Quantification of chirality based on electric toroidal monopole

Quantification of chirality based on electric toroidal monopole

22 February 2024 | A. Inda, R. Oiwa, S. Hayami, H. M. Yamamoto, H. Kusunose
This paper proposes a method to quantify chirality using electric toroidal monopoles (ETMs), a T-even pseudoscalar quantity that reflects the electronic wave functions of chiral systems. The authors analyze twisted methane (CH₄) at the quantum-mechanical level to demonstrate that the ETMs serve as a quantitative indicator of chirality. In twisted methane, the handedness of chirality corresponds to the sign of the expectation value of the ETMs. The study shows that the spin-dependent imaginary hopping between hydrogen atoms is crucial for activating chirality, while relativistic spin-orbit coupling within carbon atoms is less important. The ETMs are derived from the symmetry-adapted multipole basis (SAMB), which provides a complete basis set for describing electronic degrees of freedom. The analysis reveals that the ETMs belong to the A₂ irreducible representation in the Td symmetry and become finite when the symmetry of CH₄ is lowered from Td to D₂. The results indicate that the sign of the ETM expectation values represents the handedness of chirality. The study also highlights that the spin-dependent imaginary hopping between hydrogen atoms is the most important parameter for achieving chiral properties in twisted CH₄. The approach can be applied to other chiral molecules and crystals, providing a new way to quantify chirality at the quantum-mechanical level. The paper concludes that the ETMs offer a quantitative measure of chirality, which can deepen our understanding of chirality in materials.This paper proposes a method to quantify chirality using electric toroidal monopoles (ETMs), a T-even pseudoscalar quantity that reflects the electronic wave functions of chiral systems. The authors analyze twisted methane (CH₄) at the quantum-mechanical level to demonstrate that the ETMs serve as a quantitative indicator of chirality. In twisted methane, the handedness of chirality corresponds to the sign of the expectation value of the ETMs. The study shows that the spin-dependent imaginary hopping between hydrogen atoms is crucial for activating chirality, while relativistic spin-orbit coupling within carbon atoms is less important. The ETMs are derived from the symmetry-adapted multipole basis (SAMB), which provides a complete basis set for describing electronic degrees of freedom. The analysis reveals that the ETMs belong to the A₂ irreducible representation in the Td symmetry and become finite when the symmetry of CH₄ is lowered from Td to D₂. The results indicate that the sign of the ETM expectation values represents the handedness of chirality. The study also highlights that the spin-dependent imaginary hopping between hydrogen atoms is the most important parameter for achieving chiral properties in twisted CH₄. The approach can be applied to other chiral molecules and crystals, providing a new way to quantify chirality at the quantum-mechanical level. The paper concludes that the ETMs offer a quantitative measure of chirality, which can deepen our understanding of chirality in materials.
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