Lithium's Hartree-Fock ground state is the Slater determinant built from the…

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · Many-electron atoms and the periodic table · Many-electron atoms and the periodic table

Deriving the Spectrum, Ch Self-consistent fields and correlation

Problem

Lithium's Hartree-Fock ground state is the Slater determinant built from the spin-orbitals

1s\uparrow

1s\downarrow

2s\uparrow

, where

1s

and

2s

are self-consistent Hartree-Fock spatial orbitals. ((a)) Write the Hartree-Fock energy of this determinant in terms of the one-electron integrals

\langle 1s|h|1s\rangle

\langle 2s|h|2s\rangle

and the Coulomb and exchange integrals

J_{1s,1s}

J_{1s,2s}

K_{1s,2s}

, stating which pair of electrons contributes the exchange term. ((b)) The Hartree-Fock orbital energy of the

2s

electron is

\epsilon_{2s}=-0.1963\,E_{\rm h}

. Use Koopmans' theorem to estimate lithium's first ionisation energy in eV. The measured value is

5.392\,\mathrm{eV}

. State whether the Koopmans estimate is too large or too small, and explain the sign in terms of orbital relaxation and correlation.

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