Time-Local Equation for the Exact Optimized Effective Potential in Time-Dependent Density Functional Theory

A long-standing challenge in the time-dependent density functional theory is to efficiently solve the exact time-dependent optimized effective potential (TDOEP) integral equation derived from orbital-dependent functionals, especially for the study of nonadiabatic dynamics in time-dependent external fields. In this Letter, we formulate a completely equivalent time-local TDOEP equation that admits a unique real-time solution in terms of time-dependent Kohn-Sham and effective memory orbitals. The time-local formulation is numerically implemented, with the incorporation of exponential memory loss to address the unaccounted for correlation component in the exact-exchange-only functional, to enable the study of the many-electron dynamics of a one-dimensional hydrogen chain. It is shown that the long time behavior of the electric dipole converges correctly and the zero-force theorem is fulfilled in the current implementation.

Figure 1

Comparison between the TDOEP, TDKLI, and LHXO approximations. In TDOEP simulations, we choose $\mathrm{\Delta }t=0.001$ for the time step and $\tau =2$ for the memory loss in the modified ETD integration scheme. The external field is a sine-square pulse with frequency $\omega =0.1$, 0.2 (upper and lower panels) and length $T=400$. The left-hand side is the resulting time-dependent dipole, and the right-hand side shows the net xc force.