A new scheme for fixed node diffusion quantum Monte Carlo with pseudopotentials: Improving reproducibility and reducing the trial-wave-function bias

Systematic discrepancies between reference methods for noncovalent interactions within the S66 dataset

The accurate treatment of noncovalent interactions is necessary to model a wide range of applications, from molecular crystals to surface catalysts to aqueous solutions and many more. Quantum diffusion Monte Carlo (DMC) and coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)] are considered two widely trusted methods for treating noncovalent interactions. However, while they have been well-validated for small molecules, recent work has indicated that these two methods can disagree by more than 7.5 kcal/mol for larger systems. The origin of this discrepancy remains unknown. Moreover, the lack of systematic comparisons, particularly for medium-sized complexes, has made it difficult to identify which systems may be prone to such disagreements and the potential scale of these differences. In this work, we leverage the latest developments in DMC to compute interaction energies for the entire S66 dataset, containing 66 medium-sized complexes with a balanced representation of dispersion and electrostatic interactions. Comparison to previous CCSD(T) references reveals systematic trends, with DMC predicting stronger binding than CCSD(T) for electrostatic-dominated systems, while the binding becomes weaker for dispersion-dominated systems. We show that the relative strength of this discrepancy is correlated to the ratio of electrostatic and dispersion interactions, as obtained from energy decomposition analysis methods. Finally, we have pinpointed model systems: the hydrogen-bonded acetic acid dimer (ID 20) and dispersion-dominated uracil-cyclopentane dimer (ID 42), where these discrepancies are particularly prominent. These systems offer cost-effective benchmarks to guide future developments in DMC, CCSD(T), as well as the wider electronic structure theory community.

Accuracy of the diffusion quantum Monte Carlo method on dissociation curves of van der Waals systems with the single-Slater-Jastrow trial wavefunction

Non-covalent interactions, particularly van der Waals (vdW) interactions, are crucial in chemistry, physics, and life sciences. Accurate calculation of both the repulsive and attractive parts of vdW potentials is essential for a reliable description of dissociation curves in vdW systems. The diffuse quantum Monte Carlo (DMC) method is known to be able to calculate vdW interactions accurately at equilibrium distances. However, its performance on vdW interaction energies at non-equilibrium distances has not been investigated. In this work, dissociation curves for ten vdW interacting systems with dispersion interactions, dipole-induced dipole interactions and dipole-dipole interactions are calculated using the DMC and some low-level methods, i.e., MP2 and CCSD, to illustrate their accuracy for the attractive and repulsion potentials of vdW potentials. Our results indicate that while MP2 and CCSD provide reliable descriptions of the attractive potential in vdW interactions, they exhibit larger errors in the repulsive region. On the other hand, DMC employing a single-Slater-Jastrow trial wavefunction with the DLA approximation for the non-local part of pseudopotentials demonstrates high accuracy in both repulsive and attractive potentials. Additionally, reliable DMC results can be achieved with T-moves if the Jastrow factor is optimized carefully. These results emphasize the capability of the DMC with DLA to accurately compute vdW interaction energies at non-equilibrium distances.

Best-of-both-worlds computational approaches to difficult-to-model dissociation reactions on metal surfaces

The accurate modeling of dissociative chemisorption of molecules on metal surfaces presents an exciting scientific challenge to theorists, and is practically relevant to modeling heterogeneously catalyzed reactive processes in computational catalysis. The first important scientific challenge in the field is that accurate barriers for dissociative chemisorption are not yet available from first principles methods. For systems that are not prone to charge transfer (for which the difference between the work function of the surface and the electron affinity of the molecule is larger than 7 eV) this problem can be circumvented: chemically accurate barrier heights can be extracted with a semi-empirical version of density functional theory (DFT). However, a second important challenge is posed by systems that are prone to (full or partial) electron transfer from the surface to the molecule. For these systems the Born-Oppenheimer approximation breaks down, and currently no method of established accuracy exists for modeling the resulting effect of non-adiabatic energy dissipation on the dissociative chemisorption reaction. Because two problems exist for this class of reactions, a semi-empirical approach to computing barrier heights, which would demand that computed and experimental dissociative chemisorption probabilities match, is unlikely to work. This Perspective presents a vision on how these two problems may be solved. We suggest an approach in which parameterized density functionals are used as in the previous semi-empirical approach to DFT, but in which the parameters are based on calculations with first principles electronic structure methods. We also suggest that the diffusion Monte-Carlo (DMC) and the random phase approximation (RPA) probably are the best two first principles electronic structure methods to pursue in the framework of the approach that we call first-principles based DFT (FPB-DFT) - providing DMC and the RPA with a steppingstone towards benchmarking and future applications in computational catalysis. Probably the FPB density functional is best based on screened hybrid exchange in combination with non-local van der Waals correlation. We also propose a new electronic friction method called scattering potential friction (SPF) that could combine the advantages and avoid the disadvantages of the two main existing electronic friction approaches for describing non-adiabatic effects: by extracting an electronic scattering potential from a DFT calculation for the full molecule-metal surface system, it might be possible to compute friction coefficients from scattering phase shifts in a computationally convenient and robust fashion. Combining the FPB-DFT and SPF methods may eventually result in barrier heights of chemical accuracy for the difficult-to-model class of systems that are prone to charge transfer. This should also enable the construction of a representative database of barrier heights for dissociative chemisorption on metal surfaces. Such a database would allow testing new density functionals, or, more generally, new electronic structure approaches on a class of reactions that is of huge importance to the chemical industry. Additionally, the difficult-to-model sub-class of systems we focus on is essential to sustainable chemistry and important for a sustainable future. Adding the database envisaged to large databases already existing but mostly addressing gas phase chemistry will enable testing density functionals that have a claim to universality, i.e., to be good for all chemical systems of importance. We also make a suggestion for how to develop such a generally applicable functional, which should have the correct asymptotic dependence of the exchange contribution to the energy in both the gas phase and the metal. Finally we suggest some improvements in the representation of potential energy surfaces and in dynamics methods that would help with the validation of the proposed methods.

How Accurate Are Simulations and Experiments for the Lattice Energies of Molecular Crystals?

Molecular crystals play a central role in a wide range of scientific fields, including pharmaceuticals and organic semiconductor devices. However, they are challenging systems to model accurately with computational approaches because of a delicate interplay of intermolecular interactions such as hydrogen bonding and Van der Waals dispersion forces. Here, by exploiting recent algorithmic developments, we report the first set of diffusion Monte Carlo lattice energies for all 23 molecular crystals in the popular and widely used X23 dataset. Comparisons with previous state-of-the-art lattice energy predictions (on a subset of the dataset) and a careful analysis of experimental sublimation enthalpies reveals that high-accuracy computational methods are now at least as reliable as (computationally derived) experiments for the lattice energies of molecular crystals. Overall, this work demonstrates the feasibility of high-level explicitly correlated electronic structure methods for broad benchmarking studies in complex condensed phase systems, and signposts a route towards closer agreement between experiment and simulation.

Beyond Single-Reference Fixed-Node Approximation in Ab Initio Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of Electrons

Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.

Diffusion Monte Carlo method for barrier heights of multiple proton exchanges and complexation energies in small water, ammonia, and hydrogen fluoride clusters

Proton exchange reactions are of key importance in many processes in water. However, it is nontrivial to achieve reliable barrier heights for multiple proton exchanges and complexation energies in hydrogen-bonded systems theoretically. Performance of the fixed-node diffusion quantum Monte Carlo (FN-DMC) with the single-Slater-Jastrow trial wavefunction on total energies, barrier heights of multiple proton exchanges, and complexation energies of small water, ammonia, and hydrogen fluoride clusters is investigated in this study. Effects of basis sets and those of locality approximation (LA), T-move approximation (T-move), and determinant localization approximation (DLA) schemes in dealing with the nonlocal part of pseudopotentials on FN-DMC results are evaluated. According to our results, diffuse basis functions are important in achieving reliable barrier heights and complexation energies with FN-DMC, although the cardinal number of the basis set is more important than diffuse basis functions on total energies of these systems. Our results also show that the time step bias with DLA and LA is smaller than T-move; however, the time step bias of DMC energies with respect to time steps using the T-move is roughly linear up to 0.06 a.u., while this is not the case with LA and DLA. Barrier heights and complexation energies with FN-DMC using these three schemes are always within chemical accuracy. Taking into account the fact that T-move and DLA are typically more stable than LA, FN-DMC calculations with the T-move or DLA scheme and basis sets containing diffuse basis functions are suggested for barrier heights of multiple proton exchanges and complexation energies of hydrogen-bonded clusters.

Molecules in Environments: Toward Systematic Quantum Embedding of Electrons and Drude Oscillators

We develop a quantum embedding method that enables accurate and efficient treatment of interactions between molecules and an environment, while explicitly including many-body correlations. The molecule is composed of classical nuclei and quantum electrons, whereas the environment is modeled via charged quantum harmonic oscillators. We construct a general Hamiltonian and introduce a variational Ansatz for the correlated ground state of the fully interacting molecule-environment system. This wave function is optimized via the variational Monte Carlo method and the ground state energy is subsequently estimated through the diffusion Monte Carlo method. The proposed scheme allows an explicit many-body treatment of electrostatic, polarization, and dispersion interactions between the molecule and the environment. We study solvation energies and excitation energies of benzene derivatives, obtaining excellent agreement with explicit ab initio calculations and experiments.

Unraveling H2 chemisorption and physisorption on metal decorated graphene using quantum Monte Carlo

Molecular hydrogen has the potential to significantly reduce the use of carbon dioxide emitting energy processes. However, hydrogen gas storage is a major bottleneck for its large-scale use as current storage methods are energy intensive. Among different storage methods, physisorbing molecular hydrogen at ambient pressure and temperatures is a promising alternative-particularly in light of the advancements in tunable lightweight nanomaterials and high throughput screening methods. Nonetheless, understanding hydrogen adsorption in well-defined nanomaterials remains experimentally challenging and reference information is scarce despite the proliferation of works predicting hydrogen adsorption. We focus on Li, Na, Ca, and K, decorated graphene sheets as substrates for molecular hydrogen adsorption, and compute the most accurate adsorption energies available to date using quantum diffusion Monte Carlo (DMC). Building on our previous insights at the density functional theory (DFT) level, we find that a weak covalent chemisorption of molecular hydrogen, known as Kubas interaction, is feasible on Ca decorated graphene according to DMC, in agreement with DFT. This finding is in contrast to previous DMC predictions of the 4H2/Ca+ gas cluster (without graphene) where chemisorption is not favored. However, we find that the adsorption energy of hydrogen on metal decorated graphene according to a widely used DFT method is not fully consistent with DMC. The reference adsorption energies reported herein can be used to find better work-horse methods for application in large-scale modeling of hydrogen adsorption. Furthermore, the implications of this work affect strategies for finding suitable hydrogen storage materials and high-throughput methods.

Many-Body Methods for Surface Chemistry Come of Age: Achieving Consensus with Experiments

The adsorption energy of a molecule onto the surface of a material underpins a wide array of applications, spanning heterogeneous catalysis, gas storage, and many more. It is the key quantity where experimental measurements and theoretical calculations meet, with agreement being necessary for reliable predictions of chemical reaction rates and mechanisms. The prototypical molecule-surface system is CO adsorbed on MgO, but despite intense scrutiny from theory and experiment, there is still no consensus on its adsorption energy. In particular, the large cost of accurate many-body methods makes reaching converged theoretical estimates difficult, generating a wide range of values. In this work, we address this challenge, leveraging the latest advances in diffusion Monte Carlo (DMC) and coupled cluster with single, double, and perturbative triple excitations [CCSD(T)] to obtain accurate predictions for CO on MgO. These reliable theoretical estimates allow us to evaluate the inconsistencies in published temperature-programed desorption experiments, revealing that they arise from variations in employed pre-exponential factors. Utilizing this insight, we derive new experimental estimates of the (electronic) adsorption energy with a (more) precise pre-exponential factor. As a culmination of all of this effort, we are able to reach a consensus between multiple theoretical calculations and multiple experiments for the first time. In addition, we show that our recently developed cluster-based CCSD(T) approach provides a low-cost route toward achieving accurate adsorption energies. This sets the stage for affordable and reliable theoretical predictions of chemical reactions on surfaces to guide the realization of new catalysts and gas storage materials.

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.

Optimization of Large Determinant Expansions in Quantum Monte Carlo

We present a new method for the optimization of large configuration interaction (CI) expansions in the quantum Monte Carlo (QMC) framework. The central idea here is to replace the nonorthogonal variational optimization of CI coefficients performed in usual QMC calculations by an orthogonal non-Hermitian optimization thanks to the so-called transcorrelated (TC) framework, the two methods yielding the same results in the limit of a complete basis set. By rewriting the TC equations as an effective self-consistent Hermitian problem, our approach requires the sampling of a single quantity per Slater determinant, leading to minimal memory requirements in the QMC code. Using analytical quantities obtained from both the TC framework and the usual CI-type calculations, we also propose improved estimators which reduce the statistical fluctuations of the sampled quantities by more than an order of magnitude. We demonstrate the efficiency of this method on wave functions containing 10⁵-10⁶ Slater determinants, using effective core potentials or all-electron calculations. In all the cases, a sub-milli-Hartree convergence is reached within only two or three iterations of optimization.

Improving the Accuracy of Atomistic Simulations of the Electrochemical Interface

Atomistic simulation of the electrochemical double layer is an ambitious undertaking, requiring quantum mechanical description of electrons, phase space sampling of liquid electrolytes, and equilibration of electrolytes over nanosecond time scales. All models of electrochemistry make different trade-offs in the approximation of electrons and atomic configurations, from the extremes of classical molecular dynamics of a complete interface with point-charge atoms to correlated electronic structure methods of a single electrode configuration with no dynamics or electrolyte. Here, we review the spectrum of simulation techniques suitable for electrochemistry, focusing on the key approximations and accuracy considerations for each technique. We discuss promising approaches, such as enhanced sampling techniques for atomic configurations and computationally efficient beyond density functional theory (DFT) electronic methods, that will push electrochemical simulations beyond the present frontier.

The binding of atomic hydrogen on graphene from density functional theory and diffusion Monte Carlo calculations

In this work, density functional theory (DFT) and diffusion Monte Carlo (DMC) methods are used to calculate the binding energy of a H atom chemisorbed on the graphene surface. The DMC value of the binding energy is about 16% smaller in magnitude than the Perdew–Burke–Ernzerhof (PBE) result. The inclusion of exact exchange through the use of the Heyd–Scuseria–Ernzerhof functional brings the DFT value of the binding energy closer in line with the DMC result. It is also found that there are significant differences in the charge distributions determined using PBE and DMC approaches.

Noncovalent Interactions by the Quantum Monte Carlo Method: Strong Influence of Isotropic Jastrow Cutoff Radii

We present a paradigmatic example of a strong effect of Jastrow cutoff radii setup on the accuracy of noncovalent interaction energy differences within one-determinant Slater-Jastrow fixed-node diffusion Monte Carlo (1FNDMC) simulations using isotropic Jastrow terms and effective-core potentials. Analysis of total energies, absolute and relative errors, and local energy variance of energy differences vs the reference results suggests a simple procedure to marginalize the related biases. The presented data showcase improvements in dispersion-bounded systems within such a 1FNDMC method.

Interactions between large molecules pose a puzzle for reference quantum mechanical methods

Quantum-mechanical methods are used for understanding molecular interactions throughout the natural sciences. Quantum diffusion Monte Carlo (DMC) and coupled cluster with single, double, and perturbative triple excitations [CCSD(T)] are state-of-the-art trusted wavefunction methods that have been shown to yield accurate interaction energies for small organic molecules. These methods provide valuable reference information for widely-used semi-empirical and machine learning potentials, especially where experimental information is scarce. However, agreement for systems beyond small molecules is a crucial remaining milestone for cementing the benchmark accuracy of these methods. We show that CCSD(T) and DMC interaction energies are not consistent for a set of polarizable supramolecules. Whilst there is agreement for some of the complexes, in a few key systems disagreements of up to 8 kcal mol^-1 remain. These findings thus indicate that more caution is required when aiming at reproducible non-covalent interactions between extended molecules.