# Optimized Atomic Partial Charges and Radii Defined by Radical Voronoi Tessellation of Bulk Phase Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Method

#### 2.1. General Workflow

#### 2.2. Voronoi Tessellation and Radical Voronoi Tessellation

**radical Voronoi tessellation**[44] (also termed “power diagram” in the two-dimensional case) will be employed here. It is an extension of the classical Voronoi tessellation, where a radius ${r}_{i}$ is assigned to each Voronoi site ${p}_{i}$. This results in the definition

#### 2.3. Integrating over Voronoi Cells

#### 2.4. Charge Variance Minimization Algorithm

## 3. Results and Discussion

#### 3.1. Molecular Charges

#### 3.2. Atomic Partial Charges and Radii

#### 3.3. Basis Set Dependence

## 4. Computational Details

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

[EMIm]${}^{+}$ | 1-Ethyl-3-methylimidazolium cation |

[OAc]${}^{-}$ | Acetate anion |

BLYP | Becke–Lee–Yang–Parr density functional |

bqb | The bqb file format for lossless compression |

CG | Conjugate gradient optimization method |

DDAPC | Density derived atomic point charges (Blöchl charges) |

DFT | Density functional theory |

DFTB | Density functional based tight binding |

GTH | Goedecker–Teter–Hutter pseudopotentials |

IL | Ionic liquid |

MP2 | Møller–Plesset perturbation theory |

RESP | Restrained electrostatic potential |

ROA | Raman optical activity |

VCD | Vibrational circular dichroism |

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**Figure 1.**Flowchart of the proposed two-step optimization algorithm which yields optimized molecular and atomic partial charges as well as two sets of optimized atomic radii $\left\{{r}^{\mathrm{Mol}}\right\}$ and $\left\{{r}^{\mathrm{Atom}}\right\}$ via minimization of the charge variance.

**Figure 2.**Separation plane between two Voronoi sites A and B with radii ${r}_{\mathrm{A}}$ and ${r}_{\mathrm{B}}$ in the radical Voronoi tessellation, see Equation (5).

**Figure 3.**Schematic two-dimensional illustration of the radical Voronoi tessellation in the bulk phase of benzene. The solid black lines are iso-lines of the electron density, the dashed circles indicate the atomic radii, and the radical Voronoi cells are shown as gray solid lines with the resulting molecular boundaries drawn in blue. The position of the blue lines is exclusively determined by the first set of atomic radii $\left\{{r}^{\mathrm{Mol}}\right\}$, while the position of the gray lines is solely determined by the second set of radii $\left\{{r}^{\mathrm{Atom}}\right\}$.

**Figure 4.**Atom labeling of the molecules studied. Element colors: gray—carbon, red—oxygen, blue—nitrogen, white—hydrogen.

**Figure 6.**Influence of the basis set size on the optimized atomic partial charges in the ILW simulation. Vertical axis in units of e. For atom labels, see Figure 4.

**Table 1.**Exemplary liquid phase simulations to which the method is applied, including composition, orthorhombic cell vector, density, simulation temperature, and total electron density volumetric grid resolution.

System | Composition | Cell/pm | Density/g cm${}^{-3}$ | Temp./K | Grid Resolution |
---|---|---|---|---|---|

Benzene | 32 benzene | 1690 | 0.860 | 350 | $192\times 192\times 192$ |

Methanol | 48 methanol | 1515 | 0.735 | 350 | $180\times 180\times 180$ |

Phenol | 32 phenol | 1688 | 1.040 | 400 | $192\times 192\times 192$ |

IL | 36 [EMIm]^{+}36 [OAc] ^{−} | 2121 | 1.066 | 350 | $240\times 240\times 240$ |

ILW | 27 [EMIm]^{+}27 [OAc] ^{−}81 H _{2}O | 2158 | 1.000 | 350 | $243\times 243\times 243$ |

**Table 2.**Resulting average molecular charges and corresponding standard deviation for the five simulations from Table 1. All numbers in units of e.

Molecule | Charge | Std. Dev. |
---|---|---|

Benzene | 0 | 0.010 |

Methanol | 0 | 0.013 |

Phenol | 0 | 0.012 |

IL | ||

[EMIm]${}^{+}$ | 0.846 | 0.016 |

[OAc]${}^{-}$ | −0.846 | 0.011 |

ILW | ||

[EMIm]${}^{+}$ | 0.859 | 0.017 |

[OAc]${}^{-}$ | −0.817 | 0.015 |

Water | −0.014 | 0.015 |

Atom | Charge | Std. Dev. | ${\mathit{r}}^{\mathbf{Mol}}$ | ${\mathit{r}}^{\mathbf{Atom}}$ |
---|---|---|---|---|

Benzene | ||||

C | 0.141 | 0.007 | 173.7 | 72.9 |

H | −0.141 | 0.010 | 107.6 | 41.8 |

Methanol | ||||

C | 0.764 | 0.008 | 172.5 | 76.4 |

HC | −0.169 | 0.010 | 112.9 | 51.7 |

O | −0.609 | 0.023 | 152.9 | 81.1 |

HO | 0.350 | 0.026 | 104.0 | 28.8 |

Phenol | ||||

C1 | 0.248 | 0.014 | 170.1 | 73.8 |

C2 | 0.084 | 0.013 | 172.8 | 73.9 |

C3 | 0.154 | 0.010 | 171.6 | 72.9 |

C4 | 0.122 | 0.012 | 173.2 | 72.8 |

H2 | −0.115 | 0.011 | 107.9 | 41.3 |

H3 | −0.134 | 0.011 | 111.5 | 41.5 |

H4 | −0.130 | 0.011 | 111.4 | 40.4 |

O | −0.447 | 0.025 | 152.9 | 78.0 |

HO | 0.229 | 0.024 | 104.3 | 39.2 |

Atom | IL | ILW | |||||||
---|---|---|---|---|---|---|---|---|---|

Charge | Std. Dev. | ${\mathit{r}}^{\mathbf{Mol}}$ | ${\mathit{r}}^{\mathbf{Atom}}$ | Charge | Std. Dev. | ${\mathit{r}}^{\mathbf{Mol}}$ | ${\mathit{r}}^{\mathbf{Atom}}$ | ||

[EMIm]${}^{+}$ | |||||||||

N1 | −0.306 | 0.011 | 167.3 | 82.7 | −0.316 | 0.011 | 162.2 | 80.8 | |

C2 | 0.270 | 0.015 | 165.2 | 79.5 | 0.350 | 0.015 | 160.1 | 76.9 | |

N3 | −0.223 | 0.013 | 163.7 | 80.3 | −0.294 | 0.011 | 168.3 | 80.1 | |

C4 | 0.149 | 0.014 | 169.1 | 78.0 | 0.254 | 0.014 | 168.1 | 74.8 | |

C5 | 0.148 | 0.013 | 169.1 | 78.7 | 0.219 | 0.013 | 171.9 | 75.7 | |

C6 | 0.468 | 0.009 | 172.3 | 73.5 | 0.541 | 0.008 | 170.3 | 71.7 | |

C7 | 0.518 | 0.007 | 171.4 | 71.4 | 0.455 | 0.007 | 178.0 | 71.9 | |

C8 | 0.655 | 0.010 | 165.2 | 72.0 | 0.688 | 0.008 | 163.5 | 71.4 | |

H2 | 0.092 | 0.025 | 107.3 | 37.3 | 0.055 | 0.022 | 108.0 | 36.3 | |

H4 | 0.041 | 0.022 | 106.9 | 38.2 | −0.025 | 0.018 | 105.1 | 40.8 | |

H5 | −0.093 | 0.019 | 107.3 | 39.8 | −0.124 | 0.015 | 109.8 | 40.8 | |

H6 | −0.092 | 0.020 | 107.3 | 39.8 | −0.123 | 0.015 | 109.8 | 40.8 | |

H7 | −0.156 | 0.015 | 109.6 | 41.6 | −0.139 | 0.014 | 105.9 | 40.2 | |

H8 | −0.115 | 0.019 | 108.9 | 41.7 | −0.125 | 0.014 | 110.1 | 42.1 | |

[OAc]${}^{-}$ | |||||||||

C1’ | 0.761 | 0.015 | 162.4 | 73.4 | 0.882 | 0.018 | 165.2 | 72.7 | |

C2’ | 0.482 | 0.008 | 175.2 | 71.7 | 0.414 | 0.008 | 175.2 | 73.5 | |

H’ | −0.162 | 0.011 | 109.8 | 41.1 | −0.138 | 0.011 | 109.1 | 41.8 | |

O’ | −0.801 | 0.020 | 155.5 | 83.1 | −0.849 | 0.024 | 155.6 | 85.0 | |

Water | |||||||||

OW | −0.553 | 0.024 | 155.8 | 79.2 | |||||

HW | 0.269 | 0.025 | 108.1 | 36.0 |

**Table 5.**Optimized molecular and atomic partial charges as well as corresponding standard deviations for the ILW simulation with three different basis set sizes. All numbers in units of e. For atom labels, see Figure 4.

Atom | SZV | DZVP | TZVPP | |||
---|---|---|---|---|---|---|

Charge | Std. Dev. | Charge | Std. Dev. | Charge | Std. Dev. | |

[EMIm]${}^{+}$ | 0.948 | 0.015 | 0.859 | 0.017 | 0.841 | 0.018 |

N1 | −0.331 | 0.010 | −0.316 | 0.011 | −0.313 | 0.011 |

C2 | 0.352 | 0.015 | 0.350 | 0.015 | 0.382 | 0.015 |

N3 | −0.280 | 0.011 | −0.294 | 0.011 | −0.307 | 0.011 |

C4 | 0.241 | 0.014 | 0.254 | 0.014 | 0.257 | 0.014 |

C5 | 0.158 | 0.014 | 0.219 | 0.013 | 0.220 | 0.013 |

C6 | 0.395 | 0.009 | 0.541 | 0.008 | 0.515 | 0.008 |

C7 | 0.461 | 0.008 | 0.455 | 0.007 | 0.513 | 0.007 |

C8 | 0.619 | 0.009 | 0.688 | 0.008 | 0.705 | 0.008 |

H2 | 0.072 | 0.022 | 0.055 | 0.022 | 0.038 | 0.021 |

H4 | 0.031 | 0.019 | −0.025 | 0.018 | −0.033 | 0.018 |

H5 | −0.057 | 0.014 | −0.124 | 0.015 | −0.119 | 0.015 |

H6 | −0.057 | 0.014 | −0.123 | 0.015 | −0.118 | 0.015 |

H7 | −0.128 | 0.013 | −0.139 | 0.014 | −0.157 | 0.014 |

H8 | −0.093 | 0.013 | −0.125 | 0.014 | −0.131 | 0.014 |

[OAc]${}^{-}$ | −0.909 | 0.016 | −0.817 | 0.015 | −0.797 | 0.015 |

C1’ | 0.711 | 0.018 | 0.882 | 0.018 | 0.791 | 0.015 |

C2’ | 0.378 | 0.009 | 0.414 | 0.008 | 0.506 | 0.008 |

H’ | −0.132 | 0.010 | −0.138 | 0.011 | −0.164 | 0.011 |

O’ | −0.800 | 0.028 | −0.849 | 0.024 | −0.801 | 0.024 |

Water | −0.013 | 0.014 | −0.014 | 0.015 | −0.015 | 0.015 |

OW | −0.565 | 0.027 | −0.553 | 0.024 | −0.553 | 0.023 |

HW | 0.276 | 0.029 | 0.269 | 0.025 | 0.269 | 0.025 |

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**MDPI and ACS Style**

Brehm, M.; Thomas, M. Optimized Atomic Partial Charges and Radii Defined by Radical Voronoi Tessellation of Bulk Phase Simulations. *Molecules* **2021**, *26*, 1875.
https://doi.org/10.3390/molecules26071875

**AMA Style**

Brehm M, Thomas M. Optimized Atomic Partial Charges and Radii Defined by Radical Voronoi Tessellation of Bulk Phase Simulations. *Molecules*. 2021; 26(7):1875.
https://doi.org/10.3390/molecules26071875

**Chicago/Turabian Style**

Brehm, Martin, and Martin Thomas. 2021. "Optimized Atomic Partial Charges and Radii Defined by Radical Voronoi Tessellation of Bulk Phase Simulations" *Molecules* 26, no. 7: 1875.
https://doi.org/10.3390/molecules26071875