Beryllium's Hartree-Fock ground configuration is the closed-shell determinant .

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

Beryllium's Hartree-Fock ground configuration is the closed-shell determinant

1s^22s^2

. ((a)) Using the closed-shell Hartree-Fock energy

E_{\rm HF}=2\sum_a\langle a|h|a\rangle+\sum_{a,b}(2J_{ab}-K_{ab})

, summed over the doubly-occupied spatial orbitals

a,b\in\{1s,2s\}

, write out

E_{\rm HF}

explicitly in terms of

\langle1s|h|1s\rangle

\langle2s|h|2s\rangle

J_{1s,1s}

J_{2s,2s}

J_{1s,2s}

and

K_{1s,2s}

. ((b)) The Hartree-Fock orbital energy of the

2s

electron is

\epsilon_{2s}=-0.3092\,E_{\rm h}

. Estimate beryllium's first ionisation energy by Koopmans' theorem, in eV. The measured value is

9.323\,\mathrm{eV}

. State whether Koopmans over- or underestimates it, and explain the sign, contrasting with lithium's

2s

electron.

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