A brief introduction to the diffusion Monte Carlo method and the fixed-node approximation.

Electronic structure calculations by quantum Monte Carlo methods

We have calculated the ground state energy of the hydrogen molecule using the quantum Monte Carlo (QMC) method of solving the Schrödinger equation, without the use of the Born–Oppenheimer or any other adiabatic approximations. The wave function sampling was carried out in the full 12-dimensional configuration space of the four particles (two electrons and two protons). Two different methods were employed: the diffusion quantum Monte Carlo (DQMC) method and the Green’s function quantum Monte Carlo (GFQMC) method. This computation is very demanding because the configurations must be evolved on the time scale of the electronic motion, whereas the finite nuclear mass effects are resolved accurately only after equilibration on the much slower time scale of the nuclear motion. Thus, a very large number of iterations is required. The calculations were performed on the CM-2 Connection Machine computer, a massively parallel supercomputer. The enormous speedup afforded by the massive parallelism allowed us to complete the computation in a reasonable amount of time. The total energy from the DQMC calculations is −1.163 97±0.000 05 a.u. A more accurate result was obtained from the GFQMC calculations of −1.164 024±0.000 009 a.u. Expressed as a dissociation energy, the GFQMC result is 36 117.9±2.0 cm−1, including the corrections for relativistic and radiative effects. This result is in close agreement with accurate nonadiabatic-relativistic dissociation energies from variational calculations (corrected for radiative effects) in the range of 36 117.9–36 118.1 cm−1 and with the best experimentally determined dissociation energy of McCormack and Eyler 36 118.1±0.2 cm−1.

A benchmark study of atomization energies is reported for 22 hydrocarbons using single determinant trial functions in the diffusion Monte Carlo (DMC) variant of the quantum Monte Carlo (QMC) method. The DMC atomization energies are compared to experiment, a complete basis set approach (CBS-Q), density functional theory with the B3LYP functional, and coupled-cluster singles, doubles and perturbative triples, CCSD(T), methods. Comparison of the DMC results to experiment yields a mean absolute deviation of 1.9kcalmol−1, which is comparable to that of the B3LYP/cc-pVQZ (1.7kcalmol−1) level of theory, but less accurate than that of CBS-Q (1.1kcalmol−1). DMC performs similarly for both closed-shell and open-shell molecules with mean absolute deviations of 2.1kcalmol−1 for the former and 1.7kcalmol−1 for the latter systems. The use of experimental zero-point energies (ZPEs), rather than scaled B3LYP ZPEs, is found to have negligible effect on DMC atomization energies. The latter reported here provide a baseline from which further improvement in the calculation of DMC atomization energies, including the use of multi-determinant and other trial function improvements, can be measured.

We report on the results of both the diffusion quantum Monte Carlo (DMC) and reptation quantum Monte Carlo (RMC) methods on the potential energy curve of the helium dimer. We show that it is possible to obtain a highly accurate description of the helium dimer. An improved stochastic reconfiguration technique is employed to optimize the many-body wave function, which is the starting point for highly accurate simulations based on the DMC and RMC methods. We find that the results of these methods are in excellent agreement with the best theoretical results at short range, especially the recently developed RMC method, yield particularly accurate results with reduced statistical error, which gives very excellent agreement across the whole potential curve. For the equilibrium internuclear distance of 5.6 bohrs, the calculated total energy with RMC method is -5.807 483 599+/-0.000 000 016 hartree and the corresponding well depth is -11.003+/-0.005 K.

The applicability of Green's function (GF) and Feynman path-integral quantum Monte Carlo (QMC) methods for the simulation of cyclic networks with (4n + 2) and 4n (n = 1, 2, 3, …) electrons is analysed. Both QMC techniques are employed in simulations on the basis of the simple Huckel Hamiltonian which is exclusively defined by nearest-neighbour hopping elements. In addition we have used the Pariser-Parr-Pople (PPP) Hamiltonian to perform GF QMC simulations. The electronic energies E derived by the QMC methods are compared either with Huckel molecular orbital (HMO) results or exact configuration interaction data where (π) electronic correlations are fully taken into account. A sign problem occurs in QMC simulations of 4n annulenes. This leads to an error in the total energy in the standard formulations of the employed QMC techniques, which is enhanced with decreasing ring size. A simple modification in the QMC formalisms is suggested to avoid the numerical uncertainties caused by the sign problem in 4n annu...

This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter. In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations. In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter. In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition. In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector. In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.

A quantum Monte Carlo (QMC) benchmark study of heats of formation at 298 K and bond dissociation energies (BDEs) of 22 small hydrocarbons is reported. Diffusion Monte Carlo (DMC) results, obtained using a simple product trial wavefunctions consisting of a single determinant and correlation function, are compared to experiment and to other theory including a version of complete basis set theory (CBS‐Q) and density functional theory (DFT) with the B3LYP functional. For heats of formation, the findings are a mean absolute deviation from experiment of 1.2 kcal/mol for CBS‐Q, 2.0 kcal/mol for B3LYP, and 2.2 kcal/mol for DMC. The mean absolute deviation of 31 BDEs is 2.0 kcal/mol for CBS‐Q, 4.2 kcal/mol for B3LYP, and 2.5 kcal/mol for DMC. These findings are for 17 BDEs of closed‐shell molecules that have mean absolute deviations from experiment of 1.7 kcal/mol (CBS‐Q), 4.0 kcal/mol (B3LYP), and 2.2 kcal/mol (DMC). The corresponding results for the 14 BDEs of open‐shell molecules studied are 2.4 kcal/mol (CBS‐Q), 4.3 kcal/mol (B3LYP), and 2.9 kcal/mol (DMC). The DMC results provide a baseline from which improvement using multideterminant trial functions can be measured. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 583–592, 2005

Pseudopotential locality errors have hampered the applications of the diffusion Monte Carlo (DMC) method in materials containing transition metals, in particular oxides. We have developed locality error free effective core potentials, pseudo-Hamiltonians, for transition metals ranging from Cr to Zn. We have modified a procedure published by some of us in Bennett et al. [J. Chem. Theory Comput. 18, 828 (2022)]. We carefully optimized our pseudo-Hamiltonians and achieved transferability errors comparable to the best semilocal pseudopotentials used with DMC but without incurring in locality errors. Our pseudo-Hamiltonian set (named OPH23) bears the potential to significantly improve the accuracy of many-body-first-principles calculations in fundamental science research of complex materials involving transition metals.

A simple scheme is described for introducing the correct cusps at nuclei into orbitals obtained from Gaussian basis set electronic structure calculations. The scheme is tested with all-electron variational quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) methods for the Ne atom, the H2 molecule, and 55 molecules from a standard benchmark set. It greatly reduces the variance of the local energy in all cases and slightly improves the variational energy. This scheme yields a general improvement in the efficiency of all-electron VMC and DMC calculations using Gaussian basis sets.

We report a study of the electronic dissociation energy of the water dimer using quantum Monte Carlo techniques. We have performed variational quantum Monte Carlo and diffusion quantum Monte Carlo (DMC) calculations of the electronic ground state of the water monomer and dimer using all-electron and pseudopotential approaches. We have used Slater-Jastrow trial wave functions with B3LYP type single-particle orbitals, into which we have incorporated backflow correlations. When backflow correlations are introduced, the total energy of the water monomer decreases by about 4-5 mhartree, yielding a DMC energy of -76.428 30(5) hartree, which is only 10 mhartree above the experimental value. In our pseudopotential DMC calculations, we have compared the total energies of the water monomer and dimer obtained using the locality approximation with those from the variational scheme recently proposed by Casula [Phys. Rev. B 74, 161102(R) (2006)]. The time step errors in the Casula scheme are larger, and the extrapolation of the energy to zero time step always lies above the result obtained with the locality approximation. However, the errors cancel when energy differences are taken, yielding electronic dissociation energies within error bars of each other. The dissociation energies obtained in our various all-electron and pseudopotential calculations range between 5.03(7) and 5.47(9) kcalmol and are in good agreement with experiment. Our calculations give monomer dipole moments which range between 1.897(2) and 1.909(4) D and dimer dipole moments which range between 2.628(6) and 2.672(5) D.

There has been significant recent attention surrounding the accuracy of electronic densities produced by modern parameterized density functional approximations (DFAs). Here, we investigate the impact of using orbitals from density functional calculations in fixed-node Diffusion Monte Carlo (DMC) methods, which is common practice in the calculation of large systems. We find that the accuracy of the density is a strong indicator of the quality of the many-body nodal surface produced by a determinant of the corresponding Kohn-Sham orbitals. Functionals which produce the most accurate electronic densities also produce the lowest variational DMC energies, while functionals that produce poor densities lead to significantly higher energies. This result simplifies the process of choosing orbitals for DMC calculations of large systems and suggests that prioritizing accurate densities in the future development of DFAs would also contribute to the continued improvement of DMC.

Quantum Monte Carlo (QMC) methods are the gold standard for studying equilibrium properties of quantum many-body systems. However, in many interesting situations, QMC methods are faced with a sign problem, causing the severe limitation of an exponential increase in the runtime of the QMC algorithm. In this work, we develop a systematic, generally applicable, and practically feasible methodology for easing the sign problem by efficiently computable basis changes and use it to rigorously assess the sign problem. Our framework introduces measures of non-stoquasticity that-as we demonstrate analytically and numerically-at the same time provide a practically relevant and efficiently computable figure of merit for the severity of the sign problem. Complementing this pragmatic mindset, we prove that easing the sign problem in terms of those measures is generally an NP-complete task for nearest-neighbor Hamiltonians and simple basis choices by a reduction to the MAXCUT-problem.

Quantum Monte Carlo (QMC) methods are often used to calculate properties of many body quantum systems. The main cost of many QMC methods, for example the variational Monte Carlo (VMC) method, is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators so that the cost of constructing Slater matrices in these systems is now linear in the number of particles, whereas computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing the determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. The main contribution of this paper is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of ILUTP preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of QMC algorithms from O(n^3) per sweep to roughly O(n^2), where n is the number of particles, and a sweep is a sequence of n steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors. Our results show that preconditioned iterative solvers can dramatically reduce the cost of VMC for large(r) systems.

We review some of the recent development in quantum Monte Carlo (QMC) methods for models of strongly correlated electron systems. QMC is a promising general theoretical tool to study many-body systems, and has been widely applied in areas spanning condensed-matter, high-energy, and nuclear physics. Recent progress has included two new methods, the ground-state and finite-temperature constrained path Monte Carlo methods. These methods significantly improve the capability of numerical approaches to lattice models of correlated electron systems. They allow calculations without any decay of the sign, making possible calculations for large system sizes and low temperatures. The methods are approximate. Benchmark calculations show that accurate results on energy and correlation functions can be obtained. This chapter gives a pedagogical introduction to quantum Monte Carlo, with a focus on the constrained path Monte Carlo methods.

The ground and lowest three adiabatic excited states of methylene are computed using the variational Monte Carlo and diffusion Monte Carlo (DMC) methods using progressively larger Jastrow-Slater multideterminant complete active space (CAS) wave functions. The highest of these states has the same symmetry, (1)A(1), as the first excited state. The DMC excitation energies obtained using any of the CAS wave functions are in excellent agreement with experiment, but single-determinant wave functions do not yield accurate DMC energies of the states of (1)A(1) symmetry, indicating that it is important to include in the wave function Slater determinants that describe static (strong) correlation. Excitation energies obtained using recently proposed pseudopotentials [Burkatzki et al., J. Chem. Phys. 126, 234105 (2007)] differ from the all-electron excitation energies by at most 0.04 eV.