Variational Approach for Linearly Dependent Moving Bases in Quantum Dynamics: Application to Gaussian Functions

Area of Science:

• Quantum mechanics
• Computational chemistry
• Theoretical physics

Background:

• Solving the Schrödinger equation for complex systems requires efficient numerical methods.
• Non-orthogonal time-dependent basis sets introduce challenges due to linear dependence.
• Maintaining unitarity in quantum dynamics simulations is crucial for physical accuracy.

Purpose of the Study:

• To develop a variational method for treating linear dependence in non-orthogonal time-dependent basis sets.
• To ensure accurate and unitary time evolution in quantum dynamics simulations.
• To provide a robust framework for solving the Schrödinger equation.

Main Methods:

• Defining a linearly independent working space.
• Variational construction of the propagator over finite time steps.
• Representing time evolution via a semi-unitary transformation.

Main Results:

• The proposed method effectively handles linear dependence in non-orthogonal basis sets.
• Simulations on a quartic double-well potential show convergence to exact dynamics.
• The time evolution is demonstrated to be unitary by construction.

Conclusions:

• The variational treatment provides an accurate and unitary approach for quantum dynamics with non-orthogonal basis sets.
• This method offers a reliable solution for the challenges posed by linear dependence.
• The developed technique enhances the capabilities of solving the time-dependent Schrödinger equation.