Multi-determinant Expansions Research Articles

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.

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.

Quantum Monte Carlo calculations of energy gaps from first principles

We review the use of continuum quantum Monte Carlo (QMC) methods for the calculation of energy gaps from first principles, and present a broad set of excited-state calculations carried out with the variational and fixed-node diffusion QMC methods on atoms, molecules, and solids. We propose a finite-size-error correction scheme for bulk energy gaps calculated in finite cells subject to periodic boundary conditions. We show that finite-size effects are qualitatively different in two-dimensional materials, demonstrating the effect in a QMC calculation of the band gap and exciton binding energy of monolayer phosphorene. We investigate the fixed-node errors in diffusion Monte Carlo gaps evaluated with Slater-Jastrow trial wave functions by examining the effects of backflow transformations, and also by considering the formation of restricted multideterminant expansions for excited-state wave functions. For several molecules, we examine the importance of structural relaxation in the excited state in determining excited-state energies. We study the feasibility of using variational Monte Carlo with backflow correlations to obtain accurate excited-state energies at reduced computational cost, finding that this approach can be valid. We find that diffusion Monte Carlo gap calculations can be performed with much larger time steps than are typically required to converge the total energy, at significantly diminished computational expense, but that in order to alleviate fixed-node errors in calculations on solids the inclusion of backflow correlations is sometimes necessary.

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.

Deterministic Construction of Nodal Surfaces within Quantum Monte Carlo: The Case of FeS.

In diffusion Monte Carlo (DMC) methods, the nodes (or zeroes) of the trial wave function dictate the magnitude of the fixed-node (FN) error. In standard DMC implementations, the nodes are optimized by stochastically optimizing a short multideterminant expansion in the presence of an explicitly correlated Jastrow factor. Here, following a recent proposal, we pursue a different route and consider the nodes of selected configuration interaction (sCI) expansions built with the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm. By increasing the size of the sCI expansion, these nodes can be systematically and deterministically improved. The present methodology is used to investigate the properties of the transition metal sulfide molecule FeS. This apparently simple molecule has been shown to be particularly challenging for electronic structure theory methods due to the proximity of two low-energy quintet electronic states of different spatial symmetry and the difficulty to treat them on equal footing from a one-electron basis set point of view. In particular, we show that, at the triple-ζ basis set level, all sCI results-including those extrapolated at the full CI (FCI) limit-disagree with experiment, yielding an electronic ground state of 5Σ+ symmetry. Performing FN-DMC simulation with sCI nodes, we show that the correct 5Δ ground state is obtained if sufficiently large expansions are used. Moreover, we show that one can systematically get accurate potential energy surfaces and reproduce the experimental dissociation energy as well as other spectroscopic constants.

Energy-based truncation of multi-determinant wavefunctions in quantum Monte Carlo.

We present a method for truncating large multi-determinant expansions for use in diffusion Monte Carlo calculations. Current approaches use wavefunction-based criteria to perform the truncation. Our method is more intuitively based on the contribution each determinant makes to the total energy. We show that this approach gives consistent behaviour across systems with varying correlation character, which leads to effective error cancellation in energy differences. This is demonstrated through accurate calculations of the electron affinity of oxygen and the atomisation energy of the carbon dimer. The approach is simple and easy to implement, requiring only quantities already accessible in standard configuration interaction calculations.

Quantum Monte Carlo with very large multideterminant wavefunctions.

An algorithm to compute efficiently the first two derivatives of (very) large multideterminant wavefunctions for quantum Monte Carlo calculations is presented. The calculation of determinants and their derivatives is performed using the Sherman-Morrison formula for updating the inverse Slater matrix. An improved implementation based on the reduction of the number of column substitutions and on a very efficient implementation of the calculation of the scalar products involved is presented. It is emphasized that multideterminant expansions contain in general a large number of identical spin-specific determinants: for typical configuration interaction-type wavefunctions the number of unique spin-specific determinants Ndetσ ( σ=↑,↓) with a non-negligible weight in the expansion is of order O(Ndet). We show that a careful implementation of the calculation of the Ndet -dependent contributions can make this step negligible enough so that in practice the algorithm scales as the total number of unique spin-specific determinants, Ndet↑+Ndet↓, over a wide range of total number of determinants (here, Ndet up to about one million), thus greatly reducing the total computational cost. Finally, a new truncation scheme for the multideterminant expansion is proposed so that larger expansions can be considered without increasing the computational time. The algorithm is illustrated with all-electron fixed-node diffusion Monte Carlo calculations of the total energy of the chlorine atom. Calculations using a trial wavefunction including about 750,000 determinants with a computational increase of ∼400 compared to a single-determinant calculation are shown to be feasible. © 2016 Wiley Periodicals, Inc.

Simple formalism for efficient derivatives and multi-determinant expansions in quantum Monte Carlo.

We present a simple and general formalism to compute efficiently the derivatives of a multi-determinant Jastrow-Slater wave function, the local energy, the interatomic forces, and similar quantities needed in quantum Monte Carlo. Through a straightforward manipulation of matrices evaluated on the occupied and virtual orbitals, we obtain an efficiency equivalent to algorithmic differentiation in the computation of the interatomic forces and the optimization of the orbital parameters. Furthermore, for a large multi-determinant expansion, the significant computational gain afforded by a recently introduced table method is here extended to the local value of any one-body operator and to its derivatives, in both all-electron and pseudopotential calculations.

Performance of quantum Monte Carlo for calculating molecular bond lengths.

This work investigates the accuracy of real-space quantum Monte Carlo (QMC) methods for calculating molecular geometries. We present the equilibrium bond lengths of a test set of 30 diatomic molecules calculated using variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC) methods. The effect of different trial wavefunctions is investigated using single determinants constructed from Hartree-Fock (HF) and Density Functional Theory (DFT) orbitals with LDA, PBE, and B3LYP functionals, as well as small multi-configurational self-consistent field (MCSCF) multi-determinant expansions. When compared to experimental geometries, all DMC methods exhibit smaller mean-absolute deviations (MADs) than those given by HF, DFT, and MCSCF. The most accurate MAD of 3 ± 2 × 10(-3) Å is achieved using DMC with a small multi-determinant expansion. However, the more computationally efficient multi-determinant VMC method has a similar MAD of only 4.0 ± 0.9 × 10(-3) Å, suggesting that QMC forces calculated from the relatively simple VMC algorithm may often be sufficient for accurate molecular geometries.

Ab Initio Molecular Dynamics with Quantum Monte Carlo

Quantum Monte Carlo (QMC) is one of the most accurate electronic structure methods for ab initio many-body calculations in a broad range of electronic systems. Thanks to fast evolving massively parallel computers, much larger simulations of realistic systems become affordable. We have developed an ab initio simulation scheme at finite temperature based on molecular dynamics (MD) and QMC during the past few years. In this approach, a second order Langevin MD is employed by using a statistical evaluation of the forces acting on each atom by means of QMC. Moreover, the corresponding statistical noise is also used to drive and accelerate the dynamics of ions. The accuracy and the reliability of our ab initio MD have been studied systematically and it has been already successfully applied to study the molecular and atomic phases of liquid hydrogen under pressure and liquid water at ambient conditions. Since this ab initio method provides a better resolution of the thermodynamic properties of materials, we believe that it represents a very promising tool for the most challenging applications in physics, chemistry, and biology. Since the Car-Parrinello molecular dynamics (Car and Parrinello, 1985) was proposed three decades ago, ab initio molecular dynamics (MD) based on the density functional theory (DFT) has been widely accepted as an accurate and powerful tool to study the thermodynamic properties of systems at ambient and extreme conditions in the fields of physics, chemistry, and biology. Even though DFT is in principle exact, its accuracy, in practice, is affected by the quality of the approximation used in the exchange correlation functional term whose universal exact expression is not accessible, as the simplest approximate functionals, such as LDA, PBE, BLYP, and several other ones, so far cannot be systematically improved in a computationally efficient way. Quantum Monte Carlo (QMC) is the most promising successor of DFT in ab initio molecular dynamics. Indeed, the scaling behavior of QMC with the number of electrons is very good, from the second power (Ceperley et al., 1977) for thermodynamic properties – e.g., energy per electron, because in this case the number of samples to obtain a fixed accuracy in this quantity can be decreased when increasing the number N e of electrons, implying N 3 e /Ne = N 2 e scaling with the standard algorithm – to the fourth power of the number of electrons when chemical accuracy in the total energy is required – in this case, instead the number of samples required for fixed accuracy has to increase with N e, yielding N 3 e × Ne = N 4 e scaling. For this reason (a cost similar to DFT, albeit with a considerably larger pre-factor), QMC is also by far more efficient compared with other quantum chemistry methods based on multi-determinant expansions or many-body perturbation theory around the Hartree–Fock, such as, for instance, configuration interaction and coupled cluster theory, respectively. These methods provide a similar or even better accuracy than QMC for systems consisting of a few atoms, but they become computationally prohibitive when the system sizes increase. Another very important reason favoring the QMC algorithm is its well established efficient parallelization in current and upcoming supercomputers. QMC codes can be easily adapted to millions of CPU cores while DFT codes are still struggling with its parallel performance over several thousand cores. For the above reasons, it is probably not too far from reality to claim that QMC represents the rising star in the ab initio MD simulations. This opinion paper focuses on the construction of the QMC-based MD by showing the key ingredients and the main recent developments of the building blocks necessary for the efficient implementation of this method. There are four main components: the basics of QMC, the wavefunction optimization, the evaluation of forces, and the molecular dynamics scheme. The last three ingredients were efficiently developed within only 10 years, with substantial contribution given also by us (Casula et al., 2004; Sorella et al., 2007; Umrigar et al., 2007; Attaccalite and Sorella, 2008; Sorella and Capriotti, 2010; Luo et al., 2014; Mazzola et al., 2014; Zen et al., 2014). Thus, the QMC-based MD is no longer a dream.

Static and Dynamical Correlation in Diradical Molecules by Quantum Monte Carlo Using the Jastrow Antisymmetrized Geminal Power Ansatz.

Diradical molecules are essential species involved in many organic and inorganic chemical reactions. The computational study of their electronic structure is often challenging, because a reliable description of the correlation, and in particular of the static one, requires multireference techniques. The Jastrow correlated antisymmetrized geminal power (JAGP) is a compact and efficient wave function ansatz, based on the valence-bond representation, which can be used within quantum Monte Carlo (QMC) approaches. The AGP part can be rewritten in terms of molecular orbitals, obtaining a multideterminant expansion with zero-seniority number. In the present work we demonstrate the capability of the JAGP ansatz to correctly describe the electronic structure of two diradical prototypes: the orthogonally twisted ethylene, C2H4, and the methylene, CH2, representing respectively a homosymmetric and heterosymmetric system. In the orthogonally twisted ethylene, we find a degeneracy of π and π* molecular orbitals, as correctly predicted by multireference procedures, and our best estimates of the twisting barrier, using respectively the variational Monte Carlo (VMC) and the lattice regularized diffusion Monte Carlo (LRDMC) methods, are 71.9(1) and 70.2(2) kcal/mol, in very good agreement with the high-level MR-CISD+Q value, 69.2 kcal/mol. In the methylene we estimate an adiabatic triplet-singlet (X̃(3)B1-ã(1)A1) energy gap of 8.32(7) and 8.64(6) kcal/mol, using respectively VMC and LRDMC, consistently with the experimental-derived finding for Te, 9.363 kcal/mol. On the other hand, we show that the simple ansatz of a Jastrow correlated single determinant (JSD) wave function is unable to provide an accurate description of the electronic structure in these diradical molecules, both at variational level (VMC torsional barrier of C2H4 of 99.3(2) kcal/mol, triplet-singlet energy gap of CH2 of 13.45(10) kcal/mol) and, more remarkably, in the fixed-nodes projection schemes (LRDMC torsional barrier of 97.5(2) kcal/mol, triplet-singlet energy gap of 13.36(8) kcal/mol) showing that a poor description of the static correlation yields an inaccurate nodal surface. The suitability of JAGP to correctly describe diradicals with a computational cost comparable with that of a JSD calculation, in combination with a favorable scalability of QMC algorithms with the system size, opens new perspectives in the ab initio study of large diradical systems, like the transition states in cycloaddition reactions and the thermal isomerization of biological chromophores.

Multideterminant Wave Functions in Quantum Monte Carlo.

Quantum Monte Carlo (QMC) methods have received considerable attention over past decades due to their great promise for providing a direct solution to the many-body Schrodinger equation in electronic systems. Thanks to their low scaling with the number of particles, QMC methods present a compelling competitive alternative for the accurate study of large molecular systems and solid state calculations. In spite of such promise, the method has not permeated the quantum chemistry community broadly, mainly because of the fixed-node error, which can be large and whose control is difficult. In this Perspective, we present a systematic application of large scale multideterminant expansions in QMC and report on its impressive performance with first row dimers and the 55 molecules of the G1 test set. We demonstrate the potential of this strategy for systematically reducing the fixed-node error in the wave function and for achieving chemical accuracy in energy predictions. When compared to traditional quantum chemistry methods like MP2, CCSD(T), and various DFT approximations, the QMC results show a marked improvement over all of them. In fact, only the explicitly correlated CCSD(T) method with a large basis set produces more accurate results. Further developments in trial wave functions and algorithmic improvements appear promising for rendering QMC as the benchmark standard in large electronic systems.

A benchmark quantum Monte Carlo study of the ground state chromium dimer

AbstractWe report calculations of the ground state energy and binding curve of the chromium dimer using the variational and diffusion quantum Monte Carlo (VMC and DMC) methods. We examined various single‐determinant and multideterminant wavefunctions multiplied by a Jastrow factor as a trial/guiding wavefunction for VMC/DMC. The molecular orbitals in the single determinants were calculated using restricted or unrestricted Hartree–Fock or density functional theory (DFT) calculations where five commonly used local (SVWN5), semilocal (PW91 and BLYP), and hybrid (B1LYP and B3LYP) functionals were examined. The multideterminant expansions were obtained from the generalized valence bond and (truncated) unrestricted configuration interaction with single and double excitations (UCISD) methods. We also examined a UCISD wavefunction in which UCISD expansions were added to the UB3LYP single‐determinant reference, and their coefficients were optimized at the VMC level. In addition to the wavefunction dependence, the effects of pseudopotentials and backflow transformation were also investigated. The UB3LYP single‐determinant and multideterminant wavefunctions were found to give the variationally best DMC energies within the framework of single‐determinant and multideterminants, respectively, though both the DMC energies were higher than twice the DMC atomic energy. Some of the VMC binding curves show a flat or quite shallow well bottom, which gets recovered deeper by DMC. All the DMC binding curves have a minimum indicating a bound state, but the unrestricted ones overestimate the equilibrium bond length. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

Self-healing diffusion quantum Monte Carlo algorithms: Direct reduction of the fermion sign error in electronic structure calculations

We develop a formalism and present an algorithm for optimization of the trial wave-function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project-out a multi-determinant expansion of the fixed-node ground-state wave function and (ii) to define a cost function that relates the fixed-node ground-state and the non-interacting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave-function with better nodal structure and (b) we argue that the noise in the fixed-node wave-function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust towards the ones of the exact many-body ground state in a simulated annealing-like process. We propose a method to improve both single determinant and multi-determinant expansions of the trial wave-function. We test the method in a model system where benchmark configuration interaction calculations can be performed. Comparing the DMC calculations with the exact solutions, we find that the trial wave-function is systematically improved. The overlap of the optimized trial wave function and the exact ground state converges to 100% even starting from wave-functions orthogonal to the exact ground state. In the optimization process we find an optimal non-interacting nodal potential of density-functional-like form whose existence was predicted earlier[Phys.Rev. B {\bf 77}, 245110 (2008)]. We obtain the exact Kohn-Sham effective potential from the DMC data.

Weak Hydrogen Bonding Can Initiate Alkane C−H Bond Activation in Acidic Zeolites

Ab initio calculations at the Hartree-Fock self-consistent field/single determinant (SCF) and configuration interaction multi-determinant (CI) expansion levels have been used to show that isobutane primary C-H bond activation occurs via direct protium exchange with the zeolite surface via a weakly hydrogen-bonded complex. The calculated 15 kcal/mol activation barrier agrees with the 13.7 kcal/mol value from a recently reported experimental study (J. Am. Chem. Soc. 2006, 128, 1847-1852). Overall, the mechanism described in this contribution demonstrates that weak C-H to O hydrogen bonding leads to complexes at the zeolite acid site that can facilitate C-H bond activation.

Momentum space properties for the atoms helium to neon from energy‐optimized explicitly correlated wave functions

AbstractAtomic momentum space properties have been calculated starting from explicitly correlated wave functions. The one‐body momentum distribution and some of their radial moments and related properties for the atoms helium to neon are reported. The wave functions include a generalized Jastrow factor and in some cases a multideterminant expansion to take care of the nondynamic correlation effects such as the 2s‐2p near degeneracy. Both the variational trial wave functions and the momentum space properties have been calculated using the variational Monte Carlo method. The optimization of the free parameters of the trial wave functions is done by using a recently developed energy‐optimization technique. Most of the previous variational Monte Carlo applications to quantum chemistry rely on variance‐optimized wave functions. A comparison of momentum space properties calculated with energy‐optimized wave functions with those obtained by means of variance‐optimized wave functions shows a better performance of the former. Despite the simple parameterization of the wave functions used here, they present a good agreement with the results obtained from extensive configuration interaction wave functions. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Electronic structure of SiO2: Charge redistribution contributions to the dynamic dipoles/effective charges of the infrared active normal modes

This article applies ab initio calculations at the (i) Hartree–Fock self-consistent field single determinant and (ii) configuration interaction multideterminant expansion levels to study the diagonal components of the dynamic dipoles, Δμi/Δqj with i=j, or equivalently infrared effective charges, ei*, associated with asymmetric bond stretching νy, symmetric bond stretching νz, and out-of-plane bond rocking νx, normal mode infrared active vibrations of noncrystalline SiO2. The normal mode dynamic dipoles (hereafter, Δμ/qi) are decomposed into equilibrium charge density (ionic) and orbital variation (charge redistribution) contributions. The calculations are based on small clusters in which Si–O–Si groups are connected through O atoms to embedding Si atom terminators Si* that emulate the connectivity of these Si–O–Si groups to the SiO2 continuous random network. Values of Δμ/Δqi have been determined as a function of the Si–O–Si bond angle α at the bridging O-atom sites, and agree with values obtained from analysis of infrared spectra. Finally, the ab initio calculations are extended to noncrystalline silicon–carbon alloys and silicon nitride, and values of Δμ/Δqi are determined for infrared active vibrations associated with N- and C-atom asymmetric stretching normal mode motions. The normalized equilibrium charge density or ionic contributions of the O, N, and C atoms follow trends expected on the basis of their relative Pauling electronegativities in bonds with Si.