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Hierarchy of Terms in the Effective Hamiltonian

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Spectra and Dynamics of Small Molecules

Part of the book series: Lecture Notes in Physics ((LNP,volume 900))

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Abstract

This lecture is an introduction to effective Hamiltonians, transition selection rules, Hund’s coupling cases, pattern-forming rotational quantum numbers, and straight line plots. The goal is to create a fit model: \(\mathbf{H}^{\mathrm{eff}}\).

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Notes

  1. 1.

    If spin–orbitals are labeled 1, 2, 3, 4, …then ψ 1234 differs by one spin–orbital from ψ 1235 or ψ 1534.

  2. 2.

    Orbit-spin is not the same as spin–orbit.

  3. 3.

    N is the total angular momentum exclusive of spin, \(\mathbf{N} = \mathbf{J} -\mathbf{S}.\)

  4. 4.

    There is a special complication that arises for the body-fixed components of angular momenta that contain the rotation of the body frame (\(\mathbf{J},\ \mathbf{N} = \mathbf{J} -\mathbf{S}\), but not L or S). For these angular momenta, the body-fixed components follow “reversed angular momentum commutation rules”, \([\mathbf{A}_{i},\mathbf{A}_{j}] = -\sum _{k}\varepsilon _{\mathit{ijk}}\mathbf{A}_{k}\), and the roles of A + and A − are exchanged.

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Authors and Affiliations

  1. Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

    Robert W. Field

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  1. Robert W. Field
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Field, R.W. (2015). Hierarchy of Terms in the Effective Hamiltonian. In: Spectra and Dynamics of Small Molecules. Lecture Notes in Physics, vol 900. Springer, Cham. https://doi.org/10.1007/978-3-319-15958-4_2

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