Derivative coupling

We now return to the nuclear BO Eq. (91) in the molecular context. Consider the derivative coupling term in it, having the form... 

D. Second-Derivative Coupling Matrix TIT. Adiabatic-to-Diabatic Transformation... 

W (Rj.) is an n X n diabatic first-derivative coupling matrix with elements defined using the diabatic electronic basis set as... 

Requiring l/f (r qx) to be real, the matrix W (Rx) becomes real and skew-symmetiic (just like its adiabatic counterpart) with diagonal elements equal to zero. Similarly, W (Rx) is an n X u diabatic second-derivative coupling matrix with elements defined by... 

In the -electronic-state adiabatic representation involving real electronic wave functions, the skew-symmetiic first-derivative coupling vector mahix... 

As an example, in a four-electronic-state problem (n = 4) consider the electronic states i = 2 and f = 4 along with the first-derivative coupling vector element Wj4 (Rl) that couples those two states. The ADT matrix ui.4(qx) can... 

We want to choose the ADT matiix U(qx) that either makes the diabatic first-derivative coupling vector matrix W (Rx) zero if possible or that minimizes its magnitude in such a way that the gradient term Vr. x (Rx) in... 

The ADT matrix U(q ) obtained in this way makes the diabatic first-derivative coupling matrix that appears in the diabatic Schrodinger... 

Nevertheless, the residual first-derivative coupling term Vr does not... 

A perfect diabatic basis would be one for which the first-derivative coupling in Eq. (31) vanishes [10]. From the above mentioned... 

The ADT matrix for the lowest two electronic states of H3 has recently been obtained [55]. These states display a conical intersection at equilateral triangle geometi ies, but the GP effect can be easily built into the treatment of the reactive scattering equations. Since, for two electronic states, there is only one nonzero first-derivative coupling vector, w5 2 (Rl), we will refer to it in the rest of this... 

Figure 3. Transverse (nonremovable) pare of the ab initio first-derivative coupling vector.

Figure 4. Same as Figure 3 for transverse (nonremovable) part of the ab initio 6rst-derivative coupling vector 6, obtained using the all-Dirichlet boundary conditions. 

Fig re 5. Second-derivative coupling term defined at the end of Section TTI.D for... 

Since the second-derivative coupling matrix is only an additive teiin in Eq. (87), we can merge it with the diabatic energy matrix and define a 2 x 2 diabatic matrix... 

Assuming that the diabatic space can be truncated to the same size as the adiabatic space, Eqs. (64) and (65) clearly define the relationship between the two representations, and methods have been developed to obtain the tians-formation matrices directly. These include the line integral method of Baer [53,54] and the block diagonalization method of Pacher et al. [179]. Failure of the truncation assumption, however, leads to possibly important nonremovable derivative couplings remaining in the diabatic basis [55,182]. 

From Eqs. (30a)-(30c), the singularity in as the conical intersection is approached, is of order 1/p. Only /7, (n= 0, local diabatic representation that removes the singularity [10]. 

By using Eq. (A. 17), the first derivative coupling for the complex electronic wave function assumes the fomi [4]... 

We follow Thompson and Mead [13] to discuss the behavior of the electronic Hamiltonian, potential energy, and derivative coupling between adiabatic states in the vicinity of the D31, conical intersection. Let A be an operator that transforms only the nuclear coordinates, and A be one that acts on the electronic degrees of freedom alone. Clearly, the electronic Hamiltonian satisfies... 

---