Gradient Extremals GE

We generalize the two-dimensional floor path problem of a valley which we have analyzed in Sect. 3.1, to the n-dimensional case, ns(3N-3). In general, if n 2 we cannot visualize the PES in such a dimension. We will discuss representations of equipotential sections. In the 2D case, a section E(x,y)=const of E with a shifted 2D configuration plane gives ID figures of the section, curves of the type in Fig.l of Sect. 3.1 (E(x,y)=0 is the section with the x,y- 

which is an ordinary 2D spherical surface x +y +z =1 in the 3D space of the coordinates (x,y,z). It can be considered either as being shifted at the height one of the dimension axis of , or, as being projected as an equipotential hypersurface down into the configuration space. In the general n-dimensional configuration space, the PES is a curvilinear n-dimensional hypersurface in the (n+1) dimensions of the space of coordinates plus E, and equipotential sections of E are (n-1)-dimensional hypersurfaces. 

In this section we study the course of valley floors in an n-dimensional landscape. The illustrations are two dimensional examples. 

The projection of the floor line in the configuration space intersects every (n-1)-dimensional contour h ersurface E(x- 

Any gradient vector is always perpendicular to the contour hypersurface especially to the corresponding tangential hyperplane. This is to say that the floor line intersects any contour in that point where the slope of the line is minimal, i.e., where contours equidistant in energy are spaced farthest apart, i.e., where the valley is least steep . Basilevsky and Shamov call a corresponding walk mountaineer s algorithm. 

---

We call this line a gradient extremal (GE).16 Unless the function increases indefinitely the gradient approach zero. It is therefore reasonable to expect that by following a GE we sooner or later hit a new stationary point... 

Gradient Extremal (GE), 338 Gradient norm minimization, 333 Gradient of a function, 238 Greens function, 257 GROMOS force field, 40 Gross atomic charge, 218... 

Figure 2. Steepest descent and gradient extremals (GE) using the Miiller-Brown potential [23], steepest descent path — GE.

Reaction paths are a widely used concept in theoretical chemistry. It is evident that the invariance problem, which was mathematically solved a long time ago (cf. the report given in Ref. [1 ]), penetrates again and again the discussions in this field (see Ref. [2]). We give both the non-invariant and the invariant definitions with respect to the choice of the particular coordinate system for two important kinds of chemical reaction pathways (RP), namely, steepest descent lines (SDP) and gradient extremal (GE) curves. 

A gradient extremal (GE) of the Miiller-Brown potential (see Figure 15 [28]) is a suitable example showing how the definition of these curves works The bowl of Mini is a deep, long, and relatively straight valley. The col of SPi opens to this main valley and a steepest descent path goes downhill perpendicular to the contour lines of the floor. At the valley floor, it joins in the floor line. However, we cannot decide at which point the lines cross, because this is an asymptotic junction. 

It is the equation for a gradient extremal depending from the point x. H and g are terms given with the PES. This means, a point x lies on a gradient extremal (GE), if the gradient of E is also an eigenvector of... 

Note So far, we do not know if this extremal of g is a minimum (corresponding to a VF-GE) or a maximum, or an inflection point. With these preliminary statements, we can now give the definition of a gradient extremal [19]. 

Every point (x y) fulfills the GE equation. Thus, the whole plane itself is a 2D gradient extremal. 

CPR = conjugate peak refinement GDIIS = geometry direct inversion in the iterative subspace GE = gradient extremal LST = linear synchronous transit LTP = line then plane LUP = locally updated planes NR = Newton-Raph-son P-RFO = partitioned rational function optimization QA = quadratic approximation QST = quadratic synchronous transit SPW = self-penalty walk STQN = synchronous transit-guided quasi-Newton TRIM = trust radius image minimization TS = transition structure. 

Despite the crystalline nature of the Al, its surface is extremely rough up to the scale of lOOnm, as can be seen in Fig.2. High resolution microscopy of this interface indicates roughness down to atomic scale. Nanoprobe analysis (with probe size down to 4 nm) in the amorphous phase at increasing distances from the Al crystal gave no gradient of concentration indicating that this structure is not related to diffusion process. However, nanoprobe analysis at the Al rim combined with atomic resolution microscopy indicate that amorphous pockets of Ge exists in the Al rim. ... 

---