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11: Evaluating the Matrix Elements of N-electron Wavefunctions

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    One must be able to evaluate the matrix elements among properly symmetry adapted N-electron configuration functions for any operator, the electronic Hamiltonian in particular. The Slater-Condon rules provide this capability

    • 11.1: Configuration State Functions can Express the Full N-Electron Wavefunction
      A given electronic configuration can yield several space- and spin- adapted determinental wavefunctions referred to as configuration state functions (CSFs). These CSF wavefunctions are not the exact eigenfunctions of the many-electron Hamiltonian; they are simply functions which possess the space, spin, and permutational symmetry of the exact eigenstates. As such, they comprise an acceptable set of functions to use in, for example, a linear variational treatment of the true states.
    • 11.2: The Slater-Condon Rules Give Expressions for the Operator Matrix Elements Among the CSFs
      To form the Hamiltonian matrix elements, one uses the so-called Slater-Condon rules which express all non-vanishing determinental matrix elements involving either one- or two- electron operators (one-electron operators are additive).
    • 11.3: The Slater-Condon Rules
      The Slater–Condon rules express integrals of one- and two-body operators over wavefunctions constructed as Slater determinants of orthonormal orbitals in terms of the individual orbitals. In doing so, the original integrals involving N-electron wavefunctions are reduced to sums over integrals involving at most two molecular orbitals, or in other words, the original 3N dimensional integral is expressed in terms of many three- and six-dimensional integrals.
    • 11.4: Examples of Applying the Slater-Condon Rules
      It is wise to gain some experience using the Slater-Condon rules, so let us consider a few illustrative example problems.
    • 11.S: Evaluating the Matrix Elements of N-electron Wavefunctions (Summary)
      In all of the examples in Chapter 11, the Slater-Condon rules were used to reduce matrix elements of one- or two- electron operators between determinental functions to one- or two- electron integrals over the orbitals which appear in the determinants. In any ab initio electronic structure computer program there must exist the capability to form symmetry-adapted CSFs and to evaluate, using these rules.

    This page titled 11: Evaluating the Matrix Elements of N-electron 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.