Hyperbolic singular point

Fig. 4. 32. Potential singular point surfaces and stable node bifurcation behavior for an intrinsically hyperbolic system ...

The ZA(f) are chosen so that they tend to 1 in the vicinity of atom A but drop to zero in the direction of all other nuclei. Thus, even for integrals involving atomic basis functions of two different atoms the integrand of each contribution la has no more than one singular point. The integration can be further simplified by suitable transformations to intrinsic coordinates, e.g. elliptic-hyperbolic coordinates for diatomic molecules or spherical coordinates for polyatomic systems. 

SDtj j is hyperbolic. The reason this flip is possible is that /3 is ramified in fact there are 2 finite points, 7=0 and j = 123 at which 3 1 —y SDTi sx is respectively triply and doubly ramified, and 1 infinite point j = oo over which the /3 s are arbitrarily highly ramified. From another point of view, 3i sx admits a canonical metric with negative curvature, i.e., ds2 = dx2+dy2/y2, (if z = x+iy 6 H 3i,i is the coordinate). This induces a negatively curved metric on each SDtfj. In this metric, SUtf x has finite volume, but the metric has singularities, a) at points... 

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