Eigenfunctions equation

It is found empirically and of course is predictable theoretically that, when using a model for molecular electronic structure, the set of eigenfunction equations associated with the operators commuting with H are constraints on the action of the variation principle if Et is computed from R subject to symmetry constraints and E2 is computed in the same model with no such constraints then (2)... 

Say Y is the non-eigenfunction equation, use the homogeneous boundary condition to obtain the constant of integration. 

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. 

For the purpose of calculating wave functions which satisfy the correct physical scattering conditions, it becomes useful to define the symmetry wave functions JMsn solutions of the set of eigenfunction equations... 

Equations (4.II.6) and (4.II.7) are the eigenfunction equations for which are well known and have the solution... 

Substituting the expansion for the eigenvalue and the eigenfunction, equations (6) and (8), respectively, into the Schrodinger equation, equation (5), gives... 

The angular momentum has three components —, and — along the X-, y-, and z-axes in a Cartesian coordinate system. These components satisfy the eigenvalue-eigenfunction equation, with eigenvalue matrix, A ... 

These equations in g+Cx) and g (x) are inhomogeneous and coupled to one another by the terms in tfieir right-hand sides. If we drop those terms in the second equation, we generate the eigenvalue-eigenfunction equation,... 

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