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How good is a Jastrow trial wave function for nuclear matter?
Volume 65B, number 5
PHYSICS LETTERS
20 December 1976
H O W G O O D IS A J A S T R O W T R I A L W A V E F U N C T I O N F O R N U C L E A R M A T T E R ? * M.L. RISTIG and P. HECKING Institut f~r Theoretische Physik, UniversiRltzu K61n, 5 Kiffn 41, Germany Received 26 September 1976 - We analyse the variance of the Hamiltonian with respect to a Jastrow wave function for extended nuclear matter. Numerical results are presented for a semi-realisticcentral potential and state independent correlations which are suitably constrained for good cluster convergence. The method of correlated basis functions [1 ] has been extremely fruitful in developing a practical microscopic description of dense quantum fluids, especially of the helium liquids. Its success motivates great interest in exploring the applicability of this method to quantum systems which are only moderately dense like extended nuclear matter or finite nuclei [2]. With few exceptions [3] such studies have been confined to the Jastrow variational approach and the approximate evaluation of the expectation value E = (q'lHIxI,)/(~I'l~I')ofthe ground state energy with respect to a correlated trial state [ 4 - 6 ] . The procedure yields numerical values for the approximate energy expectation value of extended nuclear matter which differ rather strongly from the results derived by employing the standard Brueckner-Bethe-Goldstone formalism [5, 7 - 9 ] . The discrepancy calls urgently for a thorough clarification. In this context it might be very helpful to study the density matrices [10] and the distribution functions [11,8, 12] which correspond to the trial ground state assumed. Recently three additional tests of the variational approach have been proposed which might give valueable information on the merits or deficiencies of a trial state xI, = F ~ compared with the exact ground state [2]. The function • is to be taken as the ground state wave function of the nuclear system with the interactionA V= ~.A.~o(i/),~turned off. The correlation operator F is" to be assumed of the state independent form F = Ili
(1)
with respect to the trial state q'. Fluctuation (1) vanishes only if the function xI, represents an eigenstate of the Hamiltonian. Otherwise it is a positive and finite quantity for extended nuclear matter (particle number A -~ 0% while particle density P is kept constant). In this note we provide the necessary formalism to perform practical evaluations of the second moment (1). If the correlation factor f(r) is assumed to be of short range we may develop quantity (1) in a factorized Iwamoto Yamada cluster expansion [4]. Instrumental for this analysis is the generalized normalization integral 1(3) = (41exp 3(11- E0)I~). The quantity E 0 is the kinetic energy (~ITI~) and the parameter 3 serves to express the deviation (1) in the form =
1 a2
In
(2)
Research supported by the Deutsche Forschungsgemeinschaftunder Grant no. Ri 267/1. 405
Volume 65B, number 5
PHYSICS LETTERS
20 December 1976
Table 1 Variance of the Hamiltonian with respect to the optimized state independent wave function of ref. [ 7] for nuclear matter (OMYpotential). [fm-1 ]
[fm-11
[fm -t ]
1.2 1.3 1.4 1.5 1.54 1.6 1.7
2.88 2.76 2.46 2.46 2.46 2.40 3.00
2.22 2.17 2.08 2.00 2.02 2.14 2.37
0.873 0.893 1.012 0.915 0.938 1.147 0.946
0.131 0.170 0.220 0.281 0.300 0.323 0.368
[MeV]
[MeV2 ]
[MeV2 ]
[MeV2 ]
[MeV]
-10.7 -12.8 -14.6 -15.9 -16.1 -15.3 -10.3
4760 5760 6465 7482 8503 12070 23759
-313 -244 - 61 242 379 592 1166
279 393 516 648 745 1023 1650
68.8 76.9 83.2 91.5 98.1 117.0 163.0
This equation is the basic formula to develop the moment (1) in a factorized cluster expansion. Ref. [4] provides the factor cluster decomposition of the logarithm of the generalized normalization integral. Employing the result given therein we arrive at the factorized I w a m o t o - Y a m a d a expansion of deviation (1), ( A H 2) = (AH2)2 + (AH2)3 + ... + (AH2)m + ....
(3)
The term (AH2)m is the m-body cluster contribution. For finite nuclei series (3) terminates at the A-body cluster portion. For extended nuclear matter each term of eq. (3) approaches a finite value exhibiting the linked cluster property of expansion (3). To characterize the range and strength of the two-body correlations one usually introduces [4] the parameter t = ( l / A ) ~ ( i ] l f 2 ( r ) - l lij - j i ) , ii
where the sum extends over the Fermi sea, i.e., over the orbitals i, j occupied in the unperturbed state ~. This smallness parameter may be used for classifying individual contributions to the cluster terms (AH2)m, m = 2, 3, .... We may rearrange the factorized expansion (3) into terms of increasing powers in the smallness parameter, ( A H 2) = (AH2) (0) + (AH2) (1) + (AH2) (2) + ...,
(4)
where the contribution (AH2) (0) is of the order t 0, the term (AH2) (1) is of order t 1 , etc. For sufficiently small values of the parameter t classification (4) provides a sensible procedure to measure the magnitude of successive terms. Present Jastrow calculations of the expectation value of the ground state energy employ trial wave functions for which expansion (4) should be rapidly convergent. The leading term (AH2)(0) for extended nuclear matter consists in part of two-, three-, and four-body cluster contributions of expansion (3),
( AH 2 ) (0)--(
A H 2 )2(0)+ ( A H 2 )~(0)+ ( A H 2 )~(O) .
(5)
The terms (AH2)~0), (AH2)~0) are the portions of order t 0 which arise from the cluster contributions (AH2)3, (AH2)4, respectively, the clusters (AH2)m, m > 4, being at least linear in the smallness parameter. The explicit expressions of the two-, three-, and four-body portions of quantity (5) follow: (AH2)(20) = (1/2A) ~ (ijl~221i } - j i ) , ij
(AH2) (0) = ( l / A ) ~ ( i j k l [ 2 3 1 j k i ijk
ikj),
(6) (AH2)(40) = (1/4A) ~ (i]kll~241kli ] - lki] + lkji - klji) . ijkl
The sums extend over occupied single particle states li) .... being plane waves with specified spin- and isospin projections. The effective two-, three-, and four-body operators ~ 2 , ~ 3 , ~24, respectively, are given by the expressions: 406
Volume 65B, number 5
PHYSICS LETTERS
20 December 1976
Table 2 Variance of the Hamiltonian with respect to a class of Jastrow trial wave functions specified by differing values of the parameters %/~1 = tz2 constrained by the average Pauli condition at Fermi wave number k F = 1.54 fm -1 (OMY-potential) ')'
#l = u2 [fm -1 ]
~
E.~21 [MeV]
(An2)2 [MeV2 ]
1.7 1.65 1.6 1.55 1.5 1.45 1.4 1.35 1.3 1.25
3.009 2.827 2.638 2.442 2.237 2.021 1.790 1.537 1.258 0.933
0.241 0.246 0.253 0.263 0.275 0.291 0.316 0.356 0.425 0.604
- 6.7 -10.0 -12.5 -14.3 -15.2 -14.9 -13.1 - 9.3 - 2.3 + 10.6
58797 42183 29363 19 814 13215 9089 6989 6473 6963 7614
I22(12 ) = f2(r12)v2(12) + 2(h2/rn) [Vlf(rl2)O(12)l.[Vlf(rl2)] + (h2/m)2 [Af(rl2) ] 2
+ (h2/m) 2 [Af(rl2)] [ V1f(r12)]" (V1 - V2) + (h2/m)2(V1 - V2)" [ Vlf(rl2)] [Vlf(r12)]" (VI - V 2 ) ,
(7)
~23 (123) = f2 (r 12) v(12) f 2 ( r 13) o (13) + 2 (h 2/m)f2 (r 12) v (12) [ V l f ( r 13)] 2 + (h 2/m)2 [Af(rl2) ] [Af(r13)]f(r12)f(r13) + (h2/m)2(V1 - V 2 ) " [ Vlf(r12)]f(r12)f(r13) [Af(r13)] + (h2/m) 2 [Af(rl2)]f(rl3)[ Vlf(rl3)]" (V1 - V3)
(8)
+ (h2/m)2('V1 - V2)" [Vlf(rl2)]f(rl2)f(rl3) [Vlf(rl3)]" (V1 - V 3 ) , ~4(1234) = f2(r12)o(12)f2(r34)v(34 ) + 2(l~2/m)f2(r12) v(12) [ V3f(r34)] 2
(9)
+ (h2/m) 2 {(V 3 - V4)" [V3/(r34)] + [A/(r34)] ) f(r34)X f ( r l 2 ) ( [A/(rl2)] + [Vlf(rl2)]" (V1 - V 2 ) ) " The gradients appearing in eqs. ( 7 ) - ( 9 ) operate to the right or left, the action being indicated by right or left oriented arrows. Results ( 7 ) - ( 9 ) apply to spin- and isospin-dependent central potentials o (12). Tensor effects as well as state dependent correlations modify these xesults. Their incorporation would not present any insuperable difficulty. For a preliminary numerical study we employ a semi-realistic central potential which served as a test potential in earlier numerical calculations for nuclear matter [5, 7]. This potential designated OMY [13] has a state independent hard core of radius r c = 0.6 fm, an even-state attractive component of Yukawa form adjusted to produce a fit of the low-energy singlet and triplet scattering parameters, and zero attraction in odd states. The general form adopted for the correlation factor f(r) is taken from refs. [5,7], 0,
r<<,re
f(r) = {1 - exp [ - - g t l ( r - rc) ] ) {1 + 7 exp [ - / a 2 ( r -
rc)]),
(10) r ~> r c ,
subject to the average Pauli condition [4],
p y [1 - f ( r ) ] g F ( r ) dr = 0 ,
(11)
which keeps the parameters gtl, gt2, 7 within the domain corresponding to good cluster convergence. Constraint (11) involves the radial distribution function gF(r) of the non-interacting fermi sea. 407
Volume 65B, number 5
PHYSICS LETTERS
20 December 1976
Table 1 displays our numerical results for the leading term (5) of deviation (4) evaluated for the set o f parameters reported in refs. [5, 7] which optimize the ground state energy E~ 31 in three-body cluster approximation. The pair contribution is the most important portion of the t e r m ( A H 2)~(0) the t h er e - and four-body portions being a correction of less than 12% at all densities considered. We have also calculated certain pieces of the contribution (AH2) (1) appearing in eq. (4). Their values are found to be sufficiently small to justify the truncation of expansion (4) at the level (AH2) (1). Table 2 shows the variance (AH 2) with respect to differing choices of parameters of ansatz (10). The calculation has been performed at saturation density characterized by the Fermi wave number k F = 1.54 fm -1 . For simplicity we adopted the special choice It 1 =/.t 2 and approximated deviation (3) by its pair term (AH2)2 = (AH2)~0). We find that this quantity has a minimum at the parameters 3' = 1.35,/a 1 =/22 = 1.537 fm -1 . These values are very close to the optimal set of parameters (/a 1 =/a 2 , 3') which minimize the ground state energy in three-body approximation E [3] . Numerical data of this quantity are given in ref. [14]. Table 2 supplies information on the two-body approximation E ) 21 of the Jastrow ground state energy. We have also performed calculations of the fluctuations (3) employing more sophisticated state independent correlation factorsf(r) than those permitted by ansatz (10) and constraint (1 1). These refined correlations do not lower substantially the numerical values of the quantity x/(AH2)(0) shown in table 1. We expect an appreciable decrease of the magnitude of the second moment only if spin- and isospin dependence is included in the correlation operator F. We conclude from the results presented on the second moment that the trial ground state wave function of refs. [5, 7] with the optimal choice of parameters 3', ~t1, ~2 must be close to the best wave function which can be construct. ed assuming state independent spatial correlations f(r) of short range.
References [1] [2] [3] [4] [5] [6] [7] [81 [9] [10] [11] [12] [13] [14]
408
E. Feenberg, Theory of quantum fluids (Academic, New York, 1969). J.W. Clark, Crisis in nuclear matter theory (preprint, 1976). J.W. Clark, P.M. Lam and W.J. Ter Louw, Nucl. Phys. A255 (1975) 1. J.W. Clark and M.L. Ristig, in: The nuclear many-body problem, vol. 2,(Editrice Compositori, Bologna, 1973). S.-O. B~ickman, D.A. Chakkalakal and J.W. Clark, Nucl. Phys. A130 (1969) 635. V.R. Pandharipande and H.A. Bethe, Phys. Rev. C7 (1973) 1312. S.-O. B~/ckman et al., Phys. Lett. 41B (1972) 247. S. Fantoni and S. Rosati, Nuovo Cimento 25A (1975) 593. V.R. Pandharipande, R.B. Wiringa and B.D. Day, Phys. Lett. 57B (1975) 205. M.L. Ristig and J.W. Clark, Phys. Rev. B (1976), in press. E. Krotscheck and M.L. Ristig, Phys. Lett. 48A (1974) 17. E. Krotscheck and M.L. Ristig, Nucl. Phys. A242 (1975) 389. T. Ohmura, M. Morita and M. Yamada, Progr. Theor. Phys. 15 (1956) 222. J. Nitsch, doctoral dissertation 1973, University of K61n.
PHYSICS LETTERS
20 December 1976
H O W G O O D IS A J A S T R O W T R I A L W A V E F U N C T I O N F O R N U C L E A R M A T T E R ? * M.L. RISTIG and P. HECKING Institut f~r Theoretische Physik, UniversiRltzu K61n, 5 Kiffn 41, Germany Received 26 September 1976 - We analyse the variance of the Hamiltonian with respect to a Jastrow wave function for extended nuclear matter. Numerical results are presented for a semi-realisticcentral potential and state independent correlations which are suitably constrained for good cluster convergence. The method of correlated basis functions [1 ] has been extremely fruitful in developing a practical microscopic description of dense quantum fluids, especially of the helium liquids. Its success motivates great interest in exploring the applicability of this method to quantum systems which are only moderately dense like extended nuclear matter or finite nuclei [2]. With few exceptions [3] such studies have been confined to the Jastrow variational approach and the approximate evaluation of the expectation value E = (q'lHIxI,)/(~I'l~I')ofthe ground state energy with respect to a correlated trial state [ 4 - 6 ] . The procedure yields numerical values for the approximate energy expectation value of extended nuclear matter which differ rather strongly from the results derived by employing the standard Brueckner-Bethe-Goldstone formalism [5, 7 - 9 ] . The discrepancy calls urgently for a thorough clarification. In this context it might be very helpful to study the density matrices [10] and the distribution functions [11,8, 12] which correspond to the trial ground state assumed. Recently three additional tests of the variational approach have been proposed which might give valueable information on the merits or deficiencies of a trial state xI, = F ~ compared with the exact ground state [2]. The function • is to be taken as the ground state wave function of the nuclear system with the interactionA V= ~.A.~o(i/),~turned off. The correlation operator F is" to be assumed of the state independent form F = Ili
(1)
with respect to the trial state q'. Fluctuation (1) vanishes only if the function xI, represents an eigenstate of the Hamiltonian. Otherwise it is a positive and finite quantity for extended nuclear matter (particle number A -~ 0% while particle density P is kept constant). In this note we provide the necessary formalism to perform practical evaluations of the second moment (1). If the correlation factor f(r) is assumed to be of short range we may develop quantity (1) in a factorized Iwamoto Yamada cluster expansion [4]. Instrumental for this analysis is the generalized normalization integral 1(3) = (41exp 3(11- E0)I~). The quantity E 0 is the kinetic energy (~ITI~) and the parameter 3 serves to express the deviation (1) in the form =
1 a2
In
(2)
Research supported by the Deutsche Forschungsgemeinschaftunder Grant no. Ri 267/1. 405
Volume 65B, number 5
PHYSICS LETTERS
20 December 1976
Table 1 Variance of the Hamiltonian with respect to the optimized state independent wave function of ref. [ 7] for nuclear matter (OMYpotential). [fm-1 ]
[fm-11
[fm -t ]
1.2 1.3 1.4 1.5 1.54 1.6 1.7
2.88 2.76 2.46 2.46 2.46 2.40 3.00
2.22 2.17 2.08 2.00 2.02 2.14 2.37
0.873 0.893 1.012 0.915 0.938 1.147 0.946
0.131 0.170 0.220 0.281 0.300 0.323 0.368
[MeV]
[MeV2 ]
[MeV2 ]
[MeV2 ]
[MeV]
-10.7 -12.8 -14.6 -15.9 -16.1 -15.3 -10.3
4760 5760 6465 7482 8503 12070 23759
-313 -244 - 61 242 379 592 1166
279 393 516 648 745 1023 1650
68.8 76.9 83.2 91.5 98.1 117.0 163.0
This equation is the basic formula to develop the moment (1) in a factorized cluster expansion. Ref. [4] provides the factor cluster decomposition of the logarithm of the generalized normalization integral. Employing the result given therein we arrive at the factorized I w a m o t o - Y a m a d a expansion of deviation (1), ( A H 2) = (AH2)2 + (AH2)3 + ... + (AH2)m + ....
(3)
The term (AH2)m is the m-body cluster contribution. For finite nuclei series (3) terminates at the A-body cluster portion. For extended nuclear matter each term of eq. (3) approaches a finite value exhibiting the linked cluster property of expansion (3). To characterize the range and strength of the two-body correlations one usually introduces [4] the parameter t = ( l / A ) ~ ( i ] l f 2 ( r ) - l lij - j i ) , ii
where the sum extends over the Fermi sea, i.e., over the orbitals i, j occupied in the unperturbed state ~. This smallness parameter may be used for classifying individual contributions to the cluster terms (AH2)m, m = 2, 3, .... We may rearrange the factorized expansion (3) into terms of increasing powers in the smallness parameter, ( A H 2) = (AH2) (0) + (AH2) (1) + (AH2) (2) + ...,
(4)
where the contribution (AH2) (0) is of the order t 0, the term (AH2) (1) is of order t 1 , etc. For sufficiently small values of the parameter t classification (4) provides a sensible procedure to measure the magnitude of successive terms. Present Jastrow calculations of the expectation value of the ground state energy employ trial wave functions for which expansion (4) should be rapidly convergent. The leading term (AH2)(0) for extended nuclear matter consists in part of two-, three-, and four-body cluster contributions of expansion (3),
( AH 2 ) (0)--(
A H 2 )2(0)+ ( A H 2 )~(0)+ ( A H 2 )~(O) .
(5)
The terms (AH2)~0), (AH2)~0) are the portions of order t 0 which arise from the cluster contributions (AH2)3, (AH2)4, respectively, the clusters (AH2)m, m > 4, being at least linear in the smallness parameter. The explicit expressions of the two-, three-, and four-body portions of quantity (5) follow: (AH2)(20) = (1/2A) ~ (ijl~221i } - j i ) , ij
(AH2) (0) = ( l / A ) ~ ( i j k l [ 2 3 1 j k i ijk
ikj),
(6) (AH2)(40) = (1/4A) ~ (i]kll~241kli ] - lki] + lkji - klji) . ijkl
The sums extend over occupied single particle states li) .... being plane waves with specified spin- and isospin projections. The effective two-, three-, and four-body operators ~ 2 , ~ 3 , ~24, respectively, are given by the expressions: 406
Volume 65B, number 5
PHYSICS LETTERS
20 December 1976
Table 2 Variance of the Hamiltonian with respect to a class of Jastrow trial wave functions specified by differing values of the parameters %/~1 = tz2 constrained by the average Pauli condition at Fermi wave number k F = 1.54 fm -1 (OMY-potential) ')'
#l = u2 [fm -1 ]
~
E.~21 [MeV]
(An2)2 [MeV2 ]
1.7 1.65 1.6 1.55 1.5 1.45 1.4 1.35 1.3 1.25
3.009 2.827 2.638 2.442 2.237 2.021 1.790 1.537 1.258 0.933
0.241 0.246 0.253 0.263 0.275 0.291 0.316 0.356 0.425 0.604
- 6.7 -10.0 -12.5 -14.3 -15.2 -14.9 -13.1 - 9.3 - 2.3 + 10.6
58797 42183 29363 19 814 13215 9089 6989 6473 6963 7614
I22(12 ) = f2(r12)v2(12) + 2(h2/rn) [Vlf(rl2)O(12)l.[Vlf(rl2)] + (h2/m)2 [Af(rl2) ] 2
+ (h2/m) 2 [Af(rl2)] [ V1f(r12)]" (V1 - V2) + (h2/m)2(V1 - V2)" [ Vlf(rl2)] [Vlf(r12)]" (VI - V 2 ) ,
(7)
~23 (123) = f2 (r 12) v(12) f 2 ( r 13) o (13) + 2 (h 2/m)f2 (r 12) v (12) [ V l f ( r 13)] 2 + (h 2/m)2 [Af(rl2) ] [Af(r13)]f(r12)f(r13) + (h2/m)2(V1 - V 2 ) " [ Vlf(r12)]f(r12)f(r13) [Af(r13)] + (h2/m) 2 [Af(rl2)]f(rl3)[ Vlf(rl3)]" (V1 - V3)
(8)
+ (h2/m)2('V1 - V2)" [Vlf(rl2)]f(rl2)f(rl3) [Vlf(rl3)]" (V1 - V 3 ) , ~4(1234) = f2(r12)o(12)f2(r34)v(34 ) + 2(l~2/m)f2(r12) v(12) [ V3f(r34)] 2
(9)
+ (h2/m) 2 {(V 3 - V4)" [V3/(r34)] + [A/(r34)] ) f(r34)X f ( r l 2 ) ( [A/(rl2)] + [Vlf(rl2)]" (V1 - V 2 ) ) " The gradients appearing in eqs. ( 7 ) - ( 9 ) operate to the right or left, the action being indicated by right or left oriented arrows. Results ( 7 ) - ( 9 ) apply to spin- and isospin-dependent central potentials o (12). Tensor effects as well as state dependent correlations modify these xesults. Their incorporation would not present any insuperable difficulty. For a preliminary numerical study we employ a semi-realistic central potential which served as a test potential in earlier numerical calculations for nuclear matter [5, 7]. This potential designated OMY [13] has a state independent hard core of radius r c = 0.6 fm, an even-state attractive component of Yukawa form adjusted to produce a fit of the low-energy singlet and triplet scattering parameters, and zero attraction in odd states. The general form adopted for the correlation factor f(r) is taken from refs. [5,7], 0,
r<<,re
f(r) = {1 - exp [ - - g t l ( r - rc) ] ) {1 + 7 exp [ - / a 2 ( r -
rc)]),
(10) r ~> r c ,
subject to the average Pauli condition [4],
p y [1 - f ( r ) ] g F ( r ) dr = 0 ,
(11)
which keeps the parameters gtl, gt2, 7 within the domain corresponding to good cluster convergence. Constraint (11) involves the radial distribution function gF(r) of the non-interacting fermi sea. 407
Volume 65B, number 5
PHYSICS LETTERS
20 December 1976
Table 1 displays our numerical results for the leading term (5) of deviation (4) evaluated for the set o f parameters reported in refs. [5, 7] which optimize the ground state energy E~ 31 in three-body cluster approximation. The pair contribution is the most important portion of the t e r m ( A H 2)~(0) the t h er e - and four-body portions being a correction of less than 12% at all densities considered. We have also calculated certain pieces of the contribution (AH2) (1) appearing in eq. (4). Their values are found to be sufficiently small to justify the truncation of expansion (4) at the level (AH2) (1). Table 2 shows the variance (AH 2) with respect to differing choices of parameters of ansatz (10). The calculation has been performed at saturation density characterized by the Fermi wave number k F = 1.54 fm -1 . For simplicity we adopted the special choice It 1 =/.t 2 and approximated deviation (3) by its pair term (AH2)2 = (AH2)~0). We find that this quantity has a minimum at the parameters 3' = 1.35,/a 1 =/22 = 1.537 fm -1 . These values are very close to the optimal set of parameters (/a 1 =/a 2 , 3') which minimize the ground state energy in three-body approximation E [3] . Numerical data of this quantity are given in ref. [14]. Table 2 supplies information on the two-body approximation E ) 21 of the Jastrow ground state energy. We have also performed calculations of the fluctuations (3) employing more sophisticated state independent correlation factorsf(r) than those permitted by ansatz (10) and constraint (1 1). These refined correlations do not lower substantially the numerical values of the quantity x/(AH2)(0) shown in table 1. We expect an appreciable decrease of the magnitude of the second moment only if spin- and isospin dependence is included in the correlation operator F. We conclude from the results presented on the second moment that the trial ground state wave function of refs. [5, 7] with the optimal choice of parameters 3', ~t1, ~2 must be close to the best wave function which can be construct. ed assuming state independent spatial correlations f(r) of short range.
References [1] [2] [3] [4] [5] [6] [7] [81 [9] [10] [11] [12] [13] [14]
408
E. Feenberg, Theory of quantum fluids (Academic, New York, 1969). J.W. Clark, Crisis in nuclear matter theory (preprint, 1976). J.W. Clark, P.M. Lam and W.J. Ter Louw, Nucl. Phys. A255 (1975) 1. J.W. Clark and M.L. Ristig, in: The nuclear many-body problem, vol. 2,(Editrice Compositori, Bologna, 1973). S.-O. B~ickman, D.A. Chakkalakal and J.W. Clark, Nucl. Phys. A130 (1969) 635. V.R. Pandharipande and H.A. Bethe, Phys. Rev. C7 (1973) 1312. S.-O. B~/ckman et al., Phys. Lett. 41B (1972) 247. S. Fantoni and S. Rosati, Nuovo Cimento 25A (1975) 593. V.R. Pandharipande, R.B. Wiringa and B.D. Day, Phys. Lett. 57B (1975) 205. M.L. Ristig and J.W. Clark, Phys. Rev. B (1976), in press. E. Krotscheck and M.L. Ristig, Phys. Lett. 48A (1974) 17. E. Krotscheck and M.L. Ristig, Nucl. Phys. A242 (1975) 389. T. Ohmura, M. Morita and M. Yamada, Progr. Theor. Phys. 15 (1956) 222. J. Nitsch, doctoral dissertation 1973, University of K61n.