Active learning accelerates ab initio molecular dynamics on reactive energy surfaces

•Autonomous acquisition of datapoints for reactive force field training•Neural nudged-elastic band•Neural reactive molecular dynamics•Switching from entropic to thermodynamic intermediate in solvents with high polarity In silico elucidation of reaction mechanisms using density functional theory (DFT) can explain and predict experimental observations. However, because of the computational cost of accurate DFT simulations, theoretical studies are often restricted to systems with fewer than 100 atoms at a few stationary points on potential energy surfaces. These can be insufficient for describing challenging reaction mechanisms where dynamical effects are important. Herein, we report a low-cost transferable pipeline that accelerates ab initio molecular dynamics by a factor of 2,000. It consists of high-throughput DFT computation, active learning, and transfer learning to train high-quality reactive force fields based on neural networks. These force fields reproduce the underlying DFT potential energy surface and enable reactive dynamics simulations. We anticipate that our pipeline will lead to accurate and affordable mechanistic studies of complex, experimentally relevant reactive systems. Modeling dynamical effects in chemical reactions typically requires ab initio molecular dynamics (AIMD) simulations due to the breakdown of transition state theory (TST). Reactive AIMD simulations are limited to lower-accuracy electronic structure methods and weak statistics because quantum mechanical energies and forces must be evaluated at femtosecond time resolution over many replicas. We report a data-driven pipeline that allows for the treatment of dynamical effects with the same level of theory and overall cost as that of TST approaches. High-throughput ab initio calculations and autonomous data acquisition are coupled to graph convolutional neural-network interatomic potentials, allowing for inexpensive reactive AIMD simulations at quantum mechanical accuracy. We demonstrate the approach by accurately simulating post-TS dynamical effects in three distinct pericyclic reactions, including a challenging trispericyclic reaction with a complex bifurcating potential energy surface. This approach is broadly applicable to understanding dynamical effects and predicting reaction outcomes in large, previously intractable systems. Modeling dynamical effects in chemical reactions typically requires ab initio molecular dynamics (AIMD) simulations due to the breakdown of transition state theory (TST). Reactive AIMD simulations are limited to lower-accuracy electronic structure methods and weak statistics because quantum mechanical energies and forces must be evaluated at femtosecond time resolution over many replicas. We report a data-driven pipeline that allows for the treatment of dynamical effects with the same level of theory and overall cost as that of TST approaches. High-throughput ab initio calculations and autonomous data acquisition are coupled to graph convolutional neural-network interatomic potentials, allowing for inexpensive reactive AIMD simulations at quantum mechanical accuracy. We demonstrate the approach by accurately simulating post-TS dynamical effects in three distinct pericyclic reactions, including a challenging trispericyclic reaction with a complex bifurcating potential energy surface. This approach is broadly applicable to understanding dynamical effects and predicting reaction outcomes in large, previously intractable systems. Transition state theory (TST) lies at the heart of most computational chemistry approaches to chemical reactivity. Reaction rates and product selectivities can be quantified, and thus can be predicted, from the free energy differences between reactants and transition states (TS), whose atomic configurations are taken to be static points on the potential energy surface (PES). Effective approaches that combine numerical optimization and chemical intuition, such as nudged-elastic band (NEB) and eigenvector following (EV), are routinely used to determine the geometry and free energy of TSs with near-experimental accuracy. Typically, density functional theory (DFT) is used for geometries and free energy corrections, and high-level correlated methods such as CCSD(T) for electronic energies. The static nature of TST, however, fails to explain dynamical effects, such as post-transition-state bifurcation,1Hare S.R. Tantillo D.J. Post-transition state bifurcations gain momentum – current state of the field.Pure Appl. Chem. 2017; 89: 679-698Crossref Scopus (91) Google Scholar, 2Tan J.S.J. Hirvonen V. 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Mechanisms and dynamics of reactions involving entropic intermediates.Trends Chem. 2019; 1: 22-34Abstract Full Text Full Text PDF Scopus (18) Google Scholar). Studies are thus limited to affordable DFT simulations with small basis sets. The computational cost also prevents a comprehensive exploration of different conditions, such as different solvents and different substituents. Recent advances in machine learning (ML) techniques and the modularization of neural network (NN) and automatic differentiation routines7Abadi M. Agarwal A. Barham P. Brevdo E. Chen Z. Citro C. Corrado G.S. Davis A. Dean J. Devin M. et al.TensorFlow: large-scale machine learning on heterogeneous distributed systems.arXiv. 2016; https://arxiv.org/abs/1603.04467Google Scholar facilitate the interplay between physical sciences and statistical learning.8Butler K.T. Davies D.W. Cartwright H. Isayev O. Walsh A. 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We define concerted reactions as those with a time gap of < 60 fs between bond changes,3Yang Z. Jamieson C.S. Xue X.-S. Garcia-Borràs M. Benton T. Dong X. Liu F. Houk K.N. Mechanisms and dynamics of reactions involving entropic intermediates.Trends Chem. 2019; 1: 22-34Abstract Full Text Full Text PDF Scopus (18) Google Scholar First, two model systems were studied: a classic Claisen rearrangement of allyl vinyl ether (Scheme 1A) and a Diels-Alder reaction between acrylaldehyde and cis-butadiene (Scheme 1B). The approach was then tested on a complicated trispericyclic reaction involving 8,8-dicyanoheptafulvene and 6,6-dimethylfulvene (Scheme 1C), whose experimental behavior cannot be explained by TST.37Liu C.Y. Ding S.T. Cycloadditions of electron-deficient 8,8-disubstituted heptafulvenes to electron-rich 6,6-disubstituted fulvenes.J. Org. Chem. 1992; 57: 4539-4544Crossref Scopus (23) Google Scholar,38Xue X.S. Jamieson C.S. Garcia-Borràs M. Dong X. Yang Z. Houk K.N. Ambimodal Trispericyclic transition state and dynamic control of periselectivity.J. Am. Chem. Soc. 2019; 141: 1217-1221Crossref PubMed Scopus (29) Google Scholar The MD simulations revealed an interplay between solvent effects unique to each reaction that would have been difficult to elucidate using traditional approaches because of their high computational cost. Specifically, for the trispericyclic reaction, we conclude that the relationship between bifurcation and product distribution is heavily dependent on the dielectric constant of the solvent due to the charge-separated nature of the intermediate. We envisage that our pipeline will be used to produce system-specific faithful NN-based interatomic potentials trained on high-accuracy QM data. This will empower the chemistry community to investigate the dynamics of large, complex reactive systems that were previously intractable. The key components of the proposed approach are coupled loops of (1) HT simulation for rapid, parallel acquisition of DFT data; (2) training ML interatomic potentials based on graph convolutional neural networks; and (3) rapid inference to sample the configurational space with NN-based versions of NEB and reactive MD (Scheme 2). Because model uncertainty is not necessary for our AL approach, it can be categorized as membership query synthesis, as proposed by Settles.39Settles B. Active learning literature survey.http://digital.library.wisc.edu/1793/60660Date: 2009Google Scholar For the Claisen rearrangement and Diels-Alder reactions, we constructed two independent active learning cycles to acquire data for each reaction, resulting in two reaction-specific NN-based interatomic potentials. For each cycle, 14,976 and 7,927 M062X/def2-SVP calculations were performed, respectively. For the more complex trispericyclic reaction, three active learning cycles were constructed for each of the three distinct steps leading to each product. A single converged interatomic potential was trained on a total of 31,397 configurations obtained from these loops at the same level of theory. All converged interatomic potentials attained sub-kcal mol−1 accuracy on the corresponding test sets (Table S2). To further ascertain the accuracy of the models, we evaluated relative energies and thermodynamic quantities on the relevant stationary points of the three reactions (Table 1). The mean absolute error (MAE) in relative energy was 0.4 kcal mol−1, attaining sub-chemical accuracy relative to the reference M06-2X/def2-SVP method. Despite not training explicitly on Hessians, which were used to calculate various thermodynamic quantities, the MAE of the Gibbs free energy correction term, Gcorr, had a reasonable value of 2.3 kcal mol−1. The majority of the deviation came from the entropic correction term, Scorr, which is known to have large errors with the presence of low-frequency vibrations. Several quasi-harmonic entropic corrections could be applied to produce better estimates.40Ribeiro R.F. Marenich A.V. Cramer C.J. Truhlar D.G. Use of solution-phase vibrational frequencies in continuum models for the free energy of solvation.J. Phys. Chem. B. 2011; 115: 14556-14562Crossref PubMed Scopus (590) Google Scholar,41Grimme S. Supramolecular binding thermodynamics by dispersion-corrected density functional theory.Chemistry. 2012; 18: 9955-9964Crossref PubMed Scopus (894) Google ScholarTable 1Relative NN (DFT) energies and thermodynamic quantities [in kcal/mol]ClaisenSpeciesRel. EaM06-2X/def2-SVP energies relative to the respective reactants.ZPEHcorrTScorrGcorrbGcorr = ZPE + Hcorr -TScorrCL-R0.0 (0.0)73.7 (74.6)5.0 (5.2)25.5 (24.2)53.2 (55.4)CL-TS32.9 (32.4)71.0 (74.0)4.2 (4.8)24.0 (22.3)51.2 (55.9)CL-P−16.6 (−16.6)72.9 (74.5)5.0 (5.1)24.7 (24.3)53.2 (55.2)Diels-AlderDA-R0.0 (0.0)95.5 (93.4)6.0 (7.3)27.2 (29.6)74.3 (71.1)DA-TS15.2 (14.3)94.4 (95.0)6.4 (5.7)28.4 (25.9)72.4 (74.8)DA-P−49.1 (−49.2)96.2 (98.2)5.4 (5.2)25.3 (24.9)76.3 (78.5)TrispericyclicTP-R0.0 (0.0)180.8 (181.1)12.7 (12.5)56.5 (55.0)136.9 (138.7)TP-TS18.4 (7.6)181.1 (181.9)12.2 (11.9)43.9 (39.1)149.6 (154.7)TP-P1−19.5 (−19.6)184.4 (184.4)11.6 (11.6)39.5 (38.5)156.5 (157.4)TP-TS2a5.6 (5.4)182.3 (182.6)11.5 (11.4)39.3 (37.9)154.4 (156.0)TP-P2a−27.2 (−27.4)184.3 (185.1)11.3 (11.1)38.7 (37.2)156.9 (159.0)TP-TS2b6.1 (6.2)182.9 (182.6)11.3 (11.5)38.0 (38.3)156.2 (155.8)TP-P2b−23.4 (−24.1)184.4 (183.7)11.8 (11.7)40.9 (39.0)155.2 (156.5)ZPE, Hcorr, Scorr, Gcorr are abbreviations of zero-point energy, enthalpy (without ZPE), entropy, and Gibbs free energy corrections, respectively. Species abbreviations are defined in Scheme 1. T = 298.15 Ka M06-2X/def2-SVP energies relative to the respective reactants.b Gcorr = ZPE + Hcorr -TScorr Open table in a new tab ZPE, Hcorr, Scorr, Gcorr are abbreviations of zero-point energy, enthalpy (without ZPE), entropy, and Gibbs free energy corrections, respectively. Species abbreviations are defined in Scheme 1. T = 298.15 K To efficiently upgrade the theoretical accuracy and to include solvation effects, we leveraged a transfer learning (TL) strategy36Smith J.S. Nebgen B. Lubbers N. Isayev O. Roitberg A.E. Less is more: sampling chemical space with active learning.J. Chem. Phys. 2018; 148: 241733Crossref PubMed Scopus (220) Google Scholar to a larger basis set (def2-TZVPD) with the SMD solvation model (for details see Supplemental information). We used three common organic solvents (cyclohexane, chloroform, and acetone) with about 10%–20% additional training points each (Table S1). Furthermore, TL was also performed with a more accurate double-hybrid functional in vacuum, DLPNO-DSD-PBEP86-D3BJ/def2-TZVPD (abbreviated as DH/def2-TZVPD), to investigate the role of the functional. All transfer-learned potentials attained out-of-sample kcal mol−1 accuracy (Figures S1–S5 for Claisen, Figures S6–S10 for Diels-Alder, and Figures S11–S15 for trispericyclic). All reactive trajectories were initiated from the vicinities of QM-optimized TS geometries of the corresponding levels of theories in the forward and backward directions (see Computational methods for details). In the following sections, we mainly discuss the MD results obtained from the transfer-learned (SMD-)M06-2X/def2-TZVPD potentials in order to focus on solvent effects on the reaction dynamics. The foundational M06-2X/def2-SVP and the higher quality DH/def2-TZVPD potentials in vacuum are used as comparisons. All reactive MD simulations utilize the Langevin thermostat with the corresponding friction coefficient of the solvent (see Computational methods for details). In total, 500 pairs of trajectories were initiated from the vicinity of CL-TS for each solvent. Of these, the number of valid trajectories (those that resulted in CL-P on one end and CL-R on the other) ranged from 384 to 474 (Table S3). We calculated the time gap between the breaking of C–O bond and the formation of C–C bond in the Claisen rearrangement of allyl vinyl ether to γ,δ-unsaturated aldehyde as a quantitative measure of the dynamic concertedness of the reaction. The time at which the C–O bond breaks (C–C bond forms) is defined when the bond length exceeds (falls below) 1.7 Å. This value, while somewhat arbitrary, is consistent with the convention adopted in previous studies.3Yang Z. Jamieson C.S. Xue X.-S. Garcia-Borràs M. Benton T. Dong X. Liu F. Houk K.N. Mechanisms and dynamics of reactions involving entropic intermediates.Trends Chem. 2019; 1: 22-34Abstract Full Text Full Text PDF Scopus (18) Google Scholar,38Xue X.S. Jamieson C.S. Garcia-Borràs M. Dong X. Yang Z. Houk K.N. Ambimodal Trispericyclic transition state and dynamic control of periselectivity.J. Am. Chem. Soc. 2019; 141: 1217-1221Crossref PubMed Scopus (29) Google Scholar The histograms of the distribution of absolute time gaps and the corresponding C–O (breaking) and C–C (forming) bond lengths of (SMD-)M06-2X/def2-TZVPD-optimized TSs are presented in Figure 1. Regardless of the solvent environment, the percentage of trajectories that are dynamically concerted (with time gaps less than 60 fs) are 100% in vacuum and cyclohexane, and 97% and 98% in chloroform and acetone, respectively. Both the mean and the median time gaps in each solvent fall in a narrow range of 26–36 fs. We note that the distributions in chloroform and acetone are more spread out due to the increased stability of the transient charge-separated species (after C–O bond breaking but before C–O bond forming) due to the increase in polarity of the solvent. The neural reactive molecular dynamics(NRMD) results at DH/def2-TZVPD level of theory in vacuum showed no significant deviation in terms of percentage of dynamically concerted trajectories (97%), and the mean and median time gaps (37 and 35 fs, respectively). A comprehensive list of statistical quantities relating to time gaps is presented in Table S3. In the NRMD simulations of the Diels-Alder reaction, over 400 pairs of valid trajectories were generated for each solvent (Table S3). The results suggest that there is a subtle interplay between solvent polarity and concertedness. In vacuum and in the more polar solvents chloroform and acetone, over 90% of trajectories were dynamically concerted, with median time gaps of 18, 20, and 14 picoseconds (ps), although a slightly bimodal distribution was observed with another peak at around 200 ps (Figure 2). In the case of cyclohexane, however, the percentage of dynamically concerted trajectories decreased from 92% to 57% (Table S3) and the distribution was much broader and clearly bimodal with a mean gap of 86 ps. In order to disentangle the role of friction and polarity, we further performed reactive MD simulations in all four media with fixed friction values of 10−4, 10−3, and 10−2 a.u. It was observed that the trends in median time gaps and percentage of dynamically concerted trajectories across solvents arose from polarity and were independent of friction (Table S4). To elucidate the reasons for the longer time gaps in cyclohexane, we calculated the magnitude of the imaginary frequency of DA-TS at SMD-M06-2X/def2-TZVPD level of theory. The frequencies were −468.57, −484.73 and −481.50 cm−1 for cyclohexane, chloroform, and acetone, respectively, suggesting that the flatter surface around DA-TS in cyclohexane is a possible factor of the longer time gap observed. Comparing the NRMDs of M06-2X/def2-TZVPD and DH/def2-TZVPD in vacuum, the percentage of dynamically concerted trajectories slightly decreased from 93% to 89%, while the mean and median time gaps increased from 28 to 43 fs and 18 to 31 fs, respectively. This is possibly due to the inclusion of MP2 correlation and explicit Grimme’s D3 dispersion term,42Grimme S. Ehrlich S. Goerigk L. Effect of the damping function in dispersion corrected density functional theory.J. Comput. Chem. 2011; 32: 1456-1465Crossref PubMed Scopus (9713) Google Scholar,43Grimme S. Antony J. Ehrlich S. Krieg H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu.J. Chem. Phys. 2010; 132: 154104Crossref PubMed Scopus (22221) Google Scholar which can stabilize the transient charge-separated species after the formation of bond B. Experimental results for the trispericyclic reaction at room temperature in chloroform yield a product distribution of 0: 1.17: 1 for TP-P1[4+6]: TP-P2a[6+4](+tautomer): TP-P2b[8+2] in 1 day. Recently, Xue et al.38Xue X.S. Jamieson C.S. Garcia-Borràs M. Dong X. Yang Z. Houk K.N. Ambimodal Trispericyclic transition state and dynamic control of periselectivity.J. Am. Chem. Soc. 2019; 141: 1217-1221Crossref PubMed Scopus (29) Google Scholar carried out TS and 500-fs BOMD simulations of the same reaction in vacuum at the ωB97X-D/6-31G(d) level of theory. For trajectories initiated from TP-TS1, 87% of the 142 pairs of trajectories resulted in TP-P1, and 3% each for TP-P2a and TP-P2b. For trajectories initiated from TP-TS2a and TP-TS2b, the majority of the trajectories were reported to result in TP-P2a and TP-P2b, respectively. From this, it was concluded that the products TP-P2a and TP-P2b observed at the end of the experiment are a result of thermodynamic control, but no connection was made between the trajectories obtained and the experimental yield. In order to understand the role of choice of QM method and solvent on this reactive system, we simulated 500 pairs of trajectories in each medium for a significantly longer timescale of 5 ps in each direction, so as to account for the longer lifetimes of the reactive intermediate in solvents due to both polarity and friction. In addition to trajectories connecting two distinct minima, trajectories corresponding to “recrossing” and “hovering” were also observed. The “recros