Lambda doubling

Figure 2. Schematic of the energy levels for the OH molecule. The collision-induced energy-transfer transitions are denoted by double-line arrows. The rotational quantum number is denoted by K or K". Both spin doubling and lambda doubling have been suppressed for clarity.

The experimentally achievable localized excitations are typically described by one of the zero-order basis states (see Section 3.2), which are eigenstates of a part of the total molecular Hamiltonian. Localization can be in a part of the molecule or, more abstractly, in state space . The localized excitations are often described by extremely bad quantum numbers. The evolution of initially localized excitations is often more complex and fascinating than an exponential decay into a nondescript bath or continuum in which all memory of the nature of the initial excitation is monotonically lost. The terms in the effective Hamiltonian that give birth to esoteric details of a spectrum, such as fine structure, lambda doubling, quantum interference effects (both lineshapes and transition intensity patterns), and spectroscopic perturbations, are the factors that control the evolution of an initially localized excitation. These factors convey causality and mechanism rather than mere spectral complexity. 

If the adiabatic Bom-Oppenheimer approximation were exact, photochemistry and photophysical processes would be rather straightforward to describe. Molecules would be excited by the incident radiation to some upper electronic state. Once in this electronic state, the molecules could radiate to a lower electronic state, or they could decompose or isomerize on the upper electronic potential energy surface. No transitions to other electronic states would be possible. The spectroscopy of the systems would also be greatly simplified, as there would no longer be any phenomena such as lambda doubling, etc., which lifts degeneracy of some energy levels of the clamped-nucleus electronic Hamiltonian, //,. 

As to the lambda doubling, it is, as we have asserted, diagonal in a basis where parity is a good quantum number. In terms of the basis (Equation 2.15), wavefunctions of well-defined parity are given by the linear combinations... 

The net result is that for each value of m, the Hamiltonian for our lambda-doubled molecule can be represented as a 2 x 2 matrix, similar to the ones above ... 

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