Numerical integration of quantum time evolution in a curved manifold

The numerical integration of the Schrödinger equation by discretization of time is explored for the curved manifolds arising from finite representations based on evolving basis states. In particular, the unitarity of the evolution is assessed, in the sense of the conservation of mutual scalar products in a set of evolving states, and with them the conservation of orthonormality and particle number. Although the adequately represented equation is known to give rise to unitary evolution in spite of curvature, discretized integrators easily break that conservation, thereby deteriorating their stability. The Crank-Nicolson algorithm, which offers unitary evolution in Euclidian spaces independent of time-step size $dt$, can be generalized to curved manifolds in different ways. Here we compare a previously proposed algorithm that is unitary by construction, albeit integrating the wrong equation, with a faithful generalization of the algorithm, which is, however, not strictly unitary for finite $dt$.

Figure 1

Evolution of the deviation from orthonormality for the two occupied states for a collision between two He atoms using the gauge-potential (GP) integrator, comparing a time step of (a) 0.01 as and (b) 1 as. (c) Comparison of the evolution of $\langle {\psi }_2|{\psi }_2\rangle -1$ for the two integrators, GP and Tomfohr-Sankey (TS), with $dt$ = 1 as.

Figure 2

Deviation in electronic energy uptake vs time step $dt$ for the collision between two He atoms, relative to $dt$ = 0.01 as, for both the TS and GP integrators.

Figure 3

Deviation in electronic stopping power $S_e$ vs time step $dt$ for a proton traveling through graphite along the (0001) direction, for both the TS and GP integrators.

Figure 4

Comparison of converged results for both the GP and TS integrators, for the electronic stopping power $S_e$ vs velocity for a proton traveling through graphite along the (0001) direction. PSTAR data are shown for comparison [32].

Figure 5

Electronic energy as a function of position along the projectile trajectory for a collision between two He atoms, one traveling at 1 a.u. The left (right) panel shows the results for the TS (GP) integrator.