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Hellmann and Feynman Theorem Versus Diffusion Monte Carlo Experiment Solid State Communications 159 (2013) 106–109 Contents lists available at SciVerse ScienceDirect Solid State Communications journal homepage: www.elsevier.com/locate/ssc Hellmann and Feynman theorem versus diffusion Monte Carlo experiment Riccardo Fantoni n National Institute for Theoretical Physics (NITheP) and Institute of Theoretical Physics, Stellenbosch 7600, South Africa article info abstract Article history: In a computer experiment the choice of suitable estimators to measure a physical quantity plays an Received 13 February 2012 important role. We propose a new direct route to determine estimators for observables which do not Received in revised form commute with the Hamiltonian. Our new route makes use of the Hellmann and Feynman theorem and 27 January 2013 in a diffusion Monte Carlo simulation it introduces a new bias to the measure due to the choice of the Accepted 29 January 2013 auxiliary function. This bias is independent from the usual one due to the choice of the trial wave by H. Akai Available online 8 February 2013 function. We used our route to measure the radial distribution function of a spin one half Fermion fluid. & 2013 Elsevier Ltd. All rights reserved. Keywords: Hellmann and Feynman theorem Diffusion Monte Carlo Radial distribution function Jellium In a computer experiment of a many particles system, a fluid, then use our route to measure the radial distribution function of a the determination of suitable estimators to measure, through a spin one half Fermion fluid. statistical average, a given physical quantity, an observable, plays In ground state Monte Carlo simulations [1,2], unlike classical an important role. Whereas the average from different estimators Monte Carlo simulations [3–5] and path integral Monte Carlo must give the same result, the variance, the square of the simulations [6], one has to resort to the use of a trial wave statistical error can be different for different estimators. We will function [1], C. While this is not a source of error, bias,ina / S denote with O f the measure of the physical observable O and diffusion Monte Carlo simulation [2] of a system of Bosons, it is / S with ÁÁÁ f the statistical average over the probability distribu- for a system of Fermions, due to the sign problem [7]. Since this is tion f. In this communication we use the word estimator to always present in a Monte Carlo simulation of Fermions we will indicate the function O itself, unlike the more commonP use of not consider it any further while talking about the bias. N the word to indicate the usual Monte Carlo estimator i ¼ 1 Oi=N Another source of bias inevitably present in all the three of the average, where fOig is the set obtained by evaluating O over experiments, which we will not take into consideration in the a finite number N of points distributed according to f. This aspect following, is the finite size error. In the rest of the paper we will of finding out different ways of calculating quantum properties in generally refer to the bias to indicate the error (neglecting the some ways resembles experimental physics. The theoretical finite size error and the sign problem) that we make when concept may be perfectly well defined but it is up to the ingenuity defining different estimators of the same quantity which do not of the experimentalist to find the best way of doing the measure- give the same average. ment. Even what is meant by ‘‘best’’ is subject to debate. In a ground state Monte Carlo simulation, the energy has the In this communication we propose a new direct route to zero-variance principle [8]: as the trial wave function approaches determine, in a diffusion Monte Carlo simulation, estimators for the exact ground state, the statistical error vanishes. In a diffusion observables which do not commute with the Hamiltonian. Our Monte Carlo simulation of a system of Bosons the local energy of new route makes use of the Hellmann and Feynman theorem and the trial wave function, ELðRÞ¼½HCðRÞ=CðRÞ, where R denotes a it introduces a new bias to the measure due to the choice of the configuration of the system of particles and H is the Hamiltonian auxiliary function. We show how this bias is independent from assumed to be real, is an unbiased estimator for the ground state. the usual one due to the choice of the trial wave function. We For Fermions, the ground state energy measurement is biased by the sign problem. For observables O which do not commute with the Hamiltonian, the local estimator, OLðRÞ¼½OCðRÞ=CðRÞ,is inevitably biased by the choice of the trial wave function. A way n of remedy to this bias can be the use of the forward walking Corresponding author. E-mail address: [email protected] method [9,10] or the reptation quantum Monte Carlo method [11] URL: http://www-dft.ts.infn.it/~rfantoni/ to reach pure estimates. Otherwise this bias can be made of 0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.01.028 R. Fantoni / Solid State Communications 159 (2013) 106–109 107 2 l ¼ 0 leading order d , with d ¼ f0ÀC, where f0 is the ground state ext we had chosen C as the exact ground state wave function, f0, wave function, introducing the extrapolated measure, O ¼ instead of the trial wave function, C, then both the corrections 2/O S À/O S where the first statistical average, the mixed L f L f vmc would have vanished. When the trial wave function is sufficiently measure, is over the diffusion Monte Carlo (DMC) stationary close to the exact ground state function a good approximation to probability distribution f and the second, the variational measure, the auxiliary function can be obtained from the first-order over the variational Monte Carlo (VMC) probability distribution perturbation theory for l51. So the Hellmann and Feynman f vmc which can also be obtained as the stationary probability measure is affected by the new source of bias due to the choice of distribution of a DMC without branching [12]. the auxiliary function which is independent from the bias due to One may follow different routes to determine estimators such the choice of the trial wave function. as the direct microscopic route, the virial route through the use of We applied the Hellmann and Feynman route to the measure- the virial theorem, or the thermodynamic route through the use of ment of the radial distribution function (RDF) of the Fermion fluid thermodynamic identities. In an unbiased experiment the differ- studied by Paziani [16]. This is a fluid of spin one-half particles ent routes to the same observable must give the same average. interacting with a bare pair-potential vmðrÞ¼erfðmrÞ=r immersed In this communication we propose to use the Hellmann and in a ‘‘neutralizing’’ background. The pair-potential depends on the Feynman theorem as a direct route for the determination of parameter m in such a way that in the limit m-0 one recovers the estimators in a diffusion Monte Carlo simulation. Some attempts ideal Fermi gas and in the limit m-1 one finds the Jellium model. in this direction have been tried before [13,14]. The novelty of our We chose this model because it allows to move continuously from approach, in respect to Ref. [13], is a different definition of the a situation where the trial wave function coincides with the exact correction to the variational measure, necessary in the diffusion ground state, in the m-0 limit, to a situation where the correla- experiment, and, in respect to Ref. [14], the fact is that the bias tions due to the particles interaction become important, in the stemming from the sign problem does not exhaust all the bias due opposite m-1 limit. to the choice of the trial wave function. We chose as auxiliary function C0 ¼ QC, the first one of l l l We start with the eigenvalue expression ðH ÀE ÞC ¼ 0 for the Toulouse et al. [17] (their Eq. (30)) l ground state of the perturbed Hamiltonian H ¼ HþlO, by taking Z 2 X the derivative with respect to the parameter l, multiplying on the rs dOr 1 0 0 Qs,s ðr,RÞ¼ ds,si ds ,sj , ð5Þ right by the ground state at l ¼ 0, f , and integrating over the 8pVn n 0 4p 9 À 9 0 s s i,j a i r rij particles configuration to get ! 0 Z l Z l l where s and s denote the spin species, r ¼ 9r9 the separation l l @C dE dH l dR f ðH ÀE Þ ¼ dR f À C : between two particles, r the separation between particle i and j, 0 @l 0 dl dl ij si the spin species of particle i, and dOr the solid angle element of Then we note that due to the Hermiticity of the Hamiltonian the integration. The particles are in a recipient of volume V at a 3 left hand side vanishes at l ¼ 0 so that we further get density n ¼ n þ þnÀ ¼ 1=½4pða0rsÞ =3 with a0 being the Bohr R radius, a ¼ a0rs the lengths unit, and ns the density of the spin dR f OCl dEl R 0 ¼ : ð1Þ s particles. With this choice the a correction partially cancels the l dl P R dR f0C ¼ 0 ¼ 0 0 0 l l histogram estimator Is,s ðr,RÞ¼ i,j a ids,si ds ,sj dðrÀrijÞ dOr= This relation holds only in the l-0 limit unlike the more ð4pVnsns0 Þ, and one is left with a HFv estimator which goes to l / a S Rcommon formR [15] which holds for any l.
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