Single cusps

The cusps which have just been described are called single cusps in contradistinction to double cusps or points of osculation in which the curves extend to both sides of the point of contact. These are what Cayley calls tacnodes. The differential coefficient has now two or more equal roots and y has at least two equal values. The different branches of the curve have a common tangent. 

This discussion will be limited to functions of one variable that can be plotted in 2-space over the interval considered and that constitute the upper boundar y of a well-defined area. The functions selected for illustration are simple and well-behaved, they are smooth, single valued, and have no discontinuities. When discontinuities or singularities do occur (for example the cusp point of the Is hydrogen orbital at the nucleus), we shall integrate up to the singularity but not include it. 

A completely different type of property is for example spin-spin coupling constants, which contain interactions of electronic and nuclear spins. One of the operators is a delta function (Fermi-Contact, eq. (10.78)), which measures the quality of the wave function at a single point, the nuclear position. Since Gaussian functions have an incorrect behaviour at the nucleus (zero derivative compared with the cusp displayed by an exponential function), this requires addition of a number of very tight functions (large exponents) in order to predict coupling constants accurately. ... 

However, it is indeed fortunate that the IV-representability problem for the electron density p(r) greatly simplifies itself. In fact, the necessary and sufficient conditions that a given p(r) be /V-representable are actually given by Equation 4.5 above. Nevertheless, question remains Can the single-particle density contain all information about a many-electron system, at least in its ground state An affirmative answer to this question can be given from Kato s cusp condition for a nuclear site in the ground state of any atom, molecule, or solid, viz.,... 

The differences between the single-configuration wavefunctions are more clearly illustrated by comparing their plots of the intracule function h(ri2), also shown in Fig. 1. This plot reveals the absence of an electron-electron cusp for both the closed and split-shell functions, but shows that the inclusion of exp( —yri2) causes the distribution to have a minimum at ri2=0, forming a cusp (of the correct sign) at that point. This feature will be important for the description of phenomena that depend upon the coincidence probability. 

The term local density approximation (LDA) was originally used to indicate any density functional theory where die value of fixe at some position r could be computed exclusively from the value of p at diat position, i.e., the local value of p. In principle, then, the only requirement on p is that it be single-valued at every position, and it can otherwise be wildly ill-behaved (recall that there are cusps in the density at the nucleus, so some ill-behavior... 

With the exponential approximation (y 0) and the assumption that the inflow and ambient temperatures are equal, we have a stationary-state equation which links ass to tres and which involves two other unfolding parameters, 0ad and tn. Depending on the particular values of the last two parameters the (1 — ass) versus rres locus has one of five possible qualitative forms. These different patterns are shown in Fig. 7.4 as unique, single hysteresis loop, isola, mushroom, and hysteresis loop plus isola. The five corresponding regions in the 0ad-rN parameter plane are shown in Fig. 7.5. This parameter plane is divided into these regions by a straight line and a cusp, which cut each other at two points. 

This corresponds to an ordinary cusp, with an unfolding to a unique locus or a single hysteresis loop. 

Fig. 7.8. The full unfolding of the stationary-state locus (a) corresponding to a winged cusp singularity, (b) unique (c) isola (d) mushroom (e) single hysteresis loop (f) hysteresis loop + isola (g) reverse hysteresis loop (h) reverse hysteresis loop + isola.

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