The quantum Monte Carlo (QMC) is one of the most promising many-body electronic structure approaches. It employs stochastic techniques for solving the stationary Schr\" odinger equation and for evaluation of expectation values. The key advantage of QMC is its capability to use the explicitly correlated wave functions, which allow the study of many-body effects beyond the reach of mean-field methods. The most important limit on QMC accuracy is the fixed-node approximation, which comes from necessity to circumvent the fermion sign problem. The size of resulting fixed-node errors depends on the quality of the nodes (the subset of position space where the wave function vanishes) of a used wave function. In this dissertation, we analyze the nodal properties of the existing fermionic wave functions and offer new types of variational wave functions with improved nodal structure.
The quantum Monte Carlo (QMC) is one of the most promising many-body electronic structure approaches. It employs stochastic techniques for solving the stationary Schr" odinger equation and for evaluation of expectation values. The key advantage of QMC is its capability to use the explicitly correlated wave functions, which allow the study of many-body effects beyond the reach of mean-field methods. The most important limit on QMC accuracy is the fixed-node approximation, which comes from necessity to circumvent the fermion sign problem. The size of resulting fixed-node errors depends on the quality of the nodes (the subset of position space where the wave function vanishes) of a used wave function. In this dissertation, we analyze the nodal properties of the existing fermionic wave functions and offer new types of variational wave functions with improved nodal structure.
Electronic structure quantum Monte Carlo Quantum Monte Carlo (QMC) is an advanced simulation methodology for studies of manybody quantum systems. The QMC approaches combine analytical insights with stochastic computational techniques for efficient solution of several classes of important many-body problems such as the stationary Schrödinger equation. QMC methods of various flavors have been applied to a great variety of systems spanning continuous and lattice quantum models, molecular and condensed systems, BEC-BCS ultracold condensates, nuclei, etc. In this review, we focus on the electronic structure QMC, i.e., methods relevant for systems described by the electron-ion Hamiltonians. Some of the key QMC achievements include direct treatment of electron correlation, accuracy in predicting energy differences and favorable scaling in the system size. Calculations of atoms, molecules, clusters and solids have demonstrated QMC applicability to real systems with hundreds of electrons while providing 90-95% of the correlation energy and energy differences typically within a few percent of experiments. Advances in accuracy beyond these limits are hampered by the so-called fixed-node approximation which is used to circumvent the notorious fermion sign problem. Many-body nodes of fermion states and their properties have therefore become one of the important topics for further progress in predictive power and efficiency of QMC calculations. Some of our recent results on the wave function nodes and related nodal domain topologies will be briefly reviewed. This includes analysis of few-electron systems and descriptions of exact and approximate nodes using transformations and projections of the highly-dimensional nodal hypersurfaces into the 3D space. Studies of fermion nodes offer new insights into topological properties of eigenstates such as explicit demonstrations that generic fermionic ground states exhibit the minimal number of two nodal domains. Recently proposed trial wave functions based on Pfaffians with pairing orbitals are presented and their nodal properties are tested in calculations of first row atoms and molecules. Finally, backflow "dressed" coordinates are introduced as another possibility for capturing correlation effects and for decreasing the fixed-node bias.
Quantum Monte Carlo (QMC) is one of the most promising methods for solving quantum many-body problems. QMC combines known analytical properties of wave functions, results from other methods, and stochastic techniques, into a powerful tool for investigation of the electronic structure of real materials. Furthermore, the method has a favorable scaling ( O[N3] in the number of electrons), perfect scalability on parallel architectures, and can be applied to a wide range of systems. Our work has focused both on expanding the capabilities of QMC for calculation of quantities beyond energies, such as interatomic forces, and on application of the method to challenging problems in nanostructure materials research. For the calculation of interatomic forces we have implemented a finite difference correlated sampling method. The correlated sampling enables us to avoid the problem of statistical noise in evaluating the energy differences and allows us to obtain forces with an average 1% error. We have analyzed the effect of the method's main approximations on the estimate of the forces, both by theoretical arguments and by performing a variety of tests on small systems. Furthermore, we have used this newly developed method for estimating equilibrium geometries in a problem where other methods have failed. QMC enabled us to treat the electron correlation effects with high accuracy and to provide accurate predictions in a number of interesting applications. In particular, for carbon rings of C4N+2 stoichiometry we have resolved a long-standing problem regarding the competition between the aromaticity and second order Jahn-Teller effects. We have found that the dominant mechanism at small and intermediate sizes, which stabilizes the bond angle and bond length alternated geometries, is the secondorder Jahn-Teller effect while the aromatic isomer is always found to be a transition state. These high accuracy calculations which have involved multi-reference wave functions, extensive variational optimizations, and the use of massively parallel platforms demonstrate clearly the unique capabilities of the QMC approach.
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.
Simulating strongly correlated fermionic systems remains a fundamental challenge in quantum physics, largely due to the sign problem in quantum Monte Carlo (QMC) methods. We present a neural network-based variational Monte Carlo (NN-VMC) approach, leveraging a flexible neural network ansatz to represent the many-body wavefunction. Focusing on quantum dots with up to 30 electrons, we demonstrate that NN-VMC significantly reduces variational bias and achieves ground-state energies surpassing those of fixed-node diffusion Monte Carlo (DMC). A key feature is that the neural network adaptively learns and optimizes nodal structures during energy minimization. We provide qualitative insights into the nodal structure of fermionic wavefunctions by comparing the nodal structures generated by NN-VMC with those obtained from traditional trial functions. Additionally, we reveal spin-resolved radial distributions and electron density profiles, highlighting the versatility and accuracy of NN-VMC. This work underscores the potential of machine learning to advance quantum simulations and deepen our understanding of strongly correlated systems.
Describing correlated electron systems has been a major challenge in computational condensed-matter physics. Quantum Monte Carlo, a powerful computational tool for the study of correlated systems, solves electron correlation problems explicitly. It has been taken as a benchmark method for understanding the correlated systems. Instead of making approximations to Hamiltonian, QMC methods work with the wave functions, and the computational cost scales well with the system size. With the development of parallel computing, QMC calculations on large systems are becoming more and more feasible. We have investigated two correlated systems with highly accurate fixed node QMC techniques. The first system is a correlated hydrogen model system near the metal to insulator transition. We have successfully identified the transition point by calculating spin and charge properties and analyzing the low energy Hilbert space. The second one is a strongly correlated Fe/O system. Calculations on the Fe atoms, O atoms, and FeO molecules are conducted with multiple highly accurate many-body techniques. The source of errors has been disentangled by comparing the results of the many body techniques with the experimental results. For the Fe and O atoms, the calculated properties coincide well with previous experimental results. For the basis-based techniques, the performance is mainly limited by the basis set. The calculated equilibrium bond length, excitation energy and vibrational frequency of the FeO molecules are also in close agreement with the known values from previous experiments.
The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0 of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0 as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψI which hopefully approximates some ground state ψ0 of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψI coincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψI differ from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.