Climbing the Rotational Ladder to Chirality

Molecular chirality is conventionally understood as space-inversion-symmetry breaking in the equilibrium structure of molecules. Less well known is that achiral molecules can be made chiral through extreme rotational excitation. Here, we theoretically demonstrate a clear strategy for generating rotationally induced chirality: An optical centrifuge rotationally excites the phosphine molecule (${\mathrm{PH}}_3$) into chiral cluster states that correspond to clockwise ($R$ enantiomer) or anticlockwise ($L$ enantiomer) rotation about axes almost coinciding with single $\mathrm{P}─\mathrm{H}$ bonds. The application of a strong dc electric field during the centrifuge pulse favors the production of one rotating enantiomeric form over the other, creating dynamically chiral molecules with permanently oriented rotational angular momentum. This essential step toward characterizing rotationally induced chirality promises a fresh perspective on chirality as a fundamental aspect of nature.

Figure 1

Rotationally induced chirality in ${\mathrm{PH}}_3$ using an optical centrifuge in conjunction with a static electric field. Reversing the direction of the dc field produces the other rotating enantiomer.

Figure 2

Rotational energy level clustering in the ground vibrational state of ${\mathrm{PH}}_3$, c. f. Fig. 2 of Ref. [14]. (a) The energy difference $E_{Jk}-E_{Jk}^{\max }$ was plotted for each rotational energy level $E_{Jk}$ relative to the maximum energy $E_{Jk}^{\max }$ in its $J$ multiplet. (b) Schematic representation of the localized chiral cluster states, which are separated by high kinetic energy barriers. (c) Wave functions of the delocalized rotational cluster states in the representation of the localized chiral cluster states.

Figure 3

Wave packet population by $m=-J$ stationary states and rotational probability density function $P(\theta ,\chi )$, in terms of the Euler angles $\theta$ and $\chi$, at the end of the centrifuge pulse ($t=36.8\mathrm{ps}$), (a) without, and (b) with a dc field of $\pm 1\mathrm{MV}/\mathrm{cm}$ applied. Wave packet population of stationary states with odd $J$ quantum numbers or $m\ne -J$ are much smaller and not shown in the plot. The two-dimensional rotational density contour plot, shown in the middle, has been projected onto a sphere, shown on the right hand side, to illustrate the orientation of ${\mathrm{PH}}_3$ and the molecular frame relative to the rotation axes. Each high-density island (red) indicates an axis of rotation and $x$, $y$, $z$ refer to the molecule-fixed axis system. In the bottom left hand corner, the time evolution of the expectation value of the laboratory-fixed effective dipole moment $⟨{\mu }_Z⟩$ has been calculated for the $R$ and $L$ rotating enantiomers. The exhibited antiphase behavior between the rotating enantiomers confirms chirality.

Figure 4

Definition of true and false chirality. (a) If ${\mathrm{PH}}_3$ is translating along the axis of rotation (dashed arrow), space inversion ($\widehat{P}$) is not equivalent to time reversal ($\widehat{T}$) combined with any proper spatial rotation (${\widehat{R}}_{\pi }$) and the system is truly chiral. (b) If ${\mathrm{PH}}_3$ is not translating, the system exhibits false chirality as space inversion is equivalent to time reversal combined with spatial rotation.