Power Series Expansion of Collision Amplitudes. Electrostatic Potentials

By means of a conformal transformation t= K/(K2+α2)1/2 (where, in physical applications, K is the momentum change of a colliding electron) the complex K plane is mapped onto the complex t plane. With cuts from iα to i ∞ and from −iα to −i ∞, the mapping is one to one. Let I be the ionization potential from a one-electron orbital φi and W the excitation potential from φi to φj. Then if α=(2I)1/2+[2(I−W)]1/2 a region of the K plane, which contains (a) no singular point of the form factor ε=∫ exp(iK · r) φiφj*(dr) and (b) the entire real axis of K, is mapped into the unit circle in the t plane. Hence, when ε is expanded into a power series in t the series converges when t<1 and the region of convergence includes all real values of K. Moreover, the coefficients of the power series in t can be simply obtained from the coefficients of a series for ε in powers of K. This extends to the collision amplitude a result which had previously been obtained for the generalized oscillator strength [E. N. Lassettre, J. Chem. Phys. 43, 4479 (1965)]. The extension is nontrivial since the power series in K for ε contains both even and odd powers while that for oscillator strength contains only even powers of K. A particular series expansion for ε, which takes account of the behavior of ε at both large and small K, can be exploited to obtain a series expansion for an electrostatic potential function. This is possible because the Fourier transform of the potential function is proportional to ε/K2. Hence, an inversion gives the potential function. A particular expansion, which expresses the potential function in terms of a set of generalized moments, is obtained, discussed in detail, and illustrated with some simple examples.