Session W13: Quantum Monte Carlo

The Continuous Time Quantum Monte Carlo method as a cluster solver in the Dynamical Cluster Approximation

We have investigated the application of the Continuous Time Quantum Monte Carlo (CTQMC) method, based on interaction expansion, to solve the Hubbard model within the Dynamical Cluster Approximation (DCA). We show that CTQMC reproduces results obtained with the well known Hirsch - Fye method (HFQMC), including non-perturbative phenomena. We discuss the advantages and limitations of CTQMC as a cluster solver in the DCA. Since any QMC method suffers from fermion sign problem at low temperatures and large system sizes, we present the results of a detailed study of the sign problem within CTQMC, and compare it to HFQMC. We also discuss potential extensions of (CTQMC+DCA) for treating the phases with broken symmetry.   

Continuous-time method for quantum impurity models

We present a new continuous-time quantum Monte Carlo algorithm for quantum impurity problems. The method allows an efficient simulation of large cluster impurity models with density density coupling. We compare the computational effort and average sign to alternative quantum Monte Carlo approaches, such as the discrete-time Hirsch-Fye algorithm [ J. E. Hirsch and R. M. Fye, Phys. Rev. Lett. 56, 2521 (1986)] and the weak coupling solver by Rubtsov, Savkin and Lichtenstein [Phys. Rev. B 72 035122 (2005)]   

Progress in the study of Molecular Hydrogen$-$Benzene binding$^\dagger$

In this work we present a quantum Monte Carlo study of the hydrogen-benzene system where binding is very weak. We demonstrate that the binding is well described at both the VMC and DMC levels by a single determinant correlated geminal wave function$^1$ with an optimized compact basis set that includes diffuse functions. Agreement between VMC and fixed node DMC binding energies is found to be within 0.18 mHa, suggesting the calculations are well-converged with respect to the basis. This relative insensitivity to basis set and superposition error is an advantage of the QMC methods we employ. Comparison is made with a Slater-Jastrow wave function at the DMC level using a trial function comprised of PBE single-body orbitals, empirical models and previous work$^2$. The physical underpinnings of the interaction will be discussed including the role of diffuse basis functions in this system. Progress on systems where binding is expected to be more favorable for practical hydrogen storage will also be presented.\vspace{0.2 cm} \\$^1$ M. Casula, C. Attaccalite, and S. Sorella, J. Chem. Phys. 121, 7110 (2004). \\$^2$ S. Hamel and M. C\^ot\'e, J. Chem. Phys. 121, 12618 (2004).\vspace{0.2 cm} \\$^\dagger$ Supported by NSF DMR03-25939 and A6062 ArmyUMC00005071-3   

Geometry optimization with a noisy potential energy surface

Molecular and solid systems in the excited state and in the weak-binding regime (for example) are often not described well by current density functional(DFT) methods, often leading to inaccurate minimum energy structures. Quantum Monte Carlo(QMC) is a tempting method to improve on these deficiencies, since it offers a highly accurate fully correlated first principles description. However, QMC suffers from two major deficiencies: 1) forces are not easily calculated and 2) the energy is obtained with stochastic uncertainty, which makes optimization a challenging task. We examine several ways of compensating for this uncertainty while only using the reliable total energies obtained in QMC.   

Quantum Monte Carlo calculations of NiO

We describe variational and diffusion quantum Monte Carlo (VMC and DMC) calculations [1] of NiO using a 1024-electron simulation cell. We have used a smooth, norm-conserving, Dirac-Fock pseudopotential [2] in our work. Our trial wave functions were of Slater-Jastrow form, containing orbitals generated in Gaussian-basis UHF periodic calculations. Jastrow factor is optimized using variance minimization with optimized cutoff lengths using the same scheme as our previous work. [4] We apply the lattice regulated scheme [5] to evaluate non-local pseudopotentials in DMC and find the scheme improves the smoothness of the energy-volume curve. \newline \newline [1] CASINO ver.2.1 User Manual, University of Cambridge (2007). \newline [2] J.R. Trail {\it et.al.}, J. Chem. Phys. {\bf 122}, 014112 (2005). \newline [3] CRYSTAL98 User's Manual, University of Torino (1998). \newline [4] Ryo Maezono {\it et.al.}, Phys. Rev. Lett., {\bf 98}, 025701 (2007). \newline [5] Michele Casula, Phys. Rev. B {\bf 74}, 161102R (2006).   

Efficient orbital storage and evaluation for quantum Monte Carlo simulations of solids

Researchers have applied continuum quantum Monte Carlo methods to solids with great success, but thus far applications have been largely limited to crystals with simple geometry. In these simulations, three-dimensional cubic B-splines have proven to be a fast and accurate means of storing and evaluating electron orbitals. While B-splines require less memory than other spline interpolation schemes, modern cluster nodes often have insufficient memory to store the orbitals for more complex systems. We introduce three techniques, appropriate in different circumstances, to dramatically reduce the memory required for orbital storage, while retaining high accuracy: the generalized tiling of primitive-cell orbitals into a supercell of arbitrary shape, the use of nonuniform grids for localized orbitals, and the periodic replication of localized orbitals. We give examples for cubic boron nitride and w\"{u}stite (FeO), and show that these methods can reduce the memory used for orbital storage by more than two orders of magnitude. Finally, we introduce an open-source B-spline library to facilitate the incorporation of these methods into QMC simulation codes.   

Triplet pairings and fermion wave functions nodal topologies

Fixed-node quantum Monte Carlo methods rely on accurate fermion nodes of trial wave functions. Recently, we have shown that BCS wave functions possess for generic singlet ground states possess the correct minimal number of two nodal cells. This contrasts with the Hartree-Fock wave functions which exhibit higher counts of four or more nodal domains resulting in incorrect nodal topologies. We prove that for fully spin-polarized systems one can show the same effect. As a simple example, we consinder the HF wave function for the lowest quartet of S symmetry and even parity for three electrons in a Coulomb potential. The wave function of this state $^4S(1s2s3s)$ has six nodal cells corresponding to 3! reordering of the radii. We show that pfaffian with triplet pairings is the simplest wave function which has the correct topology with two nodal cells. We further expand the study to some exactly solvable models to study the exact nodal structures dependence on potentials.   

Compact and accurate quantum Monte Carlo wave functions for first-row atoms

Many-body wave functions for the first row atoms (Li to Ne) are represented as expansions in eigenstates of $\hat{L}^2,$ $\hat{L}_z,$ $\hat{S}^2,$ $\hat{S}_z,$ multiplied by a Jastrow factor. This configuration state function (CSF) expansion provides a systematic means for improving a wave function by including CSFs corresponding to higher excitations. Optimization of all wave function parameters including Jastrow, CSF and orbital coefficients as well as basis exponents, starting from a simple initial guess, results in compact and accurate wave functions (low energy and variance of local energy). Further improvements by use of backflow transformations are explored. This work aims to develop insight into selecting the relevant CSFs particularly for large systems, where it is difficult to include all CSFs to a given order.   

A DQMC study of cohesion energy of small Li clusters based on an RVB nodal structure

We have carried out a diffusion Quantum Monte Carlo study (DQMC) of the cohesion energy of small (2, 4, and 8 atom) Li clusters based on Resonating Valence Bond (RVB) wavefunctions, and compared the results to the corresponding values obtained via wavefunctions utilizing a typical Hartree-Fock (HF) nodal structure (Jastrow-Slater wavefunctions). The RVB wavefunction allows more flexibility in the nodal structure than the HF wavefunction, and yields some improvement in the cohesion energy of Li$_2$, with comparable gain for the larger clusters. Interestingly, the variance of the local energy for the variationally optimized (VQMC) RVB wavefunction is found to be significantly smaller than for the VQMC-optimized Jastrow-Slater wavefunction, resulting in faster convergence of the DQMC calculations. This would make the RVB wavefunction a promising candidate for investigating larger and more complicated clusters.   

An Improved Pressure Estimator for quantum Monte Carlo

Assaraf and Caffarel have developed a systematic method for deriving reduced variance estimators for observables for quantum Monte Carlo calculations and have applied to forces[1], the one body density and the spherical and system averaged pair density. It has yet to be applied to a thermodynamic observable. In this work we derive an expression for an improved pressure estimator for use in variational Monte Carlo and diffusion Monte Carlo calculations. We show that because the dependence of the trial wave function on the density is known, for the homogeneous electron gas this new estimator is accurate and efficient to implement. We discuss its application to many-body Hydrogen at high pressure. [1] Roland Assaraf and Michel Caffarel, J. Chem. Phys. 113, 4028 (2000)   

The Model Periodic Coulomb interaction in k-space: modelling the spherically averaged structure factor

Within Quantum Monte Carlo (QMC) calculations the Model Periodic Coulomb (MPC) interaction is a well know method to reduce finite size effects related to the long range nature of the Coulomb interaction. Recently we presented a method based on modelling the continuous-$k$ spherically averaged structure factor (SF) to understand and reduce Coulomb finite size effects. Here we show that our SF based method can be viewed as $k$-resolved MPC. This allows us to analyse the implicit assumptions that underlie MPC and what to do when these assumptions are not justified, i.e. in non-interaction Hartree-Fock systems or even surfaces. While we present data for the homogeneous electron gas the method itself is general.   

Backflow transformation improves QMC calculations of silicon self-interstitial defects

Recent advances in quantum Monte Carlo (QMC) reduce error introduced by approximations. Direct improvement of the trial wave function through backflow transformation of the electron coordinates[1] produces a wave function closer to the ground state by moving electrons out of the way of a given electron. Adding plane waves of particle position to the Jastrow factor[2] augments the accounting for interparticle correlation in QMC calculations by capturing the ``corners'' of the simulation cell neglected when the Jastrow is only a function of pair separation. Hybrid density functionals have produced better starting trial wave functions for molecules by incorporating some exact exchange to more accurately describe electron-electron interactions. We apply backflow transformation, plane-wave-expanded Jastrow factors and hybrid functional trial wave functions to QMC calculations of silicon self-interstitial defects. [1] L\'opez-R\'ios {\it et al.}, Phys. Rev. E {\bf 74}, 066701 (2006). [2] Drummond {\it et al.}, Phys. Rev. B {\bf 70}, 235119 (2004).   

Density Dependence of Fixed-Node Errors

With both variational and diffusion Monte Carlo (VMC and DMC) methods, we calculate the ground state energy of isoelectronic free ions in the first row of the periodic table for both Hartree-Fock and Configuration Interaction based trial wave functions. As it is well-known, the fixed-node DMC is exact in the limit that the fermion nodes of the trial wave function are also exact. This study is focused on understanding of the density dependence of the fixed-node error since one expects that with increasing density of electrons the errors would be more pronounced due to higher frequency of sampling of the nodal regions and/or areas with low potential energy. For this purpose we construct Hartree-Fock and multi-reference wave functions and quantify the fixed-node biases. We compare strongly bonded highly localized cations, neutral atoms and weakly bonded anions. We compare the absolute and relative sizes of fixed-node errors and their relationships to multi-reference wave functions.   

Slater determinant and pfaffian expansions for wave functions in electronic structure quantum Monte Carlo

We investigate several types of expansions in Slater determinants and pfaffians for trial wave functions in fixed-node quantum Monte Carlo. The long expansions in determinants are analyzed in order to identify the terms with the largest contributions towards decreasing the fixed-node errors. We further investigate the efficient mapping of these terms onto pfaffian expansions. We apply this technique to test the cases of molecular and atomic systems and we discuss the amounts of recovered correlation energy relative to the expansion size. Finally, following upon our previous study [1], we explore the use of multiple determinants and pfaffians for the accurate description of the wave functions of simple solids. [1] M. Bajdich et al. Phys. Rev. Lett. 96, 130201 (2006).