Conic intersections symmetry approach

In Chapter VIII, Haas and Zilberg propose to follow the phase of the total electronic wave function as a function of the nuclear coordinates with the aim of locating conical intersections. For this purpose, they present the theoretical basis for this approach and apply it for conical intersections connecting the two lowest singlet states (Si and So). The analysis starts with the Pauli principle and is assisted by the permutational symmetry of the electronic wave function. In particular, this approach allows the selection of two coordinates along which the conical intersections are to be found. 

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. 

When the molecule contains hetero atoms such as nitrogen or oxygen one may want to include also lone-pair orbitals of rr-type in the active space. Note, however, that c —> tt excitations are of another symmetry than tt tt excitations for planar systems. One can therefore often use a different active space for these two types of excitations. The CASSCF method is frequently used to study photochemical processes that involve conical intersections, intersystem crossings, etc. where simpler approaches, as for example, time-dependent (TD) DFT do not work well. Here, one is only interested in the lower excited states of different spin-multiplicities and the demands on the active space are not so high. 

In the present context, the standard BO description corresponds to diagonalizing equation (VIII. 12) in Ref. [25] for all values of the PCB. In contrast, for the GED scheme, these calculations only make sense at the stationary geometries. Our approach makes it apparent that there is no actual physical process associated with the crossings of electronic states occurring within the BO calculations. In contrast, important conical intersections associated with molecular symmetries still find a natural place in the present post-BO approach, as it incorporates the intersections of diabatic potential energy surfaces. 

Here the nonrelativistic (Coulomb, r] = 2) and relativistic (Coulomb - -spin-orbit, = 3) seams of conical intersection are compared. In Cg symmetry, there are 5 symmetry preserving internal coordinates. The seam conditions, the conditions that require X to be a point of conical intersection, define r] internal coordinates. Thus there is a many-to-one association of nonrelativistic and relativistic seam points. This many-to-one association can be reduced to one-to-one association by comparing the lowest energy point of conical intersection for a fixed-value of the H -H distance, — H ), see Fig. 8(a) for atom labelling. Below we show that this approach results in meaningful comparisons. 

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