Multideterminant Wave Functions Research Articles

Accurate Quantum Monte Carlo Forces for Machine-Learned Force Fields: Ethanol as a Benchmark.

Quantum Monte Carlo (QMC) is a powerful method to calculate accurate energies and forces for molecular systems. In this work, we demonstrate how we can obtain accurate QMC forces for the fluxional ethanol molecule at room temperature by using either multideterminant Jastrow-Slater wave functions in variational Monte Carlo or just a single determinant in diffusion Monte Carlo. The excellent performance of our protocols is assessed against high-level coupled cluster calculations on a diverse set of representative configurations of the system. Finally, we train machine-learning force fields on the QMC forces and compare them to models trained on coupled cluster reference data, showing that a force field based on the diffusion Monte Carlo forces with a single determinant can faithfully reproduce coupled cluster power spectra in molecular dynamics simulations.

Graphic Characterization and Clustering Configuration Descriptors of Determinant Space for Molecules.

Quantum Monte Carlo approaches based on stochastic sampling of determinant space have evolved to be powerful methods to compute the electronic states of molecules. These methods not only calculate the correlation energy at an unprecedented accuracy but also provide insightful information on the electronic structures of computed states, for example, the population, connection, and clustering of determinants, which have not been fully explored. In this work, we devise a configuration graph for visualizing determinant space, revealing the nature of the molecule's electronic structure. In addition, we propose two analytical descriptors to quantify the extent of configuration clustering of multideterminant wave functions. The graph and descriptors provide us with a fresh perspective of the electronic structures of molecules and can assist with further development of configuration interaction-based electronic structure methods.

Fanpy: A python library for prototyping multideterminant methods in ab initio quantum chemistry.

Fanpy is a free and open-source Python library for developing and testing multideterminant wavefunctions and related ab initio methods in electronic structure theory. The main use of Fanpy is to quickly prototype new methods by making it easier to convert the mathematical formulation of a new wavefunction ansätze to a working implementation. Fanpy is designed based on our recently introduced Flexible Ansatz for N-electron Configuration Interaction (FANCI) framework, where multideterminant wavefunctions are represented by their overlaps with Slater determinants of orthonormal spin-orbitals. In the simplest case, a new wavefunction ansatz can be implemented by simply writing a function for evaluating its overlap with an arbitrary Slater determinant. Fanpy is modular in both implementation and theory: the wavefunction model, the system's Hamiltonian, and the choice of objective function are all independent modules. This modular structure makes it easy for users to mix and match different methods and for developers to quickly explore new ideas. Fanpy is written purely in Python with standard dependencies, making it accessible for various operating systems. In addition, it adheres to principles of modern software development, including comprehensive documentation, extensive testing, quality assurance, and continuous integration and delivery protocols. This article is considered to be the official release notes for the Fanpy library.

Reference Excitation Energies of Increasingly Large Molecules: A QMC Study of Cyanine Dyes.

We revisit here the lowest vertical excitations of cyanine dyes using quantum Monte Carlo and leverage recent developments to systematically improve on previous results. In particular, we employ a protocol for the construction of compact and accurate multideterminant Jastrow-Slater wave functions for multiple states, which we have recently validated on the excited-state properties of several small prototypical molecules. Here, we obtain quantum Monte Carlo excitation energies in excellent agreement with high-level coupled cluster for all the cyanines where the coupled cluster method is applicable. Furthermore, we push our protocol to longer chains, demonstrating that quantum Monte Carlo is a viable methodology to establish reference data at system sizes which are hard to reach with other high-end approaches of similar accuracy. Finally, we determine which ingredients are key to an accurate treatment of these challenging systems and rationalize why a description of the excitation based on only active π orbitals lacks the desired accuracy for the shorter chains.

Flexible ansatz for N-body configuration interaction

We present a Flexible Ansatz for N-body Configuration Interaction (FANCI) that includes any multideterminant wavefunction. This ansatz is a generalization of the Configuration Interaction (CI) wavefunction, where the coefficients are replaced by a specified function of certain parameters. By making an appropriate choice for this function, we can reproduce popular wavefunction structures like CI, Coupled-Cluster, Tensor Network States, and geminal-product wavefunctions. The universality of this framework suggests a programming structure that allows for the easy construction and optimization of arbitrary wavefunctions. Here, we will discuss the structures of the FANCI framework and its implications for wavefunction properties, particularly accuracy, cost, and size-consistency. We demonstrate the flexibility of this framework by reconstructing popular wavefunction ansätze and modifying them to construct novel wavefunction forms. FANCI provides a powerful framework for exploring, developing, and testing new wavefunction forms.

Some recent developments in auxiliary-field quantum Monte Carlo for real materials.

The auxiliary-field quantum Monte Carlo (AFQMC) method is a general numerical method for correlated many-electron systems, which is being increasingly applied in lattice models, atoms, molecules, and solids. Here, we introduce the theory and algorithm of the method specialized for real materials and present several recent developments. We give a systematic exposition of the key steps of AFQMC, closely tracking the framework of a modern software library we are developing. The building of a Monte Carlo Hamiltonian, projecting to the ground state, sampling two-body operators, phaseless approximation, and measuring ground state properties are discussed in detail. An advanced implementation for multi-determinant trial wave functions is described, which dramatically speeds up the algorithm and reduces the memory cost. We propose a self-consistent constraint for real materials, and discuss two flavors for its realization, either by coupling the AFQMC calculation to an effective independent-electron calculation or via the natural orbitals of the computed one-body density matrix.

Representing Exact Electron Densities by a Single Slater Determinant in Finite Basis Sets.

The premise for Kohn-Sham density functional theory is that the exact electron density can be generated by a set of orbitals in a single Slater determinant. While this is ensured in a complete basis set, it has been shown that it cannot hold in small basis sets. The present work probes how accurately a reference electron density of the full-CI type can be reproduced by a set of orbitals in a single Slater determinant, as a function of the basis set used for the fitting electron density. The key finding is that the fitting error may be significant for basis sets of double- or triple-ζ quality. It is also shown that it is important that the fitting basis set includes the same basis functions as used for generating the reference electron density. The main limitation in a given basis set is the lack of higher order polarization functions. The error for practical purposes becomes insignificant for basis sets of quadruple-ζ or better quality, and this should be the choice when assessing the accuracy of exchange-correlation functionals by comparing electron densities to accurate reference results generated by wave function methods. The methodology in the present work can be used to transform an electron density from a multideterminant wave function into a set of orbitals in a single Slater determinant, and this may be useful for developing and testing new exchange-correlation functionals.

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.

Systematic comparison and cross-validation of fixed-node diffusion Monte Carlo and phaseless auxiliary-field quantum Monte Carlo in solids

Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC). The multi-determinant trial wave functions employed in both approaches are generated using the configuration interaction using a perturbative selection made iteratively (CIPSI) technique. Complete basis set full configuration interaction (CBS-FCI) energies estimated with CIPSI are used as a reference in this comparative study between DMC and AFQMC. By focusing on a set of canonical finite size solid state systems, we show that both QMC methods can be made to systematically converge towards the same energy once basis set effects and systematic biases have been removed. AFQMC shows a much smaller dependence on the trial wavefunction than DMC while simultaneously exhibiting a much larger basis set dependence. We outline some of the remaining challenges and opportunities for improving these approaches.

Quantum Monte Carlo simulations using Slater-Jastrow-backflow wave function

Diffusion Quantum Monte-Carlo computations of the ground state energies of the Li, Li−, Be, Be+, B and B+ species are reported. We use trial wave function of four types: single determinant Slater-Jastrow, single determinant Slater-Jastrow-Backflow, multi-determinant Slater-Jastrow and multi-determinant Slater-Jastrow-Backflow. The single determinant wave functions are computed by the HF method and the multi-determinant wave functions are computed by the CI and CASSCF methods. Up to 99.9% and 100% of the correlation energy are recovered for the studied species without and with the backflow transformation respectively.

Predicting Ligand-Dissociation Energies of 3d Coordination Complexes with Auxiliary-Field Quantum Monte Carlo.

Transition-metal complexes are ubiquitous in biology and chemical catalysis, yet they remain difficult to accurately describe with ab initio methods because of the presence of a large degree of dynamic electron correlation, and, in some cases, strong static correlation which results from a manifold of low-lying states. Progress has been hindered by a scarcity of high-quality gas-phase experimental data, while exact ab initio predictions are usually computationally unaffordable because of the large size of the relevant complexes. In this work, we present a data set of 34 tetrahedral, square planar, and octahedral 3d metal-containing complexes with gas-phase ligand-dissociation energies that have reported uncertainties of ≤2 kcal/mol. We perform all-electron phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) calculations utilizing multideterminant trial wave functions selected by a black box procedure. We compare the results with those from the density functional theory (DFT) with the B3LYP, B97, M06, PBE0, ωB97X-V, and DSD-PBEP86/2013 functionals and a localized orbital variant of the coupled cluster theory with single, double, and perturbative triple excitations (DLPNO-CCSD(T)). We find mean averaged errors of 1.07 ± 0.27 kcal/mol for our most sophisticated ph-AFQMC approach versus 2.81 kcal/mol for DLPNO-CCSD(T) and 1.49-3.78 kcal/mol for DFT. We find maximum errors of 2.96 ± 1.71 kcal/mol for our best ph-AFQMC method versus 9.15 kcal/mol for DLPNO-CCSD(T) and 5.98-13.69 kcal/mol for DFT. The reasonable performance of a number of DFT functionals is in stark contrast to the much poorer accuracy previously demonstrated for diatomic species, suggesting a moderation in electron correlation because of ligand coordination in most cases. However, the unpredictably large errors for a small subset of cases with both DFT and DLPNO-CCSD(T) methods leave cause for concern, especially in light of the unreliability of common multireference indicators. In contrast, the robust and, in principle, systematically improvable results of ph-AFQMC for these realistic complexes establish the method as a useful tool for elucidating the electronic structure of transition-metal-containing complexes and predicting their gas-phase properties.

Electron correlation effects in isomers of C20

The C20 molecule exists as three low-energy isomers: the monocyclic ring, a corannulene-like bowl structure, and the cage-the smallest possible fullerene. The curious structures of these isomers, along with the valuable properties and possible applications of fullerenes more generally, mean that C20 has attracted interest both experimentally and computationally. Unfortunately, previous theoretical studies have found these C20 isomers present a significant computational challenge, and widely used methods such as Density Functional Theory with different functionals have been unable to agree on even the relative ordering of the isomer energies. Even accurate high-level methods such as coupled-cluster with singles, doubles, and perturbative triples (CCSD(T)) and Diffusion Monte Carlo (DMC) have previously disagreed on the energetic ordering of these isomers. Here we re-examine the ring, bowl, and cage isomers of C20 using a more accurate DMC technique. We employ a novel method to go beyond the single-determinant DMC approaches previously used, and instead use more accurate multi-determinant trial wavefunctions. Our results show that the fullerene cage exhibits different electron correlation effects than the ring and bowl, which when taken into account leads to a reordering of their relative energies. This finally establishes agreement between DMC and relatively recent complete-basis CCSD(T) results, thereby resolving a long-standing disparity between these two high-level descriptions of the C20 isomers. The approach we use is generalisable, and could be used to provide insight into even larger systems in future.

Post-Hartree-Fock methods for Hirshfeld atom refinement: are they necessary? Investigation of a strongly hydrogen-bonded molecular crystal

Hirshfeld atom refinement (HAR) is a method for refining X-ray crystal structures that is able to provide bond lengths involving hydrogen atoms in statistical agreement with those derived from neutron diffraction data, provided the data reach 0.8 Å resolution. Rather than using tabulated spherical atomic structure factors, HAR uses “tailor made” aspherical atomic structure factors obtained directly from quantum chemical calculations. Despite the very good results obtained so far, which make HAR an emerging refinement method of modern crystallography, until now all the Hirshfeld atom refinements were exclusively based on Hartree-Fock (HF) or density functional theory (DFT) methods, but never on the so-called post-HF techniques of quantum chemistry. Post-HF methods exploit more sophisticated multi-determinant wavefunctions and, consequently, should provide more accurate electron densities for the refinements. For this reason, for the first time we have performed HARs based on two well-known post-HF strategies (MP2 and Coupled Cluster) combined with three different basis-sets (def2-SVP, def2-TZVP and def2-TZVPP). The obtained results have been afterwards analyzed and compared to those resulting from neutron and other Hirshfeld atom refinements, the latter relying on Hartree-Fock and DFT (BLYP and B3LYP) calculations in order to evaluate if the use of more sophisticated and expensive approaches of quantum chemistry can improve the performances of the HAR technique.

Prediction of many-electron wavefunctions using atomic potentials: Refinements and extensions to transition metals and large systems.

For a given many-electron molecule, it is possible to define a corresponding one-electron Schrödinger equation, using potentials derived from simple atomic densities, whose solution predicts fairly accurate molecular orbitals for single-determinant and multideterminant wavefunctions for the molecule. The energy is not predicted and must be evaluated by calculating Coulomb and exchange interactions over the predicted orbitals. Transferable potentials for first-row atoms and transition metal oxides that can be used without modification in different molecules are reported. For improved accuracy, molecular wavefunctions can be refined by slightly scaling nuclear charges and by introducing potentials optimized for functional groups. The accuracy is further improved by a single diagonalization of the Fock matrix constructed from the predicted orbitals. For a test set of 20 molecules representing different bonding environments, the transferable potentials with scaling give wavefunctions with energies that deviate from exact self-consistent field or configuration interaction energies by less than 0.05 eV and 0.02 eV per bond or valence electron pair, respectively. On diagonalization of the Fock matrix, the corresponding errors are reduced by a factor of three to less than 0.016 eV and 0.006 eV, respectively. Applications to the ground and excited states of a Ti18O36 nanoparticle and chlorophyll-a are reported.

Prediction of many-electron wavefunctions using atomic potentials: extended basis sets and molecular dissociation.

A one-electron Schrödinger equation based on special one-electron potentials for atoms is shown to exist that produces orbitals for an arbitrary molecule that are sufficiently accurate to be used without modification to construct single- and multi-determinant wavefunctions. The exact Hamiltonian is used to calculate the energy variationally and to generate configuration interaction expansions. Earlier work on equilibrium geometries is extended to larger basis sets and molecular dissociation. For a test set of molecules representing different bonding environments, a single set of invariant atomic potentials gives wavefunctions with energies that deviate from configuration interaction energies based on SCF orbitals by less than 0.04 eV per bond or valence electron pair. On a single diagonalization of the Fock matrix, the corresponding errors are reduced 0.01 eV. Atomization energies are also in good agreement with CI values based on canonical SCF orbitals. Configuration interaction applications to single bond dissociations of water and glycine, and multiple bond dissociations of ethylene and oxygen produce dissociation energy curves in close agreement with CI calculations based on canonical SCF orbitals for the entire range of internuclear distances.

Quantum Chemical Topology as a Theory of Open Quantum Systems.

Although real space regions have been widely used in theoretical chemistry, not much effort has been devoted to treat them as open quantum systems. We embrace this task here, finding closed expressions for the density operator of a quantum subsystem in real space by tracing out the degrees of freedom in its complementary region. Our results are then linked to previous knowledge. For single-determinant descriptions it is shown that the entanglement orbitals coincide with Ponec's domain natural orbitals. In general, the subsystem density operator is written as a direct sum of a fixed number of electron sectors, with weights that turn out to be equal to those found within the theory of electron distribution functions. As a computational application we show how to obtain the global first order density matrix of a subsystem and its eigensolution in a couple of toy systems. In the multideterminant wave function case, the domain natural orbitals defined through this open system approach do not coincide with those of Ponec and, contrary to the latter, have always strictly positive occupations.

Reciprocal transformation of seniority number restricted wave function.

In this work, an extension of Malmqvist's algorithm is efficiently implemented for the reciprocal transformation of a seniority number restricted multideterminant wave function constructed by nonorthogonal orbitals to an equivalent one constructed by biorthogonal orbitals. Test calculations show that the novel algorithm is capable of applying to transform quite large nonorthogonal orbital based wave function of distributing 22 electrons on 22 orbitals with the maximum seniority number sector and for the evaluation of a three-body operator, at an affordable memory cost of about 800 GB, which is only about one tenth of the size of the corresponding complete active space. Moreover, the algorithm displays improved computational efficiency, due to the compactness of seniority number restricted multi-determinant wave functions.

Phaseless Auxiliary-Field Quantum Monte Carlo on Graphical Processing Units.

We present an implementation of phaseless Auxiliary-Field Quantum Monte Carlo (ph-AFQMC) utilizing graphical processing units (GPUs). The AFQMC method is recast in terms of matrix operations which are spread across thousands of processing cores and are executed in batches using custom Compute Unified Device Architecture kernels and the GPU-optimized cuBLAS matrix library. Algorithmic advances include a batched Sherman-Morrison-Woodbury algorithm to quickly update matrix determinants and inverses, density-fitting of the two-electron integrals, an energy algorithm involving a high-dimensional precomputed tensor, and the use of single-precision floating point arithmetic. These strategies accelerate ph-AFQMC calculations with both single- and multideterminant trial wave functions, though particularly dramatic wall-time reductions are achieved for the latter. For typical calculations we find speed-ups of roughly 2 orders of magnitude using just a single GPU card compared to a single modern CPU core. Furthermore, we achieve near-unity parallel efficiency using 8 GPU cards on a single node and can reach moderate system sizes via a local memory-slicing approach. We illustrate the robustness of our implementation on hydrogen chains of increasing length and through the calculation of all-electron ionization potentials of the first-row transition metal atoms. We compare long imaginary-time calculations utilizing a population control algorithm with our previously published correlated sampling approach and show that the latter improves not only the efficiency but also the accuracy of the computed ionization potentials. Taken together, the GPU implementation combined with correlated sampling provides a compelling computational method that will broaden the application of ph-AFQMC to the description of realistic correlated electronic systems.

From quantum fragments to Lewis structures: electron counting in position space.

An electron counting technique that easily provides Lewis structures from real space analyses of general wavefunctions is proposed. We base our approach on reformulating the adaptive natural density partitioning (AdNDP) algorithm proposed by Zubarev and Boldyrev (Phys. Chem. Chem. Phys., 2008, 10, 5207) in position space through the use of domain-averaged cumulant densities, which take into account many-electron correlations. Averages are performed over the basins provided by the quantum theory of atoms in molecules. The decomposition gives rise to a set of n-center, two-electron orbitals which describe the dominant Lewis structures of a molecular system, and is available both for single- and multi-determinant wavefunctions. As shown through several examples, chemically intuitive Lewis descriptions are now available from fully orbital invariant position space descriptors. In this sense, real space methods are now in a position to compete with natural bond order (NBO) orbital procedures without the many biases of the latter.

Quantitative estimation of localization errors of 3d transition metal pseudopotentials in diffusion Monte Carlo.

The necessarily approximate evaluation of non-local pseudopotentials in diffusion Monte Carlo (DMC) introduces localization errors. We estimate these errors for two families of non-local pseudopotentials for the first-row transition metal atoms Sc-Zn using an extrapolation scheme and multideterminant wavefunctions. Sensitivities of the error in the DMC energies to the Jastrow factor are used to estimate the quality of two sets of pseudopotentials with respect to locality error reduction. The locality approximation and T-moves scheme are also compared for accuracy of total energies. After estimating the removal of the locality and T-moves errors, we present the range of fixed-node energies between a single determinant description and a full valence multideterminant complete active space expansion. The results for these pseudopotentials agree with previous findings that the locality approximation is less sensitive to changes in the Jastrow than T-moves yielding more accurate total energies, however not necessarily more accurate energy differences. For both the locality approximation and T-moves, we find decreasing Jastrow sensitivity moving left to right across the series Sc-Zn. The recently generated pseudopotentials of Krogel et al. [Phys. Rev. B 93, 075143 (2016)] reduce the magnitude of the locality error compared with the pseudopotentials of Burkatzki et al. [J. Chem. Phys. 129, 164115 (2008)] by an average estimated 40% using the locality approximation. The estimated locality error is equivalent for both sets of pseudopotentials when T-moves is used. For the Sc-Zn atomic series with these pseudopotentials, and using up to three-body Jastrow factors, our results suggest that the fixed-node error is dominant over the locality error when a single determinant is used.