The Diels–Alder (DA) reactions are one category of organic chemical reactions (specifically, a [4+2] cycloaddition) between a conjugated diene and a dienophile, which involve a dual carbon–carbon bond-forming process. Among them, the DA reactions of cyclopentadiene (CP) with acrylonitrile (ACR) or 1-4-naphthoquinone (NAP) have attracted much attention from experimental [1
] and computational scientists [4
]. It has been observed that the reaction rate is very sensitive to the solvent [2
]. Therefore, the solvation effect should be explicitly considered in order to unveil the reaction mechanism, in which the polarization effect and the reorganization of the solvent molecules may play a critical role.
Hybrid quantum mechanical/molecular mechanical (QM/MM) method, which was proposed by Warshel and Levitt in 1976 [9
], is now a mature method that can be used to study chemical reactions taking place in condensed phase such as aqueous solution or enzymatic environment [10
]. In this approach, only reactive region is treated quantum mechanically, and the remaining part is described by molecular mechanical force field. However, when the reaction barrier is much larger than
, which is pervasive for reactions under mild condition, a direct ab initio (ai
) QM/MM simulation is still notoriously time consuming, if feasible at all. Despite the continuous development in computer technology and enhanced sampling methods, investigation of reaction mechanism in solution or enzymes using direct ai
QM/MM simulations is still a daunting task.
To reduce the computational expense, Jorgenson et al. proposed a new approach for the calculations of free energy changes in chemical reactions in solution by combining gas phase QM calculations with free energy simulations (QM-FE method) [14
]. Kollman et al. extended this method to the studies of enzymatic reactions [17
]. However, in the QM-FE approach, the QM Hamiltonian and the MM Hamiltonian are separated, which is not a rigorous QM/MM approach. In other words, the impact of the solvent or the enzyme environment on the electronic structure of the QM region is not considered. However, solvent or enzyme environment often has a remarkable impact on the reaction process. Zhang and Yang [18
] were motivated by the QM-FE approach and developed a more practical method, which combines ai
QM/MM calculations with free energy perturbation (FEP) [19
]. Later, Thiel et al. named this method “QM/MM-FEP” [21
]. In this method, an efficient iterative optimization procedure was developed to determine the optimized structures and the minimum energy paths for a large-sized system on an ai
QM/MM potential energy surface. However, this still requires arduous computations.
To further reduce the computational expense, Jorgensen et al. used semi-empirical methods such as AM1 and PDDG/PM3 in the free energy calculations of these DA reactions [6
], followed by some high-level correction to the stationary points in the reaction path including the reactant, the product and the transition state (TS) [7
]. They found that the solvent-sensitivity originated from a significant nonhydrophobic component stemming from enhanced polarization of the transition state, which leads to strengthened hydrogen bonds [6
]. They also found that the DA reaction between CP and ACR was an asynchronous and concerted process [4
], while that between CP and NAP has a synchronous and concerted feature [6
]. This observation was evidenced in a two-dimensional free energy landscape for these DA reactions, and a one-dimensional reaction coordinate for these reactions would lead to potential artifacts and uncertainty in the locations of transition states, which in the end leads to an ambiguous reaction mechanism [6
]. Although semi-empirical (SE) QM/MM calculations have been widely used [12
], unfortunately, these semi-empirical methods may lead to large errors in the results due to the approximations adopted. Thus, high-level quantum mechanical methods at ab initio levels are necessary for a reliable depiction of electron redistribution during the reaction, which can be critical to the energetics such as activation and reaction free energies.
Based on the idea of probability-reweighting, Gao developed a method termed the dual-Hamiltonian method, also known as the reference-potential method, and applied it in a study of hydration free energy [30
]. Utilizing an empirical valence bond (EVB) method [32
] as the reference-potential, Warshel et al. developed a dual-Hamiltonian approach for calculating the free energy (FE) profiles of chemical or enzymatic reactions, from which an ai
FE profile can be obtained with much lower computational expense than a direct approach [26
]. Rod and Ryde calculated the activation free energy of a methyl transfer reaction in enzyme using the dual-Hamiltonian approach where the free energy was found to be overestimated [37
]. Recently, Jia et al. used the dual-Hamiltonian approach to calculate the solvation free energies of the molecules in the SAMPL4 competition by an alchemical decoupling method, which yielded the globally minimal variance for the QM/MM free energies [38
]. Liu et al. used this dual-Hamiltonian approach to calculate protein–ligand binding affinity at an ai
QM/MM level [39
In our previous work [40
], in the spirit of reference-potential method, a new method termed MBAR+wTP was proposed to obtain the ai
QM/MM FE profiles with much less computational expense. In this method, a weighted thermodynamic perturbation (wTP) [19
] correction is applied to the semi-empirical profile, which is generated by the Multistate Bennett Acceptance Ratio (MBAR) [41
] analysis of the trajectories from umbrella sampling (US) [43
]. The raw ai
QM/MM FE profile was then smoothed via Gaussian process regression [44
]. This MBAR+wTP method had been validated by calculating the FE profiles of one quasi-chemical reaction and three chemical reactions in aqueous solution. The results showed that even if the SE FE profiles deviated from the ai
ones by several kcal/mol in terms of activation free energy and reaction free energy, after the SE-to-ai
correction the FE profiles agree much better with the direct QM/MM simulated ones with errors below 1 kcal/mol.
In this work, we applied the MBAR+wTP method to calculate the FE surfaces of two Diels–Alder reactions of cyclopentadiene with either acrylonitrile or 1-4-naphthoquinone as mentioned above and investigated the applicability of this method to the study of reactions with two-dimensional (2D) reaction coordinates (RC).