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arXiv:1811.03191 · Journal of Chemical Theory and Computation

Clean and Convenient Tessellations for Number Counting Jastrow Factors

Beatrice W. Van Der Goetz, Leon Otis, Eric Neuscamman

Published Jan 8, 2019• 5 citations
arXivHTMLPDF

Abstract

We introduce a basis of counting functions that, by cleanly tessellating three-dimensional space, allows real space number counting Jastrow factors to be straightforwardly applied to general molecular situations. By exerting direct control over electron populations in local regions of space and encoding pairwise correlations between these populations, these Jastrow factors allow even very simple reference wave functions to adopt nodal surfaces well suited to many strongly correlated settings. Being trivially compatible with traditional Jastrow factors and diffusion Monte Carlo and having the same cubic per-sample cost scaling as a single determinant trial function, these Jastrow factors thus offer a powerful new route to the simultaneous capture of weak and strong electron correlation effects in a wide variety of molecular and materials settings. In multiple strongly correlated molecular examples, we show that even when paired with the simplest possible single determinant reference, these Jastrow factors allow quantum Monte Carlo to out-perform coupled cluster theory and approach the accuracy of traditional multireference methods.

DOI
10.1021/acs.jctc.8b01139

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We introduce a basis of counting functions that, by cleanly tessellating three-dimensional space, allows real space number counting Jastrow factors to be straightforwardly applied to general molecular situations. By exerting direct control over electron populations in local regions of space and encoding pairwise correlations between these populations, these Jastrow factors allow even very simple reference wave functions to adopt nodal surfaces well suited to many strongly correlated settings. Being trivially compatible with traditional Jastrow factors and diffusion Monte Carlo and having the same cubic per-sample cost scaling as a single determinant trial function, these Jastrow factors thus offer a powerful new route to the simultaneous capture of weak and strong electron correlation effects in a wide variety of molecular and materials settings. In multiple strongly correlated molecular examples, we show that even when paired with the simplest possible single determinant reference, these Jastrow factors allow quantum Monte Carlo to out-perform coupled cluster theory and approach the accuracy of traditional multireference methods.
Quantum Monte Carlo (QMC) algorithms have long relied on Jastrow factors to incorporate dynamic correlation into trial wave functions. While Jastrow-type wave functions have been widely employed in real-space algorithms, they have seen limited use in second-quantized QMC methods, particularly in projection methods that involve a stochastic evolution of the wave function in imaginary time. Here we propose a scheme for generating Jastrow-type correlated trial wave functions for auxiliary-field QMC methods. The method is based on decoupling the two-body Jastrow into one-body projectors coupled to auxiliary fields, which then operate on a single determinant to produce a multideterminant trial wave function. We demonstrate that intelligent sampling of the most significant determinants in this expansion can produce compact trial wave functions that reduce errors in the calculated energies. Our technique may be readily generalized to accommodate a wide range of two-body Jastrow factors and applied to a variety of model and chemical systems.
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We show that for both single-Slater-Jastrow and Jastrow geminal power wave functions the formal cost scaling of Hilbert space variational Monte Carlo can be reduced from fifth to fourth order in the system size, thus bringing it in line with the long-standing scaling of its real space counterpart. While traditional quantum chemistry methods can reduce costs related to the two-electron integral tensor through various tensor decomposition methods, we show that such approaches are ineffective in the presence of Hilbert space Jastrow factors. Instead, we develop a simple semi-stochastic approach that can take similar advantage of the near-sparsity of this four-index tensor. Through demonstrations on alkanes of increasing length, we show that accuracy and overall statistical uncertainty are not meaningfully affected and that a total cost crossover is reached as early as 50 electrons when using a minimal basis. Further study will be needed to assess where the crossover occurs in more compact molecular geometries and larger basis sets and to explore how in that context the crossover can be accelerated.
Accurately predicting the effects of correlations caused by strong interactions between electrons represents one of the key unsolved challenges in quantum chemistry. This thesis details theoretical developments addressing two aspects of this challenge. First, advances in Canonical Transformation (CT) theory address the task of describing weak interactions between the chemically active valence electrons and the chemically inert core electrons when the valence electrons are strongly correlated. Second, new methodologies for Jastrow factor wavefunctions are explored in an attempt to describe strong interactions amongst the valence electrons. In its original form, CT theory suffered from poorly understood numerical instabilities in strongly interacting systems and poor accuracy relative to coupled cluster theory in weakly interacting systems. Here the numeric issues are identified as consequences of CT theory's central approximation and shown to be similar to the intruder states of many-body perturbation theory. It is demonstrated that these instabilities can be avoided by placing particular restrictions on the theory's degrees of freedom. In weakly interacting systems, we show that using a quadratic, rather than linear, commutator approximation improves accuracy, and that the accuracy in any system can be improved by making use of the system's 3-body reduced density matrix. These improvements are demonstrated in a variety of systems, including N2 , H2 O, NiO, and free base porphin. Jastrow factor wavefunctions have traditionally been handled using quan- tum Monte Carlo techniques such as variational Monte Carlo (VMC). Here we explore how VMC can be made to accommodate the unconventionally large numbers of variational parameters present in many-body Jastrow factors and show that the stochastic reconfiguration algorithm can be thus improved by employing the method of conjugate gradients to solve its central linear equation. In addition to extending the reach of VMC, we have also developed nonstochastic methods for working with Jastrow factor wavefunctions that do not rely on random sampling. These methods, similar in design to those used with the coupled cluster wavefunction, produce energies very close to those of VMC in applications to the Heisenberg, spinless Hubbard, and full Hubbard model Hamiltonians.
We investigate the optimization of flexible tailored real-space Jastrow factors for use in the transcorrelated (TC) method in combination with highly accurate quantum chemistry methods, such as initiator full configuration interaction quantum Monte Carlo (FCIQMC). Jastrow factors obtained by minimizing the variance of the TC reference energy are found to yield better, more consistent results than those obtained by minimizing the variational energy. We compute all-electron atomization energies for the challenging first-row molecules C2, CN, N2, and O2 and find that the TC method yields chemically accurate results using only the cc-pVTZ basis set, roughly matching the accuracy of non-TC calculations with the much larger cc-pV5Z basis set. We also investigate an approximation in which pure three-body excitations are neglected from the TC-FCIQMC dynamics, saving storage and computational costs, and show that it affects relative energies negligibly. Our results demonstrate that the combination of tailored real-space Jastrow factors with the multi-configurational TC-FCIQMC method provides a route to obtaining chemical accuracy using modest basis sets, obviating the need for basis-set extrapolation and composite techniques.