Correlated Electron-Nuclear Dynamics with Conditional Wave Functions

The molecular Schrödinger equation is rewritten in terms of nonunitary equations of motion for the nuclei (or electrons) that depend parametrically on the configuration of an ensemble of generally defined electronic (or nuclear) trajectories. This scheme is exact and does not rely on the tracing out of degrees of freedom. Hence, the use of trajectory-based statistical techniques can be exploited to circumvent the calculation of the computationally demanding Born-Oppenheimer potential-energy surfaces and nonadiabatic coupling elements. The concept of the potential-energy surface is restored by establishing a formal connection with the exact factorization of the full wave function. This connection is used to gain insight from a simplified form of the exact propagation scheme.

Figure 1

(a) Exact (solid black line) and approximated (blue circles) nuclear probability densities renormalized as $|{\chi }_{\mathrm{rn}}(\mathbf{R},t){|}^2=|\chi (\mathbf{R},t){|}^2/\max |\chi (\mathbf{R},t){|}^2$ at four different times. (b) Arrows refer to the (two-dimensional) velocity field computed from the approximated conditional wave functions. The gray contour lines represent the corresponding electron-nuclear two-dimensional potential-energy surface. (c) First (red line), second (green line), and third (magenta line) BOPESs involved in the nonadiabatic process. Inset: Adiabatic populations as a function of time computed from the exact solution.