Extended Born-Oppenheimer equation for a three-state system.

Area of Science:

• Quantum Chemistry
• Molecular Dynamics
• Theoretical Spectroscopy

Background:

• Nonadiabatic coupling (NAC) elements are crucial for describing transitions between electronic states in molecules.
• The nuclear Schrödinger equation (SE) governs molecular motion, but its complexity increases with coupled electronic states.
• The Born-Oppenheimer (BO) approximation is fundamental but breaks down near degeneracies like conical intersections.

Purpose of the Study:

• To derive explicit forms of NAC elements for a three-state electronic system.
• To investigate the conditions for formulating the extended Born-Oppenheimer (EBO) equation.
• To analyze the behavior of NAC and adiabatic-diabatic transformation (ADT) matrices, particularly near conical intersections.

Main Methods:

• Explicitly formulating NAC elements using mixing angles of real electronic basis functions.
• Utilizing the adiabatic-diabatic transformation (ADT) to simplify the nuclear SE.
• Analyzing the curl and divergence properties of ADT and NAC matrices.
• Deriving and applying the extended Born-Oppenheimer (EBO) equation.
• Performing numerical calculations on two nuclear-coordinate-dependent three-surface BO models.

Main Results:

• Explicit forms of NAC elements were derived for three-state systems.
• Conditions for the validity of the EBO equation were established, relating to coordinate-independent ratios of mixing angle gradients.
• A curl condition with nonzero divergence was shown for ADT and NAC matrices.
• Numerical validation of the derived diabatic and EBO equations was performed, yielding transition probabilities.

Conclusions:

• The study provides a theoretical framework for handling nonadiabatic effects in three-state systems.
• The derived conditions for the EBO equation offer insights into molecular dynamics near conical intersections.
• The findings facilitate more accurate simulations of chemical reactions and spectroscopic phenomena involving multiple electronic states.