Slater-Jastrow Trial Wave Functions Research Articles

Fixed-node diffusion Monte Carlo shows promise for modeling reaction thermochemistry of hydrocarbon-based radicals.

Reliable thermodynamic and kinetic properties of free radical polymerization reactions are essential for synthesizing both primary polymeric materials and specialty polymers. The computational generation of these data from quantum chemistry requires a time-efficient method capable of capturing the essential physics. One such method, fixed-node diffusion Monte Carlo (FN-DMC) (using single Slater-Jastrow trial wavefunctions), has demonstrated the capability to recover 90%-95% of missing dynamic correlation energy for typical systems. In this study, methyl radical addition to ethylene serves as a simple model to test FN-DMC's ability to calculate enthalpies of reaction and activation energies with different time steps, antisymmetric trial wavefunctions, basis set sizes, and effective core potentials. The FN-DMC computational protocol thus defined for methyl radical addition to ethylene is subsequently benchmarked against Weizmann-1 and experimental reaction enthalpies from Lin et al.'s test set of 21 radical addition and 28 hydrogen abstraction enthalpies. Our findings reveal that FN-DMC consistently generates reaction enthalpies with chemical accuracy, exhibiting mean absolute deviation of 3.5(7) and 1.4(8) kJ/mol from the Weizmann-1 reference for radical addition and hydrogen abstraction reactions, respectively. Given its favorable computational scaling and high degree of parallelizability, we, therefore, recommend more comprehensive testing of FN-DMC with effective core potentials to address more extensive and intricate polymerization reactions and reactions with other radicals.

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.

Performance of the Diffusion Quantum Monte Carlo Method with a Single-Slater-Jastrow Trial Wavefunction Using Natural Orbitals and Density Functional Theory Orbitals on Atomization Energies of the Gaussian-2 Set.

Performance of the fixed-node diffusion quantum Monte Carlo method (FN-DMC) with a single Slater-Jastrow trial wavefunction using natural orbitals (NOs) from MP2, CCSD, CCSD(T), and CASSCF as well as density functional theory orbitals on atomization energies of the molecules in the Gaussian-2 set is investigated in this work. The effects of spin contamination and pseudopotentials (PPs) are also studied. Our results show that the DMC energy with NOs from MP2 or CCSD(T) is the lowest on average, whereas that from CASSCF is only lower than the DMC energy using Hartree-Fock orbitals. Atomization energies are generally underestimated with DMC, and mean absolute deviations of DMC atomization energies are about 2.8 kcal/mol with NOs from MP2 or CCSD(T) and about 2.7 kcal/mol with NOs from CASSCF or B3LYP orbitals. The better performance of the latter two orbitals is due to a more effective error cancellation. The accuracy of the present FN-DMC is similar to CCSD(T)/aug-cc-pVQZ on atomization energies. In addition, the error of DMC atomization energies tends to be larger for molecules with multiple bonds. DMC energies of the open-shell systems with spin-restricted orbitals are generally higher than those with spin-unrestricted orbitals if spin contamination is not serious and the atomization energies are improved to some extent. Furthermore, our results indicate that the error of the PPs is rather small at the CCSD(T) level, but could be more pronounced in DMC calculations and DMC atomization energies are improved with some newly developed PPs.

Singlet-triplet gaps in diradicals obtained with diffusion quantum Monte Carlo using a Slater-Jastrow trial wavefunction with a minimum number of determinants.

Diradicals are essential species in a wide range of chemical processes, whereas the computational study of their electronic structure often remains a challenge due to near-degeneracy of the frontier molecular orbitals. The fixed-node diffusion quantum Monte Carlo (FN-DMC) method is employed to calculate adiabatic energy gaps of some typical diradicals with the Slater-Jastrow trial wavefunction. The antisymmetrized part of the trial wavefunction is taken to be a linear combination of a minimum number of determinants using RB3LYP orbitals from the closed-shell singlet state or ROB3LYP orbitals from the triplet state. Our results show that using the two-determinant-Jastrow trial wavefunction is necessary to achieve reliable energy differences between closed-shell singlet states. The energy of the triplet state with MS = 1 is calculated to be lower than that with MS = 0 with FN-DMC even using trial wavefunctions with spin-pure states as their antisymmetrized parts and this difference is reduced with better orbitals. This indicates that the fixed-node error is smaller for the triplet state with MS = 1. Adiabatic energy gaps obtained from the present FN-DMC calculations are in reasonable agreement with available experimental values. Compared with results of the high level EOM-SF-CC method, energy gaps of FN-DMC with RB3LYP orbitals are slightly better than those using ROB3LYP orbitals and results of EOM-SF-CCSD. The present FN-DMC calculations using the simplest ansatz for the trial wavefunction can achieve reasonable results for these diradicals and they can readily be applied to large diradicals.

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.

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.

Diffusion Monte Carlo Perspective on the Spin-State Energetics of [Fe(NCH)6](2.).

The energy difference between the high spin and the low spin state of the model compound [Fe(NCH)6](2+) is investigated by means of Diffusion Monte Carlo (DMC), where special attention is dedicated to analyzing the effect of the fix node approximation on the accuracy of the results. For this purpose, we compare several Slater-Jastrow and multireference Slater-Jastrow trial wave functions. We found that a Slater-Jastrow trial wave function constructed with the generalized Kohn-Sham orbitals from hybrid DFT represents the optimal choice. This is understood by observing that hybrid functionals account for the subtle balance between exchange and correlation effects and the respective orbitals accurately describe the ligand-metal hybridization as well as the charge reorganization accompanying the spin transition. Finally the DMC results are compared with those obtained by Hartree-Fock, DFT, CASSCF, and CASPT2. While there is no clear reference value for the high spin-low spin energy difference, DMC and high level CCSD(T) calculations agree within around 0.3 eV.

Diffusion Monte Carlo for Accurate Dissociation Energies of 3d Transition Metal Containing Molecules.

Transition metals and transition metal compounds are important to catalysis, photochemistry, and many superconducting systems. We study the performance of diffusion Monte Carlo (DMC) applied to transition metal containing dimers (TMCDs) using single-determinant Slater-Jastrow trial wavefunctions and investigate the possible influence of the locality and pseudopotential errors. We find that the locality approximation can introduce nonsystematic errors of up to several tens of kilocalories per mole in the absolute energy of Cu and CuH if Ar or Mg core pseudopotentials (PPs) are used for the 3d transition metal atoms. Even for energy differences such as binding energies, errors due to the locality approximation can be problematic if chemical accuracy is sought. The use of the Ne core PPs developed by Burkatzki et al. (J. Chem. Phys. 2008, 129, 164115), the use of linear energy minimization rather than unreweighted variance minimization for the optimization of the Jastrow function, and the use of large Jastrow parametrizations reduce the locality errors. In the second section of this article, we study the general performance of DMC for 3d TMCDs using a database of binding energies of 20 TMCDs, for which comparatively accurate experimental data is available. Comparing our DMC results to these data for our results that compare best with experiment, we find a mean unsigned error (MUE) of 4.5 kcal/mol. This compares well with the achievable accuracy in CCSDT(2)Q (MUE = 4.6 kcal/mol) and the best all-electron DFT results (MUE = 4.5 kcal/mol) for the same set of systems (Truhlar et al. J. Chem. Theory Comput. 2015, 11, 2036-2052). The mean errors in DMC depend less on the exchange-correlation functionals used to generate the trial wavefunction than the corresponding mean errors in the underlying DFT calculations. Furthermore, the QMC results obtained for each molecule individually vary less with the functionals used. These observations are relevant for systems such as molecules interacting with transition metal surfaces where the DFT functionals performing best for molecules (hybrids) do not yield improvements in DFT. Overall, the results presented in this article yield important guidelines for both the assessment of the achievable accuracy with DMC and the design of DMC calculations for systems including transition metal atoms.

QMC Calculations of Total Energy and Bond Length of Some Polyatomic Organic Molecules

This paper aims at determining the total energy and bond lengths of some polyatomic organic molecules, using quantum Monte Carlo (QMC) CASINO-code. The QMC code employed the VMC and DMC methods in the computations with emphasis on DMC, and using Slater-Jastrow trial wave-function formed from Hartree-Fock orbitals. The calculated results show that our reported values are in good agreement with the experimental values of both Hehre et al., and Linus Pauling. The total energies obtained in this study are 6 significant figures more accurate than those of previous studies.

QMC Calculations of Total Energy and Bond Length of Some Polyatomic Organic Molecules

This paper aims at determining the total energy and bond lengths of some polyatomic organic molecules, using quantum Monte Carlo (QMC) CASINO-code. The QMC code employed the VMC and DMC methods in the computations with emphasis on DMC, and using Slater-Jastrow trial wave-function formed from Hartree-Fock orbitals. The calculated results show that our reported values are in good agreement with the experimental values of both Hehre et al., and Linus Pauling. The total energies obtained in this study are 6 significant figures more accurate than those of previous studies.

DMC and VMC Calculations of the Electric Dipole Moment and the Ground-State Total Energy of Hydrazine Molecule Using CASINO-Code

Quantum Monte Carlo (QMC) calculations of the electric dipole moment and ground-state total energy of hydrazine (N2H4) molecule using CASINO-code have been carried-out by employing the VMC and DMC techniques. The optimization of the Slater-Jastrow trial wave-function was done using variance-minimization scheme. The simulations require that the configurations must evolve on the time scale of the electronic motion, and after equilibration, the estimated effective time-step be obtained. In this study, the electric dipole moment of N2H4 molecule was calculated using only the DMC technique; and a value of 2.0D which is in good agreement with the experimental value of 1.85D was obtained. On the other hand, the ground-state total energy of N2H4 molecule was calculated using both VMC and DMC methods. It was observed that the result obtained from the VMC technique agrees very-well with the best theoretical value [17], but the DMC technique gave a ground-state total energy lower than all other theoretical values in literature, suggesting that the DMC result of –111.842774 ± 0.00394 a.u. may be the exact ground-state total energy of hydrazine molecule. The calculated values of electric dipole moment and ground-state total energy in this work are compared with the available experimental values and the values reported by different workers. Reasonably good agreement has been obtained between them in the required order of chemical accuracy.

DMC and VMC Calculations of the Electric Dipole Moment and the Ground-State Total Energy of Hydrazine Molecule Using CASINO-Code

Quantum Monte Carlo (QMC) calculations of the electric dipole moment and ground-state total energy of hydrazine (N2H4) molecule using CASINO-code have been carried-out by employing the VMC and DMC techniques. The optimization of the Slater-Jastrow trial wave-function was done using variance-minimization scheme. The simulations require that the configurations must evolve on the time scale of the electronic motion, and after equilibration, the estimated effective time-step be obtained. In this study, the electric dipole moment of N2H4 molecule was calculated using only the DMC technique; and a value of 2.0D which is in good agreement with the experimental value of 1.85D was obtained. On the other hand, the ground-state total energy of N2H4 molecule was calculated using both VMC and DMC methods. It was observed that the result obtained from the VMC technique agrees very-well with the best theoretical value [17], but the DMC technique gave a ground-state total energy lower than all other theoretical values in literature, suggesting that the DMC result of –111.842774 ± 0.00394 a.u. may be the exact ground-state total energy of hydrazine molecule. The calculated values of electric dipole moment and ground-state total energy in this work are compared with the available experimental values and the values reported by different workers. Reasonably good agreement has been obtained between them in the required order of chemical accuracy.

Reptation Quantum Monte Carlo calculation of charge transfer: The Na–Cl dimer

The phenomenon of ion pairing in aqueous solutions is of widespread importance in chemistry and physics, and charge transfer between the ions plays a significant role. We examine the performance of quantum Monte Carlo (QMC) calculations for describing the charge transfer behavior in a NaCl dimer. The influence of the fermion nodes is investigated by obtaining the electron density using the reptation Monte Carlo approach. The fermion nodes are given by single-particle orbitals in Slater–Jastrow trial wavefunctions. We consider the single-particle orbitals from Hartree–Fock and density functional theory calculations with several exchange-correlation approximations. Appreciable dependence of the charge transfer on the fixed-node approximation was found although the total energy was found to be rather insensitive. Our work shows that a careful examination of the fixed-node approximation is necessary for quantifying charge transfer in QMC calculations even when other properties such as reaction energetics are insensitive to the approximation.

Approaching chemical accuracy with quantum Monte Carlo

A quantum Monte Carlo study of the atomization energies for the G2 set of molecules is presented. Basis size dependence of diffusion Monte Carlo atomization energies is studied with a single determinant Slater-Jastrow trial wavefunction formed from Hartree-Fock orbitals. With the largest basis set, the mean absolute deviation from experimental atomization energies for the G2 set is 3.0 kcal/mol. Optimizing the orbitals within variational Monte Carlo improves the agreement between diffusion Monte Carlo and experiment, reducing the mean absolute deviation to 2.1 kcal/mol. Moving beyond a single determinant Slater-Jastrow trial wavefunction, diffusion Monte Carlo with a small complete active space Slater-Jastrow trial wavefunction results in near chemical accuracy. In this case, the mean absolute deviation from experimental atomization energies is 1.2 kcal/mol. It is shown from calculations on systems containing phosphorus that the accuracy can be further improved by employing a larger active space.

Dissociation energy of the water dimer from quantum Monte Carlo calculations

We report a study of the electronic dissociation energy of the water dimer using quantum Monte Carlo techniques. We have performed variational quantum Monte Carlo and diffusion quantum Monte Carlo (DMC) calculations of the electronic ground state of the water monomer and dimer using all-electron and pseudopotential approaches. We have used Slater-Jastrow trial wave functions with B3LYP type single-particle orbitals, into which we have incorporated backflow correlations. When backflow correlations are introduced, the total energy of the water monomer decreases by about 4-5 mhartree, yielding a DMC energy of -76.428 30(5) hartree, which is only 10 mhartree above the experimental value. In our pseudopotential DMC calculations, we have compared the total energies of the water monomer and dimer obtained using the locality approximation with those from the variational scheme recently proposed by Casula [Phys. Rev. B 74, 161102(R) (2006)]. The time step errors in the Casula scheme are larger, and the extrapolation of the energy to zero time step always lies above the result obtained with the locality approximation. However, the errors cancel when energy differences are taken, yielding electronic dissociation energies within error bars of each other. The dissociation energies obtained in our various all-electron and pseudopotential calculations range between 5.03(7) and 5.47(9) kcalmol and are in good agreement with experiment. Our calculations give monomer dipole moments which range between 1.897(2) and 1.909(4) D and dimer dipole moments which range between 2.628(6) and 2.672(5) D.

Quantum Monte Carlo studies on small molecules

The Variational Monte Carlo (VMC) and Fixed-Node Diffusion Monte Carlo (FNDMC) methods have been examined, through studies on small molecules. New programs have been written which implement the (by now) standard algorithms for VMC and FNDMC. We have employed and investigated throughout our studies the accuracy of the common Slater–Jastrow trial wave function. Firstly, we have studied a range of sizes of the Jastrow correlation function of the Boys–Handy form, obtained using our optimization program with analytical derivatives of the central moments in the local energy. Secondly, we have studied the effects of Slater-type orbitals (STOs) that display the exact cusp behaviour at nuclei. The orbitals make up the all important trial determinant, which determines the fixed nodal surface. We report all-electron calculations for the ground state energies of Li2, Be2, H2O, NH3, CH4 and H2CO, in all cases but one with accuracy in excess of 95%. Finally, we report an investigation of the ground state energies, dissociation energies and ionization potentials of NH and NH+. Recent focus paid in the literature to these species allow for an extensive comparison with other ab initio methods. We obtain accurate properties for the species and reveal a favourable tendency for fixed-node and other systematic errors to cancel. As a result of our accurate predictions, we are able to obtain a value for the heat of formation of NH, which agrees to within less than 1 kcal mol−1 to other ab initio techniques and 0.2 kcal mol−1 of the experimental value.

Quantum Monte Carlo study of the Ne atom and the Ne+ ion

We report all-electron and pseudopotential calculations of the ground-state energies of the neutral Ne atom and the Ne(+) ion using the variational and diffusion quantum Monte Carlo (DMC) methods. We investigate different levels of Slater-Jastrow trial wave function: (i) using Hartree-Fock orbitals, (ii) using orbitals optimized within a Monte Carlo procedure in the presence of a Jastrow factor, and (iii) including backflow correlations in the wave function. Small reductions in the total energy are obtained by optimizing the orbitals, while more significant reductions are obtained by incorporating backflow correlations. We study the finite-time-step and fixed-node biases in the DMC energy and show that there is a strong tendency for these errors to cancel when the first ionization potential (IP) is calculated. DMC gives highly accurate values for the IP of Ne at all the levels of trial wave function that we have considered.

Accurate, Efficient, and Simple Forces Computed with Quantum Monte Carlo Methods

Computation of ionic forces using quantum Monte Carlo methods has long been a challenge. We introduce a simple procedure, based on known properties of physical electronic densities, to make the variance of the Hellmann-Feynman estimator finite. We obtain very accurate geometries for the molecules H(2), LiH, CH(4), NH(3), H(2)O, and HF, with a Slater-Jastrow trial wave function. Harmonic frequencies for diatomics are also in good agreement with experiment. An antithetical sampling method is also discussed for additional reduction of the variance.

An inhomogeneous and anisotropic Jastrow function for non-uniform many-electron systems

Quantum Monte Carlo simulations of interacting electrons in solids often use Slater–Jastrow trial wave functions. The Jastrow function takes into account correlations between pairs of electrons. In simulations of solids, it is common to use a Jastrow function which is both homogeneous and isotropic. The use of a homogeneous and isotropic Jastrow factor is questionable in applications to systems with strong density inhomogeneities, such as surfaces and multilayers. By generalizing the original derivation of the RPA Jastrow factor for the homogeneous electron gas (HEG) [Proc. Phys. Soc. 77 (1961) 1182; 80 (1962) 1091] to inhomogeneous systems we derive a scheme for generating inhomogeneous and anisotropic Jastrow factors for use in non-uniform systems, from the non-interacting static structure factor and density of the system. We discuss aspects of our scheme and illustrate it with a first application to an inhomogeneous electron gas.

Inhomogeneous random-phase approximation and many-electron trial wave functions

The long-range electronic correlations in a uniform electron gas may be deduced from the random-phase approximation (RPA) of Bohm and Pines [Phys. Rev. 92, 609 (1953)]. Here we generalize the RPA to nonuniform systems and use it to derive many-electron Slater-Jastrow trial wave functions for quantum Monte Carlo simulations. The RPA theory fixes the long-range behavior of the inhomogeneous two-body terms in the Jastrow factor and provides an accurate analytic expression for the one-body terms. It also explains the success of Slater-Jastrow trial functions containing determinants of Hartree-Fock or density-functional orbitals, even though these theories do not include Jastrow factors. After adjusting the RPA Jastrow factor to incorporate the known short-range behavior, we test it using variational Monte Carlo simulations. In the small inhomogeneous electron gas system we consider, the analytic RPA-based Jastrow factor slightly outperforms the standard numerically optimized form. The inhomogeneous RPA theory therefore enables us to reduce or even avoid the costly numerical optimization process.