Propensity rules in photoelectron circular dichroism in chiral molecules. I. Chiral hydrogen

Photoelectron circular dichroism results from one-photon ionization of chiral molecules by circularly polarized light and manifests itself in forward-backward asymmetry of electron emission in the direction orthogonal to the light polarization plane. What is the physical mechanism underlying asymmetric electron ejection? How “which way” information builds up in a chiral molecule and maps into forward-backward asymmetry? We introduce instances of bound chiral wave functions resulting from stationary superpositions of states in a hydrogen atom and use them to show that the chiral response in one-photon ionization of aligned molecular ensembles originates from two propensity rules: (i) sensitivity of ionization to the sense of electron rotation in the polarization plane and (ii) sensitivity of ionization to the direction of charge displacement or stationary current orthogonal to the polarization plane. In the companion paper [Phys. Rev. A 99, 043417 (2019)], we show how the ideas presented here are part of a broader picture valid for all chiral molecules and arbitrary degrees of molecular alignment.

Figure 1

Top: contour map of ${\chi }_{\mathrm{p}}^{+}\left(\vec{r}\right)$ [Eq. (1)] on the $y=0$ plane where it only takes real values. Dashed (solid) lines indicate negative (zero or positive) contours. Bottom: isosurface $|{\chi }_{\mathrm{p}}^{+}\left(\vec{r}\right)|=0.01\mathrm{a}.\mathrm{u}.$ colored according to the phase. The chiral symbol in the upper left corner indicates the polarization of the density (vertical arrow) and the probability current in the azimuthal direction (curved arrow).

Figure 2

(a) Cut of ${\chi }_{\mathrm{c}}^{+}\left(\vec{r}\right)$ [Eq. (2)] on the $y=0$ plane. The black lines indicate the contours of $|{\chi }_{\mathrm{c}}^{+}\left(\vec{r}\right)|$, whereas the colors indicate its phase. The white arrows indicate the direction of the component of the probability current on the $y=0$ plane. (b) Isosurfaces $|{\chi }_{\mathrm{c}}^{+}\left(\vec{r}\right)|=0.011\mathrm{a}.\mathrm{u}.$ and (c) $|{\chi }_{\mathrm{c}}^{+}\left(\vec{r}\right)|=0.005\mathrm{a}.\mathrm{u}.$ colored according to the phase. (d) Trajectory followed by an element of the probability fluid $|{\chi }_{\mathrm{c}}^{+}\left(\vec{r}\right){|}^2$. The rotation around the $z$ axis is counterclockwise. The radial distance in this specific trajectory varies between 1 and 18.5 a.u.

Figure 3

Isosurfaces ${\chi }_{\rho \left(421\right)}^{+}\left(\vec{r}\right)=\pm 0.001\mathrm{a}.\mathrm{u}.$ (left) and ${\chi }_{\rho \left(421\right)}^{+}\left(\vec{r}\right)=\pm 0.008\mathrm{a}.\mathrm{u}.$ (right) [Eq. (4)] viewed along the $x$ (top) and $z$ (bottom) axes.

Figure 4

Top: cut of ${\chi }_{\mathrm{c}\left(421\right)}^{+}\left(\vec{r}\right)$ [Eq. (6)] on the $y=0$ plane. The black lines indicate the contours of $|{\chi }_{\mathrm{c}\left(421\right)}^{+}\left(\vec{r}\right)|$, whereas the colors indicate its phase. The white arrows indicate the direction of the component of the probability current on the $y=0$ plane. Bottom: isosurfaces $|{\chi }_{\mathrm{c}\left(421\right)}^{+}\left(\vec{r}\right)|=0.001\mathrm{a}.\mathrm{u}.$ (left) and $|{\chi }_{\mathrm{c}\left(421\right)}^{+}\left(\vec{r}\right)|=0.004\mathrm{a}.\mathrm{u}.$ (right) colored according to the phase.

Figure 5

Top: cut of ${\chi }_{\mathrm{c}\left(422\right)}^{+}\left(\vec{r}\right)$ [Eq. (6)] on the $y=0$ plane. The white arrows indicate the direction of the component of the probability current on the $y=0$ plane. Bottom: isosurfaces $|{\chi }_{\mathrm{c}\left(422\right)}^{+}\left(\vec{r}\right)|=0.003\mathrm{a}.\mathrm{u}.$ (left) and $|{\chi }_{\mathrm{c}\left(422\right)}^{+}\left(\vec{r}\right)|=0.005\mathrm{a}.\mathrm{u}.$ (right) colored according to the phase.

Figure 6

Isosurfaces ${\chi }_{\rho \left(422\right)}^{+}\left(\vec{r}\right)=\pm 0.001\mathrm{a}.\mathrm{u}.$ (left) and ${\chi }_{\rho \left(422\right)}^{+}\left(\vec{r}\right)=\pm 0.006\mathrm{a}.\mathrm{u}.$ (right) [Eq. (7)] viewed along the $x$ (top) and $z$ (bottom) axes.

Figure 7

Absolute value (top) and phase (bottom) of the scattering wave-function ${\psi }_{\vec{k}}^{(-)}\left(\vec{r}\right)$ evaluated in a plane containing $\vec{k}$ for $k=0.3\mathrm{a}.\mathrm{u}.r_{\parallel }$ and $r_{\perp }$ are the coordinates parallel and perpendicular to $\vec{k}$, respectively.

Figure 8

Photoelectron current along $z$ as a function of photoelectron momentum resulting from photoionization of a $p$-type chiral state [see Eq. (5)] via light circularly polarized on the $xy$ plane. (a) Diagrams indicating the electronic polarization (vertical arrow) and the electronic current (circular arrow) in the $p$-type chiral states for two opposite enantiomers and two opposite orientations. ${\widehat{R}}_x^{\pi }$ is the operator that rotates the wave function by $\pi \mathrm{rad}$ around the $x$ axis. (b) Photoelectron current [Eq. (9)] for different enantiomer, orientation, and light polarization combinations. The enantiomer and its orientation are indicated by the diagrams explained in (a), and the light polarization is indicated by the circular arrows after the plus signs. Note that the sign of $j_z$ is determined by the direction of the electronic polarization and that the magnitude of $j_z$ is determined by the relative direction between the electronic current and the light polarization. (c) Photoelectron current averaged over two opposite orientations (equivalent to the aligned case) for different combinations of enantiomer and light polarization. There is no cancellation of the asymmetry because for one orientation the bound electron corotates with the field, whereas for the opposite orientation it counter-rotates. The calculations shown are for states ${\chi }_{\mathrm{p}\left(311\right)}^{\pm }$, but the conclusions are valid for any ${\chi }_{\mathrm{p}\left(nlm\right)}^{\pm }$ state.

Figure 9

Total photoelectron current resulting from photoionization of a $p$-type chiral state [see Eq. (5)] via light circularly polarized on the $xy$ plane for different enantiomer, orientation, and light polarization combinations. Diagrams are explained in Fig. 8. Only the relative direction between the bound electronic current and the rotating electric field determines $j_r$. The calculations shown are for the states ${\chi }_{\mathrm{p}\left(311\right)}^{\pm }$, but the conclusions are valid for any ${\chi }_{\mathrm{p}\left(nlm\right)}^{\pm }$ state.

Figure 10

The same as Fig. 8 but for the $c$-type chiral states [see Eq. (6)]. The role played by the electronic polarization in the $p$-type states is replaced by the out-of-plane component of the probability current in the inner region in the $c$-type states. The results in (c) are also valid for photoionization from $\rho$-type chiral states (see the text). The calculations shown are for states ${\chi }_{\mathrm{c}\left(422\right)}^{\pm }$, but the conclusions are valid for any ${\chi }_{\mathrm{c}\left(nlm\right)}^{\pm }$ state.

Figure 11

Cut of ${\chi }_{\mathrm{c}}^{+-}\left(\vec{r}\right)$ [Eq. (11)] on the $y=0$ plane. The white arrows indicate the direction of the component of the probability current on the $y=0$ plane. The extra loops in the innermost region (cf. Fig. 4) allow for an extra degree of handedness. One handedness is associated with the big loops $(+)$ and the other with the small loops $(-)$.

Figure 12

Normalized photoelectron current [Eqs. (9) and (10)] as a function of photoelectron momentum resulting from photoionization of initial-state $|{\chi }_{\mathrm{c}}^{+-}\rangle$ in Eq. (11) (solid red line) via light left circularly polarized (rotating counterclockwise as viewed from the $+z$ direction) and alignment perpendicular to the light polarization plane. The FBA changes sign as a function of photoelectron energy $k^2/2$ reflecting the ambivalent handedness of $|{\chi }_{\mathrm{c}}^{+-}\rangle$ (see Fig. 11). The dashed lines show the corresponding currents for the “single-handed” states that make up $|{\chi }_{\mathrm{c}}^{+-}\rangle$.

Figure 13

Normalized photoelectron current [Eqs. (9) and (10)] as a function of photoelectron momentum resulting from photoionization of initial-state $|{\chi }_{\mathrm{p}\left(311\right)}^{+}\rangle +|{\chi }_{\mathrm{c}\left(311\right)}^{+}\rangle$ (solid red line) via light left circularly polarized (rotating counterclockwise as viewed from the $+z$ direction) and alignment perpendicular to the light polarization plane. The FBA changes sign as a function of photoelectron energy $k^2/2$ reflecting the $p$-like (lower dashed line) and $c$-like characters (upper dashed line) at lower and higher photoelectron energies, respectively.