Beyond Single-Reference Fixed-Node Approximation in Ab Initio Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of Electrons

Nakano K., Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.

TurboGenius: Python suite for high-throughput calculations of ab initio quantum Monte Carlo methods

Nakano K., Kohulák O., Raghav A., Casula M., TurboGenius is an open-source Python package designed to fully control ab initio quantum Monte Carlo (QMC) jobs using a Python script, which allows one to perform high-throughput calculations combined with TurboRVB [Nakano et al. J. Phys. Chem. 152, 204121 (2020)]. This paper provides an overview of the TurboGenius package and showcases several results obtained in a high-throughput mode. For the purpose of performing high-throughput calculations with TurboGenius, we implemented another open-source Python package, TurboWorkflows, that enables one to construct simple workflows using TurboGenius. We demonstrate its effectiveness by performing (1) validations of density functional theory (DFT) and QMC drivers as implemented in the TurboRVB package and (2) benchmarks of Diffusion Monte Carlo (DMC) calculations for several datasets. For (1), we checked inter-package consistencies between TurboRVB and other established quantum chemistry packages. By doing so, we confirmed that DFT energies obtained by PySCF are consistent with those obtained by TurboRVB within the local density approximation (LDA) and that Hartree-Fock (HF) energies obtained by PySCF and Quantum Package are consistent with variational Monte Carlo energies obtained by TurboRVB with the HF wavefunctions. These validation tests constitute a further reliability check of the TurboRVB package. For (2), we benchmarked the atomization energies of the Gaussian-2 set, the binding energies of the S22, A24, and SCAI sets, and the equilibrium lattice parameters of 12 cubic crystals using DMC calculations. We found that, for all compounds analyzed here, the DMC calculations with the LDA nodal surface give satisfactory results, i.e., consistent either with high-level computational or with experimental reference values.

Jastrow wave function for the spin-1 Heisenberg chain: The string order revealed by the mapping to the classical Coulomb gas

Piccioni D., Apostoli C., Becca F., Mazzola G., Parola A., We show that a two-body Jastrow wave function is able to capture the ground-state properties of the S=1 Heisenberg chain with nearest-neighbor superexchange J and single-ion anisotropy term D, in both the topological and large-D phases (with D/J≥0). Here, the optimized Jastrow pseudopotential assumes a very simple form in Fourier space, i.e., vq≈1/q2, which is able to give rise to a finite string-order parameter in the topological regime. The results are analyzed by using an exact mapping from the quantum expectation values over the variational state to the classical partition function of the one-dimensional Coulomb gas of particles with charge q=±1. Here, two phases are present at low temperatures: the first one is a diluted gas of dipoles (bound neutral pairs of particles), which are randomly oriented (describing the large-D phase); the other one is a dense liquid of dipoles, which are aligned thanks to the residual dipole-dipole interactions (describing the topological phase, with the finite string order being related to the dipole alignment). Our results provide an insightful interpretation of the ground-state nature of the spin-1 antiferromagnetic Heisenberg model.

Quantum phase diagram of high-pressure hydrogen

Monacelli L., Casula M., Nakano K., Hydrogen is the most abundant element in the Universe. However, understanding the properties of dense hydrogen is still an open challenge because—under megabar pressures—the quantum nature of both electrons and protons emerges, producing deviations from the common behaviour of condensed-matter systems. Experiments are challenging and can access only limited observables, and the interplay between electron correlation and nuclear quantum motion makes standard simulations unreliable. Here we present the computed phase diagram of hydrogen and deuterium at low temperatures and high pressures using state-of-the-art methods to describe both many-body electronic correlation and quantum anharmonic motion of protons. Our results show that the long-sought atomic metallic hydrogen phase—predicted to host room-temperature superconductivity—forms at 577(4) GPa. The anharmonic vibrations of nuclei pushes the stability of this phase towards pressures much larger than previous estimates or attained experimental values. Before atomization, molecular hydrogen transforms from a metallic phase (phase III) to another metallic structure that is still molecular (phase VI) at 410(20) GPa. Isotope effects increase the pressures of both transitions by 63 and 32 GPa, respectively. We predict signatures in optical spectroscopy and d.c. conductivity that can be experimentally used to distinguish between the two structural transitions.

TREXIO: A file format and library for quantum chemistry

Posenitskiy E., Chilkuri V.G., Ammar A., Hapka M., Pernal K., Shinde R., Landinez Borda E.J., Filippi C., Nakano K., Kohulák O., TREXIO is an open-source file format and library developed for the storage and manipulation of data produced by quantum chemistry calculations. It is designed with the goal of providing a reliable and efficient method of storing and exchanging wave function parameters and matrix elements, making it an important tool for researchers in the field of quantum chemistry. In this work, we present an overview of the TREXIO file format and library. The library consists of a front-end implemented in the C programming language and two different back-ends: a text back-end and a binary back-end utilizing the hierarchical data format version 5 library, which enables fast read and write operations. It is compatible with a variety of platforms and has interfaces for Fortran, Python, and OCaml programming languages. In addition, a suite of tools have been developed to facilitate the use of the TREXIO format and library, including converters for popular quantum chemistry codes and utilities for validating and manipulating data stored in TREXIO files. The simplicity, versatility, and ease of use of TREXIO make it a valuable resource for researchers working with quantum chemistry data.

Toward Chemical Accuracy Using the Jastrow Correlated Antisymmetrized Geminal Power Ansatz

Raghav A., Maezono R., Hongo K., Herein, we report accurate atomization energy calculations for 55 molecules in the Gaussian-2 (G2) set using lattice regularized diffusion Monte Carlo (LRDMC). We compare the Jastrow-Slater determinant ansatz with a more flexible JsAGPs (Jastrow correlated antisymmetrized geminal power with singlet correlation) ansatz. AGPs is built from pairing functions, which explicitly include pairwise correlations among electrons, and hence, this ansatz is expected to be more efficient in recovering the correlation energy. The AGPs wave functions are first optimized at the variational Monte Carlo (VMC) level, which includes both the Jastrow factor and the nodal surface optimization. This is followed by the LRDMC projection of the ansatz. Remarkably, for many molecules, the LRDMC atomization energies obtained using the JsAGPs ansatz reach chemical accuracy (∼1 kcal/mol), and for most other molecules, the atomization energies are accurate within ∼5 kcal/mol. We obtained a mean absolute deviation of 1.6 kcal/mol with JsAGPs and 3.2 kcal/mol with JDFT (Jastrow factor + Slater determinant with DFT orbitals) ansatzes. This work shows the effectiveness of the flexible AGPs ansatz for atomization energy calculations and electronic structure simulations in general.

Systematically improvable mean-field variational ansatz for strongly correlated systems: Application to the Hubbard model

A systematically improvable wave function is proposed for the numerical solution of strongly correlated systems. With a stochastic optimization method, based on the auxiliary field quantum Monte Carlo technique, an effective temperature Teff is defined, probing the distance of the ground-state properties of the model in the thermodynamic limit from the ones of the proposed correlated mean-field ansatz. In this way, their uncertainties from the unbiased zero temperature limit may be estimated by simple and stable extrapolations well before the so-called sign problem gets prohibitive. At finite Teff, the convergence of the energy to the thermodynamic limit is indeed shown to already be possible in the Hubbard model for relatively small square lattices with linear dimension L≃10, thanks to appropriate averages over several twisted boundary conditions. Within the estimated energy accuracy of the proposed variational ansatz, two clear phases are identified, as the energy is lowered by spontaneously breaking some symmetries satisfied by the Hubbard Hamiltonian: (a) a stripe phase where both spin and translation symmetries are broken and (b) a strong coupling d-wave superconducting phase when the particle number is not conserved and global U(1) symmetry is broken. On the other hand, the symmetric phase is stable in a wide region at large doping and small coupling.

High-pressure hydrogen by machine learning and quantum Monte Carlo

Tirelli A., Tenti G., Nakano K., We have developed a technique combining the accuracy of quantum Monte Carlo in describing the electron correlation with the efficiency of a machine learning potential (MLP). We use kernel regression in combination with the smooth overlap of atomic position (SOAP) features, implemented here in a very efficient way. The key ingredients are as follows: (i) a sparsification technique, based on farthest point sampling, ensuring generality and transferability of our MLPs, and (ii) the so-called Δ-learning, allowing a small training data set, a fundamental property for highly accurate but computationally demanding calculations, such as the ones based on quantum Monte Carlo. As an application we present a benchmark study of the liquid-liquid transition of high-pressure hydrogen and show the quality of our MLP, by emphasizing the importance of high accuracy for this very debated subject, where experiments are difficult in the laboratory, and theory is still far from being conclusive.

Two-dimensional t-t′ Holstein model

Araújo M.V., De Lima J.P., The competition and interplay between charge-density wave and superconductivity have become a central subject for quasi-two-dimensional compounds. Some of these materials, such as the transition-metal dichalcogenides, exhibit strong electron-phonon coupling, an interaction that may favor both phases, depending on the external parameters, such as hydrostatic pressure. In view of this, here we analyze the single-band t-t′ Holstein model in the square lattice, adding a next-nearest neighbor hopping t′ in order to play the role of the external pressure. To this end, we perform unbiased quantum Monte Carlo simulations with an efficient inversion sampling technique appropriately devised for this model. Such a methodology drastically reduces the autocorrelation time and increases the efficiency of the Monte Carlo approach. By investigating the charge-charge correlation functions, we obtain the behavior of the critical temperature as a function of t′ and, from compressibility analysis, we show that a first-order metal-to-insulator phase transition occurs. We also provide a low-temperature phase diagram for the model.

QMC study of the chiral Heisenberg Gross-Neveu universality class

Otsuka Y., Seki K., We investigate a quantum criticality of an antiferromagnetic phase transition in the Hubbard model on a square lattice with a d-wave pairing field by large-scale auxiliary-field quantum Monte Carlo simulations. Since the d-wave pairing filed induces Dirac cones in the non-interacting single-particle spectrum, the quantum criticality should correspond to the chiral Heisenberg universality class in terms of the Gross-Neveu theory, which is the same as those expected in the Hubbard model on the honeycomb lattice, despite the unit cells being different (e.g., they contain one and two sites, respectively). We show that both the two phase transitions, expected to occur on the square and on the honeycomb lattices, indeed have the same quantum criticality. We also argue that details of the models, i.e., the way of counting the total number N of fermion components and the anisotropy of the Dirac cones, do not change the critical exponents.

Space-warp coordinate transformation for efficient ionic force calculations in quantum Monte Carlo

Nakano K., Raghav A., Ab initio quantum Monte Carlo (QMC) methods are a state-of-The-Art computational approach to obtaining highly accurate many-body wave functions. Although QMC methods are widely used in physics and chemistry to compute ground-state energies, calculation of atomic forces is still under technical/algorithmic development. Very recently, force evaluation has started to become of paramount importance for the generation of machine-learning force-field potentials. Nevertheless, there is no consensus regarding whether an efficient algorithm is available for the QMC force evaluation, namely, one that scales well with the number of electrons and the atomic numbers. In this study, we benchmark the accuracy of all-electron variational Monte Carlo (VMC) and lattice-regularized diffusion Monte Carlo (LRDMC) forces for various mono-and heteronuclear dimers (1 ? Z ? 35, where Z is the atomic number). The VMC and LRDMC forces were calculated with and without the so-called space-warp coordinate transformation (SWCT) and appropriate regularization techniques to remove the infinite variance problem. The LRDMC forces were computed with the Reynolds (RE) and variational-drift (VD) approximations. The potential energy surfaces obtained from the LRDMC energies give equilibrium bond lengths (req) and harmonic frequencies (?) very close to the experimental values for all dimers, improving the corresponding VMC results. The LRDMC forces with the RE approximation improve the VMC forces, implying that it is worth computing the DMC forces beyond VMC despite the higher computational cost. The LRDMC forces with the VD approximations also show improvement, which unfortunately comes at a much higher computational cost in all-electron calculations. We find that the ratio of computational costs between QMC energy and forces scales as Z?2.5 without the SWCT. In contrast, the application of the SWCT makes the ratio independent of Z. As such, the accessible QMC system size is not affected by the evaluation of ionic forces but governed by the same scaling as the total energy one.

Erratum: Towards the solution of the many-electron problem in real materials: Equation of state of the hydrogen chain with state-of-the-art many-body methods (Physical Review X (2017) 7 (031059) DOI: 10.1103/PhysRevX.7.031059)

Motta M., Ceperley D.M., Chan G.K.L., Gomez J.A., Gull E., Guo S., Jiménez-Hoyos C.A., Lan T.N., Li J., Ma F., Millis A.J., Prokof'Ev N.V., Ray U., Scuseria G.E., This paper was published online on 28 September 2017 with a typographical error in a Grant number in the Acknowledgments. In the Acknowledgments, the third to last sentence should read as “S.?R.?W. and E.?M.?S. acknowledge support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Grant No. DE-SC0008696.” The Acknowledgments have been corrected as of 24 March 2021. The Acknowledgments are incorrect in the printed version of the journal.

Atomic forces by quantum Monte Carlo: Application to phonon dispersion calculations

Nakano K., Morresi T., Casula M., Maezono R., We report a successful application of the ab initio quantum Monte Carlo (QMC) framework to a phonon dispersion calculation. A full phonon dispersion of diamond is successfully calculated at the variational Monte Carlo (VMC) level, based on the frozen-phonon technique. The VMC-phonon dispersion is in good agreement with the experimental results, giving renormalized harmonic optical frequencies very close to the experimental values, and improving upon previous density functional theory estimates. Key to success for the QMC approach is the statistical error reduction in the atomic force evaluation. We show that this can be achieved by using well conditioned atomic basis sets and by explicitly removing the basis-set redundancy, which reduces the statistical error of forces by up to two orders of magnitude by combining it with the so-called space-warp transformation algorithm. This leads to affordable and accurate QMC-phonons calculations, which are up to 104 times more efficient than a bare force treatment, and paves the way to new applications, particularly in correlated materials, where phonons have been poorly reproduced so far.

Magnetism and Charge Order in the Honeycomb Lattice

Costa N.C., Seki K., Despite being relevant to better understand the properties of honeycomblike systems, as graphene-based compounds, the electron-phonon interaction is commonly disregarded in theoretical approaches. That is, the effects of phonon fields on interacting Dirac electrons is an open issue, in particular when investigating long-range ordering. Thus, here we perform unbiased quantum Monte Carlo simulations to examine the Hubbard-Holstein model (HHM) in the half-filled honeycomb lattice. By performing careful finite-size scaling analysis, we identify semimetal-to-insulator quantum critical points, and determine the behavior of the antiferromagnetic and charge-density wave phase transitions. We have, therefore, established the ground state phase diagram of the HHM for intermediate interaction strength, determining its behavior for different phonon frequencies. Our findings provide quantitative and qualitative descriptions of the model at intermediate coupling strengths, and may shed light on the emergence of many-body properties in honeycomblike systems.

Dirac electrons in the square-lattice Hubbard model with a d -wave pairing field: The chiral Heisenberg universality class revisited

Otsuka Y., Seki K., We numerically investigate the quantum criticality of the chiral Heisenberg universality class with the total number of fermion components N=8 in terms of the Gross-Neveu theory. Auxiliary-field quantum Monte Carlo simulations are performed for the square lattice Hubbard model in the presence of a d-wave pairing field, inducing Dirac cones in the single-particle spectrum. This property makes the model particularly interesting because it turns out to belong to the same universality class of the Hubbard model on the honeycomb lattice, which is the canonical model for graphene, despite the unit cells being apparently different (e.g., they contain one and two sites, respectively). We indeed show that the two phase transitions, expected to occur on the square and on the honeycomb lattices, have the same quantum criticality. We also argue that details of the models, i.e., the way of counting N and the anisotropy of the Dirac cones, do not change the critical exponents. The present estimates of the exponents for the N=8 chiral Heisenberg universality class are ν=1.05(5), ηφ=0.75(4), and ηψ=0.23(4), which are compared with the previous numerical estimations.

Phase diagram of the two-dimensional Hubbard-Holstein model

Costa N.C., Seki K., Yunoki S., The electron–electron and electron–phonon interactions play an important role in correlated materials, being key features for spin, charge and pair correlations. Thus, here we investigate their effects in strongly correlated systems by performing unbiased quantum Monte Carlo simulations in the square lattice Hubbard-Holstein model at half-filling. We study the competition and interplay between antiferromagnetism (AFM) and charge-density wave (CDW), establishing its very rich phase diagram. In the region between AFM and CDW phases, we have found an enhancement of superconducting pairing correlations, favouring (nonlocal) s-wave pairs. Our study sheds light over past inconsistencies in the literature, in particular the emergence of CDW in the pure Holstein model case.

The nature of the chemical bond in the dicarbon molecule

Genovese C., The molecular dissociation energy has often been explained and discussed in terms of singlet bonds, formed by bounded pairs of valence electrons. In this work, we use a highly correlated resonating valence bond ansatz, providing a consistent paradigm for the chemical bond, where spin fluctuations are shown to play a crucial role. Spin fluctuations are known to be important in magnetic systems and correspond to the zero point motion of the spin waves emerging from a magnetic broken symmetry state. Within our ansatz, a satisfactory description of the carbon dimer is determined by the magnetic interaction of two carbon atoms with antiferromagnetically ordered S = 1 magnetic moments. This is a first step that, thanks to the highly scalable and efficient quantum Monte Carlo techniques, may open the door for understanding challenging complex systems containing atoms with large spins (e.g., transition metals).

General Correlated Geminal Ansatz for Electronic Structure Calculations: Exploiting Pfaffians in Place of Determinants

Genovese C., Shirakawa T., Nakano K., We propose here a single Pfaffian correlated variational ansatz that dramatically improves the accuracy with respect to the single determinant one, while remaining at a similar computational cost. A much larger correlation energy is indeed determined by the most general two electron pairing function, including both singlet and triplet channels, combined with a many-body Jastrow factor, including all possible spin-spin, spin-density, and density-density terms. The main technical ingredient to exploit this accuracy is the use of the Pfaffian for antisymmetrizing a highly correlated pairing function, thus recovering the Fermi statistics for electrons with an affordable computational cost. Moreover, the application of the diffusion Monte Carlo, within the fixed node approximation, allows us to obtain very accurate binding energies for the first preliminary calculations reported in this study: C2, N2, and O2 and the benzene molecule. This is promising and remarkable, considering that they represent extremely difficult molecules even for computationally demanding multideterminant approaches, and opens therefore the way for realistic and accurate electronic simulations with an algorithm scaling at most as the fourth power of the number of electrons.

Ground-State Properties of the Hydrogen Chain: Dimerization, Insulator-to-Metal Transition, and Magnetic Phases

Motta M., Genovese C., Ma F., Cui Z.H., Sawaya R., Chan G.K.L., Chepiga N., Helms P., Jiménez-Hoyos C., Millis A.J., Ray U., Ronca E., Shi H., Accurate and predictive computations of the quantum-mechanical behavior of many interacting electrons in realistic atomic environments are critical for the theoretical design of materials with desired properties, and they require solving the grand-challenge problem of the many-electron Schrödinger equation. An infinite chain of equispaced hydrogen atoms is perhaps the simplest realistic model for a bulk material, embodying several central themes of modern condensed-matter physics and chemistry while retaining a connection to the paradigmatic Hubbard model. Here, we report a combined application of cutting-edge computational methods to determine the properties of the hydrogen chain in its quantum-mechanical ground state. Varying the separation between the nuclei leads to a rich phase diagram, including a Mott phase with quasi-long-range antiferromagnetic order, electron density dimerization with power-law correlations, an insulator-to-metal transition, and an intricate set of intertwined magnetic orders.

TurboRVB: A many-body toolkit for ab initio electronic simulations by quantum Monte Carlo

Nakano K., Attaccalite C., Barborini M., Capriotti L., Casula M., Coccia E., Dagrada M., Genovese C., Luo Y., Mazzola G., Zen A., TurboRVB is a computational package for ab initio Quantum Monte Carlo (QMC) simulations of both molecular and bulk electronic systems. The code implements two types of well established QMC algorithms: Variational Monte Carlo (VMC) and diffusion Monte Carlo in its robust and efficient lattice regularized variant. A key feature of the code is the possibility of using strongly correlated many-body wave functions (WFs), capable of describing several materials with very high accuracy, even when standard mean-field approaches [e.g., density functional theory (DFT)] fail. The electronic WF is obtained by applying a Jastrow factor, which takes into account dynamical correlations, to the most general mean-field ground state, written either as an antisymmetrized geminal power with spin-singlet pairing or as a Pfaffian, including both singlet and triplet correlations. This WF can be viewed as an efficient implementation of the so-called resonating valence bond (RVB) Ansatz, first proposed by Pauling and Anderson in quantum chemistry [L. Pauling, The Nature of the Chemical Bond (Cornell University Press, 1960)] and condensed matter physics [P.W. Anderson, Mat. Res. Bull 8, 153 (1973)], respectively. The RVB Ansatz implemented in TurboRVB has a large variational freedom, including the Jastrow correlated Slater determinant as its simplest, but nontrivial case. Moreover, it has the remarkable advantage of remaining with an affordable computational cost, proportional to the one spent for the evaluation of a single Slater determinant. Therefore, its application to large systems is computationally feasible. The WF is expanded in a localized basis set. Several basis set functions are implemented, such as Gaussian, Slater, and mixed types, with no restriction on the choice of their contraction. The code implements the adjoint algorithmic differentiation that enables a very efficient evaluation of energy derivatives, comprising the ionic forces. Thus, one can perform structural optimizations and molecular dynamics in the canonical NVT ensemble at the VMC level. For the electronic part, a full WF optimization (Jastrow and antisymmetric parts together) is made possible, thanks to state-of-the-art stochastic algorithms for energy minimization. In the optimization procedure, the first guess can be obtained at the mean-field level by a built-in DFT driver. The code has been efficiently parallelized by using a hybrid MPI-OpenMP protocol, which is also an ideal environment for exploiting the computational power of modern Graphics Processing Unit accelerators.