Time-dependent Bivariational Principle: Theoretical Foundation for Real-Time Propagation Methods of Coupled-Cluster...

Area of Science:

• Quantum Chemistry
• Theoretical Physics
• Computational Science

Background:

• Real-time propagation methods are crucial for simulating quantum dynamics.
• Variational techniques form the basis for many such methods.
• The time-dependent bivariational principle (TD-BIVP) is a key framework, particularly for coupled-cluster methods.

Purpose of the Study:

• To investigate the TD-BIVP from a differential geometric perspective.
• To establish a connection between TD-BIVP and classical mechanics.
• To reconstruct various time-dependent variational principles from the TD-BIVP framework.

Main Methods:

• Differential geometry applied to the TD-BIVP.
• Analysis of the real and imaginary parts of the action integral.
• Formulation of bivariational principles for real submanifolds.
• Reconstruction of time-dependent univariational principles.

Main Results:

• Two distinct classical Hamilton's equations of motion emerge from the TD-BIVP.
• Two distinct bivariational principles are derived for real approximation submanifolds.
• Conservation laws and Poisson brackets are introduced, strengthening the classical mechanics analogy.
• Established real-time propagation methods are contextualized within the TD-BIVP framework.

Conclusions:

• The differential geometric approach provides a unified view of real-time propagation methods.
• The TD-BIVP offers a powerful and generalizable framework for quantum dynamics.
• This work deepens the understanding of the relationship between quantum dynamics and classical mechanics.