# 2 Magnetic moment of the free ion

The starting point to determine the magnitude and other properties of the magnetic moment of a specific atom is the electronic configuration of the so-called *free ion*, that is the atom in a given oxidation state thought in the gas phase (symmetric environment). To the purposes of this course, it is important to keep in mind what type of interactions are already taken into account in the scheme of levels of the free ion and which interactions need to be considered at a second stage^{6}. We anticipate that Zeeman and spin-orbit interactions will be treated as perturbations.

## 2.1 Independent electrons and central potential approximation for atomic Hamiltonian

Let
\[\begin{equation}
\tag{2.1}
\mathcal{H} = \sum_{i=1}^{N_{\rm e}} \left( \frac{\hat{\mathbf p}^2_i}{2m_{\rm e}} -\frac{Ze^2}{4\pi \epsilon_0 r_i} \right) + \frac{1}{2} \sum_{i\ne j} \frac{Ze^2}{4\pi \epsilon_0} \frac{1}{|\underline{r}_i-\underline{r}_j|}
\end{equation}\]
be the atomic Hamiltonian. The term in parenthesis is the electronic Hamiltonian for an atom with \(N_{\rm e}\) electrons with nuclear charge \(Ze\) located at the origin of the coordinates \(\underline{r}_i\);
\(m_{\rm e}\) is the electron mass. The other term on the r.h.s. represents the Coulomb repulsion between electrons, the factor \(1/2\) avoiding double counting.
While the first contribution is nothing but a sum of single-electron operators, the second one contains two-electron operators, which means that it couples reciprocally the d.o.f. of different electrons. This electron-electron repulsion can be of comparable size as the interaction with the nucleus, which does not allow to consider it as a perturbation straightforwardly.
Nevertheless, we would like to approximate the Hamiltonian as a sum of single-electron Hamiltonians involving a central potential \(V(r_i)\):
\[\begin{equation}
\tag{2.2}
\mathcal{H}_0 = \sum_{i=1}^{N_{\rm e}} \mathcal{H}_{0,i}
\end{equation}\]

with
\[\begin{equation}
\tag{2.3}
\mathcal{H}_{0,i} = \frac{\hat{\mathbf p}^2_i}{2m_{\rm e}} -\frac{Ze^2}{4\pi \epsilon_0 r_i} + V_{\rm ee}(r_i)\,.
\end{equation}\]

The potential \(V_{\rm ee}(r_i)\) is the electron-electron contribution to the central potential, which results from *some* averaging procedure \(\langle\dots\rangle_j\)
over all the “positions” of the electrons \(j\) different from \(i\):

\[\begin{equation}
\tag{2.4}
V_{\rm ee}(r_i) = \frac{Ze^2}{4\pi \epsilon_0} \langle \sum_{j\ne i} \frac{1}{|\underline{r}_i-\underline{r}_j|}\rangle_j\,.
\end{equation}\]

In practice, this potential is calculated self-consistently: Starting from an approximate solution of the stationary Schrödinger equation
\[\begin{equation}
\tag{2.5}
\mathcal{H}_0 \Psi(\underline{r}_1,\underline{r}_2,\dots \underline{r}_{N_{\rm e}}) = E\, \Psi(\underline{r}_1,\underline{r}_2,\dots \underline{r}_{N_{\rm e}})
\end{equation}\]

the calculation is reiterated till a predefined convergence criterion is fulfilled. In the Hamiltonian \(\mathcal{H}_0\) the d.o.f. of electrons are not coupled with each other, namely the position of each electron is not correlated with the position of any other electron.
By solving the 3D Schrödinger equation for \(\mathcal{H}_{0,i}\) one obtains *single-electron eigenstates* for the considered atom. Due to the fact that the effective potential is a central field, each single-electron wave function can be expressed as product of a radial part by a spherical harmonic
\[\begin{equation}
\tag{2.6}
\psi_{n,l,m}(\underline{r}_i) = R_{n,l}(r_i)Y_{l,m}(\theta,\phi)\,.
\end{equation}\]

In particular, the kinetic energy can be expressed as a sum of the radial component of the linear momentum plus a contribution proportional to the angular momentum:
\[\begin{equation}
\tag{2.7}
\frac{\hat{\mathbf p}^2_i}{2m} = \frac{1}{2m_{\rm e}} \left[ (\hat{p^r}_i)^2 + \frac{\hbar^2 (\hat{\mathbf l}_i)^2}{r_i^2}\right],
\end{equation}\]
where the orbital angular momentum is defined as the operator \(\hbar \,\hat{\mathbf l}=\hat{\underline{r}} \times \hat{\mathbf p}^2\),
so that the radial part \(R_{n,l}(r_i)\) is obtained by solving the 1D Schrödinger equation
\[\begin{equation}
\tag{2.8}
\left[\frac{\hat{p^r}^2_i}{2m_{\rm e}} + V_{\rm eff}( r_i) \right]P_{n,l}(r_i) = E_{n,l} P_{n,l}(r_i)
\end{equation}\]
with \(P_{n,l}(r_i)=r_i\,R_{n,l}(r_i)\) and
\[\begin{equation}
\tag{2.9}
V_{\rm eff}( r_i)=-\frac{Ze^2}{4\pi \epsilon_0 r_i} + V_{\rm ee}(r_i) + \frac{l(l+1)\hbar^2}{2m_{\rm e} \,r_i^2} \,.
\end{equation}\]
The last term on the right is the quantum-mechanical equivalent of the *centrifigual potential* encountered in gravitational problems.
As known from basic courses, the quantum numbers \((n, l,m)\) define the *single-electron* atomic orbitals of the free ion.

Before considering the effect of a magnetic field we note that, if \(V_{\rm ee}(r_i)\) is estimated properly, the residual contribution to the electron-electron repulsion \[\begin{equation} \tag{2.10} \mathcal{H}^1=\frac{1}{2} \sum_{i\ne j} \frac{Ze^2}{4\pi \epsilon_0} \frac{1}{|\underline{r}_i-\underline{r}_j|} - \sum_{i} V_{\rm ee}(r_i) \end{equation}\] shall be small and, as such, can be treated as a perturbation.

What can we say about the strength of the Zeeman and spin-orbit interaction in comparison with the \(\mathcal{H}_{0,i}\) Hamiltonian? The spacing between atomic levels defined by the \(\mathcal{H}_{0,i}\) Hamiltonian falls typically in the range \(1-10\) eV, whilefor applied fields in the range \(1-10\) T (in ordinary laboratories magnetic fields hardly exceed this upper bound) the interaction \(\mathcal{H}_{{\rm Z}}\) is of the order of \(0.1-1\) meV \(\simeq\) \(1-10\) K. Thus, the Zeeman interaction can be safely treated as a perturbation with respect to \(\mathcal{H}_{0,i}\). We will see later that the Zeeman splitting is actually comparable with the thermal energy \(k_{\rm B}T\) and this is one of the reasons why statistical physics will be needed. The Zeeman interaction is associated with the

*paramagnetic*response of an atom.

For what concerns the spin-orbit interaction, the operator \(\xi_{\rm so}\) defined in Eq. (1.43) generally takes different values for electrons belonging to different orbitals as a result of its averaging over the radial wave function
\[\begin{equation}
\tag{2.11}
\zeta_{nl} = \int_0^{\infty} R_{nl}(r) \xi_{\rm so}(r) R_{nl}^*(r) r^2 dr \,.
\end{equation}\]
This quantity, called *spin-orbit coupling constant*, falls in the range \(10-100\) meV for unpaired 3d electrons and \(50-300\) meV for 4f electrons (see Figure).

To conclude we list on the r.h.s. of the next equation the contributions to the Hamiltonian of electrons in atoms introduced so far, ordered by descending strength from left to right \[\begin{equation} \tag{2.12} \mathcal{H} = \mathcal{H}_0 + \mathcal{H}_{\rm ee}^1 + \mathcal{H}_{\rm so} + \mathcal{H}_{\rm Z}\,. \end{equation}\]

### The role of spin

In the previous section, we have shown that the magnetic field is coupled with the orbital angular momentum (operator) of each electron, \(\mu_{\rm B}\) providing the strength of this coupling. However, an additional contribution to the Zeeman Hamiltonian is provided by the spin \(\hat{\mathbf s}_i\) of each electron:
\[\begin{equation}
\tag{2.13}
\mathcal{H}_{{\rm Z},i} = \mu_{\rm B} \,\left(\hat{\mathbf l}_i + 2 \hat{\mathbf s}_i\right) \cdot \vec B \,,
\end{equation}\]
which suggests to express the single-electron magnetic moment as
\[\begin{equation}
\tag{2.14}
\hat{\boldsymbol \mu}_i= - \mu_{\rm B} \,\left(\hat{\mathbf l}_i + 2\,\hat{\mathbf s}_i\right) \,.
\end{equation}\]
The addition of the spin contribution to the Zeeman Hamiltonian transforms the Schrödinger equation(2.5) into the Pauli equation.
The latter includes the spin contribution explicitly but it is not yet relativistic, which means that spatial and spin d.o.f. are not directly coupled.
Later on we will see that such a coupling is provided by the *spin-orbit* interaction, which we neglect for the moment. As a result, the *single-electron* wave functions will have the form
\[\begin{equation}
\tag{2.15}
\psi_{n,l,m}(\underline{r}_i) \,|\sigma_i\rangle = R_{n,l}(r_i)Y_{l,m}(\theta_i,\phi_i)\,|\sigma_i\rangle \,
\end{equation}\]

where \(|\sigma_i\rangle\) are the eigenstates of the Pauli matrix \(\hat{\boldsymbol \sigma}_i^z =2\,\hat{\mathbf s}_i^z\). Namely, as already done for the separation between radial part and angular part of the wave function, the spin contribution comes into the wave function only as multiplicative contribution.

## 2.2 Spin states of two electrons and the Pauli principle

When one considers more than one electron, restrictions to the symmetry of the total wave function apply.
Since electrons are fermions, according to the *symmetrization postulate* their total wave function has to be antisymmetric w.r.t. the exchange of any pairs of electrons. In a two-electron system the spin space defined by the direct product of the two single-particle basis

\[\begin{equation}
|\sigma_1\rangle \otimes |\sigma_2\rangle =|\sigma_1\,\sigma_2\rangle
\end{equation}\]
splits into two subsets: the spin *singlet* space
\[\begin{equation}
\chi_{0,0}=|S=0,M=0\rangle=\frac{1}{\sqrt{2}}\left[|+ -\rangle - |- +\rangle\right]
\end{equation}\]
and the spin *triplet* space
\[\begin{equation}
\begin{split}
&\chi_{1,1}=|S=1,M=1\rangle=|++\rangle \\
&\chi_{1,0}=|S=1,M=0\rangle=\frac{1}{\sqrt{2}}\left[|+-\rangle + |-+\rangle\right] \\
&\chi_{1,-1}=|S=1,M=-1\rangle=|- -\rangle
\end{split}
\end{equation}\]
The spin singlet is clearly antisymmetric, while the set of spin states defining the triplet are symmetric.
A direct consequence of the symmetrization postulate is that the spin singlet is only compatible with a spatially symmetric wave function, while the spin triplet is only compatible with a spatially antisymmetric wave function. However, in order to build an antisymmetric wave function the two electrons have to occupy two different orbitals, i.e. orbitals with different quantum numbers (\(n,l\)). If they occupy the same orbital, only a symmetric total spatial wave function can be built and, consequently, their spin wave function *must* be the singlet \(\chi_{0,0}\). This is the content of the famous **Pauli principle**.

## 2.3 He atom and intra-atomic exchange interaction

The Pauli principle imposes that the ground-state spin configuration of a He atom be the singlet: Both electrons are in the orbital with quantum number \(n=1\) and \(l=0\), i.e., the electronic configuration is \((1s)^2\) and, therefore, the spin part of the wave function has to be in the singlet state. In the first excited state, with electronic configuration \((1s)^1(2s)^1\), electrons are instead allowed to be in the triplet state as sketched in the following “truth table”:

spin part symmetrictriplet \(\uparrow\uparrow\) |
spin part antisymmetricsinglet \(\uparrow\downarrow\) |
---|---|

spatial part antisymmetric\(\Psi^{\rm T}(\underline{r}_1,\underline{r}_2)=-\Psi^{\rm T}(\underline{r}_2,\underline{r}_1)\) \(\propto\left[\psi_{1s}(\underline{r}_1)\psi_{2s}(\underline{r}_2)-\psi_{1s}(\underline{r}_2)\psi_{2s}(\underline{r}_1)\right]\) |
spatial part symmetric\(\Psi^{\rm S}(\underline{r}_1,\underline{r}_2)=\Psi^{\rm S}(\underline{r}_2,\underline{r}_1)\) \(\propto\left[\psi_{1s}(\underline{r}_1)\psi_{2s}(\underline{r}_2)+\psi_{1s}(\underline{r}_2)\psi_{2s}(\underline{r}_1)\right]\) |

These symmetry considerations about the total wave function \(\Psi(\underline{r}_1,\underline{r}_2)\) do not tell anything about the relative energy of the state \((1s)^1(2s)^1\,\chi_{1,m}\) (spin triplet) and the state \((1s)^1(2s)^1\,\chi_{0,0}\) (spin singlet). To be quantitative in this respect one needs to consider explicitly the Hamiltonian. For simplicity we will not consider the contribution of the electron-electron repulsion to the central potential. Therefore, \(\mathcal{H}_{0}\) will contain the Coulomb interaction of both electrons with the nucleus plus the kinetic energy terms of the electrons, while the perturbation will be given by
\[\begin{equation}
\tag{2.16}
\mathcal{H}^1=\frac{e^2}{4\pi \epsilon_0} \frac{1}{|\underline{r}_1-\underline{r}_2|} \,.
\end{equation}\]

After having inserted the appropriate wave functions, – i.e. solutions of the hydrogen-atom problem – one estimates a ground-state energy \(E_0^{\rm gs}=2E_{1s}=-4e^2/(4\pi \epsilon_0 a_0)\), where \(a_0\) is the Bohr radius. The first-order correction to the ground-state energy due to \(\mathcal{H}^1\) is \(E_1^{\rm gs}=+5e^2/(16\pi \epsilon_0 a_0)\). But the most interesting case concerns the two possible excited states indicated in the table. In the absence of electron-electron repulsion the energy of the singlet and triplet state would be degenerate and equal to
\[\begin{equation}
\tag{2.17}
E_0^{\rm exc}= E_{1s} + E_{2s}\,,
\end{equation}\]

where \(E_{1s}\) and \(E_{2s}\) are the energies of the corresponding single-electron states. The perturbative Hamiltonian \(\mathcal{H}^1\) removes this degeneracy. In fact, the first-order correction associated with the triplet and singlet wave functions are
\[\begin{equation}
\label{He_excited_1}
\begin{split}
&\langle \Psi^{\rm T} | \mathcal{H}^1 | \Psi^{\rm T} \rangle = Q - J \\
&\langle \Psi^{\rm S} | \mathcal{H}^1 | \Psi^{\rm S} \rangle = Q + J \,.
\end{split}
\end{equation}\]
The energies \(Q\) and \(J\) are called *Coulomb* and *Exchange* integrals and are evaluated expressing explicitly the wave functions \(\Psi^{\rm T}(\underline{r}_1,\underline{r}_2)\)
and \(\Psi^{\rm S}(\underline{r}_1,\underline{r}_2)\) to obtain
\[\begin{equation}
\label{Coulomb_exchange}
\begin{split}
&Q=\int|\psi_{1s}(\underline{r}_1)|^2\frac{e^2}{4\pi \epsilon_0} \frac{1}{|\underline{r}_1-\underline{r}_2|} |\psi_{2s}(\underline{r}_2)|^2 \,d^3r_1 d^3r_2\\
&J=\int\psi_{1s}(\underline{r}_1)\psi_{2s}(\underline{r}_2) \frac{e^2}{4\pi \epsilon_0}
\frac{1}{|\underline{r}_1-\underline{r}_2|}\psi_{1s}^*(\underline{r}_2)\psi_{2s}^*(\underline{r}_1) \,d^3r_1 d^3r_2\,.
\end{split}
\end{equation}\]
If one understands the terms \(e\,|\psi_{nl}(\underline{r}_1)|^2\) as the quantum equivalent of charge distributions, the Coulomb integral \(Q\) just represents the Coulomb repulsion between the electrostatic charge distributions of the two electrons. The integral \(J\) reflects, instead, the energy associated with an exchange of quantum states between the two electrons and has no classical equivalent. The energy of the triplet state is lower, with respect to that of the excited state by a factor \(\Delta E = 2J\). In an intuitive picture, this energy difference can be ascribed to the fact that in the singlet state the spatial wave function is symmetric and the electrons have a tendency to be close to each other, while in the triplet state the spatial wave function is antisymmetric and the electrons tend to avoid each other. This produces a higher electrostatic repulsion in the singlet state and, consequently, a higher energy. The same calculation could be repeated for the (1s)(2p) excited state and a similar result would be obtained. The main contents of this discussion are summarized in the following semiquantitative sketch

The picture depicted above is consistent with a series of transitions that only occur within a given spin subspace – the singlet or the triplet. In fact, in the electric-dipole approximation, transitions between levels associated with different spin states are forbidden (only transitions involving \(\Delta S=0\) and \(\Delta L = \pm 1\) are allowed to the first order). This scenario was actually puzzling in the early decades of the last century.
The seminal works of Heisenberg to interpret the emission spectrum of neutral He thus pinpointed the importance of spin in defining the scheme of the energy levels in atoms and the

allowed/forbidden transitions among them. Starting from those works Heisenberg developed the concept of exchange interaction in the late 1920s. Particular influential was his proposal to
express the splitting between singlet and triplet state for the occupation of the same single-electron levels, e.g. (1s)(2s) or (1s)(2p), in terms of an effective coupling between spins.
Using the fact that \((\hat{\mathbf S})^2=(\hat{\mathbf s}_1+\hat{\mathbf s}_2)^2=(\hat{\mathbf s}_1)^2+2\,\hat{\mathbf s}_1\cdot\hat{\mathbf s}_2+(\hat{\mathbf s}_2)^2\), we have shown in a previous **Assignment** that the *effective* spin Hamiltonian
\[\begin{equation}
\tag{2.18}
\mathcal{H}_{\rm exch}=
-2J\,\hat{\mathbf s}_1\cdot\hat{\mathbf s}_2
\end{equation}\]
has eigenvalues
\[\begin{equation}
\begin{split}
\langle\chi_{0,0} | \mathcal{H}_{\rm exch}|\chi_{0,0} \rangle = +\frac{3}{2} J &\qquad \text{for the singlet} \\
\langle\chi_{1,m} | \mathcal{H}_{\rm exch}|\chi_{1,m} \rangle =-\frac{1}{2} J &\qquad \text{for the triplet} \,,
\end{split}
\end{equation}\]

meaning that this operator exactly reproduces the splitting existing between the singlet and the triplet state of a given electronic configuration.
Note that this coupling between spin d.o.f. does not occur in reality: the Hamiltonian(2.18) is just an extremely useful representation, which we will largely use in the following, of the combined effect of electron-electron Coulomb repulsion and the Pauli principle. The word *effective* has to be understood in this meaning.

Generally, the exchange integral can be either positive or negative:

- \(J>0\) favors the spin triplet
- \(J<0\) favors the spin singlet.

In this specific case the exchange interaction is associated with an energy splitting of atomic levels, that is spins belong to electrons of the same atom. One speaks in this case about *intra-atomic* exchange interaction.

## 2.4 Ground-state electronic configuration in free ions

In atoms with many electrons the total contribution to the Zeeman energy of electrons filling up a complete shell is zero. Therefore, the magnetic moment of each element in the periodic table, and the relative ions, is only determined by electrons that occupy partially filled shells. These shells are formally defined *magnetic orbitals*. In the presence of several unpaired electrons, the sum of individual spin and orbital angular momenta yields different values of the quantum numbers associated with
the total spin \(S\) and the total orbital angular momentum \(L\): these quantum numbers are eventually needed to compute the atomic magnetic moment. The multi-electron configurations expressed on the basis of the total angular momenta \(L\) and \(S\) of such electrons are called Russell-Saunders terms, indicated with \(^{2S+1}L\). The first two Hund’s rules provide a recipe to determine the multiplet with lower energy out of all possible \(^{2S+1}L\) terms^{7}.

### 1\(^{st}\) Hund’s rule

Since the energy splitting between the singlet and the triplet state described for the excited states of He atoms originates from very basic physical laws – the symmetrization postulate and Coulomb interaction –
one can expect that a similar phenomenology also occurs in atoms with more than two electrons. For instance, carbon has electronic configuration \([\)He\(]\)(2s)\(^2\)(2p)\(^2\). In this case, one can imagine that
the role analogous to that of (1s)(2s) levels in He is played by the two occupied (2p) orbitals. Schematically, the doubly occupied (2p) single-electron levels with the same energy determined by \(\mathcal{H}_{0}\) can be represented as follows

where the notation \(\uparrow\uparrow\) has to be understood as the triplet state and \(\uparrow\downarrow\) as the singlet state. Therefore, two spatial wave functions with opposite symmetry hide behind these two schematic representations. Assuming that the same arguments as per the He atoms apply, we would conclude that the ground state configuration of the C atom is realized in the triplet state. Parenthetically, we note that it is not by accident that we focused directly on the outer shell of C, i.e., the one partially filled 2p shell:
the levels of the inner shells are all doubly occupied by electrons in a spin singlet, as prescribed by the Pauli principle.

Tentatively, one may try to extend these arguments to atoms with more than two electrons in the outer shell, possibly partially filled. From the excited states of He, we have learned that the pairwise Coulomb interaction between electrons takes its minimal energy when the total spatial wave function \(\Psi(\underline{r}_1,\underline{r}_2,\dots \underline{r}_{N_{\rm e}})\) is antisymmetric. Therefore, it would be useful to find a recipe that allows identifying the symmetric spin wave functions of many electron systems. A theorem by Hermann Weyl provides a great help to this purpose. It states that all the eigenfunctions of \((\hat{\mathbf S})^2\) corresponding to the same eigenvalue \(S(S+1)\) – i.e., belonging to the same spin multiplet – have the same symmetry with respect to the exchange of any pair of particles. The symmetry property of the eigenfunctions of \((\hat{\mathbf S})^2\) under permutation are called “Spinrasse”. With this result at hand it is easy to guess a spin multiplet of many electrons that is symmetric: the one for which the total spin takes its maximal value. To this multiplet, in fact, shall belong the state with all spin up \(|+++\dots+\rangle\), which is manifestly symmetric under the exchange of any particle. The Weyl theorem then warrants that all the other spin states of this multiplet will be symmetric under the exchange of any particle, including the state with all the spin down \(|---\dots-\rangle\).

We are now in the position to state the **1st Hund’s rule**: Provided there is sufficient degeneracy that non-equivalent orbital wave functions can be constructed, the configuration realizing the lowest energy state corresponds to the spin multiplet with **maximum** value of the total spin \(S\).
This is schematically represented by filling the atomic orbitals – like the ones sketched above for the 2p orbitals of the C atom – with parallel arrows till the whole shall is half-filled. Once this limit is reached, the Pauli principle forces to create spin singlets within individual single-electron levels, thus reducing the value of the total \(S\). However, one should not forget that aligning parallel arrows should be understood as creating the multiplet with the largest total spin \(S\) and not simply aligning spin parallel to each other!

It is convenient to understand the 1st Hund’s rule as resulting from an effective *intra-atomic* exchange coupling whose strength falls in the range 1-2 eV. We remark that orbital degeneracy is the key ingredient to the formation of a finite total spin and, consequently, of atomic magnetic moments.

### 2\(^{nd}\) Hund’s rule

While Hund’s first rule concerns *spin-spin* coupling,
Hund’s second rule is about *orbit-orbit* coupling. It states that, after having
established the maximum value of \(S\), the lowest energy state corresponds to the configuration that maximizes the total orbital momentum of electrons \(L\). The basis for this rule is again the electron-electron Coulomb repulsion^{8}, which is minimal for larger values of \(L\).

### Spin-orbit interaction within a Russell-Saunders term

We evaluate in this section the effect of spin-orbit coupling on a free ion with many electrons and relate it to the the 3d Hund’s rule. The total spin-orbit interaction is the sum of all the single-electron contributions given in Eq. (1.42)
\[\begin{equation}
\tag{2.19}
\mathcal{H}_{\rm so}= \sum_{i}\zeta_{nl}\,\hat{\mathbf s}_{i}\cdot \hat{\mathbf l}_{i}
\end{equation}\]
with \(i\) labelling different electrons and \(\zeta_{nl}\) indicating the spin-orbit coupling constant defined in Eq. (2.11). When evaluated within a Russell-Saunders term \(^{2S+1}L\), the Hamiltonian (2.19)
takes the simplified form

\[\begin{equation}
\tag{2.20}
\mathcal{H}_{\rm so}= \lambda\,\hat{\mathbf S}\cdot \hat{\mathbf L}
\end{equation}\]
where \(\lambda=\pm\zeta_{nl}/(2S)\). The sign plus applies to electronic configurations with less than half shell filled, the sign minus to configurations with more than half shell filled; when a shell is exactly half-filled it is \(\lambda=0\).

This change in the sign of \(\lambda\) can be understood applying to holes the arguments that we have spelled out for single-electron orbitals.
In particular, the electronic configuration (3d)\(^9\) can be thought of as equivalent to (3d)\(^1\) with hole in place of an electron. In this perspective, it is clear that the spin-orbit coupling constant \(\lambda\) becomes negative if the sign of the effective electric charge of the orbiting particle (with unpaired spin) is positive.

For free ions the total angular momentum \(\hat{\mathbf J}=\hat{\mathbf S}+\hat{\mathbf L}\) is a good quantum number and the Hamiltonian (2.20) is diagonal on the basis \(|J,M_J\rangle\), in spectroscopic notation indicated as \(^{2S+1}L_J\)^{9}. The eigenvalues are readily obtained adapting Eq. (2.21) to the total angular momenta
\[\begin{equation}
\tag{2.21}
\hat{\mathbf S}\cdot\hat{\mathbf L}=\frac{1}{2}(\hat{\mathbf J}^2-\hat{\mathbf S}^2-\hat{\mathbf L}^2)\,.
\end{equation}\]
Note that the Hilbert space \(|L,M\rangle\otimes|S,M_S\rangle\) is *reducible* with respect to rotations of \(O(3)\) and the \(|J,M_J\rangle\) multiplets are just its *irreducible* sub-sets.

### The 3rd Hund’s rule

The Russell-Saunders multiplet of a free ion corresponding to the electronic ground state is determined by the first two Hund’s rules and has a degeneracy equal to \((2S+1)\times(2L+1)\). The spin-orbit interaction partially removes this degeneracy and stands behind the 3rd Hund’s rule. According to this rule,

the multiplet \(|J,M_J\rangle\) that minimizes the energy corresponds to \(J=\mid L-S\mid\) if the shell is **less** than half filled, while it corresponds to \(J=L+S\) if the shell is **more** than half filled. This behavior reflects the change in the sign of \(\lambda\) discussed in relation to Eq. (2.20).

In summary, it should be clear that the intra-atomic exchange and the spin-orbit interactions are at the origin of the famous empirical rules suggested by Friedrich Hund in 1925. Hund’s rules determine the multiplet term \(^{2S+1}L_J\) with lowest energy (or electronic ground state) for a given electronic configuration of the free ion, such as \([Ar](4s)^2(3d)^{n_{\rm e}}\), with \(n_{\rm e}\) number of electrons in the 3d shell. The difference between the strength of spin-orbit interaction (\(10-100\) meV) and the intra-atomic exchange interaction (\(\sim\) 1 eV) is responsible for the hierarchy of Hund’s rules. In practice, Hund’s rules allow computing the magnetic moments for the so-called free ions and for rare-earths only, for reasons that will be clarified in the next chapter.

## 2.5 Zeeman interaction in free and rare-earth ions

The three Hund’s rules are determined by the symmetrization postulate and the first three inteactions on the r.h.s. Eq. (2.12) – \(\mathcal{H}_0\), \(\mathcal{H}_{\rm ee}^1\), and \(\mathcal{H}_{\rm so}\). Once the multiplet \(^{2S+1}L_J\) with lower energy has been determined, the Zeeman energy \(\mathcal{H}_{\rm Z}\) can be treated as a perturbation on this basis \(|J,M_J\rangle\). The Zeeman Hamiltonian for a ion with many electrons reads
\[\begin{equation}
\tag{2.22}
\mathcal{H}_{{\rm Z}} = \mu_{\rm B} \sum_i \left(\hat{\mathbf l}_i + g_s\, \hat{\mathbf s}_i\right) \cdot \vec B
= \mu_{\rm B}\, B\,\left(\hat{L}^z + g_s\hat{S}^z\right)
\end{equation}\]
for \(\vec B\) assumed parallel to \(z\). For generality, in this section we do not restrict ourselves to \(g_s=2\) for the electron but we allow for quantum-electrodynamic corrections that yield \(g_s = 2.0023\).

To consider (at a first-order of perturbation theory) the correction to the

energies of the ground-state multiplet \(^{2S+1}L_J\) produced by the Zeeman interaction, we need to evaluate the matrix elements
\[\begin{equation}
\label{Matrix_el_LS_1} \langle J,M_J|\hat{L}^z|J,M_J\rangle \qquad \text{and}\qquad \langle J,M_J| \hat{S}^z|J,M_J\rangle \,.
\end{equation}\]
The projection theorem of tensor algebra turns out to be very useful to this aim. Applied to the present case, the theorem guarantees the following equation:
\[\begin{equation}
\label{Matrix_el_LS_2} \langle J,M_J|\hat{L}^z|J,M_J\rangle = \frac{1}{J(J+1)} \langle J,M_J|(\hat{\mathbf J}\cdot\hat{\mathbf L})\hat{J}^z|J,M_J\rangle
\end{equation}\]
and a similar one in which \(\hat{\mathbf L}\) is replaced by \(\hat{\mathbf S}\). Using the fact that \(\hat{\mathbf L}^2=(\hat{\mathbf J} -\hat{\mathbf S})^2\) and \(\hat{\mathbf S}^2=(\hat{\mathbf J} -\hat{\mathbf L})^2\), one immediately obtains
\[\begin{equation}
\begin{split}
\label{Matrix_el_LS_3}
&\hat{\mathbf J}\cdot\hat{\mathbf L} =\frac{1}{2}\left(\hat{\mathbf J}^2 +\hat{\mathbf L}^2-\hat{\mathbf S}^2\right) \\
&\hat{\mathbf J}\cdot\hat{\mathbf S} =\frac{1}{2}\left(\hat{\mathbf J}^2 +\hat{\mathbf S}^2-\hat{\mathbf L}^2\right) \,.
\end{split}
\end{equation}\]
With this knowledge, the interested reader can verify that to first order perturbation theory the Hamiltonian(2.22) is equivalent to the following one
\[\begin{equation}
\label{Ham_Zeeman_J}
\mathcal{H}_{{\rm Z}} = g_{J} \mu_{\rm B}\, B\,\hat{J}^z
\end{equation}\]
with the Landé factor
\[\begin{equation}
\tag{2.23}
g_{J} = 1 + (g_s-1)\, \frac{J(J+1) + S(S+1)-L(L+1)}{2J(J+1)}\,.
\end{equation}\]

This means that a magnetic field removes the \((2J+1)\) degeneracy of the multiplet \(|J,M_J\rangle\) completely. The center of gravity of the Zeeman splitting is symmetric around the unperturbed energy of the multiplet. The distance between two consecutive Zeeman levels is
\[\begin{equation}
\Delta E_{\rm Z} = \mu_{\rm B}\, B\, g_{J}
\end{equation}\]
i.e., it is proportional to \(B\). These results are known as the *anomalous* Zeeman effect, counterposed to the normal Zeeman effect. The second one is observed when only the orbital angular momentum contributes to the Zeeman energy and no contribution comes from spin coordinates. This definition dates back to the end of the nineteenth century, when the spin had not been discovered yet: With today’s knowledge, there is nothing anomalous in the anomalous Zeeman effect.

Equation(2.23) reproduces the magnetic moments of rare-earth ions fairly well, but not those of transition-metal ions in the solids. In the next chapter, we will see that this is due to the partial or total quenching of the angular momentum, which typically occurs when a transition-metal ion is hosted in a solid.

This paragraph has been adapted from Chapter 6 of the book

*``Magnetism: from Fundamentals to Nanoscale Dynamics’’*, by J. St"ohr and H.~C. Siegmann (2006) (source available in the { BookChapters} folder).↩︎The energy splitting between different terms is established by the kinetic energy of electrons, their Coulomb interaction (both with the nucleus and reciprocal) and the Pauli principle.↩︎

In a semiclassical picture, one could explain this by considering that electrons orbiting in the same direction meet less often than electrons orbiting in opposite directions: In the first case electron pairs experience in average less reciprocal Coulomb repulsion.↩︎

In general, single-electron states are labeled with small letters and multi-electron states are labeled with capital letters.↩︎