A brief on perturbation theory

In other words, the diagonal elements of the perturbing Hamiltonian provide the first-order correction to the energies of the spin manifold, and the nondiagonal elements give the second-order corrections. Perturbation theory also provides expressions for the calculation of the coefficients of the second-order corrected wavefunctions l / in terms of the original wavefunctions (p) 

An—at least, theoretically—simple example is the S = 1 system in weak-field subject to a dominant zero-field interaction and a weakly perturbing electronic Zeeman interaction (similar to the S = 2 case treated above). The initial basis set is 

This matrix is diagonal in the zero-field interaction, so the zeroth-order energy levels can be directly seen to be 

The perturbing Zeeman interaction has no elements on the diagonal, so there is no first-order correction. The second-order corrections are 

This is a pretty unusual expression, and it should warn us that resonance conditions derived via perturbation theory should always be checked for their validity under actual conditions. Suppose that the zero-field intradoublet splitting, /L Cl)(lls ni))-ii,(0) 

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