Diffusion Monte Carlo Method Research Articles

Variational Monte Carlo Method with Dirichlet Boundary Conditions: Application to the Study of Confined Systems by Impenetrable Surfaces with Different Symmetries.

Variational Monte Carlo method is a powerful tool to determine approximate wave functions of atoms, molecules, and solids up to relatively large systems. In the present work, we extend the variational Monte Carlo approach to study confined systems. Important properties of the atoms, such as the spatial distribution of the electronic charge, the energy levels, or the filling of electronic shells, are modified under confinement. An expression of the energy very similar to the estimator used for free systems is derived. This opens the possibility to study confined systems with little changes in the solution of the corresponding free systems. This is illustrated by the study of helium atom in its ground state (1)S and the first (3)S excited state confined by spherical, cylindrical, and plane impenetrable surfaces. The average interelectronic distances are also calculated. They decrease in general when the confinement is stronger; however, it is seen that they present a minimum for excited states under confinement by open surfaces (cylindrical, planes) around the radii values corresponding to ionization. The ground (2)S and the first (2)P and (2)D excited states of the lithium atom are calculated under spherical constraints for different confinement radii. A crossing between the (2)S and (2)P states is observed around rc = 3 atomic units, illustrating the modification of the atomic energy level under confinement. Finally the carbon atom is studied in the spherical symmetry by using both variational and diffusion Monte Carlo methods. It is shown that the hybridized state sp(3) becomes lower in energy than the ground state (3)P due to a modification and a mixing of the atomic orbitals s, p under strong confinement. This result suggests a model, at least of pedagogical interest, to interpret the basic properties of carbon atom in chemistry.

Diffusion Monte Carlo method with symmetry-constrained variational principle for degenerate excited states

Fixed-node diffusion Monte Carlo (FN-DMC) is an accurate and useful method for estimating the wave function and the energy of the quantum ground state of a many-fermion system. However, it has been shown that difficulties with the method may occur when it is applied to a degenerate excited state because the nodal surface of the degenerate trial function is generally insufficient to impose the complete symmetry properties of the trial function on the FN-DMC wave function. As a result, the tiling theorem and the symmetry-constrained variational principle may be violated by FN-DMC when the excited state is degenerate. There are two practical consequences for the study of degenerate excited states: The FN-DMC energy may lie below the energy of the lowest stationary state that transforms according to the same degenerate irreducible representation as the trial function; and the convergence of the FN-DMC energy with improvements in the trial function may not be quadratic. In this paper a diffusion Monte Carlo method for degenerate excited states is presented. It provides a direct generalization of the FN-DMC method, and when applied to the study of degenerate excited states, it has the support of the tiling theorem and the symmetry-constrained variational principle. The method is applied to the lowest degenerate state of a simple test problem in which FN-DMC has been shown to violate both the tiling theorem and the symmetry-constrained variational principle. The numerical results support the assertion that this method for degenerate excited states satisfies both the tiling theorem and the symmetry-constrained variational principle.

Exact ground-state properties of a one-dimensional Coulomb gas

The ground state properties of a single-component one-dimensional Coulomb gas are investigated. We use Bose-Fermi mapping for the ground state wave function which permits to solve the Fermi sign problem in the following respects (i) the nodal surface is known, permitting exact calculations (ii) evaluation of determinants is avoided, reducing the numerical complexity to that of a bosonic system, thus allowing simulation of a large number of fermions. Due to the mapping the energy and local properties in one-dimensional Coulomb systems are exactly the same for Bose-Einstein and Fermi-Dirac statistics. The exact ground state energy has been calculated in homogeneous and trapped geometries by using the diffusion Monte Carlo method. We show that in the low-density Wigner crystal limit an elementary low-lying excitation is a plasmon, which is to be contrasted with the large-density ideal Fermi gas/Tonks-Girardeau limit, where low lying excitations are phonons. Exact density profiles are confronted to the ones calculated within the local density approximation which predicts a change from a semicircular to inverted parabolic shape of the density profile as the value of the charge is increased.

Binding of hydrogen on benzene, coronene, and graphene from quantum Monte Carlo calculations

Quantum Monte Carlo calculations with the diffusion Monte Carlo (DMC) method have been used to compute the binding energy curves of hydrogen on benzene, coronene, and graphene. The DMC results on benzene agree with both Møller-Plessett second order perturbation theory (MP2) and coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] calculations, giving an adsorption energy of ∼25 meV. For coronene, DMC agrees well with MP2, giving an adsorption energy of ∼40 meV. For physisorbed hydrogen on graphene, DMC predicts a very small adsorption energy of only 5 ± 5 meV. Density functional theory (DFT) calculations with various exchange-correlation functionals, including van der Waals corrected functionals, predict a wide range of binding energies on all three systems. The present DMC results are a step toward filling the gap in accurate benchmark data on weakly bound systems. These results can help us to understand the performance of current DFT based methods, and may aid in the development of improved approaches.

BCS-BEC Crossover in a Two-Dimensional Fermi Gas

We investigate the crossover from Bardeen-Cooper-Schrieffer (BCS) superfluidity to Bose-Einstein condensation (BEC) in a two-dimensional Fermi gas at T=0 using the fixed-node diffusion MonteCarlo method. We calculate the equation of state and the gap parameter as a function of the interaction strength, observing large deviations compared to mean-field predictions. In the BEC regime our results show the important role of dimer-dimer and atom-dimer interaction effects that are completely neglected in the mean-field picture. Results on Tan's contact parameter associated with short-range physics are also reported along the BCS-BEC crossover.

Solidification of Small p-H2 Clusters at Zero Temperature

We have determined the ground-state energies of para-H(2) clusters at zero temperature using the diffusion Monte Carlo method. The liquid or solid character of each cluster is investigated by restricting the phase through the use of proper importance sampling. Our results show inhomogeneous crystallization of clusters, with alternating behavior between liquid and solid phases up to N = 55. From there on, all clusters are solid. The ground-state energies in the range N = 13-75 are established, and the stable phase of each cluster is determined. In spite of the small differences observed between the energy of liquid and solid clusters, the corresponding density profiles are significantly different, a feature that can help to solve ambiguities in the determination of the specific phase of H(2) clusters.

Ground state of small mixed helium and spin-polarized tritium clusters: A quantum Monte Carlo study

We report results for the ground-state energy and structural properties of small (4)He-T↓ clusters consisting of up to four T↓ and eight (4)He atoms. These results have been obtained using very well-known (4)He-(4)He and T↓- T↓ interaction potentials and several models for the (4)He- T↓ interatomic potential. All the calculations have been performed with variational and diffusion Monte Carlo methods. It takes at least three atoms to form a mixed bound state. In particular, for small clusters the binding energies are significantly affected by the precise form of the (4)He- T↓ interatomic potential but the stability limits remain unchanged. The only exception is the (4)He(2)T↓ trimer whose stability in the case of the weakest (4)He- T↓ interaction potential is uncertain while it seems stable for other potentials. The mixed trimer (4)He(T↓)(2), a candidate for the Borromean state, is not bound. All other studied clusters are stable. Some of the weakest bound clusters can be classified as quantum halo as a consequence of having high probability of being in a classically forbidden region.

Efficient implementation of the Hellmann–Feynman theorem in a diffusion Monte Carlo calculation

Kinetic and potential energies of systems of (4)He atoms in the solid phase are computed at T = 0. Results at two densities of the liquid phase are presented as well. Calculations are performed by the multiweight extension to the diffusion Monte Carlo method that allows the application of the Hellmann-Feynman theorem in a robust and efficient way. This is a general method that can be applied in other situations of interest as well.

Back Cover: Accuracy of quantum Monte Carlo methods for point defects in solids (Phys. Status Solidi B 2/2011)

Quantum Monte Carlo approaches such as the diffusion Monte Carlo (DMC) method are among the most accurate many-body methods for extended systems. Their scaling makes them well suited for defect calculations in solids. Parker, Wilkins and Hennig (pp. 267–274) review the various approximations needed for DMC calculations of solids and the results of previous DMC calculations for point defects. They present estimates of how approximations affect the accuracy of calculations for self-interstitial formation energies in silicon. As shown on the cover image, including the corrections for the approximations yields DMC values for the X, T and H interstitial defects in silicon on par with the GW and hybrid-functional DFT (HSE) calculations. This Feature Article is part af the article series “Advanced Calculations for Defects in Solids – Electronic Structure Methods” (guest edited by A. Alkauskas, P. Deák, J. Neugebauer, A. Pasquarello, and C. G. Van de Walle) which is being published in the issues of physica status solidi (b) during 2011.

Helium Atoms Kinetic Energy at Temperature T=0

Kinetic energies for systems formed from 4He atoms at T=0 in the solid and liquid phases are reported. The “exact” results are obtained by an extension of the diffusion Monte Carlo method. The overall agreement of the results with experimental values and calculations at finite temperatures is very good.

Renormalization of the rotational constants of an ammonia molecule seeded into a 4He droplet

The solvation dynamics of an ammonia molecule seeded into a droplet of 4He containing N = 1 – 25 4He atoms are studied using the diffusion Monte Carlo method. Excited states are computed using the fixed-node procedure. The reduction of ∼3–5% in the rotational constants of the molecule is in quantitative agreement with recent experiments of Slipchenko and Vilesov (2005) [26] and disagrees with the much larger ∼25% reduction originally reported by Behrens et al. (1998) [25].

An Integro-Differential Equation for Bose–Einstein Condensates

We present an integro-differential equation describing systems with large number of bosons. The new equation includes the two-body correlations exactly into account and the kernel has a simple analytic form. The equation has been employed to obtain results for $${A\in\{10,100\}}$$ 87Rb atoms confined by an externally applied trapping potential V trap(r). Our results are in excellent agreement with those of the Potential Harmonic Expansion Method (PHEM) and the Diffusion Monte Carlo (DMC) method.

Integro-differential equation for Bose-Einstein condensates

We use the assumption that the potential for the $A$-boson system can be written as a sum of pairwise acting forces to decompose the wave function into Faddeev components that fulfill a Faddeev type equation. Expanding these components in terms of potential harmonic (PH) polynomials and projecting on the potential basis for a specific pair of particles results in a two-variable integro-differential equations suitable for $A$-boson bound-state studies. The solution of the equation requires the evaluation of Jacobi polynomials ${P}_{K}^{\ensuremath{\alpha},\ensuremath{\beta}}(x)$ and of the weight function $W(z)$ which give severe numerical problems for very large $A$. However, using appropriate limits for $A\ensuremath{\rightarrow}\ensuremath{\infty}$ we obtain a variant equation which depends only on the input two-body interaction, and the kernel in the integral part has a simple analytic form. This equation can be readily applied to a variety of bosonic systems such as microclusters of noble gasses. We employ it to obtain results for $A\ensuremath{\in}(10\ensuremath{-}100)$ $^{87}\mathrm{Rb}$ atoms interacting via interatomic interactions and confined by an externally applied trapping potential ${V}_{\mathrm{trap}}(r)$. Our results are in excellent agreement with those previously obtained using the potential harmonic expansion method (PHEM) and the diffusion Monte Carlo (DMC) method.

Accuracy of quantum Monte Carlo methods for point defects in solids

AbstractQuantum Monte Carlo (QMC) approaches such as the diffusion Monte Carlo (DMC) method are among the most accurate many‐body methods for extended systems. Their scaling makes them well suited for defect calculations in solids. We review the various approximations needed for DMC calculations of solids and the results of previous DMC calculations for point defects in solids. Finally, we present estimates of how approximations affect the accuracy of calculations for self‐interstitial formation energies in silicon and predict DMC values of 4.4(1), 5.1(1), and 4.7(1) eV for the X, T, and H interstitial defects, respectively, in a 16(+1)‐atom supercell.

Failure of Conventional Density Functionals for the Prediction of Molecular Crystal Polymorphism: A Quantum Monte Carlo Study

We have applied the diffusion Monte Carlo method, for the first time, to an organic molecular crystal (para-diiodobenzene) in order to determine the relative stability of its two well-known polymor...

Size-consistent variational approaches to nonlocal pseudopotentials: Standard and lattice regularized diffusion Monte Carlo methods revisited

We propose improved versions of the standard diffusion Monte Carlo (DMC) and the lattice regularized diffusion Monte Carlo (LRDMC) algorithms. For the DMC method, we refine a scheme recently devised to treat nonlocal pseudopotential in a variational way. We show that such scheme-when applied to large enough systems-maintains its effectiveness only at correspondingly small enough time-steps, and we present two simple upgrades of the method which guarantee the variational property in a size-consistent manner. For the LRDMC method, which is size-consistent and variational by construction, we enhance the computational efficiency by introducing: (i) an improved definition of the effective lattice Hamiltonian which remains size-consistent and entails a small lattice-space error with a known leading term and (ii) a new randomization method for the positions of the lattice knots which requires a single lattice-space.

Electronic structure of solid FeO at high pressures by quantum Monte Carlo methods

We determine equation of state of stoichiometric FeO by employing the diffusion Monte Carlo method. The fermionic nodes of the many-body wave function are fixed by a single Slater determinant of one-particle orbitals extracted from spin-unrestricted Kohn–Sham equations utilizing a hybrid exchange-correlation functional. The calculated ambient pressure properties agree very well with available experimental data. At approximately 65 GPa, the atomic lattice is found to change from the rocksalt B1 to the NiAs-type inverse B8 structure.

Diffusion Monte Carlo: Exponential scaling of computational cost for large systems

The computational cost of a Monte Carlo algorithm can only be meaningfully discussed when taking into account the magnitude of the resulting statistical error. Aiming for a fixed error per particle, we study the scaling behavior of the diffusion Monte Carlo method for large quantum systems. We identify the correlation within the population of walkers as the dominant scaling factor for large systems. While this factor is negligible for small and medium sized systems that are typically studied, it ultimately shows exponential scaling. The scaling factor can be estimated straightforwardly for each specific system and we find that is typically only becomes relevant for systems containing more than several hundred atoms.

Pressure-induced diamond toβ-tin transition in bulk silicon: A quantum Monte Carlo study

The pressure-induced structural phase transition from diamond to $\ensuremath{\beta}$ tin in silicon is an excellent test for theoretical total-energy methods. The transition pressure provides a sensitive measure of small relative energy changes between the two phases (one a semiconductor and the other a semimetal). Experimentally, the transition pressure is well characterized. Density-functional results exhibit sensitivity to the particular form of the exchange-correlation functional. Even the generally much more accurate diffusion Monte Carlo method has shown a noticeable fixed-node error. We use the recently developed phaseless auxiliary-field quantum Monte Carlo (AFQMC) method to calculate the relative energy differences in the two phases. In this method, all but the error due to the phaseless constraint can be controlled systematically and driven to zero. In both structural phases we were able to benchmark the error of the phaseless constraint by carrying out exact unconstrained AFQMC calculations for small supercells. Comparison between the two shows that the systematic error in the absolute total energies due to the phaseless constraint is well within $0.5\text{ }\text{m}{E}_{\text{h}}/\text{atom}$. Consistent with these internal benchmarks, the transition pressure obtained by the phaseless AFQMC from large supercells is in very good agreement with experiment.

The lithium–thiophene interaction: a critical study using highly correlated electronic structure approaches of quantum chemistry

The fundamental multicentric interaction of a lithium atom with a single thiophene ring is addressed. A systematic study of the interaction energy (IE) and geometry for the Li–T charge-transfer complex is done at the MP2 and CCSD(T) levels using increasingly large basis sets up to aug-cc-pVQZ (AVQZ). Basis set superposition errors (BSSE) are evaluated and shown to have a major impact on the value of the IE. The Fixed-Node Diffusion Monte Carlo (FN-DMC) method is used as an alternative basis-set-free approach to obtain what is likely to be the most accurate estimate of the IE obtained so far. While counterpoise-corrected MP2/AVQZ and CCSD(T)/AVTZ interaction energies are found to be −3.8 and −7.5 kcal/mol, the FN-DMC method yields +1.3 ± 1.7 kcal/mol. The slow convergence of the ab initio IE (and some key structural parameters) with respect to basis set quality and the discrepancy with the FN-DMC result is discussed. A visualization of the electron pairing using the electron pair localization function (EPLF) for the Li-doped versus undoped thiophene is also presented.