PyQMC: An all-Python real-space quantum Monte Carlo module in PySCF.

We describe a new open-source Python-based package for high accuracy correlated electron calculations using quantum Monte Carlo (QMC) in real space: PyQMC. PyQMC implements modern versions of QMC algorithms in an accessible format, enabling algorithmic development and easy implementation of complex workflows. Tight integration with the PySCF environment allows for a simple comparison between QMC calculations and other many-body wave function techniques, as well as access to high accuracy trial wave functions.

Quantification of electron correlation for approximate quantum calculations.

State-of-the-art many-body wave function techniques rely on heuristics to achieve high accuracy at an attainable computational cost to solve the many-body Schrödinger equation. By far, the most common property used to assess accuracy has been the total energy; however, total energies do not give a complete picture of electron correlation. In this work, we assess the von Neumann entropy of the one-particle reduced density matrix (1-RDM) to compare selected configuration interaction (CI), coupled cluster, variational Monte Carlo, and fixed-node diffusion Monte Carlo for benchmark hydrogen chains. A new algorithm, the circle reject method, is presented, which improves the efficiency of evaluating the von Neumann entropy using quantum Monte Carlo by several orders of magnitude. The von Neumann entropy of the 1-RDM and the eigenvalues of the 1-RDM are shown to distinguish between the dynamic correlation introduced by the Jastrow and the static correlation introduced by determinants with large weights, confirming some of the lore in the field concerning the difference between the selected CI and Slater-Jastrow wave functions.

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.

Excited states in variational Monte Carlo using a penalty method.

In this article, the authors present a technique using variational Monte Carlo to solve for excited states of electronic systems. This technique is based on enforcing orthogonality to lower energy states, which results in a simple variational principle for the excited states. Energy optimization is then used to solve for the excited states. This technique is applied to the well-characterized benzene molecule, in which ∼10 000 parameters are optimized for the first 12 excited states. Agreement within ∼0.2 eV is obtained with higher scaling coupled cluster methods; small disagreements with experiment are likely due to vibrational effects.

Identifying materials with charge-spin physics using charge-spin susceptibility computed from first principles.

The authors present a quantity termed charge-spin susceptibility, which measures the charge response to spin degrees of freedom in strongly correlated materials. This quantity is simple to evaluate using both standard density functional theory and many-body electronic structure techniques, enabling comparison between different levels of theory. A benchmark on 28 layered magnetic materials shows that large values of charge-spin susceptibility correlate with unconventional ground states such as disordered magnets and unconventional superconductivity.

Recent developments in the PySCF program package.

PySCF is a Python-based general-purpose electronic structure platform that supports first-principles simulations of molecules and solids as well as accelerates the development of new methodology and complex computational workflows. This paper explains the design and philosophy behind PySCF that enables it to meet these twin objectives. With several case studies, we show how users can easily implement their own methods using PySCF as a development environment. We then summarize the capabilities of PySCF for molecular and solid-state simulations. Finally, we describe the growing ecosystem of projects that use PySCF across the domains of quantum chemistry, materials science, machine learning, and quantum information science.

High Hole Mobility and Nonsaturating Giant Magnetoresistance in the New 2D Metal NaCu4Se4 Synthesized by a Unique Pathway.

The new compound NaCu4Se4 forms by the reaction of CuO and Cu in a molten sodium polyselenide flux, with the existence of CuO being unexpectedly critical to its synthesis. It adopts a layered hexagonal structure (space group P63/ mmc with cell parameters a = 3.9931(6) Å and c = 25.167(5) Å), consisting of infinite two-dimensional [Cu4Se4]- slabs separated by Na+ cations. X-ray photoelectron spectroscopy suggests that NaCu4Se4 is mixed-valent with the formula (Na+)(Cu+)4(Se2-)(Se-)(Se2)2-. NaCu4Se4 is a p-type metal with a carrier density of ∼1021 cm-3 and a high hole mobility of ∼808 cm2 V-1 s-1 at 2 K based on electronic transport measurements. First-principles calculations suggest the density of states around the Fermi level are composed of Cu-d and Se-p orbitals. At 2 K, a very large transverse magnetoresistance of ∼1400% was observed, with a nonsaturating, linear dependence on field up to 9 T. Our results indicate that the use of metal oxide chemical precursors can open reaction paths to new low-dimensional compounds.

Non-orthogonal determinants in multi-Slater-Jastrow trial wave functions for fixed-node diffusion Monte Carlo.

The accuracy and efficiency of ab initio Quantum Monte Carlo (QMC) algorithms benefit greatly from compact variational trial wave functions that accurately reproduce ground state properties of a system. We investigate the possibility of using multi-Slater-Jastrow trial wave functions with non-orthogonal determinants by optimizing identical single particle orbitals independently in separate determinants. As a test case, we compute variational and fixed-node diffusion Monte Carlo (FN-DMC) energies of a C2 molecule. For a given multi-determinant expansion, we find that this non-orthogonal orbital optimization results in a consistent improvement in the variational energy and the FN-DMC energy on the order of a few tenths of an eV. In some cases, fewer non-orthogonal determinants are required compared to orthogonal ones in order to achieve similar accuracy in FN-DMC. Our calculations indicate that trial wave functions with non-orthogonal determinants can improve computed energies in a QMC calculation when compared to their orthogonal counterparts.

Erratum: Computation of the Correlated Metal-Insulator Transition in Vanadium Dioxide from First Principles [Phys. Rev. Lett. 114, 176401 (2015)].

This corrects the article DOI: 10.1103/PhysRevLett.114.176401.

Semiconducting Ba3Sn3Sb4 and Metallic Ba7-xSn11Sb15-y (x = 0.4, y = 0.6) Zintl Phases.

We report the discovery of two ternary Zintl phases Ba3Sn3Sb4 and Ba7-xSn11Sb15-y (x = 0.4, y = 0.6). Ba3Sn3Sb4 adopts the monoclinic space group P21/c with a = 14.669(3) Å, b = 6.9649(14) Å, c = 13.629(3) Å, and ß = 104.98(3)°. It features a unique corrugated two-dimensional (2D) structure consisting of [Sn3Sb4]6- layers extending along the ab-plane with Ba2+ atoms sandwiched between them. The nonstoichiometric Ba6.6Sn11Sb14.4 has a complex one-dimensional (1D) structure adopting the orthorhombic space group Pnma, with unit cell parameters a = 37.964(8) Å, b = 4.4090(9) Å, and c = 24.682(5) Å. It consists of large double Sn-Sb ribbons separated by Ba2+ atoms. Ba3Sn3Sb4 is an n-type semiconductor which has a narrow energy gap of ∼0.18 eV and a room temperature carrier concentration of ∼4.2 × 1018 cm-3. Ba6.6Sn11Sb14.4 is determined to be a metal with electrons being the dominant carriers.

Importance of σ Bonding Electrons for the Accurate Description of Electron Correlation in Graphene.

Electron correlation in graphene is unique because of the interplay between the Dirac cone dispersion of π electrons and long-range Coulomb interaction. Because of the zero density of states at Fermi level, the random phase approximation predicts no metallic screening at long distance and low energy, so one might expect that graphene should be a poorly screened system. However, empirically graphene is a weakly interacting semimetal, which leads to the question of how electron correlations take place in graphene at different length scales. We address this question by computing the equal time and dynamic structure factor S(q) and S(q,ω) of freestanding graphene using ab initio fixed-node diffusion Monte Carlo simulations and the random phase approximation. We find that the σ electrons contribute strongly to S(q,ω) for relevant experimental values of ω even at distances up to around 80 Å. These findings illustrate how the emergent physics from underlying Coulomb interactions results in the observed weakly correlated semimetal.

Charge Density Wave and Narrow Energy Gap at Room Temperature in 2D Pb3-xSb1+xS4Te2-δ with Square Te Sheets.

We report a new two-dimensional compound, Pb3-xSb1+xS4Te2-δ, that has a charge density wave (CDW) at room temperature. The CDW is incommensurate with q-vector of 0.248(6)a* + 0.246(8)b* + 0.387(9)c* for x = 0.29(2) and δ = 0.37(3) due to positional and occupational long-range ordering of Te atoms in the sheets. The modulated structure was refined from the single-crystal X-ray diffraction data with a superspace group P1̅(αßγ)0 using (3 + 1)-dimensional crystallography. The resistivity increases with decreasing temperature, suggesting semiconducting behavior. The transition temperature (TCDW) of the CDW is ∼345 K, above which the Te square sheets become disordered with no q-vector. First-principles density functional theory calculations on the undistorted structure and an approximate commensurate supercell reveal that the gap is due to the structure modulation.

Accurate barrier heights using diffusion Monte Carlo.

Fixed nodediffusion Monte Carlo (DMC) has been performed on a test set of forward and reverse barrier heights for 19 non-hydrogen-transfer reactions, and the nodal error has been assessed. The DMC results are robust to changes in the nodal surface, as assessed by using different mean-field techniques to generate single determinant wave functions. Using these single determinant nodal surfaces, DMC results in errors of 1.5(1) kcal/mol on barrier heights. Using the large data set of DMC energies, we attempted to find good descriptors of the fixed node error. It does not correlate with a number of descriptors including change in density but does show some correlation with the gap between the highest occupied and lowest unoccupied orbital energies in the mean-field calculation.

Discovering correlated fermions using quantum Monte Carlo.

It has become increasingly feasible to use quantum Monte Carlo (QMC) methods to study correlated fermion systems for realistic Hamiltonians. We give a summary of these techniques targeted at researchers in the field of correlated electrons, focusing on the fundamentals, capabilities, and current status of this technique. The QMC methods often offer the highest accuracy solutions available for systems in the continuum, and, since they address the many-body problem directly, the simulations can be analyzed to obtain insight into the nature of correlated quantum behavior.

Diffusion Monte Carlo Perspective on the Spin-State Energetics of [Fe(NCH)6](2.).

The energy difference between the high spin and the low spin state of the model compound [Fe(NCH)6](2+) is investigated by means of Diffusion Monte Carlo (DMC), where special attention is dedicated to analyzing the effect of the fix node approximation on the accuracy of the results. For this purpose, we compare several Slater-Jastrow and multireference Slater-Jastrow trial wave functions. We found that a Slater-Jastrow trial wave function constructed with the generalized Kohn-Sham orbitals from hybrid DFT represents the optimal choice. This is understood by observing that hybrid functionals account for the subtle balance between exchange and correlation effects and the respective orbitals accurately describe the ligand-metal hybridization as well as the charge reorganization accompanying the spin transition. Finally the DMC results are compared with those obtained by Hartree-Fock, DFT, CASSCF, and CASPT2. While there is no clear reference value for the high spin-low spin energy difference, DMC and high level CCSD(T) calculations agree within around 0.3 eV.

Using local operator fluctuations to identify wave function improvements.

A method is developed that allows analysis of quantum Monte Carlo simulations to identify errors in trial wave functions. The purpose of this method is to allow for the systematic improvement of variational wave functions by identifying degrees of freedom that are not well described by an initial trial state. We provide proof of concept implementations of this method by identifying the need for a Jastrow correlation factor and implementing a selected multideterminant wave function algorithm for small dimers that systematically decreases the variational energy. Selection of the two-particle excitations is done using the quantum Monte Carlo method within the presence of a Jastrow correlation factor and without the need to explicitly construct the determinants. We also show how this technique can be used to design compact wave functions for transition metal systems. This method may provide a route to analyze and systematically improve descriptions of complex quantum systems in a scalable way.

Diffusion Monte Carlo for Accurate Dissociation Energies of 3d Transition Metal Containing Molecules.

Transition metals and transition metal compounds are important to catalysis, photochemistry, and many superconducting systems. We study the performance of diffusion Monte Carlo (DMC) applied to transition metal containing dimers (TMCDs) using single-determinant Slater-Jastrow trial wavefunctions and investigate the possible influence of the locality and pseudopotential errors. We find that the locality approximation can introduce nonsystematic errors of up to several tens of kilocalories per mole in the absolute energy of Cu and CuH if Ar or Mg core pseudopotentials (PPs) are used for the 3d transition metal atoms. Even for energy differences such as binding energies, errors due to the locality approximation can be problematic if chemical accuracy is sought. The use of the Ne core PPs developed by Burkatzki et al. (J. Chem. Phys. 2008, 129, 164115), the use of linear energy minimization rather than unreweighted variance minimization for the optimization of the Jastrow function, and the use of large Jastrow parametrizations reduce the locality errors. In the second section of this article, we study the general performance of DMC for 3d TMCDs using a database of binding energies of 20 TMCDs, for which comparatively accurate experimental data is available. Comparing our DMC results to these data for our results that compare best with experiment, we find a mean unsigned error (MUE) of 4.5 kcal/mol. This compares well with the achievable accuracy in CCSDT(2)Q (MUE = 4.6 kcal/mol) and the best all-electron DFT results (MUE = 4.5 kcal/mol) for the same set of systems (Truhlar et al. J. Chem. Theory Comput. 2015, 11, 2036-2052). The mean errors in DMC depend less on the exchange-correlation functionals used to generate the trial wavefunction than the corresponding mean errors in the underlying DFT calculations. Furthermore, the QMC results obtained for each molecule individually vary less with the functionals used. These observations are relevant for systems such as molecules interacting with transition metal surfaces where the DFT functionals performing best for molecules (hybrids) do not yield improvements in DFT. Overall, the results presented in this article yield important guidelines for both the assessment of the achievable accuracy with DMC and the design of DMC calculations for systems including transition metal atoms.

Hexagonal boron nitride and water interaction parameters.

The study of hexagonal boron nitride (hBN) in microfluidic and nanofluidic applications at the atomic level requires accurate force field parameters to describe the water-hBN interaction. In this work, we begin with benchmark quality first principles quantum Monte Carlo calculations on the interaction energy between water and hBN, which are used to validate random phase approximation (RPA) calculations. We then proceed with RPA to derive force field parameters, which are used to simulate water contact angle on bulk hBN, attaining a value within the experimental uncertainties. This paper demonstrates that end-to-end multiscale modeling, starting at detailed many-body quantum mechanics and ending with macroscopic properties, with the approximations controlled along the way, is feasible for these systems.

Towards a systematic assessment of errors in diffusion Monte Carlo calculations of semiconductors: Case study of zinc selenide and zinc oxide.

The fixed node diffusion Monte Carlo (DMC) method has attracted interest in recent years as a way to calculate properties of solid materials with high accuracy. However, the framework for the calculation of properties such as total energies, atomization energies, and excited state energies is not yet fully established. Several outstanding questions remain as to the effect of pseudopotentials, the magnitude of the fixed node error, and the size of supercell finite size effects. Here, we consider in detail the semiconductors ZnSe and ZnO and carry out systematic studies to assess the magnitude of the energy differences arising from controlled and uncontrolled approximations in DMC. The former include time step errors and supercell finite size effects for ground and optically excited states, and the latter include pseudopotentials, the pseudopotential localization approximation, and the fixed node approximation. We find that for these compounds, the errors can be controlled to good precision using modern computational resources and that quantum Monte Carlo calculations using Dirac-Fock pseudopotentials can offer good estimates of both cohesive energy and the gap of these systems. We do however observe differences in calculated optical gaps that arise when different pseudopotentials are used.