Theory of time-resolved x-ray photoelectron diffraction from transient conformational molecules

We formulate x-ray photoelectron diffraction (XPD) from molecules undergoing photochemical reactions induced by optical laser pulses, and then apply the formula to the simulation of time-dependent XPD profiles from both dissociating ${\mathrm{I}}_2$ molecules and bending $\mathrm{C}{\mathrm{S}}_2$ molecules. The dependence of nuclear wave-packet motions on the intensity and shape of the optical laser pulses is examined. As a result, the XPD simulations based on such nuclear wave-packet calculations are observed to exhibit characteristic features, which are compared with the XPD profiles due to classical trajectories of nuclear motions. The present study provides a methodology toward creating “molecular movies” of ultrafast photochemical reactions by means of femtosecond XPD with x-ray free-electron lasers.

Figure 1

Sketch of an optical laser pump–XFEL probe experimental scheme. An optical laser prepares vibrational wave packets on an excited state. They are then probed by means of XPD images with XFEL pulses.

Figure 2

Pictorial representations of the multiple-scattering series of $Z_0$, $Z_1$, and $Z_2$. See text for abbreviations.

Figure 3

Muffin-tin potential of a triatomic molecule. ${\varepsilon }_p$: photoelectron kinetic energy from the vacuum level, $V_0$: energy between the vacuum level and the muffin-tin constant, and $E_p$: photoelectron energy in the molecular region.

Figure 4

Potential energy curves of an ${\mathrm{I}}_2$ molecule. A vibrational wave packet is induced on the $B{}^3{{\mathrm{\Pi }}_u}^{+}$ state.

Figure 5

Nuclear wave-packet evolutions on the state $B{}^3{{\mathrm{\Pi }}_u}^{+}$ state of the ${\mathrm{I}}_2$ molecule. Left column: laser intensity of $I_0=1.0\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ with pulse width of $\mathrm{\Gamma }=50\mathrm{fs}$. Right column: laser intensity of $I_0=1.0\times {10}^{13}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ with pulse width of $\mathrm{\Gamma }=50\mathrm{fs}$. Upper panels: the wave-packet evolutions expressed as functions of the delay time τ and internuclear distance $R$. Lower panels: cross sections at $\tau =100,1000$, and 3000 fs of the upper panels. Vertical scales of $|{\chi }_e(\mathit{R},t){|}^2$ are normalized intensities; the norm of ${\chi }_g\left(\mathit{R}\right)$ is normalized to unity, so that the integral of $|{\chi }_e(\mathit{R},t){|}^2$ over the stretching nuclear coordinate $R$ yields the population of the electronic excited state; 26.5% for $I_0=1.0\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and $\mathrm{\Gamma }=50\mathrm{fs}$, 31.7% for $I_0=1.0\times {10}^{13}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and $\mathrm{\Gamma }=50\mathrm{fs}$.

Figure 6

Time-resolved I $3s$ XPD profiles of ${\mathrm{I}}_2$ molecules. The delay times are written on the left of the panel. Left: laser intensity of $I_0=1.0\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ with pulse width of $\mathrm{\Gamma }=50\mathrm{fs}$, middle: laser intensity of $I_0=1.0\times {10}^{13}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ with pulse width of $\mathrm{\Gamma }=50\mathrm{fs}$, and right: a classical trajectory.

Figure 7

Potential energy curves of a $\mathrm{C}{\mathrm{S}}_2$ molecule. A vibrational wave packet is induced on the ${}^1{B}_2$ state.

Figure 8

Nuclear wave-packet evolutions on the state ${}^1{B}_2$ state of the $\mathrm{C}{\mathrm{S}}_2$ molecule. Left column: laser intensity of $I_0=3.0\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ with pulse width of $\mathrm{\Gamma }=7\mathrm{fs}$. Right column: laser intensity of $I_0=3.0\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ with pulse width of $\mathrm{\Gamma }=20\mathrm{fs}$. Upper panels: the wave-packet evolutions expressed as functions of the delay time τ and S–C–S bond angle. Lower panels: cross sections at $\tau =30$, 50, 80, 130, and 160 fs of the upper panels. Vertical scales of $|{\chi }_e(\mathit{R},t){|}^2$ are normalized intensities; the norm of ${\chi }_g\left(\mathit{R}\right)$ is normalized to unity, so that the integral of $|{\chi }_e(\mathit{R},t){|}^2$ over the bending nuclear coordinate $R$ yields the population of the electronic excited state; 24.4% for $I_0=3.0\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and $\mathrm{\Gamma }=7\mathrm{fs}$, and 12.7% for $I_0=3.0\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and $\mathrm{\Gamma }=20\mathrm{fs}$.

Figure 9

Time-resolved C $1s$ XPD profiles of $\mathrm{C}{\mathrm{S}}_2$ molecules. The delay times are written on the left of the panel. Left: laser intensity of $I_0=3.0\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ with pulse width of $\mathrm{\Gamma }=7\mathrm{fs}$, middle: laser intensity of $I_0=3.0\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ with pulse width of $\mathrm{\Gamma }=20\mathrm{fs}$, and right: a classical trajectory.