ABNR

The adopted basis Newton-Raphson method (ABNR) was developed originally by M. Karplus and D.J. States. It has been used widely for the optimization of biomolecules since its original implementation in CHARMM package [53]. Even though, there has never been a paper describing the method and the corresponding implementation. Only in a very recent paper by B. R. Brooks and co-workers [144] we can find some equations that picture the most important aspects of the method.

In the ABNR minimization the Newton-Raphson scheme is only applied in a small subspace of the molecule. So the whole displacement of the geometry is a combination of a steepest descent(SD) step plus a small contribution of Newton-Raphson(NR).

$$\Delta {\bf q}_k=\Delta {\bf q}_k^{SD} + \Delta {\bf q}_k^{NR}$$ (2.94)

At the beginning of the minimization only the steepest descent component is employed ( $\Delta {\bf q}_k^{NR}=0$). After several SD steps, the last $m$ geometry displacements can be used as a basis $\{\Delta {\bf q}_{lk}\}_m$ of dimension $m$ to obtain the NR step. So at step $k$ the last $m$ geometries are used ${\bf q}_l;\quad l=k-1,k-2,\ldots,k-m$

$$\Delta {\bf q}_k^{NR}= \sum_{l=1}^m\Delta {\bf q}_{lk} c_{lk} \quad where \quad \Delta {\bf q}_{lk}={\bf q}_{k-l}-{\bf q}_k$$ (2.95)

This equation can be approximated in a Taylor expansion with respect to ${\bf q}_k$ and then the Newton-Raphson equation becomes

$$\sum_{l'=1}^m\sum_{l=1}^m \Delta {\bf q}_{kl'}\cdot[{\bf g}({\bf ... ...bf q}_{k})]c_{lk} = -\sum_{l'=1}^m \Delta {\bf q}_{kl'}\cdot {\bf g}({\bf q}_k)$$ (2.96)

Equation 1.96 is a set of $m$ equations that can be solved diagonalizing the small $m\times m$ matrix. With the coefficients $\{c_{lk}\}$ we can obtain the geometry displacement component $\Delta {\bf q}_k^{NR}$. But what incorporates the Hessian character in ABNR method is the particular steepest descent step that is computed in the following way

$$\Delta {\bf q}_k^{SD}=\frac{{\bf F}_k}{\vert{\bf F_k}\vert},\quad {\bf F}_k=-{\bf g}({\bf q}_k+\Delta {\bf q}_k^{NR})$$ (2.97)

Where this gradient vector at point ${\bf q}_k+\Delta {\bf q}_k^{NR}$ is estimated as

$${\bf g}({\bf q}_j+\Delta {\bf q}_k^{NR})\approx {\bf g}({\bf q}_k)+\sum_{l=1}^m [{\bf g}({\bf q}_{k-l})-{\bf g}({\bf q}_{k})]c_{lk}$$ (2.98)

This additional dimension added to steepest descents steps stands for an estimation update of the local Hessian. If any eigenvalue of equation 1.96 is negative the NR step is deactivated and only SD steps are performed. As in L-BFGS a subspace dimension of $m=5$ is usually enough for a good performance.