Separation of Dirac's Hamiltonian by Van Vleck transformation

What is it about?

(Uploaded 2018-04-18). The renowned Foldy-Wouthuysen transformation aims at separating the upper eigenstates for Dirac's Hamiltonian from the lower. The principle is to use a series of unitary transformations exp(g1), exp(g3), exp(g5), .... where the odd and anti-hermitian operators are chosen consecutively so that the odd parts of the transformed Hamiltonian become smaller and smaller. At a first sight this successive approach seems more simple than the alternative exp(g1+g3+g5....). However, both principles are examples of the so-called Van Vleck Transformation (VVT) for a general perturbed Hamiltonian. And over the years the theory has undergone substantial developments and the form exp(g1+g2+g3....) is now referred to as the canonical VVT. It has turned out that it has a number of attractive geometric and algebraic properties, not shared by other approaches for transforming to an effective Hamiltonians (S. Kvaal: Geometry of effective Hamiltonians, Phys. Rev. C 78, 044330 (2008) . F. Jørgensen: Geometry of the canonical VVT , Int. J. Quantum Chem. 2015 DOI: 10.1002/qua.24992). Writing Dirac's Hamiltonian as beta+v1+v2 the transform is obtained through the order 6, assuming only that the perturbation v1/v2 of first/second order is odd/even. Thereby the perturbational technique can be studied separately from the physical meaning of the terms.

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Why is it important?

As the importance of relativistic quantum chemistry has risen , the Foldy-Wouthuysen transfor-mation (FWT) and the further developed Douglas-Kroll transformation (DKT) have become central. The direct impetus for the reconsideration in the present work has been two papers [A. Wolf, M. Reiher and B. A. Hess, J. Chem. Phys. 2002, 117, 2002] [M. Reiher and A. Wolf, J. Chem. Phys. 2004, 121, 2004 ] in which extremely accurate numerical energies for a hydrogen atom with Z = 10,20, ...., 120 are compared with the known analytical results. In these the authors follow the seemingly universal tradition of using successive transformations. By the present paper I intend to draw attention to the developments within the canonical approach to Van Vleck Transformation. One example: Wolf, Reiher and Hess have introduced a genuinely new idea by which each of the successive transformations used in the DKT are modified in a way which is uniquely defined by a demand which intends to make sure that truncation at any given order of expansion has a minimal effect. I transfer this optimized approach from DKT to successive VVT in general - and thereafter find that the canonical VVT is invariant to the same optimization. That is, it has the desired optimal property from a start. Another example: Kvaal loc. cit.] has shown how the canonical VVT in particular lends itself very directly for numerical treatment by a standard program for matrix SVD.

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