OAI.NeutralAtom.generalized_outer_radii

The theorem states that, for every choice of wavefunctions Ψ_N for neutral atoms with N+1 electrons and nuclear charge Z=N+1 (each a spin-dependent complex function of N+1 positions in three-dimensional space, with spins taking two values), such that each Ψ_N is a normalized ground state, the outer radii of the atoms obey a Thomas–Fermi-type scaling law. A normalized ground state is an antisymmetric wavefunction with a weak gradient, square-integrable in each spin component together with its gradient, with finite Coulomb integrals against |Ψ|², total squared norm 1 summed over spins, and minimal energy among all such normalized functions. The energy is half the squared L² norm of the gradient plus the expectation of the Coulomb potential, which is −Z times the sum of inverse distances of electrons to the nucleus plus the sum of inverse inter-electron distances over pairs. The electron density is N+1 times the spin-summed integral of |Ψ|² over the other N positions, and the radius for m is the infimum of r≥0 such that the density mass outside the ball of radius r is at most m. For each m, upperRadius and lowerRadius are the limsup and liminf, in the extended reals, of these radii as N→∞ with N+1>m. With bTF=(81π²/2)^(1/3), the theorem states that both m^(1/3)·upperRadius(m) and m^(1/3)·lowerRadius(m) converge to bTF as m→∞ through the natural numbers.