High-order geometric integrators for the variational Gaussian approximation

Area of Science:

• Quantum mechanics
• Computational chemistry
• Theoretical physics

Background:

• The time-dependent Schrödinger equation (TDSE) governs quantum system evolution.
• Gaussian-based methods offer approximations for solving the TDSE.
• The variational Gaussian approximation (VGA) is accurate but computationally expensive.

Purpose of the Study:

• To enhance the efficiency of the variational Gaussian approximation (VGA).
• To develop high-order geometric integrators for quantum dynamics simulations.
• To assess the performance of these integrators on complex, high-dimensional systems.

Main Methods:

• Symmetric composition of a second-order symplectic integrator (Faou-Lubich).
• Development of arbitrary even-order geometric integrators.
• Application to a twenty-dimensional model of coupled Morse oscillators.

Main Results:

• High-order integrators drastically speed up convergence compared to second-order methods.
• The developed integrators are time-reversible and conserve norm and symplectic structure exactly.
• The variational method, enhanced by these integrators, captures tunneling and improves accuracy over the thawed Gaussian approximation.

Conclusions:

• The developed high-order geometric integrators significantly improve the computational efficiency of the variational Gaussian approximation.
• This approach provides an accurate and stable method for simulating quantum dynamics, even in high-dimensional systems.
• The method offers advantages over traditional numerical techniques like the Runge-Kutta method.