Exploring, Refining, and Validating the Paradynamics $\mathbf{QM}∕\mathbf{MM}$ Sampling

Abstract

The performance of the paradynamics (PD) reference potential approach in $\text{QM}∕\text{MM}$ calculations is examined. It is also clarified that, in contrast to some possible misunderstandings, this approach provides a rigorous strategy for $\text{QM}∕\text{MM}$ free energy calculations. In particular, the PD approach provides a gradual and controlled way of improving the evaluation of the free energy perturbation associated with moving from the $\text{EVB}$ reference potential to the target $\text{QM}∕\text{MM}$ surface. This is achieved by moving from the linear response approximation to the full free energy perturbation approach in evaluating the free energy changes. We also present a systematic way of improving the reference potential by using Gaussian-based correction potentials along a reaction coordinate. In parallel, we review other recent adaptations of the reference potential approach, emphasizing and demonstrating the advantage of using the $\text{EVB}$ potential as a reference potential, relative to semiempirical $\text{QM}∕\text{MM}$ molecular orbital potentials. We also compare the PD results to those obtained by direct calculations of the potentials of the mean force (PMF). Additionally, we propose a way of accelerating the PMF calculations by using Gaussian-based negative potentials along the reaction coordinate (which are also used in the PD refinement). Finally, we discuss performance of the PD and the metadynamics approaches in ab initio $\text{QM}∕\text{MM}$ calculations and emphasize the advantage of using the PD approach.

Graphical Abstract

I. INTRODUCTION

Modeling chemical processes in condensed phases using the $\text{QM}∕\text{MM}$ method became the mainstream in computer simulations.^1-5 Although recent methodological advances in electronic structure calculations and in parallel computing as well as an increase in available computer power, have opened new opportunities for predictive ab initio $\text{QM}∕\text{MM}$ simulations, evaluation of the free energies in ab initio $\text{QM}∕\text{MM}$ ($\text{QM}(\text{ai})∕\text{MM}$) studies has remained one of the major challenges in the field. One of the most promising options for performing $\text{QM}(\text{ai})∕\text{MM}$ free energy calculations has been our approach^6-8 of using a reference potential ($\text{RP}$) for the $\text{QM}∕\text{MM}$ potential and recent related versions discussed below. The most recent version of our idea of using the $\text{RP}$ is implemented in the paradynamics (PD) model.⁹ The PD model is based in a large part on a number of earlier works and ideas including (a) those which have established the way of efficient sampling of the $\text{RP}$ using the empirical valence bond¹⁰ ($\text{EVB}$) method combined with the evaluation of the free energy surface by a specialized combination^11,12 of the free energy perturbation ($\text{FEP}$¹³) and umbrella sampling (US¹⁴) protocols, with the energy gap as a reaction coordinate;^11,12 (b) those works which introduced the idea of using the $\text{EVB}$ $\text{RP}$ and the $\text{FEP}$/US approach in semiempirical⁷ and ab initio⁶ $\text{QM}∕\text{MM}$ free energy calculations; and (c) the ideas of refining the $\text{EVB}$ $\text{RP}$⁸ to minimize the energy difference between the $\text{EVB}$ $\text{RP}$ and the $\text{QM}(\text{ai})∕\text{MM}$ target potential ($\text{TP}$), and using the linear response approximation^15,16 ($\text{LRA}$) for calculating the $\text{FEP}$, while moving from the $\text{RP}$ to the $\text{TP}$ at the transition ($\text{TS}$) and reactant states ($\text{RS}$).

The PD approach has been found to be quite effective,⁹ where a typical evaluation of the activation free energy barrier by the PD approach (using the $\text{EVB RP}$) was estimated to require ~200 times less calls to the $\text{TP}$ compared to the metadynamics (MTD) studies involving direct sampling of the $\text{TP}$.¹⁷ Thus the PD approach provides a rigorous way of dealing with the high computational cost of sampling of the $\text{QM}(\text{ai})∕\text{MM}$ potential which is necessary to get an accurate free energy barrier.

In the present work, we report recent advances in expanding capabilities of the PD approach, clarifying possible misunderstandings and introducing several practical advances.

The first point that will be addressed is a misunderstanding associated with the fact that implementation of the PD scheme of moving from the $\text{RP}$ to the $\text{TP}$ has been mostly done with the $\text{LRA}$ approach. This might have led some to the assumption that our approach is limited by validity of the $\text{LRA}$ and thus does not yield an estimate of the relevant error. However, as we tried hard to clarify, the perturbation between the two potentials can be done with a full $\text{FEP}$ treatment. This point will be demonstrated here.

The second point that will be addressed in this work is our finding that the iterative PD approach⁹ of refining the $\text{EVB RP}$ has not been fully robust. In particular, in some cases, one may want to change the $\text{EVB}$ functional form rather than just refine the $\text{EVB}$ parameters. In order to overcome this problem, we developed an approach that at least in some cases can be very effective in constructing and refining the $\text{RP}$ in an alternative, more general way. This new approach involves fitting the potential energy scan (PES) along the specified reaction coordinate ($\text{RC}$) in the gas phase (or in an implicit solvent) for both the $\text{RP}$ and the $\text{TP}$ by Gaussian functions. The refined $\text{RP}$ is constructed as the original $\text{RP}$ modified with a correction that is taken as the difference between the Gaussian functions that approximate the $\text{TP}$ and the $\text{RP}$.

The original $\text{RP}$ approach⁷ has been adopted and implemented in a number of methods aimed at $\text{QM}(\text{ai})∕\text{MM}$ calculations of activation free energies in enzymes^18-20 and in aqueous solutions,²¹ as well as at evaluation of free energies of solvation,^22,23 binding,²⁴ and conformational changes.²⁵ For example, in one recent approach, a semiempirical AM1/MM potential was used as the $\text{RP}$ for the configurational sampling and for the construction of the reference free energy surface (FES). This was followed by an optimization on a DFT $\text{QM}∕\text{MM}$ potential at a number of points along the chosen $\text{RC}$, yielding the least energy reaction path. The entropic contribution was approximated by the difference between the activation free energy and the activation energy obtained on the AM1/MM $\text{RP}$. Furthermore, single point energy calculations on the optimized DFT $\text{QM}∕\text{MM}$ least energy path, using CCSD¹⁸ or MP2,¹⁹ were proposed to provide the activation free energy for a high level ab initio $\text{QM}∕\text{MM}$ surface. Another technique, introduced by Yang and co-workers,²⁶ and adopted elsewhere,²⁷ provides an estimate of the $\text{QM}(\text{ai})∕\text{MM}$ free energy profiles with a fixed solute at a predetermined $\text{RC}$ (which is determined, for instance, through iterative sequential optimization of the $\text{QM}$ and $\text{MM}$ regions independently, where in each step one of the regions is kept fixed while the second is optimized²⁶). This is followed by a single-step $\text{FEP}$ from the MM-type $\text{RP}$ to a $\text{QM}∕\text{MM}$ $\text{TP}$ (with averaging on the $\text{RP}$). The reference FES is constructed by sampling on the $\text{RP}$. The ESP-derived solute charges for the MM-type $\text{RP}$ are periodically updated by the $\text{QM}∕\text{MM}$ calculations. This method seems to be capable of capturing both the fixed solute polarization and the solvent polarization in a physically consistent way. A related approach involves single point energy evaluations^20,28 with a high level $\text{QM}∕\text{MM}$, performed at fixed solute geometries (in a similar way to that used in the work¹⁸ mentioned earlier). The energy difference between the DFT $\text{QM}∕\text{MM}$ and the high-level $\text{QM}∕\text{MM}$ was assumed to approximate the $\text{FEP}$ while moving between these two potentials.²⁷ In another recent work, a different semiempirical $\text{RP}$, $\text{PM}3∕\text{MM}$,²⁵ was used to construct the reference FES, followed by a single step $\text{FEP}$ from the $\text{RP}$ to a DFT $\text{QM}∕\text{MM}$ $\text{TP}$ with the average being calculated on the $\text{RP}$.

The overall $\text{RP}$ results, obtained in the above studies, clearly indicate that this approach is a powerful and versatile tool, which allows one to overcome the high cost of direct sampling on the $\text{QM}(\text{ai})∕\text{MM}$ potential. However, there is a great potential for improvements to some of the existing $\text{RP}$ treatments, especially considering the fact that some of them overlooked our earlier, and arguably more consistent, $\text{RP}$ treatment. For instance, one can foresee certain disadvantages to the use of minimization in exploring the multidimensional DFT $\text{QM}∕\text{MM}$ surface, as it is expensive and rather computationally inefficient. In addition to this, such a strategy can lead to multiple local minima. One has to keep in mind that the entropic contribution to the activation free energy barrier can be different for the $\text{TP}$ and the $\text{RP}$, and that the activation energy is not equal to the activation enthalpy. Another serious problem is the slow convergence of the single step $\text{FEP}$ approach when the average is taken only on the $\text{RP}$. This is especially problematic when the difference between the $\text{TP}$ and the $\text{RP}$ is significant (and their overlap in the phase space is small). In this case, sampling on the $\text{RP}$ poorly represents the relevant region of interest on the $\text{TP}$.

The above problems are largely eliminated by the PD approach mainly due to the following features: (a) using the end points $\text{LRA}$ that takes the averages both on the $\text{RP}$ and on the $\text{TP}$ and (b) using the PD refinement, which ensures faster $\text{LRA}$ convergence due to a higher overlap between the $\text{TP}$ and the refined $\text{RP}$. These points will be illustrated and discussed below.

An important element of the PD approach is the idea of refining the $\text{RP}$ so it will be as close as possible to the $\text{TP}$. Our main refinement strategy starts with evaluation of the energy gap between the $\text{RP}$ and the $\text{TP}$ for the geometries generated by MD trajectories propagated at the $\text{RS}$ and at the $\text{TS}$ on the $\text{RP}$ as well as on the $\text{TP}$. This step is followed by a least-squares minimization of the energy difference (plus other quantities). A somewhat similar refinement strategy has also been introduced in another context.²⁹ Although this approach is very effective, we found out that in cases when the improvement of the $\text{EVB}$ requires a new functional we can get an improved refinement by adding Gaussian functions to the $\text{RP}$ in a way that minimizes the difference between the $\text{RP}$ and $\text{TP}$ along the $\text{RC}$. A related idea of deforming the original potential to eliminate local minima was proposed by Scheraga and co-workers³⁰ in order to find the global minimum on a multidimensional surface. The use of a single negative Gaussian in the bias potential was also found to be useful in a MC study performed by Jorgensen and co-workers;³¹ however, it was for a rather stand-alone, specific case. A more general strategy of improved sampling by iteratively fitting the negative of the original potential with Gaussians in the local elevation method³² was formulated by van Gunsteren and co-workers. This idea was adopted, generalized in an elegant way, and popularized in the MTD approach of Parinello and co-workers.³³ As will be clarified below, unlike these approaches, ours does not require expensive iterative sampling while building the negative potential. This step is accomplished by fitting the PES, and using the negative potential approximated by Gaussians in our $\text{FEP}$/US PMF³⁴ calculations. We accelerate convergence of PMF calculations with explicit solvent models by using the $\text{EVB}$ type solvent driving potential, which forces the solvent to be polarized by the solvated solute charges along the reaction path.

In summary, we carry out two separate applications with the Gaussian functions: (a) during the PD refinement procedure, we evaluate two PESs (on the $\text{RP}$ and on the $\text{TP}$), fit them both with Gaussian functions, and take the difference as a correction to the original $\text{RP}$; and (b) as the improved sampling strategy in $\text{FEP}$/US PMF calculations, we evaluate the negative of the original potential (without any iterative approaches) and use the corresponding Gaussians to create a flat mapping potential. The sampling approach is augmented by the $\text{EVB}$-type solvent driving potential for the condensed phase calculations with explicit solvent models. At any rate, our Gaussian refinement strategy is as related to the MTD as our idea of building the $\text{RP}$ that will be as similar as possible to the $\text{TP}$. As we clarified in our previous work,⁹ this early $\text{RP}$ idea is formally identical to the MTD idea of building the negative of the $\text{TP}$ but allows for more effective implementation.

II. METHODS AND RESULTS

II.1. The Standard Paradynamics Approach.

The central idea of the PD ($\text{RP}$) approach is that the extensive configurational sampling required to calculate the $\text{QM}(\text{ai})∕\text{MM}$ free energy barrier can be done on a computationally inexpensive reference potential^6-8,16 rather than directly on the expensive target $\text{QM}(\text{ai})∕\text{MM}$ potential. Our standard approach, described in Figure 1, involves construction of the free energy surface (FES) using the $\text{EVB}$¹⁰ method and the $\text{FEP}$/US technique.^12,14 This is followed by evaluating the $\text{FEP}$ using the $\text{LRA}$ approach, while moving from the $\text{EVB}$ $\text{RP}$ to the $\text{TP}$ at the $\text{RS}$ and at the $\text{TS}$.

Figure 1.

The thermodynamic cycle used in the paradynamics approach to calculate the free energy barriers. The blue curve is the free energy surface, $\Delta g(\xi )$, obtained for the reference potential with the corresponding barrier, $\Delta g^{‡}$ (blue arrow); the red curve is the free energy surface for the target potential, for which we want to evaluate the free energy barrier (red arrow). To achieve that, we estimate the free energy of moving from the reference potential to the target potential at the $\text{RS}$ and at the $\text{TS}$ (shown by black arrows).

In considering activation barriers for chemical reactions, we follow our previous works¹⁶ and clarify the difference between the free energy profile (or the PMF), $\Delta g$, and the free energy of the system in a particular state, $\Delta G$. This is done by starting with the expression for the rate constant, $k$:^35,36

$$\begin{gathered} k=\kappa \frac{{\langle \mid \stackrel{.}{\xi }\mid \rangle }_{\text{TS}}}{2\Delta {\xi }^{‡}}\frac{\Delta {\xi }^{‡}\text{exp}(-\beta \Delta g^{‡})}{{\int }_{-\infty }^{{\xi }^{‡}}\text{exp}(-\beta \Delta g(x))\mathrm{d}x} \\ =\kappa \frac{{\langle \mid \stackrel{.}{\xi }\mid \rangle }_{\text{TS}}}{2\Delta {\xi }^{‡}}\text{exp}(-\beta \Delta G^{‡}) \end{gathered}$$ (1)

where $\kappa$ is the transmission factor, $\stackrel{.}{\xi }$ is the velocity along the $\text{RC}$, $\xi$, $\Delta {\xi }^{‡}$ is the width of the $\text{TS}$ region, $\Delta g_{\text{RS}}$ is the value of the free energy profile (or PMF) at the minimum at the $\text{RS}(x={\xi }_{\text{RS}})$, and $\Delta g^{‡}$ is its value at the $\text{TS}(x={\xi }^{‡})$. Here $\Delta G^{‡}$ is defined by

$$\text{exp}(-\beta \Delta G^{‡})=\frac{\text{exp}(-\beta (\Delta g^{‡}-\Delta g_{\text{RS}}))}{\frac{1}{\Delta {\xi }^{‡}}{\int }_{-\infty }^{{\xi }^{‡}}\text{exp}(-\beta (\Delta g(x)-\Delta g_{\text{RS}}))\mathrm{d}x}$$ (2a)

or

$$\begin{gathered} \Delta G^{‡}=\Delta g^{‡}-\Delta g_{\text{RS}} \\ +kT\text{ln}\left(\frac{1}{\Delta {\xi }^{‡}}{\int }_{-\infty }^{{\xi }^{‡}}\text{exp}(-\beta (\Delta g(x)-\Delta g_{\text{RS}}))\mathrm{d}x\right) \end{gathered}$$ (2b)

where the last term of eq 2b provides a convenient way of incorporating the entropic effect of the ground state into the $\text{TS}$-theory rate constant.³⁵ For the free energy of the $\text{RS}$, we also have

$${\mathrm{e}}^{-\beta \Delta G(\text{RS})}={\int }_{-\infty }^{{\xi }^{‡}}{\mathrm{e}}^{-\beta \Delta g(x)}\mathrm{d}x$$ (3)

This work will use the lower case $\Delta g(\xi )$ and the capital $\Delta G$, as respectively, the value of the free energy profile, or the PMF, at a particular $\text{RC}$ value (e.g., at $\text{TS}$, $\text{RS}$) and the free energy of a particular state (e.g., $\text{RS}$ or $\text{TS}$ defined by eq 3 or by $\Delta g^{‡}$).

The second key idea of the PD approach, which drastically improves its computational cost, is the use of the $\text{LRA}$ approach to calculate the free energy differences between the $\text{RP}$ and the $\text{TP}$ shown in Figure 1.

For example, in the case where the $\text{TS}$ position is identical for the two free energy surfaces, we can write¹⁶

$$\begin{gathered} \Delta \Delta g_{E_{\text{REF}}\to E_{\text{TARG}}}({\xi }_{E_{\text{TARG}}}^{‡}) \\ =-kT\text{ln}\frac{q_{E_{\text{TARG}}}({\xi }_{E_{\text{TARG}}}^{‡})}{q_{E_{\text{REF}}}({\xi }_{E_{\text{TARG}}}^{‡})} \\ =-kT\text{ln}{\langle \text{exp}\left(-\frac{E_{\text{TARG}}({\xi }_{E_{\text{TARG}}}^{‡})-E_{\text{REF}}({\xi }_{E_{\text{TARG}}}^{‡})}{kT}\right)\rangle }_{E_{\text{REF}}} \\ \cong \frac{1}{2}({\langle E_{\text{TARG}}({\xi }_{E_{\text{TARG}}}^{‡})-E_{\text{REF}}({\xi }_{E_{\text{TARG}}}^{‡})\rangle }_{E_{\text{TARG}}} \\ +{\langle E_{\text{TARG}}({\xi }_{E_{\text{TARG}}}^{‡})-E_{\text{REF}}({\xi }_{E_{\text{TARG}}}^{‡})\rangle }_{E_{\text{REF}}}) \end{gathered}$$ (4)

where the $q’\mathrm{s}$ are the partition functions at the specific $\text{RC}$ values.

Finally, the third important component of the PD approach is the refinement of the $\text{RP}$, which brings it closer to the $\text{TP}$ and ensures fast convergence of the $\text{FEP}$ calculation from the $\text{RP}$ to the $\text{TP}$. The above $\text{LRA}$ expression also provides a framework for the PD refinement procedure, where the $\text{EVB}$ parameters are refined by seeking the minimum of the least-squares function:⁹

$$\begin{gathered} 𝒯(\mathbf{p},\mathbf{r})=k_1\sum _{i=1}^N{({E_i}^{\text{EVB}}(\mathbf{p},\mathbf{r})-E_i^{\text{QM}}(\mathbf{r}))}^2 \\ +k_2\sum _{i=1}^N\sum _{j=1}^3{\left(\frac{\partial }{\partial x_j}{E_i}^{\text{EVB}}(\mathbf{p},\mathbf{r})-\frac{\partial }{\partial x_j}{E}_i^{\text{QM}}(\mathbf{r})\right)}^2 \end{gathered}$$ (5)

by either using a simplifed Newton–Rhaphson approach where we refine one $\text{EVB}$ parameter, $p^i$, at a time, where the result of the $k+1\text{-th}$ iteration is given

$${p^i}_{k+1}={p^i}_k-g[\partial 𝒯∕\partial p_i]∕{[{\partial }^2ℐ∕{\partial p}_i^2]}_k$$ (6)

or just by refining the vector of $\text{EVB}$ parameters, $\mathbf{p}$, in the optimal steepest descent approach, where the result of the $k+1\text{-th}$ iteration is given by

$${\mathbf{p}}_{k+1}={\mathbf{p}}_k-s\cdot \text{grad}𝒯({\mathbf{p}}_k)$$ (7)

The PD approach can be applied to any $\text{RP}$. However, the $\text{EVB}$ potential has been the $\text{RP}$ of choice, due to its very low computational cost, good performance in condensed phases (in particular in studies of enzymatic reactions¹⁶), physically consistent treatment of the solute–solvent electrostatic coupling, and other advantages discussed elsewhere.³⁷

While the refinement scheme given above is for $\text{EVB}$ parameters, we also provide below a benchmark example of using a more expensive (compared to $\text{EVB}$) semiempirical $\text{RP}$ to demonstrate that the PD approach is general as far as the $\text{RP}$ is concerned.

II.2. Validating the $\mathbf{LRA}$ by a Full $\mathbf{FEP}$ Treatment.

II.2.1. The Case of Two General Potentials.

Although the $\text{LRA}$ treatment of eq 4 has been effective, its overall convergence may be problematic when the difference between the $\text{RP}$ and the $\text{TP}$ is significant. This has led some to assume that the validity of the $\text{LRA}$ treatment limits application of our PD approach. However, what is missing in this assumption is the fact that $\text{LRA}$ is just the end point approximation of the full $\text{FEP}$ treatment which can be easily implemented by using

$$\Delta \Delta G_{\text{FEP}}=-kT\sum _{m=1}^{n-1}\text{ln}{\langle \text{exp}\left(-\frac{E_{m+1}-E_m}{kT}\right)\rangle }_{E_m}$$ (8)

where $n$ is the number of simulation windows, between which the perturbation parameter $\lambda$ (here weight of the $\text{TP}$) changes from 0 to 1, and $E_m$ is the mapping potential given by

$$E_m=(1-{\lambda }_m)E_{\text{REF}}+{\lambda }_m{E}_{\text{TARG}}$$ (9)

$E_{\text{REF}}$ and $E_{\text{TARG}}$ designate the $\text{RP}$ and the $\text{TP}$, respectively. Better convergence is obtained, however, if we take the average of forward and backward $\text{FEP}$ (see, e.g., ref 34):

$$\begin{gathered} \Delta \Delta G_{\text{FEP}}=\frac{1}{2}(\sum _{m=1}^{n-1}-kT\text{ln}{\langle \text{exp}\left(-\frac{E_{m+1}-E_m}{kT}\right)\rangle }_{E_m}\phantom{)} \\ -\sum _{m=2}^n-kT\text{ln}\phantom{(}{\langle \text{exp}\left(-\frac{E_{m-1}-E_m}{kT}\right)\rangle }_{E_m}) \end{gathered}$$ (10)

With the accurate results of the $\text{FEP}$ method, we can explore the validity of the end point $\text{LRA}$, which gives (see also eq 4) the relevant $\Delta \Delta G$ by¹⁵

$$\begin{gathered} \Delta \Delta G_{\text{LRA}}=\frac{1}{2}({\langle E_{\text{TARG}}-E_{\text{REF}}\rangle }_{E_{\text{REF}}} \\ +{\langle E_{\text{TARG}}-E_{\text{REF}}\rangle }_{E_{\text{TARG}}}) \end{gathered}$$ (11)

Furthermore, to demonstrate how the performance of the $\text{LRA}$ is affected by the difference between the two potentials, we also examined the $n$-points $\text{LRA}$ between the adjacent mapping potentials given by eq 9:

$$\Delta \Delta {G^n}_{\text{LRA}}=\sum _{m=1}^{n-1}\frac{1}{2}({\langle E_{m+1}-E_m\rangle }_{E_m}+{\langle E_{m+1}-E_m\rangle }_{E_{m+1}})$$ (12)

Obviously, with the large $n$, this expresion converges to the regular $\text{FEP}$.

In the present case, we start our examination of the validity of the $\text{LRA}$ treatment by evaluating the $\text{FEP}$ and $\text{LRA}$ perturbations between the $\text{PM}3∕\text{MM}$³⁸ and $\text{PM}6∕\text{MM}$³⁹ potentials of the MOPAC2009⁴⁰ package, with the correct treatment of the solute polarization^1,7 described in section II.3.3 (see also simulation 1 in the Supporting Information for additional computational details). This is done for an S_N2 reaction in water between methyl chloride and chloride (here $\text{PM}3∕\text{MM}$ is the $\text{RP}$, and $\text{PM}6∕\text{MM}$ is the $\text{TP}$).

For the $\text{FEP}$ treatment at the $\text{TS}$, we used (in addition to the $\text{QM}∕\text{MM}$ potential) the harmonic potential:

$$E_{\text{CONS}}=K{(\xi -{\xi }_0)}^2$$ (13)

where $\xi$ is the reaction coordinate defined as

$$\xi =R(C-C{l}_{\mathrm{A}})-R(C-C{l}_{\mathrm{B}})$$ (14)

Note that ${\xi }_0(x_{\text{PM}3}^{‡})={\xi }_0(x_{\text{PM6}}^{‡})=0$.

We also evaluated the free energy by the thermodynamic integration method (which is very similar to the regular $\text{FEP}$ approach), using

$$\begin{gathered} \Delta \Delta G_{\text{TDI}}={\int }_0^1\frac{\partial G}{\partial \lambda }\mathrm{d}\lambda \\ =\sum _{m=2}^n\frac{1}{2}({\langle E_{\text{TARG}}-E_{\text{REF}}\rangle }_{E_m} \\ +{\langle E_{\text{TARG}}-E_{\text{REF}}\rangle }_{E_{m-1}})\Delta \lambda \end{gathered}$$ (15)

The $\Delta \Delta G$ was evaluated for the difference between the potentials $E_{\text{REF}}$ and $E_{\text{TARG}}$, which correspond, respectively, to $E_{\text{PM}3∕\text{MM}}$ and $E_{\text{PM}6∕\text{MM}}$. These potentials were contained within the $\text{TS}$ region by

$$\begin{gathered} E_{\text{REF}}=E_{\text{PM}3∕\text{MM}}+100{(\xi -{\xi }_0{\mathsf{I}}_{{\xi }_0^{‡}=0})}^2 \\ {E}_{\text{TARG}}=E_{\text{PM}6∕\text{MM}}+100{(\xi -{\xi }_0{\mathsf{I}}_{{\xi }_0^{‡}=0})}^2 \end{gathered}$$ (16)

Similarly, for the $\text{RS}$, we have

$$\begin{gathered} E_{\text{REF}}=E_{\text{PM}3∕\text{MM}}+100{(\xi -{\xi }_0{\mathsf{I}}_{{\xi }_0^{\text{RS}}=-1.75})}^2 \\ {E}_{\text{TARG}}=E_{\text{PM}6∕\text{MM}}+100{(\xi -{\xi }_0{\mathsf{I}}_{{\xi }_0^{\text{RS}}=-1.75})}^2 \end{gathered}$$ (17)

The performance of the $\text{FEP}$ and $\text{LRA}$ calculations is compared in Figure 2, where we see remarkably good performance of the end-point $\text{LRA}$ treatment with nearly identical results to those of the full $\text{FEP}$ treatment. Note that Figure 2 gives estimates of the free energy difference between the $\text{RP}$ and the $\text{TP}$, which were modified by adding the constraining harmonic potentials in order to keep the trajectories at the specific $\text{RC}$ values ($\text{RS}$ and $\text{TS}$). Note, also, that the harmonic potentials used to obtain the sampling within the $\text{RS}$ and within the $\text{TS}$ have no bias effect at the $\text{RC}$ values (specified by the corresponding ${\xi }_0$) at which the $\text{FEP}$ was calculated, and a small effect in the vicinity of these points.

Figure 2.

Comparative performance of the $\text{LRA}$ approach and of a full multi-step $\text{FEP}$ treatment in estimating the free energy of moving from $E_{\text{PM}3∕\text{MM}}$ to $E_{\text{PM}6∕\text{MM}}$ at the $\text{TS}$ (A) and at the $\text{RS}$ (B). The estimates were obtained by: 2-point $\text{LRA}$ (black dashes); single-point $\text{LRA}$ with the average of the energy gap, $E_{\text{PM}6∕\text{MM}}-E_{\text{PM}3∕\text{MM}}$, calculated on $E_{\text{PM}3∕\text{MM}}$ (red dashes) and on $E_{\text{PM}6∕\text{MM}}$ (green dashes); forward $\text{FEP}$ (magenta squares); backward $\text{FEP}$ (cyan triangles); average $\text{FEP}$ (blue line); $n$-point $\text{LRA}$ (black triangles); TDI (orange stars). Note that $\text{FEP}$-based estimates are hardly distinguishable due to overlap (they are shown in insets).

Of course, a convincing validation requires us to show that the barrier calculated for the $\text{TP}$ using the PD approach is the same as the barrier obtained in a separate PMF calculation. In order to be able to compare the $\Delta \Delta g$ of the PD treatment to the corresponding difference between the full PMF calculations we have to take into account the fact that what we get from the PMF corresponds to the incomplete partition functions, $q’\mathrm{s}$, of eq 4, whereas the $\text{FEP}$ simulations described above sample the whole phase space. Thus, strictly speaking, in order to compare with the difference between the individual PMFs, we have to determine $\Delta \Delta g(\xi )$ through the difference between the free energy functions of the $\text{RC}$ (at its specific values with removing the effect of the constraining harmonic potential). This, in turn, can be compared to the $\Delta \Delta G$ estimates obtained by the $\text{FEP}$ treatment (eqs 8-15) between the potentials of eqs 16 and 17. This can be done with a treatment, which is a variant of the $\text{EVB}$ $\text{FEP}$/US mapping formulation.^11,12,41 Namely,

$$\begin{gathered} \Delta \Delta g_{\text{PM}3∕\text{MM}\to \text{PM}6∕\text{MM}}(\xi ) \\ =\Delta g_{\text{PM}6∕\text{MM}}(\xi )-\Delta g_{\text{PM}3∕\text{MM}}(\xi ) \end{gathered}$$ (18)

where the free energy functions are given by

$$\begin{gathered} \Delta g_{\text{PM}3∕\text{MM}}(\xi )=\Delta \Delta G_m \\ -kT\text{ln}{\langle \delta (x-\xi )\text{exp}\left(\frac{E_{\text{PM}3∕\text{MM}}-E_m}{kT}\right)\rangle }_{E_m} \end{gathered}$$ (19)

and

$$\begin{gathered} \Delta g_{\text{PM}6∕\text{MM}}(\xi )=\Delta \Delta G_m-kT \\ \text{ln}{\langle \delta (x-\xi )\text{exp}\left(\frac{E_{\text{PM}6∕\text{MM}}-E_m}{kT}\right)\rangle }_{E_m} \end{gathered}$$

where

$$E_m=(1-{\lambda }_m)E_{\text{PM}3∕\text{MM}}+{\lambda }_m{E}_{\text{PM}6∕\text{MM}}+E_{\text{CONS}}$$

In this expression, we deal with the change from $E_{\text{REF}}$ to $E_{\text{TARG}}$ (in the $\text{EVB}$ treatment, we go from $E_1$ to $E_2$) and $\Delta \Delta G_m$ is the result of moving from $E_{\text{PM}3∕\text{MM}}+E_{\text{CONS}}$ to $E_m$ by the $\text{FEP}$ procedure. The second term removes the effect of the constraining potential and selects the particular values of $\text{RC}$.

To demonstrate the point discussed above, we start by building a histogram for the MD trajectories to see the distribution of the energy gap between the two potentials, ($E_{\text{PM}3∕\text{MM}}-E_{\text{PM}6∕\text{MM}}$). As can be seen from Figure 3A, the distribution of the energy gap for the MD trajectories propagated during calculation of the free energy of moving from the $\text{RP}$ to the $\text{TP}$ at the $\text{RS}$ is centered at −3.5 kcal/mol. Similarly, the corresponding gap evaluated near the $\text{TS}$ is centered at 4 kcal/mol. These are the most probable differences between the two potentials constrained at the $\text{RS}$ and at the $\text{TS}$. Adding up these estimates gives an estimate of the total difference between the activation free energy barriers of the $\text{RP}$ and of the $\text{TP}$, which is 7.5 kcal/mol. These estimates are close to the values obtained by performing the $\text{FEP}$ between the two potentials (see Figure 2A and B).

Figure 3.

(A) The distribution of the energy gap, $E_{\text{PM}3∕\text{MM}}-E_{\text{PM}6∕\text{MM}}$, obtained from the MD trajectories runs used in $\text{FEP}$ calculations while moving from $E_{\text{PM}3∕\text{MM}}+E_{\text{CONS}}(\xi )$ to $E_{\text{PM}6∕\text{MM}}+E_{\text{CONS}}(\xi )$, where $\xi =d(C-C{l}_{\mathrm{L}})-d(C-C{l}_{\mathrm{A}})$ is the solute $\text{RC}$, in the $\text{RS}$ region and at the $\text{TS}$ regions (designated by red and green, respectively). (B) The sampling distribution of the nuclear $\text{RC}$, obtained from the MD runs used in $\text{FEP}$ calculation of moving from $E_{\text{PM}3∕\text{MM}}+E_{\text{CONS}}(\xi )$ to $E_{\text{PM}6∕\text{MM}}+E_{\text{CONS}}(\xi )$ in the $\text{RS}$ and $\text{TS}$ regions (which are designated by red and green, respectively).

Further examination of the distribution of the nuclear $\text{RC}$ is given in Figure 3B. The figure shows that the harmonic potentials effectively narrowed the sampling to the corresponding target values of the $\text{RC}$. Thus, one can evaluate the free energy perturbations at any value of the $\text{RC}$ specified by the ${\xi }_0$ parameter of the harmonic potential. While the ${\xi }_0$ parameter specifies the $\text{RC}$ value at which the free energy perturbation is estimated, the force constant controls the width of the $\text{RC}$ distribution. Note, however, that all other degrees of freedom are sampled. We also found that, while increasing the force constant further narrowed the distribution in the $\text{RC}$, it had very little effect on the distribution of the energy gap and almost no effect on the $\text{FEP}$ estimate.

At this stage, we estimated the free energy functions of the $\text{RC}$ at the $\text{RS}$ and at the $\text{TS}$ regions, $\Delta g_{\text{PM}3∕\text{MM}}(\xi )$ and $\Delta g_{\text{PM}6∕\text{MM}}(\xi )$, using eq 19 (note that this treatment removes the bias potential of eqs 16 and 17). The resulting differences between the $\text{PM}6∕\text{MM}$ and $\text{PM}3∕\text{MM}$ free energy functions, $\Delta \Delta g(\xi )$, are shown in Figure 4, e.g., ~−4 kcal/mol at the $\text{TS}$ ($\xi =0$) and ~3.5 kcal/mol at the $\text{RS}$ ($\xi =-1.75$). Thus, we demonstrated that the estimates obtained by the full $\text{FEP}$ and by the $\text{LRA}$ evaluation of the free energy of moving from the $\text{RP}$ to the $\text{TP}$ (which are modified by the harmonic potentials to keep the sampling within the $\text{RS}$ and the $\text{TS}$ regions) are in excellent agreement with the difference between the free energy functions at the corresponding $\text{RC}$ values (obtained by removing the bias potential using eq 19).

Figure 4.

The free energy functions of the solute $\text{RC}$, $\xi$, obtained at the $\text{RC}$ values corresponding to the $\text{TS}$ (A) and to the $\text{RS}$ (B), for the full mapping between $\text{PM}3∕\text{MM}$ and $\text{PM}6∕\text{MM}$. The blue line designates $\Delta g_{\text{PM}6∕\text{MM}}$ and the red line designates $\Delta g_{\text{PM}3∕\text{MM}}$, obtained as weighted averages over all mapping potentials, blue and red dots, respectively.

Finally, we compared in Figure 5 our careful evaluation to the difference between the PMFs calculated for $\text{PM}3∕\text{MM}$ and $\text{PM}6∕\text{MM}$ potentials. Each PMF was obtained (see also simulation 2 in the Supporting Information) by introducing a harmonic constraint at different $\text{RC}$ values, ${\xi }_0$, along the reaction path, and combining the simulation windows using the WHAM equations.^42-44 Inspecting the PMFs, one finds that the difference between the activation free energy barriers is about 7.5 kcal/mol, which is in perfect agreement with the results obtained by the $\text{FEP}∕\text{LRA}$ treatment (eqs 8-15) and by the $\text{FEP}$/US approach (eqs 18 and 19).

Figure 5.

The PMF obtained for the $\text{PM}3∕\text{MM}$ potential with the solute charges derived by the Mulliken analysis (red) and by the ESP-fitting (blue) as well as for $\text{PM}6∕\text{MM}$ potential with the Mulliken solute charges (green). The PMFs have been calculated by the WHAM procedure.

In summary, in the above section, we demonstrated by using two MO-type $\text{QM}∕\text{MM}$ potentials that the $\text{LRA}$ approach of eq 4 provides a very good approximation for the full $\text{FEP}$/US treatment while giving significant time savings compared to the full $\text{FEP}$ treatment. Also note that the $\text{LRA}$ approach performs remarkably well when the averages are calculated on both potentials. Finally, by evaluating both the PMFs for the $\text{TP}$ and for the $\text{RP}$ and the free energy perturbation from the $\text{RP}$ to the $\text{TP}$ at the $\text{TS}$ and the $\text{RS}$, the consistency of our approach is established.

II.2.2. Using the $EVB$ as a Reference Potential.

In this section, we will expand on the $\text{FEP-LRA}$ evaluation of $\Delta \Delta G$ for moving from the $\text{EVB}$ $\text{RP}$ to a $\text{QM}∕\text{MM}$ potential. In this case, it is useful to comment here about the evaluation of eq 4 at particular values of the $\text{RC}$, when one uses $\text{EVB}$ as the $\text{RP}$. Now, as we have shown in the previous example, the harmonic constraint on the solute $\text{RC}$ is quite efficient for estimating the $\text{FEP}$ at a particular $\text{RC}$ value for an MO-based $\text{QM}∕\text{MM}$ potential (where it is sufficient to have a constraint with $K=100\text{kcal}∕({\mathrm{A}}^2\cdot \text{mol})$ to stay at the specified $\text{RC}$). For the $\text{EVB}$ potential, we found that a more effective way to keep $\text{EVB}$ at its $\text{TS}$ (where $E_1=E_2$) involves using the approximation¹¹

$$E_{\text{EVB}}(x_{\text{EVB}}^{‡})\approx 0.5E_1+0.5E_2-H_{12}$$ (20)

where we use the fact that eq 20 is equal to the adiabatic $\text{EVB}$ at the $\text{TS}$:

$$E_{\text{EVB}}={c_1}^2{E}_1+{c_2}^2{E}_2-2c_1{c}_2{H}_{12}$$ (21)

The use of this expression is equivalent to constraining the eigenvector components to be equal (so that both of the $\text{EVB}$ diabatic states equally contribute to the adiabatic state).

Here we estimate the $\text{FEP}$ between the $\text{EVB}$ and MO-based $\text{PM}3∕\text{MM}$ potentials (see also simulation 3 in the Supporting Information) at the $\text{TS}$ using the potentials:

$$\begin{gathered} E_{\text{REF}}=E_{\text{EVB}}(x_{\text{EVB}}^{‡})+100{(\xi -{\xi }_0{\mathsf{I}}_{{\xi }_0=0})}^2\text{and} \\ {E}_{\text{TARG}}=E_{\text{PM}3∕\text{MM}}+100{(\xi -{\xi }_0{\mathsf{I}}_{{\xi }_0=0})}^2 \end{gathered}$$ (22)

and at the $\text{RS}$ using

$$\begin{gathered} E_{\text{REF}}=E_{\text{EVB}}+100{(\xi -{\xi }_0{\mathsf{I}}_{{\xi }_0=-1.5})}^2\text{and} \\ {E}_{\text{TARG}}=E_{\text{PM}3∕\text{MM}}+100{(\xi -{\xi }_0{\mathsf{I}}_{{\xi }_0=-1.5})}^2 \end{gathered}$$

The estimate of $\Delta \Delta G$ by the end-point $\text{LRA}$ approach was found to provide again a very good approximation to the full $\text{FEP}$ (see Figure 6). However, we see the importance of averaging on both the $\text{RP}$ as well as on the $\text{TP}$.⁴⁵ The $\text{FEPs}$ at the $\text{TS}$ and $\text{RS}$ are, respectively, ~2 and ~6 kcal/mol; thus, the activation free energy barrier for $\text{EVB}$ is ~4 kcal/mol higher than the $\text{PM}3∕\text{MM}$ activation free energy barrier based on this estimate.

Figure 6.

Comparative performance of the $\text{LRA}$ approach and of a full multi-step $\text{FEP}$ treatment in estimating the free energy of moving from $E_{\text{EVB}}$ to $E_{\text{PM}3∕\text{MM}}$ at the $\text{TS}$ (A) and at the $\text{RS}$ (B). The estimates were obtained by: 2-point $\text{LRA}$ (black dashes); single-point $\text{LRA}$ with the average of the energy gap, $E_{\text{PM}3∕\text{MM}}-E_{\text{EVB}}$, calculated on $E_{\text{EVB}}$ (red dashes) and on $E_{\text{PM}3∕\text{MM}}$ (green dashes); forward $\text{FEP}$ (magenta squares); backward $\text{FEP}$ (cyan triangles); average $\text{FEP}$ (blue line); $n$-point $\text{LRA}$ (black triangles); TDI (orange stars). Note that $\text{FEP}$-based estimates are hardly distinguishable due to overlap (they are shown in insets).

In the next step, we constructed (see Figure 7) the histograms of the energy gap $E_{\text{EVB}}-E_{\text{PM}3∕\text{MM}}$ for the $\text{FEP}$ calculations at the $\text{RS}$ and at the $\text{TS}$. The histograms show that the most probable difference between the $\text{TP}$ and the $\text{RP}$, $E_{\text{PM}3∕\text{MM}}-E_{\text{EVB}}$, is ~7 kcal/mol at the $\text{RS}$ and ~2 at the $\text{TS}$, which practically coincides with the full $\text{FEP}$ estimates and with $\text{LRA}$.

Figure 7.

The distribution of the energy gap between the $\text{EVB}$ potential and the $\text{PM}3∕\text{MM}$ potential, obtained while performing $\text{FEP}$ calculations at the $\text{RS}$ (green) and the distribution of the gap between the $\text{EVB}$ mapping potential and $\text{PM}3∕\text{MM}$ while doing $\text{FEP}$ at the $\text{TS}$ (red).

$$\begin{gathered} \Delta g_{\text{PM}3∕\text{MM}}(\xi )=\Delta \Delta G_m-kT \\ \text{ln}{\langle \delta (x-\xi )\text{exp}\left(\frac{E_{\text{PM}3∕\text{MM}}-E_m}{kT}\right)\rangle }_{E_m} \end{gathered}$$ (23)

$$\begin{gathered} \Delta g_{\text{EVB}}(\xi )=\Delta \Delta G_m \\ -kT\text{ln}{\langle \delta (x-\xi )\text{exp}\left(\frac{E_{\text{EVB}}-E_m}{kT}\right)\rangle }_{E_m} \end{gathered}$$

Next, using eq 23, we find that the difference between the free energy functions at the $\text{TS}$, $\xi =0$, is ~4.5 kcal/mol and at the $\text{RS}(\xi =-1.5)$ is ~5.5 kcal/mol (see Figure 8). While this is close to the value found from the $\text{FEP}$ calculation at the $\text{RS}$, the difference between the free energy functions at the $\text{TS}$ is ~3 kcal/mol higher than the corresponding $\text{FEP}$ estimate. This is probably due to use of the potential given by eq 20 in the $\text{FEP}$ calculation, since if we construct the free energy function along the $\text{RC}$ for the approximate $\text{EVB}$ potential at the $\text{TS}$ (see also Figure 9):

$$\begin{gathered} \Delta g_{\text{EVB}(x_{\text{EVB}}^{‡})}(\xi )=\Delta \Delta G_m \\ -kT\text{ln}{\langle \delta (x-\xi )\text{exp}\left(\frac{E_{\text{EVB}(x_{\text{EVB}}^{‡})}-E_m}{kT}\right)\rangle }_{E_m} \end{gathered}$$ (24)

Figure 8.

The free energy functions along the reaction coordinate $\xi =d(C-C{l}_{\mathrm{L}})-d(C-C{l}_{\mathrm{A}})$ obtained by $\text{FEP}$ calculations of the free energy for moving from the $\text{EVB}$ potential to the $\text{PM}3∕\text{MM}$ in water at the $\text{TS}$ (A) and at the $\text{RS}$ (B). The blue line designates $\Delta g_{\text{EVB}}$ (weight-averaged over the blue points), while the red line designates $\Delta g_{\text{PM}3∕\text{MM}}$ (weight-averaged over red points).

Figure 9.

The free energy functions along the reaction coordinate obtained by $\text{FEP}$ calculations of the free energy for moving from the $\text{EVB}$ mapping potential to $\text{PM}3∕\text{MM}$ in water at the $\text{TS}$. The blue line designates the $\text{EVB}$ function (weight-averaged over blue points), while the red line designates the $\text{PM}3∕\text{MM}$ function (weight-averaged over red points).

The difference between the free energy functions at the $\text{TS}$ is about 3 kcal/mol, which is close to the $\text{FEP}$ estimate of 2 kcal/mol. In any case, the correction to the $\text{PM}3∕\text{MM}$ free energy barrier based on eq 23 is about −1 kcal/mol, while on the basis of the $\text{FEP}∕\text{LRA}$ (with the approximation of $\text{EVB}$ potential at the $\text{TS}$ by eq 24) it is −4 kcal/mol.

The calculated $\text{EVB}$ free energy profile and the $\text{PM}3∕\text{MM}$ PMF are given in Figure 10. The $\text{EVB}$ free energy barrier is found to be ~1.5 kcal/mol higher than the $\text{PM}3∕\text{MM}$ free energy barrier, and this is in a perfect agreement with the results calculated in the PD model using $\text{FEP}$/US of eq 23. Alternatively, if the $\text{EVB}$ free energy profile vs the $\text{EVB}$ energy gap is taken as the reference free energy surface, for which the activation free energy barrier is ~3.5 kcal/mol higher than the barrier on the $\text{PM}3∕\text{MM}$ PMF, a good agreement with the corresponding $\text{FEP}∕\text{LRA}$ estimate of 4 kcal/mol is achieved.

Figure 10.

(A) The free energy profile along the nuclear $\text{RC}$ obtained by the $\text{FEP}$/US approach using the $\text{EVB}$ reference potential (blue dots) and PMF for the $\text{PM}3∕\text{MM}$ target potential (red line); (B) $\text{EVB}$ $\text{FEP}$/US free energy profile along the energy gap between the $\text{EVB}$ diabatic states.

The above approach evaluates only the vertical free energy changes, and thus when the $\text{TS}$ coordinates on the $\text{RP}$ and on the $\text{TP}$ are different, we have to evaluate the PMFs on both surfaces at the $\text{TS}$ region (see ref 16 for the full discussion of this case). More specifically, if the $\text{RC}$ for the $\text{QM TS}$ is known (e.g., from the gas phase or from the implicit solvent model minimization), we can evaluate the vertical transition from the $\text{QM TS}$ on the $\text{RP}$ and then use the PMF (again on the $\text{RP}$) to obtain the free energy of moving from the $\text{QM TS}$ on the $\text{EVB}$ potential to the actual $\text{EVB}$ $\text{TS}$ state. In this case, the PMF on $\text{EVB}$ can be estimated using the $\text{FEP}$/US,⁶ method as is done in Figure 10. If we do not know the exact $\text{QM TS}$, then PMF on the $\text{TP}$ is required. This can be done by one of the approaches described below only in the $\text{TS}$ region, which will still give significant time savings compared to the PMF in the full range of the $\text{RC}$. In general, there is no need to use any constraints while performing perturbation at the $\text{RS}$, since the corresponding $\text{RSs}$ are effectively sampled when the trajectories are propagated on both the $\text{RP}$ and the $\text{TP}$.

II.3. PD Refinement of the Reference Potential by Modifying Its Functional form.

II.3.1. Refining the Intramolecular Potential in the Gas Phase Using Gaussians.

In considering the PD refinement, we noted that it is possible to take advantage of the fact that evaluating the PES in the gas phase for chemical reactions is a relatively trivial operation, which is routinely performed nowadays. This makes PES an easily available source of data for our refinement procedure.

With the gas phase PES, we can simply fit the $\text{TP}$ and the $\text{RP}$ (e.g., $\text{EVB}$ potential) by a set of Gaussian functions. For instance, for the $\text{TP}$, we minimize the least-squares function, using the optimal steepest descent with analytical derivatives, with respect to the parameters of the Gaussians:

$$𝒯(\{{\alpha }_k\},\{A_k\})=\sum _{i=1}^N{({\Gamma }_{\text{TARG}}({\mathbf{r}}_i)-E_{\text{TARG}}({\mathbf{r}}_i))}^2$$ (25)

The result of this fitting is a function, $\Gamma$, which approximates the $\text{TP}$ in the range of the $\text{RC}$ where the PES was performed:

$${\Gamma }_{\text{TARG}}=\sum _k{A}_k\text{exp}(-{\alpha }_k{(x-{\xi }_k)}^2)$$ (26)

It was found that a combination of 3 Gaussians was sufficient to fit the PES for $\text{PM}3$, $\text{PM}6$, and $\text{EVB}$ (see Figure 11 and Table 1). For the details of the fitting algorithm, see also the Supporting Information. This procedure can be done for both the $\text{RP}$ and the $\text{TP}$. In the next step, we use the obtained $\Gamma$-functions to refine the original $\text{RP}$ (e.g., the $\text{EVB}$ $\text{RP}$) using:

$$E_{\text{EVB}}^{\prime }=E_{\text{EVB}}-{\Gamma }_{\text{EVB}}+{\Gamma }_{\text{QM}}$$ (27)

Figure 11.

Fitting the $\Gamma$-functions. (A) ${\Gamma }_{\text{EVB}}$ (green line) fitted to the $\text{EVB}$ gas-phase PMF/WHAM (red triangles). The $\text{EVB}$ potential used in the simulation (blue squares) was modified by removing the effect of the $\text{EVB}$ distance constraints for the $\text{RC}$ values outside the range $-1>\xi >1$. (B) $\text{PM}3$ and $\text{PM}6$ gas phase energy scans from MOPAC2009 (red stars and green triangles) and the corresponding $\Gamma$-functions.

Table 1.

Parameters for the $\Gamma$-Functions Presented in Figure 11 Composed of Three Gaussians: ${\mathit{A}}_{\mathit{i}}\mathbf{exp}(-{\alpha }_{\mathit{i}}{(\xi -{\xi }_{\mathit{i}})}^2)$

	A1, A3	A2 (kcal/mol)	${\alpha }_1$, ${\alpha }_3$	${\alpha }_2$ (A−1)	${\xi }_1({\xi }_3)$	${\xi }_2$ (A)
$\text{EVB}$	−3.92	6.98	1.58	3.66	−1.1 (1.1)	0.0
$\text{PM}3$	−4.39	6.80	2.40	2.22	−1.0 (1.0)	0.0
$\text{PM}6$	−5.25	3.35	2.89	3.70	−1.0 (1.0)	0.0
$\text{PM}{3}_{\text{cosmo}}$	−0.18	20.85	0.63	2.47	−1.2 (1.2)	0.0
$\text{PM}{6}_{\text{cosmo}}$	−1.02	16.58	1.34	2.16	−1.1 (1.1)	0.0

This refinement procedure is aimed to bring the $\text{RP}$ close to the $\text{TP}$ in a given range of the $\text{RC}$, since:

$$E_{\text{TARGET}}\approx E_{\text{REF}}^{\prime }=E_{\text{REF}}-{\Gamma }_{\text{REF}}+{\Gamma }_{\text{TARG}}$$ (28)

Now we have the refined $\text{RP}$ that provides an improved approximation for the $\text{TP}$ which is essential for good convergence of the $\text{LRA}$. This approach is particularly useful when the $\text{EVB}$ functional form is sophisticated and the refinement of its parameters is lengthy and tedious.

As a quick demonstration of the described above refinement procedure, we consider an example with an $\text{EVB}$ potential serving as the $\text{RP}$ and the $\text{PM}6$ potential serving as the $\text{TP}$ (see also simulation 4 in the Supporting Information). Figure 12 illustrates the effect of the $\Gamma$-correction refinement (using eq 27) of the original $\text{EVB}$ $\text{RP}$ on the reference free energy surface. As seen from the figure, the calculated PMF on the original $\text{EVB}$ $\text{RP}$ is 10.5 kcal/mol, whereas the PMF on the refined $\text{EVB}$ $\text{RP}$ is 8.8 kcal/mol, which is in excellent agreement with the activation free energy barrier of 8.9 kcal/mol for $\text{PM}6$ (calculated using the harmonic approximation for the entropic contribution). As can be seen, this relatively simple functional form is quite effective in reducing the difference between reference FES and the target FES.

Figure 12.

The gas-phase $\text{EVB}$ PMF obtained by using the WHAM approach and the harmonic constraints on different values of $\text{RC}$. The red line is for PMF on the original $\text{EVB}$ reference potential, while the green line is for the PMF obtained on the refined $\text{EVB}$ reference potential ($E_{\text{EVB}}-{\Gamma }_{\text{EVB}}+{\Gamma }_{\text{PM}6}$). The black stars are the free energy points calculated on $\text{PM}6$ (target potential) using the harmonic approximation for activation entropy.

II.3.2. Application of the Gamma-Correction in PMF Calculations: Derivation.

Obviously the main application of $\text{QM}∕\text{MM}$ calculations is in studies of reactions in condensed phases. However, prior to extension of the refinement approach proposed in section II.3.1 to the condensed phases, we would like to report several practical applications of this approach to calculating the PMF, which were found to be quite useful and can be also applied to condensed phase PMF calculations.

Since PMF calculations can be quite expensive, it is important to look for ways to optimize the corresponding calculations. Here we tried to exploit an element of the MTD approach (although, as will be clarified in the concluding discussion, with an entirely different philosophy). That is, as pointed out by van Gunsteren and co-workers³² and as implemented elegantly in the MTD³³ approach, flattening the original potential by iteratively adding to it the negative of the original potential, approximated by a sum of Gaussians, improves the convergence of free energy sampling. Implementation of this strategy in the MTD approach results in a high computational cost of building the negative potential; see the detailed discussion in our recent paper.⁹ However, here we bypassed this cost by using the idea of the PD refinement described in section II.3.1. Namely, we derive the negative potential by fitting the corresponding PES in the gas phase by a sum of Gaussians using eqs 25 and 26. Thus, in addition to their use in the PD refinement, the $\Gamma$-functions of eq 26 help in convergence of the PMF calculations. This is achieved by flattening the original potential by adding to it the negative of the corresponding $\Gamma$-function. That is, following the $\text{FEP}$/US approach to PMF,³⁴ we construct the mapping potentials of the form

$$E_m=(1-{\lambda }_m)E_{\text{RS}}+{\lambda }_m{E}_{\text{PS}}$$ (29)

where

$$\begin{gathered} E_{\text{RS}}=E_{\text{QM}}-{\Gamma }_{\text{QM}}+E_{\text{CONS}}({\xi }_{\text{RS}}) \\ {E}_{\text{PS}}=E_{\text{QM}}-{\Gamma }_{\text{QM}}+E_{\text{CONS}}({\xi }_{\text{PS}}) \end{gathered}$$ (30)

where one can, for instance, use $E_{\text{CONS}}$ of eq 13.

To obtain the PMF, we start with the $\text{FEP}$ approach for the corresponding change in free energy (using eqs 8 and 10). Then, the PMF is evaluated by using a modification of eq 19 ($\text{FEP}$/US approach):

$$\begin{gathered} \Delta g(\xi )=\Delta G_m \\ -kT\text{ln}{\langle \delta (x-\xi )\text{exp}\left(-\frac{E_{\text{TARG}}(x)-E_m(x)}{kT}\right)\rangle }_{E_m} \end{gathered}$$ (31)

Furthermore, the results from different simulation windows are combined by

$$\Delta \overline{g}(\xi )=\sum _{\text{frames}}\frac{N_i(\xi )}{{\sum }_{\text{frames}}{N}_i(\xi )}\Delta g_i(\xi )$$ (32)

where $N_i(\xi )$ is the number of times MD visited a particular $\text{RC}$ value, $\xi$, while propagating on the $i\text{th}$ mapping potential. The practical application of this approach to PMF calculations, when the solvent is modeled using an implicit solvent model, will be illustrated below.

II.3.3. Application of the Gamma-Correction in PMF Calculations: Practical Examples for Condensed Phase Calculations.

When moving to a condensed phase, we should take into account the fact that different $\text{QM}(\text{ai})∕\text{MM}$ potentials will have different charge distributions for the solute, and that this has to be reflected in the $\text{RP}$. A straightforward application of the $\Gamma$-function approach (described in section II.3.2) to condensed phases is to evaluate the PES with a solvent model, and subsequently derive the corresponding $\Gamma$-function that reflects the solvation effects. This strategy is obviously impractical when one deals with explicit solvent models, which are extremely expensive for $\text{QM}(\text{ai})∕\text{MM}$ studies. However, performing reasonable minimization is not expensive when one uses an implicit solvent model.

To illustrate this approach, as a first step, we examined the option of using the $\Gamma$-function which captures the solvation effect by the $\text{COSMO}$ model.⁴⁶ As a test system, we took the same S_N2 reaction described above and fitted the ${\Gamma }_{\text{COSMO}}$ function to the PES obtained using MOPAC2009⁴⁰ with $\text{PM}3+\text{COSMO}$ combination (see Figure 13). The free energy profile, $\Delta g_{\text{PM}3,\text{COSMO}}$, was obtained by the approach described in section II.3.2 with the MD simulation run on the flat potential, $E_{\text{PM}3,\text{COSMO}}-{\Gamma }_{\text{COSMO}}$. The high efficiency of sampling at the $\text{TS}$ region and its uniform distribution along the studied range of the $\text{RC}$ is shown in Figure 14.

Figure 13.

The PMFs obtained for the $\text{PM}3$ potential with explicit (green) and implicit $\text{COSMO}$ (red) solvation models. The PMFs were obtained by adding to the $\text{PM}3(\text{COSMO})$ and to the $\text{PM}3∕\text{MM}$ potentials the negative of ${\Gamma }_{\text{COSMO}}$. The figure also depicts the ${\Gamma }_{\text{COSMO}}$ (blue line) function, derived from PES performed on the $\text{PM}3$ potential with the $\text{COSMO}$ solvation model (blue triangles).

Figure 14.

The distribution of the nuclear $\text{RC}$ during MD simulation on the potential $E_{\text{PM}3∕\text{MM}}-{\Gamma }_{\text{COSMO}}$ (red) and on the potential $E_{\text{PM}3+\text{COSMO}}-{\Gamma }_{\text{COSMO}}$ (green). The histogram for $E_{\text{PM}3∕\text{MM}}-{\Gamma }_{\text{COSMO}}$ shows a poor sampling at the $\text{TS}$ for the explicit model, since ${\Gamma }_{\text{COSMO}}$ does not make the explicit potential fully flat.

The proposed approach provides an effective and powerful way that outperforms the recent attempts to obtain the FES of solute in implicit solvent models.⁴⁷ In particular, the $\text{FEP}$/US PMF allows movement along the specified reaction path in the most efficient way as far as the sampling is concerned (since the $\text{FEP}$ increments are chosen in terms of the energy difference between the two adjacent states which is what essentially matters the most for the fastest free energy convergence). The representative $\Gamma$-function flattens the original potential and within a given simulation window, and thus, it naturally allows for proper sampling in all directions perpendicular to the $\text{RC}$.

Unfortunately, the implicit solvent models have difficulties in capturing changes of the solute cavity during chemical reaction (even with a reasonable calibration) and can miss other microscopic features of solvent. Thus, we may have significant errors in estimates of the $\text{TS}$ solvation (see Table 2 and simulation 5 in the Supporting Information). Furthermore, the implicit models cannot capture the microscopic physics of protein interiors.

Table 2.

Comparing the Solvation Energies for the $\mathbf{COSMO}$ and for the Explicit SCAAS Models

	$\Delta G_{\text{AC}}(0\to Q_{\text{gas}})$, kcal/mol	$\Delta H_{\mathrm{f},\text{COSMO}}-\Delta H_{\mathrm{f},\text{GAS}}$, kcal/mol
${\mathrm{X}}_{\text{TS}}\text{PM}3$ (fixed)	−52.1	−61.1
${\mathrm{X}}_{\text{RS}}\text{PM}3$ (fixed)	−67.8	−70.4
$\Delta \Delta G_{\text{solv}}$	15.7	9.3

In an attempt to overcome the limitations of implicit solvent models, we considered the performance of the proposed approach for the explicit solvent models. We obtained the PMF with the explicit solvent model, $\Delta g_{\text{PM}3∕\text{MM}}$, (using the ESP-derived $\text{QM}$ charges) by running on the potential

$$\begin{gathered} E_m=\langle {\Psi }_{\text{polar}}\mid H_{\text{PM}3∕\text{MM}}^{\prime }\mid {\Psi }_{\text{polar}}\rangle -{\Gamma }_{\text{PM}3,\text{COSMO}} \\ +(1-{\lambda }_m)K{(\xi -{\xi }_{\text{RS}})}^2+{\lambda }_{m}K{(\xi -{\xi }_{\text{PS}})}^2 \end{gathered}$$ (33)

with the polarized $\text{PM}3∕\text{MM}$ Hamiltonian

$$h_{\text{PM}3∕\text{MM}}^{\prime }(i)=h_{\text{PM}3}(i)+\sum _{\text{MM atoms}}\frac{q_j}{r_{ij}}$$ (34)

which we have implemented in MOPAC2009, following our earlier work⁷ (see also simulation 6 in the Supporting Information). While the resulting PMF (see Figure 13) is in excellent agreement with the PMF separately obtained by the WHAM approach (see simulation 2 in the Supporting Information), the efficiency of sampling on the potential given by eq 33 at the $\text{TS}$ (shown in Figure 14) is very low. This can be explained by comparing the barriers in the explicit and the implicit solvation models. The addition of $-{\Gamma }_{\text{COSMO}}$ does not make the $\text{TS}$ region on the explicit potential fully flat due to the differences in the $\text{TS}$ solvation shown in Table 2.

The PMF on the $\text{PM}3+\text{COSMO}$ potential resulted in a lower free energy barrier than that obtained by the energy scan. Furthermore, with the explicit solvent model, we also obtained a higher free energy barrier. To explain the difference in the activation free energy between different solvent models, we calculated the solvation free energies for the gas-phase optimized $\text{PM}3\text{RS}$ and $\text{TS}$ using MOPAC2009 with the $\text{COSMO}$⁴⁶ model and by the adiabatic charging model.³⁵ From the results given in Table 2, one can see that there is a difference in solvation of the $\text{RS}$ and of the $\text{TS}$ between the explicit and implicit models, with the highest disagreement between the implicit and explicit solvent models found for the $\text{TS}$ solvation. Essentially for this S_N2 reaction, $\Delta \Delta G_{\text{SOLV}}$ is a rough approximation of the contribution to the activation free energy barrier due to the solvation difference (if we add the gas phase barrier of ~12 kcal/mol, we get roughly the $\text{PM}3∕\text{MM}$ barrier given in Figure 13). The deficiencies of the barrier obtained by this specific $\text{COSMO}$ implementation are beyond the scope of this work, since different implicit models can give different results and it is always important to compare the solvation estimates by the different models (see, e.g., the careful study of ref 48). In general, any attempts to obtain a reliable $\text{QM}∕\text{MM}$ surface should involve careful calibration of the model on the observed solvation energies (as we have been doing with the $\text{EVB}$ model for a very long time), but this is not the issue addressed in the present work.

In section II.3.3, we showed that the approach of section II.3.2 provides an extremely powerful way of evaluating the PMF with an implicit solvent model. This is significant, since the implicit solvent models are frequently used in studies of reference solution reactions, and such studies require a very tedious manual mapping in order to obtain reliable estimates.⁴⁹ Now we have provided a systematic and effective way of obtaining the relevant free energy surfaces.

Mapping with an implicit solvent model does not present major problems (except missing the nonequilibrium solvation effect⁴⁵ which would lead to underestimating the implicit solvation barrier) as well as having other well-known disadvantages of continuum models. However, in section II.3.3, we also saw that evaluation of the PMF becomes much more challenging when we deal with explicit solvent models. Here the selection of the proper mapping potential is not trivial, as we have to force the combined solute–solvent coordinate to respond to the change in the solute charge distributions. Here we can take advantage of the fact that any $\text{RP}$-based $\text{QM}∕\text{MM}$ calculation in polar environments (water or enzymes), and, in particular, in the approaches that use semiempirical MO-based $\text{QM}∕\text{MM}$ potentials as the $\text{RP}$,^19,25 can be refined using the $\Gamma$-correction of eq 28. This correction can be derived, for instance, by eqs 25 and 26 with an implicit solvent model for the original $\text{RP}$ and for the $\text{TP}$ (e.g., $\text{COSMO}$ for which the analytical gradients are available), using:

$$E_{\text{REF}}^{\prime }=E_{\text{REF}}-{\Gamma }_{\text{REF},\text{IMPL}}+{\Gamma }_{\text{TARG},\text{IMPL}}$$ (35)

Even though evaluating the PES with the implicit solvent is relatively expensive, it describes more consistently the solute changes along the reaction path in the condensed phases than the corresponding gas phase PES, since the solute polarization is captured in a physically more reasonable way by the implicit solvent models.

In view of the length of section II.3, we summarize below what was done in this section. Section II.3.1 showed how to refine the $\text{RP}$ using the $\Gamma$-correction. Section II.3.2 introduced another practical use of the $\Gamma$-correction for improving the sampling efficiency while evaluating PMF. Section II.3.3 demonstrated a successful straightforward application with an implicit solvent model and less efficient (but possible if agreement between the implicit and explicit solvation models is improved) with the explicit solvent. Moreover, eq 35 provides a practical recipe for refinement of $\text{RP}$ in the condensed phase described in section II.3.1.

II.4. Extension to the Condensed Phase Using the $\mathbf{EVB}$-Type Solvent Potential.

In this section, we review application of the methods described in section II.3 when they are combined with an idea of using the $\text{EVB}$-type solvent driving potential. This allows for implementation of eq 30 in a practical way when the difference between the implicit and explicit models is significant or when it is desirable to have transferable $\Gamma$-functions (e.g., from the gas phase to water or from water to protein). These functions can be easily derived in the gas phase (or with an implicit solvent model) and subsequently applied to calculations in condensed phases without reparameterization. The $\Gamma$-correction, fitted to the gas phase calculations, contains only information about changes in the solute along the reaction path, namely, the information about the intramolecular part of the potential. To incorporate the effect of solvent and still take advantage of the gas-phase derived $\Gamma$-function, we consider a mapping potential where the $\text{QM}$ and $\text{MM}$ regions are coupled classically by

$$\begin{gathered} E_{\text{map}}=E_{\text{PM}3,\text{gas}}-{\Gamma }_{\text{PM}3,\text{gas}}+(1-\lambda )(E_{\text{CONS}}({\xi }_{\text{RS}}) \\ +E_{\text{sS}}({\mathbf{Q}}_{\text{RS}}))+\lambda (E_{\text{CONS}}({\xi }_{\text{PS}})+E_{\text{sS}}({\mathbf{Q}}_{\text{PS}})) \end{gathered}$$ (36)

where ${\mathbf{Q}}_{\text{RS}}$ and ${\mathbf{Q}}_{\text{PS}}$ are the vectors of the $\text{QM}$ solvated charges at the $\text{RS}$ and at the PS.

$$E_{\text{sS}}({\mathbf{Q}}_k)=\sum _{i=1}^{\text{QM}}\sum _{j=1}^{\text{MM}}\left(\frac{{Q_i}^{(k)}{q}_j}{r_{ij}}+\frac{A_{ij}}{r_{ij}^{12}}-\frac{B_{ij}}{r_{ij}^6}\right)$$ (37)

Equation 37 contains the $\text{EVB}$-type solvent driving potential, which polarizes the solvent in the correct direction toward the product state by the solvated solute charges, and captures the nonequilibrium solvent effect. However, this treatment should be applied with care to S_N1 and other charge separation reactions¹¹ (since essentially the $\text{QM}$ region is described by the gas phase $\text{QM}$ potential, which is perturbed by the $\text{MM}$ force field without the correct solute polarization). This mapping potential allows separating the $\text{QM}$ and $\text{MM}$ parts for the straightforward application of the gas-phase $\Gamma$-functions, while addressing the sampling problems separately (namely, requiring that the intra $\text{QM}$ barrier is flattened with the gas-phase $\Gamma$-function and the solvent is pulled toward the $\text{TS}$ by the $\text{EVB}$-type potential).

A correct treatment of the solute polarization in the case of a MO-based $\text{QM}∕\text{MM}$ potential is obtained by:

$$E_{\text{QM}∕\text{MM}}=\langle {\Psi }_{\text{polar}}\mid H_{\text{QM}∕\text{MM}}^{\prime }\mid {\Psi }_{\text{polar}}\rangle +E_{\text{VdW}}$$ (38)

Note that this treatment was used for most of the calculations in this work.

At this stage, it is interesting to see whether we can extend the approach of eq 28 to simulations of processes in polar environments. As was mentioned above, the gas phase $\Gamma$-function only contains information about the intramolecular contributions and says nothing about the intermolecular interactions with solvent. Equation 35 with the $\Gamma$-function obtained with an implicit solvent can provide a higher-level approximation, but this requires further examination. In particular, it is interesting to examine whether such a correction (obtained by fitting the PES performed with an implicit solvent model) can be directly used to refine the $\text{RP}$ for enzymatic simulation. Similarly, application of the $\text{EVB}$-type solvent driving potential combined with eq 33 should be further explored for purposes of improving the sampling of a MO-based $\text{QM}∕\text{MM}$ $\text{RP}$ including the correct solute polarization.

A hypothetical situation of refining a general $\text{RP}$ (e.g., a MO-type $\text{QM}∕\text{MM}$ $\text{PM}3∕\text{MM}$ potential) is considered in Figure 15A. Here the original reference potential is given by

$$E_{\text{REF}}=E_{\text{PM}3,\text{gas}}+E_{\text{sS}}({\mathbf{Q}}_{\text{PM}3,\text{gas}})$$ (39)

Figure 15.

The PMFs obtained for the ${\text{CH}}_3\text{Cl}$ and ${\text{Cl}}^{-}{\mathrm{S}}_{\mathrm{N}}2$ reaction in water. (A) (red) PMF constructed for the $E_{\text{PM}3∕\text{MM}}$ potential (polarized $\text{PM}3$ Hamiltonian with the solvent polarized by the ESP charges) using the WHAM approach. (blue) PMF obtained by sampling on the flat $E_{\text{PM}3(\text{GAS})}-{\Gamma }_{\text{PM}3(\text{GAS})}$ with the $\text{EVB}$-type solvent driving potential using the $\text{FEP}$/US approach for the potential $E_{\text{PM}3(\text{GAS})}+E_{\text{sS}}(Q_{\text{PM}3})$. (B) Demonstrating the refinement of the original reference potential, $E_{\text{PM}3(\text{GAS})}+E_{\text{sS}}(Q_{\text{PM}3})$, by forcing it to approximate the $\text{PM}6∕\text{MM}$ target potential. (red) PMF for the $E_{\text{PM}6∕\text{MM}}$ potential (polarized $\text{PM}6$ Hamiltonian with the solvent polarized by the ESP charges) constructed using WHAM. (blue) The refined reference free energy surface obtained by sampling on the flat $E_{\text{PM}3(\text{GAS})}-{\Gamma }_{\text{PM}3(\text{GAS})}$ with the $\text{EVB}$-type solvent driving potential, using the $\text{FEP}$/US approach, for $E_{\text{TARG}}=E_{\text{PM}3}-{\Gamma }_{\text{PM}3}+{\Gamma }_{\text{PM}6}+E_{\text{SOLV}}(Q_{\text{PM}3})$.

(with the corresponding FES obtained after removing the bias of eq 36 using eq 31, and combining the multiple simulation windows using eq 32) (see simulation 7 in the Supporting Information). The corresponding original FES (as well as the PMF obtained using eq 38) are given in Figure 15A. To refine this reference potential for the PD calculation of the free energy barrier on the $\text{PM}6∕\text{MM}$ potential, we use

$$E_{\text{REF}}^{\prime }=E_{\text{REF}}-{\Gamma }_{\text{PM}3,\text{gas}}+{\Gamma }_{\text{PM}6,\text{gas}}$$ (40)

with the mapping potential given by eq 36. The refined reference FES (for the potential of eq 40) as well as the target $\text{PM}6∕\text{MM}$ FES (obtained using the $\text{PM}6∕\text{MM}$ version of eq 38) are given in Figure 15B. In actual $\text{QM}(\text{ai})∕\text{MM}$ studies, one can substitute the $\text{PM}6∕\text{MM}$ $\text{TP}$ (which was used here only for demonstration purposes) by an expensive $\text{QM}(\text{ai})∕\text{MM}$ potential. In such cases, it is reasonable to assume that the corresponding gas phase potential and the $\Gamma$-function of eq 40 can be substituted with the corresponding correction obtained with an implicit solvent model (e.g., $\text{COSMO}$).

Turning now to the $\text{EVB}$ $\text{RP}$, we note that this model describes the correct polarization of solute by including the effect of solvent in the $\text{EVB}$ diagonal elements.¹¹ Thus, for the $\text{EVB}$ $\text{RP}$, the $\Gamma$-correction in the gas phase is the correction for the intramolecular interactions within the solute. Moreover, the solute–solvent interaction term can be refined by using

$$\begin{gathered} E_{\text{EVB}}^{\prime }=E_{\text{INTRA}}+\Delta {\Gamma }_{\text{INTRA}}+{c_1}^2{E}_{1,\text{sS}}({\mathbf{Q}}_{\text{RS}}^{\text{TARG}}) \\ +{c_2}^2{E}_{2,\text{sS}}({\mathbf{Q}}_{\text{PS}}^{\text{TARG}}) \end{gathered}$$ (41)

In other words, in the case of the $\text{EVB}$ $\text{RP}$, the gas phase correction, $\Delta \Gamma$, refines the intramolecular part of the $\text{EVB}$ $\text{RP}$ by the relationship:

$$\Delta {\Gamma }_{\text{INTRA}}={\Gamma }_{\text{TARG}}-{\Gamma }_{\text{EVB}}$$ (42)

Since the original $\text{EVB}$ $\text{RP}$ can be partitioned as

$$\begin{gathered} E_{\text{EVB}}={c_1}^2{E}_1+{c_2}^2{E}_2-2c_1{c}_2{H}_{12} \\ =E_{\text{INTRA}}+{c_1}^2{E}_{1,\text{sS}}({\mathbf{Q}}_{\text{RS}}^0)+{c_2}^2{E}_{2,\text{sS}}({\mathbf{Q}}_{\text{PS}}^0) \end{gathered}$$ (43)

The only minor problem here can be that the intramolecular interactions reflected by the ${\Gamma }_{\text{EVB}}$ are calculated with the original vector of $\text{EVB}$ charges, ${\mathit{Q}}^0$, and one might need to recalculate the ${\Gamma }_{\text{EVB}}$ for the new set of charges to improve the accuracy of the $\Delta {\Gamma }_{\text{INTRA}}$ correction.

In conclusion, we demonstrate in Figure 16 how we refine the $\text{EVB}$ $\text{RP}$ originally derived for the $\text{PM}3$ gas-phase calculations. In other words, we refine the $\text{EVB}$ $\text{RP}$ for $\text{PM}3∕\text{MM}$ and $\text{PM}6∕\text{MM}$ as the $\text{TPs}$, see also simulation 8 in the Supporting Information.

Figure 16.

Demonstrating the refinement of the $\text{EVB}$ reference potential in condensed phases, using the correction potential $-{\Gamma }_{\text{EVB}}+{\Gamma }_{\text{TARG}}$ fitted by Gaussians plus a vector of the new $\text{EVB}$ charges derived for the target potential. The free energy profiles obtained by the $\text{EVB}$ $\text{FEP}$/US approach along the $\text{EVB}$ energy gap (A) and along the nuclear $\text{RC}$ (B). (blue) the original reference potential with parameters refined for the gas phase $\text{PM}3$; (red) the refined $\text{EVB}$ reference potential for $\text{PM}3∕\text{MM}$ target potential; (green) the refined $\text{EVB}$ reference potential for $\text{PM}6∕\text{MM}$ target potential.

III. CONCLUDING DISCUSSION

This work refines, quantifies, and validates our paradynamics model. The validation starts by comparing the $\text{LRA}$ approach to the full $\text{FEP}$ treatment for two arbitrary MO-type $\text{QM}∕\text{MM}$ potentials, and is also carried out for $\text{EVB}$ as a $\text{RP}$. The $\text{LRA}$ is found to be sufficiently accurate in evaluation of the free energy perturbation when moving from the $\text{RP}$ to the $\text{TP}$ at the $\text{TS}$ and at the $\text{RS}$. It is also found that it is sufficient to use a harmonic constraint potential in determining the free energy correction at the specific value of the $\text{RC}$. Overall, it is concluded that the calculated $\Delta \Delta G$ between the two modified potentials is a good approximation for the difference between the free energy functions, $\Delta \Delta g(\xi )$, of the $\text{TP}$ and of the $\text{RP}$ at the coordinate $\xi$. That is, it is found that

$$\begin{gathered} \Delta \Delta G_{\text{LRA}}(E_{\text{REF}}+E_{\text{CONS}}(\xi )\to E_{\text{TARG}}+E_{\text{CONS}}(\xi )) \\ \cong \Delta \Delta G_{\text{FEP}} \\ \cong \Delta g_{\text{TARG}}(\xi )-\Delta g_{\text{REF}}(\xi ) \end{gathered}$$ (44)

Several practical improvements for the PD reference potential are proposed. More specifically, a novel idea of refining the $\text{RP}$ (usually following the initial refinement of the $\text{EVB}$ parameters in the case of the $\text{EVB}$ $\text{RP}$) in a general case is put forward and tested on the ${\text{CH}}_3\text{Cl}+{\text{Cl}}^{-}$ S_N2 reaction in water. The refinement approach is based on modifying the $\text{RP}$ with the help of Gaussian functions fitted to the PESs along the $\text{RC}$ performed on the original $\text{RP}$ and on the $\text{TP}$. This technique is shown to be efficient for gas phase simulations, as well as with the implicit solvent models with no further modifications. Moreover, the extension of this model to simulation in condensed phases with the explicit solvent models is made in a reasonable way, although further validation is needed.

It should be clarified here that our approach for adding Gaussian functions shares common features with the MTD basic idea of flattening the original potential by adding to it the negative of the model potential.^32,33 However, in contrast to spending an enormous amount of time while iteratively elevating the local minima of the ab initio $\text{QM}∕\text{MM}$ potential,⁵⁰ we propose a simple approach of building the negative potential by simply evaluating the gas phase or the implicit solvent PES along the $\text{RC}$.

We note here that some $\text{RP}$ approaches, which are very similar to the PD approach (e.g., ref 51), expressed the optimistic perspective that the results obtained therein are superior to our earlier $\text{RP}$ results. What has been overlooked is that those works addressed the rather trivial challenge where the solute is fixed (this case has been already handled in our 1992 work⁷). That is, with a fixed solute, the calculations converge very fast. Nevertheless, obtaining reliable solvation free energies by ab initio calculation is still a challenge even with a fixed solute.³ Here we must point out that even a very recent innovative idea of combining lambda dynamics and metadynamics⁵² in solvation calculations is significantly less efficient than our $\text{RP}$ approach, where almost all the charging processes are done with a classical potential.⁵³

The present work demonstrated that the end point $\text{LRA}$ approach, involving averaging both on the $\text{RP}$ and on the $\text{TP}$, practically coincides with the full $n$-step $\text{FEP}$, whereas the averages taken only on the $\text{RP}$ result in a higher error. This point is important, since most of the works^24,25,51,54 that adopted our $\text{RP}$ idea have not yet moved to the end points $\text{LRA}$ treatment. Of course, the error associated with the initial $\text{LRA}$ estimate will further decrease as the $\text{RP}$ is refined to increase the overlap with the $\text{TP}$. In any case, both elements of our PD approach, the refinement of the $\text{RP}$ and the $\text{LRA}$ estimate of the $\text{FEP}$, are crucial for the efficiency and accuracy when the $\text{RP}$ is used. In fact, using the refined PD $\text{RP}$, one can sufficiently sample the $\text{TP}$ to ensure the free energy convergence (since the computationally cheap refined $\text{RP}$ has high overlap with the $\text{TP}$) as well as to use the highest possible level of theoretical treatment for the $\text{QM}$ region given limited computational resources (with evaluation of the true $\text{FEP}$). In fact, we demonstrated in our previous work⁹ that the PD approach is about 2 orders of magnitude more efficient (in terms of calling the ab initio $\text{QM}$ calculations) than MTD. These advantages of the PD approach might help in avoiding the possible corresponding artifacts encountered by some researchers in MTD $\text{QM}∕\text{MM}$ studies, which resulted in difficulties of reproducing the experimental catalytic activity of enzymes.⁵⁵ In this respect, we would like to clarify that even the extreme efforts and technically impressive (by the size of the $\text{QM}$ region) results in studies of B₁₂ enzymes⁵⁶ by MTD are not necessarily as reliable as one might tend to think considering the elegance of the approach. In fact, the conclusions⁵⁶ that the surface cannot be concerted are problematic, as the actual surface is quite flat in the diagonal range and the actual calculations involve a very short sampling time for the PMF calculations (1.5 ps for each simulation window) and a very short (5 fs) time interval between the iterative depositions of Gaussians in the MTD sampling, which might not be sufficient for the accurate elevation process. Furthermore, we would like to clarify that there is no a single experiment that actually excludes the concerted mechanism (clearly no such experiment exists in solution where there is no relevant model system with both the Co─C bond and the hydrogen donor). Here we would like to state that the strategy (e.g., ref 57) of using very careful $\text{QM}$ calculations in solution (with a higher level $\text{QM}$ model than that used in ref 56) has produced a concerted path, and that moving the solution surface to the protein environment by a calibrated $\text{EVB}$ model is expected to be more reliable (both in terms of the sampling and in terms of extrapolation of a reliable reference system) than the direct MTD in the protein site. For more solid conclusions, it is crucial to perform the MTD simulations also for the solution reaction (this is also crucial for obtaining the catalytic effect⁵⁷). Furthermore, studies of enzyme catalysis require major experience and validation in treating electrostatic effects in proteins and such validations have not yet been reported by the MTD studies, whereas this has been done reliably by reference potential $\text{QM}(\text{ai})∕\text{MM}$ calculations of $p{K}_{\mathrm{a}}’\mathrm{s}$.³ Here again, it would be interesting to see a comparison between the PD and MTD studies, and since the $\text{EVB}$ (with a very careful calibration) has already reproduced quantitatively the catalysis in B₁₂ with a concerted path, we tend to believe that the same results (or at least a flat surface) will be obtained by the PD $\text{QM}(\text{ai})∕\text{MM}$ approach with the $\text{EVB}$ $\text{RP}$.

As clarified in our previous work, the MTD is formally similar to the earlier PD approach⁹ in the sense that the $\text{RP}$ is built iteratively. However, the important difference, which results in major time savings in the PD approach, is that, while the search of the negative $\text{RP}$ is carried out blindly by directly sampling of the $\text{TP}$ in MTD calculations, PD calculations start with a $\text{RP}$ that is already close to the $\text{TP}$ and further refining it iteratively. Of course, our PD approach requires an approximate knowledge of the reaction path, but as we pointed out in detail elsewhere,⁹ we find it useful not to have a black box blind approach in studies of chemical problems. Our recipe of overcoming the blind search is quite simple. It involves the initial mapping with an implicit solvent model that can provide the main mechanistic options, the relevant $\text{RC}$, and the reaction paths. It is true that there is a conceivable advantage in fully automated black box approaches (provided they are sufficiently efficient). However, when one faces the dilemma, in principle, between the simplified “artificial intelligence” and human intellect and experience then, knowing the fundamental chemical concepts, or identifying the possible reaction path in solution using fast models, the latter has an advantage in solving complex problems. Thus, we consider the present work as another illustration of the advantage and power of the PD in $\text{QM}∕\text{MM}$ studies.

Supplementary Material

Computational details of the performed simulations and description of the implemented MOPAC-MOLARIS $\text{QM}∕\text{MM}$ interface and the script in Maple9.5 used in fitting potential energy scans with sums of Gaussians. This material is available free of charge via the Internet at http://pubs.acs.org.