Helical orbitals in electrical uni-directional molecular motors

Abstract

The generation of unidirectional motion has been a long-standing challenge in engineering of molecular motors. Here, a mechanism driving the rotation is presented based on electron current through helical orbitals on a $π$-bonded carbon chain. Such electron current through helical orbitals has been shown to be circulating around the carbon chain. It is natural to expect that the associated electronic angular momentum drives a rotation when the current is turned on. As intuitive as this relation might seem, it is also incomplete because a formal definition of helicality in terms of a physical observable has not yet been given. Such a definition is proposed here. Based on this definition, we show how helicality determines the motor's sense of rotation. We exemplify the relation between helicality and angular momentum in Hückel models of linear carbon chains (cumulenes and oligoynes). We attribute the previously reported opposite helicality sense of frontier orbitals (HOMO and LUMO) to the approximate sub-lattice symmetry. For oligoynes, this symmetry is hidden in the sense that it does not reduce to a mere labeling of atoms. Sub-lattice symmetry, combined with time-reversal invariance, allows us to derive Onsager-type reciprocal relations of various linear response coefficients, dictating e.g. an odd energy dependence of angular momentum response to voltage bias. We propose an observable consequence of the approximate sub-lattice symmetry: If the carbon chain is employed as an axle of a molecular rotor, the sense of rotation is independent on the direction of the current.

Figures (13)

• Figure 1: Top: Atomic structure of a molecular motor, containing a central rotating moiety and left and right stators (C=gray, H=white, O=red). The pair of connecting --C$\equiv$C-- chains are molecular axles and bearings. The rotor's hexagonal rings are tilted by $\approx 29^\circ$, because of the oxygen bridge. Each axle is attached to two hexagons that can be tilted with respect to each other, giving rise to helical orbitals. Bottom: isosurface of the highest occupied molecular orbital (left) and the lowest occupied molecular orbital (right) calculated in DFT, showing helical orbitals along the axles with opposite winding senses. The bottom row portrays these orbitals from a side view.
• Figure 2: Top: Diacetylene (butadiyne), a member of the oligoyne family with four carbons. Bottom: The Hückel wave-function of the $\pi$-orbitals is represented in the basis of p$_{x,y}$ orbitals on each carbon.
• Figure 3: Examples of carbon chains with different end-groups. (a) Propadiene with sp$^2$ hybridized end carbons, belongs to allenes. Longer variants, with additional double-bonded carbons inserted in the chain, are called cumulenes. (b) Propyne can be seen as an oligoyne with a substituted methyl group. (c) 2,4-hexadiyne with two methyl end-groups, belongs to oligoynes. We treat the relative angle $\theta_{\mathrm{T}}$ of the two groups as a parameter. In cumulenes and allenes, carbons are double-bonded. Single and triple bonds alternate in oligoynes.
• Figure 4: Illustration of the helicality operator. A wavefunction on a given site $n$ is a linear combination of p$_x$ and p$_y$ orbitals. Therefore, it is a p-orbital oriented along a unit vector $\bm P_n$. On the consecutive site, the unit vector $\bm P_{n+1}$ is tilted by an angle $\phi$ with respect to $\bm P_n$. The sign of $\phi$ determines the helicality of the given bond, i.e. the clockwise ($\phi>0$) or anti-clockwise winding of the $\pi$-orbital along the bond with increasing $n$. Assuming $|\phi| < \pi$, the bond helicality is given by the sign of $\bm e_z \cdot \qty(\bm P_n \times\bm P_{n+1}) = |\bm{P}_n||\bm{P}_{n+1}| \sin\phi$, where $\times$ is the cross-product. For a given wavefunction, we evaluate $\bm P_n \times\bm P_{n+1}$ as an expectation value of the local pitch density operator $\hat{h}_n$ introduced in the Eq. (\ref{['eq:hn']}).
• Figure 5: Scheme of the SL symmetric model of a carbon chain with 4 atoms in the representation of angular momentum eigenstates. On each atom (yellow box), there are two $\hat{L}_z$ eigenstates with $L_z = \pm\hbar$, labeled by the arrow. Blue (red) color denotes the emergent sub-lattice A (B) of the model. Solid lines represent the nearest-neighbor 'hopping' terms in the Hamiltonian, equal to $-t$. Dashed lines represent couplings from the end-group terms. Atoms of the same SL do not couple. The couplings effectively arrange into a ring.