Intersection-adapted coordinates

The topography of a conical intersection affects the propensity for a nonadiabatic transition. Here, we focus on the essential linear tenns. Higher order effects are described in [10]. The local topography can be detennined from Eq. (13). For T] = 3, Eq. (13) becomes, in orthgonal intersection adapted coordinates... 

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. 

The lowest order contributions to the energy are described by the conical parameters g, h, and s, k = x,y,z, or by d, A = 1,2 and s, k = x,y,z-Here and below the superscript ij is suppressed when no confusion will result. We also will use the nonrelativistic convention g - x,fi" y and h J z, where is real is parallel to. These parameters [9] are reported in Figure 4a and b. Their continuity is attributable to the use of orthogonal intersection adapted coordinates. For comparison, Figure 4a and b reports the nonrelativistic quantities g , and s, respectively. While noting that there is no unique correspondence... 

The total energy expansion is obtained by adding the terms that are equal for the two states to (15). When finite displacements are considered, the expansion of the energy around the intersection in intersection-adapted coordinates becomes ... 

Using intersection-adapted coordinates, the quadratic approximation, in other words the local harmonic approximation, of the adiabatic energy difference for a finite displacement around Qo reads thus... 

These results can be simplified by introducing scaled orthogonal intersection adapted coordinates x =, y = and z = In these coordinates,... 

To characterize the neighborhood of X it is convenient to define intersection adapted coordinates, x,y,Wi,i = f-(fV " - 2), where is the number of internal coordinates. In this cartesian coordinate system the x-and j/-axes are chosen as unit vectors along the gradients g and h that is,... 

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... 

Using intersection adapted coordinates, the topograghical parameters, pitch, as3mmetry and tilt, described in the Introduction, can be quantified. With these definitions, W [Eq. (4)] becomes, through first order... 

Intersection-Adapted Cartesian Coordinates 3.3.1. in Intersection Adapted Coordinates... 

Examples of orthogonal intersection adapted coordinates are provided in Sec. 4. 

Figures 6(a) and 6(b) report the conical parameters Sy and g, h, respectively, obtained from orthogonal intersection adapted coordinates. If orthogonal g and h had not been used g and h would vary erratically along the path, making interpolation and extrapolation impossible. Significantly, from Figs. 6(a) and 6(b), it can be seen that although these conical parameters are by no means constant, the inequalities > g > h, discussed above for Tl trans,p, 2.95), are, in a qualitative sense, typical.

---