Spin contamination for Hartree-Fock, optimized effective potential, and density functional approximations.

Area of Science:

• Quantum Chemistry
• Computational Physics
• Theoretical Chemistry

Background:

• Slater determinants are fundamental in quantum chemistry for describing multi-electron wavefunctions.
• Unrestricted electronic structure methods (e.g., Hartree-Fock, DFT) can suffer from spin contamination, affecting accuracy.
• Spin contamination arises from the difference between the exact spin operator S(2) and the approximate wavefunction.

Purpose of the Study:

• To determine the spin contamination S(con) in unrestricted single determinant approximations.
• To derive an explicit formula for spin contamination based on orbital overlaps.
• To analyze the dependence of spin contamination on system properties like unpaired electrons.

Main Methods:

• Utilized an explicit formula for Slater determinant expansion in terms of S(2) eigenstates.
• Derived an expression for S(con) using the overlap of spatial parts of spin-up and spin-down orbitals.
• Investigated the behavior of S(con) for varying orbital overlaps and number of unpaired electrons.

Main Results:

• Spin contamination S(con) was found to depend on the overlap between corresponding spin-up and spin-down orbitals.
• For large orbital overlaps (lower-order approximations), S(con) is independent of M (S(z) eigenvalue).
• For small orbital overlaps, the dominant spin state increases with the number of unpaired electrons.

Conclusions:

• The derived formula provides a quantitative measure of spin contamination in common electronic structure methods.
• Understanding spin contamination is crucial for accurate predictions in systems with multiple unpaired electrons.
• The study offers insights into the limitations of unrestricted methods and the behavior of spin states.