J. Phys. Chem. 1993,97, 49414944
4941
Bayesian Analysis Investigation of Chemical Exchange above and below the Coalescence Point K. Sbay Vines, Ronald F. Evilia, and Scott L. Whittenburg’ Department of Chemistry, University of New Orleans, New Orleans, Louisiana 70148 Received: January 25, 1993; In Final Form: March 3, 1993
The coalescence temperature for the exchange of the N-methyl substituents in N,N-dimethylacetamide (DMA) was determined to be 97 OC by Bayesian analysis, which is higher than any previously reported temperature. The FFT method is unable to determine that there are still two separate peaks for the N-methyl groups just below this temperature due to line broadening in the Fourier transformed spectrum. The decay rates for DMA over a temperature range 23-141 OC were determined by the doublet frequency model of Bayesian analysis below the coalescence temperature, 97 OC,and by the singlet frequency model at and above 97 OC. These decay rates were used with the time domain spectral equations to determine the rate of exchange at each temperature. The Arrhenius plot of the rate constant as a function of temperature was linear over this temperature range and gave the following activation parameters: E, = 19.37 f 0.03 kcal/mol, In A = 31.10 f 0.07, and AG*298~ = 18.40 f 0.03 kcallmol.
Introduction The spectral averaging effects of dynamic processes on NMR spectra are well-known and understood.’ Because chemical shift differences are typically in the few hertz to several thousand hertz range, NMR spectra are sensitive to dynamic processes in the tenths of secondsto sub-millisecondtime scale. While spectral averaging gives an immediate qualitative estimate of the time scale of the process, it can be difficult to extract reliable, quantitative information. These difficulties arise from a variety of sources, including errors in measuring the sample temperature, errors arising from the temperature dependence of the “frozen” chemical shiftsand lineshapes,as well as the uncertaintiesinherent in all fitting procedures, especially when low signal-to-noiseratio data are involved. Thus, it is common for NMR measured kinetic parameters to have substantial error bars.2 Perhaps the most extensivelystudied class of dynamic processes is that associated with the barrier to rotation about the C-N bond in amides, R , C ( O ) N ( R ) Z . ~This - ~ ~ barrier arises from the partial double bond character of the amide C-N bond as shown in resonance structure 11. Typically, the partial double bond
There are several inherent problems with the method of total line shape analysis. The method relies on numbers obtained from the frequency domain spectrum and therefore is subject to estimation errors caused by line broadening and signal-to-noise ratio limitations. Furthermore, one needs to know the chemical shifts and chemical shift temperature dependencies in the absence of exchange. Significant errors can be introduced if other temperature-dependent processes take place in addition to exchange. For example, dimerizationof N,N-dimethylformamide led to considerable variation in measured rate constants in early ~ o r k . ~IfJ the signals are complicated by coupling of the RI group with the R groups, the parameters required for total line shape analysis can be difficult to determine. The method relies on comparison of a calculated spectrum to the experimental spectrum. Visual comparison has obvious limitations. Neuman and Jonas used a least-squares procedure to find the best fit of calculated and observed parameters subject to the limitations mentioned above.* In this paper we report the use of Bayesian analysis to extract decay information directly from the time domain signal, therefore avoiding the problems associatedwith frequencydomain estimates.
Theory
I
I1
character increases the barrier to rotation about the CN bond sufficiently that the two nitrogen R groups are nonequivalent and give rise to resolved resonances at room temperature. As the temperature is raised, the exchange rate becomes faster, resulting in a decrease in the shift difference between the two signals and line broadening. When the temperature is sufficiently high to cause the rate of rotation to exceed the “frozen” shift difference, the two signals coalesce into one signal. As temperature is further increased beyond the coalescencepoint, the singlesignal sharpens, eventually reaching an averagevalue characteristic of the inherent line widths of the two exchanging sites. Early studies of this exchange process involved line shape analysis of the two signals before c o a l e ~ c e n c e . ~ ~ ~ -These ~J studies assumed the same line shape for each signal produced by the individual R groups. Later studies incorporated total line shape analysis, taking into account the shapes of each signal produced by each R group.
* To whom correspondence should be addressed.
Consider a system with two sites, A and B, in which each site has a single Larmor frequency, YA and YB, respectively, with two transverse relaxation rates R ~ =A 1/ T ~ and A R ~ =B1/ T ~ BThe . signals, iw1,2,are given by20
1 ial,2 = - - [ R ~ A R,B 2 where
+
W
+ kAB + kBA-i(YA + YB)
+
= [A2 4 k , ~ k ~ , ] ” ~
f w]
(1)
(2)
and
A = R2, - R2,
+ k,B
- kBA - i(VA - Y B )
(3) The rate constants, AB and ~ B A are , related through the mole fractions in chemical equilibrium,
kAB = X B ~ ,~ B = A X A ~ (4) The decay rate of the signal, a,comes from the real part of eq 1. The rate constant k can then be calculated by eq 1 from the signal decay rate.
0022-3654/93/2097-4941$04.00/00 1993 American Chemical Society
Vines et al.
4942 The Journal of Physical Chemistry, Vol. 97,No. 19, 1993
The room temperature decay rate, which was determined via Bayesian analysis, was used for RZAand R ~ B : (5) = RZB = %onexchanging Also, AB = keAfor our system. The rate constant for the overall exchange is R.2A
k = 2kAg = 2 k g ~
(6)
At low temperatures (before coalescence), the real part of eq 1 reduces down to 1
(Yobserved
= j[2anoncxchanging + k1
probability P(DIB,w,a,I) divided by the (normalization) marginal probability of the data given only the prior information. In application to NMR data, the amplitudes, B, are initially treated as nuisance parameters in order to locate the most probable frequencies independent of their amplitudes. Nuisance parameters may only be integrated out of the probability if the model functions are orthogonal. To make the model functions in eqs 10 and 11 orthogonal, the eigenvalues and eigenvectors of the matrix formed by all possible products of the model functions are computed.21 Bretthorst has shown that, after removal of the nuisance parameters, eq 14 can be expressed as
(7)
At high temperatures (after coalescence), the real part of eq 1 becomes where m is the number of model functions -and N is the number of data points. The sufficient statistic, hZ,is given by where Av = V A - YB is the frequency difference between the two signals in hertz at room temperature under nonexchanging conditions. For NMR data collected in the time domain, the FID, or free induction decay, is represented by
di =Ati)+ e,
(9)
Here, di is the discrete data value at time t,, e, is the value for the noise at that time, andf(ti) is the model function describing the FID in the absence of noise. Typical NMR data are acquired as both in-phase and quadrature signals, so for a single frequency in the FID, the model functions are fR(t)
+
= (B, cos(ot) B, sin(ot))e-a'
(10)
for the in-phase (or real) channel and
f i ( t ) = (B, cos(ot)
+ B , sin(wt)}e-a'
(1 1)
for the quadrature (or imaginary) channel. Here, B I and BZare amplitudes, w is the frequency in hertz, and a is the decay rate, or 1/ Tz, in hertz. If the signal is perfectly in phase, BZis zero. Under this condition the real channel would contain only the cosine function and the imaginary channel would contain only a sine function with the same amplitude as the real channel. Assuming that the data may be modeled by this frequency model function, the probability that the data contains a resonance at frequency w with amplitudeB and decay rate a is given by Bayes's Theorem, which follows directly from the product rule of probability theory, which states P(X,YII) = P(XlV,I) PCVII) (12) The probability is a continuous function describing the relative 'believability" of all possible values of the parameters. The most probable set of values of the parameters is the one that gives the maximum in the probability. A significant benefit to using probability theory in this form is that one can use marginalization to eliminate nuisance parameters and then back calculate them. Marginalization is defined as P(Xl1) = JP(X,YlI) dY
P(B,w,alI) m v , o , d )
The value hi is the projection of the orthogonal model functions onto the experimental FID, and the di are the actual data points from the experimental FID. The model functions are the closedform expressions describing the experimental data, such as eqs 10 and 11. The probability given by eq 15 is the joint marginal posterior probability for the frequencies and decay rates and is called the Student's t-distribution. The maximum of this probability occurs when thesufficient statistic h2 is maximized. Bretthorst has shown2'that h2can be evaluated from an equation which takes advantage of the fast Fourier transform, FFT. The data are first premultiplied by an exponential decay, and the FFT is performed on the data set which is zero-filled in order to achieve adequate digital resolution. In this technique, the sufficient statistic is given by
where
R, = R(w,,a) and Ii = I(o,,a)
(19) which represent the real and imaginaryparts of the discrete Fourier transform of the model functions described in eqs 10 and 11. C(0,2a) is described by the function
(13)
where y is the nuisance parameter. Bayes's Theorem as applied to NMR data is written as P(B,w,alD,I) =
and dz is given by
(14) VII) This means that the probabilityof the parameter values in question being the 'correct" ones given the data D and any prior information I is the product of the prior probability P(B,w,alI) and the direct
where w and a are in reduced units.,' The above model can be extended to the case where there are two or more resonances in the data. The nonorthogonal model functions for two frequencies in the data are fR(t)
= (B, cos(w,t)
+ B, sin(w,t)}e"" + (B, cos(w,t) + B, sin(w,t)je-a2' (21)
Bayesian Analysis Investigation of Chemical Exchange
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 4943 45
and fi(t)
= ( B cos(o,t) ~
+ B , ~ i n ( o , t ) ) e - ~+" (B4cos(w,t) + B, s i n ( ~ ~ t ) ] e(22) ~~'
48
For the two-frequency model, the probability is the same as that in eq 12, but the sufficient statistic, h2, is now a function of the two frequencies and two decay rates. The situation is simplified somewhat if the signal can be modeled as a doublet (i.e., two equal intensity signals with the same decay rate), rather than simply two different frequencies of independent amplitude and decay. The frequencyof each peak in the doublet can be specified as occurring at plus and minus AI2 about a center frequency wc. For the doublet model, h2 is represented by
52
55
-
0
For the singlet, - h2 is computed as a function of a and w . For the doublet, h2 is computed as a function of A, a,and w , and the maximum is found in terms of these variables. The values found for a at the various temperatures can be used to solve for k according to eq 7 before coalescence and eq 8 after coalescence without estimation of spectral parameters from the frequency domain. We studied the exchange of the N-methyl groups in N,Ndimethylacetamide (DMA, 111). In this system, we assume that
2
5
8
1
0
Decay (Hz)
Figure 1. Doublet frequency model Bayesian transform spectrum of the experimental N,N-dimethylacetamide FID at 58 O C . Contour plot of the sufficien_tstatisticas a function of A and a at the value for wc that maximizes hZ is shown. 50
40 "3+ :N
