# 3 Transition-metal ions in crystals

When an atom is embedded in a solid, the main difference with respect to the free ion is that electrons are also affected by the electrostatic potential created by charges outside the atom itself. The aim of *ligand-field theory* is that of evaluating the effect of neighboring atoms – referred to as *ligands* – on the energy levels of the atom under consideration. Within this theory, the effect of the ligands is taken into account by the symmetry and the strength of the electric field produced by them. In its original formulation, developed by Bethe in 1929, ligands were treated as point charges (see Solomon-Lever). With this simplification, the approach goes under the name of *crystal-field theory*. Already from the first attempts to turn the Bethe approach into a quantitative calculation, it was clear that it produced a splitting of the energy levels of transition-metal ions far smaller than the one observed in experiments. Van Vleck (1935) demonstrated that this mismatch originated from neglecting the overlap between the paramagnetic-ion orbitals and the ligand valence orbitals.

In transition metals the magnetic orbitals (i.e., the 3d orbitals containing unpaired electrons) are rather delocalized and have a comparatively high degree of covalency, namely they participate in the chemical bonding. For these reasons the magnetic d orbitals are strongly affected by the crystal-field interaction and the strength of this interaction is comparable to the intra-atomic exchange interaction (1-2 eV). In other words, the crystal-field interaction is stronger than the spin-orbit coupling. The hierarchy of contributions to the Hamiltonian of a transition-metal ion reads \[\begin{equation} \tag{3.1} \mathcal{H} = \mathcal{H}_0 + \mathcal{H}^1_{\rm ee} + \mathcal{H}_{\rm cf} + \mathcal{H}_{\rm so} + \mathcal{H}_{\rm Z}\,. \end{equation}\] As seen in Section 2, the Hamiltonian \(\mathcal{H}_0\) on the r.h.s. of Eq.(3.1) includes the kinetic energy of electrons of the atom, their Coulomb interaction with the nucleus, and the part of electron-electron repulsion that can be written as a central potential. The orbital eigenstates of the \(\mathcal{H}_0\) are the same as those of the hydrogen atom. If terms of higher order than \(\mathcal{H}_0\) are neglected in the electronic Hamiltonian \(\mathcal{H}\), the spin and orbital eigenstates can be expressed on a basis of mutually commuting operators: the hydrogen-like Hamiltonian (\(\mathcal{H}_0\)), \(\hat{\mathbf L}^2\), \(\hat{\mathbf S}^2\), \(\hat{L}^z\), \(\hat{S}^z\). These eigenstates are called Russell-Saunders terms. A perturbative treatment of the remaining contribution of the electron-electron repulsion (\(\mathcal{H}_{\rm ee}^1\)) removes part of the degeneracy and defines a ground-state multiplet consistent with the first two Hund’s rules. The next contributions on the right-hand side of Eq.(3.1) represent the crystal-field interaction (\(\mathcal{H}_{\rm cf}\)), the spin-orbit coupling (\(\mathcal{H}_{\rm so}\)) and the Zeeman energy (\(\mathcal{H}_{\rm Z}\)). In transition-metal ions the strength of interactions on the r.h.s. of Eq.(3.1) decreases from left to right. We will see that in rare-earth ions the relative position of \(\mathcal{H}_{\rm cf}\) and \(\mathcal{H}_{\rm so}\) is swapped.

## 3.1 Crystal-field theory for 3d ions

In the following pages we sketch an instructive calculation^{10}that estimates how the free-ion energy levels are split due to the presence of the ligands. As we do not aim at quantitative predictions, we will treat ligands as point charges (

*crystal-field theory*). We will restrict ourselves to placing these point charges – which mimic the effect of the orbitals of neighboring atoms – at the corners of a square (Fig.3.1) or at the vertices of an octahedron (Fig.3.2) in which a magnetic d ion is supposed to be embedded. These tutorial examples are, nevertheless, of relevance in nowadays research. To the first order in perturbation theory, the effect of crystal-field interaction on single-electron levels can be expressed through the matrix elements

\[\begin{equation} \tag{3.2} \langle l',m'| \mathcal{H}_{\rm cf}|l,m\rangle \,, \end{equation}\]

where \(l\) and \(m\) are the quantum numbers associated with \(\hat{\mathbf l}^2\) and \(\hat{l}^z\) for one electron in a d orbital, namely they represent single-electron levels. Since the principal quantum number \(n\) is the same for all the d levels, it has been omitted for simplicity of notation. The spin has also been neglected because \(\mathcal{H}_{\rm cf}\) does not affect spin coordinates directly. The advantage of using point charges is that the potential associated with the electric field generated by one ligand onto the considered 3d ion can be expanded in spherical harmonics. Taking the nucleus of the paramagnetic ion as the origin of coordinates, the Coulomb energy associated with the interaction of the electron \(i\) with the point-like ligand \(h\) of charge \(Z_h e\) is \[\begin{equation} \tag{3.3} \begin{split} \mathcal{H}_{{\rm cf},h} &=\frac{Z_he^2}{4\pi \epsilon_0} \frac{1}{|\underline{r}_i-\underline{r}_h|} =\frac{Z_he^2}{4\pi \epsilon_0} \sum_{k=0}^{\infty}\frac{4\pi}{2k+1} \sum_{q=-k}^k \left[{\color{blue}Y^*_{k,q}(\theta_h,\phi_h)}\,\frac{r_i^k}{r_h^{k+1}} Y_{k,q}(\theta_i,\phi_i)\right]\,. \end{split} \end{equation}\] The spherical harmonics that are functions of the angular coordinates of the ligand are highlighted in blue. Summing this potential over all the ligands labeled by \(h\) gives the total energy of the crystal-field Hamiltonian acting on the \(i\)-th electron: \(\mathcal{H}_{\rm cf} = \sum_h \mathcal{H}_{{\rm cf},h}\). The dependence of \(\mathcal{H}_{{\rm cf},h}\) on the angular coordinates of the \(i\)-th electron put restrictions on the possible values of \(k\). Eventually, we will have to evaluate matrix elements of the type given in Eq.(3.2), containing terms like \[\begin{equation} \tag{3.4} \langle l',m'| Y_{k,q}(\theta_i,\phi_i)|l,m\rangle \sim \int Y_{l',m'}^*(\theta_i,\phi_i) Y_{k,q}(\theta_i,\phi_i)Y_{l,m}(\theta_i,\phi_i) d\Omega_i \,, \end{equation}\]

where the integral is performed over the solid angle \(d\Omega_i = \sin\theta_i\, d\theta_i\,d\phi_i\). For d orbitals (\(l=2\)) such terms vanish for \(k>4\), therefore we can restrict ourselves to \(k\leq 4\)

^{11}.

Let us start focusing on a square planar complex, associated with the \(D_{4h}\) point group (see Fig.3.1). The symmetry elements of the group impose additional restrictions to the spherical harmonics that are allowed in the expansion of \(\mathcal{H}_{{\rm cf},h}\), so that the most general crystal-field Hamiltonian for a square planar arrangement of ligands reads \[\begin{equation} \tag{3.5} \begin{split} \mathcal{H}_{D_{4h}} &= {\color{blue}\gamma_2^0} \, \rho^2\, Y_{2,0} (\theta_i,\phi_i) + {\color{blue}\gamma_2^2} \,\rho^2\, \left[Y_{2,2}(\theta_i,\phi_i) + Y_{2,-2}(\theta_i,\phi_i)\right] \\ & + {\color{blue}\gamma_4^0} \, \rho^4\, Y_{4,0} (\theta_i,\phi_i) + {\color{blue}\gamma_4^2} \,\rho^4\, \left[Y_{4,2}(\theta_i,\phi_i) + Y_{4,-2}(\theta_i,\phi_i)\right] \\ & + {\color{blue}\gamma_4^4} \,\rho^4\, \left[Y_{4,4}(\theta_i,\phi_i) + Y_{4,-4}(\theta_i,\phi_i)\right] \,, \end{split} \end{equation}\] where \(\rho=r_i/a\) with \(a\) distance between each ligand and the nucleus of the paramagnetic ion, thought in the center of a square (Fig.3.1 right); the \(\gamma_k^q\) coefficients (highlighted in blue for convenience) depend on the ligand angular coordinates \[\begin{equation} \tag{3.6} \gamma_k^q = \frac{4\pi}{2k+1} \,\frac{Z_he^2}{4\pi \epsilon_0} \,\frac{1}{a} \, \sum_h Y_{k,q}(\theta_h,\phi_h) \end{equation}\] and have the units of an energy. For this specific geometry it is \(\theta_h=\pi/2\) for every ligand and \(\phi_h=0,\pi/2,\pi, 3\pi/2\). The value of each \(\gamma_k^q\) coefficient is obtained inserting the appropriate spherical harmonic and summing over all the ligands. Assuming the charge \(Z_he^2\) to be the same for each ligand, for instance, the coefficient \(\gamma_2^0\) is obtained as follows \[\begin{equation} \tag{3.7} \begin{split} \gamma_2^0 &= \frac{4\pi}{5} \,\frac{Z_he^2}{4\pi \epsilon_0} \,\frac{1}{a} \, \frac{1}{4}\sqrt{\frac{5}{\pi}} \sum_h \left(3\cos^2\theta_h -1\right) \\ &= -4 \sqrt{\frac{\pi}{5}} \,\frac{Z_he^2}{4\pi \epsilon_0} \,\frac{1}{a} \,. \end{split} \end{equation}\] Any other coefficient can be determined performing the sum \(\sum_h\) of the trigonometric function corresponding to the specific spherical harmonic evaluated on the four ligands. Apart from constant prefactors, this trigonometric contribution to the \(\gamma_k^q\) coefficients that are relevant for the planar \(D_{4h}\) geometry are computed in Table.

### Matrix elements of \(\mathcal{H}_{D_{4h}}\): operator-equivalence method

In this paragraph we will evaluate explicitly the matrix elements of the crystal-field Hamiltonian on the single-electron basis \(|l,m\rangle\). This task is facilitated by expressing the spherical harmonics that are functions of the electron coordinates in terms of Racah tensor operators
\[\begin{equation}
\tag{3.5}
C_k^q=\sqrt{\frac{4\pi}{2k+1}} Y_{k,q}
\end{equation}\]
where \(k\) is the rank of the tensor and \(q\) its component. Note that we have omitted the explicit dependence on the coordinate \((\theta_i,\phi_i)\) because this more abstract representation will turn out to be useful. The reader who feels uncomfortable with this notation can always think of the \(C_k^q\) operators in real space, where they are just functions of \((\theta_i,\phi_i)\) proportional to the spherical harmonics, as defined in Eq.(3.5). In terms of the \(C_k^q\) operators, the crystal-field Hamiltonian \(\mathcal{H}_{D_{4h}}\) can be expressed in a very compact way
\[\begin{equation}
\tag{3.8}
\mathcal{H}_{D_{4h}} =21Dq\left[C_4^0 + \sqrt{\frac{5}{14}} \left(C_4^4 + C_4^{-4}\right)\right] -21Dt \,C_4^0 -7Ds\,C_2^0 \,,
\end{equation}\]
in which the crystal-field coefficients \(Dq\), \(Dt\), and \(Ds\) contain the numerical factors arising from the sum over the \(\gamma_k^q\) indices associated with each ligand (see Table) and the
average of \(r_i\) with respect to the radial part of the single-electron wave function \(R_{nl}(r_i)\). We give the crystal-field coefficients for the more general case of a 3d complex associated with a \(D_{4h}\) symmetry, in which the point-charge ligands occupy the vertices of a compressed/elongated octahedron:

\[\begin{equation}
\tag{3.9}
Dq=\frac{1}{6} \frac{Z_he^2}{4\pi \epsilon_0} \frac{\bar{r^4}}{a^5}
\qquad\, Ds=\frac{2}{7} \frac{Z_he^2}{4\pi \epsilon_0} \left(\frac{\bar{r^2}}{a^3}-\frac{\bar{r^2}}{b^3}\right)
\qquad\, Dt=\frac{2}{21} \frac{Z_he^2}{4\pi \epsilon_0} \left(\frac{\bar{r^4}}{a^5}-\frac{\bar{r^4}}{b^5}\right)
\end{equation}\]
The length \(b\) represents the distance between the nucleus of the paramagnetic ion and the ligands positioned on the axis perpendicular to the \(xy\) plane (Fig.3.2 right).
The bar indicates the average over the radial coordinates. If the “\(b\)” terms are omitted, the crystal field generated by a planar square geometry is obtained (Fig.3.1). For \(b=a\), instead, one has \(Dt=Ds=0\); this limit corresponds to the perfect octahedral symmetry (\(O_h\) point group, illustrated in Fig.3.2 left).

To determine how the degenerate levels \(|l,m\rangle\) – originating from the \(\mathcal{H}_0\) Hamiltonian – are split by the effect of the crystal field, one has to calculate the matrix elements of the Racah \(C_k^q\) operators on the basis \(|l,m\rangle\). Using the Wigner-Eckart theorem and other elegant results of tensor algebra, the following equation is obtained \[\begin{equation} \langle l',m'|C_k^q|l,m\rangle = (-1)^{m'}\sqrt{(2l+1)(2l'+1)} \begin{pmatrix} l' & k & l \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l' & k & l \\ m' & q & -m \end{pmatrix} \end{equation}\] where the term in parenthesis are just numerical coefficients called 3-\(j\) Wigner symbols. Nowadays routines to compute the 3-\(j\) symbols are available in standard environments, like Mathematica and Matlab, or even from online calculators. A 3-\(j\) symbol is zero if the sum of the components in the bottom raw (\(m' + q -m\)) is not zero, which restricts the coefficients that are to be computed explicitly. In the end, only the following non-zero terms remain \[\begin{equation} \tag{3.10} \begin{split} &\langle 2,2|C_4^0|2,2\rangle = \langle 2,-2|C_4^0|2,-2\rangle = \frac{1}{21} \\ &\langle 2,1|C_4^0|2,1\rangle = \langle 2,-1|C_4^0|2,-1\rangle = -\frac{4}{21} \\ &\langle 2,0|C_4^0|2,0\rangle = \frac{2}{7} \\ &\\ &\langle 2,2|C_2^0|2,2\rangle = \langle 2,-2|C_2^0|2,-2\rangle = - \frac{2}{7} \\ &\langle 2,1|C_2^0|2,1\rangle = \langle 2,-1|C_2^0|2,-1\rangle = \frac{1}{7} \\ &\langle 2,0|C_2^0|2,0\rangle = \frac{2}{7} \\ &\\ &{\color{blue} \langle 2,-2|C_4^{4}|2,2\rangle = \langle 2,2|C_4^{-4}|2,-2\rangle = \sqrt{\frac{10}{63}}}\,. \end{split} \end{equation}\] The last ones are the only off-diagonal matrix elements. We have now all what we need to compute the desired matrix elements which are given in Table. The eigenvalues of that matrix give the correction to the energies of the five levels \(|l,m\rangle\) that are degenerate with respect to the Hamiltonian \(\mathcal{H}_0\). The crystal-field Hamiltonian \(\mathcal{H}_{D_{4h}}\) generally removes this degeneracy leaving only one pair of levels with the same energy. However, in the most symmetric case of octahedral symmetry (\(O_h\) group), the five levels \(|l,m\rangle\) are split only into two multiplets, with degeneracy 2 and 3.

## 3.2 Orbital degeneracy and quenching of \(\hat{\mathbf L}\)

The matrix \(\langle l',m'|\mathcal{H}_{\rm cf}|l,m\rangle\) in given in Table. The resulting eigenvectors are directly related to the irreducible representations of the \(D_{4h}\) group. In other words, they are determined by the symmetry of the ligands surrounding the transition-metal ion under consideration. For this reason, these eigenvectors are called*symmetry-adapted wave functions*and, for the specific case of the matrix \(\langle l',m'|\mathcal{H}_{\rm cf}|l,m\rangle\) obtained from the Racah oprators Eq.(3.10), are the so-called real d orbitals \[\begin{equation} \tag{3.11} \begin{split} &d_{x^2-y^2}=\frac{1}{\sqrt{2}}\left[|2,+2\rangle+ |2,-2\rangle \right] \\ &d_{z^2}=|2,0\rangle \\ &d_{xy}=-\frac{i}{\sqrt{2}}\left[|2,+2\rangle - |2,-2\rangle \right] \\ &d_{xz}=-\frac{1}{\sqrt{2}}\left[|2,+1\rangle- |2,-1\rangle \right] \\ &d_{yz}=\frac{i}{\sqrt{2}}\left[|2,+1\rangle+ |2,-1\rangle \right] \,. \end{split} \end{equation}\] The real d orbitals are linear combinations of single-electron orbitals \(|l,m\rangle\) with \(l=2\) and \(m=-2,-1,0,+1,+2\) (spherical harmonics in the real space). For the reader’s convenience we provide a link to a 3D representation of the real d orbitals but other versions can easily be found on the web. The perfect octahedral symmetry (\(O_h\)) corresponds to choosing \(Dt=Ds=0\) in the matrix given in Table. In this special case, that matrix has only two distinct eigenvalues: \[\begin{equation} \tag{3.12} \begin{split} & E_{e_{g}} = 6Dq \\ & E_{t_{2g}}= -4Dq\,, \end{split} \end{equation}\]

two-fold (\(e_g\)) and three-fold (\(t_{2g}\)) degenerate, respectively. The \(t_{2g}\) triplet is composed by the orbitals \(d_{xy}\), \(d_{xz}\) and \(d_{yz}\), which lie at lower energy in perfect octahedral environment (\(O_h\) symmetry group). The orbitals \(d_{z^2}\) and \(d_{x^2-y^2}\), instead, form the \(e_g\) doublet placed at higher energy (see Fig.3.3 middle).

*any*of the \(d_{xy}\), \(d_{xz}\) and \(d_{yz}\) orbitals defining the \(t_{2g}\) multiplet, the energy of these three configurations being \(E_{t_{2g}}\). When \(B\ne 0\), the matrix elements of the Hamiltonian \(\mathcal{H}_{0}+\mathcal{H}_{\rm cf}+\mathcal{H}_{\rm Z}\) evaluated within the \(t_{2g}\) multiplet read

\(\mathcal{H}_{0}+\mathcal{H}_{\rm cf}+\mathcal{H}_{\rm Z}\) | \(|d_{xy}\rangle\) | \(|d_{xz}\rangle\) | \(|d_{yz}\rangle\) |
---|---|---|---|

\(\langle d_{xy}|\) | \(E_{t_{2g}}\) | 0 | 0 |

\(\langle d_{xz}|\) | 0 | \(E_{t_{2g}}\) | \(-i\mu_{\rm B}B\) |

\(\langle d_{yz}|\) | 0 | \(i\mu_{\rm B}B\) | \(E_{t_{2g}}\) |

This Hamiltonian is trivially diagonal on the basis set
\[\begin{equation}
\begin{split}
& |d_{xy}\rangle \\
& |2, +1\rangle = -\frac{1}{\sqrt{2}}\left[|d_{xz}\rangle + i |d_{yz}\rangle \right] \\
& |2, -1\rangle =\frac{1}{\sqrt{2}}\left[|d_{xz}\rangle - i |d_{yz}\rangle \right]
\end{split}
\end{equation}\]
with ordered eigenvalues
\[\begin{equation}
\begin{split}
& E_{d_{xy}}= E_{t_{2g}} \\
& E_{2, +1} = E_{t_{2g}} + \mu_{\rm B}B \\
& E_{2, -1} = E_{t_{2g}} - \mu_{\rm B}B \,.
\end{split}
\end{equation}\]
A single electron put in the \(t_{2g}\) multiplet thus contributes to the Zemman energy as if it had an *effective* orbital momentum \(L=1\). Later on we will see that in terms of multi-electron states this configuration is indicated as \(^2T_{2g}\). With respect to the free-ion term \(^2D_{3/2}\) – associated with one electron in the d shell – the crystal field has partially quenched the orbital contribution to the magnetic moment.

Similarly to \(\hat{l}_i^z\), the components \(\hat{l}_i^x\) and \(\hat{l}_i^y\) have non-zero matrix elements within the \(t_{2g}\) multiplet. The spin-orbit interaction is, therefore, effective at the first order in perturbation theory. In reality, in transition metals the spin-orbit interaction is typically stronger than the Zeeman energy. To proceed in a proper way, we should have considered \(\mathcal{H}_{\rm so}\) at least on the same footing as \(\mathcal{H}_{\rm Z}\). A detailed calculation of this type is beyond our scope. These few elements should anyway suffice to understand that a degenerate subset of the Hamiltonian \(\mathcal{H}_{0}+\mathcal{H}_{\rm cf}+\mathcal{H}_{\rm so}\) that is split in two levels by the application of a \(\vec{B}\) field behaves as an effective spin 1/2; when the splitting consists of three levels, the degenerate subset behaves as an effective spin 1, and so on. Of course, the Landé factor may differ significantly from \(g=1\) or 2.

For completeness, we evaluate also the orbital contribution to the Zeeman energy within the \(e_g\) multiplet. Still denoting with \(i\) the electron occupying either the \(d_{x^2-y^2}\) or the \(d_{z^2}\) orbital, we need to compute the matrix elements of \(\hat{l}^z_i\) on the basis defined by this pair of orbitals. It turns out that they are all zero, namely

\[\begin{equation}
\langle d_{x^2-y^2}|\hat{l}^z_i |d_{x^2-y^2}\rangle =
\langle d_{z^2}|\hat{l}^z_i |d_{z^2}\rangle =
\langle d_{z^2}|\hat{l}^z_i |d_{x^2-y^2}\rangle =0\,.
\end{equation}\]
The same result is obtained for the \(\hat{l}^x_i\) and \(\hat{l}^y_i\) components of the angular momentum. This means that the Zeeman interaction does not remove the two-fold degeneracy of the \(e_g\) doublet and that the spin-orbit interaction is not effective at the first order in perturbation theory. The multi-electron configurations characterized by an unpaired electron or a hole in the \(e_g\) doublet are named \(E_g\) and, in this regard, behave pretty differently from the \(T_{1g}\) and \(T_{2g}\) configurations (see next paragraph).

More details about this calculation can be found in the book

*Inorganic Electronic Structure and Spectroscopy*, Vol. I: Methodology, Editors: E. I. Solomon and A. B. P. Lever (1999, John Wiley and Sons, Inc.) Chapter: “Ligand Field Theory and the Properties of Transition Metal Complexes”.↩︎For the same reason, in the case of rare-earths metals, in which the magnetic contribution is provided by partially filled f orbitals (\(l=3\)), one would have to consider \(k\leq 6\).↩︎