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Propagators for the Time-Dependent Kohn-Sham Equations: Multistep, Runge-Kutta, Exponential Runge-Kutta, and

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This study evaluates integration schemes for nonlinear time-dependent Kohn-Sham equations. A simplified fourth-order commutator-free Magnus integrator offers the best balance of accuracy, simplicity, and efficiency for electronic structure calculations.

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Area of Science:

  • Computational physics
  • Quantum chemistry
  • Electronic structure theory

Background:

  • The time-dependent Kohn-Sham (TDKS) equations describe the behavior of electrons in materials over time.
  • Unlike the time-dependent Schrödinger's equation, TDKS equations are nonlinear due to the density-dependent Hamiltonian.
  • Understanding their properties, such as symplectic structure, is crucial for accurate simulations.

Purpose of the Study:

  • To investigate and compare various numerical integration schemes for solving the TDKS equations.
  • To identify robust, simple, and efficient methods for time-dependent electronic structure calculations.
  • To assess the performance of largely overlooked propagator families.

Main Methods:

  • Examined four families of propagators: linear multistep, Runge-Kutta, exponential Runge-Kutta, and commutator-free Magnus schemes.
  • Analyzed the performance of these schemes based on a cost-versus-accuracy trade-off.
  • Focused on schemes previously underutilized in time-dependent electronic structure computations.

Main Results:

  • A simplified fourth-order commutator-free Magnus integrator emerged as the most effective scheme.
  • This integrator demonstrated superior robustness, simplicity, and efficiency.
  • Implicit multistep methods showed potential utility in specific, niche applications.

Conclusions:

  • The simplified fourth-order commutator-free Magnus integrator is highly recommended for general time-dependent electronic structure calculations.
  • While other methods like implicit multistep schemes have specialized uses, the Magnus integrator offers the best overall performance.
  • This work highlights the importance of exploring diverse numerical methods for advancing computational quantum mechanics.