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Chapter 13: Problem 14

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

Short Answer

Expert verified
After constructing the Hamiltonian and determining its matrix representation, we calculate the eigenvalues and eigenstates. This allows us to determine allowed magnetic dipole transitions and their relative intensities. The NMR spectra can then be drawn for the given conditions, illustrating the impacts of different chemical shifts and spin-spin coupling constants on the acquired spectrum.

Step by step solution

01

Construction of spin Hamiltonian

Construct the NMR spin Hamiltonian for two protons that experiences different chemical shifts and are coupled through a constant \(J\). This can be written as \(H = \hbar v_0 (\delta_A I_{Az} + \delta_B I_{Bz} + 2\pi J I_A \cdot I_B)\).
02

Represent the base states

Represent the base states as \(|m_{IA}, m_{IB}>\). These will take four forms: \(|+, +>\), \(|+, ->\), \(|-,+>\), \(|-,->\), where \(+, -\) represent the spin up and spin down states respectively.
03

Find the matrix representation of Hamiltonian

Find the matrix representation of the hamiltonian in the basis of the spin states \(|m_{IA}, m_{IB}>\). Each element of the \(4 \times 4\) matrix is obtained by taking the expectation value of the Hamiltonian between the spin states.
04

Calculate the eigenvalues and eigenstates

Calculate the eigenvalues and eigenstates of the Hamiltonian by solving the characteristic equation of the Hamiltonian matrix, and by normalization of the eigenvectors.
05

Determine the dipole transitions and intensities

Determine the allowed magnetic dipole transitions by calculating the matrix elements of \(I_{Ax} + I_{Bx}\) and also find their relative intensities which are proportional to the square of these matrix elements.
06

Draw the NMR spectra for the given conditions

Finally, draw the NMR spectra corresponding to when: (a) \(J=0\) (b) \(J \ll (\delta_A - \delta_B) v_0\) (c) \(J = (\delta_A - \delta_B) v_0\) (d) \(\delta_A = \delta_B\). The change in the patterning and intensity of the spectra with the change in the coupling constant and chemical shifts forms the signature of the spin-spin coupling in the NMR spectrum.

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Most popular questions from this chapter

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.

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

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.

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 \(40 \mathrm{Hz}\).

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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