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## Force Constant Update Algorithms

In a Newton-type geometry update procedure often only a crude approximation to the force constant matrix is available. What can be done then is to update in each iteration using information about previous coordinates and gradients. This constitutes the quasi-Newton or variable metric methods of which there are a few variants:
1. Murtagh/Sargent (MS):

2. Broyden/Fletcher/Goldfarb/Shanno (BFGS):

3. Davidon/Fletcher/Powell (DFP):

4. combined method (BFGS/DFP): If and perform DFP update, otherwise BFGS.
The meaning of the symbols above is as follows:
approximate inverse force constant matrix in the k-th iteration.s
general coordinates in the k-th iteration.
An alternative is to use update algorithms for the hessian itself:

Ehrig, Ahlrichs : Diagonal update for the hessian by means of a least squares fit

with the new estimate for the diagonal elements obtained by

and the error obtained by the regression

Another alternative is to use DIIS-like methods: structure optimization by direct inversion in the iterative subspace. (See ref. [28] for the description of the algorithm). The DIIS procedure can often be applied with good success, using static or updated force constant matrices.

Any of the algorithms mentioned above may be chosen. Recommended is the macro option ahlrichs, which leads to the following actions ( is the maximum number of structures to be included for the update, default is ):

:
geometry update by inter/extrapolation using the last 2 geometries.
:
diagonal update for the hessian as described above; DIIS-like update for the geometry.
:
BFGS-type update of the hessian and quasi-Newton update of (generalized) coordinates.
References for the algorithms mentioned above: [29,25,30,28,31,32]

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