Transition metals

Published

November 12, 2025

Transition metals have either cubic (FCC, BCC) or Hexagonal (HCP) symmetry. This perturbs the OAM of the atoms by

$s-d$ orbital mixing (no continuous rotation symmetry),
$Y_\ell^m$ mixing of different $m$ numbers.

Parity is still a good quantum number (thus no $s-p$ mixing). The available $m$-states ($d$-waves, $\ell=2$) are then $2\ell+1=5$. These $5$ states are degenerate in the single atom and it is appropriate to hybridize them to have cubic-symmetry (atoms in HCP have cubic nearest-neighbor symmetry, it is until n.n.n. that it is not cubic).

Mixing

The orbitals available are \[ \begin{align} \Phi_{x_ix_j}&=x_ix_j,\quad i\neq j,\quad x_i\in\{x,y,z\}\tag{$t_{2g}$},\\ \Phi_{x^2-y^2} &= \frac{1}{2}(x^2-y^2),\tag{$e_{g}$}\\ \Phi_{z^2} &= \frac{1}{2\sqrt{3}}(3z^2-r^2).\tag{$e_{g}$} \end{align} \]

This set of orbitals is not invariant under cubic symmetry (e.g. $\pi/2$ rotation about $z$-axis). The crystal potential/field splits and lifts the levels. However, even though there is some orbital degrees of freedom (see quenching), which gives rise to OAM and OHE, such as in kontaniGiantOrbitalHall2009

# “Autogenerated link references”