From the Hamiltonian

The eigenvalues E of It can be detemiined from the Hamiltonian matrix by solving the secular equation... 

In order to better understand the origin of the first term in (5.59) we separate from the Hamiltonian the part proportional to and average it over the equilibrium oscillators. This gives rise to an effective tunneling splitting A n,... 

The equations of motion obtained from the Hamiltonian (5) at a fixed value of the scaled energy are equivalent to ones obtained from the Hamiltonian (8). 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

It should be stressed that in this first group of works (1956-1970) there is nowhere any deviation from the Hamiltonian laws of dynamics. This feature was taken as a preliminary postulate of our work. Irreversibility appeared as an asymptotic property of the evolution of certain classes of systems. The term asymptotic refers to the large size of the system, as well as to the long time scale of observation. 

We start from the Hamiltonian (in normal order with respect to the genuine vacuum)... 

Standard Green s function techniques are used in the following [46] to describe the dynamics of the protons and the ionic displacements. The equations of motions for the retarded Green s functions [[A (q) S (q))) are obtained from the Hamiltonian Eq. 1 where the operator A denotes 0p, 0k> u or... 

It is important to understand why the J R) and K R) integrals have the sizes they do. We consider J R) first. As we have seen from Eq. (2.42), E(l, 2) is the sum of four different Coulombie terms from the Hamiltonian. If these are substituted into Eq. (2.39), we obtain... 

Of course, the function G (s) does not contain any new information in addition to g(s). The reason that two physically equivalent quantities have been introduced, is that there are two approaches to the dynamics of charge transfer, as explained earlier either one starts from the OLE (when (t) is the observable and G(s) is the fundamental quantity) or one starts from the Hamiltonian Eq. (36), when the coupling g(s) is the fundamental quantity. The work of Voth and collaborators " gives a strong indication that these two approaches are equivalent, as in the case of the position-independent friction. 

Thus it is necessary to study the stability of the state in the G well. To obtain the energy functional it is convenient to start from the Hamiltonian given by Eqs. 1-3. Assuming a stationary solution ijf t)=il/ exp -iEt/1t), and substituting Eq. 5 into the Hamiltonian, we obtain the energy functional... 

From the Hamiltonian Eqs. 1-3 we derive the equations of motion of the polaron ... 

In this way the mass polarization term may be removed from the Hamiltonian. However, the resulting electronic wave functions which are obtained are then dependent upon the nuclear masses as well as the nuclear charges and such wave functions are an inconvenient basis from which to investigate nonadiabatic processes. 

The equilibrium electronic energy is a constant, and we shall drop it from the Hamiltonian this does not affect the eigenfunctions of (6.44) and simply decreases the eigenvalues by Ue (Problem 1.6). We write the vibrational Hamiltonian as... 

While keeping in mind the general picture of nuclear relaxation in paramagnetic systems as described in Section 3.1, it is appropriate to consider first the simple case of dipolar coupling between two point-dipoles as if the unpaired electrons were localized on the metal ion. The enhancement of the nuclear longitudinal relaxation rate Rim due to dipolar coupling with unpaired electrons can be calculated starting from the Hamiltonian for the system ... 

As we have seen in the previous section, Jordan blocks appear easily in an ever so slight non-Hermitean extension of Quantum Mechanics [19]. The success in both atomic as well as molecular applications has also been noted [20-22]. Non-trivial extensions from the Hamiltonian to the Liouville picture was moreover soon realised [23]. 

The second order electronic density matrix can be constructed from the Hamiltonian matrix eigenvector given in eq(5) above to give... 

To conclude, after the canonical transformation we have two equivalent models (1) the initial model (145) with the eigenstates (160) and (2) the fictional free-particle model (154) with the eigenstates (158). We shall call this second model polaron representation. The relation between the models is established by (155)-(157). It is also clear from the Hamiltonian (148), that the operators < , d. ad. and a describe the initial electrons and vibrons in the fictional model. 

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