Ab initio quantum Monte Carlo (QMC) methods are state-of-the-art electronic structure calculations based on highly parallelizable stochastic frameworks for accurate solutions of the many-body Schrödinger equation, suitable for modern many-core supercomputer architectures. Despite its potential, one of the major drawbacks that still hinders QMC applications, especially when targeting dynamical properties of large systems or extensive datasets, is the lack of an affordable method to compute atomic forces that are consistent with the corresponding potential energy surfaces (PESs), also known as unbiased atomic forces. Recently, one of the authors in the present paper proposed a way to obtain unbiased forces with the Jastrow-correlated Slater determinant Ansatz, where the determinant part is frozen to the values obtained by a mean-field method, such as density functional theory [K. Nakano, M. Casula, and G. Tenti, Phys. Rev. B 109, 205151 (2024)]. However, the proposed method has a significant drawback for its applications: for a system with N nuclei, one requires 6N additional density functional theory (DFT) calculations to get unbiased forces, which is not negligible as the system size increases. This paper presents a way to replace the 6N DFT calculations with a single coupled-perturbed Kohn-Sham calculation, following the so-called Lagrangian technique established in quantum chemistry. This improves the computational cost and scalability of the method. We also demonstrate that the developed unbiased variational Monte Carlo (VMC) force calculation improves not only the consistency with PESs but also its accuracy, by investigating three molecules from the rMD17 benchmark set, and comparing the unbiased VMC forces with those obtained by the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] calculations. We found that the bare VMC forces are biased from the CCSD(T) ones, while the unbiased ones give values closer to those of the CCSD(T) ones. Our benchmark test also reveals that the unbiased VMC forces yield very consistent values with hybrid and meta generalized gradient approximations (e.g., ωB97X-D3BJ and ωB97M-D3BJ), but do not necessarily yield values that are very close to those of CCSD(T). Our finding paves the way to generate machine learning interatomic potentials based on VMC forces more efficiently and accurately.
Ab initio quantum Monte Carlo (QMC) methods are state-of-the-art electronic structure calculations based on highly parallelizable stochastic frameworks for accurate solutions of the many-body Schrödinger equation, suitable for modern many-core supercomputer architectures. Despite its potential, one of the major drawbacks that still hinders QMC applications, especially when targeting dynamical properties of large systems or extensive datasets, is the lack of an affordable method to compute atomic forces that are consistent with the corresponding potential energy surfaces (PESs), also known as unbiased atomic forces. Recently, one of the authors in the present paper proposed a way to obtain unbiased forces with the Jastrow-correlated Slater determinant Ansatz, where the determinant part is frozen to the values obtained by a mean-field method, such as density functional theory [K. Nakano, M. Casula, and G. Tenti, Phys. Rev. B 109, 205151 (2024)]. However, the proposed method has a significant drawback for its applications: for a system with N nuclei, one requires 6N additional density functional theory (DFT) calculations to get unbiased forces, which is not negligible as the system size increases. This paper presents a way to replace the 6N DFT calculations with a single coupled-perturbed Kohn-Sham calculation, following the so-called Lagrangian technique established in quantum chemistry. This improves the computational cost and scalability of the method. We also demonstrate that the developed unbiased variational Monte Carlo (VMC) force calculation improves not only the consistency with PESs but also its accuracy, by investigating three molecules from the rMD17 benchmark set, and comparing the unbiased VMC forces with those obtained by the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] calculations. We found that the bare VMC forces are biased from the CCSD(T) ones, while the unbiased ones give values closer to those of the CCSD(T) ones. Our benchmark test also reveals that the unbiased VMC forces yield very consistent values with hybrid and meta generalized gradient approximations (e.g., ωB97X-D3BJ and ωB97M-D3BJ), but do not necessarily yield values that are very close to those of CCSD(T). Our finding paves the way to generate machine learning interatomic potentials based on VMC forces more efficiently and accurately.
quantum Monte Carlo (QMC) is a state-of-the-art numerical approach for evaluating accurate expectation values of many-body wave functions. However, one of the major drawbacks that still hinders widespread QMC applications is the lack of an affordable scheme to compute unbiased atomic forces. In this study, we propose an efficient method to obtain unbiased atomic forces and pressures in the variational Monte Carlo (VMC) framework with the Jastrow-correlated Slater determinant ansatz or the Jastrow antisymmetrized geminal power ansatz, exploiting the gauge-invariant and locality properties of their geminal representation. We demonstrate the effectiveness of our method for H 2 and Cl 2 molecules and for the cubic boron nitride crystal. Our framework has a better algorithmic scaling with the system size than the traditional finite-difference method and, in practical applications, is as efficient as single-point VMC calculations. Thus, it paves the way to study dynamical properties of materials, such as phonons, and is beneficial for pursuing more reliable machine-learning interatomic potentials based on unbiased VMC forces. Published by the American Physical Society 2024
to develop quantum Monte Carlo (QMC) approaches for excited states and for more systematic and accurate treatment of ground states of strongly correlated sytems and heavier elements. We use the auxiliary-field quantum Monte Carlo (AFQMC) framework, which is a many-body total energy method that looks like many coupled mean-field calculations. I will discuss this method’s relations and dierences with traditional density-functional and quantum chemistry methods, and with other forms of QMC. The AFQMC method uses importance-sampled random walks in the space of nonorthogonal Slater determinants to project out the many-body ground state, or excited state. The random walks are constrained by a trial wave function (TWF) to remove the so-called sign problem, giving approximate (but non-perturbative) results. With singledeterminant TWFs from Hartree-Fock or density-functional theory (DFT), this approach can be thought of as a post-DFT method with no parameter. In this form, the accuracy is typically comparable to the quantum chemistry coupled-cluster CCSD(T) method near equilibrium geometries, and better than CCSD(T) when bonds are stretched. The computational cost of AFQMC scales as N 3 with characteristic size of the system. Its accuracy can be further improved at the cost of increased computing (releasing the constraint) or better TWF. With AFQMC, we can systematically target selected orbitals to correlate. This allows many-body calculations in solids with a frozen core, with down-folded Hamiltonians, or possibly embeded in a DFT calculations to extend scales. Results from a few on-going applications will be presented, including band structures in simple solids, magnetic phases in MnO, and Cobalt adsoption on graphene.
Quantum Monte Carlo (QMC) methods have received considerable attention over the last decades due to their great promise for providing a direct solution to the many-body Schrodinger equation in electronic systems.Thanks to their low scaling with number of particles, QMC methods present a compelling competitive alternative for the accurate study of large molecular systems and solid state calculations.In spite of such promise, the method has not permeated the quantum chemistry community broadly, mainly because of the fixed-node error, which can be large and whose control is difficult.In this Perspective, we present a systematic application of large scale multi-determinant expansions in QMC, and report on its impressive performance with first row dimers and the 55 molecules of the G1 test set.We demonstrate the potential of this strategy for systematically reducing the fixed-node error in the wave function and for achieving chemical accuracy in energy predictions.When compared to traditional quantum chemistry methods like MP2, CCSD(T), and various DFT approximations, the QMC results show a marked improvement over all of them.In fact, only the explicitly-correlated CCSD(T) method with a large basis set produces more accurate results.Further developments in trial wave functions and algorithmic improvements appear promising for rendering QMC as the benchmark standard in large electronic systems.
Solutions to the Schrodinger equation can be used to predict the electronic structure of molecules and materials and therefore infer their complex physical and chemical properties. Variational Quantum Monte Carlo (VMC) is a technique that can be used to solve the weak form of the Schrodinger equation. Applying VMC to systems with N electrons involves evaluating the determinant of an N by N matrix. The evaluation of this determinant scales as O(N3) and is the main computational cost in the VMC process. In this work, we investigate an alternative VMC technique based on the Vandermonde determinant. The Vandermonde determinant is a product of pairwise differences and so evaluating it scales as O(N2). Therefore, this approach reduces the computational cost by a factor of N. The Vandermonde determinant was implemented in PyTorch and the performance was assessed in approximating the ground state energy of various quantum systems against existing techniques. The performance is evaluated in a variety of systems, starting with the one-dimensional particle in a box, and then considering more complicated atomic systems with multiple particles. The Vandermonde determinant was also implemented in PauliNet, a deep-learning architecture for VMC. The new method is shown to be computationally efficient, and results in a speed-up as large as 5X. In these cases, the new ansatz obtains a reasonable approximation for wavefunctions of atomic systems, but does not reach the accuracy of the Hartree-Fock method that relies on the Slater determinant. It is observed that while the use of neural networks in VMC can result in highly accurate solutions, further work is necessary to determine an appropriate balance between computational time and accuracy.
With the ability to solve the many-body Schrödinger equation accurately, in principle all physics and chemistry could be derived from first principles. However, exact wave functions of realistic and interesting systems are in general unavailable because they are non-deterministic polynomial-hard to compute (Troyer, 2005). This implies that we need to rely on approximations. The variational Monte Carlo (VMC) method is widely used for ground state studies, but requires a trial wave function ansatz which must trade off between efficiency and accuracy. The method also has many common features with machine learning algorithms, and as neural networks have shown impressive power as function approximators, the idea is to use a neural network as the trial wave function guess. For fermionic systems, like electronic structure systems, the wave function needs to obey Fermi-Dirac statistics. This is typically achieved using a Slater determinant. As a neural network hardly can model this feature, our approach is to replace the single-particle functions in the Slater determinant with restricted Boltzmann machines, getting the RBM ansatz. In addition, we add further correlations via so-called Jastrow factors (Drummond, 2004). The de facto standard trial wave function ansatz for electronic structure calculations is the Slater-Jastrow ansatz, which was implemented as a reference. Our primary focus is on closed-shell circular quantum dots, where we compute the ground state energy and electron density of two-dimensional systems with up to 90 electrons and three-dimensional systems with up to 70 electrons. The energy obtained by the RBM ansatz was reasonably close to experimental results, and it gradually became closer as we added more complex correlation factors. For our most complicated Jastrow factor, the energy was found to be lower than the energy provided by the Slater-Jastrow ansatz for small dots, but for larger dots it was slightly higher. However, the one-body density profile reveals that the RBM ansatz gives more distinctly located electrons compared to the Slater-Jastrow ansatz. This can be explained by the way the RBM ansatz models the correlations. From the two-body density profile, we also observe that the repulsive interactions get more significant as we add a Jastrow factor. Based on the electron densities and the energy distribution between kinetic and potential energy, it is certain that the various methods provide different electron configurations. For low-frequency dots, the electron density becomes more localized with an additional radial peak compared to high-frequency dots. This is reminiscent of what is known as Wigner localization (Ghosal, 2007). The computational time consumption was found to be favorable for the RBM ansatz for small systems and the Slater-Jastrow ansatz for large systems. This can be explained by the exploding number of variational parameters in the RBM ansatz as the system sizes increase. The RBM ansatz with Jastrow factors were notably more computationally intensive than the other ansätze, and evidently, there is no reason to use the simplest Jastrow factor when more complicated Jastrow factors are available.
Variational quantum Monte Carlo (VMC) method applied to interacting electron systems has played a pivotal role for the studies of electronic states of atoms, molecules and condensed matters. First, the many-body wave functions optimized by the VMC method provide good guiding functions in more accurate, diffusion quantum Monte Carlo (DMC) calculations for identical systems. Second, the VMC calculations themselves can give a good upper bound for the exact ground-state energy of many-electron systems when an appropriate form for the variational wave function is employed. In addition to a conventional form for the variational wave function that is expressed by the product of the Jastrow factor and the single Slater determinant, recent investigations have found that the expression in terms of many Slater determinants can provide a very accurate description of electronic states, especially for intrinsically multi-reference systems, which corresponds to the configuration interaction (CI) expansion method in conventional quantum chemistry. Although the CI method needs an extremely large number of expansion coefficients as variational parameters, the number of parameters can be reduced efficiently by using an idea of (molecular) orbital pair correlations. The purpose of the present paper is to examine the feasibility and usefulness of this idea within the framework of VMC method with the Jastrow–Slater wave function, where the ground state of a water molecule is investigated as a test case. In this study we rely on a variational wave function composed of a product of the Jastrow and Slater parts as 1⁄4 J S; ð1Þ