Toward Accurate Hydrogen Bonds by Scalable Quantum Monte Carlo

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Toward accurate hydrogen bonds by scalable quantum Monte Carlo Matus Dubecky, Petr Jurecka, Lubos Mitas, Matej Ditte, and Roman Fanta J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.9b00096 • Publication Date (Web): 26 Apr 2019 Downloaded from http://pubs.acs.org on April 27, 2019

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Journal of Chemical Theory and Computation

Toward accurate hydrogen bonds by scalable quantum Monte Carlo Matúˇs Dubecký,∗,†,‡ Petr Jureˇcka,¶ Lubos Mitas,§ Matej Ditte,† and Roman Fanta† †Department of Physics, University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech Republic ‡ATRI, Slovak University of Technology, Paul´ınska 16, 917 24 Trnava, Slovakia ¶Regional Centre of Advanced Technologies and Materials, Department of Physical Chemistry, Palacký University Olomouc, tˇr. 17 listopadu 12, 771 46 Olomouc, Czech Republic §Department of Physics and CHiPS, North Carolina State University, Raleigh, NC 27695, USA E-mail: [email protected]

Abstract Single-determinant (SD) fixed-node diffusion Monte Carlo (FNDMC) gains popularity as a benchmark method scalable to large noncovalent systems although its accuracy limits are not yet fully mapped out. We report on an interesting example of significant SD FNDMC accuracy variations in middle-sized hydrogen-bonded dimer complexes, formic acid (FA) vs methanediol (MD), distinct by the maximum bond order (2 vs 1). While the traditional SD FNDMC schemes based on bias cancellation are capable of achieving benchmark (2%) accuracy for MD, this has not been the case for FA. We identify the leading systematic error source in energy differences and show that suitably

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designed Jastrow factors enable SD FNDMC to reach the reference accuracy for FA. This work clearly illustrates the varying accuracy of the present-day SD FNDMC at the 0.1 kcal/mol scale for a particular set of systems but also points out promising routes toward alleviation of these shortcomings, still within the single-reference framework.

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Motivation

Single-determinant (SD) fixed-node diffusion Monte Carlo 1–3 (FNDMC), an electronic structure quantum Monte Carlo (QMC) method, gains popularity as a benchmark method 4 suitable for large noncovalent systems 5–9 in cases where the traditional approaches like coupledcluster (CC) 10 reach their limits (e.g., in supramolecular complexes containing >100 carbon atoms 5,9,11 ). This is relevant in particular for exact description of dynamic electronic correlation effects important for noncovalent interactions (NI), direct description of correlation in periodic systems, advantageous low-order polynomial CPU cost scaling, and, nearly ideal parallelism of QMC approaches. 12–14 FNDMC is a position space (complete basis set) projector technique where antisymmetry of the simulated state is preserved via the nodes of the trial state ΨT . The approximate nodes induce energy bias that is usually small 15 for reasonable ΨT that is constructed from selfconsistent approaches (like, e.g., DFT 16 ) and in many cases provides competitive accuracy with compact yet effective ansätze like SD Slater-Jastrow (SJ) ΨT (from now, SD SJ FNDMC is referred to as 1FN). However, use of approximate ΨT introduces somewhat empirical component into the FNDMC methodology since the node quality and thus bias limits of a given ΨT are not a priori known. Therefore, the method performance should be tested and illustrated on a handful of representative systems to provide more clear delineation of the accuracy bounds for broader use. Non-covalent bonds represent a weak interaction limit with monomer wave functions modified only in the low-density intermolecular regions where the exchange is negligible. Since FNDMC well describes dynamic correlations, it is expected that the corresponding 2


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Figure 1: Dimer complexes of formic acid (FA), FA-derived complexes FA+ and FA’ (see Models and Methods), methanediol (MD), and, a model of the central part of the WatsonCrick adenine-thymine complex (ATm). Colorcode: white, H; gray, C; red, O; blue, N. nodal bias mostly cancels out in energy differences and the interaction energies (∆E) can be estimated with a high accuracy. 4,17–20 Indeed, this has been observed for trivially saturated closed-shell s/p complexes like dimers of FH or water where 1FN with DFT orbitals reached benchmark accuracy (absolute relative errors, RE, in energy differences not exceeding 2% 21 ) of CCSD(T)/CBS. 6,20,22–29 This is very encouraging and suggests that bias-cancellationbased 1FN method has a potential to become a complementary reference method that scales to truly large and also periodic systems. There are however examples of non-trivially saturated closed-shell s/p complexes, e.g., dimers of HCN, 27 formaldehyde 27 or formic acid, 30,31 where the present-day 1FN produces unacceptable RE of the order of 3-9%. One would expect routine use of 1FN method here, since these complexes are referred to as single-reference ones 29 within the theories where the low-rank excitations from the reference determinant are present (e.g., CCSD). It is not clear whether these cases represent a major accuracy obstacle for 1FN setting, or, whether it is possible to overcome this barrier as would be very much desired for large systems. Clearly, more research is required to offer a clear picture for representative classes in this context. In this letter, we demonstrate these nonsystematic accuracy issues of conventional 1FN approach in similar middle-sized hydrogen-bound dimers: formic acid (FA) 32,33 featuring

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two kinds of oxygen coordination (in =O and -O-H setting) vs trivially saturated methanediol (MD) complex (Fig. 1) with one type of oxygen. Our results reinforce the point that the current 1FN employing Jastrow correlating factor (J) with isotropic nucleus-universal electron-nucleus (en) and een terms is not able to reach benchmark accuracy in FA, in line with the existing literature, 30,31 whereas, it is well able to do so for trivially saturated MD complex. Our computations involving unconventional J factors and additional complexes reveal that the main contribution to the poor bias cancellation in FA stems from the locality approximation in FNDMC. We exemplify that conventional J terms, with a single parameter set per nucleus with distinct charge, perform less effectively in complexes where inequivalent configurations of the same-type atom exist, since the single parameter set per type effectively averages local variability that in turn increases the bias. By “inequivalent” we understand atoms that differ by their local molecular environment like hybridization, covalent/noncovalent bonding patterns and similar (e.g., sp2 vs sp3 -hybridized carbons in one system, or, an atom participating in a hydrogen bond vs same-type atom that does not). We verify that if the leading error in energy differences stems from such a problem, the accuracy bottleneck of 1FN may be overcome by a suitable J adjustment. Two distinct strategies are sketched out as promising routes worth of further work: i) a more elaborated yet still simple J term that takes into account local variability by using separate parameter sets for inequivalent atoms of the same type (we dub this approach as “distinct” J), and, ii) simple yet effective omission of en terms, that, instead of more variational flexibility aims to preserve bias-cancellation nature of the SD scheme (it seems that omission of en terms limits bias better than optimized but averaged ne terms do). Interestingly, such 1FN model supplied with intact DFT orbitals can ultimately reach benchmark accuracy even for FA. Further promise of these approaches is also exemplified on a few other systems with related challenges as an additional corroboration of the proposed approach. Alongside, we employ an a priori diagnostic method 34 based on correlated non-QMC natu-

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Table 1: Results summary. 1FN interaction energies ∆E (kcal/mol), their differences vs reference CCSD(T)/CBS data AE (kcal/mol), and, related absolute relative errors, RE (%). Shorthand: 2cJ/3cJ denotes 2/3-center Jastrows containing electron-electron (ee), electronnucleus (en)/ee, en, een terms; 3cJdis contains ee and different en and een terms for distinct types of the same-type nuclei; 3cJxEN contains ee and een terms (no en terms). Complex FA

MD

FA+

FA’

WA

FD

ATm

Method CCSD(T)/CBSa 1FN/3cJb 1FN/3cJc 1FN/3cJc

∆E -18.81 -20.2±0.2 -19.9±0.1 -19.6±0.1

AE

RE

-1.39 -1.09 -0.79

7.4 5.8 4.2

1FN/2cJ 1FN/3cJ 1FN/3cJdis 1FN/3cJxEN

-19.76±0.07 -19.41±0.07 -19.16±0.06 -18.85±0.09

-0.95 -0.60 -0.35 -0.04

5.0 3.2 1.9 0.2

CCSD(T)/CBS 1FN/2cJ 1FN/3cJ 1FN/3cJxEN

-13.18 -13.37±0.07 -13.12±0.06 -13.08±0.06

-0.19 0.06 0.10

1.4 0.5 0.7


-9.80 -10.24±0.07 -10.12±0.1 -9.71±0.1

-0.44 -0.32 0.09

4.5 3.3 0.9


-6.24 -6.50±0.08 -6.37±0.1 -6.20±0.1

-0.26 -0.12 0.04

4.1 2.0 0.6

CCSD(T)/CBSd 1FN/2cJ 1FN/3cJ 1FN/3cJdis 1FN/3cJxEN

-5.01 -5.20±0.04 -5.14±0.03 -5.10±0.03 -5.00±0.03

-0.20 -0.13 -0.09 0.01

3.9 2.6 1.8 0.1

CCSD(T)/CBSd 1FN/2cJe 1FN/2cJ 1FN/3cJe 1FN/3cJ 1FN/3cJdis 1FN/3cJxEN

-4.55 -4.90±0.10 -5.07±0.07 -4.88±0.06 -4.96±0.05 -4.87±0.04 -4.55±0.05

-0.35 -0.52 -0.33 -0.41 -0.32 0.00

7.7 11.3 7.3 8.9 6.9 0.1

CCSD(T)/CBS 1FN/3cJ 1FN/3cJxEN

-15.95 -16.36±0.04 -16.31±0.05

-0.41 -0.36

2.6 2.3

a

Ref. 35;

b

Ref. 30;

c

Ref. 31;

d

Ref. 36;

e

Ref. 27

ral orbital occupation numbers (NOONs) to further trace down the expected magnitude and sign of bias in 1FN ∆Es.

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Results and discussion

We begin with FA complex (Fig. 1, Tab. 1). The FNDMC data available for FA in the literature indicate that 1FN with 3-center J (3cJ) containing up to three-center een terms overbinds ∆E by as much as 4-7%. Our 1FN/2cJ (J with 2-center ee and en terms) and 1FN/3cJ results are in line with such a finding. This is indeed above the desired benchmark accuracy level of RE

Toward Accurate Hydrogen Bonds by Scalable Quantum Monte Carlo

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