Mapping operators effective Hamiltonian

Effective Operators and Classification of Mapping Operators Effective Operators Generated by Norm-Preserving Mappings Effective Operators Generated by Non-Norm-Preserving Mappings which Produce a Non-Hermitian Effective Hamiltonian... 

Consider a time-independent operator A whose matrix elements, yf a, /3 d (both expectation values and transition moments), in the space fl we wish to compute. This goal is to be achieved by transforming the calculation from 0 into one in O, resulting in an effective operator a whose matrix elements, taken between appropriate model eigenfunctions of an effective Hamiltonian h, are the desired As we now discuss, numerous possible definitions of a arise depending on the type of mapping operators that are used to produce h and on the choice of model eigenfunctions. 

Mappings Norms Conserved" Effective Hamiltonian Effective operator definition Effective Hamiltonian is the A = H Case State- Independent... 

We now determine particular classes of commutation relations that are, indeed, conserved upon transformation to state-independent effective operators. The proof of (4.1) demonstrates that the preservation of [A, B] by definition A requires the existence of a relation between K, K, or both and one or both of the true operators A or B. Likewise, there must be a relation between the appropriate wave operator, the inverse mapping operator, or both, and A, B, or both for other state-independent effective operator definitions to conserve [A, B]. All mapping operators depend on the spaces and fl. Although the model space is often specified by selecting eigenfunctions of a zeroth order Hamiltonian, it may, in principle, be arbitrarily defined. On the other hand, the space fl necessarily depends on H. Therefore, the existence of a relation between mapping operators and A, B, or both, implies a relation between H and A, B, or both. 

Q is the normal-ordered wave operator, mapping the eigenfunctions of the effective Hamiltonian onto the exact ones, = 1, It satisfies intermediate normalization,... 

The other approach is to map the polarization effect onto the protein-solvent interface in the form of induced charge, and then take the induced charge as a single electron operator in the MFCC Hamiltonian. The latter approach is employed in working with the discrete representation of the electron density by (Ji et al., 2008). Similar to the MFCC-CPCM approach, this procedure must be iterated until convergence is reached. 

For distances ry > D/B) l with D = (g di e+)p and d,- the dipole operator, the dipole-dipole interaction in Equation 12.4 between two polar molecules can be mapped onto the effective spin interaction Hamiltonian The first... 

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