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Long-range Jastrow correlations
Nuclear Physics A242 (1975) 389 - 405; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
LONG-RANGE JASTROW CORRELATIONS E. K R O T S C H E C K and M. L. RISTIG
Institut ffir Theoretische Physik, Universitiit zu K61n, 5 Kfln 41, Germany Received 14 August 1974 (Revised 22 November 1974) Abstract: We develop a promising many-body method to evaluate the equation of state for dense neutron matter and liquid helium. The ground state of the Fermi fluid is described by a conventional Jastrow ansatz. We admit the presence of short- and long-range correlations. Under this assumption we study the generating function which has been introduced by Wu and Feenberg. We employ a graphic formulation and develop the diagrammatic expansion of the generating function and the radial distribution function. If long-range correlations are assumed, the diagrams have singular parts. We give a proof that the total contribution of such diagrams to the generating function which contain two, three, and four correlation lines is of finite value. The same property is shown for a selected class of singular diagrams containing ct correlation lines (ct > 4). To verify the cancellation phenomenon we introduce a two-body function which serves graphically as an insertion into selected singular diagrams. For the remaining classes of diagrams we need three-, four-, . . . , n-body insertions. The result is cast into the form of a theorem. The cancellation rests on the exclusion principle and does not depend on the special shape of the correlation function. Finally, a generalized hypernetted-chain summation of diagrams which represent the radial distribution function is executed. The procedure includes exchange contributions and can be employed if shortand/or long-range correlations are present.
1. Introduction There has recently been an upsurge of interest in a precise description and quantitatively accurate evaluation of the equation of state for dense and superdense neutron matter 1-6). It has become clear that a rather sophisticated many-body treatment has to be developed if a reliable prediction of a solidification of this system of fermions is to be ventured 3). Most of the available work which has been done for the liquid phase in the highdensity regime employs the familiar Jastrow ansatz ~P = F ~ ,
F = 1-I f(rij),
f2(r) = 1 +r/(r),
(1)
i
to include properly short-range correlations by a suitably chosen function f ( r ) or a correlation factor t/(r) which, more generally, might be assumed to be statedependent 7- 9). The model wave function q~is taken to be the ground-state function of a gas of A independent fermions. In addition, the ground-state energy of the system has been approximated by the expectation value of the Hamiltonian 389
390
E. KROTSCHECK AND M. L. RISTIG
H --- T + V with respect to the correlated state (1): E = E K+ Ep -
+
(2)
A first generation of such studies of neutron matter employed the well established Iwamoto-Yamada expansion 10) or its factorized version (FIY expansion 11,12)) to evaluate the expectation value (2). The correlation factor has been assumed to be of short range and the expansion has been truncated at the two- or three-body level 7-9,1 a, 14). At normal nuclear density this approximation might yield reliable results for the ground-state energy 9). In the more interesting range of high density, the procedure is not sufficient and a more accurate many-body method is called for 15). To overcome this problem several groups agreed to begin by studying a simple "homework problem" which can be attacked by already existing powerful manybody techniques. In the model the neutrons interact in all states through the repulsive part of the Reid 1So potential, and Boltzmann statistics is assumed for further simplification 3, 15). Numerical resu!ts for the equation of state in hypernetted-chain (HNC) approximation 16) and Percus-Yevick approximation 17) are now available for the fluid model problem15). In addition, a paired-phonon analysis 17) has been successfully performed 1s). This procedure takes into account short- and long-range effects. Both kinds of correlations are present in the o p t i m i z e d ground state 1~, 19) described by the factor F of ansatz (1). The long-range behavior of the correlation factor is defined by the relation q(r) = cr -2,
r--* 00,
(3)
the coefficient c being independent of the distance r. In addition, elaborate improvements have been made to grasp the influence of three-phonon corrections 17) to the ground-state energy of the fluid 18). We believe that the achieved accuracy now justifies the advancement towards a more realistic "homework problem" in which the Fermi statistics of the neutrons is properly taken into account. This enterprise should of course be done on the same sophisticated level on which the Bose fluid was dealt with. Such an ambitious treatment of the Fermi fluid must clearly permit long-range correlations of the type (3) in the trial ground-state function (1) as was done for the Bose fluid. We should note, in addition, that for formal as well as practical reasons it is not possible to separate short-range from long-range correlations in a clean manner. The optimization procedure for the expectation value of the groundstate energy (2) (without subsidiary constraints) with respect to a correlated wave function (1) leads inevitably to the appearance of long-range correlations2°).
JASTROW CORRELATIONS
391
The same difficulty is well known in the theory of Bose fluids 17,19) and of quantum solids 21). In a first stage we therefore must develop a many-body method which is able to deal with short- a n d long-range correlations in a system of interacting fermions. If we admit only short-range correlations we may expand the expectation value (2) into the FlY cluster series provided the value of the smallness parameter of that theory is sufficiently small 22). But the n-body cluster terms (n > 4) diverge if we permit long-range correlations (3). Thus we have to search for a suitable rearrangement of the FlY expansion. In the present paper we rearrange the contributions to the FIY expansion of the generating function of Wu and Feenberg 17,23). We collect terms which have the same singular behavior in the presence of long-range correlations (3). To achieve that goal, most efficiently, we employ and develop further a graphic description which has been initiated by Gaudin et aL 24). Our study reveals that the collection of a finite number of divergent diagrams containing the same number of correlation lines leads to a complete cancellation of their singular portions. Thus the ordering of diagrams according to the number of correlation lines involved provides a sensible classification scheme if long-range correlations are admitted. In a further step we perform a hypernetted-chain summation of the contributions to the rearranged expansion of the radial distribution function. Ignoring the influence of Fermi statistics, the generalized H N C equations reduce to the familiar HNC result for a Bose fluid. The application of the proposed method to the homework problem would, on the other hand, tell how important the effects of differing statistics are. It is evident that the presented many-body technique can be applied to the problem of liquid aHe. A detailed comparison with the results of the established Wu-Feenberg approach 17) would be highly informative. In sect. 2 we present the factorized cluster expansion for the generating function. We introduce the graphic notation of the cluster terms in sect. 3. We summarize the diagrammatic rules and discuss two ordering schemes for the diagrams. We turn in sect. 4 to the problem of long-range correlations and study the singularities of divergent diagrams. Sect. 5 is devoted to the hypernetted-chain summation which is performed for the diagrammatic expansion of the radial distribution function. 2. Factor cluster expansion Cluster expansions of the expectation value of the ground-state energy (2) or other physical quantities with respect to a correlated wave function (1) can be derived most efficiently for a uniform extended Fermi fluid from a generating function 2a) G(fl) which has been introduced by Wu and Feenberg 25). This quantity is well known in terms of the FIY expansion which is ordered according to the number of particles (or hole lines) involved: G(fl) = (AG)2 +(AG)3 + . . . +(AG).+ ....
(4)
392
E. KROTSCHECK AND M. L. RISTIG
The contributions of the two- and three-body clusters are
ii (AG)3 = ~ ~, xijk(fl)-½ ~ Xo(fl)Xik(fl). ijk
(5)
ijk
The summations extend over occupied single-particle states inside the Fermi sea and the functions xo(fl), Xi~k(fl), ..-, are the familiar reduced cluster integrals which depend on a parameter fl suitably introduced 17, 26, 12). The explicit expressions of the four-, five-, and six-body terms of the function G(fl) are given by Wu 27). The general term can be systematically constructed, for example, by expanding the renormalized FIY expansion described in refs. 23, 28). The specific structure of the cluster integrals becomes more transparent if they are represented in terms of diagonal and non-diagonal matrix elements 23) r/il i.... i. and r/~l.... ~,,i.... ~.(2 =< n ___.4) of properly defined irreducible n-body operators taken between normalized and antisymmetrized states lil i2... in>a and ] j l j 2 . . . J.>~, Xij = ~ij,
Xi~k = tl ijk,
(6)
Xijkl = l~ijkl "31-(~ij, kl qkl, ij "~- qik, jl qjl, ik "~- qjk, it qil, jk)"
Incidentally, we should note that the introduction of these matrix elements provides a convenient definition of cluster integrals with coincident indices. Adopting special choice (1), the irreducible operators can be constructed in an elementary manner from the correlation factor t/(12) = q(r12 , fl) = f2(q2) exp [flv(rx2) ] - 1.
(7)
For our purposes the function v(r12 ) may be identified with the bare neutronneutron potential of the homework problem. The two- and three-body matrix elements are given explicitly: tlO, kl =
( ij[rl(12 )lk l >a .
thjk ' ,.,. =a.
(8)
The subscript a indicates that the matrix elements have to be taken with respect to antisymmetrized and normalized states. The assumption (1) has an important consequence for the general n-body con-
JASTROW CORRELATIONS
393
tribution of eq. (4). In the simplest case n = 3, for example, the cluster (d G)s collapses (by insertion of eq. (8) into eq. (5)) to the simple expression: (AG) 3 = ½ ~ (ijklrl(12)rl(13)ljki-ikj)
ijk + ~ ~ ( ijklrl(12)rl(23)~l(13)lijk)a.
(9)
ijk
The first term ofeq. (9) exhibits a cancellation effect which occurs in any order. Only irreducible integrals survive. This property is a special feature of the cluster expansion (4) if we assume a trial function of Jastrow type 29). A general formal proof may be found in ref. 30). The generating function may be employed to derive the FlY expansions of various physical quantities. For example, we generate the cluster expansion of the energy Ep (eq. (2)) by the straightforward prescription: Ep = ~ G(fl)l#=o.
(10)
For the investigations to come we are also interested in the radial distribution function:
g(r) = A ( A - 1)p-2N -1 ~
f~t~dr3dr4.., dra.
(11)
1
The quantity p is the particle density. It is related to the Fermi momentum kr by the expression p = v(6n2) - lk3F where the factor v gives the degeneracy (v = 2 for neutron matter or SHe, v --- 4 for nuclear matter). The constant N is the norm of the wave function ~ and ~0,~ indicates summation over all A spins (v = 2) and isospins (v = 4). Employing eqs. (10) and (4) we can extract the FIY expansion of the radial distribution function from the relation Ep = ½p2 f dr 1dr2 v(rl 2)g(rl 2)-
(12)
The result can be written in the form
g(r) = [1 + r/(r)]{gF(r) + ~ Ag.(r)}.
(13)
n=3
The function gr(r) is the familiar distribution function of a gas of independent fermions: gF(r) = 1--v-ll2(rkF),
l(x) = 3x-a(sin x - - x cOS x).
In eq. (13) the subscript n indicates the n-body cluster.
(14)
394
E. KROTSCHECK AND M. L. RISTIG
Closely related to the function (11) is the liquid structure function which is defined by S(k) = 1 + p f { y ( r ) - 1}eik"dr.
(15)
The shape of this function for small values of the momentum might be influenced by the long-range correlations (3). The structure function has been measured for liquid 3He by X-ray experiments 31). For our following discussions it is highly desirable to have an efficient formulation of the general n-body contribution to the generating function, the distribution function, etc. A graphic description provides such a powerful tool.
3. Diagrammatic expansion A simple graphic description in momentum space of the reduced cluster integrals and the generating function exists 17,26) but the specific features of the clusters assuming the ansatz (1) - are more apparent if we employ a graphic formulation in coordinate space. We adopt the diagrammatic notations of ref. 24) which originate from the Yvon-Mayer technique devised for classical systems 32). The basic elements of any diagram are (i) internal and external points (solid and open dots), (ii) correlation lines (dashed lines) and (iii) oriented exchange lines (oriented solid lines). (i) An external point labels the coordinates ri of the particle i. An internal point indicates a factor p and integration over the coordinate space of the particle involved. (ii) A correlation line connecting the points ri and r i represents a correlation factor r/(/j). (iii) An exchange line represents the exchange factor l(kF r~). These lines always form closed loops. Each distinct loop connecting p points is associated with a factor vx-P. The weight ( - 1 ) 1-P of the loop determines the sign of the diagram. Before presenting the formal rules which establish the diagrammatic expansion of the generating function let us study in some detail the two- and three-body contributions. Fig. 1 shows the graphs D~I)(1) and D~1)(2)which constitute the twobody cluster (AG)2. Here, and in the following discussion, we use the notation Dt,~)(s) for any diagram where the subscript n refers to the number of particles in-
D2(n(I)
020)(2)
Fig. 1. The diagrams D~)(1) and D~x)(2) which constitute the two-body cluster (AG)2. Superscripts refer to the number of correlation lines, subscripts refer to the number of particles involved. Both graphs have multiplicity m = 1.
JASTROW CORRELATIONS
395
volved and the superscript • indicates the number of correlation lines. The integer argument s = 1, 2 . . . . , denotes successively each graph of the set of diagrams with given numbers ~ and n. Using the scheme (i)-(iii) we easily extract from eq. (9) all five diagrams which represent the three-body term (AG)3. They are given in fig. 2. In graphic notation we may therefore write: 1
(AG)2 = ~.~ {O(21)(1) - O(i)(2)},
1
(AG)3 = 3.1 ~ - 3D (2) 3 (1)+3D (2) 3 (2)+D (3) 3 (1)-3D (3) 3 (2)+2D (3) 3 (3)}.
(16)
We stress that all diagrams are irreducible. The absolute values of the weight factors of the graphs D~I)(1), D~1)(2), D~2)(1), D~3)(1), D~3)(2) and D~3)(3) of eq. (16) may be identified with the multiplicity m of topologically distinct diagrams. But the coefficient of the graph D~2)(2) differs from the multiplicity (m = 6). This phenomenon can be explained by the existence of equivalent graphs for the uniform extended medium. Because of momentum conservation a diagram which belongs to a special class of graphs can have the same value as a topologically distinct diagram. The simplest example is shown in fig. 3. The equivalence of both graphs becomes apparent if we take notice of the relation
( ijkllrl(12)rl(34)ljkli> = ~ ( ijklrl(12)rl(13)ljki>, ijkl
(17)
ijk
where the states represent products of normalized single-particle functions. Property (17) can be expressed in a form which is more suited for our discussion:
13] 6D(2)(2)- 4!1 12D(2)(3) = ~ . 3D(32)(2).
/
(17')
\
Da (2) ( I ) /
Da(2) (2)
\
Da(3) (I)
Da(3) (2)
D3 (3) (3)
Fig. 2. The irreducible and topologically distinct graphs O~2)(1), D~2)(2), D~a)(1), D~3)(2), and D~3)(3) which constitute the three-body cluster (AG)3. Their multiplicities m are 3, 6, 1, 3 and 2, respectively.
D4(2) (3)
D3(2) (2)
Fig. 3. A simple example of two equivalent diagrams, D~2)(3) and D~2)(2), which are of the same value. Their multiplicities m are 12 and 6, respectively.
396
E. KROTSCHECK AND M. L. RISTIG
Here, we may equate the weight factors 6 and 12 with the multiplicity of the diagrams D~2)(2) and Dt42)(3). The coefficient of the graph D~2)(2) on the right hand side of eq. (1 7') is the weight which correctly appears in eq. (1 6). A general discussion of equivalent diagrams may be found in ref. 20). We are now ready to summarize the general formal rules which determine the graphic construction of the generating function. (i) The contributing diagrams are built from correlation lines and oriented exchange lines which connect irreducibly a given number of internal points. (ii) Each internal point of a diagram is joined by at least one correlation line. Two points are connected by at most one correlation line. (iii) Each point of a diagram is joined by at most one incoming exchange line. No exchange line terminates. (iv) The weight of a diagram with n internal points is given by its topologic multiplicity m divided by the number n! The sign of a diagram is determined by prescription (iii). The sum of all permitted and weighted diagrams constitutes the diagrammatic expansion of the function G(fl). For example, fig. 4 shows the contributing fourbody diagrams with two correlation lines. There are twenty-two topologically distinct diagrams with three correlation lines (without the equivalent graphs) which are permitted by the rules. To be more specific we have three three-body diagrams (D~a)(1), D~a)(2), D~3)(3) of fig. 2), eight four-body diagrams, six five-body diagrams and five six-body diagrams. Some irreducible graphs are presented in fig. 5. If we collect all weighted diagrams with n points and take into account, in addition, all associated equivalent graphs we reproduce exactly the n-body cluster
, ,
D4(a)(i)
D(3)(I)
D(a) (I)
W D4(al(t)
D4(a) (2)
Fig. 4. The four-body diagrams Dt42)(l) and D~2)(2)which contain two correlation lines. They have multiplicity m = 6.
D4(3) (2)
Ds(~) (2)
Ds(3} (2)
Fig. 5. The diagrams Dk~)(l), D~)(I), D~)(1), D(43)(2),Dg3)(2),and D~63)(2)with three correlation lines which contribute to the generating function. Their multiplicitiesm are 12, 60, 120,24, 240, and 240, respectively.
JASTROW CORRELATIONS
397
(AG), o f the generating function. Thus adopting the F l Y classification all diagrams with n points and their equivalent graphs are of the same order n. In contrast we may collect all diagrams with the same number o f correlation lines into a single term. This classification scheme establishes the power-series expansion a0, 3a) (PS expansion) G(fl) = (d G) tl) + (A G) (2) + ... + (A G) t') + ....
(18)
The superscript ~(~t = 1, 2 . . . . . ) indicates the contribution of all diagrams with ~t correlation lines. For the Cases ~ -- 1 and ~ = 2 we have
(AG)(1) = (AG) 2
=
_1n(l)tl~ '~ 2~ o, 2 ~ . t / - - 2 ~!t .n , - o2) t~-~1,
(AG)(2) = - ~ID (2) I (2) I (2) i (2) 3 (1)+~D 3 (2)+~D 4 (1)-aD 4 (2).
(19)
The explicit expression o f the term (AG) (3) is given in ref. 20). We observe that the PS expansion groups together a finite number of diagrams with differing numbers o f particles involved. The diagrammatic expansion of the function G(fl) can be used to generategraphic descriptions o f various physical quantities. For example, the interaction energy (10) may be presented in graphic form upon introduction of an interaction line (a double or helical line). This new line represents a factor f2(r)v(r). An interaction diagram contains only one interaction line. The diagram is generated according to eq. (10) from an ordinary graph contributing to the function G(fl) by replacing in turn each correlation line by an interaction line. The sum of these interaction diagrams establishes the diagrammatic expansion o f the quantity (10). The graphic expansion of the radial distribution function (11), (13) follows from the diagrammatic expansion of the interaction energy (10). According to eq. (12)
Da(ll (r, I)
Da(t) (r,2)
N D4") (r,O
D4t~I(r,2)
Fig. 6. The three- and four-bodydiagrams Dgl)(r, 1), D~t)(r,2), D(~)(r,1), and D~X)(r,2) with one correlation line which contribute to the radial distribution function. They are generated successively from the diagrams Dg2)(1),D~2)(2)of fig. 2, and diagrams Dk2)(1),D~2)(2)of fig. 4.
398
E. KROTSCHECK A N D M. L. RISTIG
we have to drop a factor ½ and to remove the interaction lines from all interaction diagrams. Adopting the PS classification we may express the result by
g(r) = [1 + ~(r)J{gF(r) + ~ Ag(~)(r)}.
(20)
The term Ag(~)(r) collects all diagrams with a correlation lines. A simple execution of the prescription for a = 1 involves the diagrams D~2)(1), D~2) (fig. 2) and D(42)(1), D(42)(2) (fig. 4) and generates the diagrams which are shown in fig. 6. With the help of these graphs we get explicitly
Ag(')(r) = -- 2D~sl)(r, 1) + 2D(1)(r, 2) + Dt41)(r, 1 ) - O~l)(r, 2).
(21)
4. Long-range correlations The FlY classification of diagrams which contribute to the generating function serves as a suitable ordering scheme if the correlations are sufficiently short ranged. The scheme is of rather limited value for the high density region where those diagrams which are classified as of higher orders become important. The FIY classification fails completely if long-range correlations (3) are assumed to be present in the Jastrow function (1). The incapacity of the FlY scheme becomes apparent if the correlation factor (7) is expressed in momentum space,
f rl(r, fl) e,k, dr = rl(k) ",. k-
k ~ O:
(22)
Consequently the diagram D~I)(1) diverges, D~)(1) = ½p2 ~drxdrztt(12 ) = lim ½pA~l(k). J
(23)
k~0
The general n-body cluster (AG)n(n >=3) contains a direct ring diagram s4'sS) (a boson diagram) which may be expressed in the form D~)(1) =
Alpn-'(2=)-3fdk~l'(k).
(24)
This graph is the most "dangerous" diagram 36) within a given cluster (AG)~. For long-range correlations (3) expression (24) diverges. There is no way to cancel the singularity of the n-body graph (24) by other divergent n-body diagrams. For example, all three-body diagrams (fig. 2) are of finite value even for long-range correlations except the singular direct ring D~3)(1). Employing eq. (10) we observe that the divergence of diagrams D~I)(1) and D~3)(1) of the generating lunction has no serious consequence for the corresponding parts of the two- and three-body cluster contributions (O/dfl)(A G)21# = o and (t3/t3fl)(A G)s [a=o
JASTROW CORRELATIONS
399
of the cluster expansion of the expectation value Ep(eq. (2)). Both terms are of finite value for short-range as well as long-range correlations. It is for this physical reason that the appearance of the divergent terms D~I)(1) and D~3)(1) may still be tolerated. But we conclude from eq. (24) that each n-body contribution (d/Ofl)(AG)nla=oto the energy E p diverges for any integer n > 4. Thus the FlY cluster scheme inevitably ceases to provide a meaningful classification of terms contributing to the expansion of the expectation value of any physical quantity. Abandoning the FlY scheme we are of course free to search for appropriate diagrams among contributions ofdifferino clusters to deal properly with the divergent parts. A careful study of this idea reveals indeed a general cancellation phenomenon of the singular portions of the divergent diagrams. The result may be cast into the form: (i) If correlations of long and short range are present in the ground-state trial function (1) the two-body diagram D~I)(1) which contributes to the generating function G(fl) diverges. The corresponding cluster contribution to any physical quantity generated by this singular graph is still of finite value. (ii) Except for the graph D~I~(1) there exists for any divergent diagram contributing to the function G(fl) a finite number Of irreducible graphs with the same number of correlation lines which also contribute to that function but cancel the singular portion. The cancellation is independent of the special shape of correlation function and is caused by the exclusion principle. The PS classification (18) offers the most convenient scheme in which to verify the presented statements. We begin with a study of contributions (AG)~ with one, two, and three correlation lines involved. There are only two diagrams with a single correlation line (fig. 1). It is evident that the divergent part of the direct diagram D~)(1) cannot be cancelled by the non-singular exchange graph D~1)(2). The four diagrams with two correlation lines (figs. 2, 4) have finite values for short- and long-range correlations. According to eq. (24) the direct ring diagram D~3)(1) (fig. 2) is the first non-trivial singular graph. An elementary inspection of all twenty-two irreducible graphs which constitute the term (AG) ~3) shows that only four diagrams become singular if correlations of the type (22) are assumed. These diagrams are, in addition to the direct ring D~3~(1), the exchange rings D~3~(1), D~3~(1)and Dr63)(1) (fig. 5). Application of the graphic rules leads to the portion (AG)~3)
1 (3) 1 (3) 1 (3) 1 (3) = gD3 (1)--gD4 (1)+~D5 (1)-gD6 (1).
(25)
It has the analytic form (AG)~3) = A-~p2(27t)-3fdk r/3(k){1 - 311 - SF(k)] + 311 - SF(k)] 2 _ [1 -- SF(k)] 3}.
(26)
The function St(k) is the familiar structure function of a gas of independent fermions 17)
400
E. KROTSCHECK AND M. L. RISTIG
SF(k)=
4 kF l 1,
~ ~
,
k < 2k v
(27)
k > 2kv.
By virtue of eqs. (14), (15) we recognize that this function is the appropriate quantity for representing the exchange factor 12(rkv) in momentum space. Execution of the elementary summation within the curly bracket of eq. (26) leads to
(AG)~a) = A~p2(En)- af dk[~l(k)Sv(k)] a.
(28)
Eq. (28) exhibits the asserted cancellation of all dangerous parts which are of order = 3. Indeed, the linear dependence (27) of the structure function on small values of the momentum cancels the singularity (22) of the long-range contributions. This property is independent of the special shape of the factor r/(r). Its proof rests solely on the existence of exchange diagrams, i.e., it is a consequence of the exclusion principle. We may generate the result (28) of the summation of divergent diagrams (25) from the direct ring D~a)(1) by a simple graphic manipulation. Introducing the Fourier transform SF(r) = 6(r)" V- lpIE(rkF)
(29)
of the structure function SF(k) we define a new graphic line which represents a factor
p-lSF(r). Insertion of this diagrammatic element at each point of the diagram D~a)(1) reproduces exactly the convolution integral (28). The prescription can be easily generalized to collect a special class of singular diagrams with ~ correlation lines (~ > 3) which contribute to the portion (AG) (~) of the generating function. We insert at each point of the direct ring D~)(1) the line representing the function p-1SF(r). The procedure collects, in addition to the direct ring, properly weighted exchange ring diagrams which are contained in the term (AG) (~). The analytic expression of the sum of these divergent graphs collapses to a convolution integral similar to eq. (28). Its integrand is given by the function [~l(k)SF(k)] ~ and is therefore non-singular. Let us now proceed to the contribution (AG) (4) of the generating function. In addition to the graph D(44)(1) and the four exchange ring diagrams which can be collected by the procedure already described, we find nine diagrams of a differing structure which have singular parts. Three of these diagrams are shown in fig. 7. They have in common a direct chain of three correlation lines. Insertion of the line representing the factor p - 1SF(r), eq. (29), into both points of the chain generates the rest of all singular diagrams which contribute to the function G(fl) in the order at = 4. We may therefore conclude that the portion (AG)~4) which collects all contributions of singular diagrams with four correlation lines is of finite value. In higher orders we find more complicated contributions of singular diagrams to
JASTROW CORRELATIONS
,k---~
04(4) (2)
\
I
~,
b--.-d
~--4
D5(4) (2)
401
/
06(4) (2)
Fig. 7. The diagrams D~*}(2), D~*~(2), and D~*>(2)with four correlation lines which contribute to the generating function. They diverge if longrange correlations are present in the correlated trial ground state.
Fig. 8. A special diagram with six correlation lines which diverges if longrange correlations are assumed.
the function G(fl) which cannot be collected by the simple insertion (29). The singular diagram of fig. 8 provides an illuminating example. It is evident that the rules of sect. 3 do not permit a diagram to be a contribution to the term (AG) {6} which is generated from the graph of fig. 8 by the described insertion technique. Instead of employing the function (29) we introduce an appropriate function ALtl~(r, fl) containing one correlation factor which defines a new line. This graphic element may serve as an insertion at each point of the direct ring D~3~(1). It generates the diagram of fig. 8 and other singular diagrams of the term (AG) {6~ which should be grouped together. An elementary study reveals that the function ALtl~(r, fl) should be defined by the properly weighted sum of the diagrams Dt44~(2), D~4~(2), D~4}(2) in which the direct chain of three correlation lines has been removed (appendix). The Fourier transform AL"~(k) of this function (for brevity we suppress the explicit reference to the parameter fl) has the important property
AIg)(k)
~ k 2,
k ~ O.
(30)
Consequently, the contribution (AG)~6) = A-~p2(Ex)-3 fdk[n(k)Al3"(k)] a
(31)
of the divergent diagrams extracted from the term (riG) t6) is non-singular. The same procedure can be applied to the direct ring diagram D~}(1) with ~ correlation lines (ot > 3). Again, it collects a class of divergent diagrams which contribute to the term (AG) t2~), thereby cancelling the singular portions. We may enlarge our class of examples if we employ the function
L(r, fl) = SF(r)+ A/3~)(r, fl)
(32)
instead of the functions on the right hand side separately. We can proceed further if we think of the function AL")(r, fl) as the first-order approximation ~ = 1 of a function
AL(r, fl) = AI31~(r, fl)+ dLt,2~(r, fl)+ ... +d/3~}(r, fl)+ ....
(33)
This function should achieve the cancellation of the singular portions of entire
402
E. KROTSCHECKAND M. L. RISTIG
classes of diagrams which contribute to the generating function. A careful investigation of the function (33) uncovers its close kinship to the generalized distribution function 17). The function AL(r, fl) is represented by that portion of the expansion (20) (it is understood that the correlation factors appearing in eq. (20) depend on the parameter fl) which is determined by the following class of graphs: (i) diagrams which have no nodal points 16), (ii) diagrams for which each external point is joined by an incoming (and outgoing) exchange line. The part AL(2)(r, fl) is given explicitly in ref. 20). A rather lengthy but elementary analysis of the twenty-six diagrams which contribute to that portion reveals that the Fourier transform AL~2)(k) has the property
k-lAIJ2)(k) -- 0,
k ~ 0.
(34)
There is every reason to believe that eq. (34) is also valid for the general term AL(~)(k) (~ > 3). The Fourier transform AL(k) of the function AL(r, fl) should therefore show the same behavior (34) as the function AL(2)(k). There are still classes of divergent diagrams which have not yet been included in our studies. These diagrams have a structure of increasing complexity and appear in higher orders ~ (7 > 4). We are convinced that a successful approach to deal with the singular portions of these graphs must employ suitably introduced threefour-, ...-body functions which serve as insertions in analogy to the function L(r, fl). At the present stage we have not attempted to tackle that problem in detail. A complete proof of the theorem in higher orders is in preparation.
5. Hypernetted-chain summation The investigation of the generating function (sect. 4) has revealed that each term (AG)(~)(ct _>_2) of its PS expansion is of finite value if short- and long-range correlations are present in the trial ground state (1). Consequently, in this case, we order the diagrams contributing to the graphic expansion of the radial distribution function also according to the numbers of correlation lines involved. This scheme could serve as a suitable means to approximate the distribution function by truncating expansion (20) at a chosen level ~ = ~o. But we can improve this simple scheme considerably if we employ the time-honored hypernetted-chain technique of classical liquids 16) and boson fluids 17,33). The formal steps along which the hypernettedchain summation proceeds (series connections, parallel connections of chains) are well described in ref. 16). They need not be repeated in any detail. For the fermion problem at hand we should of course take proper care of the exclusion principle and the cancellation phenomenon in each step of the HNC procedure. Drawing on our findings of sect. 4 these effects can be taken into account if we connect successively the chains (the simplest chain being a correlation line) by an insertion line which represents the function p- 1L(r, fl = O) - p- 1L(r). On the other hand, the parallel connection of chains is executed directly, i.e., without insertion. This prescription
JASTROW CORRELATIONS
403
avoids any over-counting problem of diagrams which contribute to the distribution function. The described procedure defines a renorrnalized correlation factor r/R(r) and its Fourier transform t/s(k) as solution to a coupled set of generalized hypernetted-chain equations, r/R(r) = f2(r) exp [N(r)] - 1, (35) N(k){1 + p~lR(k)L(k)} = pn2(k)L(k). The redundant function N(r) and its Fourier transform N(k) serve merely to present the relations (35) in a convenient form. To describe the renormalized factor graphically we introduce a renormalized correlation line (for example, a fat dashed line). The new graphic element collects diagrams of a special type as prescribed by eqs. (35); ordinary diagrams which have exchange lines attached to the external points do not contribute. The function L(k) has to be replaced by unity if we want to omit exchange effects in eqs. (35). In this case set (35) collapses to the familiar HNC equations for the quantity r/R(r) which constitutes the hypernetted-chain approximation of the radial distribution function of a Bose fluid ~7). Set (35) affects a simple relation between the long-range behavior of the correlation factor (3) and the asymptotic form of the renormalized correlation factor. A simple analysis of eqs. (35) under the assumption (3) yields the property r/R(r ) ---- c R r-
2,
r ~ oo.
(36)
The renormalized coefficient CRdepends on the constant c of eq. (3) in ar, elementary manner: cR = 4c(4- vk2c)- 1
(37)
Relation (37) has an interesting bearing on the liquid structure functions S(k) in the range of small values of the momentum. We shall report on these aspects of the present analysis in a separate publication. We may employ the quantity r/R(r) to collect successively entire classes of diagrams which establish the expansion (13) of the radial distribution function. This renormalization can be executed, in principle, by a two-step procedure. In the first stage we replace in all contributing diagrams the ordinary correlation line by a renormalized line. Thereupon we omit those graphic portions which are already included in lower orders. The result can be cast into the form
g(r) = [1 + qs(r)] {gv(r) + ~ Ag(R~)(r)}.
(38)
It is ordered according to the number a of renormalized correlation lines. The lowest-order term a = 1 may be constructed from the contribution (21) by replacing the correlation factor by the renormalized factor tt~(r). It need hardly be mentioned
404
E. K R O T S C H E C K A N D M. L. R I S T I G
that expansion (33) of the function AL(r, fl) can be renormalized in the same way 20). We should finally note that differing hypernetted-chain procedures can be devised 2. aT). We have shown in the present paper that the cancellation of singular portions within a finite sum of diagrams with given number of correlation lines is a characteristic feature of a system of fermions. This property should not be lost if rearrangements of terms are performed. For this reason the present device should be preferred. We take this opportunity to thank Prof. J. W. Clark for stimulating correspondence and conversations. Special thanks go to Prof. Chia-Wei Woo for providing us with his recent results on the homework problem prior to publication.
Appendix The function ALtl)(r, fl) may be written as a sum (for brevity we suppress in the following any reference that the expressions depend on the parameter fl), A/31)(r, 2) = AI3~)( r , 2) + AI3~)( r, 2) + A/3~)(rl 2).
(A.1)
The terms represent the analytic expressions of those diagrams which constitute the insertion as has been discussed in sect. 4. They are explicitly given by d~l)(r12) = - v - l~l(r12)12(r12 kF),
(A.2)
A/J~)(rl 2) = 2 p v - 2 f dr a l(r 12 kF)l(r l a kF)l(r 2 3 kr)r/(r23),
(A.3)
d/3~)(r12) = _ p2 v - 3 f d r 3 dr 4 l(rl 3 kr)l(r14 kF)l(r2 a kF)l(r24 kv)r/(r34). 3 The property
f
Al_Jl)(r)dr
lim A/31)(k)
0
(A.4)
(A.5)
k--*O
is a direct consequence of eqs. (A.2)-(A.4) if we employ the relation fdr
/(r 1 kF)/(Ir-- rll" kr) = v p - ll(rkr).
(A.6)
The evaluation of the dependence of the function ALtl)(k) on small values of the momentum can be performed most efficiently by introducing cylindrical coordinates. Suppressing the technical details one arrives at t3k
AI3~)(k)
~k Al3~)(k) =
O,
k ~ O,
a di3~)(k) ' ~k
k --, O.
(A.7) (A.8)
JASTROW CORRELATIONS
405
Application of the same technique yields for the second derivative the result (30) of sect. 4. References 1) V. R. Pandharipande, Nucl. Phys. A217 (1973) 1 2) V. R. Pandharipande and H. A. Bethe, Phys. Rev. C'/(1973) 1312 3) S. Chakravarty, M. D. Miller and Chia-Wei Woo, Nucl. Phys. A220 (1974) 233 4) V. Canuto and S. M. Chitre, Phys. Rev. Lett. 30 (1973) 999 5) S. Cochran and G. V. Chester, preprint (1973) 6) V. Canuto, J. Lodenquai and S. M. Chitre, preprint (1974) 7) P. J. Siemens and V. R. Pandharipande, Nucl. Phys. A173 (1971) 561 8) V. R. Pandharipande, Nucl. Phys. A174 (1971) 641; A178 (1971) 123 9) M. Miller, C. W. Woo, J. W. Clark and W. J. Ter Louw, Nucl. Phys. A184 (1972) 1 10) F. Iwamoto and M. Yamada, Progr. Theor. Phys. 17 (1957) 543 11) J. W. Clark and P. Westhaus, J. Math. Phys. 9 (1968) 131, 149 12) M. L. Ristig, W. J. Ter Louw and J. W. Clark, Phys. Rev. C3 (1971) 1504; 5 (1972) 695 13) D. A. Chakkalakal, Ph.D. thesis, Washington University, 1969 (unpublished) 14) J. Nitsch, Dissertation, Universit~it zu K61n (1974) 15) L. Shen and C.-W. Woo, preprint (1974) 16) J. M. J. van Leeuwen, J. Groeneveld and J. de Boer, Physica 25 (1959) 792; 30 (1964) 2265 17) E. Feenberg, Theory of quantum fluids (Academic, New York, 1969) 18) C.-W. Woo, private communication 19) L. Reatto and G. V. Chester, Phys. Rev. 155 (1967) 88 20) E. Krotscheck, Dissertation, Universit~it zu K61n (1974) 21) .N.R. Werthamer, Phys. Rev. A7 (1973) 254 22) J. W. Clark and M. L. Ristig, Phys. Rev. C5 (1972) 1553 23) M. L. Ristig and J. W. Clark, Nucl. Phys. A199 (1973) 351 24) M. Gaudin, J. Gillespie and G. Ripka, Nucl. Phys. A176 (1971) 237 25) F. Y. Wu and E. Feenberg, Phys. Rev. 128 (1962) 943 26) F. Calogero and C. Ciofi degli Atti, ed., The nuclear many-body problem, vol. 2 (Editrice Compositori. Bologna, 1973) p. 273 27) F. Y. Wu, J. Math. Phys. 4 (1963) 1438 28) E. Krotscheck, J. Nitsch, M. L. Ristig and J. W. Clark, Nuovo Cim. Lett. 6 (1973) 143 29) S.-O. B~ickman, J. W. Clark, W. J. Ter Louw, D. A. Chakkalakal and M. L. Ristig, Phys. Lett. 41B (1972) 247 30) S. Fantoni and S. Rosati, Nuovo Cim. 20A (1974) 179 31) E. K. Achter and L. Meyer, Phys. Rev. 188 (1969) 291 32) J. E. Mayer and M. G. Mayer, Statistical mechanics (Wiley, New York and London, 1940) 33) E. Krotscheck and M. L. Ristig, Phys. Lett. 48A (1974) 17 34) F. lwamoto, Progr. Theor. Phys. 19 (1958) 595 35) C. D. Williams, Ph.D. thesis, Washington University, 1961 (unpublished) 36) P. G. Reinhard and H. Arenhfvel, Z. Phys. 262 (1973) 1; 264 (1973) 211 37) S. Fantoni and S. Rosati, Nuovo Cim. Lett. 10 (1974) 545
LONG-RANGE JASTROW CORRELATIONS E. K R O T S C H E C K and M. L. RISTIG
Institut ffir Theoretische Physik, Universitiit zu K61n, 5 Kfln 41, Germany Received 14 August 1974 (Revised 22 November 1974) Abstract: We develop a promising many-body method to evaluate the equation of state for dense neutron matter and liquid helium. The ground state of the Fermi fluid is described by a conventional Jastrow ansatz. We admit the presence of short- and long-range correlations. Under this assumption we study the generating function which has been introduced by Wu and Feenberg. We employ a graphic formulation and develop the diagrammatic expansion of the generating function and the radial distribution function. If long-range correlations are assumed, the diagrams have singular parts. We give a proof that the total contribution of such diagrams to the generating function which contain two, three, and four correlation lines is of finite value. The same property is shown for a selected class of singular diagrams containing ct correlation lines (ct > 4). To verify the cancellation phenomenon we introduce a two-body function which serves graphically as an insertion into selected singular diagrams. For the remaining classes of diagrams we need three-, four-, . . . , n-body insertions. The result is cast into the form of a theorem. The cancellation rests on the exclusion principle and does not depend on the special shape of the correlation function. Finally, a generalized hypernetted-chain summation of diagrams which represent the radial distribution function is executed. The procedure includes exchange contributions and can be employed if shortand/or long-range correlations are present.
1. Introduction There has recently been an upsurge of interest in a precise description and quantitatively accurate evaluation of the equation of state for dense and superdense neutron matter 1-6). It has become clear that a rather sophisticated many-body treatment has to be developed if a reliable prediction of a solidification of this system of fermions is to be ventured 3). Most of the available work which has been done for the liquid phase in the highdensity regime employs the familiar Jastrow ansatz ~P = F ~ ,
F = 1-I f(rij),
f2(r) = 1 +r/(r),
(1)
i
to include properly short-range correlations by a suitably chosen function f ( r ) or a correlation factor t/(r) which, more generally, might be assumed to be statedependent 7- 9). The model wave function q~is taken to be the ground-state function of a gas of A independent fermions. In addition, the ground-state energy of the system has been approximated by the expectation value of the Hamiltonian 389
390
E. KROTSCHECK AND M. L. RISTIG
H --- T + V with respect to the correlated state (1): E = E K+ Ep -
+
(2)
A first generation of such studies of neutron matter employed the well established Iwamoto-Yamada expansion 10) or its factorized version (FIY expansion 11,12)) to evaluate the expectation value (2). The correlation factor has been assumed to be of short range and the expansion has been truncated at the two- or three-body level 7-9,1 a, 14). At normal nuclear density this approximation might yield reliable results for the ground-state energy 9). In the more interesting range of high density, the procedure is not sufficient and a more accurate many-body method is called for 15). To overcome this problem several groups agreed to begin by studying a simple "homework problem" which can be attacked by already existing powerful manybody techniques. In the model the neutrons interact in all states through the repulsive part of the Reid 1So potential, and Boltzmann statistics is assumed for further simplification 3, 15). Numerical resu!ts for the equation of state in hypernetted-chain (HNC) approximation 16) and Percus-Yevick approximation 17) are now available for the fluid model problem15). In addition, a paired-phonon analysis 17) has been successfully performed 1s). This procedure takes into account short- and long-range effects. Both kinds of correlations are present in the o p t i m i z e d ground state 1~, 19) described by the factor F of ansatz (1). The long-range behavior of the correlation factor is defined by the relation q(r) = cr -2,
r--* 00,
(3)
the coefficient c being independent of the distance r. In addition, elaborate improvements have been made to grasp the influence of three-phonon corrections 17) to the ground-state energy of the fluid 18). We believe that the achieved accuracy now justifies the advancement towards a more realistic "homework problem" in which the Fermi statistics of the neutrons is properly taken into account. This enterprise should of course be done on the same sophisticated level on which the Bose fluid was dealt with. Such an ambitious treatment of the Fermi fluid must clearly permit long-range correlations of the type (3) in the trial ground-state function (1) as was done for the Bose fluid. We should note, in addition, that for formal as well as practical reasons it is not possible to separate short-range from long-range correlations in a clean manner. The optimization procedure for the expectation value of the groundstate energy (2) (without subsidiary constraints) with respect to a correlated wave function (1) leads inevitably to the appearance of long-range correlations2°).
JASTROW CORRELATIONS
391
The same difficulty is well known in the theory of Bose fluids 17,19) and of quantum solids 21). In a first stage we therefore must develop a many-body method which is able to deal with short- a n d long-range correlations in a system of interacting fermions. If we admit only short-range correlations we may expand the expectation value (2) into the FlY cluster series provided the value of the smallness parameter of that theory is sufficiently small 22). But the n-body cluster terms (n > 4) diverge if we permit long-range correlations (3). Thus we have to search for a suitable rearrangement of the FlY expansion. In the present paper we rearrange the contributions to the FIY expansion of the generating function of Wu and Feenberg 17,23). We collect terms which have the same singular behavior in the presence of long-range correlations (3). To achieve that goal, most efficiently, we employ and develop further a graphic description which has been initiated by Gaudin et aL 24). Our study reveals that the collection of a finite number of divergent diagrams containing the same number of correlation lines leads to a complete cancellation of their singular portions. Thus the ordering of diagrams according to the number of correlation lines involved provides a sensible classification scheme if long-range correlations are admitted. In a further step we perform a hypernetted-chain summation of the contributions to the rearranged expansion of the radial distribution function. Ignoring the influence of Fermi statistics, the generalized H N C equations reduce to the familiar HNC result for a Bose fluid. The application of the proposed method to the homework problem would, on the other hand, tell how important the effects of differing statistics are. It is evident that the presented many-body technique can be applied to the problem of liquid aHe. A detailed comparison with the results of the established Wu-Feenberg approach 17) would be highly informative. In sect. 2 we present the factorized cluster expansion for the generating function. We introduce the graphic notation of the cluster terms in sect. 3. We summarize the diagrammatic rules and discuss two ordering schemes for the diagrams. We turn in sect. 4 to the problem of long-range correlations and study the singularities of divergent diagrams. Sect. 5 is devoted to the hypernetted-chain summation which is performed for the diagrammatic expansion of the radial distribution function. 2. Factor cluster expansion Cluster expansions of the expectation value of the ground-state energy (2) or other physical quantities with respect to a correlated wave function (1) can be derived most efficiently for a uniform extended Fermi fluid from a generating function 2a) G(fl) which has been introduced by Wu and Feenberg 25). This quantity is well known in terms of the FIY expansion which is ordered according to the number of particles (or hole lines) involved: G(fl) = (AG)2 +(AG)3 + . . . +(AG).+ ....
(4)
392
E. KROTSCHECK AND M. L. RISTIG
The contributions of the two- and three-body clusters are
ii (AG)3 = ~ ~, xijk(fl)-½ ~ Xo(fl)Xik(fl). ijk
(5)
ijk
The summations extend over occupied single-particle states inside the Fermi sea and the functions xo(fl), Xi~k(fl), ..-, are the familiar reduced cluster integrals which depend on a parameter fl suitably introduced 17, 26, 12). The explicit expressions of the four-, five-, and six-body terms of the function G(fl) are given by Wu 27). The general term can be systematically constructed, for example, by expanding the renormalized FIY expansion described in refs. 23, 28). The specific structure of the cluster integrals becomes more transparent if they are represented in terms of diagonal and non-diagonal matrix elements 23) r/il i.... i. and r/~l.... ~,,i.... ~.(2 =< n ___.4) of properly defined irreducible n-body operators taken between normalized and antisymmetrized states lil i2... in>a and ] j l j 2 . . . J.>~, Xij = ~ij,
Xi~k = tl ijk,
(6)
Xijkl = l~ijkl "31-(~ij, kl qkl, ij "~- qik, jl qjl, ik "~- qjk, it qil, jk)"
Incidentally, we should note that the introduction of these matrix elements provides a convenient definition of cluster integrals with coincident indices. Adopting special choice (1), the irreducible operators can be constructed in an elementary manner from the correlation factor t/(12) = q(r12 , fl) = f2(q2) exp [flv(rx2) ] - 1.
(7)
For our purposes the function v(r12 ) may be identified with the bare neutronneutron potential of the homework problem. The two- and three-body matrix elements are given explicitly: tlO, kl =
( ij[rl(12 )lk l >a .
thjk ' ,.,. =
(8)
The subscript a indicates that the matrix elements have to be taken with respect to antisymmetrized and normalized states. The assumption (1) has an important consequence for the general n-body con-
JASTROW CORRELATIONS
393
tribution of eq. (4). In the simplest case n = 3, for example, the cluster (d G)s collapses (by insertion of eq. (8) into eq. (5)) to the simple expression: (AG) 3 = ½ ~ (ijklrl(12)rl(13)ljki-ikj)
ijk + ~ ~ ( ijklrl(12)rl(23)~l(13)lijk)a.
(9)
ijk
The first term ofeq. (9) exhibits a cancellation effect which occurs in any order. Only irreducible integrals survive. This property is a special feature of the cluster expansion (4) if we assume a trial function of Jastrow type 29). A general formal proof may be found in ref. 30). The generating function may be employed to derive the FlY expansions of various physical quantities. For example, we generate the cluster expansion of the energy Ep (eq. (2)) by the straightforward prescription: Ep = ~ G(fl)l#=o.
(10)
For the investigations to come we are also interested in the radial distribution function:
g(r) = A ( A - 1)p-2N -1 ~
f~t~dr3dr4.., dra.
(11)
1
The quantity p is the particle density. It is related to the Fermi momentum kr by the expression p = v(6n2) - lk3F where the factor v gives the degeneracy (v = 2 for neutron matter or SHe, v --- 4 for nuclear matter). The constant N is the norm of the wave function ~ and ~0,~ indicates summation over all A spins (v = 2) and isospins (v = 4). Employing eqs. (10) and (4) we can extract the FIY expansion of the radial distribution function from the relation Ep = ½p2 f dr 1dr2 v(rl 2)g(rl 2)-
(12)
The result can be written in the form
g(r) = [1 + r/(r)]{gF(r) + ~ Ag.(r)}.
(13)
n=3
The function gr(r) is the familiar distribution function of a gas of independent fermions: gF(r) = 1--v-ll2(rkF),
l(x) = 3x-a(sin x - - x cOS x).
In eq. (13) the subscript n indicates the n-body cluster.
(14)
394
E. KROTSCHECK AND M. L. RISTIG
Closely related to the function (11) is the liquid structure function which is defined by S(k) = 1 + p f { y ( r ) - 1}eik"dr.
(15)
The shape of this function for small values of the momentum might be influenced by the long-range correlations (3). The structure function has been measured for liquid 3He by X-ray experiments 31). For our following discussions it is highly desirable to have an efficient formulation of the general n-body contribution to the generating function, the distribution function, etc. A graphic description provides such a powerful tool.
3. Diagrammatic expansion A simple graphic description in momentum space of the reduced cluster integrals and the generating function exists 17,26) but the specific features of the clusters assuming the ansatz (1) - are more apparent if we employ a graphic formulation in coordinate space. We adopt the diagrammatic notations of ref. 24) which originate from the Yvon-Mayer technique devised for classical systems 32). The basic elements of any diagram are (i) internal and external points (solid and open dots), (ii) correlation lines (dashed lines) and (iii) oriented exchange lines (oriented solid lines). (i) An external point labels the coordinates ri of the particle i. An internal point indicates a factor p and integration over the coordinate space of the particle involved. (ii) A correlation line connecting the points ri and r i represents a correlation factor r/(/j). (iii) An exchange line represents the exchange factor l(kF r~). These lines always form closed loops. Each distinct loop connecting p points is associated with a factor vx-P. The weight ( - 1 ) 1-P of the loop determines the sign of the diagram. Before presenting the formal rules which establish the diagrammatic expansion of the generating function let us study in some detail the two- and three-body contributions. Fig. 1 shows the graphs D~I)(1) and D~1)(2)which constitute the twobody cluster (AG)2. Here, and in the following discussion, we use the notation Dt,~)(s) for any diagram where the subscript n refers to the number of particles in-
D2(n(I)
020)(2)
Fig. 1. The diagrams D~)(1) and D~x)(2) which constitute the two-body cluster (AG)2. Superscripts refer to the number of correlation lines, subscripts refer to the number of particles involved. Both graphs have multiplicity m = 1.
JASTROW CORRELATIONS
395
volved and the superscript • indicates the number of correlation lines. The integer argument s = 1, 2 . . . . , denotes successively each graph of the set of diagrams with given numbers ~ and n. Using the scheme (i)-(iii) we easily extract from eq. (9) all five diagrams which represent the three-body term (AG)3. They are given in fig. 2. In graphic notation we may therefore write: 1
(AG)2 = ~.~ {O(21)(1) - O(i)(2)},
1
(AG)3 = 3.1 ~ - 3D (2) 3 (1)+3D (2) 3 (2)+D (3) 3 (1)-3D (3) 3 (2)+2D (3) 3 (3)}.
(16)
We stress that all diagrams are irreducible. The absolute values of the weight factors of the graphs D~I)(1), D~1)(2), D~2)(1), D~3)(1), D~3)(2) and D~3)(3) of eq. (16) may be identified with the multiplicity m of topologically distinct diagrams. But the coefficient of the graph D~2)(2) differs from the multiplicity (m = 6). This phenomenon can be explained by the existence of equivalent graphs for the uniform extended medium. Because of momentum conservation a diagram which belongs to a special class of graphs can have the same value as a topologically distinct diagram. The simplest example is shown in fig. 3. The equivalence of both graphs becomes apparent if we take notice of the relation
( ijkllrl(12)rl(34)ljkli> = ~ ( ijklrl(12)rl(13)ljki>, ijkl
(17)
ijk
where the states represent products of normalized single-particle functions. Property (17) can be expressed in a form which is more suited for our discussion:
13] 6D(2)(2)- 4!1 12D(2)(3) = ~ . 3D(32)(2).
/
(17')
\
Da(2) (2)
\
Da(3) (I)
Da(3) (2)
D3 (3) (3)
Fig. 2. The irreducible and topologically distinct graphs O~2)(1), D~2)(2), D~a)(1), D~3)(2), and D~3)(3) which constitute the three-body cluster (AG)3. Their multiplicities m are 3, 6, 1, 3 and 2, respectively.
D4(2) (3)
D3(2) (2)
Fig. 3. A simple example of two equivalent diagrams, D~2)(3) and D~2)(2), which are of the same value. Their multiplicities m are 12 and 6, respectively.
396
E. KROTSCHECK AND M. L. RISTIG
Here, we may equate the weight factors 6 and 12 with the multiplicity of the diagrams D~2)(2) and Dt42)(3). The coefficient of the graph D~2)(2) on the right hand side of eq. (1 7') is the weight which correctly appears in eq. (1 6). A general discussion of equivalent diagrams may be found in ref. 20). We are now ready to summarize the general formal rules which determine the graphic construction of the generating function. (i) The contributing diagrams are built from correlation lines and oriented exchange lines which connect irreducibly a given number of internal points. (ii) Each internal point of a diagram is joined by at least one correlation line. Two points are connected by at most one correlation line. (iii) Each point of a diagram is joined by at most one incoming exchange line. No exchange line terminates. (iv) The weight of a diagram with n internal points is given by its topologic multiplicity m divided by the number n! The sign of a diagram is determined by prescription (iii). The sum of all permitted and weighted diagrams constitutes the diagrammatic expansion of the function G(fl). For example, fig. 4 shows the contributing fourbody diagrams with two correlation lines. There are twenty-two topologically distinct diagrams with three correlation lines (without the equivalent graphs) which are permitted by the rules. To be more specific we have three three-body diagrams (D~a)(1), D~a)(2), D~3)(3) of fig. 2), eight four-body diagrams, six five-body diagrams and five six-body diagrams. Some irreducible graphs are presented in fig. 5. If we collect all weighted diagrams with n points and take into account, in addition, all associated equivalent graphs we reproduce exactly the n-body cluster
, ,
D4(a)(i)
D(3)(I)
D(a) (I)
W D4(al(t)
D4(a) (2)
Fig. 4. The four-body diagrams Dt42)(l) and D~2)(2)which contain two correlation lines. They have multiplicity m = 6.
D4(3) (2)
Ds(~) (2)
Ds(3} (2)
Fig. 5. The diagrams Dk~)(l), D~)(I), D~)(1), D(43)(2),Dg3)(2),and D~63)(2)with three correlation lines which contribute to the generating function. Their multiplicitiesm are 12, 60, 120,24, 240, and 240, respectively.
JASTROW CORRELATIONS
397
(AG), o f the generating function. Thus adopting the F l Y classification all diagrams with n points and their equivalent graphs are of the same order n. In contrast we may collect all diagrams with the same number o f correlation lines into a single term. This classification scheme establishes the power-series expansion a0, 3a) (PS expansion) G(fl) = (d G) tl) + (A G) (2) + ... + (A G) t') + ....
(18)
The superscript ~(~t = 1, 2 . . . . . ) indicates the contribution of all diagrams with ~t correlation lines. For the Cases ~ -- 1 and ~ = 2 we have
(AG)(1) = (AG) 2
=
_1n(l)tl~ '~ 2~ o, 2 ~ . t / - - 2 ~!t .n , - o2) t~-~1,
(AG)(2) = - ~ID (2) I (2) I (2) i (2) 3 (1)+~D 3 (2)+~D 4 (1)-aD 4 (2).
(19)
The explicit expression o f the term (AG) (3) is given in ref. 20). We observe that the PS expansion groups together a finite number of diagrams with differing numbers o f particles involved. The diagrammatic expansion of the function G(fl) can be used to generategraphic descriptions o f various physical quantities. For example, the interaction energy (10) may be presented in graphic form upon introduction of an interaction line (a double or helical line). This new line represents a factor f2(r)v(r). An interaction diagram contains only one interaction line. The diagram is generated according to eq. (10) from an ordinary graph contributing to the function G(fl) by replacing in turn each correlation line by an interaction line. The sum of these interaction diagrams establishes the diagrammatic expansion o f the quantity (10). The graphic expansion of the radial distribution function (11), (13) follows from the diagrammatic expansion of the interaction energy (10). According to eq. (12)
Da(ll (r, I)
Da(t) (r,2)
N D4") (r,O
D4t~I(r,2)
Fig. 6. The three- and four-bodydiagrams Dgl)(r, 1), D~t)(r,2), D(~)(r,1), and D~X)(r,2) with one correlation line which contribute to the radial distribution function. They are generated successively from the diagrams Dg2)(1),D~2)(2)of fig. 2, and diagrams Dk2)(1),D~2)(2)of fig. 4.
398
E. KROTSCHECK A N D M. L. RISTIG
we have to drop a factor ½ and to remove the interaction lines from all interaction diagrams. Adopting the PS classification we may express the result by
g(r) = [1 + ~(r)J{gF(r) + ~ Ag(~)(r)}.
(20)
The term Ag(~)(r) collects all diagrams with a correlation lines. A simple execution of the prescription for a = 1 involves the diagrams D~2)(1), D~2) (fig. 2) and D(42)(1), D(42)(2) (fig. 4) and generates the diagrams which are shown in fig. 6. With the help of these graphs we get explicitly
Ag(')(r) = -- 2D~sl)(r, 1) + 2D(1)(r, 2) + Dt41)(r, 1 ) - O~l)(r, 2).
(21)
4. Long-range correlations The FlY classification of diagrams which contribute to the generating function serves as a suitable ordering scheme if the correlations are sufficiently short ranged. The scheme is of rather limited value for the high density region where those diagrams which are classified as of higher orders become important. The FIY classification fails completely if long-range correlations (3) are assumed to be present in the Jastrow function (1). The incapacity of the FlY scheme becomes apparent if the correlation factor (7) is expressed in momentum space,
f rl(r, fl) e,k, dr = rl(k) ",. k-
k ~ O:
(22)
Consequently the diagram D~I)(1) diverges, D~)(1) = ½p2 ~drxdrztt(12 ) = lim ½pA~l(k). J
(23)
k~0
The general n-body cluster (AG)n(n >=3) contains a direct ring diagram s4'sS) (a boson diagram) which may be expressed in the form D~)(1) =
Alpn-'(2=)-3fdk~l'(k).
(24)
This graph is the most "dangerous" diagram 36) within a given cluster (AG)~. For long-range correlations (3) expression (24) diverges. There is no way to cancel the singularity of the n-body graph (24) by other divergent n-body diagrams. For example, all three-body diagrams (fig. 2) are of finite value even for long-range correlations except the singular direct ring D~3)(1). Employing eq. (10) we observe that the divergence of diagrams D~I)(1) and D~3)(1) of the generating lunction has no serious consequence for the corresponding parts of the two- and three-body cluster contributions (O/dfl)(A G)21# = o and (t3/t3fl)(A G)s [a=o
JASTROW CORRELATIONS
399
of the cluster expansion of the expectation value Ep(eq. (2)). Both terms are of finite value for short-range as well as long-range correlations. It is for this physical reason that the appearance of the divergent terms D~I)(1) and D~3)(1) may still be tolerated. But we conclude from eq. (24) that each n-body contribution (d/Ofl)(AG)nla=oto the energy E p diverges for any integer n > 4. Thus the FlY cluster scheme inevitably ceases to provide a meaningful classification of terms contributing to the expansion of the expectation value of any physical quantity. Abandoning the FlY scheme we are of course free to search for appropriate diagrams among contributions ofdifferino clusters to deal properly with the divergent parts. A careful study of this idea reveals indeed a general cancellation phenomenon of the singular portions of the divergent diagrams. The result may be cast into the form: (i) If correlations of long and short range are present in the ground-state trial function (1) the two-body diagram D~I)(1) which contributes to the generating function G(fl) diverges. The corresponding cluster contribution to any physical quantity generated by this singular graph is still of finite value. (ii) Except for the graph D~I~(1) there exists for any divergent diagram contributing to the function G(fl) a finite number Of irreducible graphs with the same number of correlation lines which also contribute to that function but cancel the singular portion. The cancellation is independent of the special shape of correlation function and is caused by the exclusion principle. The PS classification (18) offers the most convenient scheme in which to verify the presented statements. We begin with a study of contributions (AG)~ with one, two, and three correlation lines involved. There are only two diagrams with a single correlation line (fig. 1). It is evident that the divergent part of the direct diagram D~)(1) cannot be cancelled by the non-singular exchange graph D~1)(2). The four diagrams with two correlation lines (figs. 2, 4) have finite values for short- and long-range correlations. According to eq. (24) the direct ring diagram D~3)(1) (fig. 2) is the first non-trivial singular graph. An elementary inspection of all twenty-two irreducible graphs which constitute the term (AG) ~3) shows that only four diagrams become singular if correlations of the type (22) are assumed. These diagrams are, in addition to the direct ring D~3~(1), the exchange rings D~3~(1), D~3~(1)and Dr63)(1) (fig. 5). Application of the graphic rules leads to the portion (AG)~3)
1 (3) 1 (3) 1 (3) 1 (3) = gD3 (1)--gD4 (1)+~D5 (1)-gD6 (1).
(25)
It has the analytic form (AG)~3) = A-~p2(27t)-3fdk r/3(k){1 - 311 - SF(k)] + 311 - SF(k)] 2 _ [1 -- SF(k)] 3}.
(26)
The function St(k) is the familiar structure function of a gas of independent fermions 17)
400
E. KROTSCHECK AND M. L. RISTIG
SF(k)=
4 kF l 1,
~ ~
,
k < 2k v
(27)
k > 2kv.
By virtue of eqs. (14), (15) we recognize that this function is the appropriate quantity for representing the exchange factor 12(rkv) in momentum space. Execution of the elementary summation within the curly bracket of eq. (26) leads to
(AG)~a) = A~p2(En)- af dk[~l(k)Sv(k)] a.
(28)
Eq. (28) exhibits the asserted cancellation of all dangerous parts which are of order = 3. Indeed, the linear dependence (27) of the structure function on small values of the momentum cancels the singularity (22) of the long-range contributions. This property is independent of the special shape of the factor r/(r). Its proof rests solely on the existence of exchange diagrams, i.e., it is a consequence of the exclusion principle. We may generate the result (28) of the summation of divergent diagrams (25) from the direct ring D~a)(1) by a simple graphic manipulation. Introducing the Fourier transform SF(r) = 6(r)" V- lpIE(rkF)
(29)
of the structure function SF(k) we define a new graphic line which represents a factor
p-lSF(r). Insertion of this diagrammatic element at each point of the diagram D~a)(1) reproduces exactly the convolution integral (28). The prescription can be easily generalized to collect a special class of singular diagrams with ~ correlation lines (~ > 3) which contribute to the portion (AG) (~) of the generating function. We insert at each point of the direct ring D~)(1) the line representing the function p-1SF(r). The procedure collects, in addition to the direct ring, properly weighted exchange ring diagrams which are contained in the term (AG) (~). The analytic expression of the sum of these divergent graphs collapses to a convolution integral similar to eq. (28). Its integrand is given by the function [~l(k)SF(k)] ~ and is therefore non-singular. Let us now proceed to the contribution (AG) (4) of the generating function. In addition to the graph D(44)(1) and the four exchange ring diagrams which can be collected by the procedure already described, we find nine diagrams of a differing structure which have singular parts. Three of these diagrams are shown in fig. 7. They have in common a direct chain of three correlation lines. Insertion of the line representing the factor p - 1SF(r), eq. (29), into both points of the chain generates the rest of all singular diagrams which contribute to the function G(fl) in the order at = 4. We may therefore conclude that the portion (AG)~4) which collects all contributions of singular diagrams with four correlation lines is of finite value. In higher orders we find more complicated contributions of singular diagrams to
JASTROW CORRELATIONS
,k---~
04(4) (2)
\
I
~,
b--.-d
~--4
D5(4) (2)
401
/
06(4) (2)
Fig. 7. The diagrams D~*}(2), D~*~(2), and D~*>(2)with four correlation lines which contribute to the generating function. They diverge if longrange correlations are present in the correlated trial ground state.
Fig. 8. A special diagram with six correlation lines which diverges if longrange correlations are assumed.
the function G(fl) which cannot be collected by the simple insertion (29). The singular diagram of fig. 8 provides an illuminating example. It is evident that the rules of sect. 3 do not permit a diagram to be a contribution to the term (AG) {6} which is generated from the graph of fig. 8 by the described insertion technique. Instead of employing the function (29) we introduce an appropriate function ALtl~(r, fl) containing one correlation factor which defines a new line. This graphic element may serve as an insertion at each point of the direct ring D~3~(1). It generates the diagram of fig. 8 and other singular diagrams of the term (AG) {6~ which should be grouped together. An elementary study reveals that the function ALtl~(r, fl) should be defined by the properly weighted sum of the diagrams Dt44~(2), D~4~(2), D~4}(2) in which the direct chain of three correlation lines has been removed (appendix). The Fourier transform AL"~(k) of this function (for brevity we suppress the explicit reference to the parameter fl) has the important property
AIg)(k)
~ k 2,
k ~ O.
(30)
Consequently, the contribution (AG)~6) = A-~p2(Ex)-3 fdk[n(k)Al3"(k)] a
(31)
of the divergent diagrams extracted from the term (riG) t6) is non-singular. The same procedure can be applied to the direct ring diagram D~}(1) with ~ correlation lines (ot > 3). Again, it collects a class of divergent diagrams which contribute to the term (AG) t2~), thereby cancelling the singular portions. We may enlarge our class of examples if we employ the function
L(r, fl) = SF(r)+ A/3~)(r, fl)
(32)
instead of the functions on the right hand side separately. We can proceed further if we think of the function AL")(r, fl) as the first-order approximation ~ = 1 of a function
AL(r, fl) = AI31~(r, fl)+ dLt,2~(r, fl)+ ... +d/3~}(r, fl)+ ....
(33)
This function should achieve the cancellation of the singular portions of entire
402
E. KROTSCHECKAND M. L. RISTIG
classes of diagrams which contribute to the generating function. A careful investigation of the function (33) uncovers its close kinship to the generalized distribution function 17). The function AL(r, fl) is represented by that portion of the expansion (20) (it is understood that the correlation factors appearing in eq. (20) depend on the parameter fl) which is determined by the following class of graphs: (i) diagrams which have no nodal points 16), (ii) diagrams for which each external point is joined by an incoming (and outgoing) exchange line. The part AL(2)(r, fl) is given explicitly in ref. 20). A rather lengthy but elementary analysis of the twenty-six diagrams which contribute to that portion reveals that the Fourier transform AL~2)(k) has the property
k-lAIJ2)(k) -- 0,
k ~ 0.
(34)
There is every reason to believe that eq. (34) is also valid for the general term AL(~)(k) (~ > 3). The Fourier transform AL(k) of the function AL(r, fl) should therefore show the same behavior (34) as the function AL(2)(k). There are still classes of divergent diagrams which have not yet been included in our studies. These diagrams have a structure of increasing complexity and appear in higher orders ~ (7 > 4). We are convinced that a successful approach to deal with the singular portions of these graphs must employ suitably introduced threefour-, ...-body functions which serve as insertions in analogy to the function L(r, fl). At the present stage we have not attempted to tackle that problem in detail. A complete proof of the theorem in higher orders is in preparation.
5. Hypernetted-chain summation The investigation of the generating function (sect. 4) has revealed that each term (AG)(~)(ct _>_2) of its PS expansion is of finite value if short- and long-range correlations are present in the trial ground state (1). Consequently, in this case, we order the diagrams contributing to the graphic expansion of the radial distribution function also according to the numbers of correlation lines involved. This scheme could serve as a suitable means to approximate the distribution function by truncating expansion (20) at a chosen level ~ = ~o. But we can improve this simple scheme considerably if we employ the time-honored hypernetted-chain technique of classical liquids 16) and boson fluids 17,33). The formal steps along which the hypernettedchain summation proceeds (series connections, parallel connections of chains) are well described in ref. 16). They need not be repeated in any detail. For the fermion problem at hand we should of course take proper care of the exclusion principle and the cancellation phenomenon in each step of the HNC procedure. Drawing on our findings of sect. 4 these effects can be taken into account if we connect successively the chains (the simplest chain being a correlation line) by an insertion line which represents the function p- 1L(r, fl = O) - p- 1L(r). On the other hand, the parallel connection of chains is executed directly, i.e., without insertion. This prescription
JASTROW CORRELATIONS
403
avoids any over-counting problem of diagrams which contribute to the distribution function. The described procedure defines a renorrnalized correlation factor r/R(r) and its Fourier transform t/s(k) as solution to a coupled set of generalized hypernetted-chain equations, r/R(r) = f2(r) exp [N(r)] - 1, (35) N(k){1 + p~lR(k)L(k)} = pn2(k)L(k). The redundant function N(r) and its Fourier transform N(k) serve merely to present the relations (35) in a convenient form. To describe the renormalized factor graphically we introduce a renormalized correlation line (for example, a fat dashed line). The new graphic element collects diagrams of a special type as prescribed by eqs. (35); ordinary diagrams which have exchange lines attached to the external points do not contribute. The function L(k) has to be replaced by unity if we want to omit exchange effects in eqs. (35). In this case set (35) collapses to the familiar HNC equations for the quantity r/R(r) which constitutes the hypernetted-chain approximation of the radial distribution function of a Bose fluid ~7). Set (35) affects a simple relation between the long-range behavior of the correlation factor (3) and the asymptotic form of the renormalized correlation factor. A simple analysis of eqs. (35) under the assumption (3) yields the property r/R(r ) ---- c R r-
2,
r ~ oo.
(36)
The renormalized coefficient CRdepends on the constant c of eq. (3) in ar, elementary manner: cR = 4c(4- vk2c)- 1
(37)
Relation (37) has an interesting bearing on the liquid structure functions S(k) in the range of small values of the momentum. We shall report on these aspects of the present analysis in a separate publication. We may employ the quantity r/R(r) to collect successively entire classes of diagrams which establish the expansion (13) of the radial distribution function. This renormalization can be executed, in principle, by a two-step procedure. In the first stage we replace in all contributing diagrams the ordinary correlation line by a renormalized line. Thereupon we omit those graphic portions which are already included in lower orders. The result can be cast into the form
g(r) = [1 + qs(r)] {gv(r) + ~ Ag(R~)(r)}.
(38)
It is ordered according to the number a of renormalized correlation lines. The lowest-order term a = 1 may be constructed from the contribution (21) by replacing the correlation factor by the renormalized factor tt~(r). It need hardly be mentioned
404
E. K R O T S C H E C K A N D M. L. R I S T I G
that expansion (33) of the function AL(r, fl) can be renormalized in the same way 20). We should finally note that differing hypernetted-chain procedures can be devised 2. aT). We have shown in the present paper that the cancellation of singular portions within a finite sum of diagrams with given number of correlation lines is a characteristic feature of a system of fermions. This property should not be lost if rearrangements of terms are performed. For this reason the present device should be preferred. We take this opportunity to thank Prof. J. W. Clark for stimulating correspondence and conversations. Special thanks go to Prof. Chia-Wei Woo for providing us with his recent results on the homework problem prior to publication.
Appendix The function ALtl)(r, fl) may be written as a sum (for brevity we suppress in the following any reference that the expressions depend on the parameter fl), A/31)(r, 2) = AI3~)( r , 2) + AI3~)( r, 2) + A/3~)(rl 2).
(A.1)
The terms represent the analytic expressions of those diagrams which constitute the insertion as has been discussed in sect. 4. They are explicitly given by d~l)(r12) = - v - l~l(r12)12(r12 kF),
(A.2)
A/J~)(rl 2) = 2 p v - 2 f dr a l(r 12 kF)l(r l a kF)l(r 2 3 kr)r/(r23),
(A.3)
d/3~)(r12) = _ p2 v - 3 f d r 3 dr 4 l(rl 3 kr)l(r14 kF)l(r2 a kF)l(r24 kv)r/(r34). 3 The property
f
Al_Jl)(r)dr
lim A/31)(k)
0
(A.4)
(A.5)
k--*O
is a direct consequence of eqs. (A.2)-(A.4) if we employ the relation fdr
/(r 1 kF)/(Ir-- rll" kr) = v p - ll(rkr).
(A.6)
The evaluation of the dependence of the function ALtl)(k) on small values of the momentum can be performed most efficiently by introducing cylindrical coordinates. Suppressing the technical details one arrives at t3k
AI3~)(k)
~k Al3~)(k) =
O,
k ~ O,
a di3~)(k) ' ~k
k --, O.
(A.7) (A.8)
JASTROW CORRELATIONS
405
Application of the same technique yields for the second derivative the result (30) of sect. 4. References 1) V. R. Pandharipande, Nucl. Phys. A217 (1973) 1 2) V. R. Pandharipande and H. A. Bethe, Phys. Rev. C'/(1973) 1312 3) S. Chakravarty, M. D. Miller and Chia-Wei Woo, Nucl. Phys. A220 (1974) 233 4) V. Canuto and S. M. Chitre, Phys. Rev. Lett. 30 (1973) 999 5) S. Cochran and G. V. Chester, preprint (1973) 6) V. Canuto, J. Lodenquai and S. M. Chitre, preprint (1974) 7) P. J. Siemens and V. R. Pandharipande, Nucl. Phys. A173 (1971) 561 8) V. R. Pandharipande, Nucl. Phys. A174 (1971) 641; A178 (1971) 123 9) M. Miller, C. W. Woo, J. W. Clark and W. J. Ter Louw, Nucl. Phys. A184 (1972) 1 10) F. Iwamoto and M. Yamada, Progr. Theor. Phys. 17 (1957) 543 11) J. W. Clark and P. Westhaus, J. Math. Phys. 9 (1968) 131, 149 12) M. L. Ristig, W. J. Ter Louw and J. W. Clark, Phys. Rev. C3 (1971) 1504; 5 (1972) 695 13) D. A. Chakkalakal, Ph.D. thesis, Washington University, 1969 (unpublished) 14) J. Nitsch, Dissertation, Universit~it zu K61n (1974) 15) L. Shen and C.-W. Woo, preprint (1974) 16) J. M. J. van Leeuwen, J. Groeneveld and J. de Boer, Physica 25 (1959) 792; 30 (1964) 2265 17) E. Feenberg, Theory of quantum fluids (Academic, New York, 1969) 18) C.-W. Woo, private communication 19) L. Reatto and G. V. Chester, Phys. Rev. 155 (1967) 88 20) E. Krotscheck, Dissertation, Universit~it zu K61n (1974) 21) .N.R. Werthamer, Phys. Rev. A7 (1973) 254 22) J. W. Clark and M. L. Ristig, Phys. Rev. C5 (1972) 1553 23) M. L. Ristig and J. W. Clark, Nucl. Phys. A199 (1973) 351 24) M. Gaudin, J. Gillespie and G. Ripka, Nucl. Phys. A176 (1971) 237 25) F. Y. Wu and E. Feenberg, Phys. Rev. 128 (1962) 943 26) F. Calogero and C. Ciofi degli Atti, ed., The nuclear many-body problem, vol. 2 (Editrice Compositori. Bologna, 1973) p. 273 27) F. Y. Wu, J. Math. Phys. 4 (1963) 1438 28) E. Krotscheck, J. Nitsch, M. L. Ristig and J. W. Clark, Nuovo Cim. Lett. 6 (1973) 143 29) S.-O. B~ickman, J. W. Clark, W. J. Ter Louw, D. A. Chakkalakal and M. L. Ristig, Phys. Lett. 41B (1972) 247 30) S. Fantoni and S. Rosati, Nuovo Cim. 20A (1974) 179 31) E. K. Achter and L. Meyer, Phys. Rev. 188 (1969) 291 32) J. E. Mayer and M. G. Mayer, Statistical mechanics (Wiley, New York and London, 1940) 33) E. Krotscheck and M. L. Ristig, Phys. Lett. 48A (1974) 17 34) F. lwamoto, Progr. Theor. Phys. 19 (1958) 595 35) C. D. Williams, Ph.D. thesis, Washington University, 1961 (unpublished) 36) P. G. Reinhard and H. Arenhfvel, Z. Phys. 262 (1973) 1; 264 (1973) 211 37) S. Fantoni and S. Rosati, Nuovo Cim. Lett. 10 (1974) 545