Fano interference and cross-section fluctuations in molecular photodissociation

We derive an expression for the total photodissociation cross section of a molecule incorporating both direct and indirect processes that proceed through excited resonances, and show that it exhibits generalized Beutler-Fano line shapes. Assuming that the closed system can be modeled by random-matrix theory, we derive the statistical properties of the photodissociation cross section and find that they are significantly affected by the direct processes. In the limit of isolated resonances, we find that direct processes suppress the correlation hole of the cross-section autocorrelation function and lead to a maximum in the cross-section distribution.

Figure 1

(Color online) Indirect molecular photodissociation. Shown are three electronic surfaces: the ground-state surface $e_0$ (in which the ground state $\mid g⟩$ resides), a binding surface $e_1$, and a repulsive surface $e_2$. The surfaces $e_1$ and $e_2$ give rise to an effective electronic surface with a barrier near their crossing (not shown), and the bound vibrational states $\mid n⟩$ in $e_1$ become resonances. Indirect photodissociation proceeds through these resonances with a total cross section $\sigma \left(E\right)$ (shown on the left) that is a sum of Lorentzian line shapes.

Figure 2

(Color online) Top: states and transition matrix elements in the random-matrix model discussed in the text. Bottom: the photodissociation cross section $\sigma \left(E\right)$ (arb. units) for one time-reversal invariant realization of $H_0$ (circles), with $N=128$ and $\Lambda =1$. The solid lines describe fitted Beutler-Fano line shapes.

Figure 3

(Color online) Cross-section autocorrelation functions for GOE (left panel) and GUE (right panel). The symbols are from simulations of random matrices $(N=128)$ at the center of the band $(E=0)$ and for $\Lambda =10$ and ${\lambda }_c={10}^{-2}$ for all channels $c$. The autocorrelation functions without direct coupling ($\theta =0$, open squares) and with direct coupling ($\theta =1.26$, open circles) are compared to Eq. (12) (solid lines).

Figure 4

(Color online) Cross-section distributions. (a) Cross section vs energy $E$ in the absence of direct coupling $(\theta =0)$ and for one random-matrix realization with $H_0$ from the GOE. Here $N=64$, $\Lambda =10$, and ${\lambda }_c=5\times {10}^{-2}$ for all $c$. (b) Same as in (a) but in the presence of direct processes $(\theta =0.125)$. (c) Cross-section distributions at $E=0$ for $\theta =0$ (squares) and $\theta =0.125$ (circles). (d) as in (c) but for GUE. The solid lines in (c) and (d) are inverse Fourier transforms of $F_{\beta }\left(s\right)$, obtained by computing the average in Eq. (13) numerically.