By gradient methods

Atkins, D. H. F. and Garland. J. A., 1974 The measurement of deposition velocities for sulphur dioxide and particulate material by gradient method. WMO Special Environmental Rep. 3, 579-594. 

CONDITIONS FOR NONLINEAR PROGRAMMING (NLP) BY GRADIENT METHODS WITH TWO OR MORE DECISION VARIABLES... 

Conditions for Nonlinear Programming (NLP) by Gradient Methods with Two or More Decision Variables 631... 

The optimization of the orbitals by gradient methods of the kind discussed in this section forms the basis of highly efficient numerical techniques (see e.g. Werner and Knowles, 1985) in which simultaneous... 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

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. 

Conjugate Gradient methods compute the conjugate directions hj by iterative computation involving the gradient gj without... 

The term gradient method is often applied to algorithms that make use of a gradient to find stationary points. It is always possible to estimate the gradient numerically, for example by calculating the function value at points that are close together in space... 

Many gradient methods approximate the energy surface at step by a quadratic expression in terms of the coordinate vector the total energy the gradient and the Hessian... 

The conjugate gradient method is one of the oldest in the Retcher-Reeves approach, the search direction is given by... 

The diffusion coefficients of the constituent ions in ionic liquids have most commonly been measured either by electrochemical or by NMR methods. These two methods in fact measure slightly different diffusional properties. The electrochemical methods measure the diffusion coefficient of an ion in the presence of a concentration gradient (Pick diffusion) [59], while the NMR methods measure the diffusion coefficient of an ion in the absence of any concentration gradients (self-diffusion) [60]. Fortunately, under most circumstances these two types of diffusion coefficients are roughly equivalent. 

Energy minimization methods that exploit information about the second derivative of the potential are quite effective in the structural refinement of proteins. That is, in the process of X-ray structural determination one sometimes obtains bad steric interactions that can easily be relaxed by a small number of energy minimization cycles. The type of relaxation that can be obtained by energy minimization procedures is illustrated in Fig. 4.4. In fact, one can combine the potential U r) with the function which is usually optimized in X-ray structure determination (the R factor ) and minimize the sum of these functions (Ref. 4) by a conjugated gradient method, thus satisfying both the X-ray electron density constraints and steric constraint dictated by the molecular potential surface. 

The state and co-state equations (Appendix A) together with the optimality conditions are numerically solved by a gradient method as outlined in Appendix A to get the nonisoth-ermal policy. 

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