Noncollinear Spin-Flip TDDFT for Potential Energy Surface Crossings: Conical Intersections and Spin Crossings

TL;DR

The paper advances noncollinear spin-flip TDDFT within a multicollinear functional framework to accurately describe potential energy surface crossings, including conical intersections and singlet–triplet spin crossings. It introduces a robust spin-flip kernel and a penalty-function strategy to locate MECIs and MECPs without derivative couplings, and demonstrates topologically correct intersections across multiple benchmark systems with reduced spin contamination. The approach yields consistent triplet energies and captures DES-dominated states, supporting its use in nonadiabatic dynamics, while acknowledging residual spin contamination in some states and ongoing development of derivative couplings. Overall, the method offers a computationally efficient and broadly applicable route to PES crossings and nonadiabatic processes, without relying on extensive multireference methods or strict functional choices.

Abstract

We recently proposed a scheme to generalize collinear functionals to the noncollinear regime, termed the multicollinear approach. The resulting noncollinear functionals preserve spin symmetry while providing numerically stable higher-order functional derivatives. This scheme has already been applied to noncollinear spin-flip TDDFT and its analytic gradient calculations. In the present work, with the aid of the penalty function method, we employ the noncollinear spin-flip TDDFT in multicollinear scheme to locate potential energy surface crossings. We investigate two distinct types of crossings and analyze their topographical and spin characteristics near the crossing points. The first type is conical intersections, typically involving two singlet states such as the ground and first excited states. The second type involves spin crossings that occur between electronic states with different spin multiplicities, such as between singlet and triplet. These crossing regions enable ultrafast nonadiabatic transitions through either nonadiabatic coupling or spin-orbit coupling, playing a crucial role in photochemistry. Through theoretical analysis and illustrative examples, we demonstrate the advantages of noncollinear spin-flip TDDFT over conventional collinear spin-flip TDDFT or spin-conserving TDDFT. Finally, we systematically evaluate its prospects as an electronic structure method for use in nonadiabatic molecular dynamics.

Figures (8)

• Figure 1: Schematic diagram of the spin-flip-down TDDFT starting from a high-spin triplet reference state $|{\mathrm{T}(S_z=1)}\rangle$. C, O, and V stand for doubly-occupied orbitals, singly-occupied orbitals, and virtual orbitals, respectively. For the O-O type excitations, four spin-complete configurations can be generated, including closed-shell singlet $|{\mathrm{CSS}}\rangle$, doubly-excited singlet $|{\mathrm{DES}}\rangle$, open-shell singlet $|{\mathrm{OSS}}\rangle$, and spin-flip triplet $|{\mathrm{T}(S_z=0)}\rangle$. The O-V, C-O, and C-V excitations yield states that lack compensating configurations to maintain the spin multiplicity.
• Figure 2: (a) Geometries of MECIs obtained by MRCISD (gray, from Ref. nikiforov2014assessment), collinear spin-flip TDDFT with BHHLYP (green, from Ref. nikiforov2014assessment), and noncollinear spin-flip TDDFT with BHHLYP (red, this work). Structures are visualized using PyMOLPyMOL. (b) PESs of $\mathrm{S}_0$ and $\mathrm{S}_1$ near the MECI of ethylene. The twisting angles follow the convention in Ref. minezawa2009optimizing. Structures are visualized using VMDhumphrey1996vmdstone1998tachyon.
• Figure 3: Potential energy curves for ethylene (a, b) and stilbene (c, d) as a function of torsion angle, computed using collinear (a, c) and noncollinear (b, d) spin-flip TDDFT with BLYP functional. SC stands for strongly spin-contaminated states with $\langle{\hat{S}^2}\rangle\approx 1$. Geometries of twisted stilbene are generated using Molclusmolclus.
• Figure 4: (a) MECP geometries of nitroxyl calculated by different methods. (b) PESs of $\mathrm{S}_0$ and $\mathrm{T}$ states near the crossing point calculated by noncollinear spin-flip TDDFT. BLA stands for bond length alternation.
• Figure 5: (a) Chemical structure of $o$-nitrophenol. (b) Geometry of the $\mathrm{S}_1/\mathrm{T}$ crossing point calculated by noncollinear spin-flip TDDFT. The numbers indicate the bond lengths (in $\angstrom$) and dihedral angles (in $°$), with the values in parentheses corresponding to the reference results.