General functions steepest descent method

For a given Hamiltonian the calculation of the partition function can be done exactly in only few cases (some of them will be presented below). In general the calculation requires a scheme of approximations. Mean-field approximation (MFA) is a very popular approximation based on the steepest descent method [17,22]. In this case it is assumed that the main contribution to Z is due to fields which are localized in a small region of the functional space. More crudely, for each kind of particle only one field is... 

The rate of convergence of the Steepest Descent method is first order. The basic difficulty with steepest descent is that the method is too sensitive to the scaling of S(k), so that convergence is very slow and oscillations in the k-space can easily occur. In general a well scaled problem is one in which similar changes in the variables lead to similar changes in the objective function (Kowalik and Osborne, 1968). For these reasons, steepest descent/ascent is not a viable method for the general purpose minimization of nonlinear functions, ft is of interest only for historical and theoretical reasons. 

Steepest descent method for a general quadratic function. 

In the method of steepest descents one calculates the gradient at a point. The method of attack depends on whether this gradient may be calculated analytically or numerically (which requires calculations at N + 1 points for an TV dimensional surface) and one moves along this direction until the lowest point is reached when a new gradient is calculated. When one is close to the minimum and the gradient is small it is necessary to have a method which is quadratically convergent, and to calculate the general quadratic function for N dimensions numerically requires N -t 1)(A+ 2)/2 function evaluations. 

If there are no roots to (128), then the primitive stationary phase approximation implies / 0 although it is true that in such cases the value of the integral is small, one often wishes to know how small—10 2, say, or 10-4. To determine the asymptotic approximation to the integral in such cases one analytically continues (129), the mathematical apparatus for which is the method of steepest descent .59 This approach notes that although there are no real values of t which satisfy (128), there will in general be complex values which do so—provided, of course, that it is possible to analytically continue the function /(f) into the complex t-plane. The method of steepest descent then deforms the path of integration in (127) from the real f-axis,... 

Because the transition state geometry optimized in solution and the solution-path reacton path may be very different from the gas-phase saddle point and the gas-phase reaction path, it is better to follow the reaction path given by the steepest-descents-path computed from the potential of mean force. This approach is called the equilibrium solvation path (ESP) approximation. In the ESP method, one also substitutes W for V in computing the partition functions. In the ESP approximation, the solvent coordinates are not involved in the definition of the generalized-transition-state dividing surface, and hence, they are not involved in the definition of the reaction coordinate, which is normal to that surface. One says physically that the solvent does not participate in the reaction coordinate. That is the hallmark of equilibrium solvation. 

---