Finite size scaling for the atomic Shannon-information entropy

We have developed the finite size scaling method to treat the criticality of Shannon-information entropy for any given quantum Hamiltonian. This approach gives very accurate results for the critical parameters by using a systematic expansion in a finite basis set. To illustrate this approach we present a study to estimate the critical exponents of the Shannon-information entropy S approximately (lambda-lambda(c))(alpha(S) ), the electronic energy E approximately (lambda-lambda(c))(alpha(E) ), and the correlation length xi approximately mid R:lambda-lambda(c)mid R:(-nu) for atoms with the variable lambda=1/Z, which is the inverse of the nuclear charge Z. This was realized by approximating the multielectron atomic Hamiltonian with a one-electron model Hamiltonian. This model is very accurate for describing the electronic structure of the atoms near their critical points. For several atoms in their ground electronic states, we have found that the critical exponents (alpha(E),nu,alpha(S)) for He (Z=2), C (Z=6), N (Z=7), F (Z=9), and Ne (Z=10), respectively, are (1, 0, 0). At the critical points lambda(c)=1/Z(c), the bound state energies become absorbed or degenerate with continuum states and the entropies reach their maximum values, indicating a maximal delocalization of the electronic wave function.