Abstract From the electronic wavefunctions for 5d orbitals expressed in a common coordinate system, the radial part of the integrals and angular part of the hyperfine interaction integrals are evaluated and expressed in analytical forms. General formulas for the NMR chemical shifts arising from the 5d electron orbital angular momentum and the 5d electron spin dipolar-nuclear spin angular momentum interaction in zero crystal field and in weak and strong crystal-field environments of octahedral symmetry are derived. In zero crystal field the NMR chemical shift is independent of the angular coordinate. However, in a weak crystal-field environment of octahedral symmetry the ground state is split into three levels by the spin-orbit coupling and a weak crystal-field interactions but in a strong crystal-field environment of octahedral symmetry, the ground state is split into two levels by the spin-orbit coupling interaction. The NMR chemical shifts along the x, y, and z axes equal each other in a weak or strong crystal-field environment of octahedral symmetry. The 1 R 5 term contributes dominantly to the NMR chemical shift in a weak octahedral crystal field, while the major contribution arises from the 1 R 7 term in a strong octahedral crystal field. The temperature dependence analysis shows that the 1 T term contributes dominantly to the NMR chemical shift in a weak octahedral crystal field; however, the major contribution arises from the 1 T 2 term in a strong octahedral crystal field.