Improve LargeSigmaHandler

[esoteric-ephemera]

Context

The LargeSigmaHandler for VASP calculations tries to ensure that the energy contribution to the electronic free energy due to Fermi-level smearing ("smearing entropy") is not in excess of 1 meV/atom. This recommendation comes from VASP.

Currently, this only checks for order >= 1 Methfessel-Paxton smearing (ISMEAR >= 1), probably because the VASP manual doesn't make it clear that Gaussian smearing (ISMEAR = 0) is just order-0 Methfessel-Paxton.

For Fermi-Dirac smearing (ISMEAR = -1), one can simulate the effect of finite electronic temperature, so it's not a good idea to check the entropy in this case.

Changes

This PR ensures that the smearing entropy is monitored for ISMEAR >= 0, and now looks at the smearing entropy at the end of each ionic step, rather than just the last step.

Additionally, the method by which SIGMA (the smearing width) is reduced has been changed based on the work of dos Santos and Marzari. (See note below.)

The test for this handler has also been updated.

Copying @matthewkuner who brought this issue up and worked with me to get a fix together

Explanation of new smearing extrapolation

Near the SIGMA = 0, the free energy has a low-order Taylor expansion: $F(\sigma) \approx E(\sigma \to 0) + \frac{1}{2} \gamma \sigma^2$ where $F(\sigma)$ is the free energy due to smearing, $E(\sigma \to 0)$ is the electronic energy without smearing (the quantity we usually want), and $\gamma$ is related to the second derivative of the smearing entropy wrt $\sigma$.

One can then extrapolate to what the best reduced sigma value $\sigma'$ is by demanding that the smearing energy is $E_s=1$ meV/atom: $\sigma' = [2 E_s / \gamma]^{1/2} \sigma$. Practically, we also impose $\sigma' = \max(0.01, 0.8 \cdot [2 E_s / \gamma]^{1/2} \sigma)$ to ensure that $\sigma'$ is not too small.

The factor of 0.8 helps ensure that the threshold is overshot slightly. The previous LargeSigmaHandler decreased SIGMA too slowly in many cases.