G-h plane

Following this prescription, we have performed dynamics simulations with initial velocities pointing along 24 different directions in the g-h plane and with several initial kinetic-energy values between the maximum and the minimum. The results are presented in Figure 8-12. This figure shows the CN distance in pyrrole after 40 fs (radial coordinate) as a function of the initial direction (angular coordinate). Most of initial directions lead to a CN distance of around 1.5 A and do not result in non-cyclic structures. In these cases they correspond to photophysical... 

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 second of Eqs. (77a), when combined with Eq. (77b), permits the parameters p x = p,w to be determined from derivative coupling data in the g-h plane. Equation (77b), then provides a consistency check on the parameters so determined. This approach is used in Sec. 4 as part of the algorithm for locating confluences. More details concerning the use of Eqs. (77a) and (77b) can be found in Refs. 47 and 66. 

In Eqs. (54c) and (69), the parameters s, g and h are used to determine the energies and derivative couplings near a point of conical intersection. Below we compare these perturbative results with those of ab initio calculations at (trans, p, 2.95). Figures 4 depicts the g h plane for this point. From this figure it is seen that the coupling mode, y, has a" synunetry and therefore is the unique internal a" coordinate. The tuning mode x has a symmetry and tends to decrease i (C-N) and increase ZNCO. 

Figures 8 and 9 compare the g-h space for the relativistic hamil-tonian with the g-h plane for the nonrelativistic hamiltonian. Figure 8 reports Y i,2E,SE their orthogonalized counterparts, r,2E,iE i,2E,iE x,2E,iE (2.7688,1.8790,2.3375),...

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... 

Fig. 5. Branching (or g, h) plane vectors for the Cl structure of Fig. 3. The Xi and X2 vectors correspond to the derivative coupling (or non-adiabatic coupling) and gradient difference vectors between the and So states.

From equations (31a,b) the degeneracy is lifted linearly in nuclear displacements in the g-h(R ) plane, the plane defined by the vectors, g (Rjr) and h (R,). Only the g-h plane, rather than the individual vectors, g (Rx) and h (R ), is uniquely determined since c (Ri) and c (Rv) are only defined up to a rotation. This point is discussed further in Section 6.3. The numerical determination of the g-h plane is discus.sed in Sections 7 and 8. In the g-h- -(Ri) space, the orthogonal complement of the g-h plane, the degeneracy is lifted starting at second order in the nuclear displacements. The implications of this difference are profound so that the numerical determination of the g-h plane is a key computational issue and is discussed in detail below. 

In order to understand the importance of the g-h plane, its relation to the geometric phase effect, and the connection between the f - (R) and the geometric phase effect, a more careful analysis of equation (24) is required. The following analysis is based on the seminal work of Mead (see also Ref. 17). Consider (R), S (R) and //(R) to be expanded in a power series in ( R, displacements from R in the g-h plane. 

In the vicinity of a conical intersection the Cl portion of the derivative coupling is preeminent. The Cl portion of the derivative coupling corresponding to displacements in the g-h plane... 

If g and h are perpendicular and of equal length the right hand side of equation (46c) is 1. Also note that from equation (46a), the circulation of f (R) along Ce, an infinitesimal loop in the g-h plane surrounding R t. S given by ... 

However, if Ce does not contain a point of conical intersection, then the circulation of f (R) approaches 0 as p decreases to zero. Below, it will be convenient to denote a circle in the g-h plane with origin O and radius p, by C(0, p). 

Figure 1 Unit vectors in the directions h - (Rmex) and g (Rmex) (the component of g (Rmex) >n the g-h plane perpendicular to h (Rmex)), represented in terms of atomic displacements. Also shown are the Jacobi coordinates R, r, y...

While AE/j is quite small, less than 0.5 cm , for the points of conical intersection given in Table I, a numerical procedure can never produce an exact crossing point. To prove the existence of a point of conical intersection, the circulation of f fR) can be evaluated in the g-h plane. Consider again Rmex- The circulation of along Ci is 0.9995 r, proving the existence of a conical intersection at Rmex- Since A(Ci) = O.OOOSn the derivative coupling is removable to an excellent approximation inside this loop. 

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