Notes on integrators

R-K 4th order: about 4.5 times of computation time of the same step V-V.

However, it has about the same conservation of energy (a gauge of accuracy) as V-V with 1/50 time step. Overall speaking, it's about 10 times more efficient. RK4 for E = 1000 cm^-1, dt = 0.005 t₀ for t=100 ps integration gives ~ 10^-11 eV of total energy shift, for E = 500 cm^-1, it is even better (~3×10^-12 eV). Doubling dt (to 0.01 t₀) leads to ~35 times higher E_shift.

We still ought to check the symplecticity of the integrator, i.e. whether it preserves the two-form:
ω² = ∑ dp_i ^ dq_j. PS: as it turns out, the SIA4 integrator proposed by J. Candy in J. Comput. Phys, 92, 230 (1991) is a better (and symplectic) one than the plain RK4 method.