single and double projections of generator-coordinate wave functions

1.D.2 I Nuclear Physics A202 (1973) 209--224; (~) North-HollandPublishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

S I N G L E AND D O U B L E P R O J E C T I O N S

O F G E N E R A T O R - C O O R D I N A T E WAVE F U N C T I O N S

CHUN WA WONG t Laboratory for Nuclear Science and Department of Physics,

Massachusetts Institute of Technology, Cambridge, Massachusetts and

Department of Physics tt, University of California, Los Angeles, California 90024 ttt

Received 24 January 1972 (Revised 30 October 1972)

Abstract: By re-examining single- and double-projection methods for translation and rotation we find that both the Peierls-Yoccoz single projection and the Peierls-Thouless double projection of angular momentum may contain similar errors due to the inclusion of the spurious c.m. degrees of freedom. These errors are estimated by using a simple product of deformed Gaussian single-particle wave functions. Large errors appear even in heavy nuclei when the c.m. kinetic energy is included in the Hamiltonian. Other spurious effects studied have a rough A- 1 behavior. The conjugate generator coordinate in the Peierls-Thouless method is found to have no effect on angular-momentum projections. It serves a configuration-mixing role. The configurations so mixed appear to be determined partly by the choice of basis for the single-particle wave functions and partly by the nature of the first projection.

1. Introduction

Nuclear many-body wave functions (w.f.) are usually constructed of single-particle

(s.p.) w.f. generated by a shell-model potential well localized at the origin. We shall

denote these many-body shell-model w.f. by O(x), where x represents the set of s.p.

coordinates {rl}. It is sometimes desirable to construct f rom O(x) a many-body w.f.

in which the center of mass (c.m.) of the system is in a plane-wave state of m o m e n t u m

K. For this purpose Peierls and Yoccoz 1) suggested the linear combina t ion

= f e'"" (1.1 /

of shell-model w.f. ~ ( x - ~ ) for which the shell-model potential is localized at a.

The p a r a m e t e r , is usually called a generator coordinate and # ( x - a ) a generating or

generator-coordinate w.f. u). The linear combina t ion (1.1) is referred to as a single

t Alfred P. Sioan Research Fellow, 1970/72. t, Permanent address.

ttt This work is supported in part through funds provided by the Atomic Energy Commission under contract AT(I 1-1)-3069, in part by the National Science Foundation, and in part by the Alfred P. Sloan Foundation.

2O9

210 C.W. WONG

projection in the sense that momentum eigenstates ofc.m, motion are projected out by integration over the single variable (or set of variables) ~.

Study of the projection (1.1) was partly motivated by the need to project from a spherical asymmetric shell-model w.f. 4~(x) states of good angular momentum J and its projection M along a z-axis in space. Peierls and Yoccoz ~) suggested for nuclear "rotational" states a linear combination of rotated w.f. R(~) #(x), where R(£I) is the rotation operator in the Euler angles ~ = co, ~, 7, weighted by the rotational eigenfunction [~]¢~(£2)] t, a matrix element of R(~2). The projected w.f. is

PY 4~s~ ' ~(x) = PsM 4~(x), (1.2) where

P~M -- 2J8. 2+ if [~@sM(Q)]tR(I2)daI2, (1.3)

K being the projection of the angular momentum along a z-axis attached to the potential well.

Subsequently Peierls and Thouless 3) pointed out that the w.f. (l.1) is not Galilean invariant. They show that as a result the expectation value ev Pv Pv ev of the total Hamiltonian,

A

~¢~(A) = Z ti+½ Z vii, (1.4) i = z ij

does not have the correct dependence h2K2/2AM on K. Here M is the nucleon mass, A the number of nucleons in the system, t~ the kinetic energy operator for nucleon i, and v u the two-nucleon interaction. The possibility was then raised that the rotational analog (1.2) might likewise cause errors in the energies of nuclear rotational states. But since the correct rotational energies are not known a priori, the authors felt it advisable to have a method which in treating translation gives the right kinetic energy of c.m. motion.

For this purpose they suggested 3) another linear combination

~.P~X(x) = f d3Q F(Q)ei(~-Q)" x~P~Y(x), (1.5)

where A

X = (l/A) Z r , , (1.6) i = 1

and F(Q) is any function of Q. The w.f. (1.5) differs from that in (1.1) in the appearance of a second generator coordinate Q, which can be considered the momentum conjugate to the first generator coordinate at. Because of the appearance of a "conjugate pair" of generator coordinates the Peierls-Thouless projection (1.5) is called a double projection. The authors then give the analogous double orojection for rotation.

GENERATOR-COORDINATE WAVE FUNCTIONS 211

Peierls and Thouless a) further pointed out that the freedom in choosing the function F(Q) in the w.f. (1.5) can be used to improve the dynamical content of the w.f. This was done in the context of a Rayleigh-Ritz variational principle in terms of which the dynamical problem for these generator-coordinate w.f. had been formu- lated 2). The result should be similar to configuration mixing in the shell model.

Other variants of these projection methods have since been proposed. For example, Rouhaninejad and Yoccoz 4) proposed a Galilean-invariant single projection. An alternative position had also been taken that the crucial problem is simply the separation between c.m. and internal degrees of freedom 5).

Nevertheless the role of the second projection deserves further analysis. A study of the similarities and differences between translation and rotation could clarify the translation-rotation analogy. The configuration-mixing function of the second generator coordinate could also be visualized more readily when expressed in the familiar shell-model language. In this paper we attempt to analyze the effect of both the first and the second projection of generator-coordinate w.f. along these lines.

In sect. 2 the different w.f. for the translational case are introduced and compared. The importance of treating the c.m. part of the many-body Hamiltonian separately from the internal part is emphasized. An example is given, in sect. 3, to illustrate the role of the second generator coordinate in separating the different internal w.f. contained in the original Slater determinant.

Effects arising from the inclusion of the spurious c.m. degrees of freedom in angular-momentum projections are pointed out in sect. 4. Spurious components in the projected w.f. are estimated in sect. 5, using a product of deformed Gaussian s.p. wave functions. They are found to have roughly an A-1 dependence. The resulting errors in the projected energies are also found, in sect. 6, to be of the order A-1, provided that only the internal Hamiltonian is used. If the c.m. kinetic energy is also included, the errors become large even for heavy nuclei.

In sect. 7 the configuration-mixing role of the Peierls-Thouless second generator coordinate for rotation - the angular velocity to - is illustrated. An argument is given, in sect. 8, for the belief that the second or conjugate generator coordinate involves a degree of freedom different from that of the first generator coordinate.

Sect. 9 contains brief concluding remarks.

2. Single and double projections: translation

We first study the simple case of translation. This serves partly to present our point of view in a transparent way and partly to introduce our notation.

Let a given Slater determinantal w.f. be written in the form

N (~)D(x) = Z ~e'ra'(x)~ilnt( x - X), (2.1)

i=1

where X is the c.m. coordinate and x - X the set of internal coordinates. The Fourier

212 C.W. WONG

transform N

• (K, x - X ) y o.m. i., = ~, (K)~ ( x - X ) (2.2) i = 1

can be used in two different ways in many-body problems. Firstly, it is an A-body w.f. in a mixed representation which is no longer Slater determinantal. Matrix elements of operators such as the Hami l ton ian~(A) of eq. (1.4) cannot be calculated too easily unless we first transform back to ~o(x). This can be done conveniently with the help of the generator coordinate at which is the mean value of X:

• (K, x - X ) = fei"'¢'-x) i,(x-at)d3aL (2.3)

i.e., the Peierls-Yoccoz projection (1.1) of the Slater determinant

cl,~(x - at; K ) = e - '~ " xepv(x - at). (2.4)

Secondly the w.f. (2.2) is also a coordinate w.f. for just the 3 A - 3 internal degrees of freedom. To emphasize this we write

N

c ,,-,,q~i,,, X" (2.5) • ~ ( x - X ) = ~., ~ t ~ ) , ~ x - ), i = 1

where c , (Q) = ~O~.'m'(Q). (2.6)

If N = 1, different values of the parameter Q give the same internal w.f. Therefore Q is redundant. I f there are 1 < n =< N linearly independent -i~int, then a suitable choice of n values of Q will span the available internal w.f. contained in qh).

Operator matrix elements for the w.f. (2.5) can be calculated with the help of the w.f. (2.4). The operators should be internal operators such as

~t~in, = ~(A)- -~- - . . . . (2.7)

( j -c . , . being the c.m. kinetic energy operator), so that the integration over X has no ultimate effect.

We now show that the Peierls-Yoccoz 1) (py) , the Peierls-Thouless 3) (PT) and the Rouhaninejad-Yoccoz 4) (RY) generating w.f. are all expressible in terms of the internal w.f. (2.5). First, from eqs. (1.1) and (2.3),

• ~V(x) = e ' l 'xc , K(x--X ). (2.8) Secondly,

• ~T(x; Q ) = e i K ' x ~ ( x - - X ) (2.9)

can be seen to be the generating w.f. for the PT w.f. (I .5).

Finally,

--- o(x--at; K)d , , (2.10)


where *~(x-~t ; K) = e'K'~x-*'#o(x-at ) (2.11)

is the K-dependent Slater determinant introduced by RY 4). That is,

q'IV(x) = e'K'x4,~=o(x-X). (2.12)

Eqs. (2.8), (2.9), and (2.12) show that all three generating w.L are eigenfunctions of ~" . . . . with the correct eigenvalue:

3"¢'m'lf'~ame(x)) = (2AM)-lhOKZl~ame(x)). (2.13)

However, the matrix elements of aft int are not the same:

IHint(K, K) PY n a m e i n t n a m e ( , , ) Ix I*, , (,,)> ( 2 , 0 3 a ( K - K ') H '°' ' = { (Q, Q ) for case PT

~Hint(Q = 0, Q ' = 0) RY.

(2.14) Here

Hi"t(Q, Q') = <4,Q(x- X)laflntlf,~,(x- X)>. (2.15)

The appearance of K in eq. (2.14) for the PY w.f. describes a change of internal structure with K. The w.f. is therefore unacceptable wherever the two groups of degrees of freedom are known to be uncorrelated. Such is the case when the system is Galilean- invariant * - a).

Now if one is not concerned about internal structure or internal energies, but only that the c.m. wave function has the correct form exp(iK. X), then all three generating w.f. are successful 2). In all cases there has been a symmetry restoration achieved by first eliminating the dynamical variable X by suitably integrating over its average value ~t. The correct c.m. function known a priori, is next substituted.

On the other hand, if the purpose is purely to isolate all the N linearly independent internal w.f. tJ}iint contained in the original Slater determinant (2.1), then ff~Rv be- comes inadequate when N > 1. The w.f. ff~T is similarly inadequate if one value of Q is used. It takes the second projection of Peierls and Thouless a), the one involving Q, to completely extract all the internal w.f.. Furthermore the two generator coordinates clearly involve different degrees of freedom: • for the c.m., and Q for internal properties in the example studied. The roles are reversed with Slater determinants given in momentum representation.

3. Illustrative example on the role of the second projection

The effect of the second generator coordinate Q in isolating the different internal w.f. contained in a Slater determinantal w.f. is illustrated in this section. The ground- state w.f. of 4He in the independent-particle shell model is examined. Since this ground state has zero spin and isospin we use a simple product of four single-particle

2 1 4 C . W . W O N G

w.f. They are taken to be

tp(r,) = c o tpos(ri) + c t tp~ s(r,), (3.1) where

¢Pos(r) = (b2zc)-~e -~'2/b2, (3.2a)

¢p,~(r) = (b2n) -~ 1 - e -~'2/b~, (3.2b) 3b 2]

are oscillator w.f. characterized by the oscillator length b = (h/M¢°) ~, h¢o being the oscillator spacing. The expansion coefficients ci are normalized to

2 2 C o + C l = 1.

The product w.f. now contains five terms with 0, 1 . . . . . 4 nucleons respectively in the Is state. Each term has spin zero and will be called an oscillator shell-model configuration. A compact expression for this product w.f. is

S o ( x ) : - 3 , , - . . = ( b 7 0 ECo cl E l-I 1 - e x p ( - ½ E r 2 / b 2 ) • (3.3) ,=0 c(,,,) {i}, 3 b z]

The product 1-I in this equation is defined by

I-IfJ = 1; 1-[fJ = f j f ~ . . . (n factors). (3'4) {J}o {JJn

The second summation is over all the n-combinations of four distinct s.p. labels, and is symbolized by C(4, n).

The internal w.f. (2.5), defined in eq. (2.3), can be written compactly in terms of the internal variables

4

= x = E , , . ( 3 . 5 ) i = l

Three of these vectors are independent, the fourth satisfying the relation

,4-

E r; = 0. (3.6) i = 1

The functionfj ofeq. (3.4) in eq. (3.3) can be written as

f j = (~)*{1 ,2 , . -~,[r~ +2r~ (X-oO+(X-a t )2] /b2} . (3.7)

From this we obtain the internal w.f.

4-

S o ( x - X ) = const. (b2zc) -3 E " - " " CO C 1 n=O

x ~ l-I {(~)~[1--j(r'jE+2irj • V~-V~)/b2-]} exp (~-bEQ2-½ X r;2/b2) • (3.8) c ( 4 , n) {j},, i


Of the terms in the w.f. (3.8) the V~ terms change the Q-dependence without affect- ing the functions of the internal coordinates. If the r) . VQ terms are temporarily ignored, there remain five types of terms. They are just the original five oscillator shell-model configurations on which a Gartenhaus-Schwartz 6) correction for c.m. motion has been applied. For different values of Q these terms appear in different proportions. Thus the use of five suitable values of Q will permit a complete separation of these configurations.

We see then that the first projection (2.3) is equivalent to a Gartenhaus-Schwartz transformation, while the "second projection" with Q separates (and eventually mixes) the five oscillator configurations "hidden" in the original determinant.

The same result can be obtained, both in this example and (according to the discussion of sect. 2) more generally, by mixing in the conventional way suitable Slater determinants after single projections. Full equivalence between the two ap- proaches will be achieved if the resulting configurations span the same shell-model space as that of the "hidden" configurations. It may also be assumed that knowledge of the structure of the original determinant implies knowledge of this shell-model space.

Finally the terms involving r) • Vq bring in other internal states, some with zero internal angular momentum, most with non-zero values. They appear because of the displacement (by an amount at) of the center of the potential well. The states of different internal angular momentum can be isolated by integration over suitable spherical harmonics of ~.

4. Angular-momentum projections

In discussing rotation it is convenient to write the shell-model many-body w.f. • a(x) in the form

t~D(X ) = ~ PSx~ t (~I~D(X), (4.1) JMK

where P ~ is the PY single-projection operator (1.3). In analogy with eq. (2.1) we have

N' o(x) = (4.2)

i=1 L , I

where L is the c.m. orbital angular momentum and I the internal angular momentum. The physical collective states of nuclear spectra are eigenstates of ~¢t °int a n d / , not of the to ta l~ ' (A) and J. In translationl the correct c.m. energy is obtained if y¢.m. is used with exp(iK • X), irrespective of what internal w.f. might also appear. Similarly for nuclear spectra the kinematically correct result is obtained if ~(F ant is used with [~i~t(x- X)] r, irrespective of what L and J might also appear.

Unfortunately the PY single projection isolates the state (4.2) of a given total angular momentum J rather than of I. Therefore errors might appear (i) when there are L ~ 0 components in the sum (4.2), and (ii) when .~. (A) is used instead of

2 1 6 C . W . W O N G

3~ i"t. In the first case spurious components appear in the w.f. (4.2) with wrong values of L The second adds a spurious c.m. energy to the calculated nuclear spectrum. These errors can be avoided by u s i n g ~ int with internal w.f.

One suggestion s) on the construction of internal w.f. from ~o(x) is to use displaced determinants

h"

P~u'4~I)(x-at) E E ( [ ~ , ( ,~)]L[4)i,"t(x--X)]r)~r • (4.3) i = l L , I

Here P~u, can be defined with respect to the origin of a coordinate system which is either stationary (while the shell-model potential well is displaced by a¢) or displaced together with the potential well. The latter situation is simpler because then

: c.m.," x [~, ~ , ~)3~ = [~h~'m'(x- ~)3~. (4.4)

The internal w.f. is of course independent of the choice of coordinate systems. The dependence on the c.m. coordinate X - ~ can next be eliminated, using the

method of sect. 2. Finally a generating w.f. for internal w.f. of good internal angular momentum I can be obtained:

,- f f 4)IMI(X--X; QLJK) = ~. (LMIMIIJM') dao~d2Qy*u(O~)e'e'(°'-X)pSu,¢r~(X-~), MM"

(4.5)

with Q, L, J, K as generator coordinates. [The dependence on K causes the well-known K mixing, and has been investigated before 4, 7).] According to the discussion of sect. 2, these generator coordinates are either redundant, or they permit the isolation of certain internal w.f. "hidden" in ~D- These generator coordinates can be considered a second set of generator coordinates "conjugate" to the first set of I2 and ~.

5. Spurious components in projected wave functions

As discussed in sect. 4, two problems might arise in the PY single projection of angular momentum: spurious components in the projected w.f., and a spurious contribution from the c.m. part of the Hamiltonian. We expect intuitively that these errors will decrease with increasing A, the number of nucleons in the system, possibly as A - i

To obtain a more detailed statement, we consider an independent-particle shell model of deformed axially symmetric Gaussian s.p. wave functions. Let a deformation parameter ~ be defined 8) by

= =

o . . = (5.1) The average oscillator frequency ~ satisfies volume conservation:

= COo(1 + its)- ~(1 - :~e)- ~, (5.2)

G E N E R A T O R - C O O R D I N A T E WAVE F U N C T I O N S 217

co o being the common oscillator frequency when e = 0. The deformed Gaussian one-particle w.f. can be written as

[-le M ~ (2z~ - x ~ - Y2)I ' (5.3) to(r, M, ~; s) = to(r, M, ~; 0)(1 +~e)~r(1-je) ÷ exp ~_v -~-

where

= r 2 (5.4) \Tth ] 2 h '

i.e., eq. (3.29). The model to be studied is one of simple products of such w.f. in which both spin-isospin degrees of freedom and antisymmetrization are neglected. The nuclear wave function

A

~,4 = I I to,, to, -= t0(r,, M, N; 8), (5.5) t=1

has even parity and an angular-momentum projection along the body axis of K = 0. As a result it contains states of even values of the total angular momentum J.

The strength (square of the expansion coefficient) of these J-states is

c~ - 2J--~+21f d3f2 [~So(O)]*(~alR(~)l~a). (5.6a) 8re d

Because of the symmetry of ~,~ this equation can be simplified to

fo c j = (2J + 1) dx P s ( x ) ( O a l R ( a r c c o s x)l~a), (5.6b)

where Ps is a Legendre polynomial and arccos x = fl is the Euler angle of rotation about the y-axis. Since qi a is a product of identical s.p. wave functions, we find that

A

(~alR(arccos x)l~a) = I-[ (to, IR(arccos x)lto,) = [f(x)] a, (5.7) i = l

where

f ( x ) = (1 + ~e)½(1 - ]e )½(ac - ¼b2) - ~, (5.8)

a = 1 + ½e(~x 2 - ½),

b 2 = g2X2(1- -X2) ,

c = 1 - ½ e ( ~ x 2 +½). (5.9)

The funct ionf(x) is independent of ~ and M. The strengths c 2 satisfy the normaliza- tion

Z cs 2 = 1. (5.10) d

The w.f. ~a is factorable in its c.m. dependence, i.e.,

,I, a = q~(X, A M , ~; e)~ i"', (5.11)

218 C . W . W O N G

where X is the c.m. coordinate and ~int is an internal w.f.. The strength c 2 of the c.m. state of c.m orbital angular momentum L is therefore given by eq. (5.6) with A = 1. Because of the factorization (5.11) a unique internal state can be isolated for each value I of internal angular momentum. The corresponding strength c~ 2 is related to e 2 and cj 2 by the vector-coupling relation

NI NL

( ~_, c2(LOIOlJO)Z)cff=c 2, J - O , 2 . . . . , (5.12) I=0 , 2 . . . . L=0, 2 . . . .

with the number of angular momenta Ni = ~ , i = I, L, J. The strengths c~ decrease rapidly with increasing value of the angular momentum

i. The infinite system of eqs. (5.12) may be truncated without much error at the finite numbers N i in order to extract the unknowns c~ 2.

In the numerical calculations reported below we use a 24-point Gauss integration for eq. (5.6b) with the Legendre polynomials generated by forward recursion. The system of simultaneous eqs. (5.12) is solved by using the IBM subroutine D G E L G . F O R T R A N IV double-precision arithmetic is used in these steps.

Table 1 shows the results for Ni = N = 4 (L, I, J =< 6); A = 1, 4, 20, 240; e = 0.25, 1.0. I t can be seen that c 2 is always between c2= t and C2L=1, becoming increasingly

TABLE I

Strengths eL 2, ca 2, and ct z o f the c.m.) total and internal wave funct ions respectively o f given angular m o m e n t u m (L, d, or I )

e A Strengths L, J or I

0 2 4 6

0.25 1 CL 2 0.9943 0 .5685(--2) 0 .2508(--4) 0.1087 ( - -6 )

4 C.~ 2 0.9774 0 .2241(--1) 0 .1979(--3) 0.1372 ( - -5 ) c~ 2 0.9830 0 .1689(- -1) 0 .1242(--3) 0.7538 ( - -6 ) r i 3.0 3.0 2.3 2.0

20 ca 2 0.8929 0.1037 0 .3367(--2) 0.7016 ( - -4 ) c~ 2 0.8979 0.9895(-- 1) 0 .3068(--2) 0.6125 ( - -4 ) r~ 2.0(1) 2.1(1) 1.1(1) 7.9

240 ca 2 0.31047 0.47467 0.17195 0 .36577(- -1) cl 2 0.31171 0.47456 0.17121 0.36282(-- 1 ) r t 5.5(2) 4.3(3) 2.3(2) 1.2(2)

1.0 l cL 2 0.8580 0.1258 0 .1434(- -1) 0.1608 ( - -2 )

4 cj 2 0.5566 0.3461 0 .8009(- -1) 0.1446 ( - - 1 ) c~ 2 0.6400 0.2933 0 .5633(-- 1) 0.8968 ( - - 2 ) r t 3.6 4.2 2.8 2.3

The s t rengths are calculated for a p roduc t o f deformed s.p. Gauss i an wave funct ions with different number s A o f particles and two values o f the Ni lsson deformat ion pa ramete r e.

The n u m b e r a ( n ) represents a × 10 n.

G E N E R A T O R - C O O R D I N A T E W A V E F U N C T I O N S 219

closer to the former as A increases. We also tabulate, as a visual aid, the ratio

r, \ c 2 -c-------~JS=L=," (5.13)

Table 1 shows that r~ has the same order of magnitude as A, the number of particles in the system, and is roughly independent of 5. This behavior is also consistent with the structure of the integrand (5.7) in eq. (5.6b). Thus the fraction of spurious states in the projected strengths could be described as an A-1 effect.

6. Errors in projected energies

Since the strength cs 2 of J-projected states differs from the correct strength c2=j by something of the order A-1, the J-projected energies of the correct internal Hamil- tonian jg, int are likely to have errors of the same order of magnitude. In the simple model of sect. 5 we have

( ~ i n t ) j = c f 2 ~, c 2 cff(LOiOiJO)gEir~t" (6.1) L, I

Here E~ ~t is the expectation value of,-~ int in the internal state of internal angular momentum I projected from ~i,t of eq. (5.11).

Eq. (6.1) can be applied to the states of a rotational band under the assumptions that (i) the band can be described by a single Slater determinant of deformed s.p.

TABLE 2

Projected c.m. kinetic energy TL c'm" in uni ts o f the average oscillator spacing 1 ~ and, in MeV, (i) exper imental energies Et l*t (expt) o f the ground-s ta te rotat ional bands in Ne, Er, and Pu; (ii) the energies (.,~vl~t)j modif ied by spur ious c.m. componen t s conta ined in the model wave funct ion o f total angula r m o m e n t u m J ; and (iii) the J-projected total energies (v%a)j including the c.m. kinetic

energy

Case A L, I or J

0 2 4 6

c.m. 1 TL . . . . / h ~ 0.7446 1.6943 2.6446 3.5948

Ne 20 El l*t (expt) 0 1.63 4.25 8.78 (~alnt~j 0 1.55 4.00 8.14 (,g'~) j 0 2.26 5.41 10.25

Er 168 E~ i"t (expt) 0 0.0798 0.264 0,549 (,.~¢flnt) j 0 0.0793 0.262 0.544 ( ~ ) S 0 0.1223 0.346 0.668

Pu 240 E~ i"t (expt) 0 0,042 0,142 0.292 < ~aint >d 0 0.042 0.141 0.290 ( ~ ) a 0 0.068 0.192 0.364

All calculat ions are made by us ing a product o f s.p. Gauss ian wave funct ions with Ni lsson de fo rmat ion pa ramete r e = 0.25.

220 C.W. WONG

wave functions, and (ii) the strengths c 2 obtained from the model of sect. 5 are not too different from those from more physical models. By using the experimental energies as E~ nt and e = 0.25, N i = N = 4 (i.e., L, I, J < 6) theoretical values for ( ~ i , t ) j can be calculated. They are given in table 2 for the ground-state rotational bands 9) of :ONe, 16SEt and 24°pu.

Table 2 shows that (jy~i,t)j are closer together in the band than ~nt E~ by a factor of roughly (A - 1)/A. The results are quite independent of the value for e used in the model. For example, the energy (~int)~= 2 for 2°Ne varies from 1.549 MeV for

= 0.05 to 1.554 MeV for e --- 0.45. Also a calculation for 2°Ne using N = 5 (by including the 11.95 MeV I = 8 state in the band) shows no change in @¢pint)j up to the reported significant figures.

Finally the effect of the spurious c.m. kinetic energy operator 5' -c'm" is studied. In the model of sect. 5 we have

(~A= 11~" . . . . R(arccos x ) l ~ ) = (~ lR(arccos x ) l ~ l ) h ~

_ b z × {g~-¼ [(1 q-~-e)+ al (1+]~)2+ ( c - 4~) -1 { (1- je )2+ 4a 2 - ( l+]e)z}]} ' (6.2)

where a, b, c are defined in eq. (5.9). Then

2 L + l f ldx Y~ . . . . c~ Jo PL(x)(Oa=~lJ-°'~'R(arcc°sx)lO~) (6.3)

is the L-projected kinetic energy of c.m. motion. For small e, the w.f. (5.3) may be expanded to the leading power in e for each L

to give T~'m" ~ ½(~+ L ) h ~ , (6.4)

i.e., the original oscillator kinetic energies. For large ~, the exact T~ 'm' tends to be smaller than the spherical-oscillator value when e > 0; slightly larger when e < 0. For example, T~'m' /h~ = 0.677, 1.726, 2.779, 3.832, 4.886 for L = 0, 2, 4, 6, 8 when e = - I; and 0.644, 1.305, 1.970, 2.635, 3.302 when e = + 1.

With the inclusion of 3 - . . . . the J-projected energies become

/ ~ i n t k (6.5) ( ~ f ) j = ( j . . . . ) j + \ / J ,

where ( j - . . . . ).r = c ; 2 ~_, c 2 c2(LOIO[JO)2T~. "m'. (6.6)

L , I

The numerical results for the ground-state rotational bands mentioned before are given in table 2. The approximation

~ o9 o ~ 40A -~ MeV (6.7)

is used. Again the dependence on ~ is found to be weak: ( ' ~ ) J=2 on 2°Ne changes from 2.28 MeV at e = 0.05 to 2.24 MeV at e = 0.45.


Table 2 shows that the energies ( ~ ) s are now much further apart than the experimental energies El nt. The errors are quite large even in 24°pu .

The reason for such large errors can be seen in eq. (6.6). Although the products c 2 c 2 (L010JJ0) 2 for spurious terms in the sum (6.6) are of order A-1

c.m. Tc.m. they bring in c.m. excitation energies T~. - i.=o ~ ½Lh~ which are large compared to the level spacings of the internal spectrum. For example, the physical component in the J = 2 state has angular momenta (L, I ) = (0, 2). With the inclusion of the dominant spurious component (L, I ) = (2, 0) the c.m. kinetic energy increases the J-projected energy by

c ~ 2C~=o h~ h~ L= ~ - - ~ 40A - t MeV. C 2 C 2 a (6.8) L=0 I=2

Here use has been made of eq. (6.7) and the rough A- x c.m. effect in the strengths of table 1. This is a substantial correction. For example in 24°pu (9~'int)s= 2 = 0.042 MeV, (.y-c.m.)s = 2 = 0.026 MeV of which the LHS ofeq. (6.8) accounts for 0.011 MeV. [The estimate on the RHS of eq. (6.8) gives 0.03 MeV.] The results of table 2 thus emphasize the importance of removing the c.m. part of the many-body Hamiltonian in calculations of rotational energies.

We have also calculated the/-projected internal kinetic energy: A

T: at = ( E t , - - J " . . . . ) I s (6.9) i=1

by solving algebraic equations having the same structure as eqs. (5.12). Typical results for e = 0.25, A = 4 are Zilnt/h~ = 2.2337, 3.1850, 4.1355, 5.0859 for 1 = 0, 2, 4, 6, respectively; and for A = 240: T~nt/h~ = 178.34, 179.30, 180.24, 181.18 for these values of I and e. These results can be written as

T)"t = h~[¼(A - 1) + ½I + C,(e)], (6.10)

where Ct(e) is a relatively small correction due to deformation. That is, T] "t follows closely the spherical-oscillator kinetic energies. This feature is likely to be model- dependent, however, since in the higher oscillator shells there are states of different /-values with roughly the same kinetic energy.

The model of deformed Gaussian s.p. wave functions used here is not realistic for A > 4. In addition, each rotational band has been described by a single Slater determinant. In spite of these inadequacies the general conclusions on c.m. corrections are likely to be roughly independent of the model used.

7. Peierls-Thoules~ double projection of angular momentum

In sects. 2-4 the second or conjugate generator coordinate is found to cause a separation and mixing of configurations within a certain shell-model space. We now show that the second generator coordinate of the PT double projection of angular momentum 3) has the same function.

In the PT method, the first set of generator coordinates is f2, the Euler angles in eq.

222 C.W. WONG

(1.3). The second set is the angular velocity to, which now labels the Slater determinant • o(X; to). The constituent s.p. wave function ~ ( r i ; to) of particle i in state ~ is obtained from a constrained Hartree-Fock procedure 3). For this purpose, a Hamiltonian such as ~ ( A ) - t o - ( J ) is used, in analogy to ~ ( A ) - v . ( P ) for translation, v being the linear velocity and P the c.m. momentum.

We consider a model in which the s.p. basis consists of the same number M of s.p. wave functions tp~(ri) for all values of to. This means that

M

= E (7.1)

varies with to only through its expansion coefficients c~(to). For each to the PY projection on ~D(X; to) gives unique J, M states.

Now the A-body w.f. can also be described in terms of the occupation numbers (2), such as (s4p a) for 160. From the corresponding w.f. ~ ) ( x ) the PY projection again gives unique J, M states. Clearly the J-projected states from ~o(x; to) are linear combinations of those from ~ ) ( x ) summed over all possibilities (2) implied by the expansion (7.1). Furthermore, the coefficients of the linear combination are completely determined by the s.p. expansion coefficients c~x(to).

Conversely, the Constituent J-projected states from ~ ) ( x ) can be separated by using a suitable number of different values of to in ~D(X; to). Hence the second projection with to again serves a configuration-mixing role, in complete analogy with the translational case of sect. 2.

As an illustration, we consider a system of three spin-zero particles. Let M = 2, and q~z(r~) be oscillator w.f. with s.p. labels 2 = nlm = 000 and 020. The distinct occupations and J-projected states are: (sa), J = 0; (s2d), J = 2; (sd2), J = 0, 2, 4; (d3), J = 0, 2, 4, 6. The J = 6 state is unique and independent of to. There are three distinct states of J = 0 and 2, and two of J = 4. These can be separated by using three and two suitable values respectively of to in ~D(X; to).

Another interesting feature is that the PY projection gives on ly one J, M state from each ~ ) ( x ) , whereas the conventional shell model might give several for the occupation (2) when the vector coupling of s.p. angular momenta to J, M is not unique. The distinction arises because in the PY formula (1.3) all the particles (i.e., particle coordinates) are rotated simultaneously, not separately ~ 0). In fact, the PY projection was originally designed for rotational states ~).

Had the first projection been different, the same procedure (7.1) with to as the second generator coordinate would cause the mixing of (in general) a different group of states. For example, a cluster single projection operator can be defined by replacing the rotation operator R(t2) in eq. (1.3) by 1-It= 1R~(~) for F clusters of Ai nucleons each, -~u( t2 ) by a tensor product of rotation matrices, and then by inte, grating over all t2~. The second projection with to then mixes these clustering states.

Finally, corrections for spurious c.m. motion are to be made in addition to the second projection.


8. Conjugate pair of generator coordinates

In both translation and rotation, the two generator coordinates in the many-body w.f. which constitute a classical conjugate pair (Q and at, oJ and f2) do not describe the same degrees of freedom. The fact that we are only interested either in the c.m. or in the internal degrees of freedom simplifies the discussion considerably, because it permits the interpretation that the w.f. (2.5) is an internal w.f.. Such an interpretation is not allowed if the Hamiltonian contains both types of degrees of freedom. The use of the first generator coordinate to project out states of known kinematical symmetries also obscures the situation.

We would like to emphasize that it is the nature of the w.f., not a simplification of the Hamiltonian or the imposition of kinematical symmetries, which causes the two generator coordinates to play different roles.

For this purpose let the Slater determinant in the problem be expanded in the form (2.1):

N OD(X) = Z q~'m'(x)o~n'( x - X). (8.1)

i=1

The N internal w.f. are assumed to be orthonormal for simplicity. The PT generating w.f. can be written as a two-parameter family of functions 3):

ePr(x; Q, at) = e -'x" e-i"r/~Oo(x ) (8.2)

in the case of translation. Here X and P are dynamical variables, and Q and ~ are generator coordinates. The most general form of the resulting A-body w.f. is the Griffin-Hill-Wheeler (GHW) linear combination 2):

I//PT(x ) m f f d 3atd3Qf(Q, 0g)*PT(x ; Q, ,). (8.3)

Now if N = 1, only one generator coordinate is needed because an arbitrary function of X can be expressed as a linear combination of either ~b . . . . (X-a t ) or of e x p ( - iX" Q). Th w.f. of a one-particle system is an example of this.

If N > I the most general GHW linear combination obtainable from the w.f. (8.1) is

~'(x) = ~ [ff,(,),f~,'m'(x-at)d3~ 1 ¢"m(X--X). (8.4) i=1

This w.f. contains two features: (i) improvement of the c.m.w.f. ~b~'m'(x) by using the generator coordinate a; (ii) configuration mixing in the sum over i. The linear combination (8.3) achieves the same result by using the dependence on Q to separate and mix these configurations. The same analysis for rotation can be made in a way similar to that of sect. 7.

Unfortunately, the use of conjugate pairs of generator coordinates involves the following difficulty. Ordinary generator coordinates are simple parameters or labels

224 C.W. WONG

characterizing the Slater determinant or its constituent s.p. wave functions. It is not always clear how their conjugate generator coordinates can be introduced conveniently. One possibility is the use of a collective dynamical variable and its conjugate momentum, as in eq. (8.2). But surely this method has little practical value (except in simple cases like translation) because one of the original motivations for using generator coordinates is precisely to avoid the construction of collective variables 2).

9. Concluding remarks

In our analysis of the Peierls-Thouless second projection 3) of generator-coordinate

wave functions, simplification in interpretation is obtained by separating the c.m. from the internal degrees of freedom. It is then found that in translation the total energy

has the correct dependence on the c.m. linear momentum K when the internal part of the Hamiltonian is omitted. In this sense, the Peierls-Yoccoz single projection 1) already satisfies the Peierls-Thouless criterion for correctness 3).

In the case of nuclear rotational states of internal degrees of freedom we find, with the help of a simple model, that the greatest source of error is again the inclusion of a wrong Hamiltonian: the c.m. kinetic energy. The resulting errors are found to be large even in heavy nuclei. In addition, angular-momentum coupling causes other

spurious effects of c.m. motion. But these are shown to be roughly of the order A - expected of c.m. effects of kinematical origin. They are important only in light nuclei.

The interesting role of the Peierls-Thouless second projection is found to be that of configuration mixing, a role already mentioned by Peierls and Thouless 3). In

cases examined here the configurations so mixed are determined partly by the choice of basis for the single-particle w.f. and partly by the nature of thefirst projection.

The author would like to thank Professor H. Feshbach for the hospitality of the Center for Theoretical Physics, M.I.T. He is grateful to Sir Rudolf Peierls for several

very stimulating letters. He thanks M. Baranger, D. M. Brink, F. Villars, and A. Weigung for useful remarks.

References 1) R. E. Peierls and Y. Yoccoz, Proc. Phys. Soc. AT0 (1957) 381 2) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102;

J. J. Griffin and J. A. Wheeler, Phys. Rev. 108 (1957) 311 3) R. E. Peierls and D. J. Thouless, Nucl. Phys. 38 (1962) 154 4) J. Yoccoz, in Many-body description of nuclear structure and reactions, ed. C. Bloch

(Academic Press, New York, 1966); H. Rouhaninejad and Y. Yoccoz, Nucl. Phys. 78 (1966) 353

5) C. W. Wong, Nucl. Phys. A147 (1970) 545 6) S. Gartenhaus and C. Schwartz, Phys. Rev. 108 (1957) 482 7) B. Giraud and P. U. Sauer, Phys. Lett. 3013 (1969) 218 8) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 (1955) no. 16 9) Landolt-B/~rnstein Tables, new series, vol. 1, ed. K. H. Hellwege (Springer-Verlag, Berlin, 1961);

F. Ajzenberg-Selove, Nucl. Phys. A190 (1972) 1 10) M. Hamermesh, Group theory (Addison-Wesley, Reading, Mass., 1962) sect. 9-8

single and double projections of generator-coordinate wave functions

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