*Article* **Revision and Extension of a Generally Applicable Group Additivity Method for the Calculation of the Refractivity and Polarizability of Organic Molecules at 298.15 K**

**Rudolf Naef 1,\* and William E. Acree, Jr. <sup>2</sup>**


**Abstract:** In a continuation and extension of an earlier publication, the calculation of the refractivity and polarizability of organic molecules at standard conditions is presented, applying a commonly applicable computer algorithm based on an atom group additivity method, where the molecules are broken down into their constituting atoms, these again being further characterized by their immediate neighbor atoms. The calculation of their group contributions, carried out by means of a fast Gauss– Seidel fitting calculus, used the experimental data of 5988 molecules from literature. An immediate subsequent ten-fold cross-validation test confirmed the extraordinary accuracy of the prediction of the molar refractivity, indicated by a correlation coefficient R2 and a cross-validated analog Q<sup>2</sup> of 0.9997, a standard deviation σ of 0.38, a cross-validated analog S of 0.41, and a mean absolute deviation of 0.76%. The high reliability of the predictions was exemplified with three classes of molecules: ionic liquids and silicon- and boron-containing compounds. The corresponding molecular polarizabilities were calculated indirectly from the refractivity using the inverse Lorentz–Lorenz relation. In addition, it could be shown that there is a close relationship between the "true" volume and the refractivity of a molecule, revealing an excellent correlation coefficient R2 of 0.9645 and a mean absolute deviation of 7.53%.

**Keywords:** group additivity method; Gauss–Seidel diagonalization; refractivity; polarizability; ionic liquids; silanes; boranes

#### **1. Introduction**

In continuation of an earlier paper [1], which used a generally applicable atom groups additivity method for the prediction of various molecular descriptors including the refractivity and the polarizability of molecules, the present work puts the focus on the latter two descriptors, for which on the one hand, an extended number of further experimental refractivity data has been included in the atom group parameters calculation, and on the other hand, a different method for the prediction of the polarizability has been introduced, this time based on the former descriptor. The main goal of the present work was to not only increase the reliability of the atom group parameters already published in [1], but in particular to extend the number of atom groups for which as yet no parameter values have been available, with the main interest aimed at atom groups found in ionic liquids. In addition to these, parameters for a large number of additional groups with boron and silicon as central atom could be generated, thus enabling the prediction of the refractivities and polarizabilities of many boranes and silanes.

Earlier calculations of the refractivity and polarizability have been based on the bond refraction and bond polarizability, respectively, on the assumption that the molar refraction and polarizability is the sum of all the bonds in the molecule [2]. The average error between experiment and calculation was 0.7% over a number of less than 100 sample molecules. Later on, Ghose and Crippen [3] developed a method based on 110 atom types,

**Citation:** Naef, R.; Acree, W.E., Jr. Revision and Extension of a Generally Applicable Group Additivity Method for the Calculation of the Refractivity and Polarizability of Organic Molecules at 298.15 K. *Liquids* **2022**, *2*, 327–377. https://doi.org/10.3390/ liquids2040020

Academic Editor: Enrico Bodo

Received: 30 August 2022 Accepted: 7 October 2022 Published: 13 October 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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characterized by the polarizing effect of the heteroatoms and the effect of overlapping with non-hydrogen atoms, again assuming that the sum of all the atom parameters defines the molecular descriptor value. Applying a quadratic, constrained least squares technique for the evaluation of the atom type parameters for 504 molecules, they reported a correlation coefficient of 0.994 and a standard deviation of 1.269. Except for the parameters calculation approach, Ghose and Crippen's method compares closely with the present one, since their atom types follow a similar principle and therefore the present results may best be compared with theirs. Another group additivity approach was chosen by Miller [4,5] for the calculation of the molecular polarizability, whereby the atoms are defined by their state of hybridization, neglecting their neighbor atoms.

The importance of the knowledge of the refractivity and polarizability for the modeling of the dispersive and hydrophobic interactions was outlined in detail by Ghose and Crippen [3]. The attractive forces between nonpolar compounds, also known as dispersive forces, are the result of the correlated motions of their electrons. These forces are evidently closely related to the polarizability of the molecules. Their polarizability again is linearly proportional to their refractivity, given by the Lorentz–Lorenz relation R = 4/3πNα, where R is the molar refractivity, N is Avogadro's constant, and α is the polarizability. Accordingly, and in contrast to our earlier calculations of the polarizability by means of the group additivity method in [1], the present polarizabilities are directly evaluated from the molar refractivities, with the added bonus that the amount of experimental refractivity data is much larger than that of the polarity, thus enabling the prediction of molecular polarities for which in its atom groups parameter set in [1], no atom groups are defined. It has also been shown that the molecular polarizability is directly proportional to the molecular volume [6]. Hence, on combining the polarizability/volume and polarizability/refractivity correlations, there should be a direct correlation of the refractivity with the molecular volume as postulated by Ghose and Crippen [3]. It would therefore be interesting to see if there is indeed a direct correlation of the refractivity with the "true" molecular volume as applied for the prediction of the heat capacity of solids and liquids in an earlier paper [7].

#### **2. Method**

The calculations were carried out on a set of 5988 compounds for which the experimental refractivity or polarizability data have been published, collected from a database of at present 35,952 molecules in their geometry-optimized 3D conformation, encompassing pharmaceuticals, plant protectors, dyes, ionic liquids, liquid crystals, metal-organics, intermediates, and many more, including many further experimentally determined and calculated molecular descriptors. The structural presentations were standardized before storage by a special algorithm, ensuring that all six-membered aromatic ring systems are defined by six aromatic bonds in order to avoid structural ambiguities. In addition and for the same reason, the positive charge in amidinium, pyrazolium, and guanidinium fragments of the ionic liquids was manually positioned on the carbon atom between the nitrogen atoms and their C(+)-N bonds were assumed to be aromatic, which incidentally is in better conformance with the true charge distribution in these cations, as exemplified in, e.g., Figure 1 in [8]. The analogous treatment of the carboxylate and nitro groups is not necessary, as within the present concept of atom groups definitions, they are unambiguously defined.

#### *2.1. Definition of the Atom Groups*

Details of the definition of the atom groups for use in a computer-readable form were outlined in [1]. In Table 1 of [1], their namings and meanings were explained; they have been retained in all the subsequent papers including the present one. However, in order to cover the successively increasing amount of additional, structurally variable molecules, several further atom groups had to be added to the parameters list. In particular, the inclusion of ordinary salts and ionic liquids as well as a number of boron- and siliconcontaining molecules required the corresponding atom groups listed and explained in Table 1 on some examples. These new atom groups were interpreted and processed by the

computer algorithm in the same way as the remaining ones. In fact, some of these have already been applied in the calculation of the liquid viscosity of molecules in [8].


**Table 1.** Examples of charged or boron- or silicon-containing atom groups and their meaning.

#### *2.2. Calculation of the Atom Group Contributions*

As outlined in [1], the parameter values of the atom groups are evaluated in four steps: in the first step, those compounds for which the experimental refractivities are known are stored in a temporarily generated help list. In the second step, each molecule in the help list is broken down into its constituting "backbone" atoms (i.e., atoms bound to at least two directly bound neighbor atoms), their atom types and neighbor terms defined according to the rules detailed in [1], and then their occurrences counted. The third step involves the generation of an M × (N + 1) matrix, wherein M is the number of molecules, N + 1 is the complete number of atom groups occurring plus the molecules' refractivity value, and where each matrix element (i,j) receives the number of occurrences of the jth atom group in the ith molecule. The final step comprises the normalization of this matrix into an Ax = B matrix and its subsequent balancing by means of a fast Gauss–Seidel calculus [9] to receive the atom group contributions x, which are stored and shown in Table 2, together with the corresponding statistics data at the bottom in lines A to H.

**Table 2.** Atom groups and their contribution in refractivity calculations.


**Table 2.** *Cont.*


**Table 2.** *Cont.*


**Table 2.** *Cont.*


**Table 2.** *Cont.*


**Table 2.** *Cont.*


**Table 2.** *Cont.*


**Table 2.** *Cont.*


**Table 2.** *Cont.*


**Table 2.** *Cont.*


**Table 2.** *Cont.*


#### *2.3. Calculation of the Refractivity*

The calculation of the refractivity of a molecule, based on the atom group parameters compiled in Table 2, is a simple summing up of the contribution of each atom group found in a molecule, as exemplified in Table 3 for 1-butyl-3-methylimidazolium tetrafluoroborate (Figure 1), for which the experimentally evaluated refractivity value was 47.81 [10]. The parameters for the monoatomic anions found among some ILs are given under the respective "group" names "Chloride" and "Bromide". Any further halogenide anion can be taken into account analogously as soon as the experimental data of at least three representative compounds are available.


**Table 3.** Example calculation of the refractivity of 1-butyl-3-methylimidazolium tetrafluoroborate.

**Figure 1.** 1-Butyl-3-methylimidazolium tetrafluoroborate.

It goes without saying that this calculation method is limited to compounds for which each atom group is defined by a parameter value in Table 2. In addition, as the reliability of these parameter values increases with the number of independent molecules upon which they are based, only atom groups should be considered for which the number of molecules in the rightmost column of Table 2 is three or more, which are henceforth called "valid". (It could be shown by means of several cross-validation calculations that the decrease of the cross-validated standard deviation on going from three to four molecules per atom group is insignificant compared with the decrease observed when going from two to three molecules per atom group.) Consequently, the number of molecules for which the refractivity values have been calculated (lines B, C, and D in Table 2) is necessarily smaller than the number upon which the calculation of the complete set of parameters is based (line A in Table 2).

#### *2.4. Cross-Validation Calculations*

The calculations of the atom group parameters are immediately followed by a plausibility test applying a 10-fold cross-validation algorithm comprising 10 recalculations omitting in each case a different tenth of the complete set of compounds, ensuring that each compound has been used once as a test sample. The resulting training and test data are added to the molecule's datafiles. Finally, the corresponding statistics data are evaluated and collected at the bottom of Table 2. Due to the smaller number of training molecules in the cross-validation calculations and the condition that only atom group parameters should be considered in the calculation of the individual refractivities for which the number of molecules in the rightmost column is three or more, the number of molecules with calculated refractivities (lines E, F, G, and H) is again lower in the test set than in the training set (lines B, C, and D). Atom group parameters with molecule numbers below three in the rightmost column, which are accordingly at present not applicable for refractivity calculations, have deliberately been left in Table 2 for future use in this continuing project and not least in the hope that interested scientists may assist in increasing the number of "valid" groups in this parameters list by compounds carrying the underrepresented atom groups. At present, the list of elements for refractivity/polarizability calculations is limited to H, B, C, N, O, P, S, Si, and/or halogen, but is easily extendable to enable the parametrization of atom groups containing additional elements for which experimental densities and refractive indices are available.

#### *2.5. Calculation of the Polarizability*

According to the Lorentz–Lorenz relation R = 4/3πNα, N being Avogadro's number, the refractivity R of a molecule can be translated into its polarizability α by simply multiplying its refractivity value with the reciprocal value of 4/3πN, which is 0.3964, if the refractivity is expressed in mL and the polarizability in A3. Therefore, in this study, for each input experimental refractivity value, the corresponding polarizability value was also evaluated and stored as experimental value in the database, and vice versa. The latter is all the more justified as in many (if not most) cases, the polarizability value was evaluated via the refractivity value. Accordingly, the number of experimental data for these two descriptors is identical, and so is their list of atom group parameters. As a consequence, calculation of a molecule's refractivity value by means of the group additivity method, based on the refractivity parameters in Table 2, immediately enabled the calculation of its polarizability value by simply multiplying it by 0.3964.

#### **3. Sources of Refractivity and Polarizability Data**

In most cases, it was not the refractivity value itself that was published in the following references but the refractive index (nd) and the density (d) of the molecules, which then had to be translated into the refractivity (R) according to the equation R = (nd <sup>2</sup> − 1)/ (nd <sup>2</sup> + 2) × (M/d), where M is the molecular weight. The primary sources of the refractivity data for the earlier [1] as well as the present study were the comprehensive CRC Handbook of Chemistry and Physics [11] and the collective work of Ghose and Crippen [12]. Within the last 7 years since the first publication dealing with the present subject however, a large number of further papers has been collected producing additional refractivity and polarizability data which helped to extend the scope of applicability of the atom group additivity method, particularly for boron- and silicon-containing compounds and ionic liquids. In the following, they have been sorted by their dominant functional features. Within the last ca. 85 years, many papers have been published producing the refractive indices and densities to characterize various hydrocarbons [13–34], alcohols [35–42], ethers [43–47], acids [48], (ortho)esters and carbonates [49–62], acetals [63,64], ketones [65,66], peroxides [67–71], amines, hydrazines, nitriles, and nitro compounds [72–78], and various boron- [79–96], phosphorus- [97–136], and sulfur-containing compounds [137–160]. Many of the compounds mentioned so far also carried halogens [161–182,182–196]. An interesting extension to the parameters database was provided by papers presenting results of silicon-containing compounds [197–240]. Beyond the refractivity data of the various mentioned functional groups, those for a number of hetarenes and heterocycles have been published [241–260]. Another important extension that was not covered in the earlier paper [1] is the class of ionic liquids [10,261–366]. In addition, several papers have been added which contributed various subjects that could not be assigned to any specific subject of the aforementioned ones [2,367–385]. Finally, a number of papers published experimental data of the polarizability of molecules, in many cases derived from their refractivity values [6,386–395].

#### **4. Results**

#### *4.1. Refractivity*

In the paper of Ghose and Crippen [3] mentioned earlier, it was stated that the molar refractivity is directly related to the molecule's volume, expressed in the refractivity's unit "mL", their atom group parameters accordingly being associated with the volume of the molecule's constituting atoms. The present approach, on the other hand, does not care about the theoretical background of the refractivity as it is a purely mathematical method to adjust the calculated to the experimental data, and therefore the resulting atom group parameters must not be assigned with any physical meaning. Consequently, as can be seen in Tables 2 and 3, negative parameter values are not unusual.

While in the earlier paper [1], generally no limit was given concerning the deviation of the experimental data from the calculated ones for the evaluation of the atom group parameters, in the present work, the atom group parameters in Table 2 and the statistics data at its bottom (Lines A to H) are the result after a stepwise elimination of outliers, defined as their experimental value deviating from the calculated one by more than three times the cross-validated standard error Q2. The final list of discarded outliers is available in the Supplementary Materials. As a consequence, 5988 of the originally 6501 compounds with experimental data remained for the evaluation of said parameters, leaving ca. 7.9% as outliers. Due to the elimination of the outliers, the statistical data significantly improved in comparison with those in the earlier paper: not only is the present set of parameters based on a significantly larger number of molecules (5988 vs. 4300, rows A in present Table 2 vs. corresponding Table 13 of [1]) and a larger number of atom groups (562 vs. 364), but also their standard deviation (0.38 vs. 0.66, rows D) and the cross-validated deviation (0.41 vs. 0.7, rows H) drastically improved. Together with the corresponding correlation coefficients R<sup>2</sup> and Q<sup>2</sup> of 0.9997 (rows B and F in Table 2) in the present work, they compare very favorably with the correlation coefficient of 0.994 and the standard deviation of 1.269 published by Ghose and Crippen [3]. The mean absolute percentage deviation (MAPD) of the finally calculated refractivities from the experimental values of 5988 molecules is 0.76%. The increased number of "valid" atom group parameters enabled the calculation of the refractivities and polarizabilities of ca. 80% of the close to 36,000 molecules in the present database, which can be viewed as representative for the entire chemical realm. The excellent correlation between experiment and prediction is visualized in the correlation diagram of Figure 2. The corresponding histogram in Figure 3 confirms the uniform distribution of the deviations between the experimental and calculated refractivities, their experimental values ranging from 8.23 (methanol) to 271.13 (glycerol tristearate). The complete set of compounds with experimental refractivities used for the atom group parameters of Table 2 are available in the Supplementary Materials.

An interesting observation can be made with respect to the outliers in that many of them are solids. Cao et al. [396] showed that solid compounds can exhibit up to three differing refractive indices, depending on their crystal symmetry. A typical example is Ibuprofen, a non-steroidal antirheumatic, which shows the three refractive index values 1.522, 1.572, and 1.644. With its reported density of 1.119 g/cm<sup>3</sup> and a refractivity of 60.95, calculated by means of our group additivity model, we calculated a refractive index of 1.575, which is pretty close to the mean of the three experimental values. Analogous results have been found with several other outliers, confirming the assessment that in cases where the experimental refractive index strongly deviates from the predicted one, the reason might be that a specific crystal form of the compound was examined. Since these mean refractive index values usually do not represent real crystalline forms, they were not included in the group parameters optimization procedure.

Since the last paper [1] of 2015, the systematic screening of the chemical literature has provided a number of refractivities data of previously under-represented classes of molecules enabling, as mentioned earlier, a substantial increase of the number of atom group parameters in the present Table 2 compared with the one of Table 13 in [1]. In particular, three classes of compounds have experienced an extended representation and will in the following be discussed in more detail: ionic liquids and silicon- and boroncontaining compounds.

**Figure 2.** Correlation diagram of the refractivity data. Cross-validation data are superpositioned as red circles. (10-fold cross-valid.: N = 5763, Q<sup>2</sup> = 0.9997, regression line: intercept = 0.0292; slope = 0.9995, MAPD = 0.76%).

**Figure 3.** Histogram of the refractivity data. Cross-validation data are superpositioned as red bars. (σ = 0.38; S = 0.41; experimental values range from 8.23 to 271.13).

#### 4.1.1. Ionic Liquids

In the last ca. 25 years, ionic liquids (ILs) have experienced increasing attention as potential replacements for volatile solvents as they are non-volatile, non-inflammable, and can easily be recycled. Their physical properties are easily tunable by suitable choice of their cation and anion, making them favorable candidates as media for chemical syntheses. The enormous variety of potential cation–anion combinations, however, obliges one to put the focus on those candidates with the most promising properties. In the last few years, a substantial number of ILs have been synthesized and their physical properties have been examined. Based on these results, few attempts have been made so far to utilize these results for the prediction of the physical properties of as yet unknown cation–anion combinations, and if yes, then for a narrow scope within the scientists' range of experience (see, e.g., Almeida et al. [317]), or as in other cases, as in the papers of Sattari et al. [335] or of Venkatraman et al. [359], for the prediction of a specific property based on a fairly large range of ILs by either applying quantitative structure–property relationship (QSPR) technique or machine learning. The present atom group additivity approach, on the other hand, has proven its versatility in that it is able to predict a number of properties of nearly any type of compound by means of an identical algorithm, simply using the appropriate atom group parameters tables. Accordingly, based on the updated parameters tables in

this ongoing project, we have been able to calculate the heat of combustion [397] for 30 ILs with a correlation coefficient R2 of 1.0 and a mean average percentage deviation from experimental values (MAPD) of 0.21% and a standard deviation σ of 17.75 kJ/mol, the heat of vaporization [398] of 61 ILs (R<sup>2</sup> = 0.9615, MAPD = 2.12%, σ = 4.22 kJ/mol), the liquid viscosity [8] for 113 ILs (R<sup>2</sup> = 0.9830, MAPD = 3.43%, σ = 0.11 J/mol/K), the surface tension [399] of 161 ILs (R<sup>2</sup> = 0.8413, MAPD = 5.17%, σ = 2.40 dyn/cm), and the liquid heat capacity at 298 K [400] of 140 ILs (R2 = 0.9986, MAPD = 1.05%, σ = 7.50 J/mol/K). In analogy to these results, the refractivity values of 228 ILs calculated by means of the atom group parameters of Table 2 were compared with their experimental data and collected alphabetically in Table 4, revealing a MAPD of only 0.44% and a σ of 0.38. For comparison: the statistics for the 203 ILs for which, while serving as test samples in the cross-validation calculations, the test results could be calculated, yielded an only slightly inferior MAPD of 0.51% and a σ of 0.44.

**Table 4.** Calculated and experimental refractivity of ionic liquids.


#### **Table 4.** *Cont.*


#### **Table 4.** *Cont.*


#### **Table 4.** *Cont.*


4.1.2. Silanes, Silanols, Siloxanes, Silazanes, and Silicates

Silicon-containing compounds have found use in synthetic processes as intermediates as well as in commercial products, e.g., in detergents, cosmetics, deodorants, soaps, as water-resistant coatings, as defoaming agents, or as coolants. Despite the large variety of applications, the number of physico-chemical data for this class of molecules is fairly limited within the chemical realm. Nevertheless, a thorough scan of the literature of the last ca. 80 years delivered a sufficient number of data to enable the creation of a basis for the prediction of several chemical descriptors of interest based on the present atom group additivity principle. Accordingly, in analogy to the previous section, the updated parameters tables provided the group parameters for the heat of combustion [397], enabling its calculation for 99 silicon compounds with a correlation coefficient R2 of 1.0, a MAPD of 0.19% and a σ of 17.67 kJ/mol, for the heat of vaporization [398] for 106 (R<sup>2</sup> = 0.7936, MAPD = 10.91%, σ = 6.17 kJ/mol), for the surface tension [399] of 18 (R<sup>2</sup> = 0.9835, MAPD = 2.62%, σ = 0.66 dyn/cm), for the liquid heat capacity at 298 K [400] of 26 (R2 = 0.9981, MAPD = 2.32%, σ = 9.02 J/mol/K), for the solid heat capacity at 298 K [400] of 14 (R<sup>2</sup> = 0.9925, MAPD = 2.77%, σ = 18.35 J/mol/K), for the standard entropy of fusion [398] of 45 (R2 = 0.7251, MAPD = 15.09%, σ = 15.56 J/mol/K), and even for the vapor pressure at 298 K [401] of 9 silicon compounds (R2 = 0.9897, MAPD = 7.92%, σ = 0.16). In addition to these descriptors, the present work now provides the refractivity data for 351 silicon derivatives, alphabetically sampled in Table 5. They prove the reliability of the calculated values with a MAPD of only 0.39% and a standard deviation σ of 0.31, compared with their experimental values. Analogously, when used as test samples in the cv calculations, 324 of these silicon compounds yielded a MAPD of 0.47% and a σ of 0.37.

**Table 5.** Calculated and experimental refractivity of silicon-containing compounds.



**Table 5.** *Cont.*


**Table 5.** *Cont.*


**Table 5.** *Cont.*

**Table 5.** *Cont.*



**Table 5.** *Cont.*


**Table 5.** *Cont.*

#### 4.1.3. Boranes, Borines, Borazines, Boronates, and Borates

In contrast to the prior two sections, boron-containing compounds are essentially important intermediates in chemical syntheses, and therefore experimental physico-chemical data are scarce. The large number of refractivity data, on the other hand, is primarily owed to the need to characterize the newly synthetized molecules by some easily accessible physical data, such as elemental analysis, melting point, density, and refractive index. With a few exceptions (e.g., Christopher et al. [80]) however, most authors have not shown any interest in using the latter two values for the calculation of the molecules' refractivity or polarizability. The present collection of refractivity data for 137 boron compounds listed in Table 6, although perhaps of merely academic interest, nevertheless confirms—by the strong linearity over the complete set—the overall correctness of the experimental data and at the same time proves the versatility of the present group additivity approach, revealing a MAPD of just 0.46% and a σ of 0.37. Similarly, the test data of 127 of these boron derivatives, when applied as test samples in the cv calculations, resulted in a MAPD of 0.54% and a σ of 0.45.


**Table 6.** Calculated and experimental refractivity of boron-containing compounds.

**Table 6.** *Cont.*



**Table 6.** *Cont.*

#### *4.2. Polarizability*

The calculation of the molecular polarizabilities was carried out indirectly via the calculated refractivities applying the inverse Lorentz–Lorenz relation. In order to include the relatively limited number of experimentally determined polarizability data in the atom group parameters and any further calculations, they were translated into the corresponding refractivity values and henceforth treated just like the remaining experimental refractivities. Conversely, all the experimental refractivity values were analogously converted into "experimental" polarizabilities. The complete set of true and indirectly determined experimental polarizability values is compared in Figure 4 with the indirectly calculated polarizability values, mirroring the excellent correlation of Figure 2, which at first sight is not surprising as both value sets are multiplied with the same factor. However, we should not forget that the truly experimentally determined polarizability values were evaluated by various methods that differ from those for the experimental determination of the refractivity. In fact, as the histogram in Figure 5 reveals, it turned out that 23 compounds should be viewed as outliers because their experimental refractivity values deviated by more than three times the standard deviation σ of 0.15 A<sup>3</sup> from calculations. They are collected in a separate list, available in the Supplementary Materials, together with the complete set of compounds with experimental and calculated polarizabilities.

In a paper by Tariq et al. [273], the applicability of the Lorentz–Lorenz relation was questioned for ILs because it is based on the assumption of the compounds being "isotropic fluids composed of spherical and non-interacting particles" which is not given with this class of salts, since at least one of its ions is non-spherical, and they are clearly nonisotropic fluids as they consist of polar centers surrounded by non-polar moieties. These considerations are certainly justified with respect to the relationship between refractivity and polarizability of ILs. In the following section, however, we will demonstrate that the non-spherical character of the ILs is no obstacle for a reasonable correlation between molecular volume and refractivity.

**Figure 4.** Correlation diagram of the polarizability data (in A3). (N = 5763, R2 = 0.9997, regression line: intercept = 0.0115; slope = 0.9995, MAPD = 0.72%).

#### *4.3. Refractivity/Polarizability and Molecular Volume*

A paper of Brinck et al. [6] discussed the relationship between the polarizability of a molecule and its volume, arguing that physically "the polarizability α of a conducting sphere of radius R is equal to R3", its relation expressed by the equation α = 3V/4π, where V is the volume. This relation is true on condition that the electrostatic potential is uniform within this sphere, which is certainly not the case in a molecule. An approximate equation, known as the Clausius–Mossotti equation, proposes for nonpolar molecules the polarizability as being directly proportional to their volume and a function of their dielectric constant. Several approaches for the calculation of the molecular volumes have been chosen in order to assess their applicability for polarizability predictions. Gough [402], Laidig and Bader [403], and Brinck et al. [6] used various Hartree–Fock self-consistent field methods to compute the volumes of a limited number of small molecules and achieved good linearity with their polarizability, depending on the size of the contour of the electronic density defining the molecule's surface. In an earlier paper [7], we presented a fast numerical method for the calculation of the "true" molecular volume (in A3) of molecules of any size and type, including ILs, based on the atoms' Van-der-Waals radii. Since these "true" (elsewhere also called "hardcore") volumes are automatically generated on entering a new compound to the database, it was obvious to examine their potential linearity with their experimental polarizability or refractivity as far as available. In Figure 6, the correlation between the "true" molecular volume and the experimental refractivity of 6069 molecules is shown, revealing an excellent correlation coefficient R2 of 0.9645 and a MAPD of 7.53%.

Figure 7 presents the same correlation diagram, but restricted to the class of ILs, indicating that the path of prediction of their refractivity R via their molecular volume V as calculated in [7] and applying the simple linear equation R = intercept + (V × slope) provides a reliable refractivity value with a MAPD of little more than 5% within the ILs class over a large range of molecular volumes, if the atom group additivity method does not allow a calculation due to the limitations mentioned earlier. The complete list of molecules with their "true" molecular volume and experimental and volume-derived refractivity is available in the Supplementary Materials.

**Figure 5.** Histogram of the polarizability data (σ = 0.15 A3; exp. values range from 3.23 to 107.53 A3).

**Figure 6.** Correlation diagram of "true" molecular volume (in A3) [7] vs. experimental refractivity. (N = 6069, R<sup>2</sup> = 0.9645, regression line: intercept = 1.4354; slope = 0.2743, MAPD = 7.53%).

**Figure 7.** Correlation diagram of "true" molecular volume (in A3) [7] vs. experimental refractivity of ILs. (N = 247, R2 = 0.9700, regression line: intercept = <sup>−</sup>2.4557; slope = 0.2686, MAPD = 5.35%).

#### **5. Conclusions**

In several earlier papers [1,7,8,397–401], the present atom groups additivity algorithm, outlined in [1], proved its formidable versatility for the reliable prediction of up to 17 physical, thermodynamic, solubility-, optics-, charge-, and environment-related descriptors. In the present work, which is part of an ongoing project, the results of the present refractivity/polarizability calculations again demonstrate its as-yet unsurpassed accuracy and easy expandability. The nearly 6000 molecules providing their experimental refractivity or polarizability values, either directly or via their refractive index and density, enabled the calculation of a large set of atom group parameters allowing the refractivity/polarizability of nearly 80% of the compounds listed in a database of presently approaching 36,000 of nearly any molecular structure, size, and application. The big advantage of the present method is the basic possibility to calculate the refractivity simply by means of paper and pencil applying the parameters set listed in Table 2. In addition, we have shown that optional refractivity/polarizability calculations are possible via the molecular volume route—although with lower accuracy—in cases where the group additivity method is disabled.

The mentioned project's software is called ChemBrain IXL, available from Neuronix Software (www.neuronix.ch, 1.1.2015, Rudolf Naef, Lupsingen, Switzerland).

**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/liquids2040020/s1, The list of compounds used in the present work, their experimental data, and 3D structures are available online as standard SDF files, accessible for external chemistry software, under the name of "S01. Compounds List for Refractivity-Parameters Calculations.sdf". The list of the compounds used in the correlation diagrams and histograms containing their names and their experimental and calculated values are available under the corresponding names of "S02. Experimental vs. Calculated Refractivities.doc", "S03. Experimental vs. Calculated Polarizabilities.doc" and "S04. Molecular Volume vs. Refractivity Data Table.doc". Separate analogous lists are available for ionic liquids under the name of "S05. Experimental vs. Calculated Refractivities of Ionic Liquids.doc", for silicon compounds under the name of "S06. Experimental vs. Calculated Refractivities of Silicon Compounds.doc", and for boron compounds under the name of "S07. Experimental vs. Calculated Refractivities of Boron Compounds.doc". In addition, two lists containing the outliers in the calculations of the refractivity and polarizability of molecules are available under the names of "S08. Refractivity Outliers.doc" and "S09. Polarizability outliers.doc". Finally, the figures are available as .tif files and the tables as .doc files under the names given in the text.

**Author Contributions:** R.N. developed project ChemBrain and its software upon which this paper is based, and also fed the database, calculated, and analyzed the results, and wrote the paper. W.E.A.J. suggested the extension of ChemBrain's tool and contributed experimental data and the great majority of the literature references. Beyond this, R.N. is indebted to W.E.A.J. for the many valuable discussions. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data presented in this study are available in Supplementary Material.

**Acknowledgments:** R.N. is indebted to the library of the University of Basel for allowing him full and free access to the electronic literature database.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

