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## Example: beryllium

In its ground state, the beryllium atom is a "closed-shell" system (spin multiplicity: singlet) with the electron configuration (1s)^{2} (2s)^{2}.

So-called single zeta functions STO-3G were used as basic functions for the following Hartree-Fock calculation.

The following table shows the energies (in atomic units, 1a.u. = 27.21 eV). These data are the results of Be^{+}, attached to the ionized atom (spin multiplicity: doublet). The Restricted Hartree Fock Data (RHF) take into account the coupling of the singly occupied 2nd*s*-Orbitals with the two electrons of the 1*s*-Peel. By removing an electron from the 2nd*s*-Shell, the remaining 3 electrons are bound more tightly (electron relaxation after ionization) and in particular the two-particle repulsion is reduced by the missing electron, but the energy of the ion is around the calculated adiabatic ionization potential (0.254037 au = 6.91 eV) decreased. Koopman's theorem applies.

- Tab. 1
- Table: HF-calculated one- and two-electron energies, total and orbital energies (in a.u.)

Be: | Be^{+}: RHF | Be^{+}: UHF | |
---|---|---|---|

E (1e) | -19,227701 | -17,471354 | |

E (2e) | 4,875821 | 3,373511 | |

Coulomb | 7,643888 | 5,895213 | |

Exchange | -2,768067 | -2,521702 | |

E (total) | -14,351880 | -14,097843 | -14,097843 |

ε1 | -4,4840 | -5,0331 | -5.0553 (α); -5.0109 (β) |

ε2 | -0,2540 | -0,6581 | -0.6581 (α); -0.2540 (β) |

ε (${\text{p}}_{\text{z}}$) | 0,2211 | -0,1451 | -0,1885 |

ε (${\text{p}}_{\text{y}}$) | 0,2211 | -0,1451 | -0,1885 |

ε (${\text{p}}_{\text{x}}$) | 0,2211 | -0,1451 | -0,1885 |

In an unrestricted Hartree-Fock calculation (UHF), a separate eigenvalue problem is solved for each of the α and β electrons. Since the number of both electrons is different, they do not have the same energy eigenvalues; make this clear from the right column of the table above.