Electronic structure calculations without SCF?

Question

Is there an ab-initio calculation of electronic structure that does not involve self-consistent iteration? What I mean by self-consistent iteration is the iteration to solve the equation $$ \mathbf{y} = \mathbf{f}(\mathbf{y}, p), $$ for some parameters $\mathbf{y}$.

Answer 1

Yes! The Schroedinger equation for a molecule is just a differential equation, so you can solve it numerically "on a grid" or in "real-space", and there is a review paper on the topic: "Real-space numerical grid methods in quantum chemistry".

You can also simply model the wavefunction $|\psi\rangle$ with some formula, and estimate the energy using:

$$\tag{1} E \le \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi| \psi \rangle }, $$

and keep adjusting the parameters in your model for $|\psi \rangle$ until your estimate of $E$ no longer gets lower. This is called the "variational method" and does not require a self-consistent-(mean)field.

Every example in the following answers, uses the variational method without an "SCF" calculation:

• How accurate are the most accurate calculations?
• High precision helium energy
• Are there examples of ab initio predictions on small molecules without the "major approximations"?

The list may grow, without anyone remembering to add new examples to this answer, so I recommend you look at the high-precision tag!

Answer 2

One can solve SCF-like problems (e.g. KS-DFT/HF) by directly minimizing the energy with respect to the MO coefficients, rather than actually performing self-consistent iterations. This was done recently in J. Chem. Theory Comput. 2021, 17, 1, 151–169, where they modeled the energy landscape (distribution of minima and saddle points) for $\ce{H4}$. The trick to this, as described in the paper, is finding an appropriate parameterization of the MO coefficients such that your minimization procedure is always moving among feasible solutions. This sort of optimization is one way in which you can find a higher energy SCF minima, which have been used as starting points for calculating excited states of a molecule.

Answer 3

As has already been pointed out by Tyberius and Nike Dattani, a commonly-used alternative to the self-consistent field method is to use direct minimization methods for the energy, where the SCF problem is reformulated as a minimization in terms of the molecular orbitals (MOs) / MO coefficients. You can read about both approaches in our recent review paper, Molecules 25, 1218 (2020). Several recent fully numerical approaches also solve the SCF problem using iterative integral equations with the Helmholtz kernel, where you obtain updated orbitals from the current ones through numerical quadrature.

The connection between the two is that when you write the energy $E$ in terms of the orbitals ${\mathbf C}$ $$E=E(\mathbf{C})$$ you can either minimize this directly with respect to your parameters $\mathbf{C}$, or set the variation with respect to $\mathbf{C}$ to zero that gives you the SCF equations. In either approach, you have to maintain orbital orthonormality with Lagrange multipliers, which give you the orbital energies. Moreover, both approaches require an iterative solution.