Session X24: Electronic Structure: Quantum Monte Carlo

The Homogeneous Electron Gas: Beyond Fixed Nodes

The ground state energy of the homogeneous electron gas (HEG) still presents a significant challenge to Quantum Chemical methods, in spite of being a model Hamiltonian that has been studied for many decades and is often regarded as the archetypal system in solid state physics and in Fermi liquid theory. To date the only truly successful methods to yield accurate ground state energies at a range of densities have been quantum Monte Carlo techniques, in particular Diffusion Monte Carlo (DMC). Attempts to go beyond the fixed-node approximation have been met with some success, however elimination of this error all-together has not been achieved. Full Configuration Interaction (FCI) would provide an exact solution to this problem in the limit of an infinite basis set, which can be approached in a systematically improvable way. However, this is prohibitively expensive, scaling exponentially in the electron number and the size of the underlying one-electron basis with very large pre-factors. We present the application of a new method, FCI Quantum Monte Carlo, which stochastically samples the exact wavefunction producing FCI accuracy at a greatly improved computational cost, to the high-density HEG.   

Simulation of the Warm-Dense Homogeneous Electron Gas

Warm-dense matter (WDM), where both the Coulomb coupling parameter ($\Gamma \equiv q^{2}/(r_{s} k_{B} T)$) and the electron degeneracy parameter ($\Theta \equiv k_{B} T/\epsilon_{F}$) are approximately unity, exists in systems as disparate as planetary interiors and along the pathway to inertial confinement fusion. Attempts to characterize this regime through the use of Density Functional Theory (DFT) require an accurate equation of state. Here we present results for a first-principles simulation of the homogeneous electron gas (HEG) in the warm-dense regime through Restricted Path Integral Monte Carlo (RPIMC). These results could be used as a benchmark for improved functionals, as well as input for orbital-free DFT formulations.   

Improved Full Configuration Interaction Monte Carlo for the Hubbard Model

We consider the recently proposed full configuration interaction quantum Monte Carlo (FCI-QMC) method and its ``initiator'' extension, both of which promise to ameliorate the sign problem by utilizing the cancellation of positive and negative walkers in the Hilbert space of Slater determinants. While the method has been primarily used for quantum chemistry by A.Alavi and his co-workers [1,2], its application to lattice models in solid state physics has not been tested. We propose an improvement in the form of choosing a basis to make the wavefunction more localized in Fock space, which potentially also reduces the sign problem. We perform calculations on the 4x4 and 8x8 Hubbard models in real and momentum space and in a basis motivated by the reduced density matrix of a 2x2 real space patch obtained from the exact diagonalization of a larger system in which it is embedded. We discuss our results for a range of fillings and U/t and compare them with previous Auxiliary Field QMC and Fixed Node Green's Function Monte Carlo calculations. \\[4pt] [1] George Booth, Alex Thom, Ali Alavi, J Chem Phys, 131, 050106,(2009)\\[0pt] [2] D Cleland, GH Booth, Ali Alavi, J Chem Phys 132, 041103, (2010)   

Semi-stochastic full configuration interaction quantum Monte Carlo

In the recently proposed full configuration interaction quantum Monte Carlo (FCIQMC) [1,2], the ground state is projected out stochastically, using a population of walkers each of which represents a basis state in the Hilbert space spanned by Slater determinants. The infamous fermion sign problem manifests itself in the fact that walkers of either sign can be spawned on a given determinant. We propose an improvement on this method in the form of a hybrid stochastic/deterministic technique, which we expect will improve the efficiency of the algorithm by ameliorating the sign problem. We test the method on atoms and molecules, e.g., carbon, carbon dimer, N2 molecule, and stretched N2. \\[4pt] [1] Fermion Monte Carlo without fixed nodes: a Game of Life, death and annihilation in Slater Determinant space. George Booth, Alex Thom, Ali Alavi. J Chem Phys 131, 050106, (2009).\\[0pt] [2] Survival of the fittest: Accelerating convergence in full configuration-interaction quantum Monte Carlo. Deidre Cleland, George Booth, and Ali Alavi. J Chem Phys 132, 041103 (2010).   

Excited state calculations in solids by auxiliary-field quantum Monte Carlo

We have studied electronic excitations in solid systems using the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method.\footnote{S.~Zhang and H.~Krakauer, Phys.~Rev.~Lett.~{\bf 90}, 136401 (2003); W.~Purwanto, S.~Zhang, and H.~Krakauer, J. Chem. Phys. {\bf 130}, 094107 (2009).} Trial wave functions for excited states are simply constructed from the corresponding density functional theory (DFT) ground state orbitals by promoting electrons to conduction bands. The post-processing finite size (FS) correction method\footnote{F.~Ma, S.~Zhang, and H.~Krakauer, Phys.~Rev.~B {\bf 84}, 155130 (2011); H.~Kwee, S.~Zhang, and H.~Krakauer, Phys. Rev. Lett. {\bf 100}, 126404 (2008).} is applied to remove the many-body FS effects. By fitting the calculated excitation energies at various crystal momentum values, a many-body electronic band structure is obtained. Our results for prototypical semiconductors such as silicon are compared to those from the GW approximation\footnote{M.~Rohlfing, P.~Kr\"uger, and J.~Pollmann, Phys.~Rev B.~{\bf 48}, 17791 (1993).} and diffusion Monte Carlo calculations.\footnote{A.~J.~Williamson, R.~Q.~Hood, R.~J.~Needs, and G.~Rajagopal, Phys.~Rev.~B {\bf 57}, 12140 (1998).}   

Frozen core method in auxiliary-field quantum Monte Carlo

We present the implementation of the frozen-core approach in the phaseless auxiliary-field quantum Monte Carlo method (AFQMC). Since AFQMC random walks take place in a many-electron Hilbert space spanned by a chosen one-particle basis, this approach can be achieved without introducing additional approximations, such as pseudopotentials. In parallel to many-body quantum chemistry methods, tightly-bound inner electrons occupy frozen canonical orbitals, which are determined from a lower level of theory, e.g. Hartree-Fock or CASSCF. This provides significant computational savings over fully correlated all-electron treatments, while retaining excellent transferability and accuracy. Results for several systems will be presented. This includes the notoriously difficult Cr$_2$ molecule, where comparisons can be made with near-exact results in small basis sets, as well as an initial implementation in periodic systems.   

Tests on novel pseudo-potentials generated from diffusion Monte Carlo data.

Since Dmitri Mendeleev developed a table in 1869 to illustrate recurring ("periodic") trends of the elements, it has been understood that most chemical and physical properties can be described by taking into account the outer most electrons of the atoms. These valence electrons are mainly responsible for the chemical bond. In many ab-initio approaches only valence electrons are taken into account and a pseudopotential is used to mimic the response of the core electrons. Typically an all-electron calculation is used to generate a pseudopotential that is used either within density functional theory or quantum chemistry approaches. In this talk we explain and demonstrate a new method to generate pseudopotentials directly from all-electron many-body diffusion Monte Carlo (DMC) calculations and discuss the results of of the transferability of these pseudopotentials. The advantages of incorporating the exchange and correlation directly from DMC into the pseudopotential are also discussed.   

A diffusion Monte Carlo study of sign problems from non-local pseudopotentials

Difficulties can arise in simulating various Hamiltonian operators efficiently in diffusion Monte Carlo (DMC) such as those associated with non-local pseudopotentials which require the introduction of an approximate form. The locality approximation and T-moves are two widely used techniques in fixed-node diffusion Monte Carlo (FN-DMC) that provide a tractable approach for treating non-local pseudopotentials, however their use introduces an uncontrolled approximation. Exact treatment of the non-local pseudopotentials in FN-DMC introduces a sign problem with the associated Green's function matrix elements which take on both positive and negative values. Here we present an analysis of the nature of the sign problem that non-local operators introduce into the Green's function. We then consider the feasibility of running DMC simulations in which the non-local pseudopotentials are treated exactly and demonstrate the algorithm on a few molecular systems.   

Variational and diffusion quantum Monte Carlo methods for spin-orbit interactions in heavy element systems

In most electronic structure quantum Monte Carlo calculations spins of individual electrons have fixed values which are determined by the spin and spatial symmetries of the desired eigenstate. However, for heavy atoms the spin-orbit interaction becomes important and its influence on electronic structure becomes comparable to the electron exchange, correlations or other many-body effects. In such cases the simplest antisymmetric wave function based on one-particle states is a determinant of spinors, while the simplest pairing wave function is a pfaffian. We will present calculations using variational and diffusion Monte Carlo for heavy atoms systems with spin-orbit operators. We eliminate the atomic cores by effective core potentials (pseudopotentials) which are formulated so as to include also effective spin-orbit operators. We test both discrete spin sampling as well as continuous spin representation in the variational and diffusion Monte Carlo methods. Corresponding generalizations of the fixed-node/fixed-phase methods will be discussed as well.   

Progress in exact treatment of fermions at finite temperature

We will discuss some key features of the structure of permutation space for interacting Fermi systems. Exploiting these features, we will then demonstrate improved efficiency in the exact path integral Monte Carlo treatment of liquid $^3$He by using importance sampling to deemphasize the contribution of long permutation cycles to the partition function. Finally, a route to a polynomial scaling algorithm for homogeneous Fermi systems will be presented.   

An energy density estimator for quantum Monte Carlo calculations

We establish a physically meaningful representation of a quantum energy density for use in quantum Monte Carlo calculations. The energy density operator, defined in terms of Hamiltonian components and density operators, returns the correct Hamiltonian when integrated over a volume containing a cluster of particles. This property is demonstrated for a Helium-Neon ``gas,'' showing that atomic energies obtained from the energy density correspond to eigenvalues of isolated systems. The formation energies of defects or interfaces are typically calculated as total energy differences. Using a model of delta doped Silicon (where dopant atoms form a thin plane) we show how interfacial energies can be calculated more efficiently with the energy density, since the region of interest is small. We also demonstrate how the energy density correctly transitions to the bulk limit away from the interface where the correct energy is obtainable from a separate total energy calculation.   

Metal-Insulator Transition in Low Density Atomic Hydrogen

At low density BCC hydrogen undergoes a metal-insulator transition. We compute the zero temperature equation of state for the paramagnetic and anti-ferromagnetic phases using diffusion Quantum Monte Carlo. We predict the phase transition density, investigate the shape of the anti-ferromagnetic curve, and compare to previous results   

Quantum Monte Carlo of ThO2

Thorium dioxide solid is a unique optical and heat-resistant actinide material with large gap and cohesion. It is a diamagnet, unlike a number of other similar actinide oxides. We investigate the electronic structure of ThO2 using Density Functional Theory (DFT) and quantum Monte Carlo (QMC) methods. We adopt Stuttgart RLC and RSC effective core potentials (pseudopotentials) for the Th atom. In the DFT calculations, some of the properties are verified in all-electron calculations using the FLAPW techniques. Using the fixed-node diffusion Monte Carlo we calculate the ground state and several excited states from which we estimate the cohesion and the band gap. Simulation cells of several sizes are used to estimate/reduce the finite size effects. We compare the QMC results with recent DFT calculations with several types of functionals which include hybrids such as PBE0 and HSE. Insights from QMC calculations give us understanding of the correlations beyond the DFT approaches and pave the way for accurate electronic structure calculations of other actinide materials.   

Quantum Monte Carlo for the Spectroscopy of Core Excited States

X-ray absorption spectroscopy is a powerful experimental tool that is capable of delivering valuable information about very delicate aspects of electronic structure and reveals details of the local chemical environment in many systems of fundamental and applied importance. The rigorous interpretation of core-level spectra requires very accurate quantum chemical simulations. The trade-off between feasibility of treatment of large systems and consistency in description of electron correlation tremendously hinders the generation of accurate theoretical results for many experimental studies. We show that the fixed-node diffusion Monte Carlo (FN-DMC) approach can be used straightforwardly for the accurate simulation of core-level spectra. Basic methodological aspects are addressed, including the strategy for the construction of adequate trial wave functions. Examples of FN-DMC calculations of core-level spectra of water and pyrrole are presented. The possibility of the simulation of X-ray absorption spectra of solvent-solute systems is discussed.   

Density Functional Monte Carlo: an efficient way to calculate the ground state density profile of nanostructures

We present a method in which the Hohenberg-Kohn theorems are implemented directly by simulating the density profile using Bernouilli walkers and conserving the total number of particles during the Monte Carlo process. This leads to a much faster algorithm than, e.g., by solving the Kohn-Sham equations. The method is explained in detail and results are shown for a nanoshell which contains several millions of conduction electrons.