Effective Hamiltonian derivation

If we compare these equations with the projection operator expansion given in equation (7.43), we find that the expressions are identical up to and including the X2 contribution but that the 7.3 term derived here corresponds not to the X3 term in the expansion (7.43) but to its symmetrised (Hermitian) form discussed at the end of section 7.2. Since the discrepancies that arise from these two different forms are of order Xs or higher, the effective Hamiltonians derived by the two methods are identical to order X3. In the literature the Van Vleck transformation is normally implemented by use of equations (7.67) to (7.70) although the X3 contribution (7.70) has often been ignored. 

There is a further term which should be included in the effective Hamiltonian, derived in chapter 7, describing the electron spin-nuclear rotation interaction. This may be written in the form... 

The g-factors also point to the problem of excited state mixing, the values of gL and gs in particular being too far from the free electron values for comfort. The single state effective Hamiltonian, derived by perturbation theory, may be inadequate in molecules where there are several close-lying electronic states which are appreciably mixed. 

While the 0 -theory discussed in section 3.3 does not provide such averages it is essential that these can be performed in the framework of the MH model. With the effective Hamiltonian derived in section 2.4 it turns out that the moments correspond to the propagators of this theory with masses rk that reflect the fact that there is a distinct critical point associated to each moment, i.e. there is no multicritical point as in the spin models with finite numbers of components and as suggested by the d > 4 interpretation of the (j> polymer theory [39] in sect. 3.3. In extracting the scaling behavior of the moments gw or equivalently of the masses r the central quantities will be the terms linear in k in an expansion in A as suggested by Eq. (115). 

It one wants to treat an problem using a bielectronic effective Hamiltonian derived from exact calculations on H2, several questions arise concerning ... 

The total effective Hamiltonian H, in the presence of a vector potential for an A + B2 system is defined in Section II.B and the coupled first-order Hamilton equations of motion for all the coordinates are derived from the new effective Hamiltonian by the usual prescription [74], that is. 

Wesolowski and Warshel197 introduced a DFT based approach in which all short-range terms in the effective Hamiltonian (Eq. 4.25) were derived entirely from density functional theory and were involved in the construction of the Fock matrix195 196. In this approach, the H croEnv is expressed using explicit functionals of the electron density ... 

This procedure follows, in effect, the derivation of Jarzynski s identity in discrete time [2,18], as outlined in Sect. 5.5. Finally, for Hamiltonian dynamics, one can use (5.23) and calculate the work directly from the difference in total energy between trajectory start and end points. 

To this point, the formalism has been quite general, and from here we could proceed to derive any one of several single-site approximations (such as the ATA, for example). However, we wish to focus on the desired approach, the CPA. To do so, we recall that our aim is to produce a (translationally invariant) effective Hamiltonian He, which reflects the properties of the exact Hamiltonian H (6.2) as closely as possible. With that in mind, we notice that the closer the choice of unperturbed Hamiltonian Ho (6.4) is to He, then the smaller are the effects of the perturbation term in (6.7), and hence in (6.10). Clearly, then, the optimal choice for H0 is He. Thus, we have... 

That effective hamiltonian according to formula 29, with neglect of W"(R), appears to be the most comprehensive and practical currently available for spectral reduction when one seeks to take into account all three principal extramechanical terms, namely radial functions for rotational and vibrational g factors and adiabatic corrections. The form of this effective hamiltonian differs slightly from that used by van Vleck [9], who failed to recognise a connection between the electronic contribution to the rotational g factor and rotational nonadiabatic terms [150,56]. There exists nevertheless a clear evolution from the advance in van Vleck s [9] elaboration of Dunham s [5] innovative derivation of vibration-rotational energies into the present effective hamiltonian in formula 29 through the work of Herman [60,66]. The notation g for two radial functions pertaining to extra-mechanical effects in formula 29 alludes to that connection between... 

Turning now back to the single-root MR BWCC approach, we derive the basic equations for the effective Hamiltonian and cluster amplitudes in the spin-orbital form without the use of the BCH formula. We limit ourselves to a complete model space which implies that amplitudes corresponding to internal excitations (i.e. excitations within the model space) are equal to zero. In our derivation we shall work with the Hamiltonian in the normal-ordered-product form, i.e. 

This effective Hamiltonian for the interaction of two magnetic moments may also easily be derived from the one photon exchange diagram in Fig. 8.2. 

Hence, the effective Hamiltonian as relevant in ESR spectroscopy in terms of the occurring interactions (derived by Abragam and Pryce (51)) reads,... 

To illustrate the failure more quantitatively, I might quote the errors of anharmonic potential constants derived from the effective Hamiltonian analysis, which can be wrong by factors of 2-5 in individual cases, and the nature and structure of the basis states of Hea, which can be quite different in reality from the simple analysis based on writing product functions. ... 

Thus, in order to derive quantitatively meaningful wavepacket dynamics of molecular motion (and also quantitative stationary-state wave functions) from the effective Hamiltonian, it is necessary to carry out a further step in the analysis, which is highly nontrivial. We have carried out such analyses for a number of systems and I think an adequate understanding of the problem does exist nowadays and is accepted by the subcommunity most interested in this question. I like to illustrate this in the scheme from molecular spectra to molecular motion [5] (see Scheme 1). 

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