The Van Vleck Transformation: Further properties. Operators as vectors. Numerical algorithms

What is it about?

(Uploaded 2018-10-30). This Part II is a continuation of Part I: "Geometry of the canonical Van Vleck Transformation (VVT)". To indicate some highlights, let U stand for an arbitrary of the possible transformations to an effective Hamiltonian, let Ue stand for its even (block-diagonal) part, let Uc be the canonical VVT and assume that the Hilbert space is of finite dimension. Then the operators can be considered as vectors in a space with inner product and norm so that the Pythagorean Theorem, the Triangular Inequality, .... , apply. It is then found that when U varies, the distance dist(U,1) ( = norm of U-1) is stationary iff Ue is hermitian and minimal iff Ue is positive (hermitian with no negative eigenvalues). By this result, Klein's theorem of uniqueness gets a condensed form which is naturally connected to the equivalent but seemingly very different form due to Jørgensen - according to which U equals Uc iff Ue is positive. And it gets a strong likeness to a simple familiar fact: The minimal value of a real function f(x) is the smallest of those x at which f(x) is stationary - i.e. at which the derivative f'(x) is zero. Jørgensen's theorem is used in a new way as the background for an extended version of Kvaal's efficient SVD-determination of the matrices which represent the various canonical operators and vectors in a numerical treatment. A number of results are generalized to block diagonalization in case of more than two blocks.

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Why is it important?

The properties of the canonical VVT have been found by different authors over many years. The present Part II establishes a unifying theoretical background. Thus Löwdin's two orthonormaliza-tion methods have been central throughout the developments - although not always realized as such. Our Appendix B: "Singular Value Decomposition (SVD) and Polar Decomposition (PD)" is a self-contained brief compendium of general results which lead fairly directly to most of the properties of the canonical VVT - as well as (see Appendix D) to extended versions of Löwdin's methods and to a close connection between these, the Carlson-Keller theorem and a standard program for SVD of matrices. Appendix E: "From des Cloizeaux's unknowing use of Löwdin's orthonormalization methods to Kvaal's principle vectors" is a brief review where this unification also serves to clarify how Löwdin's methods were first recognized in the development of what over the years has become known as the canonical VVT.

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