Diatomic molecules Difference spectra

Until recently, only estimates of the Hartree-Fock limit were available for molecular systems. Now, finite difference [16-24] and finite element [25-28] calculations can yield Hartree-Fock energies for diatomic molecules to at least the 1 ghartree level of accuracy and, furthermore, the ubiquitous finite basis set approach can be developed so as to approach this level of accuracy [29,30] whilst also supporting a representation of the whole one-electron spectrum which is an essential ingredient of subsequent correlation treatments. 

Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

The vibration-rotation hamiltonian of a polyatomic molecule is more complicated than that of a diatomic molecule, both because of the increased number of co-ordinates, and because of the presence of Coriolis terms which are absent from the diatomic hamiltonian. These differences lead to many more terms in the formulae for a and x values obtained from the contact transformation, and they also lead to various kinds of vibrational and rotational resonance situations in which two or more vibrational levels are separated by so small an energy that interaction terms in the hamiltonian between these levels cannot easily be handled by perturbation theory. It is then necessary to obtain an effective hamiltonian over these two or more vibrational levels, and to use special techniques to relate the coefficients in this hamiltonian to the observed spectrum. 

Almost all infrared work makes use of absorption techniques in which radiation from a source emitting all infrared frequencies is passed through a sample of the material to be studied. When the frequency of this radiation is the same as a vibrational frequency of the molecule, the molecule may be vibrationally excited this results in loss of energy from the radiation and gives rise to an absorption band. The spectrum of a polyatomic molecule generally consists of several such bands arising from different vibrational motions of the molecule. This experiment involves diatomic molecules, which have only one vibrational mode. 

Vibrations of atoms in a molecule can be divided in six different forms symmetrical and antisymmetrical stretching, rocking, scissoring, twisting and wagging. Simple diatomic molecules have only one bond, and only one fundamental vibrational mode (the interatomic stretching mode) is seen in the spectrum. More complex molecules, such as hydrogels, have many bonds, and their vibrational spectrum is much more complex. 

Since the occurrence of the Raman effect depends on the change in polarizability as vibration occurs, the selection rules are different for the Raman effect than they are for the infrared spectrum. In particular, in molecules with a center of symmetry the totally symmetric vibration is Raman-active, but is forbidden in the infrared since it produces no change in dipole moment. Thus the homonuclear diatomic molecules, H2, O2, N2, show the Raman effect but do not absorb in the infrared. There is also a purely rotational Raman eff ect in these molecules. However, in this case the selection rule is A J = 2. Thus we have for the rotational Stokes lines... 

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