# Representation of the QM Subsystem for Long-Range Electrostatic Interaction in Non-Periodic Ab Initio QM/MM Calculations

^{1}

^{2}

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## Abstract

**:**

^{−3}eV from theoretical reference values.

## 1. Introduction

## 2. Theory

#### 2.1. Mechanical Embedding ai-QM/MM Calculations

#### 2.2. Electrostatic Embedding ai-QM/MM Calculations

#### 2.3. Multipolar Representation with Mulliken Charges and Dipoles

#### 2.4. Multipolar Representation with ESP Charges and Dipoles

#### 2.5. A Hybrid Scheme with Switching Functions

## 3. Computational Details

^{2}). Cartesian coordinates were saved every 10 ps along the trajectory, which resulted in 100 structures for each QM/MM benchmarking calculation.

## 4. Results and Discussion

#### 4.1. Electrostatic and Polarization Energies

#### 4.1.1. Truncation and Truncation/MMLC Models

^{−2}kcal/mol. Here, for the computation of polarization energy, QM/MM embedding schemes using switching functions performed several times better than using the Step function, again attesting to the usefulness of scaling down near-field MM charges to compensate for neglecting far-field MM charges.

#### 4.1.2. MulC and MulCD Models

^{2}kcal/mol (as shown in Plots a5, a6, b5, b6, c5, c6, d5 and d6 in Figure 2), even when large cutoffs were used for both the anionic and neutral QM systems. When the dipoles were included to interact with far-field MM charges, as in the MulCD embedding scheme, the results were much improved. However, in order to achieve a target accuracy of 0.1 kcal/mol, a large cutoff (>30 Å) and LREC/Switch functions were needed (see Plots a7, a8, b7, b8, c7, c8, d7 and d8 in Figure 2). This again confirms the inappropriateness of using Mulliken charges to account for long-range ai-QM/MM electrostatics, as discovered by Holden and Herbert [16,17].

#### 4.1.3. ESPC and ESPCD Models

^{−3}kcal/mol can be seen from Plots a12, b12, c12 and d12 in Figure 2 for computed QM/MM polarization energies using the LREC/Switch functions and a cutoff distance beyond 10 Å, which meant the QM wavefunction deviated little from the reference ones.

#### 4.2. Population Analysis and Excitation Energies

^{−2}eV from the reference values.

^{−2}eV accuracy at a 15 Å cutoff (see Plots c1, c2, d1 and d1 in Figure 3). The MulC model performed poorly for excited state calculations, especially for the enzyme system where the RMSDs of excitation energies were found to be always above 0.1 eV at all cutoff distances (see Plots d5 and d6 in Figure 3). The ESPC model displayed a similar performance to the truncation model. Surprisingly, the MulCD model delivered rather accurate excitation energies (RMSD < 10

^{−2}eV) beyond a 15 Å cutoff. The ESPCD model with LREC/Switch/Shift functions demonstrated a superior performance: the corresponding vertical excitation energies fell within 10

^{−2}eV from the reference values already at a 5 Å cutoff, and the deviation further reduced to 10

^{−3}eV beyond a 10 Å cutoff.

#### 4.3. Computational Timings

## 5. Conclusions

- The long-range QM/MM electrostatic interactions can be rather significant. At cutoff distances of 30 Å, their average contribution was found to be still around 10 kcal/mol with the anionic QM subsystem and around 1 kcal/mol with the neutral QM subsystem.
- In truncated QM/MM calculations, where only MM charges within a cutoff distance are kept, like Fang et al. [25], we found it to be beneficial to use LREC/Switch functions to scale down MM charge values to compensate for neglecting all far-field MM charges.
- The MulC model should generally be avoided because it can cause extremely large errors in QM/MM electrostatic and polarization energies. While the MulCD model offered an improvement upon the MulC model, its use is still not recommended because large distance cutoffs are needed and because the Mulliken charges on QM atoms obtained with this model still looked erroneous.
- While the ESPC model performed significantly better than MulC, it polarized the QM region only as well as the truncation model (as indicated by atomic charge populations, and vertical excitation energies) when LREC or Switch functions were used. Its only significant advantage over the truncation model occurred with the computation of QM/MM electrostatics energy with an anionic QM subsystem (the oxyluciferin anion).
- The ESPCD model yielded the best performances. At a 10 Å cutoff distance, it reproduces QM/MM electrostatics energy within 0.1 kcal/mol, polarization energy within 10
^{−3}kcal/mol and TDDT vertical excitation energy within 10^{−3}eV from the reference values. Therefore, ESPCD with LREC/Switch smoothing functions and a 10 Å cutoff would be our recommended combination for a hybrid representation for the QM subsystem in the treatment of short-range and long-range QM/MM electrostatics. - Besides avoiding discontinuity at the cutoff distance, a LREC/Switch smoothing function between the near- and far-field interactions can also lead to better results in most cases, and thus should be applied in general.

- Only the oxyluciferin systems were studied using very short MD simulations. Testing on longer simulations and more systems, including enzymatic reactions, needs to be carried out for more general conclusions.
- Our ESPCD scheme should be readily extendable to ai-QM/MM PBC calculations, where both ESP charges and dipoles are used to represent the QM subsystem in the long-range electrostatics calculations. However, this needs to be thoroughly tested.
- We used the standard ESP grid (discretized four layers of vdW surfaces) in the computation of ESP charges and dipoles. Other grids should be explored.
- Only energy values were reported. The corresponding analytical gradient has yet to be coded.
- For the MM region, the TIP3P water model and C36 protein force field were employed. Therefore, it would be interesting to see if other force fields might offer a physically better description for the QM/MM electrostatics/polarization interactions.
- Our focus was placed on QM/MM electrostatics/polarization energies. Therefore, we have not considered QM/MM vdW and charge-transfer interactions, which can also significantly affect the simulation results.
- We studied how to more quickly converge QM/MM electrostatic energy with the number of MM charges included in the short-range evaluation. Therefore, we have not addressed the equally important issue of how to achieve quicker convergence of QM/MM results with the size of the QM region.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Sample Availability: Samples of the compounds are not available from the authors. |

**Figure 1.**Structures of the (

**a**) neutral (OLU) and (

**b**) anionic (${\mathrm{OLU}}^{-}$) forms of oxyluciferin.

**Figure 2.**RMSD of electrostatic and polarization energies for neutral oxyluciferin in the (

**a**) aqueous and (

**b**) enzyme environments and anionic oxyluciferin in the (

**c**) aqueous and (

**d**) enzyme environments. All MM charges beyond a cutoff distance were removed from the QM/MM electrostatics calculation in the “Truncation” model, and the long-range electrostatics interaction between these far-field charges and the QM subsystem were described at the MM level in the “Truncation/MMLC” model. Near-field MM charges remained unchanged in the “Step” calculations, but scaled with the long-range electrostatic corrections (“LREC”), “Switch” and “Shift” calculations to ensure a continuous potential energy surface.

**Figure 3.**RMSD of Mulliken and ESP charges for the anionic oxyluciferin systems in the (

**a**) aqueous and (

**b**) enzyme environments; RMSD of excitation energies for the anionic oxyluciferin systems in the (

**c**) aqueous and (

**d**) enzyme environments.

**Table 1.**Number of near-field MM charges and CPU time (in seconds) for QM/MM electrostatic energy and force evaluations at different cutoff distances. The QM region is the oxyluciferin anion, which is described by the B3LYP/6-31+G* level of theory. The MM region is a 117 Å × 117 Å × 117 Å unit cell of TIP3P water molecules. Obtained using a single Intel Xeon E5-2650 core at 2.3 GHz.

Cutoff (Å) | # (Charges) | Time (s) | Cutoff (Å) | # (Charges) | Time (s) | ||
---|---|---|---|---|---|---|---|

Energy | Force | Energy | Force | ||||

5 | 144 | 0.1 | 0.4 | 25 | 9297 | 3.7 | 28.2 |

10 | 837 | 0.3 | 2.0 | 30 | 15,291 | 5.8 | 42.5 |

15 | 2418 | 0.9 | 5.8 | 60 | 107,535 | 39.4 | 262.7 |

20 | 5121 | 2.2 | 15.3 | Unit Cell | 166,114 | 58.2 | 403.6 |

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**MDPI and ACS Style**

Pan, X.; Rosta, E.; Shao, Y. Representation of the QM Subsystem for Long-Range Electrostatic Interaction in Non-Periodic Ab Initio QM/MM Calculations. *Molecules* **2018**, *23*, 2500.
https://doi.org/10.3390/molecules23102500

**AMA Style**

Pan X, Rosta E, Shao Y. Representation of the QM Subsystem for Long-Range Electrostatic Interaction in Non-Periodic Ab Initio QM/MM Calculations. *Molecules*. 2018; 23(10):2500.
https://doi.org/10.3390/molecules23102500

**Chicago/Turabian Style**

Pan, Xiaoliang, Edina Rosta, and Yihan Shao. 2018. "Representation of the QM Subsystem for Long-Range Electrostatic Interaction in Non-Periodic Ab Initio QM/MM Calculations" *Molecules* 23, no. 10: 2500.
https://doi.org/10.3390/molecules23102500