Estimation of the on-site Coulomb potential and covalent state in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>La</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math> by muon spin rotation and density functional theory calculations

The on-site Coulomb potential, $U$, and the covalent state of electronic orbitals play key roles for Cooper pair symmetry exotic electromagnetic properties high-${T}_{\mathrm{c}}$ superconducting cuprates. In this paper, we demonstrate a way to determine value $U$ present whole picture Cu spins in mother system La-based cuprate, ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$, by combining muon spin rotation $(\ensuremath{\mu}\mathrm{SR})$ density functional theory (DFT) calculation. We reveal local deformations ${\mathrm{CuO}}_{6}$ octahedron followed changes Cu-spin distributions caused injected muon. Adjusting DFT $\ensuremath{\mu}\mathrm{SR}$ results, minimum charge-transfer energy between upper Hubbard band O $2p$ were optimized be 4.87(4) 1.24(1) eV, respectively.