About OOPSE OOPSE News Features Download CVS Mailing Lists Documentation Authors Funding Results	Energy Minimization As one of the basic procedures of molecular modeling, energy minimization is used to identify local configurations that are stable points on the potential energy surface. There is a vast literature on energy minimization algorithms have been developed to search for the global energy minimum as well as to find local structures which are stable fixed points on the surface. We have included two simple minimization algorithms: steepest descent, (SD) and conjugate gradient (CG) to help users find reasonable local minima from their initial configurations. Since OOPSE handles atoms and rigid bodies which have orientational coordinates as well as translational coordinates, there is some subtlety to the choice of parameters for minimization algorithms. Given a coordinate set $x_{k}$ and a search direction $d_{k}$ , a line search algorithm is performed along $d_{k}$ to produce $x_{k+1}=x_{k}+$ $\lambda _{k}d_{k}$ . In the steepest descent (SD) algorithm, $$d_{k}=-\nabla V(x_{k}).$$ (5.1) The gradient and the direction of next step are always orthogonal. This may cause oscillatory behavior in narrow valleys. To overcome this problem, the Fletcher-Reeves variant [52] of the conjugate gradient (CG) algorithm is used to generate $d_{k+1}$ via simple recursion: $$d_{k+1} =-\nabla V(x_{k+1})+\gamma_{k}d_{k}$$ (5.2) where $$\gamma_{k} =\frac{\nabla V(x_{k+1})^{T}\nabla V(x_{k+1})}{\nabla V(x_{k})^{T}\nabla V(x_{k})}.$$ (5.3) The Polak-Ribiere variant [53] of the conjugate gradient ( $\gamma_{k}$ ) is defined as $$\gamma_{k}=\frac{[\nabla V(x_{k+1})-\nabla V(x)]^{T}\nabla V(x_{k+1})}{\nabla V(x_{k})^{T}\nabla V(x_{k})}%$$ (5.4) It is widely agreed that the Polak-Ribiere variant gives better convergence than the Fletcher-Reeves variant, so the conjugate gradient approach implemented in OOPSE is the Polak-Ribiere variant. The conjugate gradient method assumes that the conformation is close enough to a local minimum that the potential energy surface is very nearly quadratic. When the initial structure is far from the minimum, the steepest descent method can be superior to the conjugate gradient method. Hence, the steepest descent method is often used for the first 10-100 steps of minimization. Another useful feature of minimization methods in OOPSE is that a modified SHAKE algorithm can be applied during the minimization to constraint the bond lengths if this is required by the force field. Meta-data parameters concerning the minimizer are given in Table 5.1 Table 5.1: Meta-data Keywords: Energy Minimizer Parameters
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	Updated on January 16, 2006