Thomas–Fermi limit and leading corrections for atoms and ions

Abstract

The Z−1 perturbation expansion is used to derive a formula of the energy of an ion in the limit of large nuclear charge Z and number of electrons N of the form E=∑nZ(7−n)/3 fn(q) where q=N/Z and fn(q)=q(1−n)/3[b0n+b 1nq+b2nq2] with b1n≂[(7−n)/3]Cn−2b0n and b2n≂b0n −[(4−n)/3]Cn. The constants b0n correspond to the asymptotic expansion of the zero-order perturbation coefficient ε0(N) and the constants Cn correspond to the neutral atom binding energy E=∑nCnZ(7−n)/3. The first function f0(q), which corresponds to the Thomas-Fermi limit, is then used to obtain approximate analytical expressions for the first derivative at the origin S(q) and the radius of the ion, χ0(q), of the Thomas-Fermi screening function. The expressions for f0(q), S(q), and χ0(q) provide an excellent representation of the numerical solutions. The function f1(q) is used to show that the value of the coefficient of the leading correction to the Thomas-Fermi energy C1 is 1/2. Finally, it is shown that the description of the ratio of the total energy and the nuclear-electron attraction energy is greatly improved over the Thomas-Fermi values by including the leading corrections.