Hougens isomorphic Hamiltonian

There is a particular difficulty in the formulation of the full Hamiltonian for a linear molecule (and hence for a diatomic molecule) which was first identified by Hougen [15], The source of this difficulty is the fact that only two rotational coordinates are required to define the orientation of a diatomic molecule in laboratory space, the Euler 

Using standard methods to transform differential operators, Hougen attempted to achieve as great a separation of electronic, vibrational, rotational and spin coordinates as possible. The resultant operator representing the rotational kinetic energy has the form 

The angular momentum operator g which emerges from this treatment has the form  

Thus we see that the operator g is not strictly an angular momentum operator in the quantum mechanical sense, which is why we have assigned it a different symbol. More importantly for the present purposes, we cannot use the armoury of angular momentum theory and spherical tensor methods to construct representations of the molecular Hamiltonian. In addition, the rotational kinetic energy operator, equation (7.89), takes a more complicated form than it has for a nonlinear molecule where there are three Euler angles (rotational coordinates). 

To obtain the isomorphic Hamiltonian for a diatomic molecule, x is introduced as an independent variable and the coordinates of the particles which make up the molecule are measured in an axis system (x, y, z) whose orientation is described by the Euler angles ( -/ , 0, x) in the (X, Y, Z) axis system. We recall that we chose x to be zero in constructing the true Hamiltonian. The (x, y, z) axes are therefore obtained by rotation of the (x7, y, z ) axis system about the z (= z) axis through the angle x As a result, we have 

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The second line of this equation follows from (7.102) above. We note that the awkward sin 0 factors in (7.89) have now disappeared. As Hougen points out, the eigenfunctions of the true Hamiltonian involve one less variable and so one less quantum number than the eigenfunctions of the artificial Hamiltonian and consequently the two operators cannot be completely isomorphic. However, a simple restriction on the extra quantum number in the artificial problem identifies that part of the full artifical Hamiltonian which is isomorphic with the true operator. Since the isomorphic Hamiltonian commutes with (Jz — W-), the two operators have a set of simultaneous eigenfunctions. Equation (7.102) states that only those eigenfunctions of the isomorphic Hamiltonian which have an eigenvalue of zero for (Jz - Wz) are eigenfunctions of the true Hamiltonian. 

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