Two (or More) Magnetic Centers

Abstract

The description of the magnetic interactions is now extended to more than one magnetic center. First it is shown that the two-electron/two-orbital system can be approached from different viewpoints using (de-)localized, (non-)orthogonal orbitals. After this quantum chemical description of the magnetic interaction we discuss the more phenomenological approach based on spin operators. Starting with the standard Heisenberg Hamiltonian for isotropic bilinear interactions, the chapter discusses how biquadratic, anisotropic and four-center interactions can be accounted for within this spin formalism. Furthermore, it is shown how the microscopic electronic interaction parameters can be used to describe macroscopic properties by diagonalization of model Hamiltonians, Monte Carlo simulations and some other techniques.

Problems

3.1

Overlap: Demonstrate that $c_1/c_2$ in Eq. 3.7 is equal to $1-S_{ab}/1+S_{ab}$, where $S_{ab} = \langle {\phi _a}|{\phi _b}\rangle$ and $\phi _a$ and $\phi _b$ are the orbitals of Eq. 3.17.

3.2

From delocalized to localized: Transform the following determinants and CSFs from a delocalized to a localized orbital basis . Determine the percentage of ionic and neutral character of the wave function. Are the wave functions eigenfunctions of $\hat{S}^2$ ?

• a. $\varPhi _1 = |{g_1\overline{g}_1}|$; $\varPhi _2 = |{g_1g_2}|$; $\varPhi _3=|{g_1\overline{u}_1}|$

• b. $\varPsi _1 = (|{g_1\overline{g}_1}| + |{u_1\overline{u}_1}|)/\sqrt{2}$; $\varPsi _2 = (|{g_1\overline{g}_1}| - |{u_1\overline{u}_1}|)/\sqrt{2}$

• c. $\varPhi _4 = |{g_1u_1}|$; $\varPhi _5 = |{g_1u_1v_1}|$

• d. $\varPsi _3 = (2|{g_1u_1\overline{v}_1}|-|{g_1\overline{u}_1v_1}|-|{\overline{g}_1u_1v_1}|)/\sqrt{6}$

with $g_i = \frac{1}{\sqrt{2}}(a_i+b_i)$; $u_i = \frac{1}{\sqrt{2}}(a_i - b_i)$; $v_i = c_i$. $a_i$, $b_i$ and $c_i$ are orbitals localized on centers A, B and C, respectively.

3.3

Singlet and triplet eigenvalues: Calculate the eigenvalues of the Heisenberg Hamiltonian given in Eq. 3.31 of $\varPhi (T) = |{\alpha \alpha }|$ and $\varPhi (S) = (|{\alpha \beta }|-|{\beta \alpha }|)/\sqrt{2}$.

3.4

Extracting J -values for a three-center system: The following wave functions $\varPsi _k$ were obtained from an ab initio calculation on a system with three $S=1/2$ magnetic centers. Each magnetic orbital $\phi _i$ is localized on center i and has the same spatial part in all five wave functions .

	$\varPsi _1$	$\varPsi _2$	$\varPsi _3$	$\varPsi _4$	$\varPsi _5$
$|{\phi _1\phi _2\overline{\phi }_3}|$	$-$0.4426	$-$0.6583	0.5774	$-$0.1465	0.1135
$|{\phi _1\overline{\phi }_2\phi _3}|$	0.7706	$-$0.0661	0.5774	0.0367	$-$0.2476
$|{\overline{\phi }_1\phi _2\phi _3}|$	$-$0.3280	0.7243	0.5774	0.1098	0.1341
$|{\phi _1\overline{\phi }_1\phi _2}|$	0.0102	0.0234	0.0000	$-$0.0440	0.0017
$|{\phi _1\overline{\phi }_1\phi _3}|$	$-$0.0725	$-$0.0495	0.0000	0.1244	0.0341
$|{\phi _1\phi _2\overline{\phi }_2}|$	0.2243	$-$0.1120	0.0000	0.7653	$-$0.5685
$|{\phi _2\overline{\phi }_2\phi _3}|$	0.2017	0.1336	0.0000	$-$0.5805	$-$0.7636
$|{\phi _1\phi _3\overline{\phi }_3}|$	$-$0.0789	0.0407	0.0000	$-$0.1472	0.0147
$|{\phi _2\phi _3\overline{\phi }_3}|$	0.0076	0.0508	0.0000	0.0579	0.0127

The energies (in $\mathrm {E_h}$) are $E_1 = -27.9611962$, $E_2 = -27.9601927$, $E_3=-27.9596947$, $E_4 =-27.8326257$, $E_5 = -27.83169141$.

• a. Determine the $M_S$ quantum numbers of the determinants and identify $\varPsi _3$ as a spin eigenfunction with $S=3/2$.

• b. Extract the J-values from the energies of the lowest three states under the assumption that $J_{12}=J_{23} \ne J_{13}$ (see Eq. 3.44).

• c. Write down the determinants that span the model space of the Heisenberg Hamiltonian and determine the norm of the projections of $\varPsi _k$ on this model space.

• d. Select the three roots with the largest norm and orthogonalize the projections $\widetilde{\varPsi }_k$

• e. Construct the $3\times 3$ effective Hamiltonian and extract the different J-values by comparing with the matrix elements of the Heisenberg Hamiltonian given in Eq. 3.39.

3.5

Heisenberg twice. (a) Use the eigenvalues of Q, T and S for $\hat{H}=-J\hat{S}_1\cdot \hat{S}_2$ to compute the eigenvalues of Q, T and S for the operator $\hat{S}_1\cdot \hat{S}_2$. (b) From this, compute the eigenvalues of Q, T and S for the biquadratic operator $(\hat{S}_1\cdot \hat{S}_2)^2$ and check the validity of Eq. 3.75.

3.6

Biquadratic interactions: Do the following total energies follow the regular spacing predicted by the Heisenberg Hamiltonian? $E_Q = -139.48992180$ $\mathrm {E_h}$, $E_T =-139.49305142$ $\mathrm {E_h}$ and $E_S=-139.49443101$ $\mathrm {E_h}$. Calculate J and $\lambda$ (in meV) from the energy differences .