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The determination of individual rates from magnetization-transfer measurements
JOURNAL
OF MAGNETIC
69,92-99 ( 1986)
RESONANCE
The Determination of Individual Rates from Magnetization-Transfer Measurements MARIA GRASSI,*BRIAN E. MANN, BARRY T. PICKUP, AND CATRIONA
M. SPENCER
Department of Chemistry, The University, Shefield, S3 7HF, England
Received February 11, 1986; revised March 28, 1986 The rate equations for magnetization transfer between many sites have been solved for the general problem of one or many sites being perturbed by a selective pulse or preirradiation with a low power continuous wave frequency. Individual rate constants and relaxation times are derived, but no allowance is made for nuclear Overhauser effects.The analysisis applied to magnetization-transfermeasurementsmade on the four-site 13CNMR problem, Cr($-CsHr.)jCO)3. @ 1986 Academic press, hc. INTRODUCTION
Recently, several papers have appeared which discuss different methods of determ ining individual rate constants from magnetization-transferdata (I-4). These papers include the analysis of three-site problems (3, 4), and the use of m u ltiple-irradiation techniques (2, 5). None of these treatments is general, although the basic treatment was originally described in 1963 by For&n and Hoffman (6, 7) and subsequently applied to a wide rangeof problems. Two-site exchangeis readily analyzed,and recently a detailed comparison of the various methods has appeared (2). The m u ltiple-site exchangeproblem in the presenceof continuous irradiation was analyzed and applied to the five-site exchange in (~5-C5Hs)Mo(~-C5H5)(CO)(NO)(SKNBu~)(8), but the analysis required that each site had the same relaxation tim e and NOE effects could be neglected. The analysis of this problem was facilitated by the steady-statenature of the experimental technique. Analysis becomes far more difficult when nonequilibrium problems are discussed.Such casesare frequently encountered in organometallic and biological systems.A significant advance in the analysis of such problems has just appearedin this journal (4). A quantitative analysis of a NOESY exchangecorrelation experiment was reported. The analysis was applied to a system with three equally populated sites, Ru&H)(C0)9(MeCCHCMe). The analysis is applicable to any number of equally populated exchanging sites, but will encounter problems when the sites are not equally populated, due to the unsymmetric matrix generated, with possible difficulties of diagonalization. The treatment is also not very amenable to the incorporation of constraints on the values of the exchangerates and relaxation times. * present address:Universita degli Studi di Milano, Dipartimento di Chimica Inorganica e Metallorganica, Via Venezian 2 1, 20 133 Milano, Italy. 0022-2364186 $3.00 Copyright 0 1986 by Academic Pres, Inc. All rights of repmdwlion in any form reserved.
92
RATES FROM MAGNETIZATION
TRANSFER
93
Recently, we have been using the DANTE pulse sequenceto excite nuclei selectively, and to determine the mechanism in multisite exchange problems. So far the analysis has been qualitative or at best semiquantitative. Before this powerful approach to the determination of mechanism can be fully exploited, it is necessaryto develop a general analysis of such exchange data, which we report here. Problems of cross-relaxation and coupling are avoided by using natural abundance 13CNMR spectra. THEORY
The general form of the coupled differential equations has been derived earlier (5). For a system consisting of n magnetically nonequivalent sites, i, the equation is dM, : koMi + dt = -j=l(i+j)
i
kjiA4j + {Mi(oO) - Mi}/Tli
[II
j= I(i#j)
where TIi is the spin-lattice relaxation time for site i, kij is the first-order rate constant for exchange from site i to site j, M i is the magnetization at site i at time t, and M i( Co) is its equilibrium magnetization. Equation [l] may be rewritten in matrix notation as dM z=AMtB.
where the elements of the column matrix M are M i, A, = kji, i # j, and Ai, = - cjzi X kij - l/T,i. AS t -+ 00, dM,/dt + 0 and hence 0 = AM(a)
+ B
[31
Subtracting Eq. [3] from [2] we obtain the equation dM = A[M - M(m)]. dt
Since dM(co)/dt
[41
= 0, we can devise a homogeneous version dD -z=AD
where the column D = M(t) - M(m).
161
Equation [5] can be solved by matrix manipulation, but in cases where there are unequal site populations, the matrix A will be unsymmetric. Unsymmetric matrices cannot always be diagonalized by an orthogonal transformation. These special cases make the computation of the matrix exponential below much more difficult. Fortunately, a simple transformation renders [ 51 symmetric. We define the diagonal matrix N with diagonal elements M i( a). Equation [ 51 may be rewritten d[N-“*D] dt
which can be written as
= (N-‘/*AN+‘+-‘/*,,
94
GRASS1 ET AL.
with X = N-‘I’D and & = N-‘12AN +‘I2. It is easy to seethat & = Aji, as required, since fori#j
191
Equation [8] can be solved by defining U as the orthogonal matrix of eigenvectors and a is the diagonal eigenvalue matrix. Then Eq. [8] becomes dY -z=
aY
[lOI
where Y(t) = u+x
[Ill
The equations [lo] are now decoupled, since 3 is diagonal, and have the solutions yi = @+‘“‘y(()) 1121 where the initial condition is Y,(t) = Vi(O)
at
t= to.
[I31
Transforming back to X variables, the solution to Eq. [8] is X(t) = ei(l-@)x(o)
iI41
with the initial condition X(t) = X(0)
at
t = to.
1151
The matrix exponential can be calculated for any time t. In all problems, it is necessary to fit a series of experimentally measured magnetizations using rate constants kii and relaxation times Tli. In all cases,the constraint k..
=
Mi(oo)
'
Mj(o0)
1161
k.. "
applies, but often it is necessaryto apply additional constraints of the type kg = constant, kti = k,y, Tii = T,j, and ko + ki,, = k,fy. All these constraints are linear, and can be written as a set of matrix equations Cx = d
iI71
where C is ap X q (p < q) matrix, x is a column of q variables, (i.e., rates or relaxations), and d is a column of p constraint values. The matrix C is rectangular, but can be converted to a square matrix by defining the q X p matrix (C+)ji = C$
iI81
and multiplying [ 171 from the left by Ct C+Cx = (C+d) = f.
1191
RATES FROM MAGNETIZATION
TRANSFER
95
The matrix E = C+C
PO1
is a q X q Hermitian square matrix, while f = C+d
1211
is a q X 1 column. It follows that there is a unitary matrix V which diagonalizes E, hence VtEV = e.
WI
Equation [ 191 is multiplied by Vt to give [231
w = 9 where Vtf = g and Vtx = y, which in component form is i = 1 to q.
eiYi = gi
[241
It can be shown that p eigenvalues ei normally are nonzero, while the remaining (p - q) are zero. The zero eigenvalues correspond to free unconstrained optimization parameters, while the remainder correspond to the constraints. The yi are transformed parameters which must be used in the optimization procedure. The optimizer then works in terms of free (transformed) variables, and these are reconverted back to actual physical quantities everytime the M(t) must be calculated. In this way the linear constraints are satisfied. The program is written in FORTRAN77, and is designed to take a general number of sites and linear parameter constraints. The nonlinear least-squaresoptimization is effected in terms of the transformed free parameters yi using the simplest NAG algorithm E04FDF which does not require the computation of first or second derivatives analytically. While this simple routine was meant as a first step only, the procedure has been found to be stable and effective, so that no more sophisticated approach was justified. RESULTS AND DISCUSSION
In an earlier study using the steady state produced by continuous irradiation of one site, it had been shown that in C~(T,~-C~H~)(CO)J, [ 1,3] and possibly [ 1,2] shifts occur (8) as shown in Scheme 1. The same compound was chosen for this investigation to test the applicability of our analysis. In theory, there are four possible metal m igrations (Scheme 1). These are indicated by the rate constants Ki2, Ki3, Ki4, and Ki5. The molecule possessesfour NMR-distinct cycle-octatetraene carbon atoms. Exchange between them can be described by the rate constants ki2, ki3, kid, kz3, kz4, kj4. Examination of the relation between these two sets of rate constants shows that k,2 =
K12
+
K13
kn =
K13
+
K14
k1.t =
K14
+ KIS
k23 =
K12
+ KI,
GRASS1 ET AL.
96
,; ..__, 0I:;,[ cr(CO)
3
0..Js]lli I 3 cx(CO)
SCHEME
1. The four possible mechanisms of metal migration in Cr(q6-C8H&CO),
k2r.t=
K13
+
K14
k34 =
K12
+
K13.
Hence the rate matrix becomes k k 12 I’ k 13 C ku
hz
+
k,4
k 13
Wl
k k,z + k:: - k,3 -
h3
k
12
k k:; k 1'
’ 1
Thus although there are four possible rate constants to obtain the four possible metal migrations, the experiment can determine only three of them.
RATES FROM MAGNETIZATION
97
TRANSFER
A poor fit of the experimental data was produced with kr2, k13, k14, and the four T, values as variables. All data points were fitted with an accuracy significantly better than the noise level. The resulting derived variables were k12 = 11.1 SC’,k13 = 9.0 s-‘, k14 = 0.4 s-‘, T,, = 10.9 s, Tr2 = 1.0 s, T13 = 170.8 s, and T14 = 300.3 s. Although reasonable rates are produced, the spin-lattice relaxation times are subject to considerable errors. These errors are not apparent from the excellent least-squareserror of 0.0025. It has previously been noted that when exchange is fast compared with spinlattice relaxation, then the inversion-recovery method of Tl measurements, as used here, is subject to gross errors (IO). If the fit is repeated with four variables, k 12, k13, k14, and T,, once again all data points were fitted with an accuracy significantly better than the noise level, with a least-squares error of 0.0036. The resulting derived variables were k12 = 10.8 s-l, k,, = 9.1 s-‘, k14 = 0.4 SC’,and TI = 3.8 s. This fit is shown graphically in Fig. 1. To test the susceptibility of this analysis to false m inima and oscillating fitting, the fit was initially carried out using values of k estimated from initial gradients and T, values estimated from the total relaxation of the system. Subsequently, values of k and T, differing by up to a factor of 10 were tried, but in each case, the program rapidly produced the same result. However, if grossly inaccurate values of the rates and relaxation times are used, then the method does fail. This is no significant constraint as 0
b
-10 -+--t--A---.------.
___-----7
-20
/’
9 -40 x
//
-50
i’
-60 -70
tf f
-80 0.00
0.05 TIME
-0.20 0.15
0.10 (SECONDS)
TIME
(SECONDS) d
0.05 TIME
0.15 0.10 (SECONDS)
0.20
0.00
0.05 TIHE
0.10 (SECONDS)
,, 0.15
o.io
FIG. 1. Plots of the initial points of the “C magnetization of sites 1 to 4 of the cyclooctatetraene ring in Cr($-CsHs)(CO), after applying a selectiveapproximately 180” DANTE pulse to site 1. The + signs represent experimental points and the line is calculated on the basis of best fit for /cl*, k,3, and k14, and having one T, value for all sites. The y axis represents the percentage difference in the signal between full inversion and full recovery. (a) Magnetization at site 1; (b) magnetization at site 2; (c) magnetization at site 3; (d) magnetization at site 4.
98
GRASS1 ET AL.
approximate values of the rates can be estimated using the initial gradients, and the average T, can be accurately determined using the recovery of the sum of all the magnetizations, and following normal analytical techniques (I I). The sensitivity of the rates to T, values was tested by applying the unreasonable constraint that T,, = 2T,2 = 4T,3 = 8 T,+ The resulting derived variables were k,* = 10.9 s-‘, /& = 9.2 s-l, k,4 = 0.5 s-‘, T,, = 13.9 s, T,* = 6.9 s, T,3 = 3.5 s, and T,4 = 1.7 s. The least-squares error increased to 0.0047. Clearly the derived rates are insensitive to T, values. As at lower temperatures, when exchange is slow, the T, values are similar in the four different sites. It was assumed in all subsequent calculations that all the T, values are equal. As k,4 is small, the calculation was repeated but now with k,,, set to zero. This produced only a small increase in the least-squares error to 0.0040, and the new set of variables is k,2 = 11.1 s-‘, k,, = 9.4 s-l, and T, = 3.8 s. If it is assumed in Scheme 1 that J&, = & = 0, then Z& = 1.7 s-, and K,3 = 9.4 s-‘. As K,3 is much larger than KL2, it is reasonable to try the effect of putting K,* = 0, which requires that k,2 = k,3, and k,4 = 0. All the T, values were set equal. The leastsquares error doubled to 0.0082 and the discrepancies between several of the calculated and experimental magnetizations became larger than the noise level. The derived variables were k,* = k,3 = 10.2 s-‘, and T, = 3.8 s. Thus from Scheme 1, K,2 = 0 and KL3 = 10.2 s-‘. As in an analysis of the exchange mechanism (9), it had been suggested that, in Scheme 1, K,2 = 3KL4, when K,3 & K,2, and K ,5 = 0. When these conditions were applied, an acceptable least-squares error of 0.0039 resulted, with variables k,2 = 10.5 s-l, k,3 = 9.0 s-‘, k,4 = 0.7 s-l, and T, = 3.8 s. The derived rate constants for Scheme 1 using these assumptions are K,* = 2.2 s-,, K,J = 8.3 SK’,and K,4 = 0.7 s-‘. Recently, it has been shown that, for OS($‘-C~H~)(~~-C~H,~),the dominant mechanism for cycle-octatetraene fluxionality corresponds to [ 1, 51 shifts (12), i.e., KL5 in Scheme 1 is dominant. Thus a fit was attempted with Klz = K,4 = 0, resulting from Eqs. [25] with k,2 = k13. The least-squares error became 0.0078, with variables k,2 = k13 = 10.0 s-l, k14 = 0.4 s-‘, and T, = 3.8 s-,. The derived Scheme 1 rate constants are then K,J = 10.0 s-, and Kls = 0.4 s-‘. Finally, a fit based on only k,* and T, as variables was attempted to try a completely unreasonable fit. The least-squares error increased greatly to 0.242 and only half the points were fitted within the signaknoise ratio. Errors as large as 11% were produced between the calculated and observed magnetizations. Clearly this was not a viable fit. Although Cr(#-CsHs)(CO), has posed a number of problems in the analysis, it is clear from the above results, that the data analysis method works extremely well, and provides a reliable method to obtain individual rate constants in a multisite exchange problem, with or without constraints. The ability of the method to add constraints, to make allowance for symmetry imposed by the compound, or to test mechanistic hypotheses is extremely powerful, permitting some mechanisms to be eliminated, while other mechanisms remain. Examination of the data given above shows that, even though several mechanisms may be present, then the dominant process involves a [ 1,3] shift of the metal on the ring (Scheme 1). The fit improves considerably, when a [ 1, 21 shift is included in the mechanism, but there are only minor improvements when a [ 1, 41 or a [ 1, 51 shift is included.
RATES PROM MAGNETIZATION
TRANSFER
99
EXPERIMENTAL
The 13CNMR measurements were made on a Bruker WH400 equipped with a ten clock pulse programmer. A 5 m m dual 13C/‘H probe was used for the measurements. The selective 180” pulse was generated using the DANTE pulse sequence (13). The DANTE delay was chosen to produce sidebands as far away from exchanging carbon atom signals as possible. Typically, a series of thirty 1.3 ps pulses spaced by 0.2 ms was used. In addition to the blanks with 5 PS and 20 s delay between the selective pulse and the general 90” observing pulse, delays varying between 2.5 ms and 3.2 s were used for the experiment to obtain both reliable kinetic and relaxation data. To avoid any errors introduced by temperature or resolution drift during an overnight run, the sets of measurements were acquired in blocks of eight acquisitions and stored on disk. At the end of each set of DANTE delays, the stored spectra were retrieved from disk, an additional set of eight spectra added, and stored again on disk. Typically this was repeated ten times to give 80 acquisitions for each spectrum. The signal-tonoise ratio was improved by applying a substantial exponential multiplication to the FID to produce a 10 Hz line broadening. The temperature was determined using a thermocouple in a 5 m m NMR tube containing CHzClz and a Comark electronic thermometer, 5235. The Cr(#-CsHs)(CO), was prepared as previously described (9) and dissolved in CD&l2 under N2. ACKNOWLEDGMENTS We thank the SERC for financial support to M.G. and C.M.S. and for access to their Bruker WH400 NMR spectrometer. M.G. thanks the University of Milan for leave of absence. REFERENCES 1. K. UGURBIL, J. Mugn. Reson. 64,207 (1985). 2. C. R. MALLOY, A. D. SHERRY, AND R. L. NUNNALLY, J. Mugn. Reson. 64,243 (1985). 3. C. L. PERRIN AND E. R. JOHNSTON, J. Magn. Reson. 33,619 (1979). 4. G. E. HAWKES, L. Y. LIAN, E. W. RANDALL, K. D. SALES, AND S. AIME, J. Magn. Reson. 65, 173
(1985). 5. S. FOR&N AND R. A. HOFF~~AN, J. Chem. Phys. 40, 1189 ( 1964). 6. S. FOR&N AND R. A. HOFI%AN, J. Chem. Phys. 39,2892 (1963). 7. S. FOR&N AND R. A. HOFFMAN, Acta Chem. Stand. 17, 1787 (1963). 8. M. M. HUNT, W. G. KITA, B. E. MANN, AND J. A. MCCLEVERTY, J. Chem. Sot. Dalfon. Trans., 467
(1978). 9. J. A. GIBSON AND B. E. MANN, J. Chem. Sot. Dalton Trans., 102 1 (1979). 10. J. B. LAMBERT AND J. W. KEEPERS,J. Mugn. Reson. 38,233 (1980). 11. G. C. LEVY, “Topics in “C NMR Spectroscopy,” p. 24 1 ( 1984). 12. M. GRASSI, B. E. MANN, AND C. M. SPENCER,J. Chem. Sot. Chem. Commun., 1169 (1985). 13. G. A. MORRISAND R. FREEMAN, J. Mugn. Reson. 29,433 (1978).
OF MAGNETIC
69,92-99 ( 1986)
RESONANCE
The Determination of Individual Rates from Magnetization-Transfer Measurements MARIA GRASSI,*BRIAN E. MANN, BARRY T. PICKUP, AND CATRIONA
M. SPENCER
Department of Chemistry, The University, Shefield, S3 7HF, England
Received February 11, 1986; revised March 28, 1986 The rate equations for magnetization transfer between many sites have been solved for the general problem of one or many sites being perturbed by a selective pulse or preirradiation with a low power continuous wave frequency. Individual rate constants and relaxation times are derived, but no allowance is made for nuclear Overhauser effects.The analysisis applied to magnetization-transfermeasurementsmade on the four-site 13CNMR problem, Cr($-CsHr.)jCO)3. @ 1986 Academic press, hc. INTRODUCTION
Recently, several papers have appeared which discuss different methods of determ ining individual rate constants from magnetization-transferdata (I-4). These papers include the analysis of three-site problems (3, 4), and the use of m u ltiple-irradiation techniques (2, 5). None of these treatments is general, although the basic treatment was originally described in 1963 by For&n and Hoffman (6, 7) and subsequently applied to a wide rangeof problems. Two-site exchangeis readily analyzed,and recently a detailed comparison of the various methods has appeared (2). The m u ltiple-site exchangeproblem in the presenceof continuous irradiation was analyzed and applied to the five-site exchange in (~5-C5Hs)Mo(~-C5H5)(CO)(NO)(SKNBu~)(8), but the analysis required that each site had the same relaxation tim e and NOE effects could be neglected. The analysis of this problem was facilitated by the steady-statenature of the experimental technique. Analysis becomes far more difficult when nonequilibrium problems are discussed.Such casesare frequently encountered in organometallic and biological systems.A significant advance in the analysis of such problems has just appearedin this journal (4). A quantitative analysis of a NOESY exchangecorrelation experiment was reported. The analysis was applied to a system with three equally populated sites, Ru&H)(C0)9(MeCCHCMe). The analysis is applicable to any number of equally populated exchanging sites, but will encounter problems when the sites are not equally populated, due to the unsymmetric matrix generated, with possible difficulties of diagonalization. The treatment is also not very amenable to the incorporation of constraints on the values of the exchangerates and relaxation times. * present address:Universita degli Studi di Milano, Dipartimento di Chimica Inorganica e Metallorganica, Via Venezian 2 1, 20 133 Milano, Italy. 0022-2364186 $3.00 Copyright 0 1986 by Academic Pres, Inc. All rights of repmdwlion in any form reserved.
92
RATES FROM MAGNETIZATION
TRANSFER
93
Recently, we have been using the DANTE pulse sequenceto excite nuclei selectively, and to determine the mechanism in multisite exchange problems. So far the analysis has been qualitative or at best semiquantitative. Before this powerful approach to the determination of mechanism can be fully exploited, it is necessaryto develop a general analysis of such exchange data, which we report here. Problems of cross-relaxation and coupling are avoided by using natural abundance 13CNMR spectra. THEORY
The general form of the coupled differential equations has been derived earlier (5). For a system consisting of n magnetically nonequivalent sites, i, the equation is dM, : koMi + dt = -j=l(i+j)
i
kjiA4j + {Mi(oO) - Mi}/Tli
[II
j= I(i#j)
where TIi is the spin-lattice relaxation time for site i, kij is the first-order rate constant for exchange from site i to site j, M i is the magnetization at site i at time t, and M i( Co) is its equilibrium magnetization. Equation [l] may be rewritten in matrix notation as dM z=AMtB.
where the elements of the column matrix M are M i, A, = kji, i # j, and Ai, = - cjzi X kij - l/T,i. AS t -+ 00, dM,/dt + 0 and hence 0 = AM(a)
+ B
[31
Subtracting Eq. [3] from [2] we obtain the equation dM = A[M - M(m)]. dt
Since dM(co)/dt
[41
= 0, we can devise a homogeneous version dD -z=AD
where the column D = M(t) - M(m).
161
Equation [5] can be solved by matrix manipulation, but in cases where there are unequal site populations, the matrix A will be unsymmetric. Unsymmetric matrices cannot always be diagonalized by an orthogonal transformation. These special cases make the computation of the matrix exponential below much more difficult. Fortunately, a simple transformation renders [ 51 symmetric. We define the diagonal matrix N with diagonal elements M i( a). Equation [ 51 may be rewritten d[N-“*D] dt
which can be written as
= (N-‘/*AN+‘+-‘/*,,
94
GRASS1 ET AL.
with X = N-‘I’D and & = N-‘12AN +‘I2. It is easy to seethat & = Aji, as required, since fori#j
191
Equation [8] can be solved by defining U as the orthogonal matrix of eigenvectors and a is the diagonal eigenvalue matrix. Then Eq. [8] becomes dY -z=
aY
[lOI
where Y(t) = u+x
[Ill
The equations [lo] are now decoupled, since 3 is diagonal, and have the solutions yi = @+‘“‘y(()) 1121 where the initial condition is Y,(t) = Vi(O)
at
t= to.
[I31
Transforming back to X variables, the solution to Eq. [8] is X(t) = ei(l-@)x(o)
iI41
with the initial condition X(t) = X(0)
at
t = to.
1151
The matrix exponential can be calculated for any time t. In all problems, it is necessary to fit a series of experimentally measured magnetizations using rate constants kii and relaxation times Tli. In all cases,the constraint k..
=
Mi(oo)
'
Mj(o0)
1161
k.. "
applies, but often it is necessaryto apply additional constraints of the type kg = constant, kti = k,y, Tii = T,j, and ko + ki,, = k,fy. All these constraints are linear, and can be written as a set of matrix equations Cx = d
iI71
where C is ap X q (p < q) matrix, x is a column of q variables, (i.e., rates or relaxations), and d is a column of p constraint values. The matrix C is rectangular, but can be converted to a square matrix by defining the q X p matrix (C+)ji = C$
iI81
and multiplying [ 171 from the left by Ct C+Cx = (C+d) = f.
1191
RATES FROM MAGNETIZATION
TRANSFER
95
The matrix E = C+C
PO1
is a q X q Hermitian square matrix, while f = C+d
1211
is a q X 1 column. It follows that there is a unitary matrix V which diagonalizes E, hence VtEV = e.
WI
Equation [ 191 is multiplied by Vt to give [231
w = 9 where Vtf = g and Vtx = y, which in component form is i = 1 to q.
eiYi = gi
[241
It can be shown that p eigenvalues ei normally are nonzero, while the remaining (p - q) are zero. The zero eigenvalues correspond to free unconstrained optimization parameters, while the remainder correspond to the constraints. The yi are transformed parameters which must be used in the optimization procedure. The optimizer then works in terms of free (transformed) variables, and these are reconverted back to actual physical quantities everytime the M(t) must be calculated. In this way the linear constraints are satisfied. The program is written in FORTRAN77, and is designed to take a general number of sites and linear parameter constraints. The nonlinear least-squaresoptimization is effected in terms of the transformed free parameters yi using the simplest NAG algorithm E04FDF which does not require the computation of first or second derivatives analytically. While this simple routine was meant as a first step only, the procedure has been found to be stable and effective, so that no more sophisticated approach was justified. RESULTS AND DISCUSSION
In an earlier study using the steady state produced by continuous irradiation of one site, it had been shown that in C~(T,~-C~H~)(CO)J, [ 1,3] and possibly [ 1,2] shifts occur (8) as shown in Scheme 1. The same compound was chosen for this investigation to test the applicability of our analysis. In theory, there are four possible metal m igrations (Scheme 1). These are indicated by the rate constants Ki2, Ki3, Ki4, and Ki5. The molecule possessesfour NMR-distinct cycle-octatetraene carbon atoms. Exchange between them can be described by the rate constants ki2, ki3, kid, kz3, kz4, kj4. Examination of the relation between these two sets of rate constants shows that k,2 =
K12
+
K13
kn =
K13
+
K14
k1.t =
K14
+ KIS
k23 =
K12
+ KI,
GRASS1 ET AL.
96
,; ..__, 0I:;,[ cr(CO)
3
0..Js]lli I 3 cx(CO)
SCHEME
1. The four possible mechanisms of metal migration in Cr(q6-C8H&CO),
k2r.t=
K13
+
K14
k34 =
K12
+
K13.
Hence the rate matrix becomes k k 12 I’ k 13 C ku
hz
+
k,4
k 13
Wl
k k,z + k:: - k,3 -
h3
k
12
k k:; k 1'
’ 1
Thus although there are four possible rate constants to obtain the four possible metal migrations, the experiment can determine only three of them.
RATES FROM MAGNETIZATION
97
TRANSFER
A poor fit of the experimental data was produced with kr2, k13, k14, and the four T, values as variables. All data points were fitted with an accuracy significantly better than the noise level. The resulting derived variables were k12 = 11.1 SC’,k13 = 9.0 s-‘, k14 = 0.4 s-‘, T,, = 10.9 s, Tr2 = 1.0 s, T13 = 170.8 s, and T14 = 300.3 s. Although reasonable rates are produced, the spin-lattice relaxation times are subject to considerable errors. These errors are not apparent from the excellent least-squareserror of 0.0025. It has previously been noted that when exchange is fast compared with spinlattice relaxation, then the inversion-recovery method of Tl measurements, as used here, is subject to gross errors (IO). If the fit is repeated with four variables, k 12, k13, k14, and T,, once again all data points were fitted with an accuracy significantly better than the noise level, with a least-squares error of 0.0036. The resulting derived variables were k12 = 10.8 s-l, k,, = 9.1 s-‘, k14 = 0.4 SC’,and TI = 3.8 s. This fit is shown graphically in Fig. 1. To test the susceptibility of this analysis to false m inima and oscillating fitting, the fit was initially carried out using values of k estimated from initial gradients and T, values estimated from the total relaxation of the system. Subsequently, values of k and T, differing by up to a factor of 10 were tried, but in each case, the program rapidly produced the same result. However, if grossly inaccurate values of the rates and relaxation times are used, then the method does fail. This is no significant constraint as 0
b
-10 -+--t--A---.------.
___-----7
-20
/’
9 -40 x
//
-50
i’
-60 -70
tf f
-80 0.00
0.05 TIME
-0.20 0.15
0.10 (SECONDS)
TIME
(SECONDS) d
0.05 TIME
0.15 0.10 (SECONDS)
0.20
0.00
0.05 TIHE
0.10 (SECONDS)
,, 0.15
o.io
FIG. 1. Plots of the initial points of the “C magnetization of sites 1 to 4 of the cyclooctatetraene ring in Cr($-CsHs)(CO), after applying a selectiveapproximately 180” DANTE pulse to site 1. The + signs represent experimental points and the line is calculated on the basis of best fit for /cl*, k,3, and k14, and having one T, value for all sites. The y axis represents the percentage difference in the signal between full inversion and full recovery. (a) Magnetization at site 1; (b) magnetization at site 2; (c) magnetization at site 3; (d) magnetization at site 4.
98
GRASS1 ET AL.
approximate values of the rates can be estimated using the initial gradients, and the average T, can be accurately determined using the recovery of the sum of all the magnetizations, and following normal analytical techniques (I I). The sensitivity of the rates to T, values was tested by applying the unreasonable constraint that T,, = 2T,2 = 4T,3 = 8 T,+ The resulting derived variables were k,* = 10.9 s-‘, /& = 9.2 s-l, k,4 = 0.5 s-‘, T,, = 13.9 s, T,* = 6.9 s, T,3 = 3.5 s, and T,4 = 1.7 s. The least-squares error increased to 0.0047. Clearly the derived rates are insensitive to T, values. As at lower temperatures, when exchange is slow, the T, values are similar in the four different sites. It was assumed in all subsequent calculations that all the T, values are equal. As k,4 is small, the calculation was repeated but now with k,,, set to zero. This produced only a small increase in the least-squares error to 0.0040, and the new set of variables is k,2 = 11.1 s-‘, k,, = 9.4 s-l, and T, = 3.8 s. If it is assumed in Scheme 1 that J&, = & = 0, then Z& = 1.7 s-, and K,3 = 9.4 s-‘. As K,3 is much larger than KL2, it is reasonable to try the effect of putting K,* = 0, which requires that k,2 = k,3, and k,4 = 0. All the T, values were set equal. The leastsquares error doubled to 0.0082 and the discrepancies between several of the calculated and experimental magnetizations became larger than the noise level. The derived variables were k,* = k,3 = 10.2 s-‘, and T, = 3.8 s. Thus from Scheme 1, K,2 = 0 and KL3 = 10.2 s-‘. As in an analysis of the exchange mechanism (9), it had been suggested that, in Scheme 1, K,2 = 3KL4, when K,3 & K,2, and K ,5 = 0. When these conditions were applied, an acceptable least-squares error of 0.0039 resulted, with variables k,2 = 10.5 s-l, k,3 = 9.0 s-‘, k,4 = 0.7 s-l, and T, = 3.8 s. The derived rate constants for Scheme 1 using these assumptions are K,* = 2.2 s-,, K,J = 8.3 SK’,and K,4 = 0.7 s-‘. Recently, it has been shown that, for OS($‘-C~H~)(~~-C~H,~),the dominant mechanism for cycle-octatetraene fluxionality corresponds to [ 1, 51 shifts (12), i.e., KL5 in Scheme 1 is dominant. Thus a fit was attempted with Klz = K,4 = 0, resulting from Eqs. [25] with k,2 = k13. The least-squares error became 0.0078, with variables k,2 = k13 = 10.0 s-l, k14 = 0.4 s-‘, and T, = 3.8 s-,. The derived Scheme 1 rate constants are then K,J = 10.0 s-, and Kls = 0.4 s-‘. Finally, a fit based on only k,* and T, as variables was attempted to try a completely unreasonable fit. The least-squares error increased greatly to 0.242 and only half the points were fitted within the signaknoise ratio. Errors as large as 11% were produced between the calculated and observed magnetizations. Clearly this was not a viable fit. Although Cr(#-CsHs)(CO), has posed a number of problems in the analysis, it is clear from the above results, that the data analysis method works extremely well, and provides a reliable method to obtain individual rate constants in a multisite exchange problem, with or without constraints. The ability of the method to add constraints, to make allowance for symmetry imposed by the compound, or to test mechanistic hypotheses is extremely powerful, permitting some mechanisms to be eliminated, while other mechanisms remain. Examination of the data given above shows that, even though several mechanisms may be present, then the dominant process involves a [ 1,3] shift of the metal on the ring (Scheme 1). The fit improves considerably, when a [ 1, 21 shift is included in the mechanism, but there are only minor improvements when a [ 1, 41 or a [ 1, 51 shift is included.
RATES PROM MAGNETIZATION
TRANSFER
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EXPERIMENTAL
The 13CNMR measurements were made on a Bruker WH400 equipped with a ten clock pulse programmer. A 5 m m dual 13C/‘H probe was used for the measurements. The selective 180” pulse was generated using the DANTE pulse sequence (13). The DANTE delay was chosen to produce sidebands as far away from exchanging carbon atom signals as possible. Typically, a series of thirty 1.3 ps pulses spaced by 0.2 ms was used. In addition to the blanks with 5 PS and 20 s delay between the selective pulse and the general 90” observing pulse, delays varying between 2.5 ms and 3.2 s were used for the experiment to obtain both reliable kinetic and relaxation data. To avoid any errors introduced by temperature or resolution drift during an overnight run, the sets of measurements were acquired in blocks of eight acquisitions and stored on disk. At the end of each set of DANTE delays, the stored spectra were retrieved from disk, an additional set of eight spectra added, and stored again on disk. Typically this was repeated ten times to give 80 acquisitions for each spectrum. The signal-tonoise ratio was improved by applying a substantial exponential multiplication to the FID to produce a 10 Hz line broadening. The temperature was determined using a thermocouple in a 5 m m NMR tube containing CHzClz and a Comark electronic thermometer, 5235. The Cr(#-CsHs)(CO), was prepared as previously described (9) and dissolved in CD&l2 under N2. ACKNOWLEDGMENTS We thank the SERC for financial support to M.G. and C.M.S. and for access to their Bruker WH400 NMR spectrometer. M.G. thanks the University of Milan for leave of absence. REFERENCES 1. K. UGURBIL, J. Mugn. Reson. 64,207 (1985). 2. C. R. MALLOY, A. D. SHERRY, AND R. L. NUNNALLY, J. Mugn. Reson. 64,243 (1985). 3. C. L. PERRIN AND E. R. JOHNSTON, J. Magn. Reson. 33,619 (1979). 4. G. E. HAWKES, L. Y. LIAN, E. W. RANDALL, K. D. SALES, AND S. AIME, J. Magn. Reson. 65, 173
(1985). 5. S. FOR&N AND R. A. HOFF~~AN, J. Chem. Phys. 40, 1189 ( 1964). 6. S. FOR&N AND R. A. HOFI%AN, J. Chem. Phys. 39,2892 (1963). 7. S. FOR&N AND R. A. HOFFMAN, Acta Chem. Stand. 17, 1787 (1963). 8. M. M. HUNT, W. G. KITA, B. E. MANN, AND J. A. MCCLEVERTY, J. Chem. Sot. Dalfon. Trans., 467
(1978). 9. J. A. GIBSON AND B. E. MANN, J. Chem. Sot. Dalton Trans., 102 1 (1979). 10. J. B. LAMBERT AND J. W. KEEPERS,J. Mugn. Reson. 38,233 (1980). 11. G. C. LEVY, “Topics in “C NMR Spectroscopy,” p. 24 1 ( 1984). 12. M. GRASSI, B. E. MANN, AND C. M. SPENCER,J. Chem. Sot. Chem. Commun., 1169 (1985). 13. G. A. MORRISAND R. FREEMAN, J. Mugn. Reson. 29,433 (1978).