Modern computational quantum chemistry has developed largely from applications of the Hartree–Fock method to atoms and molecules [1–3]. A simple representation of a many-electron atom is given by a Slater determinant constructed from  occupied spin-orbitals:  , where spin-orbitals are products of the form  . A Slater determinant automatically satisfies the Pauli exclusion principle, requiring that spin-orbitals be, at most, singly occupied. The Hamiltonian for an  -electron atom, in hartree atomic units, is given by:  . The corresponding approximation to the total energy is then given by  , where  ,  and  are, respectively, the core, coulomb and exchange integrals. This Demonstration carries out a simplified Hartree–Fock computation on the ground states of the atoms He to Ne,  to 10. Assume a single Slater determinant, thus possibly shortchanging open-shell configurations. The computation involves only  ,  and  orbitals, approximated by modified Slater-type orbitals:  ,  ,  . These are normalized and mutually orthogonal, which simplifies the computation. The orbital parameters  are chosen so as to minimize the total energy, in accordance with the variational principle. You can vary these using the sliders. Actually, since the possible values of  vary over inconveniently large ranges, the sliders vary internal parameters that determine them. The values of  are shown in the graphic. The blue bar shows the exact energy of the atomic ground state (actually, the exact nonrelativistic value) in hartrees. The red bar shows the calculated energy, given the displayed values of  . You should try to adjust the three parameters to get the lowest possible energy. You can never reach the exact energy, however, since a single Hartree–Fock determinant fails to account for correlation energy, which involves instantaneous electron-electron interactions.
Following are results for optimized functions ψ1 s, ψ2 s and ψ2 p. For comparison we also include results from the best Hartree-Fock computations and the exact atomic ground-state energies. [1] S. M. Blinder, "Simplified Hartree-Fock Computations on Second-Row Atoms," https://arxiv.org/abs/2105.07018 [2] A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Mineola, NY: Dover, 1996. [3] S. M. Blinder, "Introduction to the Hartree–Fock Method," in Mathematical Physics in Theoretical Chemistry (S. M. Blinder and J. E. House, eds.), Elsevier, 2018.
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