Frontiers of stochastic electronic structure calculations.

In recent years there has been a rapid growth in the development and application of new stochastic methods in electronic structure. These methods are quite diverse, from many-body wave function techniques in real space or determinant space to being used to sum perturbative expansions. This growth has been spurred by the more favorable scaling with the number of electrons and often better parallelization over large numbers of central processing unit (CPU) cores or graphical processing units (GPUs) than for high-end non-stochastic wave function based methods. This special issue of the Journal of Chemical Physics includes 33 papers that describe recent developments and applications in this area. As seen from the articles in the issue, stochastic electronic structure methods are applicable to both molecules and solids and can accurately describe systems with strong electron correlation. This issue was motivated, in part, by the 2019 Telluride Science Research Center workshop on Stochastic Electronic Structure Methods that we organized. Below we briefly describe each of the papers in the special issue, dividing the papers into six subtopics.

Melting and density of MgSiO3 determined by shock compression of bridgmanite to 1254GPa.

The essential data for interior and thermal evolution models of the Earth and super-Earths are the density and melting of mantle silicate under extreme conditions. Here, we report an unprecedently high melting temperature of MgSiO3 at 500 GPa by direct shockwave loading of pre-synthesized dense MgSiO3 (bridgmanite) using the Z Pulsed Power Facility. We also present the first high-precision density data of crystalline MgSiO3 to 422 GPa and 7200 K and of silicate melt to 1254 GPa. The experimental density measurements support our density functional theory based molecular dynamics calculations, providing benchmarks for theoretical calculations under extreme conditions. The excellent agreement between experiment and theory provides a reliable reference density profile for super-Earth mantles. Furthermore, the observed upper bound of melting temperature, 9430 K at 500 GPa, provides a critical constraint on the accretion energy required to melt the mantle and the prospect of driving a dynamo in massive rocky planets.

Toward a systematic improvement of the fixed-node approximation in diffusion Monte Carlo for solids-A case study in diamond.

While Diffusion Monte Carlo (DMC) is in principle an exact stochastic method for ab initio electronic structure calculations, in practice, the fermionic sign problem necessitates the use of the fixed-node approximation and trial wavefunctions with approximate nodes (or zeros). This approximation introduces a variational error in the energy that potentially can be tested and systematically improved. Here, we present a computational method that produces trial wavefunctions with systematically improvable nodes for DMC calculations of periodic solids. These trial wavefunctions are efficiently generated with the configuration interaction using a perturbative selection made iteratively (CIPSI) method. A simple protocol in which both exact and approximate results for finite supercells are used to extrapolate to the thermodynamic limit is introduced. This approach is illustrated in the case of the carbon diamond using Slater-Jastrow trial wavefunctions including up to one million Slater determinants. Fixed-node DMC energies obtained with such large expansions are much improved, and the fixed-node error is found to decrease monotonically and smoothly as a function of the number of determinants in the trial wavefunction, a property opening the way to a better control of this error. The cohesive energy extrapolated to the thermodynamic limit is in close agreement with the estimated experimental value. Interestingly, this is also the case at the single-determinant level, thus, indicating a very good error cancellation in carbon diamond between the bulk and atomic total fixed-node energies when using single-determinant nodes.

A new generation of effective core potentials from correlated calculations: 4s and 4p main group elements and first row additions.

Recently, we developed a new method for generating effective core potentials (ECPs) using valence energy isospectrality with explicitly correlated all-electron (AE) excitations and norm-conservation criteria. We apply this methodology to the 3rd-row main group elements, creating new correlation consistent ECPs (ccECPs) and also deriving additional ECPs to complete the ccECP table for H-Kr. For K and Ca, we develop Ne-core ECPs, and for the 4p main group elements, we construct [Ar]3d10-core potentials. Scalar relativistic effects are included in their construction. Our ccECPs reproduce AE spectra with significantly better accuracy than many existing pseudopotentials and show better overall consistency across multiple properties. The transferability of ccECPs is tested on monohydride and monoxide molecules over a range of molecular geometries. For the constructed ccECPs, we also provide optimized DZ-6Z valence Gaussian basis sets.

A new generation of effective core potentials from correlated calculations: 3d transition metal series.

Recently, we have introduced a new generation of effective core potentials (ECPs) designed for accurate correlated calculations but equally useful for a broad variety of approaches. The guiding principle has been the isospectrality of all-electron and ECP Hamiltonians for a subset of valence many-body states using correlated, nearly-exact calculations. Here we present such ECPs for the 3d transition series Sc to Zn with Ne-core, i.e., with semi-core 3s and 3p electrons in the valence space. Besides genuine many-body accuracy, the operators are simple, being represented by a few gaussians per symmetry channel with resulting potentials that are bounded everywhere. The transferability is checked on selected molecular systems over a range of geometries. The ECPs show a high overall accuracy with valence spectral discrepancies typically ≈0.01-0.02 eV or better. They also reproduce binding curves of hydride and oxide molecules typically within 0.02-0.03 eV deviations over the full non-dissociation range of interatomic distances.

An efficient hybrid orbital representation for quantum Monte Carlo calculations.

The scale and complexity of the quantum system to which real-space quantum Monte Carlo (QMC) can be applied in part depends on the representation and memory usage of the trial wavefunction. B-splines, the computationally most efficient basis set, can have memory requirements exceeding the capacity of a single computational node. This situation has traditionally forced a difficult choice of either using slow internode communication or a potentially less accurate but smaller basis set such as Gaussians. Here, we introduce a hybrid representation of the single particle orbitals that combine a localized atomic basis set around atomic cores and B-splines in the interstitial regions to reduce the memory usage while retaining the high speed of evaluation and either retaining or increasing overall accuracy. We present a benchmark calculation for NiO demonstrating a superior accuracy while using only one eighth of the memory required for conventional B-splines. The hybrid orbital representation therefore expands the overall range of systems that can be practically studied with QMC.

A new generation of effective core potentials from correlated calculations: 2nd row elements.

Very recently, we have introduced correlation consistent effective core potentials (ccECPs) derived from many-body approaches with the main target being their use in explicitly correlated methods, while still usable in mainstream approaches. The ccECPs are based on reproducing excitation energies for a subset of valence states, namely, achieving near-isospectrality between the original and pseudo Hamiltonians. In addition, binding curves of dimer molecules were used for refinement and overall improvement of transferability over a range of bond lengths. Here we apply similar ideas to the 2nd row elements and study several aspects of the constructions in order to find the high accuracy solutions within the chosen ccECP forms with 3s, 3p valence space (Ne-core). Our new constructions exhibit accurate low-lying atomic excitations and equilibrium molecular bonds (on average within ≈0.03 eV and 3 mÅ); however, the errors for Al and Si oxide molecules at short bond lengths are notably larger for both ours and existing effective core potentials. Assuming this limitation, our ccECPs show a systematic balance between the criteria of atomic spectra accuracy and transferability for molecular bonds. In order to provide another option with much higher uniform accuracy, we also construct He-core ccECPs for the whole 2nd row with typical discrepancies of ≈0.01 eV or smaller.

QMCPACK: an open source ab initio quantum Monte Carlo package for the electronic structure of atoms, molecules and solids.

QMCPACK is an open source quantum Monte Carlo package for ab initio electronic structure calculations. It supports calculations of metallic and insulating solids, molecules, atoms, and some model Hamiltonians. Implemented real space quantum Monte Carlo algorithms include variational, diffusion, and reptation Monte Carlo. QMCPACK uses Slater-Jastrow type trial wavefunctions in conjunction with a sophisticated optimizer capable of optimizing tens of thousands of parameters. The orbital space auxiliary-field quantum Monte Carlo method is also implemented, enabling cross validation between different highly accurate methods. The code is specifically optimized for calculations with large numbers of electrons on the latest high performance computing architectures, including multicore central processing unit and graphical processing unit systems. We detail the program's capabilities, outline its structure, and give examples of its use in current research calculations. The package is available at http://qmcpack.org.

A new generation of effective core potentials for correlated calculations.

We outline ideas on desired properties for a new generation of effective core potentials (ECPs) that will allow valence-only calculations to reach the full potential offered by recent advances in many-body wave function methods. The key improvements include consistent use of correlated methods throughout ECP constructions and improved transferability as required for an accurate description of molecular systems over a range of geometries. The guiding principle is the isospectrality of all-electron and ECP Hamiltonians for a subset of valence states. We illustrate these concepts on a few first- and second-row atoms (B, C, N, O, S), and we obtain higher accuracy in transferability than previous constructions while using semi-local ECPs with a small number of parameters. In addition, the constructed ECPs enable many-body calculations of valence properties with higher (or same) accuracy than their all-electron counterparts with uncorrelated cores. This implies that the ECPs include also some of the impacts of core-core and core-valence correlations on valence properties. The results open further prospects for ECP improvements and refinements.

Quantum Monte Carlo analysis of a charge ordered insulating antiferromagnet: the Ti4O7 Magnéli phase.

The Magnéli phase Ti4O7 is an important transition metal oxide with a wide range of applications because of its interplay between charge, spin, and lattice degrees of freedom. At low temperatures, it has non-trivial magnetic states very close in energy, driven by electronic exchange and correlation interactions. We have examined three low-lying states, one ferromagnetic and two antiferromagnetic, and calculated their energies as well as Ti spin moment distributions using highly accurate quantum Monte Carlo methods. We compare our results to those obtained from density functional theory-based methods that include approximate corrections for exchange and correlation. Our results confirm the nature of the states and their ordering in energy, as compared with density-functional theory methods. However, the energy differences and spin distributions differ. A detailed analysis suggests that non-local exchange-correlation functionals, in addition to other approximations such as LDA+U to account for correlations, are needed to simultaneously obtain better estimates for spin moments, distributions, energy differences and energy gaps.

Shock Response and Phase Transitions of MgO at Planetary Impact Conditions.

The moon-forming impact and the subsequent evolution of the proto-Earth is strongly dependent on the properties of materials at the extreme conditions generated by this violent collision. We examine the high pressure behavior of MgO, one of the dominant constituents in Earth's mantle, using high-precision, plate impact shock compression experiments performed on Sandia National Laboratories' Z Machine and extensive quantum calculations using density functional theory (DFT) and quantum Monte Carlo (QMC) methods. The combined data span from ambient conditions to 1.2 TPa and 42 000 K, showing solid-solid and solid-liquid phase boundaries. Furthermore our results indicate that under impact the solid and liquid phases coexist for more than 100 GPa, pushing complete melting to pressures in excess of 600 GPa. The high pressure required for complete shock melting has implications for a broad range of planetary collision events.

Application of Diffusion Monte Carlo to Materials Dominated by van der Waals Interactions.

van der Waals forces are notoriously difficult to account for from first principles. We have performed extensive calculations to assess the usefulness and validity of diffusion quantum Monte Carlo when predicting van der Waals forces. We present converged results for noble gas solids and clusters, archetypical van der Waals dominated systems, as well as the highly relevant π-π stacking supramolecular complex: DNA + intercalating anticancer drug ellipticine. Analysis of the calculated binding energies underscores the existence of significant interatomic many-body contributions.

Crystal structures of (Mg1-x,Fe(x))SiO3 postperovskite at high pressures.

X-ray diffraction experiments on postperovskite (ppv) with compositions (Mg(0.9)Fe(0.1))SiO(3) and (Mg(0.6)Fe(0.4))SiO(3) at Earth core-mantle boundary pressures reveal different crystal structures. The former adopts the CaIrO(3)-type structure with space group Cmcm, whereas the latter crystallizes in a structure with the Pmcm (Pmma) space group. The latter has a significantly higher density (ρ = 6.119(1) g/cm(3)) than the former (ρ = 5.694(8) g/cm(3)) due to both the larger amount of iron and the smaller ionic radius of Fe(2+) as a result of an electronic spin transition observed by X-ray emission spectroscopy (XES). The smaller ionic radius for low-spin compared to high-spin Fe(2+) also leads to an ordered cation distribution in the M1 and M2 crystallographic sites of the higher density ppv structure. Rietveld structure refinement indicates that approximately 70% of the total Fe(2+) in that phase occupies the M2 site. XES results indicate a loss of 70% of the unpaired electronic spins consistent with a low spin M2 site and high spin M1 site. First-principles calculations of the magnetic ordering confirm that Pmcm with a two-site model is energetically more favorable at high pressure, and predict that the ordered structure is anisotropic in its electrical and elastic properties. These results suggest that interpretations of seismic structure in the deep mantle need to treat a broader range of mineral structures than previously considered.