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On the accuracy of the fixed-node diffusion quantum Monte Carlo method

Sebastian Manten, Arne Lüchow

Published Jun 18, 2007• 3 citations
Publisher

Abstract

Abstract In this work Manten and Liichow assessed the accuracy of allelectron fixed-node diffusion QMC calculations with nodes taken from Hartee-Fock wavefunctions with high-quality basis sets. The electronic energies associated with 17 different reactions involving a total of 20 small molecules were determined and compared with experimental values and with those calculated by Klopper et al.a using the coupled cluster method CCSD(T) with correlation consistent basis sets cc-pVDZ and cc-pVTZ. The molecules involved were limited to those containing the atoms H, C, 0, N, and F. Typical reactions were H2CO + 2H2 --+ CH4 + H2O and H2O + F --+ HOF + H. The geometries for the QMC calculations were optimized with MP2/cc-pVTZ calculations. The QMC total energies for the 20 molecules were lower than CCSD(T) values, but only valence electrons were correlated in the CCSD(T) calculations. Node location error was approximately 0.0015 hartrees for the C, N, 0, and F atoms and their hydrides, but substantial cancellation of node error is to be expected in determining enthalpy changes for reactions.

DOI
10.1093/oso/9780195310108.003.00133

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The Journal of Physical Chemistry A•Apr 5, 2019•18 citations
Abstract In this work Manten and Liichow assessed the accuracy of allelectron fixed-node diffusion QMC calculations with nodes taken from Hartee-Fock wavefunctions with high-quality basis sets. The electronic energies associated with 17 different reactions involving a total of 20 small molecules were determined and compared with experimental values and with those calculated by Klopper et al.a using the coupled cluster method CCSD(T) with correlation consistent basis sets cc-pVDZ and cc-pVTZ. The molecules involved were limited to those containing the atoms H, C, 0, N, and F. Typical reactions were H2CO + 2H2 --+ CH4 + H2O and H2O + F --+ HOF + H. The geometries for the QMC calculations were optimized with MP2/cc-pVTZ calculations. The QMC total energies for the 20 molecules were lower than CCSD(T) values, but only valence electrons were correlated in the CCSD(T) calculations. Node location error was approximately 0.0015 hartrees for the C, N, 0, and F atoms and their hydrides, but substantial cancellation of node error is to be expected in determining enthalpy changes for reactions.
The accuracy of the fixed-node diffusion quantum Monte Carlo (FN-DQMC) method is compared to the coupled cluster method CCSD(T). For a test set of 20 small molecules and 17 reactions the electronic contribution to the reaction enthalpy is calculated with the FN-DQMC method using the nodes of a Slater determinant calculated at the HF/cc-pVTZ level. By comparison with reference reaction enthalpies the FN-DQMC method is shown to be more accurate than the CCSD(T)/cc-pVDZ method and almost as accurate as CCSD(T)/cc-pVTZ. The deviation from the reference data is comparable to the CCSD(T)/cc-pVTZ deviation, but, with only two exceptions, of opposite sign.
Abstract Results of fixed-node diffusion QMC calculations for 17 small molecules made up of atoms H, C, N, 0, and F were compared in this paper to experimental values for atomization energies, bond lengths, and bond angles, as well as corresponding values from coupled cluster calculations. The QMC calculations were of the Ornstein-Uhlenbeck type,a with trial functions for importance sampling and node locations based on linear combinations of determinants with molecular orbitals composed of floating spherical Gaussian orbitals. Initial geometries were taken from density functional calculations and optimized to find the minima given by the QMC calculations. Zero-point energies were calculated using the energy gradients given by the same calculations. The calculated atomization energies for the 14 species for which experimental data were available were in excellent agreement with the experimental values and with values given by CCSD(T) with a cc-pVQZ basis set. The mean absolute deviation from experimental atomization energies was 0.16 kcal/mol for the QMC and 0.21 kcal/mol for the CCSD(T) calculations. Geometric parameters were in agreement with each other and with experimental values within their uncertainties. Overall, the results provide a solid confirmation of the accuracy of fixed-node diffusion QMC calculations.
Abstract In this paper Grossman reported pseudopotential fixed-node diffusion QMC calculations of atomization energies for the Gl set of 55 small molecules,0 a standard benchmark set for testing methods in quantum chemistry. To maintain the uniformity in the calculations, they were carried out with the same basis sets and orbital types for single-determinant trial functions and effective core potentials (with a few exceptions). Variational QMC was used to optimize the determinan Jastrow trial functions. The mean absolute deviation EMAD in atomization energy from the experimental values was found to be 2.9 kcal/mol, the maximum deviation was 14 kcal/mol (for SO2), and the next five largest deviations were 7-8 kcal/mol. A comparison with results from other methods tested with the Gl set shows the FN-DQMC method to be competitive. For EMAD the list is: FN-DQMC, 2.9 kcal/mol; DFT (LDA), 40 kcal/mol; DFT (B3LYP or B3PW91), 2.9 kcal/mol; CCSD(T)/aug-cc-pVQZ, 2.8 kcal/mol; CCSD(T)/extrapolated, 1.3 kcal/mol; semi-empirical G1 theory, 1.6 kcal/mol. Additional calculations with different trial functions indicated that node location errors could be significantly reduced with multi determinant trial functions. Computational effort was not compared, but linear scaling for FN-DQMC was mentioned as an important advantage for QMC. 0J. A. Pople, M. Head-Gordon, D. J. Fox, K. Raghavachari, and L. A. Curtiss, Gaussian-I theory - A general procedure for prediction of molecular energies, J. Chem. Phys. 90, 5622 (1989).
Fixed node diffusion Monte Carlo (FN-DMC) atomization energies are calculated for a benchmark set of 55 molecules. Using single determinant trial wave functions, comparison with experiment yields an average absolute deviation of 2.9 kcal/mol, placing this simplest form of FN-DMC roughly at the same level of accuracy as the CCSD(T)/aug-cc-pVQZ method. However, unlike perturbative wave function expansion approaches, FN-DMC is applicable to systems containing thousands of valence electrons. For the P2 molecule, a number of possible sources of error are explored in detail. Results show that the main error is due to the fixed-node approximation and that this can be improved significantly with multireference trial wave functions.
We study the efficiency, precision and accuracy of all-electron variational and diffusion quantum Monte Carlo calculations using Slater basis sets. Starting from wave functions generated by Hartree-Fock and density functional theory, we describe an algorithm to enforce the electron-nucleus cusp condition by linear projection. For the 55 molecules in the G2 set, the diffusion quantum Monte Carlo calculations recovers an average of 95% of the correlation energy and reproduces bond energies to a mean absolute deviation of 3.2 kcal/mol. Comparing the individual total energies with essentially exact values, we investigate the error cancellation in atomization and chemical reaction path energies, giving additional insight into the sizes of nodal surface errors.
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.