Zero-Variance Zero-Bias Principle for Observables in quantum Monte Carlo: Application to Forces

A simple and stable method for computing accurate expectation values of observable with Variational Monte Carlo (VMC) or Diffusion Monte Carlo (DMC) algorithms is presented. The basic idea consists in replacing the usual ``bare'' estimator associated with the observable by an improved or ``renormalized'' estimator. Using this estimator more accurate averages are obtained: Not only the statistical fluctuations are reduced but also the systematic error (bias) associated with the approximate VMC or (fixed-node) DMC probability densities. It is shown that improved estimators obey a Zero-Variance Zero-Bias (ZVZB) property similar to the usual Zero-Variance Zero-Bias property of the energy with the local energy as improved estimator. Using this property improved estimators can be optimized and the resulting accuracy on expectation values may reach the remarkable accuracy obtained for total energies. As an important example, we present the application of our formalism to the computation of forces in molecular systems. Calculations of the entire force curve of the H$_2$,LiH, and Li$_2$ molecules are presented. Spectroscopic constants $R_e$ (equilibrium distance) and $\omega_e$ (harmonic frequency) are also computed. The equilibrium distances are obtained with a relative error smaller than 1%, while the harmonic frequencies are computed with an error of about 10%

Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density

We construct improved quantum Monte Carlo estimators for the spherically- and system-averaged electron pair density (i.e. the probability density of finding two electrons separated by a relative distance u), also known as the spherically-averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth non-local operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.Comment: 13 pages, 9 figures, published versio

Pressure-energy correlations and thermodynamic scaling in viscous Lennard-Jones liquids

We use molecular dynamics simulation results on viscous binary Lennard-Jones mixtures to examine the correlation between the potential energy and the virial. In accord with a recent proposal [U. R. Pedersen et. al. Phys. Rev. Lett. 100, 015701 (2008)], the fluctuations in the two quantities are found to be strongly correlated, exhibiting a proportionality constant, Gamma, numerically equal to one-third the slope of an inverse power law approximation to the intermolecular potential function. The correlation is stronger at higher densities, where interatomic separations are in the range where the inverse power law approximation is more accurate. These same liquids conform to thermodynamic scaling of their dynamics, with the scaling exponent equal to Gamma. Thus, the properties of strong correlation between energy and pressure and thermodynamic scaling both reflect the ability of an inverse power law representation of the potential to capture interesting features of the dynamics of dense, highly viscous liquids.Comment: 5 pages, 4 figures; published version, one figure remove

Funding Universal Service Obligations with an Essential Facility: Charges vs. Taxes and Subsidies

This paper compares three schemes for funding Universal Service Obligations in network industries with an essential facility: an uplift to the network access charge, the establishment of a Universal Service (US) fund financed through a lump-sum tax and a US fund financed through a unit tax. The comparison is made under a duopoly structure with a potential entrant and an incumbent, which owns the essential facility and is responsible for universal service. The incumbent is also constrained to offer the same price on all markets. Using a social welfare criteria, we show that the US fund financed with a lump sum tax dominates the other two schemes, while the US fund with unit tax is equivalent to the access charge uplift.UNIVERSAL SERVICE OBLIGATIONS; ACCESS CHARGES; REGULATION

Failure of classical elasticity in auxetic foams

A recent derivation [P.H. Mott and C.M. Roland, Phys. Rev. B 80, 132104 (2009).] of the bounds on Poisson's ratio, v, for linearly elastic materials showed that the conventional lower limit, -1, is wrong, and that v cannot be less than 0.2 for classical elasticity to be valid. This is a significant result, since it is precisely for materials having small values of v that direct measurements are not feasible, so that v must be calculated from other elastic constants. Herein we measure directly Poisson's ratio for four materials, two for which the more restrictive bounds on v apply, and two having values below this limit of 0.2. We find that while the measured v for the former are equivalent to values calculated from the shear and tensile moduli, for two auxetic materials (v < 0), the equations of classical elasticity give inaccurate values of v. This is experimental corroboration that the correct lower limit on Poisson's ratio is 0.2 in order for classical elasticity to apply.Comment: 9 pages, 2 figure

Are polar liquids less simple?

Strong correlation between equilibrium fluctuations of the potential energy, U, and the virial, W, is a characteristic of a liquid that implies the presence of certain dynamic properties, such as density scaling of the relaxation times and isochronal superpositioning of the relaxation function. In this work we employ molecular dynamics simulations (mds) on methanol and two variations, lacking hydrogen bonds and a dipole moment, to assess the connection between the correlation of U and W and these dynamic properties. We show, in accord with prior results of others [T.S. Ingebrigtsen, T.B. Schroder, J.C. Dyre, Phys. Rev. X 2, 011011 (2012).], that simple van der Waals liquids exhibit both strong correlations and the expected dynamic behavior. However, for polar liquids this correspondence breaks down - weaker correlation between U and W is not associated with worse conformance to density scaling or isochronal superpositioning. The reason for this is that strong correlation between U and W only requires their proportionality, whereas the expected dynamic behavior depends primarily on constancy of the proportionality constant for all state points. For hydrogen-bonded liquids, neither strong correlation nor adherence to the dynamic properties is observed; however, this nonconformance is not directly related to the concentration of hydrogen bonds, but rather to the greater deviation of the intermolecular potential from an inverse power law (IPL). Only (hypothetical) liquids having interactions governed strictly by an IPL are perfectly correlating and exhibit the consequent dynamic properties over all thermodynamic conditions.Comment: 14 pages, 8 figure

Renormalization of the electron-phonon coupling in the one-band Hubbard model

We investigate the effect of electronic correlations on the coupling of electrons to Holstein phonons in the one-band Hubbard model. We calculate the static electron-phonon vertex within linear response of Kotliar-Ruckenstein slave-bosons in the paramagnetic saddle-point approximation. Within this approach the on-site Coulomb interaction U strongly suppresses the coupling to Holstein phonons at low temperatures. Moreover the vertex function does not show particularly strong forward scattering. Going to larger temperatures kT\sim t we find that after an initial decrease with U, the electron-phonon coupling starts to increase with U, confirming a recent result of Cerruti, Cappelluti, and Pietronero. We show that this behavior is related to an unusual reentrant behavior from a phase separated to a paramagnetic state upon decreasing the temperature.Comment: 4 pages, 6 figure

Interference Phenomena in Medium Induced Radiation

We consider the interference pattern for the medium-induced gluon radiation produced by a color singlet quark-antiquark antenna embedded in a QCD medium with size $L$ and `jet quenching' parameter $\hat q$. Within the BDMPS-Z regime, we demonstrate that, for a dipole opening angle $\theta_{q\bar q} \gg\theta_c\equiv {2}/{\sqrt{\hat q L^3}}$, the interference between the medium--induced gluon emissions by the quark and the antiquark is suppressed with respect to the direct emissions. This is so since direct emissions are delocalized throughout the medium and thus yield contributions proportional to $L$ while interference occurs only between emissions at early times, when both sources remain coherent. Thus, for \tqq \gg\theta_c, the medium-induced radiation is the sum of the two spectra individually produced by the quark and the antiquark, without coherence effects like angular ordering. For \tqq \ll\theta_c, the medium--induced radiation vanishes.Comment: 4 pages, 2 figures; Proceedings of the "Quark Matter 2011" conferenc