Fluxional behaviour of tricyclo[2.2.1.0^2,6]heptaphosphide trisanion: a DFT investigation

3.1 All-phosphacyclic systems

We computed the following [2,2]sigmatropic rearrangements at the DFT (B3LYP/6-31 + G*) level [rearr. (1)–(4), Scheme 1]. To understand the effect of the counterions, we investigated the [2,2]sigmatropic rearrangements of both the trisanion without counterion (1a) and with alkali metal ions (1b–d).

Scheme 1:

Computed model [2,2]sigmatropic rearrangements of alkali salts of tricyclo[2.2.1.0^2,6]heptaphosphide trisanions (1a–d) and tricyclo[2.2.1.0^2,6]heptaphosphane triradical (3).

3.2 Optimised geometries and NBO analysis

Optimised geometries of different species are given in Fig. 1.

Fig. 1:

Geometries of the tricyclo[2.2.1.0^2,6]heptaphosphide trisanion without counterion (1a), and with alkali metal ions (1b–d), tricyclo[2.2.1.0^2,6]heptaphosphane triradical (3) and the corresponding transition structures (TS_1a–d, TS₂) and products (2a–d, 4) optimised at the B3LYP/6-31 + G* level with bond lengths (Å) and Wiberg bond indices (in parentheses).

The existence of negative hyperconjugation in a neutral molecule or anion is manifested by several electronic and structural features. The electron density of the lone pair at the anionic centres (e.g. P³, P⁶ and P⁷ in structure 1a) is reduced as a result of its transfer to the σ* orbital. Secondly, the σ bonds (corresponding to the electron-density recipient σ* orbital, e.g. P¹–P⁴, P¹–P⁵ and P⁴–P⁵ in structure 1a) are elongated, whereas the bonds emanating from the anionic centres (e.g. P³–P⁴, P⁷–P¹ and P⁶–P⁵ in structure 1a) are shortened. These changes are conveniently detected by NBO analysis. The NBO analysis of the trisanion 1a reveals that the lone pairs at the P³, P⁶ and P⁷ atoms are present in p orbitals. As a result of the electron density transfer, the lone pair occupancies at the P³, P⁶ and P⁷ atoms decrease to 1.70e each, while the presence of 0.24e could be detected in σ*P¹–P⁴, σ* P¹–P⁵ and σ* P⁴–P⁵ each. Besides, the negative charge at the anionic centres, P³, P⁶ and P⁷ in 1a, is reduced to –0.61 each, while the other phosphorus atoms have a charge of –0.31 each. It is noteworthy that these values of the negative charges are very close to the values of –0.85 and –0.20, respectively, as reported by Böhm and Gleiter [14]. Furthermore, the P³–P⁴ and P⁵–P⁶ bonds are found to be much shorter (2.184 Å) than the other P–P bonds and have a higher Wiberg bond index (WBI) (1.185). On the other hand, the P¹–P⁴, P¹–P⁵ and P⁴–P⁵ bonds are elongated (2.351 Å) with a smaller WBI (0.921).

The transfer of the electron density induces second-order perturbation interactions, which are depicted along with their energies in Fig. 2.

Fig. 2:

Second-order LP → σ* interactions along with their energies in 1a.

In the transition structure, TS_1a, manifestation of the transfer of the electron density into σ* orbitals accompanied by shortening of the P³–P⁴ and P⁵–P⁶ bonds is further augmented. NBO analysis reveals the presence of the πP³–P⁴ and πP⁵–P⁶ orbitals, each occupied by about 2e and having a WBI of 1.268. The WBI of the P⁴–P⁵ bond (the bond undergoing a sigmatropic shift in this step) comes down to 0.241 only with a simultaneously developing bond between the P³ and P⁶ atoms with an equal WBI. Small electron densities of about 1e are detected each at the P³ and P⁶ atoms. An important feature to be noted here is that the P³ and P⁶ atoms expand their valence shells; as will be seen later, in the carbocyclic analogue, this type of expansion is not possible, which makes the [2,2]sigmatropic shift in that species impossible. All these features support a pericyclic mechanism of the [2,2]sigmatropic shift involving an aromatic transition structure. Schleyer and co-workers [20] developed a simple computational method to check the aromatic character of a molecule or the transition structure based on the magnetic shielding at the centre due to diatropic current induced on placing the system in an external magnetic field. The constant, called the nucleus independent chemical shift (NICS), is the negative of the computed magnetic shielding at the centre. While a high negative NICS confirms the aromatic character of the species, a low negative or a positive value indicates the non-aromatic or the anti-aromatic character, respectively. Later, it was suggested that the NICS determined at 1 Å above the centre of the ring gives the better measure of aromaticity, as in that case, σ-paratropic influences could be avoided [21]. We determined NICS(0) (at the ring critical point, RCP [22]) and NICS(1) (1 Å above RCP); these values, –30.03, NICS(0) and –13.45, NICS(1) (Fig. 3), confirm the aromatic character of the transition structure, TS_1a. Thus, negative hyperconjugation causes weakening of the σP⁴–P⁵ bond facilitating its [2,2]sigmatropic shift to the P³–P⁶ position through the pericyclic mechanism.

Fig. 3:

NICS(0) and NICS(1) values of the transition structure, TS_1a.

The optimised geometry of the product 2a formed after the [2,2]sigmatropic shift in 1a is shown in Fig. 1. It may be noted that the structural parameters of 2a are closely similar to those of 1a. Now the anionic centres are P⁴, P⁵ and P⁷. The NBO analysis of 2a also reveals that, as in 1a, electron density is transferred from the P⁴, P⁵ and P⁷ atoms to the σ*P²–P³, σ*P²–P⁶ and σ*P³–P⁶ orbitals causing weakening of the corresponding σ bonds. Furthermore, the second-order perturbative interactions in 2a are detected exactly as found in 1a. Thus, negative hyperconjugation sets in a fast degenerate [2,2]sigmatropic shift making all the phosphorus atoms in 1a equivalent (Fig. 4).

Fig. 4:

Degenerate [2,2]sigmatropic shift in the tricyclo[2.2.1.0^2,6]-heptaphosphide trisanion (1a).

To look into the existence of negative hyperconjugation, if any, in the related tricyclo[2.2.1.0^2,6]heptaphosphane triradical (3), we computed its [2,2]sigmatropic rearrangement at the same level [rearr. (5), Scheme 1]. The optimised geometries of 3, the transition structure TS₂ and the resulting product 4 are given in Fig. 1. The NBO analysis reveals much weaker interactions in this case. The electron densities at the free radical centres, P³, P⁶ and P⁷, are only slightly reduced (0.93e), with a corresponding presence of 0.06e each in the σ*P¹–P⁴, σ*P¹–P⁵ and σ*P⁴–P⁵ orbitals. Second-order perturbation energies (2.1 kcal mol^–1) are comparatively smaller. It is noteworthy that the expansion of the valence shells of the P³, P⁶ and P⁷ atoms having the unpaired electron is not required in this case.

3.3 Energetics and counterion effect

The relative activation enthalpies and the reaction enthalpies of the [2,2]sigmatropic rearrangements discussed above are given in Table 1. As expected, the activation enthalpy of the [2,2]sigmatropic rearrangement of the free heptaphosphide ion [1a, rearr. (1), Scheme 1] is rather small; it is 8.0 kcal mol^–1 in the gas phase, which increases to 9.0 kcal mol^–1 in THF, the solvent actually used experimentally. This marginal increase in the activation enthalpy may be attributed to the dipole–dipole interaction between 1a and the solvent molecules, which is expected to be stronger than the dipole–dipole interaction between the corresponding transition structure TS_1a and the solvent molecules. It has been reported earlier that the rates of the oxy-Cope rearrangement of 1,5-diene alkoxides [23] and the anionic oxy-Claisen rearrangement of enolates of α-allyloxy ketones [24] are affected by the nature of counterions; potassium salts undergo rearrangements most readily, whereas the rearrangement of lithium salts occurs most slowly. In view of this, we investigated the effect of the counterions Li⁺, Na⁺ and K⁺ on the activation enthalpy of the rearrangement of 1a. The values of the relative activation enthalpies for the [2,2]sigmatropic rearrangement of the free heptaphosphide ion [1a, rearr. (1), Scheme 1] and those for its salts with the alkali metal ions [1b–d, rearr. (2)–(4), Scheme 1] calculated at the same level are given in Table 1. It may be noted that the activation enthalpy of the [2,2]sigmatropic rearrangement of the trilithium salt of the heptaphosphide ion [1b, rearr. (2), Scheme 1] is greater than that of the free anion (1a) in the gas phase by more than 5 kcal mol^–1, which may be attributed to the dispersion of the negative charge on the anionic centres caused due to the polarising effect of the counterion Li⁺. It results in the stabilisation of 1b more than the corresponding transition state TS_1b enhancing the activation enthalpy as compared to that for the sigmatropic rearrangement of the free anion 1a. This argument is supported on two counts: first, the activation enthalpy for the rearrangement of 1b in THF is smaller than that in the gas phase by 0.6 kcal mol^–1 due to formation of the solvent-separated ion pair thereby diminishing the polarising effect of the Li⁺ ions, and secondly, the activation enthalpies decreasing in the order of Li⁺ > Na⁺ > K⁺ salts of 1a which is parallel to the decreasing order of the polarising effect of these alkali metal ions. Interestingly, our results are quite similar to the reported counterion effect of the alkali metal ions on the rates of the oxy-Cope rearrangement of 1,5-diene alkoxides [23] and the anionic oxy-Claisen rearrangement of enolates of α-allyloxy ketones [24].

As mentioned earlier, the magnitude of the negative hyperconjugation in heptaphosphane triradical (3) is much weaker than in trisanion 1a, as a result of which, the relative activation enthalpy for the [2,2]sigmatropic rearrangement of the former in the gas phase is greater (12 kcal mol^–1) than that for the latter. Furthermore, as expected, it is not affected by the solvent.

3.4 Carbocyclic systems

In view of our interest in studying electronic and energetic changes on CH/P exchange in organic molecules [25, 26], we investigated theoretically the possibility of the occurrence of a similar type of negative hyperconjugation induced [2,2]sigmatropic rearrangement in the carbocyclic analogues of 1 and 2, namely tricyclo[2.2.1.0^2,6]-heptanide trisanion (5) and tricyclo[2.2.1.0^2,6]heptane triradical (6) and computed the following model rearrangements at the same level (Scheme 2).

Scheme 2:

Computation of [2,2]sigmatropic rearrangement of tricyclo[2.2.1.0^2,6]heptanide trisanion (5) and tricyclo[2.2.1.0^2,6]heptane triradical (6).

The optimised geometries of the carbocyclic systems 5 and 6 and the transition structure involved in the [2,2]sigmatropic shift of 6 (TS₃) and the resulting product 7 are shown in Fig. 5.

Fig. 5:

Geometries of the tricyclo[2.2.1.0^2,6]heptanide trisanion (5), tricyclo[2.2.1.0^2,6]heptane triradical (6), the transition structure (TS₃) and the product (7) optimised at the B3LYP/6-31 + G* level with bond lengths (Å) and Wiberg bond indices (in parentheses).

It is found that in contrast to the P73− anion (1a), its carbocyclic analogue 5 does not undergo a [2,2]sigmatropic shift, which is also in conformity with the results reported by Böhm and Gleiter [14]. NBO studies reveal that unlike 1a, in which the lone pairs at the P³, P⁶ and P⁷ atoms are present in p orbitals, in 5, the lone pairs are present in sp³ orbitals. Negative hyperconjugation exists in this case also, but the orientation of the sp³ orbital makes the transfer of the electron density into two neighbouring σ*orbitals possible. For example, the negative charge at C³ is transferred into the σ*C²–C⁶ and σ*C⁴–C⁵ orbitals, resulting in the increased lengths of the corresponding σ bonds (Fig. 5). Similar is the case with the negative charges at the C⁶ and C⁷ centres. As shown in Fig. 6, these electron-density transfers result in second-order perturbation energy stabilisation. Despite this, 5 does not undergo a [2,2]sigmatropic shift, the reason being that unlike a P atom, the carbon atom is unable to expand its valence shell.

Fig. 6:

Second-order LP → σ* interactions along with their energies in 5 and 6.

In contrast to the C₇H₇^3– anion (5), the triradical 6 undergoes a [2,2]sigmatropic rearrangement readily, rather much more readily than its phospha-analogue, 3. The reason is obvious; valence-shell expansion is not necessary for the triradical to undergo a [2,2]sigmatropic rearrangement. The unpaired electrons at the C³, C⁶ and C⁷ atoms are present in p orbitals. In this case also, there is electron-density transfer from these orbitals to the σ* orbitals causing elongation of the corresponding σC–C bonds (Fig. 5). The σ bonds so weakened undergo a [2,2]sigmatropic rearrangement easily with a relative activation barrier of 5.7 kcal mol^–1 only. It is interesting to note that this activation energy barrier is much smaller than that for its phospha-analogue. It has been reported that the relative activation energy barriers for the [3,3]sigmatropic rearrangements of divinyl derivatives of cyclopropane and its mono-aza, mono-oxa and mono-phospha analogues are in the order: divinylcyclopropane < divinylaziridine < divinyloxirane < divinylphosphirane [27; P. Maheshwari, R. K. Bansal, to be published]. Thus, the release of the ring strain affects the relative activation energy value: the higher the strain, smaller is the activation energy barrier. In the present case also, it appears that besides negative hyperconjugation, which induces a [2,2]sigmatropic shift by weakening the σ bond, strain of the ring which opens as a result of the sigmatropic shift also affects the activation energy value. Thus the relative activation energy for the [2,2]sigmatropic rearrangement of 6 accompanied by opening of the cyclopropane ring is much lower than that for the [2,2]sigmatropic rearrangement of 3 accompanied by opening of the triphosphirane ring, as ring strain in the former is much higher than that in the latter.