Conjugate gradients

The choice of minimization method depends on two factors, the size of the systan and its conformational state. As a guiding principal, if the structure is far from the minimum then the steepest descent should be used for the first few steps, followed by a more precise conjugate gradient or Newton-Raphson methods. For detailed information, the reader is requested to refer to some of the standard textbooks (Allen and Tildesley 1987 Frenkel and Smit 1996X 

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Fletcher R and Reeves C M 1964 Function minimization by conjugate gradients Comput. J. 7 149... 

Payne M C, Teter M P, Allan D C, Arias T A and Joanopoulos J D 1992 Iterative minimization techniques for ab /M/o total energy calculations molecular dynamics and conjugate gradient Rev. Mod. Phys. 64 1045... 

Davis, M. E., McCammon, J. A. Solving the finite difference linearized Poisson-Boltzmann equation A comparison of relaxation and conjugate gradients methods.. J. Comp. Chem. 10 (1989) 386-394. 

IlyperChem supplies three types of optimi/ers or algorithms steepest descent, conjugate gradient (Fletcher-Reeves and Polak-Ribiere), and block diagonal (Newton-Raph son). 

A con jugate gradicri I method differs from the steepest descent technique by using both the current gradient and the previous search direction to drive the rn in im i/ation. , A conjugate gradient method is a first order in in im i/er. 

HyperChem provides two versions of the conjugate gradient method, Fletcher-Reeves and Bolak-Rihiere. Polak-Ribiere is more refined and is the default eh oiee in HyperChem,... 

Tbis tecbnic ne is available only for the MM-i- force field. As is true for the conjugate gradient methods, yon should noi use this algorithm when the initial interatomic forces are very large (meaning, the molecular structure is far from a inmiimim). 

Several variants of the conjugate gradients method have been proposed. The formulatior given in Equation (5.7) is the original Fletcher-Reeves algorithm. Polak and Ribiere proposed an alternative form for the scalar constant 7) ... 

Tafe/e 5.1 A comparison of the steepest descents and conjugate gradients methods for an initial refinement and a stringent minimisation. 

This study shows that the steepest descent method can actually be superior to conjugate gradients when the starting structure is some way from the minimum. However, conjugate gradients is much better once the initial strain has been removed. 

Fig. 11.38 Lag ejfects in ab initio molecular dynamics. (Figure redrawn from Payne MC, M P Teter, D C Allan, R A Arias and D ] Joannopoidos 1992. Iterative Minimisaticm Techniques for Ab Initio Total-Energy Calculations Molecular Dynamics and Conjugate Gradients. Reviews of Modern Physics 64 1045-1097.)...

This algorithm alternates between the electronic structure problem and the nuclear motion It turns out that to generate an accurate nuclear trajectory using this decoupled algoritlun th electrons must be fuUy relaxed to the ground state at each iteration, in contrast to Ihe Car-Pairinello approach, where some error is tolerated. This need for very accurate basis se coefficients means that the minimum in the space of the coefficients must be located ver accurately, which can be computationally very expensive. However, conjugate gradient rninimisation is found to be an effective way to find this minimum, especially if informatioi from previous steps is incorporated [Payne et cd. 1992]. This reduces the number of minimi sation steps required to locate accurately the best set of basis set coefficients. 

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