Second-Quantized Surface Hopping

The trajectory surface hopping method for quantum dynamics is reformulated in the space of many-particle states to include entanglement and correlation of trajectories. Used to describe many-body correlation effects in electronic structure theories, second quantization is applied to semiclassical trajectories. The new method allows coupling between individual trajectories via energy flow and common phase evolution. It captures the properties of a wave packet, such as branching, Heisenberg uncertainty, and decoherence. Applied to a superexchange process, the method shows very accurate results, comparable to exact quantum data and improving greatly on the standard approach.

Figure 1

Construction of $N$-particle states from 1-particle states for a 3-level system. (a) 1-particle state basis used in the FSSH method; (b) 6 of 9 possible 2-particle states. All particles are distinguishable. $N$-particle state energies are normalized to the number of particles. Each energy level can be occupied by any number particles from 0 to $N$, provided that the total number of particles distributed in all levels is $N$. Additional lower-energy levels are created between the original states $|1⟩$ and $|2⟩$. Utilizing only the 2-particle states denoted by black lines in the top section of part (b) is equivalent to the 1-particle FSSH scheme of panel (a).

Figure 2

Possible values of nuclear momenta in the rescaling scheme in the case of two disentangled 1-particle trajectories (as in FSSH) and one 2-particle trajectory (SQUASH).

Figure 3

Probabilities of transmission on the (a) second and (b) third states. Performance of three methods is compared: FSSH, exact quantum solution, and SQUASH with 2-particle states. SQUASH describes the superexchange transmission mechanism corresponding to the initial momenta around 5 a.u., part (a), while FSSH does not allow superexchange.