in the literature w50x. In this approach rather than starting from first principles, a phenomenological density matrix is proposed in which the parameters were determined by imposing the correct local semiclassical kinetic energy density and the projector character of the DM in an integrated form. Using this parametrized DM and the Gogny interaction, the

Table 4 The same as Table 3 but using the BDM3Y1) force 4

HF CB SLŽKS. ETFŽKS.

16

He

E ŽMeV.

² r 2 :1r 2 Žfm.

y23.30 y22.33 y19.79 y22.42

1.89 1.90 1.94 1.90

40

O

Ca

E ŽMeV.

² r 2 :1r2 Žfm.

E ŽMeV.

² r 2 :1r2 Žfm.

y115.57 y112.90 y107.94 y114.13

2.63 2.64 2.65 2.63

y364.85 y360.00 y352.45 y362.96

3.30 3.32 3.32 3.31

V.B. Soubbotin, X. Vinas ˜ r Nuclear Physics A 665 (2000) 291–317

311

Table 5 Comparison between the binding energies and root mean square radii of the 4 He, 16 O and 40 Ca nuclei obtained within the KS scheme using the semiclassical ETF, NV and CB exchange-correlation energy densities Žsee text. for the Brink–Boeker Žtop. and Gogny Žbottom. forces 4

16

He 2 :1r 2

40

O 2 :1r2

Ca

E ŽMeV.

²r Žfm.

E ŽMeV.

²r Žfm.

E ŽMeV.

² r 2 :1r2 Žfm.

NV CB ETF

y21.75 y22.39 y23.78

1.89 1.87 1.82

y97.55 y98.27 y101.61

2.73 2.72 2.69

y309.42 y310.10 y317.11

3.41 3.40 3.38

NV CB ETF

y27.41 y27.53 y29.05

1.93 1.93 1.91

y133.65 y133.80 y135.38

2.65 2.65 2.64

y399.57 y399.62 y402.66

3.36 3.36 3.35

binding energy of 16 O and 40 Ca Žincluding the Coulomb energy. are y128.3 and y337.9 MeV, respectively. These results can be compared with those obtained in our ETFŽKS. approach, y124.7 and y337.2 MeV, and with the fully quantal results, y126.0 and y338.4 MeV. In this case the ETFŽKS. results are obtained by solving the Schrodinger equation Ž50. with the exchange energy density given by Eq. Ž51.. The ¨ results obtained using the ETFŽKS. approximation are similar to those obtained with the phenomenological DM discussed previously, but clearly improve the CB results reported in Ref. w50x, which are y118.2 and y326.4 MeV for 16 O and 40 Ca, respectively.

6. Summary In this paper we have derived, for the first time to our knowledge, the Extended Thomas–Fermi approximation to the one-body density matrix up to " 2-order for a non-local single-particle Hamiltonian. The " 2 contribution can be written in terms of spherical Bessel functions combined with second-order gradients of the local density and the inverse of the effective mass as well as momentum derivatives of the latter computed at the Fermi momentum. This density matrix includes, as particular cases, results reported previously in the literature w40,41,44x for the local case. We have compared this new semiclassical approximation with former approaches, namely the Negele–Vautherin w9,10x and Campi–Bouyssy w12,13x methods. It is found that as in the case of the quantal density matrix, the Extended Thomas–Fermi approximation sums all the momentum-weighted integrals w8x, but with their quantal values replaced by their semiclassical counterparts. In this respect the Extended Thomas–Fermi approach differs from the Negele–Vautherin and Campi–Bouyssy approximations, which are truncations of the quantal density matrix. It should also be pointed out that if in the Negele–Vautherin and Campi–Bouyssy approaches the quantal momentumweighted integrals are replaced by their semiclassical counterparts, they become truncations of the Extended Thomas–Fermi density matrix. We have applied this new semiclassical approach for deriving the smoothly varying part of the Hartree–Fock energy of a nucleus in the case of effective finite-range interactions. In this case the " 2-order Extended Thomas–Fermi kinetic and exchange

312

V.B. Soubbotin, X. Vinas ˜ r Nuclear Physics A 665 (2000) 291–317

energy densities contain, in addition to the second-order gradients of the local density and inverse effective mass, new terms that account for the momentum dependence of the effective mass. The part of the Hartree–Fock ground-state binding energy which varies smoothly with the number of particles is obtained through the Strutinsky averaged method. The corresponding energies for 4 He, 16 O and 40 Ca nuclei using the Brink–Boeker and Gogny forces are obtained in a restricted variational calculation with HO wavefunctions. These Strutinsky results are compared with the pure Thomas–Fermi ŽSlater. and Extended Thomas–Fermi values obtained starting from the Strutinsky kinetic energy and particle densities. The Extended Thomas–Fermi binding energies including only " 2-order contributions are not able to reproduce the Strutinsky results and consequently cannot be used for obtaining the shell energies. We have approximately estimated the " 4-order contribution to the binding energy and verified that if this correction is taken into account, the Strutinsky values are practically recovered. The full Hartree–Fock binding energy of a nucleus can be compared with that obtained within the Kohn–Sham scheme which provides a local approximation to the non-local quantal energy density. We have applied the Kohn–Sham method using the exchange-correlation energy density obtained with the Slater, Negele–Vautherin, Campi–Bouyssy and Extended Thomas–Fermi approaches to the density matrix. Again, the Hartree–Fock and Kohn–Sham calculations are carried out in a restricted variational approach using HO wavefunctions for the same nuclei as before and with the Brink– Boeker, Gogny and M3Y ŽParis. forces. It is found that the Kohn–Sham binding energies and r.m.s. radii obtained with the exchange-correlation energy density ansatz Ž51. nicely agree with the HF calculations and improve the results obtained from the Slater, Negele–Vautherin and Campi–Bouyssy approximations to the density matrix. From our analysis presented in this paper we conclude that the Extended Thomas– Fermi approach to the density matrix for non-local forces is a powerful tool that can be applied confidently in many physical situations where the shell effects can be dropped at least to a first approximation. The quality of these semiclassical calculations is similar to that obtained with zero-range Skyrme forces and the Extended Thomas–Fermi approach at the same order of the " expansion. Our proposed Kohn–Sham scheme based on Eq. Ž51. offers a reliable approximation to the full Hartree–Fock calculations that can be easily applied to any kind of effective finite-range forces. On the other hand, effective finite-range forces of the M3Y type together with the Double Folded Model have been applied to compute the real part of the microscopic Heavy-Ion optical potential. In these calculations, the exchange part is usually obtained using the Negele–Vautherin or Campi–Bouyssy approaches to the density matrix with a semiclassical kinetic energy density w45x. As a first application of the Extended Thomas–Fermi density matrix presented here, we have analyzed the real part of the Heavy-Ion optical potential within the Double Folded Model using the Gogny and BDM3Y1) effective interactions w53x. Acknowledgements The authors are indebted to P. Schuck, M. Centelles and S.K. Patra for very useful discussions and with L. Egido for supplying us the full quantal results. This work has

V.B. Soubbotin, X. Vinas ˜ r Nuclear Physics A 665 (2000) 291–317

313

been partially supported by Grants PB95-1249 from the DGICIT ŽSpain. and 1998SGR00011 from DGR ŽCatalonia..

Appendix A The WK density matrix up to " 2 order Žassuming degeneracy g . in coordinate space for a non-local potential is given by the inverse Wigner transform of Eq. Ž32., dk r˜ WK Ž R, s . s g f˜ R,k . e i k s 3 WK Ž 2 p Ž .

H

s

gk F3 3 j1 Ž k F s . 6p

2

g" 2 16 mp

kF s g" 2

E

y

2

El

Hdk k

2

j0 Ž ks . F1 Ž R,k . d Ž l y HW .

E2

dk k 2 j0 Ž ks . F2 Ž R,k . d Ž l y HW . , 48 mp 2 El2 which written in terms of the local Fermi momentum k F with the help of

d Ž l y HW . s

Ž A.1 .

H

q

m d Ž kykF .

Ž A.2 .

" 2 k F f Ž R,k F .

and

El E kF

s

"2 k F

q Vk Ž R,k F . s

m

" 2 k F f Ž R,k F .

Ž A.3 .

m

reads

r˜ WK Ž R, s . s

gk F3 3 j1 Ž k F s . 6p 2

kF s

E j0 Ž k F s .

g q

2

E kF

m d

F2

48p q

=

q j0 Ž k F s . r 2,WK Ž R . m " k F f Ž R,k F . y

" 2 dk f Ž R,k . 2

ž

m 2

" k F f Ž R,k F

m

2

2

./

q ksk F

kF2 f Ž R,k .

kF2

" kf Ž R,k . dk f Ž R,k .

3kF1 f Ž R,k .

d

2

,

g

E 2 j0 Ž k F s .

48p 2

E k F2

Ž A.4 .

ksk F

where r 2,WK is the WK " 2-order contribution to the density in the non-local problem given in Appendix A of w23x and the inverse effective mass f ŽR,k . is defined as in Eq. Ž35.. In Eq. ŽA.4. the gradients of the non-local potential V ŽR,k . appearing in F1ŽR,k ., F2 ŽR,k . and their momentum derivatives have to be evaluated at k s k F . To do this one starts from the definition of the Fermi energy, " 2 k F2 2m

q V Ž R,k . s l ,

Ž A.5 .

V.B. Soubbotin, X. Vinas ˜ r Nuclear Physics A 665 (2000) 291–317

314

where k F is also a function of R. Now taking the gradients of Eq. ŽA.5., the spatial derivatives of the potential are transformed into gradients of the local Fermi momentum through

Ž = V . kF q

"2 k F m

f Ž R,k F . =k F s 0 ,

Ž A.6 .

"2

Ž DV . k F q

m

2

k F f Ž R,k F . D k F q f Ž R,k F . Ž =k F . q 2 k F =f Ž R,k F . =k F

qk F f k Ž R,k F . Ž =k F .

2

s0 .

Ž A.7 .

To obtain the ETF DM, one proceeds as in the local potential case. First the WK local density up to " 2-order is inverted to obtain k F , kF s k0 y

2p 2 gk 02

r 2 ,WK w k 0 x ,

Ž A.8 .

where k 0 is the zeroth order local Fermi momentum given by k 0 s Ž6p 2rrg .1r3 and the gradients of V that appear in r 2,WK w23x have been replaced by gradients of k 0 with the help of Eqs. ŽA.6., ŽA.7.. Finally, expanding k F in the WK DM equation ŽA.4. with the help of Eq. ŽA.8. one obtains the ETF density matrix written as

r˜ ETF s

gk 03 3 j1 Ž k 0 s . 6p 2

k0 s

ž

q 4 q 2 k0

yk 0

q

Ž =f . f

2

fk f

/

gk 0 s 144p 2

ž

7 q 6 k0

ž

D k0 q 6 y 2 k0

dj0 Ž k 0 s . dŽ k0 s.

q

fk f

gk 02 s 2 144p 2

/ 2

fk f

q 2 k 02

=f =k 0 f

Ž =k 0 . k0

fk k f

y k 02

q 4 k0

f k2 f2

/

=f k=k 0

2

y D k0

f

Ž =k 0 .

2

k0

q 2 k0

d 2 j0 Ž k 0 s . dŽ k0 s.

2

.

Df f

Ž A.9 .

If now the gradients of k 0 are written in terms of gradients of the density using Eqs. Ž9., Ž10. one obtains the ETF DM for a non-local potential written as a functional of the local density, which is just Ž36.. The semiclassical counterpart of the HF energy of an uncharged and spin-saturated nucleus in the ETF approximation can be obtained by replacing the quantal integrand in Eq. Ž37. by its corresponding ETF approximation as follows. The ETF kinetic energy density can be derived from the ETF DM using w15x

t ETF Ž R . s Ž 14 DR y D s . r˜ ETF Ž R, s . < ss 0 ,

Ž A.10 .

from where Eq. Ž39. is easily obtained. The direct energy is obtained using the diagonal part of the DM that in the ETF approach reduces simply to the local density r .

V.B. Soubbotin, X. Vinas ˜ r Nuclear Physics A 665 (2000) 291–317

315

The exchange potential is given by Vex ŽR, s . s yÕ Ž s . r˜ ŽR, s . and, consequently, the ETF exchange energy will be

´exETF Ž R, s . s 12 ds VexETF Ž R, s . r˜ ETF Ž R, s .

H

2 s y 12 ds Õ Ž s . r˜ ETF ˜ ETF ,0 Ž R, s . r˜ ETF ,2 Ž R, s . ,0 Ž R, s . q 2 r

H

s y 12 r 2 Ž R . dsÕ Ž s .

H

9 j12 Ž k F s .

Ž kF s.

q ds VexETF ˜ ETF ,2 Ž R, s . . ,0 Ž R, s . r

H

2

Ž A.11 . The integral over s in the " 2 contribution to the ETF exchange energy can be performed analytically taking into account the fact that in Wigner space the " 0 ETF exchange potential ŽSlater. can also be written as ETF yi k s s y ds Õ Ž s . r˜ 0 Ž R, s . j0 Ž ks . , VexETF ,0 Ž R,k . s ds Vex ,0 Ž R, s . e

H

Ž A.12 .

H

if the exchange potential is spherically symmetric in k. The k derivatives calculated at k s k 0 are easily obtained starting from Eq. ŽA.12., VexETF ˜ 0 Ž R, s . s ,0, k Ž R,k 0 . s y ds Õ Ž s . r

H

dj0 Ž ks .

s

" 2 k Ž f y 1.

d Ž ks . d 2 j0 Ž ks .

VexETF ˜ 0 Ž R, s . s 2 ,0, k k Ž R,k 0 . s y ds Õ Ž s . r

H

,

m

d Ž ks .

2

s

Ž A.13 .

ks k 0

" 2 Ž f q k f k y 1. m

. ks k 0

Ž A.14 . With the help of Eqs. ŽA.13. and ŽA.14. the integral over s of the " 2 part of ŽA.11. can be done obtaining

´ex ,2 Ž R . s

" 2 k 02 36mp 2

½

ž

ž

Ž f y 1. 9 q 6 k 0

q 3 q 2 k0

q2 k 0

Df f

fk f

/

y k0

ž

fk f

q 2 k 02

D k0 q 6 y 2 k0

Ž =f . f

fk f

2

q k 0 fk 2

/

fk k f

y k 02

=f =k 0 f

Ž =k 0 . k0

f k2 f2

/

q 4 k0

Ž =k 0 .

2

k0 =f k=k 0 f

2

y D k0

5

.

Ž A.15 .

Using this equation and replacing the gradients of k 0 by gradients of r with the help of Eqs. Ž9. and Ž10. together with the ETF kinetic energy density given by Eq. Ž39., the " 2

V.B. Soubbotin, X. Vinas ˜ r Nuclear Physics A 665 (2000) 291–317

316

contribution to the exchange energy in the ETF approximation can be recast in the form given in Eq. Ž40..

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