Optimal basis states for a microscopic calculation of intrinsic vibrational wave functions of deformed rotational nuclei

Nuclear Physics A 703 (2002) 167–187 www.elsevier.com/locate/npe

Optimal basis states for a microscopic calculation of intrinsic vibrational wave functions of deformed rotational nuclei M.J. Carvalho a,∗ , D.J. Rowe b , S. Karram b , C. Bahri b a Department of Mathematics, Physics and Computer Science, Ryerson Polytechnic University,

Toronto, ON, M5B2K3, Canada b Department of Physics, University of Toronto, Toronto, ON, M5S 1A7, Canada

Received 20 September 2001; revised 7 November 2001; accepted 15 November 2001

Abstract The primary achievement of the symplectic model is to give a realistic microscopic shell-model expression of the nuclear collective model. However, in applications of the model, one has to contend with the fact that its Hilbert spaces, like those of the shell model, are infinite dimensional. This means that truncation of the model Hilbert space to a finite-dimensional subspace is inevitable. Nevertheless, it is in principle possible to get results to any desired accuracy if a sequence of increasingly large, but finite-dimensional subspaces of the full Hilbert space can be determined so that truncation to the subspaces of the sequence leads to rapidly convergent results. We show in this paper that generator coordinate (also called coherent state) bases can be constructed and optimized to give extraordinarily rapid convergence. This makes it possible to perform accurate symplectic model calculations with realistic microscopic Hamiltonians by a method that is essentially an angular-momentum projected, multi-determinant, Hartree–Fock calculation for which powerful but highly practical methods, that make a minimal use of group theory, have been developed. It is shown that the techniques developed give a natural representation of the intrinsic states of rotational bands as beta- and/or gammavibrational wave functions. The techniques are illustrated by computation of beta-vibrational wave functions for the ground-state and one-phonon excited rotational bands of 8 Be for a microscopic Hamiltonian with a Brink–Boeker two-body interaction.  2001 Elsevier Science B.V. All rights reserved.

1. Introduction Collective states of nuclei, that are well described by phenomenological collective models, should ultimately be expressed in microscopic shell-model terms if one is * Corresponding author.

E-mail address: [email protected] (M.J. Carvalho). 0375-9474/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 1 4 5 8 - 0

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to understand the dynamics of nuclear collective motions and how they arise from interacting neutrons and protons. The challenge is to handle the huge number of shellmodel configurations needed to account for the observed collective properties, in a conventional shell-model basis and to interpret the results in physically meaningful collective terms. A major advance towards a microscopic description of collective states was achieved by the discovery that collective models invariably have an algebraic expression. By definition, the basic observables of an algebraic model span a so-called spectrum generating algebra (SGA), which is the Lie algebra for a dynamical group for the model. The Hilbert space for such a model is then the carrier space for an irrep (irreducible representation) of the model SGA. Thus, if collective model observables can be expressed as a subset of shellmodel operators, it becomes feasible to construct irreps of a collective model SGA within the space of the shell model and thereby endow collective states with microscopic wave functions. Such an embedding of collective model states in the shell-model space makes it possible, at least in principle, to exploit the full arsenal of shell-model observables to explore the microscopic dynamics of collective states [1]. The SGA of the nuclear symplectic model [2] is the noncompact symplectic algebra sp(3, R) (called sp(6, R) by some authors) whose elements are infinitesimal generators of quadrupole and monopole collective motions. The elements are realized as bilinear combinations of harmonic-oscillator creation and annihilation operators which makes it possible to expand symplectic model states in a harmonic-oscillator shell-model basis. The model then appears as an extension of Elliott’s SU(3) model [3], whose SGA is the su(3) subalgebra of sp(3, R) elements that commute with the harmonic-oscillator Hamiltonian. An important achievement of the symplectic model is its ability to link together the phenomenological and microscopic descriptions of rotations and giant vibrations in nuclei to provide a deeper understanding of nuclear collective motions. In working with an algebraic model, the standard procedure is to start by constructing an orthonormal basis for the model Hilbert space using the algebraic methods of representation theory. Then, with a Hamiltonian expressed as a low-order polynomial in the SGA, the Hamiltonian matrix can be computed and diagonalized. However, a microscopic Hamiltonian expressed as a sum H = H0 + V of a single-particle Hamiltonian and a two-nucleon interaction is not of this type. Powerful methods have been developed for implementing microscopic symplectic model calculations in an su(3) basis (cf. Escher and Draayer [4] and references therein). However, for the kinds of interactions we consider, the dimensions of the spaces needed for low-energy states to obtain accurate results in an su(3) basis, are still too large for practical purposes. This is in spite of the restriction of the shell-model space to a single sp(3, R) irrep. Generator coordinate methods [5,6] which employ finite group transformations rather than the infinitesimal transformations of the algebra, are then often easier and better. Generator coordinate bases for the symplectic model were introduced by Filippov and colleagues [7,8]. They were developed in the form we use them in Refs. [9–11]. The value of a generator coordinate basis, as we show here, is that it is possible to select optimized subsets of generator states so that accurate results can be obtained within very low-dimensional spaces. An even more important advantage, as

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shown here, is that the wave functions in a generator coordinate can be expressed in a form that gives them a meaningful physical interpretation in collective model terms. Application of the generator coordinate method to the symplectic model is a natural extension of that used for the SU(3) model, for which it is known [3] that if |Φ0  is the highest-weight state for an su(3) irrep, then the set of rotated highest-weight states
 |ΦΩ  = R(Ω)|Φ0 : Ω ∈ SO(3) , (1) where SO(3) is the rotation group, span the space of the su(3) irrep (note we use capital letters, e.g., SO(3), to distinguish a group from its Lie algebra, so(3)). Equivalently, a discrete basis of states for an su(3) irrep is given by projecting out states of good angular momentum by taking the linear combinations   2l + 1 L∗ (Ω)R(Ω)|Φ0 , (2) dΩ DMK |KLM = PLMK |Φ = 8π 2 for a range of KLM values given by the known SU(3) → SO(3) branching rules [3]. Thus, the highest-weight state for an su(3) irrep is a generator state from which a basis of (nonorthonormal) states is constructed by angular momentum projection. Moreover, the spectrum of a microscopic Hamiltonian H restricted to the Hilbert space of a single su(3) irrep can be computed from the values of the overlaps Φ0 |H |ΦΩi  for a linearlyindependent set of basis states {|ΦΩi , i = 1, . . . , N}, where N is the dimension of the irrep. Nonorthonormal basis states are similarly generated for an sp(3, R) irrep. It is known [1, 9,10] that if |Φ0  is the lowest-weight state for an sp(3, R) irrep (cf. Section 2), then the Hilbert space of the irrep is spanned by the set of states 
(3) |g = Γ (g)|Φ0 : g ∈ GL+ (3, R) , where GL+ (3, R) is the group of general linear transformations of the nucleon coordinates of positive determinant. A GL+ (3, R) matrix can be factored: g = ΩΛ,

(4)

where Ω is an SO(3) rotation and Λ is a real 3 × 3 positive symmetric matrix. It follows that an sp(3, R) irrep is spanned by a set of states
 , R(Ω)|Λ: Ω ∈ SO(3), Λ = Λ (5)  is the transpose of Λ. Thus, in parallel with the above result for an su(3) irrep, where Λ a nonorthonormal basis of states for an sp(3, R) irrep is angular-momentum projected from a suitably selected set of generator states {|Λ}. Details are given in Section 2. It is useful to note that if a state |Φ0  is of lowest-weight relative to a particular basis for the sp(3, R) Lie algebra, then the transformed state |Λ is also of lowest weight, albeit relative to a transformed basis for the algebra. For if |Φ0  is annihilated by the sp(3, R) lowering operators, e.g., Bij |Φ0  = 0, then the transformed state |Λ is annihilated by the transformed lowering operators, e.g., Bij (Λ) = Γ (Λ)Bij Γ (Λ−1 ). In fact, transformed lowest-weight states are coherent states of the symplectic group sp(3, R) and, as such,

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have many useful properties. For present purposes, the most useful property is that the states angular-momentum projected from a discrete set {|Λ} of lowest-weight states span an sp(3, R) irrep. Thus, suitably selected subsets of such coherent states provide physically useful generator states for optimally truncated symplectic model calculations. Since the Hilbert space for an sp(3, R) irrep is infinite, it is necessary, for practical purposes, to truncate to a finite-dimensional subspace. In the so-called SO(3) × D model [9], the Λ matrices, which define the generator states {|Λ}, are selected from among a subset of diagonal matrices for which Λij = δij λi . In the so-called sp(1, R) model of Arickx [12], the matrices are further restricted to a set with λ2 = λ3 = 1. Clearly the accuracy of the results depends on the choice made. A primary objective of the present investigation is therefore to consider optimal ways to select finite numbers of generator states such that, whatever number is chosen, the results are the most accurate obtainable with that number of generator states. For practical reasons, we again restrict consideration to diagonal Λ matrices, as in the SO(3) × D model. An exploration of the effects of including nondiagonal Λ matrices is a high priority for subsequent investigations. However, it is appropriate to proceed systematically to a selection of generator states among a larger class of possibilities only after the criteria are established for selection among smaller classes. We therefore consider the selection of optimal generator states first among a set with λ2 = λ3 fixed and subsequently extend the exploration to more general diagonal matrices. The effects of ignoring generator states with nondiagonal Λ matrices is discussed in the concluding remarks. The detailed investigations of this paper are restricted to the 8 Be nucleus for which many exploratory calculations are possible. Subsequent applications of what has been learned will be extended to heavier nuclei. The results for 8 Be show that the rate of convergence, as a function of the number of generator states retained, is rapid when the sequence of generator states is optimized according to the prescription outlined in Section 4. In fact, results that are close to fully converged results can already be obtained with a single generator state. This is a remarkable result with significance both for the validity of the Hartree–Fock approximation and for a rotor model interpretation of the results. Recall that a characteristic of a rigid rotor is that all states of a rotational band can be angular-momentum projected from a single intrinsic state [14] whereas for a less rigid rotor the intrinsic state may change, e.g., due to centrifugal stretching, with increasing angular momentum. Thus, it is natural to interpret a set of generator states {|Λ} of the symplectic model as a basis of intrinsic states. The results of the model then allow one to determine the extent to which the linear combinations of generator states form a single intrinsic state that remains constant within a symplectic model band and, conversely, the extent to which the intrinsic state is changed with increasing angular momentum by the inertial forces; this enables one to see if the states are characteristic of rigid or soft rotations. The basic ingredients of the symplectic model and the generator coordinate method, as applied to the symplectic model, are reviewed in Sections 2 and 3. Section 4 is concerned with a strategy for finding optimal sets of generator states for 0p–0h and 4p–4h irreps of some A = 4n nuclei. From the results presented in Section 4, it follows that the

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wave functions have a natural interpretation in terms of beta-vibrational intrinsic states of a nuclear rotor. In Section 5 the spherical shell-model content of the symplectic model states is given in an su(3) basis and used to exhibit the relationship between the rotor model and the shell model.

2. Symplectic irreps and their generator states A shell-model representation of the sp(3, R) algebra, with center-of-mass degrees of freedom removed, is spanned by the bilinear combinations Aij =

A 

† † bni bnj

n=1

Cij = Bij =

1 2

m,n=1

A  n=1

A 

A 1  † † − bmi bnj , A

A  † 1   † †  †  bni bnj + bnj bni − bmi bnj + bnj bmi , 2A m,n=1

bni bnj −

n=1

1 A

A 

bmi bnj ,

(6)

m,n=1

of harmonic-oscillator raising and lowering operators



1 xsi b0 1 xsi b0 † bsi =√ − i psi , + i psi , bsi = √ (7) h¯ h¯ 2 b0 2 b0 √ where b0 = h¯ /mω0 is the harmonic-oscillator unit of length; s is a particle index and i or j = 1, 2, 3 index the (x, y, z) spatial directions. The Cij operators span the subalgebra u(3). For convenience, the following expressions are given in harmonic units of length in which b0 = 1. The shell-model Hilbert space is a direct sum of irreducible subspaces of the symplectic algebra sp(3, R). Each irrep is characterized by a lowest-weight state, Φ0 , which satisfies the condition Bij |Φ0  = 0,

∀ i, j,

Cij |Φ0  = 0,

∀ i < j,

(8)

and is labeled by a lowest weight [f1 , f2 , f3 ] where Cii |Φ0  = fi |Φ0 ,

i = 1, 2, 3.

(9)

Equivalently, it is labeled by harmonic-oscillator quantum numbers (ν1 , ν2 , ν3 ) related to the lowest weight by fi = νi + (A − 1)/2. The harmonic-oscillator energy, E0 , of the lowest-weight state is then E0 = h¯ ω0

3 

fi .

(10)

i=1

If the harmonic-oscillator energy of a lowest-weight state is as low as it can be for a given nucleon number, i.e., if the nucleons occupy the lowest-energy single-particle

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levels available to them, subject to the antisymmetry constraints of the Pauli principle, then the lowest-weight state is said to be a 0 h¯ ω state. Symplectic irreps can also be constructed with lowest-weight states of higher harmonic-oscillator energies. An irrep with lowest-weight state having harmonic-oscillator energy E0 + nh¯ ω0 is then described as an nh¯ ω irrep. It will be noted that a lowest-weight state for a symplectic irrep in a spherical harmonic-oscillator basis, as defined above, is a highest-weight state for an su(3) irrep with the added condition that Bij |Φ0  is zero. This su(3) irrep has highest weight (λ = ν1 − ν2 , µ = ν2 − ν3 ). In this paper, we restrict consideration to even–even N = Z nuclei (the so-called 4n nuclei) for which both the spin and isospin are zero to first approximation. We also restrict consideration to the so-called leading irreps. The leading irrep among the nh¯ ω irreps for a given nucleus is defined as the one with largest f1 and smallest f3 . These irreps have largest quadrupole matrix elements, meaning that they generate rotational bands with the largest deformations. They are understood (from su(3) analyses [3]) to be the most relevant for the description of low-energy collective states. The argument is based on the fact that among the states of a given harmonic-oscillator shell-model energy, the collective correlation energy most lowers the energies of those states with the largest deformation [13]. Thus, restriction to leading irreps is justified on physical grounds. It leads to considerable simplification because the wave functions for the corresponding lowestweight states are Slater determinants of single-particle wave functions. Note that it cannot be assumed that the low-lying physical states of a nucleus are associated with nh¯ ω irreps with energies increasing sequentially with n = 0, 1, 2, . . . . For example, the ground state of 16 O is believed to have largest overlap with the 0 h¯ ω closed-shell state. But the first excited state is understood to be a 4 h¯ ω 4-particle–4-hole state which is brought below the 2 h¯ ω states, for example, by its large deformation. The interesting observation is that, in 16 O, both the lowest and next lowest-energy symplectic irreps are understood to be described by leading sp(3, R) irreps. As remarked above, the lowest-weight state for a leading sp(3, R) irrep in a spherical harmonic-oscillator basis, is a determinant of spherical harmonic-oscillator single-particle states. In fact, it is a product of four determinants of spatial wave functions: one for each of the four spin–isospin states of a single nucleon. A spatial wave function for a singleparticle spherical harmonic-oscillator state is characterized in a Cartesian basis by a triplet of integers m ≡ (m1 , m2 , m3 ) and, for a harmonic oscillator of frequency ω0 , it is given to within a norm factor by  1 2 x1 +x22 +x32

φm (x) = Hm1 (x1 )Hm2 (x2 )Hm3 (x3 )e− 2

,

(11)

where x ≡ (x1 , x2 , x3 ) is a set of Cartesian (x, y, z) coordinates for a nucleon (in harmonic√ oscillator units of b0 = h¯ /mω0 ) and Hmi is a Hermite polynomial. This expression is simplified by the fact that, if a determinantal state is a lowest-weight state for an sp(3, R) irrep, then it is not possible to lower the indices (m1 , m2 , m3 ) of any occupied single-particle state by the replacement of any one of the integers mi by mi = mi − 2 without obtaining the indices of another already occupied single-particle state and causing the determinant to vanish. Because of this property, every Hermite polynomial in the

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expression of an occupied single-particle state can be replaced by its component of highest degree. In other words, a lowest-weight state for a leading irrep has wave function given, to within a norm factor, by a Slater determinant of single-particle wave functions of the form  1 2 x1 +x22 +x32

m φ˜ m (x) = x1m1 x2m2 x3 3 e− 2

.

(12)

Now observe that, if the wave function for the lowest-weight state |Φ0  is a determinant of a set of single-particle wave functions {φ˜ m }, then the wave function for the generator coordinate state |Λ = Γ (Λ)|Φ0  (cf. Eq. (3)) is, to within a norm factor, a determinant of Λ } where the transformed single-particle wave functions {φ˜ m Λ φ˜ m (x) = φ˜m (xΛ),

(13)

and xΛ is the vector with components  (xΛ)i = xj Λj i .

(14)

j

In particular, for a diagonal matrix, Λij = δij λi , the wave functions are given, to within a norm factor, by  1 2 2 λ1 x1 +λ22 x22 +λ23 x32

Λ (x) ∝ (λ1 x1 )m1 (λ2 x2 )m2 (λ3 x3 )m3 e− 2 φ˜ m

or, including the term

1 m2 m3 λm 1 λ2 λ3

,

(15)

in the norm factor, by

 1 2 2 λ1 x1 +λ22 x22 +λ23 x32

m Λ (x) ∝ x1m1 x2m2 x3 3 e− 2 φ˜ m

.

Of particular interest is the lowest-energy generator state. This is the state |Λ which minimizes the energy expectation E(Λ) =

Λ|H |Λ , Λ|Λ

(16)

for a microscopic Hamiltonian  1 H= Ti − Tc.m. + Vij , 2 i

(17)

ij

where Tc.m. is the center-of-mass kinetic energy. For present purposes the interaction potential will be taken as the Brink–Boecker two-body interaction [15]. For a leading irrep, the lowest-energy generator state is a Slater determinant and a solution of the Hartree–Fock self-consistent field equations. Moreover, it follows from the scaling relationships that |Λ is the ground state for a system of nucleons in a deformed harmonic-oscillator potential with frequencies (ω1 , ω2 , ω3 ) related to the parameters of Λ by ωi = αi ω0

with αi = λ2i .

(18)

We have studied the dependence of the microscopic energy (16) on the parameters (ω1 , ω2 , ω3 ) and found, as expected, that there is a unique set that yields a minimum energy generator state for each irrep. Table 1 shows, for each (leading) symplectic irrep

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Table 1 Values of α = (α1 , α2 , α3 ) and h¯ ω0 for minimum energy generator states of several 4n nuclei Nucleus

Configuration of lowest-weight state hω ¯ 0

4 He 25.8 8 Be 20.5 8 Be 20.5 12 C 12 C 16 O 16 O 20 Ne 20 Ne 24 Mg 24 Mg 24 Mg

17.9 17.9 16.3 16.3 15.1 15.1 14.2 14.2 14.2

(000)4 (000)4 (100)4 (000)4 (200)4 (000)4 (100)4 (010)4 (000)4 (100)4 (200)4 (000)4 (100)4 (010)4 (001)4 (000)4 (100)4 (010)4 (200)4 (000)4 (100)4 (010)4 (001)4 (200)4 (000)4 (100)4 (010)4 (200)4 (110)4 (000)4 (100)4 (010)4 (001)4 (200)4 (110)4 (000)4 (100)4 (010)4 (200)4 (110)4 (020)4 (000)4 (100)4 (010)4 (200)4 (110)4 (101)4

0p–0h 0p–0h 4p–4h 0p–0h 4p–4h 0p–0h 4p–4h 0p–0h 4p–4h 0p–0h 4p–4h 4p–4h

sp(3, R) labels [f1 , f2 , f3 ] [3/2, 3/2, 3/2] [15/2, 7/2, 7/2] [23/2, 7/2, 7/2] [19/2, 19/2, 11/2] [35/2, 11/2, 11/2] [23/2, 23/2, 23/2] [39/2, 23/2, 15/2] [43/2, 27/2, 27/2] [51/2, 35/2, 19/2] [55/2, 39/2, 31/2] [55/2, 55/2, 23/2] [63/2, 39/2, 31/2]

(α1 , α2 , α3 )

Energy (MeV)

(0.815, 0.815, 0.815) −27.86 (0.475, 0.924, 0.924) −40.36 (0.252, 0.684, 0.684) −6.01 (0.659, 0.659, 1.081) −65.30 (0.325, 1.042, 1.042) −54.93 (0.816, 0.816, 0.816) −104.78 (0.572, 0.887, 1.298) −81.52 (0.605, 0.923, 0.923) −124.30 (0.547, 0.782, 1.287) −105.31 (0.628, 0.825, 1.013) −162.31 (0.623, 0.623, 1.330) −133.66 (0.508, 0.779, 0.942) −109.81

Minimum energies are given for the Brink–Boeker interaction for 0p–0h and 4p–4h irreps of the 4n nuclei considered. Also given are the associated configurations and irrep labels.

of the 4n nuclei considered, the corresponding α = (α1 , α2 , α3 ) for a given choice of h¯ ω0 . The results confirm the expectation that, to a good approximation, this α-set is such that the products fi αi for i = 1, 2, 3 are essentially equal. The deformations of the generator states that give the lowest energies after projecting out a particular angular momentum component differ somewhat from these values, especially in light nuclei. For 8 Be, for example, there is a substantial change from the values (0.475, 0.924, 0.924) in Table 1 to the values (0.395, 1.050, 1.050) which give the lowest-energy L = 0 state after projection. This is a much larger change than expected for heavier rotational nuclei. The difference in the deformation of the lowest-energy state before and after projection is an indication of the rigidity of the generator state as an intrinsic state of a rotational band. Evidently, the ground-state rotational band of 8 Be is relatively soft.

3. The Generator Coordinate Method (GCM) Let {|k ≡ |Λ(k); k = 1, 2, 3, . . .} denote a linearly independent set of generator coordinator states and let   2L + 1 L∗ |kKLM = (Ω) dΩ (19) R(Ω)|k DMK 8π 2 denote the corresponding (nonnormalized) angular-momentum-projected states. The norms and overlaps of these states are given by  L  L∗ (20) NkK,lK  ≡ kKLM|lK LM = DKK  (Ω) k|R(Ω)|l dΩ and the Hamiltonian matrix elements are

M.J. Carvalho et al. / Nuclear Physics A 703 (2002) 167–187 L  HkK,lK  ≡ kKLM|H |lK LM =

175

 L∗ DKK  (Ω) k|H R(Ω)|l dΩ.

(21)

It follows that the closest approximation to the eigenstates of the Hamiltonian within the linear span of a finite set of states {|kKLM} is given by the solutions of the matrix equation   L L ∀ k, K. (22) HkK,lK  − EαL NkK,lK  flK  (αL) = 0, lK 

Methods for evaluating the overlaps k|R(Ω)|l and k|H R(Ω)|l are described in the papers of Vassanji et al. [10]. The essential observation that makes these evaluations relatively simple is that, for generator states obtained by deforming highest-weight states of leading su(3) irreps, as described in Section 2, both the generator states {|k} and the rotated generator states {R(Ω)|k} are Slater determinants of single-particle states. 4. Selection of optimal sets of generator states For L = 0, there is a single L-projected state, |k000, for each generator state; for higher L there may be a multiplicity of states, {|kKLM}, distinguished by the K quantum number. However, we first consider the selection of optimal sets of generator states within the restricted class of axially symmetric states, i.e., states for which α2 = α3 . The analysis is then considerably simplified because, for axially symmetric generator states, the K quantum number is zero for all values of L. Thus, each generator state |k generates a unique sequence of angular momentum states {|kLM: L = 0, 2, 4, . . .}. We consider now the selection of a set of N generator states, for various values of N . Optimal sets can be defined according to different variational criteria. The variational criteria we use are based on energy minimization. However, since the objective is to derive a band of rotational states, the energy to be minimized is not unique. We could, for example, choose to minimize the ground-state energy. This would give the best ground state obtainable with a given number of generator states. However, it would not necessarily give the lowest energies for excited states. Another possibility is to minimize the sum of the lowest 0+ , 2+ , and 4+ energies. Other criteria are possible. In principle, one could seek sets that minimize the energies for each value of L separately. However, having different generator states for each angular momentum would make the task of interpreting the results more difficult. Furthermore, it is important to recognize that there is always a trade off between the amount of effort and computer time one is willing to put into finding a truly optimal set of N generator states versus settling for a slightly less than optimal set and increasing N . With the goal of getting the best results for a given amount of computer time, we seek a strategy for quickly identifying sets of generators states that are close to being optimal according to an appropriate criterion of energy minimization. In seeking an efficient strategy, we first considered the idealized problem of finding the best approximation to a gaussian wave function  1 x2 F (x) = √ exp − 2 (23) 2w w

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of half width w that can be obtained with a sum of N Gaussians of unit half width (ω0 = 1). Thus, we consider the values of {ai } and {Ci } that give the best approximation F (x) ≈

N 

Ci f (x − ai ),

with f (x) = e−x

2 /2

.

(24)

i

For a finite value of N , this problem is solved rather easily by noting that f is the groundstate wave function for a harmonic-oscillator Hamiltonian  1 (25) H = b † b + bb† , 2 and F is the ground-state wave function for the dilated harmonic oscillator H (σ ) = eσ (b

† b† −bb)/2

H e−σ (b

† b† −bb)/2

,

(26)

with w = eσ . Moreover, f (x − a) is the value of the wave function at x of a displaced harmonic-oscillator ground state, i.e., the coherent state √



D(a) = ea(b† −b)/ 2 |0, (27) whereas F is the wave function for the so-called squeezed coherent state



S(σ ) = eσ (b† b† −bb)/2 |0.

(28)

Thus, the overlap and Hamiltonian matrix elements between states of different ai are readily determined by standard coherent state methods to be given by

    (29) Nij = D(ai ) D(aj ) = exp −(ai − aj )2 /4 ,



Hij = D(ai ) H (σ ) D(aj )      1 2(1 + ai aj ) cosh(2σ ) − ai2 + aj2 sinh(2σ ) exp −(ai − aj )2 /4 . (30) 4 The parameters {ai } and {Ci } can then be determined such that the lowest energy of the eigenvalue equation  (Hij − ENij )Cj = 0, (31) =

j

in minimized. The√lowest energies and the optimal {ai } sets that fit a gaussian function of half width w = 2 are shown in Table 2. It is seen that the ground-state energy, which has the harmonic-oscillator value of 0.5, is already accurate to within 2% when the Gaussian of √ half width 2 is approximated by a sum of just two suitably separated unit-half-width Gaussians. A plot of the functions shows that, in this case, a combination of three Gaussians is barely distinguishable by eye from the function fitted. Not surprisingly, a larger number of small-width Gaussians is required for a good fit to a large-width Gaussian. Table 3 shows the energies and optimal√{ai } sets obtained in fits of unit-half-width Gaussians to a Gaussian of half width w = 11. It is interesting to note that, for N < 2w, a good first guess for an optimal set of Gaussians appears to

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Table 2 Lowest-energy eigenvalue of Eq. (31) for optimized {ai } sets for different values of N when w2 = 2

177

Table 3 Lowest-energy eigenvalue of Eq. (31) for optimized {ai } sets for different values of N when w2 = 11

N

{ai }

E

N

{ai }

E

1 2 3 4 5

{0} {±0.8} {0, ±1.4} {±0.6, ±1.9} {0, ±1.3, ±3.4}

0.625 0.509 0.5006 0.50007 0.50002

1 2 3 4 5 6 7

{0} {±1.0} {0, ±2.2} {±1.0, ±3.1} {0, ±2.0, ±4.1} {±0.9, ±2.8, ±4.9} {0, ±1.9, ±3.8, ±5.8}

2.779 1.289 0.825 0.640 0.558 0.524 0.510

be a set that is uniformly spaced at intervals equal to their widths; i.e., equal to two, for unit-half-width Gaussians. However, for larger values of N it is more profitable to space the Gaussians more closely. A rule of thumb is that, to fit a Gaussian of width w with a linear combination of N Gaussians of width w0 , N should be of the order of 2w/w0 , when w > w0 , to obtain an energy expectation accurate to about 5%. Now consider the L = 0, 2, and 4 lowest states for 8 Be that can be projected from a linear combination of N axially symmetric (Slater determinant) generator states. For N = 1, we  with Λ ij = δij λ¯ i and λ¯ 2 = λ¯ 3 , for which the energy take the generator state |Λ,     Λ0|H |Λ0  = E0 Λ ,  Λ0  Λ0|

(32)

of the L = 0 state projected from it is a minimum. The values of α¯ i = λ¯ 2i determined in this calculations are respectively α¯ 1 = 0.395 and α¯ 2 = α¯ 3 = 1.0504. For N > 1 the generator states are expressed in terms of a parameter a by |Λ(a) with Λ(a)ij = δij λ(a)i and λ(a)21 = α¯ 1 + a,

λ(a)22 = α¯ 2 = λ(a)23 = α¯ 3 .

(33)

Our first discovery was that, for N = 2, the energy of the L = 0 ground state obtained with generator states {|Λ(−a), |Λ(+a)} was insensitive to the choice of values for the parameter a and remained close to its minimum value for all a values in the range [0.03, 0.08]. The same behaviour persisted at the N = 3 level with generator states {|Λ(−a), |Λ(0), |Λ(+a)} and |a|
0.25. Thus, in view of these observations, we took for N = 2, generator states {|Λ(−a), |Λ(+a)} with an a value (within the above mentioned range of a values) chosen to minimize the energy of the lowest L = 2 state. (Note that if we had chosen for a the value that gave a minimum for the L = 2 state in an unrestricted range of values then, unless we also changed the value of α¯ 1 , the energy of the L = 0 state would have been noticeably higher than the one previously obtained.) Similarly, to generate a basis set of order three  |Λ(−a), |Λ(+a)} and selected the value of a we took the generator states {|Λ(0) = |Λ, that yielded a minimum, now, for the energy of the lowest L = 4 state while keeping the energies of the L = 0 and L = 2 states close to their minimum values. The generator states

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selected in this manner were effective at giving rapidly converging energy eigenvalues, especially for the lowest L = 0, 2 and 4 states. However, their selection required subjective judgements and was time consuming. We therefore considered if we could do as well or better with simple criteria based on the study of harmonic-oscillator (gaussian) functions described at the beginning of this section. To see if generator states could be reasonably represented as Gaussians, we plotted the overlaps L, Λ(−a)|L, Λ(a) , (34) L, Λ(−a)|L, Λ(−a)1/2 L, Λ(a)|L, Λ(a)1/2 (for L = 0, 2 or 4), as functions of a and compared them with the corresponding overlaps for harmonic-oscillator states with gaussian wave functions of half width w0 , for which

  2 2 (35) Dw0 (−a) Dw0 (a) = e−a /w0 . N (a) =

It is found that the overlaps of the lowest L = 0, 2 and 4 states of 8 Be are practically indistinguishable from one another and qualitatively similar to those of harmonic-oscillator ground states with w0 ≈ 0.22 (cf. Fig. 1). Now, if the L = 0 ground state, to be fitted by a superposition of angular-momentum projected generator states, could be represented by a Gaussian of half width w then, for w  w0 , we should expect, according to the above analysis of gaussian fits, that the optimal values of a for N = 2 should be a ≈ ±w0 . Exploratory results (cf. Fig. 2) confirm that the ground state can be represented by a Gaussian to a good approximation but that w < 1.1w0 . On the basis of gaussian fits with such an w/w0 ratio, we selected values for the parameter a and obtained the energies for the L = 0, 2 and 4 states shown in Table 4. First observe that the N -generator states selected contain all the states of the (N − 2) subsets. Thus, all the absolute (as opposed to relative) energies of the N = 1, 3, 5, and 7

Fig. 1. The overlap N (a) (dashed curve), cf. Eq. (34), for α¯ 1 = 0.395 and α¯ 2 = α¯ 3 = 1.0504, 2 2 compared with the overlap e−a /w0 for a gaussian wave function of width w0 ≈ 0.22.

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Fig. 2. Representations of the symplectic model L = 0 ground states, obtained with 1 (dotted line), 5 (dot-dashed line) and 7 (continuous line) generator states, as superpositions of corresponding numbers of Gaussians of half width w0 = 0.22 weighted by the appropriate coefficients of the L = 0 eigenvector. Table 4 Energy eigenvalues (in MeV) of the states L = 0, L = 2 and L = 4 projected from linear combinations of N generator states N

{a}

1 {0} 2 {±0.066} 3 {0, ±0.13} 4 {±0.066, ±0.19} 5 {0, ±0.13, ±0.26} 6 {±0.066, ±0.19, ±0.26} 7 {0, ±0.13, ±0.26, ±0.36} Expt.

EL=01

EL=21

EL=41

EL=02 EL=22 EL=42

−47.60 −47.59 −47.61 −47.64 −47.66 −47.67 −47.67

3.18 3.10 3.12 3.14 3.15 3.15 3.16 3.04 ± 0.03

11.54 11.27 11.14 11.14 11.15 11.16 11.16 11.3 ± 0.3

17.68 17.23 17.17 17.16 17.15 17.13 20.2

19.44 19.14 19.15 19.15 19.15 19.13 22.2

24.67 24.23 24.07 24.05 24.06 24.06 25.5

The values α¯ 1 = 0.395 and α¯ 2 = α¯ 3 = 1.0504 are kept constant throughout the calculation. Absolute energies are shown for the ground state; for the excited states, the energies tabulated are excitation energies relative to the ground state. Experimentally measured energies are also shown for comparison.

sequence decrease monotonically with increasing N . Similarly, the energies of all states of the N = 2, 4, 6 sequence decrease monotonically. However, a few of the N -generator states are actually slightly higher in energy than their (N − 1) counterparts. For example, the N = 2 ground state is slightly higher in energy than the N = 1 ground state. This is possible because the generator states selected are those that would minimize the energies of the related harmonic-oscillator problem. Thus, they are close to but are not quite identical to the optimal generator states for 8 Be. We observed, however, that the sums of lowest 0+ , 2+ , and 4+ absolute energies decrease or stay the same (to the accuracy shown) as N → N + 1.

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Moreover, the energies of the lowest projected L = 0, 2 and 4 states converge rapidly with the number of generator states selected according to the procedure adopted; this confirms that the procedure is meaningful. Indeed, the excitation energies of the members of the ground-state band would appear to be already equal at N = 1 to their limiting values, to within the experimental energy bars, given in Table 4 [16]. By a careful choice of the a parameters, it was also found that the N = 7 results could be obtained with one or two fewer generator states. However, finding an optimal set proved to be more time consuming than following a simple prescription, such as the one described above, and settling for a slightly larger value of N . The results of the diagonalization in the generator bases have a natural interpretation in terms of beta-vibrational wave functions as follows. With each generator state represented by a Gaussian of half width w0 = 0.22, the eigenfunctions of the nuclear Hamiltonian can be plotted as superpositions of these Gaussians. Fig. 2 shows the resulting functions obtained for the L = 0 ground state with combinations of 1, 5 and 7 generator states represented by Gaussians of half width w = 0.22 and weighted by the appropriate coefficients of the eigenvectors. The corresponding functions for the first excited L = 0 state are shown in Fig. 3. Note, however, that although this band has all the characteristics of a one-phonon beta-vibrational band it occurs at a relatively high energy compared to the

Fig. 3. Representations of the first excited symplectic model L = 0 states, obtained with 2 (dotted line), 5 (dot-dashed line) and 7 (continuous line) generator states, as superpositions of corresponding numbers of Gaussians of half width w0 = 0.22 weighted by the appropriate coefficients of the L = 0 eigenvector.

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low energies customarily expected for beta vibrational excitations. One can think of it as a giant beta-vibrational band. Several characteristics of the results are worth noting. The first is the remarkable rapidity with which they converge when the parameters are optimized in the prescribed manner. Already with only one generator state, the results for the intrinsic state of the ground-state band are close to those of the best calculation (with seven generator states) that we have done. This reflects the fact that the final (seven gaussian) wave function is close to that of a single Gaussian of width w = 1.08w0 which is only marginally wider than that of a single generator state of width w0 . Parallel results are obtained for the giant beta band, which appears when there is a minimum of two generator states. A second characteristic is the close resemblance of the functions obtained to those of harmonic-oscillator ground and first excited states, respectively. A more realistic representation of the wave functions could be obtained by finding non-gaussian wave functions, to represent more precisely the basic generator states, with shapes adjusted to give the calculated overlaps, shown in Fig. 1. Clearly, a better function would fall off more rapidly with increasing a than a Gaussian. The optimal function also need not be symmetric about its mean value. It is possible that the (negative going) large α1 behaviour of the wave functions, seen in Figs. 2 and 3, might change if this were done.

5. The wave functions in a shell-model basis It is instructive to expand the symplectic model wave functions obtained by the above described generator coordinate method in a standard su(3) shell-model basis. This is easy to do using recently developed analytical angular momentum projection techniques [14,17]. Following the notations of Ref. [14], the contributions of higher spherical harmonicoscillator shells to a state |βLM of angular momentum L, projected from a normalized (generator) state of a deformed axially symmetric harmonic oscillator, is obtained from the expansion in an su(3) basis 

 fnL (β) Nn (λ + 2n, 0)LM , (36) |βLM = n
0

where, in the above notation, β is related to the oscillator strength by e2β =

ω2 α¯ 2 , = ω1 α¯ 1 + a

(37)

and Nn = f1 + f2 + f3 + 2n,

λ = f1 − f2 ,

µ = f2 − f3 .

(38)

Analytical expressions are given for the fnL (β) coefficients as a function of β in Ref. [14]. For the 8 Be symplectic model calculations reported here, f1 = 15/2 and f2 = f3 = 7/2. Fig. 4 shows histograms of the coefficients

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Fig. 4. Spherical SU(3) expansion coefficients f¯nL for the N = 1, 3, 5, and 7 ground state and for the N = 2, 3, 5, and 7 first excited L = 0 state with the parameter values listed in Table 4.

f¯nL =



Ck (L)fnL (βk ),

(39)

k

as a function of n for the superpositions of generator coordinate states obtained in the above symplectic model calculations for the ground and first excited L = 0 states; k indexes the generator coordinate basis. Histograms are shown for states projected from linear combinations of N = 1, 3, 5, and 7 generator states for the ground state, and for N = 2, 3, 5, and 7 generator states for the excited L = 0 state. The histograms show the contributions to the wave functions coming from the various 2nh¯ ω-oscillator shells. A notable characteristic of the figures is that the wave functions change relatively little with increase in the number of generator states. As observed in the previous section, this

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Fig. 5. Spherical SU(3) expansion coefficients f¯nL of the N = 7 eigenstates of L = 0, 2, 4 for the ground and beta-vibrational band with the parameter values listed in Table 4.

is because the wave functions are already close to their fully converged limits for just a few (optimally chosen) generator coordinate states. It is also of interest to compare the wave functions for different values of L. Fig. 5 shows that, for the ground-state band, the dominant component for the lowest L = 0 state comes from the 0 h¯ ω shell, while for the lowest L = 2 and L = 4 states, they come from 2 h¯ ω (n = 1) and 4 h¯ ω (n = 2) shells, respectively. Parallel results are shown for the giant beta band. For heavier deformed nuclei, it is expected that the corresponding histograms will change much less rapidly with angular momentum indicating a much greater degree of rigidity of the intrinsic states for their rotational bands [18].

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6. Discussion A remarkable conclusion of this investigation is that extraordinarily good approximations to a complete symplectic model diagonalization can be obtained with states angular momentum projected from linear combinations of just a few deformed Slater determinants. Indeed, states of the ground-state band projected from a single Slater determinant are already surprisingly good. This result has important implications for the validity of angular momentum projected Hartree–Fock theory and for the unified-model description of the intrinsic states of rotational bands. It goes a long way to substantiating the intuitive belief that the mean field accounts for the dominant component of the interactions between nucleons. The central (spherically symmetric) component of the mean field is known from the success of the spherical-shell model to be of paramount importance. However, the failure of the spherical-shell model to explain the huge electric quadrupole transitions and moments observed in deformed nuclei, without substantial major-shell mixing, is a direct indication that more is needed. The extra ingredient is provided by the unified model and the deformed-shell model which provide simple microscopic descriptions of nuclear rotational states in terms of independent particles in a deformed mean field. These approximations have their foundations in the Hartree–Fock variational principle. However, their justification relies primarily on physical intuition and computational convenience. In contrast, the symplectic model of nuclear rotational states makes no a priori assumption that its states should be projected from one or even a combination of a small number of independent-particle states. Thus, the discovery that they can be so obtained, is significant. A second highly significant result is a demonstration that symplectic calculations in a generator coordinate basis have natural interpretations both as vibrational wave functions, in collective model language, and as superpositions of su(3) states in a shell-model basis. We have shown that, from a collective model perspective, the generator coordinate method of determining symplectic model wave functions, provides intrinsic states for rotational bands with a natural representation as vibrational wave functions. We have found that the overlaps of Slater determinant states of different (axially symmetric) deformation, with respect to the β deformation parameter, are very close to the overlaps of gaussian wave functions. Thus, an intrinsic state of a symplectic model rotational band, when expanded as a linear superposition of (Slater determinant) generator states, has an immediate representation as a β-vibrational wave function. It is then observed that the width of this vibrational wave function is comparable to that of a single Slater determinant. It is also worth noting that, although the overlaps of differently deformed generator states are close to those of displaced Gaussians, it is possible to seek out other basic functions of β whose overlaps are much closer to those of the generator states. The representation of linear combinations of generator states as linear combinations of such functions would then give a better representation of the beta-vibrational character of the results. Such a strategy could be particularly useful in searching for a representation of more general intrinsic states as functions of both beta and gamma coordinates. As expected, a (one-phonon) beta-vibrational band emerges in the present calculations at a giant-resonance excitation energy of some 17 MeV. As is well known, there is no low-

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energy beta band in the symplectic model (without mixing of irreps) any more than in the SU(3) model. To obtain E2 transitions between different low-energy K = 0 bands, in the shell model, it is necessary to mix the K = 0 bands that arise from different symplectic model irreps. However, although such mixing inevitably occurs, there does not appear to be much compelling experimental evidence to indicate that it is of the collective coherent type assumed in the phenomenological collective model [19]. Low-energy K = 2 bands, commonly referred to as gamma bands, do occur within the symplectic model just as they do in the SU(3) model. Moreover, mixing of symplectic bands to account for the K = 0 and K = 2 and other excited bands observed, for example, in heavy rotational nuclei, can be understood as arising from spin–orbit, pairing, and other symplectic symmetry breaking interactions [20]. The pseudo-symplectic model has an effective way of taking into account these interactions when describing heavy nuclei [21] (cf. concluding paragraph). In a sequel to this work, we intend to explore the contributions from axially asymmetric generator states. The inclusion of such generator states is undoubtedly necessary for optimal symplectic model calculations. Their study will also further the understanding of the nature of gamma vibrational fluctuations in axially symmetric and triaxial rotational model states. Also we intend to seek more realistic representations of intrinsic vibrational wave functions by expanding in wave functions that reproduce the overlaps of different generator states more accurately and are more appropriately adapted to the boundary conditions and volume elements than Gaussians. One result of such a more careful expansion would be to remove the small components of the beta wave functions seen in Figs. 2 and 3 at unphysical negative values of α1 . In view of the very promising results obtained in this preliminary analysis, such a more careful treatment is clearly worth pursuing. From a shell-model perspective, our calculations demonstrate a way of rapidly approaching the fully converged symplectic model results that would be obtained if there were no necessity to restrict the calculations to a truncated Hilbert space. Figs. 4 and 5 give a sense of the dimensions of a comparable spherical-shell model calculation. They show that with just a few generator states one can obtain results that would require spherical su(3) basis states from approximately 10 major harmonic-oscillator shells (states of up to 20 h¯ ω). Admittedly, the present results are obtained in a highly restricted space of only axially symmetric states; in spherical-shell model language this corresponds to a space of only (λµ) su(3) irreps with µ = 0. When this constraint is relaxed, the shell-model space of states below 20 h¯ ω becomes enormous. Nevertheless, we expect that near converged symplectic model states will still be obtainable with a relatively small number of axially symmetric and triaxial generator states. Finally, a comment needs to be made about the restriction to generator states {|Λ} with Λ diagonal. It is known that a 3 × 3 general linear transformation can be factored: g = ΩΛ = ΩDω,

(40)

where Ω and ω are SO(3) rotations and D is a diagonal matrix. When the linear transformation g is applied to the lowest-weight state of a symplectic irrep to produce

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a generator state, the rotation Ω whose infinitesimal generators are the total angular momenta, is associated with the rotational degrees of freedom of the nucleus, the diagonal transformation D is associated with monopole, beta, and gamma vibrational degrees of freedom, and the rotation ω is associated with intrinsic (so-called vortex spin) degrees of freedom. Suppressing the generator coordinates associated with ω, by projecting from a set of generator states with Λ diagonal, corresponds to neglecting dynamical vortex spin–rotational interactions. In conventional collective model terminology, it would be described as a strong-coupling approximation in which the vortex spin of the intrinsic state is kept aligned with the intrinsic axes of the deformed nucleus. Note that picking a single generator state with the Λ matrix diagonal similarly results in the neglect of dynamical rotational–vibrational interactions. It does not imply a neglect of the vibrational degrees of freedom. On the contrary, our results show that the beta-vibrational wave function of the ground-state rotational band is given rather well by the N = 1 results shown in Fig. 2. However, it does suppress the possibility that the shape of the intrinsic state can change with increasing angular momentum. In contrast, when there are many (fixed) generator states the intrinsic state can change with angular momentum as a result of the different generator states being weighted differently. We conclude that restricting to generator states with a single D matrix suppresses centrifugal stretching interactions whereas restricting to a single (e.g., unit) ω matrix suppresses the vortex spin–rotational decoupling effects of the Coriolis force. On the basis of symplectic model calculations in a spherical shellmodel basis (where all symplectic model degrees of freedom are included), we surmise that the effects of neglecting the ω degrees of freedom in generator coordinate calculations will be relatively minor. This is not because we believe vortex spin to be unimportant. On the contrary, we believe it to be essential for understanding nuclear moments of inertia. However, we think that the vortex spin must be strongly coupled to the nuclear shape, as in the strong coupling of a particle with spin to an even rotor core. For, if the vortex spin and rotational degrees of freedom were only weakly coupled, the energy-level spectrum that would emerge from a symplectic model calculation would be very different from that of the simple rotational model states that, in fact, emerge and are in accord with observed rotational states. Clearly, investigating the effects of including a range of ω values is a high priority for future investigations. It should be emphasized that a classification of shell-model states by means of symplectic and SU(3) symmetry is complete. Moreover, the important symmetry breaking interactions that mix symplectic model irreps can be accommodated within the generator coordinate scheme by including generator states from other irreps. However, as always the limitation to what can be included in a shell model with realistic interactions depends very much on the choice of basis. The present calculations demonstrate that an extraordinarily good basis for an infinite-dimensional symplectic model calculation can be generated by angular momentum projection from a very small number of determinantal generator states. This suggests that it could be profitable to investigate the selection of optimal generator states for mixed representation calculations. As a first step, one might consider generator states of the pseudo-symplectic model [21] which was designed to take into account the spin–orbit splitting of single-particle levels that are known to control the shell structure of

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heavy nuclei. One could also work with generator states of the Nilsson model type [22] and even include generator states of mixed nucleon number to generate bands of states with good angular momentum and nucleon number by projection of both. Clearly the generator coordinate method is able to capitalize on the many physical insights gained from the phenomenological collective model, the unified model, the projected Hartree–Fock and Hartree–Fock–Bogolyubov, and algebraic models. The major contribution of the algebraic approach to such a unified theory, is that it provides a formally exact framework for an investigation of optimal generator states, and makes it possible to understand precisely what degrees of freedom are included and, conversely, what are omitted.

Acknowledgements The authors are indebted to A. Williamson for setting up the Maple programs and to N. Noormohamed for collecting data. This work had the financial support of NSERC, and grants from Ryerson Polytechnic University and the Ontario Ministry.

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