Second-order methods

Due to the large number of variables in wavefiinction optimization problems, it may appear that fiill second-order methods are impractical. For example, the storage of the Hessian for a modest closed-shell wavefiinction with 500... 

Chaban G, Schmidt M W and Gordon M S 1997 Approximate second order methods for orbital optimization of SCF and MCSCF wavefunctlons Theor. Chim. Acta 97 88... 

Errors are proportional to At for small At. The trapezoid rule is a second-order method. 

The calculations are straightforward, and no explicit velocity is needed. The storage requirement is modest, and the precision is modest (it is a second-order method). Note that one must start the calculation with values of Jr) at times t and t - At. 

The steepest-descent method does converge towards the expected solution but convergence is slow in the vicinity of the minimum. In order to scale variations, we can use a second-order method. The most straightforward method consists in applying the Newton-Raphson scheme to the gradient vector of the function/to be minimized. Since the gradient is zero at the minimum we can use the updating scheme... 

Jones and Finch (1984) provide an extensive comparison of both first- and second-order methods on several process examples. 

These results confirm the observation of Mihailovic and Rosina that second-order methods cannot characterize the correlations induced by two-body forces. 

In this formula, p. stands for gTd This is certainly an eigensystem equation. However, we must add a small correction to H, and moreover, this correction is not known in advance. It turns out that this correction can be left out, unless it is important that the linear equation is exactly solved. This is not necessary if the object is to find a good step for a macroiteration. Moreover, it turns out that, in such a context, the discrepancy introduced between this method and the exact NR-steps has the same asymptotic dependence as the error. Therefore, the method is still a second-order method with this modification, and there is no way to say a priori that this method is better or worse than the exact NR-iterations. This method is called the augmented Hessian (AH-)method. It is seen to be equivalent to a Newton-Raphson using a shifted hessian. This can be very advantageous, since this shift tends to keep the step down, and to keep the shifted hessian positive definite, when one is far from a solution. The size of... 

We shall in this chapter discuss the methods employed for the optimization of the variational parameters of the MCSCF wave function. Many different methods have been used for this optimization. They are usually divided into two different classes, depending on the rate of convergence first or second order methods. First order methods are based solely on the calculation of the energy and its first derivative (in one form or another) with respect to the variational parameters. Second order methods are based upon an expansion of the energy to second order (first and second derivatives). Third or even higher order methods can be obtained by including more terms in the expansion, but they have been of rather small practical importance. 

The choice of optimization scheme in practical applications is usually made by considering the convergence rate versus the time needed for one iteration. It seems today that the best convergence is achieved using a properly implemented Newton-Raphson procedure, at least towards the end of the calculation. One full iteration is, on the other hand, more time-consuming in second order methods, than it is in more approximative schemes. It is therefore not easy to make the appropriate choice of optimization method, and different research groups have different opinions on the optimal choice. We shall discuss some of the more commonly implemented methods later. 

To minimize the image function we use a second-order method since in each iteration the Hessian is needed anyway to identify the mode to be inverted (the image mode). Line search methods cannot be used since it is impossible to calculate the image function itself when carrying out the line search. However, the trust region RSO minimization requires only gradient and Hessian information and may therefore be used. In the diagonal representation the step Eq. (5.8) becomes... 

There are several methods that we can use to increase the order of approximation of the integral in eqn. (8.72). Two of the most common higher order explicit methods are the Adams-Bashforth (AB2) and the Runge-Kutta of second and fourth order. The Adams-Bashforth is a second order method that uses a combination of the past value of the function, as in the explicit method depicted in Fig. 8.19, and an average of the past two values, similar to the Crank-Nicholson method depicted in Fig. 8.21, and written as... 

In some cases, the major disadvantage of second-order methods may be the programming effort required to derive explicit expressions for the Hessian elements, whose number increases as the square of the number of parameters. In Sect. 3.6, a simplified form of the Hessian matrix is derived by considering the particular form of the least-squares objective functions. 

A second order method with respect to both space and time can be derived by approximating the diffusion equation at timestep tn+i/2 = tn + At/2 and employing a centered-difference scheme for the time derivative, too. Considering the Taylor series in t at constant x = Xi ... 

The algorithm consists of the judicious application of one of two integration formulas to each equation in the system and the choice of formula is based on the time constant for each equation evaluated at the beginning of each chemical time step. Species with time constants too small are treated by the stiff method and the remaining species are treated by a classical second order method. The algorithm is characterized by a high degree of stability, moderate accuracy and low overhead which are very desirable features when applied to reactive flow calculations. 

It must be taken into account that second order MCSCF procedures, as the AH and other exact or approximate second order methods, converge quadratically when close to the final solution, but with a very small radius of convergence. More than this, when the MO-CI coupling is not included one finds linear convergence even with second order methods. For example, see Table II of Werner s paper /14/. 

In this paper the authors produced a class of non-linear explicit second-order methods for solving one-dimensional periodic initial value problems (IVPs). These methods are P-stable (based on the definition given by Lambert and Watson ) and they have phase-lag of high-order. The authors introduce a special vector operation such that the obtained methods to be extended to be vector-applicable directly. With this extension the produced methods can be applied to multi-dimensional problems. 

In 52 the author develops a symplectic exponentially fitted modified Runge-Kutta-Nystrom method. The method of development was based on the development of symplectic exponentially fitted modified Runge-Kutta-Ny-strom method by Simos and Vigo-Aguiar. The new method is a two-stage second-order method with FSAL-property (first step as last). 

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