Update expressions and initial Hessians

The Hessian is not usually calculated explicitly but is obtained by updating an initial estimate. The calculation of an expensive initial Hessian can be also avoided by suitable estimates obtained from empirical, molecular mechanics or semiempirical methods[134].

In any case the initial Hessian must have the same number of negative eigenvalues than the stationary point we want to find.

For quasi-Newton-Raphson and RFO methods, at every step, the approximated Hessian matrix is updated from the information of previous steps.

$${\bf B}_{k+1}={\bf B}_0+\sum_{i=0}^k[{\bf j}_i{\bf u}_i^T +{\bf u... ..._i^T -({\bf j}_i^T \Delta {\bf q}_i){\bf u}_i {\bf u}_i^T ] \qquad k=0,1,\ldots$$ (2.87)

Where ${\bf j}_i={\bf D}_i-{\bf A}_i$, ${\bf D}_i={\bf g}_{i+1}-{\bf g}_i$, ${\bf A}_i={\bf B}_i\Delta {\bf q}_i$, ${\bf u}_i={\bf M}_i\Delta {\bf q}_i / (\Delta {\bf q}^T_i{\bf M}_i\Delta {\bf q}_i)$. Different election of the ${\bf M}_i$ matrix leads to different update Hessian matrix formula [135]. In particular, for the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update ${\bf M}_i=a_i{\bf B}_{i+1}+b_i{\bf B}_i$ for some selected positive definite scalars $a_i$ and $b_i$. For the Powell update case the matrix ${\bf M}_i$ is equal to unit matrix ${\bf I}$.

These expressions give an approximation to the proper Hessian matrix. However, in NR equation (equation 1.75) we need the inverse rather than the Hessian. In this case we can use the update expression for its inverse. But we must never forget that while working with the Hessian matrix we can know and control the number of positive eigenvalues and then, the order of stationary point reached. When the inverse of the Hessian is used this information is lost. This is why we will prefer RFO technique rather that pure NR because an explicit control of eigenvalues is performed without being necessary a full diagonalization of a Hessian or Augmented Hessian matrix.

The BFGS update is not acceptable to locate first-order saddle points because this update is positive definite. While several update formulae are used to locate minimum energy structures, the Powell formula is a more suitable update to find transition state structure [136].