Effective Hamiltonians relation

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

Applying the adiabatic approximation, we restrict the representation of the Hamiltonian to the reduced base (89). Within this base, the Hamiltonian that describes an undamped H bond involving a Fermi resonance may be split into effective Hamiltonians whose structure is related to the state of the fast and bending modes ... 

The effective Hamiltonian /7 °f, related to the ground states 0 ) and [0]) of the fast and bending modes, is the Hamiltonian of a quantum harmonic oscillator characterizing the slow mode ... 

We shall give here a brief summary of our previous work [71,72] that was concerned with the introduction of the relaxation phenomenon within the adiabatic treatment of the Hamiltonian (77), as was done in the undamped case by Witkowski and Wojcik [74]. Following these authors, we applied the adiabatic approximation and then we restricted the representation of the Hamiltonian to the reduced base (89). Within this base, the Hamiltonian that describes a damped H bond involving a Fermi resonance may be split into effective Hamiltonians whose structure is related to the state of the fast and bending modes ... 

In light of the above comments, the effective Hamiltonian Hc for the chemisorbed system is related to the pre-chemisorption Hamiltonian Hs of (6.41) by... 

Dunham achieved inter-relations between term coefficients l i through use of an intermediate radial function V(J ) in an effective hamiltonian for motion of atomic nuclei of this form. 

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

Obviously, the situation will be more difficult when additional approximations have to be employed, e.g., the divide-and-conquer scheme, an effective Hamiltonian or a perturbation approach. A central open question is which electronic states of the fragments have to be included so that reliable results, as compared with supermolecular calculations, can be provided. In any case, accounting for just one state per base pair will yield only semi-quantitative results. A careful analysis of this point is highly desirable for DNA-related systems. 

Substituting this last relation into the Hamiltonian Eq. 17 we obtain an effective Hamiltonian for stationary states. The fact that this Hamiltonian and that of Eq. 10 represent the same system requires that there be a relation between g - on the one hand and and k on the other. Requiring also that because Q is a screening charge, we can obtain this relation. 

In ESR spectroscopy the terms in the effective Hamiltonian are typically expressed by virtue of effective coupling matrices or tensors, whereas in this review we shall relate them to their corresponding terms in the Breit-Pauli Hamiltonian. The effective coupling matrices parametrize the electronic structure of the molecule under study and can be calculated from the Breit-Pauli Hamiltonian by employing a suitable representation of the molecular wave function. 

The proposed approach to static cooperative Jahn-Teller effect is based on the exact effective Hamiltonian (7), acting in the reduced space of active one-centre distortions only. It involves effective force constants, which are analytically related to the parameters of the full microscopic Hamiltonian. Direct electronic interactions between sites, such as orbital-dependent electrostatic and exchange interactions [28], can be added to the effective Hamiltonian without modifying it. This approach proves to be especially efficient in the case of strong Jahn-Teller distortions, when the effects of second-order Jahn-Teller coupling become important. 

In this new representation /// given by Eq. (C.l), the effective Hamiltonians of the slow mode (46) corresponding to the ground state 0 ) of the fast mode is unmodified. On the other hand, the slow mode effective Hamiltonian (47), related to the situation where the fast mode has jumped into its first excited state 1 ), becomes diagonal. The II and III representations of these Hamiltonians are given in Table VI and the details of the calculations are reported in Appendix C. 

Next, it may be seen that the effective Hamiltonians in representations /// are related to those in representation II by equations similar to those given by Eqs. (C.3) and (C.4),... 

On the other hand, the slow mode effective Hamiltonian (47), related to the situation where the fast mode has jumped into its first excited state 1 ), becomes... 

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