The self-consistent field method utilized for solving the Hartree-Fock (HF)
problem and the closely related Kohn-Sham problem, is typically thought of as
one of the cheapest methods available to quantum chemists. This intuition has
been developed from the numerous applications of the self-consistent field
method to a large variety of molecular systems. However, as characterized by
its worst-case behavior, the HF problem is NP-complete. In this work, we map
out boundaries of the NP-completeness by investigating restricted instances of
HF. We have constructed two new NP-complete variants of the problem. The first
is a set of Hamiltonians whose translationally invariant Hartree-Fock solutions
are trivial, but whose broken symmetry solutions are NP-complete. Second, we
demonstrate how to embed instances of spin glasses into translationally
invariant Hartree-Fock instances and provide a numerical example. These
findings are the first steps towards understanding in which cases the
self-consistent field method is computationally feasible and when it is not.Comment: 6 page
