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

Loïc Joubert-Doriol1

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This study introduces a variational method to handle linear dependence in quantum dynamics using non-orthogonal basis sets. The approach ensures accurate and unitary time evolution for solving the Schrödinger equation.

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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.