NAC-TDDFT: Time-Dependent Density Functional Theory for Nonadiabatic Couplings.

Area of Science:

• Computational Chemistry
• Theoretical Molecular Physics

Background:

• First-order nonadiabatic coupling matrix elements (fo-NACMEs) are crucial for understanding electronically nonadiabatic processes in molecules.
• Time-dependent density functional theory (TDDFT) is a common method for large systems, but formulating accurate fo-NACMEs within TDDFT is challenging due to the absence of many-electron wave functions.

Purpose of the Study:

• To critically analyze various existing time-dependent density functional theory (TDDFT) formulations for calculating first-order nonadiabatic coupling matrix elements (fo-NACMEs).
• To identify the most theoretically sound and practically viable method for computing fo-NACMEs in molecular systems.

Main Methods:

• Comparative analysis of different NAC-TDDFT variants: Hellmann-Feynman-like expression, auxiliary/pseudo-wave function (AWF)-based, equation-of-motion (EOM)-based, and time-dependent perturbation theory (TDPT)-based formulations.
• Numerical validation using azulene as a test case.
• Assessment of computational efficiency compared to standard density functional theory (DFT) and TDDFT energy gradient calculations.

Main Results:

• The Hellmann-Feynman-like expression is computationally demanding and impractical for large basis sets.
• AWF variants are theoretically questionable, especially for excited-state couplings, despite agreement with EOM and TDPT under the Tamm-Dancoff approximation.
• The TDPT variant is rigorous but suffers from numerical instabilities; the EOM variant is found to be the most reliable and practical choice for calculating fo-NACMEs between ground and excited states, and between two excited states.

Conclusions:

• The equation-of-motion (EOM) variant of time-dependent density functional theory (TDDFT) is the recommended approach for calculating first-order nonadiabatic coupling matrix elements (fo-NACMEs) in both theoretical and practical applications.
• The EOM-NAC-TDDFT implementation is computationally comparable to analytic energy gradients in DFT and TDDFT.
• Future work should focus on extending EOM-NAC-TDDFT to spin-adapted open-shell systems and incorporating spin-orbit couplings.