# A Test of Various Partial Atomic Charge Models for Computations on Diheteroaryl Ketones and Thioketones

## Abstract

**:**

## 1. Introduction

_{a}variations in series of molecules [21,22]. Since the definition of partial atomic charges is not strict, various models of partial atomic charges can, however, differ significantly in the reliability of their predictions. Therefore, an evaluation of the performance of a partial atomic charge model for a given problem is necessary before using it for making reliable predictions [23,24,25,26,27,28].

**1a**), di(thiophen-2-yl)methanone (

**2a**), di(selenophen-2-yl)methanone (

**3a**), di(1H-pyrrol-2-yl)methanone (

**4a**), di(1-methylpyrrol-2-yl)methanone (

**5a**) and their thiocarbonyl analogs (

**1b**–

**5b**). The structural skeletal formulas of

**1a**–

**5a**and

**1b**–

**5b**are shown in Figure 1. The thioketones

**1b**–

**5b**have recently been synthesized by the O/S exchange in the corresponding ketones by treatment with Lawesson’s reagent [29]. In the first stage of the present work, various models of partial atomic charge are tested in terms of their ability to reproduce the molecular dipole moments of

**1a**–

**5a**and

**1b**–

**5b**. Next, the efficiency of the models for predicting the conformational behavior of

**1a**–

**5a**and

**1b**–

**5b**is studied. The prediction of the conformational behavior will be restricted here to the determination of a preferred conformer and the sequence of higher-energy conformers.

## 2. Computational Details

**1a**–

**5a**and

**1b**–

**5b**are taken from our previous works [30,31,32] in which Becke’s three-parameter hybrid exchange functional combined with the correlation functional of Lee, Yang and Parr (B3LYP) [33,34,35] and the def2-QZVPP basis set [36] were used to optimize the geometries of the isolated molecules of

**1a**–

**5a**and

**1b**–

**5b**. It was also established there that for each of the compounds its three conformations could be formed by the rotation of heteroaryl substituents about the single C-C bonds linking these substituents with the C atom of carbonyl/thiocarbonyl group. The resulting conformers of

**1a**–

**5a**and

**1b**–

**5b**are denoted by the prefixes cc, ct and tt, indicating the spatial arrangement of ring heteroatoms with respect to the O/S atom of carbonyl/thiocarbonyl group (see Figure 2).

**Figure 2.**Three conformations of the investigated diheteroaryl ketones and thioketones. The B3LYP/def2-QZVPP-optimized conformers of

**2a**are shown as an example.

**1a**–

**5a**and

**1b**–

**5b**, their molecular wave functions/electron densities are calculated at three levels of theory, that is, HF/def2-QZVPP [36,37,38], B3LYP/def2-QZVPP [33,34,35,36] and MP2/def2-QZVPP [36,39]. These molecular wave functions/electron densities are the starting point for deriving partial atomic charges from various models belonging to the three groups mentioned in the introduction. The partial atomic charges on all atoms of each conformer are determined by means of ten models, namely Mulliken, NPA, GDMA, AIM, Hirshfeld, CM5, MKS, CHELP, CHELPG and HLY. Then, the magnitude of the dipole moment μ of the conformer is approximated by the following formula:

**1a**–

**5a**and

**1b**–

**5b**. Therefore, the dipole moments calculated by the ten models of partial atomic charge are compared with the corresponding dipole moments obtained from the regular quantum chemical calculations involving the full molecular electron density (to be strict, these dipole moments are calculated as an expectation value of the appropriate quantum operator, which is well defined for the dipole moment).

**1a**–

**5a**and

**1b**–

**5b**. On this basis, the conformational behavior of these compounds can be predicted. Interaction energies in the pairs of partial atomic charges (E

_{C}(i,j)) are calculated using classical Coulomb’s law. These interaction energies are summed up over all pairs of partial atomic charges within each conformer (Σ

_{i}

_{>j}E

_{C}(i,j)) to roughly estimate the energy E

_{elst}associated with electrostatic effects occurring in the conformer. The conformers of each compound can be ordered with respect to their E

_{elst}values. In the resulting sequence, the lower (that is, the more negative) the value of E

_{elst}is obtained for a conformer, the more stable the conformer is. Such a procedure taking only E

_{elst}into consideration assumes that the electrostatic effects play an important role in governing the conformational behavior of investigated compounds. These effects, indeed, contribute significantly to the ordering of conformers for diheteroaryl ketones and thioketones [30,32].

## 3. Results and Discussion

**1a**–

**5a**and

**1b**–

**5b**. Figure 3 shows the mean signed error (MSE) and root mean square error (RMSE) in the μ values approximated by the partial atomic charges derived from each model with respect to the reference results obtained from the full-density calculations. Additionally, the values of MSE and RMSE are determined for three levels of theory. When comparing here the approximated values of μ with the corresponding reference values, both the former and the latter are obtained from the molecular wave functions/electron densities generated at the same level of theory. The approximated and reference μ values used for the calculations of the MSE and RMSE presented in Figure 3 can be found in Tables S1–S6 in Supplementary Materials. The MSE provides information about systematic errors occurring in the approximated values of μ, whereas the RMSE allows us to rank the accuracy of individual models for reproducing the reference values of μ.

**Figure 3.**MSE and RMSE (in Debyes) in the μ values approximated by 10 models of partial atomic charge for 30 conformers of

**1a**–

**5a**and

**1b**–

**5b**. The errors are calculated relative to the reference results obtained from the full density at the HF/def2-QZVPP, B3LYP/def2-QZVPP and MP2/def2-QZVPP levels of theory.

**1a**–

**5a**and

**1b**–

**5b**. The CHELP and CHELPG models demonstrate larger RMSE values but both are still fairly successful in reproducing the μ values from the full density. This seems to be in line with what was previously reported for the MKS and CHELPG models [46]. The former turned out to be superior to the latter for the representation of dipole moments in ionic liquids. One of the reasons for the good performance of the MKS, CHELP, CHELPG and HLY models in reproducing μ for the conformers of

**1a**–

**5a**and

**1b**–

**5b**is that the molecular shape of these conformers is not very complex (the heteroaryl substituents are planar although they are most often not coplanar with one another) and almost all their atoms are near their molecular van der Waals surfaces. This practically eliminates the occurrence of the so-called “buried atoms” for which the electrostatic potential fitting is inaccurate, and thus, the resulting partial atomic charges are poorly determined [47].

**1a**–

**5a**and

**1b**–

**5b**. Of three levels considered in this work, the μ values obtained from the B3LYP/def2-QZVPP full density turned out to be closest to experiment for a test set of simple molecules being the building blocks of

**1a**–

**5a**and

**1b**–

**5b**(see section S2 in Supplementary Materials). On this basis, the μ values calculated using the B3LYP/def2-QZVPP full density [32] are also assumed to be the most realistic for the conformers of

**1a**–

**5a**and

**1b**–

**5b**. These values are now used as the only reference results for the calculations of the MSE and RMSE in the μ values approximated by the ten models that, in turn, operate on the HF/def2-QZVPP, B3LYP/def2-QZVPP and MP2/def2-QZVPP wave functions/electron densities. The calculated MSE and RMSE values are presented graphically in Figure 4. For the B3LYP method, its results shown in this figure are obviously identical to those depicted in Figure 3.

**Figure 4.**MSE and RMSE (in Debyes) in the μ values approximated by 10 models of partial atomic charge for 30 conformers of

**1a**–

**5a**and

**1b**–

**5b**. The errors are calculated relative to the reference results obtained from the full density at the B3LYP/def2-QZVPP level of theory.

**1a**–

**5a**and

**1b**–

**5b**. The E

_{elst}energy calculated using the partial atomic charges determined for each conformer is considered here to be a simple measure of electrostatic effects stabilizing the conformer. It is convenient to express the E

_{elst}values obtained for the conformers of each compound with respect to the E

_{elst}energy of the conformer that is most favorable (in other words, with respect to the conformer possessing the lowest E

_{elst}energy). The resulting relative electrostatic energy ΔE

_{elst}is equal to zero for the preferred conformer while it is positive for less stable conformers. A part of the ΔE

_{elst}values characterizing the cc-, ct-, and tt-conformers of

**1a**–

**5a**and

**1b**–

**5b**is given in Table 1 and Table 2. The tabulated values have been calculated using selected models that have operated on the molecular wave functions/electron densities computed at the B3LYP/def2-QZVPP level of theory (a complete set of results obtained from all ten models of partial atomic charge, as well as at the HF/def2-QZVPP, B3LYP/def2-QZVPP and MP2/def2-QZVPP levels of theory can be found in Tables S8–S13 in Supplementary Materials). Additionally, the relative electron energies ΔE and full-density dipole moments μ obtained from the previous regular calculations at the B3LYP/def2-QZVPP level [30,32] are also presented in Table 1 and Table 2. The values of ΔE allow us to establish the reference orderings of conformers for

**1a**–

**5a**and

**1b**–

**5b**.

**Table 1.**Relative electron energies (ΔE in kcal/mol), dipole moments obtained from the full density (μ in Debyes) and relative electrostatic energies (ΔE

_{elst}in kcal/mol) for

**1a**–

**5a**in their three conformations. All the results are calculated at the B3LYP/def2-QZVPP level of theory.

Conformer | ΔE ^{a} | μ ^{b} | ΔE_{elst} | |||||
---|---|---|---|---|---|---|---|---|

Mulliken | NPA | AIM | Hirshfeld | MKS | HLY | |||

cc-1a | 2.14 | 4.73 | 13.78 | 8.80 | 35.42 | 1.81 | 0.07 | 0.00 |

ct-1a | 0.00 | 3.91 | 4.13 | 0.00 | 0.95 | 1.10 | 0.47 | 19.65 |

tt-1a | 0.34 | 2.91 | 0.00 | 0.67 | 0.00 | 0.00 | 0.00 | 25.15 |

cc-2a | 0.00 | 4.03 | 0.93 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

ct-2a | 0.77 | 3.37 | 0.30 | 5.71 | 5.29 | 0.68 | 44.47 | 64.19 |

tt-2a | 1.85 | 2.69 | 0.00 | 10.99 | 10.58 | 1.47 | 50.28 | 69.59 |

cc-3a | 0.00 | 3.65 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

ct-3a | 1.41 | 3.21 | 1.01 | 7.60 | 10.11 | 0.15 | 53.46 | 68.89 |

tt-3a | 2.93 | 2.81 | 1.42 | 16.44 | 22.96 | 0.05 | 69.25 | 87.14 |

cc-4a | 0.00 | 0.41 | 0.00 | 0.00 | 5.38 | 0.00 | 0.00 | 0.00 |

ct-4a | 3.82 | 3.31 | 10.99 | 7.79 | 0.00 | 1.75 | 6.79 | 1.51 |

tt-4a | 8.99 | 5.31 | 9.77 | 11.96 | 8.28 | 3.34 | 26.92 | 25.70 |

cc-5a | 0.00 | 0.13 | 0.00 | 0.00 | 3.35 | 0.00 | 0.00 | 0.00 |

ct-5a | 5.37 | 3.55 | 8.00 | 1.81 | 3.30 | 0.50 | 36.19 | 44.15 |

tt-5a | 9.63 | 5.29 | 28.21 | 5.54 | 0.00 | 1.36 | 99.21 | 106.26 |

_{elst}show that the energy of electrostatic interactions between partial atomic charges is able to give us some indication of the preferred conformations for the investigated compounds, especially for those whose preferred conformation is characterized by either the largest or the smallest values of μ. The most energetically stable conformers of

**1a**and

**1b**possess molecular dipole moments whose values lie in the middle between the values obtained for the other two conformers. In such case the E

_{elst}energy usually turns out to be insufficient to identify the preferred conformation. Although the preference of the ct-conformation can be inferred from ΔE

_{elst}calculated using the NPA and Mulliken partial atomic charges, none of the two sets of partial atomic charges leads to ct-conformation preference simultaneously for

**1a**and

**1b**. Nevertheless, the NPA model is recognized to be most successful in predicting the preferred conformers of

**1a**–

**5a**and

**1b**–

**5b**in terms of E

_{elst}. The Mulliken, AIM and Hirshfeld models lead to a slightly greater number of incorrect indications of preferred conformers than the NPA model does. Furthermore, this model always performs best, irrespective of the level of theory used to generate the molecular wave functions. The electrostatic potential-based models are generally less accurate in identifying the preferred conformers of

**1a**–

**5a**and

**1b**–

**5b**than the models belonging to the two remaining classes. Of the electrostatic potential-based models, only MKS and HLY are able to correctly indicate the preferred conformers for more than half of the investigated compounds.

**Table 2.**Relative electron energies (ΔE in kcal/mol), dipole moments obtained from the full density (μ in Debyes) and relative electrostatic energies (ΔE

_{elst}in kcal/mol) for

**1b**–

**5b**in their three conformations. All the results are calculated at the B3LYP/def2-QZVPP level of theory.

Conformer | ΔE ^{a} | μ ^{b} | ΔE_{elst} | |||||
---|---|---|---|---|---|---|---|---|

Mulliken | NPA | AIM | Hirshfeld | MKS | HLY | |||

cc-1b | 1.46 | 4.69 | 7.79 | 0.00 | 0.00 | 0.80 | 51.02 | 42.63 |

ct-1b | 0.00 | 4.09 | 0.00 | 1.35 | 2.22 | 0.62 | 10.35 | 11.94 |

tt-1b | 0.31 | 3.35 | 0.75 | 10.90 | 36.07 | 0.00 | 0.00 | 0.00 |

cc-2b | 0.00 | 4.13 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

ct-2b | 0.99 | 3.62 | 1.02 | 2.84 | 2.09 | 0.48 | 37.25 | 49.47 |

tt-2b | 2.22 | 3.08 | 1.23 | 6.27 | 4.48 | 1.15 | 35.82 | 51.96 |

cc-3b | 0.00 | 3.80 | 0.00 | 0.00 | 0.00 | 0.25 | 0.00 | 0.00 |

ct-3b | 1.44 | 3.46 | 2.09 | 3.29 | 1.51 | 0.27 | 56.32 | 57.83 |

tt-3b | 3.00 | 3.17 | 3.05 | 8.54 | 5.66 | 0.00 | 65.31 | 67.72 |

cc-4b | 0.00 | 1.54 | 0.64 | 0.00 | 0.00 | 0.00 | 27.94 | 31.98 |

ct-4b | 3.92 | 3.96 | 5.86 | 1.59 | 11.12 | 1.22 | 37.82 | 41.82 |

tt-4b | 9.41 | 5.66 | 0.00 | 0.01 | 38.50 | 1.95 | 0.00 | 0.00 |

cc-5b | 0.00 | 1.23 | 0.00 | 0.00 | 0.00 | 0.00 | 180.49 | 180.68 |

ct-5b | 3.05 | 4.44 | 10.21 | 2.66 | 0.01 | 0.91 | 117.01 | 113.67 |

tt-5b | 5.04 | 5.62 | 17.64 | 2.61 | 7.42 | 1.60 | 0.00 | 0.00 |

_{elst}to the identification of the most stable conformation, this quantity allows us to order conformers relative to their E

_{elst}values and the resulting orderings of conformers for

**1a**–

**5a**and

**1b**–

**5b**can be compared with the orderings based on ΔE. The evident relationship between μ and ΔE for

**2a**–

**5a**and their thiocarbonyl counterparts suggests that electrostatic effects are responsible to a certain extent for the ordering of individual conformers relative to their energetic stability. We focus here only on the successful reproduction of conformer orderings and not on the quantitative correlation between individual non-zero ΔE

_{elst}and ΔE values. The values of ΔE

_{elst}should not be compared directly with ΔE because the former are merely a crude approximation of intramolecular electrostatic effects. The electrostatic effects are undoubtedly important but not the sole factor affecting the stability of the investigated conformers. Therefore, the values of ΔE

_{elst}are usually far from the corresponding ΔE values. The values of ΔE

_{elst}obviously inherit the deficiencies of the applied model of partial atomic charge. The ΔE

_{elst}values calculated using the AIM partial atomic charges are usually large for

**1a**–

**3a**and

**1b**–

**3b**, whereas the Hirshfeld partial atomic charges lead to very small ΔE

_{elst}values for the majority of the investigated compounds. This is due to the fact that AIM partial atomic charges generally tend to adopt large absolute values [23], while the Hirshfeld model assigns partial atomic charges that are very small in magnitude [52].

_{elst}) in the investigated compounds, although some inconsistencies with the orderings predicted by ΔE are found. The inconsistencies in the conformer orderings yielded by the NPA model occur mostly for diheteroaryl thioketones. The AIM and Hirshfeld models give less satisfactory results than those of the NPA model. On the other hand, they reproduce the reference orderings of conformers for a greater number of the investigated compounds than the HLY model does. Of the electrostatic potential-based models, HLY produces the conformer orderings that fit best to the corresponding reference orderings indicated by ΔE. All the aforementioned findings are common to the three levels of theory used to generate the wave functions/electron densities of the conformers. This is additionally supported by the results presented in Table 3. This table shows the percentage similarity of the conformer sequences deduced from ΔE

_{elst}to the reference conformer orderings determined in terms of ΔE. The percentage similarity has been obtained through comparing the conformer orderings of all investigated compounds.

**Table 3.**Percentage similarity of the conformer sequences predicted by ΔE

_{elst}to the reference sequences determined using ΔE. The values of ΔE

_{elst}are calculated by 10 partial atomic charge models operating on the wave functions/electron densities calculated at three levels of theory.

Model | Level of Theory | ||
---|---|---|---|

HF | B3LYP | MP2 | |

Mulliken | 50 | 70 | 50 |

NPA | 83 | 73 | 83 |

GDMA | 57 | 53 | 57 |

AIM | 67 | 70 | 80 |

Hirshfeld | 73 | 70 | 70 |

CM5 | 37 | 43 | 47 |

MKS | 60 | 60 | 50 |

CHELP | 23 | 33 | 20 |

CHELPG | 57 | 43 | 53 |

HLY | 60 | 67 | 63 |

## 4. Conclusions

**1a**–

**5a**and

**1b**–

**5b**. These wave functions/electron densities have been calculated at three different levels of theory. The ten models were tested in order to assess their usefulness in performing effective computations on diheteroaryl ketones and thioketones. More specifically, our test assesses the models’ abilities (1) to approximate the magnitude of μ for the conformers of

**1a**–

**5a**and

**1b**–

**5b**, and (2) to correctly determine the conformers’ orderings through the estimation of the electrostatic interaction between partial atomic charges within the conformers.

**1a**–

**5a**and

**1b**–

**5b**. These models are able to reproduce very accurately the reference μ values obtained from the full density, and they perform well for the molecular wave functions calculated at all three levels of theory. Among the models that are not based directly on the molecular electrostatic potential, the CM5 model offers a reasonable accuracy in approximating the values of μ. From the subsequent results of our test it can be concluded that the most successful estimation of the intramolecular electrostatic effects governing the conformational behavior of

**1a**–

**5a**and

**1b**–

**5b**is provided by the partial atomic charges derived from the NPA model. Besides the designation of the most successful model for the determination of the conformational behavior, this part of the test also shows that the simple approach utilizing the calculation of E

_{elst}by means of NPA partial atomic charges is a surprisingly effective yet still qualitative tool to anticipate the energetic orderings of the conformers of

**1a**–

**5a**and

**1b**–

**5b**. It also implies an important role of intramolecular electrostatic effects in determining the conformational behavior of the investigated compounds.

## Supplementary Files

Supplementary File 1## Acknowledgments

## Conflicts of Interest

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Matczak, P.
A Test of Various Partial Atomic Charge Models for Computations on Diheteroaryl Ketones and Thioketones. *Computation* **2016**, *4*, 3.
https://doi.org/10.3390/computation4010003

**AMA Style**

Matczak P.
A Test of Various Partial Atomic Charge Models for Computations on Diheteroaryl Ketones and Thioketones. *Computation*. 2016; 4(1):3.
https://doi.org/10.3390/computation4010003

**Chicago/Turabian Style**

Matczak, Piotr.
2016. "A Test of Various Partial Atomic Charge Models for Computations on Diheteroaryl Ketones and Thioketones" *Computation* 4, no. 1: 3.
https://doi.org/10.3390/computation4010003