Hamiltonian correlation crystal field

The one-electron crystal field Hamiltonian does not take into account electron correlation effects. For some systems, it has been useful to augment the crystal field Hamiltonian with additional terms representing the two-electron, correlated crystal field. The additional terms most commonly used (see, for example, Peijzel et al., 2005b Wegh et al., 2003) are from the simplified delta-function correlation crystal field model first proposed by Judd (1978) that assumes electron interaction takes place only when two electrons are located at the same position (hence the name delta-function ). This simplified model, developed by Lo and Reid (1993), adds additional terms, given as,... 

The correlation crystal-field extension to the Hamiltonian becomes more important when many high-energy states are observed. For the 4P 4f transitions, many new energy lev-... 

The term correlated crystal field designates the two-electron part of the crystal field Hamiltonian. Although a concept of the 1970s, correlation is a keyword of the late 1980s and 1990s. 

Searching along the lines of selective modification of the radial wavefunctions, Newman (1970) and Judd (1977b) found another source of correlated crystal field in the exchange forces between 4f electrons with similarly directed spins. They introduce a correction of the type s, S (where s,- is the zth electron spin and S the total spin) as a multiplicative factor to apply to the operators. The effective crystal field Hamiltonian therefore becomes ... 

There is a remarkable discontinuity in magnitude of the crystal field parameters, especially the sixth-rank parameters (k = 6), between the first and second half of the lanthanide series in systems like LaFs and LaCb (Camall et al. 1989). According to Judd (1979) the drop in the k = 6 parameters in going Ifom LaCl3 Eu to LaCbiTb " is an indication for the need to include two-electron operators in the crystal-field Hamiltonian (which is done in correlation crystal-field theory, see sect. 4.5). 

To remove these discrepancies, the Hamiltonian is extended to account for the effects of the electron correlation on the anisotropic crystal-field interactions in the 4f shell. Two-body operators are necessary to describe the correlation effects. The expression correlation crystal field is used for the two-electron part of the crystal-field perturbation. A review of the theoretical development of correlation crystal-field theory is given by Garcia and Faucher (1995). A practical problem for taking the correlation effects into account is the enormous number of parameters required. 

A general correlation crystal-field (CCF) parametrization accounts for all possible CCF operators, whereas SCCF and LCCF are restricted models (Newman and Ng 1988, Reid 1992). The Hamiltonian can be extended with the term... 

A new treatment for S = 7/2 systems has been undertaken by Rast and coworkers [78, 79]. They assume that in complexes with ligands like DTPA, the crystal field symmetry for Gd3+ produces a static ZFS, and construct a spin Hamiltonian that explicitly considers the random rotational motion of the molecular complex. They identify a magnitude for this static ZFS, called a2, and a correlation time for the rotational motion, called rr. They also construct a dynamic or transient ZFS with a simple correlation function of the form (BT)2 e t/TV. Analyzing the two Hamiltonians (Rast s and HL), it can be shown that at the level of second order, Rast s parameter a2 is exactly equivalent to the parameter A. The method has been applied to the analysis of the frequency dependence of the line width (ABpp) of GdDTPA. These results are compared to a HL treatment by Clarkson et al. in Table 2. 

Fig. 3.5/ and 6d manifolds in an octahedral field and their correlation with free ion one-electron levels. Note that the crystal field lowers the energy of the 6d levels (relative to 5J) prior to splitting. Bethe notation is used for the Kramers s doublets and Mulliken notation for the spin-free Hamiltonian states, as usual. 

The shortcoming of the mean field method is that it admits no correlation between the motions of the individual particles. This correlation can be introduced by means of the random phase approximation (RPA) or time-dependent Hartree (TDH) method. In order to formulate this method, we introduce excitation operators (Ep) which replace f) p by when applied to the mean field ground state of the crystal when applied to any other state, they yield zero. Then, we write the Hamiltonian as a quadratic form in the excitation operators (Ep)+ and their Hermi-tean conjugates Ep... 

As we can see from this simple but general consideration of a multilevel molecule, nonlinear electronic polarizations occur naturally in all materials illuminated by an optical field. The differences among the nonlinear responses of different materials are due to differences in their electronic properties (wave functions, dipole moments, energy levels, etc.) which are determined by their basic Hamiltonian Hq. For Uquid crystals in their ordered phases, an extra factor we need to take into account are molecular correlations. 

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