Minima and Transition State Optimization

A faster convergence can be obtained by using methods based on a second order expansion of the energy such as the Newton-Raphson-based methods [47]. In these methods, the search direction is defined as (see also Eq. (2.74)) 

Two problems need to be addressed to make this class of methods robust and efficient. The first one is to improve the search direction and the size of the step taken along this direction. This can be done by adding a shift parameter A in the Newton-Raphson step 

The shift parameter can be used to ensure that the optimization proceeds downhill even if the Hessian has negative eigenvalues. In addition, it can be chosen such that the step size is lower or equal to a predefined threshold. Popular methods using a shift parameter are the rational function optimization (RFO) [48] and Trust Radius (TR) methods [49, 50]. A finer control on the step size and direction can be achieved using an approximate line search method, which attempts to fit a polynomial function to the energies and gradients of the best previous points [51]. 

A second important issue is the calculation of the Hessian matrix which can be a computationally expansive task. A method to avoid such calculation is to start with an approximate Hessian, e.g. empirically determined or calculated at a lower level of theory, and to update the Hessian during the optimization using only energies and 

The above methods can be modified to enforce convergence to a transition state. In this case, the quality of the initial geometry and Hessian are more crucial than for the optimization of a minimum. 

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