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Helical Electronic Transitions of Spiroconjugated Molecules

The two perpendicularly oriented π-systems of allene mix into helical molecular orbitals (MOs) when the symmetry of the molecule is reduced. However, the π-π ∗ transitions of allenes are linear combinations of two excitations that always consist of both helicities; consequently, the electronic transitions are not helical. Here, we examine the electronic structure of spiroconjugated molecules, which have the same parent symmetry as allene but with different relative orientation of the two π-systems. We show how the π-mixing in spiropentadiene is analogous to the helical π-mixing in allene. However, in spiroconjugated systems only half the π-MOs become helical. Due to this difference, the π-π ∗ transitions in substituted spiropentadiene come in near-degenerate pairs where the helicity is symmetry protected, and consequently there is no significant mixing between excitations involving MOs of opposite helicity. This inherent helicity of the π-π * transitions is verified by computation of the change of electron density. These transitions have big rotatory strengths where the sign correlates with the helicity of the transition. The electronic helicity of spiroconjugated molecules thus manifests itself in observable electronic and optical properties. ABSTRACT: The two perpendicularly oriented π-systems of allene mix into helical molecular orbitals (MOs) when the symmetry of the molecule is reduced. However, the π-π ∗ transitions of allenes are linear combinations of two excitations that always consist of both helicities; consequently, the electronic transitions are not helical. Here, we examine the electronic structure of spiroconjugated molecules, which have the same parent symmetry as allene but with different relative orientation of the two π-systems. We show how the π-mixing in spiropentadiene is analogous to the helical π-mixing in allene. However, in spiroconjugated systems only half the π-MOs become helical. Due to this difference, the π-π ∗ transitions in substituted spiropentadiene come in near-degenerate pairs where the helicity is symmetry protected, and consequently there is no significant mixing between excitations involving MOs of opposite helicity. This inherent helicity of the π-π * transitions is verified by computation of the change of electron density. These transitions have big rotatory strengths where the sign correlates with the helicity of the transition. The electronic helicity of spiroconjugated molecules thus manifests itself in observable electronic and optical properties. Tricyclic spiro-analogues.

Our chemical intuition is often challenged because one simple electronic structure model cannot perfectly account for every type of molecular property. 1 The electrohelicity effect that has been described in chiral allenes, as well as longer cumulenes and polyynes, is a particular fascinating and ambiguous case. 2 In these linear molecules, their otherwise orthogonal p-systems can mix and form helical molecular orbitals (MOs). [2][3][4] However, an MO is not an observable property of a molecule. But although purely a theoretical construct for the purpose of computing properties of molecules, MOs have successfully provided insight into the fundamental nature of electrons in molecules, 5 most famously the selection rules for pericyclic reactions. [6][7][8][9] The understanding of helical p-conjugation through MOs may provide new chemical intuition. 2 However, while the molecules where the effect appear are single-handed chiral, both helicities are present in the electronic structure. The chirality associated with the electronic structure is thus more complex than that of the of molecular structure. 10 This dual-helicity may limit the possibility of experimentally observable effects arising from the helicity of any specific MO, but some relations to properties have been proposed in recent years. 2,[11][12][13][14][15][16][17][18][19][20] The helicity of the electronic structure in allene (Scheme 1, left) is mediated by the p-orbitals on its central sp-hybridized carbon. As discussed in detail in prior publications, an initial choice is made for the orientation of these two p-basis functions. 4,8,21 By convention, we orient allene relative to its two mirror-planes. Shown in Figure 1a, the px and py basis functions are thereby symmetry-adapted to the sub-groups of the D2d point group that have mirror-plane symmetry, the most notable being C2v. However, the px and py basis functions can be rotated 45° around the allenic axis by forming linear combinations of the two, thus making a coarctate orbital system; i.e., one where there is a continuous p-overlap between the carbon termini despite their relative orientation of 90°. 4,22,23 Shown in the right column of Figure 1a, this basis is symmetry-adapted to the subgroups of D2d that retain its rotation axes, such as D2. These two bases are connected by a unitary transformation and the choice between them is arbitrary. 4, 21 Scheme 1. Allene and Spiro-type Molecules.
In spiroconjugated molecules, two p-systems are separated by an sp 3 -hybridized carbon atom. 24, 25 The two p-systems are oriented 90° relative to each other and interact by a well-described through-space interaction. 26 Spiropentadiene is the simplest spiroconjugated molecule, constituting two ethene units fused together through a fifth spiro-carbon as shown in Scheme 1. Similar to allene, the parent spiropentadiene has D2d symmetry with the same number of p-electrons. 27 Shown in Figure  1b, we can make an analogous basis set rotation of the p-basis functions on the central carbon atom of spiropentadiene. These p-orbitals on the formally saturated spiro-carbon will participate in the s-bonds of the molecule and are normally considered in the form of sp x hybrid orbitals; however, the p-orbitals, which mathematically are part of the carbon basis set, can mediate the possible through-bond coupling between the two p-systems. This through-bond effect has been described in the related saturated triangulanes (Scheme 1), where helical s-orbitals appear along the full length of the molecular backbone. 28, 29 The p-orbitals in Figure 1b are available as relay orbitals on the spirocarbon, and is equivalent to the way s-orbitals mediate throughbond coupling between the p-systems in larger tricyclic analogues of spiroconjugated molecules (Scheme 1) as described by Gleiter and co-workers. 30, 31 While this is conceptually similar to allene, the two p-systems are oriented different spatially in spiropentadiene. As we shall see, this difference has a profound effect on the electronic structure and p-p* transitions. This letter proceeds as follows. We revisit the electronic structure and p-p* transitions of allene and its 1,3-disubstituted derivates. 1,3-disubstituted allene constitutes the simplest case of a molecule with helical MOs, and we show how the symmetry of the parent allene dictates that their electronic transitions cannot exhibit clear helicity. In contrast, we demonstrate that in spiroconjugated molecules both the electronic structure and p-p * transitions have distinct helicity associated to them because the helicity is symmetry-protected. These transitions show helical change of electron density and have high rotatory strengths where the sign depends on the helicity of the transition. Computations are carried out using density functional theory (DFT) as implemented in Gaussian 16. 32 Excited states and optical properties are computed using time-dependent DFT. [33][34][35] Like the longer cumulenes and polyynes, allene is routinely described as having two orthogonal p-systems lying in each their plane. 36,37 The highest occupied MO (HOMO) and HOMO-1 are degenerate p-orbitals as shown in Figure 2. However, when the symmetry of the molecule is reduced from D2d to C2 by substituting two of the hydrogens, the p-systems mix and helical p-MOs are formed. 2,4 In the case of R-1,3-dimethyallene the HOMO is an M-helix and the HOMO-1 is a Phelix. In D2d-allene, the resulting p®p* transitions between the two degenerate occupied and two degenerate unoccupied p-MOs consist of the four possible linear combinations of excitations between the two sets of MOs. As outlined in Figure 3, the four transitions are non-degenerate and belong to A2, B1, A1, and B2 irreducible representations. 38 Of these four, only the B2 transition is electric-dipole allowed, and it turns out to be the S0®S4 transition. Reducing the symmetry of allene by substituents does not change much and the first three transitions remain quasi-forbidden in dimethylallene. The optical properties of substituted allenes have been explored experimentally in great detail. [39][40][41][42][43][44] The transitions are all linear combinations of the possible excitations between the occupied pP/M and unoccupied pP/M*, with almost equal weighting. Oscillator strengths and magnitude of the MO weights will of course be method dependent to some extent; a qualitatively similar result is achieved at the CASPT2(8,8)/cc-pvdz level ( Figure S1). In spiroconjugated molecules the two p-systems mix through-space. Still, the p-electrons cannot be considered fully shared between the two rings in carbon-based spiroconjugated systems. 48-50 The simplest one, spiropentadiene, is a fairly unstable motif due to bond strain, 51 but has nonetheless been synthesized with some variation. [52][53][54][55][56] Shown in the left column of Figure 4a, its HOMO and HOMO-1 are degenerate and each have clear p-character on one side of the molecule. The LUMO and LUMO+1 are non-degenerate and the two p-segments mix due to through-space p-overlap, which is characteristic of spiroconjugated systems. 24-26 Shown in the right column of Figure  4, the disubstitution we performed on allene has a similar effect on spiropentadiene. The symmetry is reduced from D2d to C2 and the degenerate px and py mix through-bond via the central carbon atom into helical pP and pM. The unoccupied px+y* and px-y* are largely unchanged and do not mix further.  In D2d-spiropentadiene, py and px are degenerate while px+y* and px-y* are non-degenerate. Consequently, the four p®p* transitions will be two degenerate pairs belonging to the E representation as verified by the direct products of the E with A2 and B1 representations in Figure 5a. 38, 57, 58 All four p®p* transitions are electric-dipole allowed, and this does not change when the symmetry is reduced by substituents. In dimethylspiropentadiene the electronic transitions become non-degenerate, but do not split by much energetically. Different from dimethylallene, the transitions are not equally weighted linear combinations. Take the two lowest energy MO excitations; pP®px+y* belong to the B irreducible representation while pM®px+y* belong to the A irreducible representation (A ´ A = A). Because they are symmetry-protected, these transitions cannot mix and do not form a superposition. As listed in Figure 5b, there is configuration interaction with higher energy excitations of same symmetry. However, excitations into px-y* contribute less than 10% to the S0®S1 and S0®S2 transitions. A qualitatively similar result is achieved at the CASPT2(8,12)/cc-pvdz level ( Figure S2).
All four p-p* transitions have clear helicity associated to the change of electron density as shown in Figure 5b. We also verify schematically in Figure S5 that all excitations from a helical to a linear MO will yield helical change of electron density. The through-bond interaction of the two p-systems mediated by the spiro-carbon thus manifests itself in the electron density. The first four electronic transitions all have notable rotatory strengths, in particular the strong S0®S3 and S0®S4 transitions. Each pair of near-degenerate transitions consist of opposite helicity in the change of electron density, and have opposite sign of the rotatory strength. Circularly polarized light will thus be more likely to absorb due to transitions of a specific helicity. This underlines the connection between the helicity of the electronic structure and the chiroptical response of the molecule. It is possible that a helical change of electron density and the rotatory strength of a transition are fundamentally related. The overall optical activity of dimethylspiropentadiene will be small unless the near-degenerate transitions can be split energetically, as opposite sign in the rotatory strengths cancels the overall chiroptical response of the molecule. The substituent strategy we used to split the helical frontier MOs of allene and longer odd-carbon cumulenes appears to be ineffective in spiropentadiene and larger spiroconjugated molecules. 20 We instead explore strained allenes and spiropentadienes by applying a cyclically-connected butadiyl-substiuent to the molecules as illustrated for spiropentadiene in Figure 6. This alters the energetics of the frontier MOs considerably as the dihedral and bond angles of the allenic and spiro-segments are strained, thereby unwinding or overwinding the helices. 3, 17 While both molecules show increased chiroptical response, cf. Figure S6 and S7, the strained R-1,4-butadiylspiropentadiene sees a major change compared to the dimethylspiropentadiene as shown in the simulated electronic circular dichroism (ECD) spectrum in Figure  6. The big rotatory strengths of the S0®S3 and S0®S4 transitions now give rise to two opposite peaks in the simulated ECD spectrum. It is likely that the extra strain in these species will destabilize them. We include them as a computational proof-ofconcept for how the electronic structure of spiroconjugated molecules can be manipulated to maximize the chiroptical response. The detailed description we have provided of spiropentadiene is in principle valid for larger spiroconjugated motives and tricyclic analogues, such as those shown in scheme 2. 58-60 In larger spiroconjugated systems, such as spirononatetraene, there are more p-electrons and therefore the MO symmetries switch. Furthermore, the helicity is less obvious in higher energy MOs due to the sharper twist. The HOMO-2 and HOMO-3 of dimethylspirononatetraene are clearly helical ( Figure S8), but will be less relevant for observable properties. In tricyclic spiro-analogues where the parent system has D2d symmetry will also have similar mixing of its two p-systems; albeit there will be extra nodal planes due to the extra s-bonds ( Figure S10 and S11. 30, 31, 61 Helical p-mixing in larger spiroconjugated motives will be subject to further investigations.

Scheme 2. Larger D2d-symmetry molecules.
In summary, We have explored how helical p-orbitals manifest themselves in the electronic structure of allene, spiropentadiene, and spirononatetraene. In the electronic transitions of allene the helicity of the p-MOs cancel out as all transitions are linear combinations of excitations involving MOs of both helicities. In spiropentadiene, the HOMO-1 and HOMO become helical in similar fashion to the p-mixing in allene, but the LUMO and LUMO+1 remain rectilinear p-MOs. The p ® p* transitions are dominated by a single configuration, which is symmetry-protected from mixing with excitations of opposite helicity. These transitions have a clear helicity associated to the change of electron density, and the rotatory strength of each transition has opposite sign from the near-degenerate transition of opposite helicity. The electrohelicity effect in spiropentadiene thus manifests itself in the optical properties of the molecule. Our results underline the continued need for improving the chemical intuition for chiroptical properties of molecules. 62 Finally, these results suggest that helical MOs can manifest in observable properties, and, as has been noted elsewhere, 63 an understanding of the electrohelicity effect beyond mean-field theory is desired. E. Spirononatetraene S12 F. Tricyclic Spiro-analogues S14 References S15 S2

A. Computational Details
All molecular structures were optimized in their electronic ground state using density functional theory (DFT) at the at the wB97X-D/Def2-TZVP level of theory using the Gaussian program package. 1-3 Structural optimizations were completed according to the tight criteria. Integrals were evaluated on the ultrafine grid. Frequency analysis was done to ensure that all molecules have been optimized to local energy minima. All optimized structures are included as supporting files in xyz format. Molecular orbitals (MOs) and electron densities were rendered using Avogadro with iso-value = 0.02. [4][5] Atomic orbitals were visualized using wxMacMolPlt. 6 Excited-state computations were carried out using linear response time-dependent DFT at the wB97X-D/Def2-TZVP level of theory as implemented in the Gaussian program package. 3 Chiroptical properties were assessed using this method; for more detail on the approach for computing molecular optical activity we refer to the extensive reviews by Autschbach. 7-8 Conformation-resolved UV-Vis and electronic circular dichroism (ECD) spectra were generated using Gaussview by assuming a gaussian line-shape with the default parameter for the standard deviation, σ=1/3099.6 nm −1 . 9 Optical rotation dispersion (ORD) spectra were made by computing the specific rotation on a grid of wavelengths between 400 and 700 nm with 1 nm grid-spacing.

C. Change of Density Schematics
We are interested in understanding the patterns of Dr which is appears in from ab initio computations. To assess this, we schematically draw up each type of occupied MO (p x , p y , p M , p P ) and unoccupied MO (p x *, p y *, p M *, p P *). Given the unoccupied p-MOs will have one extra nodal plane, we can schematically predict qualitatively where there will be a big depletion of electron density. In helical MOs the extra nodal plane is expressed as extra torsion along the helical axis.
This torsion is quite systematic, and we refer to earlier work for a detailed assessment. 13 In Figure S4, this assessment is carried out for allene. As one may expect, any single excitation  Oscillator strengths, f, are dimensionless. Rotatory strengths, R, are in 10 40 erg·esu·cm/Gauss. Depletion of density is in orange. Iso-value = 0.002. c) Simulated UV-Vis, ECD, and ORD spectra. S12

E. Spirononatetraene
Like spiropentadiene and allene, spirononatetraene has D2d symmetry. 14-15 While its electronic structure is similar to spiropentadiene, spirononatetraene has four more p-electrons and therefore the MO symmetries switch. Thus the HOMO-2 and HOMO-3 (px and py), and the LUMO and LUMO+1 (px and py) are degenerate, while the HOMO and HOMO-1 (px+y and px-y) are nondegenerate and delocalize via through-space overlap, as shown in Figure S8. Upon disubstitution the HOMO-2 and HOMO-3 (pP and pM) mix and show clear helicity. The LUMO and LUMO+1 (pP* and pM*) also become helical, which can be seen by close inspection of the nodal planes (there are none between the two p-systems). However, their much sharper twist means the helicity is less obvious. The HOMO-2 and HOMO-3 (pP and pM) are clearly helical, but will be less relevant for observable properties. An overview of the electronic transitions are provided in Figure S9. Most of the helicity is lost in the transitions due to the weak helicity in (pP* and pM*).