Direct minimization

A finite difference formula is used to estimate the second derivatives of the coordinate vector with respect to time and S is now a function of all the intermediate coordinate sets. An optimal value of S can be found by a direct minimization, by multi-grid techniques, or by an annealing protocol [7]. We employed in the optimization analytical derivatives of S with respect to all the Xj-s. 

DM (direct minimization) an algorithm for forcing SCF calculations to converge... 

A drop of water that is placed on a hillside will roll down the slope, following the surface curvature, until it ends up in the valley at the bottom of the hill. This is a natural minimization process by which the drop minimizes its potential energy until it reaches a local minimum. Minimization algorithms are the analogous computational procedures that find minima for a given function. Because these procedures are downhill methods that are unable to cross energy barriers, they end up in local minima close to the point from which the minimization process started (Fig. 3a). It is very rare that a direct minimization method... 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

In-plant reclamation refers to the sand reclamation process in a foundry facility, which directly minimizes the generation of spent foundry sand. Sand reclamation includes physical, chemical, or thermal treatment of foundry sands so they may be safely substituted for new sand in molding and core-making mixes. 

To work in the opposite direction, minimization of the Landau free energy with respect to the order parameter, dip/dV = 0, clearly yields... 

Going back to Eq. (8), the problem now is that one must minimize Ev with respect to the y/i, instead of directly minimizing it with respect to p. This can, however, be achieved by solving Eq. (9), called the Kohn-Sham equation ... 

Direct minimization of the energy as a functional of the p-RDM may be achieved if the p-particle density matrix is restricted to the set of Al-represen-table p-matrices, that is, p-matrices that derive from the contraction of at least one A-particle density matrix. The collection of ensemble Al-representable p-RDMs forms a convex set, which we denote as P. To define P, we first consider the convex set of p-particle reduced Hamiltonians, which are... 

As shown in the second line, like the expression for the energy as a function of the 2-RDM, the energy E may also be expressed as a linear functional of the two-hole reduced density matrix (2-HRDM) and the two-hole reduced Hamiltonian K. Direct minimization of the energy to determine the 2-HRDM would require (r — A)-representability conditions. The definition for the p-hole reduced density matrices in second quantization is given by... 

We have estimated each of the parameters in Eq. (2) in a unified manner by combining the strengths of several previous molecular modeling studies. We used the force field of Karasawa and (ioddard" to model the atomic potential energy surface and to describe the charge distribution at the atomic level. This force field includes the effects of electronic polarization via the shell model of electronic polarization, originally developed by Dick and Overhauser." By direct minimization of total crystal free energy with respect to both the atomic and shell... 

In a recent paper Shapiro Shapley [4] have considered the problem of the uniqueness of equilibrium of systems of reactions in several phases in great detail. The computation of equilibrium compositions by direct minimization of the Gibbs free energy function has proved a valuable tool in the discussion of very complex systems and it is important to show that this minimum is unique and achieved under the same conditions that satisfy the mass action laws. This is what Shapiro Shapley have done Sellers has suggested some improvements [5]. 

In direct condensing, turbine exhaust steam is conveyed through a trunk to the air-cooled coils, where cooling air passing over the finned coil surfaces condenses the steam. Steam enters the coil section and condenses as it travels downward, with steam and condensate flowing in the same direction, minimizing pressure loss and increasing the heat transfer coefficient. 

Step 1. For a function to be minimized, determine its ideal solution and anti-ideal solutions by directly minimizing and maximizing the objective function. 

In Section 2 we outlined briefly how the ordinary closed-shell SCF problem could be recast as a problem in direct minimization. In this section we consider the problem in a little more detail and also consider its generalization, particularly in respect of incorporating constraints and finding the relevant gradient expressions. 

Direct Minimization Methods in Quantum Chemistry follows that, to second order, the change in R is... 

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