Muon spin rotation experiments involve muons that experience zero-point vibration at their implantation sites. Quantum-mechanical calculations of the host material usually treat the muon as a point impurity, ignoring its zero-point vibrational energy that, however, plays a role in determining the stability of calculated implantation sites and estimating physical observables. As a first-order correction, the muon zero-point motion is usually described within the harmonic approximation, despite the anharmonicity of the crystal potential. Here we apply the stochastic self-consistent harmonic approximation, a quantum variational method devised to include anharmonic effects in total energy and vibrational frequency calculations, in order to overcome these limitations and provide an accurate ab initio description of the quantum nature of the muon. We applied this full quantum treatment to the calculation of the muon contact hyperfine field in textbook-case metallic systems, such as Fe, Ni, Co including MnSi and MnGe, improving agreement with experiments. Our results show that there are anharmonic contributions to the muon vibrational frequencies with the muon zero-point energies above 0.5 eV. Finally, in contrast to the harmonic approximation, we show that including quantum anharmonic fluctuations, the muon stabilizes at the octahedral site in bcc Fe.
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