Description of excited states in systems with short-range correlations

I•.C }

Nuclear Physics A242 (1975) 376 - 388; (~) North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher

DESCRIPTION OF EXCITED STATES IN SYSTEMS WITH SHORT-RANGE CORRELATIONS + J. DA PROVIDI~NCIA

Laboratorio de Fisica da Universidade, Coimbra, Portugal and C. M. SHAKIN

Institute for Nuclear Theory and Department of Physics, Brooklyn College, Brooklyn, New York 11210 Received 15 July 1974 (Revised 10 December 1974) A~traet: It is shown that the cluster expansion formalism previously developed as a basis for a theory of the ground state of correlated systems may be generalized in such a way as to also provide a basis for the description of excited states of closed or open-shell systems. In particular, the possibility of using the cluster expansion formalism in combination with the generator coordinate method is discussed.

1. Introduction

In a series of works 1- 3) we have discussed a non-perturbative theory of the ground state of a closed-shell Fermi system, based on a wave function possessing a correlation structure similar to that proposed by Jastrow t+. In ref. 1) we discussed the theory of correlated, finite systems in which one systematically neglects the contribution of three-body cluster terms. This work contains a derivation of the BetheGoldstone equation for finite systems which includes modified occupation probabilities and self-consistent potentials for occupied states. It was also shown that a careful treatment of the Pauli principle and occupation probabilities allows for the application of variational techniques in the calculation of the energy of the system. In refs. 2, a) we extended our considerations to the study of the three-body clusters. In particular, it was shown that starting with a correlated wave function of general form one could make direct correspondence with the diagrammatic theory of Brueckner, Bethe, and Goldstone. It was also demonstrated that the expression for t Supported in part by the National Science Foundation and by Project CF-1, Institutio de Alta Cultura, Lisbon, Portugal. tt In this work we do not discuss the detailed relationship of our wave function to that of Jastrow. This has been done, in part, in ref. 4), where references to related work of other authors may be found. 376

SHORT-RANGE CORRELATIONS

377

the three-body cluster contribution to the energy is the same as that obtained by Bethe. This body of work is closely related to the works of ref. 5), where a similar correlated wave function is used as a starting point. However, there exist some differences in the method of derivation of the fundamental equations and in the partial summations made. The equations derived in ref. 5) have received extensive numerical study and various applications have been made to calculation of the ground state properties of light nuclei 6). A natural extension of the above non-perturbative theory is to be found in the study of the excited states of correlated systems. In the present paper we wish to show how the techniques previously developed may be applied to the study of the excited states of closed-shell or open-shell nuclei. In sect. 2 we consider a closed-shell nucleus having a few holes, and in sect. 3 we rederive a related result, obtained previously, for an energy-weighted average of spectroscopic factors. In sect. 4 we consider a closed core plus a few particles and finally in sect. 5 we relate our method to the generator coordinate method. 2. Closed-shell nuclei having a few holes

We begin by discussing the case of a closed core with a few holes. We use [~c) to denote the uncorrelated core of N particles, N

14~o) = [-[ a~10).

(2.1)

i=1

We introduce a correlated core via the relation (2.2) where, in general, N

S = ~ S (n),

(2.3)

n=l

and where S ~") is an n-body operator. We define the operators S ~") such that

S ~a) = ~, a~s,., iai,

(2.4)

m,i

1 S(2) = 27 E atmatnfran, i j a j a i , mnij

etc. We denote occupied states in ]~c) by i,j . . . . , and non-occupied states by m, n . . . . . We also define n-particle uncorrelated states, Ii) = a~*[0),

(2.5)

lij) = ati a~lO ),

(2.6)

378

J. DA PROVIDI~NCIA AND C. M. SHAKIN

etc., and the unnormalized, but correlated states, [~i) = eS[i>,

(2.7)

Iq/iS> = eSlij),

(2.8)

etc.

= Ki,j - $1j •

:

=

Kij,k I

=

hij,k I

I

hi,j I

Fig. 1. Diagrammatic representation of the cluster integrals defined in eqs. (2.9)-(2.12).

We also introduce the cluster integrals, (see fig. 1), xi, s = <~kii~bj>, Kik, jl = <~bikl~/ j l ) - - l~i, SXk , l ~- l('i, l l~k, S . . . . .

hi, s =

<~,lHl~s>,

hik, S, = <~biklHl~sl>--h~,srCk, l--hk, trci, s+ hi,,xk, s + hk, sxi, l.

(2.9) (2.10) (2.11) (2.12)

Suppose we now wish to describe the states of a system consisting of a few holes in a dosed core. We are led to the following eigenvalue problems. For one-hole states we have

~ [r/s, i - Eyj, J C s = 0,

(2.13)

J

and for two-hole states

~, [r/kt, 'S-- E~:ki, o]Ck, = 0.

(2.14)

kl

Here t,

7j,, =- <~cla~asl~¢>/<~cl~¢>,

(2.15)

7ki, o = < ~'cla~aJ a t akl ~ > / < ~V¢l~ > ,

(2.16)

r/s,i = ( ~ J a ~ ( H -

g ¢ ) a j [ ~ > / ( ~ l ~¢>,

(2.17)

r/kt, iS = < tP¢la~a~( n -- ~¢)a I ak[ ~,>/(~ol~eo >,

(2.18)

e~ -- <~eolHl~e~)t(~cl'eo).

(2.19)

t We note that because of the correlations contained in I~c>, the states aA~c>, a t a k J ~ > , etc., contain multi-particle-hole components in addition to their leading configurations.

"

"~J,k'~j,I

4"

-'~J,I ~j,k

i

4-

J

+

+ -..

4"

I

....

i

: " ~ i , k ")'j,l - " ~ i , l "~'j,k

=

4"

k

I

4. • . .

k

4-

I

4". • •

Fig. 4. The quantity 71,j is represented by a heavy line in the top of this figure. Partial summations may be made in the cluster expansion for the quantity 71j,k~resulting in eq. (2.21) which is indicated in the lower part of the diagram.

~iJ,kl

~'i,j

i

Fig. 2. Diagrammatic representations of the cluster expansions for the quantities defined in eqs. (2.15), (2.16), containing elements (filled circles) defined in fig. 1.

-~ij,kl

~i,j

i

''~i,k

@-..

+

'~j,I + ~ j , l ' ~ i , k - ~ i , I

+

~"l,k-

+

...

7j,k

+

~i,I

+

.-

+ (

I

+

"'"

+...

:

-

4-

k

~i,k"~j,I

i

+

I

4"

* ~j,l'~i,k

• ..

+

"

4-

~i,l"~j,k

+

-

÷

• • •

~j,k'~f'i,I

+

• . .

Fig. 5. Diagrammatic representations ofeqs. (2.22), (2.23). The heavy lines, open and filled circles have the meanings given in figs. 1 and 4.

~ij,kl

"Zi,j

Fig. 3. Diagrammatic representations of the cluster expansions for the quantities defined in eqs. (2.17), (2.18), containing elements defined in fig. 1.

"~ij,kl

"~i,j

Z

-]

m

©

Z

-]

0

380

J. DA PROVIDI~NCIA AND C. M. SHAKIN

The quantity 7j, sis, of course, the same one which has been introduced in ref. x). We notice that eq. (11) of ref. ~), for the expectation value g = ( 7 ' l n l ~ ) / ( ~ U l T ) , is approximate, in the sense that the quantity multiplying h~j,k~has been factorized. The correct expression for the core energy, ~o, involves the quantity 7k~,ij and is given by, gc = ~ hi, ;)'j,i + ¼ ~ hu, kt 'Ykt, ij dr- . . . . ij

(2.20)

ijkl

The most convenient way to represent the cluster expansions for the quantities Tj,~, 7kt, ij . . . . , q~,i, qk~,~;,'" ", is by means of diagrams containing open external lines, as indicated in figs. 2 and 3. The quantities (7-'cla~najltt'c>/(7'olT'o) and (Tcla~a~HazaklT~)/(TolT~) are represented by linked diagrams. A diagram with external lines is linked if no piece of it is disconnected either from the open dot, representing an energy cluster integral, or from at least one of the external lines. The quantities q~,~, ~/o,k~ are represented by linked diagrams in which the open dot is not disconnected from at least one of the external lines. We observe that the quantity 70, k~-(7~,k~j,~-7~, ~)'j,k) is represented by diagrams such that the external lines are not disconnected from each other. A similar observation applies to the quantity ?~ij, k l - - [ q i , kTj, l ' ~ q j , 17i, k--?]i,l~j,k--?~j,k]~i,l] •

It is natural to choose the orbitals such that S ~) = 0. Then we have xi,~ = 6i~, and the diagrams representing our cluster expansions simplify. In any case, partial summations are readily performed in the cluster expansions represented by figs. 2 and 3. Following ref. x) we shall use a thick line to denote a factor Yl,j. Our cluster expansions are then represented by figs. 4 and 5. We may write, taking into account the most important terms only, ~ij, kl = 7i, kTj, l--Ti, 17j, k "~- ~ 7i,a~)j,b~ab, cd~)c,k~)d,l "~- "" ", abed

~i,j =

-- ~,, kl

~i. khk. z~'Z.j-- ~

])i. khkg, lh~l,j]~h,g "It- "" "'

(2.21) (2.22)

ghkl

?~ij, kl = ~i,k~)j,l ~-~j,l]~i,k--~i,l~j,k--?~j,k~i,l -1- ~ ])i,a~)j, bhab, cd~c, kTd, l'~- . . . . abed

(2.23)

Notice the ( - ) signs in eq. (2.22), which are related, in the usual fashion, to the number of fermion lines. 3. Theorem for energy-weighted averages of spectroscopic factors We find it instructive to present a new proof of this theorem 7), entirely based on our diagrammatic cluster expansion, including the external line extension developed

SHORT-RANGE CORRELATIONS

381

here. For this purpose it is convenient to define I~¢) as the ground state of H for N particles. We assume, that S (1) = 0, and that S (2) is determined by the requirement that 8¢ be a minimum, as in ref. 1). Using a convenient matrix notation and taking into account the diagram.s explicitly represented in fig. 5, we may write, ( ~ c l a ~ ( H - ¢¢)a~ I~eo>/<~e¢l~c>

=

- [Tx

hi 71 +tr2 71 h127172--tr271 X1271 Y2h272 --tr2 tra71/£127172h237273+ -..]j,i-

(3.1)

In ref. 1) the following expression has been derived for the single-particle energy matrix e I ,

e 1 = h I +tr2 [h12-(gl "~-/~2)/£12172 "

(3.2)

This may be re-written in the form el(1 +tr2/£1272) = e I 711 = h 1 +tr2h1272 -tr2g2r,1272, where the equation 71 = I1 -tr2/£127172 has been used. We may also write el = h171+tr2h127172-tr2e2/£127172 = hi 71 q- tr2 h12 7172 - tr2/£12 71 ~2 h272 --

tr2 tr3/£12 71 72 h2a 72 73 "~- . . . .

so that we finally have (~P¢la~(H-8~)ajlW¢>/(~¢l~

= -1"71el]~,~ = - ~ 7i, kek.~. k

(3.3)

It has been shown in ref. 1) that the matrix 71 el is hermitian and can be diagonalized. This result is probably valid under more general conditions, i.e., by taking into account a wider class of diagrams, provided el is accordingly modified. 4. A closed core plus a few particles

This problem is more difficult than the problem of a core with a few holes, because the matrix elements (~P¢[amHa~l~Pc) are infinite if the two-body interaction contains a hard core. The difficulty may be circumvented by allowing for correlations between the valence particles and the core. This may be done in the following way. We denote by i, j, k, ..., single-particle states occupied in I~c). By m, n . . . . . we denote singleparticle states not occupied in I ~ ) . We now introduce the notation #, v , . . . , for valence states. Finally, we represent by ~',j, ..., single-particle states which are either occupied in [~c) or are valence states, and we represent by rh, a . . . . . non-occupied states which also are not valence states. We are therefore led to a new correlation operator g of the form = ½ ~ a ~ a ~ f ~ , r i a r a . j + .... (4.1) mnlJ

382

J. DA PROVIDI~NCIA A N D C. M. S H A K I N

It should be observed that S commutes with the a r and therefore with the a u . T h e matrix elements f~s,,7 (between occupied or valence states and non-occupied, nonvalence states) should describe well the correlation structure of the system obtained by filling up all valence states, [~u; V) = eg[-[ a,~[O) = eg]-[ 1-[ a~a/t[o). f

u

(4.2)

i

There is no need to introduce in S' a one-body part, if the orbitals are conveniently chosen. With the help of the formalism previously developed, it is now a simple matter to compute matrix elements of H between states containing valence particles correlated to the core, with the desired accuracy in the cluster integrals. We keep the definitions previously introduced, I~¢) -- [-[ a~lO), i

I~o) = eglq)¢),

(4.3)

and we introduce the following new definitions I~u¢; # ) = ega,~l~c),

(4.4)

I~u¢; #v) = eSa~a~[~).

(4.5)

Suppose, for simplicity, that we wish to describe states with one valence particle. Then we may still write (~c;/~lnl~c;/~) = ~(~)hr' yy~}+ ½ ~U,hv7, ~rY~,}Y~,}+ ....

(4.6)

where the matrix ~7,z o,~ulsatisfies the equation

~

= 6r,, j

~

...... ~"~-

--/_, ~hk

n.~,~h~o, iYh, k~

(4.7)

....

The symbol ~ ) means that the sums extend over all occupied states and over the valence state/~. (The sum does not extend over valence states other than/~.) The exact definition of the matrix y ~ is 7~,.~ = (kuc; #la~a,q~c; # ) " (~o; ul~'o; ~)

(4.8)

The quantity (~u c ; pique ; # ) is, of course, easily related to (kuo]~). Indeed we have (~l~)

(~'o; ul~o; ~)

_ (~c;

#la*~a.l~t'c;P )

(~'o; .1~'~; u )

= v<")

""'"

(4.9)

SHORT-RANGE CORRELATIONS

383

Similarly we may write,

(tP~; lav[a~Ha,q%; l~v> ----

~,~v)o~(t,v)_ Z.~ x-~(uv).,(uv)u /i,~ ug, ~i .,(uv) ~'~i, )" -

~

t~) ~,.~ -~)h--~ . ~ .~~ . ~~ + ....

(4.10)

~hta

where --

(~c;

~/Y]lI/c; ~AY)

=

~

,"3

;'r,)'~'j,~" r 2

~

v~7~t

Tj, k l ~ ' k , :

/l,'j

....

(~o; pvla~a,q~¢; #v) = ~, J - ~X'~v)'~,~hrg, J~'9"U'~k'rh,-- ....

(4.11)

(4.12)

The symbol ~t~v) means that the sums extend over all occupied states and over the valence states # and v (the sums do not extend over valence states other than # and v). The norm <~¢ ;/~vl~u¢ ;/~v) may be computed from the relation

CeoI~eo>

_~.

<~o; p vlaut a t~a~a,l~c, • #v)

~

,~(~v),~(t~v) ~ ~(~v),~(/avj

~v, v/]~, jl

//~, v iv,/~

~__ e,,~ z~. h ~ , Urk, u zt, ~ + ....

(4.13)

~hkt

If we want to describe states with one valence particle, we are led to consider the following eigenvalue equation:

Y [ 5 e ¢ ; ~lnl~uo; v> - E <~o; p[~¢; v ) [ Cv = 0.

7L

<%1~o>

(4.14)

<~oI%> J

We are now in a position to compute, in terms o f our cluster integral, the quantities involved in this equation. Indeed, for v = # we have only to use eqs. (4.6) and (4.9). F o r v 4 # we may write <~¢;/~IHI~¢; v>

(4.15)

Now we have only to use eqs. (4.10) and (4.13). Our formalism is therefore appropriate if we wish to perform a shell-model calculation, using a realistic interaction. To extend the techniques proposed here so as to enable the introduction o f two-particle-one-hole, and higher components, into the one-particle states, presents no problem in principle. Similarly, a particlehole calculation may be set up, and one-particle-two-hole states may be added to one-hole states.

384

J. D A PROVIDI~NCIA A N D C. M. S H A K I N

5. The generator coordinate method A different approach is also possible for the description of excited states (of the particle-hole type) with our cluster expansion formalism. This has already been the subject of a publication s), but we believe it is worthwhile to present a more detailed account of our method here. The problem is to obtain expressions for the kernels to be used in the generator coordinate method. We define for the discussion of this section, N

1¢) = ]-[ a~10), i=1

1~) = eSl¢),

(5.1)

I~') = eS'l¢). Notice that 1~) and I~P') are defined with the same I~) but with S ¢ S'. Here it is important to keep the one-body parts of S and S'. Our cluster expansion remains valid for the overlaps (~'[~e) and (~P'IHI~) provided we define appropriate asymmetrical cluster integrals. We write S = S(1)-Jt-S(2)-[-...,

(5.2)

S ' = S '(1)-~- S '(2)-~ . . . . N

N

S m = y~ s~,

s ' m = y~ s;,

i=1

i=1

St2) = ½ E f~j,

(5.3)

S'~2~ = E fi), etc.

i, j

U

(The operators S and S' may also be written using the formalism of second quantization.) Then we define

~,,j = (~k~lCj) = (il(l+s'~tXl +sl)lj), g,j.~ = (¢,jlO~,)x , . , g,.~gj.,+ x ~ , ~ "'

~

(5.4)

= (ijl{(1 + s,'*-, s2,t+s,lts,2,+f;t2)

x(l+s~+s~+s~sz+f~2)-(l+s'~*Xl+s'2*Xl+sIXl+s~)}lkl), Bi,s = (0'ilnl¢~) = (ile)lho911j),

= (ijl{09'~to/2to)'~(tl+t2+v12)o)lo)20912-09'~tco'2*(tl+tE)O)lo92}lkl),

(5.5) (5.6)

(5.7)

where a short-hand notation has been used, 0)1 = l + s 1, o912 = 1+f12,

o9'1 = l + s ' l ,

(5.8)

o9~2 = 1+f1'2.

(5.9)

SHORT-RANGE CORRELATIONS

385

The correlation operatorf~ 2 (S(2)) is a function of the deformation operator st (S {~)) determined by minimizing the expectation value,

= tr 1 h171+½tr I tr 2 h 1 2 ~ 1 ~ 2

,

(5.10)

with respect to variations of f12. (Similarly, f12 is determined by minimizing <~'IHI~">/C/"I~/">.) Because S (1) =~ 0, the equation in ref. 1) for 71 has to be modified. The correct equation is, in matrix notation, ~1 = /£1 1 --K1 1 tr2 ~12~1~2"

(5.11)

Following the methods of ref. ~), the following equations are obtained:

{[)'1 Y2]kl,ij kl

-[71Y2(e1+ e2)]u, ij} -- O, ht - e l ~1 +tr2 [h12-(e1 + e2)~c12]Y2 = 0.

(5.12) (5.13)

(The Lagrange multiplier e I refers to the equation I z = K 1 7 1 ÷ t r 2 ~ 1 2 ~ l ) ~ 2 = ~1 K1 + tr2 ~172 ~:12.) We have analogous equations for f~2- To avoid long expressions, an obvious matrix notation is used throughout. For instance, the brackets, [ ], in eqs. (5.12) and (5.13) are to be understood as follows: [~lT2]kl, ij = ~ k , i ~ l , j ,

[~1 '~2(81÷ 82)]kl, ij = E [~k, gF'g,i~l,j ÷ ~k, i~l,g~'g,j]" g

We now introduce the notation y~, e~ in order to denote that these quantities refer to the deformation operator s~. We therefore have the following quantities: ~1, / / e~, h~, ..., ~,~, e 1 , h 1 . . . . . 91,7;~, .... We may now write, <~'[H[~) -- l t r I (e~ + e'l ) + ~1t r l (~1--~xeft)91+1tr1 (~1--e~1)91

<~"I~>

+¼ tr, tH [~12 - - ~ 1 2 ( ~ ÷ 8"~'2)]9192 ÷¼ trl tr2 [~'12-(e'~ + e~)~1219, 92.

(5.14)

We have here used eq. (5.11) in the form trl [11 -R191 - t r 2 ~1291 92]el = trl e*l[I1-917cl -tr29192/£12] = 0. With the help of eq. (5.12) we may write [~12 -- ~12(etl "[- ~t2)]ij, kl :


('O2 0912] kl>

+

- [(ijls'~ts'2t(tl + t2)t01 CoElkl> - ~.
386

J. D A P R O V I D t ~ N C I A A N D C. M, S H A K I N

with a similar expression for 1~12- (/~1 At-e2)~12 • Neglecting terms of higher order than the second in si, s~., we have [ ~ 1 2 - - ~ 1 2 ( F . l ÷ e$2)]ij, kl ,~

( ijl(l + s'lt + s'2t)v~2~l (D2(D121kl) + (ijl(s'l*t2 ÷ t~ s2'* )A 21k/).

In order to proceed further we must compute the matrices el, e~, Yl. A good starting point is, probably, to approximate

~1 ~ ~ ~ 4 °~, 71 ~ 7~o),

(5.15)

where e~°) and y~o) refer to sl = 0. In order to better understand the structure of the equations obtained, we assume further that the matrices y~o) and e~°) are diagonal, eto) .to) 6 i,j ~ bi ij~

,to)

. to~6

i,j ~ ~i

(5.16)

ij"

We may then replace eq. (5.12) by the following one, (mnl[t~ + t 2 +/)12 - -

E(k0 ) -

EIO)](D1(DE(D~E[kl) = 0.

(5.17)

This is an equation of the Bethe-Goldstone type. We may write, with a diagonal tl, [mn)(mnlvl2ld/u) I~kk,) = (l + sl + s2)lkl) + ,.~ etkO)+ e l o ) _ t m _ t ' where t~]kl) =

°)1(D2(DI2[kl) •

(5.18)

(5.19)

Eq. (5.18) is formally solved as follows: [~kkl) = { 1 + - Q v + Q e- v Q--e v+e

...}(1WSlWS2)lkl) = I2(l+sl+s2)lkl).

(5.20)

The operator t2 is, of course, a function of dk°~+ e}°), i.e., f2 = t2(dk°) + etkO)). With the help of all these results, we now write eq. (5.14) in the form (for an infinite system, so that terms like (ij[s'ltt2f12[ij) vanish) (~e'lHl'e) ~,.~ ~ ~i ~to) + ~ (~'i, i -~". . , , ,~co)~,
oto)+ ~(~'i , i - ~,~,,,o, oto)~o,~o) (5.21) Z o, ,r, i

+1 y, E (/jl(1 +sT + ~h*)~l~a(1 +sl + sglij)~%~°~. i

j

The terms linear in s~, s~, disappear if we impose the condition (m[t~[i) + ~ [(mk[V~E[~k) + (mk[(tl + t2)A2[ik)]~tk °) = O. k

SHORT-RANGE CORRELATIONS

387

For an infinite system, this condition follows from momentum conservation. For a finite system, this condition means that the minimum of < ~ [ H [ ~ > / < ~ I ~ > is obtained for S ~ = 0 [see ref. 1), footnote 5]. We may finally write, with the help of eqs. (5.4) and (5.6),

.,, , - o~ , .

•

<~'lq'>

i

T 2 Y- Z ~10~ °)

i

, j

+ Y' [ - ~l°qTl °~ i

+ ½ Y~ ~°)~ °'. (5.22) ij We have, therefore, been able to expand <~U'lHl~>/<~'lq/> in powers of s~ and s,. This expression contains, however, an error, due to the replacement a~ ~ ~t ~ ~o). If we compute ~1- a~o) we find s.,,J'- ~,.~!°!J~ + ~ 7~°~ J

-

~

(o)

t

-

~i, k.

(5.23)

k

This correction, however, would appear multiplied by the small quantity (1-?¢,°J), so it may be neglected. For consistency, the occupation factors yl°) should also be replaced by unity in the terms involving s~, sl. We have, explicitly, <~'IHI~> <~'1~'>

- ~o+ ys',.*,s,..,(t,.-~?'~im

Y

s,., t21nJ"> '* :., i
ijmn

+ ~ s,,,is,,j.

(5.24)

ijmn

We need now an expression for the overlap . We know that log <~[~u> is given by a sum over linked diagrams, involving only normalization cluster integrals (black dots). The change in log due to a small change in Xl, Xl~ . . . . , is therefore given by 6 log <'/'lq'> = tr~ 6xt ~21 + 1 tr~ tr 2 6K1271 ~)2"~-

....

We have finally, log < ~"1~> = log <~¢°)l~t°)> + tr * (~, _ ~-,"t°)x~'t°),~t ~.¢o),~,{o).,¢o) +½ tr, tr 2 czt,~t2--~12]/1 ¥2 •

(5.25)

With the help of eqs. (5.4), (5.5) and (5.20) we may also write

log <~'1~> -- log + ~] yl °' i

+ ~ ~ [ + i

j

+ < qls~t(f~+~ - 1)s, Iq> + ]~:l°)~ °).

(5.26)

388

J. DA PROVIDI~NCIA AND C. M. SHAKIN

Neglecting the occupation factor y~0~, for consistency with eqs. (5.24), this may be written log ( ~ ' [ ~ ) = log (tPt°~lIPt°>)+ ~ s~*ism,i im

+ ~ [s~,
+ Z [s'=*,,sn.,(mjl(t2tf2-1)lnj)+s',*.,sn, j(mJl(t~tt~-l)lin)] •

(5.27)

ijmn

The Hill-Wheeler equation,

f

l-(~'lHI ~ ) - E(~'I ~)]F(sm, i) [I dsm. i = 0,

(5.28)

m, i

leads essentially to the Tamm-Dancoff equations, with some renormalization effects due to the correlations. If in eq. (5.27) we neglect the small terms involving the quantity f2* t 2 - 1, we obtain

f

{8o+ ~

s~,sn.;Am,,,j-e } exp ( ~ s~is,,,,,)F(s,, ) l-I ds~,j = 0,

rainj

mi

(5.29)

nj

where

Ami,. J = (tm--elO))3m~6ij+60 ~ (mklv12Qlnk)+(njlv12t21ni).

(5.30)

k

The solution of the eigenvalue problem of eq. (5.29) is equivalent to the diagonalization of the Tamm-Dancoff matrix, eq. (5.30). References 1) 2) 3) 4) 5)

J. da Provid~ncia and C. M. Shakin, Phys. Rev. C4 (1971) 1560 J. da Provid~ncia and C. M. Shakin, Phys. Rev. C5 (1972) 53 J. da Provid6ncia and C. M. Shakin, Phys. Rev. C6 (1972) 1455 C. M. Shakin, Phys. Rev. C4 (1971) 684 F. Coester, Nucl. Phys. 7 (1958) 421 ; Lectures in theoretical physics, vol. 2, ed. K. T. Mahanthappa and W. E. Brittin (Gordon and Breach, New York, 1969); F. Coester and H. Kiimmel, Nucl. Phys. 17 (1960) 477; H. Kiimmel, in Lectures on the many-body problem, ed. E. R. Caianello (Academic Press, New York, 1962); Nucl. Phys. A176 (1971) 205; H. K/immel and K. H. Liihrmann, Nucl. Phys. A191 (1972) 525; K. H. Liihrmann and H. Kiimmel, Nucl. Phys. A194 (1972) 225; H. Kiimmel, Phys. Rev. C8 (1973) 1144 6) H. Kiimmel and J. G. Zabolitsky, Phys. Rev. C7 (1973) 547; J. G. Zabolitzky, Phys. Lett. 47B (1973) 487; M. Fink, M. Gari and H. Hebach, Phys. Lett. 49B (1974) 20; J. G. Zabolitzky, Z. Phys. 267 (1974) 161 ; Nucl. Phys. A228 (1974) 272, 285 7) C. M. Shakin and J. da Provid~ncia, Phys. Rev. Lett. 27 (1971) 1069 8) J. da Provid~ncia and C. M. Shakin, Z. Naturf. 28a (1973) 394