11.3.5: Applications of Tanabe-Sugano Diagrams
- Page ID
- 377930
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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 Tanabe-Sugano diagrams can be used to interpret absorption spectra and gain insight into the properties of a coordination complex. For example, you could use the appropriate diagram to predict the number of transitions, assign the identity of a specific transition, or calculate the value of \(\Delta\) for a specific metal complex.
- Determine the \(d\)-electron count of the metal ion of interest.
- Choose the appropriate Tanabe-Sugano diagram: this is the one matching the \(d\)-electron count of the metal ion. There is a full list of Tanabe-Sugano diagrams in the Resources Section.
- Acquire an electronic spectrum of the metal complex and identify \(\lambda_{max}\) for spin-allowed (strong intensity) and spin forbidden (weak intensity) transitions.
- Convert wavelength (\( \lambda_{max} \)) to energy (E) in wavenumbers (\(cm^{-1}\)) and generate energy ratios relative to the lowest-energy allowed transition. (i.e. \(\frac{E_2}{E_1}\) and\(\frac{E_3}{E_1}\)).
- Using a ruler, slide it across the printed Tanabe-Sugano diagram until the E/B ratios between lines is equivalent to the ratios found in step 4.
- Solve for B using the E/B values (y-axis, step 4) and Δoct/B (x-axis, step 5) to yield the ligand field splitting energy, \(Delta\) (Sometimes this is labeled as \(10D_q\), and it is useful to know that \(\Delta=10D_q\)).
A Cr3+ metal complex has strong transitions and \(\lambda_{max}\) at
- 431.03 nm,
- 781.25 nm, and
- 1,250 nm.
Determine the \(Δ_{oct}\) for this complex.
Solution
- Cr has 6 electrons. Cr3+ has three electrons, so it has a d-configuration of d3
- Locate the d3 Tanabe-Sugano diagram
- Convert to wavenumbers:
\[\dfrac{10^7(nm/cm)}{1250\; nm}= 8,000\; cm^{-1}\]
\[\dfrac{10^7(nm/cm)}{781.25\; nm}= 13,600\; cm^{-1}\]
\[\dfrac{10^7(nm/cm)}{431.03\; nm}= 23,200\; cm^{-1}\]
- Allowed transitions are \(\ce{^4T_{1g}} \leftarrow \ce{ ^4_{\,}A_{2g}}\), \(\ce{^4T_{1g} \leftarrow ^4_{\,}A_{2g}}\) and \(\ce{^4T_{2g}\leftarrow ^4_{\,}A_{2g}}\).
Transition | Energy cm-1 | Ratios to lowest |
---|---|---|
\(\ce{^4T_{1g}} \leftarrow \ce{ ^4_{\,}A_{2g}}\) | 23,200 | 2.9 |
\(\ce{^4T_{1g} \leftarrow ^4_{\,}A_{2g}}\) | 13,600 | 1.7 |
\(\ce{^4T_{2g}\leftarrow ^4_{\,}A_{2g}}\) | 8,000 | 1 |
- Sliding the ruler perpendicular to the x-axis of the d3 diagram yields the following values:
Δoct/B | 10 | 20 | 30 | 40 |
---|---|---|---|---|
Height E(ν3)/B | 29 | 45 | 64 | 84 |
Height E(ν2)/B | 17 | 30 | 40 | 51 |
Height E(ν1)/B | 10 | 20 | 30 | 40 |
Ratio E(ν3)/E(ν1) | 2.9 | 2.25 | 2.13 | 2.1 |
Ratio E(ν2)/E(ν1) | 1.7 | 1.5 | 1.33 | 1.275 |
- Based on the two tables above it should be assessed that the Δoct/B value is 10. B is found by dividing E by the height.
Energy cm-1 | Height | B |
---|---|---|
23,200 | 29 | 800 |
13,600 | 17 | 800 |
8,000 | 10 | 800 |
- Next multiply Δoct/B by B to yield the Δoct energy. \[10 \times 800 = 8000\; cm^{-1}=Δ_{oct}\]
Each problem is of varying complexity as several steps may be needed to find the correct Δoct/B values that yield the proper energy ratios.