d2dz2

d2dz2 (also written d_z^2 or d(2z^2−x^2−y^2)) refers to one of the five d atomic orbitals. In the standard cubic Cartesian basis the orbital is defined by the angular part proportional to 3cos²θ−1, where θ is the angle measured from the z axis. The spatial distribution of the d_z^2 orbital consists of two lobes along the z axis separated by a toroidal donut shape around the equatorial plane. This leads to one nodal surface that is a circle in the xy plane, while the remainder of the orbital remains non‑zero along the z direction.

The quantum mechanical description of the d2dz2 orbital uses principal quantum number n≥3, azimuthal quantum number l=2, and magnetic quantum number m_l=0. Because m_l=0, the angular part has no dependence on the azimuthal angle φ, giving the above axial symmetry. The radial part of the orbital is governed by the usual hydrogenic radial equations for d orbitals, with n determining the number of radial nodes.

In molecular chemistry, the d_z^2 orbital plays a significant role when the symmetry of a ligand field is approximately axial. For example, in a linear transition metal complex or a square–planar complex with axial ligands, the d_z^2 orbital can participate in σ‑bonding or accept electron density from axial ligands. Due to its lobes along the z axis, this orbital can overlap with ligand orbitals oriented along the symmetry axis, whereas the other d orbitals (d_xy, d_xz, d_yz, d_x²–y²) are typically involved in π‑bonding or other σ‑interactions in the equatorial plane.

The d_z^2 orbital is also important in spectroscopy and ligand field theory. Its energy relative to the other d orbitals depends on the strength and geometry of the ligand field; high‑field ligands often raise its energy differently from that of the other d orbitals. Consequently, the splitting pattern of d orbitals in crystal field theory and the resulting electronic configurations of transition metal complexes can be influenced significantly by the behaviour of the d2dz2 orbital.