transition-orbitals

While there is quenching, the orbitals are split into two degenerate groups (\(t_{2g}\) and \(e_g\)). However, one can rotate the \(t_{2g}\) orbitals into each other. These “rotations” (\(t_{2g}-p\) isomorphism) allow such orbitals to have OAM (and thus are slightly less quenched). Degeneracy lifting within both groups is possible, but is not on the scale of the original cubic-symmetry lifting (i.e. \(|E_{t_{2g}}-E_{e_g}|\))

\(s\) and \(p\)-orbitals

The overlap too strongly to be considered by a Tight-Binding picture. This is why there is little contribution to the DOS (as well from \(p\) orbitals). These are similar to a free-electron gas (which scales the Fermi energy with lattice constant \(a\) as \(\epsilon_F\propto a^{-2}\))

\(d\)-orbitals

Can be described by a Tight-binding picture, i.e. \[ \begin{aligned} u_{\mathbf{k}}(\mathbf{r})=\sum_je^{-i\mathbf{k}\cdot(\mathbf{r}-\mathbf{R}_j)}\sum_{\beta} f_\beta\Phi_\beta,\\ \psi_\mathbf{k}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_\mathbf{k}(\mathbf{r}) \end{aligned} \] with \(\beta\in\{xy,xz,yz,x^2-y^2,z^2\}\). In other words, \(H\) is a \(5\times 5\) block Hamiltonian. Each band will contain \(2\times (2\ell +1)\) degeneracies (spin and OAM).

Any compound with a non-symmetric ocupancy of the \(t_{2g}\) orbitals exhibits OAM and deviates from the quenching arguments.

Most of the DOS contribution comes from the \(d\)-orbitals

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