State independence effective Hamiltonian formalisms

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,... 

The commutation relation between two arbitrary operators is not conserved upon transformation to effective operators by any of the definitions. Many state-independent effective operator definitions preserve the commutation relations involving // or a constant of the motion, as well as those involving operators which are related to P in a special way, for example, A with [P, 4] = 0. Many state-dependent definitions also conserve these special commutation relations. However, state-dependent definitions are not as convenient for formal and possibly computational reasons. The most important preserved commutation relations are those involving observables, since, as discussed in Section VII, they ensure that the basic symmetries of the system are conserved in effective Hamiltonian calculations. 

In (1) the Coulomb interaction between the conduction states is neglected. These states are rather extended (see fig. 1) and the Coulomb integrals are not very large. For such states the local spin density (LSD) approximation of the spin density functional formalism (Kohn and Sham 1965) has been rather successful (von Barth and Williams 1983). In this scheme the electrons are formally treated as independent and correlation effects are included in an effective one-particle potential. By using 6fc s in (1) which are obtained from a LSD approximation or deduced from experiment, we may incorporate some interaction effects implicitly in the Hamiltonian (1). In this way chemical information about the compound considered may also be incorporated. Relativistic effects on the band structure are also included in this approach. 

The excited-state density matrix. We now apply the general formalism developed in the previous paragraphs to the particular case of the Hanle effect. In these experiments the excited atoms are subjected to a static external magnetic field B whose direction is chosen as the axis of quantization. In the absence of hyperfine structure the time-independent Hamiltonian for the system becomes... 

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