Diffusion Quantum Monte Carlo Calculations Research Articles

Quantum Monte Carlo study of three-dimensional Coulomb complexes: Trions and biexcitons, hydrogen molecules and ions, helium hydride cations, and positronic and muonic complexes

Three-dimensional (3D) excitonic complexes influence the optoelectronic properties of bulk semiconductors. More generally, correlated few-particle molecules and ions, held together by pairwise Coulomb potentials, play a fundamental role in a variety of fields in physics and chemistry. Based on statistically exact diffusion quantum Monte Carlo calculations, we have studied excitonic three- and four-body complexes (trions and biexcitons) in bulk 3D semiconductors as well as a range of small molecules and ions in which the nuclei are treated as quantum particles on an equal footing with the electrons. We present interpolation formulas that predict the binding energies of these complexes either in bulk semiconductors or in free space. By evaluating pair distribution functions within quantum Monte Carlo simulations, we examine the importance of harmonic and anharmonic vibrational effects in small molecules.

Point defect formation energies in graphene from diffusion quantum Monte Carlo and density functional theory

Density functional theory (DFT) is widely used to study defects in monolayer graphene with a view to applications ranging from water filtration to electronics to investigation of radiation damage in graphite moderators. To assess the accuracy of DFT in such applications, we report diffusion quantum Monte Carlo (DMC) calculations of the formation energies of some common and important point defects in monolayer graphene: monovacancies, Stone-Wales defects, and silicon substitutions. We find that standard DFT methods underestimate monovacancy formation energies by around 1 eV. The disagreement between DFT and DMC is somewhat smaller for Stone-Wales defects and silicon substitutions. We examine vibrational contributions to the free energies of formation for these defects, finding that vibrational effects are non-negligible. Finally, we compare the DMC atomization energies of monolayer graphene, monolayer silicene, and bulk silicon, finding that bulk silicon is significantly more stable than monolayer silicene by 0.7522(5) eV per atom.

Diffusion quantum Monte Carlo method on diradicals using single- and multi-determinant-Jastrow trial wavefunctions and different orbitals.

In this work, the diffusion quantum Monte Carlo (DMC) method is employed to calculate the energies of singlet and triplet states for a series of organic diradicals and diatomic diradicals with π2 configuration. Single-determinant-Jastrow (SDJ) trial wavefunctions for triplet states, two-determinant-Jastrow (2DJ) trial wavefunctions for the singlet states, and multi-determinant-Jastrow (MDJ) trial wavefunctions are employed in DMC calculations using restricted open-shell B3LYP (ROB3LYP) orbitals, complete-active-space self-consistent field (CASSCF) orbitals, state-average CASSCF orbitals, or frozen-CASSCF orbitals. Our results show that DMC energies using either SDJ/2DJ or MDJ with ROB3LYP orbitals are close to or lower than those with the other orbitals for organic diradicals, while they are not very sensitive to the employed orbitals for diatomic diradicals. Furthermore, using MDJ can reduce DMC energies to some extent for most of the investigated organic diradicals and some diatomic diradicals. The importance of MDJ on DMC energies can be estimated based on the percentage of main determinants in the CASCI wavefunction. On the other hand, singlet-triplet gaps can be calculated reasonably with DMC using MDJ with a mean absolute error of less than 2 kcal/mol with all these orbitals. CASCI wavefunctions using density functional theory orbitals are preferred in constructing MDJ trial wavefunctions in practical DMC calculations since it is easier to obtain such wavefunctions than CASSCF methods.

Enhancing the Accuracy of Ab Initio Molecular Dynamics by Fine Tuning of Effective Two-Body Interactions: Acetonitrile as a Test Case.

Grimme's dispersion-corrected density functional theory (DFT-D) methods have emerged among the most practical approaches to perform accurate quantum mechanical calculations on molecular systems ranging from small clusters to microscopic and mesoscopic samples, i.e., including hundreds or thousands of molecules. Moreover, DFT-D functionals can be easily integrated into popular ab initio molecular dynamics (MD) software packages to carry out first-principles condensed-phase simulations at an affordable computational cost. Here, starting from the well-established D3 version of the dispersion-correction term, we present a simple protocol to improve the accurate description of the intermolecular interactions of molecular clusters of growing size, considering acetonitrile as a test case. Optimization of the interaction energy was performed with reference to diffusion quantum Monte Carlo calculations, successfully reaching the same inherent accuracy of the latter (statistical error of ∼0.1 kcal/mol per molecule). The refined DFT-D3 model was then used to perform ab initio MD simulations of liquid acetonitrile, again showing significant improvements toward available experimental data with respect to the default correction.

Quantitative electron-electron interaction using local field factors from quantum Monte Carlo calculations

The effective electron-electron interaction in electron gas depends on both the density and spin local field factors. Variational diagrammatic quantum Monte Carlo calculations of the spin local field factor are reported. These are used together with the charge local field factor from previous diffusion quantum Monte Carlo calculations to quantitatively present the full effective spin-dependent electron-electron interaction in three-dimensional electron gas. Very simple quadratic formulas are presented for the local field factors that quantitatively produce all of the response functions of the electron gas at metallic densities. Exchange and correlation become increasingly important at low densities. At the compressibility divergence at ${r}_{s}=5.25$, both the direct (screened Coulomb) term and the charge-dependent exchange term in the electron-electron interaction at $q=0$ are separately divergent. However, due to large cancellations, their difference is finite, well behaved, and much smaller than either term separately. As a result, the spin contribution to the electron-electron interaction becomes an important factor. The static electron-electron interaction is repulsive as a function of density but is less repulsive for electrons with parallel spins. The effect of allowing a deformable, rather than rigid, positive background is shown to be as quantitatively important as exchange and correlation. As a simple concrete example, the electron-electron interaction is calculated using the measured bulk modulus of the alkali metals with a linear phonon dispersion. The net electron-electron interaction in lithium is attractive for wave vectors $0\ensuremath{-}2{k}_{F}$, which suggests superconductivity, and is mostly repulsive for the other alkali metals.

Diffusion Monte Carlo method on small boron clusters using single- and multi- determinant-Jastrow trial wavefunctions.

Multireference character in some small boron clusters could be significant, and a previous all-electron fixed-node diffusion quantum Monte Carlo (FN-DMC) calculation with the single-determinant-Jastrow (SDJ) trial wavefunction shows that the atomization energy (AE) of B4 + is overestimated by about 1.4 eV compared with the coupled cluster method with single, doubles, and perturbative triples [CCSD(T)] results. All-electron FN-DMC calculations and those with the pseudopotential (PP) using SDJ and multi-determinant-Jastrow (MDJ) trial wavefunctions with B3LYP orbitals as well as CC calculations at different levels are carried out on Bn Q (n = 1-5, Q = -1, 0, 1) clusters. The obtained FN-DMC energies indicate that the node error of the employed SDJ trial wavefunction in all-electron calculations is different from that with the PP for some clusters. The error of AEs and dissociation energies (DEs) from all-electron FN-DMC calculations is larger than that with the PP when the SDJ trial wavefunction is employed, while errors of CC methods do not depend on whether the PP is used. AEs and DEs of the boron clusters are improved significantly when MDJ trial wavefunctions are used in both all-electron calculations and those with the PP, and their error is similar to that of CCSD(T) compared with CCSDT(Q) results. On the other hand, reasonable adiabatic electron detachment energies (ADEs) and ionization potentials (AIPs) are achieved with FN-DMC using SDJ trial wavefunctions and MDJ is less effective on ADEs and AIPs. Furthermore, the relative energy between two structures of B9 - is predicted reliably with FN-DMC using the SDJ trial wavefunction and the effect of MDJ is negligible, while density functional theory results using different exchange-correlation functionals differ significantly.

Perspectives on van der Waals Density Functionals: The Case of TiS2.

The van der Waals interaction is of foundational importance for a wide variety of physical systems. In particular, van der Waals forces lie at the heart of potential device technologies that may be realized from the functional organization of layered two-dimensional (2D) nanomaterials. For intermediate to large-scale applications modeling, van der Waals density functionals have become the de facto choice for first-principles calculations. In particular, the vdW-DF family of functionals have provided a systematic approach to this theoretically challenging problem. While much progress has been made, there remains room for improvement in the microscopic description of vdW forces from these density functionals. In this work, we compute benchmark results for the binding energy and the electronic density response to binding in TiS2 via accurate diffusion quantum Monte Carlo calculations. We compare these benchmark data to results obtained from local, semilocal, and van der Waals functionals. In particular, we gauge the quality of the original vdW-DF/vdW-DF2 functionals, as well as updated variants such as vdW-DF-C09, vdW-DF-optB88, vdW-DF-optB86b, and vdW-DF2-B86R. We find a close relationship between the accuracy of predicted interlayer separation distances and binding energies for TiS2, with the vdW-DF-optB88 functional performing very well in terms of both quantities. In general, the more recently developed functionals are systematic improvements over older ones. However, when considering the response of the electron density to binding, we find that local-density approximation (LDA) and PBEsol generally outperform the vdW-DF functionals in describing the interlayer charge accumulation with vdW-DF-C09 variants performing the best overall.

(INVITED) First-principles calculations of quantum transitions at local centers

Fundamental quantum transitions at local centers are the basis for numerous technologies, including LED, laser, photodetection, and quantum computing. However, accurate theoretical calculations of basic parameters related to these processes remain challenging. In this paper, we review recent developments in first-principles calculations of electronic transitions at local centers in solid-state materials. Specifically, we discuss (1) promising results on hybrid density functional calculations of self-trapped excitons in low-dimensional metal halides and 5d-4f transitions at rare-earth dopants in solids as well as (2) the more challenging case of intra-d-shell transitions at transition-metal ions, which require a higher-level many-body theory to better account for the electron correlation effect. Diffusion quantum Monte Carlo calculations of Mn4+ emissions in oxides and halides show excellent agreement with experimental results, demonstrating the potentially predictive power in calculating d-d transitions at transition-metal ions and other optical transitions at local centers.

Accurate spectroscopic properties by diffusion quantum Monte Carlo calculations

The capability of Diffusion Quantum Monte Carlo (DMC) to produce high quality potential energy curve (PEC) was evaluated. H2+, HeH+ and LiH PECs were built by all-electron fixed-node DMC calculations. Trial wave functions were obtained from Hartree-Fock (HF) (H2+), MCSCF and CI (HeH+ and LiH) calculations multiplied by Jastrow factor. The quality of these generated PECs was analyzed throughout equilibrium distance, dissociation energy, vibrational energies and rovibrational spectroscopic constants (ωe, ωexe, ωeye, αe, γe and Be). The Discrete Variable Representation (DVR) and the Dunham approaches were used to determine the rovibrational spectroscopic constants. The PECs and the aforementioned properties were also obtained by the following methods: MCSCF/aug-cc-pV5Z (LiH), CCSD(T)/aug-cc-pV5Z (HeH+ and LiH) and HF (H2+ and HeH+) levels. The results of these DMC computations, specially the DMC-DVR procedure, are the most accurate among others DMC calculations available in the literature for these systems. They suggest that DMC can be used to achieve accurate PECs to produce spectroscopic properties with the same level of accuracy of theoretical benchmarks and experimental data of the literature.

Variational and diffusion quantum Monte Carlo calculations with the CASINO code.

We present an overview of the variational and diffusion quantum Monte Carlo methods as implemented in the casino program. We particularly focus on developments made in the last decade, describing state-of-the-art quantum Monte Carlo algorithms and software and discussing their strengths and weaknesses. We review a range of recent applications of casino.

Diffusion quantum Monte Carlo calculations with a recent generation of effective core potentials for ionization potentials and electron affinities

Pseudopotentials are an essential ingredient in diffusion quantum Monte Carlo (DMC) calculations to increase efficiency substantially. A new generation of effective core potentials (ccECP) has been recently developed for DMC calculations. In this paper, performance of DMC using ccECP potentials on total energies, ionization potentials (IPs) and electron affinities (EAs) of some second- and third-row atoms and molecules is investigated systematically with different types of trial wave functions. DMC results are compared with those of high-level coupled-cluster methods extrapolated to complete basis set limit (CC-CBS). Error of ccECP potentials on IPs and EAs is also evaluated through a comparison with those from all-electron calculations. Our results show that mean errors in DMC energies with the ccECP potentials are smaller than those with the pseudopotentials developed by Burkatzki, Filippi, and Dolg (BFD), when the same type of trial wave functions is adopted. Mean absolute deviations (MADs) on IPs of DMC compared with those of CC-CBS are about 1.6 kcal/mol with single-determinant-Jastrow trial wave functions, and 1 kcal/mol with multideterminant-Jastrow trial wave functions using either ccECP or BFD potentials. MADs on EAs with DMC using the ccECP potentials are about 1 kcal/mol and slightly larger than those with the BFD potentials. Our results show that ccECP potentials are able to provide reliable IPs and EAs in DMC calculations. Accuracy of IPs and EAs from DMC calculations using ccECP potentials is similar to that with the BFD potentials, although mean error in total DMC energies with ccECP potentials is smaller. Furthermore, error of ccECP potentials in DMC calculations on IPs and EAs is smaller than that of BFD potentials compared with all-electron results.

Tail-regression estimator for heavy-tailed distributions of known tail indices and its application to continuum quantum Monte Carlo data.

Standard statistical analysis is unable to provide reliable confidence intervals on expectation values of probability distributions that do not satisfy the conditions of the central limit theorem. We present a regression-based estimator of an arbitrary moment of a probability distribution with power-law heavy tails that exploits knowledge of the exponents of its asymptotic decay to bypass this issue entirely. Our method is applied to synthetic data and to energy and atomic force data from variational and diffusion quantum Monte Carlo calculations, whose distributions have known asymptotic forms [J.R. Trail, Phys. Rev. E 77, 016703 (2008)PLEEE81539-375510.1103/PhysRevE.77.016703; A. Badinski et al., J. Phys.: Condens. Matter 22, 074202 (2010)JCOMEL0953-898410.1088/0953-8984/22/7/074202]. We obtain convergent, accurate confidence intervals on the variance of the local energy of an electron gas and on the Hellmann-Feynman force on an atom in the all-electron carbon dimer. In each of these cases the uncertainty on our estimator is 45% and 60 times smaller, respectively, than the nominal (ill-defined) standard error.

Quantum Monte Carlo Studies of CO Adsorption on Transition Metal Surfaces

The adsorptions of CO molecule on various fcc(111) surfaces (Rh, Ir, Pt, and Cu) have been studied by diffusion quantum Monte Carlo (DMC) calculations, and the results show that the top site is the most stable adsorption site on all the four surfaces, in agreement with experiments. In particular, the site preference including the bridge site for CO/Pt(111) is predicted, i.e., the top site is most preferred followed by the bridge site while the hollow sites are much less favorable, in accordance with the existing experimental observations of the bridge-site adsorption, yet never on the hollow sites. Compared to the DMC results, density functional theory (DFT) calculations with the generalized-gradient approximation (GGA) predict very similar adsorption energies on the top site, but they overestimate those on the bridge and hollow sites. That is, although the nonlocal exchange-correlation contribution is small for the single-coordinated top-site adsorption, it is essential and required to be properly included for a correct description of the higher coordinated bridge- and hollow-sites adsorptions. These altogether explain why the top site adsorption for CO on Rh, Pt, and Cu(111) surfaces was not predicted correctly by the previous standard local or semilocal DFT calculations.

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.

Accuracy of the Heyd-Scuseria-Ernzerhof hybrid functional to describe many-electron interactions and charge localization in semiconductors

Hybrid functionals, which mix a fraction of Hartree-Fock (HF) exchange with local or semilocal exchange, have become increasingly popular in quantum chemistry and computational materials science. Here, we assess the accuracy of the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional to describe many-electron interactions and charge localization in semiconductors. We perform diffusion quantum Monte Carlo (DMC) calculations to obtain the accurate ground-state spin densities of the negatively charged (SiV)$^-$ and the neutral (SiV)$^0$ silicon-vacancy center in diamond, and of the cubic silicon carbide (3C-SiC) with an extra electron. We compare our DMC results with those obtained with the HSE functional and find a good agreement between both methods for (SiV)$^-$ and (SiV)$^0$, whereas the correct description of 3C-SiC with an extra electron crucially depends on the amount of HF exchange included in the functional. Also, we examine the case of the neutral Cd vacancy in CdTe, for which we assess the performance of HSE against the many-body \emph{GW} approximation for the description of the position of the defect states in the band gap.

Diffusion quantum Monte Carlo and density functional calculations of the structural stability of bilayer arsenene.

We have studied the structural stability of monolayer and bilayer arsenene (As) in the buckled (b) and washboard (w) phases with diffusion quantum Monte Carlo (DMC) and density functional theory (DFT) calculations. DMC yields cohesive energies of 2.826(2) eV/atom for monolayer b-As and 2.792(3) eV/atom for w-As. In the case of bilayer As, DMC and DFT predict that AA-stacking is the more stable form of b-As, while AB is the most stable form of w-As. The DMC layer-layer binding energies for b-As-AA and w-As-AB are 30(1) and 53(1) meV/atom, respectively. The interlayer separations were estimated with DMC at 3.521(1) Å for b-As-AA and 3.145(1) Å for w-As-AB. A comparison of DMC and DFT results shows that the van der Waals density functional method yields energetic properties of arsenene close to DMC, while the DFT + D3 method closely reproduced the geometric properties from DMC. The electronic properties of monolayer and bilayer arsenene were explored with various DFT methods. The bandgap values vary significantly with the DFT method, but the results are generally qualitatively consistent. We expect the present work to be useful for future experiments attempting to prepare multilayer arsenene and for further development of DFT methods for weakly bonded systems.

Stability of trions in coupled quantum wells modeled by two-dimensional bilayers

We report variational and diffusion quantum Monte Carlo calculations of the binding energies of indirect trions and biexcitons in ideal two-dimensional bilayer systems within the effective-mass approximation, and with a Coulomb $1/r$ interaction between charge carriers. The critical layer separation at which trions become unbound has been studied for various electron-hole mass ratios, and found to be over an order of magnitude larger than the critical layer separation for biexcitons.

Erratum: Binding energies of trions and biexcitons in two-dimensional semiconductors from diffusion quantum Monte Carlo calculations [Phys. Rev. B 95, 081301(R) (2017)

Received 24 August 2017DOI:https://doi.org/10.1103/PhysRevB.96.119902©2017 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBand gapBiexcitonsExcitonsPhysical SystemsQuantum wellsSemiconductorsTransition metal dichalcogenidesTechniquesDiffusion quantum Monte CarloCondensed Matter, Materials & Applied Physics

Full Wave Function Optimization with Quantum Monte Carlo and Its Effect on the Dissociation Energy of FeS.

Diffusion quantum Monte Carlo calculations with partial and full optimization of the guide function are carried out for the dissociation of the FeS molecule. For the first time, quantum Monte Carlo orbital optimization for transition metal compounds is performed. It is demonstrated that energy optimization of the orbitals of a complete active space wave function in the presence of a Jastrow correlation function is required to obtain agreement with the experimental dissociation energy. Furthermore, it is shown that orbital optimization leads to a 5Δ ground state, in agreement with experiments but in disagreement with other high-level ab initio wave function calculations which all predict a 5Σ+ ground state. The role of the Jastrow factor in DMC calculations with pseudopotentials is investigated. The results suggest that a large Jastrow factor may improve the DMC accuracy substantially at small additional cost.

A benchmark study of Li2+, Li2−, LiH+ and LiH−: Quantum Monte-Carlo and coupled-cluster computations

For the first time, very accurate potential energy curves (PEC) have been computed by the all electron Diffusion Quantum Monte-Carlo (DMC) and CCSD(T) methods for the ground-state of the Li2+, Li2−, LiH+ and LiH− ions. In addition to the molecular moieties, Li+, Li−, H− ions have also been studied by the aforementioned methods. The employed trial wave functions for the DMC calculations are a multiplication of multi-configuration expansions which are prepared by the Complete Active Space Self-Consistent Field method (CASSCF) with the aug-cc-pVTZ and the aug-cc-pCVTZ basis sets and the Jastrow factor. The CCSD(T) computations with all electrons correlated, have been performed by the cc-pV5Z, aug-cc-pCVQZ and aug-cc-pV6Z basis sets. In addition, the correlation energies for all of the studied atomic and molecular species have been calculated. For the studied molecular species, the Hartree-Fock energies in the complete basis set limit (ECBS) have not been reported. In this work for each of the molecular systems, the most negative energy which has been computed by the cc-pV5Z or aug-cc-pCVQZ basis sets has been used. The obtained PECs by the DMC and CCSD(T) methods are used to calculate the vibration-rotation energy levels and spectroscopic constants.