Performance of the Diffusion Quantum Monte Carlo Method with

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Performance of the Diffusion Quantum Monte Carlo Method with Single-Slater-Jastrow Trial Wavefunction Using Natural Orbitals and DFT Orbitals on Atomization Energies of the G2 Set Ting Wang, Xiaojun Zhou, and Fan Wang J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b01933 • Publication Date (Web): 05 Apr 2019 Downloaded from http://pubs.acs.org on April 8, 2019

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The Journal of Physical Chemistry

Performance of the Diffusion Quantum Monte Carlo Method with Single-Slater-Jastrow Trial Wavefunction Using Natural Orbitals and DFT Orbitals on Atomization Energies of the G2 Set Ting Wang, Xiaojun Zhou and Fan Wang*

Institute of Atomic and Molecular Physics, Key Laboratory of High Energy Density Physics and Technology, Ministry of Education, Sichuan University, Chengdu, P. R. China

*Corresponding author: [email protected] 1

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ABSTRACT 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 DFT orbitals on atomization energies of the molecules in the G2 set is investigated in this work. 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, while that from CASSCF is only lower than the DMC energy using HF orbitals. Atomization energies are generally underestimated with DMC and mean absolute deviations of DMC atomization energies are about 2.8kcal/mol with NOs from MP2 or CCSD(T) and about 2.7kcal/mol with NOs from CASSCF or B3LYP orbitals. Better performance of the latter two orbitals is due to a more effective error cancellation. Accuracy of the present FN-DMC is similar to CCSD(T)/aug-cc-pVQZ on atomization energies. In addition, 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 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.

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I. INTRODUCTION The diffusion quantum Monte Carlo method (DMC) is one of the most popular quantum Monte Carlo (QMC)1-3 methods in electronic structure calculations for molecules and solids to achieve highly accurate results. Besides its high accuracy, the most appealing aspects of the DMC method are its relatively low computational scaling and high parallel efficiency compared with other traditional correlated wavefunction methods. Computational effort for each step of a DMC calculation scales as N2~3, where N is the number of particles in the system3, although the prefactor is rather large. Furthermore, QMC calculations can take advantage of the largest supercomputers effectively and an almost perfect parallel scale was achieved for computations up to more than 100,000 cores4. These aspects render its possible applications to large systems with high accuracy. DMC has been applied with great success to a large variety of problems such as bond dissociation energies5-6, barrier heights7-8, reaction energies9, atomization energies10, and noncovalent interactions11. A trial wavefunction is introduced in DMC calculations to improve statistical efficiency through importance sampling. Fixed-node approximation is generally employed in DMC calculations to avoid the Fermion sign problem in which the node of the wavefunction for a desired eigenstate is constrained to be identical to that the trial wavefunction, i.e. the fixed node diffusion quantum Monte Carlo method (FN-DMC)12.The related bias, i.e. the fixed-node bias, scales quadratically with the nodal displacement error1. The main challenge of the DMC method is to improve the quality of the nodes of the trial wave function, which controls accuracy of the final DMC results. Practical FN-DMC approach available for large systems relies on compact yet effective trial wavefucntion, which is often chosen to be an explicitly correlated single Slater determinant (SD), i.e. a single-Slater-Jastrow wavefunction13. 3



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However, error of the SD nodes may vary from one system to another, which would result in nonsystematic error cancellation and lead to inaccurate FN-DMC energy differences11.

To

improve

accuracy

of

DMC

results,

more

complicated

anti-symmetrized trial wavefunctions such as multi-determinant wave function14, valence

bond

(VB)

wavefunction15,

Pfaffian

wavefunction16,

backflow

wavefunction17-18, or antisymmetrized germinal power wavefunction19 are required and lower energies will be obtained with these trial wavefunctions. DFT or Hartree-Fock (HF) orbitals are usually employed in the SD in DMC calculations. In some systems, DMC results with HF orbitals are similar to those with DFT orbitals, for example, in the dissociation energy of the water dimer18 and stacking interaction energy in B-DNA21. In other cases such as transition-metal-oxide molecules22 and atomization energies of some diatomic molecules in the first and second rows23, DFT orbitals have been shown to provide more accurate DMC results than HF orbitals. It is also possible to optimize the orbitals in the presence of the Jastrow factor in variational QMC (VMC) calculations24-25. On the other hand, one can use the natural orbitals (NOs) in DMC calculations. Grossman and Mitas26 proposed to use NOs that diagonalize the one-electron density matrix of a multi-configuration Hartree-Fock wavefunction in their DMC calculations. NOs were believed to be optimal orbitals that generate fast convergence for the configuration-interaction expansion27, although it was found that this is not necessarily the case28. Nevertheless, it has been proved that a SD will have the largest overlap with a given wavefunction if it is composed of the corresponding NOs29. Improved DMC results using natural orbitals compared with those using HF orbitals are achieved for reaction barriers and heat of formation of three organic reactions30. In addition, Brueckner orbitals obtained from coupled-cluster calculations are employed 4


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in the SD in DMC calculations by Deible and Jordan31 and accuracy of this method on atomization energies of some molecules is similar to that with DFT orbitals and better than that using HF orbitals. Another crucial aspect of DMC calculations concerns the treatment of core electrons. Kinetic and potential energy changes rapidly when an electron moves near a bare nuclear and most of the computational effort will be spent on averaging out large fluctuations of energies. Computational cost of DMC calculations scales as Z5.5-6.5, where Z refers to the nuclear charge32.Efficiency of DMC calculations is increased tremendously when the core electrons are replaced by pseudopotentials(PPs). Unlike the case in traditional quantum chemistry calculations, PPs are required to be nonsingular at the nucleus and such PPs have been developed particularly for DMC calculations33-37. Atomization energies of 55 neutral molecules containing one to three heavy atoms in the Gaussian-2(G2) set38 are one of the most popular benchmark sets to test electronic structure methods. Atomization energies of these molecules have been calculated

previously

using

DMC

with

either

single-Slater-Jastrow

trial

wavefunction10, 24, 39 or multi-determinant Slater-Jastrow trial wavefunction24, 40. A mean absolute deviation (MAD) on atomization energies of 2.9kcal/mol is obtained using single-Slater-Jastrow trial wavefunction with natural orbitals from the MCSCF wavefunction and the Stevens–Basch–Krauss pseudopotentials10. On the other hand, an MAD of 3.2kcal/mol is achieved by Nemec et al.39 with the HF determinant in all-electron DMC calculations. Petruzielo et al.24 investigated performance of FN-DMC on these atomization energies carefully with the Burkatzki-Filippi-Dolg pseudopotentials (BFD) and different basis sets. MADs of 3.0kcal/mol and 2.1kcal/mol are reported using single-Slater Jastrow trial wavefunction with HF 5



orbitals and optimized orbitals from VMC, respectively. The MAD is reduced to 1.2kcal/mol when a complete active space Slater-Jastrow wavefunction is employed. An MAD of 0.8kcal/mol is further achieved using multi-determinant wavefunction chosen from CISDTQ calculations together with a Jastrow factor as the trial wavefunction40. This indicates that DMC results can be improved systematically by including more determinants in the trial wavefunction. Some techniques have been designed to deal efficiently with trial wavefunctions containing a large number of determinants14, 40 and highly accurate DMC results are obtained. However, it becomes expensive to obtain such a trial wavefunction for large molecules and to apply such trial wavefunctions to those molecules in DMC calculations. In this work, we propose to investigate performance of DMC method on atomization energies of the G2 set using the single-Slater-Jastrow trial wavefunction with NOs. NOs from MCSCF were usually employed previously. However, multireference character in the G2 molecules is generally insignificant and dynamic correlation should be taken good care of to achieve accurate results. DMC results using NOs from MP2, CCSD, CCSD(T) and CASSCF will be presented and compared with those using HF orbitals or B3LYP41-42 orbitals. In addition, effects of spin contamination as well as PPs on atomization energies will also be investigated. This paper is organized in the following manner: methods and computational details are given in Sec. II. DMC results using different NOs as well as using HF and DFT orbitals are presented in Sec. III. Influence of these orbitals on the atomization energies and relation between the errors and spin contamination, as well as PPs will be analyzed in detail in this section. Conclusion will be drawn in Sec. IV. II. METHODS AND COMPUTATIONAL DETAILS  The trial wavefunction Ψ T ( x ) introduced in DMC calculations to exploit 6


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importance sampling usually takes the following form:    ΨT (x) = Φ ( x )e J ( x ) .

(1)

 Φ ( x ) in Eq. (1) is an anti-symmetrized wave function, such as a Slater determinant, a  multi-determinant wave function or a VB wave function, and J ( x ) is the Jastrow factor13, which is a symmetric function with respect to electron exchange. The trial wavefunction adopted in this work takes the single-Slater-Jastrow form using the NOs in the SD. NOs are eigenfunctions of the one-electron density matrix D defined as the following: D pq = Ψ a +p aq Ψ ,

(2)

where Ψ is a correlated wavefunction, p, q are indices for a set of orthonormalized orbitals which are usually chosen to be HF orbitals. NOs from MP2, CCSD, CCSD(T) as well as from CASSCF wavefunction are employed in the trial wavefunction. It should be noted that the density matrix in MP2, CCSD and CCSD(T) are introduced in calculating properties and energy gradients43 and contribution of orbital relaxation is included in the density matrix. |Ψ> and

Performance of the Diffusion Quantum Monte Carlo Method with

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