Determination of Energetics of Fluxional Molecules by NMR*

OpenStax-CNX module: m50237 1

Determination of Energetics of

Fluxional Molecules by NMR*

Andrew Metzger

Andrew R. Barron

This work is produced by OpenStax-CNX and licensed under the

Creative Commons Attribution License 4.0�

1 Introduction to �uxionality

It does not take an extensive knowledge of chemistry to understand that as-drawn chemical structures donot give an entirely correct picture of molecules. Unlike drawings, molecules are not stationary objects insolution, the gas phase, or even in the solid state. Bonds can rotate, bend, and stretch, and the moleculecan even undergo conformational changes. Rotation, bending, and stretching do not typically interfere withcharacterization techniques, but conformational changes occasionally complicate analyses, especially nuclearmagnetic resonance (NMR).

For the present discussion, a �uxional molecule can be de�ned as one that undergoes an intramolecularreversible interchange between two or more conformations. Fluxionality is speci�ed as intramolecular todi�erentiate from ligand exchange and complexation mechanisms, intermolecular processes. An irreversibleinterchange is more of a chemical reaction than a form of �uxionality. Most of the following examplesalternate between two conformations, but more complex �uxionality is possible. Additionally, this modulewill focus on inorganic compounds. In this module, examples of �uxional molecules, NMR procedures,calculations of energetics of �uxional molecules, and the limitations of the approach will be covered.

2 Examples of �uxionality

2.1 Bailar twist

Octahedral trischelate complexes are susceptible to Bailar twists, in which the complex distorts into atrigonal prismatic intermediate before reverting to its original octahedral geometry. If the chelates are notsymmetric, a ∆ enantiomer will be inverted to a Λ enantiomer. For example not how in Figure 1 with theGaL3 complex of 2,3-dihydroxy-N,N`-diisopropylterephthalamide (Figure 2) the end product has the chelateligands spiraling the opposite direction around the metal center.

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Figure 1: Bailar twist of a gallium catchetol tris-chelate complex. Adapted from B. Kersting, J. R.Telford, M. Meyer, and K. N. Raymond, J. Am. Chem. Soc., 1996, 118, 5712.

Figure 2: Substituted catchetol ligand 2,3-dihydroxy-N,N`-diisopropylterephthalamide. Adapted fromKersting, B., Telford, J.R., Meyer, M., Raymond, K.N.; J. Am. Chem. Soc., 1996, 118, 5712.

2.2 Berry pseudorotation

D3h compounds can also experience �uxionality in the form of a Berry pseudorotation (depicted in Fig-ure 3), in which the complex distorts into a C4v intermediate and returns to trigonal bipyrimidal geometry,exchanging two equatorial and axial groups . Phosphorous penta�uoride is one of the simplest examplesof this e�ect. In its 19FNMR, only one peak representing �ve �uorines is present at 266 ppm, even at lowtemperatures. This is due to interconversion faster than the NMR timescale.



Figure 3: Berry pseudorotation of phosphorus penta�uoride.

2.3 Sandwich and half-sandwich complexes

Perhaps one of the best examples of �uxional metal complexes is (π5-C5H5)Fe(CO)2(π1-C5H5) (Figure 4).

Not only does it have a rotating η5 cyclopentadienyl ring, it also has an alternating η1 cyclopentadienylring (Cp). This can be seen in its NMR spectra in Figure 5. The signal for �ve protons corresponds to themetallocene Cp ring (5.6 ppm). Notice how the peak remains a sharp singlet despite the large temperaturesampling range of the spectra. Another noteworthy aspect is how the multiplets corresponding to the otherCp ring broaden and eventually condense into one sharp singlet.

Figure 4: Structure of (π5-C5H5)Fe(CO)2(π1-C5H5). Reprinted with permission from M. J. Bennett

Jr., F. A. Cotton, A. Davison, J. W. Faller, S. J. Lippard, and S. M. Morehouse, J. Am. Chem. Soc.,

1966, 88, 4371. Copyright: American Chemical Society (1966).



Figure 5: Variable temperature NMR spectra of (π5-C5H5)Fe(CO)2(π1-C5H5). Reprinted with permis-

sion from M. J. Bennett Jr., F. A. Cotton, A. Davison, J. W. Faller, S. J. Lippard, and S. M. Morehouse,J. Am. Chem. Soc., 1966, 88, 4371. Copyright: American Chemical Society (1966).

3 An example procedure

Sample preparation is essentially the same for routine NMR. The compound of interest will need to bedissolved in an NMR compatible solvent (CDCl3 is a common example) and transferred into an NMR tube.Approximately 600 µL of solution is needed with only micrograms of compound. Compounds should be atleast 99 % pure in order to ease peak assignments and analysis. Because each spectrometer has its ownprotocol for shimming and optimization, having the supervision of a trained specialist is strongly advised.Additionally, using an NMR with temperature control is essential. The basic goal of this experiment is to�nd three temperatures: slow interchange, fast interchange, and coalescence. Thus many spectra will beneeded to be obtained at di�erent temperatures in order to determine the energetics of the �uctuation.

The process will be much swifter if the lower temperature range (in which the �uctuation is much slowerthan the spectrometer timescale) is known. A spectra should be taken in this range. Spectra at highertemperatures should be taken, preferably in regular increments (for instance, 10 K), until the peaks ofinterest condense into a sharp single at higher temperature. A spectrum at the coalescence temperatureshould also be taken in case of publishing a manuscript. This procedure should then be repeated in reverse;that is, spectra should be taken from high temperature to low temperature. This ensures that no thermalreaction has taken place and that no hysteresis is observed. With the data (spectra) in hand, the energeticscan now be determined.



4 Calculation of energetics

For intramolecular processes that exchange two chemically equivalent nuclei, the function of the di�erencein their resonance frequencies (∆v) and rate of exchange (k) is the NMR spectrum. Slow interchange occurswhen ∆v � k, and two separate peaks are observed. When ∆v � k, fast interchange is said to occur, andone sharp peak is observed. At intermediate temperatures, the peaks are broadened and overlap one another.When they completely merge into one peak, the coalescence temperature, Tc is said to be reached. In thecase of coalescence of an equal doublet (for instance, one proton exchanging with one proton), coalescencesoccurs when ∆v0t = 1.4142/(2π), where ∆v0 is the di�erence in chemical shift at low interchange and wheret is de�ned by (5), where ta and tb are the respective lifetimes of species a and b. This condition only occurswhen ta = tb, and as a result, k = ½ t.

1

t=

1

ta+

1

tb(5)

For reference, the exact lineshape function (assuming two equivalent groups being exchanged) is given bythe Bloch Equation, (5), where g is the intensity at frequency v,and where K is a normalization constant

g (v) =Kt (va + vb)

2

[0.5 (va + vb)− v]

2

+ 4π2t2 (va − v)2

(vb − v)2

(5)

4.1 Low temperatures to coalescence temperature

At low temperature (slow exchange), the spectrum has two peaks and ∆v � t. As a result, (5) reduces to(5), where T2a' is the spin-spin relaxation time. The linewidth of the peak for species a is de�ned by (5).

g (v)a = g (v)b =KT2a

1 + T 22a (va − v)

2

(5)

(∆va)1/2 =1

π

(1

T2a+

1

ta

)(5)

Because the spin-spin relaxation time is di�cult to determine, especially in inhomogeneous environments,rate constants at higher temperatures but before coalescence are preferable and more reliable.

The rate constant k can then be determined by comparing the linewidth of a peak with no exchange (lowtemp) with the linewidth of the peak with little exchange using (5), where subscript e refers to the peak inthe slightly higher temperature spectrum and subscript 0 refers to the peak in the no exchange spectrum.

k =π√2

[(∆ve)1/2 − (∆v0)1/2

](5)

Additionally, k can be determined from the di�erence in frequency (chemical shift) using (5), where ∆v0 isthe chemical shift di�erence in Hz at the no exchange temperature and ∆ve is the chemical shift di�erenceat the exchange temperature.

k =π√2

(∆v2

0 −∆v2e

)(5)

The intensity ratio method, (5), can be used to determine the rate constant for spectra whose peaks havebegun to merge, where r is the ratio between the maximum intensity and the minimum intensity, of themerging peaks, Imax/Imin

k =π√2

(r +

(r2 − r

)1/2 )−1/2(5)



As mentioned earlier, the coalescence temperature, Tc is the temperature at which the two peaks corre-sponding to the interchanging groups merge into one broad peak and (5) may be used to calculate the rateat coalescence.

k =π∆v0√

2(5)

4.2 Higher temperatures

Beyond the coalescence temperature, interchange is so rapid (k � t) that the spectrometer registers thetwo groups as equivalent and as one peak. At temperatures greater than that of coalescence, the lineshapeequation reduces to (5).

g (v) =KT2[

1 + πT2 (va + vb − 2v)2] (5)

As mentioned earlier, determination of T2 is very time consuming and often unreliable due to inhomogeneityof the sample and of the magnetic �eld. The following approximation ((5)) applies to spectra whose signalhas not completely fallen (in their coalescence).

k =0.5π∆v2

(∆ve) 1/2

− (∆v0)1/2 (5)

Now that the rate constants have been extracted from the spectra, energetic parameters may now becalculated. For a rough measure of the activation parameters, only the spectra at no exchange and coalescenceare needed. The coalescence temperature is determined from the NMR experiment, and the rate of exchangeat coalescence is given by (5). The activation parameters can then be determined from the Eyring equation((5)), where kB is the Boltzmann constant, and where ∆H� - T∆S� = ∆G�.

ln

(k

T

)=

∆H‡

RT− ∆S‡

R+ ln

(kBh

)(5)

For more accurate calculations of the energetics, the rates at di�erent temperatures need to be obtained. Aplot of ln(k/T) versus 1/T (where T is the temperature at which the spectrum was taken) will yield ∆H�,∆S�, and ∆G�. For a pictorial representation of these concepts, see Figure 6.



Figure 6: Simulated NMR temperature domains of �uxional molecules. Reprinted with permissionfrom F. P. Gasparro and N. H. Kolodny, J. Chem. Ed., 1977, 4, 258. Copyright: American ChemicalSociety (1977).

4.3 Diverse populations

For unequal doublets (for instance, two protons exchanging with one proton), a di�erent treatment is needed.The di�erence in population can be de�ned through (6), where Pi is the concentration (integration) of speciesi and X = 2π∆vt (counts per second). Values for ∆vt are given in Figure 7.

∆P = Pa − Pb =

[X2 − 2

3

]3/2(1

X

)(6)



Figure 7: Plot of ∆vt versus ∆P. Reprinted with permission from H. Shanan-Atidi and K. H. Bar-Eli,J. Phys. Chem., 1970, 74, 961. Copyright: American Chemical Society (1970).

The rates of conversion for the two species, ka and kb, follow kaPa = kbPb (equilibrium), and becauseka = 1/ta and kb = 1/tb, the rate constant follows (7).

ki =1

2t(1−∆P) (7)

From Eyring's expressions, the Gibbs free activation energy for each species can be obtained through (7)and (7)

∆G‡a = RTcln

(kTc

hπ∆v0x

X

1−∆P

)(7)

∆G‡b = RTcln

(kTc

hπ∆v0x

X

1−∆P

)(7)

Taking the di�erence of (7) and (7) gives the di�erence in energy between species a and b ((7)).

∆G‡ = RTcln

(Pa

Pb

)= RTcln

(1 + P

1− P

)(7)



Converting constants will yield the following activation energies in calories per mole ((7) and (7)).

∆G‡a = 4.57Tc

[10.62 + log

(X

2p (1−∆P)

)+ log (Tc/∆v)

](7)

∆G‡b = 4.57Tc

[10.62 + log

(X

2p (1−∆P)

)+ log (Tc/∆v)

](7)

To obtain the free energys of activation, values of log (X/(2π(1 + ∆P))) need to be plotted against ∆P(values Tc and ∆v0 are predetermined).

note: This unequal doublet energetics approximation only gives ∆G� at one temperature, and amore rigorous theoretical treatment is needed to give information about ∆S� and ∆H�.

5 Example of determination of energetic parameters

Normally ligands such as dipyrido(2,3-a;3′,2′-j)phenazine (dpop') are tridentate when complexed to tran-sition metal centers. However, dpop' binds to rhenium in a bidentate manner, with the outer nitrogensalternating in being coordinated and uncoordinated. See Figure 8 for the structure of Re(CO)3(dpop')Cl.This �uxionality results in the exchange of the aromatic protons on the dpop' ligand, which can be observedvia 1HNMR. Because of the complex nature of the coalescence of doublets, the rate constants at di�erenttemperatures were determined via computer simulation (DNMR3, a plugin of Topspin). These spectra areshown in Figure 9.

Figure 8: The structure of Re(CO)3(dpop')Cl. Reprinted with permission from K. D. Zimmera, R.Shoemakerb, and R. R. Ruminski, Inorg. Chim. Acta., 2006, 5, 1478. Copyright: Elsevier (2006).



Figure 9: experimental and simulated 1HNMR spectra for Re(CO)3(dpop')Cl. Reprinted with permis-sion from K. D. Zimmera, R. Shoemakerb, and R. R. Ruminski, Inorg. Chim. Acta., 2006, 5, 1478.Copyright: Elsevier (2006).

The activation parameters can then be obtained by plotting ln(k/T) versus 1/T (see Figure 10 for theEyring plot). ∆S� can be extracted from the y-intercept, and ∆H� can be obtained through the slope ofthe plot. For this example, ∆H�, ∆S� and ∆G�. were determined to be 64.9 kJ/mol, 7.88 J/mol, and 62.4kJ/mol.



Figure 10: Eyring plot of ln(k/T) versus 1/T for Re(CO)3(dpop')Cl. Adapted from K. D. Zimmera,R. Shoemakerb, and R. R. Ruminski, Inorg. Chim. Acta, 2006, 5, 1478. Copyright: Elsevier (2006).

6 Limitations to the approach

Though NMR is a powerful technique for determining the energetics of �uxional molecules, it does have onemajor limitation. If the �uctuation is too rapid for the NMR timescale (< 1 ms) or if the conformationalchange is too slow meaning the coalescence temperature is not observed, the energetics cannot be calculated.In other words, spectra at coalescence and at no exchange need to be observable. One is also limited bythe capabilities of the available spectrometer. The energetics of very fast �uxionality (metallocenes, PF5,etc) and very slow �uxionality may not be determinable. Also note that this method does not prove any�uxionality or any mechanism thereof; it only gives a value for the activation energy of the process. As a sidenote, sometimes the coalescence of NMR peaks is not due to �uxionality, but rather temperature-dependentchemical shifts.

7 Bibliography

• H. S. Gutowsky and C. J. Ho�mann, J. Chem. Phys., 1951, 19, 1259.• H. Nakazawa, K. Kawamura, K. Kubo, and K. Miyoshi, Organometallics, 1999, 18, 2961.• C. Raynaud, L. Maron, J. P. Daudey, and F. Jolibois. ChemPhysChem, 2006, 7, 407.• F. P. Gasparro and N. H. Kolodny, J. Chem. Ed., 1977, 4, 258.• B. Kersting, J. R. Telford, M. Meyer, and K. N. Raymond, J. Am. Chem. Soc., 1996, 118, 5712.• M. J. Bennett Jr., F. A. Cotton, A. Davison, J. W. Faller, S. J. Lippard, and S. M. Morehouse, J. Am.

Chem. Soc., 1966, 88, 4371.• H. Shanan-Atidi and K. H. Bar-Eli, J. Phys. Chem., 1970, 74, 961.• D. J. Press, N. M. R. McNeil, A. Rauk, and T. G. Back, J. Org. Chem., 2012, 77, 9268.• B. D. Nageswara Rao, Meth. Enzymol., 1989, 176, 279.• K. D. Zimmera, R. Shoemakerb, and R. R. Ruminski, Inorg. Chim. Acta, 2006, 5, 1478.


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Determination of Energetics of Fluxional Molecules by NMR*

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