Intersection Optimization

The practical algorithm,47-51 which was first implemented in the Gaussian 94 program package,62 can be described as follows. For minimization of E2 - E1 in the Xj and x2 plane, we have  

Clearly f will go to zero when E2 = Et, independently of the magnitude of. Note, however, that the gradient will also go to zero if Et is different from E2 but the two surfaces are parallel (i.e., Xj, the gradient difference vector, has zero length). In this case the method would fail. This situation will occur for a Renner-Teller-like degeneracy, for example. Of course, in this case, the geometry can be found by normal unconstrained geometry optimization. 

If we now define the projection of the gradient of E2 onto the n - 2 orthogonal complement to the plane x, x2 as 

---

Run a final state-averaged calculation at the fuUy-optimized conical intersection using the 4-31G basis set and P to predict the energies of the two states and view the configuration coefficients. (This step will not be necessary if you chose to use P for the final conical intersection optimization job you ll find the relevant output in the CAS output for the final optimization step, preceding the table giving the stationary point geometry.)... 

Figure 13 General behavior of a conical intersection optimization procedure. This contrived example was started from an almost planar geometry (much further from the optimum geometry than normal practice). The curve shows the rapid approach to the degenerate situation followed by minimization (retaining the degeneracy).

The optimization of a conical intersection is a constrained optimization in the space orthogonal to the two degeneracy-lifting coordinates, Xj and Xg. The optimization has an additional requirement, namely, the energy difference between the two states at the conical intersection is zero. Thus, the gradient used in our conical intersection optimization algorithm is the sum of two gradients ... 

In Sec. 3 we will mainly concentrate on the discussion of the structure (both geometrical, electronic and topographical) of the chemically relevant (low-lying) conical intersections that have been documented in a number of basic organic chromophores. As we will further point out in Sec. 3.1, the majority of these structures have been computed via conical intersection optimizations and do not strictly correspond to the conical intersection points located at the very bottom of the excited-state path. Nevertheless, their general features are representative of the chemically relevant segment of the intersection space and can be used for the mechanistic rationalization of different photoinduced molecular processes. 

Conical intersection optimization using state-averaged CASSCF. 

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

The potential surfaces of the ground and excited states in the vicinity of the conical intersection were calculated point by point, along the trajectory leading from the antiaromatic transition state to the benzene and H2 products. In this calculation, the HH distance was varied, and all other coordinates were optimized to obtain the minimum energy of the system in the excited electronic state ( Ai). The energy of the ground state was calculated at the geometry optimized for the excited state. In the calculation of the conical intersection... 

Using Poiseuille s formula, the calculation shows that for concentric-tube nebulizers, with dimension.s similar to those in use for ICP/MS, the reduced pressure arising from the relative linear velocity of gas and liquid causes the sample solution to be pulled from the end of the inner capillary tube. It can be estimated that the rate at which a sample passes through the inner capillary will be about 0.7 ml/min. For cross-flow nebulizers, the flows are similar once the gas and liquid stream intersection has been optimized. 

In this exercise, we will examine a small part of this process. We will predici ihe relative energies of the three states at the ground state geometry, and we will locate the conical intersection. We ve provided you with an optimized ground state (cis) structure and a starting structure for the conical intersection in the files 9 06 gs.pclb and 9 06 ci.pdb, respectively. 

We hope that the preceding discussions have developed the concept of a conical intersection as being as real as many other reactive intermediates. The major difference compared with other types of reactive intermediate is that a conical intersection is really a family of structures, rather than an individual structure. However, the molecular structures corresponding to conical intersections are completely amenable to computation, even if their existence can only be inferred from experimental information. They have a well-defined geometry. Like the transition state, the crucial directions governing dynamics can be determined andX2) even if there are now two such directions rather than one. As for a transition structure, the nature of optimized geometries on the conical intersection hyperline can be determined from second derivative analysis. 

---