Accurate Intermolecular Interaction Energies Using Explicitly

Subscriber access provided by UNIV OF NEW ENGLAND ARMIDALE

Quantum Electronic Structure

Accurate Intermolecular Interaction Energies Using Explicitly Correlated Local Coupled-cluster Methods [PNO-LCCSD(T)-F12] Qianli Ma, and Hans-Joachim Werner J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b01098 • Publication Date (Web): 09 Jan 2019 Downloaded from http://pubs.acs.org on January 10, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

Accurate Intermolecular Interaction Energies Using Explicitly Correlated Local Coupled-cluster Methods [PNO-LCCSD(T)-F12] Qianli Ma and Hans-Joachim Werner∗ Institut für Theoretische Chemie, Universität Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany E-mail: [email protected]

Abstract We present benchmark results for the A24, S66, and X40 sets of intermolecular interaction energies obtained with our recently developed PNO-LCCSD(T)-F12 method. Using the aug-ccpVQZ-F12 basis set and tight domain options, the root mean square (RMSD) and maximum (MAXD) deviations from the currently best CCSD(T)/CBS estimates for the S66 set amount only to 0.02 kcal mol−1 and 0.06 kcal mol−1 , respectively. The corresponding triple-ζ (aug-ccpVTZ-F12) results are similarly accurate, and even with double-ζ (aug-cc-pVDZ-F12) basis sets the RMSD and MAXD deviations amount only to 0.05 and 0.11 kcal mol−1 , respectively. Preliminary PNO-LCCSD(T)-F12 calculations on the X40 set of intermolecular interactions of halogenated molecules yield interaction energies in reasonable agreement with the original CCSD(T)/CBS estimates. The PNO-LCCSD(T)-F12 method does not rely on error cancellations as the popular ∆CCSD(T) approach, and can yield comparable or better accuracy at a fraction of the cost. This accuracy is of importance for studying reactions involving large molecules, in which intramolecular noncovalent interactions are important and no counterpoise corrections are possible.

1

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 Introduction Noncovalent interactions are of great importance in large molecular systems and in molecular biology, as they affect the three-dimensional molecular structure and properties. They also have a significant influence on the energetics of chemical reactions. The accurate prediction of such interactions is one of the greatest challenges in computational chemistry, since electron correlation effects play a crucial role. In particular, π-stacked complexes such as the benzene dimer are solely bound by dispersion interactions, which are pure correlation effects. In order to study such interactions, and to test the accuracy of various electronic structure methods, several benchmark sets of molecular dimers have been proposed and extensively studied. 1–8 For an excellent recent review ˇ c, Riley and Hobza, 2,3,7 which see ref 9. Among the most popular ones is the S66 benchmark of Rezáˇ covers many different interaction motifs and has been extensively studied by several authors. 10–14 Great efforts have been spent on estimating the complete basis set (CBS) limit for the “gold standard” coupled cluster with single and double excitations, and perturbative treatment of triple excitations [(CCSD(T)] method, which has been shown to work very well for intermolecular interactions of small systems. 8,9,15 However, due to the steep cost scaling of CCSD(T) with molecular size, it is only possible for the smallest systems to estimate the CBS limit directly at this level. Therefore, it has become usual to employ composite techniques 16–18 (sometimes called focal point approximations 19–23 ) in which one estimates the MP2/CBS limit using extrapolation techniques, and then adds CCSD and triples corrections obtained with smaller basis sets. For the intermolecular interaction energies of small systems these schemes are known to work well. 8,9 However, for larger systems the computations of the CCSD and (T) contributions are only possible with small basis sets (triple-ζ at best), and are still very expensive. Such basis sets are insufficient for describing the “high level corrections”, and the success of these calculations relies on the cancellation of the basis-set errors in CCSD(T) calculations and the intrinsic errors of MP2. The error compensation is known to break when one uses accurate extrapolated CCSD energies and adds (T) corrections computed with smaller basis. 9 If one considers applications for realistic larger molecular systems in which intramolecular noncovalent interactions are important, the MP2, CCSD, and (T) relative 2


Page 2 of 28

Page 3 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


correlation energies can be in very different ratios, and the composite approach may lead to larger errors. Also counterpoise (CP) corrections 24 for basis set superposition effects (BSSE) are not possible for such systems. The basis set problem can be much alleviated by including explicitly correlated terms (F12 methods). As first shown by Marchetti and Werner 25,26 for the S22 benchmark, 1 this yields highly accurate interaction energies already at the triple-ζ level, provided that diffuse functions are included in the basis. More recently, this was confirmed by several other studies. 10–14,27,28,28–30 There are a number of different approximations for computing the CCSD-F12 energy efficiently, 31–34 but for triple-ζ and larger basis sets the differences of the results are small. When using F12 methods, it is also important to apply the complementary auxiliary basis set (CABS) singles correction 31 in order to minimize errors of the Hartree-Fock contribution. Otherwise these can be larger than the errors of the correlation contribution. The steep scaling wall can be overcome by exploiting the short-range nature of dynamical correlation using local pair and domain approximations. Local correlation methods are particularly valuable if they are combined with explicit correlation techniques, since the F12 terms also correct for part of the domain errors. 35–37 The currently most accurate and efficient local coupled cluster methods are based on pair natural orbitals (PNOs). 28–30,38–41 For a more extensive bibliography see a recent review. 30 In the current work we apply the recently developed PNO-LCCSD(T)-F12 method 28–30,41 to the A24, 5 S66, 3 S66x8, 13 and X40 4 benchmarks in order to test its performance. Our method is well parallelized and can be applied in a black box manner to molecules with up to about 300 atoms and 10000 basis functions. In contrast to canonical methods, calculations with triple-ζ and quadruple-ζ basis sets are readily possible for all benchmark sets. For the A24 benchmark, which comprises only rather small molecules, our results are compared to highly accurate CCSD(T)/CBS estimates of ref 5. In this case we also compare the performance of different F12 approximations. For the S66 set our results are compared to the latest canonical CCSD(T)/CBS estimates of Kesharwani et al., 14 and it is shown that comparable accuracy can be achieved. As a reference, we have chosen the

3



“silver” values from Table 6 in ref 14, since these correspond to the same computational model for all 66 dimers. For a subset of 18 dimers, improved “gold” values are also presented in ref 14. In most cases these differ only slightly from the “silver” values, and the statistics using both reference sets are very similar.

2 Computational details The geometries of the A24, S66, and X40 dimers were taken from the supporting information of the original publications. 3–5 All calculations were carried out using the aug-cc-pVnZ-F12 basis sets 42,43 (n=D,T,Q) for heavy atoms. For the hydrogen atoms the standard cc-pVnZ-F12 sets 42 without additional diffuse functions were employed, since otherwise the basis sets became too linearly dependent. In the following, these mixed basis sets are denoted aVnZ-F12, or simply DZ, TZ, and QZ. Unless otherwise noted, the PNO-LCCSD(T)-F12 calculations were carried out with default and “domopt=tight” options, as defined in our previous paper 30 and implemented in the Molpro2018 package. 44 The tight options adopt more conservative domain approximations for higher accuracy. Default settings were employed for pair and projection approximations. 30 As originally proposed by Knizia et al., 32 the perturbative triples corrections 29 were scaled by the ratio Ecorr (MP2F12)/Ecorr (MP2) to account approximately for the basis set incompleteness error of the triples energy (which is not directly affected by the F12 treatment). The scaling factor determined for the dimer was also used for the monomers in order to keep size consistency. 26 In the following, the scaled triples correction is denoted (T*). Approximation F12b based on Ansatz “3*A(FIX, NOX)” was used in all LCCSD-F12 calculations. FIX means that fixed F12 amplitudes are used, which are obtained from the first-order cusp 2 conditions. 45,46 NOX means that contributions of integrals over F12 are neglected (this has no effect

in closed-shell calculations with 3*A approximation). For more details of these approximations see refs 29–32,47. The RI basis sets used for the S66 and S66x8 benchmarks in TZ and QZ calculations

4


Page 4 of 28

Page 5 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


were the union of the orbital basis and the corresponding cc-pVnZ-F12/OPTRI sets. 48 In all other calculations, the cc-pVnZ-F12+/OPTRI sets 49 were used (see below). The geminal exponent was −2 chosen to be 0.9 a−2 0 for aVDZ-F12 and 1.0 a0 for the larger basis sets. Local RI approximations

with default domains as described in ref 30 were used. The density fitted Hartree-Fock (DF-HF) calculations used the aug-cc-pVnZ/JKFIT auxiliary basis sets from the Molpro basis set library, which were derived from the original cc-pVnZ/JKFIT sets 50 by adding for each angular momentum a diffuse function in an even-tempered manner. All integrals in the PNO-LCCSD(T)-F12 calculations (except for the Fock matrix) were computed using local density fitting (LDF) and the aug-cc-pVnZ/MP2FIT auxiliary basis sets. 51 Default local fitting domains were constructed as described in refs 29,52. For the aVDZ-F12 calculations, the triple-ζ JKFIT and MP2FIT sets were used, in order to minimize errors caused by the density fitting approximations. The Hartree-Fock values were improved by adding the CABS singles corrections. 31 These were computed without local approximations using the cc-pVnZ-F12+/OPTRI CABS basis sets, 49 which differ from the original cc-pVnZ-F12/OPTRI sets by additional tight functions in order to improve the CABS correction. This correction is most important for the aVDZ-F12 basis set and is in most cases ≤ 0.001 kcal mol−1 in the calculations with the aVQZ-F12 basis set. For the bromine and iodine atoms in the X40 set we used the effective core potentials ECP10MDF and ECP28MDF, respectively, along with the aug-cc-pVTZ-PP and aug-cc-pVQZ-PP basis sets. 53,54 Corresponding auxiliary basis were used for density fitting. 55 We used the Molpro aug-cc-pVnZ/JKFIT basis sets as the RI basis for bromine atoms, and the def2-nZVPP/JKFIT basis sets 56 for iodine atoms. Only the valence electrons were correlated. All interaction energies were CP corrected, but for comparison some CP-uncorrected values were also computed. A sample input is provided in the supplementary material.

5



3 Results 3.1

The A24 benchmark

In this section we consider the A24 benchmark which consists of dimers with up to 13 atoms, representing various type of intermolecular interactions. Due to the small sizes of these dimers, most accurate calculations are possible for this set, and basis set, higher excitations, core-correlation, and relativistic effects have previously been studied. 5 Sirianni et al. 27 have presented estimated CCSD(T)/CBS values denoted A24B, which were extrapolated from aVQZ/aV5Z or larger basis sets results (including the CP correction). Here we will use these values as the reference in evaluating our computational model. The authors have also studied the convergence with the basis-set size for various approximate CCSD-F12 treatments using the aug-cc-pVnZ and cc-pVnZ-F12 basis. It was found that with scaled triples, all F12 treatments result significantly improved basis-set convergence, and the aug-cc-pVnZ basis sets are more appropriate for the computation of the intermolecular interaction energies.

RMSD from A24B / kcal mol−1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.12

CCSD(T*)-F12b/3C/AVnZ CCSD(T*)-F12b/3*A/aVnZ-F12 CCSD(T*)-F12b/3C/aVnZ-F12 CCSD(T)-F12b/3*A/aVnZ-F12 CCSD(T)-F12b/3C/aVnZ-F12

0.10 0.08 0.06 0.04 0.02 0.00

2

3 basis-set cardinal number

4

Figure 1: Root mean square deviations (RMSD) from the A24B reference values for various computation models. The AVnZ results are taken from ref 27. All other calculations were performed with the canonical program without local approximations. In the present work, we use the aug-cc-pVnZ-F12 basis, 42,43 which was developed recently in order to improve the unsatisfactory behavior of the original cc-pVnZ-F12 basis sets for the computation of intermolecular interaction energies. Also, our PNO program employs F12 Ansatz 6


Page 6 of 28

Page 7 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3*A rather than the more common and formally more rigorous Ansatz 3C. Ansatz 3*A avoids double RI approximations for which local approximations cannot be easily applied without a significant loss of accuracy or efficiency. Special PNO approximations have been proposed to treat these RIs, 11,57,58 but we find the 3*A approach much simpler and of superior accuracy (see below). Our results for the individual molecules are presented in Tables S1-S6 of the supporting information. Figure 1 illustrates the performance of the CCSD(T)-F12b/3*A(FIX) computational model used in our PNO methods without any local approximations. For comparison, results obtained with Ansatz 3C(FIX) are also shown. We first consider the results without scaled triples, as represented by the dotted lines in Figure 1. It can be seen that both F12 approaches give fast basis-set convergence, with the QZ result having RMS errors of ∼0.01 kcal mol−1 . Despite being a theoretically less rigorous treatment, the 3*A method gives smaller basis-set errors for all three basis-sets due to systematic error cancellations. This is consistent with previous findings. The scaled triples treatment significantly reduces the overall basis-set errors in both F12 methods, and RMS errors of < 0.01 kcal mol−1 are already achieved with the TZ basis with the 3*A method. For comparison, we also show the RMS errors of the interaction energies computed with the aug-cc-pVnZ basis sets (denoted AVnZ). The results are very similar to those computed with the corresponding aug-cc-pVnZ-F12 basis sets for this benchmark set. We now consider the errors caused by the local approximations in the PNO calculations. Table 1 shows the statistical deviations of the LCCSD(T*)-F12 results from the corresponding canonical ones using the same F12 approach and the same basis sets. It can be seen that the local errors decrease with increasing basis-set size, and the “tight” domain options further reduce the local errors roughly by a factor of 2 when TZ and QZ basis sets are used. As expected, the local errors are somewhat larger when triple excitations are included, but overall the triples do not deteriorate the accuracy much. Table 2 shows the error statistics of PNO-LCCSD(T*)-F12 results using various basis sets and settings relative to the A24B reference values. With “tight” domain options the local method yields results within 0.1 kcal mol−1 or less of the CCSD(T)/CBS limit, provided that at least TZ basis sets

7



Page 8 of 28

Table 1: Statistical Deviations of PNO-CCSD-F12, (T), and PNO-LCCSD(T*)-F12 Results for the A24 Benchmark Relative to the Corresponding Canonical Results using the Same Basis Sets Domopt Basis CCSD-F12: default aVDZ-F12 default aVTZ-F12 default aVQZ-F12 tight aVDZ-F12 tight aVTZ-F12 tight aVQZ-F12

RMSD MAD MEAN MAXD 0.044 0.020 0.016 0.066 0.018 0.011

0.031 0.007 0.016 0.014 0.013 0.012 0.028 −0.020 0.010 −0.005 0.008 0.001

0.151 0.053 0.039 0.241 0.076 0.024

(T) Correction: default aVDZ-F12 default aVTZ-F12 default aVQZ-F12 tight aVDZ-F12 tight aVTZ-F12 tight aVQZ-F12

0.018 0.019 0.019 0.007 0.008 0.008

0.015 0.016 0.016 0.006 0.006 0.007

0.014 0.016 0.016 0.005 0.006 0.007

0.057 0.052 0.051 0.018 0.022 0.022

CCSD(T*)-F12: default aVDZ-F12 default aVTZ-F12 default aVQZ-F12 tight aVDZ-F12 tight aVTZ-F12 tight aVQZ-F12

0.059 0.039 0.035 0.068 0.020 0.016

0.046 0.024 0.032 0.031 0.029 0.029 0.033 −0.014 0.014 0.002 0.013 0.008

0.162 0.110 0.092 0.246 0.073 0.042

Table 2: Statistical Deviations of PNO-LCCSD(T*)-F12 Results Relative to the A24B Benchmark Domopt default default default tight tight tight

Basis aVDZ-F12 aVTZ-F12 aVQZ-F12 aVDZ-F12 aVTZ-F12 aVQZ-F12

RMSD 0.077 0.039 0.031 0.073 0.020 0.013

8

MAD MEAN MAXD 0.061 0.045 0.211 0.031 0.030 0.114 0.025 0.025 0.088 0.047 0.007 0.246 0.013 0.001 0.072 0.010 0.004 0.038


Page 9 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


are used. For these small systems, the local errors are the primary source of errors, thanks to the very small basis-set errors of the F12/3*A method. It should be noted that both basis-set errors and the local errors are extensive, as are the intrinsic errors of the CCSD(T) method. Therefore, the errors are expected to increase with system size. On the other hand, the cost advantage of local methods allows carrying out calculations for large systems using quality basis sets, which would be prohibitively expensive with canonical methods. Thus, it is expected that for large systems overall more accurate results can be obtained with local methods.

3.2

The S66 and S66x8 benchmarks

The computed CP-corrected interaction energies obtained with tight domain settings are presented in Table S7 of the supporting information for the three basis sets. In the last three columns the deviations from the “silver” results of ref 14 are given. Corresponding statistical data are given in the upper part of Table 3, which include root mean square deviations (RMSD), mean absolute deviations (MAD), mean signed deviations (MEAN), and maximum deviations (MAXD). In the lower part of Table 3 the statistical measures relative to the current CP-corrected PNO-LCCSD(T*)F12/aVQZ-F12 results are presented. Both references yield rather similar deviations. The calculations with default domains are not sufficiently accurate, yielding RMS and MAX deviations of about 0.1 and 0.25 kcal mol−1 , respectively, using either TZ or QZ basis sets. This is expected from our previous results for the S22 benchmark, 30 where it was shown that the error is mainly due to the domain errors that appear in the (T) correction. While the domain and basis-set errors of the PNO-LCCSD-F12 energy are significantly reduced by the F12 treatment, 35–37 this is not the case for the triples correction, which depends linearly on the doubles amplitudes. Therefore, large PAO and PNO pair domains in LCCSD calculations are necessary to obtain an accurate triples energy. Accordingly, large triples domains are also necessary in accurate calculations of intermolecular interaction energies. Much better results are obtained with “tight” domain settings. In this case the RMS deviations from the reference values amount to only 0.024 and 0.020 kcal mol−1 for the TZ and QZ basis sets, 9



Page 10 of 28

Table 3: Statistical Deviations of Local PNO Results Relative to the “Silver” Values (Upper Part) and the aVQZ-F12 Values of the Current Work (Lower Part) Method Domopt Basis Deviations relative to “silver” values: CCSD(T*)-F12 CCSD(T*)-F12 CCSD(T*)-F12

RMSD

MAD MEAN MAXD System(MAXD)

default default default

aVDZ-F12 aVTZ-F12 aVQZ-F12

0.148 0.106 0.108

0.133 0.093 0.094

0.133 0.093 0.094

0.319 26 uracil–uracil (π–π) 0.254 26 uracil–uracil (π–π) 0.232 26 uracil–uracil (π–π)

CCSD(T*)-F12(nocp) default CCSD(T*)-F12(nocp) default CCSD(T*)-F12(nocp) default


0.074 0.069 0.102

0.056 0.055 0.088

0.006 0.054 0.088

0.258 21 AcNH2 –AcNH2 0.187 26 uracil–uracil (π–π) 0.226 26 uracil–uracil (π–π)

CCSD(T*)-F12 CCSD(T*)-F12 CCSD(T*)-F12

tight tight tight


0.044 0.024 0.020

0.037 0.025 0.018 −0.009 0.016 −0.001

0.117 20 AcOH–AcOH 0.057 26 uracil–uracil (π–π) 0.058 60 ethyne–AcOH (OH–π)

CCSD(T*)-F12(nocp) tight CCSD(T*)-F12(nocp) tight CCSD(T*)-F12(nocp) tight


0.152 0.058 0.021

0.120 −0.118 0.049 −0.049 0.017 −0.003

0.388 21 AcNH2 –AcNH2 0.149 26 uracil–uracil (π–π) 0.043 36 neopentane–neopentane

aVTZ-F12 + BFa 0.034 d-aug-cc-pVTZb 0.035

0.025 −0.023 0.027 −0.026

0.093 26 uracil–uracil (π–π) 0.088 41 uracil–pentane

CCSD(T*)-F12 CCSD(T*)-F12

tight tight

Deviations relative to CCSD(T*)-F12/aVQZ-F12 (tight): CCSD(T*)-F12 CCSD(T*)-F12 CCSD(T*)-F12

default default default


0.147 0.105 0.107

0.134 0.094 0.096

0.134 0.094 0.096

0.348 26 uracil–uracil (π–π) 0.284 26 uracil–uracil (π–π) 0.262 26 uracil–uracil (π–π)

CCSD(T*)-F12(nocp) default CCSD(T*)-F12(nocp) default CCSD(T*)-F12(nocp) default


0.076 0.067 0.101

0.055 0.056 0.090

0.007 0.055 0.090

0.257 21 AcNH2 –AcNH2 0.217 26 uracil–uracil (π–π) 0.256 26 uracil–uracil (π–π)

CCSD(T*)-F12 CCSD(T*)-F12 CCSD(T*)-F12

tight tight tight


0.039 0.015 0.000

0.030 0.027 0.011 −0.008 0.000 0.000

0.108 60 ethyne–AcOH (OH–π) 0.041 28 benzene–uracil (π–π) 0.000 (Reference)

CCSD(T*)-F12(nocp) tight CCSD(T*)-F12(nocp) tight CCSD(T*)-F12(nocp) tight


0.152 0.056 0.011

0.118 −0.117 0.049 −0.047 0.006 −0.001

0.412 17 uracil–uracil (BP) 0.121 28 benzene–uracil (π–π) 0.067 60 ethyne–AcOH (OH–π)

a b

Including bond functions as described in ref 59. aug-cc-pVTZ basis for hydrogen atoms.

10


Page 11 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


respectively, and the maximum errors are about 0.06 kcal mol−1 . The mean unsigned relative errors for TZ and QZ are both close to 0.5%. This can be compared to the intrinsic error of the frozen core CCSD(T) method, which has been estimated to be 2% for interaction energies of small organic molecules. 9 All statistical measures for the TZ and QZ basis sets are similar, while the individual results differ slightly, as seen in Table S7 in the supporting information. This indicates that the TZ results are already very close to be converged with respect to the basis set. On the other hand, the remaining errors reflect the limit of the accuracy that can be achieved in these calculations. These errors are largely due to the local approximations. The major ones are the domain and projection approximations, which slightly reduce the computed interaction energies, and the approximations in the F12 projector that slightly overestimate the interaction energies. Furthermore, the triples scaling is an approximation and introduces the largest uncertainty in our computational model (see below). All these effects are individually of the order of a few hundreds of a kcal mol−1 , and therefore it is very difficult to further improve the results. Using the TZ basis set, we also carried out calculations occ occ with even tighter PNO thresholds T pno_lmp2 = T pno_cc = 10−9 . Surprisingly, this slightly deteriorated

the statistics; in particular, the CP corrections significantly increased. This indicates that the “tight” occ occ PNO domains (T pno_lmp2 = T pno_cc = 10−8 ) of intramolecular pairs are still mainly located on the

monomers, which reduces basis set superposition (BSSE) effects. 60,61 The BSSE can be further reduced by restricting the PAO orbital domains to the individual monomers. However, this did not improve the accuracy, since then contributions of ionic excitations are missing. 61 Table 4 shows the individual contributions to the interaction energies for the π-stacked benzene and uracil dimers. These are among the most difficult cases, because the binding is solely due to dispersion, i.e., correlation effects. In case of the uracil dimer, the correlation contribution amounts to about 10 kcal mol−1 , of which ≈1.9 kcal mol−1 is due to the triples. Similar data for all 66 dimers are presented in Table S8 of the supporting information. Table 4 demonstrates that the PNO-LMP2-F12 and PNO-LCCSD-F12 interaction energies are nearly independent of the basis set. Statistical data showing the basis set effect for different

11



Table 4: Interaction Energies (kcal mol−1 ) for the π-stacked Benzene and Uracil Dimers at Different Computational Levelsa Method DF-HF+CABS PNO-LMP2-F12 PNO-LCCSD-F12 (T) Scaling factor (T*) (T[34] ) PNO-LCCSD(T*)-F12 PNO-LCCSD(T[34] )-F12 a

Benzene dimer (24) TZ QZ 3.961 3.960 −4.698 −4.710 −1.467 −1.466 −1.124 −1.146 1.079 1.038 −1.212 −1.189 -1.162 −2.679 −2.655 −2.629 −2.627

Uracil dimer (26) TZ QZ 0.378 0.377 −11.227 −11.212 −7.796 −7.800 −1.779 −1.826 1.087 1.041 −1.933 −1.902 -1.861 −9.729 −9.701 -9.657 −9.661

Tight domain options have been used in all calculations. All values include the CP correction and all F12 results the CABS singles correction.

methods are shown in Table 5. It is seen that the DF-HF values with CABS singles correction are nearly independent of the basis set. The dramatic effect of the F12 correction is demonstrated for LMP2, where the errors are reduced by one order of magnitude. Similarly small errors are obtained with LCCSD-F12. For these methods the F12 contributions not only correct for the basis set incompleteness error, but also for domain errors. However, these correcting effects are missing in the triples contribution. Both the basis set and the domain errors reduce the absolute values of the (always negative) triples contributions, and therefore the PNO-LCCSD(T)-F12 interaction energies without any correction should generally be to small. The triples scaling should account approximately for the basis set incompleteness error. The scaling factors are typically 1.08 and 1.04 for the TZ and QZ basis sets, respectively (somewhat larger for hydrogen bonded systems, e.g. 1.103 and 1.048 for the H2 O dimer). It is found that the scaled triples contributions are smaller for QZ than for TZ, which indicates that the scaling somewhat overestimates the basis set effect. An alternative possibility is to extrapolate the TZ and QZ triples contributions. We have used the usual 1/n3 extrapolation, 62,63 where n is the cardinal number. The resulting extrapolated triples contributions are denoted (T[34] ). They are consistently smaller than the scaled ones. It should be

12


Page 12 of 28

Page 13 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


noted, however, that some unavoidable uncertainty in this extrapolation arises from the fact that the domains for the two basis sets may be nonequivalent. The scaled and extrapolated QZ results differ by at most 0.04 kcal mol−1 (RMSD: 0.016 kcal/mol). It is difficult to decide which approach is more accurate. We prefer the scaled values, since the somewhat too large scaling might compensate for the remaining domain errors. Overall, however, the statistical deviations from the reference values are very similar, as seen in Table 5. On the average, our PNO-LCCSD(T*)-F12 interaction energies are slightly more negative than the “silver” reference values, while the ones with extrapolated triples contributions are more positive, as indicated by the signs of the MEAN deviations in Tables 3 and 4. Table 5: Differences of TZ and QZ CCSD-F12 (tight) Interaction Energies and Statistical Deviations of Local Coupled-Cluster Interaction Energies from the “Silver” Reference Valuesa Method RMSD DF-HF/TZ − DF-HF/QZ 0.004 (DF-HF+CABS)/TZ − (DF-HF+CABS)/QZ 0.001

MAD MEAN MAXD 0.003 0.002 0.019 0.001 0.000 0.004

MP2/TZ − MP2/QZ MP2-F12/TZ − MP2-F12/QZ LCCSD-F12/TZ − LCCSD-F12/QZ

0.144 0.015 0.008

0.126 0.126 0.009 −0.005 0.005 0.001

0.411 0.057 0.031

LCCSD-F12/TZ + (T[34] ) LCCSD-F12/QZ + (T[34] )

0.024 0.025

0.019 0.019

0.012 0.012

0.061 0.059

CCSD(T*)-F12/TZ CCSD(T*)-F12/QZ

0.024 0.020

0.018 −0.009 0.016 −0.001

0.057 0.058

a

All values include the CP correction and all F12 results the CABS singles correction.

In order to separate the basis-set errors and the local errors we also performed some canonical CCSD(T*)-F12 calculations for the S66 benchmark. Table 6 shows the deviations of canonical CCSD(T*)-F12 results from the reference values reported in ref 14. Similar to the A24 benchmark, the 3*A F12 method outperforms the 3C method. The difference is most significant for the small DZ basis set, with which the more rigorous 3C method gives a MAXD of more than 0.4 kcal mol−1 . Clearly, even with explicit correlation DZ is not a sufficient basis set for accurate computations of intermolecular interaction energies. The difference between the 3*A and 3C F12 treatments is reduced significantly when the TZ basis set is used. The statistical errors shown in Table 6 can 13



Page 14 of 28

Table 6: Statistical Deviations of Canonical CCSD(T*)-F12 Interaction Energiesa for the S66 Benchmark from the Reference Values of Ref 14 F12 Ansatz Method RMSD MAD MEAN MAXD Full set, relative to “silver” values 3C aVDZ-F12 0.128 0.103 0.102 0.411 3*A aVDZ-F12 0.035 0.026 −0.002 0.110 Subset of 18 dimers, relative to “gold” values 3C aVDZ-F12 0.158 0.132 0.132 3C aVTZ-F12 0.025 0.018 0.016 3*A aVDZ-F12 0.028 0.021 0.021 3*A aVTZ-F12 0.014 0.012 −0.012 a see footnote a of Table 5

0.414 0.078 0.085 0.027

Table 7: Statical Deviations of Local PNO-LCCSD(T*)-F12 Results for the S66 Benchmark Relative to Canonical CCSD(T*)-F12/3*A Results for the Same Basis Sets Domopt Method RMSD MAD MEAN MAXD Full set default aVDZ-F12 0.156 0.136 0.135 0.429 tight aVDZ-F12 0.040 0.032 0.027 0.111 Subset of 18 dimers default aVDZ-F12 default aVTZ-F12 tight aVDZ-F12 tight aVTZ-F12

0.100 0.091 0.034 0.019

14

0.087 0.079 0.027 0.015

0.083 0.079 0.017 0.012


0.211 0.198 0.095 0.042

Page 15 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


be compared with those shown in Table 7, which reflect the pure local errors in PNO calculations. Using the “tight” domain settings, the local errors decrease with the basis-set size, and are very comparable to the basis-set errors of the underlying CCSD(T*)-F12 method. We further performed some calculations with mid-bond function as described in ref 59 added to the TZ basis. The statistics of the deviations to the “silver” values are shown in Table 3. It can be seen that the addition of the mid-bond functions leads to systematically larger (more negative) interaction energies, and increases the RMSD from “silver” values by 0.01 kcal mol−1 . These error statistics are similar to those obtained with the doubly augmented d-aug-cc-pVTZ 64 basis sets (aug-cc-pVTZ for hydrogen atoms), although the individual values vary. It is difficult to judge whether these values are closer to the basis-set limit. However, both treatments have limitations in applications for extended systems, in which mid-bond functions are more difficult to define properly and doubly augmented basis sets can be strongly linear dependent. Of course, the error statistics depend on the reference values, which may have similar uncertainties. For example, with the TZ basis the largest error (−0.057 kcal mol−1 ) occurs for the stacked uracil dimer (system 26). Here a negative sign indicates a larger (more negative) interaction energy. If, however, the “14 carat gold” value of ref 14 is used as reference, the error is reduced to −0.035 kcal mol−1 , and the error of the QZ calculation from −0.030 to −0.008 kcal mol−1 . The corresponding extrapolated triples contribution reported in ref 14 amounts to 1.936 kcal mol−1 (fully CP corrected) or 1.951 kcal mol−1 (half CP corrected). These values are even larger than our scaled triples contributions (see Table 4). It is also interesting to note that the values of ref 14 are ˇ c et al. 2 The RMS and MAX consistently more positive (less attractive) than the earlier ones of Rezáˇ deviations between these two data sets amount to 0.039 and 0.078 kcal mol−1 . Based on the results presented above, we believe the PNO-LCCSD(T*)-F12 method with the TZ basis set and “tight” domain settings is a very cost-effective approach for computing intermolecular interaction energies. Using this approach, we have also computed the interaction energies for the S66x8 benchmark, in which the intermolecular distances are scaled by factors F between 0.9 and 2.0. We restricted our calculations to the first 5 scaling factors, i.e., 0.9, 0.95, 1.0, 1.05, and 1.10.

15



To ensure smooth potentials, the domains for each molecule were determined with F = 1.0 and kept frozen for the other distances. The resulting interaction energies for the 5 distances were interpolated using 4th order polynomials. Since for some systems the interaction energies for F = 0.9 are quite far from the ones at the minima (for the benzene dimer even positive), we also fitted the values at the other 4 distances using 3rd order polynomials. In most cases the fitted scaling factors and energies agree well, but there are some exceptions. More points closer to the minima would be needed to obtain accurate fits. Nevertheless, as shown in Tables S9 and S10 in the supporting information, in most cases the fitted energies agree closely with those obtained with the S66 geometries (which were also obtained by interpolation). In a few cases, however, the energies obtained at the S66 structures were slightly more negative than the interpolated ones (for example for dimer 61, pentane–AcOH). The reason for these deviations is still unclear. The spline-fitted potential energy curves for the benzene and uracile dimers are shown in Figure S1 in the supporting information. The red dots, which are placed at the minima of the curves, indicate the interaction energies computed at the S66 structures, and these values fit perfectly to the curves. To check the effect of domain freezing, we recomputed the interaction energies for F = 0.90 and F = 1.10 using the domains determined at these structures. The effect was found to be small but not negligible. The RMS deviations from the values with restarted domains for amount to 0.038 kcal/mol and 0.013 kcal/mol, respectively. It should be noted that domain freezing is not suitable for computing global potential energy surfaces, since the electronic structure and thus the domains may change during a reaction. 65,66 In particular this concerns the PAO domains, which depend on real space criteria. This problem can be overcome by domain merging procedures. 66 However, simple domain freezing is useful in calculations of energy derivatives using finite difference methods or for fitting the PES in the vicinity of stationary points. In the current case, no big changes of the electronic structure are expected for different F-values, but nevertheless the calculations with restarted domains may be somewhat less accurate than those for F = 1.0, which were used to determine the frozen domains.

16


Page 16 of 28

Page 17 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.3

The X40 benchmark

Finally, we present some preliminary results on the X40 set. 4 The set consists of 40 molecular dimers in which halogen atoms (F, Cl, Br, I) participate in a variety of types of noncovalent interactions. Table 8 shows the statistical data demonstrating the basis-set effect. Computed interaction energies Table 8: Differences of TZ and QZ PNO local Interaction Energies and Statistical Deviations of Local Coupled-Cluster Interaction Energies from Original X40 Valuesa,b Method DF-HF/TZ (DF-HF+CABS)/TZ

Reference RMSD DF-HF/QZ 0.013 (DF-HF+CABS)/QZ 0.004

MAD MEAN MAXD System(MAXD) 0.010 0.007 0.038 24 iodobenzene–trimethylamine 0.003 0.001 0.017 36 HI–methanol

LMP2/TZ LMP2-F12/TZ LCCSD-F12/TZ

LMP2/QZ LMP2-F12/QZ LCCSD-F12/QZ

0.156 0.022 0.024

0.136 0.136 0.015 −0.001 0.018 0.007

0.410 38 HCl–methylamine 0.070 11 trifluorobenzene–benzene 0.060 19 chlorobenzene–acetone

LCCSD-F12/TZ LCCSD-F12/QZ

X40 CCSD/CBS X40 CCSD/CBS

0.146 0.130

0.113 0.102

0.106 0.093

0.363 30 trifluoroiodomethane–benzene 0.336 30 trifluoroiodomethane–benzene

0.047 0.035 0.034

0.032 0.017 0.026 −0.003 0.024 0.001

0.177 30 trifluoroiodomethane–benzene 0.119 30 trifluoroiodomethane–benzene 0.114 30 trifluoroiodomethane–benzene

LCCSD(T*)-F12/TZ X40 CCSD(T)/CBS LCCSD(T*)-F12/QZ X40 CCSD(T)/CBS LCCSD(T[34] )-F12/QZ X40 CCSD(T)/CBS a b

Using tight domain options. All values are CP corrected and all F12 results include the CABS singles correction. Dimer 30 (hexafluorobenzene–benzene) is excluded from the statics. We were unable to perform QZ calculations for the dimer due to basis-set linear dependence. The LCCSD(T*)-F12/TZ interaction energy differ from the X40 “CCSD(T)/CBS” value by 0.06 kcal mol−1 .

can be found in Table S11 in the supporting information. Overall, the test set shows a larger basis-set dependence than found for the S66 set. This applies to both canonical HF calculations and local correlation calculations. F12 is still found to significantly reduce the basis set errors, and similarly small deviations (∼ 0.02 kcal mol−1 RMS) between TZ and QZ results are observed for LMP2-F12 and LCCSD-F12. The statistical deviations of the LCCSD-F12 and LCCSD(T)-F12 interaction energies from the ˇ c et al. 4,67 are also shown in table 8. The LCCSD-F12 results original X40 reference values of Rezáˇ deviates noticeably from the estimated “CCSD/CBS” values. The deviations are significantly larger than the expected basis-set or local errors. When (T) is included, however, the agreement between our results and the original X40 “CCSD(T)/CBS” ones is reasonably good, with RMSD of 0.047 kcal mol−1 with TZ basis sets and 0.034 kcal mol−1 with QZ. Scaled triples and extrapolated triples 17



Page 18 of 28

give very similar statistical deviations. Table 9: Interaction Energies (kcal mol−1 ) for the CF3 I–Benzene Complex at Different Computational Levels Level X40a Canonical F12/TZ PNO F12/TZ PNO F12/QZ MP2 −5.039 −5.052 −4.999 −5.036 CCSD −3.382 −3.067 −3.006 −3.044 CCSD(T)b−3.915 −3.820 −3.738 −3.796 a b

“CBS” values from ref 67. Scaled triples in F12 calculations.

The largest deviation of our computed interaction energies from the original X40 reference comes from the CF3 I–benzene dimer, which is dominated by the halogen–π interactions. To further investigate the relatively large deviations we performed canonical CCSD(T*)-F12/TZ calculations for the dimer using Ansatz 3*A. The results are shown in Table 9. It is clearly seen that the local errors are relatively large for CF3 I–benzene. They amount to 0.082 kcal mol−1 at CCSD(T*)/TZ level. The local errors are expected to be slightly smaller with the QZ basis. On the other hand, we observe 0.315 kcal mol−1 deviation of the X40 “CCSD/CBS” values from the CCSD-F12/TZ values, which is well beyond the expected basis-set error of the latter calculation. This is an example for the typical behavior of composite approaches without explicit correlation. For the original X40 set, the “CCSD/CBS” values were approximated by 4 corr ECCSD/CBS = EHF/QZ + EMP2/[34] + ∆ECCSD/TZ ,

(1)

where ∆ECCSD/TZ = ECCSD/TZ − EMP2/TZ . The first two terms on the right-hand side of eq (1) give the “MP2/CBS” value, which is in good agreement with the MP2-F12/TZ value but significantly overestimates the interaction energy. TZ calculations without explicit correlation underestimate the positive ∆ECCSD , leading to a CCSD/CBS estimate that is too negative. This error is compensated by the underestimated negative (T) contribution from TZ calculations, bringing the ∆ECCSD(T)/TZ values closer to ∆ECCSD(T)/CBS ones and in turn giving reasonable CCSD(T)/CBS estimations. Thus, the composite approach relies on error cancellations of the order of several tens of a kcal mol−1 . This 18


Page 19 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


is not the case in our local calculations with tight domains, and therefore the interaction energies computed with TZ or larger basis sets should be close to the real CBS values at MP2, CCSD, and CCSD(T) levels. For this reason, we believe our QZ results have comparable or higher accuracy than the original X40 values. Finally, it should be pointed out that inner-shell correlation effects may be important for the heavier p-block elements. 34,68,69 This will be investigated in future work.

4 Conclusions In conclusion, the current local calculations yield comparable accuracy with the best current canonical estimates obtained with composite schemes at a small fraction of the cost. This accuracy is much better than that of previous local calculations 70,71 and contradicts the conclusion in ref 9 that average errors of about 0.1 kcal mol−1 cannot be avoided with local methods. The better performance of the current method is due to more accurate pair approximations, 41,52 tighter domain options, and the F12 treatment. Note that already our default PNO domains are larger than the “tightPNO” domains in the DLPNO-CCSD(T) method of Neese an co-workers. 71 The PNO-LCCSD(T*)-F12 method represents a well defined computational model, and can be used in a black box manner also for larger systems. For example, it has previously be applied to the coronene dimer, yielding a CP-corrected interaction energy of 20.0 kcal mol−1 . 30 However, more benchmarks are necessary to test its performance for other difficult cases. Work in this direction is in progress. All calculations presented in this paper can be done on a single computing node. The most demanding calculations for the S66 set were the ones for the uracil dimer (system 26). For this, a single PNO-LCCSD(T*)-F12 calculation (tight domain options, convergence threshold 10−7 Eh , without CP correction) with TZ and QZ basis sets took 4.0 h and 18.3 h, respectively, on a single node with two Intel Xeon E5-2650 v4 @ 2.20GHz processors (using 20 cores). With 2 or 4 nodes, the latter times are reduced to 9.2 h or 5.1 h respectively. Less accurate calculations with default

19



domain settings are 2-3 times faster.

Supporting information s66_si.pdf

Detailed interaction energies for the A24, S66, and X40 benchmark sets, and the

computed potential energy curves for the benzene and uracil dimers.

s66_si.zip

CSV (comma separated values) files containing the computed interaction energies for

the A24, S66, and X40 benchmark sets.

s66_input.txt

A sample Molpro input for computing CP-corrected interaction energies using the

PNO-LCCSD(T*)-F12 method.

References ˇ (1) Jureˇcka, P.; Šponer, J.; Cerný, J.; Hobza, P. Benchmark database of accurate (MP2 and CCSD(T) complete basis set limit) interaction energies of small model complexe, DNA base pairs, and amino acid pairs. Phys. Chem. Chem. Phys. 2006, 8, 1985–1993. ˇ (2) Rezáˇ c, J.; Riley, K. E.; Hobza, P. Extensions of the S66 data set: More accurate interaction energies and angular-displaced nonequilibrium geometries. J. Chem. Theory Comput. 2011, 7, 3466–3470. ˇ (3) Rezáˇ c, J.; Riley, K. E.; Hobza, P. S66: A well-balanced database of benchmark interaction energies relevant to biomolecular structures. J. Chem. Theory Comput. 2011, 7, 2427–2438. ˇ (4) Rezáˇ c, J.; Riley, K. E.; Hobza, P. Benchmark calculations of noncovalent interactions of halogenated molecules. J. Chem. Theory Comput. 2012, 8, 4285–4292. (5) Rezac, J.; Hobza, P. Describing Noncovalent Interactions beyond the Common Approximations: How Accurate Is the “Gold Standard,” CCSD(T) at the Complete Basis Set Limit? J. Chem. Theory Comput. 2013, 9, 2151–2155. 20


Page 20 of 28

Page 21 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ˇ (6) Sedlak, R.; Janowski, T.; Pitoˇnák, M.; Rezáˇ c, J.; Pulay, P.; Hobza, P. Accuracy of quantum chemical methods for large noncovalent complexes. J. Chem. Theory Comput. 2013, 9, 3364– 3374. ˇ (7) Rezáˇ c, J.; Riley, K. E.; Hobza, P. Erratum to “S66: A well-balanced database of benchmark interaction energies relevant to biomolecular structures”. J. Chem. Theory Comput. 2014, 10, 1359–1360. ˇ c, J.; Dubecký, M.; Jurecka, P.; Hobza, P. Extensions and applications of the A24 data set (8) Rezáˇ of accurate interaction energies. Phys. Chem. Chem. Phys. 2015, 17, 19268–19277. ˇ c, J.; Hobza, P. Benchmark calculations of interaction energies in noncovalent complexes (9) Rezáˇ and their applications. Chem. Rev. 2016, 116, 5038–5071. (10) Pavoševi´c, F.; Neese, F.; Valeev, E. F. Geminal-spanning orbitals make explicitly correlated reduced-scaling coupled-cluster methods robust, yet simple. J. Chem. Phys. 2014, 141, 054106. (11) Schmitz, G.; Hättig, C.; Tew, D. P. Explicitly correlated PNO-MP2 and PNO-CCSD and their application to the S66 set and large molecular systems. Phys. Chem. Chem. Phys. 2014, 16, 22167–22178. (12) Brauer, B.; Kesharwani, M. K.; Martin, J. M. L. Some observations on counterpoise corrections for explicitly correlated calculations on noncovalent interactions. J. Chem. Theory Comput. 2014, 10, 3791–3799. (13) Brauer, B.; Kesharwani, M. K.; Kozuch, S.; Martin, J. M. L. The S66x8 benchmark for noncovalent interactions revisited: explicitly correlated ab initio methods and density functional theory. Phys. Chem. Chem. Phys. 2016, 18, 20905–20925. (14) Kesharwani, M. K.; Karton, A.; Sylvetsky, N.; Martin, J. M. L. The S66 non-covalent interactions benchmark reconsidered using explicitly correlated methods near the basis set limit. Aust. J. Chem. 2018, 71, 238–248. (15) Pittner, J.; Hobza, P. CCSDT and CCSD(T) calculations on model H-bonded and stacked complexes. Chem. Phys. Lett. 2004, 390, 496–499.

21



(16) Koch, H.; Fernández, B.; Christiansen, O. The benzene–argon complex: A ground and excited state ab initio study. J. Chem. Phys. 1998, 108, 2784–2790. (17) Hobza, P.; Šponer, J. Toward true DNA base-stacking energies: MP2, CCSD(T), and complete basis set calculations. J. Am. Chem. Soc. 2002, 124, 11802–11808. (18) Sinnokrot, M.; Valeev, E.; Sherrill, C. Estimates of the ab initio limit for pi-pi interactions: The benzene dimer. J. Am. Chem. Soc. 2002, 124, 10887–10893. (19) East, A. L. L.; Allen, W. D. The heat of formation of NCO. J. Chem. Phys. 1993, 99, 4638– 4650. (20) Marshall, M. S.; Burns, L. A.; Sherrill, C. D. Basis set convergence of the coupled-cluster correction, δCCSD(T) : Best practices for benchmarking non-covalent interactions and the attendant MP2 revision of the S22, NBC10, HBC6, and HSG databases. J. Chem. Phys. 2011, 135, 194102. (21) Hohenstein, E. G.; Sherrill, C. D. Wavefunction methods for noncovalent interactions. WIREs Comput. Mol. Sci. 2012, 2, 304–326. (22) Sherrill, C. D. Energy component analysis of π interactions. Acc. Chem. Res. 2013, 46, 1020–1028. (23) Burns, L. A.; Marshall, M. S.; Sherrill, C. D. Comparing counterpoise-corrected, uncorrected, and averaged binding energies for benchmarking noncovalent interactions. J. Chem. Theory Comput. 2014, 10, 49–57. (24) Boys, S. F.; Bernardi, F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 1970, 19, 553–566. (25) Marchetti, O.; Werner, H.-J. Accurate calculations of intermolecular interaction energies using explicitly correlated wave functions. Phys. Chem. Chem. Phys. 2008, 10, 3400. (26) Marchetti, O.; Werner, H.-J. Accurate calculations of intermolecular interaction energies using explicitly correlated coupled cluster wave functions and a dispersion weighted MP2 method. J. Phys. Chem. A 2009, 113, 11580–11585. (27) Sirianni, D. A.; Burns, L. A.; Sherrill, C. D. Comparison of explicitly correlated methods for

22


Page 22 of 28

Page 23 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


computing high-accuracy benchmark energies for noncovalent interactions. J. Chem. Theory Comput. 2017, 13, 86–99. (28) Ma, Q.; Schwilk, M.; Köppl, C.; Werner, H.-J. Scalable electron correlation methods. 4. Parallel explicitly correlated local coupled cluster with pair natural orbitals (PNO-LCCSDF12). J. Chem. Theory Comput. 2017, 13, 4871–4896. (29) Ma, Q.; Werner, H.-J. Scalable electron correlation methods. 5. Parallel perturbative triples correction for explicitly correlated local coupled cluster with pair natural orbitals. J. Chem. Theory Comput. 2017, 14, 198–215. (30) Ma, Q.; Werner, H.-J. Explicitly correlated local coupled-cluster methods using pair natural orbitals. WIREs Comput. Mol. Sci. 2018, e1371. (31) Adler, T. B.; Knizia, G.; Werner, H.-J. A simple and efficient CCSD(T)-F12 approximation. J. Chem. Phys. 2007, 127, 221106. (32) Knizia, G.; Adler, T. B.; Werner, H.-J. Simplified CCSD(T)-F12 methods: Theory and benchmarks. J. Chem. Phys. 2009, 130, 054104. (33) Hättig, C.; Tew, D. P.; Köhn, A. Accurate and efficient approximations to explicitly correlated coupled-cluster singles and doubles, CCSD-F12. J. Chem. Phys. 2010, 132, 231102. (34) Werner, H.-J.; Knizia, G.; Manby, F. R. Explicitly correlated coupled cluster methods with pair-specific geminals. Mol. Phys 2011, 109, 407–417. (35) Werner, H.-J. Eliminating the domain error in local explicitly correlated second-order MøllerPlesset perturbation theory. J. Chem. Phys. 2008, 129, 101103. (36) Adler, T. B.; Werner, H.-J. Local explicitly correlated coupled-cluster methods: Efficient removal of the basis set incompleteness and domain errors. J. Chem. Phys. 2009, 130, 241101. (37) Adler, T. B.; Werner, H.-J.; Manby, F. R. Local explicitly correlated second-order perturbation theory for the accurate treatment of large molecules. J. Chem. Phys. 2009, 130, 054106. (38) Riplinger, C.; Pinski, P.; Becker, U.; Valeev, E. F.; Neese, F. Sparse maps-A systematic infrastructure for reduced-scaling electronic structure methods. II. Linear scaling domain

23



based pair natural orbital coupled cluster theory. J. Chem. Phys. 2016, 144, 024109. (39) Pavoševi´c, F.; Peng, C.; Pinski, P.; Riplinger, C.; Neese, F.; Valeev, E. F. SparseMaps — A systematic infrastructure for reduced scaling electronic structure methods. V. Linear scaling explicitly correlated coupled-cluster method with pair natural orbitals. J. Chem. Phys. 2017, 146, 174108. (40) Guo, Y.; Riplinger, C.; Becker, U.; Liakos, D. G.; Minenkov, Y.; Cavallo, L.; Neese, F. Communication: An improved linear scaling perturbative triples correction for the domain based local pair-natural orbital based singles and doubles coupled cluster method [DLPNOCCSD(T)]. J. Chem. Phys. 2018, 148, 011101. (41) Schwilk, M.; Ma, Q.; Köppl, C.; Werner, H.-J. Scalable electron correlation methods. 3. Efficient and accurate parallel local coupled cluster with pair natural orbitals (PNO-LCCSD). J. Chem. Theory Comput. 2017, 13, 3650–3675. (42) Peterson, K. A.; Adler, T. B.; Werner, H.-J. Systematically convergent basis sets for explicitly correlated wavefunctions: The atoms H, He, B-Ne, and Al-Ar. J. Chem. Phys. 2008, 128, 084102. (43) Sylvetsky, N.; Kesharwani, M. K.; Martin, J. M. L. The aug-cc-pVnZ-F12 basis set family: Correlation consistent basis sets for explicitly correlated benchmark calculations on anions and noncovalent complexes. J. Chem. Phys. 2017, 147, 134106–14. (44) Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M.; and others, MOLPRO, 2018.1 a package of ab initio programs. 2018; see http://www.molpro.net. (45) Ten-no, S. Explicitly correlated second order perturbation theory: Introduction of a rational generator and numerical quadratures. J. Chem. Phys. 2004, 121, 117–129. (46) Ten-no, S. Initiation of explicitly correlated Slater-type geminal theory. Chem. Phys. Lett. 2004, 398, 56–61. (47) Werner, H.-J.; Adler, T. B.; Manby, F. R. General orbital invariant MP2-F12 theory. J. Chem. Phys. 2007, 126, 164102.

24


Page 24 of 28

Page 25 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(48) Yousaf, K. E.; Peterson, K. A. Optimized auxiliary basis sets for explicitly correlated methods. J. Chem. Phys. 2008, 129, 184108. (49) Shaw, R. A.; Hill, J. G. Approaching the Hartree–Fock limit through the complementary auxiliary basis set singles correction and auxiliary basis sets. J. Chem. Theory Comput. 2017, 13, 1691–1698. (50) Weigend, F. A fully direct RI-HF algorithm: Implementation, optimized auxiliary basis sets, demonstration of accuracy and efficiency. Phys. Chem. Chem. Phys. 2002, 4, 4285–4291. (51) Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. J. Chem. Phys. 2002, 116, 3175–3183. (52) Schwilk, M.; Usvyat, D.; Werner, H.-J. Communication: Improved pair approximations in local coupled-cluster methods. J. Chem. Phys. 2015, 142, 121102. (53) Peterson, K. A.; Figgen, D.; Goll, E.; Stoll, H.; Dolg, M. Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group 16–18 elements. J. Chem. Phys. 2003, 119, 11113–11123. (54) Peterson, K. A.; Shepler, B. C.; Figgen, D.; Stoll, H. On the spectroscopic and thermochemical properties of ClO, BrO, IO, and their anions. J. Phys. Chem. A 2006, 110, 13877–13883. (55) Hättig, C.; Schmitz, G.; Koßmann, J. Auxiliary basis sets for density-fitted correlated wavefunction calculations: weighted core-valence and ECP basis sets for post-d elements. Phys. Chem. Chem. Phys. 2012, 14, 6549–6555. (56) Weigend, F. Hartree–Fock exchange fitting basis sets for H to Rn. J. Comput. Chem. 2008, 29, 167–175. (57) Hättig, C.; Tew, D. P.; Helmich, B. Local explicitly correlated second- and third-order MøllerPlesset perturbation theory with pair natural orbitals. J. Chem. Phys. 2012, 136, 204105. (58) Tew, D. P.; Hättig, C. Pair natural orbitals in explicitly correlated second-order møller–plesset theory. Int. J. Quantum Chem. 2013, 113, 224–229. (59) Podeszwa, R.; Bukowski, R.; Szalewicz, K. Potential energy surface for the benzene dimer

25



and perturbational analysis of π−π interactions. J. Phys. Chem. A 2006, 110, 10345–10354. (60) Saebø, S.; Tong, W.; Pulay, P. Efficient elimination of basis set superposition errors by the local correlation method: Accurate ab initio studies of the water dimer. J. Chem. Phys. 1993, 98, 2170–2175. (61) Schütz, M.; Rauhut, G.; Werner, H.-J. Local treatment of electron correlation in molecular clusters: Structures and stabilities of (H2 O)n , n = 2-4. J. Phys. Chem. A 1998, 102, 5997–6003. (62) Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water. J. Chem. Phys. 1997, 106, 9639–9646. (63) Halkier, A.; Helgaker, T.; Jørgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K. Basis-set convergence in correlated calculations on Ne, N2 , and H2 O. Chem. Phys. Lett. 1998, 286, 243–252. (64) Woon, D. E.; Dunning, J., Thom H. Gaussian basis sets for use in correlated molecular calculations. IV. Calculation of static electrical response properties. J. Chem. Phys. 1994, 100, 2975–2988. (65) Russ, N. J.; Crawford, T. D. Potential energy surface discontinuities in local correlation methods. J. Chem. Phys. 2004, 121, 691–696. (66) Mata, R.; Werner, H.-J. Calculation of smooth potential energy surfaces using local electron correlation methods. J. Chem. Phys. 2006, 125, 184110. (67) BEGDB: Benchmark Energy and Geometry Database, http://www.begdb.com/ (accessed October 1, 2018). (68) Peterson, K. A.; Yousaf, K. E. Molecular core-valence correlation effects involving the post-d elements Ga–Rn: Benchmarks and new pseudopotential-based correlation consistent basis sets. J. Chem. Phys. 2010, 133, 174116. (69) Peterson, K. A.; Krause, C.; Stoll, H.; Hill, J. G.; Werner, H.-J. Application of explicitly correlated coupled-cluster methods to molecules containing post-3d main group elements. Mol. Phys. 2011, 109, 2607–2623.

26


Page 26 of 28

Page 27 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(70) Rolik, Z.; Szegedy, L.; Ladjánszki, I.; Ladóczki, B.; Kallay, M. An efficient linear-scaling CCSD(T) method based on local natural orbitals. J. Chem. Phys. 2013, 139, 094105. (71) Liakos, D. G.; Sparta, M.; Kesharwani, M. K.; Martin, J. M. L.; Neese, F. Exploring the accuracy limits of local pair natural orbital coupled-cluster theory. J. Chem. Theory Comput. 2015, 11, 1525–1539.

27


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

errors rel. CCSD(T)/CBS (kcal mol-1)


S66 Benchmark of intermolecular interaction energies using PNO-LCCSD(T*)-F12 aug-cc-pVQZ-F12

aug-cc-pVTZ-F12 dimer number

Figure 2: For Table of Contents Only

28


Page 28 of 28

Accurate Intermolecular Interaction Energies Using Explicitly

Recommend Documents