Vector gradient

At any geometry g.], the gradient vector having components d EjJd Q. provides the forces (F. = -d Ej l d 2.) along each of the coordinates Q-. These forces are used in molecular dynamics simulations which solve the Newton F = ma equations and in molecular mechanics studies which are aimed at locating those geometries where the F vector vanishes (i.e. tire stable isomers and transition states discussed above). 

Here g is the gradient vector at q , The minimizer q + 8q is exact on a quadratic surface... 

For a quadratic surface, the gradient vector is a linear function of the coordinates. An alternative way of using... 

Minimizing the square of the gradient vector under the condition c/ = I yields the following linear system of equations... 

Calculations using the semiempirical PM3 method with standard convergence criteria of 0.0003 aii on the maximum component of the gradient vector and either an energy change from the previous cycle of < 10 hartree or a maximum predicted displacement for the next step of < 0.0003 au. 

R F W Bader s theory of atoms in molecules [Bader 1985] provides an alternative way to partition the electrons between the atoms in a molecule. Bader s theory has been applied to many different problems, but for the purposes of our present discussion we will concentrate on its use in partitioning electron density. The Bader approach is based upon the concept of a gradient vector path, which is a cuiwe around the molecule such that it is always perpendicular to the electron density contours. A set of gradient paths is drawn in Figure 2.14 for formamide. As can be seen, some of the gradient paths terminate at the atomic nuclei. Other gradient paths are attracted to points (called critical points) that are... 

Gradient vector paths around formamide. The paths terminate at atoms or at bond critical points (indicated by squares). 

For multi-dimensional potential energy surfaces a convenient measure of the gradient vector is the root-mean-square (RMS) gradient described by... 

Steepest Descent is the simplest method of optimization. The direction of steepest descent, g, is just the negative of the gradient vector ... 

A more sophisticated version of the sequential univariate search, the Fletcher-Powell, is actually a derivative method where elements of the gradient vector g and the Hessian matrix H are estimated numerically. 

Now imagine that we rotate the molecule about the internuclear axis. The curved contour will trace out a surface. If we draw a unit outward normal vector to this surface, it will be everywhere perpendicular to the gradient vector (because the gradient vector points along the trajectory). 

There are two common choices for the error vector it can either be a geometry or a gradient vector, the latter being preferred in more recent work. ... 

It is well known from the calculus that the gradient vector points in the direction of maximum increase of the function/.84... 

Vector VS(k) contains the first partial derivatives of the objective function S(k) with respect to k and it is often called the gradient vector. For simplicity, we denoted it as g(k) in this chapter. 

Thus, the s 1 element of the gradient vector g(k) of the objective function S(k) is given by the following equation... 

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