Questioning the Central Bond in [1.1.1]Propellane: Insights from Probability Density Analysis

Abstract

In the 1980s, the investigation of [1.1.1]­propellane started to attract a lot of attention. The key question is the nature of the stabilization of its singlet state relative to its triplet state, sparking an intense debate about the existence of a central bond. In this work, this case is revisited with orbital-independent probability density analysis (PDA), which is based on the measurable many-electron probability density. The most probable electron positions and their exchange paths are studied for singlet [1.1.1], [2.1.1], [2.2.1], and [2.2.2]­propellane as well as the triplet state in [1.1.1]­propellane. The results are compared to those in ethane, cyclopropane, bicyclo[1.1.0]­butane, and bicyclo[1.1.1]­pentane. By analyzing the differences between singlet and triplet [1.1.1]­propellane in comparison with other molecules, we conclude that the notion of a central bond is not justified and the stabilization of singlet [1.1.1]­propellane arises solely from unusual many-electron exchanges involving the wing bonds.

Introduction

First introduced in 1916, Lewis structures laid the foundation of the chemical language. In their simplest form, single electrons are depicted as dots and electron pairs as lines. Using these rules, basic molecules can be sketched on paper, with covalent bonds represented as lines between atoms. For most molecules, the existence of such bonds is indisputable; however, this is not the case for [1.1.1]­propellane.

Before it was first synthesized in 1982 by Wiberg and Walker, theoretical studies had already predicted that the singlet state would be significantly more stable than the triplet state. , After its synthesis, [1.1.1]­propellane was found to be relatively stable, resisting rapid (presumably free radical) polymerization in contrast to [2.2.1]­propellane. This stability may be attributed to the high activation energy required for (triplet) diradical formation. , Subsequently, [1.1.1]­propellane, more precisely the interaction between the two bridgehead carbon atoms (C₁–C₂), began to be studied in great detail.

While the C₁–C₂ bond was discussed to result from the overlap of the smaller lobes of sp ^x hybrid orbitals of the C₁ and C₂ atoms, Jackson and Allen found the HOMO to be nonbonding and little charge density between the bridgehead atoms. They explained the stability of the singlet state and the fact that the C₁–C₂ bond distance is comparable to that of a C–C σ bond by a mechanism they called a “σ-bridged π bond”. This bond is best described by a three-center, two-electron bond composed of the p orbitals of the bridgehead carbon atoms and the sp³ hybrid orbitals of the methylene bridges.

One of the first multiconfigurational calculations on [1.1.1]­propellane was done in 1987 by Feller and Davidson using the two-configurational SCF procedure, equivalent to a CASSCF­(2,2) calculation. They pointed out that the substantial thermodynamic stabilization of [1.1.1]­propellane is incompatible with a diradical structure, allowing both a direct C₁–C₂ bond or a stabilization via the methylene wings.

Later, upon studying the electronic densities, Wiberg, Bader, and Lau found bond critical points between the C₁ and C₂ atoms for [1.1.1]­propellane but not for bicyclo[1.1.1]­pentane. Other early studies are found in the review by Wiberg. More recently, these bond critical points have also been found in experiments carried out on propellane derivatives.

As reported by Polo and co-workers, an analysis with the electron localization function (ELF) of the electronic density obtained from a DFT calculation shows a significant ionic contribution. They defined three groups of propellanes: larger propellanes ([3.2.2], [3.3.2], and [3.3.3]) with a typical C₁–C₂ σ bond, medium propellanes ([2.2.1], [3.2.1], [2.2.2], and [3.3.1]) with a lower, depleted basin population between the C₁ and C₂ atoms, and small propellanes ([1.1.1], [2.1.1], and [3.1.1]). For small propellanes, two resonance forms are found: one with an electron pair at either the C₁ or C₂ atom located outside of the cage and the second form with a C₁–C₂ bond. The weight of the latter increases in the series of [1.1.1], [2.1.1], and [3.1.1]­propellanes.

Wu, Shaik, Hiberty, and co-workers coined the C₁–C₂ interaction in [1.1.1]­propellane an inverted bond referring to the outward-directed sp ^x lobes of the C₁–C₂ bond. They calculated an in situ bond dissociation energy (BDE) by using a nonbonding quasiclassical reference state. The difference between this reference and the ground-state energy yields the in situ BDE. For [1.1.1]­propellane, this energy is 119 kcal mol^–1, which is only 20 kcal mol^–1 lower than the energy for the C–C bond in ethane with 139 kcal mol^–1. In addition to quantifying the BDE, they concluded that the inverted bond in [1.1.1]­propellane is a charge-shift bond. The positive Laplacian of the electronic density at the bond critical points of the bond and the depleted ELF basin are all properties of a charge shift bond, as defined in the valence bond (VB) framework. By definition, for a charge shift bond, more than 50% of the BDE results from covalent ionic resonance, as is the case in propellane. A more recent study by Shaik and co-workers, investigating C–C bonds in different propellane molecules, shows a trend, where the charge shift character decreases with increasing ring sizes.

Most of this literature tends toward a C₁–C₂ bond in propellane. However, in 2020, Laplaza and co-workers systematically studied the inverted C–C bond. As model systems, they chose ethane and CH₃–CCH, where they decreased the HCC angle and compared the BDE values for these different angles. They found that the bond energy decreases rapidly with the inversion angle and tends to zero, if not negative values, with the same angular constraints as in propellane. Moreover, they calculated the dynamic orbital forces and also found the HOMO of propellane to be antibonding. They concluded that the C₁–C₂ bond can be reduced to a π-type, or banana bond, a view criticized by the valence bond community.

In a recently published study, Cooper and co-workers calculated the isotropic magnetic shielding for different positions in [1.1.1]­propellane and compared this to the values for cyclopropane and bicyclo[1.1.0]­butane among others. The shielding activity inside the [1.1.1]­propellane cage, when compared to the other systems, points to a significant covalent bonding interaction inside the cage, which can however not be partitioned into contributions from specific C–C bonds.

Most of the methods used in these studies rely on either electron density or analysis of orbitals. Both are one-electron functions where the possibly important many-electron perspective is either integrated out to calculate the electronic density or only indirectly covered by the use of orbitals. With probability density analysis (PDA), the many-electron probability density |Ψ|² is investigated directly, thus giving insight into many-electron effects, which are the glue binding the pieces in the propellane puzzle.

PDA can be understood as the 3N-dimensional analog of Bader’s quantum theory of atoms in molecules (QTAIM), with N being the number of electrons. , While in Bader’s theory, the maxima of the electronic density, usually representing the atomic nuclei, are studied, in PDA, the maxima of |Ψ|² are investigated. These maxima are, see examples below, the most likely positions of all electrons simultaneously. Electrons are delocalized quantum particles, and in PDA, a maximum of |Ψ|² represents all electron arrangements with the same maximum, similar to the quantum atom in Bader’s theory represented by the nucleus. After clustering of similar maxima, each cluster can usually be assigned to a Lewis resonance structure with a weight. These weights are in good agreement with the inverse weights calculated from valence bond theory. , Moreover, for each cluster, spin correlations between pairs of electrons can be calculated, taking values between −1 for opposite-spin and +1 for same-spin correlation. Recently, Martín Pendás and co-workers used PDA to investigate the electron positions maximizing |Ψ|² for atoms in different states and correlated these maxima with maxima found in molecules. ,

In addition to the maxima, the paths connecting them result in important information. In QTAIM, the nuclei (representing atoms) are connected via bond paths defined by bond critical points, which are saddle points of the electronic density. In PDA, the exchange of electrons, as given by a maximum of |Ψ|², is characterized by a maximum probability path with a delocalization critical point, defined as a first-order saddle point. To facilitate interpretation, the probability potential Φ replaces the probability density |Ψ|². ,

$$\mathrm{\Phi }=-\frac{1}{2}\ln |\mathrm{\Psi }{|}^2$$ 1

This definition changes the maxima of |Ψ|² to minima of Φ and turns the multiplicative nature of probabilities into additive probability barriers. To clarify the importance of these barriers and their connection to covalent bonding, let us consider two orbitals, ϕ_A, ϕ_B, each occupied by one electron. ϕ_A and ϕ_B could be two 1s orbitals in the case of H₂ or two sp³ hybrids in the case of the C–C bond in ethane. With electrons 1 in ϕ_A and 2 in ϕ_B, the resulting wave function is ϕ_A(1)­ϕ_B(2). The maximum of |Ψ|² has obviously electron 1 at A and (simultaneously) electron 2 at B; both electrons are localized. This wave function violates the indistinguishability of the electrons (assumed having opposite spin). The quantum chemically correct wave function

$$\mathrm{\Psi }(1,2)={\varphi }_{\mathrm{A}}(1){\varphi }_{\mathrm{B}}(2)+{\varphi }_{\mathrm{B}}(1){\varphi }_{\mathrm{A}}(2)$$

adds a new equivalent maximum with electron 1 at B and 2 at A and a probability barrier describing the 1 ↔ 2 exchange. For H₂, this is the Heitler-London wave function and for ethane, this is part of the valence bond wave function (see Figure ). At large distances of A and B, Ψ nearly vanishes for an electron position at the midpoint between A and B, resulting in a huge value of Φ or localized electrons (but indistinguishable). At the equilibrium distance, the exchange of the electrons is characterized by the value Φ_min at one of the minima and a finite value of Φ_sp at the saddle point, allowing the definition of the probability barrier for the exchange

$$\mathrm{\Delta }\mathrm{\Phi }={\mathrm{\Phi }}_{\mathrm{s}\mathrm{p}}-{\mathrm{\Phi }}_{\min }$$ 2

2.

Qualitative representation of the two-center two-electron bond in ethane with a valence bond wave function and the corresponding bond electron exchange. Plot of the probability density |Ψ|² and probability potential Φ for the concerted two-electron exchange in a two-center two-electron exchange.

The probability barrier assigns a measure of importance to an exchange path, thus characterizing the delocalization of bond electrons and the resonance stabilization of the bond. Most importantly, PDA is not tied to valence bond theory but is applicable to all wave functions. In an MO treatment, for instance, the wave function is similar to additional ionic arrangements at A and B. For a more detailed analysis, see ref .

Why does it make sense to analyze |Ψ|² when the resonance stabilization can be directly calculated, e.g., with valence bond codes? The advantage lies in the fact that partitioning of the quantum system into fragments like ϕ_A and ϕ_B in the above example is not necessary (it might be ill-defined). Moreover, the electron exchange in the example is constructed as a two-center two-electron exchange. How can a many-electron exchange be identified? It should be noted that the many-electron wave function is not given by the molecular or atomic orbitals but by one or more Slater determinants built from the orbitals. The determinant is responsible for the exchange energy contributions in the Hartree–Fock theory and may also result in many-electron effects in |Ψ|².

PDA provides a first-principles approach to investigating chemical bonding in molecules by identifying electron maxima and characterizing exchange paths through their probability barriers. Equipped with this orbital-independent toolset, propellane molecules in different states, bicyclo[1.1.0]­butane, bicyclo[1.1.1]­pentane, and cyclopropane (Figure ), are examined and compared with the C–C bond in ethane, the archetype of a covalent two-center-two-electron (2c2e) bond, from a many-electron perspective. It is demonstrated that the special stabilization of [1.1.1]­propellane is due to cooperative exchange paths of five or six electrons.

1.

Geometries for different propellane molecules as well as bicyclo[1.1.0]­butane, bicyclo[1.1.1]­pentane, and cyclopropane (1 and 2, 3–6 CASSCF­(2,2)/def2-TZVP, and 7 HF/def2-TZVP).

Theoretical Methods

For molecular geometries, the experimental geometries from the NIST Computational Chemistry Comparison and Benchmark Database were selected for ethane, cyclopropane, and bicyclo[1.1.0]­butane. The geometry of bicyclo[1.1.1]­pentane was optimized at the HF/def2-TZVP level. For all other molecules, the geometries were optimized by using CASSCF­(2,2) with the def2-TZVP basis set. For [1.1.1]­propellane, the CASSCF­(2,2)/def2-TZVP geometry is in good agreement with the experimental one; this has also been reported in ref .

As in previous works, the wave functions used for all molecules are Jastrow-correlated linear combinations of Slater determinants. Uncorrelated wave functions show maxima of |Ψ|² having opposite spin electrons at the same position, which is unphysical and results from the lack of electron correlation. The antisymmetric parts of the wave functions were generated with CASSCF­(2,2) calculations for all propellanes and bicyclo[1.1.0]­butane, in order to allow for a smooth transition between the closed shell and the diradical. For all other molecules, a single Slater determinant from a Hartree–Fock calculation was used. The dynamic electron–electron correlation is accounted for with a Jastrow factor. For all wave functions, a generic sm555 Jastrow correlation-function , was used and optimized with the amolqc software. All calculations were performed using the all-electron triple-ζ Slater-type basis set TZPae. All CASSCF and Hartree–Fock calculations were performed using orca. , For using the Slater-type basis set in orca, each basis function was expanded into 14 Gaussian functions. ,

The probability density analysis was performed with amolqc and inPsights. The minima of the probability potential Φ were obtained with a combination of gradient descent and the L-BFGS methods. To obtain the clusters, their weights, and spin correlations, only the electrons closest to the region around the C₁ and C₂ atoms were clustered. For the saddle points search, Newton’s method and a systematic algorithm covering all of |Ψ|² were employed. The methods used to do the PDA are laid out in more detail in the Supporting Information.

For the visualization, the molara software was used.

Results and Discussion

Only all-electron basis sets were employed; therefore, for each electron arrangement that will be shown, there are always two electrons located at the carbon nuclei and at least one electron at the hydrogen nucleus. Because these core electrons are not relevant to the chemistry discussed in this work, they will be neglected. Molecules 5 and 6 contain hydrogen atoms having 2 electrons in their core. They will be labeled as negatively charged hydrogen atoms. Since they do not play an important role in describing the central C–C interaction, this will not be discussed any further. The “bonds” connecting the atoms in the following representations are only meant to be a guide for the eye but do not imply a chemical bond.

Maxima of |Ψ|²

The C–C bond in ethane is the archetype of a covalent two-electron, two-center (2c2e) bond. All C–H bonds and the C–C bond each have a pair of opposite spin electrons, equivalent to the Lewis structure. This C–C bond motif has one electron located in the proximity of one atom, and the other electron, with the opposite spin, is close to the second atom. It has a weight of 100% (Figure ).

3.

Electron positions (red and blue spheres, spin up and spin down, respectively) globally maximizing |Ψ|² for the molecules 1–7 and ethane. [1.1.1]­propellane 3s represents its singlet state and 3t its triplet state. The white and gray circles represent the hydrogen and carbon atoms. The green lines connect pairs of opposite-spin-correlated electrons and the purple line connects pairs of same-spin-correlated electrons. For molecules 5 and 6, some of the C–H bonds have a negatively charged hydrogen atom.

For cyclopropane 1 and bicyclo[1.1.0]­butane 2, the C₁–C₂ bond electrons form this typical 2c2e bond motif, which was found to be dominant in 1 and in 2 with weights of 100 and 97%, respectively. All other C–C bonds also adhere to this 2c2e bond motif. Note the bent bond motif. The C–C maximum positions in the three-rings and the methylene bridges in the propellane are bent outside the C–C line, consistent with other theoretical treatments of bent bonds.

For the singlet [1.1.1]­propellane, a different motif is observed for the C₁–C₂ electrons. Both electrons lie outside the propellane cage and not between the C₁ and C₂ atoms. This arrangement will be labeled the inverted bond motif. This motif can be best explained by electron–electron repulsion: First, all C–H bonds are normal 2c2e electron bonds, as well as the C–C bonds along the propellane wings. This leaves two valence electrons that are not accounted for. Since all 2c2e bonds are already occupied and due to the repulsion of all other electrons, the two missing electrons are found outside of the propellane scaffold at the C₁ and C₂ atoms. These electron positions can also be explained as maxima of four sp³ hybrid orbitals from a qualitative valence bond perspective. The two missing electrons would thus occupy the larger lobes of the sp³ hybrid orbitals, which would lie outside of the scaffold due to the inversion angle.

Although this inverted motif is not found in 1, it contributes the missing 3% in 2 and dominates the electron arrangements in singlet [1.1.1]­propellane with ∼100%. The normal 2c2e bond motif was not found in 3s; only the motifs where one of both electrons is located in the cage were found but with a negligible weight. For the triplet state 3t, the inverted arrangement is found to be 100%. In this case, however, the two electrons that are left after occupying all 2c2e bonds have the same spin.

For the larger propellanes 4 and 5, the inverted bond motif remains the dominant one with approximately 100% weight. All bonds in [2.2.2]­propellane 6, including the C₁–C₂ bond, show the normal 2c2e motif with a weight of around 100%. Lastly, in bicyclo[1.1.1]­pentane 7, there is no C₁–C₂ bond motif, as all valence electrons are involved in the 3 methylene bridges and the C–H bonds. All electron positions maximizing the probability density are displayed in Figure .

Spin Correlations

After clustering the maxima of |Ψ|², the spin correlations between all electrons can be determined. A correlation of −1 indicates that whenever one electron is spin-up, its correlated counterpart is spin-down, while a correlation of +1 signifies that both electrons have the same spin. Intermediate values between these extrema reflect varying probabilities of electrons having either the same or opposite spins. For normal 2c2e bonds, a spin correlation of exactly −1 is observed. In all of the propellane molecules presented in this work, the C₁–C₂ electrons show a spin correlation of −1 in the singlet state and +1 in the triplet state. However, this does not necessarily indicate a bond. A singlet spin correlation is also observed for H₂ at very large distances, as discussed above.

As explained earlier, all other bonds in the propellane molecules have 2c2e bonds. Since all 2c2e bonds involve an opposite-spin electron pair, the C₁–C₂ bond electrons must form either an opposite-spin pair in the singlet state or a same-spin pair in the triplet state. For molecule 7, this is not the case, because the electrons on the C₁–H and C₂–H bonds can exchange their spins. Therefore, there is only a negligible correlation between the C₁ and C₂ electrons. These spin correlations are represented by green and purple lines (−1 and +1, respectively) connecting correlated electron pairs in Figure .

In addition to perfect ±1 correlations, singlet propellane molecules 3–5 show a significant singlet spin correlation of ∼−0.1 between the C₁–C₂ electrons and the adjacent methylene wing-bond electrons (Figure ). This correlation is not present for the triplet wave functions or the bicyclo[1.1.1]­pentane molecule. The electrons involved in the 2c2e electron bonds on the wing-bonds can exchange their positions via a probability barrier; therefore, they are completely correlated. The correlation between the C₁–C₂ bond electrons and the wing-bond electrons arises from the fact that moving one of the C₁–C₂ bond electrons to the wing-bond would require exchanging the pair of valence electrons at the methylene bridge as well as switching the other C₁–C₂ bond electron with one of the wing-bond electrons. Otherwise, one of the 2c2e bonds would contain two electrons with the same spin, which is never found. From this, it follows that such a correlation cannot exist for the triplet wave functions as there will always be one pair of electrons with the same spin, which prevents them from being exchanged (see the wing-bond graphics in Figure ). The size of the bridge also plays a role because the correlation is not found for the ethylene bridges, which will be further discussed in the next section. These spin correlations suggest that the wing bonds play a crucial role in stabilizing the singlet state compared to the triplet state.

4.

Electron positions (red and blue spheres, spin up and spin down, respectively) and spin correlations in the global maximum of |Ψ|² for singlet and triplet [1.1.1]­propellane for one C–C–C wing-bond (lower graphics) as well as for the complete molecules 3s, 4, and 5. The white and gray circles represent the hydrogen and carbon atoms. The green lines connect pairs of opposite-spin-correlated electrons and the purple line connects pairs of same-spin-correlated electrons. The dotted green line represents a spin correlation of ∼−0.1. For molecule 5, one of the C–H bonds has a negatively charged hydrogen atom. The blue arrows between the electrons indicate possible exchanges and the red arrow indicates a forbidden exchange.

Maximum Probability Paths

Electron Exchanges on Two-Electron Two-Center Bonds

The possibility of exchanging the positions of both bond electrons is characteristic for the covalent bond and due to resonance (or interference) of the fragments (see the discussion above). For the standard covalent 2c2e bond in ethane, this means connecting a maximum of |Ψ|² via a saddle point with another maximum where the spins of the bond electrons are exchanged (Figure ). The barrier of this maximum probability path can be interpreted as the likelihood of exchange. This exchange is found with varying barrier heights for all 2c2e bonds for molecules 1–7. They can all be put into one of four categories: The C₁–C₂ bond, the bond connecting one of the bridgehead carbon atoms with the carbon atom on the methylene bridge (C₁–CH₂ –C₂), the same connection with a carbon atom of an ethylene bridge (C₁–CH₂ –CH₂–C₂), and the C–C bond on the ethylene bridge (C₁–CH₂–CH₂ –C₂). The electron arrangements and the probability barriers of these bond electron exchanges are shown in Figure and Table . For the C₁–C₂ bond, a 2c2e bond motif can be found only in ethane, 1, 2, and 6. All other studied molecules do not feature this motif. The lowest barrier is found in ethane and cyclopropane, and the value found in 2 and 6 is ∼1.5 times higher. This correlates with the strain introduced by the increase in the inversion of the bond. It has already been shown that increasing the inverted character of a 2c2e C–C bond reduces its dissociation energy. This fits the larger barrier, resulting in less likely exchange and thus smaller delocalization and less stabilization.

5.

Electron positions (red and blue spheres, spin up and spin down, respectively) for two minima and one saddle point of Φ for ethane. The white and gray circles represent the hydrogen and carbon nuclei. There is one electron located on each hydrogen nucleus as well as two electrons on each carbon nucleus. The arrows represent the eigenvector corresponding to the negative eigenvalue of the Hessian.

6.

Electron positions (red and blue spheres, spin up and spin down, respectively) for four saddle points of |Ψ|² for [1.1.1]­propellane and [2.1.1]­propellane. They serve as a definition for the bonds discussed in Table . The white and gray circles represent the hydrogen and carbon atoms. There is one electron located in each hydrogen nucleus as well as two electrons in each carbon nucleus. The arrows represent the eigenvector corresponding to the negative eigenvalue of the Hessian.

1. Probability Barriers for the Two-Center Two-Electron Bond Electron Exchange for Different Bonds in Ethane and Molecules 1–7 .

molecule	bond	ΔΦ
ethane	C–C	0.29
1	C1–C2	0.29
2	C1–C2	0.43
	C1–CH2 –C2	0.26
3s	C1–CH2 –C2	0.18
3t	C1–CH2 –C2	0.24
4	C1–CH2 –C2	0.18
	C1–CH2 –CH2–C2	0.19
	C1–CH2–CH2 –C2	0.24
5	C1–CH2 –C2	0.17
	C1–CH2 –CH2–C2	0.18
	C1–CH2–CH2 –C2	0.23
6	C1–C2	0.46
	C1–CH2 –CH2–C2	0.21
	C1–CH2–CH2 –C2	0.28
7	C1–CH2 –C2	0.25

The C₁–CH₂ –C₂ bond, present in all molecules but 6, shows notable variations across the different molecules. For 1, 2, 3t, and 7, the barriers are all close to the one found in ethane but all marginally lower. Interestingly, the barriers in 3s, 4, and 5, which are all singlet propellane molecules with an inverted bond motif, are significantly lower compared to the standard bond in ethane. Moreover, the bond electrons for these 2c2e exchanges could be located at either one of both bond atoms, with equal probability for homopolar bonds. However, in this case, they prefer to be close to the bridgehead carbon atom with a barrier difference of 0.04 between both saddle points in singlet [1.1.1]­propellane. By simply counting the electrons in the proximity of each carbon atom, this saddle point can be interpreted as a C⁺C^– ionic saddle point.

This anionic tendency of the bridgehead carbon atom is in agreement with the ELF results of a single determinant wave function of Polo and co-workers. The results presented here were obtained by using a two-determinant CASSCF­(2,2) wave function. However, when retaining only the determinant where the a₂ orbital is doubly occupied and removing the determinant with a doubly occupied a₁ orbital (Figure ), both the ionic saddle points and the ionic C₁ C₂ and C₁ C₂ maxima are recovered. Such significant changes are expected in topological analyses. To ensure comparability, all results were calculated at the same theoretical level. These findings highlight that a single-determinant wave function may be insufficient to accurately describe this system, a point previously discussed by Feller and Davidson. The energy difference between the CASSCF­(2,2) wave function and the HF wave function is 31 mE _h, indicating light multireference character (compared to 1 mE _h for ethane). Moreover, although a PDA charge shift strength cannot be determined because of the central inverted bond motif, the anionic tendency of the electron exchanges can be interpreted as a charge shift contribution. This contribution, however, cannot be quantified in the current PDA framework as it does not fit the current 2c2e bond definition but would arise from concerted electron exchanges across the wing bonds.

7.

a₁′ and a₂″ orbitals for singlet [1.1.1]­propellane obtained from the active space of a CASSCF­(2,2)/TZPAE//CASSCF­(2,2)/def2-TZVP calculation. The dominant determinant, where the a₁′ orbital is doubly occupied, has a linear coefficient of 0.977.

The barriers of the electron exchange on the C₁–CH₂ –C₂ bond lead to the conclusion that the wing bond electrons can move more freely, that is, they are more delocalized for singlet propellanes than for the triplet [1.1.1]­propellane and 7. Lastly, for ethylene bridges, although the C₁–CH₂ –CH₂–C₂ exchange barrier is lower compared to that for ethane, the C₁–CH₂–CH₂ –C₂ barrier is almost equally as high as the standard C–C bond exchange. The C₁–CH₂–CH₂ –C₂ bond can thus be considered a normal 2c2e bond. According to these results, all bonds adjacent to one of the bridgehead carbon atoms in the singlet propellanes feature relatively low 2c2e bond electron exchange barriers. This leads to increased electron delocalization across the propellane wings for singlet states.

Concerted Many-Electron Exchanges across the Wing Bonds

What about the C₁–C₂ bond electrons in the propellane molecules? As a result of the lack of a 2c2e bond motif, no direct electron exchange was found. Although it is impossible to exchange these electrons for the triplet propellane, because of the antisymmetry, it must be possible for singlet systems. Therefore, a more complex exchange path of the C₁ and C₂ bond electrons must exist. As explained earlier, it follows from the spin correlations across the singlet [1.1.1]­propellane wings that it is impossible to exchange an opposite spin electron pair across two adjacent bonds without changing all other adjacent opposite spin pairs of the same bridge. Not surprisingly, such concerted six-electron exchange paths (Figure a and b) were found, with barriers around three and four times the height of the barrier in ethane, respectively. These many-electron exchanges constitute the first step in such a more complex exchange path of the C₁ and C₂ bond electrons because it places these electrons on the wing bond. Once they are there, they can continue to the other central carbon atom via the 2c2e exchanges on the wing bonds. While for b, only one wing bond is involved, for a, two wing bonds are involved. However, a is similar to b, because by flipping the electrons from the second wing bond onto the other wing bond, an equivalent saddle point to b can be constructed. In addition to this path, another many-electron exchange path that includes the five same spin electrons from two wing bonds with a barrier in between the two other barriers is found (Figure c). Despite the higher barriers compared to the 2c2e exchange, these many-electron exchanges can only be found for the singlet propellanes with at least one methylene wing bond and not for triplet [1.1.1]­propellane. It has to be noted, that the existence and the barriers of these multielectron exchanges do not have a direct connection to the 2c2e electron exchanges on the wing bonds. For ethylene bridges, such an exchange would involve four pairs of electrons with opposite spins, making it highly improbable to the extent that it was not observed. This marks a significant difference, that correlates with the energetic stabilization in singlet propellane and the decrease in stabilization when introducing ethylene bridges.

8.

Electron positions (red and blue spheres, spin up and spin down, respectively) for three saddle points of |Ψ|² for [1.1.1]­propellane. Saddle points (a) and (b) represent two six-electron exchanges involving two and one wing bond, respectively, and (c) represents a five same-spin electron exchange. The white and gray circles represent the hydrogen and carbon atoms. There is one electron located in each hydrogen nucleus as well as two electrons in each carbon nucleus. The arrows represent the eigenvector corresponding to the negative eigenvalue of the Hessian.

σ-Aromaticity

In previous work, Hückel’s 4n + 2 rule was reformulated to a 2­(2n + 1) rule using insight gained with PDA. The idea is simple: cyclic permutations of an odd number of same spin electrons have a positive sign and thus do not change the sign of the wave function. This enables independent rotations of same spin electron rings, for example in benzene. Surprisingly, a concerted many-electron exchange involving five same-spin electrons was identified in the singlet [1.1.1]­propellane (Figure c). While rotations of three same-spin electron triangles have already been reported in other molecules and are commonly found, rotations involving more than three electrons have not been observed previously.

Saddle point c involves the electrons of two wing bonds as well as one of the C₁–C₂ electrons. For the triplet [1.1.1]­propellane, the rotation would consist of an even number of same-spin electrons, which is not possible due to antisymmetry. Furthermore, if one methylene and one ethylene bridge were involved, the number of the same spin electrons would also be even. For two ethylene bridges, such an exchange would involve seven electrons, making it too unlikely. Moreover, a concerted three-electron ring rotation was found for cyclopropane (Figure ).

9.

Electron positions (red and blue spheres, spin up and spin down, respectively) for one saddle point of |Ψ|² for cyclopropane. The white and gray circles represent the hydrogen and carbon atoms. There is one electron located in each hydrogen nucleus as well as two electrons in each carbon nucleus. The arrows represent the negative eigenvalue corresponding to the negative eigenvalue of the Hessian.

Although, the concept of σ-aromaticity is generally accepted, the identification of cyclopropane as a σ-aromatic molecule is under debate. , We do not want to join this debate; however, these concerted ring rotations need to be noted as they mark a distinct difference between singlet and triplet [1.1.1]­propellane. Additionally, they are also present in cyclopropane and hint at an aromatic contribution in the stabilization of singlet [1.1.1]­propellane.

This finding is further supported by the work of Karadakov, Stewart, and Cooper, who observed significant shielding activity within the scaffold of singlet [1.1.1]­propellane using nuclear independent chemical shift calculations (NICS).

Conclusions

The so-called “inverted bond” was only partly recovered with PDA. Only the electron positions in the global maxima of |Ψ|² resemble this interpretation. Most importantly, however, this does not imply a bond between these atoms because the C₁ and the C₂ electrons have no alternative placements; thus, their position and spin correlation can both be explained without the need for a chemical bond.

This inverted motif also results in the inability to calculate the PDA charge shift strength for this “inverted bond”. The lack of electron arrangements in the central region of the [1.1.1]­propellane scaffold does not preclude a charge shift stabilization, but according to these results, it cannot be reduced to a localized central bond. It would rather be a part of the concerted electron delocalization across the wing bonds.

For PDA, the most significant differences all originate from the wing bond electrons. Spin correlations between the C₁ and C₂ electrons and all other wing bond electrons were found for the methylene bridges. These correlations result from the antisymmetry of the wave function and pairwise electron exchanges; therefore, they do not exist in triplet [1.1.1]­propellane.

Furthermore, concerted many-electron exchanges were found for the singlet [1.1.1]­propellane wave function involving up to six electrons, whereas none were observed for the triplet wave function. They can be rationalized by even and odd numbers of the same-spin electrons and by the previously published 2­(2n + 1) rule for aromaticity. These same-spin electron ring rotations, that involve wing bond electrons, hint at an aromatic contribution to the stabilization of the singlet state. This σ-aromaticity will be part of the future studies.

To conclude, in the heavily strained [1.1.1]­propellane molecule, it is hard to assign any favorable interaction to a local chemical bond, which is in agreement with recent literature. Therefore, it is evident that for exotic stabilizations or bonding situations, we must venture away from the common chemical bond picture and accept the complexity in describing these cases.

In this work, the complete topology of the all-electron probability density has been characterized. No electron arrangements or exchanges for the central region of [1.1.1]­propellane were found that would fit the normal description of a 2c2e bond. All of the differences between the singlet and triplet state exclusively involve the electrons of the wing bonds. It follows that these differences must fully account for the energy gap between the singlet and triplet states and thus the lack of diradical character in [1.1.1]­propellane. Therefore, the stabilization of the singlet state over the triplet state in [1.1.1]­propellane cannot be reduced to the presence of a central bond but arises from cooperative delocalizations associated with the wing bonds.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.5c03794.

• Computational methods and algorithms; description of the CASSCF active space; discussion of molecular geometries; and Cartesian coordinates of optimized structures (PDF)

Roles: Conceptualization, M.V.H. and A.L.; data curation, M.V.H; formal analysis, M.V.H; funding acquisition, M.V.H. and A.L; investigation, M.V.H; methodology, M.V.H and A.L.; project administration, A.L.; resources, A.L.; software, M.V.H and A.L.; supervision, A.L.; validation, M.V.H; visualization, M.V.H; writingoriginal draft, M.V.H; and writingreview and editing, M.V.H and A.L. All authors have read and agreed to the published version of the manuscript.

The authors declare no competing financial interest.