Conical intersections elements

Elements of the matrix —are usually small in the vicinity of a conical intersection and can be added to to give a corrected diabatic energy matrix. As can be seen, whereas in Eq. (15) contains both the singular... 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

To study the two isolated conical intersections, we have to treat two-state curl equations that are given in Eq. (26). Here, the first 2 x 2 x mahix contains the (vectorial) element, that is, X012 and the second 2 x 2 x mahix contains X023- As before each of the non-adiabatic coupling terms, X012 and X023 has the following components ... 

Figure 11. Results for the C2H molecule as calculated along a circle surroiinding the A -2 A conical intersection. Shown are the geometry, the non-adiabalic coupling matrix elements i(p((p J 2) and the adiabatic-to-diabadc transformation angles y((p J2) as calculated for T] (=CC distance) = 1.35 A and for three values (j 2 is the CH distance) (a) and (i>) = 1.80 A (c) and (tf) = 2.00 A (c) and (/) = 3.35 A. (Note that q = r2.)...

Figure 12, Results for the C2H molecule as calculated along a circle surrounding Che 2 A -3 A conical intersection, The conical intersection is located on the C2v line at a distance of 1,813 A from the CC axis, where ri (=CC distance) 1.2515 A. The center of the circle is located at the point of the conical intersection and defined in terms of a radius < . Shown are the non-adiabatic coupling matrix elements tcp((p ) and the adiabatic-to-diabatic transformation angles y((p i ) as calculated for (ii) and (b) where q = 0.2 A (c) and (d) where q = 0.3 A (e) and (/) where q = 0.4 A. Also shown are the positions of the two close-by (3,4) conical intersections (designated as X34).

The closer the trajectory approaches the conical intersection, the smaller Cy becomes. Since the nonadiabatic transitions are expected to take place in the close vicinity of the conical intersection, the nonadiabatic transition direction can be approximated by the eigenvector of the Hessian d AV/dRidRj corresponding to its maximum eigenvalue. Similar arguments hold for nonadiabatic transitions near the crossing seam surface, in which case the nondiagonal elements of the diabatic Hamiltonian of Eq. (1) should be taken as nonzero constant. 

As stated in the introduction, we present the derivation of an extended BO approximate equation for a Hilbert space of arbitary dimensions, for a situation where all the surfaces including the ground-state surface, have a degeneracy along a single line (e.g., a conical intersection) with the excited states. In a two-state problem, this kind of derivation can be done with an arbitary t matrix. On the contrary, such derivation for an N > 2 dimensional case has been performed with some limits to the elements of the r matrix. Hence, in this sence the present derivation is not general but hoped that with some additional assumptions it will be applicable for more general cases. 

In this diabatic Schrodinger equation, the only terms that couple the nuclear wave functions Xd(R-/v) are the elements of the W RjJ and zd q%) matrices. The —(fi2/2p)W i(Rx) matrix does not have poles at conical intersection geometries [as opposed to W(2 ad(R>.) and furthermore it only appears as an additive term to the diabatic energy matrix cd(q>.) and does not increase the computational effort for the solution of Eq. (55). Since the neglected gradient term is expected to be small, it can be reintroduced as a first-order perturbation afterward, if desired. 

The import of diabatic electronic states for dynamical treatments of conical intersecting BO potential energy surfaces is well acknowledged. This intersection is characterized by the non-existence of symmetry element determining its location in nuclear space [25]. This problem is absent in the GED approach. Because the symmetries of the cis and trans conformer are irreducible to each other, a regularization method without a correct reaction coordinate does not make sense. The slope at the (conic) intersection is well defined in the GED scheme. Observe, however, that for closed shell structures, the direct coupling of both states is zero. A configuration interaction is necessary to obtain an appropriate description in other words, correlation states such as diradical ones and the full excited BB state in the AA local minimum cannot be left out the scheme. 

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