Review of the RGF procedure by Quapp and Hirsch

In the RGF method$^{1}$ the gradient criterion for stationary points

$$\nabla E({\bf x}) = {\bf0}$$ (1)

is reduced by one equation leading to

$$\frac{\partial E({\bf x})}{\partial x^i} = {\bf0}, \quad i=1,...,k\!\!\vert,...,N,$$ (2)

where the vector x represents the nuclear coordinates. The omitted coordinate k defines the search direction.
This was generalized by the projector equation$^{2}$

$${\bf P_r} \nabla E({\bf x}) = {\bf0} ,$$ (3)

where ${\bf P_r} {\bf r} = {\bf0}$ with a fixed search direction r. Note that Eq.(2) is a special case of Eq.(3) with the corresponding $(n-1)\times$$n$ projector matrix

$$\begin{array}{llr} {\bf P_{r}} \ = & \ \left( \begin{arra... ...d \ 1 \quad k-1 \ \ k \ k+1 \quad \ n \\ \end{array}\nonumber$$

thus, ${\bf P_r}$ is built by the unit vectors orthogonally to the search direction, where here again the k th unit vector is missing. Another possibility to define ${\bf P_r}$ is to use the dyadic product in

$${\bf P_r} = {\bf I}_N \ - \ {\bf r}^T {\bf r} , \\ \nonumber$$

where ${\bf I}_N$ is the unit matrix. It is an $N\times$$N$ matrix of rank $N-1$.
The reduced gradients of Eq.(3) define curves connecting stationary points. Starting from a given point (e.g. a minimum) one follows a selected curve to reach the saddle point of interest.
A predictor-corrector method is used for tracing these curves. Assuming a curve of points x(t) fulfilling the N - 1 conditions of Eq. (3), the tangent x$^{\prime }$ to the curve is given as

$$\frac{d}{dt} {\bf P_r} \nabla E({\bf x}(t)) = {\bf0} = {\bf... ...t)) = {\bf P_r} {\bf H}({\bf x}(t)) \frac{d{\bf x}(t)}{dt} ,$$ (4)

or

$${\bf P_r} {\bf H}({\bf x}(t)) {\bf x}^{\prime } = {\bf0} .$$ (5)

This is a homogeneous system of N - 1 linear equations where the coefficients are the elements of the Hessian matrix H projected by ${\bf P_r}$. Eq. (5) is solved using a QR decomposition.
In the predictor step the sequence of points x$_m$ and x$_{m+1}$ is computed as

$${\bf x}_{m+1} = {\bf x}_{m} + \frac{StL}{\vert\vert{\bf x}_m^{\prime }\vert\vert} {\bf x}_m^{\prime }$$ (6)

where StL is a steplength parameter.
Since the new point does not satisfy exactly the condition defined in Eq. (3), a corrector step is added in which Eq. (3) is solved using a Newton-Raphson-like method. In general, the linear combination r of internal coordinates is used as initial search direction. The use of the metric in internal coordinates is explained in the appendix of Ref.$^{3}$ A quite better combination of predictor and corrector steps is applied in Ref.4.

1 Quapp W, Hirsch M, Imig O, and Heidrich D, J. Computat. Chem. 19, 1087 (1998)
2 Quapp W, Hirsch M, and Heidrich D, Theoret. Chem. Acc. 100, 285 (1998).
3 Hirsch M, Quapp W, and Heidrich D, Phys. Chem. Chem. Phys. 1, 5291 (1999).
4 Hirsch M, Quapp W, J. Computat. Chem. 23, 887-894 (2002)

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