What is it about?
(Uploaded 2018-04-18). The classic article "Zur Quantentheorie der Molekeln" by Born and Oppemheimer (Ann. der Physik, 1927 , 84, p. 457) is honoured as the first which describes the separation of electronic motion, nuclear vibrations, and molecular rotation. The perturbation expansions, however, are extremely complicated and they only show that the rotational levels are determined by an eigenvalue problem on the space of the Euler angles for the chosen axes. After some very careful analysis, including a dramatic wrong interpretation in prevoius paper, C. Eckart (Phys. Rev. 1935, 47, p. 552) found that these axes had to be orientated by the condition which since has carried his name. Anything else implies a Coriolis interaction which requires a number of extra perturbation calculations before the expected rigid rotor Hamiltonian appears. The present work is not about why that is so.
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Why is it important?
Eckart's condition is on the instantaneous vectors for how much each nucleus is displaced from its equilibrium position. He - and subsequent authors - have somehow missed the fairly simple observation that the condition is fulfilled if the mass weighted sum of their squares attains minimum. I have, however, added more than that. Eckart found that his condition is fulfilled if the axes are found by orthonormalizing three vectors, thereafter known as the "Eckart vectors". I have recognized Eckart's orthonormalization as an example of Löwdin's symmetric and pointed out that if we have one solution to Eckart's condition we get another by rotating its axes 180 degrees around any of the axis obtained by applying Löwdin's canonical orthonormalization to the 3 Eckart vectors instead of the symmetric. Furthermore: Invoking the Carlson-Keller theorem I have given a simple explicit expression for the minimal mass weighted sum.
Perspectives
Such fine brief semi-popular explanations of Eckart's condition as in the classic books by Landau and Lifschitz ( "Mechanics" 2. ed. 1969, §24 and "Quantum Mechanics" 2. ed. 1965, § 104) could easily be supplemented with the least square form pointed out in the present work. This does not explain a minimal Coriolis interaction - but it is in an immediately understood accordance with small vibrational displacements from equilibrium. Actually a deeper problem is hidden behind this simplicity if some nuclei are identical particles. Here eigenstates can be demanded to be invariant under the inversion-permutation group. The NH3 molecule, for instance, has two different reference (equilibrium-) configurations which cannot both be orientated so that all displacements are small. And when the appropriate normal coordinate based on one reference configurations increases sufficiently, the N-nucleus moves through the plane of the H-nuclei, where the other is needed to obtain that all displacements are small. These problems are far too encompassing for a brief article like the present. I must refer to the book P. R. Bunker: Molecular Symmetry and Spectroscopy ( 1979, Academic Press. p. 352 in particular). But I think that my discussion of the several solutions to Eckart's conditions could turn out to be useful a theory on the same mathematical rigorous level as that of the Wilson-Howard-Watson Hamiltonian ( J.K.G. Watson, Mol. Phys. 1968, 15, p. 469).
Dr Flemming Jørgensen
Nygårdsvej 43, 4700 Næstved http://www.naestved-gym.dk/
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This page is a summary of: Orientation of the Eckart frame in a polyatomic molecule by symmetric orthonormalization, International Journal of Quantum Chemistry, July 1978, Wiley,
DOI: 10.1002/qua.560140106.
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