Vibrons electron-vibron Hamiltonian

A convenient definition of second-order vibronic RFs useful in the context of the FC approximation can be found in Ref. [4] based on earlier work [5]. The RFs will be expressed in the form K j rm 0 / ) for electronic perturbations V of symmetries 1 and rm where the symmetry label Af G T 0 /] . The electronic perturbation Hamiltonian within an orbital triplet 7] can be written as ... 

We see that the electron-vibron Hamiltonian (145) is equivalent to the free-particle Hamiltonian (154). This equivalence means that any quantum state IV a), obtained as a solution of the Hamiltonian (154) is one-to-one equivalent to the state ) as a solution of the initial Hamiltonian (145), with the same matrix elements for any operator... 

The full Hamiltonian is the sum of the free system Hamiltonian H the intersystem electron-electron interaction Hamiltonian He, the vibron Hamiltonian Hy including the electron-vibron interaction and coupling of vibrations to the environment (dissipation of vibrons), the Hamiltonians of the leads Hr, and the tunneling Hamiltonian Ht describing the system-to-lead coupling... 

The vibronic Hamiltonian in the one-electron model is H = Hq + V. The kernels of these operators are... 

This two-dimensional potential energy surface is the lower energy solution obtained from the diagonalization of the potential energy operators in the E 0 e vibronic Hamiltonian acting within a ( 0>, >) electronic basis ... 

Computationally, the present approach rests on the QVC coupling scheme in conjunction with coupled-cluster electronic structure calculations for the vibronic Hamiltonian, and on the MCDTH wave packet propagation method for the nuclear dynamics. In combination, these are powerful tools for studying such systems with 10-20 nuclear degrees of freedom. (This holds especially in view of so-called multilayer MCTDH implementations which further enhance the computational efficiency [130,131].) If the LVC or QVC schemes are not applicable, related variants of constructing diabatic electronic states are available [132,133], which may extend the realm of application from the present spectroscopic and photophysical also to photochemical problems. Their feasibility and further applications remain to be investigated in future work. 

The vibronic coupling in the radical and radical cation of aromatic hydrocarbons is studied by photoionizing the corresponding anion and neutral molecules, respectively. The vibronic Hamiltonian of the final states of the ionized species is constructed in terms of the dimensionless normal coordinates of the electronic ground state of the corresponding (reference) anion or neutral species. The mass-weighted normal coordinates ) are obtained by diagonalizing the force field and are converted into the dimensionless form by [68]... 

To analyze the vibronic structures of the X, A and B electronic states Ph we constructed a vibronic Hamiltonian in a diabatic electronic basis which treats the nuclear motion in the X state adiabatically, and includes the nonadiabatic coupling between the A and B electronic states. The Hamiltonian terms of the dimensionless normal coordinates of the electronic ground state (XMi) of phenide anion is given by [19]... 

The model diabatic vibronic Hamiltonian of the Do — Di — D2 electronic manifold can be expressed in terms of dimensionless normal coordinates of N as [20]... 

The Hamiltonian in (36) operates on the ground state vibronic wavefunctions and I g which can be expressed as products of electronic functions and an expansion of the two dimensional vibrational states cpi of appropriate symmetry [41], The linear combinations are found using the E g e and H2 g e vector coupling coefficients (the Hi g e coefficients are of course trivial) following the same procedure as used to construct the vibronic Hamiltonian in (15). It may then be readily shown that for strong linear coupling, p 0 and q 1/2 [10,41]. Note however, for second order coupling q can take values less than 1/2 [42]. 

We shall write the vibrational-electronic (vibronic) Hamiltonian of the system as... 

We have chosen a nonorthogonal bond lenghts-bond angle vibronic Hamiltonian H [6], which has a complicated kinetic term but gives the best convergence and assignment of the NO2 spectrum. The nonadiabatic states In) and levels have been calculated by using diabatic electronic states e) and two vibrational FBR,... 

To illustrate the gauge invariant reference section for MAB, let us revisit the linear + quadratic E< e Jahn-Teller effect, which is known to exhibit a nontrivial MAB structure. There, the symmetry induced degeneracy of two electronic states (E) is lifted by their interaction with a doubly degenerate vibrational mode (e). In the vicinity of the degeneracy point at the symmetric nuclear configuration, this may be modeled by the vibronic Hamiltonian [39]... 

Here f denotes the transition operator and E the energy transferred to the system (e.g. the photon energy in optical absorption). The initial state l kj) (with energy Ei) can, but need not, belong to the set ) of vibronic states (with energies E ) which constitute the set of final states given by the solutions of the vibronic Hamiltonian. In the applications to be discussed below, l i) will be the zero vibrational level of the electronic ground state which is assumed to be vibronicaUy uncoupled from the excited states. 

Evolving under the influence of the vibronic Hamiltonian H, the time-dependent wavepacket 14 (t)) will change in position, shape and electronic composition. A generally used, though necessarily incomplete, measure of these changes is provided by the so-called autocorrelation function (see, for example. Refs. 53 and 54)... 

As discussed in Section 8.2.1, when nonadiabatic couplings cannot be neglected, the BO approximation is not reliable and coupled electronic states must be considered simultaneously with their interactions. For small systems, several full-dimensional approaches based on the vibronic or spin-rovibronic wavefunctions and taking into account simultaneously at least two electronic states have been developed [2, 100-104]. To quote some examples, the full vibronic Hamiltonians have been derived and employed for linear tetra-atomic molecules showing Renner-Teller interactions [103] or CXaY-like molecules of Csv symmetry showing Jahn-Teller interactions [104]. In the following, we will present the computational approaches based on the full rovibronic Carter-Handy Hamiltonian [100], developed for triatomic molecules and expressed in internal coordinates, which allows us to take into account up to three interacting electronic states [2, 100, 101]. 

Figure 10.2 Absorption spectrum of adenine dimer (blue dashed line) and monomer (red solid line) obtained at pure electronic level (a) and at vibronic level (b) by adopting the vibronic Hamiltonian discussed in Section 10.3.1.3. It has been computed from the Fourier transform of the autocorrelation function obtained propagating a doorway state. The latter is a delocalized exciton state obtained mixing the two localized exciton states with equal weights.

A special situation is encountered in the formation of a K-shell vacancy in systems with several equivalent corehole sites.Owing to the localization of the core orbitals in space, there will always exist several near-degenerate electronic states which can interact through vibrational modes of suitable symmetry. In this case, however, the vibronic Hamiltonian can be diagonalized by transforming to a suitable diabatic representation. These diabatic electronic states correspond to core holes localized on the equivalent sites. From the dynamical point of view, we are dealing here with a multidimensional weakly avoided crossing. From the structural... 

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