The born-oppenheimer theory of nuclear rotations and vibrations

Nuclear Physics A473 (1987) 539-563 North-Holland, Amsterdam

THE BORN-OPPENHEIMER NUCLEAR ROTATIONS AND

THEORY OF VIBRATIONS*

Felix M.H. VILLARS Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 USA Received

15 June 1987

Abstract: In this paper it is demonstrated that there is a fruitful analogy between nuclear and molecular collective rotations and vibrations. In order to implement this analogy, and to construct the analog of the factorable molecular Born-Oppenheimer trial function, one must transform into a representation, in which the nuclear collective “position” operators and the residual intrinsic operators are diagonal. In this representation, the trial function is assumed factorable. Using projection operator techniques, this trial function can be transformed back into a particle coordinate representation, so that conventional many-body techniques can be used. The collective operators are “chosen” only to the extent needed to identify the “type” of collective process that one intends to describe, by stating their transformation properties under the standard symmetry operations in coordinate-, spin- and isospin space: In the present case, the collective operators form a symmetric, second rank tensor, are isoscalar, and thus describe rotations, and size and quadrupole shape oscillations. A more detailed determination of these operators is based on the dynamical argument of minimal coupling between collective and intrinsic modes of excitation. One then finds that in the harmonic limit, vibrations have frequencies given by RPA type equations, and rotational moments of inertia are determined by self-consistent cranking type prescriptions.

1. Introduction In a previous

paper ‘) (to be referred

to as I), a systematic

method

was developed

to describe and display the interrelation between “collective” and “intrinsic” dynamics in nuclear structure. This method was called the Born-Oppenheimer approach, and is based on the far-reaching analogies between molecular and nuclear rotation-vibration spectra. These analogies are first, that in both systems bands of collective states exist. In both, these present but a small segment of the total set of eigenstates of the respective hamiltonians. Second, in both systems are these collective states not so much an expression of a general separability of degrees of freedom, but rather of the existence of a set of states, in which rapid motion of part of the system is evolving within a slowly varying global structure, described in terms of “collective” variables. These slowly moving collective degrees of freedom then take on the role of time-dependent “external” parameters in the equations of motion for the rapidly evolving “intrinsic” variables. * This work is supported in part by funds provided contract # DE-AC02-76ER03069. 0375-9474/87/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

by the US Department

B.V.

of Energy

(D.O.E.)

under

540

F. M. H. ViZIars/ Born-O~~en~~~~er theory

the molecular case, this structural situation finds its expression in the wellknown Born-Oppenheimer (BO) ansatz “) for the approximate eigenfunctions In

In this expression, (x&(xn)) is the electronic wave function; its structure is parametrically dependent on the “instantaneous” positions xN of the nuclei in the molecule. f(xN) is the wave function describing molecular rotation-vibration. In paper I, the possibility of a similar decomposition of V in the case of nuclear dynamics was demonstrated. Before this could be implemented, a seemingly fundamental difference between the nuclear and the molecular case had to be dealt with: in the latter, the collective dynamics (rotations and vibrations) is readily expressed by means of the relative position coordinates of the molecular nuclei; these are in this case indeed the collective variables of the system. The electronic states depend parametrically on these coordinates, and in turn provide the “mean field” in which the relative motion of the molecular nuclei takes place. In the case of nuclear dynamics, no ready-made collective variables present themselves; they have to be constructed from the particle degrees of freedom at hand. The basic premise of the nuclear BO method is the assumption that suitable collective operators do exist, and can be constructed in some approximation. The complementary “intrinsic” variables are no longer in one-to-one correspondence with particle degrees of freedom, and the need to deal with them explicitly must be avoided. This is achieved by expressing the intrinsic wave functions in the configuration space of particle coordinates and eliminating the redundant collective space by means of suitable projection operators. Sect. 2 will introduce the collective operators, and the structure of the nuclear BO wave function. In sect. 3, a variational expression for the mean energy will be constructed, which will display the contributions of collective and intrinsic dynamics to the total energy. This expression is based on an expansion in terms of increasing powers of the collective momenta. All terms beyond the second power are neglected in this presentation. An attempt to develop the prerequisites which justify this approximation will be presented in an appendix. Sect. 4 will introduce a “body-fixed” set of coordinate axes and provide the basis for separating collective motion into vibration and rotation. The adjusted variational expression for the mean energy will be presented in sect. 5. Criteria for determining the collective operators will be presented in this section, and prescriptions for the collective mass parameters, and force constants for collective vibrations in the harmonic approximation are given. These prescriptions are based on minimizing the coupling of collective and intrinsic dynamics (and hence optimally justifying the use of the simple BO trial function). This leads to RPA-type prescriptions for the force- and mass-parameters of collective oscillation, and for a self-consistent cranking-type prescription for the rotational moments of inertia.

F. M. H. Pillars f Born-Oppenheimer

541

theory

Sect. 6 will sketch the application of the general results of sect. 5 to some specific cases: Spherical and axially symmetric intrinsic equilibrium structure, with and without net intrinsic angular momentum along the symmetry axis. Sect. 7 will summarize the results, put them in context with alternative approaches to describe the same phenomena, and present an agenda for further investigations and applications.

2. The collective operators and the BO trial function In this section, collective operators will be introduced to describe rotation and vibration of nuclei. The appropriate set of operators will consist of the elements of a symmetric Cartesian tensor Qrxp. (Greek subscripts will be used throughout to designate Cartesian components relative to a space-fixed frame of reference; later, in sect. 4, Latin subscripts will be introduced to designate components in a “bodyfixed” frame). Beyond the implied transformation properties, the only assumptions about the Q will be that (i) They are one-body operators, even under time reversal, isoscalars, and generally non-local (ii) The different components Qnp of Q commute with each other. (iii) They have a continuous, degenerate spectrum of eigenvalues: these will be written as q,+. There exists then a transformation matrix (x15; q) to a representation in which the operators Qap are diagonal, so that

LdQ&;

d=

W4Q,,lx’Wl5;

d = bit;

q)qc+.

(2-l)

In this expression, x stands for the set of all particle position, spin and isospin variables, q for the collective variables, and 5 for the set of residual “intrinsic” variables, which need not be specified. The momenta canonically conjugate to the Qrrp also form a symmetric tensor Kap, with the commutation relations

Qysl= (W){&,&z + -&A,,1.

i[&,

(2.2)

Using symmetrized subscripts: (cup) = (per), and the notation 1 if (@)= ($) %W,(yV = * otherwise. I Eq. (2.2) may be written as irrc,,, These commutation

(I+&,). Q,J =i ‘8(ol,&3f,(y8)

(2.3)

relations can be implemented most easily in the representation

542

F. M. H. Viilars ,I Born- Oppenheimer

?heory

in which Q is diagonal: (xIK&

9) =

dx’(xl&,lx’)(x’15;

=$i(l+-S,,)

2

a

9)

(x15; 9)

= ~*&I!% 4),

(2.4)

thus defining a derivative operator k+ on the function (~1s; q). We turn now to the structure of the BO wave function. In the spirit of the BO ansatz, this wave function, in the lowest approximation, must be a product of an “intrinsic” function (51x(q)) and a collective amplitude f(q). Transformed back into the x-representation, this reads

dq ~~CdS; q)@lx(q)M(q) .

(2.5)

As required, the intrinsic amplitude is assumed to depend parametrically on q. This amplitude is the analog of the molecular electronic wave function, and to the q-dependence of f&y(q)> corresponds to the dependence of the electronic wave function on the relative position coordinates of the molecular nuclei. These are in fact the collective coordinates, by means of which molecular rotation and vibration are described. Clearly, the most general trial function may be constructed by means of a sum of terms of type (2.5). Of course, it is exactly the idea behind the BO method that such a sum is in fact not necessary, that the simple product provides a useful approximation. This is well-understood in the molecular case. In the nuclear case, the adequacy of this approximation will be shown to depend on the proper choice of the collective operators, Q. In order to have a useful trial function, we must avoid the explicit introduction of “intrinsic” variables 5. We are, therefore, led to replace the quantity 1 d&x/t; q) (tjx(q)> appearing in (2.5) by a function (x/4(q)), preceded by a projection operator which projects Q onto one of its eigenvalues q; 6( Q - q) = &,a~(Qap -gap). Thus, the trial function (xlw) will have the form

=

dq(xlS(Q- qM4qN_f(q)

(2.6)

with ldq standing for jnas_a dq,,. Clearly, this is equivalent to representing the intrinsic wave function (&dq)) by means of (x]+(q)) in the form

(51x(q))= i dx’(& qlx’)(x’l#(q)).

E M. El. ViNars / Born-Oppenheimer

theory

543

For applications, the function (x(4(q)) will be chosen to be a mean field, independent particle or quasiparticle wave function; its q-dependence will be such that (2.7) The projection

introduces

a weight function into the expression

for the norm

(WV:

with

w(q)= (4MQ - c?)ldJ,)~

(2.9)

3. The mean energy Given a trial function, the next and crucial step is to obtain an expression for the mean energy. Given the structure (2.6) of the trial function, the hamiltonian will appear sandwiched between two projection operators: &8( Q, - qs)N~,6( Qs qs), where s = 1 , . . 6 stands for the independent tensor components 11, 22, 33, 12=21, 13 =31, 32=32. One then has to pull the operator H either to the right or to the left. Here we illustrate the result of pulling to the right. (Pulling to the left will generate the adjoint expression; eventually the average of the two will be formed.) This leads to

The curly bracket may be expanded into multiple commutators, with the result that

In the expression for HI q), the q-derivatives can be transferred onto the amplitudes (xlqb(q))f(q). 0 n returning to the notation q. + qap, one finds the operators kap, defined in eq. (2.4), operating on the above amplitudes:

It remains to be investigated whether this series may in fact be broken off at the double commutator. It will be shown, in an appendix, that this is indeed the case,

E M. H. Villars / Born-Oppenheimer

544

theory

contingent on certain collectivity properties of the operators Q in relation to the amplitude (xl 4 (q)), which will be defined on that occasion. With the expression (3.1), the projection operators in the expression for (?FIHI W) are now adjacent to each other:

ns(Q~-q45)n,s(Q,-4:)=rl:s(q,-qS)s(Qs-q,). r s s Thus, the mean value of H may be written as

(PIHI’Y)= j d4/“(s){(#is(Q-4)His)+(~l~(Q-q)

5 [K iQi&aaI#)

In this expression, the operators ka,, operate on the collective amplitude f(q), but also on the q-dependent intrinsic wave function. This gives rise to terms of type d/cYq(xl+(q)), which are normally omitted in the lowest order molecular BO approximation 3). Here, in the nuclear case, they are important and fulfill two functions: (i) They provide the proper expression for the actual intrinsic hamiltonian, which is H minus some ~ounte~erms which assure that the collective kinetic energy is not counted twice. (ii) They establish the correct formal coupling terms between collective and intrinsic motion. The subsequent approximate elimination of these terms by a suitable choice of the operators Qaa is part of the dynamic determination of these operators. An additional comment is in order here; Expression (3.2) was obtained by pulling H through the projection operators to the right. Pulling to the left provides the adjoint expression, and the average of the two defines a hermitian effective energy operator. In either case, one will have to deal with mean values of the type (#lS(Q-q) Oplr$} and (gblOp S(Q-q)l+), with Op standing for a one-, two- or few-body operator. A general theorem “) allows such expressions to be written in the form

(4lS(Q-q) ~pl~~=~~l~~Q-s~l~~~~l~~+~.~~l~~= ~~q)~ci3~O~+L.~.l4) where L.C. stands for “linked correction terms”. These terms can be obtained to any desired approximation as a series in inverse powers of the dispersion

(dd(Q- d’l44. 4. The principal axes transformation By the choice of the set of collective operators as components of a second rank symmetric tensor, we have in fact singled out a means to describe collective rotations and vibrations. Other types of collective excitations thus remain buried in the intrinsic dynamics. They may be recovered either by RPA techniques, or by extending the

F. M. H. Villars / Born-Oppenheimer

formalism

developed

the center-of-mass will be followed Separation

here so as to include motion

them. So, for instance,

could be included

in this manner.

up here, so as not to clutter

of the collective

terms is obtained

by means

degrees

545

theory

of

None of these options

up the picture.

of freedom,

of the principal

the separation

into rotational

axes transformation

and vibrational of the tensor

Q,,: (4.1)

qup =C e,,(fl)eP,(a)q,. (1

In this expression, the q. are the principal moments, and the e,,(n) are rotation matrices, parametrized by a set n of three Euler angles. They obey the orthogonality relations C eolaepa = a,, (I

,

&&?

c

=

01

6

ab.

(4.2)

Along with (4.1) we must now express the operators karpin terms of the operators conjugate to the qa and 0. A first step to this goal is the observation that the operator (4.3) is an angular momentum, obeying the usual relations [LZmp,Za,] = iZpy. It then follows from (4.1) and (4.2) that the projection of TOP on the principal axes can be expressed

in terms of the qa and the k,,: =%b

In addition,

=

1

enaepb=%p

=

pa

we define an operator

2(

%

-

qb)

1

&&bkp

(4.4)

.

by (4.5)

pa = C emaePakP.

a@ This allows the following

expression

for kap: (4.6)

As a check on these relations,

one may verify that the commutation [&, [-%b

applied

to (4.4) and (4.1) indeed [ikOP,

(4.7a)

qbl’fkb, ) eTc 1 =

-

i&&b

+

isbceyo

lead to the correct

d

relations

,

(4.7b)

expression

= t(&,b + &dpy) .

(4.8)

We must now be reminded that kap operates both on the collective amplitude f(q) and on the parameter q in (x14(q)). The effect of the first operation is readily apparent if we write f( q) as a product (or sum of products) of a Wigner D-function and an oscillator amplitude: f(4)

=;

%4K(~)&&)~

(4.9)

546

E M. H. Wars

/ Born-Oppenheimer

theory

(The definition of the DhK used here is that of Bohr and Mottelson “). To describe the action on I4( q)), it will be useful to first establish the action of the total angular momentum

Jap= C {xirxPij3 - xipPiu I + C siap

(4.10)

I

1

of the nucleus on the trial function (xlY). As in the case of evaluating NIP), the main point is to pull J to the right of the projection operators, This leads to the expression Jna n a( QS- qs) =

dp_ e’CP,(Q,-q,){e-‘~~~Q~~~~e+‘CasQa). I

The curly bracket reduces to J+, + iE: ps[J,p, Qs]; this latter commutator

is a linear function of the Q, determined by the tensorial nature of Q. The p, become derivatives on qs, and an elementary calculation shows that .Jap

I

dq g(Q - q)l#(q)}~(~)

=

s

dq SCQ - q1I-b +=%,%#4qMq),

(4.11)

the operator J&a having been defined in (4.3). The importance of this result lies in the fact that it shows the way to express the total angular momentum as an entirely collective operator. For this, it is sufficient that the intrinsic wave function (xl+(q)) should have the property (XI&3 +z+?~l~(q)) The way to assure this property is to make (xl 4( and spin variables x6 =C xj,e,,(a), a

(4.12)

= 0. q))

dependent only on the position

& =C Sinea,, CI

,

(4.13)

which are scalars under the operation {f+L!?}. Residually, (d/4(q)) will still depend on the principal moments qa, which describe the shape of the nucleus. With this, it follows that the total angular momentum of the nucleus is expressed as dqS(Q-q)l~(q,))CgK(qa)~~pD~K(LR).

(4.14)

Having established this basic feature of the trial function 1P), we may now return to the expression (3.2) for the mean energy. Several steps must be taken to develop this expression further. First, since the amplitude (x19) depends only on the variables X$ and ~6 defined in (4.13), one may write all operators occurring in the various matrix elements (4si * * * [#J) in (3.2) in the variables (4,13), which differ from the original xi, and s;, by an external rotation, e,,(n). Now, the hamiltonian is a scalar and hence form

F. M. H. Villars / Born-Oppenheimer

invariant:

H(xi,,

pia) = H(x6,

pi,); the product

547

theory

of projection

operators

is invariant,

too: n S(QaP -%7p)=ath asp

~(Qbb-%lJ

=rI s(Qh,-%XI& a

the QL, are the rotated

In this expression,

collective

8(Qhb).

(4.15)

operators

Qhb= C Qapeuaepb

(4.16)

aP

and may be expressed in the variables (4.13). This procedure the rotated operators kab, which by eq. (4.5) are given by

introduces

into (3.2)

’ qb) %b +&bPa .

kob=(l-hb)

(4.17)

2(%-

As mentioned before, we must consider the action of these operators on the collective variables qa and 0 both in f(q) and in (x/4(q)): Acting on f(q), Tab is a rotation operator on the Wigner D-function DaK(0) and ~)a a momentum operator +/$I, on g(q,). As regards their action on (x14(q)), consider with a function of the variables (4.13), consider

= This demonstrates angular

&I

Tab. Since we deal relation [z&,, xk]:

(4.18)

*

sob is equal to the negative

of the

+

(4.19)

operator Jab = c I

which represents

-[Jab,

that acting on 4, the operator

momentum

that Jab satisfies

first the operator the commutation

the total angular the “normal”

{&db

-

X;bda

momentum

commutation

%[ab,)

as seen in the rotated relations

frame.

Notice

[ Jz3, J3,] = [J1, J2] = iJ3, etc.

We are left with the operators pC acting on (xJc$(q))=(xl,l4(q,)). In the case where this amplitude represents an independent particle or quasiparticle wave function, the operation -pc = +ia/aq, is equivalent to the action of a one-particle operator (xL]GClx:!a) acting on the wave function

pcWl44q)) = -$%+#4d) C

This operator G, represents an approximate indeed, by eq. (2.7) and (4.20) one has QL;

= -WlGMq)) canonical

~(~lQ6.l~)=~(~.)=(dl[~c,,Q6,116)=s,,. C

C

.

conjugate

(4.20) to the operator

(4.21)

548

F. M. H. Wlars / Born-Oppenheimer

theory

There remains, as a final step in the transition from the original variables qnp to dq(,,, in terms of the new variables. One finds qa and R, to express the volume element d7 =naSP

d7=dfiI-IdqaA(q),

(4.22)

a

where da is the usual angular element d@ sin B d# d+, and A(q) is (4.23)

A(q)=(q,-q2)(q2-q2)(q3-q2).

It is easily verified that the operator k,@, given by eq. (4.5) in terms of L&, and pc, is hermitian with respect to the volume element (4.22).

5. The collective energy

In this section, we rewrite the expression for the mean energy, making use of the new representation for the collective operators established in sect. 4, eqs. (4.1) and (4.5). We will then address the question of the determination of the operators Qab. (Primes on Qbb will be dropped from here on.) To simplify the presentation, and improve the transparency on the expressions tht will ultimately be obtained, we will at the very outset implement two approximations: (i) All linked correction terms arising from the projection process will be neglected. (ii) Expressions representing the collective mass operators will be replaced by their expectation values over the intrinsic state. Corrections to these approximations may in principle be obtained, but will not be considered here. Approximation (i) implies that all mean values of type (#lS(Q-q)Opl+) will be written as w(q)(#lOpl#). Approximation (ii) will be implemented by the substitutions

(5.1) NW %I, iYbl+ (d[[ff, iY,l, %lM>

=F, a

[[K iYJ, Qd + (d4KK iY,l, iQd4) = 0.

(5.3)

That the mean values of these operators are diagonal in a, b will be shown shortly. With this, the mean energy may now be written as the sum of three terms, representing internal (including collective potential) energy, collective kinetic energy, and “coupling” energy

F. M. H. viilars / Born-Oppenheimer theory

549

individually given as follows; Gnt=

(W-C

I

n

~QaalG)+C(G2,)/~Mz

([H,

(5.4)

{C([H, iQcwl-2)

dTw(q).Pq

E CoupI= j

+I

(

[H, iY,l-$-

d-(q) 0

> I

=%f(4) *

a

(5.6)

This is the place to present the general idea of how to proceed with these expressions. This will be done somewhat sketchily; more details will be presented later in connection with specific cases. (Unless stated otherwise, we also assume the case of an even nucleus.) Basically, a set of nested variational principles must now be applied to this energy expression, both for the determination of the “intrinsic” state I4), the collective amplitude f(q), as well as the operators Qab. About these, little has been said so far, except for stating their tensorial nature, giving the relation [.&I obcl

=

i E

abdQde

+

i E

acdQbd

(5.7)

and eqs. (2.7), which more precisely read:

(#(q~lQ=~l~(q))= @a,.

(5.8)

In order to enforce this latter property, assume that (p is determined by a constrained variational principle (5.9) Let qz be the values of qa obtained as h, + 0. Then small deviations na = will be generated by a trial function l#GO+ 7)) = (I- i t: %@a+ * - ~lld(s”l)

q. - qs

(5.10)

and to first order in n, the variational principle (5.9) will reduce to -& (&(4’)1[&

Gzl+; kbQb&#J(40)) = 0>

(5.11)

550

F. M. H. Villars/ Born-Oppenheimer theory

where kab = (&,/~?n~)l,+. We shall retain eq. (5.11) as one of the relations that will determine, partially at least, the three operator pairs Q_, G,. In conjunction with eq. (5.4), we obtain a vibrational potential energy {#(qO+ ~)~~~~(q~+ ~))-(~(~*)~~~~(qO))

= $ ; ~~*~~~~+* ’ -

(5.12)

with kab given by the expression kab = {~(q~)~[[~

Gl, Gl/#(q”)).

(5.13)

Eq. (5.12) represents the potential energy of collective oscillation on the assumption that the additional terms in the expression (5.4) for Ei,t can be evaluated at the value q” of q. This appears plausible; more about the nature of these terms will be said below. Consider now the expression (5.5), called here the coupling energy. A term like t#l[N, 6Ll- GJMM~ is zero if 4 is even under time reversal; however, its variation S/S4 is not necessarily zero. This finding indicates that a more general “ansatz” for the trial function 1P), in which the simple product [4)f(q) used in expression (2.6) for (Yr) is replaced by the sum 14)f(q)+C c,,l&)fn(q), (the 4, representing non-collective excitations), would lead to non-vanishing terms in the coupling energy, of type C cn(fbnl[Ht n

KLI -s

(I

141)1 d~~(q)~(~~).

(5.14)

Notice that such an expression indeed describes a transfer of energy between collective and intrinsic modes of excitation. So it is seen that the coupling energy in eq. (5.5) appears to be zero only by virtue of the simple ansatz (2.6) for the trial function, and not for any dynamic reason. To achieve this latter criterion, the term (5.14) would have to be zero. So let us postulate that this be the case; the residual freedom in choosng the operators Qaa and G, should make this possible. The condition can be stated as the variational principle $H+KK

iQaal-Gpi/MJ#~I=O.

(5.15)

Together with eq. (4.21), this provides an expression for the collective inertia M,: +#[I%

i&al,

@bbli#)=$-

cl

(5.16)

Let us observe also that eqs. (5.11) and (5.15) represent a set of coupled RPA-type equations which determine the excitation energies of the oscillatory modes “). A similar argument for minimizing the coupling energy and justifying the simple form (2.6) of the trial function requires that the coefficient of Zaf in Eq. (5.6) should

F. M. H. Villars / 3orn-O~~enhe~~er

551

theory

be zero, and have zero variation. This gives (5.17) To appreciate the significance [J,, iY,]. For example Cr,,

iv,,

of this condition,

= K-4, Q231-_ 92-93

consider

the commutators

Qz - Qx 92-93

and

.yIJJ”~QJ_

1J

2

I,1

It

93-91

412

93-41’

thus follows that (+I[J,, iY&b> = 1, and (#l[J1, iY,]l~#~b) = 0. In general then

~#4C-C, ~Ytll~~= &, -

(5.18)

Using this result together with (5.17) shows that the moments of inertia 0, are given by the expression $= (1

(44Rti Xl,

iU,lld)

.

(5.19)

In conjunction with (5.17), this just represents the well-known self-consistent cranking formula 7). With this we also have now at hand some results that suggest a second look at the expression (5.4) for the internal energy. Using eq. (5.15), we see that the term -En ([H, iY,]J,> is seen to be equal to -C, {~~)/~~. The total internal energy is therefore given by (5.20) Recaliing the normalization of %@: (?P/ P) = f dr w(q)f*f= 1, it is seen that the internal energy is equai to (#J(H/~#J)minus two terms which represent the spurious collective kinetic energy contained in the unprojected mean value of H As a final remark in this rather sketchy section, we observe that the eqs. (5.11) and (5.13) determining the operator pairs QLm, G, leave them invariant under a resealing transformation

Therefore, the quantities qa = interpretations as “deformations”.

so far have no immediate geometric They can, however, be given such a meaning by

(#I/ QJ4)

F. M. EL Villars/ B#r~-Oppemhei~er theory

552

fixing the as yet undetermined scale by means of a “measuring operator” which a convenient choice would be D~=CX:~*.

D,,,

for

(5.21)

I

One would then define the qa be means of the relations; qa = (#jD&$). This normalizes the operators G, by the condition ($I[ iG,, D_]]d> = 1. Notice that in the past ‘), the operators De, = C &XL have been used to ptay the role of the operators C&b,but with this choice, eq. (5.17) cannot be satisfied; if eq. (5.19) is nevertheless used to determine the moments of inertia, one finds the well-known “irrotational flow” values 9), @ =* a

Glb--d2

(abc = 123 or

cycl.)

.

5.22)

(qb+qc)

6. Special cases: The parameters q, p, y ; symmetries In this section, we describe the additional steps that need to be taken in order to construct the collective energy spectrum. In doing this, we must distinguish several quite distinct situations. Let the point of departure be the observation that at the equilibrium positions q:, the principal axes will be well-defined only in the tri-axial case, where the qz are all differing from each other by amounts exceeding the RMS zero-point amplitudes m. If this is not the case, then the orientation of the axes is determined by the “instantaneous” values of the excursions qa, and rotational motion is not decoupled from the collective oscillations. We will therefore have to look into three distinct cases, which require a somewhat different treatment: (i) Spherical equilibrium shape; all qz are equal. (ii) Axially symmetric equilibrium shape: qy = q; f qt. (iii) Triaxial case; all q”, distinct from each other.

6.1. SPHERICAL

EQUILIBRIUM

SHAPE:

PARAMETERS

4, p, y

For this particular case, the representation of the collective amplitudes in terms of principal moments qO and Euler angles 0 may seem perverse. A more natural change of representation would be to go from the original qap to spherical tensors qzm. This step will be developed in a separate paper. Here, it will in fact prove useful and insightful to work things out in the “unnatural” representation. We first observe that in the spherical case, where qa = q”+ r],, the jacobian A(q) in the volume element dr = d0 n dqJ (q) takes the form A(q) = (s, - q~)~q~- q&q2-- qJ = (rli - ~~)(~~ - n&n2 - 7j3) .

(6.1)

This indicates that the variables qa are not suitable for this case. We need variables, in which the jacobian is factorable. This is achieved by a transformation from the

F. M. H. Viilars / Born-Oppenheimer q.

to a set of new variables

q, p,

qa=q+&cos

y,

defined yO;

theory

by yO= y-$-a.

has a jacobian d(ql, q2, q3)/d(q, p, y) = p. Also, sin 3 y, so that the new volume element (171-~72)(~71-~73)(rlz-77~)=J~P3 This

taken

553

transformation

(6.2) one has may be

as

(6.3) It was already pointed out in sect. 2 that the unsymmetric procedure of pulling the hamiltonian H to the right through the projection operators n S( QS - qs) would generate an expression for the energy in terms of operators which are not quite hermitian, a situation to be corrected by averaging with the adjoint expression. Applied to the vibrational kinetic energy, this gives (with p,frepresenting - iaf/dq=):

MO is the common value of the masses M,, eq. (5.16), as 4(qa) + 4(q”). One easily verifies that the laplacian derivative on A is zero. At this point, pC may now be written as an operator on the new variables q, p, y: PC=-i

12+&os I 3 aq

dy‘ap

Z(7);).

(6.5)

Using the relations C cos’ yC = 1 sin2 yC = $ and 1 cos yC = C sin yC = C sin 2 yc = 0, one finds for T$b the final expression

Tvib=

&0

sin3yi{ilzl’+ do4 dP dyP41

1$1*++1-$‘}.

(6.6)

This is in full agreement with the expression originally obtained by Bohr lo), with the only significant difference that the collective mass MO is not obtained from an assumed collective flow pattern, but rather gives by eq. (5.16), with 4 approximated by 4(q”). The potential energy of oscillation is obtained from eqs. (5.12) and (6.2). Because of the spherical symmetry of 4(q”), the matrix kob must have the simple form: kob = k&, + K( 1 - yab). With 7, = (q - q”) +&I cos yo, we find a potential energy (6.7)

F. M. H. Villars / Born-Oppenheimer

554 6.2. ROTATIONAL

ENERGY

IN THE

SPHERICAL

theory

LIMIT

The rotational energy is based on eqs. (5.17), (18), (19). In the spherical limit we must carefully identify the approximations used, in particular, one must distinguish between

matrix

elements

taken

. .}+(q’).

4(q”+rl)={1-iCcsG+~ >O= 40) for

4(q”N.

7% e

to +(q’),

with respect

important

In what follows,

properties

Q*3/(%-!I3)=

check

the contents

Q*3/(712-%),

L= (#dHH,iQ2A 0,

=

we will use the shorthand:

Ja~r$o)=O(alla).

of eq. (5.17).

and writing

to 4(q)

of 4. are

(40lC?~A4~) = 4’ (all a) , Let us now

or with respect

Reverting

the moment

iQ2Jl4)

to the notation

of inertia

Y1 =

as

1

=

b12-d2

(6.8)

M23( 92 - VA2

(6.9)

we may write eq. (5.17) in the form (for a = 1) (6.10) The second term in this equation appears to be singular not so, thanks to the second eq. (6.8), which leads to (4(q0+

in ( n2 - n3). In fact, this is

~Ml#4q”+7)) = i C ~(cP~l[4,Gll40). a

What is the operator [Jr, G,]? If G, were the diagonal element r,, of a symmetric tensor Tab, then we would have [J, , G,] = 0, [J, , G,] = 2iT,, , and [Jr, G3] = -2ir,, . Now there is no such tensor r,, in general; however, in operating on a spherical state such as c$~, one has [Jr, GMo)=O,

[

-4,C

(I

G,

I

I+o)=O,

since both G, and 1 G,, on 4. generate a state with 5, = 0. Thus, we have a relation of the form: -i Caqa[J1, G,]]~,) = 2(n2- n3)J’23140). This shows that in eq. (6.9), the lowest order term (in powers

of 7) are of zeroth

order,

and read (6.11)

The consistency

of this equation

Wr2,,

is certified

Q231)o = WI,

G21,

by observing

that

Q23100

= (IN.4,Q2J, Gll)o= (ILi(Q33 - Q2AG211)o= Hence,

the mass M23 is given by

&=(llIK

iQd

Q2Jl)0.

1 .

F. M. H. Viilars / Born-Oppenheimer

theory

555

It is seen to be equal to twice the mass MO in the kinetic energy of oscillation, as follows: Since ei”JIIqb,)= I&), and e-“‘1QZ3 e+iEJl = QZ3cos 2~ + ( QS3- Q2J sin (2~)/2, one has, choosing E = $r:

&= NH; =

iQ231,~Q*Jl)o

$(I[W,i(Q33 - Q**)l, i(Q33- Q22)ll~o

=HCw¶iQ**l, ~Q22ll~o = &.

0

So the moment of inertia 0, is seen to be equal to 2Mo( n2 - n2)*. Using the general relations nll - nb = q sin (y-&c), (abc) = cycl. (123), one has finally the result 0, = 4Mop2 sin2 ya.

(6.12)

Thus, the rotational kinetic energy in this case of a spherical, J = 0 equilibrium state &,, is given by 1 dr c (%f)*(Zf) 2M0 s C 4P2sinz yC .

Em = -

(6.13)

There is a set of well-known symmetry conditions lo) on the collective wave function f(q) arising from the requirement that the amplitude (xllF) be invariant under a relabelling of the principal axes as positive I-, 2- and 3-axes. For any particular labelling, we have to consider the amplitude

WM>_I-(9a) = (44); g,(q,P, Yma4(.n)

(6.14)

and its change under the 24 symmetry operations that accomplish the relabelling. On the assumption made here that (x’/&,) is invariant, the results of Bohr’s analysis apply here: The expression (6.14) must contain the pair K, -K, and invariance mandates that K must be even; if it is zero, then 1 must be even, too; if it is non-zero (~-vibration), then I = K, K + 1, K +2,. . . . Fu~he~ore, g must be periodic in y, with period $r; it must have the reflection symmetry g(7)

= e iVK’2g(+ y)

(6.15)

and therefore only the sector 0 d y s 2rr/6 needs to be considered. 6.3. THE

CASE OF AXIAL

SYMMETRY

This describes a situation where the unconstrained variational principle (eq. (5.9) with A + 0) allows solutions #o(q) = &(qz), with 4: = q!j f: 4:. We will at once assume strong deformation, for which /qz - qT/ is large compared to the zero point amplitudes of shape oscillations. The unconstrained, unprojected intrinsic wave function (x’!#~) will then be an eigenfunction of &, with eigenvalue 0: J,~c$~)= f21$,).

556

F. M. H. Wars

/ Born-Oppenheimer

theory

Consider first the case 0 = 0, along with assumed invariance under reflection of the three-axis: em’“4I&) = I&). This case is quite similar to the spherical case, with the following exceptions: (i) The moments of inertia 0, = @ are determined by eqs. (5.17) and (5.18), giving the self-consistent cranking value. By contrast, O3 is determined as in the spherical case. (ii) The potential energy oFosc~~latio~ now depends on q, /3 and ySThese variables, however, are not necessarily “normal” coordinates anymore. In order to keep the expression for the kinetic energy of vibration simple, one will modify the definition (6.2) so as to have (for the sector with y*=O):

-/?“+vQ sin y)}m

rl,=(4,-4~)=c(4-4°)-~{P

cos y

r/2=(q2-4~)=[(4-q0)-J~CP

cos y - p”-Jsp

sin

,

r}]AiZFI

,

r13=(q3-4~)=[(q-qa)-t-{8cos r-POhbwJz.

(6.16)

This assures that the kinetic energy of vibration retains the form (6.6) with MO the average mass. The jacobian A(g), eq. (6.2) can be approximated by A(q)=Iql-~72)(q~-q~)‘=P

(6.17)

sin ~(q:-&)‘~

The leading terms in the expression for the potential energy U = $xab k~~~~~~then take the form (6.18) but are supplemented by coupling terms, determined by the constants knb, which in the case of axial symmetry depend on four independent parameters. Consider now the case of a I~OYI-zero intrinsic angular momentum component LJ along the three-axis: ..7,]#>= A?]+). Unlike the R =Q case, the wave function (x’]Qt) is now no longer invariant even under the restricted symmetry operations RI = e’“iez and Rz = eifF”‘z3. In applying these operations to the amplitude ‘M, observe that L&(x’]+) = -(x’]&]+) (as explained in sect. 4; see Wf&g(q, B, Y)D eq. (4.12)), that under RZt g(q, 8, r) goes into g(q+ j3, -y) and DLK acquires a factor ei(*‘2)K, so that the original amplitude is transformed into Mx’l4Mq,

P, Y)@)MK

= ei(rr’2)(K-n)(x’19)g(q, P, -Y)&~K

.

(6.19)

This establishes the symmetry of the oscillation amplitude g under the operation y”+--y: g(7)

= e- W2)W-R)g(+y)

_

(6.20)

A repeated application Rz just gives a factor (-l)i”-“‘, showing invariance to require that K -D be even. The operation R, leaves fg3 j3, y) unchanged, produces a factor (--I) rfK from DhK; its action on (x’f$~), can be used to define the amplitude (x’l4t)_n); in doing

F. M. H. Viliars / Born-Oppenheimer

557

theory

this we must distinguish between even- and odd-nuclei. For A even, we have integer values of K, 0, and we can define (6.21)

R,(x’l+,n)= (x’le-.i”J’l$fl> = (-l)“(x’]+_n).

An even-A trial function that is invariant under repeated operations of RI and R, thus will contain the combination {(~‘~~~}~~~ +(-I)‘(x’l~-n)~~,-,)g(g, In the odd-A case, onIy R:l#)= of (~‘l+_.~) replacing (6.21) is R,(x’/#~)

I$), (-1)”

+(-I)‘+“*

(6.22)

IS . ambiguous, and the proper definition

= (x’le-i?TJZl&) = i e-ivR(x’l#_n)

and the proper invariant combination {(X’l&)DLk

P, r) ’

(6.23)

is now (x’l~-*)~L,-f&(~,

P, Y) *

(6.24)

These considerations show that the initially stated goal of using only the simplest form of the BO trial function, a single product, cannot in all cases be achieved. In the case 0 # 0, a conjugate pair of trial functions must be introduced. Clearly the mean energy is degenerate with respect to this pair. The real issue however is how to deal with interference terms within the context of the projection techniques introduced here. Such terms, that is, matrix elements of type (& [Opld-,) will occur in the expression for the mean energy, and for matrix eIements of collective transition operators, provided 10) is not too large. Their role in the case Ial= 4 is well-known ‘l). Another important point that must be considered here concerns the expression for the rotational energy in the case of axial symmetry with IGl# 0. As we know, the inverse moment of inertia l/O3 is singular in this case; if 101 f 0, there are three terms in the expression for the mean energy that are singular as l/O,, even for #n(qo). They can be read off from the expressions (X4), (5.5) and (5.6) for the mean energy, which contains the following terms in l/O,:

This can be written as (K - Lt)“lf(q)l”/2@ 3, supplemented by terms which vanish for q = q*. So in this case the rotational energy is given by (6.25) The latter term: (K - 0)‘/2& enters as a potential energy term into the equations for the ~-vibrations I’). So for f2 f 0, the collective wave function depends on the parameter K - f2, and it is consistent with this that for n # 0, the symmetry condition (6.20) should replace (6.15).

558

F. M. H. Wars / Born-Oppenheimer

The moment

of inertia

for 0, in the spherical

0, itself is obtained

theory

in the same manner

that was sketched

case. In place of eq. (6.10), we have now the relation

(with

0, = (171- 772)2/M2):

Notice again that this expression is non-singular in (ni - n2). With this, we conclude this rather sketchy presentation of some special cases. They should be sufficient to illustrate the main features of the nuclear BornOppenheimer method. Some of the issues not touched upon so far will be raised in the next section; this will also be the occasion to compare this approach with more conventional methods for dealing with nuclear rotation-vibration.

7. Conclusions

and assessment

In this last section, several important questions and issues will be taken up. First, it will be useful to identify the characteristic features of the method presented here, and to make comparisons with other approaches. Then one will want to confront the question of the general “usefulness” of this method. This includes the question of the ease with which the formalism could be implemented to obtain actual quantitative results; however, there is more than this aspect to usefulness. An attractive - in the author’s mind - feature of the approach presented here is that we do not deal with a model of the system, but with the system itself, described by the most suitable hamiltonian anyone may choose. What occurs in the implementation of the BO concept is that various approximations are made. These are all well-recognizable, and can be tested in principle to establish the importance of correction

terms that might arise. Such approximations

are: The BO trial function

itself; the truncation of the multiple commutator series which is the basis for sorting out collective and intrinsic terms in the hamiltonian; then the approximate evaluation of matrix elements containing projection operators by neglecting the linked correction terms, and finally,

of course,

the use of mean field wave functions

to describe

the intrinsic dynamics, and the associated incomplete characterization of the collective operators. What does emerge, however, is a certain formal structure, at whatever level of approximation one may choose to consider. It would seem that this formal structure should in fact be the best point of departure for the appreciation of various models of collective dynamics, and to provide insight into what features of the “real” system are properly taken care of in a model. An interesting feature of the present formalism is its completely quantum mechanical aspect. This, as well as the use of projection operators are obviously not exclusive to the Born-Oppenheimer approach as presented here. All methods that construct quantized rotational states by angular momentum projection 13,14) share these two features. In fact, the BO approach provides an alternative to the

F. M, H. Villars / Barb-Oppenheimer

theory

559

usual angular momentum projection, whose analog in the case of linear momentum projection is well-known. This analogy is, in fact, worth a bit of elaboration. In the momentum problem, eigenstates pK or the total linear momentum operator P can be obtained by momentum projection from a spatially localized state 4: W, = fi(P--IX)&; if it is assumed that (&,~P~&J=O, then this procedure is in fact the exact analog of the original Peierls-Yoccoz 13) method of angular momentum projection. There are serious objections to this procedure I’) and the proper version of it has the form Fvr, = 6(P-

I() eiKR&

(7.1)

where R is the CM position operator R = C xi/N, and it is assumed again that &, has zero mean momentum: (&lPl&J = 0. Expression (7.1) is the result obtained by a variational optimization of (b after momentum projection 14). It may be more explicitly written in the form (XilFK)=(Xi(Ij(P-K)

=e

eiKR14J

iKR(xjJ13(P)l~o> = eiKR dr(x, - r/#+J,

(7.2)

a form which displays the factorization into a plane wave in R, and a translationally invariant “intrinsic” amplitude. There is an alternative prescription for the latter: instead of j dr(xi - rl$~>,one may choose the expression (x1 -RI+). The corresponding amplitude (XilVK) then has the form (xjI~JIK)=eiKR(Xi-RI&))=

I

dr(x,lS(R-r)l+(r))eiK’,

(7.3)

where the notation (xil~(r))=(xi--r~~) h as b een used. This expression (7.3) is exactly of the form (2.4) of the BO trial function. Thus, the BO formalism in a way exploits the possibility of an *‘angular momentum analog” of the eigenfunction (7.3) of the linear momentum, whereas the more standard procedure for obtaining angular momentum states is based on the analog of expression (7.2). Interestingly, these two expressions (properly normalized), do not at all give the same values for matrix elements of operators that cause a large momentum transfer IK’- Kl. There exists the possibility of a corresponding discrepancy between the two forms for the case of large angular momentum transfers, but this has not been explored yet. It is also worth observing that in standard angular momentum projection (with variation after projection), no collective operators are actually identified. The decoupling of collective and intrinsic motion is implemented in an indirect manner only, by finding a mean field trial function which will minimize the energy for a given value of the angular momentum. In the present approach, this decoupling is effected in a much more direct and explicit way, and in this sense the method is closer in spirit to the approach of Marumori ‘$) and coworkers than to the angular momentum projection techniques.

560

F. M. H. Villars / Born-Oppenheimer

theory

A difficulty seems to arise when it comes to a comparison with methods which generate a semiclassical description of collective rotation by means of a rotating (cranked)

mean

trial function

field wave function.

in the rotating

In this approach,

system is obtained

the stationary

from the variational

mean

field

principle

“). (7.4)

The effective hamiltonian (“routhian”) is no longer time-reversal invariant. is also a feature of the effective hamiltonians that arise in the usual “variation angular momentum projection” procedure 18). By contrast, the effective hamiltonian responsible for the variational

in the present determination

This after

approach of C$ is

contained in eq. (5.4) or (5.20) and is seen to be invariant under time reversal. Eq. (5.9) expresses the simplest possible variational principle for 4. The moments of inertia that we obtain in this case corresponds to those obtained in the cranking approach in the limit o + 0. However, eq. (5.9) is a very approximate variational principle, based on neglecting all the terms besides (H) in the expressions (5.4) or (5.20) for the intrinsic energy. These neglected terms become important for large values of the angular momentum quantum number I. Indeed, it will be shown in a subsequent paper, that eq. (5.9) is, in fact, suitable only for small values of I; starting from an axially symmetric case for even nuclei, one finds that for larger values of Z, a “symmetry breaking” solution, with a non-vanishing value of (.Z) in a direction perpendicular to the original symmetry axis will have a lower energy than the time reversal invariant one. It remains to be shown how the simple BO ansatz for the trial function, eq. (2.5), is to be generalized to describe the “backbending” region, where two distinct intrinsic structures have to be described simultaneously. The calculation of matrix elements of tensor operators collective

excitation

is straightforward.

What

needs

between

states of different

to be elaborated

yet is the

construction of joint eigenfunctions of linear and angular momentum, within the framework of the Born-Oppenheimer approach. All experience gained so far with this method appears to indicate that it represents a useable dealing

and therefore,

with nuclear

in fact, useful

rotation-vibration

alternative

to the more

familiar

ways of

dynamics.

Appendix In this appendix we examine the conditions under which the truncation of the multiple commutator series (3.1) might be valid. To keep the notation simple, we will consider the case of a single collective operator Q. The procedure will be to carry the expansion just one step further, so as to include triple commutators, and to examine the contribution of the additional terms that will show up in eqs. (5.4), (5.5) and (5.6) for internal-, collectiveand coupling-energy. The following

F. M. H. Villars / horn-Oppenheimer

theory

561

expressions will be found:

Eint=

J

d7w(q)F{W)-([K

iQIG)+(G2)/2M-(W, iQ1G3)/6Wf(q), (A-1) (A.21 (A-3)

A term in p3f(q) has not been written down. It will be useful to start with the coupling term (A.3). In sect. 5, this coupling was minimized by requiring that the one-particle part of the operator [H, iQ] be equal to the operator G divided by the collective mass M. We may retain this relation here, since some relation between the operators Q and G must be set up. This allows the definition of a new one-particle operator B as the commutator B = [G, iQ] = M[[H,

iQ& , iQ] = C b,aja,,

TS

(A.41

the indices P, s, . . . referring to single-panicle orbitals, and the subscript 1 to the one-particle part. AI1 triple commutators in (A.l) to (A.3) are expressed by means of the operator B. Notice that the condition corresponding to (4.21), ([G, iQ]) = 1, can be expressed here by means of the matrices b,s as follows: ;b,,,=-Cb,,=l

(A.51

P

and p referring to hole- and particle-states, respectively. With these preliminaries, we may now look at the additional operator [B, iQ]G’ in the coupling energy (A.3). Just as the other terms in (A.31, this operator will have non-vanishing matrix elements between the intrinsic ground state (f, and associated particle-hole excited states Sbph.A prominent such matrix element is h

i(b,, - b~~)q~~tG’) and this must be compared now a necessary condition average sense - the term implemented by requiring question, summed over all This gives

(A.@

with the matrix element g,, of the operator G. Clearly for the validity of truncation will be that - in some (A.6) be small compared to gph. This notion can be that the square magnitudes of the matrix elements in particle-hole excitations, satisfy the required inequality.

or

c Ib,,-bh,12/qph12(G2)~l.

ph

(A.71

562

F. M. H. Villurs / Born-Oppenheimer

theory

Now, from the commutation relation ([G, Q]) = 1, we expect that the product CphIqph12(G2) = ((0 -(Q))*)(G’) = (AQ2)(G2) will be order unity. Hence for (A.7) to hold, it is required that lb,, - &I2 be small compared to 1 for all pairs ph for which lq,,,,[’ is significantly non-zero. At the same time, eq. (A.5) needs to be satisfied. What this implies is best shown by a schematic illustration, as follows. We define effective numbers I$ and Nh of particle- and hole-states, respectively, by the definitions

Consider now a case where for a set (S) of Np particle states and A$ hole states, we have approximately the relations

I

@Q2)

Iqph12=%Nh Using the crude approximation

0

if h-4A) E 6) otherwise

64.8)

where the collective mass M is given by (A.9)

and correspondingly,

the matrix elements bpp and bhh are given by bhh=x,

’

bpp=

1 -N,,

one finds that condition (A.7), which is necessary for the validity of truncation of the commutator series, reduces to (A.1 1) This condition on the operator Q expresses its collectivity in terms of the effective number of hole states and particle states that are engaged to express its properties. Examples of operators which do or do not satisfy this condition are well-known from solutions of the RPA equations with stylized nuclear hamiltonians. References 1) F.M.H. Villars, Nucl. Phys. A420 (1984) 61 2) M. Born and J.R. Oppenheimer, Ann. of Phys. 84 (1927) 457; M. Kotani, K. Ohno and K. Kamaya, Handbuch der Physik, Vol. 37/2, S. Fhigge, ed. (Springer, Berlin, 1961) 3) J.C. Slater, Quantum theory of molecules and solids, vol. 1 (McGraw-Hill, New York, 1963); HA. Bethe and R. Jackiw, Intermediate quantum mechanics, 2nd edition (Benjamin, Reading, 1968) 4) C. Bloch, Nucl. Phys. 7 (1958) 451 5) A. Bohr and B.R. Mottelson, Nuclear structure, voi. I (Benjamin, Reading, 1969)

F. M. H. Villars / Born-Oppenheimer

theory

563

6) D.J. Rowe, Nuclear collective motion (Ch. 13, 14) (Methuen, 1970) 7) D.J. Thouless, Nucl. Phys. 21 (1960) 225; D.J. Thouless and J.G. Valatin, Nucl. Phys. 31 (1962) 211 8) F.M.H. Villars, Nucl. Phys. 3 (1957) 240; F.M.H. Villars and G. Cooper, Ann. of Phys. 56 (1970) 224 9) A. Bohr, Rotational states of atomic nuclei (Ejnar Munksgaard Forlag, Copenhagen, 1954) 10) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26 No. 14 (1952) 11) A. Bohr and B.R. Mottelson, Matt. Fys. Medd. Dan. Vid. Selsk. 27 No. 16 (1953) 12) J.M. Eisenberg and W. Greiner, Nuclear physics, 2nd edition, vol. 1, “Nuclear Models” (Ch. 6) (North-Holland, Amsterdam, 1975) 13) R.E. Peierls and Y. Yoccoz, Proc. Phys. Sot. (London) A70 (1957) 381 14) P. Ring and P. Schuck, The nuclear many-body problem (Ch. 11) (Springer, Berlin, 1980) 15) D.H. Zeh, Z. Phys. 202 (1967) 38; R.E. Peierls and D.J. Thouless, Nucl. Phys. 38 (1962) 154; R. Rouhaninejad and J. Yoccoz, Nucl. Phys. 78 (1966) 353 16) T. Marumori, F. Sakata, T. Une and Y. Hashimoto, in Time-dependent Hartree-Fock and beyond, ed. K. Goeke and P.G. Reinhard, (Springer, Berlin, 1982), p. 308 17) D.R. lnglis, Phys. Rev. 96 (1954) 1059; 97 (1955) 701; 1. Ragnarsson, S.G. Nilsson and R.K. Sheline, Phys. Reports 45C (1978) 1; M.J.A. de Voigt, J. Dudek and Z. Szymanski, Rev. Mod. Phys. 55 (1983) 949 18) A. Kamlah, Z. Phys. 216 (1968) 52; P. Ring, R. Beck and H.J. Mang, Z. Phys. 231 (1970) 10; F.M.H. Villars and N. Schmeing-Rogerson, Ann. of Phys. 63 (1971) 443; J.O. Corbett, Nucl. Phys. Al93 (1972) 401