Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics.

Area of Science:

• Physics, Chemistry, Mathematics

Background:

• Classical mechanics, quantum mechanics, and classical thermodynamics are cornerstones of chemical dynamics and kinetics.
• These theories are primarily studied at the non-relativistic approximation.
• A unified geometrical framework can enhance understanding across these fields.

Purpose of the Study:

• To unveil the common geometrical (symplectic) structures in classical mechanics, quantum mechanics, and classical thermodynamics.
• To demonstrate how physical states and observable quantities are represented within extended phase spaces.
• To explore the application of these geometrical principles in numerical methods and deep learning.

Main Methods:

• Utilizing extended phase spaces and Lagrangian submanifolds to depict physical states of integrable dynamical systems.
• Solving Hamilton's equations of motion and variational equations for trajectory integration and analysis of constants of motion.
• Applying high-order finite difference methods and exploring Hamiltonian neural networks for solving complex equations.

Main Results:

• Physical states are represented by Lagrangian submanifolds in extended phase spaces.
• Eigenvalues equal to one in the symplectic fundamental matrix indicate constants of motion.
• Geometrical quantum mechanics shows equivalence between Schrödinger and Hamilton's equations in specific spaces.
• Hamiltonian geometry provides conditions for human-machine collaboration in deep learning.

Conclusions:

• A common Hamiltonian geometry underlies classical mechanics, quantum mechanics, and thermodynamics.
• This unified geometric approach facilitates the study of chemical dynamics and kinetics.
• Hamiltonian geometry is crucial for developing advanced computational methods and AI in physics and chemistry.