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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Quantum Monte Carlo methods are among the most accurate algorithms for predicting properties of general quantum systems. We briefly introduce ground state, path integral at finite temperature and coupled electron-ion Monte Carlo methods, their merits and limitations. We then discuss recent calculations using these methods for dense liquid hydrogen as it undergoes a molecular/atomic (metal/insulator) transition. We then discuss a procedure that can be used to assess electronic density functionals, which in turn can be used on a larger scale for first principles calculations and apply this technique to dense hydrogen and liquid water.

With the increasing computational power and the greater access to large clusters seen during the last decade, simulation methods have become an increasingly useful tool for many fields of science, including chemistry, materials science, condensed matter physics, and biophysics. In this article we explore some of the future impact of Quantum Monte Carlo in the field of first principles simulation (FPS). By this we mean reliable simulation methods that can be performed on condensed matter systems in the absence of detailed experimental information on those systems. Starting with the general Hamiltonian in

The first member of the hierarchy uses semi-empirical interatomic potentials among effective atoms considered as point particles, the best known of which is the Lennard-Jones potential. Such potentials are routinely used in the vast majority of simulations (soft condensed matter, biophysics, materials science) and are reviewed in a different contribution to this issue [

Density Functional Theory (DFT) in the Kohn-Sham formulation maps the problem of many interacting electrons in the external field of the nuclei onto a system of non-interacting electrons in external field, a one body problem, and adds electronic correlation through an exchange-correlation functional. A breakthrough in the usefulness and popularity of simulations occurred with the development of the first-principle molecular dynamics (FPMD) approach by Car and Parrinello [

In the past decade there has been an explosion of new DFT exchange-correlation functionals with various characteristics. The reason is the difficulty of making systematic improvements to the functional or judging the accuracy of a functional. If the DFT functional is considered as “variable” then how does the user, in the absence of experimental data, decide on the functional? In the case of finite molecular systems, the availability of high-level quantum chemistry methods, like Coupled-Cluster theory offers a possible path towards the improvement of approximated functionals in DFT, for example by minimizing errors in a training set between DFT and Coupled Cluster theory results at various level of accuracy (with Single, Double or Triple excitations). In fact, many exchange-correlation functionals contain optimizable parameters that are obtained from calculations on finite molecular systems (exceptions to this include LDA, PBE, among others), where results of quantum chemistry methods are routinely used as a references. In solids, accurate calculations using many-body methods are computationally expensive, which has limited their use in the development of density functionals. While there has also been considerable developments in other correlated approaches for bulk systems, such as the many-body Green’s function methods (GW approximation and Bethe-Salpeter equation), and Dynamical Mean Field Theory (DMFT), they are more expensive and still leave questions of accuracy. For reasons of space, we do not discuss these approaches further.

The third approach in our hierarchy is the use of Quantum Monte Carlo (QMC) methods, which are generalizations of the classical Monte Carlo techniques to quantum statistical physics and fundamentally based on imaginary-time path integrals. For a class of systems (bosons and systems in one dimension) such techniques provide an exact computational method. For general problems, though not exact, they are highly accurate

Later in this paper we give an example of work in progress in this direction where QMC is used to directly rank various DFT functionals. We suggest that this benchmark quality data could be used to improve directly the best functionals. One can then envision using the highest ranked functional to develop intermolecular potentials that would then be of higher quality. Ercolessi

First principles simulation methods entirely based on QMC have also been developed in the last decade. These are the Coupled Electron-Ion Monte Carlo method [

The article is organized as follows. We first describe in Section 2 the various QMC methods. Section 3 is devoted to few applications of QMC. In Section 3.1 we present a QMC study of high pressure phases of hydrogen. This is followed in Section 3.2 by a description of the use of these methods to provide quantitative information on the accuracy of various DFT functionals. Finally we close with a discussion in Section 4.

In this section, we review some of the Quantum Monte Carlo methods used in the first principles modeling of condensed matter systems. Under normal conditions of temperature and pressure, such systems are described to a high degree of accuracy by the non-relativistic Hamiltonian for a collection of electrons and ions. We will use atomic units throughout the paper, where Planck’s constant _{e}_{B}_{0} = 1 with _{B}_{h} = 315, 775 K = 27.2114 eV. Note that, in these units, the energy of a hydrogen atom is 0.5_{h}, the binding energy of a hydrogen molecule is 0.17_{h}, the unit of length is the Bohr Radius _{0} = 0.0529 nm, and the molecular equilibrium bond length is 1.4_{0}. The Hamiltonian of the systems reads
_{n}_{e}_{e}_{I}_{I}_{I}_{I}_{e}_{e}_{n}_{el}_{e}_{s}_{0}, where 4^{3}/3 = ^{−1}. Given

Finding the eigenvalues and eigenfunctions of the Hamiltonian in

The following ground state methods seek to evaluate expectation values of physical observables taken over the ground state wavefunction _{0}(_{0}(

Variational Monte Carlo (VMC) is conceptually the simplest of the ground-state QMC methods. It works by approximating the true ground-state wavefunction _{0}(_{T}_{T}_{0}(_{T}_{0}(_{L}_{T}_{T}

A popular approach for fermionic problems is to assume a Slater-Jastrow wavefunction. This type of wavefunction possesses the correct fermionic antisymmetry, and symbolically is given by Ψ_{T}^{J(R)}. Here, _{ij}_{j}_{i}_{j}

VMC can be improved if we consider classes of trial wavefunctions Ψ_{T}_{1}, . . ., _{m}

VMC has some advantages that keep it in use. First, it is usually computationally cheaper than more accurate QMC methods (to be discussed later). VMC can also include several different types of electron correlations (various forms of electronic wave functions). Lastly, it doesn’t suffer from a sign problem. However, it is at heart an approximate method, and does depend on the choice of trial wavefunction.

Projector methods attempt to stochastically project out the exact many-body ground state, allowing us to sample this distribution for Monte Carlo integration. The “projector”, or imaginary-time Green’s function _{β′→β} _{0}(_{T}

For efficiency reasons, it is better to use the “importance-sampled” Schrödinger’s equation [_{T}_{T}^{2} and _{L}_{T}

The solution of _{T}_{T}^{−1}. In the short-time approximation (_{e}_{I}^{M}

In diffusion Monte Carlo (DMC) [_{1}, . . ., _{M}

As in classical diffusion, we would then simulate _{i}_{i}_{i}_{i}_{B}_{i},R_{i}, τ_{T}

Branching diffusion Monte Carlo [_{i}_{i}

The simulation is run by initializing the starting ensemble according to _{T}^{2}. Assuming _{0}(_{T}_{0}(

It is important to note that since we are sampling _{0}(

Reptation Monte Carlo (RMC) is based on the path-integral representation of the projector. Assuming that _{T}_{0}〉:

Using the short-time approximate Green’s function at the beginning of this section, we can recast this expectation value in a more traditional path-integral form:

Here, _{0}, . . . ,_{M}^{S}^{[X]}/_{s}

The versatility of reptation Monte Carlo comes from how Π[

Most practical implementations use what’s known as the “bounce algorithm” [

RMC is appealing for two reasons. It gives us the same level of accuracy for the energy as DMC but correlated sampling between different configurations can be done without approximation. This is particularly useful in methods like the Coupled Electron-Ion Monte Carlo. RMC also gives us the ability to sample expectation values over the pure distribution, as seen below:
_{0}(^{2}, whereas the ends are distributed according to the mixed distribution

The previous projector methods we mentioned are in principle _{0}(

By far, the most common means of alleviating the sign-problem in both DMC and RMC is applying the “fixed-node” approximation [_{0}(_{T}_{T}

Like DFT, fermionic QMC typically has scaling between ^{3}) and ^{4}) depending on the property computed and the trial function. Here N is the number of particles. In contrast, popular quantum chemistry methods like Moller-Plesset Perturbation Theory, coupled-cluster, or configuration interaction, scale at least like ^{7}). This makes QMC one of the few accurate many-body theories that is able to treat bulk systems.

Unlike DFT, whose scaling prefactor is governed by the solution of a generalized eigenvalue problem, Monte Carlo methods, in general, have statistical error bars which reduce as the inverse of the square root of the sampled configurations as a consequence of the central limit theorem. This makes quantum Monte Carlo significantly more expensive than DFT to reach chemical accuracy, though it has a smaller uncontrolled bias. The necessity for a much smaller time step in projector monte carlo than in VMC can make projector monte carlo about an order of magnitude more expensive for the same statistical uncertainty.

The cost of a single N-particle monte carlo step in VMC and projector monte carlo methods are determined by the evaluation of the trial wavefunction. For bosonic trial wavefunctions with pair-wise correlations, these calculations scale like ^{2}) per N-particle step. If these correlations are short-ranged, linear scaling can be achieved.

For fermionic trial wavefunctions, the computational cost is determined by the evaluation of single-particle orbitals and by the evaluation of a Slater determinant. The scaling of orbital evaluations depends on whether the electrons are localized since evaluating localized orbitals can be done in constant time. For plane waves basis sets, the cost scales like ^{3}) per N-particle step. In theory, the cost of the determinant evaluation could be brought down by almost a factor of N if the Slater determinant is sparse, however, the crossover point is prohibitive (greater than 3000 particles for a model system) [^{3–4}) depending on whether one uses backflow or not.

Next, we summarize path integral methods. These methods are similar to DMC but can treat systems at non-zero temperature: a many-body density matrix replaces the trial wave function. Concerning first principles simulations the path integral method can be used either to simulate the properties of thermal electrons or to simulate the zero point effects of light nuclei or both. For electronic simulations there are two major problems. First, the energy scale of electrons is 1 Hartree or above, thus to reach ambient temperature requires very long paths. Second, since electrons are fermions, antisymmetrization and hence the sign problem is inevitable. For a more complete overview of the method and its application to fermion systems, see [

To begin, we define the many particle density matrix for a system in equilibrium with an external reservoir at inverse temperature _{B}T^{(1)}, . . . ,^{(N)}) with ^{(i)} specifying the spacial coordinates of the ^{th}

Finally, in order to account for the particle statistics of the simulated system, we must sum over permutations _{t}

For fermions, negative terms enter in this sum, leading to a sign problem. As was done in the previous discussion of DMC, one way to circumvent this issue is to impose a nodal constraint [_{R★β}_{★}_{★}_{β}_{★}) as the set of all continuous paths _{t}_{t}, R_{★},

We have thus turned the sign-full expression for the density matrix into one which includes only terms of a single sign, allowing efficient computation. However, because

The nodal error, arising from using an approximate restriction is problematic since it is uncontrollable. The finite temperature variational principle is through the free energy, as opposed to the internal energy in the ground state. Thus one possible solution is to parameterize the nodal ansatz, and then minimize the free energy by varying the parameters. This will require a thermodynamic integration, in general. Systems analyzed to date suggest that the nodal error arising from the free-particle ansatz is small since the correlation from the interacting potential is fully taken into account.

Even when quantum particles can be considered distinguishable, as for instance light nuclei in condensed phases, there could be substantial physical effects arising from their quantum behavior, _{n}

A frequent use of path integrals for nuclei occurs when DFT is used to integrate out the electronic degrees of freedom. However, one wants to use the DFT energy surface for the properties of the quantum nuclei in equilibrium, using the path integral method. To perform the path integration, it is advantageous to use molecular dynamics instead of Monte Carlo since that will allow the electronic wave functions to evolve smoothly in time, and thus reduce the time to convergence in solving the DFT self-consistency conditions. M. Ceriotti,

The QMC methods described so far, when applied to an ion-electron system, treat all particles on the same footing, either both in the ground state [

The Coupled Electron-Ion Monte Carlo method (CEIMC) is a QMC method based on the BO approximation [

In CEIMC the difference of BO energies of two nearby nuclear configurations in a MC attempted step, as obtained by an electronic QMC run, is affected by statistical noise which, if ignored, results in a biased nuclear sampling. To cope with this situation either the statistical noise needs to be reduced to a negligible value by long electronic calculations (very inefficient), or the Metropolis acceptance/rejection scheme needs modifications to cope with noisy energy differences. The latter strategy is implemented in the Penalty Method [

When nuclear quantum effects are included using a path integral representation (see §2.3), the relevant inverse temperature in the penalty method is the imaginary time discretization step _{s}^{−1} ≃ 4800 K is enough to reach convergence of the thermodynamics properties, which allows to study systems at low temperature with a limited number of time slices (≤50).

In CEIMC many-body nuclear moves are preferred to single-body moves. The reason is that even if only few nuclei are moved the entire electronic calculation must be repeated, by far the most expensive part of the method. For this reason we sample nuclear configuration by a smart Monte Carlo method [

The main ingredient of CEIMC is the electronic QMC engine used to compute the BO energy. As mentioned a very important aspect for the efficiency of CEIMC is the noise level which is related to the variance of the local energy. In ground state QMC (see §2.1) the “zero variance principle” applies: if the trial wave function is an eigenfunction of the Hamiltonian, the local energy is no longer a function of the electronic coordinates and a single calculation provides the exact corresponding eigenvalues. Therefore by improving the trial wave function and approaching the exact ground state, the variance of the local energy decreases to zero. In connection with CEIMC, this is important not only for the accuracy of the BO energy but also for the efficiency of the nuclear sampling since the extra rejection due to the noise is reduced for a more accurate trial wave function.

To go beyond VMC accuracy in CEIMC we have implemented Reptation QMC method (RMC) [

In order to improve the efficiency of CEIMC while keeping the RMC accuracy, we have recently developed a method, based on a peculiar thermodynamic integration, to estimate the free energy of the system with RMC based BO energy from the knowledge of the free energy of the system with VMC based BO energy [

For high pressure hydrogen we have developed a quite accurate trial function of the Slater-Jastrow, single determinant, form. The Jastrow part has an electron-proton and electron-electron Random Phase Approximation (RPA) term plus two-body and three-body empirical terms depending on few variational parameters. The Slater determinants (one for each spin state) are built with single electron orbitals obtained by a self-consistent DFT solution. We have recently integrated the PWSCF-DFT solver [

In view of the large variability of DFT results from different exchange-correlation approximations in the dissociation region of high pressure hydrogen (see next section), one interesting question is about the sensitivity of the trial wave function to the particular form of the adopted Kohn-Sham orbitals in the Slater determinant. This is particularly relevant since the form of the orbitals determine the nodal surface of the trial wave function, the ultimate limit in the accuracy of fermionic QMC. On the one hand one could hope to further improve the quality of the trial wave function by varying the type of orbitals, on the other hand a large sensitivity to the form of the Kohn-Sham orbitals will signal a too constrained form of the wave function, probably with a large room for improvements. The recent technical advance of the CEIMC code, namely the integration of PWSCF, allowed us to test several different types of orbitals: standard local (LDA) and semilocal (GGA-PBE) approximation, a non-local functional devised to reduce the self-interaction error and improve the description of the electronic correlation in DFT (HSE [_{s}_{s}

We note that for all structures and at all densities LDA, PBE and HSE orbitals provides trial functions of the same quality (differences are of the order of 0.2 mH/atoms = 90K). Instead the trial function with orbitals from vdW-DF2 functional provides higher energies, by roughly 0.4 mH/at with values up to 1.4 mH/atom (≃630 K). This first result is quite indicative that our trial function is flexible and general enough to be very little sensitive to the form of the orbitals. In order to check whether the observed differences from vdW-DF2 orbitals could be due to optimization problems only, we performed a complete RMC study for a single case, namely the Pbcn structure at _{s}^{−1} was used, which is fairly typical in this sort of calculation. No further time step error extrapolation study has been performed. In ^{2}. This plot for all studied cases is shown in the right panel of _{0} = −0.5350(2)), while the vdW-DF2 orbitals provides a higher value (_{0} = −0.5342(2)). The fact that the RMC projection is not able to remove the difference observed at the VMC level means that the nodes from the vdW-DF2 are less accurate than for the other kind of orbitals, which instead, despite their differences, provide essentially the same nodal structure. Finally we note that our old implementation of LDA orbitals provides a less accurate determination of the energy with correspondingly larger variance.

Hydrogen is the simplest element of the periodic table and also the most abundant element in the Universe. Because of its simple electronic structure, it has been instrumental in the development of quantum mechanics and remains important for developing ideas and theoretical methods. In the next section we explore its use in developing DFT functionals. Its phase diagram at high pressure has received considerable attention from the first-principles simulation community due to its critical importance in many fields like planetary science, high pressure physics, astrophysics, inertial confinement fusion, among many others [

The ground state structure of crystalline hydrogen across the pressure-induced molecular dissociation has been studied by DMC [_{s}

Temperatures lower than ∼10,000 K cannot be easily reached by RPIMC without reducing the level of accuracy. However, most of the interesting phenomena in high pressure hydrogen, like molecular dissociation under pressure, metallization, solid-fluid transition, a possible liquid-liquid phase transition and its interplay with melting, the various crystalline phases and the transition to the atomic phases [

CEIMC has been applied to investigate the WDM regime of hydrogen and helium and benchmark FPMD [

The last estimate however is for a metastable liquid state obtained by an instantaneous quenching of the fluid at higher temperature, while it is expected that the equilibrium state at 300 K and ∼290 GPa be crystalline (of unknown structure) [

Although our results demonstrate the power of CEIMC in predicting the physical properties of hydrogen, its use is still quite demanding in terms of computer time, a fact that limits its applicability. This is particularly true when a much larger exploration of external conditions is needed to clarify the physics. For example, to study the crystalline state of the molecular system and clarify the molecular-atomic transition mechanism in the solid state, it is necessary to consider a large number of candidate structures, some of which have very large unit cells (the recently proposed Pc structure for phase IV of molecular hydrogen [

Within the Born-Oppenheimer approximation at low temperatures, the only interaction between ions and electrons comes through the potential energy surface _{0}(_{0}(_{DFT}_{DFT}

The phase diagram of hydrogen at high pressure has been extensively explored using first-principles simulations with DFT [

Since sufficient experimental data is not available to validate the quality of functionals in the high-pressure high-temperature regime of the phase diagram, we used fixed-node diffusion Monte Carlo (DMC) to benchmark the accuracy of several DFT functionals over a range of densities near the liquid-liquid phase transition at a temperature of T = 1000 K. Henceforth, we will refer to densities using the parameter _{s}_{s}_{s}_{s}_{s}_{DFT}

All QMC calculations were performed with the QMCPACK [^{−1} and 6000 walkers were used. The orbitals were obtained from DFT using the Quantum Espresso software package [_{c}_{0}, DMC calculations were performed with the Coulomb potential. Based on the scale of the energy differences, we found a statistical error of 0.02 mHa/particle to be sufficient for present purposes. Since we were interested in measuring the spread of energy errors in this presentation, constant energy offsets were removed from our error assessments. This means that we did not have to include energetic finite size effects, although more detailed assessments will certainly call for this.

An example of the comparison between QMC and DFT is given in _{DFT}_{DFT}_{DMC}_{s}_{s}

To better quantify and compare the quality of functionals, we have computed the mean absolute error (MAE) from data similar to that shown in _{func}_{DFT}_{DFT}

There are several interesting features in

Water plays a central role in many scientific fields [

Atomistic simulations have the potential to resolve these issues, particularly using first-principles methods. Providing an accurate theoretical description has been a central topic and open challenge in physical chemistry for many decades. Despite considerable focus over the last decade, to date DFT has proven insufficient for the accurate description of liquid water [

In this section, we present QMC calculations of configurations of molecules extracted from PIMD simulations of liquid water. QMC has been shown to be a reliable benchmark in the study of small water clusters [^{−1} was found to be sufficiently small to produce accurate total energies and approximately 4800 walkers were used in the DMC calculations. Casula’s T-moves [

DFT calculations were performed with both Quantum Espresso (QE) [

We present calculations for 3 different configuration sets. The first two sets, which we called ^{3}, with the vdW-DF2 functional, which has been recently shown to provide an accurate description of the structure of water when combined with a path integral representation [

Direct first-principles simulations with QMC accuracy of condensed phases systems are nowadays possible but restricted so far to the simplest first few elements of the periodic table, namely hydrogen, helium and their mixtures. Even for those simple systems, challenges are present and the computational demand is large. Nonetheless, CEIMC predictions for the liquid-liquid phase transition in hydrogen remains today the target for less accurate but faster DFT-based FP methods. While much work remains to be done in developing QMC-based FP methods, the calculations presented here show one possible use of accurate many-body calculations: using QMC to benchmark the accuracy of DFT functionals. Not only does this allow us to make a judgment of the quality of a functional before its use in first-principles simulations, but it also shows us a path for the systematic improvement of the functionals by adjusting free parameters to minimize the errors. DFT users will often point to experimental data to validate the quality of a chosen functional. What we have shown is that we can use highly-accurate QMC methods to benchmark functionals around the liquid-liquid transition of hydrogen from first-principles. In addition, this set of reference energies for the bulk system can be used to optimize the free parameters in the DFT functional to minimize the errors, and in the limit of a large data set, reproduce the quality of the more accurate many-body method in first-principles calculations using DFT. This approach will be increasingly necessary as we continue to explore matter under extreme pressures, since experimental data is often insufficient or nonexistent at geophysical/planetary scales. It will also be necessary for other situations where DFT functionals have difficulties, such as near metal-insulator transitions.

Let us consider a more general point. We suggest that, in general, it is superior to use total energies to find an interatomic potential (force field). The traditional approach is to fit experimental data, for example, the melting temperature of ice, the density of water versus temperature,

Water and hydrogen show an additional complication of using experimental data: namely because of the importance of quantum zero-point effects of the protons, fitting of the experimental data becomes particularly problematic. A common approach is to do a simulation of the classical system and assume that the effective classical system includes the effects of zero-point energy; clearly this then becomes quite approximate since the zero-point effects are not small. A complication is that the interatomic potential that results can become temperature and density dependent with all known pathologies related to the use of state dependent potentials [

One aspect in determining good force fields is to find an appropriate basis set to parameterize the force field. Traditionally, these have contained few functions with very few parameters, e.g., the Lennard Jones potential with only two parameters: and

Miguel Angel Morales was supported by the U.S. Department of Energy at the Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, by LDRD Grant No. 13-LW-004 and by the Basic Energy Science (BES), DOE through the Predictive Theory and Modeling for Materials and Chemical Science program, D. M. C. and R. C. were supported by the DOE grant DE-NA0001789 and C. P. by the Italian Institute of Technology (IIT) under the SEED project grant number 259 SIMBEDD Advanced Computational Methods for Biophysics, Drug Design and Energy Research. Computer resources have been provided by the US DOE INCITE program, Lawrence Livermore National Laboratory through the 7th Institutional Unclassified Computing Grand Challenge program and PRACE Project No. 2011050781.

The authors declare no conflict of interest.

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Variational energy of four different crystalline molecular structures versus _{s}_{2}/c upper-left panel, Cmca-12 upper-right panel, P63m lower-left panel and Pbcn lower-right panel. Energies from wave functions with different orbitals relatives to the energy with LDA orbitals: PBE orbitals (red triangles), HSE orbitals (green closed circles) and vdW-DF2 orbitals (blue closed squares).

Pbcn structure of molecular hydrogen at _{s}

Proton-proton radial distribution function at various densities along the isotherm T = 600 K. Comparison between classical nuclei (red continuous line) and quantum nuclei (blue dashed line) for hydrogen nuclear mass. It is evident the molecular dissociation with increasing density.

Histograms of Δ_{DFT}_{s}_{DFT}

Mean absolute error of energy/atom

Mean absolute error in the total energy between DMC and DFT with various exchange correlation functionals for a supercell containing water molecules. Results presented correspond to calculations using the PAW formulation with VASP. X-D, where X represents a given density functional, designates results using the empirical dispersion corrections of Grimme