Exploiting a Resolution of the Identity

The peculiarity that is involved in the calculation of the DKH Hamiltonians derives from the fact that some terms in the Hamiltonians are of the form pV... Vp [compare Eq. (12.56)]. Hence, no momentum operators occur between the potential energy operators, and a new matrix representation would be needed for such terms. Even worse, the higher the order, the more complicated are the terms that arise. Hess solution to this problem was the introduction of a resolution of the identity (RI), 

Formally, we have not yet introduced any approximation. But the evaluation of the operator sequence on the right-hand side of Eq. (12.77) requires a translation into products of operator matrices, 

Note that this also implicitly implies that we never encounter a (single) linear momentum between two V operators and that there is always a (single) 

An important aspect of DKH theory is that only standard operator matrices for the nonrelativistic kinetic energy and external potential are required for the evaluation of the DKH Hamiltonian, which is after all remarkable. Only one non-standard matrix is needed, which can, however, be calculated with little additional effort. This is the matrix representation of pVp, 

Latin indices i and j refer to basis functions in p -space, whereas the Greek indices p and v denote basis functions in the original position space as before. This notation will also be adopted in section 12.5.4. The matrix representation of the DKH kinetic energy is now easily obtained. 

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