Schmidt orthogonalized effective Hamiltonians

This will be the general expression of the Schmidt-orthogonalized effective Hamiltonian. If P/x a IP, the Bloch effective Hamiltonian will be non-Hermitian. The relative advantages of the Bloch, des Cloizeaux, Schmidt-orthogonalized effective Hamiltonians, or of the intermediate Hamiltonians has never been tested, especially in transfers to large systems. 

First one can build up other effective Hamiltonians based on hierarchized orthogonalization procedures. The Gram-Schmidt procedure is recommended if one starts from the best projected wavefunctions of the bottom of the spectrum. Thus one can obtain a quite reliable effective Hamiltonian with well behaved wavefunctions and good transferability properties (see Section III.D.2). The main drawback of this approach is that the Gram-Schmidt method, which involves triangular matrices, does not lead to simple analytical expressions for perturbation expansions. A partial solution to these limitations is brought about by the new concept of intermediate Hamiltonian,... 

The Gram-Schmidt procedure provides a simpler ordu onalization scheme that leads to a hermitian effective Hamiltonian. Since and we already orthogonal to the other projections, we only have to worry about and This means that the coefficients of 4 are defined by that is, if I l = a ( afe + 6a ) -I- p ( aa -P bb )... 

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