Derivation of the effective Hamiltonian

In this chapter we introduce and derive the effective Hamiltonian for a diatomic molecule. The effective Hamiltonian operates only within the levels (rotational, spin and hyperfine) of a single vibrational level of the particular electronic state of interest. It is derived from the Ml Hamiltonian described in the previous chapters by absorbing the effects of off-diagonal matrix elements, which link the vibronic level of interest to other vibrational and electronic states, by a perturbation procedure. It has the same eigenvalues as the Ml Hamiltonian, at least to within some prescribed accuracy. 

The motivation for constructing the effective Hamiltonian is one of economy and perhaps even of feasibility. It reproduces the eigenstates of the vibronic state of interest but with a much smaller representation than that of the Ml Hamiltonian. The effective Hamiltonian provides a naMal resMg poM in the journey from experiment to theory. It permits data to be Med in an unprejudiced fashion, the parameters being determined by statistical criteria only. These parameters in turn can be interpreted in terms of various theoretical models for the electronic states, and provide a point of comparison for ab initio calculations. A soundly based effective Hamiltonian makes allowance for all possible admixtures of electronic states the relative importance of the perturbations by these different states is determined by a detailed comparison of the parameter values with theoretical predictions. In this way, the task of data fitting is clearly separated from that of theoretical interpretation. 

It took several decades for the effective Hamiltonian to evolve to its modem form. It will come as no surprise to learn that Van Vleck played an important part in this development for example, he was the first to describe the form of the operator for a polyatomic molecule with quantised orbital angular momentum [2], The present formulation owes much to the derivation of the effective spin Hamiltonian by Pryce [3] and Griffith [4], Miller published a pivotal paper in 1969 [5] in which he built on these ideas to show how a general effective Hamiltonian for a diatomic molecule can be constructed. He has applied his approach in a number of specific situations, for example, to the description of N2 in its A 3 + state [6], described in chapter 8. In this book, we follow the treatment of Brown, Colbourn, Watson and Wayne [7], except that we incorporate spherical tensor methods where advantageous. It is a strange fact that the standard form of the effective Hamiltonian for a polyatomic molecule [2] was established many years before that for a diatomic molecule [7]. 

Derivation of the effective Hamiltonian by degenerate perturbation theory general principles 

It is always possible to divide the total Hamiltonian Xinto a major part 3C (the zeroth-order Hamiltonian) and a perturbation XX  

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The above derivation of the effective Hamiltonian is only complete when, for some reasons, the uniform strains of the crystal are not relevant. This is clearly the case for crystals with low concentration of Jahn-Teller impurities. Contrary to that, bulk deformations often arise in its low-symmetry structural phases of Jahn-Teller crystals [2,11]. The uniform strains describing the bulk deformations of the crystal cannot be reduced to a combination of phonon modes, as it was first pointed out by... 

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