The model Hamiltonian

The one- and two-electron integrals appearing in the second-quantized form of the Hamiltonian carry all information about the specific features of the quantum system. The one-electron integrals are defined as 

The two-electron integrals in the Lowdin basis are then assumed to be given by 

In order to solve the eigenvalue problem in Eq.(31) we need to evaluate Hamiltonian matrix elements between the TV-electron basis functions bSM. Combining Eqs.(ll), (12), (14), (16) and (17) we get 

As one can see from Eq.(36), three classes of matrix elements over the orbital variables appear in Eq.(37)  

Matrix elements associated with the Coulomb integrals 7w. These are expressed in terms of the orbital occupation number operators. Since is an eigenfunction of Epp we get 

We report here for completeness of the model Hamiltonian previously introduced in Ref. [1] to describe the physics of the holes in the hu HOMO of C60 fullerene  

The electron-vibron (e-v) couplings griA are conveniently expressed in units of the corresponding harmonic vibron quantum of energy h,coiA. In this calculation we adopt the numerical values of the e-v coupling parameters, listed in Table 1, from the density functional (DF) calculation of Ref. [6], and a second calculation [9] yields couplings in substantial accord with those of Table 1. The numerical factors kA = 51/2, kG = (f)1/2, = 1 in //e v have been introduced for compatibility with the normalization of Ref. [6], 

The classical single-mode JT stabilization energies Es are tabulated for both D5 and D3 distortions, for one hole in the HOMO 

One of the tabulated parameters (e.g. Fx) is a linear combination of the five others 

It should be noted that U differs from the usual definition of the Hubbard XJ, involving the lowest multiplet in each -configuration f/ n = E mn(n + 1) + Elmn(n — 1) — 2 mm(w). This second definition is inconvenient here, since it depends wildly on n. 

---

Equations (16)-(20) show that the real adiabatic eigenstates are everywhere smooth and continuously differentiable functions of Q, except at degenerate points, such that E (Q) — E, [Q) = 0, where, of com se, the x ) are undefined. There is, however, no requirement that the x ) should be teal, even for a real Hamiltonian, because the solutions of Eq. fl4) contain an arbitrary Q dependent phase term, gay. Second, as we shall now see, the choice that x ) is real raises a different type of problem. Consider, for example, the model Hamiltonian in Eq. (8), with / = 0 ... 

If the states are degenerate rather than of different symmetry, the model Hamiltonian becomes the Jahn-Teller model Hamiltonian. For example, in many point groups D and so a doubly degenerate electronic state can interact with a doubly degenerate vibrational mode. In this, the x e Jahn-Teller effect the first-order Hamiltonian is then [65]... 

After integrating over the electronic coordinates and x, the model Hamiltonian (15) is represented by the matrix whose elements are... 

Thus, for a particular value of the good quantum number K, the only possible values for I are /f A. The matrix representation of the model Hamiltonian in the linear basis, obtained by integrating over the electronic coordinates and is thus... 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

Determination of the paiameters entering the model Hamiltonian for handling the R-T effect (quadratic force constant for the mean potential and the Renner paiameters) was carried out by fitting special forms of the functions [Eqs. (75) and (77)], as described above, and using not more than 10 electronic energies for each of the X H component states, computed at cis- and toans-planai geometries. This procedure led to the above mentioned six parameters... 

For the model Hamiltonian used in this study it was assumed that the bond stretching and angle i)ending satisfactorily describe all vibrational motions... 

Above mathematics shows that the changes in the model Hamiltonian (1) that do not involve the exciton-phonon coupling terms, - for instance inclusion the exciton-exciton (electron-electron) interaction, lead only to the respective change of in Eqs.(16). 

The symmetric coupling case has been examined by using Sethna s approximations for the kernel by Benderskii et al. [1990, 1991a]. For low-frequency bath oscillators the promoting effect appears in the second order of the expansion of the kernel in coj r, and for a single bath oscillator in the model Hamiltonian (4.40) the instanton action has been found to be... 

The third term, Uqt, in Eq. (27) is due to the partial electron transfer between an ion and solvents in its immediate vicinity. The model Hamiltonian approach [33], described in Section V, has shown that Uqt (= AW in Ref. 33) per primary solvent molecule, for an ion such as the polyanion, can also be expressed as a function of E, approximately a quadratic equation ... 

The reason for pursuing the reverse program is simply to condense the observed properties into some manageable format consistent with quantum theory. In favourable cases, the model Hamiltonian and wave functions can be used to reliably predict related properties which were not observed. For spectroscopic experiments, the properties that are available are the energies of many different wave functions. One is not so interested in the wave functions themselves, but in the eigenvalue spectrum of the fitted model Hamiltonian. On the other hand, diffraction experiments offer information about the density of a particular property in some coordinate space for one single wave function. In this case, the interest is not so much in the model Hamiltonian, but in the fitted wave function itself. 

A more general description of the effects of vibronic coupling can be made using the model Hamiltonian developed by Koppel, Domcke and Cederbaum [65], The basic idea is the same as that used in Section III.C, that is to assume a quasidiabatic representation, and to develop a Hamiltonian in this picture. It is a useful model, providing a simple yet accurate analytical expression for the coupled PES manifold, and identifying the modes essential for the non-adiabatic effects. As a result it can be used for comparing how well different dynamics methods perform for non-adiabatic systems. It has, for example, been used to perform benchmark full-dimensional (24-mode) quantum dynamics calculations... 

The model Hamiltonian (37) obtained from Eq.(32) contains solute oscillators linearly perturbed by its coupling with the solvent as well as bilinear terms that break down a total separability between solute and solvent ... 

The model Hamiltonian H in Eq.(9) preserves isospin symmetry if the condition... 

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

In what follows we introduce the model Hamiltonian using both diabatic and adiabatic representations. Adopting diabatic electronic basis states /j ), the molecular model Hamiltonian can be written as [162, 163]... 

To specify the model Hamiltonian in the adiabatic representation, we introduce adiabatic electronic states... 

---