CCL: real-valued versus complex-valued molecular orbitals

I'm not sure whether I've understood your questions correctly. But here are my responses:
  1. The eigenfunctions of a Hermitian operator form a complete set. So if the Hamiltonian for the system is Hermitian, then in the limit of a complete basis (which is, of course, not realized in practice), an appropriate expansion in such a basis (whether real or complex) should give the true ground state.
  2. "that using complex valued orbitals would not reach a lower energy state" - this is forbidden by the variation theorem, the validity of which does not rely upon the real or complex nature of the basis.
  3. "Are there any kinds of chemical bonds that exist for complex-valued molecular orbitals that do not exist for real-valued molecular orbitals?" - A linear combination of degenerate eigenfunctions of a Hermitian operator will also be an eigenfunction with the same eigenvalue. So an appropriate linear combination of degenerate complex orbitals can be constructed to give a real orbital with the same energy. For example, if the Hamiltonian is real, psi and psi* will be degenerate, and psi+psi* will give a real orbital with the same energy.
  4. The symmetry labels sigma, pi, delta,... are strictly valid only for linear molecules. Since molecular orbitals typically extend over multiple centers, the appropriate symmetry designations to use for the molecular orbitals will be the irreducible representations of the point group of the molecule.
N. Sukumar
Professor of Chemistry
Director, Center for Informatics
Shiv Nadar University, India
https://chemistry.snu.edu.in/people/faculty/n-sukumar

"The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe."
- Phillip W. Anderson, in "More is Different" (1972)


On Tue, Sep 22, 2020 at 11:54 AM Thomas Manz thomasamanz**gmail.com <owner-chemistry() ccl.net> wrote:
Dear colleagues,

In my research, I've encountered a question about real-valued versus complex-valued molecular orbitals. Normally, most software programs that perform DFT, Hartree-Fock, coupled cluster, or configuration interaction type quantum chemistry computations on molecules use real-valued molecular orbitals. 

My first question is whether there exists any theorem that shows this will converge to the ground state (i.e., that using complex valued orbitals would not reach a lower energy state) when the Hamiltonian does not contain any spin-orbit coupling or applied magnetic field? In other words, when the multi-electronic Hamiltonian is comprised of the normal terms: electron kinetic energy, nuclear-electron potential energy, nuclear-nuclear potential energy, electron-electron Coulomb & exchange-correlation energies.

My second question is regarding periodic DFT calculations for which many software programs use complex-valued molecular orbitals. Are their any kinds of chemical bonds that exist for complex-valued molecular orbitals that do not exist for real-valued molecular orbitals? Are their any new bonding symmetries made possible for the complex-valued orbitals that cannot exist for the real-valued orbitals? For real-valued molecular orbitals, the primary covalent bond-orbital symmetries are sigma, pi, delta, and phi. Do complex-valued orbitals enable any additional bond-orbital symmetries?

I sincerely appreciate any insights into this topic you can provide.

Tom