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I have seen an article[1] that describes a HF approach to working with exotic muonic atoms. I am curious about how far DFT has been extended for modeling muonic chemistry. Are there psuedopotentials avilable in any code for muonic substitutions into an element for example or is this something DFT fails to capture correctly?

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Available implementations of DFT are capable of studying positive muons ($\mu^{+}$) because from an electronic point of view they are the same as protons (H$^{+}$). This effectively means that all calculations involving protons can be straight-forwardly re-interpreted as calculations involving muons (with the caveats detailed below). In this context, two situations are commonly studied: (i) a positive muon, which is equivalent to adding a positively charged proton (charged defect), (ii) muonium, a situation in which the positive muon binds to an electron (neutral defect). Although in principle it is straight-forward to simulate positive muons in DFT, here are a few things to consider:

  1. Defect calculations. Muons in matter are essentially defects, so simulations must take into account all subtleties that arise in defect calculations. For example, when using codes with periodic boundary conditions, one should consider the spurious defect-defect interaction (which is particularly severe for charged defects) by converging the simulation with respect to supercell size and/or applying appropriate correction schemes. This is entirely equivalent to what one would do to study protons. The canonical reference for studying defects in materials is here.
  2. Muon site. As with any defect calculation, an important question is what the structure of the defect is. In the case of muons, which are typically implanted in a material, the question is: where is the muon? The answer is not trivial and may require a full structure prediction exercise comparing the relative energies of multiple potential muon sites. This is entirely equivalent to what one would do to study protons. A recent paper studying muon sites can be found here.
  3. Quantum zero-point motion. Up to this point, the discussion has been entirely equivalent between $\mu^{+}$ and H$^+$. This is because these two particles are entirely equivalent from an electronic point of view. Where they differ is in their mass, and this is reflected in the vibrational contribution to the energy, which is mostly dominated by quantum fluctuations (zero-point motion) for light particles such as muons or protons. This means that to simulate muons one must carefully consider the vibrational contribution. A reasonable starting point would be to treat vibrations at the harmonic level of theory, but the light mass of the muon will almost always require the inclusion of anharmonic terms too. Another early interesting idea for simulating the vibrational properties of muons is a "double" Born-Oppenheimer approximation, in which the degrees of freedom of the system are separated into three (rather than the usual two) groups: electrons (small mass), muons (intermediate mass), and the rest of atoms in the system (large mass), as described here.

Once you have characterized the energetics of the muon as described above, many DFT codes also allow you to calculate the hyperfine tensor, describing the interaction between the electron spins and the muon spin. This is relevant for example in muon spin relaxation experiments, where muons are used to study the local magnetic structure of materials.

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    $\begingroup$ I will leave the bounty open for a bit to give someone a chance to write more of a negative muonic answer, but this is a very good answer. I did not realize before now that the positive muon would be this different from a proton (while still being almost like an isotope). $\endgroup$ Aug 29 '20 at 17:14

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