The limiting behaviour of atomic wave functions for large atomic number

Abstract

If P(nl; r) is the ( nl ) radial wave function in an atom of atomic number N , and P H ( nl; r ) is the corresponding wave function of hydrogen, then, for a given configuration and for large N N -1/2 P(nl; r) = P H (nl;Nr) +(1/ N ) Q(nl;Nr) + O (1/ N 2 ) The equations for the functions Q(nl; Nr) have been set up and solved for a number of ( nl ) values and configurations of up to twenty-eight electrons. From the solutions, the limiting values as N -> oo of certain screening numbers o( nl ) have been determined, so that estimation of o( nl ) for atoms of atomic number higher than any for which calculation of wave functions frag been carried out becomes a process of interpolation instead of extrapolation. It is found that for given configuration o( nl ) is nearly linear in the mean radius r over the whole range from N ->oo to the neutral atom. For a given value of r/r(nl) , r 1/2 P N (nl; r) is nearly linear in r. The r derivatives of this function at r = 0 can also be evaluated from the Q(nl; Nr) functions.