First-principles methodology for determining the angular momentum of excitons

We develop a methodology for extracting the Kohn-Sham angular momentum of excitons in realistic systems from time-dependent density functional theory. For small systems the exciton populations can be calculated analytically, which allows us to test the methodology for a three-arm ${\mathrm{H}}_2$ molecular ring and a pair of such rings. For larger systems the developed methodology opens a venue to determine the angular momentum of excitons by first principles calculations. A chain of twenty three-arm ${\mathrm{H}}_2$ molecular rings and a triphenylphosphine molecule are investigated as illustrative examples. It is demonstrated that the angular momentum is conserved during the absorption of twisted light.

Figure 1

Arms of (a) a three-arm ${\mathrm{H}}_2$ molecular ring and (d) a triphenylphosphine molecule. Corresponding wave functions are shown in (b),(c) and (e),(f), respectively.

Figure 2

RT-TDDFT applied to a three-arm ${\mathrm{H}}_2$ molecular ring illuminated by circularly polarized light with spin 1 (purple) and spin $-1$ (red): (a) $\overline{\mathrm{XAM}}$ as obtained from Eq. (9). (b) Population of the excited state with $\mathrm{AM}=1$ (purple) and $\mathrm{AM}=-1$ (red). The structure of the three-arm ${\mathrm{H}}_2$ molecular ring (radius 2 Å; H-H bond length 0.74 Å) is shown in the inset.

Figure 3

RT-TDDFT applied to a pair of three-arm ${\mathrm{H}}_2$ molecular rings with 20 Å separation (see inset) when the first ring is illuminated by circularly polarized light with spin 1 (purple) and spin $-1$ (red): (a) $\overline{\mathrm{XAM}}$ as obtained from Eq. (9). (b) Populations of the four excited states with $\mathrm{AM}=1$ (purple) and $\mathrm{AM}=-1$ (red). The total population is shown in black.

Figure 4

RT-TDDFT applied to a chain of twenty three-arm ${\mathrm{H}}_2$ molecular rings with 20 Å separation: $\overline{\mathrm{XAM}}$ as obtained from Eq. (9) when the first ring is illuminated by circularly polarized light with spin 1 (purple) and $-1$ (red).

Figure 5

RT-TDDFT applied to a triphenylphosphine molecule: (a) $\overline{\mathrm{XAM}}$ as obtained from Eq. (9) for illumination by circularly polarized light with spin 1 (purple) and $-1$ (red) as well as for light polarized perpendicular to the rotational axis of the molecule (black). (b) Fourier transform of the dipole moment.

Figure 6

RT-TDDFT applied to a three-arm ${\mathrm{H}}_2$ molecular ring illuminated by circularly polarized light with spin $-1$: (a) $\overline{\mathrm{XAM}}$ calculated by Eq. (9) (purple) compared to analytical results (green). (b) Energy alignment and involved excitations.