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Solution of the four-nucleon Schrödinger equation
Nuclear Physics A356 (1981) 114- 128; @ North-Holland Publishing Co., Amrterdom Not to be reproduced by photoprint or microfilm without written permission from the publisher
SOLUTION
OF THE FOUR-NUCLEON
SCHRiiDINGER
EQUATION
J. G. ZABOLITZKY Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA and Institut fiir Theoretische Physik, Ruhr-Universitit Bochum, D-4630 Bochum I, West Germany
M. H. KALOS Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA Received 13 August 1980 Abstract: We present a solution of the Schriidinger equation for the case of four nucleons interacting via the Malfliet-Tjon potential. The Schriidinger equation is integrated by means of the Green function Monte Carlo method. The adaption of this method to treat finite systems is given. For the four-nucleon case, the ground-state eigenvalue is found to be - 31.3 f 0.2 MeV and the rms point mass radius 1.36kO.01 fm. The point mass distribution is given and does not exhibit a central depression. The method is also applied to the problem of four helium atoms interacting via the Lennard-Jones potential.
1. Introduction In recent years the ground state of the four-nucleon system has been computed by a variety of authors using diverse techniques. For separable two-particle interactions different integral equation approaches have been exploited le4). Hyperspherical expansions have been used for very soft local two-particle interactions ‘). For Yukawa-type local interactions Tjon 6, ‘) has solved the Yakubovsky integral equations *) employing Hilbert-Schmidt separable expansion or unitary pole expansion schemes. Almost twenty years ago a variant of the Green function Monte Carlo technique was used for a purely attractive, bounded two-particle interaction ‘). The coupled-cluster method has been applied to a variety of local interactions lo). But for ref. ‘), any of these calculations involving local interactions had to rely on some kind of expansion scheme, either separable expansions of subchannel amplitudes or a cluster expansion. In the present work we solve exactly - subject only to statistical sampling errors, which may be made arbitrarily small - the 114
J. G. Zabolitzky, M. H. Kales 1 Four-nucleon Schriidinger equation
four-nucleon
Schrodinger equation for the Malfliet-Tjon
115
interaction ii)
D(r) = (7.39 exp (- 3.11 r)/r- 2.93 exp (- 1.55 r)/r)hc,
(1)
with the interparticle distance I in fm. However, the present method is applicable for any local, central state-independent interaction D(r),as will be demonstrated by applying it to the problem of four helium atoms bound together by the Lennard-Jones potential v(r) =
40.88(x-”
- xe6)OK,
x = r/2.556 A.
(2)
In sects. 2 and 3 we describe the Green function MonteCarlo(GFMC)
method ’ 2*’ 3), Sect. 5 contains the adaption of the GFMC method to the present problem of finite systems. In sects. 6 and 7 we present our numerical results for the a-particle and in sect. 8 for the four helium-4 atoms problem. Sect. 9 contains a summary and outlook for further use of the GFMC method. Sect. 4 is devoted to methods checking the accuracy of the calculation.
2. The Green function Monte Carlo method For a local, central, state-independent wave function may be written as
interaction the four-nucleon
ti = $(rr, r2, 13, r,)det (air,) = $(R) det (ciri),
ground-state
(3)
where I,@) is symmetric with respect to interchange of particle coordinates, and the determinant of spin and isospin functions is antisymmetric. Throughout this paper R is used to denote the 1Zdimensional coordinate vector of the four particles. The Schriidinger equation then reduces to ( - V2 + WWW)
= W(R),
(4)
where v2 = iv;, i=l
J’(R)=
$ ,c,v(rij) + Vo 1 0. I
The energy scale has been shifted by a constant V, in order to have the potential always positive; this is a sufficient, but not necessary condition for the Green function, to be introduced in eq. (6), to be positive 13).No reference to spin/isospin coordinates occurs. The problem is equivalent to the problem of four interacting, spinless bosons. The boundary condition appropriate for a finite system is that the wave function should vanish whenever any of the coordinates approaches infinity. With the same boundary condition, the full Green function is defined by (-v:+UR,))G(R,R,)
= a(R-R,),
(6)
116
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
where G(R, R,) is positive and a symmetric function of its arguments. For a nondegenerate ground state it is evident that the the iteration sequence, @“+“(R)
= E,
s
G(R, R,)@‘“‘(R,)dR,,
(7)
will converge to the ground-state wave function for an arbitrary trial energy E, > 0, since
@‘“‘(R) = 1 EIQ”C,~,(R), a
if the initial guess
@co)(R) = c W,(R), a
where tj,(R) are the complete, orthonormal eigenfunctions of the hamiltonian (4). In eq. (8) the term with the largest factor EJE, will become dominant which will occur for the smallest value of E,, i.e., the ground state. Since the lowest-energy symmetric wave function will have no nodes, the groundstate wave function may be chosen real and positive. The same holds true of the initial guess and of the Green function G(R, R,) so that all quantities in eq. (7) are positive ’ 3). Therefore, at any step of the iteration (7), the current iterate Q@)(R)may be represented by a collection of configurations {Ri, i = 1,. . ., M}, where M is some number conveniently chosen to be of the order of a few hundred. The function is represented by this collection of configurations in the sense that the density of configurations in the interval dR at R is proportional to @(“)(R)dR. The iteration of eq. (7) may now readily be executed via a Monte Carlo algorithm: For any configuration R,, of the representation of @“‘(R.,), sample new configurations from the (unnormalized) probability distribution P(R) = W(R,
R,).
(9)
These new configurations descending from all the old ones are called a new generation and represent the next iterate, @(“+ ‘j(R) . Sampling from an unnormalized probability distribution is to be interpreted as forming the normalization JV =
s
dR E,G(R, R,)
and sampling JV configurations on the average. The dR, integration in eq. (7) is performed implicitly by collecting the descendants of all the configurations R, representing @“‘(RI) in order to form the new generation. If the trial energy E, is chosen close to the ground-state eigenvalue E, the size of the generations, i.e. .N =
s
c#+“‘(R)dR
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
117
will vary only slowly with the generation number n. By adjusting the trial energy it is possible to adjust the generation size for computational convenience. From the growth or decay of generations the ground-state eigenvalue may be estimated by @“)(R)dR s
E, = E,
qP+ “(R)dR ’ f ifn is sufficiently large, i.e. the other components in eq. (8) are negligible. The variance of the growth estimator for the ground-state eigenvalue may be decreased by using ’ 3, M 9’” + “(R)dR cs ;==’
EO=E,
(10) @n+i+ “(R)dR
zs
i=l
At first sight the proposed algorithm seems impossible to execute since the Green function G(R, R,) is not known. However, inspection of the algorithm shows that it is not necessary to know the full Green function but it is sufficient to construct an algorithm to sample configurations from the distribution (9) for given R,. We proceed now to describe such an algorithm. For a given configuration R, a finite domain D(R,) may be chosen so that the popotential function V(R) is bounded from the above throughout the domain D(R,) by the constant U(R,). We define a function G, by t-S:
+ WW,,
&I
= W,
- R,),
(11)
subject to the boundary condition that G,(R,, R,) = 0 whenever R, is on the boundary or outside of D(R,). Multiplying eq. (6) by G,(R,, R,,), eq (11) by G(R, R,) and integrating both equations over R, yields upon subtraction of the two equations G(R, R,) = G,(R, R,)+
+
- aG’~’ s WRo) (
s
R”) G(R, R,)dR, >
W- WWW,,
Ro)W,
WR,.
(12)
D(Ro)
In the second term on the r.h.s. the volume integral has been converted to an integral over the surface lB(R,) introducing the normal derivative a/an and use has been made of the boundary condition for G,. Eq. (12) is a linear integral equation for G in terms of G,. Since the choice of domain D(R,) is at our disposal it may be chosen in any convenient way so as to facilitate the computation of G,. If D(R,) is chosen to be a Cartesian product of three-dimensional spheres or cubes centered at R,, G, is known analytically, and may be sampled by means of straightforward techniques.
118
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schrlidinger equation
Since, by construction, all the terms and kernels in eq. (21) are non-negative, again an interpretation in terms of probabilities is possible and thereby the construction of a random walk sampling ’ 2*’ 3, configurations R for given R,.
3. Importance sampling In order to accelerate the calculation let us introduce an importance function I,+(R). This function should embody as much physical knowledge as possible about the ground-state wave function we are looking for. It will be used to guide the random walk to spend more time in favourable regions of 12-dimensional space and less time in unfavourable regions where no contributions to a next generation are to be expected. Of course we will not introduce any approximation but just try to rewrite eqs. (7) and (12) in a special way in order to reduce the variance of any expectation, e.g., the energy. It can be shown i3) that if I,+,is the exact ground-state wave function, then the energy can be estimated with zero variance from eq. (10). Not knowing the exact function, we use a variational wave function of Jastrow-type for $,(R). Introducing the new unknown function W)
= WW,(R),
eq. (7) may be rewritten as
R~)/~~(R~)]~F~)(R~). P+~)(R)= @"+~)(R)~,(R) = i7,[$,(R)G(R, (13) s Similarly, eq. (12) is identical to WW(R
&,)/ICI,(&) = c(R, R,) = WW,(R, +
s s
%,)/ICI,(&)
~IC/IW-‘~G~(R~, &))II(I,&)]~W, R,)dR,
aD(Ro)
+
(u- Wh))CWW&,
R,)I$,(R,)]&R,R,)dR,.
WW
It is seen that the transformed Green function G is sampled as easily as the orginal one, and that the iteration (13) will converge to a density g(R) = WW,(R), where e(R)is the ground-state wave function. It is seen also that the configurations R, of the random walk eq. (14) get weighted with J/,(R,). That is, in regions where t,b,(R) is large we expect that I,+(R) is also large and the random walk spends more time there than it would otherwise. Conversely, regions where t,b(R) is small tend to be less densely populated.
J. G. Zabolitzky, M. H. Kales 1 Four-nucleon SchrBdinger equation
119
4. Evaluation of expectations and error estimates One possible way to estimate the ground-state energy from a GFMC calculation has been given in eq. (lo), the so-called growth estimator. It is obvious that Q,may be replaced by 4 in eq. (10). Another useful estimator for the ground-state energy is the variational one 13),
a quantity similar in spirit to the model energy of ref. i4). The variational estimator also has the property that its variance may be made very small. The expression in brackets in eq. (15) would be a constant if I& were the eigenfunction of the hamiltonian. Like the growth estimator the variational estimator would then have zero variance being the expectation of a constant. In our calculation the variational estimator proved to have a significantly smaller variance than the growth estimator. The agreement between the growth and variational estimators provides an useful test on the accuracy of the method. Another useful test of the accuracy of the method is simply to use different importance functions $P By deliberately selecting an importance function which predicts inaccurately some physical property and thereby straining the GFMC method one may reassure oneself about the convergence to the true ground state in spite of misleading guidance via such Jl,. S~ultan~~ly, eq. (15) will yield the same energy for different importance functions if and only if the Schrodinger equation has actually been solved. Otherwise, eq. (15) may assume arbitrary values for arbitrary variations in 4% It is more difficult to estimate the error of any quantity than the quantity itself. If (Xi> is a set of statistically ~de~ndent samples, the standard deviation of the mean,
may be estimated from 1 i (Xi--X)2 d = [ N(N--1) i=r
1 *.
07)
However, in the present calculation generation averages like eq. (15) are not statistically independent, since one generation produces the next one by the random walk of eq. (14). It may be assumed that after a s~~entIy large number of generations statistical independence is achieved. Therefore, the “independent samples” (XJ in eqs. (16) and (17) are taken to be 20-generation averages in the present calculation.
120
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equution
Since in the calculation started from a trial &‘i the energy was observed to relax in much less then 20 generations this is a reasonable procedure. The error bars on our numerical results quoted below are then calculated precisely according to eq. (17).
5. Adaptation to finite systems So far we have described the GFMC method as it has been used for infinite systems. For a finite system, there are two points which need special attention: the boundary condition and the c.m. motion. The boundary condition does not pose any problem. As opposed to the treatment for an infinite system, where configurations outside a given basic cell are moved back inside in order to ensure periodic boundary conditions, one just lets the random walk, eq. (14), evolve without any restriction. If the domains D are chosen so that the union of all possible domains D covers the full 12-dimensional space, eq. (14) is seen to satisfy eq. (6) for arbitrary R, R, by simple insertion. Since no restriction will be imposed on the random walk (14), any configuration, i.e. any point in 1Zdimensional space may be reached. However, as we will use as the zeroth generation a population drawn from a localized state where all particles are reasonably close to each other, the probability to reach large values for any interparticle separation decreases with that value increasing, i.e. the wave function will tend to zero for large interparticle separation. The hamiltonian eq. (4) is translationally invariant. The ground-state wave function will therefore also be translationally invariant. Since in the lowest-energy state no kinetic energy should be associated with the c.m. of the nucleus, it will be in the zeromomentum plane-wave state. In the language of random walks the c.m. of the nucleus will diffuse freely throughout all space. This is precisely what happens if one iterates eqs. (7) and (12), i.e. not using importance sampling. When employing an importance function and executing eqs. (13) and (14), care must be taken not to bind the c.m. to a fured point in space by means of the importance function. Though not mathematically incorrect - eqs. (13) and (14) are correct for any nonzero importance function - this would lead to a slow convergence of the method, i.e. a large variance. The problem is overcome most easily by employing a translationally invariant importance function. In the present scheme this is feasible since for any given configuration R the c.m. is trivial to evaluate. Specifically we take the importance function to be of an extended Jastrow form, namely
The two-body factors Lj describing particle correlations, as well as the one-body factors vi describing the surface of the droplet, are parametrized and obtained from
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
121
variational calculations. Eq. (18) explicitly depends only on internal coordinates of the four-particle system and is independent of the c.m. coordinate. For ease of notation in most equations throughout the remainder of this paper the origin of the coordinate system will be put at the c.m. so that we need not distinguish between single-particle coordinates and internal coordinates. 6. Numerical results In order to get some idea of what a useful importance function should look like, and to generate a reasonable zeroth generation, we performed a variational calculation employing the translationally invariant Jastrow-type trial wave function &(R)
=
n
exp[-+ue
-brij] y exp [ -&fd-*(ri--$C
i
rj)*],
(19)
j
with a, b and d variational parameters. The energy expectation value,
may then be evaluated by means of the method of Metropolis et al. Is). In table 1 TABLE1 Variational energies and rms point mass radii of the a-particle with three different variational wave functions (we use h*/m = 41.47 MeV fm*) No.
a
1
5
2 3
6 6.5
,frIF 1) 4 8 4.5
f:)
(
1.2 1.7 1.25
rJW
($V)
(fm)
-28.8kO.2 - 17.4kO.2 -28.4kO.2
1.32kO.01 1.81 kO.01 1.37kO.01
we present some variational results for selected sets of parameter values. No attempt was made at a careful minimization of the energy, but the first line of table 1 represents a rough minimum of the energy. Since the Metropolis method samples configurations R from the distribution I(/gR) it is easy to obtain the point mass density, t,#r, r2, r3, r,)d*3 d3r2 d3r3 d3r, P(r) =
s
,
41~ &R)dR s
where the radius is taken with respect to the c.m. of the nucleus.
121)
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
122
The rms point mass radius is then given by
(22)
r; = 471 p(r)r4dr, s
which may also be found in table 1. Note that the variational method will not determine the radius very accurately : the energy difference between the first and last line of table 1 is statistically not significant, but the radii are considerably different. Using any of the variational functions defined above as importance functions we performed GFMC calculations using &R) as the zeroth generation. Since the first excited state is about 20 MeV above the ground state 6, and probably has a very small overlap with the trial function, convergence of the iteration (7) to the ground state should be very fast. In fact we could not - with statistical significance - discern any trends in the growth or variational energy estimators beyond the first or second generation. Nevertheless, we discarded the first 100 generations and then averaged over 800-1000 generations to obtain energies and density distributions. The generation size was kept in the range of 20&400 configurations. The computer time required totals about three hours on a CDC 6600. The resulting energies and radii for three independent calculations employing the three different importance functions of table 1 are given in table 2. The first importTABLE2 Growth and variational estimators for the Schrijdinger eigenvalue for the a-particle employing the three variational functions of table 1 as importance functions No.
a
1 2 3
5 6 6.5
b
(fm- ‘) 4 8 4.5
gest - Growth estimator.
fi)
(i&)
(
1.2 1.7 1.25
-31.6kO.4 -31.6+0.4 -30.7+0.6
-31.1 kO.2 -31.1+0.2 -31.5k0.2
r,(GFMC) (fm) 1.36kO.01 1.35 f 0.01 1.37+0.01
vest - Variational estimator.
ante function yields slightly too small a radius in table 1, whereas the second importance function was deliberatly chosen to have much too large a radius. The third one was adjusted to produce about the correct radius. It is seen from table 2 that in all three cases the final radii agree within the statistical error bar. The same holds true for all the energy estimators in table 2. We are therefore reassured that the introduction of importance sampling does not bias the final results even if extremely misleading importance functions are used. The short-range behaviour of the two-body correlation factor f(rij) is known exactly since at short distances the l/r behaviour of the Yukawa potential dominates the Schrodinger equation. This gives the well-known cusp condition r6, l’). For the inter-
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
123
action (1) the cusp condition reads lim d In f’(r) = 20. r+O
dr
With the parametrization of eq. (19) the cusp condition becomes ab = 20. It is seen from the parameter values in tables 1 and 2 that only the first importance function satisfies the cusp condition whereas the others seriously violate it. From the agreement of final results it is seen that this misdescription of the short-range behaviour also does not lead to biased results. Furthermore, since eq. (15) yields energies independent of the importance function I,$ we are reassured to have found a solution $ to the Schrijdinger equation. Otherwise eq. (15) would generally produce different results for different functions tii.
7. Density distributions It is more difficult to obtain the ground-state density distribution than the energy. The density sampled by the Monte Carlo algorithm with importance sampling, eq. PiC [hi”: a12
0.10
O.OE
0.06
0.04
0.02
0
Fig. 1. Point mass density distributions from the variational wave functions of.table 1
124
J. G. Zabolitzky,
M. H. Kaios ] hour-nucleon
SchrGdinger equation
(13), is W)i,W), that is, the product of the ground-state wave function and the importance function. In order to obtain the density distribution we would need to sample the square of the ground-state wave function, ti2W). There exist various techniques to obtain this density 13). The simplest one which we will employ here is to make use of the importance function being close to the groundstate wave function and correcting for the difference to first order, VW) = II/W+ 4% $‘(R) = 2~(R)II/,(R)--Ii/:(R) = 2+(1)+A)-(ll/+A)~
= $‘-A’,
(23)
which is correct but for the term of order A2, where A is the small difference between
Fig. 2. Point mass density distributions from the Schriidinger solution, from eq. (23), for the three importance functions of table 2.
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schrcidinger equation
125
the importance function and the exact ground-state wave function. The square of the importance function is sampled easily by means of the method of Metropolis et al. ’ 5). Inserting eq. (23) in eq. (21) the point mass density distribution is easily evaluated. The radii given in table 2 were then evaluated using eq. (22). In fig. 1 we give the point mass densities from the variational calculation, eq. (21). The densities differ considerably because of the widely varying parameters of the trial functions. In fig. 2 we give the extrapolated densities using eq. (23) for the three importance functions defined above and used in fig. 1. It is seen that in spite of the widely varying importance functions these densities agreevery well. Statistically significant deviations are found only when employing importance function no. 2 which is quite drastically wrong, as may be seen comparing figs. 1 and 2 and tables 1 and 2. It is therefore to be expected that the linear extrapolation eq. (23) is inaccurate for this case. It is nonetheless rather surprising and gratifying how small the error in eq. (23) is considering the huge discrepancy between the variational densities, fig. 1. However for some large values of the radial variable the density becomes slightly negative indicating the inappropriateness of eq. (23) for importance functions as unreasonable as no. 2. This is reflected in the slightly smaller rms radius for case 2 in table 2. With decreasing radius, i.e. distance from the c.m. of the nucleus the statistical error bars in fig. 2 increase. These error bars were calculated exactly like the energy error bars discussed in sect. 4. The increase is due to the volume element rz which leads to fewer points for small values of the radius and therefore increased statistical errors. There is no trace, however, in fig. 2 of any central depression in the point mass density.
8. Helium4 atoms droplet Precisely the same calculation as described above for nucleons interacting via the potential eq. (1) has been performed for the case of four helium-4 atoms interacting via the Lennard-Jones potential, eq. (2). The importance function used is (distances in A) $,(R) = fi exp [ -$2.6/r)5] fi (1 +exp (8.4lr,-ti i
i=l
rjl-1.2))-‘. j=t
With this importance function, the energy eigenvalue for the ground state is found to be - 0.39 +O.Ol K from the variational estimator and - 0.35 f 0.03 K from thegrowth estimator. The system is bound extremely weakly as may be seen by comparing to the bulk liquid binding energy of about 7 K per particle. The calculation converges much more slowly than in the case of the a-particle because of the continuous spectrum starting at zero energy. It was found necessary to discard the fiist hundred generations and then average over many hundred generations to obtain the results. For the
126
J. G. Zabolitzky, M. H. Kales 1 Fow-nucleon Schrcidinger equation
‘He
x GFMC 0 VAR
Fig. 3. Variational and Schriidinger point mass density distributions for the four helium-4 atom droplet.
same reason we estimated variances from quasi ~dependent groups of 50 generations. The variational and Schrodinger point mass density distributions are given in fig. 3. Variationally the system is unbound by a few tenths of a Kelvin. Because of the very loose binding the radius is very large: 5.37 kO.05 A. We are aware of only one other calculation for the tetramer of 4He, namely that of Nakaichi et af. ‘O),who used a variational procedure with a trial function constructed from solutions of scattering equations. Using a value of k2/m = 12.02 K *A’, they report an energy of - 0.43 K and a radius of 5.42 A for the Lennard-Jones interaction. Our calculations used h2/2m = 12.12 K +A2. Correcting their result for this difference by first-order perturbation theory, we get - 0.39 K, identical with our result. Therms radii of the two calculations agree well. This particular variational calculation is therefore seen to be very good, demonstrat~g its power even for this very challenging four-body problem.
J. G. Zabolitzky, M. H. Kaios / Four-nucleon Schriidinger equation
127
9. Summary and outlook T’he ground-state problem of four particles interacting with central, state-independent forces is susceptible to exact solution by means of the Green function Monte Carlo method, subject only to statistical errors which may be made arbitrarily small. The amount of computer time used is rather modest. It is obvious that the ground-state problem for a many-boson system not exceeding a few hundred particles may be solved using precisely the same techniques and computer codes and is by no means outside the scope of present-day computers. At present, extension of the techniques described to a larger number of fermions is more difficult because the coordinate-space wave function is no longer symmetric or positive. Another problem is posed by the complicated nature of nuclear interactions currently in use involving, among others, spin-spin, tensor and spin-orbit parts. Both of these problems are currently under investigation. Meanwhile the results presented in this paper could be used as benchmarks for other few-body methods, like the coupled-cluster method lo), which are more general at the expense of being less rigorous. Comparisons of this kind are presently in progress. At the same time approximate calculations will be used as guides to physically reasonable importance functions to be used in the more complicated problems. Observables other than the ones considered in the present paper may be evaluated by similar techniques, like momentum distributions or two-body correlation functions ‘*). These may again be used to verify other, more approximate calculations lg). We both enjoyed the hospitality of the Physics Division of the Argonne National Laboratory during the initial stages of this work. Thanks are due in particular to B.D. Day for numerous discussions. One of us (J.G.Z.) wishes to thank the Courant Institute for its hospitality as well as the “Freunde der Ruhr-Universitgt Bochum” for financial support. This work was supported also by the USDOE under contract No. AC02-79ER10353.
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J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
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SOLUTION
OF THE FOUR-NUCLEON
SCHRiiDINGER
EQUATION
J. G. ZABOLITZKY Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA and Institut fiir Theoretische Physik, Ruhr-Universitit Bochum, D-4630 Bochum I, West Germany
M. H. KALOS Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA Received 13 August 1980 Abstract: We present a solution of the Schriidinger equation for the case of four nucleons interacting via the Malfliet-Tjon potential. The Schriidinger equation is integrated by means of the Green function Monte Carlo method. The adaption of this method to treat finite systems is given. For the four-nucleon case, the ground-state eigenvalue is found to be - 31.3 f 0.2 MeV and the rms point mass radius 1.36kO.01 fm. The point mass distribution is given and does not exhibit a central depression. The method is also applied to the problem of four helium atoms interacting via the Lennard-Jones potential.
1. Introduction In recent years the ground state of the four-nucleon system has been computed by a variety of authors using diverse techniques. For separable two-particle interactions different integral equation approaches have been exploited le4). Hyperspherical expansions have been used for very soft local two-particle interactions ‘). For Yukawa-type local interactions Tjon 6, ‘) has solved the Yakubovsky integral equations *) employing Hilbert-Schmidt separable expansion or unitary pole expansion schemes. Almost twenty years ago a variant of the Green function Monte Carlo technique was used for a purely attractive, bounded two-particle interaction ‘). The coupled-cluster method has been applied to a variety of local interactions lo). But for ref. ‘), any of these calculations involving local interactions had to rely on some kind of expansion scheme, either separable expansions of subchannel amplitudes or a cluster expansion. In the present work we solve exactly - subject only to statistical sampling errors, which may be made arbitrarily small - the 114
J. G. Zabolitzky, M. H. Kales 1 Four-nucleon Schriidinger equation
four-nucleon
Schrodinger equation for the Malfliet-Tjon
115
interaction ii)
D(r) = (7.39 exp (- 3.11 r)/r- 2.93 exp (- 1.55 r)/r)hc,
(1)
with the interparticle distance I in fm. However, the present method is applicable for any local, central state-independent interaction D(r),as will be demonstrated by applying it to the problem of four helium atoms bound together by the Lennard-Jones potential v(r) =
40.88(x-”
- xe6)OK,
x = r/2.556 A.
(2)
In sects. 2 and 3 we describe the Green function MonteCarlo(GFMC)
method ’ 2*’ 3), Sect. 5 contains the adaption of the GFMC method to the present problem of finite systems. In sects. 6 and 7 we present our numerical results for the a-particle and in sect. 8 for the four helium-4 atoms problem. Sect. 9 contains a summary and outlook for further use of the GFMC method. Sect. 4 is devoted to methods checking the accuracy of the calculation.
2. The Green function Monte Carlo method For a local, central, state-independent wave function may be written as
interaction the four-nucleon
ti = $(rr, r2, 13, r,)det (air,) = $(R) det (ciri),
ground-state
(3)
where I,@) is symmetric with respect to interchange of particle coordinates, and the determinant of spin and isospin functions is antisymmetric. Throughout this paper R is used to denote the 1Zdimensional coordinate vector of the four particles. The Schriidinger equation then reduces to ( - V2 + WWW)
= W(R),
(4)
where v2 = iv;, i=l
J’(R)=
$ ,c,v(rij) + Vo 1 0. I
The energy scale has been shifted by a constant V, in order to have the potential always positive; this is a sufficient, but not necessary condition for the Green function, to be introduced in eq. (6), to be positive 13).No reference to spin/isospin coordinates occurs. The problem is equivalent to the problem of four interacting, spinless bosons. The boundary condition appropriate for a finite system is that the wave function should vanish whenever any of the coordinates approaches infinity. With the same boundary condition, the full Green function is defined by (-v:+UR,))G(R,R,)
= a(R-R,),
(6)
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J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
where G(R, R,) is positive and a symmetric function of its arguments. For a nondegenerate ground state it is evident that the the iteration sequence, @“+“(R)
= E,
s
G(R, R,)@‘“‘(R,)dR,,
(7)
will converge to the ground-state wave function for an arbitrary trial energy E, > 0, since
@‘“‘(R) = 1 EIQ”C,~,(R), a
if the initial guess
@co)(R) = c W,(R), a
where tj,(R) are the complete, orthonormal eigenfunctions of the hamiltonian (4). In eq. (8) the term with the largest factor EJE, will become dominant which will occur for the smallest value of E,, i.e., the ground state. Since the lowest-energy symmetric wave function will have no nodes, the groundstate wave function may be chosen real and positive. The same holds true of the initial guess and of the Green function G(R, R,) so that all quantities in eq. (7) are positive ’ 3). Therefore, at any step of the iteration (7), the current iterate Q@)(R)may be represented by a collection of configurations {Ri, i = 1,. . ., M}, where M is some number conveniently chosen to be of the order of a few hundred. The function is represented by this collection of configurations in the sense that the density of configurations in the interval dR at R is proportional to @(“)(R)dR. The iteration of eq. (7) may now readily be executed via a Monte Carlo algorithm: For any configuration R,, of the representation of @“‘(R.,), sample new configurations from the (unnormalized) probability distribution P(R) = W(R,
R,).
(9)
These new configurations descending from all the old ones are called a new generation and represent the next iterate, @(“+ ‘j(R) . Sampling from an unnormalized probability distribution is to be interpreted as forming the normalization JV =
s
dR E,G(R, R,)
and sampling JV configurations on the average. The dR, integration in eq. (7) is performed implicitly by collecting the descendants of all the configurations R, representing @“‘(RI) in order to form the new generation. If the trial energy E, is chosen close to the ground-state eigenvalue E, the size of the generations, i.e. .N =
s
c#+“‘(R)dR
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
117
will vary only slowly with the generation number n. By adjusting the trial energy it is possible to adjust the generation size for computational convenience. From the growth or decay of generations the ground-state eigenvalue may be estimated by @“)(R)dR s
E, = E,
qP+ “(R)dR ’ f ifn is sufficiently large, i.e. the other components in eq. (8) are negligible. The variance of the growth estimator for the ground-state eigenvalue may be decreased by using ’ 3, M 9’” + “(R)dR cs ;==’
EO=E,
(10) @n+i+ “(R)dR
zs
i=l
At first sight the proposed algorithm seems impossible to execute since the Green function G(R, R,) is not known. However, inspection of the algorithm shows that it is not necessary to know the full Green function but it is sufficient to construct an algorithm to sample configurations from the distribution (9) for given R,. We proceed now to describe such an algorithm. For a given configuration R, a finite domain D(R,) may be chosen so that the popotential function V(R) is bounded from the above throughout the domain D(R,) by the constant U(R,). We define a function G, by t-S:
+ WW,,
&I
= W,
- R,),
(11)
subject to the boundary condition that G,(R,, R,) = 0 whenever R, is on the boundary or outside of D(R,). Multiplying eq. (6) by G,(R,, R,,), eq (11) by G(R, R,) and integrating both equations over R, yields upon subtraction of the two equations G(R, R,) = G,(R, R,)+
+
- aG’~’ s WRo) (
s
R”) G(R, R,)dR, >
W- WWW,,
Ro)W,
WR,.
(12)
D(Ro)
In the second term on the r.h.s. the volume integral has been converted to an integral over the surface lB(R,) introducing the normal derivative a/an and use has been made of the boundary condition for G,. Eq. (12) is a linear integral equation for G in terms of G,. Since the choice of domain D(R,) is at our disposal it may be chosen in any convenient way so as to facilitate the computation of G,. If D(R,) is chosen to be a Cartesian product of three-dimensional spheres or cubes centered at R,, G, is known analytically, and may be sampled by means of straightforward techniques.
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J. G. Zabolitzky, M. H. Kales / Four-nucleon Schrlidinger equation
Since, by construction, all the terms and kernels in eq. (21) are non-negative, again an interpretation in terms of probabilities is possible and thereby the construction of a random walk sampling ’ 2*’ 3, configurations R for given R,.
3. Importance sampling In order to accelerate the calculation let us introduce an importance function I,+(R). This function should embody as much physical knowledge as possible about the ground-state wave function we are looking for. It will be used to guide the random walk to spend more time in favourable regions of 12-dimensional space and less time in unfavourable regions where no contributions to a next generation are to be expected. Of course we will not introduce any approximation but just try to rewrite eqs. (7) and (12) in a special way in order to reduce the variance of any expectation, e.g., the energy. It can be shown i3) that if I,+,is the exact ground-state wave function, then the energy can be estimated with zero variance from eq. (10). Not knowing the exact function, we use a variational wave function of Jastrow-type for $,(R). Introducing the new unknown function W)
= WW,(R),
eq. (7) may be rewritten as
R~)/~~(R~)]~F~)(R~). P+~)(R)= @"+~)(R)~,(R) = i7,[$,(R)G(R, (13) s Similarly, eq. (12) is identical to WW(R
&,)/ICI,(&) = c(R, R,) = WW,(R, +
s s
%,)/ICI,(&)
~IC/IW-‘~G~(R~, &))II(I,&)]~W, R,)dR,
aD(Ro)
+
(u- Wh))CWW&,
R,)I$,(R,)]&R,R,)dR,.
WW
It is seen that the transformed Green function G is sampled as easily as the orginal one, and that the iteration (13) will converge to a density g(R) = WW,(R), where e(R)is the ground-state wave function. It is seen also that the configurations R, of the random walk eq. (14) get weighted with J/,(R,). That is, in regions where t,b,(R) is large we expect that I,+(R) is also large and the random walk spends more time there than it would otherwise. Conversely, regions where t,b(R) is small tend to be less densely populated.
J. G. Zabolitzky, M. H. Kales 1 Four-nucleon SchrBdinger equation
119
4. Evaluation of expectations and error estimates One possible way to estimate the ground-state energy from a GFMC calculation has been given in eq. (lo), the so-called growth estimator. It is obvious that Q,may be replaced by 4 in eq. (10). Another useful estimator for the ground-state energy is the variational one 13),
a quantity similar in spirit to the model energy of ref. i4). The variational estimator also has the property that its variance may be made very small. The expression in brackets in eq. (15) would be a constant if I& were the eigenfunction of the hamiltonian. Like the growth estimator the variational estimator would then have zero variance being the expectation of a constant. In our calculation the variational estimator proved to have a significantly smaller variance than the growth estimator. The agreement between the growth and variational estimators provides an useful test on the accuracy of the method. Another useful test of the accuracy of the method is simply to use different importance functions $P By deliberately selecting an importance function which predicts inaccurately some physical property and thereby straining the GFMC method one may reassure oneself about the convergence to the true ground state in spite of misleading guidance via such Jl,. S~ultan~~ly, eq. (15) will yield the same energy for different importance functions if and only if the Schrodinger equation has actually been solved. Otherwise, eq. (15) may assume arbitrary values for arbitrary variations in 4% It is more difficult to estimate the error of any quantity than the quantity itself. If (Xi> is a set of statistically ~de~ndent samples, the standard deviation of the mean,
may be estimated from 1 i (Xi--X)2 d = [ N(N--1) i=r
1 *.
07)
However, in the present calculation generation averages like eq. (15) are not statistically independent, since one generation produces the next one by the random walk of eq. (14). It may be assumed that after a s~~entIy large number of generations statistical independence is achieved. Therefore, the “independent samples” (XJ in eqs. (16) and (17) are taken to be 20-generation averages in the present calculation.
120
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equution
Since in the calculation started from a trial &‘i the energy was observed to relax in much less then 20 generations this is a reasonable procedure. The error bars on our numerical results quoted below are then calculated precisely according to eq. (17).
5. Adaptation to finite systems So far we have described the GFMC method as it has been used for infinite systems. For a finite system, there are two points which need special attention: the boundary condition and the c.m. motion. The boundary condition does not pose any problem. As opposed to the treatment for an infinite system, where configurations outside a given basic cell are moved back inside in order to ensure periodic boundary conditions, one just lets the random walk, eq. (14), evolve without any restriction. If the domains D are chosen so that the union of all possible domains D covers the full 12-dimensional space, eq. (14) is seen to satisfy eq. (6) for arbitrary R, R, by simple insertion. Since no restriction will be imposed on the random walk (14), any configuration, i.e. any point in 1Zdimensional space may be reached. However, as we will use as the zeroth generation a population drawn from a localized state where all particles are reasonably close to each other, the probability to reach large values for any interparticle separation decreases with that value increasing, i.e. the wave function will tend to zero for large interparticle separation. The hamiltonian eq. (4) is translationally invariant. The ground-state wave function will therefore also be translationally invariant. Since in the lowest-energy state no kinetic energy should be associated with the c.m. of the nucleus, it will be in the zeromomentum plane-wave state. In the language of random walks the c.m. of the nucleus will diffuse freely throughout all space. This is precisely what happens if one iterates eqs. (7) and (12), i.e. not using importance sampling. When employing an importance function and executing eqs. (13) and (14), care must be taken not to bind the c.m. to a fured point in space by means of the importance function. Though not mathematically incorrect - eqs. (13) and (14) are correct for any nonzero importance function - this would lead to a slow convergence of the method, i.e. a large variance. The problem is overcome most easily by employing a translationally invariant importance function. In the present scheme this is feasible since for any given configuration R the c.m. is trivial to evaluate. Specifically we take the importance function to be of an extended Jastrow form, namely
The two-body factors Lj describing particle correlations, as well as the one-body factors vi describing the surface of the droplet, are parametrized and obtained from
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
121
variational calculations. Eq. (18) explicitly depends only on internal coordinates of the four-particle system and is independent of the c.m. coordinate. For ease of notation in most equations throughout the remainder of this paper the origin of the coordinate system will be put at the c.m. so that we need not distinguish between single-particle coordinates and internal coordinates. 6. Numerical results In order to get some idea of what a useful importance function should look like, and to generate a reasonable zeroth generation, we performed a variational calculation employing the translationally invariant Jastrow-type trial wave function &(R)
=
n
exp[-+ue
-brij] y exp [ -&fd-*(ri--$C
i
rj)*],
(19)
j
with a, b and d variational parameters. The energy expectation value,
may then be evaluated by means of the method of Metropolis et al. Is). In table 1 TABLE1 Variational energies and rms point mass radii of the a-particle with three different variational wave functions (we use h*/m = 41.47 MeV fm*) No.
a
1
5
2 3
6 6.5
,frIF 1) 4 8 4.5
f:)
(
1.2 1.7 1.25
rJW
($V)
(fm)
-28.8kO.2 - 17.4kO.2 -28.4kO.2
1.32kO.01 1.81 kO.01 1.37kO.01
we present some variational results for selected sets of parameter values. No attempt was made at a careful minimization of the energy, but the first line of table 1 represents a rough minimum of the energy. Since the Metropolis method samples configurations R from the distribution I(/gR) it is easy to obtain the point mass density, t,#r, r2, r3, r,)d*3 d3r2 d3r3 d3r, P(r) =
s
,
41~ &R)dR s
where the radius is taken with respect to the c.m. of the nucleus.
121)
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
122
The rms point mass radius is then given by
(22)
r; = 471 p(r)r4dr, s
which may also be found in table 1. Note that the variational method will not determine the radius very accurately : the energy difference between the first and last line of table 1 is statistically not significant, but the radii are considerably different. Using any of the variational functions defined above as importance functions we performed GFMC calculations using &R) as the zeroth generation. Since the first excited state is about 20 MeV above the ground state 6, and probably has a very small overlap with the trial function, convergence of the iteration (7) to the ground state should be very fast. In fact we could not - with statistical significance - discern any trends in the growth or variational energy estimators beyond the first or second generation. Nevertheless, we discarded the first 100 generations and then averaged over 800-1000 generations to obtain energies and density distributions. The generation size was kept in the range of 20&400 configurations. The computer time required totals about three hours on a CDC 6600. The resulting energies and radii for three independent calculations employing the three different importance functions of table 1 are given in table 2. The first importTABLE2 Growth and variational estimators for the Schrijdinger eigenvalue for the a-particle employing the three variational functions of table 1 as importance functions No.
a
1 2 3
5 6 6.5
b
(fm- ‘) 4 8 4.5
gest - Growth estimator.
fi)
(i&)
(
1.2 1.7 1.25
-31.6kO.4 -31.6+0.4 -30.7+0.6
-31.1 kO.2 -31.1+0.2 -31.5k0.2
r,(GFMC) (fm) 1.36kO.01 1.35 f 0.01 1.37+0.01
vest - Variational estimator.
ante function yields slightly too small a radius in table 1, whereas the second importance function was deliberatly chosen to have much too large a radius. The third one was adjusted to produce about the correct radius. It is seen from table 2 that in all three cases the final radii agree within the statistical error bar. The same holds true for all the energy estimators in table 2. We are therefore reassured that the introduction of importance sampling does not bias the final results even if extremely misleading importance functions are used. The short-range behaviour of the two-body correlation factor f(rij) is known exactly since at short distances the l/r behaviour of the Yukawa potential dominates the Schrodinger equation. This gives the well-known cusp condition r6, l’). For the inter-
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
123
action (1) the cusp condition reads lim d In f’(r) = 20. r+O
dr
With the parametrization of eq. (19) the cusp condition becomes ab = 20. It is seen from the parameter values in tables 1 and 2 that only the first importance function satisfies the cusp condition whereas the others seriously violate it. From the agreement of final results it is seen that this misdescription of the short-range behaviour also does not lead to biased results. Furthermore, since eq. (15) yields energies independent of the importance function I,$ we are reassured to have found a solution $ to the Schrijdinger equation. Otherwise eq. (15) would generally produce different results for different functions tii.
7. Density distributions It is more difficult to obtain the ground-state density distribution than the energy. The density sampled by the Monte Carlo algorithm with importance sampling, eq. PiC [hi”: a12
0.10
O.OE
0.06
0.04
0.02
0
Fig. 1. Point mass density distributions from the variational wave functions of.table 1
124
J. G. Zabolitzky,
M. H. Kaios ] hour-nucleon
SchrGdinger equation
(13), is W)i,W), that is, the product of the ground-state wave function and the importance function. In order to obtain the density distribution we would need to sample the square of the ground-state wave function, ti2W). There exist various techniques to obtain this density 13). The simplest one which we will employ here is to make use of the importance function being close to the groundstate wave function and correcting for the difference to first order, VW) = II/W+ 4% $‘(R) = 2~(R)II/,(R)--Ii/:(R) = 2+(1)+A)-(ll/+A)~
= $‘-A’,
(23)
which is correct but for the term of order A2, where A is the small difference between
Fig. 2. Point mass density distributions from the Schriidinger solution, from eq. (23), for the three importance functions of table 2.
J. G. Zabolitzky, M. H. Kales / Four-nucleon Schrcidinger equation
125
the importance function and the exact ground-state wave function. The square of the importance function is sampled easily by means of the method of Metropolis et al. ’ 5). Inserting eq. (23) in eq. (21) the point mass density distribution is easily evaluated. The radii given in table 2 were then evaluated using eq. (22). In fig. 1 we give the point mass densities from the variational calculation, eq. (21). The densities differ considerably because of the widely varying parameters of the trial functions. In fig. 2 we give the extrapolated densities using eq. (23) for the three importance functions defined above and used in fig. 1. It is seen that in spite of the widely varying importance functions these densities agreevery well. Statistically significant deviations are found only when employing importance function no. 2 which is quite drastically wrong, as may be seen comparing figs. 1 and 2 and tables 1 and 2. It is therefore to be expected that the linear extrapolation eq. (23) is inaccurate for this case. It is nonetheless rather surprising and gratifying how small the error in eq. (23) is considering the huge discrepancy between the variational densities, fig. 1. However for some large values of the radial variable the density becomes slightly negative indicating the inappropriateness of eq. (23) for importance functions as unreasonable as no. 2. This is reflected in the slightly smaller rms radius for case 2 in table 2. With decreasing radius, i.e. distance from the c.m. of the nucleus the statistical error bars in fig. 2 increase. These error bars were calculated exactly like the energy error bars discussed in sect. 4. The increase is due to the volume element rz which leads to fewer points for small values of the radius and therefore increased statistical errors. There is no trace, however, in fig. 2 of any central depression in the point mass density.
8. Helium4 atoms droplet Precisely the same calculation as described above for nucleons interacting via the potential eq. (1) has been performed for the case of four helium-4 atoms interacting via the Lennard-Jones potential, eq. (2). The importance function used is (distances in A) $,(R) = fi exp [ -$2.6/r)5] fi (1 +exp (8.4lr,-ti i
i=l
rjl-1.2))-‘. j=t
With this importance function, the energy eigenvalue for the ground state is found to be - 0.39 +O.Ol K from the variational estimator and - 0.35 f 0.03 K from thegrowth estimator. The system is bound extremely weakly as may be seen by comparing to the bulk liquid binding energy of about 7 K per particle. The calculation converges much more slowly than in the case of the a-particle because of the continuous spectrum starting at zero energy. It was found necessary to discard the fiist hundred generations and then average over many hundred generations to obtain the results. For the
126
J. G. Zabolitzky, M. H. Kales 1 Fow-nucleon Schrcidinger equation
‘He
x GFMC 0 VAR
Fig. 3. Variational and Schriidinger point mass density distributions for the four helium-4 atom droplet.
same reason we estimated variances from quasi ~dependent groups of 50 generations. The variational and Schrodinger point mass density distributions are given in fig. 3. Variationally the system is unbound by a few tenths of a Kelvin. Because of the very loose binding the radius is very large: 5.37 kO.05 A. We are aware of only one other calculation for the tetramer of 4He, namely that of Nakaichi et af. ‘O),who used a variational procedure with a trial function constructed from solutions of scattering equations. Using a value of k2/m = 12.02 K *A’, they report an energy of - 0.43 K and a radius of 5.42 A for the Lennard-Jones interaction. Our calculations used h2/2m = 12.12 K +A2. Correcting their result for this difference by first-order perturbation theory, we get - 0.39 K, identical with our result. Therms radii of the two calculations agree well. This particular variational calculation is therefore seen to be very good, demonstrat~g its power even for this very challenging four-body problem.
J. G. Zabolitzky, M. H. Kaios / Four-nucleon Schriidinger equation
127
9. Summary and outlook T’he ground-state problem of four particles interacting with central, state-independent forces is susceptible to exact solution by means of the Green function Monte Carlo method, subject only to statistical errors which may be made arbitrarily small. The amount of computer time used is rather modest. It is obvious that the ground-state problem for a many-boson system not exceeding a few hundred particles may be solved using precisely the same techniques and computer codes and is by no means outside the scope of present-day computers. At present, extension of the techniques described to a larger number of fermions is more difficult because the coordinate-space wave function is no longer symmetric or positive. Another problem is posed by the complicated nature of nuclear interactions currently in use involving, among others, spin-spin, tensor and spin-orbit parts. Both of these problems are currently under investigation. Meanwhile the results presented in this paper could be used as benchmarks for other few-body methods, like the coupled-cluster method lo), which are more general at the expense of being less rigorous. Comparisons of this kind are presently in progress. At the same time approximate calculations will be used as guides to physically reasonable importance functions to be used in the more complicated problems. Observables other than the ones considered in the present paper may be evaluated by similar techniques, like momentum distributions or two-body correlation functions ‘*). These may again be used to verify other, more approximate calculations lg). We both enjoyed the hospitality of the Physics Division of the Argonne National Laboratory during the initial stages of this work. Thanks are due in particular to B.D. Day for numerous discussions. One of us (J.G.Z.) wishes to thank the Courant Institute for its hospitality as well as the “Freunde der Ruhr-Universitgt Bochum” for financial support. This work was supported also by the USDOE under contract No. AC02-79ER10353.
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J. G. Zabolitzky, M. H. Kales / Four-nucleon Schriidinger equation
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