Configuration Interaction for the Helium Isoelectronic Series
Configuration interaction (CI) provides a systematic method for improving on single-configuration Hartree–Fock (HF) computations [1, 2]. This Demonstration considers the two-electron atoms in the helium isoelectronic series. The HF wavefunction is the optimal product of one-electron orbitals approximating the ground-state configuration. An improved representation of the ground state can be obtained by a superposition containing excited electronic configurations, including , , , …, with relative contributions determined by the variational principle.[more]
Shull and Löwdin  represented the total wavefunction in the form
where is the angle between and , while is the Legendre polynomial of degree . Note that the functions contain internal angular dependence but can still represent atomic states, such as and , with . In the computations presented in this Demonstration, we consider only the , , and contributions to CI. The contribution is taken as the HF function , which can be very closely approximated using double-zeta orbitals
The contribution is represented by the orthogonalized Slater-type function
Together, these two contributions can closely approximate the -limit to the CI function, as defined by Shull and Löwdin. The and contributions are represented using simple Slater-type orbitals:
All the relevant matrix elements of the Hamiltonian are then computed; for example,
and so forth. All energies are expressed in Hartree atomic units: .
Plots of the radial distribution function for each component configuration are also shown on the left. The relative magnitudes are not to scale.[less]
Snapshot 1: the single-configuration Hartree–Fock result
Snapshot 2: approximation to the -limit; a more accurate computation gives hartrees
Snapshot 3: result using optimized parameters; reaching is considered a milestone
 S. M. Blinder, "The Hartree–Fock Approximation," Mathematical Physics in Theoretical Chemistry (S. M. Blinder and J. E. House, eds.), Amsterdam: Elsevier, forthcoming.
 Wikipedia. "Configuration Interaction." (Feb 14, 2018) en.wikipedia.org/wiki/Configuration_interaction.
 H. Shull and P.-O. Löwdin, "Superposition of Configurations and Natural Spin Orbitals: Applications to the He Problem," Journal of Chemical Physics, 30(3), 1959 pp. 617–626. doi:10.1063/1.1730019.