Diabatic basis set

The components of the two vectors ( 1 i 2X when multiplied by the electronic (diabatic) basis set ( cj>i), 14b)), form the corresponding electronic adiabatic basis... 

In chemical reactions there is an electronic reordering in which some bonds are broken to form new ones. A full description of a chemical process thus requires the understanding of the electronic change involved since it will determine the main forces appearing along the process. Using the electronic states of reactants and products as a diabatic basis set representation, the reactions take place when... 

If simultaneously W k = 0 and <

= 0 in a region of M, then the nuclei move in the field of force the potential of which is given by the energy Wm of a single state of the electronic subsystem and there is no difference between the description of the electronic subsystem in the adiabatic or diabatic basis sets, i.e., Wam = W, . The corresponding behavior of the polyatomic system is referred to as electronically adiabatic ... 

The dimension of the intersections of potential energy surfaces in the diabatic basis set may be larger than or equal to that of the adiabatic surfaces. 

A certain electronic transition in a polyatomic system can be equally described either within the adiabatic or within the diabatic basis set as the transitions can be induced either by the dynamic or by the static coupling,... 

Here // m(R) are matrix elements of Cfc calculated in the diabatic basis set. 

The Hamiltonian in Eq. [26] is usually referred to as the diabatic representation, employing the diabatic basis set <1), Hamiltonian matrix is not diagonal. There is, of course, no unique diabatic basis as any pair obtained from (]), by a unitary transformation can define a new basis. A unitary transformation defines a linear combination of cj) and < >b which, for a two-state system, can be represented as a rotation of the (]), basis on the angle /... 

There are several fundamental reasons why the GMH and adiabatic formulations are to be preferred over the traditionally employed diabatic formulation. The definition of the diabatic basis set is straightforward for intermolecular ET reactions when the donor and acceptor units are separated before the reaction and form a donor-acceptor complex in the course of diffusion in a liquid solvent. The diabatic states are then defined as those of separate donor and acceptor units. The current trend in experimental design of donor-acceptor systems, however, has focused more attention on intramolecular reactions where the donor and acceptor units are coupled in one molecule by a bridge.The direct donor-acceptor overlap and the mixing to bridge states both lead to electronic delocalization, with the result that the centers of electronic localization and localized diabatic states are ill-defined. It is then more appropriate to use either the GMH or adiabatic formulation. 

The time evolution of the electronic wave function can be obtained in the adiabatic or in the diabatic basis set. At each time step, one evaluates the transition probabilities between electronic states and decides whether to hop to another siu-face. When hopping occurs, nuclear velocities have to be adjusted to keep the total energy constant. After hopping, the forces are calculated from the potential of the newly populated electronic state. To decide whether or not to hop, a Monte Carlo technique is used Once the transition probability is obtained, a random number in the range (0,1) is generated and compared with the transition probability. If the munber is less than the probability, a hop occurs otherwise, the nuclear motion continues on the same surface as before. At the end of the simulation, one can analyze populations, distribution of nuclear geometries, reaction times, and other observables as an average over all the trajectories. 

A related question is whether there would be differences in reactivity between different fine structure states of O( P) due to the accidental cancellation that causes state 1 to be largely decoupled from state 4. The presence or absence of this effect in 3D TSH calculations would be crucially dependence on the underlying basis set (adiabatic or diabatic) used. For example, this would not show up if the diabatic basis set that we used to set up our problem were used, but it should in an adiabatic basis. However, the signatures of this effect are subtle, as it mostly leads to a suppression of reactivity in state 2 compared to state 1, but there is already suppression close to threshold due to the thicker barrier. 

A diabatic basis set can be defined as eigenfunctions of the quantum subspace Hamiltonian for one fixed value Zq ... 

Note that is the vector matrix of derivative couplings in the adiabatic electronic basis and the gauge transformation (R) is the unitary transformation matrix connecting the adiabatic and diabatic basis sets. In the above example of two real electronic states, Eq. (32) is identical to Eq. (30a) where x is set to zero ... 

The above shows that, in the Condon approximation, which is rigorously valid in the purely diabatic basis set adopted in Eqs. (1) and (2), the total... 

To solve this equation one needs an integrable expression for both / 12(f) and for D t). We used for the former the well-known Rabi expression which is valid for constant potential coupling Hu and for a constant energy gap. Note that the Rabi expression is derived with a quasi-diabatic basis set. Therefore the coupling appears through the potential coupling term (and X>i,2 = 0). 

However, irrespective of the method used to treat the collision dynamics, there remains a fundamental difficulty with the molecular model, arising from the incapacity of a finite adiabatic (or diabatic) basis set to represent correctly the asymptotic conditions. In practice, the effect (the so-called translation effect) is not serious at low eV energies when the transition is well localized at the curve crossing. 

As it is well known (see, e.g.. Chapter 8), the F functions diverge when two adiabatic PESs touch each other (as in conical intersections). To avoid numerical problems, it is then often convenient to look for diabatic electronic states, exhibiting a very smooth dependence on nuclear coordinates (ideally no dependence) in such a way that the off-diagonal couplings involving derivatives are very small (ideally null). They are replaced by a potential energy coupling Vy(Q). In the diabatic basis set Eq. 10.16 then... 

By analogy with the elimination of the electronic couplings which leads to the definition of the adiabatic states, one may follow Smith and define the diabatic basis set as one where the derivative couplings vanish. Because of equation (8), we... 

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