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Projection of angular momentum and the generator coordinate method for light nuclei
PROJECTiON OF ANGULAR ~~~~M
AND
THE GENERATOR
COURDINATE METHOD FOR LIGHT p3uCLEl M. V. MIHAILGVIC t, E. KUJAWSKI Center for Theore%icalPhysJcs, Department of Physics and Astronomy, University of Maryland, College Park, Muryland 20742 tt and J. LESJAK
Received 4 September 1970 Abstract: A method for studying collective states in light nuclei not describable in terms of a single intrinsic Slater determintit is presented. It is based on the generator coordinate method and makes use of “generating functions” obtained by projecting angular momentum from intrinsic deformed states. A useful procedure for projecting angular momentum from a generator coordinate wave function is presented. As an illustration the above method is applied to ‘ONe,
In recent years attempts have been made to describe states of Id-2s nuclei using the Hartree-Fock method. By performing the projection of angular momenta many states are understood as states belonging to the rotational band of the deformed ground state determinants “). However, there are states which cannot be obtained by projecting the angular momenta from a single determinants function. One example is ?3i where two minima exist, one ~orres~n~ng to the obIate shape and the other to the prolate shape of the Hartree-Fock field. The spectrum obtained by projection of angular momentum from either Hartree-Fock solution does not resemble the experimental one “). For such nuclei the generator coordinate method (GCM) due to Hill and Wheeler “) offers the possibility to go beyond the Hartree-Fock method and at the same time preserve the pictorial description of nuclei, such as the shape and collective motion “). The GCM is a variational method based on the many-body nuclear Harn~~to~ia~, and it yields an integral homogeneous equation for the wave functions. Consider a family of N-particle functions @([a~],p), where [xl denotes all coordinates of the N particles and p denotes a set of parameters, p = (pl, pz, . . ., p,). A trial t Fulbright-Hays Program Fellow on leave of absence from Nuclear Institute “J. Stefan,” Ljubljana, Yugoslavia. tt Research supported by the National Science Foundation under Grant NSF GU2061. 2.52
GENERATOR
eigenfunction
COORDINATE
METHOD
253
of the N-particle system can be generated by the linear combination
The weight function f(p) is determined from the variational principle
where H is the Hamiltonian equation for f(p):
of the system. Eq. (2) reduces to the following integral
s
{H(P,P’>-WP, p’))f(p’)W = 0,
(3)
where WP9 P’> = <@([xl, PI9 NCXI, P')>, H(P, P'> = <@(Cd> P>, H@i(Cxl, P'>>.
(4)
The lowest eigenvalue E corresponds to the ground state energy, and the other eigenvalues to the excited states. The integral (1) generally generates a set of states !P,( [xl) which forms a subspace .%,+ of the complete Hilbert space ti. Diagonalization of the Hamiltonian H in the subspace Z+ is equivalent to the solution of the integral equation (3). The GCM will give an exact solution if the integral (1) generates all N-particle functions; usually it does not and an approximate solution is obtained. The accuracy of the solution depends on the choice of @([xl, p). Usually one takes simple functions for the generator functions @([xl, p), but even then an exact calculation is hard to perform. Most frequently the overlaps (4) are approximated ‘) by Gaussian functions, @P,
H(p, p’) = P,
P’)
= exp
9,
C-(P P-P’
-P’)~/~~~I, exp
[--(P-P’)~P~I,
where P, is a quadratic polynominal in the variables +(p+p’) and (p-p’). It has been shown by Brink and Weiguny “) that this approximation leads to a model Hamiltonian of the Bohr-Mottelson type for coupled harmonic vibrations. Although this approximation seems plausible its validity has not been shown, and furthermore Villars ‘) has given examples where it does not hold, proving in such a way that the Gaussian form is not a general property of the overlap functions (4). More recently it has been shown that the overlaps are not Gaussians if the GCM function (1) is generated by means of a N-projected BCS function, @&([x], d) [ref. “)I. In this paper a method and a procedure are presented which can be applied to the
254
M. V. MIHAILOVIC
et al.
description of collective states in light nuclei. Our work is based on the GCM and it avoids the Gaussian approximation (5). A similar method was applied to describe pairing vibrations “). The success of the Nilsson-Mottelson “) model and HartreeFock calculations in describing the properties of light nuclei suggest that a good generator coordinate function @“([xl, p) may be obtained by projecting angular momentum from a Slater determinant in which the parameter p describes the shape of the singleparticle field (i.e. deformation parameters #?, 7). We then use such functions for the generator functions. The method is described more fully in sect. 2. The central part of this paper, sect. 3, is devoted to a method of projecting angular momenta from a sum of non-orthogonal Slater determinants such as generated by function (1). The subspace for the projection of angular momentum is spanned by the states exp (0, J+)@( [xl, p). If @([xl, p) is a Slater determinant, the transformed states exp (eJ+)@( [xl, p) are also Slater determinants. This makes the calculation of the overlap integrals easier than in the basis spanned by the states J:@( [xl, p) as proposed by Kelson lo). Finally, to illustrate the above method we apply it to calculate the spectrum of “Ne which we consider as an I60 inert core plus four particles in the Id-2s shell. The GMC function for “Ne is taken as a linear combination of Slater determinants corresponding to different deformations. The results are presented in sect. 4. 2. Description of the method We now give the mathematical formulation of the GCM as applied to light nuclei using generating functions @“(p, [xl) obtained by projecting angular momentum from a Slater determinant, where the parameters p are related to the shape of the nucleus. For convenience we take as the single-particle basis used to build our Hilbert subspace of N-particle states the harmonic-oscillator functions J/j(X), (j = 1, . . . ,L), and introduce the single-particle functions 4,(x) which depend on the shape of the nucleus through the coefficients C”j
4n(‘>=j~lc*j+j(x)s
n
=
1,. . ., N.
(6)
The N-particle Slater determinant Qi, built from 4, then depends on (k) coefficients Cnj @,=
@~,(c,1,c,,,...;[~]>.
(7)
If the constants Cnj are taken as variational parameters in the generator function @=, the GCM leads to complete mixing of the configurations - the shell-model job. If the weight function f(p) in eq. (1) is restricted to a delta function, the GCM leads to the Hartree-Fock solution in the subspace defined by the set {+J. In both cases a formulation of the problem in the form of the GCM provides no advantage over the standard formulations. The GCM will, however, be very useful and simple if the
GENERATOR
COORDINATE
METHOD
255
coefficients Cnj can be expressed in terms of a single parameter which describes nuclear properties such as shape and degree of pairing. (At times it may be more convenient to introduce additional parameters.) The multiparametric function [x]) is then replaced by a single parametric function @(t, [xl) with @(C11,Gz,...;
where s,j are constants defining the system, and 1 is a parameter related to the shape of the single-particle potential such as the deformation parameter q of the Nilsson single-particle functions. In this way eq. (3) reduces to a one-dimensional integral equation which can then be solved numerically. The subspace X4 spanned by the multiparametric functions (7) can be the same as the subspace 8, spanned by the single-parametric functions @(t, [x]) if the Cnj are properly chosen. This requirement is fulfilled if the C~j are such that the vectors
are IinearIy independent for d different values of tj where d is the dimension of the subspace X+. This condition can be often satisfied “) using rather simple forms for the functions C(E,j, t). For small differences in the parameter t the wave functions do not change much so that one may get a reasonably good wave function using a finite number of values for the parameter. The integral equation (3) then reduces to an algebraic eigenvalue problem of the form
C (Hijj
EOij)fj = 0.
It can be reduced to a conventional eigenvalue problem using the procedure described below. As is well known, the lowest eigenvalues so obtained should be lower than the energies corresponding to each input state. (i) The overlaps
areformed and the matrix Oij is diagonalized. The values ti are chosen in a wide enough range around the minima of (QJ, H@J)/(@J, @‘> so that it spans a space large enough to include the physically important states. (ii) The subspace in (i) is truncated to the subspace &@&of dimension d’ K d by rejecting all vectors with eigenvalues which are a few orders of magnitude smaller than the largest one. We thereby avoid the difficulties which are associated with states with small eigenvalues for the overlap matrix since for such states a small error in the matrices Oij or Nij can then lead to a large error in the energy ll). It is further hoped that these states are not physically important. (iii) The Hamiltonian is then diagonalized in the subspace #;P.
M, V. MIHAILOVIC et al.
256
3. Projection of anguIar momentum
We now present a method for projecting angular momentum from a generator coordinate wave function generated by axially symmetric wave functions GGPaM, where 34 is the projection of the angular momentum along the axis of symmetry and ct distinguishes among the functions @ with the same M. Let P” denote the angular momentum projection operator, P’IP,,
= ip,,, = \aJM>.
jPaMmay be then expressed as Jmnr Q,aM =JE&R&u. The weight functions&
in the trial function YJM =
~f:%/na
satisfy the secular equation
-~~{~~~~~~)~f~= 0.
~~(a~~l~~~~~)
In order to proceed we then have to calculate the overlap integrals
H$ =
(aJMIH~~Ji%f),
O;l, = (aJi%fIfiJM). 3.1. SYSTEM OF EQUATIONS FOR Ua19*AND ffaBJ
We introduce the non-unitary operator exp (6J+), where 19may be a complex parameter and we then consider transformed vectors
1
b
(~-~)~(~+~+k)!
J”,k = k! [ (J+M)&hW-k)!
1 *
*
The overlaps functions O&S) and H,,(O) of the vectors exp (8J+)j@~&? and exp (BJ+)\@SMg>can then be expressed in terms of the overlaps for states of good angular momentum jaJM>.
o,,(@)= Jma,
J-M,
= JZJ,
&
J,, = oM z J=Jo
J--M@ k;. ek+k’bJkr,k bJM#
+ klBJj+G + k’)
011
GENERATOR
COORDINATE
METHOD
257
where
Jo = max (IM,I, IMsl),
MI = max CM,,MB),
M = IK-M/d, M, = min (M,, M&.
Proceeding in a similar manner for H,,(0), we obtain &&3) = <@&,I exp @J-)H exp @&M&J =
Jmax
J-MI
J=J,,
k=O
oMc
c
@kbJMlkbJM~k+bfH:/3’
(12)
We then see that the matrix elements O& and H& defined by eq. (lo), are the coefficients of the polynominals O,,(e) and I&(0) given by eqs. (11, 12). In order to calculate O& and Hi:, , we must now evamate the left-hand side of (11) and (12). The 1.h.s. of (11,12) is obtained by expressing exp (@J+)I@jam>in terms of the transformed single-particle states exp (&I+)$, and calculating the overlaps Oprsand Hols as polynominals in 8. This is described in the next section. By equating the coefficients of equal powers of 8 we obtain a linear system of equations for O&laand Hi’. The above procedure is especially simple if GcrM is a Slater determinant since the state exp (&F+)I@~M>is then again a Slater determinant, and is the reason we choose the operator exp (&.I,) to calculate the overlaps O& and H$ . We think that our method is more practical than the one given by Kelson lo) who calculates the integrals O$ and H$ from a system of linear equations, the input of which are the integrals (Ip&,[ J”J$ jtDsM8>and (GorMJJk HJ!f.1CPBMa>. The explicit evaluation of the above overlaps from the known Slater determinants and the resulting set of triangular equations is much more involved than in our method because 52 is a many-body operator. Another method, based on the Hill-Wheeler integral “) to project angular momentum, is to use the transformation eieJY which would eliminate the linear system of equations for O$ and JZ$; however, as is well known, the evaluation of the matrix elements is then much more involved t. 3.2. CALCULATION
OF O,,(e)
AND lQ@,
We now discuss the evaluation of the overlap functions OZs and Has in terms of the single-particle basis states and its development in a power series in e2. We Iimit ourselves to the case where @, is a Slater determinant. It is convenient to formulate the problem in second-quantized form. Let G: (indices m, n, p, q) create a particle in state (p, which is defined by (6), and consequently is not a state of good angular momentum:
t This was pointed out to us by M. K. Banerjee.
258
M. V. MIHAILOVIC
where #i is the harmonic-oscillator minant CD.:is then given by
et al.
single-particie state In,Ejimi). The Slater deter-
I@&,> = II KDV. m The Hamiltonian may then be expressed in either basis, but it is most convenient to evaluate the interaction matrix elements for the harmo~c-oscillator single-particle states. We also assume that the Ha~ltonian is isospin independent. This assumption is not essential but simplifies somewhat the resulting equations since it allows us to treat the neutrons and protons as almost independent. The evaluation of the overlap functions is greatly simplified when use is made of the following matrix properties. If Nij is a non-singular matrix then its lirst and second minors may be expressed in terms of the inverse matrix I’): n$j’ = det (N)N~‘~ p~$)kr= det (N)[N~“Nl~l -NkjlNlil]. This then restricts us to use non-orthogonal generating functions, but by properly choosing them it need not reduce the available space. In order to simplify the notation we let N,, be the matrix the elements of which are the overlaps of single particIe states NFn, = (~~{~~)I#~(~~)>. It clearly splits into two independent neutron and proton submatrices, N=
N(P) I 0
0 N(n)
!
where the upper indices p and n refer to the proton and neutron parts, respectively. The same holds for the transpose inverse matrix, M = (NT)-‘* Proceeding as indicated the polynominal as follows: lioa(@) = C ~j(e)[~ij(e)~“(~)
H,@(0) in eq. (12) may then be written
+ Qd W&W
+;~~~~~~~C~~~(o)p,,(s)-~ids)~jdBI1(D,(B) +tQik(e)Qjr(e)-Qil(e)Qjk(e)l(Dp(e)lon(e> (13) In the above equation o,(e)
= det (NCP)(0)),
D,(e) = det (N(n~(e)),
GENERATOR
where N(p’(0) and I
COORDINATE
are the tr~sformed
G&G P; 0) =
259
METHOD
overlap matrices with elements
exp (@+)lp,ttl,)> = C Cd(%)Gtj(?~)(exP tej+Miy exp tsj+)tij>* ij
The pofynominals Pii
and Q,j(e) are catculated as indicated below
(14) The kinetic and potential energy matrix elements in the transformed harmonic-oscillator basis are given by T,jCe) V,jkd@>
=
=
@j+>k
(@+MilV (@+Mjl
VI
exp
(&+Mj>,
ew
t@+Mk
exp
single-particle
(@+)A>.
The above numerical calculations involving the matrix polynominals P(6) and Q(e) are greatly simplified by making use of harmonic analysis 13). Consider a polynominal in 0 F(e) = i
Ckek.
k=O
Setting B = eiP we obtain
P(8 = e”“) = k$oCk(cos ktp+ i sin kq) and harmonic analysis can then be used to determine the coeEcients C, by evaluating the polynominal P(f?) at the 2ra+ 1 points 0 = eiV ; (p. = 0,
rp,, . . ., (P~“+~= 2n.
All operations involving the polynominals are then performed in terms of their expansion coefficients. We thereby eliminate the need to recompute D(0) and M(B) at each desired point which would have made the above calculation somewhat impractical. As we have already stated, using the above method eqs. (12, 13) give rise to linear systems of coupled equations with the unknowns being O$ and H$. Comparing equal powers of 8 we then obtain the following type of equations Jz
B~kx~ I3
=
Ck
where BJk and C, are determined as described in subsects. 3.1 and 3.2, respectively.
260
M. V. MIHAILOVIC et al.
4. Illustration: spectrum of “Ne
To illustrate the method we have calculated the low-lying states of “Ne treating it as a 160 inert core with four nucleons in the Nilsson single-particle level No. 6. It is known that the experimental spectrum of “Ne agrees relatively well with the spectrum obtained by projecting angular momentum from the Hartree-Fock solution, which in the language of the GCM means that either eq. (9) has only one solution for each value of angular momentum J, or the excited solutions have very high energies. However, Abgrall et al. I’) have recently shown that the order in which the Hartree-Fock procedure and the projection of angular momentum are performed is very important, even for Z”Ne . By treating the deformation parameter q as a variational parameter for each projected state, they found that it varies sizeably indicating that a single Slater determinant does not constitute a good intrinsic state. In terms of our scheme, we then use generating functions spanning the above space, and the problem is solved consistently. We then proceeded as described below. (i) Hamiltonian. For the spherical single-particle energies we used the energies given by Kelson ‘): &d+= 4.8 MeV, sdt = 9.9 MeV, .eZs+= 5.3 MeV. The two-body interaction used is the one proposed by Rosenfeld r “) -r/a V(r) = -V,[W+BP,+MP,+HP,]
e r/a
with a/b = 0.08, where b is the harmonic-oscillator
parameter,
V, = -42.5
MeV,
w = -0.13,
M = +0.93,
H = -0.26,
B = +0.46.
(ii) Basis. The generator coordinate function &(r~; [xl) is a Slater determinant made up of Nilsson single-particle orbitals where q is Nilsson deformation parameter. Since the Nilsson orbital No. 6 contains the jj orbitals d,, d, and 2s,, the function @Jr~;[x]) has components in the subspace spanned by the spherical jj function of the Id-2s shell. We used the following values of the parameter q in order to span the subspace for diagonalizing the Hamiltonian: 2.75,4.50, 6.40 and 8.00. It is worthwhile to note at this point that an improper choice of values of the parameter may lead to a poor representation for the description of some low-lying states. Such sets are characterized by small eigenvalues of the overlap matrix. (iii) Eigenvalue problem. We then solved the eigenvalue problem corresponding to (3) using the procedure described in sect. 2. The diagonalization of the overlap matrix O&,(e), which is given by eq. (1 I), provided 1 or 2 vectors with overlap eigenvalues larger than 10m3 x (largest eigenvalue). The solution of eq. (9) consequently provided 1 or 2 states of the form given by eq. (8) for each J” = O+, 2+, 4+, 6+. (iv) Results. The results of the diagonalization are shown in fig. 1. The results of the projection of angular momentum from the single determinants @(vi, [xl) are given on the left-hand side, and the GCM solution on the right. It was not the intention of this calculation to reproduce the experimental spacing between levels, and for this reason, neither the force nor the single-particle
GENERATOR
COORDINATE
METHOD
261
energies wer3 varied. This calculation was performed to show that for a reasonable choice of parameters, the qualitative features of the solution are present. There are two sources of numerical errors: (i) the calculation of the polynomials of the overlaps pij(e) given by eq. (14) in terms of the Fourier analysis becomes difficult, especially as the number of particles becomes large; (ii) the matrix elements
‘ON8 IO +
6-
0+ 8 -2+=
_+ 0-
6+ _
4+ --
+ 4 -I)
2-+
2+--
4-c z $
6-
& 3 3 4-
6+-
+ 6-
4+2+0+I
+ 4-
+ 22o-+
01 2
+ o-
I
I
4
6
+ o1
I%-r
GCM
Fig. 1. Levels of *ONe obtained by projecting angular momentum from single Slater determinants with q = 2.75, 4.50, 6.40, 8.0 and the corresponding generator coordinate method solution. Vij~, in thejj-basis and m-scheme are not in~~ndent and if they are not calculated accurately enough the errors accumulate rapidly when they come in the sum (13) in which the polynomials Pij(6) also appear. We remark that this second source of error is not intrinsic to our formulation, but rather arises from the use of a specific representation for calculating the potential matrix elements, The calculation we presented was performed on a IBM 360/44 computer, which has low accuracy for this sort of problem and, for this reason we have not tried to deduce quantitative conclusions. For future nuclear structure, calculations of more interesting nuclei ( 160, 28Si) double precision seems necessary.
5. Conclusion In this work we have presented what we think is a useful method for studying collective states in light nuclei. Being based on the generator coordinate technique
262
M. V. MIHAILOVIC
ef al.
it provides a picturesque connection between the shell model and collective model, while working in a space of dimension much smaller than the shell model. The essential part of this work was the presentation of an operational method for projecting angular momentum from an arbitrary sum of Slater determinants which is necessary for studying complex collective states which cannot be described using a single deformed state. Finally, we propose to undertake further studies of the 2s-ld shell for which we think the above method is well suited. The authors would like to thank M. K. Banerjee, M. Rosina and G. Stephenson for several interesting discussions. We express our appreciation to the University of Maryland Cyclotron Group for the use of their IBM 360/44 and their assistance. One of us (M.V.M.) also wishes to thank the Center for Theoretical Physics at the University of Maryland for its hospitality. References 1) I. Kelson, Phys. Rev. 132 (1963) 2189; I. Kelson and C. A. Levinson, Phys. Rev. 134 (1964) B269; J. Bar-Touv and I. Kelson, Phys. Rev. 138 (1965) B1035; G. Ripka, Advances in Nuclear Physics, Vol. 1, eds. M. Baranger and E. Vogt (Plenum Press, 1968) 2) S. Das Gupta and M. Harvey, Nucl. Phys. A94 (1967) 602 3) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102 4) R. E. Peierls and J. Yoccoz, Proc. Phys. Sot. A70 (1957) 381; R. E. Peierls and D. J. Thouless, Nucl. Phys. 38 (1962) 154 5) J. J. Griffin and J. A. Wheeler, Phys. Rev. 108 (1957) 311; B. Jancovici and D. H. Schiff, Nucl. Phys. 58 (1964) 678 6) D. M. Brink and A. Weiguny, Nucl. Phys. A120 (1968) 59 7) F. Villars, Proc. of the Int. School of Physics “Enrico Fermi” Course 36 (Academic Press, 1966) 8) D. Justin, M. V. Mihailovic and M. Rosina, Phys. Lett. 29B (1969) 458 9) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 1, No. 8 (1959) 10) I. Kelson, Nucl. Phys. 89 (1966) 387; M. I. Friedman and I. Kelson, Nucl. Phys. Al44 (1970) 209 11) D. Brink, Proc. of the Int. School of Physics “Enrico Fermi” Course 36 (Academic Press, 1966) 12) P. 0. Lbwdin, Rev. Mod. Phys. 36 (1964) 966 13) Handbook of Mathematical Functions, eds. by M. Abramowitz and I. S. Stegun (National Bureau of Standards, 1966) 14) Y. Abgrall, G. Baron, E. Caurier and 0. Monsonego, Nucl. Phys. A131 (1969) 609 15) L. Rosenfeld, Nuclear Forces (North-Holland, Amsterdam, 1948) p. 233
AND
THE GENERATOR
COURDINATE METHOD FOR LIGHT p3uCLEl M. V. MIHAILGVIC t, E. KUJAWSKI Center for Theore%icalPhysJcs, Department of Physics and Astronomy, University of Maryland, College Park, Muryland 20742 tt and J. LESJAK
Received 4 September 1970 Abstract: A method for studying collective states in light nuclei not describable in terms of a single intrinsic Slater determintit is presented. It is based on the generator coordinate method and makes use of “generating functions” obtained by projecting angular momentum from intrinsic deformed states. A useful procedure for projecting angular momentum from a generator coordinate wave function is presented. As an illustration the above method is applied to ‘ONe,
In recent years attempts have been made to describe states of Id-2s nuclei using the Hartree-Fock method. By performing the projection of angular momenta many states are understood as states belonging to the rotational band of the deformed ground state determinants “). However, there are states which cannot be obtained by projecting the angular momenta from a single determinants function. One example is ?3i where two minima exist, one ~orres~n~ng to the obIate shape and the other to the prolate shape of the Hartree-Fock field. The spectrum obtained by projection of angular momentum from either Hartree-Fock solution does not resemble the experimental one “). For such nuclei the generator coordinate method (GCM) due to Hill and Wheeler “) offers the possibility to go beyond the Hartree-Fock method and at the same time preserve the pictorial description of nuclei, such as the shape and collective motion “). The GCM is a variational method based on the many-body nuclear Harn~~to~ia~, and it yields an integral homogeneous equation for the wave functions. Consider a family of N-particle functions @([a~],p), where [xl denotes all coordinates of the N particles and p denotes a set of parameters, p = (pl, pz, . . ., p,). A trial t Fulbright-Hays Program Fellow on leave of absence from Nuclear Institute “J. Stefan,” Ljubljana, Yugoslavia. tt Research supported by the National Science Foundation under Grant NSF GU2061. 2.52
GENERATOR
eigenfunction
COORDINATE
METHOD
253
of the N-particle system can be generated by the linear combination
The weight function f(p) is determined from the variational principle
where H is the Hamiltonian equation for f(p):
of the system. Eq. (2) reduces to the following integral
s
{H(P,P’>-WP, p’))f(p’)W = 0,
(3)
where WP9 P’> = <@([xl, PI9 NCXI, P')>, H(P, P'> = <@(Cd> P>, H@i(Cxl, P'>>.
(4)
The lowest eigenvalue E corresponds to the ground state energy, and the other eigenvalues to the excited states. The integral (1) generally generates a set of states !P,( [xl) which forms a subspace .%,+ of the complete Hilbert space ti. Diagonalization of the Hamiltonian H in the subspace Z+ is equivalent to the solution of the integral equation (3). The GCM will give an exact solution if the integral (1) generates all N-particle functions; usually it does not and an approximate solution is obtained. The accuracy of the solution depends on the choice of @([xl, p). Usually one takes simple functions for the generator functions @([xl, p), but even then an exact calculation is hard to perform. Most frequently the overlaps (4) are approximated ‘) by Gaussian functions, @P,
H(p, p’) = P,
P’)
= exp
9,
C-(P P-P’
-P’)~/~~~I, exp
[--(P-P’)~P~I,
where P, is a quadratic polynominal in the variables +(p+p’) and (p-p’). It has been shown by Brink and Weiguny “) that this approximation leads to a model Hamiltonian of the Bohr-Mottelson type for coupled harmonic vibrations. Although this approximation seems plausible its validity has not been shown, and furthermore Villars ‘) has given examples where it does not hold, proving in such a way that the Gaussian form is not a general property of the overlap functions (4). More recently it has been shown that the overlaps are not Gaussians if the GCM function (1) is generated by means of a N-projected BCS function, @&([x], d) [ref. “)I. In this paper a method and a procedure are presented which can be applied to the
254
M. V. MIHAILOVIC
et al.
description of collective states in light nuclei. Our work is based on the GCM and it avoids the Gaussian approximation (5). A similar method was applied to describe pairing vibrations “). The success of the Nilsson-Mottelson “) model and HartreeFock calculations in describing the properties of light nuclei suggest that a good generator coordinate function @“([xl, p) may be obtained by projecting angular momentum from a Slater determinant in which the parameter p describes the shape of the singleparticle field (i.e. deformation parameters #?, 7). We then use such functions for the generator functions. The method is described more fully in sect. 2. The central part of this paper, sect. 3, is devoted to a method of projecting angular momenta from a sum of non-orthogonal Slater determinants such as generated by function (1). The subspace for the projection of angular momentum is spanned by the states exp (0, J+)@( [xl, p). If @([xl, p) is a Slater determinant, the transformed states exp (eJ+)@( [xl, p) are also Slater determinants. This makes the calculation of the overlap integrals easier than in the basis spanned by the states J:@( [xl, p) as proposed by Kelson lo). Finally, to illustrate the above method we apply it to calculate the spectrum of “Ne which we consider as an I60 inert core plus four particles in the Id-2s shell. The GMC function for “Ne is taken as a linear combination of Slater determinants corresponding to different deformations. The results are presented in sect. 4. 2. Description of the method We now give the mathematical formulation of the GCM as applied to light nuclei using generating functions @“(p, [xl) obtained by projecting angular momentum from a Slater determinant, where the parameters p are related to the shape of the nucleus. For convenience we take as the single-particle basis used to build our Hilbert subspace of N-particle states the harmonic-oscillator functions J/j(X), (j = 1, . . . ,L), and introduce the single-particle functions 4,(x) which depend on the shape of the nucleus through the coefficients C”j
4n(‘>=j~lc*j+j(x)s
n
=
1,. . ., N.
(6)
The N-particle Slater determinant Qi, built from 4, then depends on (k) coefficients Cnj @,=
@~,(c,1,c,,,...;[~]>.
(7)
If the constants Cnj are taken as variational parameters in the generator function @=, the GCM leads to complete mixing of the configurations - the shell-model job. If the weight function f(p) in eq. (1) is restricted to a delta function, the GCM leads to the Hartree-Fock solution in the subspace defined by the set {+J. In both cases a formulation of the problem in the form of the GCM provides no advantage over the standard formulations. The GCM will, however, be very useful and simple if the
GENERATOR
COORDINATE
METHOD
255
coefficients Cnj can be expressed in terms of a single parameter which describes nuclear properties such as shape and degree of pairing. (At times it may be more convenient to introduce additional parameters.) The multiparametric function [x]) is then replaced by a single parametric function @(t, [xl) with @(C11,Gz,...;
where s,j are constants defining the system, and 1 is a parameter related to the shape of the single-particle potential such as the deformation parameter q of the Nilsson single-particle functions. In this way eq. (3) reduces to a one-dimensional integral equation which can then be solved numerically. The subspace X4 spanned by the multiparametric functions (7) can be the same as the subspace 8, spanned by the single-parametric functions @(t, [x]) if the Cnj are properly chosen. This requirement is fulfilled if the C~j are such that the vectors
are IinearIy independent for d different values of tj where d is the dimension of the subspace X+. This condition can be often satisfied “) using rather simple forms for the functions C(E,j, t). For small differences in the parameter t the wave functions do not change much so that one may get a reasonably good wave function using a finite number of values for the parameter. The integral equation (3) then reduces to an algebraic eigenvalue problem of the form
C (Hijj
EOij)fj = 0.
It can be reduced to a conventional eigenvalue problem using the procedure described below. As is well known, the lowest eigenvalues so obtained should be lower than the energies corresponding to each input state. (i) The overlaps
areformed and the matrix Oij is diagonalized. The values ti are chosen in a wide enough range around the minima of (QJ, H@J)/(@J, @‘> so that it spans a space large enough to include the physically important states. (ii) The subspace in (i) is truncated to the subspace &@&of dimension d’ K d by rejecting all vectors with eigenvalues which are a few orders of magnitude smaller than the largest one. We thereby avoid the difficulties which are associated with states with small eigenvalues for the overlap matrix since for such states a small error in the matrices Oij or Nij can then lead to a large error in the energy ll). It is further hoped that these states are not physically important. (iii) The Hamiltonian is then diagonalized in the subspace #;P.
M, V. MIHAILOVIC et al.
256
3. Projection of anguIar momentum
We now present a method for projecting angular momentum from a generator coordinate wave function generated by axially symmetric wave functions GGPaM, where 34 is the projection of the angular momentum along the axis of symmetry and ct distinguishes among the functions @ with the same M. Let P” denote the angular momentum projection operator, P’IP,,
= ip,,, = \aJM>.
jPaMmay be then expressed as Jmnr Q,aM =JE&R&u. The weight functions&
in the trial function YJM =
~f:%/na
satisfy the secular equation
-~~{~~~~~~)~f~= 0.
~~(a~~l~~~~~)
In order to proceed we then have to calculate the overlap integrals
H$ =
(aJMIH~~Ji%f),
O;l, = (aJi%fIfiJM). 3.1. SYSTEM OF EQUATIONS FOR Ua19*AND ffaBJ
We introduce the non-unitary operator exp (6J+), where 19may be a complex parameter and we then consider transformed vectors
1
b
(~-~)~(~+~+k)!
J”,k = k! [ (J+M)&hW-k)!
1 *
*
The overlaps functions O&S) and H,,(O) of the vectors exp (8J+)j@~&? and exp (BJ+)\@SMg>can then be expressed in terms of the overlaps for states of good angular momentum jaJM>.
o,,(@)=
J-M,
= JZJ,
&
J,, = oM z J=Jo
J--M@ k;. ek+k’bJkr,k bJM#
+ klBJj+G + k’)
011
GENERATOR
COORDINATE
METHOD
257
where
Jo = max (IM,I, IMsl),
MI = max CM,,MB),
M = IK-M/d, M, = min (M,, M&.
Proceeding in a similar manner for H,,(0), we obtain &&3) = <@&,I exp @J-)H exp @&M&J =
Jmax
J-MI
J=J,,
k=O
oMc
c
@kbJMlkbJM~k+bfH:/3’
(12)
We then see that the matrix elements O& and H& defined by eq. (lo), are the coefficients of the polynominals O,,(e) and I&(0) given by eqs. (11, 12). In order to calculate O& and Hi:, , we must now evamate the left-hand side of (11) and (12). The 1.h.s. of (11,12) is obtained by expressing exp (@J+)I@jam>in terms of the transformed single-particle states exp (&I+)$, and calculating the overlaps Oprsand Hols as polynominals in 8. This is described in the next section. By equating the coefficients of equal powers of 8 we obtain a linear system of equations for O&laand Hi’. The above procedure is especially simple if GcrM is a Slater determinant since the state exp (&F+)I@~M>is then again a Slater determinant, and is the reason we choose the operator exp (&.I,) to calculate the overlaps O& and H$ . We think that our method is more practical than the one given by Kelson lo) who calculates the integrals O$ and H$ from a system of linear equations, the input of which are the integrals (Ip&,[ J”J$ jtDsM8>and (GorMJJk HJ!f.1CPBMa>. The explicit evaluation of the above overlaps from the known Slater determinants and the resulting set of triangular equations is much more involved than in our method because 52 is a many-body operator. Another method, based on the Hill-Wheeler integral “) to project angular momentum, is to use the transformation eieJY which would eliminate the linear system of equations for O$ and JZ$; however, as is well known, the evaluation of the matrix elements is then much more involved t. 3.2. CALCULATION
OF O,,(e)
AND lQ@,
We now discuss the evaluation of the overlap functions OZs and Has in terms of the single-particle basis states and its development in a power series in e2. We Iimit ourselves to the case where @, is a Slater determinant. It is convenient to formulate the problem in second-quantized form. Let G: (indices m, n, p, q) create a particle in state (p, which is defined by (6), and consequently is not a state of good angular momentum:
t This was pointed out to us by M. K. Banerjee.
258
M. V. MIHAILOVIC
where #i is the harmonic-oscillator minant CD.:is then given by
et al.
single-particie state In,Ejimi). The Slater deter-
I@&,> = II KDV. m The Hamiltonian may then be expressed in either basis, but it is most convenient to evaluate the interaction matrix elements for the harmo~c-oscillator single-particle states. We also assume that the Ha~ltonian is isospin independent. This assumption is not essential but simplifies somewhat the resulting equations since it allows us to treat the neutrons and protons as almost independent. The evaluation of the overlap functions is greatly simplified when use is made of the following matrix properties. If Nij is a non-singular matrix then its lirst and second minors may be expressed in terms of the inverse matrix I’): n$j’ = det (N)N~‘~ p~$)kr= det (N)[N~“Nl~l -NkjlNlil]. This then restricts us to use non-orthogonal generating functions, but by properly choosing them it need not reduce the available space. In order to simplify the notation we let N,, be the matrix the elements of which are the overlaps of single particIe states NFn, = (~~{~~)I#~(~~)>. It clearly splits into two independent neutron and proton submatrices, N=
N(P) I 0
0 N(n)
!
where the upper indices p and n refer to the proton and neutron parts, respectively. The same holds for the transpose inverse matrix, M = (NT)-‘* Proceeding as indicated the polynominal as follows: lioa(@) = C ~j(e)[~ij(e)~“(~)
H,@(0) in eq. (12) may then be written
+ Qd W&W
+;~~~~~~~C~~~(o)p,,(s)-~ids)~jdBI1(D,(B) +tQik(e)Qjr(e)-Qil(e)Qjk(e)l(Dp(e)lon(e> (13) In the above equation o,(e)
= det (NCP)(0)),
D,(e) = det (N(n~(e)),
GENERATOR
where N(p’(0) and I
COORDINATE
are the tr~sformed
G&G P; 0) =
259
METHOD
overlap matrices with elements
exp (@+)lp,ttl,)> = C Cd(%)Gtj(?~)(exP tej+Miy exp tsj+)tij>* ij
The pofynominals Pii
and Q,j(e) are catculated as indicated below
(14) The kinetic and potential energy matrix elements in the transformed harmonic-oscillator basis are given by T,jCe) V,jkd@>
=
=
@j+>k
(@+MilV (@+Mjl
VI
exp
(&+Mj>,
ew
t@+Mk
exp
single-particle
(@+)A>.
The above numerical calculations involving the matrix polynominals P(6) and Q(e) are greatly simplified by making use of harmonic analysis 13). Consider a polynominal in 0 F(e) = i
Ckek.
k=O
Setting B = eiP we obtain
P(8 = e”“) = k$oCk(cos ktp+ i sin kq) and harmonic analysis can then be used to determine the coeEcients C, by evaluating the polynominal P(f?) at the 2ra+ 1 points 0 = eiV ; (p. = 0,
rp,, . . ., (P~“+~= 2n.
All operations involving the polynominals are then performed in terms of their expansion coefficients. We thereby eliminate the need to recompute D(0) and M(B) at each desired point which would have made the above calculation somewhat impractical. As we have already stated, using the above method eqs. (12, 13) give rise to linear systems of coupled equations with the unknowns being O$ and H$. Comparing equal powers of 8 we then obtain the following type of equations Jz
B~kx~ I3
=
Ck
where BJk and C, are determined as described in subsects. 3.1 and 3.2, respectively.
260
M. V. MIHAILOVIC et al.
4. Illustration: spectrum of “Ne
To illustrate the method we have calculated the low-lying states of “Ne treating it as a 160 inert core with four nucleons in the Nilsson single-particle level No. 6. It is known that the experimental spectrum of “Ne agrees relatively well with the spectrum obtained by projecting angular momentum from the Hartree-Fock solution, which in the language of the GCM means that either eq. (9) has only one solution for each value of angular momentum J, or the excited solutions have very high energies. However, Abgrall et al. I’) have recently shown that the order in which the Hartree-Fock procedure and the projection of angular momentum are performed is very important, even for Z”Ne . By treating the deformation parameter q as a variational parameter for each projected state, they found that it varies sizeably indicating that a single Slater determinant does not constitute a good intrinsic state. In terms of our scheme, we then use generating functions spanning the above space, and the problem is solved consistently. We then proceeded as described below. (i) Hamiltonian. For the spherical single-particle energies we used the energies given by Kelson ‘): &d+= 4.8 MeV, sdt = 9.9 MeV, .eZs+= 5.3 MeV. The two-body interaction used is the one proposed by Rosenfeld r “) -r/a V(r) = -V,[W+BP,+MP,+HP,]
e r/a
with a/b = 0.08, where b is the harmonic-oscillator
parameter,
V, = -42.5
MeV,
w = -0.13,
M = +0.93,
H = -0.26,
B = +0.46.
(ii) Basis. The generator coordinate function &(r~; [xl) is a Slater determinant made up of Nilsson single-particle orbitals where q is Nilsson deformation parameter. Since the Nilsson orbital No. 6 contains the jj orbitals d,, d, and 2s,, the function @Jr~;[x]) has components in the subspace spanned by the spherical jj function of the Id-2s shell. We used the following values of the parameter q in order to span the subspace for diagonalizing the Hamiltonian: 2.75,4.50, 6.40 and 8.00. It is worthwhile to note at this point that an improper choice of values of the parameter may lead to a poor representation for the description of some low-lying states. Such sets are characterized by small eigenvalues of the overlap matrix. (iii) Eigenvalue problem. We then solved the eigenvalue problem corresponding to (3) using the procedure described in sect. 2. The diagonalization of the overlap matrix O&,(e), which is given by eq. (1 I), provided 1 or 2 vectors with overlap eigenvalues larger than 10m3 x (largest eigenvalue). The solution of eq. (9) consequently provided 1 or 2 states of the form given by eq. (8) for each J” = O+, 2+, 4+, 6+. (iv) Results. The results of the diagonalization are shown in fig. 1. The results of the projection of angular momentum from the single determinants @(vi, [xl) are given on the left-hand side, and the GCM solution on the right. It was not the intention of this calculation to reproduce the experimental spacing between levels, and for this reason, neither the force nor the single-particle
GENERATOR
COORDINATE
METHOD
261
energies wer3 varied. This calculation was performed to show that for a reasonable choice of parameters, the qualitative features of the solution are present. There are two sources of numerical errors: (i) the calculation of the polynomials of the overlaps pij(e) given by eq. (14) in terms of the Fourier analysis becomes difficult, especially as the number of particles becomes large; (ii) the matrix elements
‘ON8 IO +
6-
0+ 8 -2+=
_+ 0-
6+ _
4+ --
+ 4 -I)
2-+
2+--
4-c z $
6-
& 3 3 4-
6+-
+ 6-
4+2+0+I
+ 4-
+ 22o-+
01 2
+ o-
I
I
4
6
+ o1
I%-r
GCM
Fig. 1. Levels of *ONe obtained by projecting angular momentum from single Slater determinants with q = 2.75, 4.50, 6.40, 8.0 and the corresponding generator coordinate method solution. Vij~, in thejj-basis and m-scheme are not in~~ndent and if they are not calculated accurately enough the errors accumulate rapidly when they come in the sum (13) in which the polynomials Pij(6) also appear. We remark that this second source of error is not intrinsic to our formulation, but rather arises from the use of a specific representation for calculating the potential matrix elements, The calculation we presented was performed on a IBM 360/44 computer, which has low accuracy for this sort of problem and, for this reason we have not tried to deduce quantitative conclusions. For future nuclear structure, calculations of more interesting nuclei ( 160, 28Si) double precision seems necessary.
5. Conclusion In this work we have presented what we think is a useful method for studying collective states in light nuclei. Being based on the generator coordinate technique
262
M. V. MIHAILOVIC
ef al.
it provides a picturesque connection between the shell model and collective model, while working in a space of dimension much smaller than the shell model. The essential part of this work was the presentation of an operational method for projecting angular momentum from an arbitrary sum of Slater determinants which is necessary for studying complex collective states which cannot be described using a single deformed state. Finally, we propose to undertake further studies of the 2s-ld shell for which we think the above method is well suited. The authors would like to thank M. K. Banerjee, M. Rosina and G. Stephenson for several interesting discussions. We express our appreciation to the University of Maryland Cyclotron Group for the use of their IBM 360/44 and their assistance. One of us (M.V.M.) also wishes to thank the Center for Theoretical Physics at the University of Maryland for its hospitality. References 1) I. Kelson, Phys. Rev. 132 (1963) 2189; I. Kelson and C. A. Levinson, Phys. Rev. 134 (1964) B269; J. Bar-Touv and I. Kelson, Phys. Rev. 138 (1965) B1035; G. Ripka, Advances in Nuclear Physics, Vol. 1, eds. M. Baranger and E. Vogt (Plenum Press, 1968) 2) S. Das Gupta and M. Harvey, Nucl. Phys. A94 (1967) 602 3) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102 4) R. E. Peierls and J. Yoccoz, Proc. Phys. Sot. A70 (1957) 381; R. E. Peierls and D. J. Thouless, Nucl. Phys. 38 (1962) 154 5) J. J. Griffin and J. A. Wheeler, Phys. Rev. 108 (1957) 311; B. Jancovici and D. H. Schiff, Nucl. Phys. 58 (1964) 678 6) D. M. Brink and A. Weiguny, Nucl. Phys. A120 (1968) 59 7) F. Villars, Proc. of the Int. School of Physics “Enrico Fermi” Course 36 (Academic Press, 1966) 8) D. Justin, M. V. Mihailovic and M. Rosina, Phys. Lett. 29B (1969) 458 9) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 1, No. 8 (1959) 10) I. Kelson, Nucl. Phys. 89 (1966) 387; M. I. Friedman and I. Kelson, Nucl. Phys. Al44 (1970) 209 11) D. Brink, Proc. of the Int. School of Physics “Enrico Fermi” Course 36 (Academic Press, 1966) 12) P. 0. Lbwdin, Rev. Mod. Phys. 36 (1964) 966 13) Handbook of Mathematical Functions, eds. by M. Abramowitz and I. S. Stegun (National Bureau of Standards, 1966) 14) Y. Abgrall, G. Baron, E. Caurier and 0. Monsonego, Nucl. Phys. A131 (1969) 609 15) L. Rosenfeld, Nuclear Forces (North-Holland, Amsterdam, 1948) p. 233