Effective Hamiltonian formalisms

We recall that the effective Hamiltonian formalism considers a model space Mo together with a target space M,... 

Note that all the above expressions characterize the effective Hamiltonian formalism per se, and are independent of a particular form of the wave operator U. Indeed, this formalism can be exploited directly, without any cluster Ansatz for the wave operator U (see Ref. [75]). We also see that by relying on the intermediate normalization, we can easily carry out the SU-Ansatz-based cluster analysis We only have to transform the relevant set of states into the form given by Eq. (16) and employ the SU CC Ansatz,... 

If we now consider the generalized Silverstone-Sinanoglu strategy discussed above, then the most natural way to proceed is by way of an effective hamiltonian formalism. We introduce a single (state-universal) wave-operator O, whose action on 4 s produce the functions 4>k, defined by eq. (6.1.1) or (6.1.7), and write Schrodinger equations for 4[Pg.327]

We shall make use of the effective Hamiltonian formalism [14] that enables us to isolate effects of interest from irrelevant complications. We divide the electronic Hamiltonian into a strong part H° and a weak part H, and we shall suppose that H° is simple enough to be solved exactly. The Hamiltonian including the cubic field and interelectronic repulsion only is the usual choice for H in the case of the 3d group ions. Then H should include all other interactions (spin-orbit coupling, lower symmetry fields, electron-phonon interaction, external fields, strain etc). The most important assumption is that the perturbations, described by the H Hamiltonian (in particular the JT interaction) must be smaller relative to the initial splitting due to H°. In the case of the 3d metal ions the assumption is usually well justified. 

Both VU and SU MR CC methods employ the effective Hamiltonian formalism the relevant cluster amplimdes are obtained by solving Bloch equations and the (in principle exact) energies result as eigenvalues of a non-Hermitian effective Hamiltonian that is defined on a finite-dimensional model space Mq. An essential feature characterizing this formalism is the so-called intermediate or Bloch normalization of the projected target space wave functions I f, ) with respect to the corresponding model space configurations 1, ), namely = 8 (for details, see, e.g. Refs. [172,174]). 

An Effective Hamiltonian Formalism for the Computation of Diabatic Surfaces. 

In the effective Hamiltonian formalism just reviewed, the diabatic state energies are obtained as the diagonal matrix elements of the effective Hamiltonian, while the resonance interaction between product-like and reactant-like diabatic surfaces is obtained as the off-diagonal matrix elements. The adiabatic states are obtained as the eigenvalues. Thus, the effective Hamiltonian corresponds to the projection of the full Cl Hamiltonian onto the subspace of the (product-like and reactant-like) Heitler-London and no-bond configurations. We now wish to comment briefly on the physical interpretation of the effective Hamiltonian computed via Eq. (17). 

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