C. J. Umrigar

A diffusion Monte Carlo algorithm with very small time-step errors

We propose modifications to the simple diffusion Monte Carlo algorithm that greatly reduce the time‐step error. The improved algorithm has a time‐step error smaller by a factor of 70 to 300 in the energy of Be, Li2 and Ne. For other observables the improvement is yet larger. The effective time step possible with the improved algorithm is typically a factor of a few hundred larger than the time step used in domain Green function Monte Carlo. We also present an optimized 109 parameter trial wave function for Be which, used in combination with our algorithm, yields an exceedingly accurate ground state energy. A simple solution to the population control bias in diffusion Monte Carlo is also discussed.

Quantum Monte Carlo methods in physics and chemistry

Preface. 1. Basics, Quantum Monte Carlo and Statistical Mechanics M.P. Nightingale. 2. Stochastic Diagonalization H. de Raedt, et al. 3. World-Line Quantum Monte Carlo R.T. Scalettar. 4. Variational Monte Carlo in Solids S. Fahy. 5. Variational Monte Carlo Basics and Applications to Atoms and Molecules C.J. Umrigar. 6. Calculations of Exchange Frequencies with Path Integral Monte Carlo: Solid 3He Adsorbed on Graphite B. Bernu, D. Ceperley. 7. Static Response of Homogeneous Quantum Fluids by Diffusion Monte Carlo G. Senatore, et al. 8. Equilibrium and Dynamical Path Integral Methods: An Introduction J.D. Doll, et al. 9. Diffusion Monte Carlo L. Mitas. 10. Fermion Monte Carlo M.H. Kalos, F. Pederiva. 11. Quantum Monte Carlo in Nuclear Physics J. Carlson. 12. Reputation Quantum Monte Carlo: A Round-Trip Tour from Classical Diffusion to Quantum Mechanics S. Baroni, S. Moroni. 13. Quantum Monte Carlo for Lattice Fermions A. Muramatsu. 14. Phase Separation in the 2D Hubbard Model: A Challenging Application of Fixed-Node QMC G.B. Bachelet, A.C. Cosentini. 15. Constrained Path Monte Carlo for Fermions Shiwei Zhang. 16. Serial and Parallel Random Number Generation M. Mascagni. 17. Fixed-Node DMC for Fermions on a Lattice: Application to Doped Fullerides E. Koch, et al. 18. Index.

Multiconfiguration wave functions for quantum Monte Carlo calculations of first‐row diatomic molecules

We use the variance minimization method to determine accurate wave functions for first‐row homonuclear diatomic molecules. The form of the wave function is a product of a sum of determinants and a generalized Jastrow factor. One of the important features of the calculation is that we are including low‐lying determinants corresponding to single and double excitations from the Hartree–Fock configuration within the space of orbitals whose atomic principal quantum numbers do not exceed those occurring in the Hartree–Fock configuration. The idea is that near‐degeneracy correlation is most effectively described by a linear combination of low‐lying determinants whereas dynamic correlation is well described by the generalized Jastrow factor. All the parameters occurring in both the determinantal and the Jastrow parts of the wave function are optimized. The optimized wave functions recover 79%–94% of the correlation energy in variational Monte Carlo and 93%–99% of the correlation energy in diffusion Monte Carlo.

Natural Orbital Functional for the Many-Electron Problem

The solution of the quantum mechanical many-electron problem is one of the central problems of physics. A great number of schemes that approximate the intractable many-electron Schrodinger equation have been devised to attack this problem. Most of them map the manybody problem to a self-consistent one-particle problem. Probably the most popular method at present is density functional theory (DFT) [1] especially when employed with the generalized gradient approximation (GGA) [2,3] for the exchange-correlation energy. DFT is based on the Hohenberg-Kohn theorem [4] which asserts that the electronic charge density completely determines a manyelectron system and that, in particular, the total energy is a functional of the charge density. Attempts to construct such a functional for the total energy have not been very successful because of the strong nonlocality of the kinetic energy term. The Kohn-Sham scheme [5] where the main part of the kinetic energy, the single particle kinetic energy, is calculated by solving one-particle Schrodinger equations circumvented this problem. The difference between the one-particle kinetic energy and the many-body kinetic energy is a component of the unknown exchange-correlation functional. The exchangecorrelation functional is thus a sum of a kinetic energy contribution and a potential energy contribution, and partly for this reason it does not scale homogeneously [6] under a uniform spatial scaling of the charge density. It has been known for a long time that one can also construct a total energy functional using the firstorder reduced density matrix. Several discussions of the existence and the properties of such a functional can be found in the literature [7‐ 10]. However, no explicit functional has ever been constructed and tested on real physical systems. An important advantage of this approach is that one employs an exact expression for the many-body kinetic energy. Only the small non-HartreeFock-like part of the electronic repulsion is an unknown functional [9]. We propose in this paper an explicit form of such a functional in terms of the natural orbitals. The high accuracy of this natural orbital functional theory (NOFT) is then established by applying it to several atoms and ions. If C is an arbitrary trial wave function of an N-electron system, the first- and second-order reduced density matrices [11,12], g1 and g2, are

Optimization of quantum Monte Carlo wave functions by energy minimization

We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.

Full optimization of Jastrow–Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules

We pursue the development and application of the recently introduced linear optimization method for determining the optimal linear and nonlinear parameters of Jastrow-Slater wave functions in a variational Monte Carlo framework. In this approach, the optimal parameters are found iteratively by diagonalizing the Hamiltonian matrix in the space spanned by the wave function and its first-order derivatives, making use of a strong zero-variance principle. We extend the method to optimize the exponents of the basis functions, simultaneously with all the other parameters, namely, the Jastrow, configuration state function, and orbital parameters. We show that the linear optimization method can be thought of as a so-called augmented Hessian approach, which helps explain the robustness of the method and permits us to extend it to minimize a linear combination of the energy and the energy variance. We apply the linear optimization method to obtain the complete ground-state potential energy curve of the C(2) molecule up to the dissociation limit and discuss size consistency and broken spin-symmetry issues in quantum Monte Carlo calculations. We perform calculations for the first-row atoms and homonuclear diatomic molecules with fully optimized Jastrow-Slater wave functions, and we demonstrate that molecular well depths can be obtained with near chemical accuracy quite systematically at the diffusion Monte Carlo level for these systems.

Relationship of Kohn-Sham eigenvalues to excitation energies

In Kohn-Sham density functional theory, only the highest occupied eigenvalue has a rigorous physical meaning, viz., it is the negative of the lowest ionization energy. Here, we demonstrate that for finite systems, the unoccupied true Kohn-Sham eigenvalues las opposed to the those obtained from the commonly used approximate density functionals) are also meaningful in that good approximations to excitation energies can be obtained from them. We argue that the explanation for this observed behavior is that, at large distances, the Kohn-Sham orbitals and the quasiparticle amplitudes satisfy the same equation to order 1/r(4)

A critical assessment of the Self-Interaction Corrected Local Density Functional method and its algorithmic implementation

We calculate the electronic structure of several atoms and small molecules by direct minimization of the Self-Interaction Corrected Local Density Approximation (SIC-LDA) functional. To do this we first derive an expression for the gradient of this functional under the constraint that the orbitals be orthogonal and show that previously given expressions do not correctly incorporate this constraint. In our atomic calculations the SIC-LDA yields total energies, ionization energies and charge densities that are superior to results obtained with the Local Density Approximation (LDA). However, for molecules SIC-LDA gives bond lengths and reaction energies that are inferior to those obtained from LDA. The nonlocal BLYP functional, which we include as a representative GGA functional, outperforms both LDA and SIC-LDA for all ground state properties we considered.

Comparison of exact and approximate density functionals for an exactly soluble model

We consider a model, given by two interacting electrons in an external harmonic potential, that can be solved analytically for a discrete and infinite set of values of the spring constant. The knowledge of the exact electronic density allows us to construct the exact exchange–correlation potential and exchange–correlation energy by inverting the Kohn–Sham equation. The exact exchange–correlation potential and energy are compared with the corresponding quantities, obtained for the same densities, using approximate density functionals, namely the local density approximation and several generalized gradient approximations. We consider two values of the spring constant in order to study the system in the low correlation case (high value of the spring constant) and in the high correlation case (low value of the spring constant). In both cases, the exchange–correlation potentials corresponding to approximate density functionals differ from the exact one over the entire spatial range. The approximate correlation...

Energy and Variance Optimization of Many-Body Wave Functions

We present a simple, robust, and efficient method for varying the parameters in a many-body wave function to optimize the expectation value of the energy. The effectiveness of the method is demonstrated by optimizing the parameters in flexible Jastrow factors that include 3-body electron-electron-nucleus correlation terms for the NO2 and decapentaene (C10H12) molecules. The basic idea is to add terms to the straightforward expression for the Hessian of the energy that have zero expectation value, but that cancel much of the statistical fluctuations for a finite Monte Carlo sample. The method is compared to what is currently the most popular method for optimizing many-body wave functions, namely, minimization of the variance of the local energy. The most efficient wave function is obtained by optimizing a linear combination of the energy and the variance.