Modern NMR Spectroscopy in Education - American Chemical Society

Chapter 13

NMR Exchange Spectroscopy

Downloaded by CORNELL UNIV on November 24, 2012 | http://pubs.acs.org Publication Date: August 16, 2007 | doi: 10.1021/bk-2007-0969.ch013

Holly C. Gaede Department of Chemistry, Texas A&M University, College Station, TX 77843

Two-dimensional exchange spectroscopy (EXSY) is a useful approach for studying dynamic processes with NMR. This method may be used in undergraduate laboratories as an alternative to the established techniques of line shape analysis or saturation transfer. In this chapter, the theory is briefly outlined for all three methods, and their experimental approaches are compared.

Introduction Dynamic NMR involves the study of samples that undergo chemical or physical changes with time. Because no sample is truly rigid, all NMR is dynamic NMR. The timescale of motions that can be observed ranges from nanoseconds to minutes, depending on the sampled experimental observable, such as chemical shift, relaxation rate, or coupling constant. The accessible timescale includes important molecular motions, including cis-trans isomerization and boat-chair cyclohexane intraconversions. Timescales for molecular motions and NMR experiments that can be used to study them are summarized in Figure 1. Besides molecular motions, important dynamic processes that occur in the available time scale include keto-enol tautomerization and the formation of intermolecular complexes. The focus of this chapter will be on the exchange of two species, A and B, between two different environments. Examples of this kind of dynamic equilibrium include internal rotations about a bond, ring puckering, and proton exchange.

176

© 2007 American Chemical Society

In Modern NMR Spectroscopy in Education; Rovnyak, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2007.

177

Timescale (s) 3

10

3

6

10-

10°

10-

1 0

db

Chemical kinetics hindered

Ε

rotation about a bond


Free

Rotating Frame Relaxation Measurements

c

Ε

-12

proton transfer

ring flips

Real time monitoring

Ί α) Ο Χ

10

Bond librations

Protein unfolding

g

-9

EXSY

α>

Relaxation Measurements

Lineshape analysis

5

X

Figure 1. Timescales for molecular motions and NMR experiments that c used to study them. The NMR parameters measured are reflective of the environment. If the parameter measured is chemical shift, the exchange between two environments is considered slow if k«\S -δ \ \ intermediate if -δ \; and fast if A

Β

Β

k » \δ - δ I, where the chemical shifts of the two environments are δ and δ . Chemical shift differences in NMR spectrometers available in most wellequipped undergraduate labs (*H resonancefrequenciesof 200 - 400 MHz) are typically 10 - 500 Hz, meaning that fast exchange occurs in systems that are exchanging with rate constants k^;10 s", corresponding to lifetimes of milliseconds or less. Since chemical shift differences are magnetic field dependent, a system that is in the fast exchange regime on a low field instrument may enter the slow exchange regime when studied at a higher field. Under slow exchange, each chemical shift is observed distinctly. In contrast, under fast exchange, only a single chemical shift is observed at the population-weighted average position. Α

B

Α

3

δ

Sobs = Ρλ Α

Β

1

+0-^)^5»

0)

where p is thefractionalpopulation of A. Figure 2 shows the expected appearance of NMR spectra under slow, intermediate, and fast exchange. Note that different NMR parameters give windows into different time regimes. Averaged H coupling constants are observed for all but the slowest exchange processes, as H coupling constants have differences on the order of only 10 Hz, making 100 ms the upper lifetime limit of fast exchange. Relaxation times, A

!

L


178


Slow exchange

0

Hz

-100

-200

Figure 2. Simulated NMR spectra for a system with two resonances separ by 200 Hz undergoing mutual site exchange at different rates.

with differences on the order of 1 ms in large systems, are averaged only in systems with lifetimes -0.1 ms or less. One intuitive method for extracting the rate constant of an exchange process is through two-dimensional NMR exchange spectroscopy (EXSY), where the exchange network is visually apparent in the spectrum. In fact, one of the first published two-dimensional spectra was a study of the exchange of N,Ndimethylacetamide methyl protons.(l) The EXSY experiment serves as a nice introduction to two-dimensional NMR, as its mechanism of coherence transfer is more intuitive than that in COSY or NOESY experiments. This chapter will compare this 2D approach to two other approaches used to measure exchange in undergraduate laboratories: lineshape fitting (2) and saturation transfer.(3) For ease of comparison, the same prototypical system will be used as an example throughout, the cis-trans isomerization of N,Ndimethylacetamide. This system is an example of mutual site exchange because the departing methyl group is replaced by an equivalent one. The partial double bond character


179

>

H C


3

(CH ) 3

A

/ ΝJ

\ (CH ) 3

>

(CH ) 3

H C 3

B

B

/ Ν1

\ (CH ) 3

A

of the amide bond yields a high internal rotation barrier, and the *H NMR spectrum is simple, just three well-resolved singlets under slow exchange. Beyond teaching NMR principles, these experiments can be used to illustrate concepts in many subdisciplines of chemistry. These experiments could be used in physical chemistry to illustrate basic kinetic principles and give the students experience at manipulating the Eyring equation. In organic chemistry these experiments could be used to illustrate the nature of rotation about a double bond.(^) Because these amides serve as simple models for peptide bonds, these studies could be used to explain the conformations found in proteins. Ideally, any of these experimental approaches could be combined with a computational study of the rotational barrier.(5)

Lineshape Analysis As outlined above, chemical exchange processes involve characteristic changes in the NMR lineshape. For the simple case of mutual site exchange for two equally populated states, the Bloch equations may be used to derive these lineshapes. More general cases will require a density matrix approach, which is found in an excellent recent review as well as classic texts.(tf-S) A spin-1/2 nucleus in a static magneticfieldB has a magnetization, M , along the same axis. There are two horizontal magnetization components, u and v, that oscillate about this axis at the larmorfrequencyω = χΒ , where γ is the gyromagnetic ratio of the nucleus. In the NMR experiment, there is an additional B! radiofrequency field. The Bloch equations give the time behavior of the magnetizations. 0

z

0

du ί \ = { -ω)νat ωο

0

u — 1

(2)

2

— = ^ - ( ^ - 4 - 7 at i

(3)

2

With a complex magnetization defined as, M = u + iv, the Bloch equation then becomes


180 ^-

Μ -ί{ω -ω)Μ-^

= ίγΒι

ζ

0

(4)

Ι

at

2

In the case of mutual site exchange between two sites A and Β at a rate of k, the Bloch equations for A and Β become: ^f- = iyB,M - i{m -a>)M -^ zA

A

A


Clt

+

kM -kM , B

(5)

A

T-y

= ίγΒ Μ χ

at

ζΒ

- ί{ω - ω)Μ - ^ - kM - kM Τ Β

Β

+

A

(6)

B

Ί

These equations are known as the McConnell equations after Η. M. McConnell, who first derived them in 1958.(P) Here it is assumed that M ^ M ^ M z since the B! field is so small as not to disturb the magnetization along z. Note that some derivations define Ι/ΙΜ^Μ^Μ^, and as a consequence the results that follow differ by a factor of 2. At equilibrium, dM /dt = dM /dt = 0 . Neglecting the T terms, the following expression for lineshape may be obtained, recalling that the observable part of the magnetization is the imaginary part of M +M at steady-state. A

B

2

A

ν = γΒ Μ Χ

B

*{»Λ-»ΒΫ

Ζ

2

2

A

Β

2

* k {(œ -ω)+{ω -ω))

(

7

)

2

+{ω -ω) {ω -ω)

Β

Α

Β

In this equation, the units are in radians per second. Using ω=2πν, the absorption lineshape can be given in frequency units.

**(V -VBY

g{v)=yB M ]

(8)

A

z

This equation is valid as long as linebroadeningfromprocesses other than exchange is negligible. Historically, equation 8 was approximated under certain limiting conditions, and k could be estimatedfromthe spectrum. In particular, coalescence, where the appearance of the spectrum changedfromtwo separate peaks to one flat-topped peak, was sought. At coalescence thefirstand second derivative of the lineshape is zero, and k = π(ν - v )/yÎ2 • A laboratory experiment published nearly 30 years ago uses this approach tofindk for the cisC()alesence

Α

B


181 T

trans isomerization of Ν,Ν-dimethylacetamide, where AG was found to be 71 kJ/mole. (Figure 3) These approximations are generally severe, and today a better approach is a full-curve fitting of equation 8.(70) Optimal values for k, v , andv can be obtained from simulations in which deviations between the calculated and experimental spectrum are minimized. Computational packages are available for this fitting (e.g. WinDNMR(77)). Alternatively,fittingroutines can be written with symbolic programs such as MathCad or Mathematica. B


A

V, HfctalotivetoTMS)

Figure 3. Effect of temperature on line shapes and values for k, in the ci isomerization ofΝ,Ν-dimethylacetamide. Used with permission from the Journa of Chemical Education, Vol. 54, No. 4, 1977, pp. 258-261; copyright© 19 Division of Chemical Education, Inc.


182 Saturation Transfer Another method for evaluating the rate of exchange exploits the fact that irradiation of resonance A will cause changes in the intensity of resonance Β in the case of slow exchange. The rate equations for the lower spin state populations [A] and [B] are given by


M - . -

k

W

m

, . M .

( 9 )

Similarly, the rate equations for the upper spin-state populations [A*] and [B*] are given by 4ϋ1 = - φ * ] dt

+

φ * ] = - « . dt

(10)

Spin-lattice relaxation processes that keep the upper and lower spin state populations in equilibrium in the absence of chemical exchange are given by dM dt

A

M -M T

=

0A

dM dt

A

B

XA

M -M T

=

0B

B

XB

where the net magnetizations are given by M = [À\- [À *] and M - [β]- [B *], and MOA and M B are the magnetizations at equilibrium. Combining equations 9-11 gives the change of net magnetization of A and Β due to spin-lattice relaxation and chemical exchange A

B

0

^L,- ( -M,h^MZ^L.

(,2)

k Uj

dt dM

Modern NMR Spectroscopy in Education - American Chemical Society

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