Intersection coordinate subspace

The expression avoided state touching (crossing) can refer to one of two situations (Salem et al., 1975). First, an avoided touching can be encountered along paths that pass close to a real state touching but miss the tip of the cones and do not quite reach the required values of Xy and Xj to enter the (F - 2)-dimensional intersection coordinate subspace. An example would be a path that does not quite preserve the high symmetry that is characteristic of some conical intersections. Second, it can be encountered upon improvement of a simpler quantum mechanical approximation in which a... 

It is instructive to examine further the approximate semi-classical form for R7 shown above because, when viewed as a rate of transition between two intersecting energy sur ces, one anticipates that connection can be made with the well known Landau-Zener theory (10). For a non-linear molecule with N atoms, the potentials (Q) depend on 3N-6 internal degrees of fi eedom (for a linear molecule, Vj f depend on 3N-5 internal coordinates). The subspace S... 

The general statements regarding the transversality of the intersections of submanifolds are stated in terms of the properties of the intersections of their associated tangent hyperplanes. These are vector subspaces, a plane or line spanned by a corresponding set of gradient vectors, not necessarily at the origin of the coordinate system. Such a vector subspace is called an affine subspace. 

In addition to the analysis of the topology of a conical intersection, the quadratic expansion of the Hamiltonian matrix can be used as a new practical method to generate a subspace of active coordinates for quantum dynamics calculations. The cost of quantum dynamics simulations grows quickly with the number of nuclear degrees of freedom, and quantum dynamics simulations are often performed within a subspace of active coordinates (see, e.g., [46-50]). In this section we describe a method which enables the a priori selection of these important coordinates for a photochemical reaction. Directions that reduce the adiabatic energy difference are expected to lead faster to the conical intersection seam and will be called photoactive modes . The efficiency of quantum dynamics run in the subspace of these reduced coordinates will be illustrated with the photochemistry of benzene [31,51-53]. 

Fig. 2 Representative cuts through the potential energy surfaces of Bz+ (upperpanel or a) and its mono fluoro derivative, F-Bz+ (lowerpanel or b). The upper panel shows the results for the linear vibronic coupling model, while in the lower one the quadratic coupling terms are also included. In both panels the effective coordinate connects the centre of the Franck-Condon zone to the minimum of the intersection seam between the A and C states of F-Bz" ", and between the X and B states of the parent cation (within the subspace of JT active coordinates)...

It can be seen from (26) that a is independent of 0,. If we set either 02 or 03 equal to zero, ead and jad are planes in the corresponding three-dimensional subspaces and intersect along a straight line in the e, 0j plane. In addition, a is restricted to the two values tt/4 and 3ir/4 for 02 = 0 and the different two values 0 and tt/2 for 0, = 0. If, on the other hand, we set Q, = 0, efd and ef become the lower and upper manifold of a circular double cone whose vertex is the origin and whose axis of symmetry is the e axis. In a coordinate system in which this axis is replaced by an e/b axis (in order to make its units the same as those of the Qk), the cone s half-angle is 45°. The intersection... 

Therefore, the hypersurface E+ intersects with the hypersurface E-, and their common part, i.e. the intersection set, are all those vectors of the n-dimensional space that fulfil the condition = 0 and (2 = 0- The intersection represents a (n — 2)-dimensional subspace of the n-dimensional space of the nuclear configurations. When we withdraw from the point (0,0, fa, 4,..., sN-e) by changing the coordinates f 1 and/or 2> a difference between E+ and E- appears. For small increments df j the changes in the energies E+ and E- are proportional to d i and for E+ and differ in sign. This means that the hypersurfaces E+ and E-as functions of f j (at 2 = 0 and fixed other coordinates) have the shapes shown in Fig. 6.14.a. For 2 the situation is similar, but the cone may differ by its angle. From this it follows that... 

As an illustration. Fig. 6 depicts a set of three intersections occurring in the benzene radical cation. One is of the conical type, whereas at the other two crossings no interaction occurs, at least not in the subset of coordinates chosen for the drawing. The situation changes, however, in different vibrational subspaces, which imderlines the complex topology of the surfaces in the pertinent higher-dimensional coordinate space. 

These computations establish conclusively that there is a seam of low-energy conical intersections, primarily in the Gi. Gsfl. QiOa subspace of normal-coordinate space. Here, Vi and V6 are two of the five totally synunetric normal modes, while V[o<, is the (single) mode of appropriate symmetry (82, X Bou = Big) to couple the 5 and S2 excited states in first order. Two other totally symmetric modes, vga and i>9u, turn out to be of secondary importance, while the coupling strength of the remaining mode of this symmetry, V2, is found to be negligible. The existence of a conical intersection has already been inferred from lower-level ab initio data, as well as from a combination of semiempirical calculations and adjustments of parameters to reproduce the spectral profile. " This conical intersection dominates the short-time photophysics of 5 - and 52-excited pyrazine, as documented below for absorption and resonance Raman spectra. 

Neumann and Wigner proved, in their seminal work in 1929 that, for a molecular system with N internal nuclear coordinates (N = 3N — 6), two electronic surfaces become degenerate in a subspace of dimension N — 2. To illustrate this dimensionality rule, consider two intersecting adiabatic electronic states, /i and 2- These two states are expanded in terms of two diabatic states i, which are diagonal to all the remaining electronic states and to each other, ... 

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