Hamiltonian operators simulation techniques

The present review is devoted to two main approaches that may lead in a controlled way from the exact Hamiltonian to simplified Hamiltonians, which are more easy to handle and on which deductive derivations may be easier to draw. One approach proceeds through projections of some exact wavefunc-tions into a relevant reduced subspace and leads to the ejfective Hamiltonian methodology. The techniques will be described in Section II.A, and their applications in Section III. The other procedure may be considered as a simulation of the considered exact Hamiltonian by a simpler Hamiltonian, the efficiency of the simulation being measured through a reduced distance, taken on a small subspace. This procedure, described in Section II.B, leads to the definition of pseudo-Hamiltonians or pseudo-operators. The corresponding applications are reviewed in Section IV. 

The term pseudo-Hamiltonian should be used when they are obtained through simulation techniques, i.e. minimization of the distance between the exact and pseudo-operators in a reduced subspace. 

To summarize this section one should say that an effective Hamiltonian treatment of the core electron effect faces a contradiction between the necessity to use extended valence basis sets for the extraction and the risk of appearance of core excited intruder states. One should also recognize that this approach leads to p-electron operators for atoms involving p valence electrons and seems much more difficult to handle than the monoelectronic core pseudopotentials extracted by simulation techniques and discussed in Section IV of the present contribution. As a counterpart one should mention that this core effective Hamiltonian would be much superior, since it would include for instance the core-valence correlation effects which play such an important role in alkali- or alkaline-earth-containing molecules. 

This technique has provided a potential for the carbon atom which is universal in the sense that it can be transferred with good accuracy in the whole series of hydrocarbon compounds. The main advantage of simulating the Fock operator by the kinetic operator plus the sum of atomic potentials is to produce approximate valence Fock Hamiltonians which can be solved with... 

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