Solid State Nuclear Magnetic Resonance, 1(1992) 211-215
211
Elsevier Science Publishers B.V., Amsterdam
Time-domain calculation of chemical exchange effects in the NMR spectra of rotating solids M.J. Duer a b
a
and M.H . Levitt
b
Department of Chemistry, University of Cambridge, Lensficld Road, Cambridge CB2 1EII~ UK Physical Chemistry, Arrhenius Laboratory, Unirersity of Stockholm, Stockholm, S-106 91, Sweden
(Received 4 August 1992; accepted 20 Augu st 1992)
Abstract A simple method is described for the calculation of magic-angle-spinning (MAS) spectra of solids in the presence of chemical exchange. The method converges quickly, allowing rapid calculation of the spectra, Calculated spectra are given for molecular motion involving 2 and 6 sites. Keywords : solid-state NMR; chemical exchange; magic-angle spinning; simulation ; spinning sidebands
Introduction Solid-state NMR has long been an important technique for probing molecular dynamics in solids. The NMR spectra depend on the anisotropic interactions such as chemical shifts and quadrupole couplings, and are sensitive to the modulation of these interactions by molecular motion. Four characteristic timescales may be identified: (i) very slow motions (typically with correlation times 1 ms ~ T e ~ 10 s) which may be detected through transfer of spin populations, as in two-dimensional exchange experiments [1-5] or 2H spin' alignment sequences [6,7]; (ii) intermediate timescale motions (with T; I of the order of the anisotropy of the interactions, 10 j1.S ::; Te ~ 1 ms) which induce characteristic changes in the lineshape of the NMR spectra, especially broadening of certain peaks; (iii) rapid motions (Te ::; 10 j1.s) which cause a partial averaging of the anisotropic interactions; and (iv) very rapid
Correspondence to: Dr. MJ. Duer, Department of Chemistry, Un iversity of Cambridge, Lensfield Road, Cambridge CB2
lEW, UK.
motions (T e ~ 100 ns) which may induce relax ation. In this paper, we consider the effect of intermediate timescale motions [regime (ii)] on the magic-angle-spinning (MAS) spectra of isolated spin-l/2 systems, a typical case being 13C in organic solids in the presence of strong I H decoupling. Although a more sensitive probe of molecular motion is undoubtedly the lineshape of the static solid, MAS spectra are often more convenient, since the overlapping peaks from different molecular sites are resolved into separate sets of spinning sidebands. MAS spectra therefore give a readily-available probe of the effect of molecular motion or chemical exchange at different parts of the molecule or in different regions of the sample. Although it is well known that motion causes broadening of the MAS sidebands, there have been very few quantitative studies of the influence of molecular motion on MAS spectra. This may in part be due to the difficulty of simulating MAS spectra in the presence of chemical exchange/molecular motion. The first analysis of the problem was by Maricq and Waugh [8] and later by Suwelack et al. [9]. Two models were considered: (i) rapid motion
0926-2lJ.t0/92/S05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
212
M.I. Duer, AUI. LeL'itt/ Solid State Nllc/. Magn. Rcson. I (I992) 211-215
described by a diffusion equation and treated by second-order relaxation theory, and (ii) infrequent jumps between a small number of molecular sites, treated by the stochastic LiouviIIe equation [10]. The second of these approaches is most similar in philosophy to the majority of simulations performed for static solids, and can indeed accommodate virtually any model of molecular motion, providing a sufficient number of sites is included. However, both studies [8,9] only considered spectra obtained by "synchronous sampling", i.e., collecting data once every rotational period. This corresponds to the sum of all sidebands in the Fourier transformed spectrum, and obscures any details of differential sideband broadening, so losing valuable clues as to the precise nature of the motion. Schmidt et al. [11,12] calculated full sideband patterns in the presence of molecular motion. They handled the problematic time-dependence of the chemical shifts by using a Floquet representation of the stochastic LiouviIIe equation. Although this method is formally elegant, it proves cumbersome in practice. The infinite-dimensional Floquet matrix must be approximated by a finite matrix of a dimension given by the number of exchanging sites, multiplied by an integer many times larger than the number of sidebands in the spectrum. Since the spectra are of most interest in the slow-spinning limit, where there are a large number of spinning sidebands, this dimension is very large if more than a few sites are involved. Diagonalization can therefore be slow. In this communication, a more direct method of calculation in the time-domain is described. It is an obvious extension of the stochastic LiouviIIe formalism described by Suwelack et al. [9] and allows rapid calculation in any regime of motional correlation time and spinning period. Some calculated MAS lineshapes are presented.
Theory
Consider a single crystallite containing many independent one spin-l/2 systems (such as 13C in an organic solid in the presence of proton decoupling) each undergoing Markovian exchange be-
tween N different molecular sites. At any time t, the instantaneous resonance frequency for a spin in site j is given by w}t}: in a rotating solid, w}t} is periodically time-dependent, with period = 2../ w .. where W r is the rotation frequency. The frequencies Wj also depend on the magnitude and orientation of the shielding anisotropy, and the orientation of the crystal with respect to the rotor axis. These dependencies are given by many authors [8,13-17]. According to the stochastic LiouviIIe equation, the time evolution of the complex magnetization vectors M/(t} associated with the N sites may be represented in matrix form,
"r
dM+(t) = {ill(t) dt
+ W}At+(t)
( 1)
where M+(t} is an N-dimensional vector whose components are the transverse magnetizations Mf(t} associated with the individual sites, ll(t} is a N X N-dimensional, diagonal matrix, with elements wj(t), and W is a kinetic matrix describing the exchange between the sites: the element Hjk is given by the jump rate from site k to site j. In the absence of sample rotation, II is timeindependent, and eqn. (1) may be integrated to yield
M+(t) =L(t)At+(O)
(2)
where
L( t) = exp] (ill + W)t}
(3)
This can be solved by diagonalizing (ill + W). However, this solution is not applicable if II is time-dependent. The superoperator (ill(t) + W) is then homogeneous in the sense that its values at two different times do not necessarily commute [8]. Schmidt et al. circumvented this difficulty by employing a Floquet expansion [11,12], exploiting the periodicity of ll(t} [8,17]. A more direct method is simply to integrate eqn. (1) numerically over one rotor period, and thereby derive L(t} directly for all times 0:::;; t :::;; "r' For example, one rotor period may be divided into 11 equal periods, .Jt = "r/lI. Consider the equallyspaced time points, t m = (111 - 1/2) .Jt, where 111 is an integer, 111 = 0, 1, 2, ... ,11. The superopera-
M.I. Ducr, M.lI. Lcrin f Solid State Nucl. Magn. Reson. 1 (J992) 211-215
213
tor matrix L at times m at may be estimated through the iterative scheme L( l1l.6.t) == exp{[ (i a; t",)
+ W ).6.t ]}
XL((111 - 1).6.)
(4)
L(O) = 1
O.85kHz
The incremental propagator exp{[Ci n(t,,,) + JJ').6.t]} is calculated by diagonalizing Ci nUll,) + W) and assuming that nu) is effectively time-independent over each short interval .6.t. The accuracy of this procedure is determined by the number of increments II. In practice, the number of steps is increased until convergence is established. Once L(r) has been estimated over one rotor period, the values at all subsequent times may be derived from
L(t +MTr) =L(Tr)'lfL(t)
O.98kHz
(5)
The NMR spectrum is thus given by
s+(t) = I'L(t) 'M+(O)
(6)
where At+(O) is a vector containing the initial magnetizations at the N sites. If the N sites are populated equally in thermal equilibrium, and non-selective excitation is used, M+(O) is equal to 1.
1.10kHz
For powder spectra, the signal s + (t) must be .averaged over a large ensemble of crystallite orientations. It is then advantageous to use the following symmetry relationship between the evolution matrices L(r) of crystallites with orientations related by a rotation through 'Y around the rotor axis:
L(t; 'Y) =L(t +'Y/wr; O)L('Y/wr; O)-t
(7)
This can be used to reduce greatly the number of matrix diagonalizations required for powder aver-
Fig. 1. Simulations of the 79.9 MHz 13C magic-angle spinning for dimethylsuphone, undergoing two-site exchange at the indicated spinning speeds. The chemical shift anisotropy (I v .l - VIII> used in the calculations was 56 ppm, with 1) = O. The exchange rate is 1.8 kHz ("c = 0.6 ms). Spectral frequencies are relative to the isotropic chemical shift at 0 Hz.
1.40kHz
...- - . - - . -
4
2
0
-I
I
-2-4
frequency I kHz
214
MJ. Duer, AUf. Levitt / Solid State Nucl. Magn. Reson. 1 (1992) 211-215
aging. A similar computation scheme has been used in rotational resonance calculations [18]. It is of interest to compare roughly the number of numerical operations required by the direct time-domain method and the Floquet procedure of Schmidt et al. [11,12]. The numerical intensiveness of both procedures depends upon the product I wJ~nisoTr I= ' x where w~niso is the anisotropic ' J time-dependent part of the shielding frequency. In the slow-spinning limit, (x > 1), x is given approximately by the number of visible sidebands in the spectrum. In the time-domain method, each rotor period must be divided into 11 equal segments .1t, such that I wj"iS°.1t I « 1. The integer 11 should therefore exceed x by several factors. The calculation involves successive diagonalization of 11 different N X N matrices. In the Floquet method on the other hand, the calculation involves a single diagonalization of one matrix of dimension 11'N X 11'N, where 11' should again exceed x by a factor of five or so. Although the convergence criteria for the two methods is probably different, we may anticipate that 11':::::: 11 for comparable accuracy. Since the number of numerical operations required to diagonalize a matrix increases with approximately the third power of the dimension, the direct time-domain method may be expected to be much faster in most cases. Furthermore, the 11 successive diagonalizations may readily be vectorized in parallel processing algorithms. Figure 1 shows simulations of 13e magic-angle spinning of dimethylsulphone undergoing two-site exchange at four different spinning speeds. The calculation assumed two-site hopping for the methyl carbons with an angle of 1080 between the unique axes of the 13e shielding tensors. The fid for each spectrum had 400 calculated points and was the sum of contributions from 4096 crystallite orientations. The fid was multiplied by a decaying exponential with a time constant of 1.5 ms prior to Fourier transformation. The other parameters used in the calculation are given in the figure legend. These time-domain simulations compare well with the experimental results and Floquet simulations given in Fig. 4 of ref. 11. Figure 2 compares some spectra calculated using this method for two- and six-site molecular
2ms
0.2ms
G7J.lS
+20
o
-20
+20
o
v/kHz
vi kHz
(a)
(b)
-20
Fig. 2. Calculated spectra for (a) two-site, and (b) six-site molecular hopp ing processes. See text for det ails. The chemical shift anisotropy was 12 kHz, the asymmetry parameter 1/ was zero, and the spinning speed was 2 kHz in both cases. The molecul ar hopping correlation times, T e , ar e indicated. Spectral frequencies are relat ive to the isotropic chemical shift at 0 Hz.
hopping, for spins with axially symmetric shielding tensors. The two-site process consists of 180 flips with the unique axes of the shielding tensors 0
MJ. Duer, M'H. Lecitt I Solid State Nucl. Magn. Reson. 1 (J992) 211-215
for each nucleus jumping through 120° in each jump. In the six-site process, the nucleus can hop with equal probability to any of six sites that define the vertices of a hexagon, i.e., 60°, 120°, and 180° are all equally probable. The principal z-axis for the shielding tensor in each site is along the radius from the centre of the hexagon to the vertex. This latter model could be a model for hexamethylbenzene in a strong collision limit. Again, 4096 crystallite orientations were used to calculate 400 fid points for each spectrum, and each fid was multiplied by a decaying exponential prior to Fourier transformation. All calculations were performed with our computer program, CARLA [19]. In conclusion, we have demonstrated a simple time-domain numerical procedure for calculating MAS spectra in the presence of multiple-site exchange, which is expected to be computationally more efficient than the Floquet method for large numbers of exchanging sites.
References 1 J. Jeener, n.H. Meier, P. Bachmann and R.R. Ernst, Chem. Phys., 71 (1979) 4546.
215
2 C. Schmidt, B. B1umich and H.W. Spiess, J. Magn. Reson., 79 (1988) 269. 3 S. Wefing and H.W. Spiess, J. Chon. Phys., 89 (1988) 1219. 4 S. Wefing, S. Kaufmann and H.W. Spiess, J. Chem. Phys., 89 (1988) 1234. 5 H.W. Spiess, Chem. Rev., 91 (1991) 1321. 6 M. Lausch and H.W. Spiess, J. Magn. Reson., 54 (1983) 466. 7 H.W. Spiess, J. Chem. Phys., 72 (1980) 6755. 8 M.M. Maricq and J.S. Waugh, J. Chem. Phys., 70 (1979) 3300. 9 D. Suwelack, W.P. Rothwell and J.S. Waugh, J. Chem. Phys., 73 (1980) 2559. 10 R. Kubo, in I. Prigagine and S.A. Rice (Eds.), Stochastic Processes in Chemical Physics, Advances in Chemical Physics, Vol. 15, Wiley, New York, 1969. 11 A. Schmidt, S.O. Smith, D.P. Raleigh, J.E. Roberts, R.G. Griffin and S. Vega, J. Chem. Pltys., 85 (1986) 4248. 12 A. Schmidt and S. Vega, J. Chem. Phys., 87 (1987) 6895. 13 A. Abragarn, Principles of Nuclear Magnetism, Clarendon press, Oxford, 1961. 14 M. Mehring, in P. Diehl, E. Fluck and R. Kosfeld (Eds.), Principles of High. Resolution NMR in Solids, SpringerVerlag, Berlin, 1983. 15 M.S. Greenfield, A.D. Ronemus, R.L. Void, R.R. Void, P.D. Ellis and T.E. Raidy, J. Magn. Reson., 72 (1987) 89. 16 H.W. Spiess, Chem. Pliys.; 6 (1974) 217. 17 J. Herzfeld and A.E. Berger, J. Chem. Phys., 73 (1980) 6021. 18 M.H. Levitt, D.P. Raleigh, F. Creuzet and R.G. Griffin, J. Chem. Phys., 92 (1990) 6347. 19 M.J. Duer, CARLA - a FORTRAN program.
211
Elsevier Science Publishers B.V., Amsterdam
Time-domain calculation of chemical exchange effects in the NMR spectra of rotating solids M.J. Duer a b
a
and M.H . Levitt
b
Department of Chemistry, University of Cambridge, Lensficld Road, Cambridge CB2 1EII~ UK Physical Chemistry, Arrhenius Laboratory, Unirersity of Stockholm, Stockholm, S-106 91, Sweden
(Received 4 August 1992; accepted 20 Augu st 1992)
Abstract A simple method is described for the calculation of magic-angle-spinning (MAS) spectra of solids in the presence of chemical exchange. The method converges quickly, allowing rapid calculation of the spectra, Calculated spectra are given for molecular motion involving 2 and 6 sites. Keywords : solid-state NMR; chemical exchange; magic-angle spinning; simulation ; spinning sidebands
Introduction Solid-state NMR has long been an important technique for probing molecular dynamics in solids. The NMR spectra depend on the anisotropic interactions such as chemical shifts and quadrupole couplings, and are sensitive to the modulation of these interactions by molecular motion. Four characteristic timescales may be identified: (i) very slow motions (typically with correlation times 1 ms ~ T e ~ 10 s) which may be detected through transfer of spin populations, as in two-dimensional exchange experiments [1-5] or 2H spin' alignment sequences [6,7]; (ii) intermediate timescale motions (with T; I of the order of the anisotropy of the interactions, 10 j1.S ::; Te ~ 1 ms) which induce characteristic changes in the lineshape of the NMR spectra, especially broadening of certain peaks; (iii) rapid motions (Te ::; 10 j1.s) which cause a partial averaging of the anisotropic interactions; and (iv) very rapid
Correspondence to: Dr. MJ. Duer, Department of Chemistry, Un iversity of Cambridge, Lensfield Road, Cambridge CB2
lEW, UK.
motions (T e ~ 100 ns) which may induce relax ation. In this paper, we consider the effect of intermediate timescale motions [regime (ii)] on the magic-angle-spinning (MAS) spectra of isolated spin-l/2 systems, a typical case being 13C in organic solids in the presence of strong I H decoupling. Although a more sensitive probe of molecular motion is undoubtedly the lineshape of the static solid, MAS spectra are often more convenient, since the overlapping peaks from different molecular sites are resolved into separate sets of spinning sidebands. MAS spectra therefore give a readily-available probe of the effect of molecular motion or chemical exchange at different parts of the molecule or in different regions of the sample. Although it is well known that motion causes broadening of the MAS sidebands, there have been very few quantitative studies of the influence of molecular motion on MAS spectra. This may in part be due to the difficulty of simulating MAS spectra in the presence of chemical exchange/molecular motion. The first analysis of the problem was by Maricq and Waugh [8] and later by Suwelack et al. [9]. Two models were considered: (i) rapid motion
0926-2lJ.t0/92/S05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
212
M.I. Duer, AUI. LeL'itt/ Solid State Nllc/. Magn. Rcson. I (I992) 211-215
described by a diffusion equation and treated by second-order relaxation theory, and (ii) infrequent jumps between a small number of molecular sites, treated by the stochastic LiouviIIe equation [10]. The second of these approaches is most similar in philosophy to the majority of simulations performed for static solids, and can indeed accommodate virtually any model of molecular motion, providing a sufficient number of sites is included. However, both studies [8,9] only considered spectra obtained by "synchronous sampling", i.e., collecting data once every rotational period. This corresponds to the sum of all sidebands in the Fourier transformed spectrum, and obscures any details of differential sideband broadening, so losing valuable clues as to the precise nature of the motion. Schmidt et al. [11,12] calculated full sideband patterns in the presence of molecular motion. They handled the problematic time-dependence of the chemical shifts by using a Floquet representation of the stochastic LiouviIIe equation. Although this method is formally elegant, it proves cumbersome in practice. The infinite-dimensional Floquet matrix must be approximated by a finite matrix of a dimension given by the number of exchanging sites, multiplied by an integer many times larger than the number of sidebands in the spectrum. Since the spectra are of most interest in the slow-spinning limit, where there are a large number of spinning sidebands, this dimension is very large if more than a few sites are involved. Diagonalization can therefore be slow. In this communication, a more direct method of calculation in the time-domain is described. It is an obvious extension of the stochastic LiouviIIe formalism described by Suwelack et al. [9] and allows rapid calculation in any regime of motional correlation time and spinning period. Some calculated MAS lineshapes are presented.
Theory
Consider a single crystallite containing many independent one spin-l/2 systems (such as 13C in an organic solid in the presence of proton decoupling) each undergoing Markovian exchange be-
tween N different molecular sites. At any time t, the instantaneous resonance frequency for a spin in site j is given by w}t}: in a rotating solid, w}t} is periodically time-dependent, with period = 2../ w .. where W r is the rotation frequency. The frequencies Wj also depend on the magnitude and orientation of the shielding anisotropy, and the orientation of the crystal with respect to the rotor axis. These dependencies are given by many authors [8,13-17]. According to the stochastic LiouviIIe equation, the time evolution of the complex magnetization vectors M/(t} associated with the N sites may be represented in matrix form,
"r
dM+(t) = {ill(t) dt
+ W}At+(t)
( 1)
where M+(t} is an N-dimensional vector whose components are the transverse magnetizations Mf(t} associated with the individual sites, ll(t} is a N X N-dimensional, diagonal matrix, with elements wj(t), and W is a kinetic matrix describing the exchange between the sites: the element Hjk is given by the jump rate from site k to site j. In the absence of sample rotation, II is timeindependent, and eqn. (1) may be integrated to yield
M+(t) =L(t)At+(O)
(2)
where
L( t) = exp] (ill + W)t}
(3)
This can be solved by diagonalizing (ill + W). However, this solution is not applicable if II is time-dependent. The superoperator (ill(t) + W) is then homogeneous in the sense that its values at two different times do not necessarily commute [8]. Schmidt et al. circumvented this difficulty by employing a Floquet expansion [11,12], exploiting the periodicity of ll(t} [8,17]. A more direct method is simply to integrate eqn. (1) numerically over one rotor period, and thereby derive L(t} directly for all times 0:::;; t :::;; "r' For example, one rotor period may be divided into 11 equal periods, .Jt = "r/lI. Consider the equallyspaced time points, t m = (111 - 1/2) .Jt, where 111 is an integer, 111 = 0, 1, 2, ... ,11. The superopera-
M.I. Ducr, M.lI. Lcrin f Solid State Nucl. Magn. Reson. 1 (J992) 211-215
213
tor matrix L at times m at may be estimated through the iterative scheme L( l1l.6.t) == exp{[ (i a; t",)
+ W ).6.t ]}
XL((111 - 1).6.)
(4)
L(O) = 1
O.85kHz
The incremental propagator exp{[Ci n(t,,,) + JJ').6.t]} is calculated by diagonalizing Ci nUll,) + W) and assuming that nu) is effectively time-independent over each short interval .6.t. The accuracy of this procedure is determined by the number of increments II. In practice, the number of steps is increased until convergence is established. Once L(r) has been estimated over one rotor period, the values at all subsequent times may be derived from
L(t +MTr) =L(Tr)'lfL(t)
O.98kHz
(5)
The NMR spectrum is thus given by
s+(t) = I'L(t) 'M+(O)
(6)
where At+(O) is a vector containing the initial magnetizations at the N sites. If the N sites are populated equally in thermal equilibrium, and non-selective excitation is used, M+(O) is equal to 1.
1.10kHz
For powder spectra, the signal s + (t) must be .averaged over a large ensemble of crystallite orientations. It is then advantageous to use the following symmetry relationship between the evolution matrices L(r) of crystallites with orientations related by a rotation through 'Y around the rotor axis:
L(t; 'Y) =L(t +'Y/wr; O)L('Y/wr; O)-t
(7)
This can be used to reduce greatly the number of matrix diagonalizations required for powder aver-
Fig. 1. Simulations of the 79.9 MHz 13C magic-angle spinning for dimethylsuphone, undergoing two-site exchange at the indicated spinning speeds. The chemical shift anisotropy (I v .l - VIII> used in the calculations was 56 ppm, with 1) = O. The exchange rate is 1.8 kHz ("c = 0.6 ms). Spectral frequencies are relative to the isotropic chemical shift at 0 Hz.
1.40kHz
...- - . - - . -
4
2
0
-I
I
-2-4
frequency I kHz
214
MJ. Duer, AUf. Levitt / Solid State Nucl. Magn. Reson. 1 (1992) 211-215
aging. A similar computation scheme has been used in rotational resonance calculations [18]. It is of interest to compare roughly the number of numerical operations required by the direct time-domain method and the Floquet procedure of Schmidt et al. [11,12]. The numerical intensiveness of both procedures depends upon the product I wJ~nisoTr I= ' x where w~niso is the anisotropic ' J time-dependent part of the shielding frequency. In the slow-spinning limit, (x > 1), x is given approximately by the number of visible sidebands in the spectrum. In the time-domain method, each rotor period must be divided into 11 equal segments .1t, such that I wj"iS°.1t I « 1. The integer 11 should therefore exceed x by several factors. The calculation involves successive diagonalization of 11 different N X N matrices. In the Floquet method on the other hand, the calculation involves a single diagonalization of one matrix of dimension 11'N X 11'N, where 11' should again exceed x by a factor of five or so. Although the convergence criteria for the two methods is probably different, we may anticipate that 11':::::: 11 for comparable accuracy. Since the number of numerical operations required to diagonalize a matrix increases with approximately the third power of the dimension, the direct time-domain method may be expected to be much faster in most cases. Furthermore, the 11 successive diagonalizations may readily be vectorized in parallel processing algorithms. Figure 1 shows simulations of 13e magic-angle spinning of dimethylsulphone undergoing two-site exchange at four different spinning speeds. The calculation assumed two-site hopping for the methyl carbons with an angle of 1080 between the unique axes of the 13e shielding tensors. The fid for each spectrum had 400 calculated points and was the sum of contributions from 4096 crystallite orientations. The fid was multiplied by a decaying exponential with a time constant of 1.5 ms prior to Fourier transformation. The other parameters used in the calculation are given in the figure legend. These time-domain simulations compare well with the experimental results and Floquet simulations given in Fig. 4 of ref. 11. Figure 2 compares some spectra calculated using this method for two- and six-site molecular
2ms
0.2ms
G7J.lS
+20
o
-20
+20
o
v/kHz
vi kHz
(a)
(b)
-20
Fig. 2. Calculated spectra for (a) two-site, and (b) six-site molecular hopp ing processes. See text for det ails. The chemical shift anisotropy was 12 kHz, the asymmetry parameter 1/ was zero, and the spinning speed was 2 kHz in both cases. The molecul ar hopping correlation times, T e , ar e indicated. Spectral frequencies are relat ive to the isotropic chemical shift at 0 Hz.
hopping, for spins with axially symmetric shielding tensors. The two-site process consists of 180 flips with the unique axes of the shielding tensors 0
MJ. Duer, M'H. Lecitt I Solid State Nucl. Magn. Reson. 1 (J992) 211-215
for each nucleus jumping through 120° in each jump. In the six-site process, the nucleus can hop with equal probability to any of six sites that define the vertices of a hexagon, i.e., 60°, 120°, and 180° are all equally probable. The principal z-axis for the shielding tensor in each site is along the radius from the centre of the hexagon to the vertex. This latter model could be a model for hexamethylbenzene in a strong collision limit. Again, 4096 crystallite orientations were used to calculate 400 fid points for each spectrum, and each fid was multiplied by a decaying exponential prior to Fourier transformation. All calculations were performed with our computer program, CARLA [19]. In conclusion, we have demonstrated a simple time-domain numerical procedure for calculating MAS spectra in the presence of multiple-site exchange, which is expected to be computationally more efficient than the Floquet method for large numbers of exchanging sites.
References 1 J. Jeener, n.H. Meier, P. Bachmann and R.R. Ernst, Chem. Phys., 71 (1979) 4546.
215
2 C. Schmidt, B. B1umich and H.W. Spiess, J. Magn. Reson., 79 (1988) 269. 3 S. Wefing and H.W. Spiess, J. Chon. Phys., 89 (1988) 1219. 4 S. Wefing, S. Kaufmann and H.W. Spiess, J. Chem. Phys., 89 (1988) 1234. 5 H.W. Spiess, Chem. Rev., 91 (1991) 1321. 6 M. Lausch and H.W. Spiess, J. Magn. Reson., 54 (1983) 466. 7 H.W. Spiess, J. Chem. Phys., 72 (1980) 6755. 8 M.M. Maricq and J.S. Waugh, J. Chem. Phys., 70 (1979) 3300. 9 D. Suwelack, W.P. Rothwell and J.S. Waugh, J. Chem. Phys., 73 (1980) 2559. 10 R. Kubo, in I. Prigagine and S.A. Rice (Eds.), Stochastic Processes in Chemical Physics, Advances in Chemical Physics, Vol. 15, Wiley, New York, 1969. 11 A. Schmidt, S.O. Smith, D.P. Raleigh, J.E. Roberts, R.G. Griffin and S. Vega, J. Chem. Pltys., 85 (1986) 4248. 12 A. Schmidt and S. Vega, J. Chem. Phys., 87 (1987) 6895. 13 A. Abragarn, Principles of Nuclear Magnetism, Clarendon press, Oxford, 1961. 14 M. Mehring, in P. Diehl, E. Fluck and R. Kosfeld (Eds.), Principles of High. Resolution NMR in Solids, SpringerVerlag, Berlin, 1983. 15 M.S. Greenfield, A.D. Ronemus, R.L. Void, R.R. Void, P.D. Ellis and T.E. Raidy, J. Magn. Reson., 72 (1987) 89. 16 H.W. Spiess, Chem. Pliys.; 6 (1974) 217. 17 J. Herzfeld and A.E. Berger, J. Chem. Phys., 73 (1980) 6021. 18 M.H. Levitt, D.P. Raleigh, F. Creuzet and R.G. Griffin, J. Chem. Phys., 92 (1990) 6347. 19 M.J. Duer, CARLA - a FORTRAN program.
