Abstract
Many ground state studies of 4 He using a shadow wave function with an inverse fifth power McMillan particle-particle correlation function have yielded radial distribution functions with misplaced peaks. It has been conjectured that this is due to the specific choice of the McMillan correlation function.
However, beyond the use of fully optimized two-particle correlation functions, there has been little study of simple alternatives that can correct this defect. In this work we show that the remedy is surprisingly simple. When a shadow wavefunction with an inverse seventh power particle-particle correlation function is used to study 4 He, it gives a correctly peaked radial distribution function, lowers the energy at all liquid and solid densities, and produces an excellent equation of state. 05.30.Fk, 21.65.+f, Typeset using REVT E X 1
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(DMC). The advance of EMC have seemingly oblivated the need for the "brutish" and bias-laiden method of VMC. However, the introduction of the shadow wavefunction by Vitiello et al 8 has added new subtlety and refinement to this approach. It has been shown that shadow wavefunctions can describe both the liquid and solid phase of 4 He with excellent energy, while simultaneously maintaining translational and Bose sysmmetry. Since any VMC calculation is an order of magnitude less computationally demanding than corresponding EMC calculations, the method of shadow wavefunctions remained economically and conceptionally appealing.
However, it has been noted for some time that shadow wavefunctions with the McMillan inverse fifth power particle-particle correlation function do not give a correct radial distribution function for bulk liquid 4 He. All the peaks are misplaced as in the original McMillan calculation. While there have been continued improvements on the form of two-particle correlation, leading to fully optimized correlations expressible in terms of a basis state, there has been no explorations of simpler alternatives to cure this defect. In this work, we shown that a simple replacement of the inverse fifth power by that of an inverse seventh power significantly improves simultaneously, the ground state energy, the equation of state and the radial distribution function.
The first VMC calculation of the groundstate properties of liquid and solid 4 He was carried out by McMillan 3 who employed a trial wave function with an inverse fifth power of the particle separation as the two-body correlation function. In this early study the potential between 4 He atoms is taken to be the two-body Lennard-Jones (LJ) potential, 
with a Gaussian particle-shadow correlation function
where R ≡ {r 1 , r 2 , . . . , r N } and S ≡ {s 1 , s 2 , . . . , s N } represent the set of particle and shadow coordinates. The particle-particle correlation function Φ(R) and the shadow-shadow corre- Both wavefunctions, however, produce a radial distribution function at equilibrium density whose main peak is shifted outward by about 0.1Å as compared with the experimental value. The same misplacement is also observed at the GFMC freezing density. Such a misplacement can be corrected by optimizing the two-body correlations through the method of basis state expansion 14 . (The peak height is still underestimated, however.)
In this work, we show that this crucial defect can be simply corrected by a better choice of the inverse power (from 5 to 7) in McMillan's form of the particle-particle correlation function.
Without reoptimizing the M+AS wavefunction's shadow parameters, but only varying the variational parameter b with m = 7, we obtained lower energies than those of M+AS at all liquid and solid densities, in an amount ranging from 0.1 to 0.3K o . This choice of the wavefunction in our work, referred to as M7+AS, allows us to improve the quality of the shadow wavefunction while retaining the same level of simplicity as before.
For a system of N Helium atoms interacting via two-body forces only, the Hamiltonian has the formĤ
where v(r ij ) is the Aziz HFDHE2 potential. The expectation value of the Hamiltonian can be expressed as
The local energy is written as
and does not depend on Φ S (S) sinceĤ acts only upon the variables describing the system of real particles.
The probability p(R, S, S ′ ) is given by
To evaluate the expectation value of the Hamiltonian we use the Metropolis Monte Carlo algorithm 15 to sample the probability density p(R, S, S ′ ) from the 9N dimensional configuration space of the particles and two sets of shadow coordinates. In these computations, the Metropolis steps are subdivided in two parts. In the first, one attempts to move real particle coordinates at random inside cubical boxes of side length ∆. In the second, analogous attempts to move shadow coordinates are made inside cubical boxes of side length ∆ ′ .
After we attempt to move all the shadow coordinates of set {S}, the same is done for those in set {S ′ }. The parameters ∆ and ∆ ′ were adjusted so that the acceptance ratio for both particle and shadow moves was nearly 50%.
We compute the ground state variational energy, the radial distribution function g(r), and the static structure factor S(k). These quantities are spherical averages and have been computed for both the real particles and the shadow coordinates. The radial distribution function is defined by
where the angular brackets denote an average with respect to |Ψ(R)| 2 and ρ is the particle density. The static structure factor is obtained from the average
where ρ k is given by ρ k = N j=1 exp(−ikr j ). By using this procedure S(k) is computed for a discrete set of k values where the smaller wave vector compatible with the periodic boundary condition of the system is k = 2π/L ( L is the side of the simulation box ).
All simulations presented in this work have been done with N = 108 atoms of 4 He in a cubic box with periodic boundary conditions. To enforce periodicity, the two-body interaction potential v(r) smoothly goes to zero at a cutoff distance, r c = L/2, equal half the side of the simulation box. We actually use a slightly modified two-body interaction
A correction ∆V = (ρ/2) d 3 rg(r)∆v(r) was then added to the computed potential energy,
where the radial distribution function g(r) comes from the simulation and is taken equal to 1 for r > r c . The shadow-shadow pseudopotential u S (s) was modified according to the same prescription as v(r), while the particle-particle inverse power McMillan pseudopotential u(r)
and its first two derivatives were slightly modified near the edge of the simulation box in order to go smoothly to zero, by using a third degree polynomial fit to the pseudopotential near the edge of the simulation box.
All calculations start from a perfect fcc crystal. Our runs consisted of a total of about 5.5 · 10 5 passes during each of which an attempt was made to move particles and shadows.
We allowed about 50 · 10 3 passes for equilibration followed by about 5 · 10 5 passes which comprise the equilibrated random walk.
In Table I we show the energy per particle obtained from the M7+AS shadow wave- In Table II we show the values of the optimum variational parameters b, C, τ , and α for the M7+AS shadow wavefunction at different densities ρ in the liquid phase.
The energy per particle for the M7+AS shadow wavefunction after simulations with N = 108 particles at some densities in the solid phase is shown in Table IV shows the values of the optimum variational parameters b, C, τ , and α for the M7+AS shadow wavefunction at different densities ρ in the solid phase. We fit our equation of state in the liquid phase to a cubic polynomial of the form shadow wavefunction are in good agreement with both GFMC and experimental results 16 .
In the solid phase we used the same parametrization as reported for the M+AS case. We fited the energy to a cubic polynomial of the form
where the specific density ρ S σ 3 =0.4486 is taken from the GFMC calculation. The values of the parameters to this fit are shown in Table VI , together with other results from the literature. Again the M7+AS shadow wavefunction shows a good agreement with GFMC with the exception of a discrepancy in the coefficient C. In Fig 1 we plot the equation of state for 4 He liquid as obtained by using the values of the fitting parameters reported in Table V . As already known, one notes that both M+AS and M7+AS wavefunctions give a better equation of state than the shadow wavefunction with fully optimized Jastrow particle-particle correlations (OJ+AS), although the OJ+AS wavefunction gives somewhat lower energies.
It has been argued 13 that possible causes of such behavior are the incomplete determination of the coefficients in the basis-set expansion for the OJ+AS wavefunction, or the missing full reoptimization of the shadow parameters. Indeed the recent VMC calculations with a fully optimized shadow wavefunction 17 confirm this latter possibility.
In Fig. 2 , we show the radial distribution function g(r) obtained at the GFMC equilibrium density ρσ 3 = 0.365. Our maximum of g(r) is obtained at the same position r max as the GFMC value and it is clear tha there is no shifting of our curve to larger values of r. Our variational peak g(r max ) is a little smaller than the GFMC value. Fig. 3 shows our results for the radial distribution function g(r) determined at the GFMC freezing density ρσ 3 = 0.438.
Statistical errors in the GFMC g(r) near the maximum g(r max ) are large, so a detailed comparison with GFMC is not possible. It appears that the position of our maximum of g(r) compares very well with the GFMC value, but the variational peak g(r max ) is again smaller. The trend seen at the GFMC equilibrium and freezing densities is repeated at all other densities.
In Fig. 4 we show S(k) at the equilibrium density ρσ 3 = 0.365. The experimental The energies are given in Kelvin per particle. M+MS refers to a shadow wavefunction 11 with McMillan fifth power-law pseudopotential (m=5) for both particle-particle and shadow-shadow pseudopotentials. M+AS refers to a shadow wavefunction 13 with a rescaled Aziz HFDHE2 shadow-shadow pseudopotential and a McMillan fifth power-law particle-particle pseudopotential (m=5). M7+AS
refers to a shadow wavefunction with a rescaled Aziz HFDHE2 shadow-shadow pseudopotential and a McMillan seventh power-law particle-particle pseudopotential (m=7) as used in this work. 13 incorporates a fully optimized Jastrow particle-particle pseudopotential.
The GFMC result is taken from Kalos at al 6 . The experimental equation of state (Exp) is taken from Roach et al 16 . 
ρ s σ 3 = 0.4486 is taken from the GFMC result. 

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