Semiclassical Scattering on Conical Intersections

We apply a semiclassical approach to the scattering problem of a vibrational wave packet in the vicinity of a conical intersection of electronic energy surfaces and derive analytical expressions for the scattering matrix. The latter are valid when the scattering length that scales as $\sqrt{\hslash }$ is small and a wave packet passes through the scattering region with a constant velocity. The analytical results are in excellent agreement with numerical simulations for a realistic set of parameters.

Figure 1

(a),(b) Potential energy surfaces as a function of transverse coordinates in the vicinity of the CI. A slice of the initial WP at $x=a$ with velocity $v$ directed along the $z$ axis is shown in (a) the adiabatic basis set and (b) the diabatic basis set (the diagonal components only), where red and green denote the $({\psi }_1,0{)}^T$ and $(0,{\psi }_2{)}^T$ surfaces, respectively [see Eq. (7)]. (c)–(f) Numerical simulations of the WP scattering on the CI for $g_s={10}^{-2}$ represented in the diabatic basis set. The WP is propagating in the $z$ direction (identical in all panels). The origin of the coordinate system indicates the position of the CI. The WP component colors refer to the respective surfaces in (b), and the color brightness increases with the amplitude. The distance between the WP center and the CI center is (c) $z=-215$, (d) $z=0$, (e) $z=40$, and (f) $z=300$.

Figure 2

(a) Asymptotic amplitude of the WPs transmitted through the CI. Points denote numerical results; solid lines represent analytical results [Eq. (11)] adjusted to the WP spreading while propagating away from the scattering region. The inset shows associated quantum yield. (b) Proposed scheme for classical large scale MD incorporating analytic treatment of nonadiabatic unavoided surface crossings.