4.21: Calculating the AB Proton NMR Using Tensor Algebra
- Page ID
- 150937
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The purpose of this tutorial is to deviate from the usual matrix mechanics approach to the ABC proton nmr system in order to illustrate a related method of analysis which uses tensor algebra. For a discussion of the traditional approach for the ABC system visit http://www.users.csbsju.edu/~frioux/nmr/Speclab4.htm. This site also provides general information on the quantum mechanics of nmr spectroscopy.
\[ \begin{matrix} \text{Nuclear spin and identity operators:} & I_x = \frac{1}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} & I_y = \frac{1}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} & I_z = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} & I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \text{Chemical shifts:} & \nu_A = 250 & \nu_B = 300 & \text{Coupling constant:} & \text{Jab} = 10 \end{matrix} \nonumber \]
Hamiltonian representing the interaction of nuclear spins with the external magnetic field in tensor format:
\[ \begin{matrix} \hat{H}_{mag} = \nu_A \hat{I}_z^A - \nu_B \hat{I}_z^B = - \nu_A \hat{I}_z^A \otimes \hat{I} + \hat{I} \otimes \left( - \nu_B \hat{I}_z^B \right) & \text{where, for example,} & \nu_A = g_n \beta_n B_z(1- \sigma_A) \end{matrix} \nonumber \]
Implementing the operator using Mathcad's command for the tensor product, kronecker, is as follows.
\[ H_{mag} = - \nu_A \text{kronecker}(I_z,~I) - \nu_B \text{kronecker}(I,~I_z) \nonumber \]
Hamiltonian representing the interaction of nuclear spins with each other in tensor format:
\[ \hat{H}_{spin} = Jab \left( \hat{I}_x^A \otimes \hat{I}_x^B + \hat{I}_y^B \otimes \hat{I}_y^B + \hat{I}_z^A \otimes \hat{I}_z^B \right) \nonumber \]
Implementation of the operator in the Mathcad programming environment:
\[ H_{spin} = Jab \left( \text{kronecker}(I_x,~I_x) + \text{kronecker}(I_y,~I_y) + \text{kronecker}(I_z,~I_z) \right) \nonumber \]
The total Hamiltonian spin operator is now calculated and displayed.
\[ H = H_{mag} + H_{spin} \nonumber \]
\[ \begin{matrix} \begin{array} \alpha \alpha \alpha & & \alpha \beta & & \beta \alpha & & \beta \beta \end{array} \\ \begin{pmatrix} -272.5 & 0 & 0 & 0 \\ 0 & 22.5 & 5 & 0 \\ 0 & 5 & -27.5 & 0 \\ 0 & 0 & 0 & 277.5 \end{pmatrix} \begin{array} \alpha \alpha \alpha \\ \alpha \beta \\ \beta \alpha \\ \beta \beta \end{array} \end{matrix} \nonumber \]
Calculate and display the energy eigenvalues and associated eigenvectors of the Hamiltonian.
\[ \begin{matrix} i = 1 .. 4 & E = \text{sort(eigenvals(H))} & C^{<i>} = \text{eigenvec}(H,~E_i) \end{matrix} \nonumber \]
\[ \begin{matrix} \text{augment}(E,~C^T )^T = \begin{pmatrix} -272.5 & -27.995 & 22.995 & 277.5 \\ 1 & 0 & 0 & 0 \\ 0 & -0.099 & 0.995 & 0 \\ 0 & 0.005 & 0.099 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{array} \alpha \alpha \alpha \\ \alpha \beta \\ \beta \alpha \\ \beta \beta \end{array} \end{matrix} \nonumber \]
The nmr selection rule is that only one nuclear spin can flip during a transition. Therefore, the transition probability matrix for the AB spin system is:
\[ \begin{matrix} T = \begin{pmatrix} ' & \alpha \alpha & \alpha \beta & \beta \alpha & \beta \beta \\ \alpha \alpha & 0 & 1 & 1 & 0 \\ \alpha \beta & 1 & 0 & 0 & 1 \\ \beta \alpha & 1 & 0 & 0 & 1 \\ \beta \beta & 0 & 1 & 1 & 0 \end{pmatrix} & T = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{pmatrix} \end{matrix} \nonumber \]
Calculate the intensities and frequencies of the allowed transitions.
\[ \begin{matrix} i = 1 .. 4 & j = 1 .. 4 & I_{i,~j} = \left[ C^{<i>} \left( TC^{<j>} \right) \right]^2 & V_{i,~j} = \text{if} \left( I_{i,~j} > .001,~ \left| E_i - E_j \right|,~ 0 \right) \end{matrix} \nonumber \]
\[ \begin{matrix} \text{Intensity matrix:} & I = \begin{pmatrix} 0 & 0.8 & 1.2 & 0 \\ 0.8 & 0 & 0 & 0.8 & \\ 1.2 & 0 & 0 & 1.2 \\ 0 & 0.8 & 1.2 & 0 \end{pmatrix} & \text{Frequency matrix:} & V = \begin{pmatrix} 0 & 244.5 & 295.5 & 0 \\ 244.5 & 0 & 0 & 244.5 \\ 295.5 & 0 & 0 & 254.5 \\ 0 & 305.5 & 254.5 & 0 \end{pmatrix} \end{matrix} \nonumber \]
Display the calculated AB nmr spectrum:
