Quantum theory has successfully explained the mechanics of much of the microscopic world.However, Schrödinger's equations are difficult to solve for many-particle systems.Meanfield theories such as Hartree-Fock and density functional theory account for much of the total energy of electronic systems but fail on the crucial correlation energy that predicts solid cohesion and material properties.Monte Carlo methods solve differential and integral equations with error independent of the number of dimensions in the problem.Variational Monte Carlo (VMC) applies the variational principle to optimize the wave function used in the Monte Carlo integration of Schrödinger's time-independent equation.Diffusion Monte Carlo (DMC) represents the wave function by electron configurations diffusing stochastically in imaginary time to the ground state.Approximations in VMC and DMC make the problem tractable but introduce error in parameter-controlled and uncontrolled ways.The many-electron wave function consists of single-particle orbitals.The orbitals are combined in a functional form to account for electron exchange and correlation.Plane waves are a convenient basis for the orbitals.However, plane-wave orbitals grow in evaluation cost with basis-set completeness and system size.To speed up the calculation, polynomials approximate the plane-wave sum.Four polynomial methods tested are: Lagrange interpolation, pp-spline interpolation, B-spline interpolation and B-spline approximation.The polynomials all increase speed by an order of the number of particles.B-spline approximation most consistently maintains accuracy in the seven systems tested.However, polynomials increase the memory needed by a factor of two to eight.B-spline approximation with a separate approximation for the Laplacian of List of TablesTable Page 5.1 DMC number of steps vs. statistical uncertainty . . . . . . . . . . . . . . .5.2 Percent VMC time evaluating plane waves in Si . . . . . . . . . . . . . . . .5.3 Orbital, energy and fluctuation error in three metals . . . . . . . . . . . . .5.4 Orbital, energy and fluctuation error in two semiconductors . . . . . . . . .5.5 Orbital, energy and fluctuation error in two insulators . . . . . . . . . . . .6.1 DFT, VMC and DMC total & formation energies vs. plane-wave cutoff . .6.2 DMC energy vs. configuration population . . . . . . . . . . . . . . . . . . .6.3 DMC energy vs. time step . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4 DMC total and formation energies at L-and Baldereschi points . . . . . . .6.5 DMC total and formation energies with finite-size corrections . . . . . . . .6.6 DMC total and formation energies vs. number of atoms . . . . . . . . . . .6.7 DMC total and formation energies with different pseudopotential methods .6.8 DFT & DMC total energies with different exchange-correlation potentials .6.9 DMC formation energies for varying Jastrow factors and pseudopotentials .6.10 DMC error summary . . . . . . . . . . . . . . .
Quantum theory has successfully explained the mechanics of much of the microscopic world.However, Schrödinger's equations are difficult to solve for many-particle systems.Meanfield theories such as Hartree-Fock and density functional theory account for much of the total energy of electronic systems but fail on the crucial correlation energy that predicts solid cohesion and material properties.Monte Carlo methods solve differential and integral equations with error independent of the number of dimensions in the problem.Variational Monte Carlo (VMC) applies the variational principle to optimize the wave function used in the Monte Carlo integration of Schrödinger's time-independent equation.Diffusion Monte Carlo (DMC) represents the wave function by electron configurations diffusing stochastically in imaginary time to the ground state.Approximations in VMC and DMC make the problem tractable but introduce error in parameter-controlled and uncontrolled ways.The many-electron wave function consists of single-particle orbitals.The orbitals are combined in a functional form to account for electron exchange and correlation.Plane waves are a convenient basis for the orbitals.However, plane-wave orbitals grow in evaluation cost with basis-set completeness and system size.To speed up the calculation, polynomials approximate the plane-wave sum.Four polynomial methods tested are: Lagrange interpolation, pp-spline interpolation, B-spline interpolation and B-spline approximation.The polynomials all increase speed by an order of the number of particles.B-spline approximation most consistently maintains accuracy in the seven systems tested.However, polynomials increase the memory needed by a factor of two to eight.B-spline approximation with a separate approximation for the Laplacian of List of TablesTable Page 5.1 DMC number of steps vs. statistical uncertainty . . . . . . . . . . . . . . .5.2 Percent VMC time evaluating plane waves in Si . . . . . . . . . . . . . . . .5.3 Orbital, energy and fluctuation error in three metals . . . . . . . . . . . . .5.4 Orbital, energy and fluctuation error in two semiconductors . . . . . . . . .5.5 Orbital, energy and fluctuation error in two insulators . . . . . . . . . . . .6.1 DFT, VMC and DMC total & formation energies vs. plane-wave cutoff . .6.2 DMC energy vs. configuration population . . . . . . . . . . . . . . . . . . .6.3 DMC energy vs. time step . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4 DMC total and formation energies at L-and Baldereschi points . . . . . . .6.5 DMC total and formation energies with finite-size corrections . . . . . . . .6.6 DMC total and formation energies vs. number of atoms . . . . . . . . . . .6.7 DMC total and formation energies with different pseudopotential methods .6.8 DFT & DMC total energies with different exchange-correlation potentials .6.9 DMC formation energies for varying Jastrow factors and pseudopotentials .6.10 DMC error summary . . . . . . . . . . . . . . .
Variational and diffusion Monte Carlo (VMC and DMC) are widely used techniques in the Monte Carlo study of many-body quantum systems. Through efficient use of ran- dom sampling to solve many-dimensional integrals, they can be used to reliably solve the Schrödinger equation. Within the wider ecosystems of electronic structure calculation, quantum chemistry, and quantum simulation, quantum Monte Carlo (QMC) methods are used in tandem with density functional theory (DFT) and other computational methods to deliver insight into a vast array of condensed matter systems. Of particular interest to this work are systems in the intermediate correlation regime, where methods such as simple band theory and the Hubbard model break down. This thesis proposes and tests an explicit three-body backflow term to improve the accuracy of VMC and DMC calculations by incorporating higher-order correlation in order to reduce fixed-node er- ror in DMC, finding statistically significant improvements in ground state energies and improved variance in a water molecule. Then, QMC is used to calculate accurate charge densities for weak and intermediately correlated materials, to test commonly used dens- ity functional approximations in a controlled environment, providing insight into where these approximations perform well, and where they fail. Finally, DMC is used to generate ground state energy data for a dipolar Fermi gas of dressed Rydberg atoms interacting in the intermediate correlation regime. This data is parametrised in terms of interaction strength, for use in generating exchange-correlation functionals, to enable the DFT study of inhomogeneous systems of dressed Rydberg atoms.
Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).
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.
Two distinct types of Quantum Monte Carlo (QMC) calculations are applied to electronic structure problems such as calculating potential energy curves and producing benchmark values for reaction barriers. First, Variational and Diffusion Monte Carlo (VMC and DMC) methods using a trial wavefunction subject to the fixed node approximation were tested using the CASINO code.[1] Next, Full Configuration Interaction Quantum Monte Carlo (FCIQMC), along with its initiator extension (i-FCIQMC) were tested using the NECI code.[2] FCIQMC seeks the FCI energy for a specific basis set. At a reduced cost, the efficient i-FCIQMC method can be applied to systems in which the standard FCIQMC approach proves to be too costly. Since all of these methods are statistical approaches, uncertainties (error-bars) are introduced for each calculated energy. This study tests the performance of the methods relative to traditional quantum chemistry for some benchmark systems. References: [1] R. J. Needs et al., J. Phys.: Condensed Matter 22, 023201 (2010). [2] G. H. Booth et al., J. Chem. Phys. 131, 054106 (2009).
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
Since the form of PN is the same as that occurring in the statistical mechanics of the classical gas (replace f 2 ( r ij ) by exp[- V ( r ij /kT ]), we can use the same integration techniques that are used in the classical problem. W. L. McMillan, 1965 Summary Building on the random walk methods developed in the previous chapter, we show how to compute properties of many-body trial wavefunctions using a random walk. This method, called variational Monte Carlo, is the simplest stochastic quantum many-body technique. Whereas mean-field methods are usually limited to single determinants, variational Monte Carlo can treat any correlated trial function, as long as its values are computable. We discuss how to optimize such trial wavefunctions, how to compute their momentum distribution, how to use non-local pseudopotentials, how to compute excited states, and how to correct for the finite size of the simulation cell. Deterministic quantum methods have difficulties. For example, the Hartree–Fock method assumes the wavefunction is a single Slater determinant, neglecting correlation. If one expands as a sum of determinants, it is very difficult to have the results size-consistent since the number of determinants needed will grow exponentially with the system size. As we have seen, the DMFT method introduced in Ch. 16 assumes locality. In Ch. 6 we discussed general properties of many-body wavefunctions. Using Monte Carlo methods, we can directly incorporate correlation into a wavefunction, without having to make any further approximations other than the form of the correlation factors. In many cases the energy and other properties are very close to the exact results. Some of the usual restrictions on the form of the many-body wavefunction are not an issue in variational Monte Carlo. The most important generalization of the HF wavefunction is to put correlation directly into the wavefunction via the “Jastrow” factor. At next order, one can use the “backflow” wavefunction, in which correlation is also built into the determinant. The variational Monte Carlo method (VMC) was first used by McMillan [44] to calculate the ground-state properties of superfluid 4 He. One of the key problems at that time was whether the observed superfluid properties were a consequence of Bose condensation.