Anal. Chem. 1995, 67, 4309-4315
Applicability of Factor Analysis in Solid State NMR Jeff M. Koonst and Paul D. Ellis*l*
Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina 29208, and Environmental Molecular Sciences Laboratory, Pacific Northwest Laboratory, Richland, Washingfon 99352
Factor analysis has been used to deduce the composition of multicomponent magic angle spinning NMR spectra comprised of overlapping isotropic peaks. The technique has been used to determine the number of constituents present and was combined with a target transformation minimization procedure to identify the component MAS spectra. The new analysis procedure is compared to the conventional least-squares approach and is found to be superior in all cases. An analysis of a chemical shift-dominated (Z= ' / z ) magic angle spinning (MAS) NMR spectrum yielding the complete chemical shift tensor has been previously Such accounts focus primarily on fitting the spinning side band intensities arising from (1) a single chemical component or (2) a mixture of several chemically dissimilar species, Le., where the component spectra have a large dispersion in the isotropic chemical shifts. However, when the sample of interest is a mixture of chemically similar components having nearly identical isotropic chemical shifts, the analysis of the MAS line shape is typically more complex. Extracting NMR parameters and concentrations solely on the basis of a fit to side band intensity is no longer applicable because the side band intensity is affected by the relative separation of each isotropic peak. In this case, a more detailed least-squares procedure involving the overlap of the experimental and calculated line shapes is required to fully account for the observed intensities. However, in practice, this method often fails to converge to the global minimum or returns with uncertainties in the NMR parameters which are unacceptable. Such difficulties arise because the contribution of the NMR parameters to the side band intensity is often blurred by intensity modulations introduced by the relative concentration of each component and the amount of peak overlap. In the present study, we have developed a technique based on abstract factor analysis (FA) target transformation (lT),4 similar to established methods in EPR5 which overcomes the difficultiesencountered when fitting MAS spectra with overlapping isotropic peaks. The technique gains the advantage by reducing the impact of the overlap and concentration modulation on the side band intensity. We have used the FA-'IT procedure to reproducibly recover, from a variety of initial guesses, the +
University of South Carolina.
* Pacific Northwest Laboratory. (1) Herzfeld, J.; Burger, A. E. J. Chem. Phys. 1980,73, 6021-6030. (2) Marchetti, P. S.; Bank, S.; Bell, T. W.; Kennedy, M. A,; Ellis, P. D. J. A m . Chem. SOC.1989,1 1 1 , 2063-2066. (3) Marchetti, P. S.; Ellis, P. D.; Bryant, R G. J. A m . Chem. SOC.1985,107, 8191-8196. (4) Malinowski, E. R. Factor Analysis in Chemisty, 2nd ed.; John Wiley and Sons: New York, 1991. (5) Moens, P.; De Volder, P.; Hoogewijs, R.; Callens. F.; Verbeeck, R J. Magn. Reson., Ser. A 1993,101, 1-15.
0003-2700/95/0367-4309$9.00/0 0 1995 American Chemical Society
complete chemical shift tensor and relative concentration for the individual components in a series of multicomponent mixtures containing salts of cadmium sulfate. In all cases, both the precision and the accuracy of the extracted NMR parameters and relative concentrations are increased over the least-squares approach, which considers the overlap as well as side band intensity. EXPERIMENTAL SECTION
3CdS04.8HzO (Fisher Scientific Co., Lot 732035) was recrystallized by slow evaporation of a solution containing 50 g of 3CdS044HzO and 150 g of glass-distilled water. Large opaque crystals began to form within 10 days. Excess HzO had completely disappeared within 2 weeks, at which point the crystals were stored in an air-tight bottle. To generate samples with different amounts of 3CdS04.8HzO and CdS04.Hz0, the recrystalliied 3CdS044HzO was dehydrated under a nitrogen atmosphere using a ramp-and-hold temperature program (start 273 K, rate 20 deg/ min, hold 5 min) . Samples used for the NMR measurements were prepared with the temperature program described; the final temperatures were (a) 273, (b) 300, (c) 325, (d) 350, (e) 375, and (0400 K. All '13Cd NMR work was performed at a field strength of 7.05 T ( W C ~= 66.53 MHz) using magic angle spinning, cross polarization (CP),s and proton decoupling ('H field strength was 41 KHz). The 90" pulse width was 4.8 ps. Sample rotation was controlled at 3 KHz in all cases with a precision of fl Hz. Care was taken to optimize the Hartmann-Hahn match condition7 in order to achieve an integrated intensity for the A and B sites of 3CdS044H20 close to 2:l. All NMR experiments carried out on intermediate compositions were performed under these conditions, assuming the CP dynamics are somewhat constant for the different salts of cadmium present. The recycle delay was determined to be 5 s for 3CdS044HzO and 10 s for CdSO4*H20. All intermediate compositions were collected using a 10 s recycle delay. An acquisition time of 0.16 s was used in every case. The number of transients was adjusted in each experiment to achieve a signal-to-noise ratio that was on the same order as that obtained from 32 transients on 3CdSO44HzO; the samples respectively required (b) 64,(c) 512, and (d-0 4096 transients. AU measurements were made at room temperature, and all chemical shifts are reported with respect to external 1.0 M CdCl2. All data processing and calculations were performed using the RZX solid state analysis package developed in our l a b o r a t ~ r y .The ~ ~ ~theo(6) Pines, A; Gibby, M. E.; Waugh, J. S. J. Chem. Phys. 1973,59, 569-590. (7) Hartman, S. R; Hann, E. L. Phys. Rev. 1962,128,2042-2053. (8) Koons, J. M.; Hughes, E.; Ellis, P. D. Anal. Chim. Acta 1993,283, 10451058. (9) Koons, J. M.; Hughes, E.; Cho, H. M.; Ellis, P. D. J. Magn. Reson., Ser. A 1995,114,12-23.
Analytical Chemistry, Vol. 67, No. 23, December 1, 1995 4309
retical FIDs were generated by calculating 384 points for a single rotor revolution and propagating the result to obtain a 16K point FID. Both experimental and calculated FIDs were zero-filled to 64K, exponentially multiplied with a 5 Hz Lorentzian function, and Fourier transformed. Baseline offsets were removed in the frequency domain by iteratively subtracting the average of the later 10%of the data from the entire spectrum until the difference was < on successive interactions. All calculations were performed using either a 4xR4400/150 Silicon Graphics Onyx or a VAX Station 3540. INTRODUCTION TO FACTOR ANALYSIS AND TARGET TRANSFORMATION
Factor analysis is based on modeling an experimental data point as a linear sum of terms called factors.'O A factor is determined, in general, by a multistep procedure. First, a matrix of experimental data is converted into a covariance matrix, which is subsequently diagonalized, yielding a representation of the data set in terms of factors. The next step involves determining the number of factors in the sum required to reproduce the data. The abstract factors are then rotated into physically meaningful parameters by target transformation. Finally, after each factor is identified, all experimental observations can be described in terms of their components, and the composition of the experimental system can be deduced. For example, suppose magic angle spinning NMR is used to follow the course of the reaction A 0 B (NA + NB = 1). Also assume that five spectra were collected Corresponding to different reaction times and that each line shape is sensitive to the decrease of A and increase of B. To describe the MAS data with the factor model, a data matrix D is constructed in which the rows correspond to the five experimental observations and the columns of a given row comprise the respective NMR spectra. At reaction time i, the spectrum intensity at frequency k can be expressed as expressed as
where S and L are the contributions to the observed intensity arising from the composition and concentration, respectively. In the terminology of factor analysis, S is referred to as the scores and L as the loadings. Both composition and concentration factor matrices can be determined solely from the knowledge of the matrix of experimental observations. Following Malinowski! a square covariance matrix is constructed,
z =D ~ D
(2)
u = DQ D = U Q ~
In eqs 3 and 4, the property of the matrix Q being Hermitian, Q-' = QT, was utilized. By comparing eq 4 with eq 1, it can be concluded that L = QT and S = DQ. The next step in the analysis is to determine the true size of the factor space q. By including q factors in the sum, the data matrix will be reproduced within experimental In theory, the number of nonzero eigenvalues corresponds to the size of the factor space (the rank of the covariance matrix). However, in practice, experimental error can bias the magnitude of the eigenvalues and lead to erroneous conclusions." A wide variety of methods have been published to help estimate the true size of the factor space; refs 12-14 provide an excellent summary. We have utilized the Scree test'j to estimate q. In this method, the residual variance is plotted versus the number of factors used in the reproduction of the data matrix. When the true factor size is identified, the residual variance abruptly approaches zero. M e r q has been determined, the model given in eq 1 becomes a
(5)
where D, S , and I; denote the truncated data, composition, and concentration matrices, respectively. At this point in the analysis, the scores and loadings are not recognizable in terms of chemical quantities. Instead, they are mathematical solutions to the data reproduction. Further manipulation is required to relate the abstract factors to component factors fundamental to the system. In terms of the present problem, the abstract factor spectra must be converted into the component magic angle spinning line shapes composing the data matrix. Searching for the real components is accomplished by target transformation. Each of the basic factors is located individually by repeatedly testing spectra suspected as being a primary component of the data matrix. We use a nonlinear leastsquares minimization procedure and a theoretical line shape function (see Appendix A) to perform the target transformation. The least-squares problem arising in the target transformation is defined by the residual difference between the ith point in the theoretically generated test line shape y(xJ and the corresponding point in the reproduced spectrum based on the optimal linear combinations of q abstract factor spectra 5ji,
where
which is subsequently diagonalized by a matrix Q to given eigenvalues A: Q-~ZQ= Q - ~ D ~ D=QQ ~ D ~ D=Q@Q)~DQ= 2 =
uTu
where (10) Weiner, P. H.; Malinowski, E. R.; Levinstone, A. R. J. Phys. Cbem. 1970, 74, 4537-4542.
4310
(7)
and xi (3)
Analytical Chemistry, Vol. 67, No. 23, December 1, 1995
(4)
= [dol, dcsi,..., q C d ] are
the nonlinear parameters defining
(11) Malinowski, E. R Anal. Cbem. 1977,49, 612-617. (12) Malinowski, E. R Anal. Chem. 1977,49, 606-612. (13) Harman, H. H. Modem Factor Analysis; University of Chicago Press: Chicago, IL, 1967. (14) Lawley, D. N.; Maxwell, M. A. Factor Analysis as a Sfatistical Technique; Butterworths and Co. Ltd.: London, 1963. (15) Cattell. R. B. Multivariate Bebau. Res. 1966,1, 245-251.
the calculated MAS line shape, and ti) is the ith point in the transformation vector corresponding to the lth basic factor. The solution of eq 6 is found by minimizing the residual norm with respect to the nonlinear parameters q and the transformation vector TI,
&JJ= min{IIF(x,,T,)II~XI,TIEm
(8)
where
and R" is the set of real numbers. The minimization is performed in two steps. First, the nonlinear parameters defined by q are determined, and the theoretical magic angle spinning line shape given by y(xJ is calculated as stated in Appendix A. Second, the optimal transformation vector TI is determined by solving the normal equations airising from the linear least-squares problem defined by eq 8:
T( = (S'S)-'Sy(xJ
(10)
After all factors have been identified, the final step in factor analysis is to regenerate the concentration and data matricies based on the factor solution. This is accomplished by first reconstructing the concentration dependence of the data matrix as follows: = R-lL
(11)
where R is a matrix comprised of all target vectors TIdetermined from the previous step. Likewise, the reproduced data matrix is given by
jj=SE
(12)
FACTOR ANALYSIS OF CADMIUM SULFATE MAS SPECTRA
The data matrix subjected to abstract factor analysis is shown in Figure 1 and consists of the magic angle spinning spectra obtained after heating 3CdS04.8HzO with the temperature program described and held at final temperatures of (a) 273, @) 300, (c) 325, (d) 350, (e) 375, and (0 400 K The size of the matrix was (6 x 65 536). By inspecting the data matrix in addition to the expanded region near the isotropic peaks of the cadmium salts shown in Figure 2, we can estimate the minimum size of the factor space. As 3CdSOd3H20 is heated for progressively longer periods of time, it can be seen that the intensity of the peak shown in Figure 2 is decreasing while maintaining the same highly symmetric Lorentzian profile. This is consistent with the disap pearance of a single component with an isotropic shift close to -78 ppm. However, Figure 2 shows that for the same heat treatment, the peak width at half-height has increased by nearly a factor of 2 on comparing spectrum a with f; this result clearly shows the existence of at least two separate cadmium species with similar isotropic shifts. The intermediate spectra in Figure 2 also show a deviation from the Lorentzian line profile caused by the change in concentration of the different cadmium salts as a
b
-1bO
-;0
-1'50
P P Figure 1. Experimental data matrix consisting of the magic angle spinning spectra obtained after heating 3CdS04.8H20 with the temperature program described and held at the final temperatures of (a) 273, (b) 300, (c) 325, (d) 350, (e) 375, and (f) 400 K.
function of heating time. A similar phenomenon is evident on all side bands associated with the isotropic peaks near -90 ppm. From the analysis of the isotropic peaks, it can be deduced that there are at least three components in the data matrix. After the experimental data were processed in the manner described, the data covariance matrix was computed according to eq 2 and diagonalized using EISPACK routine RG,l6 and the score and loading matrices were calculated as previously indicated. The number of abstract factors of the data was determined to be three by the Scree test.I5 Likewise, Table 1 indicates that 99.992% of the cumulative variance is accounted for by including three abstract factors in the reproduction. All such information is consistent with the trend observed from the isotropic peaks. In addition, this result agrees well with published IR data, which indicate at most three cadmium species present in a thermal gravimetric study of 3CdS0d3Hz0.17 Figure 3 shows the three abstract factors of the data. The scores and loadings matrices were subsequently truncated to retain the three significant dimensions before proceeding with target transformation. Target transformation was performed with the reduced scores matrix according to the two-step minimization procedure that was (16) Dongarra, J. J.; Garbow, B. S. Uw's Guide for the Use of EISPACK Subroutines: A Set of Subroutinesfor M u t k Eigenudues; Center for Numerical Analysis: The University of Texas at Austin, TX, 1973. (17) Minic, D. M.; Sueic, M. V.; Petranovic. N. A. Muter. Chem. Phys. 1984, 12, 389-396.
Analytical Chemistty, Vol. 67, No. 23, December 7, 7995
4311
,
I
-75.9
-77.0
, I
-78.1
-88.0
,
,
40.0
-92.0
PPm
I
Figure 2. Expansion of the isotropic peaks of the data matrix shown in Figure 1. Table I. Eigenvalues, Percent Varlance, and Cummulative Variance for Each Abstract Factor
factor 1 2 3 4 5 6
eigenvalue 4.8402 x 1.0044 x 7.1136 x 4.7432 x 6.3271 x 4.4809 x
lo6 106 lo4 lo-’
10-I
percent variance
cummulative variance
81.8131 16.9765 1.2024 0.0000 0.0000 0.0000
81.8131 98.7896 99.9920 99.9920 99.9920 99.9920
outlined. The MINPACK routine LMDIF18Jgwas used to generate refined values for the nonlinear parameters during the course of the target transformation. The linearly defined transformation vector was determined by eq 10, using the matrix inversion routine DGEDI in the LINPACK library.zO Each of the basic factors was identified individually by starting an optimization consisting of the three independent elements of the chemical shift tensor 60, &I, and 8 3 3 (with arbitrary initial values). In most cases, the optimization converged to a parameter set defining a component spectrum that was “closest to” the initial guess used. This can be understood because all of the underlying component line shapes are defined by the same theoretical function, differing only in the numerical values of the NMR parameters. However, all three component line shapes were easily recovered using a wide range of initial values. In addition, a surprisingly small number of optimizations diverged or were trapped by local minima. Both (18) More. J. J. In Lecture Nofes in Mathematics; Watson, G. A, Ed.; SpringerVerlag: Berlin, 1977; Vol. 630, p 105. (19) JMore, J. J.; Garbow, B. S.; Hillstrom, K. E. USDOE No. AVL-80.74; US. Department of Energy, U S . Government Printing Office: Washington, DC, 1980; p 888. (20) Dongarra, J. J.; Moler, C. B.; Bunch, J. R.; Stewart, G. W. LINPACK Users’ Guide; Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, 1979.
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I
Analytical Chemistry, Vol. 67,No. 23,December 7, 7995
0
I
I
-50
-100
I
-150
PPm
Figure 3. The first three abstract factors of the data matrix. Note: the second and third factors (b and c) are plotted on the same scale as the first factor (a).
conditions were extremely easy to identify because the reproduced spectra were not at all similar to the input line shapes. This is quite unlike the conventional least-squares approach, which can more easily deceive the user into accepting a local minimum. Figure 4 shows the component spectrum recovered by the target transformation procedure for the type I cadmium environment. Likewise, the procedure reproduced the other two component line shapes with comparable quality. Table 2 lists all of the relevant NMR parameters that were determined for all components. Furthermore, the tensors extracted by the FA-’lTprocedure agree well with previously published results kom single crystal studiesz1 and off-angle magic angle spinning experiment^.^^^^^ Regeneration of the Column Matrix. Since cross polarization was used to generate the cadmium magnetization and no study of the associated dynamics was attempted, the column matrix cannot be interpreted as the concentration of the component spectra (likewise, the intensity factor recovered by any other technique is equally meaningless). We ran all experiments under identical cross polarization conditions and have assumed the CP dynamics are somewhat constant for the different salts of cadmium present. Therefore, the column matrix was regenerated as previously described, and the results are plotted in Figure 5. Each of the three curves shows the relative intensity contribution under these CP conditions for each primary component spectrum toward the observed experimental line shape Figure 1) as a function of the heating time. (21) Honkonen, R S.; Doty, F. D.; Ellis, P. D. J. Am. Chem. SOC.1983,105, 4163-4168. (22) DuBois, P.; Gerstein, B. C. J. Am. Chem. SOC.1981,103, 3282-3286. (23) Cheung, T. T.; Worthington, L. E.; Murphy, P. D.; Gerstein, B. C. J . Magn. Reson. 1980,41, 158-168.
d
I
0 ' 1
1
1
I
0
-50
-100
-150
1
PPm Figure 4. Target transformation of the type I cadmium environment: (a) the target magic angle spinning line shape, which minimizes eq 8 , and (b) the reproduced side band envelope calculated by eq 7.
Table 2. Final Parameters and Error Estimates for the Chemical Shift Tensors of All Three Cadmium Components Obtained Using the FA-TT Procedure*
type 1 value error 60 (ppm) 611 622 633
(ppm) (ppm)
(ppm)
Au (ppm) rl
-90.5 -122.2 -116.4 -32.9 86.4 0.1
1.4 2.3 5.1 1.8 3.3 0.1
type I1 error
value
-77.9 -118.3 -110.3 -5.2 109.1 0.1
1.2 2.5 4.8 2.0 3.4 0.1
type I11 value error -91.2 -166.9 -108.3 -0.1 137.5 0.6
1.3 2.5 4.9 1.5 3.1 0.1
Chemical shifts reported with respect to external 1.0 M CdC12. Type I, site A of 3CdS044HzO; type 11, site B of 3CdS0443H~O;type 111, CdS04.HzO. (I
Conventional Nonlinear Least-Squares Approach. To demonstrate the superiority of the FA-'IT procedure, MINPACK routine LMDIF was used to individually fit the magic angle spinning spectra in Figure 1 to the multicomponent line shape function given in Appendix B. AU initial guesses used in this procedure were identical to those utilized to search for the basic factors using target transformation. In this case, however, all of the cadmium shift tensors and relative concentrations must be specilied simultaneously in the fit- Unfortunately, the least-squares procedure cannot (as readily) provide the number of component spectra that constitute each observation in the data matrix. We have therefore used the information from abstract factor analysis
\
I
4
2
1
3 4 Data Matrix Observation
5
6
Figure 5. Concentration weighting, given by eq 11, of each reproduced component line shape required to reproduce the observed data matrix eq 12.
to predetermine the number of component spectra present in MAS line shapes. Therefore, the numbers of parameters required to describe the individual spectra are (a) 7, @-e) 11, and (0 3 parameters, respectively. Assuming the bulk of the time is spent evaluating the line shape function, it can be seen immediately that in the worst case, the amount of computational effort has doubled; Le., the target transformation effectively fits three spectra each, requiring a single function evaluation, while the nonlinear leastsquares approach fits six spectra, each requiring three function evaluations. In addition to the increase in computing time, the results of the optimization were mostly unacceptable. In general, regardless of the initial guess used, the results of fitting spectra a and f from Figure 1 agreed well with the chemical shift tensors and relative intensities obtained using FA-'lT procedure, whereas fitting line shapes b-e with this method caused the optimization to diverge and produce results which did not visually fit the line shapes. In addition, this method produced parameter sets defined by a local minimum or gave estimates for the uncertainties in the parameters which were up to 10 times larger than the one produced by the FA-'lT method. Reformulated Least-Squares Problem. We redefined the least-squares problem to convert the nonlinear dependence of the concentration, as shown in eq B1, to a more proper linear dependence. The derivation is given in Appendix B. Although this technique simplilies the response surface for the nonlinear part of the problem and does reduce the computation complexity by n - 1 function evaluations, almost no advantage was gained by this approach. Similar results were obtained from fitting Analytical Chemistry, Vol. 67, No. 23, December 1, 1995
4313
spectra a and f with this method, while the same type of divergent behavior was seen in the analysis of the remaining spectra. In this case, however, it also becomes much more difficult to estimate the covariance between the individual effects of concentration and chemical shift anisotropy, which further discouraged its use.
P,(~,s)= (3 cos2p
-
1)(3 cos2 e - 1) -
vcs(-cos 2y sin2 P
+ 3 cos 2 y cos26 sin2 P)
pl(vC~ = 3 sin 2 a sin2 P sin2 e vcs(cos 2y sin 2 a sin2 e
+ cos2P cos 2y sin 2 a sin2 e + 2 cos 2 a cos P sin 2y sin2 e)
CONCLUSIONS
We have developed a technique based on abstract factor analysis with target transformation to reliably extract the complete chemical shift tensor and the relative intensity factor from magic angle spinning spectra with overlapping isotropic peaks. Further, it has been shown that the FA-'IT technique is superior to conventional least-squares methods in analyzing this type of data, both in the reliability of the extracted parameters and in computational effort.
p2(vC~ = 3 cos 2 a sin2 P sin2 e -
ACKNOWLEDQMENT
P4(vcJ= -6 cos a sin 2p sin 2 8 -
The authors would like to thank Dr. S. L. Morgan at the University of South Carolina for his advice on Chemometric techniques. This work was supported in part by the National Institutes of Health (Grant GM-26295). J.M.K. is supported by the Associated Western Universities, Inc., Northwest Division, under Grant DE-FG0689ER-75522 with the US. Department of Energy. J.M.K. and P.D.E. are also supported by Paciiic Northwest Laboratory, which is a multiprogram national laboratory operated by Battelle Memorial Institute for the US.Department of Energy under contract DE-AC0676RLO-1830.
vcs(cos 2 a cos 2 y sin2 e
+ cos 2 a cos2P cos 2 y sin2 e 2 cos /3 sin 2 a sin 2y sin2 6)
p3(vC3= -3 sin 2 a sin2 P sin2 e -
cos 2y sin 2 a sin2 e - cos2 P cos 2 y sin 2 a sin2 e 2 cos 2 a cos P sin 2y sin2 e) vcs(2 cos a c o s 2y sin 2p sin 20 4 sin a sin P sin 2y sin 20) (A3)
An expression for the frequency of the chemical shift contribution can be derived from eq A2,
APPENDIX A
Following Maricq and W a ~ g h , 2we~ derive the MAS Hamiltonian for the chemical shift interaction of a spin '/2 species. Using standard techniques, we transform the PAS of the chemical shift interaction f i s t into the rotor axis system and then into the laboratory frame as follows:25
and the resultant free induction decay is given as
G(xJ = Cexp(imAtG(d,,$) m
with the nonvanishing spatial components p00 = 60, ~ Z O= (3/2)1/26,,, p2+2 = -1/2dcsvcs,and spin tensors Too = Z a o TZO = (2/3)1/2Za~,T2+2 = 1 given in ref 26. Further, the average Hamiltonian defined over one rotational period is given by
where xi = [ d ~ / , d ~ upon ~ ~ , which ~ ~ ~ Jthe , Ith component line shape depends. Further, the sum over m is evaluated for all dwell periods. The Fourier transform of eq A5 is used as the target function in eq 6:
APPENDIX B
Transforming eq A5 into the multicomponent, nonlinear leastsquares fitting function, where the concentration weighting for the first line shape is assigned to unity, is given as with (24) Maricq, M. M.; Waugh, J. S. J. Chem. Phys. 1979,70, 3300-3316. (25) Zannoni, C. In NMR ofLiquid Crystals; Emsley, J. D., Ed.; NATO AS1 Series C141; Reidel: Holland, 1985; pp 1-31. (26) Spiess. H. W. In NMR Basic Principles ond Progress; Diehl, P., Fluck, E., Kosfeld, R., Eds.; Springer-Verlag: Berlin, 1978; Vol. 15.
4314 Analytical Chemistry, Vol. 67, No. 23,December 1, 1995
and X
= [xlx2..xq1,where xi = [ w ~ d 0 ~ , 6 ~corresponding ~ ~ , ~ ~ ~ ~ 1 ,to
the NMR parameters describing the Zth individual component spectra composing the observed line shape. The nonlinear leastsquares problem is defined as
Further, W = [w~w3...w,J.The NMR parameters for each X are determined using MINPACK routine LMDIF, and each individual concentrations can be obtained by the solution of the following set of normal equations,
with m
033)
where Y, corresponds to the ith point of the experimental spectrum. Likewise, rar)i is the ith point of the function given by eq B1. To avoid the nonlinear dependence of concentration as written in eq B1, the FID can be evaluated for each of the q components at unit concentration, and the original intensity weights can be subsequently handled in the frequency domain, so that the leastsquares problem can be written as
where& = WI%G(XI))a n d j = Y - %G(xJ), and all other symbols cany the previous meaning.
Received for review May 24, 1995. Accepted August 24, 1995.e
where
and
AC950499T
Abstract published in Advance ACS Abstracts, October 1, 1995.
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4315
