Convergence of effective hamiltonian expansion and partial summations of folded diagrams

I *s I

Nuclear Physics A235 (1974) 171-189;@ Not

North-Holland Publishing Co., Amsterdam

to be reproduced by photoprint or microfilm without written permission from the publisher

CONVERGENCE OF EFFECTIVE HAMILTONIAN EXPANSION AND PARTIAL SUMMATIONS OF FOLDED DIAGRAMS E. M. KRENCIGLOWA and T. T. S. KU0 Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11790 t Received 22 April 1974 (Revised 29 July 1974) Starting from a time-dependent formulation of the energy-independent effective hamiitonian, a sequence of partial summations for the folded-diagram series is defined and a connection between the energy-independent and energy-dependent effective hamiltonians is shown. The partial summations are shown to be convergent in the presence of intruder states which are weakly coupled to the model space.

Abstract:

1. Introduction

The question of convergence of the expansion for the effective hamiltonian in practical nuclear applications has been raised by Schucan and Weidenmiiller ‘, ‘), in view of known low-lying intruder states and the formal analytic properties of the effective hamiltonian that reproduces the lowest eigenvalues of the full hamiltonian. The question of convergence was also raised by Barrett and Kirson “) who examined the third order, and part of the fourth order, in the G-matrix terms of the effective interaction. However, it is well known that approximating the effective hamiltonian by the bare G-matrix plus core polarization, as originally done by Kuo and Brown 4), in fact works quite well. This represents somewhat of a perplexing situation. The Qbox formulation of the folded-diagram series 5*6, constructs the effective hamiltonian in essentially two steps: one, the Q-box itself, when expanded in powers of the interaction, essentially represents an expansion of the Q-space eigenvalues about some unperturbed value(s) and two, the folded diagrams essentially represent an expansion of some of the eigenvalues of the full hamiltonian about some unperturbed value(s). Hence, a resolution of possible divergences into two possible sources. Approximation schemes to the Q-box have been suggested by Kirson ‘) and Krenciglowa, Kuo, Osnes and Giraud “). It is the second step at which this presentation is principally directed. We will show that, by means of a sequence of partial summations, the foldeddiagram series may be summed exactly. In sect. 2, the notation and problem are defined. Sect. 3 represents a collection of previous results that may not be well known but are necessary for the development t Work supported by the U. S. Atomic Energy Commission under contract AT (ll-l)-3001. 171

172

E. M. KRENCIGLOWA

AND

T. T. S. KU0

of the partial summations for the effective hamiltonian in sect. 4. Also, in sect. 4, a formal connection between the starting point for the derivation of the folded-diagram series as given by Brandow “) and the time-dependent derivation of Kuo, Lee and Ratcliff ‘) is made; the folded-diagram series in the time dependent formulation is explicitly summed exactly, by means of a sequence of partial summations, into a form expressed in terms of energy-dependent interactions of Bloch and Horowitz lo). In sect. 5, we apply the partial summations and folded-diagram series to two simple model hamiltonians. We find that. the partial summations can converge in the presence of intruder states and that convergent partial summations yield eigenvalues of the full hamiltonian which correspond to eigenvectors with the largest overlap with the Pspace. We also show that the problem of intruder states can be consistently dealt with within the framework of the folded-diagram series and the partial summations by means of some appropriate P-space modifications. 2. Notation The notation we use will be almost identical to the notation adopted by Kuo, Lee and Ratcliff “) and Ratcliff, Lee and Kuo “). In studying truncations of a Hilbert space, the notions of a projection operator P onto the model space and its complement Q are introduced. We denote the model space of active state labels by A, B, C, . . . and passive state labels by CI,/3, y, . . . . Then the P- and Q-space projection operators are defined by P = c IA)<& A

(4a)

Q = c laX4,

= 0,

a

and PfQ=l

defines the Hilbert space we are dealing with. The nuclear hamiltonian has been traditionally divided into two pieces H = H,+H,, Ho - TfU, H,

= V-U,

and the study of a truncation of the full hamiltonian usually requires that the model space be degenerate with respect to Ho. Thus, traditionally Ho is chosen first and this choice is then used to define the P-and Q-spaces. Then, the many-particle states IA) and Icr} are eigenstates of Ho and we denote the corresponding eigenvalues as &A and E,, respectively, Ho(A) = sM>~

Holx)

= &ala>, EA

f

Ea.

173

CONVERGENCE

In diagrams, active states are denoted by lines and passive states by railed lines. Dots denote H, interactions. 3. Previous results In this section,

we collect results from three sources (Kuo et al., Ratcliff et al. and The formal structure of the energy-independent

be 11), two of which are unpublished.

effective hamiltonian (H&r) is systematically analysed by Ratcliff et al. operator The starting point for their derivation of Heff is the time-development representation. The key result is the decomposition (U(0, -00)) in th e interaction theorem and the central ideas developed are the notions of a Q-box (0) and a generalized folding operation. The Q-box is defined as a collection of one or more HI interactions and the fermion lines between these interactions which together form an irreducible, linked part of a valence diagram. Irreducible, here, means that the intermediate states between two successive H, interactions must be passive. The Q-box was given the formal definition

= +
s* A

&4)IA>Y

(1)

0

where the subscript L on the projection operator Qr specifies that the intermediate states must not only lie outside the model space but must also result in linked-valence diagrams. The energy-independent effective hamiltonian is defined as H&f = H, +N,

U&O, - co).

(2)

If the P-space is d-dimensional, the eigenvalues of (AIH,,,IB) coincide with d eigenvalues + of H and the eigenvectors of (AIH,,JB) coincide with the projections of the corresponding eigenvectors of H onto the model space. The operator Uo(0, -co) stems from a factorization of U(0, - co). This factorization is known as the decomposition theorem and is stated as U(O, -

cQ)LQ= c U,(O,-+9QwJ(O, I3

-co&o

The vector Ue(O, - co)lB) is defined by a series whose terms are grouped or classified according to the number of Q-boxes or, equivalently, the number of generalized folds. The vector U,(O, - co)lB> is defined ‘* 6, in fig. 1 where II@) represents core excitations and the railed semi-circle is essentially the Q-box without the last H1 interaction at t In the derivation of HCffby Kuo et al., it is assumed that the time-development operator can be analytically continued into the complex time-plane. This leads to the statement that the lowest-d eigenvectors of H having non-zero overlap with so called “parent states” can be constructed by appropriate application of the time-development operator and that H., reproduces the eigenvalues corresponding to these eigenvectors. However, we find, numerically, that the Herr obtained by Partial summations (subsect. 4.2 and sect. 5) yields eigenvalues of H which correspond to eigenvectors with the largest overlap with the P-space.

174

E. M. KRENCIGLOWA

Fig. 1. Dia~ammatic

AND T. T. S. KU0

structure of the vector U&O, - oo)jB>,

t = 0. The symbol, J, denotes a generalized fold. The notion of a generalized fold 5, “) is illustrated here by means of example. Consider two irreducible collections of energy diagrams, C1 and C,, not necessarily the same. The generalized fold is defined in fig. 2 where the times t; , tl are the times of the first and last H1 interactions of C, and tz is the time of the last Hs interaction of C, . The irreducible collections C, and C, also carry internal time labels in~cating the times of the other HZ interactions. The only restriction on the time labels of C, in diagram (c) of fig. 2 is t, c t,; a11 possibfe ordering of the internal time labels of C, relative to the internat time labels of C, are included in diagram (c) of fig. 2 subject to the constraint, t, < 1,. The right-hand side of fig. 2 provides a means of evaluating a generalized folded diagram.

(0) t+ tz< f,

(c)

(b)

t**t;
t;ctt

, fzcf,

Fig. 2. De~tio~ of a once generalized fold for two irreducible collections of energy diagrams, C1 and C,. The time constraints for the various terms are indicated and the state label P indicates a summation over all states in the P-space.

Thus, Heff is formally defined by eq. (2) and was analyzed, in detail, for 0-, l- and 2-valence nucleons by Ratcliff et al. For the case of 2-valence nucleons, the final result was a clean separation of the 0-, l- and 2-body contributions to &, the O-body and l-body terms being, respectively, the exact core energy and exact (ex~rimental) one-body energies. Of more interest to this work is the structural simplicity of this formulation. It was noted that a formulation of Hen in terms of Q-boxes represents, at least, a clean computational procedure. The schematic structure of H,,, is given in fig. 3 and we denote this as N,, = dE,fF,i-F1+F,+F,+

...,

(31

where I;;, denotes the n-generalized folded contribution to HePf and d.E, is the exact core energy minus the unperturbed core energy. The n-generalized folded contribu-

17s

CONVERGENCE

Fig. 3. Diagrammatic

representation

of (ASH&>.

tion can be expressed in terms of the Q-box and its derivatives as follows:

where

Here, we have explicitly assumed that the F-space is degenerate with respect to H,, and defined &A= Ed = . . . = z+. The number of terms in F,, increases rapidly with n; F4 has fourteen terms. We state a set of rules (11) for obtaining the terms of F,, nl_ 1: (i) The total number of Q-boxes, differentiated or not, is nf 1. (ii) The first Q-box must be at least once differentiated. (iii) The last Q-box must not be differentiated. (iv) Each individual Q-box can be differentiated up to n times subject to the constraints; (a) The sum of the powers of the energy derivatives is ~2.(b) If a Q-box is differentiated k times, there must be at least k undifferentiated Q-boxes to its right, not necessarily in succession. (v) With each mth order energy derivative, associate a factor of (nt!)-I. Thus a general term of F,, appears as

and F,, is the sum of all terms with the above form. Each term is characterized by the combination (ml,m,, m3, ...m,,), and all combinations subject to the above rules are allowed. This factorization of the terms of F,, into a sum of products of Q-boxes and its derivatives is accomplished by the introduction and formal definition of a generalized folding operation. This factorization is not necessarily possible for individual diagrams that contribute to F,,. The once-folded class of diagrams in fig. 4 carries the time constraints, t: < t2 < t,, where t; is the time of the first HI interaction of the first Q-box, t2 is the time of the last HI interaction of the second Q-box and t, is the time of the last HI interaction (this is the time associated with the HI in eq. (2)). To

176

E. M. KRENCIGLOWA

AND T. T. S. KU0

Fig. 4. Diagrams contained in Ft. The time constraints are t ‘1 < tZ < t,. The sum of diagrams (a) and (b) exhibits the character of the overaIl factorization of Fl = &e.

illustrate the character of the factorization of P, , we turn our attention to the diagrams Xabelfed (a} and (b) of fig. 4. Diagrams (a) or (b), in~vidually, do not exhibit any factorization. However, the sum (a)+(b) factorizes. This is explicitly shown in two ways. First, the sum of diagrams (a) and (b) constitute a generalized fold and the definition of a generalized fold ‘, “) yields immediately -&9+@)1

=c~B(~INIIP~~PIH~l~><~lr.ilia>(ul~,I~>

-tr~C-&,,c&B-FC)c~B-~~)}-ll

x ~(is_d(&~ld(&~_&~))_’

=-g<“’

1

[x

&B--&C

=

=


"tk"","'"l B

B

_

pfl~J;'"l] B

ic>(ci~~l~(ni~iIB) B

a

--&a

~q(&S)--(&C)llC)
$
p

C

[~~~&)I

cd+)P>, E=Ep

where q, analogous to the Q-box for the purposes of this example, is defined as

We assume that the model space is degenerate and interpret

CONVERGENCE

177

as a derivative. Second, if we use the usual diagrammatic rules, then

Here we note, again, that

Ed =

Ed

=

. . .

=

Ed.

4. Partial summation of the folded-diagram series 4.1. VALENCE

PART

In this section, we examine the structure of the valence part of the effective interaction. We write the valence part of the effective interaction as &-t-U, = Fo+F,+F,-t

1. ‘,

where F,, n 2 1, is the n-generalized folded contribution composed of sums of terms that are products of n-i- 1 Q-boxes, differentiated or not, as described previously. It is convenient to indicate the time constraints, hence terms, of F, diagrammatically. In fig. 5, we indicate the diagrammatic structure of the terms of E; and F2. All time references indicated in fig. 5 are either the time of the first or last R, interaction of a given Q-box. The time constraint arrow fkes the time associated with the tail of the arrow to lie within the time boundaries of the Q-box referred to by the head of the arrow. We define a partial summation, S, , of those terms in the effective valence interaction (uV) which have the time references to the first Q-box only. Thus, we have S, = ~~~~~~~~~)~, where the Q-box and its derivatives are evaluated at aP, Q = e^(srJ). The diagrammatic structure of S, is given in fig. 6. Now, S1 does not contain those terms in v,. which have derivatives on Q-boxes other than the first Q-box. Define S2 as a partial summation of the S, S, = &- *&&fS,)l.

178

E. M. KRENCIGLOWA

fj h

AND T. T. S. KU0

TIME

DIAGRAMMATIC EXPRESSION

CONSTRIANTS

ALGEBRAIC EXPRESSION

t2-s t,

-6,G

t, ---

____f2

ti ---

t;

iz

<

t’, <

f,ct,

ti

C

t3Ctj

t;<

t2 < t,

t, --_ t 3

tp ts
G,ij,d

* __ tt

Fig. 5. Examples of generalized time constraints. constraints and corresponding

Diagrammatic representation of generalized time algebraic expression are listed.

The diagrammatic structure of S, is given in fig. 6. Here, the time associated with the last interaction of any S, is constrained to lie within the time boundaries of the first Q-box only. It is clear that the time constraints for each term of S, are well-defined and distinct and thus S, can be expressed solely in terms of Q-boxes with distinct

Fig. 6. Diagrammatic definition of the partial sums S1 and S,. Note that all time constraint arrows reference the first Q-box only and S, is defined in terms of SI.

CO~ERGENCE

179

time constraints. That is, all the diagrams contained in S, are also contained in 0,. Thus, if we defined Sfii-1= then

all

e+i&?&,~~

(4)

diagrams contained in S,, 1 are also contained in ZJ,,. This suggests that if

exists, then

The suggestion is based, first, on the “counting of terms” argument presented above and, second, on the observation that the final form, S, , is of the form of Brandow’s result “).

We stress that, here, we have been dealing with the valence part of the effective interaction and that the Q-box is defined with a projection operator (Qr) in eq. (1). Finally, S, is not analogous to Brandow’s 7Y, since ?V=, for finite n, depends on the exact energy. 4.2. EFFECTIVE

WAMILTONIAN

In this section, we show that when the folded diagram series for He,, is convergent, it can be summed up to all orders and is equivalent to the Bloch-Horowitz, energydependent effective hamiltonian. We view the expression for S, as a suggestion for the following assertions. Here we take ihe particle vacuum as the “core”. Then, the Q-box becomes

where Q is just the Q-space projection operator. Then

Q(e)=

H, fH,

-_.-L-H,

z--QHQ

(54

is a solution for Q(E). Ifs is an eigenvalue of f3, then Q(e) is just the energy-dependent interaction and eq. (5) is just the Bloch-Horowitz equation. The assertion is that if

exists, then

180

E. M. KRENCIGLOWA

AND

T. T. S. KU0

where S,, is defined by eq. (4) and & by eq. (5) with E = sP . Eq. (6) has been given by Hoffmann et al. 12) and quoted by Schucan and Weidenmfiller ‘) with references to Brandow “) and Des Cloizeaux 13). In the following, S, is explicitly summed to a closed form. Assume that the P-space is d dimensional and that /1c/J is an eigenvector of H,,, with corresponding eigenvalue Ei. We assume that the P-space is degenerate with respect to HO, therefore I$i> is an eigenvector of S, with eigenvalue AE, = Ei -r+. We have Q^= PHI P+PH1 & = ;

gl &E),_

QA-‘QH,

P,

= PH, QA-I(

- A-l)‘QH,

P:

where A = +-- QHQ. We want to explicitly sum the right-hand side of S, = Qfl$(SrnY = PHI P+ -f PH, QA-l(-A-i)zQHl z=o

But, we have S,l$r) l$J, we obtain

P(S,)‘.

(7)

= dEil~i>. Therefore, if we multiply eq. (7) from the right by

[pH,p+l~oPH~QA-'(-A-')'eH~P(AE,)'Illiri)

SmIrki> =

=

[PHI P+PH1Q(AEi+A)-‘QH1

= [PHI P +PHi

P]l$i)

Q(Ep+ AEi-QHQ)-‘QHl

P]ltii)*

Thus,

sml$i> =

Q@i)Itii>3

CHo+Sm]ltii> = Eil$i> = [Ho+Q(EJIltii>,

(8) (9)

where the r.h.s. is identically the energy-dependent effective hamiltonian for the energy E, operating on the corresponding eigenvector. Eq. (8) is true for any l\c/i>, i = 1, 2, . . ., d. Let U be a matrix with I~i> as its columns u = (l+1>, W2>> - - *, I$,>>. Then, by eq. (8) or S, = (W,)ItiA = UAEU-I,

cw2)I$2>~ . * -9&(whf>)~-l

(10.1) (10.2)

CONWRGENCE

181

is a column vector and AE is a diagonal matrix with elements where each $(E#J AE,, AE,, . . ., AE,. Thus, the final result is that I& = Ho + S, and the eigenvalues and eigenvectors of He, have been shown, explicitly, to be solutions of the BlochHorowitz equation. That S, has the form given is obvious, at least in retrospect. The partial summations are defined recursively and with the above prescription, S II+1 can be summed into closed form immediately. Let h@‘> be an eigenvector of J!?e+S,, with eigenvalue Ept and Ufn) be a matrix with ]I,@“) as its columns, then S

“+1

=

(Q(Ep)/$I”)), (2(Ey)I$‘,“‘),. . ., ~(E~))13/~)))(u(n))-1.

(11)

This procedure preserves the notions of an active state and passive state (or Pand Q-space) and it is of interest to note that in the derivation of the Q-box formulation of H,, many general results are formulated in terms of active and passive states. With a state fo~ulation of &, we can choose the P-space states first and define He so that the P-space is degenerate with respect to He. Specifically, assume that the full hamiltonian, H, in the full space, P+Q, is given (that is, the P-space has been chosen). Now, define

where EP, E, are constants. The model space is thus defined to be degenerate with respect to Ho. In this separation of H = lYe+ HI, only the matrix elements of H, and HI between the many-particle states in the PS Q space are defined. This is not a statement about the traditional introduction of a one-body potential U which is then used to determine the many-nucleon matrix elements (A/T+ U(B) or {AIH- U]B>, say. It is clear that with this definition of Ho and HI, .zpis a parameter introduced to start the partial summations S, and that H,, = He -i-S, is, in fact, independent of sP, 5. Model applications and discussion In this section, we discuss truncations of the Hilbert space for two exactly solvable model hamiltonians, labelled H’ and H”I. The results will be based on the state formulation of the expansion for He, (i.e., the state formulation entails the notions of passive and active states without any specific reference to the many-body nature of the problem). Heft is computed by partial s~mations and by the folded-~a~arn series and the results are compared. Our results will show that the problem of an intruder state can be dealt with consistently within the framework of the foldeddiagram series and the partial summations. The full hamiltonian is separated into He and H, parts

H, = H-Ho,

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E. M. KR~NCIGLOWA

AND T. T. S. KU0

TABLE

1

Matrix elements of model hamiltonian State labels

I*>

12)

-10.74

-0.61 -13.54

::i (St

H’ ((ilH’lj>

-71
13)

14)

3.23 1.74

-0.35 0.57

1.75 -0.24

-8.14 -1.66

0.19 0.74 -0.08

-10.41

Here the matrix elements of Ho are -10, to 1.

-10,

-5,

-5, 0 and the strength parameter

IV

x is equal

and we will be interested in truncations to the full ha~ltonian H(x) = H,+xEI, = (H,+dP)+(xH,-LIP), as a function of x and A. The strength parameter x allows us to vary the interacting hamiltonian Hi and the shift parameter A allows us to vary the separation between the Q-space and P-space ~pert~bed spectrum. The matrix elements of H’ are the KuoBrown bare G-matrix elements plus the single particle energies for A = 18, T = 0, J” = 1’ and the full Hilbert space, of dimension 5, is taken to be the s-d shell. The matrix elements of H’ are given in table’ 1. Truncations in the s-d shell have been studied by Barrett, Halbert and McGrory 14) who explicitly construct E&r by a technique analogous to eq. (10.2). The second model hamiltonian (H”) is from Hoffmanetall’) an d is g iven in table 2. References to a particular hamiltonian, H’ or Hi’, will be exhibited with superscripts, I or II, respectively. To start, the F-space is chosen as p = l~X~l+l2X21 for both hamiltonians. The Q-box is obtained by eq. (5.2) and is evaluated at the TABLE

2

Matrix elements of model hamiltonian State labels

(11 (21 (31 <41

B” (

= >

11)

12)

13)

14)

1

5 26

0 5

5 0 I 4

-2

Here the matrix elements of Ho are 1, 1, 3, 9 and the strength parameter x is equal to 1.

183

CONVERGENCE

‘%mperturbed energy”, sp+ A. The partial summations are evaluated according to eq. (11) and in this presentation, the maximum value for n in S, is 1000. We found that

if the partial summations converged to Heff , say, for a given x, Ho and A they converge to the same He, for the same x and H,, but different A. If the partial summations are divergent for a given x, He and A, they remain divergent for different choices of A. However, divergent sequences of partial s~mations for He,, sometimes yielded one convergent eigenvalue and corresponding projected eigenvector (recall, the P-space is 2-dimensional). This can be understood from the structure of P&r, in equation (10.1). When convergent, the partial sums yield an Herr which reproduces the eigenvalue and corresponding projected eigenstates of the full hamiltonian which have the largest overlap with the P-space. Some of these points are illustrated in figs. 7 and 8, where the eigenvalues of various operators related to H”(x) are plotted as a function of x, 0 5 x 5 0.3. Here A = - 11 and H, is the same as in HofIman et al. 1“) and is the diagonal matrix with matrix elements (1, 1, 3,9>. Only the eigenvalues of the first two partial sums are shown but partial sums of higher order were evaluated to establish convergence. In fig. 8, we see that different pairs of eigenvalues of H”(x) are reproduced by the partial summations; the pairs of eigenvalues (E, , E,), (El, Es) and (E, , I&) are reproduced as x increases. The eigenvectors corresponding to these

6.0-

J

Fig. 7. Eigenvalues of operators relating to Nrt as a function of x. The exact eigenvaluesof P(x) are indicated by solid lines and IabelIedas E3, Ez , Es, and Ed. Short dashed lines indicate Q-space eigenvalues (QH”(x)Q) and long dashed lines indicate eigenvahzesof HO-&).

184

E. M. KRENCIGLOWA

AND T. T. S. KU0

8.0 -

6.0 -

0

0.1

0.2

X

0.3

Fig. 8. Eigenvalues of the first two partial summations for H$ as a function of X. The dashed lines indicate the eigenvalues of Ho+Sl(x) and solid lines indicate the eigenvalue of H,,+S,(x). The bent arrows and labels E;, i = 1,2, 3, 4 indicate regions for which the eigenvalues of Ha+&(x) reproduce the exact eigenvalues &, i = 1,2,3 or 4, to the scale of the graph.

pairs were observed to have the largest overlap with the P-space. The total probability of an eigenvector (IYJ) of H”( x ), corresponding to the eigenvalues reproduced by the partial summations, to be in the P-space is (YilPlYi>/(YilY~>

2 0.89

except for neighborhoods of Ax z 0.02 about the singularities at x x 0.06 and 0.26. The details of the eigenvalues of He + S1 in fig. 8 are not shown for 0.0628 < x < 0.0632; the two eigenvalues of Ho+S, do not cross or coincide in this region of x. The region, 0.262 < x < 0.264, is also not shown in fig. 8 and we describe the behavior of the eigenvalues of Ho +S1 in this region as follows. In fig. 8, the two eigenvalues of Ho + Sl for x s 0.262 are labelled (a) and (b). As x increases from 0.262, the eigenvalues corresponding to curves (a) and (b) of fig. 8 approach each other, coincide, become complex conjugates and then coincide again. As x increases further, one eigenvalue of Ho+ S, emerges in fig. 8 as the curve labelled (d). The other eigenvalue goes more negative, experiences a discontinuity, becomes positive and emerges in fig. 8 as the curve labelled (c). This discontinuity arises because S,(x) has a pole when the upper eigenvalue of Ho + &(x) coincides with the upper eigenvalue

CONVERGENCE

18.5

01 QH(x)Q. The eigenvalues of H,, + S, at some points in the interval 0.262 5 x S 0.264 are given in table 4. The partial summations for H” fail to converge for 0.9 < x < 1.3 but one eigenvalue and eigenvector does converge to the largest eigenvalue and corresponding projected eigenvector of H”(x). For example, at x = 1.0, the second partial sum yields an eigenvalue of 27.8229 whereas the corresponding exact eigenvalue is 27.8232. The total probability that the eigenvector corresponding to the largest eigenvalue be in the P-space is greater than 0.95 whereas this probability is less than 0.5 for the remaining eigenvectors for 0.9 < x < 1.3. The folded-diagram series for H,‘if, eq. (3) with AEc = 0, was evaluated up to and including F4 with A = 0 and it fails to converge for x 2 0.06. The partial sums for H’(x) fail to converge for x $ 0.8 and reproduce the lowest two eigenvalues of H’(x) for x < 0.8. H, was the diagonal matrix with matrix elements (- 10, - 10, - 5, - 5, 0). The folded-diagram series, with A = 0, evaluated through F4 diverges for x 2 0.3. The region of convergence for the folded-diagram series may perhaps be extended by choosing a different A but an important observation here is the potentiality of a large difference in x between the regions of convergence for H,,,(x) when evaluated by partial summations and when evaluated by perturbation (the folded-diagram series). The nature of the singularities in H,,,(x) have been studied carefully by Weidenmiiller and collaborators ‘, 2, l2 ), and several schemes have been offered to deal with the singularities. The partial summations presented can handle some of the singularities as exhibited by fig. 8. If the partial summations fail to converge, then so will the folded-diagram series. Moreover, the divergence of the folded-diagram series may merely reflect a poor choice of E+, rather than the more profound entity, the intruder state, which is the usual label for the cause of the singularity. That the partial summations for Herr are convergent, can be given the interpretation that the separation of the full space, into a particular P- and Q-spaces chosen, in fact, represents an approximate decoupling of the full hamiltonian, and conversely, if the partial sums diverge. From fig. 8, it is seen that the decoupling need not coincide with the lowest eigenvalues of the full hamiltonian. The question of convergence of the partial summations for Herr then reduces to the rather simple and obvious statement that, for a given choice of P-space of dimension d, there exist d eigenvectors of the full hamiltonian, lYi>, i = 1,2, . . ., d, that are predominantly in the P-space. That is, Iyi> -

pIyi>,

or equivalently,

l N l9

i = 1, 2, . . ., d,

(12) is a reasonable approximation. This is simply a statement of the implicit assumption that we are in fact dealing with a perturbation on these eigenvectors and is neither new nor profound. For the model hamiltonians presented, a typical number for the quality of approximation (12) is

/( yil l”i) 2 Oe8

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E. M. KRENCIGLOWA

AND

T. T. S. KU0

for convergence of the partial sums in a few iterations. In terms of fig. 8, condition (12) is not satisfied for 0.05 5 x 5 0.09 and 0.24 5 x 5 0.28. The precise reason for failure of the partial summations to converge is not clear to us. It is useful to consider the properties of H(x) 5 He +xH, . The partial sums converge within a few iterations if approximation (12) is good. Near crossing points such as x x 0.06 and 0.26 in fig. 8, the full eigenvectors corresponding to the crossing have comparable probability in both the P- and Q-spaces. The crossing point near x E 0.06, for example, can be interpreted as a mixing of an intruder state, corresponding to the lowest QHQ eigenvalue which originates from E, at x = 0 in fig. 7, into the model space. The mixing of intruder states into the model space is strong near a crossing point and the partial sums do not converge within a few iterations. However, the mixing of intruder states into the model space need not be strong only near a so called crossing point but can extend over a region of x, obviously, and we do find regions of x where S1 e0 ,, had not converged. In as much as the partial summations represent exact summations of classes of folded-diagrams, then approximation (12) must also be reasonable if the folded-diagram series is used to evaluate H,,. Within the framework of the partial summations presented here and the Q-box perturbation analysis of refs. 5, 6), the P-space can be chosen at our discretion to suit particular needs. The conventional choice for the P-space leads to a linked-valence, connected perturbation expansion in the Q-box and its derivatives 5, “) or in H,(9). That the conventional choice for the P-space represents a good choice in practical nuclear applications has been questioned by Weidenmiiller and collaborators, in view of experimentally observed low-lying nuclear energy levels whose structure is believed to be predominantly outside the conventional P-space. These low-lying states are labelled as intruder states and are essentially a consequence of the overlapping of the PHP and QHQ spectra. Various schemes to deal with intruder states have been presented and evaluated by Weidenmtiller and collaborators. The partial summations of Heff can handle intruder states, if, in fact, approximation (12) is reasonable for the particular P-space chosen. In the following, we assume that we are interested in the effective hamiltonian that reproduces the lowest eigenvalues. If the partial summation for He,, fail to converge, then assumption (12) is probably not reasonable. This can be translated to say that there is an intruder state and this intruder state is strongly coupled to the P-space by H. The presence of an intruder state is a manifestation of the choice of P-space. The intruder state can be removed by making a choice for the P-space consistent with approximation (12) but this can be difficult, apriori. It is clear that in order to make a better choice for the P-space, some knowledge of the structure of the intruder state is necessary. A new P-space can be defined to incorporate the knowledge of the intruder state, in some manner. This sort of procedure was mentioned by Hoffmann et al. 12) within the different context of modification of the single particle hamiltonian. We label the procedure as a P-space modification and contend that the consistency of the procedure can be verified. The P-space modification can be teste,d by performing the partial summations

CONVERGENCE

187

but the goal is to have the folded-diagram series converge as a test for the P-space modification. This goal is desirable because it will demand a high quality of the truncation [approximation (12)] but suffers from the drawback that non-convergence may merely reflect an inappropriate choice of .sp. Also, given that the folded-diagram series appears to converge to a given order, vt+ 1, in Q-box, convergence (or lack of ~~ve~rgence) to higher order in Q-box can be readily tested by evaluating equation (7) to finite order in I, I S n., since Qi, I I n, have already been evaluated. In the following, we discuss, by means of example, an application of P-space modifications for the effective hamiltonian for H’ given m table 1. We choose not to change the dimension of the P-space in this example although we do have the freedom to increase or decrease the dimensionality of the P-space. The original P-space is as discussed previously P = If><2IIt is clear, from inspection of table I, that the state 13) is strongly coupled to P-space. Recall, that for this P-space the partial sums fail to converge for 2 ,> At x % 0.8, a PH’P and QH’Q eigenvalue cross. The state corresponding to lowest eigenvalue of QH’Q, is the intruder state here. It is not difficult to obtain intruder state but we choose to illustrate the procedure by making the guess for intruder state, ]I),

Then, the ([3>, 14)) subspace of the full hamiltonian, unitary transformation

Hr, is transformed

the 0.8. the the the

by the

That is, the full hamiltonian, H’, is appropriately transformed so that the guess for the intruder state becomes a basis state for the representation of H’. The hamiltonian matrix is then transformed by the unitary transformation that diagonalizes the 3dimensiona subspace {Il>, ]2>, ]I>) of fir. The projection operator formed from the two eigenvectors corresponding to the lowest two eigenvalues of H’ in the subspace (]I), ]2), ]I>} is defined to be th e new P-space. The new P-space is, thus, 2-dimensional and each P-space state is a linear combination of I l), ]2), ]I>. The net transformation of the full hamiltonian in this procedure is, of course, unitary. For the old P-space, was 0.66 and 0.86 for the eigenvectors of the lowest two eigenvahtes, whereas, for the new P-space this ratio was 0.91. The partial summations and folded-diagram series are computed as outlined previously. The results for a diagonal and an off-diagonal matrix element of HePf for the new P-space are given in table 3. In table 3, the sp dependence of the partial summations and folded-diagram series is illustrated. The choices for sp are the eigenvalues of PHP. The partial summations converge for both values of ap. The folded-diagram series appears nicely-convergent for &pequal to the smallest PHF eigenvalue. It is clear from table 3 that

188

E. M. KRENCIGLOWA

AND T. T. S. KU0

TABLE 3

Convergence

of a diagonal and off-diagonal matrix element of the effective hamiltonian after P-space modification ep = -14.892 diagonal matrix element

0

-15.546 -15.428 - 15.455 -15.448 -15.450

1 2 3 4 exact

ep = -11.635

off-diagonal matrix element

-15.546 -15442 - 15.456 - 15.44s - 15.450

for W’

0.525 0.844 0.737 0.770 0.760

-15.449

diagonal matrix element

0.525 0.692 0.745 0.759 0.762

-16.187 - 15.348 -15.466 -15.447 -15.450

0.762

-16.187 -13.887 - 19.854 -0.903 -68.141

off”diagona1 matrix element

1.059 0.705 0.777 0.759 0.764

- 15.449

1.059 0.276 2.053 -3.426 15.828

0.762

Tlte ep dependence of the partial summations (He+&) and folded-diagram series (C Fr) is illustrated. The corresponding exact matrix elements are shown. Note that the folded-diagram series for ep = - 11.635 is diverging.

choosing .spas the largest PHP eigenvalue is not a good choice for the folded-dia~am series and illustrates that divergence of the folded-diagram series may merely reflect an inappropriate choice of +_ Choosing tip has been discussed in a slightly different context, by Schucan and Weidenmfiller “). The partial summations and the foldeddiagram series (for sp = - 14.892) reproduce the lowest two eigenvalues and corresponding projected eigenvectors. The convergent results in table 3, themselves, verify the consistency of the P-space modification. The particular technique for a P-space modifi~tion presented was for illustration and not meant to be the onIy technique, clearly. TABLE

Eigenvalues of &+S~(X)

4

for H” for some values of x in the interval 0.262 5 x 2 0.264 Eigenvalues of Ho + S1 (x)

x

0.999 2.534 2.545% 0.8141 -4.340* -4.368 -8.991 -43.401 88.799 14.605

0.262 0.26253760 0.26253761 0.263 0.26327519 0.26327520 0.2633 0.2634 0.2635 0.264 Note the complex conjugate eigenvalues

{i =

dli).

5.349 2.555 4.775 x lo-% 2.986i 4.943 X lo-? -4.312 -2.362 -1.176 -0.776 -0.133

CONVERGENCE

189

In concluding this presentation, we stress that discussion on the folded-diagram series has been solely for HO that is degenerate with respect to the P-space and that discussion has been directed at the state formulation as opposed to a linked-valence and connected formulation. In practical perturbation calculations, a linked-valence and connected formulation is used and H,, is separated into a sum of effective hamiltonian operators of given particle rank. For example, for “0, H,,, is separated as Herr = &@-body) + K&-body) + H&2-body), (13) when I60 is taken as a closed core, and each term in eq. (13) can be calculated separately. In practice, H,,, (2-body) is calculated using degenerate perturbation theory and H,,, (l-body) is taken from experiment. However, Heff (l-body) can be calculated and the unperturbed hamiltonian (H,) used to compute Heff (l-body) need not coincide with the unperturbed hamiltonian used to calculate H,, (2-body). Thus, in practical applications, the procedure for calculating H,,, is not fully equivalent to degenerate perturbation theory and therefore the implication, on practical application, of our results for the degenerate folded-diagram series, particularly divergence in the presence of an intruder state, must be judged with this in mind. We would like to thank Dr. E. Osnes for numerous interesting discussions in this area, and Dr. K. F. Ratcliff for very informative and enlightening discussions on the problem of intruder states. Note added in proo$ We kindly thank M. R. Anastasio and J. W. Hockert for pointing out that the partial summation, eq. (1 l), is more simply derived by noting that S,+,l$$“‘) is a Taylor series. References 1) 2) 3) 4) 5) 6) 7) 8) 9)

10) 11) 12) 13) 14) 15)

T. H. Schucan and H. A. Weidenmtiller, Ann. of Phys. 73 (1972) 108 T. H. Schucan and H. A. Weidenmtiler, Ann. of Phys. 76 (1973) 483 B. R. Barrett and M. W. Kirson, Nucl. Phys. Al48 (1970) 145 T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 T. T. S. Kuo, S. Y. Lee and K. F. Ratcliff, Nucl. Phys. Al76 (1971) 65 K. F. Ratcliff, S. Y. Lee and T. T. S. Kuo, unpublished M. W. Kirson, Ann. of Phys. 66 (1971) 624 E. M. Krenciglowa, T. T. S. Kuo, E. Osnes and B. Giraud, Phys. Lett. 478 (1973) 322 B. H. Brandow, Rev. Mod. Phys. 39 (1967) 771; B. H. Brandow, Proc. Int. School of Physics “Enrico Fermi”, Course 36, Varenna, 1965, ed. C. Bloch (Academic Press, New York, 1966) C. Bloch and J. Horowitz, Nucl. Phys. 8 (1958) 91 S. Y. Lee, thesis, SUNY, Stony Brook 1972, unpublished H. M. Hoffmann, S. Y. Lee, J. Richert, H. A. Weidenmiiller and T. H. Schucan, preprint J. Des Cloizeaux, Nucl. Phys. 20 (1960) 321 B. R. Barrett, E. C. Halbert and J. B. McGrory, in Symposium on correlations in nuclei, Balatonfiired, Hungary, September 3-8, 1973 H. M. Hoffmann, S. Y. Lee, J. Richert, H. A. Weidenmtiller and T. H. Schucan, Phys. Lett. 45B (1973) 421