  In muon spin rotation experiments the positive implanted muon vibrates with
large zero point amplitude by virtue of its light mass. Quantum mechanical
calculations of the host material usually treat the muon as a point impurity,
ignoring this large vibrational amplitude. As a first order correction, the
muon zero point motion is usually described within the harmonic approximation,
despite the large anharmonicity of the crystal potential. Here we apply the
stochastic self-consistent harmonic approximation, a quantum variational method
devised to include strong 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, significantly improving agreement with experiments. Our results show that
muon vibrational frequencies are strongly renormalized by anharmonicity.
Finally, in contrast to the harmonic approximation, we show that including
quantum anharmonic fluctuations, the muon stabilizes at the octahedral site in
bcc Fe.
