Curvature, critical principal

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 Hessian matrix H(r) is defined as the symmetric matrix of the nine second derivatives 82p/8xt dxj. The eigenvectors of H(r), obtained by diagonalization of the matrix, are the principal axes of the curvature at r. The rank w of the curvature at a critical point is equal to the number of nonzero eigenvalues the signature o is the algebraic sum of the signs of the eigenvalues. The critical point is classified as (w, cr). There are four possible types of critical points in a three-dimensional scalar distribution ... 

This behaviour of p(r) is to be contrasted with that displayed at the midpoint of the carbon-carbon axis. In the planes shown in Fig. 2.1(a) and (b), p(r) has the appearance of a saddle at this point, while in the plane perpendicular to and bisecting the carbon-carbon axis, p(r) is a maximum at this same point. In this case p(r) is a maximum in one particular plane. Knowledge of p(r) in one or two dimensions is insufficient to characterize its three-dimensional form. What is needed is a method of summarizing in a precise manner the principal topological features of a charge distribution. This information is provided by the curvatures of p(r) at its critical points (Collard and Hall 1977 Smith et al. 1977). 

An eigenvalue and its associated eigenvector of the Hessian of p (a principal curvature and its associated axis) at a critical point define a onedimensional system. If the eigenvalue or curvature is negative, then p is a maximum at the critical point on this axis and a gradient vector will approach and terminate at this point from both its left- and right-hand side as illustrated in Fig. 2.6 for the case (1, — 1), a system of rank 1 and signature... 

Extrema in the Laplacian of p are classified by rank and signature in the same way as are critical points in the charge density. The critical points in the Laplacian occur where V(V p) = 0 and the eigenvalues of the Hessian of V p are the principal curvatures of V p at the critical point. The topological discussions will always refer to the negative of the Laplacian, the quantity... 

The critical voltage Vc required for stable electrospraying can be estimated through a balance between the capfllary stress, of order y// , where/ lO -lO m is the principal curvature of the meniscus, of the same order as the radial capillary dimension, and the electric stress where is the liquid permittivity, V the applied voltage, and d the electrode separation ... 

From a thermodynamic perqiective, it is immaterial whether r or P is used to probe the range of critical pore sizes. The only relevant variables in the Kelvin equation, in addition to file radius of curvature, are yaad and it should be possible to correct for the T variations of these parameters. In principal, the pore size distribution should not depend upon the combination of T and P used to probe a particular critical pore size. For exanqile, sufqiose that... 

---