Are there singularities in the equations of Physical Chemistry, similar to those that appear in Classical Electromagnetism, General Relativity, Cosmology and Aerodynamics? What would be their physical implication, interpretation and consequences?
Of course, yes. To give an example, from quantum chemistry, let z be a perturbation parameter and suppose that we computed E(z) function by using Rayleigh-Schrödinger perturbation series (n>3 order). This function involves singularity points in the complex z plane which are responsible for the convergence of the perturbation series. In terms of Schrödinger equation (and conformal mapping) these singularities manifests some changes in molecular geometry or in basis set. Such points may connect the ground state to an excited state of the molecule (generally to a low lying excited state or the one as its lowest symmetry). Some other points behave as critical points upon dissociation. MP4 and CCSD(T) type computations reveal such phenomena.
Of course, yes. To give an example, from quantum chemistry, let z be a perturbation parameter and suppose that we computed E(z) function by using Rayleigh-Schrödinger perturbation series (n>3 order). This function involves singularity points in the complex z plane which are responsible for the convergence of the perturbation series. In terms of Schrödinger equation (and conformal mapping) these singularities manifests some changes in molecular geometry or in basis set. Such points may connect the ground state to an excited state of the molecule (generally to a low lying excited state or the one as its lowest symmetry). Some other points behave as critical points upon dissociation. MP4 and CCSD(T) type computations reveal such phenomena.
As far as seen, singularities exist everywhere in natural sciences. For a simple example, let us consider the electrostatic Poisson equation in spherically symmetric geometry. The highest-order (second-order) term in it is the usual one, whereas the lowest-order (first-order) term is a special one arising due to the non-planar (curvature) geometric effects. Clearly, the origin of our reference system is a singularity, to be removed in principle, for formulational analysis.
Presence of "singularity" usually signifies "phase transition". It is obvious in statistical physics, condensed matter physics, etc. Geometrically, herein, it implies a coordinate transition from "planar" to "nonplanar" system, and so on.
Singularities have nothing to do with Physical Chemistry in particular, they can be anywhere Math, Physics, Chemistry, Statistics, Economics, even in social sciences.
Point is that if you have a function f(x), then it is always possible to expand it in powers of x, that is in a power series, something like f(x)=∑cn xn where n runs from minus infinity to plus infinity. Coefficients cn determine whether there is a singularity in f(x) at x=0. If for you are interested to know whether there is a singularity at x=a, then shift the origin at x=a and expand it in power series. If a coefficient for negative n is non vanishing then this function has a singularity.
As an example if f(x)=1/(1+x3) then you can expand it as, ∑ (-1)n x 3n for |x|=0
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