The diabatic representation

The diabatic derivative coupling matrix can be expressed in the adiabatic representation as 

Assuming that a suitable quasidiabatic representation has been derived, the TDSE in the diabatic representation can be written in matrix form as 

Formally, we can write the unitary transformation from the adiabatic to the diabatic basis as 

Because it is computationally more convenient, most researchers choose the diabatic rather than the adiabatic representation if they try to fit experimental data. In addition to the two diagonal elements of the potential matrix, and V22 5 one merely needs a third potential (surface) which provides the coupling between the diabatic states (Shapiro 1986 Guo and Schatz 1990a,b Dixon, Marston, and Balint-Kurti 1990). 

So far we have invoked the time-independent formulation to describe electronic transitions. In the same manner as described in Section 4.1 we can also derive the time-dependent picture of electronic transitions, using either the adiabatic or the diabatic representation. In the following we feature the latter which is more convenient for numerical applications (Coalson 1985, 1987, 1989 Coalson and Kinsey 1986 Heather and Metiu 1989 Jiang, Heather, and Metiu 1989 Manthe and Koppel 1990a,b Broeckhove et al. 1990 Schneider, Domcke, and Koppel 1990 Weide, Staemmler, and Schinke 1990 Manthe, Koppel, and Cederbaum 1991 Heumann, Weide, and Schinke 1992). 

The set of coupled equations (15.11) represents an example of time-dependent close-coupling as described in Section 4.2.3. It is formally equivalent to (4.25), for example, and can be solved by exactly the same numerical recipes. The dependence on the two stretching coordinates R and r is treated by discretizing the two nuclear wavepackets on a two-dimensional grid and the Fourier-expansion method is employed to evaluate the second-order derivatives in R and r. If we additionally include the rotational degree of freedom, we may expand each wavepacket in terms of 

In this section we will elucidate the influence of nonadiabatic coupling on the dissociation dynamics for two systems which have been extensively studied both by experiment and by theory in the last decade CH3I and H2S. 

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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,... 

In [66], we have reported inelastic and reactive transition probabilities. Here, we only present the reactive case. Five different types of probabilities will be shown for each transition (a) Probabilities due to a full tri-state calculation carried out within the diabatic representation (b) Probabilities due to a two-state calculation (for which T] = 0) performed within the diabatic representation (c) Probabilities due to a single-state extended BO equation for the N = 3 case (to, = 2) (d) Probabilities due to a single-state extended BO equation for the N = 2 case (coy =1) (e) Probabilities due to a single-state ordinary BO equation when coy = 0. 

By substituting the expression for the matrix elements in Eq. (B.21), we get the final form of the Schrddinger equation within the diabatic representation... 

H3 (and its isotopomers) and the alkali metal triiners (denoted generally for the homonuclears by X3, where X is an atom) are typical Jahn-Teller systems where the two lowest adiabatic potential energy surfaces conically intersect. Since such manifolds of electronic states have recently been discussed [60] in some detail, we review in this section only the diabatic representation of such surfaces and their major topographical details. The relevant 2x2 diabatic potential matrix W assumes the fomi... 

Appendix C On the Single/Multivaluedness of the Adiahatic-to-Diahatic Transformation Matrix Appendix D The Diabatic Representation Appendix E A Numerical Study of a Three-State Model Appendix F The Treatment of a Conical Intersection Removed from the Origin of Coordinates Acknowledgments References... 

In the previous section, we discussed the calculation of the PESs needed in Eq. (2.16a) as well as the nonadiabatic coupling terms of Eqs. (2.16b) and (2.16c). We have noted that in the diabatic representation the off-diagonal elements of Eq. (2.16a) are responsible for the coupling between electronic states while Dp and Gp vanish. In the adiabatic representation the opposite is true The off-diagonal elements of Eq. (2.16a) vanish while Du and Gp do not. In this representation, our calculation of the nonadiabatic coupling is approximate because we assume that Gp is negligible and we make an approximation in the calculation of Dp. (See end of Section n.A for more details.)... 

Figure 20. The (So —> S2) absorption spectrum of pyrazine for reduced three- and four-dimensional models (left and middle panels) and for a complete 24-vibrational model (right panel). For the three- and four-dimensional models, the exact quantum mechanical results (full line) are obtained using the Fourier method [43,45]. For the 24-dimensional model (nearly converged), quantum mechanical results are obtained using version 8 of the MCTDH program [210]. For all three models, the calculations are done in the diabatic representation. In the multiple spawning calculations (dashed lines) the spawning threshold 0,o) is set to 0.05, the initial size of the basis set for the three-, four-, and 24-dimensional models is 20, 40, and 60, and the total number of basis functions is limited to 900 (i.e., regardless of the magnitude of the effective nonadiabatic coupling, we do not spawn new basis functions once the total number of basis functions reaches 900).

We have used the above analysis scheme for all single- and two-surface calculations. Thus, when the wave function is represented in polar coordinates, we have mapped the wave function, 4,ad(, t) to Tatime step to use in Eq. (17) and as the two surface calculations are performed in the diabatic representation, the wave function matrix is back transformed to the adiabatic representation in each time step as... 

When the coupling f is much larger than kT, the diabatic representation is no longer valid. The quantum treatment cannot be limited to... 

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

A measure for the electronic coherence of the wave function are the nondiagonal elements of the electronic density matrix, which, for example, in the diabatic representation are given by (k k )... 

Sometimes it is useful to employ a diabatic representations for the fast variable quantum states, rather than the adiabatic representation. In this work we define a diabatic representation as one for which < /j V /i > = 0, where the superscript d indicates the fast variable states in the diabatic representation, There are off-diagonal matrix elements of the fast variable Hamiltonian, Vy(r) = < rf /j >, in this representation. In contrast, the off-diagonal elements of If are all zero in the adiabatic representation, since the /j are eigenfimction of in this case. 

The theoretical description of photochemistry is historically based on the diabatic representation, where the diabatic models have been given the generic label desorption induced by electronic transitions (DIET) [91]. Such theories were originally developed by Menzel, Gomer and Redhead (MGR) [92,93] for repulsive excited states and later generalized to attractive excited states by Antoniewicz [94]. There are many mechanisms by which photons can induce photochemistry/desorption direct optical excitation of the adsorbate, direct optical excitation of the metal-adsorbate complex (i.e., via a charge-transfer band) or indirectly via substrate mediated excitation (e-h pairs). The differences in these mechanisms lie principally in how localized the relevant electron and hole created by the light are on the adsorbate. 

M. Lombardi What is not needed is the validity of the adiabatic approximation, that is, that there is no transition between adiabatic states. But the geometric phase is defined by following states along a path in parameter space (here nuclear coordinates) with some continuity condition. In the diabatic representation, there is no change of basis at all and thus the geometric phase is identically zero. Do not confuse adiabatic basis (which is required) and adiabatic approximation (which may not be valid). 

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