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How simple can HNC be — and still be right?
Volume 68B, number 4
PHYSICS LETTERS
20June1977
HOW SIMPLE CAN HNC BE - AND STILL BE RIGHT? A. KALLIO ’ University of Oulu, Oulu, Finland
and R.A. SMITH State University of New York, Stony Brook, New Y&k 1 I 794
Received 25 March 1977 Revised 22 April 1977 The hypernetted chain equation for boson systems is used to eliminate the Jastrow correlation function in an expression for the energy of the system. Variation of this integral leads to an equation forT= & which bears strong resemblance to the euqation used in lowest order methods, but which incorporates all clusters summed by HNC. This equation is used to study the properties of lowest-order methods, which are found to be wanting. A simple extension of the method to fermion systems gives large estimates for many-body corrections to lowest-order Fermi calculations and is in good agreement with full fermion HNC calculations.
Variational calculations with Jastrow wave functions have been used extensively in dense matter calculations. In the most sophisticated and reliable calculations, the distribution function is expanded as a sum of linked clusters, ,and some subset of these clusters are summed, as in the hypernetted chain (HNC) approximation. In the lowest order theories (LOCV), only the two-body contribution is included and rather strong constraints on the correlation are necessary to ensure plausible results. In this letter, we transform the boson HNC expansion to a form which resembles quite closely the lowest order equations; the essential step is removal of the Jastrow correlation factor! For boson systems, we obtain an equation for the optimal distribution function which is similar to ones presently solved for LOCV calculations. This can be useful in calculating ground state energies and in addition enables us to make rather critical comments about prescriptions for various LOCV constraints. We compare the prescriptions LOCVl of Pandharipande [l] and LOCV2 of Owen
* Work performed in part under the auspices of the U.S.E.R.D.A. Contract E(ll-l)-3001. ’ Present address: Department of Physics, State University of New York Stony Brook, New York 11794.
et al. [2-41 for consistency and reliability. For fermion systems a rather simple approximation is developed to allow an estimate of the many-body corrections to LOCV. For an infinite system of mass m bosons interacting with potential u, the trial wavefunction \Lv = Pi
(1)
V21n(f2)]dr
at density n. If the wave function is good, we expect this to be a good estimate of the ground state energy. The radial distribution g which appears in eq. (1) satisfies the HNC equation [5] which we write as
1
=-n
eik.r
s
(S(k) --TV--
-
1)2
&
(2)
(2n)3
where E(r) represents the sum of all elementary in the cluster expansion for g, and S(k)=
1 +nS[g(r)-
is the corresponding
l]eik”& liquid structure
diagrams
(3) function
(which 315
Volume 68B, number 4
PHYSICS LETTERS
20 June 1977
Table 1 The total energy (TOT) per boson (MeV) calculated with two constraint prescriptions (LOCVI, LOCV2) is divided into its twobody (LOW) and many-body (MBC) contributions. An HNC calculation with f chosen using d = 2ro is also shown. The potential is the short range repulsive core of the Reid 3S1 potential. n[fmm3]
LOCVI
0.1 0.3 0.5 0.7 0.9
LOCV2
ELOCV
EMBC
ETOT
ELOCV
EMBC
ETOT
36.5 155 311 496 701
1.3 5 10 13 17
37.8 160 321 509 718
27.5 119 248 407 590
6.8 20 32 40 49
34.3 139 280 447 639
must satisfy S(k) > 0 to be physical). If we neglect the elementary diagrams and substitute for ln(f2) in eq. (1) using eq. (2) we obtain the result
v9. VT+ T2(u- V2u,) dr EyIN=;j[g
(4)
where 9 % dg. Imposition of boundary conditions on 9 at large and short distances leads to the Euler-Lagrange equation for 9 :
A2
[
--gv2+(utU)
1
Y-=0
A2 1 k2 (s - I)2(2S -s U=-4mn S2
(5) + I> eik-r
dk
(6) (2703 eq. (5), equivalent to eq. (5.94) of Feenberg [6] and contained in eq. (2.18) of Pokrant [7], determines teh optimal distribution function once boundary conditions are.specified. Pokrant has solved the equation in momentum space for 4He. We have not obtained a consistent solution in coordinate space, however we do have some useful results. First, one may rather easily parametrize g and adjust the parameters to minimize eq. (4). This gives energies which are as good as those that can be obtained by other choices of the variational wave function, such as that of Pandharipande and Bethe [8]. Detailed numerical calculations show that our procedure is stable provided one chooses g’s for which S(k) is physical. In addition, we observed that the ground-state energy is rather insensitive to the mediumand long-range behavior of g. In short, the energy expression (4) is not plagued by the Emery difficulty [9]. 316
EHNC
33.3 139 275 432 602
Second, the expression (4) provides a rather strong check on LOCV theories for boson systems, where 9 is approximated by an fcomputed with some prescribed constraints. In LOCVl, eq. (6) is solved for 7 using the approximation u = -M(r - d), where X is chosen to satisfy f’(d) = 0 and d is constrained so that nJedY2dr = 1. The LOCV2 prescription requires 9 < 1 and takes u = -tA(r - d) if the resulting 9 satisties S(k) > 0. Otherwise, u = -M(d - r) - uO(r - d) and X is the minimum value required to satisfy the inequality on S(k). These constraints are intended to make the higher order cluster diagrams small; i.e., the FF2u1 term in eq. (4) should give a small contribution to the energy. We may judge the consistency of the constraint by comparing this many-body cluster FIMBc, to the twobody term ELovc. We may further judge the adequacy of the trial g by seeing how close the total variational energy, including the Emc, comes to the minimum value obtainable. In other words, ideally the g chosen should be a fair approximation to the best g and the many-body corrections should be small. For numerical stability, we actually calculated EhlBC using the equivalent expression Em,=-8-n
A2
s
dk
(243
k2(S-
1)3
(7)
s
From this form, it is not surprising that EMBC is generally positive. Tables 1 and 2, respectively, show the results of these calculations for the repulsive core of the Reid 3S, potential (“u2”) and for the full central Reid 3S1 potential [lo]. In table 1, for the purely repulsive potential, we see that EMBc (LOCVl) is roughly a third of the EM,,
Volume
68B, number
PHYSICS
4
LETTERS
20 June
1977
Table 2 The total energy (TOT) per boson (MeV) calculated with two constraint prescriptions (LOCVl, LOCVZ) is divided into its twobody (LOCV) and many-body (MBC) contributions. Also shown are the results obtained by minimizing a parametrizedg in eq. (4). The potential is the central piece of the Reid 3Sr potential. n[fme3]
0.1 0.3 0.5 0.7 0.9
LOW1
PRESENT
LOCVZ
ELOCV
EMBC
ETOT
ELOCV
EMBC
ETOT
EMBC
ETOT
-12.1 -24.9 -15.1 14.9 64.7
0.8 4.3 8.3 12.6 16.5
-11.3 -20.6 - 6.7 27.4 81.1
-12.3 -36.8 -49.9 -44.2 -18.8
1.0 17.8 29.4 41.1 50.5
-11.3 -18.8 -20.5 - 3.1 31.7
1.1 8.7 22.9 37.3 45.7
-11.1 -25.8 -26.0 - 9.6 22.5
(LOCVZ) and thus the LOCVl is more consistent than LOCV2 is about trunctaing the cluster expansion. The truncation error of LOCV2 is about 10%. On the other hand, the approximate distribution function of LOCV2 is more adequate in that the total energy is closer to that obtained by an HNC calculation with a good trial function. While the LOCV2 energy is not too far from that obtained with HNC, this cannot by itself justify the method. Again in table 2, for the full 3S, potential, the Emc (LOCVl) = 3 Emc (LOCV2). The corresponding EIMBc’s in tables 1 and 2 are also very similar. Table 2 also gives the Em and total energy obtained by varying parameters in a trial g. The total energies thus obtained are very close to those obtained by Smith [ 1 l] using methods presented in ref. [8]. As a result of cancellations in ELOCV, the Emc are comparable to the ELocv. We note that the g given by LOCV2 does not do too badly, if the MBC term is included. Also, the Em, (LOCV2) and EMBC (direct minimization) are comparable. On this basis, we are not entitled to completely rule out the possibility of a consistent and adequate LOCV prescription. However, there is ample evidence to support a strong feeling of pessimism. To be sure, we have excluded elementary diagrams and picked a particular (Jackson-Feenberg [ 121) expression for the kinetic energy. One could obtain similar results with other choices, however. In addition, if elementary diagrams are important, there is no justification for two-body truncation anyway. Let us now fix our sights on fermion systems. The situation here is much more complicated because the one integral equation for the boson g is replaced by
the more complicated set of fermion intergral equations given by Fantoni and Rosati [FR,13]. To estimate the size of corrections for fermion systems, we approximated the full fermion distribution function as g&)
=&)
(
1 - $r2(k,r)
)
;
@I = 5 iI
(8)
where g is obtained from fusing eq. (2), and s is the degeneracy factor. We also approximated the energy per particle by 3
A2k;
EVIN’T -g fi2
1
+I
&k2 (SF- I)@ -
- fib nI-(# SF(k)
=1 +nJ(g&)
1j2
s - l)eik”dr
(9)
which is obtained by dropping the W(ch) term from an expression of FR and using our approximate &, which neglects G,, Ghh and Gdd chains. Our ,!&nC for fermion systems is estimated by the second integral of eq. (9). In table 3, we compare energies obtained by numerical minimization of eq. (9) with parametrized g’s with those obtained with solution of the full set of FR equations. The g was constrained to satisfy S(k) > 0. Again, we see that the Emc are quite large (very close in fact to the corresponding Emc in the boson problem). The total energy is in surprisingly good agreement with that obtained using the FHNC equations of FR with an f chosen according to ref. [8] with d = 2r0. In the fermion case, it is a little harder to directly compare our results with those of the LOCVl and 317
Volume 68B, number 4
PHYSICS LETTERS
Table 3 The total energy (TOT) per fermion (MeV) calculated using eq. (9) is divided into one- and two-body (LO) and many-body (MBC) contributions. Also shown is the FHNC result for s = 4 obtained by solving the integral equations of Fantoni and Rosati [ 121. The potential is the central part of the Reid 3Sr potential. We have required S(k) > 0. n[fmm3]
ELO
EMBC
ETOT
EFHNC
0.1 0.3 0.5 0.7 0.9
+ 3.8 - 4.6 -10.2 3.0 22.2
1.0 7.8 22.5 34.7 56.2
4.8 3.1 12.3 37.7 78.3
3.9 2.7 12.5 37.2 76.4
LOCV2 prescriptions, because of the k- and Z-dependence which lurks there. The magnitude of our Em, makes it clear, though, that many-body corrections are extremely important. In summary, we have obtained an appealing form for the energy of a boson system. A variational equation for the optimal distribution function has a form similar to that used in lowest-order calculations. We have not obtained consistent solutions to this equation yet, but investigation continues (at the least, given g one can solve eq. (5) for the potential u). The contribution of many-body clusters is conveniently separated from the two-body term computed in LOCV calculations. Two LOCV prescriptions are shown to have large ElllBC or give poor distribution functions or both. Numerical minimization of the energy using a parametric form of g gives results consistent with other methods. For fermion systems, a very simple approximation enables us to estimate the energy of a fermion system by eq. (9). Again, the lowest-order and MBC contributions are calculated separately. Numerical minimiza-
318
20 June 1977
tion yields good results compared with more standard methods involving fewer approximations. The many body contributions are large which strongly suggests that LOCV results for fermion systems are unreliable. There are still several interesting questions. Can a method be found to solve eq. (5) for g, given u? Are the EmC sensitive primarily to the short range repulsion and insensitive to the particle statistics as one might infer from the tables? We would like to thank K. Lantto and A.D. Jackson for helpful comments on the manuscript, including methods for solven eq. (5) and a preprint by Lantto and Siemens on the optimal distribution function for fermion systems.
References 111 V.R. Pandharipande, Nuclear Physics Al78 (1971) 123. 121 J.C. Owen, R.F. Bishop, J.M. Irvine, Physics Letters 59B (1975) 1. 131 J.C. Owen, R.F. Bishop, J.M. Irvine, Physics Letters 66B (1977) 25. I41 J.C. Owen, R.F. Bishop, J.M. Irvine, Nuclear Physics A274 (1976) 108. 151 J.M.J. van Leeuwen, J. Groeneveld, J. de Boer, Physica 25 (1959) 792. I61 E. Feenberg, Theory of Quantum Fluids (Academic Press, New York, 1969) 117. 171 M.A. Pokrant, Phys. Rev. A6 (1972) 1588. 181 V.R. Pandharipande, H.A. Bethe, Phys. Rev. C7 (1973) 1312. PI V.J. Emery, Nuclear Physics 6 (1958) 585. [lOI R.V. Reid, Annals of Physics 50 (1968) 411. illI R.A. Smith, Physics Letters 63B (1976) 369. 1121 H.W. Jackson, E. Feenberg, Annals of Physics 15 (1961) 266. 1131 S. Fantoni, S. Rosati, Nuovo Cimento 25A (1975) 593.
PHYSICS LETTERS
20June1977
HOW SIMPLE CAN HNC BE - AND STILL BE RIGHT? A. KALLIO ’ University of Oulu, Oulu, Finland
and R.A. SMITH State University of New York, Stony Brook, New Y&k 1 I 794
Received 25 March 1977 Revised 22 April 1977 The hypernetted chain equation for boson systems is used to eliminate the Jastrow correlation function in an expression for the energy of the system. Variation of this integral leads to an equation forT= & which bears strong resemblance to the euqation used in lowest order methods, but which incorporates all clusters summed by HNC. This equation is used to study the properties of lowest-order methods, which are found to be wanting. A simple extension of the method to fermion systems gives large estimates for many-body corrections to lowest-order Fermi calculations and is in good agreement with full fermion HNC calculations.
Variational calculations with Jastrow wave functions have been used extensively in dense matter calculations. In the most sophisticated and reliable calculations, the distribution function is expanded as a sum of linked clusters, ,and some subset of these clusters are summed, as in the hypernetted chain (HNC) approximation. In the lowest order theories (LOCV), only the two-body contribution is included and rather strong constraints on the correlation are necessary to ensure plausible results. In this letter, we transform the boson HNC expansion to a form which resembles quite closely the lowest order equations; the essential step is removal of the Jastrow correlation factor! For boson systems, we obtain an equation for the optimal distribution function which is similar to ones presently solved for LOCV calculations. This can be useful in calculating ground state energies and in addition enables us to make rather critical comments about prescriptions for various LOCV constraints. We compare the prescriptions LOCVl of Pandharipande [l] and LOCV2 of Owen
* Work performed in part under the auspices of the U.S.E.R.D.A. Contract E(ll-l)-3001. ’ Present address: Department of Physics, State University of New York Stony Brook, New York 11794.
et al. [2-41 for consistency and reliability. For fermion systems a rather simple approximation is developed to allow an estimate of the many-body corrections to LOCV. For an infinite system of mass m bosons interacting with potential u, the trial wavefunction \Lv = Pi
(1)
V21n(f2)]dr
at density n. If the wave function is good, we expect this to be a good estimate of the ground state energy. The radial distribution g which appears in eq. (1) satisfies the HNC equation [5] which we write as
1
=-n
eik.r
s
(S(k) --TV--
-
1)2
&
(2)
(2n)3
where E(r) represents the sum of all elementary in the cluster expansion for g, and S(k)=
1 +nS[g(r)-
is the corresponding
l]eik”& liquid structure
diagrams
(3) function
(which 315
Volume 68B, number 4
PHYSICS LETTERS
20 June 1977
Table 1 The total energy (TOT) per boson (MeV) calculated with two constraint prescriptions (LOCVI, LOCV2) is divided into its twobody (LOW) and many-body (MBC) contributions. An HNC calculation with f chosen using d = 2ro is also shown. The potential is the short range repulsive core of the Reid 3S1 potential. n[fmm3]
LOCVI
0.1 0.3 0.5 0.7 0.9
LOCV2
ELOCV
EMBC
ETOT
ELOCV
EMBC
ETOT
36.5 155 311 496 701
1.3 5 10 13 17
37.8 160 321 509 718
27.5 119 248 407 590
6.8 20 32 40 49
34.3 139 280 447 639
must satisfy S(k) > 0 to be physical). If we neglect the elementary diagrams and substitute for ln(f2) in eq. (1) using eq. (2) we obtain the result
v9. VT+ T2(u- V2u,) dr EyIN=;j[g
(4)
where 9 % dg. Imposition of boundary conditions on 9 at large and short distances leads to the Euler-Lagrange equation for 9 :
A2
[
--gv2+(utU)
1
Y-=0
A2 1 k2 (s - I)2(2S -s U=-4mn S2
(5) + I> eik-r
dk
(6) (2703 eq. (5), equivalent to eq. (5.94) of Feenberg [6] and contained in eq. (2.18) of Pokrant [7], determines teh optimal distribution function once boundary conditions are.specified. Pokrant has solved the equation in momentum space for 4He. We have not obtained a consistent solution in coordinate space, however we do have some useful results. First, one may rather easily parametrize g and adjust the parameters to minimize eq. (4). This gives energies which are as good as those that can be obtained by other choices of the variational wave function, such as that of Pandharipande and Bethe [8]. Detailed numerical calculations show that our procedure is stable provided one chooses g’s for which S(k) is physical. In addition, we observed that the ground-state energy is rather insensitive to the mediumand long-range behavior of g. In short, the energy expression (4) is not plagued by the Emery difficulty [9]. 316
EHNC
33.3 139 275 432 602
Second, the expression (4) provides a rather strong check on LOCV theories for boson systems, where 9 is approximated by an fcomputed with some prescribed constraints. In LOCVl, eq. (6) is solved for 7 using the approximation u = -M(r - d), where X is chosen to satisfy f’(d) = 0 and d is constrained so that nJedY2dr = 1. The LOCV2 prescription requires 9 < 1 and takes u = -tA(r - d) if the resulting 9 satisties S(k) > 0. Otherwise, u = -M(d - r) - uO(r - d) and X is the minimum value required to satisfy the inequality on S(k). These constraints are intended to make the higher order cluster diagrams small; i.e., the FF2u1 term in eq. (4) should give a small contribution to the energy. We may judge the consistency of the constraint by comparing this many-body cluster FIMBc, to the twobody term ELovc. We may further judge the adequacy of the trial g by seeing how close the total variational energy, including the Emc, comes to the minimum value obtainable. In other words, ideally the g chosen should be a fair approximation to the best g and the many-body corrections should be small. For numerical stability, we actually calculated EhlBC using the equivalent expression Em,=-8-n
A2
s
dk
(243
k2(S-
1)3
(7)
s
From this form, it is not surprising that EMBC is generally positive. Tables 1 and 2, respectively, show the results of these calculations for the repulsive core of the Reid 3S, potential (“u2”) and for the full central Reid 3S1 potential [lo]. In table 1, for the purely repulsive potential, we see that EMBc (LOCVl) is roughly a third of the EM,,
Volume
68B, number
PHYSICS
4
LETTERS
20 June
1977
Table 2 The total energy (TOT) per boson (MeV) calculated with two constraint prescriptions (LOCVl, LOCVZ) is divided into its twobody (LOCV) and many-body (MBC) contributions. Also shown are the results obtained by minimizing a parametrizedg in eq. (4). The potential is the central piece of the Reid 3Sr potential. n[fme3]
0.1 0.3 0.5 0.7 0.9
LOW1
PRESENT
LOCVZ
ELOCV
EMBC
ETOT
ELOCV
EMBC
ETOT
EMBC
ETOT
-12.1 -24.9 -15.1 14.9 64.7
0.8 4.3 8.3 12.6 16.5
-11.3 -20.6 - 6.7 27.4 81.1
-12.3 -36.8 -49.9 -44.2 -18.8
1.0 17.8 29.4 41.1 50.5
-11.3 -18.8 -20.5 - 3.1 31.7
1.1 8.7 22.9 37.3 45.7
-11.1 -25.8 -26.0 - 9.6 22.5
(LOCVZ) and thus the LOCVl is more consistent than LOCV2 is about trunctaing the cluster expansion. The truncation error of LOCV2 is about 10%. On the other hand, the approximate distribution function of LOCV2 is more adequate in that the total energy is closer to that obtained by an HNC calculation with a good trial function. While the LOCV2 energy is not too far from that obtained with HNC, this cannot by itself justify the method. Again in table 2, for the full 3S, potential, the Emc (LOCVl) = 3 Emc (LOCV2). The corresponding EIMBc’s in tables 1 and 2 are also very similar. Table 2 also gives the Em and total energy obtained by varying parameters in a trial g. The total energies thus obtained are very close to those obtained by Smith [ 1 l] using methods presented in ref. [8]. As a result of cancellations in ELOCV, the Emc are comparable to the ELocv. We note that the g given by LOCV2 does not do too badly, if the MBC term is included. Also, the Em, (LOCV2) and EMBC (direct minimization) are comparable. On this basis, we are not entitled to completely rule out the possibility of a consistent and adequate LOCV prescription. However, there is ample evidence to support a strong feeling of pessimism. To be sure, we have excluded elementary diagrams and picked a particular (Jackson-Feenberg [ 121) expression for the kinetic energy. One could obtain similar results with other choices, however. In addition, if elementary diagrams are important, there is no justification for two-body truncation anyway. Let us now fix our sights on fermion systems. The situation here is much more complicated because the one integral equation for the boson g is replaced by
the more complicated set of fermion intergral equations given by Fantoni and Rosati [FR,13]. To estimate the size of corrections for fermion systems, we approximated the full fermion distribution function as g&)
=&)
(
1 - $r2(k,r)
)
;
@I = 5 iI
(8)
where g is obtained from fusing eq. (2), and s is the degeneracy factor. We also approximated the energy per particle by 3
A2k;
EVIN’T -g fi2
1
+I
&k2 (SF- I)@ -
- fib nI-(# SF(k)
=1 +nJ(g&)
1j2
s - l)eik”dr
(9)
which is obtained by dropping the W(ch) term from an expression of FR and using our approximate &, which neglects G,, Ghh and Gdd chains. Our ,!&nC for fermion systems is estimated by the second integral of eq. (9). In table 3, we compare energies obtained by numerical minimization of eq. (9) with parametrized g’s with those obtained with solution of the full set of FR equations. The g was constrained to satisfy S(k) > 0. Again, we see that the Emc are quite large (very close in fact to the corresponding Emc in the boson problem). The total energy is in surprisingly good agreement with that obtained using the FHNC equations of FR with an f chosen according to ref. [8] with d = 2r0. In the fermion case, it is a little harder to directly compare our results with those of the LOCVl and 317
Volume 68B, number 4
PHYSICS LETTERS
Table 3 The total energy (TOT) per fermion (MeV) calculated using eq. (9) is divided into one- and two-body (LO) and many-body (MBC) contributions. Also shown is the FHNC result for s = 4 obtained by solving the integral equations of Fantoni and Rosati [ 121. The potential is the central part of the Reid 3Sr potential. We have required S(k) > 0. n[fmm3]
ELO
EMBC
ETOT
EFHNC
0.1 0.3 0.5 0.7 0.9
+ 3.8 - 4.6 -10.2 3.0 22.2
1.0 7.8 22.5 34.7 56.2
4.8 3.1 12.3 37.7 78.3
3.9 2.7 12.5 37.2 76.4
LOCV2 prescriptions, because of the k- and Z-dependence which lurks there. The magnitude of our Em, makes it clear, though, that many-body corrections are extremely important. In summary, we have obtained an appealing form for the energy of a boson system. A variational equation for the optimal distribution function has a form similar to that used in lowest-order calculations. We have not obtained consistent solutions to this equation yet, but investigation continues (at the least, given g one can solve eq. (5) for the potential u). The contribution of many-body clusters is conveniently separated from the two-body term computed in LOCV calculations. Two LOCV prescriptions are shown to have large ElllBC or give poor distribution functions or both. Numerical minimization of the energy using a parametric form of g gives results consistent with other methods. For fermion systems, a very simple approximation enables us to estimate the energy of a fermion system by eq. (9). Again, the lowest-order and MBC contributions are calculated separately. Numerical minimiza-
318
20 June 1977
tion yields good results compared with more standard methods involving fewer approximations. The many body contributions are large which strongly suggests that LOCV results for fermion systems are unreliable. There are still several interesting questions. Can a method be found to solve eq. (5) for g, given u? Are the EmC sensitive primarily to the short range repulsion and insensitive to the particle statistics as one might infer from the tables? We would like to thank K. Lantto and A.D. Jackson for helpful comments on the manuscript, including methods for solven eq. (5) and a preprint by Lantto and Siemens on the optimal distribution function for fermion systems.
References 111 V.R. Pandharipande, Nuclear Physics Al78 (1971) 123. 121 J.C. Owen, R.F. Bishop, J.M. Irvine, Physics Letters 59B (1975) 1. 131 J.C. Owen, R.F. Bishop, J.M. Irvine, Physics Letters 66B (1977) 25. I41 J.C. Owen, R.F. Bishop, J.M. Irvine, Nuclear Physics A274 (1976) 108. 151 J.M.J. van Leeuwen, J. Groeneveld, J. de Boer, Physica 25 (1959) 792. I61 E. Feenberg, Theory of Quantum Fluids (Academic Press, New York, 1969) 117. 171 M.A. Pokrant, Phys. Rev. A6 (1972) 1588. 181 V.R. Pandharipande, H.A. Bethe, Phys. Rev. C7 (1973) 1312. PI V.J. Emery, Nuclear Physics 6 (1958) 585. [lOI R.V. Reid, Annals of Physics 50 (1968) 411. illI R.A. Smith, Physics Letters 63B (1976) 369. 1121 H.W. Jackson, E. Feenberg, Annals of Physics 15 (1961) 266. 1131 S. Fantoni, S. Rosati, Nuovo Cimento 25A (1975) 593.