7 - Derivation of the effective Hamiltonian

Summary

Introduction

The Born–Oppenheimer approximation is an important linch pin in the description of molecular energy levels. It reveals the difference between electronic and nuclear motions in a molecule, as a result of which we expect the separation between different electronic states to be much larger than that between vibrational levels within an electronic state. An extension of these ideas shows that the separation between vibrational levels is correspondingly larger than the separation between the rotational levels of a molecule. We thus have a hierarchy of energy levels which reveals itself in the electronic, vibrational and rotational structure of molecular spectra. This gradation in the magnitude of the different types of quanta also provides the inspiration for an energy operator known as 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 full 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 full Hamiltonian, at least to within some prescribed accuracy.