Matta, Boyd. The quantum theory of atoms in molecules
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38214 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry
Table 14.2 Electron population, atomic energy, localization, and delocalization indices of carbon atom in ethane (au) from QCISD/ 6-311þþG(d,p) calculations.[a]
Angle |
N(C) |
E(C) |
l(C) |
D(C) |
d(C, CO) |
|
|
|
|
|
|
0 |
5.9258 |
37.87012 |
4.2044 |
1.7214 |
0.8422 |
60 |
5.9189 |
37.87419 |
4.1930 |
1.7259 |
0.8535 |
a Ref. [26].
during the process is given by the shift of the charge centroid, given by minus the dipole moment of the methyl group, 0.001 au away into the nonbinding region of the molecule.
In the same process, the number of electrons located on a carbon atom increases by 0.0114 e whereas its total delocalization decreases by 0.0044 e. The main contribution to the latter, 0.0113 e, comes from the lower CaC delocalization (Fig. 14.2). In addition, the electrons a carbon shares with an hydrogen atom remain almost unchanged. The QCISD values of l(C) are 4.2044 e and 4.1930 e for the eclipsed and staggered conformations, respectively. The main change in delocalization is that between C1 and C2, increasing in the staggered conformation by 0.0113 electron pairs (ep). The delocalization between C1 and the H atoms bonded to it decreases in the staggered conformation by 0.0017 ep whereas the CaH bonds remain almost unchanged. The delocalization between C1 and the hydrogen atoms bonded to C2 increases in the staggered conformation by 0.0009 ep. The amounts of electron localization of a carbon atom, l ¼ 100[l(C1)/ N(C1)], are 70.95% and 70.84%, in each conformation. The total delocalization of carbon to the other basins, D(C1) ¼ N(C1) l(C1), is 1.7214 e for the eclipsed and 1.7259 e for the staggered conformation. The amount of electron delocalization of C1 into the other basins in the staggered conformation, 100[d(C1, X)/2N(C1)], is 7.21% with C2 and 7.02% with each of the three hydrogen atoms bonded to C1; the rest of the delocalization, 0.88%, occurs with the hydrogen atoms bonded to C2.
It is also interesting to note that whereas Karplus-type behavior (discussed in Section 14.2) is observed for electron delocalization between vicinal hydrogen atoms, this is not so for the energy barrier. In contrast, CaC delocalization is in accordance with the features of the barrier – during rotation toward the eclipsed arrangement the electrons shared between the carbon atoms become more localized on the respective basins thus, reducing the electron–nuclear attraction and increasing the nuclear repulsion between them.
14.3.1.2 Rotational Barrier of 1,2-Disubstituted Ethanes
This subsection presents results from a study of the conformational behavior of the molecules XCH2CH2X, where X ¼ CH3; Cl, or F. It has been shown exper-
14.3 Conformational Equilibria 383
imentally that whereas the antiperiplanar (app) conformation is the most stable for the conformers of both butane [20] and 1,2-dichloroethane [29], the gauche (sc) conformation is the most stable for 1,2-difluoroethane [29], a behavior known as the gauche e ect [30].
The highest of the rotational barriers of butane occurs when the two methyl groups eclipse each other and has been attributed to the action of van der Waals repulsion between them. In this instance the anti conformer is the one observed experimentally, with an anti-gauche energy di erence that lies between 0.67 and 0.97 kcal mol 1 in the gas phase [31]. For 1,2-dichloroethane the anti conformer is the most stable; the energy di erence between this and the gauche conformer is estimated to be between 0.9 and 3 kcal mol 1, depending on the experimental technique used [32]. The highest of the barriers in this molecule has been attributed to dipole–dipole repulsion between the CaCl bonds [33].
The gauche e ect occurs when X is an electron-withdrawing substituent, for example F or O, and has been found in many molecules in addition to 1,2- disubstituted ethanes, although 1,2-difluoroethane is regarded as a typical example. Microwave, Raman, infrared, and nuclear magnetic resonance (NMR) studies indicate that the gauche conformer is the most stable for this molecule and the anti–gauche energy di erence has been found to be between 0.6 and 0.9 kcal mol 1 [34]. The rotational barrier for 1,2-difluoroethane has been determined computationally using several levels of theory [22, 33, 35]. A value of 0.8 kcal mol 1 for the anti–gauche energy di erence has been reported at the MP2/ ANO level but even the RHF approximation, whenever a su ciently large basis set is used, accounts for the gauche e ect of the molecule [36].
Several explanations have been proposed for the gauche e ect. On the basis of analysis of the evolution of the bent-bond CaC trajectory of the molecule during rotation, Wiberg and coworkers [37] concluded that a destabilizing interaction in the anti rotamer makes the gauche conformer the most stable. The gauche e ect has also been explained in terms of enhanced sCaH ! s CaF hyperconjugative interactions in the gauche conformer [38]. Alternatively, increased orbital overlap between the HOMO molecular orbitals of two interacting CFH2 radicals when the F atoms are in a gauche arrangement has been used as an explanation [36].
As for ethane the CaC bond distance of molecules XCH2CH2X undergoes the most relevant changes during rotation for which, according to Table 14.3, there is a well defined trend as a function of X – it is shorter for the anti than for the gauche conformation of butane; it is nearly equal for both conformers of 1,2- dichloroethane; and is shorter for the gauche conformation of 1,2-difluoroethane as a consequence of the gauche e ect.
The atomic energies of the central carbon atoms of the molecules, shown in Fig. 14.3a, indicate that whereas for butane and 1,2-dichloroethane these atoms are more stable in the app conformation, for 1,2-difluoroethane they are more stable at 90 degrees. In addition, the relative substituent energies shown in Fig. 14.3b indicate that X has a stabilizing e ect when the substituents are eclipsed, except for Cl, for which this e ect is observed when this atom becomes eclipsed with an H atom. The behavior of the substituent energy is in clear contradiction
14.3 Conformational Equilibria 385
with any explanation of the barrier based on atomic or functional group repulsion in the molecules; in the same way as for ethane, the main source of the barrier is the increase of E(C).
An explanation of the gauche e ect can be provided by QTAIM analysis of the process. A lack of coincidence between the angle at which the gauche conformer is observed (70 ) and the E(C) minimum for the three molecules is found. Whereas for butane and 1,2-dichloroethane the carbon atoms are more stable at 80 , for 1,2-difluoroethane the angle is 90 , as a result of the balance among all atomic contributions. Note that for 1,2-difluoroethane the most important energy contributions to this balance come from both the central carbon atoms and the gauche H atom. Of these, the former has a stabilizing and the latter a destabilizing e ect; at 70 , the H atom that is not app to the vicinal F atom has the least destabilizing e ect and the carbon atom has the most stabilizing e ect. As a consequence, the gauche e ect in 1,2-difluoroethane is observed.
Electron delocalization between the central carbon atoms follows the same trend as in ethane; d(C, C0) is smaller for the eclipsed conformations (Figs 14.3c and 14.3d). In particular, this delocalization for the gauche conformer follows the order CH3 < Cl < F, in agreement with the gauche e ect for FCH2CH2F. CaX delocalization follows the reverse order, being larger for the eclipsed conformations. d(F, F0) also undergoes important changes during internal rotation – it is maximum at both the F/F eclipsed and anti conformations and minimum for a torsion angle of ca. 100 (Fig. 14.4). d(F, H) is smaller and maximum when F and H are at their anti and syn conformations, respectively. This does not support the classic hyperconjugative model as the origin of the gauche e ect, however, because the change of d(F, H) contributes only 0.006 e but the change of d(C, C0) contributes 0.013e to the gauche conformer. From these results, the relative con-
Fig. 14.4 d(F, H) and d(F, F0) delocalization indices. B3LYP/6-311þþG (2d,2p) wavefunctions were used.
386 14 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry
former stability can be associated with electron delocalization between the central C atoms.
14.3.2
Anomeric E ect on Heterocyclohexanes
A relevant aspect of the QTAIM to experimental chemists is the rigorous description of bonding it provides in terms of rðrÞ, because in this theory the existence of a bond path is both a necessary and su cient condition for the existence of a chemical bond [39]; this provides chemists with an important tool for analysis of covalent, shared, and polar interactions. This subsection shows the relevance of the bond-path trajectories as a guide for the design of new compounds.
The participation of weak intramolecular interactions in a conformational process can be illustrated by the anomeric e ect, defined as the thermodynamic preference of an electronegative substituent to assume the axial position when it acquires a position a to an annular heteroatom. The anomeric e ect occurs in the SaCaP(O) segment when the diphenylphosphinoyl group is attached to position 2 of 1,3 dithiane (Scheme 14.1) [40]. In this example, an atypical hydrogen bridge in the CHaOP group is essential for preference of the substituent.
Scheme 14.1 Conformational equilibrium of 2-diphenylphosphinoyl-1,3-dithiane.
In 1982 Juaristi et al. described, for the first time, the anomeric e ect at the SaCaP(O) segment [41], with a value of 2.64 kcal mol 1, one of the largest values yet reported for this e ect. Later, the enthalpic nature of this type of e ect was established [42–44]. This phenomenon ba ed the scientific community – weak anomeric e ects were expected because of the low electron-donating nature of the atoms involved (from the second row of the periodic table) [45, 46].
One of the models used to describe the anomeric e ect relies on hyperconjugation [47]. X-ray di raction data of the axial conformer do not, however, show the
14.3 Conformational Equilibria 387
Fig. 14.5 Molecular graphs of (a) 1-ax and (b) t-butyl cyclohexane calculated at the B3LYP level with the 6-31G(d,p) and 6-311G(2d,2p) basis sets, respectively. Paths connecting CCPs are shown as green lines. BCPs, RCPs, and CCPs are shown as small red, yellow, and green spheres.
bond shortening and lengthening patterns required by this model – the CaS bond should be shorter and the CaP bond longer in the axial conformer than in the equatorial conformer, but this cannot be observed (Fig. 14.5a) [48]. An alternative explanation proposed a through-space electronic interaction between the S and P atoms. An interaction between 3p(S) and 3d(P) orbitals was then proposed. According to Graczyk [49], if the SS and SP nonbonding distances are ca 3.00 A˚ and the maximum radial extent of the phosphorus 3d orbital is 2.43 A˚ [50], either no or very small 3p–3d overlap is possible. Schleyer et al. questioned the relevance of 3p–3d interactions on theoretical grounds [51].
Mikolaczyk suggested a rationalization for the origin of the anomeric e ect in the SaCaP segment in terms of an interaction between the oxygen atom of the phosphinoyl group and the hydrogen atoms at the 4,6-syn-diaxial positions [52]. Such a hypothesis is based on the observation that the distance between these hydrogen atoms and the oxygen atom is shorter than the sum of their van der Walls radii. On the other hand, interpretation of microwave data led Mikolajczyk et al. [53] to conclude that the hydrogen bridge on the CHaOP segment is of no relevance.
Calculation of the electronic properties of 2-dimethylphosphinoyl-1,3-dithiane (1) at the B3LYP/6-31G(d,p) level established that the axial conformer in which the O atom is on the ring pointing towards the axial H atoms at positions 4 and 6 is the most stable of four possible conformers (Table 14.4). Computational results obtained from experimental data establish that for evaluation of the conformational energy it is possible to replace the phenyl groups of the experimentally studied compound by methyl groups [40].
Interestingly, conformers A and B of compound 1 (see numbering scheme in Table 14.4) acquire the same atomic arrangement as that found in the SaCaP
38814 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry
Table 14.4 Relative energy and HbðrÞ of the CHaOP interaction of conformers A–D, in kcal mol 1 at B3LYP/6-31G(d,p) level.
Compound/Conformer |
A |
B |
C |
D |
|
|
|
|
|
1: X ¼ S, Y ¼ CH2, Z ¼ H |
0.0/(0.011, 0.10) |
6.37 |
5.83 |
4.45 |
2: X ¼ CH2, Y ¼ CH2, O, Z ¼ H |
1.49/(0.009, 0.009) |
3.81 |
0.0 |
0.35 |
3: X ¼ S, y ¼ S, Z ¼ H |
0.0/(0.039, 0.039) |
7.61 |
6.38 |
5.01 |
4: X ¼ SO2, Y ¼ CH2, Z ¼ H |
0.0/(0.036, 0.036) |
11.62 |
8.53 |
5.00 |
5: X ¼ S, Y ¼ CH2, Z ¼ F |
2.96 |
0.39 |
0.0 |
0.21 |
anomeric segment, so that if any stereoelectronic e ect was to participate in molecular stabilization, it should remain constant. Rotation of the phosphinoyl group results in destabilization of 6.37 kcal mol 1, however. In conformer B, the methyl group points towards the center of the ring causing, the molecular energy to increase. If conformer B of compound 2 is used as a reference in which cyclohexane is taken as the basic system, an energy increase of only 3.81 kcal mol 1 relative to the minimum value of C is observed. This is similar to the e ect experienced by a tert-butyl group on cyclohexane in which, as illustrated in Fig. 14.5b, the presence of hydrogen–hydrogen bond paths precludes assignment of repulsive character to 1,3-syn diaxial interactions in the molecule [54]. The energy increase can be a consequence of the loss of CHaOP interactions and of an increase of methyl group interactions now oriented over the dithiane ring. This would lead to a di erence of 2.56 kcal mol 1 that can be attributed to both CHaOP interactions and stabilization energy of 1.28 kcal mol 1 for each of them.
Figure 14.5a shows the critical points of the molecule. The presence of two CHaOP bonds yields a molecular structure similar to that of adamantane and enables the formation of three additional rings besides that of 1,3-dithiane. A total of four ring critical points (RCPs) and one cage critical point (CCP) are therefore generated. It is interesting to note that as the two BPCs have di erent rbðrÞ values – one being stronger than the other [40]. In this instance the associated bond paths are curved; they are, therefore, longer than the geometrical bond distances. Bond-distance analysis shows the di erence between the two interactions. The trajectory with rbðrÞ ¼ 0:011 au corresponds to a bond length of 4.726 au whereas for the other, rbðrÞ ¼0.010 au, corresponds to a trajectory length of 4.799 au.
Replacement of the S atoms with methylene groups, 2, causes rbðrÞ to decrease [55]. Introduction of methylene groups might be expected to cause a decrease of
14.3 Conformational Equilibria 389
acidity of the H atoms at positions 4 and 6, a weaker CHaOP bond, and, as a consequence, the equatorial conformer should be preferred. In addition, introduction of a third S atom at position 5 of 1,3-dithiane, 3, should increase the acidity of the CaH group, with a concomitant increase of rbðrÞ. The experimental conformational preference of 1,3,5 trithiane is 1.43 kcal mol 1 [48], a larger value than that for 1.
Something similar happens when the amount of oxidation of the sulfur atoms is increased, as shown for molecule 4. In addition, on these molecules there is no possibility of the sulfur atoms having any stereoelectronic interaction with the P atom. The preference for the axial position is, nevertheless, preserved because the acidity of the H atoms of interest is larger than for 1,3-dithiane [55].
It is also relevant that the conformational preference of the substituent at position 2 of the axial conformation with a gauche arrangement of the oxygen atom increases with rbðrÞ for the bridge CHaOP BCP.
The strongest interaction is produced when the S atom is oxidized to its corresponding sulfone. It is, unfortunately, not possible to confirm these results experimentally for 1,3-dithiane, because of the impossibility of obtaining the disulfone while at the same time keeping the chair conformation of the 1,1,3,3-tetraoxa-1,3- dithiane ring. Because of this, dimethylphosphinoyl(methylsulfanyl)methane, 6, was studied, taking into consideration that the acidity of the hydrogen atom that is a to S can be modulated to modify its oxidation state and, depending on the results obtained, to perform the synthesis.
Scheme 14.2 shows the six stable conformers of compound 6. Three have a gauche arrangement of the CaSaCaP segment and three have the anti conformation. Of these, conformer 6-A, in which the O atom approaches an H atom of the methyl group, is the most stable. The corresponding interaction energy coincides with the value obtained from analysis of 2-diphenylphosphinoyl-1,3-dithiane
Scheme 14.2 Minimum-energy conformers in the potential-energy surface of dimethylphosphinoyl(methylsulfanyl)methane (6). Adapted from Ref. [40].
390 14 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry
Fig. 14.6 RCPs and BCPs on conformer 6-A. Critical are points labeled as in Fig. 14.5.
(1.8 kcal mol 1). The energy of conformer 6-C is 3.92 kcal mol 1 higher than that of 6-A. The di erence is less for 6-B, because the CaS and PaO bonds are periplanar, probably because of a stabilizing sOaP ! s CaS stereoelectronic interaction. Of the three anti conformers, only 6-E (the most stable of the gauche series) preserves such interaction.
There is a CHaOP BCP with rbðrÞ ¼ 0:019 au on conformer 6-A and an associated RCP (Fig. 14.6). The bond path length is 4.7049 au and the geometric length is 4.6524 au.
Scheme 14.3 shows the results from optimization of the geometry of dimethylphosphinoyl (methylsulfonyl)methane, 7. Only four conformers can be found, three with a gauche and only one with an anti arrangement of the CaSaCaP segment. From this, only the latter keeps an antiperiplanarity that enables ! s CaS interactions. The energy minimum corresponds to the conformer for which the CHaOP interaction is possible. Two nonobvious BCPs are found
for compound 7-A, one for the POaHC trajectory of interest and another for the SOaCH trajectory of one of the methyl groups with a sulfonyl O atom.
The BCP typical of the CHaOP interaction can be found on 7-A with rbðrÞ ¼ 0:0152 au. A second bond path is evident for SOaHC with rbðrÞ ¼ 0:0111 au.
Scheme 14.3 Conformers of minimum energy in the potential-energy surface of dimethylphosphinoyl(methylsulfonyl)methane (7). Adapted from Ref. [40].
14.4 Aromatic Molecules 391
That the energy di erence is so large compared to the other conformers led to the proposal it be synthesized; which was accomplished with a 70% global yield from chloromethyl methyl sulfide. The NMR spectrum of 6 contain a singlet signal at 2.22 ppm for the methyl group and a doublet at 3.22 ppm, with 2JHaP ¼ 13:0 Hz, for the methylene group. The signals for the aromatic rings are complex and centered at ca 7.5 and 7.8 ppm. In contrast, the spectrum for compound 7 contains a triplet signal centered at 3.226 ppm and 4JHaH ¼ 0:75 Hz for the methyl group, and a double of quartets centered at 4.518 ppm and 4JHaH ¼ 0:75 y 2JHaP ¼ 9:0 Hz for methylene. Calculation of the coupling constants of the di erent hydrogen atoms of conformer 7-A leads to the conclusion a W-type coupling is responsible for the additional multiplicity. This shows that the CHaOP bridge exists and results in slower methyl rotation, thus enabling the observed 4JHH coupling.
It has been proposed from theoretical studies that CHaObC interactions in purinic and pyrimidic pairs are repulsive in nature [56]. The analysis described herein enables characterization of the attractive CHaOP interaction relevant to the anomeric e ect of the SaCaPaO segment, however, just as predicted by chemical intuition and emphasized theoretically [27].
14.4
Aromatic Molecules
14.4.1
Electronic Structure of Polybenzenoid Hydrocarbons
The molecular and electronic structures and intermolecular interactions of aromatic molecules have been successfully explained by the QTAIM in terms of rðrÞ, its Laplacian, ‘2rðrÞ, and the kinetic energy density, HðrÞ. Bond paths have been reported [57] between ortho H atoms in angular polybenzenoid hydrocarbons (PBHs), thus providing evidence for the presence of hydrogen–hydrogen bonding as in tert-butylcyclohexane [54], shown in Fig. 14.5b. Bader et al. [58] analyzed the properties of rðrÞ of benzene and several protonated derivatives and Wiberg [59] reported good correlations among the properties of rbðrÞ with bond distances and with the p component of Fulton bond index of PBHs. Howard and Krygowski [60] also reported good correlations for the values of rðrÞ at the RCPs in their study of aromaticity descriptors.
Matta et al. [61] further investigated the relationship between electronic and geometric data with the delocalization index. The correlations obtained imply that for the PBHs analyzed there is a nonobvious relationship between the oneelectron information contained in rbðrÞ and two-electron properties such as d(C, C0) for bonded C atoms and HbðrÞ. This finding allows information about the pair-density to be obtained from rðrÞ that might not be readily available, although formally accessible, experimentally. In other words, the main features of chemical bonding found from d(C, C0) can also be obtained from the electron density of these molecules, as shown by the patterns of the electron isodensity