A Unified Formulation for ⟨Ŝ²⟩ in Two-Component TDDFT.

Area of Science:

• Computational Chemistry
• Quantum Chemistry
• Theoretical Chemistry

Background:

• Two-component linear-response time-dependent density functional theory (TDDFT) offers a unified framework for various excitation types.
• Accurate calculation of spin properties in excited states is crucial for understanding molecular behavior and spectroscopy.
• Existing methods may not fully capture the nuances of spin dynamics in complex electronic excitations.

Purpose of the Study:

• To develop a general formalism for evaluating the expectation value of spin squared (⟨Ŝ²⟩) for electronically excited states within two-component TDDFT.
• To analyze the contributions to ⟨Ŝ²⟩ in excited states, distinguishing between the reference state and the excitation process.
• To provide a theoretical framework that can be applied to different reference states, including collinear and noncollinear ones.

Main Methods:

• Derivation of a general formalism for ⟨Ŝ²⟩ within two-component TDDFT.
• Specialization of the formalism for collinear reference determinants, connecting to conventional spin-conserving and spin-flip TDDFT.
• Systematic comparison of derived working equations with existing theoretical approaches.
• Implementation of two-component TDDFT calculations using two-component DFT, unrestricted Kohn-Sham (UKS), and restricted open-shell Kohn-Sham (ROKS) reference states.

Main Results:

• The formalism successfully evaluates ⟨Ŝ²⟩ for excited states within the two-component TDDFT framework.
• ⟨Ŝ²⟩ in excited states is shown to originate from the initial reference state (⟨Ŝ²⟩₀) and additional changes (Δ⟨Ŝ²⟩) due to the excitation.
• The two-component formalism naturally decomposes into spin-conserving and spin-flip components for collinear reference states.
• Calculations using UKS and ROKS reference states demonstrate the practical applicability of the developed method.

Conclusions:

• The presented formalism provides a robust theoretical foundation for calculating spin properties in electronically excited states using two-component TDDFT.
• Understanding the distinct contributions to ⟨Ŝ²⟩ enhances the interpretation of excitation processes and spin dynamics.
• The method offers improved accuracy and a unified treatment for various excitation scenarios in computational chemistry.