Diabatic vibronic Hamiltonian

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]... 

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. 

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]... 

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,... 

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... 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

X + P )/4, which by construction varies between 0 (system is in /i)) and 1 (system is in /2)). Describing, as usual, the nuclear motion through the position X, the vibronic PO can then be drawn in the (A dia,v) plane. Here, the subscript dia emphasizes that we refer to the population of the diabatic states which are used to define the molecular Hamiltonian H. For inteipretational purposes, on the other hand, it is often advantageous to change to the adiabatic electronic representation. Introducing the adiabatic population A ad. where Nad = 0 corresponds to the lower and A ad = 1 to the upper adiabatic electronic state, the vibronic PO can be viewed in the (Nad,x) plane. Alternatively, one may represent the vibronic PO as a curve N d i + (1 — A ad)IFi between the... 

In the following, we summarize the pertinent results of our analysis of Refs. [50-53] where we applied the LVC Hamiltonian Eq. (1) in conjunction with a 20-30 mode phonon distribution composed of a high-frequency branch corresponding to C=C stretch modes and a low-frequency branch corresponding to ring-torsional modes. In all cases, the parametrization of the vibronic coupling models is based on the lattice model of Sec. 3.1 and the complementary diabatic representation of Sec. 3.2. 

Diabatic electronic states (previously termed crude adiabatic states ) are defined as slowly varying functions of the nuclear geometry in the vicinity of the reference geometry [9-11]. The final vibronic-coupling Hamiltonian is obtained by adding the nuclear kinetic-energy operator which is assumed to be diagonal in the diabatic representation. 

Diabatic states are obtained from a similar approach, except that additional term (or terms) in the Hamiltonian are disregarded in order to adopt a specific physical picture. For example, suppose we want to describe a process where an electron e is transferred between two centers of attraction, A and B, of a molecular systems. We may choose to work in a basis of vibronic states obtained for the e-A system in the absence of e-B attraction, and for the e-B system in the absence of the e-A attraction. To get these vibronic states we again use a Born-Oppenheimer procedure as described above. The potential surfaces for the nuclear motion obtained in this approximation are the corresponding diabatic potentials. By the nature of the approximation made, these potentials will correspond to electronic states that describe an electron localized on A or on B, and electron transfer between centers A and B implies that the system has crossed from one diabatic potential surface to the other. 

The concept of diabatic electronic states plays an important role in the theoretical description of strongly vibronically coupled systems. Contrary to the usual adiabatic electronic states, they are not eigenstates of the electronic Hamiltonian which thus acquires off-diagonal matrix elements in... 

The computation of the inter-state coupling constants being defined as first derivatives of off-diagonal elements of the electronic Hamiltonian in the diabatic representation (see Eq. (11)), appears at first sight to be more difficult. It can be shown, however, by analj ng the adiabatic PE functions associated with the vibronic-coupling Hamiltonian (13) that the A -"" can be determined from second derivatives of the adiabatic energies with respect to the nontotally symmetric coordinate Qj. For an electronic two-state system, the following simple formula results ... 

As mentioned above, vibronic-coupling model Hamiltonians constructed by low-order Taylor expansions of the diabatic PE functions in terms of normal coordinates are particularly suitable for the calculation of low-resolution spectra of polyatomic molecules. In resonance Raman spectroscopy, for example, the usually extremely fast electronic dephasing in polyatomic s tems limits the time scale for the exploration of the excited-state PE surface by the nuclear wave packet to about 10 jjj... 

The vibronic problem of the Hamiltonian (23) is easily solved in closed form if either A i = 0 or A 2 = 0 in this case the potential surfaces (24), taken as diabatic surfaces, represent displaced harmonic oscillators and lead to a Poisson distribution for the vibronic line structure of electronic spectra. 

A very useful starting point for the study of non-adiabatic processes, which are common in photochemistry and photophysics, is the vibronic coupling model Hamiltonian. The model is based on a Taylor expansion of the potential surfaces in a diabatic electronic basis, and it is able to correctly describe the dominant feature resulting from vibronic coupling in polyatomic molecules a conical intersection. The importance of such intersections is that they provide efficient non-radiative pathways for electronic transitions. Not only is the position and shape of the intersection described by the model, but it also predicts which nuclear modes of motion are coupled to the electronic transition which takes place as the system evolves through the intersection. 

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