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For many applications with heavy metals, pseudopotentials can be used to include some amount of relativistic effects. But for what sort of systems does it become necessary to actually use a relativistic method (e.g Dirac Hartree Fock)? Is this there active research into material modeling with a relativistic formalism?

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    $\begingroup$ Great question! $\endgroup$ May 1 '20 at 22:49
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When to include relativistic corrections or modeling of any kind in computational methods is a rather complex one. Full Dirac methods as you asked about (DHF) recapture two important factors, so called scalar relativistic effects, and spin effects. I'll elaborate on each and when including them is important.

Scalar Relativistic Effects - This largely describes the so called "mass/velocity" relationship. You may be familiar that as a massive object moves with great velocity it gains mass (so that its velocity can never truly reach or surpass the speed of light). In the case of the electron, its effective velocity is a function of $Z_\text{eff}$ (in atomic units $Z_\text{eff}/c \approx v$ or for say hydrogen $1/137 = 0.007c$). This is why psuedo-potentials can easily approximate this effect, generally the largest $Z_\text{eff}$ is for electrons near the core (1s 2s 2p etc.) but valence electrons experience only a small percentage of the $Z_\text{eff}$ due to screening. The caveat is you may realize is that d and f orbitals experience much less screening, but are valence orbitals for the transition metals, lanthanides, and actinides. For lighter transition metals the effect is still minor but by Actinides especially, valence electrons can be moving $0.3-0.5c$. However, once your valence becomes a p or s again, you no longer have such a significant valence effect. Thus, lead can be fine with a pseudo-potential, where uranium would not be.

TL;DR : If you're worried about scalar relativistic effects you're looking at lanthanides and actinides

Spin effects - (this relates to the Pauli principle and electrons having spin$= \pm 1/2$). This is a much more complicated issue. Since spin effects are proportional to total atomic spin (or molecular) it depends on the charge and occupancy of your species. For example, molecular oxygen (ground state triplet) has very large spin effects (on the order of 0.5-1.0 kcal/mol), yet molecular nitrogen has almost none. Conversely, Ni(II) (ground state triplet) has abnormally small spin effects, even though it has a similar total spin, the considerations of the orbitals themselves also plays a role. Here is where computational chemists generally start to use (the often reviled) heuristic considerations. In the case of worrisome spin effects, you should either rely on experimental evidence that they are important or otherwise do extensive research to determine if they might play a role.

TL;DR: Spin effects are a guessing game, but if this is your area of research, endeavor to be an expert on which things have large spin effects in general.

Finally, materials modeling with relativity! The main consideration here is that DHF and other relativistic methods are QM methods with worse (roughly squared of the base scaling of a non-relativistic method, e.g. DHF is $O(N^{4-6})$ since HF is $O(N^{2-3})$ on most computers, though a book might tell you HF is formally $O(N^4)$, with computational tricks it's cheaper than that) scaling. Since materials modeling with QM methods is (to my knowledge) still in its adolescence, this makes full or even partial relativistic materials modeling more or less cutting edge science.

TL;DR : Send me a copy of your publication if you manage to figure out relativistic materials modeling on a non-super computer.

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  • $\begingroup$ Very well explained answer $\endgroup$
    – Thomas
    May 2 '20 at 21:43

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