A Combined Energy-Force Fitting Procedure to Develop DFT-Based

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A Combined Energy-Force Fitting Procedure to Develop DFT-based Force Fields Andrea Gabrieli, Marco Sant, Pierfranco Demontis, and Giuseppe B. Suffritti J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b08163 • Publication Date (Web): 21 Oct 2016 Downloaded from http://pubs.acs.org on October 25, 2016

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The Journal of Physical Chemistry

A Combined Energy-Force Fitting Procedure to Develop DFT-based Force Fields Andrea Gabrieli, Marco Sant,∗ Pierfranco Demontis, and Giuseppe B. Suffritti Dipartimento di Chimica e Farmacia, Universit` a degli Studi di Sassari, Via Vienna 2, 07100 Sassari, Italy E-mail: [email protected] Phone: +39079229496. Fax: +39079229559

Abstract

Introduction

A general method to develop accurate force fields from density functional theory (DFT) computations in periodic systems is here presented. The novelty of the method consists in the inclusion of both potential energy and forces in the same fit, by using an automated procedure to balance the relative weight of the two quantities. A thorough analysis of the method capabilities is carried out by modelling the dispersion interactions of argon adsorbed in ZIF-8. While a pure energy fit leaves the parameters of some atoms kinds underdetermined, a pure forces fit gives well converged results but fails in properly reproducing the potential energy of the system. The optimal solution is found when a small contribution of forces is included in the energy fit: this allows to fuse together the best features of the two fits, giving converged results and good reproduction of both energy and forces. The force field parameters for various DFT functionals, namely, DFT-D2, DFT-D3, vdW-DF2, and rVV10 are derived, and the corresponding isotherms are compared to experimental values. For the system here investigated the best agreement with experiments is found for the DFT-D2 functional.

Nanoporous materials, such as zeolites, have found broad application in the chemical industry, primarily as catalysts and molecular sieves. 1–3 More recently, Metal Organic Frameworks (MOFs), 4,5 a new class of nanoporous materials, have been synthesized and are receiving considerable attention from the scientific community thanks to their peculiar properties, 6 which can be tuned by functionalizing the organic linker 7 according to the intended applications. 8,9 Among these, very promising are: hydrogen storage, 10–15 carbon capture and sequestration, 16–19 drug delivery and biomedical applications. 20–26 The number of hypothetical MOFs is enormous and it is not possible to try experimentally all the possibilities. 27 Computer simulations, then, have become the standard tool to investigate these systems. Grand Canonical Monte Carlo (GCMC) and Molecular Dynamics (MD) simulations, in fact, allow the study of the adsorption and diffusion of sorbate molecules within porous materials. The accuracy of these computations depends on the force field (FF) employed. A common straightforward approach is to rely on generic FFs, 27–32 such as UFF 33 and Dreiding, 34 to model the system interactions. Unfortunately, it has been shown that these FFs give results that in many cases are unsatisfactory. 35–37 A different approach, is to derive the FFs

∗

To whom correspondence should be addressed

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from ab-initio computations. 37–39 The highest level of theory can be reached only for small cluster models, 40–43 a drawback of this approach is that the obtained results may not be transferable to the full periodic structure. 35,39,44 One possibility, then, is to rely on density functional theory (DFT) computations, which are less computationally expensive and thus allow to extend the size of the investigated systems. 45–47 A thorough review of first-principles methods for deriving FFs in nanoporous materials can be found in Ref. 39. Some notable applications of such methods are here briefly reported. McDaniel et al. 48 proposed an ab-initio method based on Symmetry Adapted Perturbation Theory (SAPT) for developing a FF to study the adsorption of CO2 within ZIF-8 and ZIF-71. A key point of this methodology is that it involves separate terms explicitly accounting for exchange, electrostatic, induction and dispersion interaction, thus limiting the possibility of error cancellation during the fitting procedure. Excellent agreement with experiments, for both materials, has been found. In a subsequent work 49 the generality of the method and the transferability of the parameters has been successfully shown. Lin et al. 36 developed an efficient approach to derive accurate FFs from periodic DFT computations. Their method involves computing the energies only on specific paths, those that are more relevant to the pairwise interaction between the approaching atoms. Moreover the use of a clever energy decomposition scheme, together with a self-consistent optimization loop, enables the fitting of only two parameters at the same time thus limiting the computational cost. The adsorption and dynamic properties of CO2 inside Mg-MOF-74 have been accurately estimated and the transferability of the methodology to CO2 in Zn-MOF-74 has been verified as well. In a recent work, the method has also been used to make accurate predictions for H2 O, CO2 , and CH4 in MOFs possessing strong-binding open-metal sites. 50 Kulkarni and Sholl 35 presented a general method for developing FFs from periodic DFT calculations. To show the performance of their

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approach, they modeled the interactions of short and long alkanes in MIL-47(V), obtaining good predictions for diffusion and adsorption. They concluded that generic FFs, even for simple sorbent-sorbate systems, could be improved using first-principles derived FFs. Demir et al. 51 modeled the interaction of Ar and Xe in several MOFs by fitting DFT data computed with various functionals. Such data were obtained over random and GCMC generated configurations. Their FFs included both van der Waals (vdW) and polarization terms, and the latter were found to give a negligible contribution to the interactions (about 1%). Comparing DFT-based isotherms to the experimental ones they found that none of the DFT functionals investigated can reproduce the experimental data accurately for all MOFs and sorbates involved, for this reason they concluded that “none of them is ideal for deriving FFs for a wide range of MOFs”. The choice of the proper functional is, then, of uttermost importance. In this work, the possibility of improving the development of DFT-based FFs by performing a combined energy and force fit is explored. An automated procedure to find the optimal balance between the two contributions is also proposed. The method is applied to a prototypical system, argon adsorbed in ZIF-8, whose properties are governed by dispersion interactions, thus allowing to focus on the investigated quantity. A thorough analysis of the strength of the method is carried out, and the performances of various DFT functionals in reproducing the experimental data are evaluated. The organization of the article is as follows: in the first section, the computational methods used throughout the work and the theoretical background of the fitting procedure are presented, while in the second section the method is applied to argon in ZIF-8 to evaluate its capabilities together with the properties of the generated FFs. The article ends with a summary of the conclusions drawn throughout the paper.


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Methodology

Force field optimization. The common approach followed to generate the reference abinitio data is to keep the framework fixed at the experimental crystallographic positions or at the DFT relaxed positions. In a previous work it has been observed that a perfectly symmetric structure can lead to underdetermination in generating atomic charges in periodic crystals. 77 On this basis, here various uncorrelated configurations of the empty framework have been obtained by taking 67 frames from a 20000 steps (10 ps) long BornOppenheimer MD (BOMD) trajectory, already available from previous computations 76 with the PBE-D3 functional. Even though this approach seems time consuming, in the context of a complete FF parameterization (comprising of bonded and nonbonded interactions) such data are already available. If that is not the case, the uncorrelated configurations can be generated with a classical MD simulation using a generic force field. Each crystal configuration is then loaded with 30 argon atoms and, keeping fixed the framework at the given positions, the sorbate is thermalized and a classical MD simulation is performed to generate 30 independent snapshots. For this task the NAMD 79 simulation package has been employed, performing a 5 · 105 steps (500 ps) NVT MD run at a temperature of 300 K. The force field used, termed A069, is reported in the Supporting Information (SI). The aim of this task is solely to obtain a set of uncorrelated configurations, for this reason it would be a waste of resources to perform it via BOMD. The number of sorbate atoms is about one half of the experimental maximum, 80,81 in order to have a more representative sample of typical working conditions. Processing of the system coordinates and production of molecular images have been realized through the VMD 82 package. At this point, a set of 2010 configurations of the framework loaded with the sorbate are available, and can be used for the generation of the reference DFT data (termed full dataset, from here on). The interactions between sorbate and framework, for a given configuration,

DFT computations. All Density Functional Theory (DFT) 52,53 computations in this work have been performed through the CP2K open source code, 54–60 relying on the Gaussian and Plane Waves (GPW) 61 method. Various functionals have been used, namely PBE-D2, 62,63 PBE-D3, 64,65 vdW-DF2 66–69 and rVV10. 70,71 Goedecker-Teter-Hutter (GTH) pseudopotentials, 72–74 and GTH basis sets 57 have been employed. The basis sets for each atom kind are triple-ζ with two sets of polarization functions (TZV2P), except for Zn (TZVP) in accordance to previous works. 44,75–78 The charge density cutoff is 700 Ry. For D2 and D3 the pair interactions were computed up to a distance of 21.17 ˚ A. For vdW-DF2 and rVV10 the cutoff of the FFT grid used in the calculation of the nonlocal vdW functional was set to 400 Ry and 240 Ry, respectively. The system is fully periodic, and it is comprised of a single unit cell. The convergence criterion for the SCF (largest element of the wave function gradient) is 10−7 . Throughout this work, where not otherwise stated, the DFT reference data are computed with the PBE-D3 functional. Force field. The classical functional form, adopted throughout this work, to model the vdW interaction is the 12-6 Lennard-Jones: σ 12 σ 6 − . (1) ULJ = 4ǫ r r Even thought more accurate representations exist, this choice gives a good compromise between computational speed and accuracy, for this reason it is commonly found in most classical computational packages. In this work the focus is on the adsorption properties of the system, in this context a typical approach is to held the framework atoms positions fixed. 39 For this reason, the parameters here optimized are only those involved in the framework-sorbate interactions, while the sorbate-sorbate interactions are modeled with parameters taken from the literature.


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are performed by interfacing the code with the LAMMPS 85 python library. The program has been tested by fitting a reference classical trajectory generated with a known FF. The program has been able to recover the parameters with an accuracy of 10−6 .

are obtained as: IFS = UF+S − UF − US

(2)

where UF+S is the DFT potential energy of the entire (i.e., framework plus sorbate) system, UF is the DFT potential energy of the empty framework for the given configuration, and US is the corresponding energy for the sorbate alone. The same approach is applied also to forces. Relevant quantities are obtained by repeating the DFT computations over the same trajectory: once for the entire system, then for the empty framework (after removing the sorbate atoms), and finally for the sorbate alone (after removing the crystal atoms). The fit consists in minimizing the following figure of merit: χ2 = [(1 − w)RRMSE ]2 + [wRRMSF ]2 ,

Long range treatment. It has been shown that a reliable fit of the vdW interactions is possible only if the cutoff of the classical simulation matches the refernce interaction range. 86 In this work, for DFT-D2 and DFT-D3 the vdW cutoff has been set to a value which ensures the interaction convergence (while for vdW-DF2 and rVV10 this is inherent to the method). In order to attain the same range of interactions in the classical simulation a tail correction for the energy has been used, checking that this approach gives practically the same result of a large (i.e., > 21 ˚ A) cutoff, with a considerable computational gain. The validity of the latter approximation should be checked case by case.

(3)

where: sP

− ui ) 2 , 2 i Ui sP P − f )2 i Pn (F Pi,n 2 i,n , RRMSF = i n Fi,n

RRMSE =

i (U Pi

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(4)

GCMC computations. The optimized parameters are subsequently used to compute sorption isotherms, and these are compared to experimental results. For the sorbate-sorbate interaction, the default choice is to use the parameterization of Ref. 87 (ǫ = 0.2464 kcal/mol and σ = 3.38 ˚ A), obtained by fitting the experimental vapor-liquid equilibrium curve. The isotherms are computed using the RASPA2 code. 88,89 The framework structure employed for this task is the one provided in the code database. The system is made of 2 × 2 × 2 unit cells, with a cutoff for the vdW interactions of 16 ˚ A. The interactions are truncated and a long range correction is added. A total of 30000 GCMC cycles are performed (5000 equilibration and 25000 production), where a cycle is comprised, on average, of a trial move for each sorbed molecule. The moves are made, with equal probability, by translation and insertion/deletion. To speedup the computation a pre-tabluated energy grid is used, with a spacing of about 0.2 ˚ A. For comparison with experimental results, the excess loading is computed as in Ref. 90.

are the roots of the relative mean square errors for energies and forces, respectively. Here U and F are the reference energies and forces, and u and f are the model ones. The index i runs over the system configurations, while n runs over the system atoms. The denominators are the magnitudes of each contribution and act as normalization terms. A weight w is introduced in the merit function to balance the contribution of forces to the fit. To evaluate the quality of the fitted parameters, the RRMSs are also computed over a validation set of 990 independent configurations (eval dataset), and such RRMSs are termed EVALs to avoid confusion. A Python program, which relies on the nonnegative least squares 83 (nnls) and the sequential least squares programming (slsqp) solvers, as implemented in the SciPy 84 minimize module, has been developed to perform the fit. The evaluation of the model energies and forces


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Table 1: Lennard-Jones ǫ (kcal/mol) parameters for 6 independent blocks of 300 configurations (EF1 trough EF6) and for the EF full dataset. Corresponding RRMSs (computed on the fitted dataset) and EVALs (computed on the evaluation dataset) values are reported at the bottom. Type

EF1

EF2

EF3

EF4

EF5

EF6

EF full

C1-Ar C2-Ar C3-Ar H2-Ar H3-Ar N-Ar Zn-Ar

0.1255 0.1375 0.1223 0.1051 0.1992 0.1967 0.2018 0.2001 0.1860 0.2002 0.1881 0.1961 0.0461 0.0453 0.0434 0.0457 0.0530 0.0464 0.0533 0.0476 0.1025 0.1137 0.111 0.1293 0.1828 0.0175 0.0894 0.0000

0.0816 0.1995 0.1923 0.0432 0.0495 0.1437 0.0000

0.0759 0.1955 0.1930 0.0445 0.0506 0.1429 0.0810

0.1131 0.1994 0.1939 0.0450 0.0497 0.1229 0.0285

RRMSE RRMSF

0.0180 0.429

0.0167 0.427

0.0174 0.430

0.0179 0.433

0.0168 0.429

0.0179 0.432

0.0175 0.431

EVALE EVALF

0.0174 0.435

0.0177 0.429

0.0173 0.436

0.0175 0.433

0.0177 0.431

0.0178 0.428

0.0175 0.432

In view of this, where not explicitly stated, all following fits are performed over the first 300 configurations of the full dataset, using the energy-force procedure.

ing from the energy-force fit over the first block of 300 configurations, performed with free and fixed σ, against reference PBE-D3 data is reported in Figure 9. It emerges that the free σ fit gives slightly better RRMSs with respect to the previously discussed results, i.e., RRMSs for energy (forces) are 0.0161 (0.410) vs 0.0180 (0.429) for the free and fixed σ cases, respectively. This behavior is obvious because an increase in the number of parameters naturally leads to a better fit of the reference data, not necessarily improving the underlying physical meaning of the model. The corresponding isotherms are reported in the SI, together with the convergence analysis of the parameters. Considering the fact that the fit performed with fixed σ is already at the limits of its capabilities (i.e., possible underdetermination of some parameters), and that the LJ σ values taken from the literature are known with reasonable accuracy, it becomes clear that there is no need to increase the complexity of the fit adding more free parameters. This is in line with the observation of Fang et al. that “there is a tradeoff between adding parameters and retaining physical simplicity and relevance”. 39

Dependence on the initial sample. The influence of the force field used for the sample generation is here evaluated via an iterative procedure, where the parameters obtained after one fit are used to generate the dataset for the next one, in accordance with the approach followed by Kulkarni and coworkers. 35 Such procedure is performed for two different initial sets of classical parameters, namely A069 and RG. The RG set is obtained by mixing, through LorentzBerthelot rules, the Garc´ıa-Pérez et al. 87 argon parameters with the ZIF-8 parameters taken from the RASPA2 database 89 (see SI for numerical values). In Figure 8 the evolution of the corresponding isotherms is reported. It can be seen that the difference between the isotherms, coming from the two initial models, decreases as the number of iterations increases, with convergence attained already after the first iteration. Fitting of σ. Another aspect to be considered is the effect of the inclusion in the fit of the LJ σ. A comparison of the energies com-

Other functionals. Here the fitting procedure is employed to parametrize different DFT


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Conclusions

guess and stopping the optimization before this spoils the results, still keeping the fit accuracy within the chosen limits. This procedure is here successfully applied to the fitting of the DFT-D2 (rVV10) dataset. In this case, keeping the RRMSs within 3% from their minimum the Zn value, for example, moves from 0.6560 (0.9822) kcal/mol to 0.1446 (0.1818) kcal/mol. The resulting isotherms are practically indistinguishable from those coming from the highest accuracy fit, and are reported in SI together with the numerical values of the parameters.

In this work a procedure to develop accurate classical force fields starting from density functional theory calculations has been presented. Peculiar in this approach is the simultaneous fit of both potential energy and forces. An automated procedure, aimed at finding the optimal balance between the two contributions, has been introduced. In particular, here the focus is on the modeling of the framework-sorbate dispersion interactions. With this in mind, the method has been applied to a prototypical system, i.e., argon in ZIF-8, whose properties are dominated by such interactions. The quality of pure energy (PE), pure force (PF), and energy-force (EF) fits has been investigated, and a thorough analysis of the convergence of these fits with respect to the size of the dataset has been performed. It emerged that some of the PE parameters, which reproduce at best the DFT energies, are underdetermined. On the other hand, the PF parameters, which are more converged than the PE, fail in reproducing the DFT energies, due to a shift inherent to the underlying physics of the problem. The best compromise is found with the EF fit, where the parameters convergence is improved with respect to PE, and both energies and forces are well reproduced. This has been assessed by computing both the sorption isotherms and the self-diffusion coefficients for the three parameters sets. It has been shown that for isotherms, ruled by energies, EF results are close to those of PE, while for selfdiffusivity, ruled by forces, EF results are close to those of PF. The convergence of the EF fit is reached already with 300 configurations. This is confirmed by the fact that fits performed over six independent samples of such size yield isotherms that are practically indistinguishable to the one coming from the fit of the full dataset. The latter behavior is also true for the RRMSs computed over the evaluation dataset. Two distinctive aspects of the presented approach are, on the one hand, the use of a loaded crystal, and on the other hand, the use of uncor-

Sorbate-sorbate interactions. The common practice of dealing with the sorbatesorbate interactions by using well-established literature values is here relaxed in order to explore the feasibility of a “full” DFT-based FF. This ultimately tells which results a DFT simulation would give if it were possible to follow the system on large time and space scales. The fitted parameter is once again the LJ well depth ǫ, with the σ fixed at 3.38 ˚ A. 87 After some tests, it has been found that the optimal (i.e., avoiding possible error cancellation) approach is to fit separately the sorbate-framework interactions and the sorbate-sorbate interactions, where the latter means to fit the US term of Eq. 2. The argon-argon ǫ resulting from the energy-force fitting of the first 300 PBE-D3 configurations (EF1) is 0.2790 kcal/mol. This value is larger than the 0.2464 kcal/mol of Ref. 87 used to compute the isotherms previously shown in this work. The same overestimating behavior is found also for the other investigated functionals, with the fit giving: 0.2961 kcal/mol for PBE-D2, 0.2682 kcal/mol for rVV10, and 0.3347 kcal/mol for vdW-DF2. In the SI, isotherms computed with these new values are depicted and compared to those shown in Figure 10. The overbinding is not surprising in view of the work of Tran and Hutter, 103 where the interaction energies of rare gas dimers have been reported for various functionals and found to be systematically larger with respect to very accurate reference values.


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References

related configurations for the framework. The former is motivated by the inclusion of forces in the fit, which improves the statistics proportionally to the number of atoms, while the latter comes from the observation that in the generation of partial charges this practice reduces the underdeterminacy of “buried” atoms. In this work it has not been possible to clearly quantify the added value of these approaches, and this will require further investigations. The ability of different DFT functionals to reproduce the experimental isotherm has been assessed. For argon in ZIF-8, it has been found that the best performance comes from the DFTD2, followed by DFT-D3, rVV10, and vdWDF2 functionals. A recursive procedure to refine the parameters which attain unphysical values after the fit has been proposed. This approach has been successfully employed for Zn, C1 and N atom types in DFT-D2 and rVV10 fits. More physically sound parameters have been obtained, without altering the fit accuracy nor the corresponding isotherms. Finally, the performance of a “full” DFTbased force field, comprised of both frameworksorbate and sorbate-sorbate interactions, has been investigated. It has been found that all resulting isotherms overestimate the loading with respect to those obtained modeling the sorbatesorbate interaction with values taken from the literature.

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Acknowledgement Support by Italian Ministero dell’Istruzione, dell’Università e della Ricerca, by Università degli Studi di Sassari, by Istituto Nazionale per la Scienza e Tecnologia dei Materiali (INSTM), and by Fondazione Banco di Sardegna is gratefully acknowledged. Cybersar Project managed by the Consorzio COSMOLAB is acknowledged for CPU time allocation.

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