Vibronic coupling, diabatic representation

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]. 

Density functional theory, direct molecular dynamics, complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 404-411 Density operator, direct molecular dynamics, adiabatic systems, 375-377 Derivative couplings conical intersections, 569-570 direct molecular dynamics, vibronic coupling, conical intersections, 386-389 Determinantal wave function, electron nuclear dynamics (END), molecular systems, final-state analysis, 342-349 Diabatic representation ... 

To apply the mapping formalism to vibronically coupled systems, we identify the / ) with electronic states and the h m with operators of the nuclear dynamics. Hereby, the adiabatic as well as a diabatic electronic representation may be employed. In a diabatic representation, we have [cf. Eq. (1)]... 

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. 

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 we saw in Sec. 2.1, CW spectra can be calculated by time-independent (Tl) or time-dependent (TD) methods. In an adiabatic electronic representation, the vibronic couplings diverge along the Cl locus. Therefore, a diabatic electronic representation strongly simplifies the calculations, coupling the electronic species with well-behaved potential terms. 

In Chap. 2, the basic concepts relevant for the description of photochemical processes are presented. The molecular Schrddinger equation and the Bom-Oppenheimer approximation are first introduced. Then, the notions of vibronic coupling and conical intersection are discussed and the diabatic representation for the electronic states is introduced. Finally, a review of the methodology used in this thesis for molecular electronic structure calculations, and their use in the exploration of potential energy surfaces, is presented. 

We have constructed several linear vibronic coupling model Hamiltonians augmented with diagonal quadratic terms for the non-totally symmetric modes. The total Hamiltonian of the molecule in the diabatic representation reads... 

A linear vibronic coupling model Hamiltonian [33], augmented with a diagonal quadratic term along the vioa mode [31] is adopted for the molecular Hamiltonian. Its matrix representation in the basis of the diabatic electronic states reads... 

Divergent couplings ai e a nuisance for the computational treatment of the nuclear dynamics. In cases of exact or near degeneracy of electronic potential-energy surfaces it is therefore preferable to introduce an alternative electronic representation, the so-called diabatic (or quasi-diabatic) representation, which avoids singular coupling elements. The basic concept of diabatic states has been introduced in early descriptions of atomic collision processes and vibronic-coupling phenomena in molecular spectroscopy. ... 

The coupled equations in the diabatic representation, equation (6), provide Ae point of departure for the present discussion of vibronic dynamics in polyatomic systems. The diabatic representation reflects more clearly than the Born-Oppenhei-mer adiabatic representation the essential physics of curvecrossing problems and is thus an essential tool for the construction of appropriate model Hamiltonians. 

Let us start with the Hamiltonian for an N-mode system described by the quadratic vibronic coupling model. For a two state conical intersection in the diabatic representation it has the form... 

The enhancement of the total S value by the CT state coupling can be understood using the simplest possible model of the system two excited electronic states and a single vibrational mode. The first state [Qy exciton state) is assumed to carry all the oscillator strength while the second state,(CT state) is dark. In the diabatic representation, the effective Hamiltonian in the linear vibronic coupling model of the excited state surface can be written as " " ... 

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