Many body trial wave functions are the key ingredient for accurate Quantum Monte Carlo estimates of total electronic energies in many electron systems. In the Coupled Electron-Ion Monte Carlo method, the accuracy of the trial function must be conjugated with the efficiency of its evaluation. We report recent progress in trial wave functions for metallic hydrogen implemented in the Coupled Electron-Ion Monte Carlo method. We describe and characterize several types of trial functions of increasing complexity in the range of the coupling parameter $1.0 \leq r_s \leq1.55$. We report wave function comparisons for disordered protonic configurations and preliminary results for thermal averages.
Deep Dive into Trial wave functions for High-Pressure Metallic Hydrogen.
Many body trial wave functions are the key ingredient for accurate Quantum Monte Carlo estimates of total electronic energies in many electron systems. In the Coupled Electron-Ion Monte Carlo method, the accuracy of the trial function must be conjugated with the efficiency of its evaluation. We report recent progress in trial wave functions for metallic hydrogen implemented in the Coupled Electron-Ion Monte Carlo method. We describe and characterize several types of trial functions of increasing complexity in the range of the coupling parameter $1.0 \leq r_s \leq1.55$. We report wave function comparisons for disordered protonic configurations and preliminary results for thermal averages.
Modern ab initio simulation methods for systems of electrons and nuclei mostly rely on Density Function Theory (DFT) for computing the electronic forces acting on the nuclei, and on Molecular Dynamics (MD) techniques to follow the real-time evolution of the nuclei. Despite recent progress, DFT suffers from well-known limitations [1,2]. As a consequence, current ab initio predictions of metallization transitions at high pressures, or even the prediction of structural phase transitions, are often only qualitative. Hydrogen is an extreme case [3,4,5], but even in silicon, the diamond/β-tin transition pressure and the melting temperature are seriously underestimated [6].
An alternative route to the ground-state properties of a many-electrons system is the Quantum Monte Carlo method (QMC) [7,2]. In QMC, a manybody trial wave function for the electrons is assumed and the electronic properties are computed by Monte Carlo methods. Bosonic details of the trial wave functions are automatically optimized by projecting the trial state onto the ground state with the same nodal structure. Hence for fermions, QMC is a variational method with respect to the nodes of the trial wave function and a systematic, often unknown, error remains [7,2]. Over the years, the level of accuracy of the fixed-node approximation has been improved [8,9,10,11] such that, in most cases, fixed-node QMC methods have proven to be more accurate than DFT-based methods, on one side, and less computationally demanding than correlated quantum-chemistry strategies (such as coupled cluster method) [2] on the other side. Recently we have developed an ab-initio simulation method, the Coupled Electron-Ion Monte Carlo (CEIMC) method, based entirely on Monte Carlo algorithms, both for solving the electronic problem and for sampling the ionic configuration space in the Born-Oppenheimer approximation [12]. A Metropolis Monte Carlo simulation of the ionic degrees of freedom (represented either by classical point particles or by path integrals) at fixed temperature is performed based on the electronic energies computed during independent ground state Quantum Monte Carlo calculations. Application of CEIMC has been limited so far to high pressure hydrogen for several reasons: a) hydrogen is the simplest element of the periodic table, and the easiest to cope with since the absence of the additional separation of energy scales between core and valence electrons as in heavier elements; b) it is an important element since most of the matter in the universe consists of hydrogen; c) its phase diagram at high pressure in the interesting region where the metallization occurs is still largely unknown because present experiments are not able to reach the relevant pressures. We have investigated the very high pressure regime where all molecules are dissociated and the system is a plasma of fully ionized protons and electrons [13], and we have studied the pressure-induced molecular dissociation transition in the liquid phase [14]. In both studies the CEIMC results were not in agreement with previous Car-Parrinello Molecular Dynamics (CPMD) calculations [15,16]. While we have evidence now that the discrepancy in the fully ionized case is removed by taking a more accurate trial wave function in the QMC, in the second study more accurate CEIMC calculations predict a continuous molecular dissociation with increasing pressure at variance with CPMD where a first order molecular dissociation transition was observed by increasing pressure at constant temperature [16]. More recently, using constant volume Born-Oppenheimer Molecular Dynamics rather then constant pressure CPMD, a continuous dissociation transition has been reported from DFT-GGA studies [17].
In the present paper we discuss in some details the various trial wave functions we have implemented for hydrogen and we compare their accuracy at various densities. We will not review the details of the CEIMC method which have been described at length in reference [12]. We just mention that in metals huge finite size effects, caused by the discrete nature of the reciprocal space, can be alleviated by averaging electronic properties over the overall undetermined phase of the many-body wave function (Twist Average Boundary Conditions). Results reported in this work are obtained by this method (see also ref. [18] for recent development). Section 2 will be devoted to describing the different trial wave functions and some details on their efficient implementation. In Section 3 we will report numerical comparisons among the various trial functions. Finally, in Section 4 we collect our conclusions and perspectives.
The trial wave functions we have adopted for hydrogen are of the simple Slater-Jastrow form. We have considered spin unpolarized hydrogen only. For each spin state, a single determinant of one electron orbitals φ k (r) is used to account for the fermionic symmetry of the many-body wave function. A Jastrow factor e -U is then
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