# Can we relate approximate Hartree-Fock orbitals to true solutions in the basis-set limit?

Assume that we've "solved" the RHF Hartree-Fock equations in some finite basis, that is, we have arrived at a self-consistent set of coefficients $$c_{\mu{}i}$$ such that each spatial molecular orbital is expanded as $$\chi_i = \sum_{\mu=1}^K c_{\mu{}i}\varphi_i$$. By construction, this set of orbitals is orthonormal, and forms a collection of eigenfunctions for the Fock operators in the given basis, so $$f_i |\chi_i\rangle = \varepsilon_i |\chi_i\rangle$$ where $$\varepsilon_i$$ is the $$i$$th orbital energy.

Question: what, if anything, can we say about these orbitals in terms of their relationship to the "true" orbitals in the basis set limit, denoted $$\chi_i^\mathrm{BS}$$? Here, I mean those orbitals that would be obtained if the calculation were to be extended to an (infinite) complete basis set. (I realise that the assumed existence of a single unique set of true orbitals may not be justified...)

For example, is it possible to estimate, bound, or otherwise describe the overlap $$\langle \chi_i | \chi_i^\mathrm{BS} \rangle$$ between an approximate solution and the true solution? How about matrix elements such as $$\langle \chi_i | \mathcal{H} | \chi_i^\mathrm{BS} \rangle$$, where $$\mathcal{H}$$ is the electronic Hamiltonian, or indeed any self-adjoint operator?

I suspect/hope that some kind of a statement towards the first part might be possible, given that the Roothan-Hall equations are effectively a collection of Galerkin equations. One of the standard hand-waving statements about solutions obtained using Galerkin methods is that they are in some sense projections of the true solution onto finite-dimensional subspaces, and have residual vectors that are "orthogonal" to that subspace(for a potentially complicated definition of "orthogonal").

(I'm also interested in answers that would apply in any of the HF settings, i.e. UHF, ROHF, etc.)

• This is a very interesting question. The answer also has many possible practical uses that have not been explored. For instance, extrapolation from the finite basis to the complete basis is often done to estimate the CBS energy, but as far as I'm aware this has rarely, if ever, been attempted for the orbitals themselves. I suppose this is because one generally cares less about the actual HF orbitals than observables associated with them. I'll be interested to see any answers. – jheindel Oct 7 '20 at 16:26