Diffusion Quantum Monte Carlo Simulations Research Articles

Quantum Monte Carlo: Reduction of time‐step error with the wave function correction method

AbstractWe compare the accuracy and precision of the wave function correction method to that of traditional diffusion quantum Monte Carlo (traditional DQMC) simulations. Direct calculation of corrections to trial wave functions reduces time step error for Be and Ne electronic ground state energies when compared to traditional DQMC results. The fixed‐node method is utilized for these fermionic systems, so error due to incorrect node locations is not addressed. This correction DQMC method achieves more accurate energies at larger time steps, promising decreased computational cost to reach a given accuracy. For an optimized Be trial wave function, the statistical uncertainty per unit of computational effort is also reduced relative to traditional DQMC. Correction DQMC utilizes walker feed integrals and growth factors, as reported previously. We illustrate the method in a simple harmonic oscillator example, and we report high accuracy energies for Be and Ne at several time steps.

The Breakup of a Helium Cluster After Removing Attractive Interaction Among a Significant Number of Atoms in the Cluster

The breakup of a quantum liquid droplet is examined through a 4He cluster by removing the attractive tail in the interaction between some of the atoms in the system with the diffusion quantum Monte Carlo simulation. The ground-state energy, kinetic energy, cluster size, and density profile of the cluster are evaluated against the percentage of the atoms without the attractive tail. The condition for the cluster to lose its ability to form a quantum liquid droplet at zero temperature is found and analyzed. The cluster is no longer able to form a quantum liquid droplet when about two-thirds of pairs of attractive interaction are removed. The findings are helpful to the current studies on the formation of quantum liquid droplets from cold atoms.

Quantum Monte Carlo study on the structures and energetics of cyclic and linear carbon clusters Cn ( n = 1,...,10)

Using fixed-node diffusion quantum Monte Carlo (DMC) simulation we investigate the structural properties and energetics of the linear and cyclic carbon clusters ${\mathrm{C}}_{n}$ for $n\ensuremath{\le}10$. We calculate the binding energy, the electron correlation energy, the dissociation energy, and the second difference in energy. We also present an analysis of the structural properties of the clusters. It is found that the bond lengths, binding energies, and dissociation energies obtained from the DMC calculations are in excellent agreement with the available experimental results. The electron-correlation contribution to the binding energy indicates that in the case of the linear isomers, the clusters of odd-number size are relatively more favored than their neighbors of even-number size, whereas for the cyclic isomers, we do not observe the oscillation pattern. In the range of cluster size under investigation, we find that the electron-correlation impact in the binding energy of the cyclic clusters is larger than that of the corresponding linear ones, varying from $30%$ to $40%$ of the binding energy values. The electron correlation is also essential to the stability of the clusters enhancing by up to $52%$ their dissociation energy. A comparative analysis of the dissociation energy and second difference in energy indicates that the linear isomers ${\mathrm{C}}_{3}$ and ${\mathrm{C}}_{5}$ are the most stable ones.

Analysis of the ionization potentials of small superalkali lithium clusters based on quantum Monte Carlo simulations

In order to investigate the electronic correlation effects in the ionization potentials of small lithium clusters, we obtain accurate ground-state energies using the diffusion quantum Monte Carlo simulation. We decompose both the adiabatic and vertical ionization potentials of the clusters into physical components to analyze the nature of the superalkali characteristics of the lithium clusters. We find that the electron-correlation contribution to the ionization potential is mostly canceled out by the orbital relaxation with net contribution less than 0.5 eV. The superalkali characteristics of the clusters is basically determined by the electrostatic and exchange interactions.

Andreev–Bashkin effect in superfluid cold gases mixtures

We study a mixture of two superfluids with density–density and current–current (Andreev–Bashkin) interspecies interactions. The Andreev–Bashkin coupling gives rise to a dissipationless drag (or entrainment) between the two superfluids. Within the quantum hydrodynamics approximation, we study the relations between speeds of sound, susceptibilities and static structure factors, in a generic model in which the density and spin dynamics decouple. Due to translational invariance, the density channel does not feel the drag. The spin channel, instead, does not satisfy the usual Bijl–Feynman relation, since the f-sum rule is not exhausted by the spin phonons. The very same effect on one dimensional Bose mixtures and their Luttinger liquid description is analysed within perturbation theory. Using diffusion quantum Monte Carlo simulations of a system of dipolar gases in a double layer configuration, we confirm the general results. Given the recent advances in measuring the counterflow instability, we also study the effect of the entrainment on the dynamical stability of a superfluid mixture with non-zero relative velocity.

Trion formation in a two-dimensional hole-doped electron gas

The interaction between a single hole and a two-dimensional, paramagnetic, homogeneous electron gas is studied using diffusion quantum Monte Carlo simulations. Calculations of the electron-hole correlation energy, pair-correlation function, and the electron-hole center-of-mass momentum density are reported for a range of electron--hole mass ratios and electron densities. We find numerical evidence of a crossover from a collective Mahan exciton to a trion-dominated state in a density range in agreement with that found in recent experiments on quantum well heterostructures.

Quantum monte carlo study of the energetics of small hydrogenated and fluoride lithium clusters.

An investigation of the energetics of small lithium clusters doped either with a hydrogen or with a fluorine atom as a function of the number of lithium atoms using fixed-node diffusion quantum Monte Carlo (DMC) simulation is reported. It is found that the binding energy (BE) for the doped clusters increases in absolute values leading to a more stable system than for the pure ones in excellent agreement with available experimental measurements. The BE increases for pure, remains almost constant for hydrogenated, and decreases rapidly toward the bulk lithium for the fluoride as a function of the number of lithium atoms in the clusters. The BE, dissociation energy as well as the second difference in energy display a pronounced odd-even oscillation with the number of lithium atoms. The electron correlation inverts the odd-even oscillation pattern for the doped in comparison with the pure clusters and has an impact of 29%-83% to the BE being higher in the pure cluster followed by the hydrogenated and then by the fluoride. The dissociation energy and the second difference in energy indicate that the doped cluster Li3 H is the most stable whereas among the pure ones the more stable are Li2 , Li4 , and Li6 . The electron correlation energy is crucial for the stabilization of Li3 H. © 2016 Wiley Periodicals, Inc.

Condensed phase of Bose-Fermi mixtures with a pairing interaction

We study the condensed phase of a Bose-Fermi mixture with a tunable pairing interaction between bosons and fermions with many-body diagrammatic methods and fixed-node diffusion Quantum Monte Carlo simulations. A universal behavior of the condensate fraction and bosonic momentum distribution with respect to the boson concentration is found to hold in an extended range of boson-fermion couplings and concentrations. For vanishing boson density, we prove that the bosonic condensate fraction reduces to the quasiparticle weight Z of the Fermi polaron studied in the context of polarized Fermi gases, unifying in this way two apparently unrelated quantities.

Binding energies of small lithium clusters: A comparison of different theoretical calculations

In order to compare different calculations used in theoretical studies on lithium clusters, we have calculated the binding energies of small lithium clusters using all electron diffusion quantum Monte Carlo (DMC) simulation, coupled-cluster (CCSD(T)) approach, as well as density functional theory (DFT) with different exchange-correlation (xc) functionals. The obtained DMC results are in very good agreement with available experimental data and with CCSD(T) calculations. However, the DFT results depend strongly on the approximations for the xc functional. Furthermore, considering the DMC result as a benchmark, we obtain the electron correlation energy of the clusters and quantify its contribution to the binding energies.

A quantum Monte Carlo study on electron correlation in all-metal aromatic clusters MAl4− (M = Li, Na, K, Rb, Cu, Ag and Au)

Using fixed-node diffusion quantum Monte Carlo (FN-DMC) simulation we investigate the electron correlation in all-metal aromatic clusters MAl4(-) (with M = Li, Na, K, Rb, Cu, Ag and Au). The electron detachment energies and electron affinities of the clusters are obtained. The vertical electron detachment energies obtained from the FN-DMC calculations are in very good agreement with the available experimental results. Calculations are also performed within the Hartree-Fock approximation, density-functional theory (DFT), and the couple-cluster (CCSD(T)) method. From the obtained results, we analyse the impact of the electron correlation effects in these bimetallic clusters and find that the correlation of the valence electrons contributes significantly to the detachment energies and electron affinities, varying between 20% and 50% of their total values. Furthermore, we discuss the electron correlation effects on the stability of the clusters as well as the accuracy of the DFT and CCSD(T) calculations in the present systems.

Quantum Monte Carlo study of high pressure solid molecular hydrogen

We use the diffusion quantum Monte Carlo (DMC) method to calculate the ground-state phase diagram of solid molecular hydrogen and examine the stability of the most important insulating phases relative to metallic crystalline molecular hydrogen. We account for finite-size errors by combining the use of twist-averaged boundary conditions with corrections obtained using the Kwee–Zhang–Krakauer functional in density functional theory. To study the closure of the band gap with increasing pressure, we perform quasi-particle many-body calculations using the GW method. In the static approximation, our DMC simulations indicate a transition from the insulating Cmca-12 structure to the metallic Cmca structure at around 375 GPa. The GW band gap of Cmca-12 closes at roughly the same pressure. In the dynamic DMC phase diagram, which includes the effects of zero-point energy in the quasi-harmonic approximation, the Cmca-12 structure remains stable up to 430 GPa, well above the pressure at which the GW band gap closes. Our results predict that the semimetallic state observed experimentally at around 360 GPa (2012 Phys. Rev. Lett. 108, 146402) may correspond to the Cmca-12 structure near the pressure at which the band gap closes. The dynamic DMC phase diagram indicates that the hexagonal-close-packed P63/m structure, which has the largest band gap of the insulating structures considered, is stable up to 220 GPa. This is consistent with recent x-ray data taken at pressures up to 183 GPa (2010 Phys. Rev. B 82 060101), which also reported a hexagonal-close-packed arrangement of hydrogen molecules.

Atomic ground states in strong magnetic fields: Electron configurations and energy levels

Using a combination of a fast two-dimensional-Hartree-Fock-Roothaan method and highly accurate fixed-phase diffusion quantum Monte Carlo simulations, we analyze the electronic structure and calculate the energy of the ground states of atoms with nuclear charges $Z=2--26$ in very strong magnetic fields $B={10}^{7}--5\ifmmode\times\else\texttimes\fi{}{10}^{8}\phantom{\rule{0.28em}{0ex}}\mathrm{T}$, relevant for astrophysical problems, e.g., the thermal emission of strongly magnetized isolated neutron stars.

Quantum Monte Carlo study of small aluminum clusters Aln(n=2–13)

Using fixed node diffusion quantum Monte Carlo (FN-DMC) simulations and density functional theory (DFT) within the generalized gradient approximations, we calculate the total energies of the relaxed and unrelaxed neutral, cationic, and anionic aluminum clusters, Al${}_{n}$ ($n=1--13$). From the obtained total energies, we extract the ionization potential and electron detachment energy and compare with previous theoretical and experimental results. Our results for the electronic properties from both the FN-DMC and DFT calculations are in reasonably good agreement with the available experimental data. A comparison between the FN-DMC and DFT results reveals that their differences are a few tenths of electron volt for both the ionization potential and the electron detachment energy. We also observe two distinct behaviors in the electron correlation contribution to the total energies from smaller to larger clusters, which could be assigned to the structural transition of the clusters from planar to three-dimensional occurring at $n=4$ to 5.

Four electrons interacting pairwise in the limit of infinitesimal confining potentials: Especially the quintet spin state

In very recent work, Amovilli and March have considered the Hookean atom with four electrons. In particular, they demonstrated the cross-over from a triplet Pg ground state at large harmonic force constant k to a quintet Su configuration for weak confinement, by means primarily of diffusion quantum Monte Carlo simulation. Here, we focus on such a quintet state for again four spin half fermions but now interacting via a harmonic pair force −K rij. The spatial quintet Ψ is then available, with its eigenvalue. Progress on calculating low-order spinless density matrices is recorded. Finally, this prompts us to examine the Amovilli–March results when extrapolated to small k≠0. We display as k→0+ approximate results for the four-electron Hookean atom model, including fitted analytic forms for kinetic and Coulomb energy as a function of k as k→0+.

Resonating-valence-bond ground state of lithium nanoclusters

We have performed diffusion quantum Monte Carlo simulations of Li clusters showing that resonating-valence-bond (RVB) pairing correlations between electrons provide a substantial contribution to the cohesive energy. The RVB effects are identified in terms of electron transfers from $s$-like to $p$-like character, constituting a possible explanation for the breakdown of the Fermi-liquid picture observed in recent high-resolution Compton scattering experiments for bulk Li.

Population size bias in descendant-weighted diffusion quantum Monte Carlo simulations

We consider the influence of population size on the accuracy of diffusion quantum Monte Carlo simulations that employ descendant weighting or forward walking techniques to compute expectation values of observables that do not commute with the Hamiltonian. We show that for a simple model system, the d-dimensional isotropic harmonic oscillator, the population size must increase rapidly with d in order to ensure that the simulations produce accurate results. When the population size is too small, expectation values computed using descendant-weighted diffusion quantum Monte Carlo simulations exhibit significant systematic biases.

Remedy for the fermion sign problem in the diffusion Monte Carlo method for few fermions with antisymmetric diffusion process

We suggest an exact approach to help remedy the fermion sign problem in diffusion quantum Monte Carlo simulations. The approach is based on an explicit suppression of symmetric modes in the Schrödinger equation by means of a modified stochastic diffusion process (antisymmetric diffusion process). We introduce this algorithm and illustrate it on potential models in one dimension (1D) and show that there it solves the fermion sign problem exactly and converges to the lowest antisymmetric state of the system. Then, we discuss extensions of this approach to many-dimensional systems on examples of quantum oscillator in 2D-20D and a toy model of three and four fermions on harmonic strings in 2D and 3D. We show that in all these cases our method shows a performance comparable to that of a fixed-node approximation with an exact node.

The accuracy of diffusion quantum Monte Carlo simulations in the determination of molecular equilibrium structures

For a test set of 17 first-row small molecules, the equilibrium structures are calculated with Ornstein-Uhlenbeck diffusion quantum Monte Carlo simulations guiding by trial wave functions constructed from floating spherical Gaussian orbitals and spherical Gaussian geminals. To measure performance of the Monte Carlo calculations, the mean deviation, the mean absolute deviation, the maximum absolute deviation, and the standard deviation of Monte Carlo calculated equilibrium structures with respect to empirical equilibrium structures are given. This approach is found to yield results having a uniformly high quality, being consistent with empirical equilibrium structures and surpassing calculated values from the coupled cluster model with single, double, and noniterative triple excitations [CCSD(T)] with the basis sets of cc-pCVQZ and cc-pVQZ. The nonrelativistic equilibrium atomization energies are also presented to assess performance of the calculated methods. The mean absolute deviations regarding experimental atomization energy are 0.16 and 0.21 kcal/mol for the Monte Carlo and CCSD(T)/cc-pCV(56)Z calculations, respectively.

Electron affinities with diffusion quantum Monte Carlo for C2 and BO molecules

To verify the performance of the fixed-node diffusion quantum Monte Carlo method in electron affinities calculations, the adiabatic electron affinities of C(2) and BO molecules calculated by the fixed-node Ornstein-Uhlenbeck diffusion quantum Monte Carlo simulations guiding by trial wave functions constructed from the floating spherical Gaussian orbitals and spherical Gaussian geminals are presented in this work. The random walk based results, 3.264(43) and 2.507(32) eV for C(2) and BO, respectively, are compared with the available best experimental determined values, 3.269(6) and 2.508(8) eV for C(2) and BO, respectively, and the results of other theoretical calculations.

Random walks based approach to study on molecular structures and torsional barrier heights of hydrogen peroxide

The optimized structures at the equilibrium, cis and trans configurations and the barrier heights for the cis and trans saddle points for hydrogen peroxide, obtained using the diffusion quantum Monte Carlo simulations are reported. The Monte Carlo calculated torsional barriers, Δ E trans = 386(26) cm −1 and Δ E cis = 2526(41) cm −1 are in very good agreement with experimental values, Δ E trans = 385/387 cm −1 and Δ E cis = 2488/2563 cm −1.