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I'm just curious as to know if there are any examples in physical chemistry or condensed matter physics where the Dirac equation is preferable to the Shrodinger equation for making predictions on the material at hand?

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    $\begingroup$ This question (v1) seems like a list question. $\endgroup$
    – Qmechanic
    Commented Sep 18, 2016 at 23:12

3 Answers 3

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Graphene is a material that needs the Dirac equation for example. The electron band structure of this material has a closed gap some electrons have "mass=0", that can only we treated with the dirac equation. I dont know if this affects the chemical properties but it sure effects the electric ones.

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    $\begingroup$ Also used in relativistic quantum chemistry, which I had never heard of before, and don't really mind if I never hear of it again. $\endgroup$
    – user108787
    Commented Sep 18, 2016 at 23:34
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    $\begingroup$ You are rigth, hope I never cross paths with it. $\endgroup$
    – Victor
    Commented Sep 18, 2016 at 23:37
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Yes, relativistic effects become important for reactions involving hydrogen, for instance. There has been a sizable literature in the physical chemistry community on variations of the diagonalization of the Dirac equation (see e. g. the review of Reiher, Wiley Interdiscip. Rev. Comp. Mol. Sci. 2, 139–149, 2012).

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Besides some quite peculiar applications already cited in the first answers, it is worth mentioning the most prominent and frequent case: the description of inner core electrons in heavy atoms.

Core electrons are increasingly confined in small regions around the nuclei, and their average kinetic energy increases with the atomic number $Z$. As a matter of fact, within one-electron approximations, they require a description in terms of the Dirac equation. The proper relativistic treatment of inner core states is important even in a frozen-core approximation at the basis of the pseudopotential method. Indeed, the pioneering paper Bachelet, G. B.; Hamann, D. R.; Schlüter, M. (1982), "Pseudopotentials that work: From H to Pu", Physical Review B, 26, pp. 4199–4228, contained the first list of pseudopotentials obtained by solving Dirac's equation.

It is difficult to underestimate the importance of the relativistic treatment of the inner core states. Without relativity, these states would be more extended, and more polarizable, and many observable properties would be quite different from the actual ones.

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