Exponential integrators in time-dependent density-functional calculations

The integrating factor and exponential time differencing methods are implemented and tested for solving the time-dependent Kohn-Sham equations. Popular time propagation methods used in physics, as well as other robust numerical approaches, are compared to these exponential integrator methods in order to judge the relative merit of the computational schemes. We determine an improvement in accuracy of multiple orders of magnitude when describing dynamics driven primarily by a nonlinear potential. For cases of dynamics driven by a time-dependent external potential, the accuracy of the exponential integrator methods are less enhanced but still match or outperform the best of the conventional methods tested.

Figure 1

The time-averaged error of various methods for integrating the TDKS equations when electrons are initialized in an excited state. The time interval considered was between $T_i=10$ a.u. and $T_f=100$ a.u. Time evolution operator approaches are shown above, while IMEX, IF, and ETD approaches are shown below. CN is shown in the latter for comparison.

Figure 2

The time-dependent energy (above) and norm (below) of the IFRK4 integration method using time step sizes of 0.5 and 0.7 a.u. as compared to a benchmark calculation.

Figure 3

The time-averaged error of various methods for integrating the TDKS equation over a long time period. The time interval considered was between $T_i=10$ a.u. and $T_f=1000$ a.u. Time evolution operator approaches are shown above while IMEX, IF, and ETD approaches are shown below. CN is shown in the latter for comparison.

Figure 4

The energy as a function of time for various methods. The time evolution operator methods (Taylor and CN) drift slowly away from the exact solution; however, they overlap one another. The divergence of such methods from the converged solution is more noticeable for longer simulations.

Figure 5

The time-averaged error of various methods for integrating the TDKS equation when electrons are driven by an external electric field of strength 0.1 a.u. The time-dependent potential from the electric field was included in the linear part. The time interval considered was between $T_i=10$ a.u. and $T_f=100$ a.u.

Figure 6

The time-averaged error of various methods for integrating the TDKS equation when electrons are driven by an external electric field of strength 0.1 a.u. The time-dependent potential from the electric field was included in the nonlinear part. The time interval considered was between $T_i=10$ a.u. and $T_i=100$ a.u. CN is shown for comparison.

Figure 7

The time-averaged error of various methods for integrating the TDKS equation when electrons are driven by an external electric field of strength 0.1 a.u. The time-dependent potential from the electric field was included in the linear part. The time interval considered was between $T_i=10$ a.u. and $T_f=100$ a.u. CN is shown for comparison.

Figure 8

The difference of the norm from unity for various methods when electrons are driven by an external electric field of strength 0.1 a.u. The time-dependent potential from the electric field was included in the linear part. The time interval considered was between $T_i=10$ a.u. and $T_f=100$ a.u.

Figure 9

The time-averaged error of various methods for integrating the TDKS equation when electrons are driven by a strong external electric field of strength 1.0 a.u. The time-dependent potential from the electric field was included in the linear part. The time interval considered was between $T_i=10$ a.u. and $T_f=100$ a.u. SPO is shown for comparison. The second-order Runge-Kutta method is excluded from the top figure due to it being unstable for each choice of time step size.

Figure 10

The time- and orbital-averaged error for model B using various methods for integrating the TDKS equation when electrons are driven by an external electric field of strength 0.1 a.u. The time interval considered was between $T_i=10$ a.u. and $T_f=100$ a.u. Time evolution operator approaches are shown above while IMEX, IF, and ETD approaches are shown below. CN is shown in the latter for comparison.

Figure 11

The time-averaged error of orbital 1 for model B, using various methods for integrating the TDKS equation when electrons are driven by an external electric field of strength 0.1 a.u. The time interval considered was between $T_i=10$ a.u. and $T_f=100$ a.u. Time evolution operator approaches are shown above while IMEX, IF, and ETD approaches are shown below. CN is shown in the latter for comparison.

Figure 12

The time-averaged error of orbital 2 for model B, using various methods for integrating the TDKS equation when electrons are driven by an external electric field of strength 0.1 a.u. The time interval considered was between $T_i=10$ a.u. and $T_f=100$ a.u. Time evolution operator approaches are shown above while IMEX, IF, and ETD approaches are shown below. CN is shown in the latter for comparison.

Figure 13

The time- and orbital-averaged error for model B using various methods for integrating the TDKS equation when electrons are driven by an external electric field of strength 1.0 a.u. The time interval considered was between $T_i=10$ a.u. and $T_f=100$ a.u. Time evolution operator approaches are shown above while IMEX, IF, and ETD approaches are shown below. SPO is shown in the latter for comparison. The second-order Runge-Kutta method is excluded from the top figure due to it being unstable for each choice of time step size.