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10: Angular Momentum and Group Symmetries of Electronic Wavefunctions

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    Electronic wavefunctions must also possess proper symmetry. These include angular momentum and point group symmetries

    • 10.1: Angular Momentum Symmetry and Strategies for Angular Momentum Coupling
      Any acceptable model or trial wavefunction for a multi-electron systme should be constrained to also be an eigenfunction of the symmetry operators that commute with H. If the atom or molecule has additional symmetries (e.g., full rotation symmetry for atoms, axial rotation symmetry for linear molecules and point group symmetry for nonlinear polyatomics), the trial wavefunctions should also conform to these spatial symmetries.
    • 10.2: Electron Spin Angular Momentum
      Individual electrons possess intrinsic spin characterized by angular momentum quantum numbers \(S\) and \(m_s\). The proper spin eigenfunctions must be constructed from antisymmetric (i.e., determinental) wavefunctions.
    • 10.3: Coupling of Angular Momenta
      The simple "vector coupling" method applies to any angular momenta. If the angular momenta are "equivalent" in the sense that they involve indistinguishable particles that occupy the same orbital shell, the Pauli principle eliminates some of the expected term symbols. For linear molecules, the orbital angular momenta of electrons are not vector coupled, but the electrons' spin angular momenta are vector coupled. For non-linear polyatomic molecules, on spin angular momenta is vector coupled.
    • 10.4: Atomic Term Symbols and Wavefunctions
      When coupling non-equivalent angular momenta (e.g., a spin and an orbital angular momenta or two orbital angular momenta of non-equivalent electrons), one vector couples using the fact that the coupled angular momenta range from the sum of the two individual angular momenta to the absolute value of their difference.
    • 10.5: Atomic Configuration Wavefunctions
      To express, in terms of Slater determinants, the wavefunctions corresponding to each of the states in each of the levels, one proceeds as follows in this section.
    • 10.6: Inversion Symmetry
      One more quantum number, that relating to the inversion (i) symmetry operator can be used in atomic cases because the total potential energy V is unchanged when all of the electrons have their position vectors subjected to inversion (i.e., ir=-r ). This quantum number is straightforward to determine.
    • 10.7: Review of Atomic Cases

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