QTAIM-DI-VISAB computational study on the so-called nonclassical bicyclobutonium cation

QTAIM-DI-VISAB analyses at the CCSD and B3PW91 levels were used to characterize the bonding of the cyclopropylcarbinyl ( 1 ) and the so-called ‘nonclassical’ bicyclobutonium and 1-methylbicyclobutonium cations, 2 and 3 as well as the transition state for rearrangement of 1 to 2 . These analyses involved obtaining QTAIM molecular graphs and delocalization indexes (DIs) for pairs of atoms that were correlated with the proximities of atomic basins (VISAB). This study showed that the supposed nonclassical bicyclobutonium and 1-methylbicyclobutonium cations do not exhibit penta-coordinate carbons at their equilibrium geometries as has been claimed. Both species are best described as distorted cyclobutyl cations that exhibit a single ring critical point in the topology of the charge density.


Introduction
The cyclopropylcarbinyl cation (1) and its isomer, the so-called nonclassical bicyclobutonium cation (2) as well as its 1-methyl analogue (3) whose structures are shown using the usual dashed-line formalism have been the focus of numerous experimental and computational studies over the span of several decades with the latest being the work of Olah and coworkers. 1The dashed-line structures of 2 and 3 have gradually been replaced with ORTEP-type/solid-line structures, shown as 4, the implication being that C3 is a penta-coordinate atom in both representations.

Computational methods
Our previous experiences with DFT calculations on carbocations clearly showed that the B3PW91 hybrid functional is superior to B3LYP in computing the geometries of delocalized, socalled nonclassical species. 7,8,9,10We obtained additional support for this finding by carrying out calculations on O-protonated 2,2-dimethyloxirane -studied by Carlier et al. 15 and described as a particularly challenging computational problem -to compare results from B3LYP, B3PW91, PBE1PBE, and CCSD calculations at the 6-311G(d,p) level as implemented in G03. 16The results presented in an earlier publication 13 clearly established that B3PW91 and PBE1PBE are expected to be superior to B3LYP for obtaining equilibrium geometries in cases where relatively weak polar bonds are involved.In this study, cation geometries were optimized at B3PW91 and MP2 levels with a range of basis sets -6-311+G(2d,p), cc-pVTZ, and aug-cc-pVTZ with OPT=TIGHT and OPT=VERYTIGHT and wave functions obtained.The OPT=VERYTIGHT B3PW91 calculations on 2 were carried out with INT=ULTRAFINE.On the other hand, CCSD(full) calculations with OPT=TIGHT were carried out first with 6-31+G(d,p) and then with the 6-311+G(2d,p) basis set.It should be pointed out that while MP2(frozen core) geometry optimizations with the cc-pVTZ -as reported by Olah et al. 1 -and aug-cc-pVTZ basis sets converged, this was not the case at MP2(full); we found that none of the cc-pVTZ and aug-cc-pVTZ MP2(full) calculations converged.Consequently, the MP2 calculations were discounted and not used in this study.Moreover, it appears that MP2 often tends to overestimate the stability of 'nonclassical' structures. 17While 1 and 2 were viewed as unsymmetrical species by Casanova et al. 4 , we found that OPT=TIGHT calculations yielded geometries that were extremely close to the C S structures.Consequently we fixed the symmetry -in our case using Chemcraft 18 -to C S as did Olah et al. 1 Selected inter-nuclear distances of 1-C S , 2-C S , and 3-C S obtained at CCSD(full)/6-311+G(2d,p) with OPT=TIGHT are displayed as Figure 1(a), (c), and (d), respectively.Frequency calculations were carried out on the stationary points to confirm them as energy minima or transitions states.B3PW91 and CCSD energies and thermochemical data are collected in Table 1.QTAIM analyses of the wave functions to investigate the topologies of the electron densities were carried out with AIM 2000 19 and the obtained molecular graphs are shown in Figure 2 to 4. AIMALL_08 20 was used to integrate atomic basins, obtain atomic populations, calculate total charges as well as atomic overlap matrices required for DI calculations.Values of ρ(r c ) at selected bond critical points are collected in Table 2.That the total charges of the cations obtained at the various levels of theory differed by less than 1% from the expected value of 1.0 confirmed the quality and validity of the QTAIM data (Table 3).This conclusion was supported by the fact that E elec (ΣE(Ω)), the sum of the atom energies E(Ω) obtained with AIMALL for 1, TS-1→2, 2, and 3 differed from the G03 molecular energy E elec (Table 3) by less than 0.45 kcal mol -1 .The program LI-DICALC 21,22,23 was used to obtain DIs; selected values for pairs of atoms are collected in Table 4. Isosurface plots of the density of atomic basins (Figure 5 -Figure 8) were obtained with AIM 2000 at a contour value of 0.005 which includes > 95% of the electrons using a mesh grid size of 0.125 and plotted with a sphere size of 0.15.Selected thermochemical data for 1-C S , TS-1→2, 2-C S , and 3-C S are collected in Table 1.As was found previously 1 cations 1-C S , and 2-C S are very close in energy: at B3PW91/aug-cc-pVTZ, 2-C S is lower in energy than 1-C S based on the ZPE-corrected difference in energy ∆E 0 (-0.24 kcal mol -1 ) and ∆H 298 (-0.54 kcal mol -1 ).On the other hand 1-C S was found to be lower in energy than 2-C S based on ∆G 298 (+0.37 kcal mol -1 ) at B3PW91/aug-cc-pVTZ(tight) and ∆E elec (+0.18 kcal mol -1 ) at CCSD(full)/6-311+G(2d,p)(tight).As mentioned in the computational methods section we chose to discount MP2 calculations because the optimizations did not converge at the MP2(full)/cc-pVTZ or MP2(full)/aug-cc-pVTZ levels.Given the expected prohibitive length of CCSD(full)/6-311+G(2d,p) transition state and frequency calculations we studied TS-1→2 only at the B3PW91/aug-cc-pVTZ(tight) level.As seen in the data collected in

Equilibrium geometrical and molecular structures
Selected equilibrium internuclear distances of 1-C S , TS-1→2, 2-C S , and 3-C S are collected in Figure 1.Of importance is the fact that the C1-C3 distance of 3 is 0.071 Å greater than the C1-C3 distance of 2 at CCSD(full)/6-311+G(2d,p). Selected QTAIM data for cations at CCSD(full)/6-311+G(2d,p)(tight) and B3PW91/aug-cc-pVTZ(tight) are collected in Table 2.The molecular graph of 1 obtained at CCSD(full)/6-311+G(2d,p)(tight) is displayed as Figure 2(a).The Poincaré-Hopf relationship (NumNACP + NumNNACP -NumBCP + NumRCP -NumCCP = 1) is satisfied.The C2-C3 and C2-C4 bond paths (BPs) are highly curved and the BCPs are in close proximity to the RCP; this means that rearrangement to another molecular structure by coalescence of a BCP and the RCP, as for example through TS-1→2, would require small nuclear displacements and a small activation energy in keeping with the very low barrier calculated for the rearrangement of 1 to 2. Its molecular structure does not exhibit BCPs between C1|C4 nor C2|C4, and no RCPs.

QTAIM-DI-VISAB Analyses 1-C S
Selected atomic basins of 1-C S obtained at the CCSD(full)/6-311+G(2d,p)(tight) level are displayed as Figure 5(a) and (b); Figure 5(a) shows that the C1 and C3 basins do not significantly impinge on each other in accord with the fact that they that exhibit a miniscule DI of 0.087: there is little delocalization of electrons between the C1 and C3 basins.As expected the DI (0.141) is somewhat larger at B3PW91/aug-cc-pVTZ.The C1-C2 bond exhibits considerable double bond character -ρ(r) cp is 0.3435 and the DI 1.317.On the other hand the C2-C3 and C2-C4 bonds are very weak -ρ(r) cp is 0.1656 and DI 0.590 at CCSD(full)/6-311+G(2d,p).The relative areas of the atomic surfaces shared with C2 seen in Figure 5(a) are in accord with the relative strengths of the C1-C2 and C2-C3 bonds.These values are proportionally larger (see Table 2 and Table 3) at B3PW91/aug-cc-pVTZ.We observed this behaviour -relatively large DIs but no BPs -previously for the so-called 7norbornyl cation 13 and in the case of trimethylsilyl(carbene) and trimethylgermyl(carbene) .

Conclusions
This study shows that the so-called nonclassical bicyclobutonium cation exhibits the molecular structure/graph of a distorted cyclobutyl cation at its equilibrium geometry.It documents another successful application of the QTAIM-DI-VISAB method in establishing the true nature of the bonding in hyper-coordinated so-called nonclassical carbocations; this approach obviates the need for using arbitrary dotted-line or solid-line representations of bonding in hypercoordinate species.

Figure 5 (
b) shows the C2 basin illustrating its flattened surfaces shared with the C3 and C4 basins.The C3|C4 basins exhibit a DI of 0.986, close to the DI expected for a single bond.

Figure 7 (
Figure7(a) shows the C1|C3 basins of 2-C S at CCSD(full)/6-311+G(2d,p)(tight) that do not exhibit a BCP/BP.Nevertheless there is a high degree of delocalization of electrons between these basins; the DI is high at 0.443; 0.588 at B3PW91/aug-cc-pVTZ.Figure7(b) and (c) display the C1 and C3 basins and clearly show the 'distortion' induced by their proximity.The DI for the C2|C4 pair is 0.066 at CCSD(full)/6-311+G(2d,p)(tight) and 0.096 at B3PW91/aug-cc-pVTZ.We observed this behaviour -relatively large DIs but no BPs -previously for the so-called 7norbornyl cation13 and in the case of trimethylsilyl(carbene) and trimethylgermyl(carbene) .12

12
Figure7(a) shows the C1|C3 basins of 2-C S at CCSD(full)/6-311+G(2d,p)(tight) that do not exhibit a BCP/BP.Nevertheless there is a high degree of delocalization of electrons between these basins; the DI is high at 0.443; 0.588 at B3PW91/aug-cc-pVTZ.Figure7(b) and (c) display the C1 and C3 basins and clearly show the 'distortion' induced by their proximity.The DI for the C2|C4 pair is 0.066 at CCSD(full)/6-311+G(2d,p)(tight) and 0.096 at B3PW91/aug-cc-pVTZ.We observed this behaviour -relatively large DIs but no BPs -previously for the so-called 7norbornyl cation13 and in the case of trimethylsilyl(carbene) and trimethylgermyl(carbene) .12

Figure 8 (
Figure 8(a) shows the C1|C3 basins of 3-C S at CCSD(full)/6-311+G(2d,p)(tight) that do not exhibit a BCP/BP.As seen in the case of 2-C S there is a high degree of delocalization of electrons between these basins; the DI is quite large (0.338) but smaller than in the case of 2-C S ; at B3PW91/aug-cc-pVTZ the value is 0.419.Figure 7(b) and (c) display the C1 and C3 basins that clearly show how they impinge on each other their surfaces are flattened.The DI for the C2|C4 pair is 0.065 at CCSD(full)/6-311+G(2d,p)(tight) and 0.084 at B3PW91/aug-cc-pVTZ.
Figure 8(a) shows the C1|C3 basins of 3-C S at CCSD(full)/6-311+G(2d,p)(tight) that do not exhibit a BCP/BP.As seen in the case of 2-C S there is a high degree of delocalization of electrons between these basins; the DI is quite large (0.338) but smaller than in the case of 2-C S ; at B3PW91/aug-cc-pVTZ the value is 0.419.Figure 7(b) and (c) display the C1 and C3 basins that clearly show how they impinge on each other their surfaces are flattened.The DI for the C2|C4 pair is 0.065 at CCSD(full)/6-311+G(2d,p)(tight) and 0.084 at B3PW91/aug-cc-pVTZ.