Neural Wavefunction Calculations of μSR Spectra with Quantum Muons and Protons

Abstract

Accurate prediction of muon hyperfine constants is useful for interpreting muon spin spectroscopy data, yet standard methods such as density functional theory (DFT) compute muon-electron pair density functions, and thus hyperfine constants, by treating the muon as a fixed classical particle. This work uses the variational quantum Monte Carlo method with neural-network trial wavefunctions, a highly accurate and flexible approach recently applied to other quantum chemical problems. The muon can be treated classically or included in the many-particle electron-muon wavefunction, in which case the fully quantum mechanical pair density is obtained directly. We calculate muon hyperfine constants in muoniated methyl and ethyl radicals for both quantum mechanical and fixed classical muons. The hyperfine constants obtained from our fixed-muon calculations in the methyl and ethyl radicals differ from the corresponding DFT results significantly, highlighting the limitations of DFT even when the muon is treated classically. The results with quantum muons are closer to experiment after accounting for environmental effects. These findings suggest that explicitly calculating the quantum mechanical muon-electron pair density improves the accuracy of muon hyperfine constant predictions.

Figures (5)

• Figure 1: Electron spin density and muon density of the muoniated methyl radical (CH$_\text{2}$Mu) with a quantum muon, projected into the XY, XZ, and YZ planes. The molecular structure is largely coplanar with the XY plane. The magnitude and sign of the electron spin density are indicated by blue-red colors, while the magnitude of the muon density is indicated in green. The electronic spin density is concentrated near the carbon atom, while the muon density is delocalized and centered some distance away.
• Figure 2: 10% isosurfaces of the muon and proton densities in the ground state of the muoniated methyl radical (CH$_\text{2}$Mu) obtained from a full quantum Psiformer simulation. The proton density is indicated in red and the muon density in green. The densities were accumulated in $5.12 \times 10^{−4} \;a_0^{\,3}$ cubic bins and smoothed using a Gaussian convolution with $\sigma = 0.06 a_0$. Contour projections of the binned function are shown on the axes. The quantum muon is more delocalized than the protons due to its lower mass.
• Figure 3: Translationally averaged muon-electron spin density for the muoniated methyl radical (CH$_\text{2}$Mu) in the classical muon, quantum muon, and full quantum systems. The muon-electron spin density is extrapolated to obtain $\Delta\rho(r=0)$ using a quadratic fit in the log domain. The extrapolated value and its statistical error determine the isotropic muon hyperfine constant $A_\upmu$ and its uncertainty.
• Figure 4: Electron spin density and muon density of the muoniated ethyl radical (C$_\text{2}$H$_\text{4}$Mu) with a quantum muon, projected into the XY, XZ, and YZ planes. The magnitude and sign of the spin density are indicated by blue-red colors while the magnitude of the muon density is indicated in green. The electronic spin density is concentrated near the carbon atom at $X\approx-1.5 a_0$, while the muon density is delocalized some distance from the other carbon atom at $X\approx1.5 a_0$.
• Figure 5: Translationally averaged muon-electron spin densities of the muoniated ethyl radical (C$_\text{2}$H$_\text{4}$Mu) in the classical muon and quantum muon systems. The muon-electron spin density is extrapolated to obtain $\Delta\rho(r=0)$ using a quadratic fit in the log domain. The extrapolated value and its statistical error determine the isotropic muon hyperfine constant $A_{\upmu}$ and its uncertainty.