Seam space

From the preceding analysis, it is seen that the coordinate space neai R can be usefully partitioned into the branching space described in tenns of intersection adapted coordinates (p, 9, ) or (x,y,z) and its orthogonal complement the seam space spanned by a set of mutually orthonormal set w, = 4 — M . From Eq. (27), spherical radius p is the parameter that lifts the degeneracy linearly in the branching space spanned by x, y, and z. 

Therefore, the hypersurface E+ intersects with the hypersurface E-, and their common part (i.e., the seam space) are all those vectors of the n-dimensional space that fulfill the condition 1=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... 

This is called the conical intersection (see Fig. 6.18b). The cone opening angle is in general different for different points of the seam space (see Fig. 6.18c). 

Although points of conical intersection are quite numerous, since the seam space has dimension - 77, when viewed from the full internal coordinate... 

The pair (x, y) define the branching plane or g-h plane. The remainder of the intersection adapted coordinate system, w , i = l-(Ar " — 2), spans the seam space. These — 2 mutually orthonormal vectors need only be orthogonal to the branching space. It is also convenient to define... 

The seam space of W [Eq. (8)] is the set of points with x = y = 0. These points are isolated. By isolated we mean that there exists a small volume surrounding the seam such that the only degeneracies in that volume are the seam points themselves. While this is the case for W, higher order terms can alter this situation. To illustrate, we consider the following simple extension of W which replaces W, see Eq. (4), with = hy + byw, so that... 

The degenerate perturbation method described in Sec. 3 assumes/requires that the degeneracy be lifted at first order. In this case, first order means that displacements exclusively in the seam space, zeroth order in displacements in the branching space [Elqs. (58)-(59)] do not lift the degeneracy. In a triatomic system, the seam would locally be a straight line perpendicular to the g h plane, and globally the seam would be approximated by a... 

Yarkony DR (2005) Escape from the double cone optimized descriptions of the seam space using gateway modes. J Chem Phys 123 134106... 

The first two present new approaches the seam space nudged elastic band method (SS-NEB) and the double Newton-Raphson algorithm using composed steps (DNR-CS). The third report describes an automated search method to find minimum energy CIXs, the so-called SMF/ ADDF approach. 

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