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11: Evaluating the Matrix Elements of N-electron Wavefunctions

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  • Page ID
    60568
    • Jack Simons and Jeff Nichols
    • University of Utah and Oak Ridge National Laboratory

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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
      This page covers configuration state functions (CSFs), which are wavefunctions resembling eigenstates but not exact ones for the many-electron Hamiltonian. The N-electron wavefunction \(\Psi\) is represented as a linear combination of CSFs in a configuration-interaction (CI) expansion, aiding variational methods in quantum chemistry.
    • 11.2: The Slater-Condon Rules Give Expressions for the Operator Matrix Elements Among the CSFs
      This page covers the formation of the \(H_{K,L}\) matrix through Slater-Condon rules, which focus on one- and two-electron operators. It explains how these rules simplify the evaluation of matrix elements between determinants by examining differing spin-orbitals and accounting for permutations.
    • 11.3: The Slater-Condon Rules
      This page explains the Slater–Condon rules, which simplify the computation of integrals involving one- and two-body operators for N-electron wavefunctions in Slater determinant form. It highlights how integrals can be expressed using sums of individual one- and two-electron integrals, addressing the differences in wavefunctions and their impact on these integrals.
    • 11.4: Examples of Applying the Slater-Condon Rules
      This page covers the application of Slater-Condon rules for calculating total energy contributions in atomic systems, particularly Carbon's \(^3P\) level. It discusses electric dipole matrix elements, the equivalence of \(2p\) orbitals, and configuration interaction (CI) calculations for beryllium.
    • 11.S: Evaluating the Matrix Elements of N-electron Wavefunctions (Summary)
      This page details Chapter 11, which focuses on the Slater-Condon (SC) rules that simplify the calculation of matrix elements for one- and two-electron operators in quantum chemistry. It explains how these rules enable the conversion of complex matrix elements into simpler integrals over molecular orbitals using configuration state functions (CSFs).

    This page titled 11: Evaluating the Matrix Elements of N-electron Wavefunctions was last modified on Fri, 28 Apr 2023 07:39:43 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jack Simons and Jeff Nichols via source content that was edited to the style and standards of the LibreTexts platform.