Fixed-node Diffusion Monte Carlo Research Articles

Many-body theory calculations of positron binding to hydrogen cyanide

Positron bound state properties in hydrogen cyanide are studied via many-body theory calculations that account for strong positron-electron correlations including positron-induced polarization, screening of the electron–positron Coulomb interaction, virtual-positronium formation and positron–hole repulsion. Specifically, the Dyson equation is solved using a Gaussian basis, with the positron self-energy in the field of the molecule calculated using the Bethe–Salpeter equations for the two-particle and particle–hole propagators. The present results suggest near cancellation of screening corrections to the bare polarization, and the non-negligible role of the positron–hole interaction. There are no existing measurements to compare to for HCN. Previous configuration interaction (CI) and fixed-node diffusion Monte Carlo (FN-DMC) calculations give positron binding energies in the range 35–44 meV, most of which used a single even-tempered basis centred near the nitrogen atom. Using a similar single-centre positron basis we calculate a positron binding energy of 41 meV, in good agreement. However, we find that including additional basis centres gives an improved description of the positron wave function near the nuclei, and results in a converged binding energy in the range 63–73 meV (depending on geometry and approximation to the positron–molecule correlation potential used).Graphical

Quantifying the Impact of Halogenation on Intermolecular Interactions and Binding Modes of Aromatic Molecules.

Halogenation of aromatic molecules is frequently used to modulate intermolecular interactions with ramifications for optoelectronic and mechanical properties. In this work, we accurately quantify and understand the nature of intermolecular interactions in perhalogenated benzene (PHB) clusters. Using benchmark binding energies from the fixed-node diffusion Monte Carlo (FN-DMC) method, we show that generalized Kohn-Sham semicanonical projected random phase approximation (GKS-spRPA) plus approximate exchange kernel (AKX) provides reliable interaction energies with mean absolute error (MAE) of 0.23 kcal/mol. Using the GKS-spRPA+AXK method, we quantify the interaction energies of several binding modes of PHB clusters ((C6X6)n; X = F, Cl, Br, I; n = 2, 3). For a given binding mode, the interaction energies increase 3-4 times from X = F to X = I; the X-X binding modes have energies in the range of 2-4 kcal/mol, while the π-π binding mode has interaction energies in the range of 4-12 kcal/mol. SAPT-DFT-based energy decomposition analysis is then used to show that the equilibrium geometries are dictated primarily by the dispersion and exchange interactions. Finally, we test the accuracy of several dispersion-corrected density functional approximations and show that only the r2SCAN-D4 method has a low MAE and correct long-range behavior, which makes it suitable for large-scale simulations and for developing structure-function relationships of halogenated aromatic systems.

Towards the ground state of molecules via diffusion Monte Carlo on neural networks

Diffusion Monte Carlo (DMC) based on fixed-node approximation has enjoyed significant developments in the past decades and become one of the go-to methods when accurate ground state energy of molecules and materials is needed. However, the inaccurate nodal structure hinders the application of DMC for more challenging electronic correlation problems. In this work, we apply the neural-network based trial wavefunction in fixed-node DMC, which allows accurate calculations of a broad range of atomic and molecular systems of different electronic characteristics. Our method is superior in both accuracy and efficiency compared to state-of-the-art neural network methods using variational Monte Carlo (VMC). We also introduce an extrapolation scheme based on the empirical linearity between VMC and DMC energies, and significantly improve our binding energy calculation. Overall, this computational framework provides a benchmark for accurate solutions of correlated electronic wavefunction and also sheds light on the chemical understanding of molecules.

Isotope Effects on Ground and Excited States of Ethyl Cation, H+(C2H4)

The structure and spectra of ethyl cation, H+(C2H4), and its deuterated analogues are investigated using diffusion Monte Carlo (DMC). These calculations all show that the ground state wave function for H+(C2H4) is localized near the minimum energy configuration in which the excess proton is in a bridging configuration, although the amplitude of the vibrational motions of the bridging proton is large. Deuteration of the bridging proton reduces the amplitude of this motion, while deuteration of only the ethylenic hydrogen atoms in H+(C2D4) has little effect on the amplitude of the motion of the bridging proton. Excited states that are accessed by spectroscopically observed transitions in H+(C2H4) are calculated using fixed-node DMC. The calculated and measured frequencies for the states with one quantum of excitation in the ethylenic CH stretching vibrations show good agreement. We also explore the excited state with one quantum of excitation in the proton transfer vibration of the bridging proton and obtain a frequency of 616 cm-1 for H+(C2H4). This frequency increases to 629 cm-1 in H+(C2D4). Deuteration decreases this frequency to 491 and 495 cm-1 in D+(C2H4) and D+(C2D4), respectively. The effects of partial deuteration on the frequencies of the CH stretching vibrations, and the corresponding probability amplitudes are also explored. Finally, we report the vibrationally averaged rotational constants for the four isotopologues of ethyl cation considered in this study.

Coupled Cluster Benchmarking of Large Noncovalent Complexes in L7 and S12L as Well as the C60 Dimer, DNA-Ellipticine, and HIV-Indinavir.

In this work, we report the benchmark binding energies of the seven complexes within the L7 data set, six host-guest complexes from the S12L data set, a C60 dimer, the DNA-ellipticine intercalation complex, and the largest system of the study, the HIV-indinavir system, which contained 343 atoms or 139 heavy atoms. The high-quality values reported were obtained via a focal point method that relies on the canonical form of second-order Møller-Plesset theory and the domain-based local pair natural orbital scheme for the coupled cluster with single double and perturbative triple excitations [DLPNO-CCSD(T)] extrapolated to the complete basis set (CBS) limit. The results in this work not only corroborate but also improve upon some previous benchmark values for large noncovalent complexes albeit at a relatively steep cost. Although local CCSD(T) and the largely successful fixed-node diffusion Monte Carlo (FN-DMC) have been shown to generally agree for small- to medium-size systems, a discrepancy in their reported binding energy values arises for large complexes, where the magnitude of the disagreement is a definite cause for concern. For example, the largest deviation in the L7 data set was 2.8 kcal/mol (∼10%) on the low end in C3GC. Such a deviation only grows worse in the S12L set, which showed a difference of up to 10.4 kcal/mol (∼25%) by a conservative estimation in buckycatcher-C60. The DNA-ellipticine complex also generated a disagreement of 4.4 kcal/mol (∼10%) between both state-of-the-art methods. The disagreement between local CCSD(T) and FN-DMC in large noncovalent complexes shows that it is urgently needed to have the canonical CCSD(T), the Monte Carlo CCSD(T), or the full configuration interaction quantum Monte Carlo approaches available to large systems on the hundred-atom scale to solve this dilemma. In addition, the performances of cheaper popular computational methods were assessed for the studied complexes with respect to DLPNO-CCSD(T)/CBS. r2SCAN-3c, B97M-V, and PBE0+D4 work well in large noncovalent complexes in this work, and GFN2-xTB performs well in π-π stacking complexes. B97M-V is the most reliable computationally efficient approach to predicting noncovalent interactions for large complexes, being the only one to have binding errors within the so-called 1 kcal/mol "chemical accuracy". The benchmark interaction energies of these host-guest complexes, molecular materials, and biological systems with electronic and medicinal implications provide crucial reference data for the improvement of current and future lower-cost methods.

Diffusion Monte Carlo Study on Relative Stabilities of Boron Nitride Polymorphs

Although Boron nitride (BN) is a well-known compound widely used for engineering and scientific purposes, the phase stability of its polymorphs, one of its most fundamental properties, is still under debate. The ab initio determination of the ground state of the BN polymorphs, such as hexagonal and zinc-blende, is difficult because of the elusive Van der Waals interaction, which plays a decisive role in some of the polymorphs, making quantitative prediction highly challenging. Hence, despite multiple theoretical studies, there has been no consensus on the ground state yet, primarily due to contradicting reports. In this study, we apply a state-of-the-art ab initio framework - fixed-node diffusion Monte Carlo (FNDMC), to four well known BN polymorphs, namely hexagonal, rhombohedral, wurtzite, and zinc-blende BNs. Our FNDMC calculations show that hBN is thermodynamically the most stable among the four polymorphs at 0 K as well as at 300K. This result agrees with the experimental data of Corrigan~{\it et al.} and Fukunaga. The conclusions are consistent with those obtained using other high-level methods, such as coupled cluster. We demonstrate that the FNDMC is a powerful method to address polymorphs that exhibit bonds of various forms. It also provides valuable information, like reliable reference energies, when reliable experimental data are missing or difficult to access. Our findings should promote the application of FNDMC for other van der Waals materials.

A quantum Monte Carlo study of systems with effective core potentials and node nonlinearities

We study beryllium dihydride (BeH$_2$) and acetylene (C$_2$H$_2$) molecules using real-space diffusion Monte Carlo (DMC) method. The molecules serve as perhaps the simplest prototypes that illustrate the difficulties with biases in the fixed-node DMC calculations that might appear with the use of effective core potentials (ECPs) or other nonlocal operators. This is especially relevant for the recently introduced correlation consistent ECPs (ccECPs) for $2s2p$ elements. Corresponding ccECPs exhibit deeper potential functions due to higher fidelity to all-electron counterparts, which could lead to larger local energy fluctuations. We point out that the difficulties stem from issues that are straightforward to address by upgrades of basis sets, use of T-moves for nonlocal terms, inclusion of a few configurations into the trial function and similar. The resulting accuracy corresponds to the ccECP target (chemical accuracy) and it is in consistent agreement with independent correlated calculations. Further possibilities for upgrading the reliability of the DMC algorithm and considerations for better adapted and more robust Jastrow factors are discussed as well.

Diffusion Monte Carlo evaluation of disiloxane linearisation barrier.

The disiloxane molecule is a prime example of silicate compounds containing the Si-O-Si bridge. The molecule is of significant interest within the field of quantum chemistry, owing to the difficulty in theoretically predicting its properties. Herein, the linearisation barrier of disiloxane is investigated using a fixed-node diffusion Monte Carlo (FNDMC) approach, which is one of the most reliable ab initio methods in accounting for the electronic correlation. Calculations utilizing the density functional theory (DFT) and the coupled cluster method with single and double substitutions, including noniterative triples (CCSD(T)) are carried out alongside FNDMC for comparison. It is concluded that FNDMC successfully predicts the disiloxane linearisation barrier and does not depend on the completeness of the basis-set as much as DFT or CCSD(T), thus establishing its suitability.

Probing the ground-state structural transition in small lithium clusters by quantum Monte Carlo simulations.

The ground-state structural transition in small lithium clusters Lin (n = 4 - 6) is analyzed based on the many-body expansion of the interaction energy using the total energy calculated by the fixed-node diffusion Monte Carlo (FN-DMC) simulations. The results show that the transition from 2D to 3D structure occurs through an intricate competition of attractive and repulsive interaction energies. As the structure dimensionality increases from 2D to 3D, the electron-correlation contribution to the interaction energy in the isomer of the ground-state structure is always the largest.

CrI3 revisited with a many-body ab initio theoretical approach

${\mathrm{CrI}}_{3}$ has recently been shown to exhibit low-dimensional, long-range magnetic ordering from few layers to single layers of ${\mathrm{CrI}}_{3}$. The properties of ${\mathrm{CrI}}_{3}$ bulk and few-layered systems are uniquely defined by a combination of short-range intralayer and long-range interlayer interactions, including strong correlations, exchange, and spin-orbit coupling. Unfortunately, both the long-range van der Waals interactions, which are driven by dynamic, many-body electronic correlations, and the competing strong intralayer correlations, present a formidable challenge for the local or semilocal mean-field approximations employed in workhorse electronic structure approaches like density-functional theory. In this paper we employ a sophisticated many-body approach that can simultaneously describe long- and short-range correlations. We establish that the fixed-node diffusion Monte Carlo (FNDMC) method reproduces the experimental interlayer separation distance of bulk ${\mathrm{CrI}}_{3}$ for the high-temperature monoclinic phase with a reliable prediction of the interlayer binding energy. We subsequently employed the FNDMC results to benchmark the accuracy of several density-functional theory exchange-correlation approximations.

Fundamental gap of fluorographene by many-body GW and fixed-node diffusion Monte Carlo methods.

Fluorographene (FG) is a promising graphene-derived material with a large bandgap. Currently existing predictions of its fundamental gap (Δf) and optical gap (Δopt) significantly vary when compared with experiment. We provide here an ultimate benchmark of Δf for FG by many-body GW and fixed-node diffusion Monte Carlo (FNDMC) methods. Both approaches independently arrive at Δf ≈ 7.1 ± 0.1 eV. In addition, the Bethe-Salpeter equation enabled us to determine the first exciton binding energy, Eb = 1.92 eV. We also point to the possible misinterpretation problem of the results obtained for gaps of solids by FNDMC with single-reference trial wave functions of Bloch orbitals. We argue why instead of Δopt, in the thermodynamic limit, such an approach results in energy differences that rather correspond to Δf, and we also outline conditions when this case actually applies.

Toward a systematic improvement of the fixed-node approximation in diffusion Monte Carlo for solids-A case study in diamond.

While Diffusion Monte Carlo (DMC) is in principle an exact stochastic method for ab initio electronic structure calculations, in practice, the fermionic sign problem necessitates the use of the fixed-node approximation and trial wavefunctions with approximate nodes (or zeros). This approximation introduces a variational error in the energy that potentially can be tested and systematically improved. Here, we present a computational method that produces trial wavefunctions with systematically improvable nodes for DMC calculations of periodic solids. These trial wavefunctions are efficiently generated with the configuration interaction using a perturbative selection made iteratively (CIPSI) method. A simple protocol in which both exact and approximate results for finite supercells are used to extrapolate to the thermodynamic limit is introduced. This approach is illustrated in the case of the carbon diamond using Slater-Jastrow trial wavefunctions including up to one million Slater determinants. Fixed-node DMC energies obtained with such large expansions are much improved, and the fixed-node error is found to decrease monotonically and smoothly as a function of the number of determinants in the trial wavefunction, a property opening the way to a better control of this error. The cohesive energy extrapolated to the thermodynamic limit is in close agreement with the estimated experimental value. Interestingly, this is also the case at the single-determinant level, thus, indicating a very good error cancellation in carbon diamond between the bulk and atomic total fixed-node energies when using single-determinant nodes.

Taming the fixed-node error in diffusion Monte Carlo via range separation.

By combining density-functional theory (DFT) and wave function theory via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixed-node diffusion Monte Carlo (FN-DMC) energies with compact multi-determinant trial wave functions. In particular, we combine here short-range exchange-correlation functionals with a flavor of selected configuration interaction known as configuration interaction using a perturbative selection made iteratively (CIPSI), a scheme that we label RS-DFT-CIPSI. One of the take-home messages of the present study is that RS-DFT-CIPSI trial wave functions yield lower fixed-node energies with more compact multi-determinant expansions than CIPSI, especially for small basis sets. Indeed, as the CIPSI component of RS-DFT-CIPSI is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional CIPSI calculation. Importantly, by performing various numerical experiments, we evidence that the RS-DFT scheme essentially plays the role of a simple Jastrow factor by mimicking short-range correlation effects, hence avoiding the burden of performing a stochastic optimization. Considering the 55 atomization energies of the Gaussian-1 benchmark set of molecules, we show that using a fixed value of μ = 0.5 bohr-1 provides effective error cancellations as well as compact trial wave functions, making the present method a good candidate for the accurate description of large chemical systems.

Fractional Charge by Fixed-Node Diffusion MonteCarlo Method.

Fixed-node diffusion MonteCarlo (FNDMC) method is a stochastic quantum many-body approach that has a great potential in electronic structure theory. We examine how FNDMC total energy E(N) satisfies exact constraints, linearity and derivative discontinuity, versus fractional electron number N, if combined with mean-field trial wave functions that miss such features. H and Cl atoms with fractional charge reveal that FNDMC method is well able to restore the piecewise linearity of E(N). The method uses ensemble and projector ingredients to achieve the correct charge localization. A water-solvated Cl^{-} complex illustrates superior performance of FNDMC method for charged noncovalent systems.

Quantum Monte Carlo simulation for the many-body decomposition of the interaction energy and electron correlation of small superalkali lithium clusters.

Using the fixed-node diffusion Monte Carlo (FN-DMC) method, we calculate the total energy of small lithium clusters Lin (n = 2-6) to obtain the many-body decomposition of the interaction energy of 2- up to 6-body interactions. The obtained many-body decomposition of the interaction energy shows an alternating series with even and odd terms of attractive and repulsive contributions, respectively. The two-body attractive interactions guarantee the stability of the Li2, Li3, and Li4 clusters. For larger clusters Li5 and Li6, the 4-body attractive interactions are required for their stabilization once the strength of the 3-body repulsive interactions overwhelms that of the 2-body attractive ones. With increasing the cluster size, the additive and nonadditive contributions to the interaction energy increase linearly in magnitude but with different slopes for the two-dimensional (2D) planar and three-dimensional (3D) cagelike clusters. The significant increment in nonadditive effects from the 4-atom to the 5-atom cluster has driven the structural transition from 2D to 3D. Combining the FN-DMC calculations with the Hartree-Fock many-body decomposition of the interaction energy, we extract the correlation effects, showing that an odd-even competition pattern in the many-body repulsive and attractive interactions is crucial for the stabilization of the clusters.

Toward Accurate Hydrogen Bonds by Scalable Quantum Monte Carlo.

Single-determinant (SD) fixed-node diffusion Monte Carlo (FNDMC) gains popularity as a benchmark method scalable to large noncovalent systems, although its accuracy limits are not yet fully mapped out. We report on an interesting example of significant SD FNDMC accuracy variations in middle-sized hydrogen-bonded dimer complexes, formic acid (FA) vs methanediol (MD), distinct by the maximum bond order (2 vs 1). While the traditional SD FNDMC schemes based on bias cancellation are capable of achieving benchmark (2%) accuracy for MD, this has not been the case for FA. We identify the leading systematic error source in energy differences and show that suitably designed Jastrow factors enable SD FNDMC to reach the reference accuracy for FA. This work clearly illustrates the varying accuracy of the present-day SD FNDMC at the 0.1 kcal/mol scale for a particular set of systems but also points out promising routes toward alleviation of these shortcomings, still within the single-reference framework.

Influence of pseudopotentials on excitation energies from selected configuration interaction and diffusion Monte Carlo

Due to their diverse nature, the faithful description of excited states within electronic structure theory methods remains one of the grand challenges of modern theoretical chemistry. Quantum Monte Carlo (QMC) methods have been applied very successfully to ground state properties but still remain generally less effective than other non-stochastic methods for electronically excited states. Nonetheless, we have recently reported accurate excitation energies for small organic molecules at the fixed-node diffusion Monte Carlo (FN-DMC) within a Jastrow-free QMC protocol relying on a deterministic and systematic construction of nodal surfaces using the selected configuration interaction (sCI) algorithm known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). Albeit highly accurate, these all-electron calculations are computationally expensive due to the presence of core electrons. One very popular approach to remove these chemically-inert electrons from the QMC simulation is to introduce pseudopotentials (also known as effective core potentials). Taking the water molecule as an example, we investigate the influence of Burkatzki-Filippi-Dolg (BFD) pseudopotentials and their associated basis sets on vertical excitation energies obtained with sCI and FN-DMC methods. Although these pseudopotentials are known to be relatively safe for ground state properties, we evidence that special care may be required if one strives for highly accurate vertical transition energies. Indeed, comparing all-electron and valence-only calculations, we show that using pseudopotentials with the associated basis sets can induce differences of the order of 0.05 eV on the excitation energies. Fortunately, a reasonable estimate of this shift can be estimated at the sCI level.

Non-orthogonal determinants in multi-Slater-Jastrow trial wave functions for fixed-node diffusion Monte Carlo.

The accuracy and efficiency of ab initio Quantum Monte Carlo (QMC) algorithms benefit greatly from compact variational trial wave functions that accurately reproduce ground state properties of a system. We investigate the possibility of using multi-Slater-Jastrow trial wave functions with non-orthogonal determinants by optimizing identical single particle orbitals independently in separate determinants. As a test case, we compute variational and fixed-node diffusion Monte Carlo (FN-DMC) energies of a C2 molecule. For a given multi-determinant expansion, we find that this non-orthogonal orbital optimization results in a consistent improvement in the variational energy and the FN-DMC energy on the order of a few tenths of an eV. In some cases, fewer non-orthogonal determinants are required compared to orthogonal ones in order to achieve similar accuracy in FN-DMC. Our calculations indicate that trial wave functions with non-orthogonal determinants can improve computed energies in a QMC calculation when compared to their orthogonal counterparts.

Excitation energies from diffusion Monte Carlo using selected configuration interaction nodes.

Quantum Monte Carlo (QMC) is a stochastic method that has been particularly successful for ground-state electronic structure calculations but mostly unexplored for the computation of excited-state energies. Here, we show that within a Jastrow-free QMC protocol relying on a deterministic and systematic construction of nodal surfaces using selected configuration interaction (sCI) expansions, one is able to obtain accurate excitation energies at the fixed-node diffusion Monte Carlo (FN-DMC) level. This evidences that the fixed-node errors in the ground and excited states obtained with sCI wave functions cancel out to a large extent. Our procedure is tested on two small organic molecules (water and formaldehyde) for which we report all-electron FN-DMC calculations. For both the singlet and triplet manifolds, accurate vertical excitation energies are obtained with relatively compact multideterminant expansions built with small (typically double-ζ) basis sets.

Noncovalent Interactions by Fixed-Node Diffusion Monte Carlo: Convergence of Nodes and Energy Differences vs Gaussian Basis-Set Size

Convergence of fixed-node (FN) shape and FN diffusion Monte Carlo (FNDMC) interaction energies is studied vs the Gaussian basis set saturation level in HF and CH4 dimers and one-determinant Slater-Jastrow trial wave functions (ΨT). The tested 25 distinct basis sets obtained by stepwise trimming of aug-VDZ and aug-VTZ bases suggest minimum basis set requirements to achieve reasonable results. A single selected trimmed basis set, about 2 times smaller in size than aug-VTZ, is extensively tested on a set of 12 noncovalent complexes including formic acid dimer, benzene-methane, or coronene-H2. The results indicate that equivalent noncovalent FNDMC energy differences are available at costs lower than assumed before. Additional insights from electron density differences and comparison of dimer vs monomer ΨT nodes explain this observation.