Towards a Quantitative Understanding

Abstract

Taking a binuclear copper complex as model system, the isotropic magnetic coupling is decomposed into different contributions. Perturbative expressions of the main contributions are derived and illustrated with numerical examples. An effective Hamiltonian is constructed that incorporates all important electron correlation effects and establishes a connection between the complex N-electron wave functions and the simpler qualitative methods discussed in the previous chapter. Subsequently an outline is given of the analysis of the coupling with a single determinant approach and the biquadratic and four-center interactions are decomposed. The chapter closes with the recently proposed method to extract DFT estimates for these complex interactions.

Problems

5.1

Zeroth-order description. Write down the matrix of the model space that only considers neutral determinants expressed in local orbitals. Diagonalize the matrix and calculate the singlet-triplet energy difference. What is the state of lowest energy?

5.2

Construction of the CAS(2,2)CI matrix in the symmetry adapted CSF basis. The CASCI matrix given in Eq. 5.4 uses the four $M_S=0$ determinants as basis. The matrix can be greatly simplified by a basis set change using symmetry adapted CSFs.

• a. Write down the four symmetry adapted CSFs that arise from the linear combinations of the four $M_S=0$ determinants. The expressions of the states after configuration interaction given in Eq. 5.5 may give a hint on the CSFs.

• b. Calculate the energy expectation values of the four CSFs and place them on the diagonal of the matrix.

• c. Identify the CSFs as singlet or triplet spin eigenfunctions and label them by gerade/ ungerade spatial symmetry, assuming that the system has an inversion center. How many off-diagonal elements have non-zero value?

• d. Calculate the remaining matrix elements to complete the CAS(2,2)CI matrix.

5.3

Spin contamination of the BS state. The relaxation of the magnetic orbitals of the BS determinant in the field of the frozen ROKS core orbitals introduces spin contamination. The amount of spin contamination can be determined analytically by rewriting Eq. 5.26 in terms of spin adapted CSFs instead of the neutral and ionic valence bond structures.

• 1. Which term in Eq. 5.26 is an eigenfunction of $\hat{S}^2$. Give the eigenvalue of this term.

• 2. The two other terms have to be written in the form of the singlet $\big (|{a\overline{b}}|+|{b\overline{a}}|\big )$ and triplet $\big (|{a\overline{b}}|-|{b\overline{a}}|\big )$ CSFs. Use the trigonometric relations $\sin ^2\phi \,+\,\cos ^2\phi = 1$ and $\sin ^2\phi \,-\,\cos ^2\phi = \cos (2\phi )$. Hint: Add and subtract $\frac{1}{2}\cos ^2\alpha |{b\overline{a}}| + \frac{1}{2}\sin ^2\alpha |{a\overline{b}}|$ and split the first two terms of Eq. 5.26 in halfs. Then, order the terms to form the given trigonometric relations.

• 3. Calculate $\langle {\varPhi _{BS}}|{\hat{S}^2}|{\varPhi _{BS}}\rangle$ using the above derived expression $\varPhi _{BS} = (|{a\overline{b}}|+|{b\overline{a}}|)/2 + (|{a\overline{b}}|-|{b\overline{a}}|)\cos (2\alpha )/2 + (|{a\overline{a}}|+|{b\overline{b}}|)\sin \alpha \cos \alpha$.

5.4

Kinetic exchange by second-order perturbation theory. Make a second-order estimate of the singlet energy taking into account the interaction between $S=\frac{1}{\sqrt{2}}\left( |{a\overline{b}}|+|{b\overline{a}}|\right)$ and the ionic states $I_1 = \frac{1}{\sqrt{2}}\left( |{a\overline{a}}|+|{b\overline{b}}|\right)$ and $I_2 = \frac{1}{\sqrt{2}}\left( |{a\overline{a}}|-|{b\overline{b}}|\right)$. The energy of the triplet, $T=\frac{1}{\sqrt{2}}\left( |{a\overline{b}}|-|{b\overline{a}}|\right)$, equals $E_{re\textit{f}} -K_{ab}$ and should be taken as reference .

5.5

Biquadratic exchange versus $\mathbf {t_{13}/t_{24}}$. Express the perturbative estimate of $\lambda$ (Eq. 5.56) in terms of $t_{13}$ and $t_{24}$ with a common denominator for the two terms. Determine for which values of $t_{13}$ and $t_{24}$ the biquadratic exchange vanishes and for which values it can be expected to be maximal .

5.6

Estimating $\mathbf {J_r}$. Accurate calculations on a polynuclear paramagnetic compound with four $S=\frac{1}{2}$ magnetic centers indicate that the only significant interactions are the following bilinear isotropic interactions: $J_{12} = J_{34} = -25.1$ meV, and $J_{23} = J_{14} = -39.5$ meV. Nevertheless, the experimental temperature dependence of the magnetic susceptibility could not be fitted satisfactorily with these values. Provide a perturbational estimate for the four-center cyclic exchange to improve the fitting (extra data: $U=5.3$ eV).