A New Understanding Of Lambda Doubling

Gordon, Robert J.

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https://hdl.handle.net/2142/116597 Copy

Description

Title

A New Understanding Of Lambda Doubling

Author(s)

Gordon, Robert J.

Contributor(s)

Field, Robert W.

Issue Date

2022-06-23

Keyword(s)

Fundamental physics

Abstract

Lambda-doubling is the splitting of rotational levels that have the same quantum numbers and differ only in their parity. This phenomenon has been understood for nearly a century to be caused by an asymmetric perturbation by an energetically remote electronic state acting on otherwise degenerate rovibronic states. The terms in the Hamiltonian responsible for this perturbation are ${\cal{\hat{H}}}^c=-B({\hat{J}}^+{\hat{L}}^- + {\hat{J}}^-{\hat{L}}^+)+(B +\frac{1}{2} A)({\hat{L}}^+{\hat{S}}^- + {\hat{L}}^-{\hat{S}}^+),$ where ${\hat{J}}^{\pm}, {\hat{L}}^{\pm}$ and ${\hat{S}}^{\pm}$ are ladder operators for total, orbital, and spin angular momenta, and $B$ and $A$ are the rotational and spin-orbit coupling constants. The time-honored method for calculating the level-splitting is to calculate off-diagonal matrix elements of this operator that connect macroscopic terms of the form $|^{2S+1}\Lambda_{\Omega}\rangle_{e,f}$, using second-order perturbation theory (the Van Vleck transformation) to determine the energy difference between states of $e$- and $f$-symmetry. We have discovered that neglect of the microscopic electronic structure of the molecule may lead to incorrect values of the level-splitting and erroneous assignment of the parity of some of the levels. The breakdown of the macroscopic method lies in its failure to recognize that the rotational component of $ {\cal{\hat{H}}}^c$ contains two-electron operators, whereas the spin-orbit component is a one-electron operator. In addition, the macroscopic formulation fails to account for exchange symmetry of electrons in partially-filled spin-orbitals. We have shown that the macroscopic formulation gives correct results for inhomogeneous ($\Omega$-changing) perturbations and fails for homogeneous ($\Omega$-preserving) perturbations produced by the rotational part of the Hamiltonian. The breakdown is especially marked for the splitting of $^2\Pi_{\frac{1}{2}}$ by $^2\Sigma_{\frac{1}{2}}^{\pm}$ states, for which both homogeneous and inhomogeneous perturbations are involved.

Publisher

International Symposium on Molecular Spectroscopy

Type of Resource

text

Language

eng

Handle URL

https://hdl.handle.net/2142/116597

DOI

https://doi.org/10.15278/isms.2022.RM02

Copyright and License Information

Copyright 2022 held by the authors

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