Multi-electron atom: example Be atom

Multi-electron atom: example Be atom

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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 2nds-Orbitals with the two electrons of the 1s-Peel. By removing an electron from the 2nds-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+: RHFBe+: UHF
E (1e)-19,227701-17,471354
E (2e)4,8758213,373511
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 (pz)0,2211-0,1451-0,1885
ε (py)0,2211-0,1451-0,1885
ε (px)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.

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