Quadratic region

It uses a linear or quadratic synchronous transit approach to get closer to the quadratic region of the transition state and then uses a quasi-Newton or eigenvalue-following algorithm to complete the optimization. 

It is clear that if the method does descend into a quadratic region, then fl may be chosen as zero in step (ii) and an exact minimum will be found at step (iv). The incorporation of step (iv) ensures that the algorithm is stable (if a minimum exists along p) so that, barring accidents, we can be sure of eventually entering a quadratic region of the function. 

Tables VII and VIII examine the nature of the Hessian matrix for CO and NH as the geometry varies during the optimization. In both cases the starting geometry is far from the quadratic region about the minima. After the first geometry update in HjCO (cycle 2) the Hessian eigenvalues are within 10% of their final...

It is this short-time quadratic region that makes it possible to suppress the decay by repeated measurements. Such monitoring will be done as shown in Fig. 23. With this modification, the moment of emission can be inferred with accuracy 8L = L/N without disturbing the signal photons. Now, the probability of having... 

The calculation of g and H(U°) requires closed form expressions for dp x x (1,1 r)/dUai and d2plf° lA>)/dUaidUbj [145]. Also required are F and KrVtS, integrals already available in standard DFT programs. We can now from (50) obtain the optimal step within the quadratic region... 

Solving for Ax gives the step that leads to the minimum in the local quadratic region,... 

Outside the quadratic region the root mean square value of the residual. 

---