The Thomas-Fermi energy is the leading term of an expansion for large
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Atomic, Molecular & Optical Physics · Many-electron atoms and the periodic table
Deriving the Spectrum, Ch Self-consistent fields and correlation
Problem
The Thomas-Fermi energy ETF=−0.7687Z7/3Eh is the leading term of an expansion for large Z, and the terms after it are smaller by powers of Z−1/3. Non-relativistic Hartree-Fock gives −128.547Eh for neon (Z=10) and −21866.8Eh for radon (Z=86).
((a)) For each atom find the gap Δ=EHF−ETF and the fractional error Δ/∣EHF∣. Show that if Δ grows as Z2, the fractional error falls roughly as Z−1/3, and test this on the two atoms.
((b)) Assume Δ=cZ2. Find c separately from neon and from radon, and say what their disagreement tells you.
((c)) Now assume Δ=cZ2−dZ5/3, one more power of Z−1/3. Solve the two atoms' equations together for c and d, and compare with c=21 (Scott, 1952) and d=0.2699 (Schwinger, 1981), both derived from the theory itself.
((d)) The innermost electrons see nearly the bare potential −Z/r, with hydrogen-like levels −Z2/2n2 holding 2n2 electrons each. Fill the lowest N shells exactly, then count the same electrons as a continuum in n, as a local Fermi-gas model does, and show that the continuum count puts them lower in energy by Z2/2Eh for large N. Explain why this makes the first correction positive and of order Z2.