Small Basis Set Allowing the Recovery of Dispersion Interactions with

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Small Basis Set Allowing the Recovery of Dispersion Interactions with Double-Hybrid Functionals Juan Sanz García, Eric Bremond, Marco Campetella, Ilaria Ciofini, and Carlo Adamo J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b01203 • Publication Date (Web): 28 Mar 2019 Downloaded from http://pubs.acs.org on March 28, 2019

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Journal of Chemical Theory and Computation

Small Basis Set Allowing the Recovery of Dispersion Interactions with Double-Hybrid Functionals Juan Sanz Garcia1, Éric Brémond2, Marco Campetella1, Ilaria Ciofini1*, and Carlo Adamo1,3* Chimie ParisTech, PSL Research University, CNRS, Institute of Chemistry for Life and Health Sciences, 11, rue Pierre et Marie Curie, F-75005 Paris, France, Univ Paris Diderot, Sorbonne Paris Cité, ITODYS, UMR CNRS 7086, 15 rue J.-A. de Baïf, F-75013 Paris, France; and Institut Universitaire de France, 103 Boulevard Saint Michel, F-75005 Paris, France.

Abstract Taking advantage of the compensation between Basis Set Superposition Error and Basis Set Incompleteness Error, a new basis is developed to improve the performances of Double Hybrid (DH) functionals in reproducing interaction energies in weak noncovalent systems. Using a self-consistent formula, containing only energy terms computed for dimers and the corresponding monomers at the same level of theory, the exponents of the more external functions of the Def2-SVPD basis were optimized on three systems extracted from the S22 set. The transferability of the obtained basis set, called DH-SVPD, was then tested on 5 benchmark sets, and it is assessed by considering 6 DH functionals, eventually corrected with empirical dispersion corrections (for a total of 11 methods). Our results show that this simple approach is able to provide accurate results for noncovalent interactions energies of all the considered systems, and, in particular, to recover the performances obtained by coupling the DH functionals with empirical dispersion corrections.

1) Chimie ParisTech; 2) Université Paris Diderot ; 3) IUF *Corresponding authors: [email protected], [email protected]

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1.Introduction Modern computational approaches based on Density Functional Theory (DFT) provide valuable answers for most of the chemical problems to which they are applied, and this success well correlates with their widespread use in the last two decades1. However, even if their domain of applicability is constantly growing, DFT approaches face some challenges that cannot be ignored, in view of their relevance in Chemistry and Physics2,3. Among others, dispersion interactions are particularly difficult to treat4-6, albeit recent exchange-correlation functionals bring significant improvements7. If dedicated DFT approaches are able to correctly reproduce the properties of van der Waals complexes (see references 8-10 and 11 for a review), these results are sometimes obtained at the expense of other properties12. A more effective way relies on the addition of a pairwise empirical potential to correct the original exchange-correlation functional without any further modification13,14. Different variants, more or less sophisticated, have been developed, each of them bringing its own empirical features and improving the accuracy on the noncovalent interaction energies (see references 15 to 18 for some examples). Simple models for three-body interactions have been also proposed, in other to better reproduce such weak interactions19. The success of these approaches rests on their simplicity and on the quality of the obtained results for weak noncovalent interactions, while other properties are, usually, not degraded20,21. Albeit tailored for DFT applications, empirical potentials have been also proposed for Hartree-Fock (HF)22 or post-HF approaches, such as second-order Moller-Plesset (MP2)23 or Coupled Cluster24. As a matter of fact, these empirical dispersion corrections have become the reference in the field and any new method should be compared with them in term of accuracy and simplicity. In this general context, Double Hybrid (DH) functionals have been developed to overcome specific limitations of more traditional hybrid functionals25. Firstly introduced by Truhlar26 and then formalized by Grimme27, we consider here as DHs those functionals based on the inclusion of a perturbative (MP2) correction computed in a basis of Kohn-Sham orbitals to the correlation energy term28,29, which has a significant beneficial effect on a number of properties, including also noncovalent interactions30. Even better results, but limited by high computational cost, can be obtained with methods casting high-level correlated methods31,32. Still, the subsequent inclusion of an empirical correction is required for increased accuracy33,34, albeit its contribution is significantly lower than that observed for conventional hybrid functionals35 or MP2 approach24. This behavior is related to the overestimation of the dispersion effects at intermediate distance which is significantly reduced in DHs24. Such as additional dispersion correction is more beneficial for empirical DH functionals than for the non-empirical 2 ACS Paragon Plus Environment

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ones36. To the first family can be ascribed the empirical DHs whose internal parameters are determined through a fitting procedure, such as B2PLYP27, while the second group encompasses functionals fully determined on theoretical constraints, such has our PBE0-DH37 and PBE-QIDH38 models. Indeed, such functionals are derived from the parent PBE0 approach39,40, where the HF exchange mixing parameter was not obtained from any sorting of fitting procedure but rationalized on perturbation theoretical grounds40. This general situation for DHs is particularly frustrating since, if it could be expected an effective compensation on weak interactions between the overbinding MP223,24 and the underbinding DFT contributions, the inclusion of empirical corrections with non-empirical DH approaches is contradictory with the philosophy used for their development. This inconsistency motivated us to explore possible alternatives. In all quantum chemical approaches based on the expression of molecular orbitals (or density) as a linear combination of basis functions (e.g. atomic orbitals) there is an intrinsic error coming from the use of a finite base. This effect is particularly relevant when small to medium size Gaussian Type Orbitals (GTO) are used, so that basis set optimization is a common practice to approach the variational energy limit using a limited number of basis functions41. A number of basis sets have been then developed within HF or post-HF frameworks, such as the Def2 basis of Ahlrichs and co-workers42, of particular relevance for this work, or the correlation consistent basis sets of Dunning43, just to mention some of the most used. These basis sets are systematically employed also for DFT calculations, although some bases have been specifically developed (see for instance reference 44). Basis sets were also optimized for accurate reproduction of specific properties, such as polarizability45, EPR46 or NMR47 spectroscopic constants. A partial optimization of the basis set to better reproduce weak noncovalent interactions at the post-HF level, is also a common practice24,48-50. The choice of the basis is a major computational parameter also in the case of DFT, where the parameters defining the empirical exchange-correlation functionals could quantitatively depend on the basis set used to optimize them51. In other words, empirical functionals are more accurate when used with the (usually modest) basis set considered during the parametrization process than with other, smaller or larger, basis sets. This could be particularly true for weak interactions involving density domains mainly described by diffuse GTO functions. In this context, we have investigated to which extend a limited optimization of a small basis set, namely Def2-SVPD, could induce an improvement in the evaluation of interaction energies in small and medium-size noncovalent systems, with a particular attention to non-empirical DHs. The aim of our study is to provide a robust, transferable and reproducible computational 3 ACS Paragon Plus Environment


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method (DH + specific basis set) able to provide results comparable with those obtained with empirical corrections at a minimal computational cost and without adding any computational variables beyond the usual choice of the basis set. 2. Computational details Some years ago52, Varandas reported that the minimization of following expression

[

(𝐸 ― 𝐸0) ― (𝐽 + 𝐾) 2

ℱ = (𝐸 ― 𝐸0) + (𝐽 + 𝐾)

]

(1)

can be used as a criterion for the optimization of the exponent of an atomic basis set tailored for the reproduction of weak-interaction energies. In Eqn. (1), E is the total energy of the molecular aggregate (dimer), J and K are the corresponding Coulomb and exchange energies and E0 is the total energy of the isolated fragments. Indeed, the minimization of the function ℱ basically allows for the optimization of the interaction energy of a dimer as expressed at perturbation theory level52. Therefore, the exponents of a basis set which minimizes ℱ for a given Hamiltonian (here the Kohn-Sham Hamiltonian) will be the optimal for the description of the interaction energy. This optimization procedure does not consider any external reference data for the interaction energy, but only the perturbation energy expression of the interaction energy of a dimer. We have used Eqn. (1) to develop an optimized basis set, called DH-SVPD, starting from the small Def2-SVPD basis set42, using the SD-Box algorithm53. The quality and the transferability of this basis were then verified on the S2254, S6655 and L756 datasets. Intermolecular energy profiles were explored for the S22x5 and S66x8 sets54,55. Some calculations were also carried out using the original Def2-SVPD and Def2-QZVP basis42 sets for comparison purposes. Beyond the non-empirical PBE0-DH37 and PBE-QIDH38 functionals the following DHs have been considered: B2PLYP27, B2GPPLYP57, mPW2PLYP58 and DSDPBEP8659. In some cases, these functionals have been coupled to the D3(BJ)60 or to the NL61 dispersion corrections and even to the Axilrod-Teller-Muto (ATM) model for three body interactions19. Most of the calculations were performed with the program Gaussian62, while those requiring the DSDPBEP86 functional or NL and ATM corrections were carried out with the program ORCA (version 4.0.1)63.


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3. Results and discussions 3.1 Basis set optimization As briefly mentioned in the introduction, an accurate modeling of dissociation energy of weak interacting systems with electronic structure methods and finite atomic basis sets faces two sources of error, namely Basis Set Superposition Errors (BSSE) and Basis Set Incompleteness Error (BSIE), this latter also known as Basis Set Convergence Error64. Typically, BSSE and BSIE, having opposite sign65, tend to cancel out so that unbalanced effects could be obtained if just one of these errors is corrected. This effect is, of course, more evident when small basis sets are used and, therefore, large basis sets are usually considered to avoid this problem. However, the counterpoise correction to BSSE is not necessarily recommended for small or medium basis set, especially at (short) equilibrium distances65,66. Furthermore, the so-called half-counterpoise approach, computed as the average of BSSE corrected and uncorrected energy, has been developed to obtain accurate interaction energy with small basis sets at postHF level67,68. This approach rests on observed behavior of the noncovalent interaction energy, whose value at basis set limit can be approached from below using the BSSE corrected energy and from above using the uncorrected one67. The average of the two value then naturally appears a balanced solution. Somehow based on similar ideas, cheap computational methods have also been developed using a smart combination of BSSE, BSIE and empirical corrections in order to reach an acceptable accuracy in the evaluations of interaction energies in noncovalent systems69-71. In this context, we have partially optimized the Def2-SVPD basis set in order to exploit the error compensation between BSSE and BSIE so to recover a correct description of the energetic in weakly interacting systems using DHs. Such optimization procedure was carried out by the minimization the ℱ function, defined by Eqn. 1, with respect to the exponent of one p-function and one d-function for C, N and O atoms together with a s-function and p-function for H atom of the Def2-SVPD basis. These optimizations were carried out considering the PBE-QIDH energies of 3 systems, namely the benzene dimer parallel displaced (for C and H basis), the ammonia (for N basis) and the water (for O basis) dimers at their equilibrium geometries as reported in the S22 set54. The optimized exponents are collected in Table I together with the standard values of the Def2-SVPD basis. This new basis set will be hereafter labeled as DHSVPD. As expected, most of the optimized exponents are greater than the originals, thus leading to more contracted functions and, hence, a reduced BSSE. Since the Def2-SVPD energies are significantly too negative thanks to BSSE (i.e. overbound systems), the reduction of this error 5 ACS Paragon Plus Environment


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leads to a significant improvement. Indeed, the interaction energy in the benzene dimer parallel displaced is -4.72 kcal/mol when using the original Def2-SVPD basis set and -2.57 kcal/mol with the DH-SVPD basis. This latter value is very close to the -2.73 kcal/mol obtained at CCSD(T)/CBS level54. The reduction of the BSSE in going from Def2-SVPD to DH-SVPD is -1.9 kcal/mol, a value that well matches with the difference in the computed dissociation energies (-2.15 kcal/mol). Similar conclusions hold for the other two systems, water and ammonia dimers, used for training of the oxygen and nitrogen basis. The so-obtained basis has been validated considering its transferability to other systems and other DH functionals as described in the following paragraphs. 3.1 Validation on two standard benchmarks: the S22 and S66 sets The first test set considered to validate the DH-SVPD is the S22 dataset developed by Hobza and coworkers few years ago54. This dataset is composed by the energies of 22 supramolecular weakly-interacting systems, including 7 H-bond dimers, 8 dispersion complexes and 7 other systems defined as “mixed influence complexes”. The complete list of these systems is given in Supplementary Information (SI) together with the details of the statistics reported in Table II. Of note, three of these complexes were already considered in the basis set optimization with the PBE-QIDH functional. Starting the discussion from the PBE-QIDH results, the data reported in Table II clearly shows a significant improvement in performance for this functional using the optimized basis (DH-SVPD), the Mean Absolute Error (MAE) decreasing from 0.93 kcal/mol (Def2-QZVP basis) to 0.62 kcal/mol (-35%). More interesting the gap between results obtained using the empirical dispersion correction and the optimized DH-SVPD basis set significantly decreases for the PBE-QIDH functional. Indeed, the PBE-QIDH-D3(BJ) approach gives a MAE of 0.43 kcal/mol with the Def2-QZVP basis set to be compared to a MAE of 0.62 kcal/mol computed with PBE-QIDH/DH-SVPD. In other words, a reduction of -33% is observed when using the optimized basis set. The same holds for the NL empirical correction is considered (see Table II). Interesting, the error obtained with the PBE-QIDH-D3(BJ) functional and the small standard Def2-SVPD basis set is significantly larger than that computed with the Def2-QZVP basis (2.55 vs. 0.43 kcal/mol) or the optimized basis (1.30 kcal/mol). These results suggest that a particular attention is need in the choice of the basis set to be used when these empirical corrections are used with DHs, as already reported in literature72. Looking more in details to the different subsets of the S22 benchmark (Tables S102 to S149), the quality of the optimized basis set can be further evidenced. Indeed, the MAEs of two subsets 6 ACS Paragon Plus Environment

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(dispersion and mixed influence complexes) decrease in going from the Def2-QZVP to Def2SVPD basis set. The larger effect is observed, as expected for dispersion complexes (from 1.78 to 0.46 kcal/mol; see Tables S124 and S140), while a lower variation is found for mixed influence complexes (from 0.57 to 0.33 kcal/mol). An opposite behavior is instead observed for hydrogen bonding systems, which show an error increase (from 0.34 to 1.06 kcal/mol). A similar behavior is, however, found when empirical correction is included with the larger Def2QZVP basis set. Indeed, in this case the error on the H-bonded complexes also increases from 0.34 to 0.62 kcal/mol (see Tables S124 and S125). Noteworthy the error at PBE-QIDHD3(BJ)/Def2-QZVP level of theory for dispersion complexes is 0.51 kcal/mol to be compared to a MAE of 0.46 kcal/mol for the PBE-QIDH/DH-SVPD model. In order to check the transferability of the basis set, five more DH functionals were considered and the obtained results are collected in Table II. For all these DHs the same global trends of PBE-QIDH can be found. Indeed, the error significantly decreases in going from the Def2QZVP basis to the optimized one, with variations ranging from -70% (B2GPPLYP) to -44% (PBE0-DH). As a consequence, a good agreement is found between the results obtained with the optimized DH-SVPD basis and those evaluated with the larger Def2-QZVP basis and empirical corrections. For instance, the MAE values obtained at the DSDPBE86 level are 0.46 and 0.27 kcal/mol for these two basis sets. Interestingly, in going from Def2-QZVP to Def2-SVPD to DH-SVPD basis, a systematic error decrease is observed for all the functionals not including dispersion corrections (i.e. B2GPPLYP, mPW2PLYP, DSDPBEP86), while the largest error is observed with the Def2-SVPD basis for all functionals corrected for dispersion. The exceptions are represented by B2PLYP and PBE0-DH, where the lowest error is obtained with the unoptimized Def2-SVPD basis. Looking in more in details to the data reported in Table S100, S108, S132 and S140, it appears that this behavior, more evident for B2PLYP than PBE0-DH, is ruled by the error on the 7 dispersion complexes of the S22 set, showing a significantly larger error for the DH-SVPD basis than for the Def2-SVPD one, while similar results are obtained for the mixed influence complexes and even better value for hydrogen bonding systems. Table II collects also the interaction energies computed by adding a three-body term, EABC(ATM)19, to the dispersion corrections. It has been suggested that the inclusion of threebody effects on top of pairwise potentials, such as those used for the D3 corrections, could lead to an overestimation of the interaction energies73. Our results, obtained for different functional/potential combinations do not confirm this expectation. Indeed, for all the considered cases, the variation of the MAE upon the addition of the EABC(ATM) correction leads to a 7 ACS Paragon Plus Environment


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variation on the second digit (around 0.05 kcal/mol). A larger variation (from 2.30 to 2.49 kcal/mol) is found only for the B2PLYP-D3(BJ) functional upon the additional three-body correction. These data are in line with recent finding.74 Table III collects the MAEs computed for the S66 dataset55, which contains 22 hydrogen bonding systems, 10 -stacking systems, 14 London dispersion complexes and 20 mixed influence complexes for which reference energies were computed at the CCSD(T) level. As a first general trend, it should be noticed that the MAEs obtained for the S66 set are lower than those already discussed for the S22 set. In particular, there is significant variation for the PBE-QIDH functional which now shows an error of 0.39 kcal/mol with the DH-SVPD basis set, very close to that obtained with the D3(BJ) correction and the Def2-QZVP basis (0.33 kcal/mol). In other words, the two approaches are (almost) equivalent. All the others trends already discussed for the S22 set are maintained with the S66 set. Notably, there is a systematic error decrease for all the functionals that are not corrected with an empirical dispersion term in going from the Def2-QVZP to the DH-SVPD basis set. Furthermore, the results obtained with this latter basis are comparable to those obtained with the larger basis and dispersion corrections. For instance, DSDPBEP86 has a MAE of 0.49 kcal/mol with the DH-SVPD basis set and of 0.19 kcal/mol when coupled with the empirical dispersion correction. The largest difference is found for B2LYP, showing a MAE of 0.73 kcal/mol for DH-SVPD and 0.19 kcal/mol with the D3(BJ) correction. Of course, the agreement obtained with the considered functionals is worse than that found for PBE-QIDH, that is the functional used for basis set optimization, but in all cases the improvement is still sizable. In this sense, it should be considered that BSIE plays a role in any optimization process, including the determination of parameters in empirical functionals, as recently underlined by Jensen51. Without pushing this concept to the limit where the basis set is ad hoc optimized for each functional, it should be underlined that our DH-SVPD basis has a (very) good transferability to other DHs which leads to improve numerical performances for all the considered functionals not empirically corrected for dispersion. This is also demonstrated by the standard deviation (SD) computed for all the functionals with the different basis (last two lines in Tables II and III). The DH-SVPD has the lowest SDs on both S22 and S66 datasets, thus suggesting a greater transferability than the other two basis sets. 3.2 Exploring energy profiles: S22x5, S66x5 data set. Up to now, we have checked the quality of our basis set considering systems at a given equilibrium structure, as usually done in standard benchmarks of DFT methods. However, this 8 ACS Paragon Plus Environment

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approach does not guarantee the accuracy of the proposed basis set in the case of molecular rearrangement far from the energy minima. Indeed, it has been recently pointed out that dispersion corrections could not correctly describe potential energy profiles outside equilibrium structures75,76. In order to investigate this point, we have considered the S22x5 and S66x8 data sets54,55. These sets collect reference CCSD(T)/CBS values for 5 and 8 points, respectively, along the dissociation energy profile of the respective weakly interacting dimers (22 and 66, respectively). The results obtained with the three different basis set and twelve functionals are collected in Table II and III. Starting from the smallest set, a general decrease of all the deviations can be observed independently from the functional or basis set used, thus suggesting an overall correct behavior for all the considered approaches. However, this effect is smaller for the functionals where empirical dispersions are used in conjunction with the large Def2QZVP basis set or with our optimized basis set. This trend suggests that these two categories of methods are also characterized by a particularly accurate shape of the dissociation profiles in regions far from the energy minima. Concerning the basis set, the MAE for the PBE-QIDH functional decreases in going from Def2QZVP to the DH-SVPD basis (from 0.62 to 0.43 kcal/mol) for the S22x5 dataset. A similar behavior is obtained for all the other uncorrected functionals with B2GPPLYP showing the lowest deviation (0.32 kcal/mol). The error obtained at PBE-QIDH/DH-SVPD represents the 6% of the mean interaction energy, and 4% is the error computed for B2GPPLYP with the same basis set. Such values should be compared with those obtained using the same functionals including dispersion correction terms and the Def2-QZVP basis set. A value of 0.28 kcal/mol is computed at PBE-QIDH-D3(BJ) level and of 0.19 kcal/mol at the DSDPBEP86-D3(BJ) level, corresponding, respectively, to 4% and 3% of the mean dissociation energy. For both functionals, a difference of about 0.13 kcal/mol is found in favor of the dispersion corrected approaches. Interestingly, DSDPBEP86 shows an error (of 0.35 kcal/mol) lower than the PBEQIDH one, thus strengthening the transferability of the DH-SVPD basis set. As before, the errors obtained coupling functionals with a dispersion correction term and the optimized basis are significantly higher with respect to those obtained with uncorrected functionals. They are, however, lower than those obtained with the Def2-SVPD basis, thus pointing out, once again, the incompatibility between small basis and dispersion corrections. In order to have a clear picture of the influence of the basis set and of the dispersion correction, Figure 2 and 3 report the dissociation profile for the stacked benzene dimer and methane dimers, as extracted from the S22x5 data sets. The curves of Figure 2 clearly show that, at PBE-QIDH level, the large Def2-QZVP basis set predicts a shallow energy minimum, while a deep well is 9 ACS Paragon Plus Environment


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obtained with the small Def2-SVPD basis. The optimized DH-SVPD basis has an intermediate behavior, leading to the observed very good agreement with the reference data all along the energy profile for the PBE-QIDH functional. This general behavior is observed with all the functionals and it yields to the trend already discussed for the MAEs, a decrease of the average error obtained in going from Def2-QZVP to Def2-SVPD to DH-SVPD (see Table II). A similar behavior is also observed for the profiles obtained with the B2PLYP functional, with the optimized basis showing a dissociation profile intermediate between those obtained using the Def2-QZVP and the Def2-SVPD basis. Nonetheless, for this functional the largest basis provides a dissociation profile with a very shallow minimum, a behavior that is not corrected by the optimized basis. Only the un-optimized Def2-SVPD basis shows a clear energy minimum, thus explaining the apparent discrepancy found in the MAEs’ order in the case of the B2PLYP functional. Analyzing Figure 3, the effect of the D3(BJ) dispersion correction term clearly appears. Indeed, the added potential further deepens the potential well leading to a better agreement with the reference values for the curve obtained with the Def2-QZVP basis. Since the DH-SVPD basis already provides a correct profile, the addition of an attractive potential increases the binding character of the potential and thus deteriorates the overall behavior. These conclusions, based on the use of the PBE-QIDH functional, can be extended also to all other functionals. These trends concerning the effect of basis set and dispersion corrections as well as functional forms are retained also in the case of the S66x8 dataset. To avoid boring discussions, interested readers are referred to the data reported in Table II, III and in the SI. In short, our results suggest that the optimized basis set retains its characteristics of accuracy also outside the energy minima and for all the considered functionals not including dispersion correction. This basis set is also more accurate than its parent small basis set when coupled with empirical dispersion, independently of the functional used. 3.3 Larger reference systems: the L7 dataset The last benchmark was realized on the L7 data set, which contains noncovalent dimers and trimers, whose size range from 48 to 112 atoms, representative of different weak interactions in Chemistry56. While the size of the studied systems has the merit to highlight the eventual accumulation of error, it also limits the accuracy of the theoretical method used to generate reference data. The QCISD(T)/CBS approach was originally used as reference56, but these data were upgraded to DLPNO-CCSD(T)/CBS level34. This method had a large impact in the domain of post-HF 10 ACS Paragon Plus Environment

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calculations of large systems77, but its accuracy depends on a number of internal parameters7880.

For instance, the energy for the parallel-stacked octadecane dimer (CBH) is -11.06 kcal/mol

at QCISD(T)/CBS level56, a value close to that obtained at DLPNO-CCSD(T)/CBS level using a “Normal” setting of the internal computational parameters (11.64 kcal/mol)34 and to that found with the CCSD(T)-F12/cc-pVDZ approach (11.13 kcal/mol)81. A “Tight” threshold at the DLPNO-CCSD(T)/CBS level gives instead 9.98 kcal/mol74. This variability well shows the difficulties related to post-HF calculations variables (including basis set effects) and the difficulty of an appropriate choice of the internal computational parameters. Due to these differences in reference data we have opted to discuss the results with respect to two different reference sets: the original one (QCISD(T)/CBS) and that obtained with the DLPNO-CCSD(T)/CBS approach using a “Normal” threshold, which has been already used for comparison purposes34,36,82. The results obtained with the PBE-QIDH functional are reported in Table IV together with the two references data sets and literature values obtained at B2PLYP-D3(BJ) level. The MAE for the PBE-QIDH/DH-SVPD approach is 1.90 kcal/mol with respect to the DLPNOCCSD(T)/CBS data set and 1.25 kcal/mol if the QCISD(T)/CBS reference is considered. These values are significantly higher than those obtained for the B2PLYP-D3(BJ) method, namely 0.78 and 0.79 kcal/mol, respectively. However, this latter method does not include three-body interactions (EABC). If these effects seem to be negligible in small molecules, such as those present in the S22 and S66 dataset, they could play a not negligible role in larger systems as those considered in the L774, so that they should be considered in order to have a fairer comparison with uncorrected DH functionals. When three-body interactions are added the MAEs increase to 1.53 and 2.08 kcal/mol with respect to the CCSD(T) and QCISD(T) reference, respectively. The agreement with the PBE-QIDH/DH-SVPD results is then evident. Furthermore, it should also be considered, that the magnitude of the three-body corrections changes according to the model used, and other potentials predict larger contributions on the L7 molecules74. These deviations computed at the B2PLYP-D3-EABC(ATM) level should be, therefore, considered more as a lower bond than true reference values. More in general, the errors obtained with the PBE-QIDH/DH-SVPD model are close to those provided by very sophisticated perturbative treatments, such as XSAPT(KS). This method includes a long-range functional, tuned on every single system of the L7 set, a D3 potential and a EABC correction (1.36 kcal/mol for XSAPT(KS)+aiD3+EATM(TS) see reference 74 for details).



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Finally, it must be noticed that the same deviations (1.9 kcal/mol) has been obtained with the PBE-QIDH+D3+EABC model36, thus indicating that the main goal of our work, namely the replacement of empirical dispersion with a properly optimized basis set, has been reached. 3.4 Self-consistent vs. interaction energy optimization At this point, a question naturally arises about the possibility of obtaining the same results with a basis set optimized on the dissociation energy, that is by minimizing the error with respect to the reference CCSD(T)/CBS value. In this case, the optimized values obtained for exponents of the p and d functions of the C atom are 0.15080365500 and 0.32292947900, respectively. These values are virtually the same than those obtained by minimizing Eqn. 1, while slightly different values are obtained for the s and p functions of the H atom (0.63354498770 and 0.09697452215, respectively). The mean absolute deviation, computed by using the systems of the S22 dataset containing only C and H atoms (6 dimers), between the interaction energies obtained with these basis functions and the DH-SVPD basis set is only of 0.06 kcal/mol. Despite the intrinsic interest of this result, a substantial difference between the two approaches should be remarked: the minimization of Eqn.(1) involves only energy terms of the dimer and monomer computed at the same level of theory, so that it is a fully self-consistent procedure while the optimization of the basis set with respect to reference interaction energy is a minimization of an error

requiring external (and accurate) reference values and, thus,

intrinsically of lower general character and transferability. 4. Conclusions Using a function derived from the perturbative expression of the interaction energy for weakly interacting systems, a new split-valence basis set, DH-SVPD, to be used together with doublehybrid functionals has been developed. The basis set optimization has been carried out by considering only energy quantities of three noncovalent dimers and their monomers, without making use of any external reference value. The optimal performances obtained with the new basis set have been analyzed in term of compensation between Basis Set Superposition and Basis Set Incompleteness Errors. However, this error compensation is transferable and robust, as showed by the results obtained on different systems, including S22x5, S66x8 and L7 data sets, and several double hybrids functionals. When the PBE-QIDH functional is considered, the results observed with the DH-SVPD basis are very close, if not equivalent, to those obtained with empirical dispersion corrections and larger Def2-QZVP basis. This will allow to restore a full non-empirical DFT approach, namely 12 ACS Paragon Plus Environment

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the PBE-QIDH/DH-SVPD model, suitable for noncovalent interactions, without making use of any empirical parameters, beyond the compulsory choice of the basis set.

Acknowledgement This manuscript is in memoriam of Dr. Gaston Berthier (Paris), which inspired this work and with whom IC and CA had the pleasure to share illuminating discussions on weak interactions and basis sets. This work has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (CoG STRIGES grant agreement No 648558). EB gratefully acknowledges the GENCI-CINES for HPC resources (Projects AP010810360 and A0040810359). Supporting information The Supporting Information is available free of charge on the ACS Publications website: Raw data of Tables II and III.



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Table I. Optimized exponents of the DH-SVPD basis set. These exponents replace the corresponding exponents of the original Def2-SVPD basis set, which are also reported for comparison. All other exponents are kept as in the original basis set. atom

function

H

s

C

original

optimized

function

original

optimized

0.12194962000

0.4617867850

p

0.11704099050 0.07913402419

p

0.15268613795

0.1508036550

d

0.11713185140

0.3229294790

N

p

0.21954348034

0.1918861474

d

0.16697708112

0.3095443259

O

p

0.06900227635

0.09892123211

d

0.17992024323

0.2373752428


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

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Table II. Mean Absolute Errors (MAE, kcal/mol) on the S22 and S22x5 datasets, obtained with different functionals and basis sets. The raw data are reported in the Supplementary Information. S22 S22x5 Def2-QZVP Def2-SVPD DH-SVPD Def2-QZVP Def2-SVPD DH-SVPD B2PLYP 1.79 0.57 0.84 1.17 0.51 0.62 B2PLYP-D3(BJ) 0.30 2.30 1.15 0.21 1.67 0.89 B2PLYP-D3(BJ)-EABC(ATM) 0.29 2.49 1.28 0.23 1.65 0.85 B2GPPLYP 1.35 1.16 0.42 0.88 0.82 0.32 B2GPPLYP-D3(BJ) 0.23 2.56 1.21 0.19 1.72 0.83 ABC B2GPPLYP-D3(BJ)- E (ATM) 0.28 2.50 1.16 0.23 1.67 0.79 mPW2PLYP 1.38 0.97 0.67 0.89 0.76 0.51 DSDPBEP86 0.83 1.91 0.46 0.55 1.30 0.35 DSDPBEP86-D3(BJ) 0.27 2.99 1.53 0.19 1.99 1.03 ABC DSDPBEP86-D3(BJ)- E (ATM) 0.23 2.94 1.48 0.18 1.95 0.99 PBE0-DH 1.72 0.88 0.97 1.09 0.56 0.63 PBE-QIDH 0.93 1.75 0.62 0.62 1.20 0.44 PBE-QIDH-D3(BJ) 0.43 2.55 1.30 0.28 1.74 0.92 PBE-QIDH-D3(BJ)- EABC(ATM) 0.49 2.50 1.25 0.32 1.70 0.88 PBE-QIDH-NL 0.44 2.69 1.45 0.29 1.83 1.01 PBE-QIDH-NL- EABC(ATM) 0.49 2.64 1.40 0.32 1.79 0.97 SDa

0.55

0.78

0.37

0.34

0.50

0.24

a) Standard Deviation (SD) of the column.


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Table III. Mean Absolute Errors (MAE, kcal/mol) on the S66 and S266x5 datasets, obtained with different functionals and basis sets. The raw data are reported in the Supplementary Information.

B2PLYP B2PLYP-D3(BJ) B2PLYP-D3(BJ)- EABC(ATM) B2GPPLYP B2GPPLYP-D3(BJ) B2GPPLYP-D3(BJ)- EABC(ATM) mPW2PLYP DSDPBEP86 DSDPBEP86-D3(BJ) DSDPBEP86-D3(BJ)- EABC(ATM) PBE0-DH PBE-QIDH PBE-QIDH-D3(BJ) PBE-QIDH-D3(BJ)- EABC(ATM) PBE-QIDH-NL PBE-QIDH-NL- EABC(ATM) SDa

Def2-QZVP 1.61 0.19 0.19 1.25 0.19 0.24 1.17 0.78 0.19 0.18 1.52 0.90 0.33 0.38 0.31 0.36 0.51

S66 S66x8 Def2-SVPD DH-SVPD Def2-QZVP Def2-SVPD DH-SVPD 0.45 0.73 1.20 0.35 0.50 2.08 1.12 0.14 1.54 0.96 2.03 1.19 0.15 1.50 0.91 0.86 0.35 0.93 0.65 0.31 2.05 1.11 0.15 1.53 0.87 2.00 1.06 0.19 1.49 0.82 0.81 0.47 0.85 0.64 0.44 1.51 0.49 0.59 1.13 0.41 2.44 1.22 0.14 1.81 1.09 2.39 1.36 0.15 1.77 1.05 0.61 0.73 1.11 0.44 0.55 1.21 0.39 0.66 0.92 0.34 1.91 1.05 0.22 1.45 0.85 1.86 1.00 0.27 1.41 0.80 2.03 1.17 0.22 1.53 0.93 1.97 1.12 0.26 1.49 0.89 0.64

0.33

0.38

0.47

0.26

a) Standard Deviation (SD) of the column.



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Table IV. Dissociation Energies (kcal/mol) and Mean Absolute Errors (MAE, kcal/mol) on the L7 dataset, obtained with different functionals and basis sets. See Figure 1 for system labels.

CBH

PBE-QIDH-D3+EABC / PBE-QIDH/ B2PLYP-D3/ B2PLYP-D3+EABC/ CCSD(T)/ QCISD(T)/ Def2-QZVPa DH-SVPD Def2-QZVPb Def2-QZVPb CBSb CBSc -8.10 -11.91 -10.28 -9.59 -11.64 -11.06

GGG

-1.40

-2.77

-1.56

-0.76

-1.68

-2.4

C3A

-16.48

-17.82

-17.82

-16.66

-17.98

-18.19

C3GC

-28.48

-30.56

-30.17

-27.95

-29.80

-31.25

PHE

-24.91

-27.06

-25.12

-23.61

-22.81

-25.76

C2C2PD

-21.61

-20.86

-23.72

-22.05

-24.81

-24.37

GCGC

-11.98

-16.06

-13.19

-12.20

-13.21

-14.37

MAE(CCSD(T))

1.88

1.90

0.78

1.53

MAE(QCISD(T))

2.07

1.25

0.79

2.08

a) ref. 36; b) ref. 34; c) ref. 56


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Figure captions Figure 1. Sketches of the supramolecular complexes constituting the L7 dataset. Color code: C in gray, H in white, N in blue and O in red. Acronyms: CBH, octadecane dimer, GGG, stacked guanine trimer, GCGC, stacked guanine-cytosine dimer, PHE, three phenylalanine residues, C3A, stacked circumcoronene–adenine dimer, C3GC, a stacked circumcoronene and guanine– cytosine dimer, C2C2PD, a parallel displaced stacked coronene dimer. Figure 2. Potential energy profiles computed for the staked benzene dimer extracted from the S22x5 data set. CCSD(T) values are taken from reference 55., Figure 3. Potential energy profiles computed for the methane dimer extracted from the S22x5 data set. CCSD(T) values are taken from reference 55.



GGG

GCGC

C3A

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PHE

C3GC

C2C2PD

CBH

Figure 1


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5.0

B2PLYP/def2-QZVP B2PLYP/def2-SVPD B2PLYP/DH-SVPD CCSD(T)/CBS QIDH/def2-QZVP QIDH/def2-SVPD QIDH/DH-SVPD

4.0 3.0 2.0

E (kcal/mol)

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


1.0 0.0 -1.0 -2.0 -3.0 -4.0 -5.0 0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

drel Figure 2



0.6 0.4 0.2

E (kcal/mol)

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

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0.0 -0.2

QIDH/def2-QZVP QIDH-D3(BJ)/def2-QZVP QIDH/DH-SVPD CCSD(T)/CBS QIDH/def2-SVPD QIDH-D3(BJ)/DH-SVPD

-0.4 -0.6 -0.8 0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

drel Figure 3


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