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18: Multiconfiguration Wavefunctions

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    The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept.

    • 18.1: Optimization of the Energy for a Multiconfiguration Wavefunction
      For a multiconfigurational wavefunction the Slater-Condon rules can be used to decompose the energy in terms of one- and two-electron integrals. The resulting set of equations are called the CI-secular equations. and the Fock equations.
    • 18.2: The Single-Determinant Wavefunction
      The simplest trial function of the form given above is the single Slater determinant function, where the configuration Interaction part of the energy minimization is absent. The resulting Schrodinger equations within an orbital approximation can be substituted with canonical Hartree-Fock equations that explicitly include exchange energy terms.
    • 18.3: The Unrestricted Hartree-Fock Spin Impurity Problem
      As formulated above in terms of spin-orbitals, the Hartree-Fock (HF) equations yield orbitals that do not guarantee that Ψ possesses proper spin symmetry.
    • 18.4: Atomic Orbital Basis Sets
      The basis orbitals commonly used in the LCAO-MO-SCF process fall into two classes Slater-type orbitals and Cartesian Gaussian-type orbitals. STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals cannot efficiently be performed when STOs are employed in polyatomic molecule calculations. Such integrals can routinely be done when GTOs are used, which is the fundamental advantage that lead to the dominance of these functions in quantum chemistry.
    • 18.5: The LCAO-MO Expansion
      The new F operator then gives new φ_i and ε_i via solution of the new Fφ_i = ε_iφ_i equations. This iterative process is continued until the φ_i and ε_i do not vary significantly from one iteration to the next, at which time one says that the process has converged. This iterative procedure is referred to as the Hartree-Fock self-consistent field (SCF) procedure because iteration eventually leads to coulomb and exchange potential fi
    • 18.6: The Roothaan Matrix SCF Process
      The Roothaan SCF process is carried out in a fully ab initio manner in that all one- and two-electron integrals are computed in terms of the specified basis set; no experimental data or other input is employed. It is possible to introduce approximations to the coulomb and exchange integrals entering into the Fock matrix elements that permit many of the requisite elements to be evaluated in terms of experimental data or in terms of a small set of 'fundamental' orbital-level coulomb interaction in
    • 18.7: Observations on Orbitals and Orbital Energies
      A so-called generalized Brillouin theorem (GBT) arises when one deals with energy optimization for a multiconfigurational variational trial wavefunction for which the orbitals and CI mixing coefficients are simultaneously optimized. This GBT causes certain Hamiltonian matrix elements to vanish, which, in turn, simplifies the treatment of electron correlation for such wavefunctions. This matter is treated in more detail later in this text.

    This page titled 18: Multiconfiguration Wavefunctions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jack Simons via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.