Hamiltonian formulation and symplectic split-operator schemes for time-dependent density-functional-theory equations of electron dynamics in molecules

We revisit Kohn-Sham time-dependent density-functional theory (TDDFT) equations and show that they derive from a canonical Hamiltonian formalism. We use this geometric description of the TDDFT dynamics to define families of symplectic split-operator schemes that accurately and efficiently simulate the time propagation for certain classes of DFT functionals. We illustrate these with numerical simulations of the far-from-equilibrium electronic dynamics of a one-dimensional carbon chain. In these examples, we find that an optimized 4th order scheme provides a good compromise between the numerical complexity of each time step and the accuracy of the scheme. We also discuss how the Hamiltonian structure changes when using a basis set to discretize TDDFT and the challenges this raises for using symplectic split-operator propagation schemes.

Automated summary

This study revisits the equations of Kohn-Sham time-dependent density-functional theory (TDDFT) and demonstrates their derivation from a canonical Hamiltonian formalism. By using a geometric description, families of symplectic split-operator schemes are defined to accurately and efficiently simulate the time propagation for specific classes of DFT functionals. Numerical simulations are conducted to illustrate the approach, focusing on the far-from-equilibrium electronic dynamics of a one-dimensional carbon chain. The optimized 4th order scheme is found to provide a good compromise between numerical complexity and accuracy of the simulation.