# Do Better Quality Embedding Potentials Accelerate the Convergence of QM/MM Models? The Case of Solvated Acid Clusters

^{1}

^{2}

^{*}

## Abstract

**:**

^{+}) acids clusters containing 160 and 480 water molecules using configurations sampled from molecular dynamics simulations. Consistently, the ωB97X-D/EFP model performed the best when using a minimal QM region size. The performance for the other potentials appears to be highly sensitive to the charge character of the acid/base pair. Neutral acids display the expected trend that semi-empirical methods generally perform better than TIP3P; however, an opposite trend was observed for the cationic acids. Additionally, electronic embedding provided an improvement over mechanical embedding for the cationic systems, but not the neutral acids. For the best performing ωB97X-D/EFP model, a QM region containing about 6% of the total number of solvent molecules is needed to approach within 10 kJ mol

^{−1}of the pure QM result if the QM region was chosen based on the distance from the reaction centre.

## 1. Introduction

## 2. Experimental Design and Methods

^{−}) and cationic acids (HB

^{+}/B) in water clusters containing up to 480 solvent molecules. In this notation, MM and QM′ refer to the embedding potential used to model the environment around the QM region. The test set consists of two examples from each class (HA = HCOOH and C

_{6}H

_{5}OH) and (HB

^{+}= CH

_{3}NH

_{3}

^{+}and H-imidazole

^{+}) as shown in Figure 1. Within each class, the acids differ in their extent of charge delocalisation, i.e., HCOO

^{−}and CH

_{3}NH

_{3}

^{+}represent relatively charge-concentrated ions while C

_{6}H

_{5}O

^{−}and protonated imidazole are significantly more diffuse (delocalised). Using the HA/A

^{−}system as an example, the deprotonation energy is defined in this study as the energy change (at 0 Kelvin without zero-point vibrational correction) associated with the following reaction:

^{−}HA(H

_{2}O)

_{m}→ A

^{−}(H

_{2}O)

_{m}+ H

^{+}

^{−1}Å

^{−2}) was applied to restrain the centre of mass of the acid to the origin. The simulations were performed for an isothermal-isobaric (NPT) ensemble, using a 1 fs integration time step and rigid bonds for water. The Langevin thermostat was used to maintain constant temperature of 300 K, and the Nosé-Hoover method was used to maintain constant pressure at 1 bar. Long-range electrostatic interactions were calculated with the particle-mesh Ewald algorithm, the Lennard-Jones interactions were truncated at 12 Å, and a switching function was applied at 10 Å. Visualisation and analysis of the MD trajectories were performed using the VMD program [41]. In-house Tcl and Perl scripts were used to extract geometries of the solvated acid clusters and for generating the ONIOM input files.

^{H}(s + m) + E

^{L}(full) − E

^{L}(s + m)

^{H}(s + m) becomes E

^{H+ptchg}(s + m), and setting the charges of the QM subsystem to zero in the MM calculations [42].

^{H}, and is obtained as the sum of two reaction energies ΔE

^{H}(s + m) and ΔE

^{L}(full, m), where m is the number of water molecules in the QM region. Note that ΔE

^{L}(full, m) is formally an isodesmic (proton exchange) reaction and therefore would benefit from cancellation of errors associated with the lower-level method ‘L’ or embedding potential. As such, the agreement between the full QM (ΔE

^{H}) result versus the QM/MM approximation (ΔE(m)) is directly measured by the difference between ΔE

^{L}(full, m) and ΔE

^{H}(full, m).

## 3. Results and Discussion

_{6}H

_{5}OH/C

_{6}H

_{5}O

^{−}where there is an inversion between QM-only and PM6 for the two clusters), we will base most of our discussion on the 160-water cluster data. For the purpose of our discussion, we refer to the rate of convergence to mean how quickly the errors decay to below a cut-off value of 10 kJ mol

^{−1}.

^{−}systems (Figure 4), the performance of the EFP method is comparable if not slightly better than the semi-empirical PM7 and DFTB methods, particularly for the smallest QM region (m = 10) where its mean absolute errors are 11.7 and 8.9 kJ mol

^{−1}for the HCOOH/HCOO

^{−}and C

_{6}H

_{5}OH/C

_{6}H

_{5}O

^{−}systems, respectively. In the 480-water cluster models, the EFP retains a similar level of accuracy; however, the errors remain relatively insensitive to the QM region size, and only start decaying towards zero after about 300 water molecules are included in the QM region. For the remaining potentials, we see the expected behaviour on the basis of treatment of intermolecular interactions (and computational cost) that the rate of convergence is fastest for the semi-empirical DFTB and PM7 methods, followed by TIP3P and QM-only methods. The only exception is the ONIOM(ωB97X-D/6-31G(d):PM6) method which has an exceedingly large error of about 40 kJ mol

^{−1}for ΔE(10) in both systems. Another unexpected behaviour is that electronic embedding did not improve the convergence relative to mechanical embedding, even though it includes polarisation of the QM region by the MM subsystem. The first finding is not so surprising since the PM6 method is known to perform poorly for predicting intermolecular interactions, whilst the PM7 method contains corrections for dispersion and hydrogen bonds that are present extensively in the solvated clusters [22]. The poorer performance of ONIOM-EE versus ONIOM-ME models has been reported, but this is likely to be system-dependent as indicated by the studies from Ryde [19] and Ochsenfeld [17], and from this study (vide infra).

^{−1}for the CH

_{3}NH

_{3}

^{+}and H-imidazole

^{+}systems, respectively. The corresponding errors for the 480-water cluster are 9.0 and 10.0 kJ mol

^{−1}, respectively. In contrast to the HA/A

^{−}systems, the rate of convergence is faster for the TIP3P model followed by PM7 and DFTB and QM-only methods. In these systems, electronic embedding also provided a modest improvement in the convergence compared to mechanical embedding. Another notable difference is that for small QM region sizes (m < 50), the errors in the cationic acids appear to be significantly higher compared to the HA/A

^{−}systems. For the smallest QM region (m = 10), the errors for the worst performing QM-only method are about 169 and 205 kJ mol

^{−1}for 160-water clusters of imidazole and methylamine, respectively (compared to ~30–40 kJ mol

^{−1}for the HA/A

^{−}systems). The larger errors in the cationic systems suggest that water may not shield cations as well as it shields anions. This is supported by experimental gas phase studies of ion-water clusters of monoatomic cations (e.g., Li

^{+}, Na

^{+}, and K

^{+}) which have clustering free energies that are significantly more exergonic compared to the monoatomic anions in the same period (F

^{−}, Cl

^{−}, and Br

^{−}) despite their larger ionic radii [43]. This may also explain the slower rate of convergence for the PM6, PM7, and DFTB potentials in the cationic systems relative to the neutral acids.

^{−}systems is particularly interesting. To better understand these results, we refer to the ONIOM reaction scheme in Figure 3, where it becomes clear that differences in the ONIOM models employing different embedding potentials are due to the ΔE

^{L}(full, m) contribution. In particular, the accuracy of the ONIOM models relies on the ability of the embedding potential to reproduce ΔE

^{L}(full, m) relative to the ωB97X-D/6-31G(d) values. Table 1 presents ΔE

^{L}(full, 10) values of two random snapshots for the CH

_{3}NH

_{3}

^{+}/CH

_{3}NH

_{2}and C

_{6}H

_{5}OH/C

_{6}H

_{5}O

^{−}systems. Formally, ΔE

^{L}(full, m) corresponds to an isodesmic proton exchange reaction and should, therefore, be less sensitive to the choice of electronic structure method due to systematic error cancellation. This is evident from the good agreement between the ωB97X-D and HF values (within 5 kJ mol

^{−1}) shown in Table 1. Thus, what is really needed is an embedding potential that display similar systematic errors to the QM level of theory for describing the isodesmic proton exchange reaction. Accordingly, the TIP3P model outperformed DFTB and PM7 due to better cancellation of errors for the cationic acids.

^{−1}of the pure QM result. The results are summarised in Table 2 for the 160-water clusters, and the values in parenthesis refer to the unsigned errors associated with the minimal (m = 10) QM region calculations. As shown, the EFP method consistently performs the best for all four systems and its use is recommended when using a small QM region. In terms of the minimal QM region needed for the error in the QM/MM approximation falling below 10 kJ mol

^{−1}, the semi-empirical PM7 and DFTB methods work best for neutral acids, while the TIP3P (with electronic embedding) is recommended for cationic acids. In this paper, the QM region is radially expanded based on distance from the reaction centre, and the data in Table 2 are probably upper bound values of the minimal QM region needed to obtain this level of accuracy. In the future, it would be interesting to examine the use of systematic QM region determination methods, e.g., charge shift analysis [44], to see if this results in a significant reduction in the size of minimal QM region.

## 4. Summary and Conclusions

^{L}(full, m) (Figure 3) relative to the QM values that is critical for the accuracy of the resulting ONIOM model. We showed that the extent of error cancellation is not necessarily better when using semi-empirical potentials compared to fixed charge force fields, and is highly sensitive to the nature of the system (cationic vs. neutral acids). Also, the pairing of a QM with MM (or QM′) method can affect the accuracy of the model relative to the pure QM calculation of the full system. For example, for the purpose of reproducing the corresponding full QM results, the PM7/TIP3P combination is likely to result in much smaller errors compared to HF/TIP3P or ωB97X-D/TIP3P models. In keeping with the findings of this work, it should be emphasised that these conclusions may not hold beyond the systems examined herein, but it confirms that the good performance of a particular QM/MM procedure may not be readily transferable to other systems, and the better quality potentials do not necessarily lead to improved performance.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Snapshots of the solvated clusters (m = 480) are collected from a molecular dynamics simulation of the solute in a periodic box of water. For a chosen snapshot, a series of quantum mechanics/molecular mechanics (QM/MM) models are set up by radially increasing the size of the QM region (balls and sticks) from the reaction centre. ΔE(m) refer to the QM/MM simulation with m water molecules in the QM region. Errors in QM/MM models are measured from absolute deviations from the pure QM calculation of the full system, ΔE = ΔE (480).

**Figure 4.**Error convergence profiles for the various QM/MM and QM/QM′ models as a function of QM region size for 160-water cluster (

**top**) and 480-water cluster (

**bottom**) of neutral acids.

**Figure 5.**Error convergence profiles for the various QM/MM and QM/QM′ models as a function of QM region size for 160-water cluster (

**top**) and 480-water cluster (

**bottom**) of cationic acids.

**Table 1.**ΔE

^{L}(full, 10) values determined at various levels of theory for two snapshots. Values in parenthesis refer to relative values.

Theory | CH_{3}NH_{3}^{+} | Phenol | ||
---|---|---|---|---|

Frame 0 | Frame 10 | Frame 0 | Frame 10 | |

ωB97X-D/6-31G(d) | 296.0 (0.0) | 206.7 (0.0) | −31.2 (0.0) | −46.9 (0.0) |

AMBER/TIP3P | 258.2 (−37.9) | 173.3 (−33.4) | −5.1 (26.2) | −1.3 (45.7) |

PM7 | 240.9 (−55.2) | 172.9 (−33.7) | −18.1 (13.2) | −24.8 (22.1) |

DFTB | 226.5 (−69.5) | 156.5 (−50.1) | −18.9 (12.4) | −32.6 (14.3) |

HF/6-31G(d) | 296.9 (0.9) | 204.8 (−1.9) | −28.7 (2.5) | −51.4 (-4.4) |

**Table 2.**The minimal quantum mechanics (QM) region size (m) needed to approach within 10 kJ mol

^{−1}of the pure QM result on the full system (160-water cluster). Values in parenthesis refer to error associated with smallest (m = 10) QM region.

Method | HCOOH | C_{6}H_{5}OH | CH_{3}NH_{3}^{+} | H-Imidazole^{+} |
---|---|---|---|---|

QM-only | >150 (28.0) | 150 (35.0) | 150 (204.9) | 150 (168.8) |

TIP3P | 60 (28.0) | 50 (32.9) | 60 (32.5) | 50 (14.6) |

TIP3P-EE | 70 (30.3) | 70 (32.5) | 30 (21.0) | 30 (11.0) |

EFP | 40 (11.7) | 10 (8.9) | 10 (5.1) | 10 (7.5) |

PM6 | 80 (43.1) | 100 (39.1) | 110 (79.9) | 110 (71.0) |

PM7 | 30 (14.9) | 40 (17.0) | 70 (36.1) | 60 (22.3) |

DFTB | 30 (10.4) | 40 (11.4) | 70 (45.2) | 70 (35.0) |

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**MDPI and ACS Style**

Ho, J.; Shao, Y.; Kato, J.
Do Better Quality Embedding Potentials Accelerate the Convergence of QM/MM Models? The Case of Solvated Acid Clusters. *Molecules* **2018**, *23*, 2466.
https://doi.org/10.3390/molecules23102466

**AMA Style**

Ho J, Shao Y, Kato J.
Do Better Quality Embedding Potentials Accelerate the Convergence of QM/MM Models? The Case of Solvated Acid Clusters. *Molecules*. 2018; 23(10):2466.
https://doi.org/10.3390/molecules23102466

**Chicago/Turabian Style**

Ho, Junming, Yihan Shao, and Jin Kato.
2018. "Do Better Quality Embedding Potentials Accelerate the Convergence of QM/MM Models? The Case of Solvated Acid Clusters" *Molecules* 23, no. 10: 2466.
https://doi.org/10.3390/molecules23102466