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Relaxation studies using saturated bandshapes in high-resolution NMR. III. Analysis of coupled multispin systems using the simple line approximation
JOURNAL
OF MAONE~C
RESONANCE
32,325-337 (1978)
Relaxation StudiesUsing Saturated Bandshapesin High-ResolutionNMR. III. Analysis of Coupled Multispin SystemsUsing the Simple Liue Approximation B. W. GOODWIN AND R. WALLACE Department of Chemistry, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2
Received January 8, 1978; revised March 20, 1978 In many cases it is possible to distinguish between competing relaxation mechanisms in coupled multispin systems through observation of differential effects in their saturated bandshapes. Unfortunately, however, it is very diilkult to extract more quantitative relaxation information, owing to the complexity of the behavior of relaxation effects and the relative insensitivity of the bandshapes to the relaxation parameters. In this report, a quantitative analysis technique is discussedwhich is based on the simple line approximation applied to saturated bandshapes. A computer program has been developed which greatly simplifies such analyses whenever the spin system under study gives rise to a sulIicient number of “simple” transitions. The utility of the technique and the program has been tested using simulated spectra of some three and four spin-) systems of nuclei, for relaxation by the random field and intramolecular dipokdipole relaxation mechanisms.Some results of these calculations are displayed, and they serveto illustrate the efficacy of this approach to quantitative relaxation analysis. INTRODUCTION
It is quite obvious from previous discussions in the literature that the major problem in the study of relaxation in multispin systems is the extraction of an accurate set of relaxation parameters from experimental spectra which display a weak and complex dependence on those parameters. A variety of high-resolution experimental methods have been developed (see, for example, Refs. (I-16), as well as those using pulised NMR techniques (Z7-20)). The present report limits its consideration to the high-resolution progressive saturation approach. The progressive saturation method has given rise to two general analysis procedures, w&h both procedures applied to the relaxation study of AB systems of spin-+ nuclei. The first such procedure, due to Harris and Worvill (IO), required doping of the sample in order to make the random field (RF) mechanism dominate the relaxation. Subsequent analysis was achieved by an iterative variation of the RF mechanism parameters (and other relevant high-resolution NMR parameters), with the object of matching calculated bandshapes with (digitized) experimental bandshapes at different strengths of the saturating rf field. This approach is probably restricted to similar types of simplified experiments, however, because of the large number of 325
002%2364/78/0323-325$02.00/O
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parameters which must be varied and because of the complexity of the calculations which are necessary to yield convergent results. A more recent study by Hartmann (15) involved a similar iterative approach applied to an AB spin system in which both the RF and intramolecular dipole-dipole (DD) mechanisms provide substantial contributions to the relaxation processes. In that procedure, an attempt was made to “orthogonalize” the effects of the various relaxation parameters on the spectral bandshape, with a separate iteration scheme on some subset of the parameters applied over a selected part of the bandshape. The procedure also produced some quantitative results; however, practical considerations based on increasing bandshape complexity will probably limit applications to systems of two spin-+ nuclei. In our previous research into this problem (26), we noted that it was often feasible to obtain an accurate qualitative description of relaxation effects using simplifications which derive from an application of the simple line approximation. This approximation requires that a spectrum consist of nondegenerate and nonoverlapping transitions, in which case a saturated bandshape may be approximated by a sum of “simple” transitions, each given by a Lorentzian curve with “effective” spin-spin and spinlattice relaxation times which are characteristic of the transition. These effective relaxation times may be related to the various relaxation mechanism parameters through the Redfield relaxation matrix, which in turn may be readily calculated using a computational algorithm previously discussed (16). These developments suggested an alternative procedure for quantitative relaxation analysis, namely, the iterative fitting of a set of calculated effective relaxation times to experimentally determined effective relaxation times. The experimental effective relaxation times for a given transition may be readily determined using several of its progressively saturated lineshapes and assuming a Lorentzian curve; in addition many of the experimental problems such as phase error, inhomogeneity, chemical shift variations, sweep rate effects, etc., can be easily accounted for during this stage. The next stage of the analysis is the iterative fitting, which can now be concentrated on fitting a set of relaxation times by varying the complete set of relaxation parameters. We have developed a computer program which can be used to determine relaxation parameters iteratively using a set of experimental effective relaxation times as the convergence criteria. The program was first tested on first-order spectra which obeyed the requirements of the simple line approximation, with computed bandshapes used to obtain a set of “experimental” effective relaxation times to serve as input to the program. These results were generally successful, in that the relaxation parameters used to produce the simulated spectra could be reproduced by the iterative fitting of the input “experimental” relaxation times. A similar test procedure using more complicated spin systems was then applied; in particular, an AB2 system for which two of the transitions exhibit considerable overlap was studied. In this case, with only the nonoverlapping effective relaxation times used as input, the program once again gave a consistent set of relaxation parameters, and in fact it was determined that other selected effective relaxation times could be ignored as well, while retaining converged and consistent iterations. This last result follows, since the problem is, in fact, overdetermined; that is, there are more known pieces of data (the effective relaxation times) than unknown
TI-IE SIMPLE
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APPROXIMATION
327
pieces (the relaxation parameters). For the above ABa system, for example, the nulmber of unique relaxation parameters for both the RF and DD mechanisms is eight, and there are available sixteen effective relaxation times (eight spin-spin and eight spin-lattice) for the iterative fitting. In principle only eight of the relaxation times are required for convergence, although any “extra” data should be included to ensure accurate results. In practice, since a given relaxation time may be quite insensitive to some of the relaxation parameters, it may be necessary to have an overdetermined problem, or at least to make some judicious choice of the relaxation times which are used. In the following section, the simple line approximation is briefly summarized, and some of the required definitions are given. Also presented is the approach used to solve the overdetermined problem and the iteration problem; this approach is very close to the technique used in LACOON3 (27) except for additional difficulties encountered with the calculation of the required Jacobian. The discussion displays selected results of program tests, and some practical considerations are mentioned. THEORY
1. The Simple Line Approximation
The simple line approximation has been described elsewhere in the literature (see, for example, Ref. (I)), and we give here only a brief summary of the pertinent equations and definitions. The starting point is the equation of motion of the spin density matrix p, fif!? = i[p, HO -t- I&(t)] dt
-I-
PI
where H,, is the high-resolution NMR spin Hamiltonian; l&(t) represents the Hamiltonian describing the (weak) coupling of the spin system with the detecting rf field, which is assumed to be of the form 24, cos wti; and the last term in Eq. [l] describes the process by which the spin system relaxes toward thermal equilibrium through couplings with its surroundings. Invoking the steady-state approximation, we are interested in solutions for which the diagonal elements Pkk are constant in time, and for which the off-diagonals Pk[ vary as exp( f iwt). For an off-diagonal, [l:] simplifies to
- iqd(w, - WI) COSwt +
PI
where xk[ N- z exp(iwt), q = fi/ZkT, d = y&(I+)k,, and wk is an eigenvalue of Ho. The matrix x is equal to the difference p - p’, where p” N (I- qH,J/Z is the spin density matrix at thermal equilibrium. Equation [2] has been considerably simplified by the introduction of the simple line approximation, which allows only one transition (that with eigenfrequency ]wk - wl] N W) to be resonant; therefore, only the two elements xkl and xlk need be considered from among all the off-diagonals.
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This approximation results in a simple expression for the relaxation for example, @x/Wm1, ;
= -2
relax
m.n
&tnnXmn
= -&d&l
= -
terms
’ 0 T2 k, ‘lc”
Equation [3] serves as a functional definition of the effective spin-spin relaxation time for the (nondegenerate and nonoverlapping) transition associated with element xrr. In the unsaturated limit (i.e., as HP tends to Zero), Xkk 2: xl1 N 0, and Eqs. [2] and [3] yield a simple Lorentzian expression for Xk[. For the unsaturated case, however, suitable expressions must be found for all of the diagonal elements, including those which do not appear explicitly in [2]. This is achieved by considering the equations of motion from [l] ; for example,
%
= 0 = - 2 &~,,,,,,Xmm - 2 Imag(zd*)
9
E o = -2 &,,,~mm
m
and n
(n # k, 0,
where d* is the complex conjugate of d. There are only two equations of type [4a] for the diagonals x,& and xII; all other diagonals have eqUatiOnS Of motion Of type [4b]. These equations in fact form a set of simultaneous equations1 of the form AX = b,
[51 where A is a matrix of elements of the Redfield relaxation matrix R, x is treated as a column vector, and b is a column vector with b, = -2 Imag(zd*)@,, - St,,,), where a,,,,,is the Dirac delta function. Equation [5] has the formal solution xkk
-
XII
=
A - ‘(bk
-
b,)
=
161
4 Ima&d*)(T&,
which serves as a functional definition of the effective spin-lattice relaxation time for the transition Ik) + 11). It is now possible to derive a useful expression for the absorption mode signal of a saturated bandshape near the transition of interest. Substitution of [3] and [6] into Eq. [2] yields the final expression (I) sew
=
wkl)
cc wdwk
-
+‘%(~+~-)kz~
1 +
@,)2T2
2
+
;2~2(1
P
+
1-j
kl
T
T
1 2
9
L71
where dw = wk - wL - w, y is the gyromagnetic ratio of the nuclear species undergoing resonance, (I+I-)kl is the transition intensity, and the subscripts (kl) on Tl and T2 have been omitted for convenience. The last half of [7] is the desired expression; it states that each transition in some set of nondegenerate and nonoverlapping transitions can be described as a simple Bloch-type transition, except for the introduction of the term (I+I-) and the qualifi1 Actually, one of the equations must be replaced by z,, x,,,, = 0 in order to obtain an independent set of equations.
THE
SIMPLE
LINE
APPROXIMATION
329
cation of T1 and Ta as effective spin-lattice and spin-spin relaxation times. In the following sections it is assumed that [7] is a good representation of the spin systems under study. Consequently it will be possible to define effective Tl and Ta values and also to extract these relaxation times through a simple analysis of the lineshapes fo:r each transition. Validation of the assumption is shown by the consistency of the calculated results. 2. Details of the Iterative Process For this discussion, we assume that some set of effective spin-spin and/or spinlattice relaxation times is available from experimental data, and it is necessary to use a fitting procedure in order to obtain the relaxation parameters corresponding to those relaxation times. The basic algorithm used is very similar to the least-squares approach used by the authors of LACOON3 (21) (see also Ref. (19)) for the analysis of high-resolution NMR spectra for chemical shifts and coupling constants. In our case, we define a function E as E = 2 (TLbP”- T:‘c)2,
m
PI
where T,,, is an effective spin-spin or spin-lattice relaxation time, and the index m runs over all such relaxation times to be used in the fitting. Each of the calculated relaxation times can be expressed as a complex function of a set of relaxation parameters {Pk}; in our studies the two mechanisms of interest are the DD and RF mechanisms. The least-squares solution to the problem in [8] is found by searching for a minimum in E, i.e., by locating the roots of
qg=o,
PI
w:here the index k runs over all the relevant relaxation parameters. Following the Newton-Raphson method, we assume that a linear change dp, produces a linear change AT,,,, or AT,,, = 2
AP,.
It is required that TObS m
_
TCalC
m
-
- p$+~tc.
In matrix notation, and using the arguments in Ref. (21), we obtain R = DA,
1124
where R is a column vector of residuals, D is the Jacobian of partial differentials (D,, = aT,@p,), and A is a column vector of the changes in the parameters which is to be determined. The system of equations in [12a] can be solved only if the numbe:r of observed values Tgbbsis greater than or equal to the number of unknown parrameters pr. For the overdetermined situation, it is convenient to form the normal set of equations (17)
330
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DtR = DtDA,
Wbl
where Dt is the transpose of D. The matrix product DtD is then symmetric and square, somewhat simplifying the method of solution of Eq. [17]. In practice, solution of [12b] for the corrections A will not satisfy Eq. [9], partly because Eq. [lo] is not exact and partly because of experimental errors. This problem is solved by an iterative scheme, whereby the parameter set (pk} is corrected by the amounts (Ap,}, and Eq. [12b] is solved again, using the new values for the residuals R and the differentials D. This process is repeated until convergence results, which ideally should occur when R equals the zero vector. In practice, however, the presence of experimental errors will result in a converged R which is nonzero and in a set of determined parameters pk which have an attached uncertainty. Our iterative process deviates from that of Ref. (21) inasmuch as an analytical expression for the Jacobian aT,&k is not known; we are forced to use numerical methods to estimate it. Another problem also occurs, owing to the effects of symmetry, which may require that some of the relaxation parameters be identical (cf. Ref. (16)). For these cases, special attention must be paid to the construction of the estimates for D, since it is possible to set up an indeterminate system of equations of the form of [12b]. 3. Details of the Program The program which was developed for the relaxation analysis will perform three types of related calculations. For the first option, a set of high-resolution NMR parameters is input, with the output consisting of a list of all possible eigenfrequencies (including multiple quantum transitions) and intensities for the specified spin system. This calculation is performed by creating the high-resolution Hamiltonian in the spin product representation, with subsequent diagonalization using the Jacobian method. Associated with each eigenfrequency is an identification number, which is used with the following options to identify the effective relaxation times. For the second calculation option, the input includes a set of relaxation parameters for the RF and DD mechanisms (the program will also accommodate the scalar type II mechanism, which is not discussed further here), and the output consists of a list of calculated effective spin-spin and spin-lattice relaxation times corresponding to the relaxation parameters chosen, assuming the simple line approximation. These calculations are performed by generating the required elements of the Redfield relaxation matrix in the spin product representation, transforming the matrix to the eigenbasis, and then using the definitions in Eqs. [3] and [6], respectively. Generation of the elements of the Redfield relaxation matrix is achieved as outlined in Ref. (26). The third calculation option performs the iterative analysis. The input for this option includes a list of the experimental spin-spin and/or spin-lattice effective relaxation times to be used as convergence criteria, and a list of the relaxation parameters which are to be varied during the iterations. The effect of symmetry on the relaxation is included by requiring that specified relaxation parameters be varied identically. It is also required that the total number of independently varied relaxation parameters be less than or equal to the number of experimental effective relaxation times which are supplied. For this calculation option, the program proceeds as out-
THE
SIMPLE
LINE
APPROXIMATION
331
lined previously; namely, the set of equations [12a] or [12b] is solved for the vector d of corrections which are to be applied to an input list of estimates to the relaxation parameters (our experience with the program has indicated that the accuracy of the input estimates is usually not too critical to the resulting rate or precision of convergence and that when divergence occurs, it occurs within a few iterations). The m,atrix of partial differentials is calculated numerically by evaluating the relaxation times Tgbb'using the supplied relaxation parameter estimates, and by evaluating several sets of relaxation times Tt which have the same relaxation parameters as used for T",b"except for parameter Pk, which is increased by ca. 0.5%. The element Dmkis then approximated using the relation D,k = (T," - T0,bs)/0.005Pk. The new set of relaxation parameters calculated is subsequently used as a starting point for the next round of computation, and the iterations are continued until the rms deviation between the calculated effective relaxation times of two consecutive iterations is less than a predetermined value, or the total number of iterations performed exceeds a specified value, or until a divergence pattern is detected. This calculation option also attempts to note which of the relaxation parameters have a weak or very weak influence on the calculated relaxation times. The final output is a list of the best relaxation parameters found, along with an estimate of their accuracy and a list of the calculated spin-spin and spin-lattice relaxation times corresponding to those parameters. Examples of the required execution times on an IBM 370/58 are typically 30 set for twenty iterations on a three-spin-$ system of nuclei involving fitting of six relaxation parameters using eight effective relaxation times. Similar calculations on a four-spin system require approximately 10 times more cpu time. DISCUSSION
The theory and the associated program described previously have been tested on a number of three- and four-spin-& systems with relaxation by the RF and DD relaxation mechanisms. We present here the details of some tests performed on an AB1 spin system, although we would like to mention that similar results have been obtained with other systems, and it is expected that more complex spin systems as well as other re:laxation mechanisms would follow a similar pattern of analysis. It should also be noted that our “experimental” spectra are, in fact, spectra calculated using the programs described in (26). While this approach allowed testing to be performed in the absence of experimental problems, it should be pointed out that the method of relaxation analysis should minimize many such difficulties (e.g., phase problems). Figure 1 contains sample plots of our “experimental” ABa spectra at various strengths of the saturating rf field. As noted previously (26), relaxation effects are more obvious when the rf field is sufficiently strong to compete effectively with the relaxation processes. For these spectra, it is apparent that the transitions numbered 1 and 5 should always fail the requirements of the simple line approximation. It also appears that the other single quantum transitions (and possibly some of the double quantum transitions, although we did not consider them here) can be approximated by the si:mple line approximation over the lower ranges of the rf field. In our testing of the analysis procedure, we used various combinations of the transitions numbered 3, 9, 14, 20, 11, and 21 as the source of our experimental effective relaxation times.
332
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l,DQ,5
8nT
+20
+io
0
-10 Hz
FIG. 1. Computed absorption mode bandshapes for an AB1 system of three spin-$ nuclei, using the program mentioned in Ref. (16), at rf field strengths ranging from 2 to 32 nT. Highresolution NMR parameters used were J.,e = 8.08 Hz, and v., - vB = 12.89 Hz. Relaxation is due to a combination of the random field and intermolecular dipoledipole mechanisms with FA = 0.447 (rad/sec)l’a, FB = 0.316 (rad/sec)l’a, C,. = 0, C.,* = 1.0, DAB= 0.27 (rad/sec)l’*, DaB* = 0.052 (rad/sec)l’a, CABAx* = -0.125, and CABBa* = 0.625. The frequency increases from right to left, and the zero on the frequency axis corresponds to ~a. The numbers appearing above the transitions are those used by the program to identify the transitions. The labels DQ denote double quantum transitions.
THE SIMPLE LINE APPROXIMATION
333
To determine these effective spin-spin and spin-lattice relaxation times, the last half of Eq. [7] was employed as the starting point:
~afw+~-) ‘(‘“)
= 1 + AwaT; + yaH;(I+I-)TITa’
Two different approaches were used, as described below (see also Ref. (22)).
(a) The half-height linewidth approach uses the equation
where (2/Ta)set is the measured half-height linewidth. The slope and intercept of a plot of (2/Ta)Eaatvs yaHi will yield the required effective relaxation times. (b) The peak height method uses the equation
where S(0) is the measured peak height. A linear plot of HP/S(O) vs yaHg will also give the required effective relaxation times (or relative values if the constant k cannot be determined). Sample plots using these two approaches are presented in Figs. 2 and 3, where the source of half-height linewidths and peak heights was a series of spectra which includes some of those shown in Fig. 1. It will be noticed that at higher values of the rf field there is a tendency for the plotted half-height linewidths to fall above the corresponding straight line which was drawn. This tendency is due to the fact that overlap eventually occurs as HP becomes larger, and consequently the premises of the simple line approximation do not apply. Qualitatively, the effect of overlap is to increase the measured half-height linewidths, so that the points are always above the “best” straight line. The fact that a straight line obtains for the lower values of HP, however, indicates that the use of the simple line approximation is valid over at least part of the plotted range of Hp. It would be expected that similar points in Fig. 3 would tend to fall below the “best” straight lines, since the effect of overlap would be to increase the peak height. These deviations are not present in Fig. 3, however, because overlap will affect half-height linewidths before it affects peak heights. Consequently the peak height plots should be valid over a wider range of HP, as is displayed by Figs. 2 and 3. The determination of effective spin-spin and spin-lattice relaxation times has a number of attractions (relative to other methods of relaxation analysis using highresolution NMR) from an experimental viewpoint: for example, phase problems can be minimized since only one transition, not an entire spectrum, need be recorded at one time. Similarly, the effects of magnetic field inhomogeneity could be minimized, for example, by subtracting an estimate of the inhomogeneity half-height linewidth from the measured half-height linewidths before attempting straight line plots. The effect of inhomogeneity on the plots in Fig. 2 would be to produce a curve which is above the straight line at lower values of HP and which would approach the straight line asymptotically at higher values of Hp. An extrapolation of the linear portion of such a curve compared with the actual intercept (also extrapolated) would yield a
334
GOODWIN AND WALLACE
40
30
w3 2.c
20
IO
P Y * tip* (sec.)-* FIG. 2. Plots of the square of the half-height linewidth vs the square of the rf field strength for some of the transitions of the ABI spin-& systemdescribedin Fig. 1.
good estimate of the inhomogeneity half-height‘ linewidth. As a result of these considerations, it would appear that the method of analysis based on the simple line approximation has a number of benefits which cannot be duplicated by other methods of analysis. The plots in Figs. 2 and 3 were used to obtain a set of spin-spin and spin-lattice relaxation times for the indicated transitions. A number of different sets of relaxation times were obtained by varying the number of points used to determine the best straight line (using the least-squares method). Still other sets of relaxation times
335
THE SIMPLE LINE APPROXIMATION
1.0
2.0
3.0
y 2 Hp2 (sec.)-2 FIG. 3. Plots of &/S(O), where S(0) is the peak height, vs the square of the rf field strength for some of the transitions of the AB1 spin-i systemdescribed in Fig. 1.
were obtained by choosing a subset of the available relaxation times; for example, all effective spin-spin relaxation times and any two effective spin-lattice relaxation times might comprise such a subset. In several test sets, some of the relaxation times were purposely modified to give an approximation of data containing random errors. The program was then tested using these different sets of data, and usually converged results were obtained which agreed with the starting relaxation parameters used to generate the spectrum.
GOODWIN
336
AND
WALLACE
Occasionally the iterations diverged for reasons which were not always obvious. For many such cases, however, it was possible to determine that one of the following problems was the cause. (a) The supplied effective relaxation times were not sufficiently accurate. In Fig. 2, for example, if the best straight lines included points well beyond the value of yaHz = 2, the “experimental” relaxation times became very inaccurate compared with the actual (calculated) relaxation times. Generally, the program required that the supplied Tl and T2 values be correct to within about 10 to 15% of their actual values. (b) A supplied subset of relaxation times was a poor choice, with all quite insensitive to one or more of the relaxation parameters. Generally all available relaxation times should be used, and quite probably multiple quantum transitions would provide useful data (cf. Ref. (IS)). (c) The program initially converged toward a false solution and then diverged. This is a common failing of the Newton-Raphson approximation, and it is most easily solved by a new choice of starting values for the relaxation parameters. (d) Some of the relaxation parameters had very little influence on any of the relaxation times available (for example, C,,; cf. Ref. (H)), and consequently an oscillation pattern would occur with very slow convergence properties. It was often useful to fix these relaxation parameters initially and to vary them with the others when a better starting estimate was available. The program output includes a note identifying the relaxation parameters that fall into this class. In spite of the aforementioned difficulties, it was usually possible to obtain converged relaxation parameters which were within approximately 5% of the starting values. A typical set of experimental relaxation times used is shown in Table 1. This TABLE EXPERIMENTAL AND ACTUAL SPIN-SPIN TIMES FOR THE SPIN SYSTEM
Transition number
1 and SPIN-LATTICE RELAXATION SHOWN IN FIG. 1
Experimental
o
Actual
Tl
T2
Tl
TQ
3
1.07
9 14
0.90 1.05
0.91 0.81 0.98
1.032 0.859 1.026
0.920 0.813 0.985
20
0.73
0.72
0.715
0.725
11 21
1.16 0.97
1.12 0.91
1.122 0.933
1.130 0.918
a “Experimental” values were obtained from a least-squares fitting of the points below ylHz = 2 set-a in Fig. 2 and were used as convergence criteria in a test of the method of relaxation analysis described.
set of data leads to converged values (the rms deviation was less than 0.05) for the six relaxation parameters FA, FB, CAB, CBB,, DAB, DBB’ (the parameters Cm,, and C ABBB’ were fixed as described in (16) for this particular case; other test results were obtained with fitting of all four of the DD parameters), which agreed with the values stated in the caption for Fig. 1 to within a total rms deviation of less than 0.1. It
THE SIMPLE
LINE
APPROXIMATION
337
might be noted from Table 1 that the calculated and experimental relaxation times are not identical, which is attributable in part to the breakdown of the simple line approximation. CONCLUSIONS
We believe that we have shown that the method of relaxation parameter analysis based on the simple line approximation is feasible, and that we now have a computer program which will perform the required analysis, given a sufficiently accurate set of experimental relaxation times for the convergence criteria. Furthermore the complexity of the computations in any relaxation analysis would indicate that the method described is the only feasible high-resolution approach to date for large spin systems. We have performed some tests on four-spin-+ systems of nuclei, with typical execution times of 5 min, and the program has the capability of handling up to six spin-+ nuclei. The program currently performs relaxation analyses for the random field, intermolecular dipole-dipole, and scalar type II relaxation mechanisms, although other mechanisms could be accommodated. Copies of the program will be made available through the ATLAS NMR Programme Library.2 ACKNOWLEDGMENTS The authors would like to acknowledge the assistance of Dr. R. K. Harris (University of East Anglia) in the preliminary planning of this work. One of us (R.W.) would also like to acknowledge funding by the National Research Council of Canada. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
REFERENCES ,4. ABRAGAM, “Principles of Nuclear Magnetism,” Oxford Univ. Press, London, 1961. :R. K. HARRIS AND K. J. PACKER, J. Chem. Sot. (London), 4736 (1961). :R. M. LYNDEN-BELL, Proc. Roy. Sot. Ser. A 286,337 (1965). IR. M. LYNDEN-BELL, in “Progress in Nuclear Magnetic Resonance Spectroscopy” (J. W. Emsley, J. Feeney, and L. H. SutclifIe, Eds.), Vol. 2, p. 163, Pergamon, Elmsford, N.Y., 1967. IR. FREEMAN, S. WIITEKOEK, AND R. R. ERNST, J. Chem. Phys. 52,1529 (1970). IB. D. N. RAO, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 271, Academic Press, New York, 1970. IR. A. HOFFMAN, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 87, Academic Press, New York, 1970. IB. GE~TBLOM AND 0. HARTMANN, J. Mugn. Reson. 8,230 (1972). R. K. HARRIS AND K. M. WORVILL, Chem. Phys. Lett. 14,598 (1972). R. K. HARRIS AND K. M. WORVILL, J. Magn. Reson. 9, 394 (1973). D. G. HUGHES, M. R. SMITH, AND D. H. M. SWITZER, J. Chem. Phys. 60,489O (1974). 1~. K. HARRIS, N. C. PYPER, ANLI K. M. WORVILL, J. Magn. Reson. 18, 139 (1975). K. M. WORVILL, I. Magn. Reson. 18,217 (1975). I?. MEAKIN AND J. P. JESSON,J. Mugn. Reson. 19, 37 (1975). 0. HARTMANN, J. Magn. Reson. 22, 125 (1976). 13. GOODWIN, R. WALLACE, AND R. K. HARRIS, J. Magn. Reson. 22,491 (1976); B. GOODWIN, AND R. WALLACE, J. Magn. Reson. 23,465 (1976). II. FREEMAN, S. WITTEKOEK, AND R. R. ERNST, J. Chem. Phys. 52, 1529 (1970). ‘W. M. M. BC&E, Mol. Phys. 29, 1673 (1975). 1~. L. VOLD AND R. R. VOLD, J. Chem. Phys. 66, 1202 (1977). I,. G. WERBELOW AND D. M. GRANT, in “Advances in Magnetic Resonance” (J. S. Wsugh, Ed.), Vol. 9, p. 189, Academic Press, New York, 1977. S. CASTELLANO AND A. A. BOTHNER-BY, J. Chem. Phys. 41, 3863 (1964).
’ Science Research Council ATLAS Computer Laboratory, NMR Program Library, c/o R. K. Harris, School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, Great Britain.
OF MAONE~C
RESONANCE
32,325-337 (1978)
Relaxation StudiesUsing Saturated Bandshapesin High-ResolutionNMR. III. Analysis of Coupled Multispin SystemsUsing the Simple Liue Approximation B. W. GOODWIN AND R. WALLACE Department of Chemistry, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2
Received January 8, 1978; revised March 20, 1978 In many cases it is possible to distinguish between competing relaxation mechanisms in coupled multispin systems through observation of differential effects in their saturated bandshapes. Unfortunately, however, it is very diilkult to extract more quantitative relaxation information, owing to the complexity of the behavior of relaxation effects and the relative insensitivity of the bandshapes to the relaxation parameters. In this report, a quantitative analysis technique is discussedwhich is based on the simple line approximation applied to saturated bandshapes. A computer program has been developed which greatly simplifies such analyses whenever the spin system under study gives rise to a sulIicient number of “simple” transitions. The utility of the technique and the program has been tested using simulated spectra of some three and four spin-) systems of nuclei, for relaxation by the random field and intramolecular dipokdipole relaxation mechanisms.Some results of these calculations are displayed, and they serveto illustrate the efficacy of this approach to quantitative relaxation analysis. INTRODUCTION
It is quite obvious from previous discussions in the literature that the major problem in the study of relaxation in multispin systems is the extraction of an accurate set of relaxation parameters from experimental spectra which display a weak and complex dependence on those parameters. A variety of high-resolution experimental methods have been developed (see, for example, Refs. (I-16), as well as those using pulised NMR techniques (Z7-20)). The present report limits its consideration to the high-resolution progressive saturation approach. The progressive saturation method has given rise to two general analysis procedures, w&h both procedures applied to the relaxation study of AB systems of spin-+ nuclei. The first such procedure, due to Harris and Worvill (IO), required doping of the sample in order to make the random field (RF) mechanism dominate the relaxation. Subsequent analysis was achieved by an iterative variation of the RF mechanism parameters (and other relevant high-resolution NMR parameters), with the object of matching calculated bandshapes with (digitized) experimental bandshapes at different strengths of the saturating rf field. This approach is probably restricted to similar types of simplified experiments, however, because of the large number of 325
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parameters which must be varied and because of the complexity of the calculations which are necessary to yield convergent results. A more recent study by Hartmann (15) involved a similar iterative approach applied to an AB spin system in which both the RF and intramolecular dipole-dipole (DD) mechanisms provide substantial contributions to the relaxation processes. In that procedure, an attempt was made to “orthogonalize” the effects of the various relaxation parameters on the spectral bandshape, with a separate iteration scheme on some subset of the parameters applied over a selected part of the bandshape. The procedure also produced some quantitative results; however, practical considerations based on increasing bandshape complexity will probably limit applications to systems of two spin-+ nuclei. In our previous research into this problem (26), we noted that it was often feasible to obtain an accurate qualitative description of relaxation effects using simplifications which derive from an application of the simple line approximation. This approximation requires that a spectrum consist of nondegenerate and nonoverlapping transitions, in which case a saturated bandshape may be approximated by a sum of “simple” transitions, each given by a Lorentzian curve with “effective” spin-spin and spinlattice relaxation times which are characteristic of the transition. These effective relaxation times may be related to the various relaxation mechanism parameters through the Redfield relaxation matrix, which in turn may be readily calculated using a computational algorithm previously discussed (16). These developments suggested an alternative procedure for quantitative relaxation analysis, namely, the iterative fitting of a set of calculated effective relaxation times to experimentally determined effective relaxation times. The experimental effective relaxation times for a given transition may be readily determined using several of its progressively saturated lineshapes and assuming a Lorentzian curve; in addition many of the experimental problems such as phase error, inhomogeneity, chemical shift variations, sweep rate effects, etc., can be easily accounted for during this stage. The next stage of the analysis is the iterative fitting, which can now be concentrated on fitting a set of relaxation times by varying the complete set of relaxation parameters. We have developed a computer program which can be used to determine relaxation parameters iteratively using a set of experimental effective relaxation times as the convergence criteria. The program was first tested on first-order spectra which obeyed the requirements of the simple line approximation, with computed bandshapes used to obtain a set of “experimental” effective relaxation times to serve as input to the program. These results were generally successful, in that the relaxation parameters used to produce the simulated spectra could be reproduced by the iterative fitting of the input “experimental” relaxation times. A similar test procedure using more complicated spin systems was then applied; in particular, an AB2 system for which two of the transitions exhibit considerable overlap was studied. In this case, with only the nonoverlapping effective relaxation times used as input, the program once again gave a consistent set of relaxation parameters, and in fact it was determined that other selected effective relaxation times could be ignored as well, while retaining converged and consistent iterations. This last result follows, since the problem is, in fact, overdetermined; that is, there are more known pieces of data (the effective relaxation times) than unknown
TI-IE SIMPLE
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pieces (the relaxation parameters). For the above ABa system, for example, the nulmber of unique relaxation parameters for both the RF and DD mechanisms is eight, and there are available sixteen effective relaxation times (eight spin-spin and eight spin-lattice) for the iterative fitting. In principle only eight of the relaxation times are required for convergence, although any “extra” data should be included to ensure accurate results. In practice, since a given relaxation time may be quite insensitive to some of the relaxation parameters, it may be necessary to have an overdetermined problem, or at least to make some judicious choice of the relaxation times which are used. In the following section, the simple line approximation is briefly summarized, and some of the required definitions are given. Also presented is the approach used to solve the overdetermined problem and the iteration problem; this approach is very close to the technique used in LACOON3 (27) except for additional difficulties encountered with the calculation of the required Jacobian. The discussion displays selected results of program tests, and some practical considerations are mentioned. THEORY
1. The Simple Line Approximation
The simple line approximation has been described elsewhere in the literature (see, for example, Ref. (I)), and we give here only a brief summary of the pertinent equations and definitions. The starting point is the equation of motion of the spin density matrix p, fif!? = i[p, HO -t- I&(t)] dt
-I-
PI
where H,, is the high-resolution NMR spin Hamiltonian; l&(t) represents the Hamiltonian describing the (weak) coupling of the spin system with the detecting rf field, which is assumed to be of the form 24, cos wti; and the last term in Eq. [l] describes the process by which the spin system relaxes toward thermal equilibrium through couplings with its surroundings. Invoking the steady-state approximation, we are interested in solutions for which the diagonal elements Pkk are constant in time, and for which the off-diagonals Pk[ vary as exp( f iwt). For an off-diagonal, [l:] simplifies to
- iqd(w, - WI) COSwt +
PI
where xk[ N- z exp(iwt), q = fi/ZkT, d = y&(I+)k,, and wk is an eigenvalue of Ho. The matrix x is equal to the difference p - p’, where p” N (I- qH,J/Z is the spin density matrix at thermal equilibrium. Equation [2] has been considerably simplified by the introduction of the simple line approximation, which allows only one transition (that with eigenfrequency ]wk - wl] N W) to be resonant; therefore, only the two elements xkl and xlk need be considered from among all the off-diagonals.
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This approximation results in a simple expression for the relaxation for example, @x/Wm1, ;
= -2
relax
m.n
&tnnXmn
= -&d&l
= -
terms
’ 0 T2 k, ‘lc”
Equation [3] serves as a functional definition of the effective spin-spin relaxation time for the (nondegenerate and nonoverlapping) transition associated with element xrr. In the unsaturated limit (i.e., as HP tends to Zero), Xkk 2: xl1 N 0, and Eqs. [2] and [3] yield a simple Lorentzian expression for Xk[. For the unsaturated case, however, suitable expressions must be found for all of the diagonal elements, including those which do not appear explicitly in [2]. This is achieved by considering the equations of motion from [l] ; for example,
%
= 0 = - 2 &~,,,,,,Xmm - 2 Imag(zd*)
9
E o = -2 &,,,~mm
m
and n
(n # k, 0,
where d* is the complex conjugate of d. There are only two equations of type [4a] for the diagonals x,& and xII; all other diagonals have eqUatiOnS Of motion Of type [4b]. These equations in fact form a set of simultaneous equations1 of the form AX = b,
[51 where A is a matrix of elements of the Redfield relaxation matrix R, x is treated as a column vector, and b is a column vector with b, = -2 Imag(zd*)@,, - St,,,), where a,,,,,is the Dirac delta function. Equation [5] has the formal solution xkk
-
XII
=
A - ‘(bk
-
b,)
=
161
4 Ima&d*)(T&,
which serves as a functional definition of the effective spin-lattice relaxation time for the transition Ik) + 11). It is now possible to derive a useful expression for the absorption mode signal of a saturated bandshape near the transition of interest. Substitution of [3] and [6] into Eq. [2] yields the final expression (I) sew
=
wkl)
cc wdwk
-
+‘%(~+~-)kz~
1 +
@,)2T2
2
+
;2~2(1
P
+
1-j
kl
T
T
1 2
9
L71
where dw = wk - wL - w, y is the gyromagnetic ratio of the nuclear species undergoing resonance, (I+I-)kl is the transition intensity, and the subscripts (kl) on Tl and T2 have been omitted for convenience. The last half of [7] is the desired expression; it states that each transition in some set of nondegenerate and nonoverlapping transitions can be described as a simple Bloch-type transition, except for the introduction of the term (I+I-) and the qualifi1 Actually, one of the equations must be replaced by z,, x,,,, = 0 in order to obtain an independent set of equations.
THE
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APPROXIMATION
329
cation of T1 and Ta as effective spin-lattice and spin-spin relaxation times. In the following sections it is assumed that [7] is a good representation of the spin systems under study. Consequently it will be possible to define effective Tl and Ta values and also to extract these relaxation times through a simple analysis of the lineshapes fo:r each transition. Validation of the assumption is shown by the consistency of the calculated results. 2. Details of the Iterative Process For this discussion, we assume that some set of effective spin-spin and/or spinlattice relaxation times is available from experimental data, and it is necessary to use a fitting procedure in order to obtain the relaxation parameters corresponding to those relaxation times. The basic algorithm used is very similar to the least-squares approach used by the authors of LACOON3 (21) (see also Ref. (19)) for the analysis of high-resolution NMR spectra for chemical shifts and coupling constants. In our case, we define a function E as E = 2 (TLbP”- T:‘c)2,
m
PI
where T,,, is an effective spin-spin or spin-lattice relaxation time, and the index m runs over all such relaxation times to be used in the fitting. Each of the calculated relaxation times can be expressed as a complex function of a set of relaxation parameters {Pk}; in our studies the two mechanisms of interest are the DD and RF mechanisms. The least-squares solution to the problem in [8] is found by searching for a minimum in E, i.e., by locating the roots of
qg=o,
PI
w:here the index k runs over all the relevant relaxation parameters. Following the Newton-Raphson method, we assume that a linear change dp, produces a linear change AT,,,, or AT,,, = 2
AP,.
It is required that TObS m
_
TCalC
m
-
- p$+~tc.
In matrix notation, and using the arguments in Ref. (21), we obtain R = DA,
1124
where R is a column vector of residuals, D is the Jacobian of partial differentials (D,, = aT,@p,), and A is a column vector of the changes in the parameters which is to be determined. The system of equations in [12a] can be solved only if the numbe:r of observed values Tgbbsis greater than or equal to the number of unknown parrameters pr. For the overdetermined situation, it is convenient to form the normal set of equations (17)
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DtR = DtDA,
Wbl
where Dt is the transpose of D. The matrix product DtD is then symmetric and square, somewhat simplifying the method of solution of Eq. [17]. In practice, solution of [12b] for the corrections A will not satisfy Eq. [9], partly because Eq. [lo] is not exact and partly because of experimental errors. This problem is solved by an iterative scheme, whereby the parameter set (pk} is corrected by the amounts (Ap,}, and Eq. [12b] is solved again, using the new values for the residuals R and the differentials D. This process is repeated until convergence results, which ideally should occur when R equals the zero vector. In practice, however, the presence of experimental errors will result in a converged R which is nonzero and in a set of determined parameters pk which have an attached uncertainty. Our iterative process deviates from that of Ref. (21) inasmuch as an analytical expression for the Jacobian aT,&k is not known; we are forced to use numerical methods to estimate it. Another problem also occurs, owing to the effects of symmetry, which may require that some of the relaxation parameters be identical (cf. Ref. (16)). For these cases, special attention must be paid to the construction of the estimates for D, since it is possible to set up an indeterminate system of equations of the form of [12b]. 3. Details of the Program The program which was developed for the relaxation analysis will perform three types of related calculations. For the first option, a set of high-resolution NMR parameters is input, with the output consisting of a list of all possible eigenfrequencies (including multiple quantum transitions) and intensities for the specified spin system. This calculation is performed by creating the high-resolution Hamiltonian in the spin product representation, with subsequent diagonalization using the Jacobian method. Associated with each eigenfrequency is an identification number, which is used with the following options to identify the effective relaxation times. For the second calculation option, the input includes a set of relaxation parameters for the RF and DD mechanisms (the program will also accommodate the scalar type II mechanism, which is not discussed further here), and the output consists of a list of calculated effective spin-spin and spin-lattice relaxation times corresponding to the relaxation parameters chosen, assuming the simple line approximation. These calculations are performed by generating the required elements of the Redfield relaxation matrix in the spin product representation, transforming the matrix to the eigenbasis, and then using the definitions in Eqs. [3] and [6], respectively. Generation of the elements of the Redfield relaxation matrix is achieved as outlined in Ref. (26). The third calculation option performs the iterative analysis. The input for this option includes a list of the experimental spin-spin and/or spin-lattice effective relaxation times to be used as convergence criteria, and a list of the relaxation parameters which are to be varied during the iterations. The effect of symmetry on the relaxation is included by requiring that specified relaxation parameters be varied identically. It is also required that the total number of independently varied relaxation parameters be less than or equal to the number of experimental effective relaxation times which are supplied. For this calculation option, the program proceeds as out-
THE
SIMPLE
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APPROXIMATION
331
lined previously; namely, the set of equations [12a] or [12b] is solved for the vector d of corrections which are to be applied to an input list of estimates to the relaxation parameters (our experience with the program has indicated that the accuracy of the input estimates is usually not too critical to the resulting rate or precision of convergence and that when divergence occurs, it occurs within a few iterations). The m,atrix of partial differentials is calculated numerically by evaluating the relaxation times Tgbb'using the supplied relaxation parameter estimates, and by evaluating several sets of relaxation times Tt which have the same relaxation parameters as used for T",b"except for parameter Pk, which is increased by ca. 0.5%. The element Dmkis then approximated using the relation D,k = (T," - T0,bs)/0.005Pk. The new set of relaxation parameters calculated is subsequently used as a starting point for the next round of computation, and the iterations are continued until the rms deviation between the calculated effective relaxation times of two consecutive iterations is less than a predetermined value, or the total number of iterations performed exceeds a specified value, or until a divergence pattern is detected. This calculation option also attempts to note which of the relaxation parameters have a weak or very weak influence on the calculated relaxation times. The final output is a list of the best relaxation parameters found, along with an estimate of their accuracy and a list of the calculated spin-spin and spin-lattice relaxation times corresponding to those parameters. Examples of the required execution times on an IBM 370/58 are typically 30 set for twenty iterations on a three-spin-$ system of nuclei involving fitting of six relaxation parameters using eight effective relaxation times. Similar calculations on a four-spin system require approximately 10 times more cpu time. DISCUSSION
The theory and the associated program described previously have been tested on a number of three- and four-spin-& systems with relaxation by the RF and DD relaxation mechanisms. We present here the details of some tests performed on an AB1 spin system, although we would like to mention that similar results have been obtained with other systems, and it is expected that more complex spin systems as well as other re:laxation mechanisms would follow a similar pattern of analysis. It should also be noted that our “experimental” spectra are, in fact, spectra calculated using the programs described in (26). While this approach allowed testing to be performed in the absence of experimental problems, it should be pointed out that the method of relaxation analysis should minimize many such difficulties (e.g., phase problems). Figure 1 contains sample plots of our “experimental” ABa spectra at various strengths of the saturating rf field. As noted previously (26), relaxation effects are more obvious when the rf field is sufficiently strong to compete effectively with the relaxation processes. For these spectra, it is apparent that the transitions numbered 1 and 5 should always fail the requirements of the simple line approximation. It also appears that the other single quantum transitions (and possibly some of the double quantum transitions, although we did not consider them here) can be approximated by the si:mple line approximation over the lower ranges of the rf field. In our testing of the analysis procedure, we used various combinations of the transitions numbered 3, 9, 14, 20, 11, and 21 as the source of our experimental effective relaxation times.
332
GOODWIN
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l,DQ,5
8nT
+20
+io
0
-10 Hz
FIG. 1. Computed absorption mode bandshapes for an AB1 system of three spin-$ nuclei, using the program mentioned in Ref. (16), at rf field strengths ranging from 2 to 32 nT. Highresolution NMR parameters used were J.,e = 8.08 Hz, and v., - vB = 12.89 Hz. Relaxation is due to a combination of the random field and intermolecular dipoledipole mechanisms with FA = 0.447 (rad/sec)l’a, FB = 0.316 (rad/sec)l’a, C,. = 0, C.,* = 1.0, DAB= 0.27 (rad/sec)l’*, DaB* = 0.052 (rad/sec)l’a, CABAx* = -0.125, and CABBa* = 0.625. The frequency increases from right to left, and the zero on the frequency axis corresponds to ~a. The numbers appearing above the transitions are those used by the program to identify the transitions. The labels DQ denote double quantum transitions.
THE SIMPLE LINE APPROXIMATION
333
To determine these effective spin-spin and spin-lattice relaxation times, the last half of Eq. [7] was employed as the starting point:
~afw+~-) ‘(‘“)
= 1 + AwaT; + yaH;(I+I-)TITa’
Two different approaches were used, as described below (see also Ref. (22)).
(a) The half-height linewidth approach uses the equation
where (2/Ta)set is the measured half-height linewidth. The slope and intercept of a plot of (2/Ta)Eaatvs yaHi will yield the required effective relaxation times. (b) The peak height method uses the equation
where S(0) is the measured peak height. A linear plot of HP/S(O) vs yaHg will also give the required effective relaxation times (or relative values if the constant k cannot be determined). Sample plots using these two approaches are presented in Figs. 2 and 3, where the source of half-height linewidths and peak heights was a series of spectra which includes some of those shown in Fig. 1. It will be noticed that at higher values of the rf field there is a tendency for the plotted half-height linewidths to fall above the corresponding straight line which was drawn. This tendency is due to the fact that overlap eventually occurs as HP becomes larger, and consequently the premises of the simple line approximation do not apply. Qualitatively, the effect of overlap is to increase the measured half-height linewidths, so that the points are always above the “best” straight line. The fact that a straight line obtains for the lower values of HP, however, indicates that the use of the simple line approximation is valid over at least part of the plotted range of Hp. It would be expected that similar points in Fig. 3 would tend to fall below the “best” straight lines, since the effect of overlap would be to increase the peak height. These deviations are not present in Fig. 3, however, because overlap will affect half-height linewidths before it affects peak heights. Consequently the peak height plots should be valid over a wider range of HP, as is displayed by Figs. 2 and 3. The determination of effective spin-spin and spin-lattice relaxation times has a number of attractions (relative to other methods of relaxation analysis using highresolution NMR) from an experimental viewpoint: for example, phase problems can be minimized since only one transition, not an entire spectrum, need be recorded at one time. Similarly, the effects of magnetic field inhomogeneity could be minimized, for example, by subtracting an estimate of the inhomogeneity half-height linewidth from the measured half-height linewidths before attempting straight line plots. The effect of inhomogeneity on the plots in Fig. 2 would be to produce a curve which is above the straight line at lower values of HP and which would approach the straight line asymptotically at higher values of Hp. An extrapolation of the linear portion of such a curve compared with the actual intercept (also extrapolated) would yield a
334
GOODWIN AND WALLACE
40
30
w3 2.c
20
IO
P Y * tip* (sec.)-* FIG. 2. Plots of the square of the half-height linewidth vs the square of the rf field strength for some of the transitions of the ABI spin-& systemdescribedin Fig. 1.
good estimate of the inhomogeneity half-height‘ linewidth. As a result of these considerations, it would appear that the method of analysis based on the simple line approximation has a number of benefits which cannot be duplicated by other methods of analysis. The plots in Figs. 2 and 3 were used to obtain a set of spin-spin and spin-lattice relaxation times for the indicated transitions. A number of different sets of relaxation times were obtained by varying the number of points used to determine the best straight line (using the least-squares method). Still other sets of relaxation times
335
THE SIMPLE LINE APPROXIMATION
1.0
2.0
3.0
y 2 Hp2 (sec.)-2 FIG. 3. Plots of &/S(O), where S(0) is the peak height, vs the square of the rf field strength for some of the transitions of the AB1 spin-i systemdescribed in Fig. 1.
were obtained by choosing a subset of the available relaxation times; for example, all effective spin-spin relaxation times and any two effective spin-lattice relaxation times might comprise such a subset. In several test sets, some of the relaxation times were purposely modified to give an approximation of data containing random errors. The program was then tested using these different sets of data, and usually converged results were obtained which agreed with the starting relaxation parameters used to generate the spectrum.
GOODWIN
336
AND
WALLACE
Occasionally the iterations diverged for reasons which were not always obvious. For many such cases, however, it was possible to determine that one of the following problems was the cause. (a) The supplied effective relaxation times were not sufficiently accurate. In Fig. 2, for example, if the best straight lines included points well beyond the value of yaHz = 2, the “experimental” relaxation times became very inaccurate compared with the actual (calculated) relaxation times. Generally, the program required that the supplied Tl and T2 values be correct to within about 10 to 15% of their actual values. (b) A supplied subset of relaxation times was a poor choice, with all quite insensitive to one or more of the relaxation parameters. Generally all available relaxation times should be used, and quite probably multiple quantum transitions would provide useful data (cf. Ref. (IS)). (c) The program initially converged toward a false solution and then diverged. This is a common failing of the Newton-Raphson approximation, and it is most easily solved by a new choice of starting values for the relaxation parameters. (d) Some of the relaxation parameters had very little influence on any of the relaxation times available (for example, C,,; cf. Ref. (H)), and consequently an oscillation pattern would occur with very slow convergence properties. It was often useful to fix these relaxation parameters initially and to vary them with the others when a better starting estimate was available. The program output includes a note identifying the relaxation parameters that fall into this class. In spite of the aforementioned difficulties, it was usually possible to obtain converged relaxation parameters which were within approximately 5% of the starting values. A typical set of experimental relaxation times used is shown in Table 1. This TABLE EXPERIMENTAL AND ACTUAL SPIN-SPIN TIMES FOR THE SPIN SYSTEM
Transition number
1 and SPIN-LATTICE RELAXATION SHOWN IN FIG. 1
Experimental
o
Actual
Tl
T2
Tl
TQ
3
1.07
9 14
0.90 1.05
0.91 0.81 0.98
1.032 0.859 1.026
0.920 0.813 0.985
20
0.73
0.72
0.715
0.725
11 21
1.16 0.97
1.12 0.91
1.122 0.933
1.130 0.918
a “Experimental” values were obtained from a least-squares fitting of the points below ylHz = 2 set-a in Fig. 2 and were used as convergence criteria in a test of the method of relaxation analysis described.
set of data leads to converged values (the rms deviation was less than 0.05) for the six relaxation parameters FA, FB, CAB, CBB,, DAB, DBB’ (the parameters Cm,, and C ABBB’ were fixed as described in (16) for this particular case; other test results were obtained with fitting of all four of the DD parameters), which agreed with the values stated in the caption for Fig. 1 to within a total rms deviation of less than 0.1. It
THE SIMPLE
LINE
APPROXIMATION
337
might be noted from Table 1 that the calculated and experimental relaxation times are not identical, which is attributable in part to the breakdown of the simple line approximation. CONCLUSIONS
We believe that we have shown that the method of relaxation parameter analysis based on the simple line approximation is feasible, and that we now have a computer program which will perform the required analysis, given a sufficiently accurate set of experimental relaxation times for the convergence criteria. Furthermore the complexity of the computations in any relaxation analysis would indicate that the method described is the only feasible high-resolution approach to date for large spin systems. We have performed some tests on four-spin-+ systems of nuclei, with typical execution times of 5 min, and the program has the capability of handling up to six spin-+ nuclei. The program currently performs relaxation analyses for the random field, intermolecular dipole-dipole, and scalar type II relaxation mechanisms, although other mechanisms could be accommodated. Copies of the program will be made available through the ATLAS NMR Programme Library.2 ACKNOWLEDGMENTS The authors would like to acknowledge the assistance of Dr. R. K. Harris (University of East Anglia) in the preliminary planning of this work. One of us (R.W.) would also like to acknowledge funding by the National Research Council of Canada. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
REFERENCES ,4. ABRAGAM, “Principles of Nuclear Magnetism,” Oxford Univ. Press, London, 1961. :R. K. HARRIS AND K. J. PACKER, J. Chem. Sot. (London), 4736 (1961). :R. M. LYNDEN-BELL, Proc. Roy. Sot. Ser. A 286,337 (1965). IR. M. LYNDEN-BELL, in “Progress in Nuclear Magnetic Resonance Spectroscopy” (J. W. Emsley, J. Feeney, and L. H. SutclifIe, Eds.), Vol. 2, p. 163, Pergamon, Elmsford, N.Y., 1967. IR. FREEMAN, S. WIITEKOEK, AND R. R. ERNST, J. Chem. Phys. 52,1529 (1970). IB. D. N. RAO, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 271, Academic Press, New York, 1970. IR. A. HOFFMAN, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 87, Academic Press, New York, 1970. IB. GE~TBLOM AND 0. HARTMANN, J. Mugn. Reson. 8,230 (1972). R. K. HARRIS AND K. M. WORVILL, Chem. Phys. Lett. 14,598 (1972). R. K. HARRIS AND K. M. WORVILL, J. Magn. Reson. 9, 394 (1973). D. G. HUGHES, M. R. SMITH, AND D. H. M. SWITZER, J. Chem. Phys. 60,489O (1974). 1~. K. HARRIS, N. C. PYPER, ANLI K. M. WORVILL, J. Magn. Reson. 18, 139 (1975). K. M. WORVILL, I. Magn. Reson. 18,217 (1975). I?. MEAKIN AND J. P. JESSON,J. Mugn. Reson. 19, 37 (1975). 0. HARTMANN, J. Magn. Reson. 22, 125 (1976). 13. GOODWIN, R. WALLACE, AND R. K. HARRIS, J. Magn. Reson. 22,491 (1976); B. GOODWIN, AND R. WALLACE, J. Magn. Reson. 23,465 (1976). II. FREEMAN, S. WITTEKOEK, AND R. R. ERNST, J. Chem. Phys. 52, 1529 (1970). ‘W. M. M. BC&E, Mol. Phys. 29, 1673 (1975). 1~. L. VOLD AND R. R. VOLD, J. Chem. Phys. 66, 1202 (1977). I,. G. WERBELOW AND D. M. GRANT, in “Advances in Magnetic Resonance” (J. S. Wsugh, Ed.), Vol. 9, p. 189, Academic Press, New York, 1977. S. CASTELLANO AND A. A. BOTHNER-BY, J. Chem. Phys. 41, 3863 (1964).
’ Science Research Council ATLAS Computer Laboratory, NMR Program Library, c/o R. K. Harris, School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, Great Britain.