Critical point saddle

Here, pb is the bond critical point (saddle point in three dimensions, a minimum on the path of the maximum electron density). In Eq. (44), and A.2 are the principal curvatures perpendicular to the bond path. The parameters A and B in Eq. (45) determined using various basis sets are given in Bader et al. [83JA(105)5061]. Convenient parameters in the quantitative analysis of a conjugation effect are the relative 7r-character tj (in %) of the CC formal double or single bonds determined with reference to the bond of ethylene (90MI2) ... 

The expansion is done around the principal axes so only tliree tenns occur in the simnnation. The nature of the critical pomt is detennined by the signs of the a. If > 0 for all n, then the critical point corresponds to a local minimum. If < 0 for all n, then the critical point corresponds to a local maximum. Otherwise, the critical points correspond to saddle points. 

The types of critical points can be labelled by the number of less than zero. Specifically, the critical points are labelled by M. where is the number of which are negative i.e. a local minimum critical point would be labelled by Mq, a local maximum by and the saddle points by (M, M2). Each critical point has a characteristic line shape. For example, the critical point has a joint density of state which behaves as = constant x — ttiiifor co > coq and zero otherwise, where coq corresponds to thcAfQ critical point energy. At... 

For a given pair of valence and conduction bands, there must be at least one and one critical points and at least tluee and tluee critical points. However, it is possible for the saddle critical points to be degenerate. In the simplest possible configuration of critical points, the joint density of states appears as m figure Al.3.19. 

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

Each maximum, minimum or saddle point occurs at a so-called critical point Tc, where the gradient vanishes. The nature of the critical point is determined by the eigenvalues of the Hessian. All the eigenvalues are real at the critical point, but some of them may be zero. The rank co of the critical point is defined to be the number of non-zero eigenvalues. The signature o is the sum of the signs of the eigenvalues, and critical points are discussed in terms of the pair of numbers (w, o). 

Points on the zero-flux surfaces that are saddle points in the density are passes or pales. Should the critical point be located on a path between bonded atoms along which the density is a maximum with respect to lateral displacement, it is known as a pass. Nuclei behave topologically as peaks and all of the gradient paths of the density in the neighborhood of a particular peak terminate at that peak. Thus, the peaks act as attractors in the gradient vector field of the density. Passes are located between neighboring attractors which are linked by a unique pair of trajectories associated with the passes. Cao et al. [11] pointed out that it is through the attractor behavior of nuclei that distinct atomic forms are created in the density. In the theory of molecular structure, therefore, peaks and passes play a crucial role. 

Figure 6.12 Classification of all types of extremum or critical point that can occur in one-, two-, and three-dimensional functions a one-dimensional function can possess only a maximum or a minimum a two-dimensional function has maxima, minima, and one type of saddle point a three-dimensional function may have maxima, minima, and two types of saddle point. The arrows schematically represent gradient paths and their direction. At a maximum all gradient paths are directed toward the maximum, whereas at a minimum all gradient paths are directed away from the minimum. At a saddle point a subset of the gradient paths are directed toward the saddle point, whereas another subset are directed away from the saddle point (see Box 6.2 for more details).

The four maxima and the saddle point are critical points in the function L(r) analogous to the maxima and saddle points in p(r) discussed in Chapter 6. Every point on the sphere of maximum charge concentration of a spherical atom is a maximum in only one direction, namely, the radial direction. In any direction in a plane tangent to the sphere, the function L does not change therefore the corresponding curvatures are zero. When an atom is part of... 

TABLE 7. Calculated relative energies at the Cl level (in kcalmol )21S (ATM = absolute true minima, TM = true minima, SP = saddle point, CP2 = critical point of index 2)... 

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

Fig. 7.1 The electron density p(t) is displayed in the and

The nature of a critical point is denoted as (rank, signature). For example, a minimum in p is designated by (3, + 3) and a maximum by (3,-3). The two remaining types of critical points, (3,-1) and (3, +1) are saddle points, called bond critical point and ring critical point, respectively. 

At a critical point, Vp(r) equals zero because each of the three contributions to Eq. (6.18) are zero. The classification of critical points is based on the second derivatives, which as noted above are all negative for a density maximum, but have different signs for saddle points and minima of the distribution. 

Passes or saddle points two curvatures are negative, and, at rc, p is a maximum in the plane defined by the axes corresponding to the negative curvatures p is a minimum at rc along the third axis which is perpendicular to this plane. The (3, — 1) critical points are found between every pair of nuclei considered linked by a chemical bond. 

The saddle point between two atoms is a (3, — 1) critical point. The saddle point is the origin of the gradient vectors along the direction in which the density is a minimum. The gradient vectors in this direction link the (3, — 1) critical point with the atoms, and constitute the bond path, connecting the atoms. In the plane perpendicular to the bond path at the (3, — 1) critical point, the gradient vectors terminate as illustrated for the two-dimensional case in Fig. 6.7. 

A (3, 1) critical point indicates a saddle point at p = 0. A typical... 

The electron density is a continuous function that is experimentally observable, hence uniquely defined, at all points in space. Its topology can be described in terms of the distribution of its critical points, i.e. the points at which the electron density has a zero gradient in all directions. There are four kinds of critical point which include maxima (A) usually found near the centres of atoms, and minima (D) found in the cavities or cages that lie between the atoms. In addition there are two types of saddle point. The first (B) represents a saddle point that is a maximum in two directions and a minimum in the third, the second (C) represents a saddle point that is a minimum in two direction and a maximum in the third. One can draw lines of steepest descent connecting the maxima (A) to the minima (D), lines whose direction indicates the direction in which the electron density falls off most rapidly. Of the infinite number of lines of steepest descent that can be drawn there exists a unique set that has the property that, in passing from the maximum to the minimum, each line passes successively through a B and a C critical point. This set forms a network whose nodes are the critical points and whose links are the lines of steepest descent connecting them. 

To carry out the proof, we note that the LMA equations are equivalent to imposing an extremum on some function of the concentrations—the free energy F for v = const or the thermodynamic potential for p = const. It is in precisely this way, as is well known, that the LMA may be derived from general thermodynamic principles. We will solve the problem if we prove that the surface F or, under the conditions imposed on the concentration and for constant v or p, has one and only one minimum, and does not have either maxima or any other critical points (so-called minimax, or saddle points). 

Near a critical point, the parent p coexists with another phase that is only slightly different if, as we assume here, the free energy function is smooth, these two phases are separated—in p-space—by a hypothetical phase which has the same chemical potentials but is (locally) thermodynamically unstable. [This is geometrically obvious even in high dimensions between any two minima of f p)—p p, at given p, there must lie a maximum or a saddle point, which is the required unstable phase. ] Now imagine connecting these three phases by a smooth curve in density space p(e). At the critical point, all three phases collapse, and the variation of the chemical potential around p e = 0) = p must therefore obey... 

A fourth type of approach relies on analyzing the overall electron density, rather than the density contributed by individual orbitals. A particularly popular analysis of this type is the atoms in molecules analysis (AIM).6 Variation of the density through space can be shown to map onto the bonding pattern. The presence of a bond between two atoms is revealed by the presence of a saddle point, or bond critical point, in the density near the bond midpoint. 

Fig. 5.41 The distribution of the electron density (charge density) p for a homonuclear diatomic molecule X2. One nucleus lies at the origin, the other along the positive z-axis (the z-axis is commonly used as the molecular axis). The xz plane represents a slice through the molecule along the z-axis. The —p = f(x, z) surface is analogous to a potential energy surface E = /(nuclear coordinates), and has minima at the nuclei (maximum value of p) and a saddle point, corresponding to a bond critical point, along the z axis (midway between the two nuclei since the molecule is homonuclear)...

This transition-state-like point is called a bond critical point. All points at which the first derivatives are zero (caveat above) are critical points, so the nuclei are also critical points. Analogously to the energy/geometry Hessian of a potential energy surface, an electron density function critical point (a relative maximum or minimum or saddle point) can be characterized in terms of its second derivatives by diagonalizing the p/q Hessian([Pg.356]

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