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Chapter 13: Problem 10

In tetrahedral complexes of \(\mathrm{Ti}^{3+}\left(\text { configuration } \mathrm{d}^{1}\right)\), a tetragonal distortion removes the degeneracy of the d-orbitals almost completely. The lowest energy orbital is \(\mathrm{d}_{z},\) and the \(\mathrm{d}_{x z}-\) and \(\mathrm{d}_{y z}\) -orbitals, which remain degenerate, are at an energy \(\Delta E\) above it. Find an expression for the \(g\) -values when the field is applied along the \(x-, y-,\) and \(z\) -axes of the complex, and estimate their values. Take \(\Delta E / h c \approx 1.0 \times 10^{4} \mathrm{cm}^{-1} \text {and } \zeta=154 \mathrm{cm}^{-1}.\)

Short Answer

Expert verified
The z-value, g_z, corresponds to the energy levels of the \(d_z\) orbital and will be different from the x and y g-values, which correspond to the energies of the degenerate \(d_{xz}\) and \(d_{yz}\) orbitals. The exact numerical values depend on the specific values of \(\Delta E\) and \(\zeta\) given.

Step by step solution

01

Calculation of g-values

The g-values are obtained by using the spin-orbit coupling constant (\(\zeta\)) and the splitting energy (\(\Delta E\)). Each g-value is calculated with respect to its corresponding axis. In our case, we know the values of \(\Delta E\) and \(\zeta\) and the fact that the electron is in the \(d_z\) orbital. Use the following formula to calculate the g-values:\[g = 1 + \frac{\zeta}{\Delta E}\]we will compute this formula for all axes x, y and z correspondingly.
02

Conversion of energy units

Before performing the calculations, it's important to note that the units of \(\zeta\) and \(\Delta E\) must match. In this case, the values are given in units of \(\mathrm{cm}^{-1}\). Therefore, no conversion is necessary.
03

Compute g-values

We now plug the values for \(\zeta\) and \(\Delta E\) into the formula and calculate the g-values for each axis.

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Most popular questions from this chapter

Show that the energy of dipolar interaction of two electron spin magnetic moments may be expressed as \(S \cdot D \cdot S,\) where \(S=s_{1}+s_{2}\) and \(S \cdot D \cdot S=\sum_{i, j} S_{i} D_{i j} S_{j}\) with \(i, j=x, y,\) and \(z .\) Hint. The energy is proportional to \(s_{1} \cdot s_{2}-3 s_{1} \cdot\left(r r / r^{2}\right) \cdot s_{2} .\) Expand this expression in terms of its Cartesian components and employ relations such \(\operatorname{as} s_{1 x}^{2}=\frac{1}{4} \hbar^{2}, S_{x}^{2}=2 s_{1 x} s_{2 x}+\frac{1}{2} \hbar^{2},\) etc.

An electron occupies one of a doubly degenerate pair of d-orbitals, and its orbital angular momentum corresponds to \(\Lambda=+2 .\) Compute an expression for the current density and plot it for a \(3 \mathrm{d}\) Slater atomic orbital on carbon (take \(Z^{*} \approx 1\) ).

Consider a nitrogen monoxide molecule (nitric oxide, \(\mathrm{NO})\) in which the unpaired electron occupies a \(2 \mathrm{p} \pi^{*}\) -orbital formed from a linear combination of the nitrogen and oxygen \(2 p\) -orbitals. For simplicity, take the molecular orbital to be \(\left(1 / 2^{1 / 2}\right)\left(\psi_{\mathrm{N}}-\psi_{\mathrm{O}}\right) ;\) we have ignored the overlap integral. Consider a plane containing both nuclei. Plot contours of the magnitude of the diamagnetic current density taking the p-orbitals to be Slater atomic orbitals: note that this produces a broadside view of the current density.

Write the NMR spin hamiltonian for a molecule containing two protons, one in an environment with chemical shift \(\delta_{\mathrm{A}}\) and the other with chemical shift \(\delta_{\mathrm{B}} .\) Let them be coupled through a constant \(J\). Evaluate the matrix elements of the hamiltonian for the states \(\left|m_{\mathrm{IA}} m_{\mathrm{IB}}\right\rangle,\) and construct and solve the \(4 \times 4\) secular determinant for the eigenvalues and eigenstates. Determine the allowed magnetic dipole transitions (they correspond to matrix elements of \(I_{\mathrm{Ax}}+I_{\mathrm{Bx}}\) ), and find their relative intensities. Draw a diagram of the spectrum expected when (a) \(J=0\) (b) \(J \ll\left(\delta_{\mathrm{A}}-\delta_{\mathrm{B}}\right) v_{0}\) (c) \(J=\left(\delta_{\mathrm{A}}-\delta_{\mathrm{B}}\right) v_{0}\) (d) \(\delta_{\mathrm{A}}=\delta_{\mathrm{B}},\) where \(v_{0}\) is the spectrometer frequency. Hint. Construct the matrix of the hamiltonian and evaluate its eigenvalues and eigenvectors. Intensities are proportional to the squares of the matrix elements of \(I_{\mathrm{Ax}}+I_{\mathrm{B} x}\).

Estimate the spin-spin coupling constant for the molecule \(^{1} \mathrm{H}^{2} \mathrm{H}\). Hint. Use eqn 13.110 with a simple LCAO-MO. Take \(\Delta E^{(\mathrm{T})}=10 \mathrm{eV} .\) Express \(J\) as a frequency. The experimental value is \(40 \mathrm{Hz}\).
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