About Klein's theorem of uniqueness and the canonical Van Vleck Transformation.

What is it about?

(Uploaded: 2018-04-18). A perturbed Hamiltonian H=Ho+V can be transformed to an effective Hamiltonian Heff in various ways. When Klein wrote his "Degenerate perturbation theory" (J. Chem. Phys. 1974, 61, p. 786) a particularly important approach to a hermitian Heff was the one des Cloizeaux obtained by adjusting a foregoing due to Bloch (which is very simple, but gives a non-hermitian Heff). Klein characterized des Cloizeaux' uniquely as the transformation which -in a precise least square sense - changes the eigenvectors of H=Ho+V minimally. However, to apply this intuitively very satisfactory result one must enter some not so easy variational calculations. And Klein made some mistakes when he did so to show that his characteristic is fulfilled by the canonical Van Vleck transformation - a designation he invented for the same reason. In the present brief article I clarify the mistakes and point out that the identity Klein aimed is correct none the less. Unaware of Klein's calculation I had found the identity he aimed at much more directly from my own very different theorem of uniqueness in my "Effective Hamiltonians" ( Mol. Phys. 1975, 29, p. 1137)

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Why is it important?

The transformation to which Klein tried to apply his unique characteristic is actually the "Level shift transformation" described in sec. 11 of my "Effective Hamiltonian". This is due to Primas whose approach reminds strongly of Van Vleck's - but differs in a detail which makes a most sub-stantial difference. So, had Klein's variational calculations been correct, my unique characteristic would have been wrong. As one can see from the disclosure of the error, it is rather treacherous.

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