What Is the Real-Time Atomistic Mechanism Behind Chirality-Induced Spin Selectivity in Donor-Chiral Bridge-Acceptor Molecules?

Technical Summary: Real-Time Atomistic Mechanism of CISS in Donor-Chiral Bridge-Acceptor Molecules

Problem Statement
Chirality-induced spin selectivity (CISS) has been experimentally confirmed in photo-excited donor–chiral bridge–acceptor (D-Bχ-A) molecules, specifically the PXX-(S,R)NMI2–NDI system, via time-resolved electron paramagnetic resonance (TREPR) measurements. However, the microscopic mechanism driving this effect in axially chiral systems remains unresolved. Existing theoretical frameworks face significant limitations: one-dimensional helical chain models fail to account for transverse spin polarization components observed in experiments, while phenomenological two-level models lack real-time dynamical resolution and detailed structural mapping. Furthermore, previous studies often relied on metallic contacts or heavy-atom substrates, obscuring the intrinsic molecular contributions. A critical gap exists in understanding how axial chirality generates both longitudinal and transverse spin polarizations without external magnetic fields or spin-superexchange transfer, and how specific geometric parameters of the chiral bridge dictate these outcomes.

Methodology
The authors develop a three-dimensional, real-space time-dependent quantum dynamics model to investigate the PXX-(S,R)NMI2–NDI molecule with full atomistic resolution.

• Geometric Spin-Orbit Coupling (SOC): Instead of relying on intrinsic SOC of light atoms, the model introduces a geometric SOC mechanism arising from the curvature of the electron transport trajectory along the twisted intramolecular pathway. The SOC strength at each atomic site is explicitly determined by local torsion angles and interplanar spacings derived from the optimized ground-state geometry.
• Hamiltonian Construction: The continuous geometric SOC is discretized onto the atomic lattice of the binaphthyl-type bridge dimer. The Hamiltonian incorporates Frenet–Serret frame vectors (tangent, normal, binormal) to define the local acceleration and momentum, yielding a geometric SOC term that scales with the gradient of electron momentum along the curved path.
• Non-Abelian Correction: To address the transverse polarization, the model introduces a non-Abelian curvature correction. This accounts for the non-commuting nature of spin rotations as the electron traverses the left and right units of the chiral bridge, generating a second-order geometric phase proportional to the cross product of accumulated curvature vectors.
• Dynamics and Analysis: The model propagates time-dependent charge-state populations and spin densities. It utilizes a density-matrix formulation to map single-electron transport to two-electron singlet-triplet observables, allowing for a direct comparison with TREPR data. Systematic variations of the bridge's dihedral angle ($\alpha$), in-plane twist ($\beta$), and folding angle ($\gamma$) are performed to isolate geometric determinants.

Key Contributions and Results

1. Magnitude of Geometric SOC: The study demonstrates that the geometric SOC coupling strength along the twisted pathways exceeds the intrinsic SOC of light atoms by one to two orders of magnitude. This provides a sufficient driving force for observable high spin polarizations without requiring heavy atoms or external fields.
2. Dual Origin of Spin Polarization: The model successfully reproduces the coexistence of two distinct polarization components:
   • Chirality-Independent Component ($SP_x$): Arises from the twisted pathways within each individual NMI unit. It persists even in achiral (coplanar) configurations and is symmetric with respect to enantiomers.
   • Chirality-Dependent Component ($SP_y, SP_z$): Arises from the non-Abelian curvature correction generated by the twisted pathway between the two NMI units. This component is intrinsic to axial chirality, vanishes in the achiral limit, and reverses sign upon enantiomeric interchange.
3. Definition of the Chiral Axis: The paper provides a rigorous mathematical definition of the chiral axis direction. It is not a predefined molecular coordinate but is uniquely determined by the orientation of the non-Abelian geometric SOC vector (the cross product of curvature vectors from the left and right units). For the specific system studied, this vector aligns with the $z$-axis for dihedral variations and the $y$-axis for folding variations.
4. Quantitative Agreement with Experiment: The calculated polarization components, their relative magnitudes (30%–40% chirality-dependent fraction), and their time evolution quantitatively match TREPR measurements. The model resolves the apparent dichotomy between axial chirality and transverse spin selectivity, showing that transverse signals are intrinsic to the axial chiral structure rather than a transfer from the axial direction.
5. Unified Framework: By employing a quantum-channel mapping within a density-matrix framework, the authors establish the strict equivalence between vector spin polarization (expectation values of spin angular momentum) and population polarization (asymmetric occupancy of Zeeman-split levels), linking single-electron transport dynamics to the two-electron spin-state observables detected in experiments.

Significance
The paper claims to offer a self-consistent and general physical picture of the CISS effect in axially chiral molecules. By bypassing the need for heavy-atom contacts or extrinsic mechanisms like the radical-pair mechanism (which is shown to be insufficient on nanosecond timescales), the geometric SOC framework explains the origin of both longitudinal and transverse spin polarizations. The work provides explicit theoretical guidance for the design of chiral spintronic devices by identifying specific geometric descriptors (dihedral angles, inter-ring coupling) that control spin polarization yield. It resolves longstanding ambiguities in CISS interpretation by demonstrating that the transverse component is an intrinsic feature of the three-dimensional transport pathway in axially chiral systems, defined by non-Abelian geometric phases.