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Benchmark quantum Monte Carlo calculations

Jeffrey C. Grossman

Published Jun 18, 2007• 3 citations
Publisher

Abstract

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).

DOI
10.1093/oso/9780195310108.003.00137

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arXiv:0908.2041•The Journal of Chemical Physics•Jan 21, 2010•81 citations
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.
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.
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.
Quantum Monte Carlo (QMC) methods have received considerable attention over the last 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 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 multi-determinant 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.
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.
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.