A Quantum Theory Atoms in Molecules Study about the Inductive Effect of Substituents in Methane Derivatives.

The substituent effect on the covalent character of C-H bonds in methane derivatives is evaluated by means of local descriptors based on the topology of the electron density. Halogens, -OH, -SH, =O, =S, -NO2, -NH2, and -OCH3 increase the covalent character of the remaining C-H bonds, while alkaline metals (-Li and -Na) result in the opposite trend. This study proposes that the inductive effect is due to polarization changes driven by substituent charges.

Introduction

The inductive effect concept is fundamental in organic chemistry and has been used to rationalize trends observed due to the presence of substituents, which can be labeled as electron-withdrawing or electron-donating groups. For example, one can cite its role in explaining the acidity of organic acids along with the regioselectivity in electrophilic substitution reactions.[1] The traditional definition of the inductive effect describes the charge transmission process through bonds, a phenomenon that is assumed to decay with distance and is directly related to the electronegativity of substituents. However, another transmission mechanism is also mentioned under the field effect nomenclature, which refers to the substituent effect propagation through space. Anyway, these two mechanisms are not easily separable, and the generic inductive effect terminology usually encompasses both them.[2]

Several theoretical studies in the literature have employed the molecular electrostatic potential (MEP) to quantify substituent effects. For example, the MEP values at carbon nuclei were considered by Galabov et al. for addressing substituent effects in monosubstituted benzenes.[3] The differences of the MEP values calculated at the carbon nuclei enabled the evaluation of the electron-withdrawing or electron-donating ability of several substituents in mono- and multi-substituted derivatives of benzene.[4] This previous study also demonstrated the additive effect of substituents. Furthermore, the MEP minimum (Vmin) was employed to investigate the inductive effect along σ bonds by using 4-substituted bicyclo[2.2.2]octane carboxylic acids, their anions, and 4-substituted quinuclidines.[5] The Vmin values showed excellent correlations with inductive substituent constants. Previously, Vmin was also considered to assess the substituent effects in amines.[6] Although studies like these examples are important for providing predictive tools regarding substituent effects, proving that these effects are directly reflected in electronic structure quantities, they do not help significantly in elucidating the transmission mechanism of inductive effects.

Alternatively, Fourré and coworkers investigated the inductive effect in substituted hydrocarbons such as ethane, pentane, bicyclo[2.2.2]octane, and bicyclo[1.1.1]pentane by means of modern topological analyses based on the electron localization function.[1] They noticed that this effect is short-ranged, affecting essentially the carbon atom directly bonded to the substituent and suggesting that any substituent effect propagating farther away from the first bond is caused instead by variations on the solvation shell due to the substituents. In addition, these authors also mentioned that fluorine increases the covalent character of the adjacent C–C bonds, while lithium provides the opposite outcome. Finally, this study also shows that the inductive substituent effect on C–C bonds varies linearly according to the number of fluorine atoms in the series C2H6–F (x = 0–6).

The Quantum Theory of Atoms in Molecules (QTAIM) offers a formal protocol for partitioning molecular properties into atomic contributions, which is based on topological features of the electron density, ρ(r).[7,8] Therefore, the values determined at bond critical points (BCPs) for ρ(r) and its Laplacian, ∇2ρ(r), along with the total energy density, Hb, are used as local parameters to evaluate the covalent/ionic character of interatomic interactions. Larger electron densities and more negative ∇2ρ(r) and Hb values at BCPs are expected for bonds with an increasingly larger covalent character.[9]

A QTAIM study was carried out by Smith et al. in bicyclo[1.1.1]pentane-1-carboxylic acid derivatives with substituents such as −Li, −CH3, −H, −NH2, −OH, −F, −CF3, −NF2, −CN, and −NO2.[2] They discussed that the inductive effect propagates via atomic dipole moment manipulation, which is consistent with the field mechanism, but is propagated along bond paths, in agreement with the traditional inductive effect definition. Our research group also used QTAIM to evaluate the inductive effects of substituents on the covalent character of C–H, C–F, and C–Cl bonds in small organic compounds (methane, fluoro-, chloro- and chlorofluoromethanes).[10] Therefore, the presence of halogens increased the covalent character of these bonds, and an explanation was proposed in terms of polarization effects due to the partial atomic charges of these substituents in such molecules. According to this proposal, the polarization distorts the remaining electron cloud of carbon away from the halogens, leading to an accumulation of electron density along the other bonds.

Here, we decided to extend the aforementioned study of inductive effects on the covalent character of C–H bonds in methane derivatives[10] by considering several substituents (−CH3, −F, −Cl, −Br, −OH, −SH, =O, =S, −NO2, −NH2, −OCH3, −Li, and −Na) and also examining a semiquantitative descriptor that reinforces the explanation previously proposed in terms of polarization due to the atomic charges of substituents.

Computational Details

The Gaussian 09 package[11] was used in electronic structure calculations with the B2PLYP double-hybrid functional[12] and the cc-pVQZ basis sets.[13−17] The required numerical integrations were performed using an ultrafine grid. Thus, the equilibrium geometries were initially determined (tight convergence criteria were employed), and the generalized electron densities obtained at these structures were posteriorly investigated using the AIMAll program.[18] The equilibrium structures presented here were plotted using Gauss View 4.1.2.[19]

The electrostatic potential at the carbon nucleus due to the charge of substituents (VC) is attained by means ofwhere qi is the QTAIM charge of an atom i, and riC refers to the internuclear distance from i to carbon. In the sequence, the electrostatic potential is also evaluated at two additional points along the C–H bonds under study, which are placed at 0.01 Å from each side of the carbon nucleus (Δr = 0.01 Å). This provided V– and V+ values, which are associated with the potentials obtained by eq at the points before and after the carbon nucleus along the C–H line toward hydrogen. Finally, the electric field at the carbon nucleus ascribed to the substituent charges (EC) is obtained by means of a two-point numerical expression for derivatives of the electrostatic potential, that is

Results and Discussion

The equilibrium geometrical parameters obtained in B2PLY/cc-pVQZ calculations are displayed in Table S1 (Supporting Information). The mean absolute deviations (MADs) with respect to the experimental data[20−24] are 0.0085 Å and 0.57o for bond lengths and bond angles, respectively. The dipole moment magnitudes obtained in B2PLY/cc-pVQZ calculations are also displayed in Table S1 along with the available experimental values,[20] and the MAD between them is 0.045 D (the maximum absolute error, MAE, is only 0.11 D). Previous accurate theoretical estimates for CH3Li and CH3Na[25,26] are also in satisfactory agreement with the dipole moments of B2PLY/cc-pVQZ calculations (see Table S1).

Table shows the local descriptors of covalent character in C–H bonds along with the electric field at the carbon nucleus due to the QTAIM charge of substituents (EC), which is determined along the C–H line toward hydrogen. The electron density (ρ), electron density Laplacian (∇2ρ), and energy density (Hb) values at C–H BCPs are presented. There are some molecules with two distinct hydrogen atoms bonded to carbon, as can be seen in Figure S1 (Supporting Information), and the corresponding quantities are listed in Table . Good linear correlations are found between pairs of these covalent character descriptors, with regression coefficients (R2) between 0.985 and 0.995, as one can see in Figure , which illustrates the relation between the electron density and its Laplacian (R2 = 0.995).

Table 1

Electric Field at Carbon Nuclei (E) along with Electron Density (ρ), Electron Density Laplacian (∇2ρ), and Energy Density (Hb) at the BCP of C–H Bonds According to B2PLYP/cc-pVQZ Calculationsa

	Ec (eV/Å)	ρ (a.u.)	∇2ρ (a.u.)	Hb (a.u.)
CH4	0.000	0.286	–1.113	–0.332
CH3F	–1.859	0.299	–1.242	–0.354
CH2F2	–3.594	0.311	–1.368	–0.376
CHF3	–6.619	0.319	–1.464	–0.395
CH3Cl	–0.486	0.297	–1.229	–0.354
CH2Cl2	–1.170	0.306	–1.325	–0.372
CHCl3	–2.074	0.312	–1.400	–0.388
CH3Br	–0.265	0.297	–1.226	–0.354
CH2Br2	–0.736	0.304	–1.310	–0.370
CHBr3	–1.435	0.310	–1.372	–0.384
CH2O	–5.880	0.291	–1.190	–0.336
CH2S	0.885	0.297	–1.244	–0.354
CH3Li	1.769	0.270	–0.959	–0.304
CH3Na	1.016	0.275	–1.003	–0.313
CH2ClF	–2.422	0.309	–1.350	–0.375
CHClF2	–5.022	0.317	–1.444	–0.393
CHFCl2	–3.491	0.315	–1.423	–0.390
CH3CH3	0.381	0.287	–1.110	–0.332
CH3OCH3b	–1.421	0.297	–1.222	–0.351
CH3OCH3c	–2.545	0.290	–1.148	–0.334
CH3OHc	–2.474	0.292	–1.168	–0.339
CH3OHb	–0.939	0.297	–1.222	–0.351
CH3SHb	–0.151	0.290	–1.159	–0.340
CH3SHc	0.098	0.292	–1.174	–0.344
CH3NH2c	–1.289	0.292	–1.163	–0.341
CH3NH2b	–2.159	0.287	–1.115	–0.330
CH3NO2b	0.058	0.300	–1.262	–0.360
CH3NO2c	0.098	0.295	–1.221	–0.351

The EC values along the C–H line are due to substituent charges.

For methyl hydrogen atoms on the symmetry plane (see Figure S1).

For methyl hydrogen atoms out of the symmetry plane (see Figure S1).

Figure 1

Electron density against its Laplacian at the BCP of C–H bonds according to B2PLY/cc-pVQZ calculations.

In sequence, we analyzed the covalent character variation of C–H bonds according to the local descriptors. As one can notice, the hydrogen replacement by groups such as −F, −Cl, −Br, −OH, −SH, =O, =S, −NO2, −NH2, and −OCH3 results in covalent character increments of the remaining C–H bonds. This effect seems to depend on the halogen electronegativities, providing covalent character increments according to the sequence −Br < −Cl < −F. The effect of halogens also shows some additivity patterns, resulting in deviations of, at most, 37.5%, if one uses the values from covalent descriptors in monosubstituted halomethanes to predict the respective descriptor result in methane derivatives containing more than one halogen atom. On the other hand, substituents such as −Li and −Na show the opposite trend, that is, they result in covalent character decrements of the remaining C–H bonds. Finally, the methyl substituent is not capable of causing significant covalent character variations of C–H bonds, as can be demonstrated as one compares methane and ethane. It is important to mention here that our findings are in good agreement with the conclusions attained by Fourré et al.,[1] who have found analogous covalent character variations for C–C bonds according to the replacement of hydrogens attached to such carbons by electron-withdrawing (fluorine) and electron-donating (lithium) substituents. The additivity of the inductive effect from fluorine atoms on the covalent character of C–C bonds was also noticed by these authors.

Therefore, these results suggest that a polarization effect associated with the atomic charges of these substituents may be responsible for the inductive effect. In order to test this hypothesis, we calculated the electric field at the carbon nucleus along the C–H line due to substituent charges (Ec), as seen in Table . Figure illustrates the relation between a local descriptor of the covalent character in C–H bonds, the electron density at the BCPs, and Ec values. As one can notice, the points seem to be distributed along a nonlinear curve (a quadratic polynomial is displayed for comparison).

Figure 2

Electron density at the BCP of C–H bonds against the electric field at the carbon nucleus along the C–H line due to substituent charges according to B2PLY/cc-pVQZ calculations.

The point ascribed to CH2O (Ec = −5.880 eV/Å and ρ = 0.291 a.u.) presents the largest discrepancy with respect to the quadratic curve shown in Figure . However, a probable reason for this discrepancy is that the C–H bond in CH2O is the longest one in all the set of compounds investigated here, 1.1020 Å (see Table S1). In fact, longer C–H bonds are expected to present lower electron densities at their BCPs than more common C–H bonds due to the superposition of more diffuse regions of the respective atomic electron densities during the chemical bond formation. An additional calculation performed for CH2O by changing only the C–H bond length to that value seen in methane provided ρ = 0.301 a.u. and Ec = −5.822 eV/Å. Hence, although the new results are in better agreement with the quadratic line seen, there are certainly other discrepancy sources. For example, one could argue that the field due to substituents must be refined by including some corrections due to nonspherical atomic basins of these substituents. Therefore, we also calculated the electrostatic field contributions due to atomic dipoles and atomic quadrupoles of substituents from QTAIM, which are shown in Figures S2–S4 (Supporting Information). However, we noticed that the inclusion of atomic dipoles provokes a slight decrease in the regression coefficient for fitting the quadratic polynomial curve. Furthermore, the addition of atomic quadrupoles leads to a much worse quadratic regression.

A possible explanation for the poorer quadratic regression coefficients obtained when higher-order atomic multipoles of substituents are taken into account is that different portions of substituent electron clouds may be more or less effective for inducing polarizations, leading to these substituent effects. For instance, the polarization effect of nonbonding substituent electron pairs over the C–H covalent character may not be so effective. Therefore, the contribution of these nonbonding electrons will be almost negligible for the derived atomic charge values (explaining the better regressions seen for the field obtained solely from such charges). However, the nonbonding electron pairs of substituents will contribute much more to the electric field at the carbon nucleus as the atomic dipoles and quadrupoles of these substituents are taken into consideration, and this would lead to worse regressions if the polarization effects of these nonbonding electrons are less effective. Anyway, further investigations are required to confirm this hypothesis.

In summary, the pattern observed in Figure supports our initial idea of the inductive effect due to the field produced by substituent charges. Therefore, the inductive effect propagates through space (field effect) but depends on the presence of chemical bonds along the way, which are polarized by the field ascribed to the substituent charges. In this way, our study also shows some accordance with the conclusions attained by Smith and coworkers regarding the propagation mechanism of the inductive effect.[2] The explanation proposed here for inductive effects, that is, polarization due to the field generated by substituent charges, is also closely related to the so-called charge transfer–counter polarization mechanism preconized by Bader and Matta.[27] In that study, simple physical arguments are mentioned to argue that atomic polarizations (dipolar, quadrupolar, and so on) should certainly arise as long as electronic charge transfers occur during chemical bond formation, leading to partial atomic charges. The atomic dipolar polarizations produce a molecular dipole moment contribution in an opposite direction with respect to the dipole moment contribution associated solely with atomic charges, as generally observed in the QTAIM investigations of several systems.

In addition, the contour maps of electron density are plotted with AIMAll[18] for some mono- and multi-substituted methane derivatives. Figure shows a set of representative monosubstituted compounds, LiCH3, NaCH3, CH3F, and CH3Cl, along with methane. It is easy to notice how chlorine and fluorine induce a distortion of the electron cloud of carbon in the direction of hydrogen when compared to methane (for example, see the line associated with electron densities equal to 0.30 a.u.). On the other hand, the opposite effect is also noticeable for lithium and sodium. Figure shows the fluoromethanes. In agreement with our previous discussions, larger displacements of the electron density toward hydrogen are clearly evidenced as more fluorine atoms are encountered in the structure.

Figure 3

Contour maps of the electron density at the HCX (X = Li, Na, H, Cl, and F) plane of methane and monosubstituted derivatives obtained according to B2PLY/cc-pVQZ calculations (the most external lines are associated with 0.02, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, and 0.40 a.u., respectively). Bond paths and BCPs (small green balls) are also presented.

Figure 4

Contour maps of the electron density at the HCF plane of fluoromethanes (CH3F, CH2F2, and CHF3 from left to right) obtained according to B2PLY/cc-pVQZ calculations (the most external lines are associated with 0.02, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, and 0.40 a.u., respectively). The bond paths and BCPs (small green balls) are also presented.

Conclusions

This study shows that three local descriptors of the covalent/ionic character at the BCPs of C–H bonds in methane derivatives are linearly correlated with each other. Moreover, substituents such as halogens, −OH, −SH, =O, =S, −NO2, −NH2, and −OCH3 provoke increments in the covalent character of the remaining C–H bonds, while alkaline metals (−Li and −Na) result in the opposite trend. The inductive effect of halogens also shows some additivity patterns.

Finally, we examined the relationship between the electric field at the carbon nucleus due to substituent charges, which was determined along the C–H line toward hydrogen, and the local descriptors of the covalent character studied. A nonlinear correlation curve seems to be observed, supporting the idea proposed in our previous study,[10] that is, the inductive effect is due to polarizations caused by the partial charges of substituents. Therefore, the inductive effect propagates through space (field effect), but the transmission depends on the presence of chemical bonds along the path, which will be polarized by the field.