d-wave superconductivity in quintuple-layer bulk superconductor La_6Ni_5O_{12}

The recent realization of bulk superconductivity in the quintuple-layer nickelate La_6Ni_5O_{12} opens new opportunities to probe the pairing symmetry of the reduced Ruddlesden--Popper (RP) nickelate family. Using an effective five-layer Hubbard model and variational Monte Carlo calculations, we find that a spin-singlet d_{x^2-y^2}-wave state is energetically favored, while extended-s-wave pairing is strongly suppressed. Based on the optimized superconducting state, we further calculate several experimentally accessible observables. The tunneling density of states exhibits a characteristic V-shaped low-energy spectrum, the superfluid density shows a linear low-temperature depletion, the Knight shift decreases below T_c, and the spin-lattice relaxation rate follows 1/T_1Tpropto T^2, together providing consistent signatures of a clean nodal superconductor. Extending the same framework to related reduced RP nickelates, we find a robust d-wave tendency in infinite-layer LaNiO_{2}, while trilayer La_4Ni_3O_{8} at its nominal filling lies close to the boundary between d-wave superconductivity and a non-superconducting state. These results establish bulk La_6Ni_5O_{12} as a promising platform for determining the pairing symmetry of nickelate superconductors and suggest a common d_{x^2-y^2}-wave pairing tendency across the reduced RP nickelate family.