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Chapter 13: Problem 4

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.

Short Answer

Expert verified
The contours of the magnitude of the diamagnetic current density for a nitrogen monoxide molecule are plotted by first understanding the formation of the molecular orbitals, then calculating the diamagnetic current density, and finally plotting it on a plane containing both nuclei.

Step by step solution

01

Understand the Orbital Formation

The \(2 \mathrm{p} \pi^{*}\) -orbital where the lone electron resides is made up of the nitrogen and oxygen \(2 p\) -orbitals. Here, the molecular orbital is written as $\left(1 / 2^{1 / 2}\right)\left(\psi_{\mathrm{N}}-\psi_{\mathrm{O}}\right)$ for simplicity and the overlap integral is ignored.
02

Calculate the Diamagnetic Current Density

The diamagnetic current density (\(j_d\)) is calculated using the formula: \(j_d = -\nabla \times (r \times j)\). This gives a measure of how much diamagnetic current is produced in the molecule. Here, \(j\) is the current density, \(r\) is the position vector, and \(\nabla \times\) is the curl operator.
03

Plot the Contours of Current Density

After calculating the diamagnetic current density, the contours of its magnitude are plotted on a plane containing both the nitrogen and oxygen nuclei. This gives a graphical representation of how the current density is distributed in the molecule.

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Most popular questions from this chapter

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 \(\Delta E / h c \approx 1.0 \times 10^{4} \mathrm{cm}^{-1} \text {and } \zeta=154 \mathrm{cm}^{-1}.\)

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 \(40 \mathrm{Hz}\).

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 \(I_{\mathrm{Ax}}+I_{\mathrm{B} x}\).

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 \(Z^{*} \approx 1\) ).
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