Questions: Electron Correlation in Multi-Electron Atoms

Question 1

The Hartree-Fock energy of helium is −2.862 hartree; the exact ground-state energy is −2.904 hartree. The correlation energy is −0.042 hartree. What does the negative sign tell you?

• A. Electron correlation destabilizes the atom — electrons repel each other, raising the energy
• B. The exact energy is lower than Hartree-Fock because correlated electrons avoid each other, reducing their mutual repulsion
• C. Hartree-Fock overestimates correlation, so the correction is subtracted
• D. The sign is a convention and has no physical meaning

Answer: (B)

The negative correlation energy means the true ground-state energy is lower (more stable) than Hartree-Fock predicts. The physical reason is that real electrons are correlated — they actively avoid each other. When electron 1 is on the left, electron 2 is more likely to be on the right. This instantaneous avoidance means the electrons spend less time close together than the mean-field picture assumes, reducing their average mutual repulsion energy. The Hartree-Fock energy is always an upper bound to the true energy; correlation always stabilizes. Option A inverts this logic — correlation reduces repulsion, it does not increase it.

Question 2

The Hartree-Fock method explicitly includes electron-electron repulsion in its energy expression. Why does it still miss the correlation energy?

• A. It ignores the kinetic energy contribution from electron motion
• B. It treats each electron as moving in the average field of all other electrons, missing the instantaneous, position-dependent avoidance between electrons
• C. It uses an incomplete basis set that cannot represent the true wavefunction
• D. It only applies to two-electron systems, making it inaccurate for larger atoms

Answer: (B)

Hartree-Fock does include electron repulsion, but only on average. Each electron sees a smeared-out electrostatic cloud representing the mean position of all other electrons. In reality, electron positions are instantaneously correlated — when one electron ventures left, the other is preferentially on the right. This instantaneous avoidance cannot be captured by any mean-field theory, no matter how large the basis set. The missing physics is the dynamic correlation between electrons, not a computational limitation. Basis set incompleteness is a separate, additional source of error.

Question 3

The correlation energy for any atom or molecule is always negative — the exact non-relativistic energy is always lower than the Hartree-Fock energy in a complete basis set.

• T. True
• F. False

Answer: True

This is a rigorous result from the variational principle. The Hartree-Fock wavefunction — a single Slater determinant — is a restricted trial wavefunction. The exact wavefunction minimizes energy over all possible wavefunctions, and since the exact wavefunction has more freedom, it always finds at least as low an energy as Hartree-Fock. Equality holds only if the exact wavefunction happens to be a single Slater determinant (which is approximately true only for hydrogen-like atoms). For any multi-electron system with electron-electron interactions, the correlation energy E_corr = E_exact − E_HF is always negative.

Question 4

The Hartree-Fock method ignores electron-electron repulsion mostly, which is why post-HF methods are needed to obtain accurate energies.

• T. True
• F. False

Answer: False

This is a fundamental misconception. The Hartree-Fock method explicitly includes electron-electron repulsion — it computes the energy of each electron in the average Coulomb field created by all other electrons. That is the defining feature of the self-consistent field calculation. What Hartree-Fock misses is not repulsion itself but the instantaneous correlation between electron positions (the mean-field approximation). Post-HF methods (configuration interaction, coupled cluster, Møller-Plesset perturbation theory) correct for this correlation, not for an absent repulsion term.

Question 5

Explain in physical terms why the true ground-state energy of a multi-electron atom is lower than the Hartree-Fock prediction.

Model answer: In Hartree-Fock, each electron moves in the smooth average field of all other electrons. In reality, electrons are point charges that repel each other instantaneously — when one electron approaches, others move away. This instantaneous avoidance means electrons spend less time in close proximity than the mean-field picture assumes. Less proximity means less repulsion energy on average, which lowers the total electronic energy below the Hartree-Fock prediction. The exact wavefunction encodes this correlated motion; the Hartree-Fock wavefunction, being a product of independent orbitals, cannot.

This physical picture describes 'dynamic correlation' — the dominant contribution for most closed-shell molecules. There is also 'static correlation,' which matters when the system is near-degenerate (e.g., bond breaking), where a single Slater determinant is qualitatively wrong even before considering the fine details of electron avoidance. The correlation energy is chemically significant — tens to hundreds of kJ/mol — because even small deviations from the mean-field picture accumulate across all pairs of electrons in the system.