Diabatization

Luckhaus D 2000 6D vibrational quantum dynamics generalized coordinate discrete variable representation and (a)diabatic contraction J. Chem. Phys. 113 1329—47... 

When relaxation of the internal motion is slow compared with the fast relative speed v., then T is expanded in temrs of the known unperturbed (diabatic) ortironomral eigenstates j(V> ... 

The Landau-Zener transition probability is derived from an approximation to the frill two-state impact-parameter treatment of the collision. The single passage probability for a transition between the diabatic surfaces H, (/ ) and R AR) which cross at is the Landau-Zener transition probability... 

In the remainder of this section, we will follow this simplifying (and problematic) assumption, and postulate that, upon the adiabatic to diabatic transfonnation, the Scln-ddinger equation has the fomi ... 

To see physically the problem of motion of wavepackets in a non-diagonal diabatic potential, we plot in figure B3.4.17 a set of two adiabatic potentials and their diabatic counterparts for a ID problem, for example, vibrations in a diatom (as in metal-metal complexes). As figure B3.4.17 shows, if a wavepacket is started away from the crossing point, it would slide towards this crossing point (where where it would... 

Figure B3.4.17. When a wavepacket comes to a crossing point, it will split into two parts (schematic Gaussians). One will remain on the same adiabat (difFerent diabat) and the other will hop to the other adiabat (same diabat). The adiabatic curves are shown by fidl lines and denoted by ground and excited die diabatic curves are shown by dashed lines and denoted 1, 2.

The problem of branching of the wavepacket at crossing points is very old and has been treated separately by Landau and by Zener [H, 173. 174], The model problem they considered has the following diabatic coupling matrix ... 

The simplest approach to simulating non-adiabatic dynamics is by surface hopping [175. 176]. In its simplest fomi, the approach is as follows. One carries out classical simulations of the nuclear motion on a specific adiabatic electronic state (ground or excited) and at any given instant checks whether the diabatic potential associated with that electronic state is mtersectmg the diabatic potential on another electronic state. If it is, then a decision is made as to whedier a jump to the other adiabatic electronic state should be perfomied. 

The ultimate approach to simulate non-adiabatic effects is tln-ough the use of a fiill Scln-ddinger wavefunction for both the nuclei and the electrons, using the adiabatic-diabatic transfomiation methods discussed above. The whole machinery of approaches to solving the Scln-ddinger wavefiinction for adiabatic problems can be used, except that the size of the wavefiinction is now essentially doubled (for problems involving two-electronic states, to account for both states). The first application of these methods for molecular dynamical problems was for the charge-transfer system... 

Baer M 1975 Adiabatic and diabatic representations for atom-molecule collisions treatment of the collinear arrangement Chem. Rhys. Lett. 35 112... 

Mead C A and Truhlar D G 1982 Conditions for the definition of a strictly diabatic electronic basis for molecular systems J. Chem. Rhys. 77 6090... 

Sadygov R G and Yarkony D R 1998 On the adiabatic to diabatic states transformation in the presence of a conical intersection a most diabatic basis from the solution to a Poisson s equation. I J. Chem. Rhys. 109 20... 

Here the transition state is approximated by the lowest crossing pomt on the seam intersecting the diabatic (non-interacting) potential energy surfaces of the reactant and product. The method was originally developed... 

McDouall J J W, Robb M A and Bernard F 1986 An efficient algorithm for the approximate location of transition structures in a diabatic surface formalism Chem. Phys. Lett. 129 595... 

The two surface calculations by using the following Hamiltonian matrix ai e rather stiaightfoiTvard in the diabatic representation... 

Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

It is important to note that the two surface calculations will be carried out in the diabatic representation. One can get the initial diabatic wave function matrix for the two surface calculations using the above adiabatic initial wave function by the following orthogonal transformation,... 

At this point, it is important to note that as the potential energy surfaces are even in the vibrational coordinate (r), the same parity, that is, even even and odd odd transitions should be allowed both for nonreactive and reactive cases but due to the conical intersection, the diabatic calculations indicate that the allowed transition for the reactive case ate odd even and even odd whereas in the case of nomeactive transitions even even and odd odd remain allowed. 

If we consider the tiansformation (P = A[Pg.64]

In this section, we prove that the non-adiabatic matiices have to be quantized ( similar to Bohr-Sommerfeld quantization of the angulai momentum) in order to yield a continous, uniquely defined, diabatic potential matrix W(i). In another way, the extended BO approximation will be applied only to those cases that fulfill these quantization rules. The ADT matrix A(s,so) transforms a given adiabatic potential matiix u(i) to a diabatic matiix W(s, so)... 

However, in order to obtain a uniquely defined diabatic potential matrix, it is not necessary for the A matiix to be uniquely defined throughout CS. Still, we ignore this difficulty and go ahead to derive A by a direct integration of Eq. (62),... 

As the D matrix has to be a unit matrix in order to get a continuous, uniquely defined diabatic matrix, the following integral is quantized as ... 

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