Geometrical derivation of collective operators and paths in TDHF

ANNALS

OF PHYSICS

138,319-352

(1982)

Geometrical Derivation of Collective Operators and Paths in TDHF B. G. GIRAUD DPh.

T, CEN-Saclay,

91191

Gif sur Yvette

Cedex,

France

AND

J. LE TOURNEUX Laboratoire

de Physique Nuclkaire, UniversitP Qkbec, Canada

de Montrhal,

Received October 1, 1980

It is first found that the intrinsic parity of an operator under time reversal and the interpretation of the operator as coordinate- or momentum-like in a TDHF calculation are not simply related. This is because the TDHF particle-hole basis is, in general, complex. The TDHF equation is then reformulated in the plane tangent to the Slater determinant manifold. This plane is spanned by the particle-hole basis. The particle-hole matrix elements of the Hartree-Fock Hamiltonian define the energy gradient vector in this tangent plane. This gradient is real when the Slater determinant is real. A TDHF calculation initiated from a real determinant induces, during the first infinitesimal time step, a purely imaginary variation of this determinant along the gradient. The gradient is thus identified with the matrix elements of a boost operator. The next infinitesimal time step defines, in turn, a displacement operator. These operators are retained as collective if the TDHF path is stable under changes of velocities. Various criteria are found for this stability condition. The theory cannot be applied straightforwardly to translations and rotations for there is no energy gradient to generate coordinate operators. Particle-hole matrix elements of boost operators can be obtained, however, by a multiplication by i of the matrix elements of displacement operators, since the latter are known explicitly. It is finally found that the rotation of a wavefunction is contradictory with angular momentum conservation in general. Conservation can be ensured by a rotation of the density only and a more elaborate evolution of the velocity field.

1. INTRODUCTION A theory of collective motion is, by definition, an attempt to reduce the number of degrees of freedom of a many-body system. The strong experimental evidence in favor of collective nuclear phenomena has led to a large body of theoretical models [ I] where a small number of collective degreesof freedom, introduced in a more or less a priori manner, give a good description of the phenomena under study. More 319 0003.4916/82/020319-34%05.00/O Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

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ambitiously, starting from a Schrodinger equation involving the degrees of freedom of all the nucleons, one may try to show that in many cases of interest the resulting dynamical predictions can be summarized by a description in terms of a small number of collective parameters only. This a posteriori display of collective features is usually referred to as a microscopic approach, as opposed to the macroscopic introduction of collective variables at the very beginning of the theory. So many theories of collective motion have been proposed [l-15], that it would be difficult to compile an exhaustive list of them. Besides differing by the extent to which they are phenomenological, they can also differ by the techniques they use. Some rely on time-dependent, other on time-independent wavefunctions. Still others, bypassing the use of any wavefunction, are based on operator algebra. Some are compatible with the superposition principle and quantized all the way; others are restricted to Slater determinants and may need requantization and/or restoration of broken symmetries. Adiabaticity is often implied, or used explicitly, on the ground that collective behavior goes with a large inertia. Nevertheless, the adiabatic assumption is no longer justified in some instances of collective motion such as fast rotations or fast relative translations of two nuclei. In spite of this diversity, a few features are common to most microscopic theories of collective motion. It is very often assumed, for instance, that the collective coordinates and momenta are symmetric one-body operators. Such sums of single-nucleon observables are expected to be very sensitive to situations where all nucleons move in a coherent way, and should lead naturally to the type of behavior (e.g., large multipole transitions) that characterizes experimentally, collective motion. At the same time, microscopic theories make extensive use of Slater determinants, since these are particularly suitable for calculations with symmetric operators and readily display the coherence of the component single-particle wavefunctions they are made of. Recent technological progress [ 161 in solving the time-dependent Hartree-Fock (TDHF) equations make the use of Slater determinants even more attractive. Indeed, there already exist TDHF solutions sufficiently collective in nature to suggest comparison with experiment. Such solutions, however, have not been shown to reproduce accurately the evolution yielded by an exact solution of the time-dependent Schrodinger equation [ 171. Furthermore, the adiabatic limit of the TDHF equations allows a stimulating interpretation of their solutions in terms of generalized, classical coordinates and momenta [7, 8, 121. A classical interpretation of collective solutions thus seems to be at hand. Whatever approach one uses, the following three questions are bound to arise in a microscopic theory: (1) (2) (3)

What are the collective operators? What are the collective paths or subspaces? What is the collective Hamiltonian?

Some detailed and fairly successful microscopic theories have been developed in order to answer these questions. Briefly, the collective Hamiltonian is the projection

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of the original, many-body Hamiltonian on the collective subspace, or a requantization of the energy function of that many-body Hamiltonian along the collective path. This reduces the solution of the last problem to that of the second, which comes down, in a large class of adiabatic theories, to a constrained variational principle. One then looks for a subspace, or a path, which tends to minimize the energy for each given expectation value of a collective operator. This cannot be done before the first problem is solved. As a matter of fact, this hierarchy of problems is unnecessary. If the collective path or subspace were known in advance, the collective operator could be any operator whose expectation values would label the wavefunctions of interest in a suitable manner. The relative importance of the first two problems would thus be reversed. This is the motivation for the present work. It is here assumed that some sequences of Slater determinants yielded by TDHF calculations with appropriate initial conditions are good candidates for a collective path. As suggested in Ref. [S], collectivity will be associated with the stability of the path with respect to initial conditions, especially velocities. Clearly, if the nuclear dynamics is governed by a reduced number of degrees of freedom, small velocity fluctuations should not lead to the excitation of neglected noncollective degrees of freedom. This stability condition must be satisfied by any collective path. In an attempt to make the theory as general as possible, the Slater determinants under consideration will not be assumed, in an a priori manner, to be close to static Hartree-Fock (HF) solutions. Large-amplitude collective motion should indeed carry the system far away from its static state. Some residual self-consistency, such as that induced by a constrained variational principle, may turn out to be important, but this should be a result of the theory, not an a priori condition. As a by-product of the path, coordinate- and momentum-like collective operators will be defined from the algebra of generators that induce boosts and displacements along or near the path. These generators will obviously be related to the TDHF Hamiltonian, since the latter determines the path through the time derivative of the Slater determinant. The theory proposed in this paper goes as follows. In Sections 2 and 3 two examples are discussed, namely, translation and elastic scattering, where both the collective path and the collective operators are known. Section 4 opens with a heuristic discussion of the initial evolution of a Slater determinant starting from rest. The more general case of TDHF solutions having a finite velocity is then considered, and for situations where this velocity is low enough to warrant an adiabatic expansion, expressions are derived for the displacement operator and the time evolution of the boost operator. In Section 5, collective boost and displacement operators are defined as those which are associated with an orbit following the bottom of a valley on the energy surface. Since no TDHF solution yields such an orbit because of velocity-dependent effects, these collective operators are obtained through a stop-and-go procedure in which the system is artificially stopped as soon as it has acquired an infinitesimal velocity. The normalization of the collective operators is chosen in such a way as to facilitate the definition of adiabatic and/or large-

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amplitude motion. A condition for collectivity is derived from the requirement that the path be stable with respect to velocity fluctuations. We study in Section 6 the possibility of taking into account the velocity-dependence of .the TDHF equations that is neglected in the stop-and-go analysis. Various generalizations are discussed in Section 7 and the conclusions are presented in Section 8.

2. A FIRST SOLUBLE MODEL: GALILEAN MOTION OF A NUCLEUS We will first reexamine the well-known Galilean covariance of TDHF, and show how it leads naturally to a definition of the particle-hole matrix elements of the operator generating collective translations. Let us consider, for the sake of simplicity, a system of two different nucleons. In this soluble model, the average potential of TDHF reduces to a local Hartree potential when the nuclear interaction is itself local. In the static case, each nucleon is in a real, self-consistent orbital $ which is a solution of the nonlinear equation - fd#(r)

+ 4(r) J dr’V(r

- r’) 4*(r’)

= +(r),

(2-l)

where h and the nucleon mass m have been taken equal to unity, and E is the lowest Hartree eigenvalue. The spatial part of the corresponding static Slater determinant then reduces to

where rp and r, are the proton and neutron coordinates, respectively, To induce a translation, one defines, as usual, the non-self-consistent orbital v/(r, t) = exp[--i(e - k*/2 + k . p)t] exp(ik . r) d(r), where p E -iv is the single-particle minant then becomes

momentum.

(2.3)

The spatial part of the Slater deter(2.4)

WI = v(r, 9 0 v@, v 0. One can readily check that w is a solution of the TDHF

equation (2.5a)

where the single-particle

operator h is given by h = - id + 1 dr’V(r

- r’) 1y(r’, t)l*.

(2.6)

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Indeed, Eqs. (2.1), (2.3) and (2.6) yield h[ Y]w = (E + k*/2)~ + exp[-i(E - k2/2 t k . p)t] exp(zk . r)(-zk . V@) (2.7)

= (E - k*/2 t k . p)y,

which is equal to iLAy/&. For this particular solution, the equation of evolution (2.5a) may thus be rewritten in the equivalent form i$=(e--k2/2+k.p)v.

(2.5b)

Except for an overall change of phase, w evolves through translation. Quite obviously, the proton and the neutron evolve identically. It should be noticed that the spreading of the wave packet usually induced by the action of the Laplacian on $ is here cancelled by the self-consistent nature of 4, as expressed in Eq. (2.1). One can immediately write down an equation for the evolution of Y,

iE=Hy/ at ’

(2.8a)

HE h(r,) + h(r,).

(2.9)

where

Just as Eq. (2.5a) was rewritten in the equivalent form (2.5b) by letting h act on w, Eq. (2.8a) may be rewritten as ig=(2c-k’)Ytk.PY,

(2.8b)

where P=pp,+p,.

(2. IO)

Both the total HF Hamiltonian Z-Zand the total momentum P are symmetric onebody operators that could be expressed through the formalism of second quantization. Let ] Zf) denote a particle-hole configuration with respect to Y(t), ] i) and ( Z) being single-particle states tilled and empty, respectively, in Y(t). Quite obviously, such a particle-hole basis will vary with time since the vacuum with respect to which it is defined is itself time-dependent. The equivalence of (2.8a) and (2.8b) entails (Z~]k.P]Y)=(Zi]H(Y),

(2.11)

or, in terms of matrix elements of one-body operators, k~(Z(p]i)=(Z~h[Y]Ii).

(2.12)

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Thus the particle-hole elements of the component of P along the direction of motion are proportional to those of the HF Hamiltonian. Through such a relation, the

particle-hole matrix elements of the operator generating collective translations may be regarded as being defined in terms of quantities coming out of a numerical TDHF calculation. This is the point of view we will adopt later when we attempt to define collective operators that are not known a priori, in situations where numerical TDHF results are available. It is instructive to consider the complex conjugate of w. I(/*@, t) = exp[i(s - k*/2 -k . p)t] exp(-zk . r) d(r), as well as the corresponding check easily that

spatial part Y* of the Slater determinant. h[!P*] = h[Y].

(2.13) One may (2.14)

Now, the complex conjugation has reversed the velocities without changing the positions, and the analogue of Eq. (2.71 is h[!P*]v*=[s-kk2/2-k.p]y/*.

(2.15)

Throughout this paper the phrase “time-reversal” will be avoided, since nowhere is t changed into -t. “Time-reversal” will be replaced by plain complex conjugation and operators will be called real or imaginary, accordingly, as their matrix elements on a real basis are real or imaginary. This convention should avoid quite a bit of confusion. A wavefunction description in coordinate representation will be adopted, and all unessential complications such as those connected with spin, odd particle number or phases for spherical harmonics will be ignored. The simple example considered in this section shows how misleading it may be to classify operators as coordinate- or momentum-like according as they are even or odd under complex conjugation: although the usual kinetic energy operator is real, it induces a translation when acting on suitable complex functions, thereby acquiring the properties of an imaginary, momentum-like operator. Clearly, this arises from the complex character of w and would not occur with real static wavefunctions. 3. A SECOND SOLUBLE MODEL: SCATTERING OF Two NUCLEI

ELASTIC

The situation considered in this section is that of two nuclei A and a colliding with an impact parameter large enough to ensure that only the tail of their mutual interaction will be felt. Only elastic scattering occurs then and a TDHF calculation shows that the nuclei follow trajectories compatible with those predicted through Ehrenfest’s theorem for a folded potential reduced to the direct, local interaction term. In such a case, it is legitimate to neglect antisymmetrization between the nuclei and replace the Slater determinant Y(t) for the whole system by the product P(t) of two determinants YA(t) and !-Pa(t)for A and a, respectively.

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Let R, and P, be the operators corresponding to the center-of-mass position and expectation values momentum of A, respectively, with the corresponding rA(t) = ( YA(f)l R, 1YA(t)) and pA(t) = ( YA(r)l P, 1YA(t)). In the case considered here, the result of a TDHF calculation may be expressed as

I

YA,

(3.1)

where vA is an overall phase of Y,.,. This result is fairly obvious since it simply means that YA(f) evolves as a result of translations and accelerations induced by P, and R, , respectively. The rates drA/dt and dp,/dt at which these changes occur are numerical coeffkients coming out of the TDHF equations. It should be noted that P, and R, are one-body operators. Their action during an infinitesimal time dt involves only particle-hole matrix elements relating YA(t) to YA(f + dt), another Slater determinant. Similarly, the evolution of YQ(t) reads

an obvious notation. The product function Y= YA !YOtherefore evolves according to the equation (3.3) with q = flA + q,. In the center-of-mass frame of reference, the trajectories obey the conservation laws pa + pA = 0 and Nor0 + N,, rA = 0, N, and NA being the mass numbers of a and A, respectively. It is therefore convenient to define the operator R,, = R, - R, and its conjugate operator E,, = (N,., P, - ?I,P,)/(y, + NA), as well as their expectation values raA = ($1 RaA 1 Y) and paA = (Y 1P,, I Y), which characterize, respectively, the relative position and the relative momentum of the two nuclei. This allows us to rewrite (3.3) as

Thus, the evolution of Y results from a succession of infinitesimal translations and accelerations induced by fi,, and R,, , respectively. If now one restores full antisymmetrization between A and a and considers the Slater determinant Y thus derived from Y, one may wonder whether there exist some conjugate operators RaA and Pa,,, that generate the equation of motion (3.5) 5951 I38/2-8

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This is easily shown to be the case when the overlap between a and A is negligible. The operators can then be written as

N,+NA dr r 6(r R,,= c

ri) - +

i=l

j dr r 6(r - ri)

(3.6)

A

region a

region A

and

jdrd(r-ri)

dr6(r-ri)-Na region a

vi, I

(3.7)

region A

where by “integration over region a (or A),’ we mean an integration restricted to that region of space where nucleus A (or a) does not go. The nature of these operators is not trivial. Since the sums in (3.6) and (3.7) run over all nucleons, R,, and P,, are symmetric one-body operators that can be expressed in the formalism of second quantization, which is obviously not the case for the asymmetric operators R,, and ',A * Of course, as operators acting in the Fock space of many-body wavefunctions, they make sense only in the subspace of states for which N, t NA nucleons are divided into two nonoverlapping fragments of N, and NA nucleons. Being one-body operators, just as R,, and PaA, they can only relate the state Y to the particle-hole states 1Ii> built upon Y, and the evolution described by (3.5) will take place through the admixture of the latter. The dynamical significance of R,, and PaA is not trivial either. Let us consider, for instance, the term dr,,/dt s P,, in (3.5). Although A and a separately appear to undergo an infinitesimal translation at each instant, the scalar product dr,,/dt . P,, induces a rotation of the dumbbell A + a, (except in the case of head-on collisions), the flow definitely remaining irrotational, however. If the reaction plane is perpendicular to the z axis, only two degrees of freedom are involved, namely, the x and y components of R,,. According to the terminology of Ref. [8], this is a typical “lasagna” situation. It may be worth pointing out that dr,,/dt and dp,,/dt now change with time, both in length and orientation, which was not the case for k in the last section. Since dr,,/dt and dp,,/dt are not parallel in general, the operators dpoA /dt . R,A and dr,,/dt a P,, do not involve the components of the position and of the momentum along the same direction. Thus they are not naturally conjugate, although the relation ‘fPaA R -.‘%A ‘IA, dt dt’

p oA

I

. ‘iPa. =‘F’T

would look analogous to a canonical commutator, for.

&A

(3.8)

if appropriate scaling were allowed

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Had it not been possible to express the evolution of the system through the operators RaA and P,, , one could have solved numerically the TDHF equations, thereby obtaining at every instant t the corresponding HF Hamiltonian H[ Y(t)] through which the system evolves: i$+z[Y(t)]Y.

(3.9)

An identification of Eqs. (3.5) and (3.9) is thus in order. The present situation is quite analogous to that encountered in the previous section. In both cases the particle-hole matrix elements of the HF Hamiltonian are found to be equal to those of specific combinations of collective operators. In Section 2, however, only one collective operator was involved, namely, k . P, whereas here we have to deal with a linear combination of two operators, dr,, /dt . P,, and dpaA/dt . R,, . At first sight, the task of disentangling the matrix elements of each one of these operators looks rather hopeless. Indeed, the fact that R,, and P,, are purely real and imaginary operators does not seem to be of much help, since Y(t) and the particle-hole basis associated with it are complex. As a matter of fact, it is still possible to disentangle the matrix elements of both operators, because the wavefunction Y and its associated particle-hole basis may be written as

I Vt>) = exp(ip,, . L) I ydt>)

(3. IO)

IIi>= exr-@,,-R,,)14,ioh

(3.11)

and

where 1PO) and / I&,) are real except, possibly, for a global phase, which is irrelevant for the present argument. It then follows immediately from the commutation relations of R,, and Pa,,, and from the orthogonality of I Y,,) and IZ,iJ, that

(3.12) The particle-hole matrix elements of draA/dt . P,, and dpaA/dt . RAa are thus equal to the imaginary and real parts of (Zil H( !Zr) I V), respectively. It must be stressed that such a result could be obtained only because RaA and PaA are canonically conjugate operators. This will usually not be the case of the operators generating accelerations and translations for an arbitrary collective motion. How then can one extract the particle-hole matrix elements of such operators from TDHF solutions, in spite of the fact that such solutions are complex?

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4. GENERALIZATION As a first step toward an answer to this question, let us consider the initial TDHF evolution of a real Slater determinant !PO which differs from a static HF solution QO. One may associate with Y0 a one-body density matrix p,,, which is real, and a HF Hamiltonian (4.1) where T and V are the usual one-body kinetic energy and two-body potential energy operators, while P indicates that the trace is computed with antisymmetrized matrix elements. According to the TDHF evolution equation, after an infinitesimal time interval dt the Slater determinant !PO receives a purely imaginary increment dY=-idt

Since Y,,, as a real determinant,

W,Y,,.

carries no average veiocity, the imaginary

(4.2) increment

dY brings velocities into Y,, + dY. If the Slater determinant evolves in a collective manner during dt (possible criteria for collectivity will be discussed in the next

section), the following identification

is thus natural:

Restricted as it is to the evolution of an initially real determinant, this equation defines the boost operator .P generating collective velocities, that is naturally induced by Yb. It generalizes Eqs. (2.8a) and (2.8b) as well as (3.5) and (3.9) to the case of the initial stage of an arbitrary collective motion starting from rest. On the real basis ] I, i,) associated with ul,, the particle-hole matrix elements of Z? and W, are proportional; no attempts will be made, at this stage, to define Z@ more completely than up to a multiplicative rate (-dn/dt). A first remark is now appropriate. On the one hand, the TDHF calculations considered in Sections 2 and 3 were initiated by exciting stationary real orbitals #(r) into complex single-particle states 4(r) exp[iq(r)]. Introducing such a factorizable phase did not prevent the Laplacian from acting on 4(r) and, as was pointed out after Eq. (2.5b), TDHF solutions could propagate without spreading. On the other hand, in the present section the excitation of the initial state arises from the fact that the real single-particle wavefunctions w,,(r) deviate from stationary orbitals O(r). Spreading is therefore not hindered any more and might interfere with collective motion. As a second remark, it will be pointed out that the particle-hole matrix elements of W, define a gradient of the average energy E[Yol=

(Yol~l~cJ*

The geometrical meaning of this gradient may be seen as follows.

(4.4)

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FIG. 1. The sphere of normalized vectors in the Hilbert space. A normalized linear combination of two vectors V, and V, does define a third vector.

FIG. 2. The same sphere. The full line is the subset .Y of Slater determinants. A normalized linear combination of two determinants does not, in general, define a third determinant.

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FIG. 3. Blowup of .Y’, which is actually a manifold. The tangent plane is spanned by particle-hole vectors /If). The heavy line is the subset ,YA of real determinants.

In the Hilbert space of many-body wavefunctions, the normalized Slater determinants make a manifold 9 which is not a linear subspace, since a sum of determinants is usually not a Slater determinant, even when no normalization is imposed on them (see Figs. 1 and 2). For each Slater determinant Y there exists therefore a plane tangent to this manifold, this plane being defined by the limit of the differences between Y and all the Slater determinants in its immediate vicinity. As a vector subspace, the tangent plane is thus spanned by the particle-hole basis Iii) corresponding to the determinant Y at which the plane is tangent to 9. Let us now consider a state !PO belonging to the restricted manifold YO c 9 of real determinants (see Fig. 3). Its infinitesimal real variatioh

entails a displacement energy variation

on the energy surface generated by (4.4). The corresponding (4.6)

takes its largest value when the displacement is parallel to the gradient of E. Clearly, this occurs when the coefftcients dCloioare proportional to (I, f, I Z 1Y,,), namely, to

FIG. 4. Blowup of PO, which is actually a manifold. The energy surface is shown by contours on .Y& The dotted line is the gradient line following the flattest direction of a valley.

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(I,, 1 W, 1i,). Thus the particle-hole matrix elements of W, define a vector in the plane tangent to PO, and this vector is parallel to the gradient of the energy (see Fig. 4). It must be stressed that the tangent plane is a complex vector subspace. Although the (natural) particle-hole basis corresponding to a real determinant Y,, is real, two independent types of steps are obviously possible in the tangent plane: those which add to Y,, a purely real increment, as exemplified by Eq. (4.5), and those which correspond to a purely imaginary d!P (see Fig. 5). The step described by Eq. (4.2) is in the direction of the energy gradient, but with a purely imaginary amplitude. As a consequence of this step the determinant gains a collective velocity of order dt. Simultaneously its real part undergoes a change of order dt*. The energy associated with this real part plays the role of a collective potential energy and decreases by an amount of order dt*. A collective kinetic energy proportional to the square of the collective velocity, therefore of order dt*, compensates for the loss of potential energy. The total energy is conserved, as always in TDHF. It is thus clear that the manifold of Slate? determinants plays somewhat the role of a phase space [8]. The determinant ‘y, can be interpreted as a classical point in that space, with a position and no velocity. Under the action of a collective force, this point is accelerated through W, and acquires a velocity which yields a displacement. In order to investigate the displacement operator, one has to go beyond the preceding heuristic discussion and to consider the evolution of a TDHF solution Y(t) corresponding to an arbitrary collective motion. A complex Slater determinant Y(t) can be written as [8, 181 Y(t) = exp[in(t) 2(t)]

YJt),

(4.7)

where the Slater determinant !PO is real, as well as the c-number 7cand the Hermitian one-body operator 9. Quite obviously, the latter acts as a boost operator. Since it is multiplied by n(t), it may be normalized, once for all, in an arbitrary manner. The

FIG. 5. The tangent plane for a given real determinant Yb. A real step along the basisvector jlOiO) converts yb into another real determinant. An imaginary step along /lOi,) converts !POinto a complex determinant, which contains velocities.

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TOURNEUX

time dependence of both !PO and 9 will be parametrized through a common monotonic function A(t). The rate of change of !PO will thus be given by

where the action on !PO of the displacement operator 9 has been defined as .Y=i$.

(4.9)

The choice of a parametrization A(t) as well as the normalization of 9 will be discussed in Section 5. The possibility of extracting matrix elements of the collective operators from those of the HF Hamiltonian does not necessarily require an adiabatic approximation, as could be seen in the last two sections, where no such approximation was made. For an arbitrary collective motion, however, one cannot exclude a priori the possibility that the collective operators vary with time, and in order to prevent the formalism from becoming quite unwieldy, we will resort to an adiabatic approximation. This will reduce the complications due to the nonlinearity of TDHF, which induces velocity-dependent forces, and should be a suitable approximation whenever the collective kinetic energy is quadratic in the velocities. The operator 7~9 will thus be assumed to be “small,” in a sense specified in the following section, and the normalization of 5% will be chosen so that rr will be a small number. Expanding the right-hand side of (4.7) in powers of 71and substituting the resulting expression into the TDHF equation, one gets, to second order in rr, [-7~9

+ i] !Po - [S? + x.5? - jn2k5P3 + i7r7S2] !Po = [Wo+7c2(W,+iW,S?-~Wo~2)+in(Wo~-iW,)]

You,. (4.10)

The operators W, =iTr

P[LZ,p,,]

(4.11)

and W,=-;Tr

p[S,

[S?,p,,]]

(4.12)

arise from the expansion of the HF Hamiltonian H[ !?‘I in powers of IL. In the derivation of Eq. (4.10), 7i was considered to be of order zero in 1~,while &’ and e0 were considered to be of order one. It will be shown indeed below that e0 is of first order. Since *,, is proportional to & it seems legitimate to conclude that 1 is a firstorder term. The proportionality between 6’ and 1 then implies that A? is equally of first order. On the other hand, 7i is essentially the time derivative of a momentum and there is no reason why the assumption of small velocities should entail that of small

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accelerations at all times. In the first approximation, the magnitude of the acceleration is expected to be determined by the restoring forces that can be derived from a static energy surface. It must be borne in mind that Eq. (4.10) is meant to be used in the particle-hole space associated with Y, [Ii) = exp(ilr.9)

11,i,).

(4.13)

Taking the scalar product of Eq. (4.10) with the expansion of (Iii in powers of 71,one gets, to second order, after separating the real and imaginary parts,

(4&l %J=&~oI~([~09~l

-W>I%)

(4.14)

and

=-&i,l

{~,+~*(~,+~[~,,~l-f[[~,,~],~])}IY~Y,).

The first of these equations, which gives the rate of displacement following expression for the operator 9, XT = 7r(Wl + i[ Iv,, 9’1).

(4.15) of Y,,, yields the (4.16)

On the other hand, the second one gives the rate of change of x.9,

~(n(t)~)=-[w,+n’(w,+i[w,,.a]-i[[wu,.~],~~])]. (4.17) A few comments are in order. Since Eqs. (4.16) and (4.17) hold in the particle-hole space only, namely in the tangent plane corresponding to Y,,, no cnumber terms dq,./dt and dqp/dt were included. Since 9 and 9 depend on Yb, which is a function of time, they are likely to be, themselves, time-dependent and to govern only the local evolution of the wavefunction. When the operator 9 is timedependent, Eq. (4.17) may be considered as the differential equation that defines it. The possibility that it be time-dependent is discussed in Section 5. It will finally be noticed that according to their definition 9 and 9 are purely real and imaginary, respectively, and both Hermitian. They are thus endowed with some at least of the usual properties of coordinate and momentum operators. Although we saw at the end of Section 2 that this kind of argument must be used with caution, there is no reason for concern here since Eqs. (4.16) and (4.17) are meant to be used in the real particle-hole basis associated with YO. The price we have to pay for disentangling the complexities arising from Y being complex appears in the supplementary commutators involving 9 in Eqs. (4.16) and (4.17); we can no longer define 9 and the rate of change of n9 in terms of the HF Hamiltonian only. Although the

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supplementary terms in Eq. (4.17) are of second order in rr, their importance is far from being negligible, even in the presently considered adiabatic situation, whenever P,, happens to be in the vicinity of a static HF solution @J,,,where the particl*hole matrix elements of W, become vanishingly small. Only the adiabatic approximation was used in deriving Eqs. (4.16) and (4.17). No attempt has been made so far to endow .5%’and .P with any collective nature, and nothing indeed prevents us from defining two families of operators 9 and .V according to Eqs. (4.16) and (4.17) along any adiabatic TDHF trajectory. Now if a nucleus possesses a collective degree of freedom, it should be possible to select a privileged set of boost and displacement operators associated with this degree of freedom. How this can be done will be discussed in the next section.

5. STOP-AND-GO

ANALYSIS

OF COLLECTIVE

MOTION

The possibility of a collective motion is often associated with the existence of a valley in the energy surface built through Eq. (4.4) upon the manifold of real Slater determinants .Y’;. When the energy E varies little along a certain line of .YO and increases sharply away from that line, there is a one-dimensional valley on the energy surface (see Fig. (4)); when it varies sharply in all directions but two, there is a twodimensional valley, and so on. Now, the solutions of a TDHF calculation are complex and belong to .Y rather than ,Vh. The set of determinants through which the system evolves may thus be considered as a trajectory in 9’. In view of Eq. (4.7), however, one may associate with each complex Slater determinant Y a real one Y,,. Similarly, one may associate with a trajectory in .Y’ its projected orbit in 9,, namely the set of corresponding real determinants YO. Just as 9 plays the role of a phase space, Y0 plays that of a configuration space. It will be noticed that a valley is usually flat in at least seven directions, since the most general determinant describes a system with a center of mass localized in space and a triaxial deformation localized in orientation. Any translation or rotation of Y,, thus generates a determinant with exactly the same energy. The energy surface is thus perfectly flat in the six corresponding directions plotted on Y,. What is interesting is to discover a seventh direction along which the gradient of the energy is small for dynamical rather than kinematical reasons. Translations are completely decoupled from other collective motions. It is thus legitimate to disregard them by restricting the present theory to a subset of 9’ that contains only determinants for which the nucleus has an arbitrary, but fixed once for all, average position in usual space. Although rotations are coupled to other degrees of freedom through Coriolis and centrifugal forces, they will also be discarded temporarily for the sake of simplicity. This is achieved by further restricting Y to a subset of determinants for which the nucleus, if deformed, has an arbitrary, but fixed once for all, average orientation in usual space. The remaining directions in the tangent plane then correspond, for instance, to vibration, scission or flexion modes.

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In this section, we will concentrate on the case in which a one-dimensional valley is left after elimination of translations and rotations. If the collective kinetic energy of the system remains small as compared with the depth of the valley, the orbits, namely, the sequences of determinants Y,, of interest, will stick closely to the line following the bottom of the valley, any deviation away from the latter being rapidly corrected by the steepness of the surface. This is what Baranger and Vineroni [8 ] described as a “spaghetto” situation. Clearly, it corresponds to a stability property of the orbits with respect to variations of both the longitudinal and transverse velocities. The evolution of various complex determinants Y associated with a given initial Y,,, namely, the evolution of the system starting from a given point with various velocities, will yield a bundle of trajectories which, once projected on ,P,,, will constitute a bundle of orbits closely bunched together along the bottom line of the valley as long of course as all the trajectories remain adiabatic. In a sense, the bottom line of the valley may be considered as an average of the various orbits in this bundle. Let us now come back to the boost and displacement operators .2 and 9 defined in the previous section. Clearly, there will be a set of such operators at each point along each trajectory in a bundle. In order to define collective boost and displacement operators .Zc and ,Pc, it seems natural to choose the operators .2 and 9 yielded by a trajectory whose projection follows the bottom of the valley. As a matter of fact, no orbit follows exactly the bottom of a valley. Indeed if one lets the system evolve from a real initial state Y,,, it builds up a certain velocity, even if the motion remains adiabatic, and velocity-dependent forces carry it somewhat away from the bottom of the valley. Starting from a slightly different real state Y0 + Sul, leads to a slightly different orbit. In order to follow the bottom of the valley, one has to prevent the system from building up velocity. This can be achieved by resorting to a ‘fstop-andgo” procedure, namely, by letting the system evolve from rest for an infinitesimal length of time, stopping it, letting it evolve again, stopping it again, and so on. To carry out this programme, we shall let the system start from rest at time t, from an arbitrary point Y&) on a valley line, to the exclusion of the lowest point, which corresponds to a static HF solution QO. Since rr = 0, the only force felt by the system is, according to Eq. (4.17) along the gradient line W, following the valley, and the collective boost operator reduces to 9$)

= -w:“(n)/7i,(n),

(5.1)

where &,(A) denotes the value of 15 when the system leaves the point Y,(n) with vanishing velocity. In order to stress the fact that Eq. (5.1) determines only the particle-hole matrix elements of 2Pc,, the notation (5.2) has been introduced, where p. and u. = 1 -p. project, respectively, on the single-particle states that are occupied and unoccupied in Yo. Quite obviously, an operator is not completely defined when only its particle-hole matrix elements are known.

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However, it was shown in Refs. [8] and [18] that while the decomposition (4.7) is not unique, it can be performed, in the case of an adiabatic motion, through an operator having vanishing particle-particle and hole-hole matrix elements. It will be assumed in this section that 9 obeys such a restriction. Since we let the system start from rest at time t, both n(t), which acts as a momentum, and X(t), which acts as a velocity, vanish. In order to extract the collective displacement operator Yc through the stop-and-go procedure one must thus consider Eq. (4.16) at time t + dt. Then n(t + dt) = io(t) dt and i(t + dt) =X,(t) dt, with I,, denoting the value of 1 when the system leaves Y,,(A) from rest. In the limit dt + 0, one gets

where WI, is given by Eq. (4.11) with 9 = 2’c, The stop-and-go procedure is thus seen to remove the velocity-dependent terms from Eq. (4.17) and to leave in Eq. (5.3) only the displacement operator naturally associated with the gradient of force Wih through SC. This is what one would have intuitively expected to result from an averaging of the operators defined by Eqs. (4.16) and (4.17) over a bundle of close trajectories. The operators 9c and Yc yielded by the stop-and-go analysis are proportional to those obtained by Villars [ 131, and Goeke and Reinhard [ 151 in their adiabatic TDHF theory. This is hardly surprising in view of the fact that their operators are defined in such a way as to satisfy the TDHF equations to first order. So far, the operators s9, and PC have only been defined up to the multiplicative factors 72, and ti,/x,, respectively. This arbitrariness may be removed by making a choice for the parametrization l(t) and the normalization of LZ~. In the following, the parameter 1, which a priori could be any monotonic label along the orbit, will be taken to verify the relation

This choice corresponds to the natural metric of the Hilbert space, since it entails that the length element dl along the orbit is given by dl* = dA*. The parameter A then is nothing but a measure of the length along the orbit. When the basis of single-particle states associated with each real yb(lz) along an orbit is orthonormal, the quantity (Y. 19 ] You,)vanishes and 9 can connect Y. with particle-hole configurations only. Consequently, Eq. (5.4) is equivalent to the normalization condition x Wo(~)l 9 l~0(wl’ = 1,

(5.5)

ido

where, again, i,(A) and lo(A) denote, respectively, occupied and unoccupied states in ‘PO(A). In an analogous manner, we shall normalize the boost operator according to

c l(Zo@lmn)l ~o(4)12= 1. ido

(5.6)

COLLECTIVE

This normalization

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condition does not warrant that .gC and Lq are weakly conjugate

a condition which in turn would ensure that x and L satisfy canonical equations [ 13 1. However, it enables one to formulate easily an adiabaticity criterion by requiring that (~1 should remain much smaller than unity. Similarly, the normalization condition (5.4) allows one to define a motion as being of large amplitude when it involves variations of J larger than unity. The collective boost operator &%‘C,Eq. (5.1), will obey condition (5.6) if (5.8)

the f sign being chosen as follows. Whenever the orbit passes through a static HF solution QO, the energy gradient Wi” vanishes and reverses its direction. If 7i., similarly changes sign whenever it vanishes, the matrix elements of the operator .$‘c = -W{h/?20 may be viewed as the components of a unit vector that slides along the bottom line of a valley without ever reversing its direction. Thus ,2’,(L) does not vary in a discontinuous fashion when passing across a static HF solution. When applied to the collective displacement operator .< given by Eq. (5.3), the normalization condition (5.5) yields

Jj$$ = [X I( 0

W,,W + i[ KM

ido

%@>IIMW12] -“‘.

(5.9)

This ratio has been given a positive value, since it has the nature of an effective mass. Indeed, if 71 and i behave as a momentum and a velocity, respectively, they are expected to be proportional, 71=&q/i,

(5.10)

with ,U playing the role of an effective mass. Now, in the stop-and-go analysis, 7i., = ~~(1) I,, which is precisely Eq. (5.9) if the right-hand side of the latter is identified as pug(d). In view of the discussion of last paragraph, the operator .2C is always different from zero. There is thus no reason for the norm I( WY,” + i[ W,, .GPc],l””I( to vanish at stationary HF solutions. As a matter of fact, the stop-and-go analysis that has just been presented enables one to compute a path, starting from any real Slater determinant. Indeed, given a PO(L), Eqs. (5.1) and (5.8) yield the operator SC(L) which in turn allows one to construct the displacement operator S,(J) through Eqs. (5.3) and (5.9). Once the latter is known, one can compute Yo(L + dJ> through Eq. (4.9) and repeat the whole process again. Clearly, a path generated in this fashion does not necessarily follow the bottom of a valley and one needs a set of criteria in order to determine whether one is dealing with a collective situation or not.

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A first criterion comes immediately to mind. If !PO evolves along the bottom of a valley, it follows a gradient line defined by IV,,ph. The displacement operator must therefore be parallel to this gradient at each point, (5.11) (WI W&) I W)) = A (Z&l Z(A) I W)). If the proportionality constant li is interpreted as a Lagrange multiplier, Eq. (5.11) may be viewed as a static HF equation with a variable constraint
(5.12)

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the condition d&/A = 0 is seen to imply the constancy of the matrix elements (I,,] 2?c ] i,) as long as 9 and & are canonically conjugate or, at least, as long as their commutator is a constant, or a particle-particle or hole-hole operator. Now, there is no reason in general to expect this stringent condition to be realized, and it is hard to see what the constancy of .I@ would imply for its particle-hole matrix elements over a changing basis. Let us mention finally, that it may be possible to determine the collectivity of a path by computing its curvature. Preliminary calculations [ 191 suggest that the closer this curvature is to fi, the more collective the motion is.

6. VELOCITY-DEPENDENCE

EFFECTS

The stop-and-go analysis presented in last section gives a path Y,,(A) along which the displacement and boost operators .c(A) and
(6.1)

The physical meaning of this ansatz must be made quite clear. Starting with a succession of YO’s and c%?c’sthat are labelled once for all by the position A along the path, one looks for TDHF solutions that differ from each other only by the functions n(t) and A(t) that characterize the time evolution of the system. Since the latter can move back and forth along the path (in a vibrational motion, for instance), the function A(t) will no longer represent the distance travelled by the system, but its position along the path at time t. Various solutions will correspond to different initial positions A(O) and momenta ~(0). The main physical assumption is that the boost operator 2, may be taken to depend only on the position A and that within a certain range of values of a(t), this approximation will yield tolerably accurate solutions of the TDHF equations up to second order, namely, of Eqs. (4.14) and (4.15). When ,%?= SC, the first of these, together with Eqs. (4.8), (5.3) and the relation yields II =poi. This establishes, by comparison with PO= 7io/Jo, immediately Eq. (5.10), that the “dynamic” effective mass p(A) is equal to ~~(1) when the TDHF equations are expanded up to second order only. On the other hand, Baranger and Veneroni [8] have stressed that in the vicinity of a static HF solution the matrix elements of Wg” remain small, so that it may be insufficient to retain only the first term on the right-hand side of Eq. (4.15) because it is of zeroth order. We shall therefore make no attempt to decompose Eq. (4.15) order by order, but examine,

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rather, the extent to which it can be satisfied globally. Now, for .R = SC, Eq. (4.15) becomes

=-(I,1 {wo+~‘(w2,+i[W,,,.,gc]-f[[K,,.%],.%])} Ii,),

(6.2)

where IV,, and W,, are given by Eqs. (4.11) and (4.12) with .9 = ~9~. Once 1 has been eliminated through the relation x =,u,,i, Eq. (6.2) reduces to a differential equation for n, all the coefficients of which can be computed easily. Now, it must be realized that Eq. (6.2) should hold for each particle-hole pair (&lo). For this to be the case, the vector (I,, SC Ii,,), which is already parallel to the vector, (I,1 W, Ii,), must also be parallel to the vector

(I,Ivli,)=(I,I

(,u;‘$+

W2,+i[W,,,.~~]-f[[W,..~~],~*,])

Ii,,). (6.3)

This condition is easily seen to be equivalent to the assumption- that Sip, depends only on 1, and not on 7~. Indeed, Eq. (6.2) describes various trajectories with different initial conditions. Clearly, this parallelism is unlikely to be realized in general, but one may hope that it will hold approximately in highly collective situations. It is then possible to satisfy Eq. (6.2) in an average manner if one multiplies it by (i,l SC [lo), sums over the pair of indices (i,, I,) and uses Eq. (5.1) and the normalization condition (5.6). The resulting equation is 7i = li, -an*,

(6.4)

where

a = C GoI,gc I~oX~ol2)lid.

(6.5 >

ido

The extent to which the time-dependent solutions (6.1) obtained in this fashion reproduce exact TDHF solutions may be considered as a measure of the collectivity of the motion. There may be situations where the collectivity is not strong enough to ensure that the ansatz (6.1) which has just been described provides the best approximation to adiabatic TDHF solutions. There may also be situations where one is interested in knowing the coordinate .R that acts as a boost operator, so as to elucidate the nature of the collective motion. One may then proceed as follows in order to optimize approximate solutions to the second-order TDHF equations (4.14) and (4.15). Of course, a similar procedure can be carried out to higher orders. This new approach starts again from the decomposition (4.7) but the assumption made so far that the boost operator has vanishing particle-particle and hole-hole matrix elements will now be relaxed. Indeed, when the physical operator .R is

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replaced in Eq. (6.1) by an operator whose only nonvanishing matrix elements are of the particle-hole type, the latter is not related to .R by a simple particle-hole projection. If one wishes to gain direct knowledge about the physical collective coordinate, it is therefore advisable to work with a complete unprojected operator .9?. The adiabatic expansion in powers of 71is made easier if one uses the systematic expansion (6.6a) ti(t) = a&l) + a,(l)?T + a*(A) 7r2, i(t) = b,(A) + b,()L)n + b*(1) x2, which has been carried up to second order only. It will be noticed that knowledge of 1 and rz at a given time specifies the initial conditions of a TDHF trajectory and yields the values of x and ti at that time. One may expand, in the same way, .!qt) = %?&) + .%?l(ll)n + 3J*(A) x2,

(6.7a)

.T(t) = .%(A) + .<(/l)r

(6.7b)

+ .%(A) 712.

For various reasons of simplicity such an expansion will not be carried out here and we will restrict ourselves to an ansatz of the form Y(f) = exp[irr(t) .‘X(n(t))] YO(J(t)). The n-independence of ul, and .W follows from truncating Eqs. (6.7) to zeroth order and implies that a family of approximate trajectories can be generated through some average path !PJn) and boost operator .a@). This is the same philosophy as was adopted in the beginning of this section, but now 9, .8, ti and i will be chosen in such a way as to optimize the solution of Eqs. (4.14) and (4.15). It may be worth mentioning that in the language of differential geometry the wavefunctions (6.8) form a fiber bundle whose base and fiber are the set {Y&)} and exp[ir&‘(A)], respectively. Our ansatz assumes that the generator 9 of the fiber depends on the position L only, and not on the momentum rc. As a final simplification it will be assumed that the L-dependence of 9 is negligible. As was mentioned at the end of Section 5, there are situations where this is expected to be a reasonable approximation. Such a stable boost operator may be viewed as resulting from some average over 1, just as the truncation of (6.7a) to zeroth order could be interpreted as the result of an average over trajectories. The use of a constant .‘Z immediately raises the question of its normalization. Quite obviously, the normalization condition (5.6), which involves a changing particle-hole basis, will usually be satisfied by a constant .R at one (or at a finite number of) value (s) of ;1 only, unless the commutator of 9’ and .P happens to have vanishing particle-hole matrix elements. In this special case only one can satisfy the condition (5.6) for all values of 1 along an orbit, as a direct consequence of Eq. (5.12). Although such a situation seems rather unlikely, the possibility of its occurrence may deserve a numerical investigation in certain cases. One may of course always normalize .R by satisfying Eq. (5.6) at some arbitrary point 1, but it is difficult to say 595/138/2-9

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anything in favour of such a prescription which is all the more arbitrary because L@ is no longer a pure particle-hole operator. Furthermore, such a normalization does not easily provide even a vague justification for the expansion in powers of II. As a measure of the extent to which Eqs. (4.14) and (4.15) are satisfied, we introduce the error function a(s?,.F’, 1, 71,x, ti, !P& = 0))

+1~~~1~~+~1,+~2(~2+~[~,,~1-~[[~,,~],~1)1~,)12},

(6.9)

where 75and x are given by Eqs. (6.6). It is this error function, averaged over a set of trajectories, that will have to be minimized. It must be stressed that Eqs. (4.14) and (4.15) should be regarded as identities with respect to IL, since the equations of motion for 7t and x that are under study should be valid, at a given value of 1, for a whole family of adiabatic TDHF trajectories. Bearing this in mind, one sees immediately that the contribution of the first term in (6.9) vanishes if 6, = b, = 0

(6.10a)

and b,(Z,l~~lh,)=

(&,I W, +il~o,~l

I&,),

(6.10b)

which is strictly equivalent to Eq. (4.16). The normalization condition (5.5) yields for side of Eq. (5.9), with the stop-and-go particle-hole operator 3?C replaced by the full .R of the present ansatz. The contribution from the second term of Eq. (6.9) cannot be made to vanish, but it will be minimized if the coefficients appearing in the expansion of ti, Eq. (6.6a), take the values b, an expression analogous to the right-hand

a,=N-’

(6.1 la)

a, =o

(6.1 lb)

where

N= - C I(442 lAJ12 fdo

will usually be a function of the position L as was discussed above.

(6.1 Id)

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The equations of motion (6.6) reduce to 7f= a,(A) + uz@)7r2,

(6.12a)

x = b,(A)?,

(6.12b)

the coefficients a,, a, and b, being known functions of A. Now it is trivial to check that A and R do not obey Hamilton’s equations, since (6.13) One may, however, introduce a new variable <(A) and new functions a,, a2 and /?, such that (6.14) and so on. The equations of motion then become

7i= %(G + a203

7r2

(6.15a) (6.15b)

and nothing equation

prevents one from choosing for r(A) a solution

of the differential

(6.16) so that

a7i -=--* ai a71 ay The pair (<, rr) may be regarded as a pair of canonically conjugate variables as long as the above equations are valid, namely, as long as the second-order adiabatic expansion is justified. This implies, amongst other things, that Liouville’s theorem holds. In order that the optimization process be completed, the optimal operator .2 and initial Slater determinant !PO(t = 0) must still be determined. This can be done in such a way as to minimize the integral of the error function 8, Eq. (6.9), over a set of trajectories, (6.18) 7 being the function B minimized

at each point of each trajectory by the equation of

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path (6.IOb) and the equations of motion (6.15). Admittedly, such an undertaking is quite difficult, unless one has in advance a pretty definite idea of the class of operators amongst which the optima1 ,W is likely to be found. One might have reasons to believe, for instance, that a linear combination of the generators 5Pi of a Lie group, .R

=c

ri.5Fi,

(6.19)

1

is a reasonable ansatz. One of the coefficients t, could then be used to normalize 2 arbitrarily, and the remaining ones could be determined so as to minimize the integral (6.18) over a set of trajectories. It may be worth pointing out a few aspects of the physical meaning of the averaging process (6.18). The set of approximate trajectories that it involves is built on an orbit defined through Eq. (6.10b), an initial Y0 and a certain 2. It is implicitly assumed that the sequence of Slater determinants along the orbit is parametrized in a one-to-one manner by {. Quite obviously any of these determinants can be used as an initial Y,, if one generates the orbit by using Eq. (6.10b) backwards and forwards. The value of the integral (6.18) should thus be regarded as a property of the orbit to which Y,,(O) belongs, rather than as a property of Y,,(O) itself. Within the bounds set by the hypothesis of adiabaticity, any point in the (<, rc) plane may be considered as the starting point of a trajectory. The integration over r and 71 in (6.18) performs a double task: it sums over a bundle of trajectories the integral of the error function along each of these trajectories. It is thus a measure of the extent to which the set of actual TDHF solutions having the same starting points would remain in that bundle. The choice of the domain over which the average (6.18) is computed obviously involves a large measure of arbitrariness. The crudest choice would consist in choosing a rectangular domain of integration in the (<, n) plane. Such a procedure is, however, likely to take into account very unequal sections of the various trajectories. This can be avoided by integrating rather over the domain comprised between two trajectories extending over the same length of time. In a more sophisticated manner, one may take a small cloud of points in the (<, rc) plane as a collection of starting points, and integrate the error function along the trajectories evolving from these starting points during a certain length of time. Liouville’s theorem then ensures that the weight to be given to the cloud is constant in time. The optimal orbit may be defined as the one which minimizes the integral (6.18) as a function of the parameters rI that enter the expansion (6.19) of ZP, and as a function of the parameters of YO(0) (bearing in mind the fact that variations of YO(0) along a given orbit leave the value of (6.18) unchanged, as was pointed out above). Quite obviously, the value of the average (6.18) will depend on the domain of integration that has been chosen, but one may hope that the optimal orbit is not too sensitive to this choice. Of course it would be more satisfactory to have explicit expressions for this minimization process. At the present stage of the theory, however, we can only advocate a numerical treatment of the problem.

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7. FURTHER GENERALIZATIONS

The “spaghetto” situation decribed in Sections 5 and 6 allows for one-dimensional paths only. If, however, collective motion is intepreted geometrically as taking place along the bottom line of a valley in the energy surface, one may expect that two (or more) collective degrees of freedom will be of interest when the valley is flat in two (or more) directions. Let again Y0 be a real determinant, IV,, the HF Hamiltonian it generates, and X, a generator defining, in the tangent plane, a direction which is different from that defined by Wth, and for which the energy surface is as flat as possible (see Fig. 6). For instance, 1, might correspond to an infinitesimal translation or rotation of Yy,, but this would be an extreme case, since the energy surface is perfectly flat along these directions. Such a case of “degenerate lasagna” will be considered later in this section. In order to stick to dynamical rather than kinematic flatness it may be simpler to associate W, and X,, with combinations of /3 and y vibrations, in the case of a very soft deformed nucleus for instance. The ansatz of interest for the fiber bundle is now the following generalization of Eq. (6.11, Y = exp[-i0X]

exp[-irW{h]

!Po,

(7.1)

where 0 and 72are real. It will be noticed that the fiber is two-dimensional. Accordingly, one expects Y,, to depend on two real parameters, I, and 1,. In other words, one now considers a fiber bundle of dimension 4, with a base of dimension 2, like the fiber. The program of the last two sections can be generalized. One has to find under which conditions this bundle of dimension 4 is approximately closed under the TDHF equations of motion.

i FIG. 6. The tangent plane for Y’,, again, with a direction given by the energy gradient (W,) and another one (X,,) along which the energy changes as little as possible. Steps defined by W,, and X,, can be real, and then mean a step along the gradient line for W,, or a translation if X0 correspond to translations. Or the steps can be purely imaginary and then mean a dynamical collective velocity (for IV,,) or a translational velocity (for X,).

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It is interesting to illustrate this “lasagna” situation through the example of Section 3, namely, the elastic scattering of two nuclei. Let the reaction plane be defined once for all. In the limit of a purely central (real optical) interaction between the nuclei, the collective mode corresponding to W$” is obviously radial. On the other hand the energy surface is perfectly flat with respect to the nonradial mode in which the system undergoes a global rotation in the reaction plane. If at time f = 0 the system is prepared in a state exp(-inI@) YO, namely with a radial velocity, quite obviously only radial motion will take place (at least as long as the nuclei do not overlap and excite inelastic modes, which would lie beyond the scope of the present model). This will remain a “spaghetto” situation whatever the value of rc may be. If, however, a nonradial velocity component is added at t = 0 through the boost operator -i&X,,, one gets the usual set of angular-momentum-dependent trajectories. With suitable initial conditions, the determinants YO(t) generated by time evolution make a two-parameter family corresponding to all relative positions of the two nuclei in the reaction plane (except, once again, for the overlap region, of course). The radial spaghetto is thus embedded in a well-defined lasagna. Actually, this special case exhibits some extreme properties. For instance, it does not require adiabaticity (one must of course use exponentials in nonadiabatic situations). As a matter of fact, it is even desirable that the nonradial intensity 0 be large enough to prevent overlap between the nuclei, otherwise the model could be salvaged only by stopping the clock before head-on collision occurs. Furthermore, there might be some special properties of the potential or kinematic conditions which would make the system very interesting from the optical point of view. Orbits might stay close to each other for a long time, thus forming a spaghetto, then become unstable and go apart, making something like a French horn, and finally converge again. The case of orbiting, frequently discussed in the heavy-ion literature, corresponds to the pinching of a family of orbits, pertaining to a certain window of initial kinematical conditions, into basically only one line, namely, the bottom line of the attractive part of the potential. In other words, orbiting can reduce a piece of lasagna into a piece of spaghetto. It would be easy to multiply provocative examples of behavior which are not easy to classify. A development of optics in the Slater manifold is likely to be in order. Clearly, when searching for lasagne one cannot treat on the same footing the identification of the two fiber generators W$” and X,,. Indeed, Wg” is the only generator provided by the stop-and-go analysis, while X0 must be taken into account because the two-dimensional bundle (6.1) of the last two sections does not remain approximately closed under TDHF evolution. It is most likely that X0 may be identified by performing the RPA along the failing spaghetto and selecting the lowest RPA mode that yields a direction significantly different from that of eh. We shall close this section by discussing the kinematical case of translations and rotations. They correspond to RPA modes of vanishing frequency and define six directions in the tangent plane, along which the energy surface is flat. In the case of translations, as exemplified in Section 2, although the dimension of the fiber has been increased by a new generator X0, namely, a center-of-mass coor-

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dinate operator R, the Hartree-Fock Hamiltonian remains equal to IV,. In this extreme case, IV, vanishes. Only the commutator [ W,, R] contributes to the collective momentum operator 9 as defined by Eq. (4.16). This observation stresses again the need to analyze the interplay between operators and real or complex particle-hole bases. In any case the problem of translations is here trivial, for one knows in advance that the algebra of generators which will be chosen to enrich to dimension of the fiber from 1 to 4 will contain the center-of-mass coordinates R. In the same way, the generators of the base of the bundle will be completed by the total center-of-mass momentum P. It may happen that static HF solutions QO, and real determinants Yd close to QO, have little center-of-mass spuriosity. In such a case the center-of-mass motion may factorize out, at least approximately, in a OS state. It can be stressed that RYO and iPY,, are then the same, for they both correspond to the well-known 0~ center-of-mass spurious spectroscopic level. Instead of having different generators R and P to enrich the dimension of the fiber and base, respectively, one can then conclude that only one set of generators is sufficient, provided one allows complex numbers 0 in the ansatz, Eq. (7.1). (This may induce deviations of phases and norms, but the relevant corrections to the formalism are trivial.) This remark about complex coefftcients for the generators is of some use in the case of rotations. The generators of the base of the bundle must now obviously include J, the total angular momentum. (Actually one should subtract from J the angular momentum of the center of mass.) The generators of the fiber should include, accordingly, the Euler angle operators n which are conjugate to J in some sense. It is well known, however, that 52 cannot easily be expressed in a convenient form. Therefore one may use J with complex coefficients, or look for an alternative for Q. The fact that the components of J do not commute with each other brings an interesting remark. In the case of translations, it is seen from the equations of motion that the wavefunction evolves under the action of P only, not of R. It will be shown that, on the contrary, J cannot act alone. Indeed if only J were involved, one would have ig=

(“.I+$)

ul,

(7.2)

where w is the real instantaneous rotation velocity, just a numerical vector depending # on time. Then (7.3)

It is known, however, that the left-hand side of Eq. (7.3) vanishes in TDHF calculations. Thus either o is parallel to the expectation value of J, which is only a restricted rotation mode, or Eq. (7.2) must be generalized.

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One way is to introduce o and tf complex. Another way is to use the form (7.4)

where Q must be a nonscalar tensor operator, contracted into a scalar by means of p, a numerical tensor of the same rank. The set of commutators [J, Q] do not vanish and actually give back linear combinations of Q. Then Eq. (7.4) becomes (7.5)

where B is a tensor calculated by tensor algebra. In the second term, it is coupled with Q to yield a vector which can now compensate w x (J) in such a way as to ensure the vanishing of d(J)/&. For the sake of simplicity it is desirable to take Q as a one-body tensor. If furthermore Q is local, one sees from Eq. (7.1) that Q brings changes of phases inside Y, namely, transforms single-particle orbitals w(t) into orbitals with locally modified phases, v(r) exp[$‘q(r)]. This has the advantage that velocities are modified, but densities are kept unchanged by Q. It is possible to interpret Eq. (7.5) as a rotation of the nuclear density, while the velocity field evolves in a more complicated way. As regards a practical choice of Q, the local quadrupole tensor is the simplest choice a priori, since of course the dipole is excluded because of center-of-mass spuriosity. It is seen that, if complex numbers are rejected from the fiber bundle ansatz for rotations, one may try to enrich the dimension of the fiber by 5 and the dimension of the base by 3 only. This lack of balance is the price one has to pay for the use of the quadrupole operator as a substitute for Euler angles.Of course, one expects that two out the five degrees of freedom thus introduced in the fiber will play the role of frozen p- and y-deformation parameters, but this has to be checked in practical examples. Alternatively, the TDHF dynamics may force two new generators Y0 and YY into the base in order to keep the bundle closed. This is also an interesting problem. In any case, one benefits from angular momentum conservation in a nontrivial way: precession, wobbling and coupling to vibrations are allowed in a microscopic theory where only one-body operators appear. 8. DISCUSSION AND CONCLUSION

Several new considerations emerge from the present study. The elementary example of Section 2, in the first place, shows that a real operator T acting on a complex wavefunction actually reduces to the product of a purely imaginary operator P and a real vector k. This follows from the nonlocality of T and from the self-consistency of the wavefunction. Such a result may be somewhat fortuitous, although the implications of self-consistency, more particularly the resulting lack of spreading, are probably far-reaching if not yet fully understood. In any case, this elementary example shows that the fact that T is intrinsically real is misleading. Only its

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particle-hole matrix elements, which can be complex, are relevant to the definition of an evolution operator. The example discussed in Section 3 is also instructive. It displays how the particle-hole matrix elements of the TDHF Hamiltonian reduce to a subtle linear combination of those of the instantaneous collective coordinate and momentum. It may happen that these instantaneous operators can be related to fixed, timeindependent operators. Time-dependent linear combinations are then involved and their coefficients dr,,/dt and dp,,/dt might be governed by equations derived from the TDHF equation of motion. The instantaneous operators, however, are not likely to be easily related to fixed operators in the general case. Both in Sections 2 and 3 the Slater determinants under consideration were closely related to static HF solutions. This restriction was dropped in the following sections. Formulated for arbitrary Slater determinants the theory gained much generality as well as the capability to incorporate the results of constrained HF calculations, to the extent that the latter are compatible with the sequences of determinants yielded by TDHF. Although the interest of constrained Hartree-Fock calculations is well established they are open to the risk of fluctuations playing an enormous role, leading for a large class of determinants to a spreading of wave packets, thus destroying the possibility of collective behavior for nucleons which become scattered (the so-called “Saturn ring effect” 1201). Th e resulting collapse of the energy surface must obviously be prevented at all costs before the present theory may be applied. The generalized theory undertaken in Section 4 is based upon the concept of an energy surface having a significant structure. Usually, the energy surface is defined as a contour plot of the energy as a function of each real determinant !PO, and it is interpreted as a collective potential energy although it includes a kinetic contribution. Each determinant !PO may be considered as a point of a continuous set YO, which is itself a subset of a more general set 9 of points, the complex determinants Y. Quite obviously, it is possible to extend the energy surface as a map of the energy plotted upon 9 rather than 5$. The average energy then includes a collective kinetic energy, and the determinants Y for which this average energy is constant form a subset of Y which can be called equienergetic. Since TDHF conserves energy, it yields families of determinants which stay amongst fixed equienergetic subsets of .Y. These subsets may be further reduced by the usual conservation laws, but nothing definite can be said about the ergodicity of the paths. The manifold 9 is thus quite analogous to a phase space. It must be stressed that neither 9 nor 9; make a vector space, because of the nonlinearities which are built in restrictions when Slater determinants are involved. Also, for a system of N nucleons, the dimension of Y(,YO) is larger than 6N(3N), since determinants actually depend on an infinite number of parameters. Fluctuations of an observable can completely modify a determinant without changing the expectation value of the observable itself. As a consequence, both Y and 9, are, in principle, infinite-dimensional manifolds, whose curvature properties have to be fully understood. It may happen, in practice, that the theory could be fairly sensitive to the finite number of parameters retained to generate 9 and
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As discussed in Section 4, the TDHF equation of motion is actually an equation in the tangent plane of the manifold 9. A reformulation of TDHF in the language of differential geometry is therefore likely to be of some interest in the future. Our analysis of collective motion.started from the traditional view that collectivity is associated with the existence of a deep valley in the energy surface built upon yO. This entails that adiabatic orbits starting from a given point !PO on that valley follow closely its bottom line. We thus looked for a collective path such that all trajectories starting from a point on this path with an arbitrary velocity (within the limits set by adiabaticity) would yield, once projected on yO, orbits following this path as a first approximation. It must be stressed that TDHF solutions do not fulfill this condition. Indeed, the TDHF equations may be written as two coupled first-order equations for !PO and M, so that Y,,(t) obeys a second-order equation and is specified by both Y,,(O) and Y,,(O). In order to loose this velocity-dependence of the path we resorted to a stop-and-go analysis. Although the resulting boost and displacement operators turned out to be proportional to those of Villars [ 131 and Goeke and Reinhard [ 151, we feel that this analysis throws some instructive light on the passage from TDHF orbits, an infinite number of which can evolve from an initial point You,,to the unique collective path that may pass through a given Y,,. As pointed out by Goeke and Reinhard [ 151, the displacement operator ye, Eq. (5.3), yields a first-order differential equation for the collective path. Quite obviously such an equation enables one to get a path from any initial YO, and not all paths derived in this fashion can be considered as collective. One must therefore formulate some criteria through which collective paths can be identified. We suggest two possible criteria. The first one, which is static, consists in checking the extent to which the displacement operator 9, is parallel to the gradient along the bottom line of the valley. Clearly, fulfillment of this condition warrants the following classical feature: starting with zero velocity from a point Y,, on the valley, the system will start moving along that valley. The second criterion is, in a way, a dynamic version of the first one. It consists in comparing actual TDHF solutions with time-dependent ones given by the ansatz (6.1), where the boost and displacement operators are frozen to be those yielded by the stop-and-go analysis. Such a comparison displays the extent to which a family of TDHF solutions is collective in the sense we have used this term throughout this paper, namely, the extent to which adiabatic trajectories starting from a given point Y0 with various velocities will have the same projected orbit on PO. The stop-and-go analysis (or equivalently the first-order analysis of Refs. [ 131 and [ 151) yields only the particle-hole part of the boost and acceleration operators. One may define them as being purely of the particle-hole type. We exploited this possibility to normalize them in such a way that the definition of adiabatic and/or large amplitude motion is made easier. Unquestionably, the use of a boost operator that is purely of the particl+hole type presents a drawback since it does not allow us to elucidate the physical nature of the collective coordinate which boosts the system. We sketched in Section 6 a method whereby one may work with a complete opertor 9 and up to an arbitrary order in the adiabatic parameter z This analysis is based on a systematic expansion of all

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quantities in powers of 71 and can even cope with situations where our definition of collectivity begins to break down with the boost operator becoming n-dependent. We suggest to determine the best boost operator and the best path by minimizing the error with which the TDHF equations are solved over a set of trajectories. Section 7 dealt with situations where the energy surface is flat in more than one dimension, special attention being paid to the kinematical flatness associated with translations and rotations, It was suggested that in certain situations the boost operator may be replaced by the better-known translation operator with a complex coefficient. This approach may turn out to be useful in the case of rotations, where the Euler angles are not easily expressible in terms of the coordinates of the particles. Finally it was pointed out that as a consequence of the noncommutativity of the components of J, TDHF solutions for rotating nuclei cannot evolve under the action of J alone, except for the restricted modes of motion where the instantaneous velocity is parallel to the expectation value of J. It was suggested that such a situation can be remedied by including an extra term in the evolution operator, which has the effect of modifying the velocity field, while leaving the density constant. To summarize the main result of this paper we have shown that there exists a welldefined set of necessary conditions, in order to extract collective features from TDHF. This set of conditions stems from a rigorous expansion with respect to a unique adiabaticity parameter, without additional assumptions concerning the nature of the collective forces and/or operators.

ACKNOWLEDGMENTS The partial acknowledged.

support of the France-Quebec

exchange program during this work

is gratefully

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