From Orbital Models to Accurate Predictions

Abstract

Basic understanding and qualitative prediction of the isotropic magnetic coupling between two magnetic centers can be obtained with two well-established valence-only models. This chapter discusses the Kahn–Briat and Hay–Thibeault–Hoffmann models, which have been (and still are) of fundamental importance for understanding the basics of magnetism in polynuclear transition metal complexes. After shortly presenting the basic model for magnetism in organic radicals, we review the most evident magnetostructural relations and then move to the accurate prediction of the magnetic coupling. An overview of the most widely used quantum chemical methods is given, including wave function based methods and approaches within the spin-unrestricted setting such as density functional theory. The last part of the chapter is dedicated to the calculation of the interactions beyond the isotropic magnetic coupling.

Problems

4.1

A master student wants to study the energy splitting $E_S-E_T$ in a planar [Cu$_2$F$_6$]$^{2-}$ model system, since experimental studies of similar di Cl-bridged Cu$^{\mathrm {II}}$ dimers suggested that $E_S-E_T$ depends strongly on the Cu–Cl–Cu angle $\theta$. She performs RHF calculations on the triplet state in order to predict $E_S-E_T$ with the HTH model. She produces a Table of results, where the gerade and ungerade open shell orbitals are denoted 1 and 2, respectively.

 

$\theta$	$\frac{J_{11}+J_{22}}{2}-J_{12}$ [K]	$K_{12}$ [$\mathrm {E_h}$]	$\varepsilon _1 - \varepsilon _2$ [$\mathrm {E_h}$]
85$^{\circ }$	24	0.4324	$-$0.0078
90$^{\circ }$	20	0.4376	$-$0.0025
95$^{\circ }$	22	0.4419	0.0034
100$^{\circ }$	26	0.4456	0.0094
105$^{\circ }$	32	0.4483	0.0150

 

Compute J (in K) for $\theta = 85^{\circ }\ldots 105^{\circ }$ using the HTH model. Do you observe a strong dependence of the coupling on the angle? Can the same conclusions be drawn when only considering the orbital energies?

4.2

Quantifying the counter-complementarity effect. Standard optimization of the molecular orbitals of a magnetic complex with two magnetic centers bridged by two different ligands normally leads to magnetic orbitals with contributions on both ligands (as $\varphi _5$ and $\varphi _6$ in Fig. 4.9). This makes it very hard to quantify the counter-complementary effect of the two ligands. Design a computational strategy to determine quantitatively the reduction of the magnetic coupling through ligand 1 by the counter-complementary effect of ligand 2. Hint: Many quantum chemical programs can divide the whole system into fragments.

4.3

Broken symmetry approach. The magnetic coupling of three binuclear TM complexes has been studied with DFT. The following results were obtained for the HS and BS determinants. (a) Calculate the magnetic coupling parameter J with the Yamaguchi equation (Eq. 4.85) and compare the outcomes to the alternative relations of Noodleman (Eq. 4.86) and Ruiz (Eq. 4.87).

 

TM	Energy [$\mathrm {E_h}$]		$\langle {{\hat{S}}^2}\rangle$
	$\varPhi _{HS}$	$\varPhi _{BS}$	$\varPhi _{HS}$	$\varPhi _{BS}$
Cu$^{\mathrm {2+}}$	$-$4061.7435920	$-$4061.7442381	2.0035	0.9957
Ni$^{\mathrm {2+}}$	$-$3797.4742498	$-$3797.4767694	6.0083	1.9931
Mn$^{\mathrm {2+}}$	$-$3082.7586297	$-$3082.7630415	30.0086	4.9936

 

(b) Calculate J in the Cu complex combining Eqs. 4.67 and 4.78 using $\rho ^{\alpha -\beta }$ is 0.6864 and 0.6757 for HS and BS, respectively.