[Paper Review] The existence and unambiguity of the principal axis system of the EPR tensors

Stanislav Komorovský|arXiv (Cornell University)|May 11, 2023

Electron Spin Resonance StudiesBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

This paper rigorously establishes the existence and unambiguity of the principal axis system for electron paramagnetic resonance (EPR) tensors—specifically the g-tensor and hyperfine coupling tensor—by proving that these tensors possess two distinct sets of principal axes: one in real space and one in fictitious spin space. The study demonstrates that only the eigenvalues of the G-tensor ($\mathbf{G} = \mathbf{g}\mathbf{g}^\top$) and the sign of the determinant of the g-tensor are observable, using group-theoretic arguments and minimal assumptions applicable to the Dirac–Coulomb–Breit Hamiltonian and any spatial symmetry.

ABSTRACT

Although the role of the electron paramagnetic resonance (EPR) g-tensor and hyperfine coupling tensor in the EPR effective spin Hamiltonian is discussed extensively in many textbooks, certain aspects of the theory are missing. In this text we will cover those gaps and thus provide a comprehensive theory about the existence of principal axes of the EPR tensors. However, an important observation is that both g- and a-tensors have two sets of principal axes -- one in the real and one in the fictitious spin space -- and, in fact, are not tensors. Moreover, we present arguments based on the group theory why only eigenvalues of the G-tensor, $\mb{G} = \mb{g}\mb{g}^{\!\mathsf{T}}$, and the sign of the determinant of the g-tensor are observable quantities (an analogical situation also holds for the hyperfine coupling tensor). We keep the number of assumptions to a minimum and thus the theory is applicable in the framework of the Dirac--Coulomb--Breit Hamiltonian and for any spatial symmetry of the system.

Motivation & Objective

• To close theoretical gaps in the understanding of the principal axis system of EPR tensors, particularly the g-tensor and hyperfine coupling tensor.
• To rigorously prove the existence and unambiguity of principal axes for EPR tensors under minimal physical assumptions.
• To clarify that the g-tensor and hyperfine coupling tensor are not true tensors but possess dual principal axis systems in real and fictitious spin space.
• To identify the physically observable quantities—specifically eigenvalues of $\mathbf{G} = \mathbf{g}\mathbf{g}^\top$ and the sign of $\det(\mathbf{g})$—using group-theoretic reasoning.
• To extend the validity of the effective spin Hamiltonian to systems with arbitrary spatial symmetry and under the Dirac–Coulomb–Breit framework.

Proposed method

• Uses the Hartree atomic unit system and applies group theory to analyze time-reversal symmetry and its implications on tensor structure.
• Applies the eigenvalue equation $\mathbf{A}\mathbf{C}_k = e_k\mathbf{C}_k$ to define principal axes, with indices not summed when repeated on both sides.
• Employs the matrix identity $\mathbf{W}^\top\mathbf{W} = \mathbf{W}\mathbf{W}^\top = \mathbf{1}$ to prove orthonormality of basis vectors in real and spin spaces.
• Derives the effective spin Hamiltonian $\mathbf{H}^{\mathrm{eff}} = \frac{1}{2c}B_u g_{uv}\mathbf{S}_v$ from the full quantum mechanical Hamiltonian under weak spin-orbit coupling.
• Applies the time-reversal operator $\mathcal{K}$ to show that Kramers pairs are degenerate and norm-preserving, ensuring consistency in the spin space formalism.
• Uses the exponential parametrization of SU(2) group elements via $\vec{\mathbf{S}} = \frac{1}{2}\vec{\bm{\sigma}}$ to fully describe doublet systems.

Experimental results

Research questions

• RQ1 What conditions guarantee the existence and uniqueness of the principal axis system for the EPR g-tensor and hyperfine coupling tensor?
• RQ2 Why are the g-tensor and hyperfine coupling tensor not true tensors, and what are the implications of their dual principal axis systems in real and fictitious spin space?
• RQ3 Which components of the g-tensor are physically observable, and why are only the eigenvalues of $\mathbf{G} = \mathbf{g}\mathbf{g}^\top$ and the sign of $\det(\mathbf{g})$ observable?
• RQ4 How does time-reversal symmetry constrain the structure of the EPR tensors and their principal axes?
• RQ5 To what extent is the effective spin Hamiltonian $\mathbf{H}^{\mathrm{eff}} = \frac{1}{2c}B_u g_{uv}\mathbf{S}_v$ valid across different electronic multiplicities and relativistic regimes?

Key findings

• The principal axis system of the EPR g-tensor and hyperfine coupling tensor exists and is unambiguous under the assumption of time-reversal symmetry and a valid effective spin Hamiltonian.
• The g-tensor and hyperfine coupling tensor possess two distinct sets of principal axes: one in real space and one in fictitious spin space, indicating they are not true tensors in the conventional sense.
• Only the eigenvalues of the matrix $\mathbf{G} = \mathbf{g}\mathbf{g}^\top$ are observable, not the individual components of the g-tensor.
• The sign of the determinant of the g-tensor is an observable quantity, which distinguishes between different topological configurations of the tensor.
• The theory is valid for any spatial symmetry and applies to the Dirac–Coulomb–Breit Hamiltonian, making it broadly applicable to ab initio quantum chemical methods.
• For doublet systems, the effective spin Hamiltonian fully describes the system under any relativistic effects, due to the exponential parametrization of SU(2) group elements.