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Optimized Multiple Quantum MAS Lineshape
Simulations in Solid State NMR
William J. Brouwer a,∗ Michael C. Davis a Karl T. Mueller a
aDepartment of Chemistry, Pennsylvania State University
Abstract
The majority of nuclei available for study in solid state Nuclear Magnetic Resonancehave half-integer spin I > 1/2, with corresponding electric quadrupole moment. Assuch, they may couple with a surrounding electric field gradient. This effect producesanisotropic line broadening in spectra for distinct chemical species in polycrystallinesolids. In Multiple Quantum Magic Angle Spinning (MQMAS) experiments, a sec-ond frequency dimension is created, devoid of quadrupolar anisotropy. As a result,the bary centers of peaks in the high resolution dimension are functions of isotropicquadrupole and chemical shifts alone. However, for complex materials, these param-eters take on a stochastic nature due in turn to structural and chemical disorder.Lineshapes may still overlap in the isotropic dimension, complicating the task ofassignment and interpretation. A distributed computational approach is presentedhere which permits optimal simulation of the MQMAS spectrum, generated by ran-dom variates from model distributions of isotropic chemical and quadrupole shifts.In this manner, local chemical environments for disordered materials may be char-acterized and via a re-sampling approach, error estimates for parameters produced.
Key words: Nuclear Magnetic Resonance, Multiple Quantum Magic AngleSpinning, OpenMP, Sobol sequence, quasi-random numbers, simulated annealing,distribution functions, quadrupole interaction.
1 Introduction1
Since the discovery of Nuclear Magnetic Resonance (NMR), there has been2
great interest in the study of quadrupolar nuclei. These nuclei have an electric3
quadrupole moment Q which couples with a non-zero electric field gradient [8].4
∗ Corresponding AuthorEmail address: [email protected] (William J. Brouwer).
Preprint submitted to Journal of Computational Chemistry 16 July 2008
Depending on the magnitude of these quantities, the quadrupole interaction is5
quite often the most significant perturbation to the Zeeman energy levels. At-6
tention here is restricted to first and second order quadrupole effects, each of7
which is proportional to a second rank tensor term P2(θ) [41,36]. In addition,8
the second order quadrupole perturbation is proportional to a fourth rank term9
P4(θ)1 . These terms depend explicitly on the angle θ between crystallite orien-10
tations and the static, applied magnetic field of NMR. As a result, anisotropic11
frequency dependence is introduced, promoting overlap between lineshapes12
arising from distinct chemical sites in powdered solids. Additionally, an ap-13
preciable second order isotropic shift occurs; the bary center of quadrupole14
lineshapes is subsequently changed from the chemically shifted value. The15
characteristic features of quadrupole spectra provide valuable local bonding16
information and hence extensive work has been devoted to both resolving17
individual chemical sites, as well as lineshape simulation. With regard to res-18
olution, the issue has been addressed over the course of time by a number of19
experimental approaches. Early in the development of solid state NMR, Magic20
Angle Spinning (MAS) [18] was proposed, which reduces or eliminates sec-21
ond rank interaction terms and therefore broadening associated with the first22
order quadrupole interaction. However, if the magnitude of the quadrupole23
interaction is significant, spinning sideband manifolds arising from satellite24
frequency transitions may still obscure spectra in one dimension [37]. Double25
Rotation (DOR) [5] is an extension of the Magic Angle Spinning technique,26
where by virtue of sample spinning at two angles, on the time average first and27
second order quadrupole anisotropy are alleviated. In order to enhance reso-28
lution beyond that available in one dimension, a natural extension was made29
to two dimensional experiments [6] where quadrupolar nutation frequencies30
(and thus underlying parameters) could be extracted via simulation. Dynamic31
Angle Spinning (DAS) [33,32] is successful in eliminating the effects of both32
second and fourth rank tensor terms, and thus also second order quadrupole33
broadening. More recently, Multiple Quantum Magic Angle Spinning (MQ-34
MAS) [2,47] and Satellite Transition Magic Angle Spinning (STMAS) [25,24]35
have become popular owing to mechanical simplicity. These procedures in-36
volve collecting data as a function of two independent time intervals in the37
pulse sequence [51] under Magic Angle Spinning conditions. Within these ex-38
periments, directly observable single quantum coherence frequency transitions39
are correlated with multiple quantum transitions [62], which evolve between40
pulses and are selected via an appropriate phase cycle, figure 1.41
During data processing, a so-called ‘shearing transformation’ is applied af-42
ter the Fourier transform in the direct dimension. This takes place before a43
Fourier transform in the second dimension, in order to create a high resolu-44
tion spectrum in the indirect dimension, devoid of anisotropy. Alternatively,45
a high resolution axis may be created during the experimental acquisition us-46
1 Relevant expressions are listed in Appendix A
2
Fig. 1. (a) Schematic of a 1D NMR experiment; p1 is the pulse duration, and thefree induction decay is recorded during the acquisition time period t2. Coherenceswhich have a change in magnetic quantum number ∆m equal to ±1 are detected,as supported by the electric dipole selection rule. (b) Schematic of a 2D NMRexperiment; acquisition takes place during t2 and time period t1 is varied to create asecond, indirect dimension. Multiple quantum ∆m 6= ±1 as well as single quantumcoherences evolve during this period. Ultimately, a particular coherence transferpathway (CTP) is selected in an MQMAS experiment via a phase table; phases ofpulses are varied in a prescribed manner to ensure the desired CTP amplitude is amaximum.
ing the split-t1 method [14]. Figure 2 displays the 27Al MQMAS spectrum47
of the large-pore aluminophosphate VPI-5 as a contour plot, alongside the48
MAS (F2) frequency dimension projection. Clearly resolved are two distinct49
chemical sites in the tetrahedral coordination region of the aluminum spectral50
window.51
From the bary centre of peaks along the high resolution axis, isotropic shifts are52
deduced which are a function of both isotropic chemical and quadrupole shifts.53
In turn, the isotropic quadrupole shift is a function of both the quadrupole54
coupling constant Cq and asymmetry parameter η. The importance of these55
quantities lies in the fact that they are directly related to the electric field56
gradient tensor, and thus the details of the local bonding environment. In57
order to unequivocally determine both Cq and η, simulation of experimental58
spectra is necessary. Work in this area was initially devoted to static line-59
shapes [11,52] and has since been extended to spinning solids. Several exam-60
ples of the latter include extraction of quadrupolar parameters using spinning61
sideband manifolds [30,1] and the use of the stochastic Liouville von-Neumann62
equation to incorporate molecular motion [27]. In the last decade, simulation63
3
-1 -0.8 -0.6 -0.4
-1.6
-1.2
-0.8
-0.4
-1 -0.8 -0.6 -0.4
F2 (kHz) F2 (kHz)
F1
(kH
z)
#2
#1
(a) (b)
Fig. 2. (a)27Al MAS spectrum of tetrahedral region within VPI-5 (b) Contour plotof MQMAS spectrum of same compound; clearly resolved are two chemical siteswith distinctive lineshapes and hence local environments.
tools including SIMPSON [35] and GAMMA [54] have been created. Cal-64
culations performed by these packages can take into account radio frequency65
pulse powers and durations, time delays, expected NMR parameters and other66
variables to provide a system response. There also exists lineshape modeling67
tools such as DMFIT [15] which provides simulation capabilities for a large68
variety of experiments and interactions. In the case of disordered chemical69
environments [23,9,57,56], calculations of powdered lineshapes for MQMAS70
becomes a formidable task. This is due to the fact that parameters relevant71
to simulation take on a distributed nature [49,22]. Whether using a high level72
of theory, inversion methods [67], or the use of table-lookup in calculating73
powder patterns, computational demands become excessive. The focus of this74
paper is devoted to the optimized simulation of multiple quantum magic an-75
gle spinning spectra, in the presence of low to significant disorder. This is76
accomplished using quasi-random numbers sampled from model distributions77
of isotropic chemical shift and quadrupole coupling constant. Simulated an-78
nealing is used to optimize the non-convex cost function. In distinction to79
spectrum-inversion approaches [19], the method proposed here also gives in-80
sight into the asymmetry parameter and is highly amenable to distributed81
computing [26].82
2 Theory83
2.1 Quadrupole Interaction84
The density operator is a popular means of describing experimental NMR. The85
matrix representation of this operator in a particular basis is diagonalized86
and the time evolution propagated using an average Hamiltonian [61]. The87
important assumption in this approach is the synchronization of the Magic88
4
Angle Spinning (MAS) speed with evolution periods in the pulse sequence.89
A more general theory for time-dependent Hamiltonians is given via Floquet90
theory [7,59,55] and will not be treated here. The quadrupole interaction in91
terms of irreducible tensor operator notation [39,40,53] may be expressed as:92
HQ = NQ
2∑
u=−2
(−1)uV2,−2K(2,u)
93
where NQ =eQ
2I(2I − 1)~(1)94
Tensor V contains electric field gradient terms, and tensor K spin angular95
momentum operators; expressions for these are given in appendix A. It is as-96
sumed for the remainder of this work that the quadrupole interaction may be97
treated as a weak perturbation on the Zeeman levels. One begins by consid-98
ering the time evolution of the operator in the interaction representation (ie.,99
the rotating frame):100
HQ(t) = exp(iHzt)HQ exp(−iHzt)101
= NQ
2∑
u=−2
(−1)uV2,−uK(2,u) exp(−iuωot), (2)102
where ω0 is the Larmor frequency. The Magnus expansion [38] is employed103
in order to find the average value of the Hamiltonian, assuming that the104
Hamiltonian may be considered piece-wise constant during short time inter-105
vals. Ignoring highly oscillating and non-secular terms (retaining only those106
that commute with the Zemman Hamiltonian), one finds to first and second107
order for the quadrupole interaction:108
H(1)Q =
1
tL
tL∫
0
HQ(t)dt = NQ1√6[3I2
z − I(I + 1)]V2,0 (3)109
H(2)Q =
−i
2tL
tL∫
0
dt
t∫
0
dt′[HQ(t), HQ(t′)] =110
−N2Q
ω0
(
1
2V2,−1V2,1[4I(I + 1) − 8I2
z − 1]111
+1
2V2,−2V2,2[2I(I + 1) − 2I2
z − 1])
Iz (4)112
5
For the purposes of determining frequency shifts, these expressions are sim-113
plified using higher rank tensors e.g.,114
H(2)Q =
−N2Q
ω0
(
1
70
√7W4,0[17L(3,0) − 6L(1,0)]115
+1√35
W2,0[3L(3,0) + L(1,0)] − 1
10
√2W0,0[3L
(3,0) − 4L(1,0)
)
(5)116
The tensor W contains components of the electric field gradient, whilst tensor117
L is a function of spin operators. Explicitly, the tensor W is related to the118
tensor V via the Clebsch-Gordon coefficients,119
Wj,M =∑
m1,m2
〈j1j2m1m2|JM〉Vj1,m1Vj2,m2 (6)120
and the L are given by:121
L(1,0) =1
5
√10[I(I + 1) − 3
4]Iz (7)122
L(3,0) =1
5
√10[3I(I + 1) − 5I2
z − 1]Iz (8)123
The latter are not normalized; in terms of normalized tensor operators P (1,0)124
and P (3,0) one may write:125
L(1,0) =
√
2
5[I(I + 1) − 3
4]P (1,0) (9)126
L(3,0) = −2P (3,0) (10)127
Therefore128
H(2)Q =
−N2Q
ω0
−17
5√
7W4,0P
(3,0) − 3
35
√
16
5[I(I + 1) − 3
4]129
×W4,0P(1,0) − 6√
35W2,0P
(3,0) +
√14
35[I(I + 1) − 3
4]130
×W2,0P(1,0) +
3√
2
5W0,0P
(3,0) +4
5√
5[I(I + 1) − 3
4]W0,0P
(1,0)
)
(11)131
If one denotes distinct energy levels by r, c, then the shift to the standard132
Zeeman frequency may be determined as,133
ωr,c = 〈r|(H(1)Q + H
(2)Q |r〉 − 〈c|(H(1)
Q + H(2)Q |c〉134
6
which is the sum of first and second order contributions:135
ω(1)r,c + ω(2)
r,c (12)136
To first order, the shift is:137
ω(1)r,c = NQ
√
3
2(r2 − c2)V2,0, (13)138
which for a symmetric transition (r = −c) is zero. To second order:139
ω(2)r,c =140
−N2Q
ω0(r − c)
1
70
√
35
2W4,0A
(4) +1
28
√14W2,0A
(2) − 1√5W0,0A
(0)
(14)141
where expressions for constants A, which depend upon I, r, c, are given in142
appendix A.143
2.1.1 Static Crystal144
It is convention to express frequencies in terms of the principal axis system,145
the frame of reference where the tensor of interest is diagonal. Referring to146
figure 3, the transformation for V2,0 from the principal axis system is rather147
simple in the case of a static single crystal:148
V2,0 =2∑
u=−2
V PAS2,u D
(2)u,0(α, β, φ) =149
=1
2
√6eq[
1
2(3 cos2 β − 1) +
1
2η sin2 β cos 2α], (15)150
where D is the Wigner rotation matrix. One may then write the 1st order151
quadrupole interaction as:152
H(1)Q =
1
4(3I2
z − I(I + 1))ΩQ[3 cos2 β − 1 + η sin2 β cos 2α] (16)153
with154
ΩQ = eqNQ =e2qQ
2I(2I − 1)~=
Cq
2I(2I − 1)155
Therefore, apart from orientation angles, quadrupole frequencies are com-156
pletely specified in terms of parameters η and Cq, which are sufficient to de-157
scribe the electric field gradient tensor, since in the PAS the tensor is diagonal158
7
(a) (b)
Fig. 3. Euler angles for: (a) the Principal Axis System (PAS) in the crystal frame(b) VAS/rotor relationship to static magnetic field B0.
and satifies Laplace’s equation. Tensor components W4,0, W2,0 and W0,0 of the159
electric field gradient transform as;160
W2x,0 =x∑
u=−x
WPAS2x,2u D
(2x)2u,0(α, β, φ) (17)161
Hence for the second order quadrupole frequency:162
w(2)r,c = −r − c
2ω0Ω2
Q
0,1,2∑
k
A(2k)(I, r, c)k∑
u=−k
B2k,2u(η)D(2k)2u,0(α, β, φ) (18)163
with164
B0,0(η) = −15(η2 + 3), B2,0(η) = 1
14(η2 − 3),
B2,±2(η) = 114
η√
6, B4,0(η) = 1140
(η2 + 18),
B4,±2(η) = 3140
η√
(10), B4,±4(η) = 14√
70η2
165
2.1.2 Magic Angle Spinning166
Referring again to figure 3, in performing sample or variable angle spinning167
(VAS) at angle θr to the static field, with angular velocity ωr, an additional168
step in transforming between PAS and the rotor frame is necessary:169
V VAS2,u =
2∑
j=−2
V PAS2,j D
(2)j,u(α, β, φ), (19)170
V2,0 =2∑
u=−2
V VAS2,u D
(2)u,0(ωrt, θr, φr) (20)171
8
For this case, the first order Quadrupole Interaction:172
ω(1)VASr,c =173
NQ1
2
√6(r2 − c2)
2∑
u=−2
D(2)u,0(ωrt, θr, φr)
2∑
j=−2
V PAS2,j D
(2)j,u(α, β, φ) (21)174
Similarly, in determining the second order shift, one finds:175
WVAS2x,u =
x∑
j=−x
WPAS2x,2j D
(2x)2j,u(α, β, φ) (22)176
W2x,0 =2∑
u=−2x
xWVAS2x,u D
(2x)u,0 (ωrt, θr, φr) (23)177
and hence:178
ω(2)VASr,c = −r − c
2ω0Ω2
Q
2∑
x=0
A(2x)(I, r, c)×179
2x∑
u=−2x
D(2x)u,0 (ωrt, θr, φr)
x∑
j=−x
B2x,2j(η)D(2x)2j,u(α, β, φ) (24)180
Now D(2x)u,0 is proportional to exp(−iuωrt) and thus spinning sidebands are181
observed in the frequency domain. In the high spinning speed approximation,182
one assumes that the only non-zero term is for u = 0, thus:183
ω(1)VAS′
r,c = νQ1
2
√6(r2 − c2)d
(2)0,0(θr)
2∑
j=−2
V PAS2,j D
(2)j,0 (α, β, φ) (25)184
ω(2)VAS′
r,c = −r − c
2ω0Ω2
Q
2∑
x=0
A(2x)(I, r, c)d(2x)0,0 (θr)×185
x∑
j=−x
B2x,2j(η)D(2x)2j,0 (α, β, φ) (26)186
= −r − c
2ω0Ω2
QA(0)(I, r, c)B0,0(η) + A(2)(I, r, c)d(2)0,0(θr)[B2,0(η)d
(2)0,0(β)187
+2B2,2(η)d(2)2,0(β) cos 2α] + A(4)(I, r, c)d
(4)0,0(θr)[B4,0(η)d
(4)0,0(β)188
+2B4,2(η)d(4)2,0(β) cos 2α + 2B4,4(η)d
(4)4,0(β) cos 4α] (27)189
9
Further, under the Magic Angle Spinning condition of P2(cos θr) = 0, there190
remains for the first and second order quadrupole interactions:191
ω(1)fast MASr,c = 0, (28)192
ω(2)fast MASr,c =193
ωisor,c − r − c
2ω0Ω2
QA(0)(I, r, c)B0,0(η) + A(4)(I, r, c)[B4,0(η)d(4)0,0(β)+194
2B4,2(η)d(4)2,0(β) cos 2α + 2B4,4(η)d
(4)4,0(β) cos 4α]P4(cos θm). (29)195
This expression 2 will be used to describe frequencies in the direct dimension196
for which r − c = −1 as well as the indirect dimension. The MQMAS experi-197
ments performed in this work use the triple quantum transition, ie., r− c = 3.198
2.2 Lineshape Simulation and Optimization199
Since the introduction of MQMAS experiments, there have been significant200
improvements in excitation efficiency and coherence transfer using Double201
Frequency Sweep (DFS) [4,3] and Fast Amplitude Modulation [47,48]. The202
discovery of the rotatory resonance phenomena has been exploited particu-203
larly for low gamma nuclei [65,60] and there have been improvements made204
in sensitivity based around the inclusion of signal intensity from additional205
coherence transfer pathways [64,43]. The Z-filter [28] method ensures that206
amplitudes for echo and anti-echo pathways are co-added with equal inten-207
sity under States [17] acquisition, providing after phase correction a purely208
absorptive 2-D spectra. Therefore, using the second order perturbation theory209
expression for multiple quantum transition frequencies given in eq. 29, one210
may model the general 2D correlation spectrum between the r ↔ c quan-211
tum transition (indirect frequency dimension) and central transition (direct212
frequency dimension r − c = −1) as [34,13,8]:213
F (f1, f2) = (1 − ǫ)λ1
λ21 + (f1 − f1m)2
λ2
λ22 + (f2 − f2m)2
214
+ǫ1
2πλ1λ2e
(
−(f1−f1m)2
2λ21
+−(f2−f2m)2
2λ22
)
(30)215
Where 2πf1m = ω(2)r,c and 2πf2m = ω
(2)−1 are the indirect and direct second216
order quadrupole frequency expressions. This model is pertinent to the dipole217
2 The isotropic chemical shift δcsiso is implicit within equation 29; ωiso
r,c = (r−c)δcsisoω0
10
Table 1Powder averaging schemes
Method α β wj
Planar Grid 2πkNα
π(j+0.5)Nβ
sin(βj)
Spherical Grid 2πkNα
arccos(
1 − 2j+1Nβ
)
1
Planar ZCW 2π(jMz mod Nz)Nz
(j+0.5)πNz
sin(βj)
Spherical ZCW 2π(jMz mod Nz)Nz
arccos(
1 − 2j+1Nz
)
1
broadened lineshape (broadening factors λ1, λ2) of a single crystallite orien-218
tation with unique isotropic chemical shift δcsiso, asymmetry parameter η and219
quadrupole coupling constant Cq. As such, it provides a suitable kernel for a220
more general lineshape intensity function, weighted by crystallite angle distri-221
bution G(α, β) and probability density P (δisocs , Cq, η):222
I(f1, f2) =223
L∑
j
Aj
∫
δisocs ,Cq,η
∫
α,β
Pj(δisocs , Cq, η)G(α, β)Fj(f1, f2)d[α, β]d[δiso
cs , Cq, η] (31)224
where L is the number of chemical sites and Aj the individual site amplitude.225
There are two aspects to a numerical evaluation of this expression, including226
powder averaging over the crystallite orientations specified by angles α and227
β. In addition, contributions to the overall spectrum from random variates228
Cq, δcsiso, and η are weighted by a multi-variate probability distribution function.229
Powder averaging in magnetic resonance is an example of a problem in broader230
quantum mechanics, evaluating integrals over the unit sphere [21,20]. There231
exists several reviews in the literature with regard to powder averaging in232
magnetic resonance [50,46]. It is assumed that the equally probable crystallite233
orientations within a powder have been equally irradiated. Table one lists234
various schemes for performing powder averaging under these assumptions,235
where the integral over angles is replaced by a sum:236
F (f1, f2) =
∑Ni wiFi(α, β)∑N
i wi
(32)237
with various choices for weights wi and angles α, β.238
239
In the Planar grid or Alderman-Solum-Grant scheme [16], as well as for the240
Spherical Grid method, α and β are varied independently with Nα and Nβ241
steps and k = 0...Nα − 1, j = 0...Nβ − 1. For the Zaremba-Conroy-Wolfsberg242
method [66,12,63], Nz and Mz are chosen to satisfy Mz = F (m) and Nz =243
F (m + 2), where F (m) the mth Fibonacci number. The latter method has244
11
been employed for the simulations in this work, and is anticipated to be op-245
timal for fast MAS and few crystallite contributions [46]. To incorporate the246
multivariate distribution in isotropic chemical shift and quadrupole param-247
eters, variables are sampled from a model distribution and a Monte Carlo248
simulation performed. In general, statistical distributions may be symmetric249
or asymmetric. The nature of the model distribution used in the simulation250
is directly related to the underlying chemical and/or structural disorder. Tra-251
ditional random number generators which create variates according to dis-252
tributions such as these are usually one of two types. They may be of the253
acceptance/rejection type, or rely on transformations of the uniform distri-254
bution, eg., the Box-Muller method for normal-distributed variables [45]. The255
latter was used here for ease of adaptation to a parallel programming envi-256
ronment. By creating variates after this fashion and converting the integral of257
eq. 31 to a summation, the integral is solved via a Monte Carlo approach. By258
the law of large numbers, Monte Carlo approximations converge to the true259
value in the limit as the samples N approach infinity. In reality, convergence is260
slow, and the error in using pseudo random numbers is O(N−1/2). This situa-261
tion is improved via using quasi-random numbers such as the Sobol sequence,262
which have an error O((log N)kN−1) for k dimensions [42]. For the purposes263
of this work, attention is restricted to the multi-variate normal distribution264
with density:265
p(x1, ...xn) =1
(2π)n|Σ|1/2exp
(
−1
2(x − µ)T Σ−1(x − µ)
)
(33)266
where random variates x1, ...xn may represent quadrupole coupling constant267
Cq, asymmetry parameter η and isotropic chemical shift δcsiso parameters (ie.,268
n = 3). Other symbols have their usual meaning; Σ is the covariance matrix269
with determinant |Σ| and µ is the vector of mean values. There is a well estab-270
lished method for generating normally distributed variates which is employed271
here, specifically:272
(1) The Cholesky decomposition of AAT = Σ is calculated, providing matrix273
A.274
(2) A vector Z of normal random variates are created via the Box-Muller275
transform276
(3) Multivariate parameters X with the desired properties are generated from277
X = µ + AZ278
For the remainder of this work, attention is restricted to bi-variate distribu-279
tions in Cq and δcsiso, parametrized by µx, σx, µy, σy, ρ, using single values of280
asymmetry parameter η per chemical site. Using the theory outlined thus far,281
an experimental spectrum may be simulated and attempts made to optimize282
the simulation parameters. Figure 4 is a plot of the cost function obtained by283
varying only chemical shifts in a fit to the two site, VPI-5 tetrahedral region.284
12
Fig. 4. Cost function, sum of squared difference between simulated and experimentalVPI-5 MQMAS spectrum, as a function of the two isotropic chemical shifts
The global minima is toward the center of the plot, within a larger area285
containing local minima. Simulated annealing [31] is a stochastic method for286
global optimization suited to non-convex cost functions. The method is analo-287
gous to the metallurgical process of annealing. The application to the current288
problem ensures that the iterative procedure avoids being trapped within local289
minima. The overall algorithm applied here is as follows:290
(1) Least squares cost function generation, the trace of the Grammian: Ei =Trace(A−291
B) × (A − B)T where A − B is a matrix of residuals, the difference be-292
tween simulated A and experimental absorption spectra B. If this is the293
initial step, a generalized temperature is defined T ≈ Ei294
(2) Each unconstrained parameter x is changed by a random amount ±∆x,295
sampled from the uniform distribution [0, 1). The corresponding energy296
Ef is calculated as before.297
(3) If Ef < Ei, the change is accepted, else,298
(4) Parameter changes are accepted or rejected in the traditional Metropo-299
lis [44] scheme, using the probabilistic factor: e−(Ef−Ei)/T300
(5) The process is repeated and the temperature lowered according to some301
schedule, until such time as convergence is reached.302
13
In order to give confidence intervals for parameters303
φ = λk1, λ
k2, ǫ
k, µkx, σ
kx, µ
ky, σ
ky , ρ
k, ηk,Ak; k = 1, .., L304
optimized in the simulation, strictly speaking the measurement or MQMAS305
experiment in conjunction with simulations ought to be repeated and statistics306
created from fitted data. However, owing to the considerable time multiple ex-307
periments and simulations requires, a more suitable approach to error analysis308
is found in statistical re-sampling, such as jackknifing or bootstrapping [29].309
In the original jackknife approach, φ−i is defined as the least squared estimate310
of parameter φ when the ith data point of n total is removed from the set.311
Pseudo values are created,312
Pi = nφ − (n − 1)φ−i (34)313
with average P and variance matrix VP :314
P = φJ = n−1n∑
i=1
Pi (35)315
nVP =1
n − 1
n∑
i=1
(Pi − P )(Pi − P )T (36)316
In the present application, this method implies n+1 non-linear optimizations317
which is still far too time consuming. Fox et al [58] propose a solution in the318
form of an approximate jackknife, which requires instead a single non-linear319
optimization, via a Taylor expansion of the least squares estimate equation320
for φi, assuming it is a stationary point for the sum of the residuals. In this321
method, an estimate of the variance matrix is given by:322
V = (ZT Z)−1n∑
j=1
zjzTj r2
j (ZT Z)−1 (37)323
where:324
zi = ∇f(xi, φ) =
∂
∂φ1
f(xi, φ)...∂
∂φl
f(xi, φ)
T
φ=φ
(38)325
ZT = (z1, ..., zn) (39)326
and ri is the vector of residuals. The model as presented here consists of ten327
free parameters per chemical site (ie., l=10), so in the case of N chemical328
sites, this corresponds to the creation of a 10N × 10N variance matrix from329
the quantities listed here. These are evaluated at best-fit parameters φ, using330
the partial derivatives as listed in Appendix B.331
14
3 Numerical Results332
Fig. 5. (a) Annealing schedule versus iterations; temperature is rapidly annealedand reannealed (b) Variation of energy and best energy value versus iteration. Inaddition to the algorithm outlined, a separate heuristic is applied whereby every psteps, the parameter values are reset to their best values to date corresponding to thestored best energy value (c) Simulated MQMAS spectrum. Comparing with fig. 2b,while the general shape and peak positions have been re-produced, small differencesarising from distributed values are apparent. The number of powder increments usedwas 1597 (d) Comparison of simulated and experimental F2 frequency projection
The aforementioned theory was implemented in C, using a number of func-333
tions from the GNU Scientific Library (GSL), as well as the math and stan-334
dard libraries. A single application was written which performs calculations335
of frequency equation 29 as a function of each powder angle α, β according336
to the ZCW scheme. For each frequency dimension Sobol sequences are gen-337
erated and used to create bi-variate distributions of isotropic chemical shift338
and quadrupole coupling constant according to the given algorithm. Finally,339
summation over powder angles and variates are performed using the kernel of340
eq. 30. The results of initial simulations pointed to single values of asymme-341
try parameter being sufficient for distinct chemical sites. A single OpenMP342
pragma was used to parallelize inner frequency loops,343
#pragma omp parallel for private(h,i)344
15
using the private declaration on loop indices to prevent a race condition oc-345
curring between separate threads. The OpenMP application programming in-346
terface is essentially a set of libraries and associated compiler directives which347
permits shared memory processing (SMP) on machines with the appropri-348
ate hardware. In order to perform optimization of the simulation parameters,349
the simulated annealing algorithm was implemented in the OCTAVE script-350
ing language. This allowed for tuning of heuristic parameters, particularly the351
annealing schedule and size of random fluctuations taken by individual param-352
eters per iteration. In addition, parameter values corresponding to the lowest353
energy obtained are stored every iteration and used for occasional resets. In354
order to determine the appropriate number of crystallite orientations neces-355
sary for a simulation as well as sample numbers from the distribution, the356
MQMAS spectrum of the tetrahedral region within simple-crystalline model357
compound VPI-5 was simulated and results are displayed in figure 5.358
Fig. 6. (a)27Al 3QMAS experimental spectrum, hydrous aluminosilicate. Thir-teen equally spaced contours are drawn from 10 to 90% of the total intensity(b)Simulation for the same
It is anticipated that the number of crystallite orientations required for ad-359
equate convergence in a particular summation will increase with linewidth,360
which in turn is proportional to the quadrupole coupling constant. Fitting to361
a crystalline model compound provides a good means of determining the min-362
imum number of crystallite orientations required for a comparable linewidth.363
16
Table 2Results for simulation of hydrous aluminosilicate MQMAS spectrum
Site # δcsiso(Hz) Cq(MHz) η Area
µ σ µ σ
1 -1399 127 2.9 0.9 0.52 0.39
2 -1163 333 3.3 0.8 0.52 0.49
3 -747 167 3.6 1.7 0.23 0.12
Table 3Jackknife parameter error estimates for simulation of hydrous aluminosilicate MQ-MAS spectrum
Site # δcsiso (%) Cq(%) η (%) Area (%)
µ σ µ σ
1 0.5 5.1 3.8 4.5 0.7 12.2
2 2.1 4.9 4.7 6.6 1.1 21.1
3 2.9 8.6 15.7 14.8 7.2 31
Convergence or lack thereof is more easily observed in a crystalline system as364
compared to a more disordered material, which is devoid of the characteristic365
features. In this case, 1597 angle pairs (F17) were minimal for quadrupole cou-366
pling constants in the range less than 4MHz, as determined from 27Al (spin367
I = 5/2) MQMAS of VPI-5. The Second Order Quadrupole Effect (SOQE) pa-368
rameters 3 determined from the simulation for the tetrahedral region of VPI-5369
were 2.6 and 1.15 MHz, which compare favorably with literature values [10].370
Using the same number of crystallite angles, optimized simulations were per-371
formed for the tetrahedral region within a hydrated aluminosilicate sample,372
using 200 samples for each of three bi-variate distributions and results are373
displayed in figure 6 and table 2. The Gaussian/Lorentzian ratio, correlation374
coefficient and broadening constants in both dimensions were constrained to375
0.5, 0, and 100 Hz respectively and 1000 simulated annealing iterations were376
performed. The experimental spectra displays regions of both order (narrow,377
horizontal peaks) and disorder (broad, indistinct). In order to test the validity378
of the simulated, optimized model, jackknife parameter error estimates were379
determined and are presented in table 3.380
381
382
3 SOQE = Cq
√
1 + η2
3
17
The chemical sites with narrow distributions (assigned here to crystalline al-383
bite and zeolitic material) have corresponding parameters with least error.This384
may be attributed to a number of factors, in this case most likely to the lower385
signal to noise ratio of the disordered region, assigned here to amorphous albite386
glass. For chemical sites with larger quadrupole coupling constants, there is387
also the possibility that due to experimental excitation deficiency, the second388
order perturbation frequency expression breaks down. Finally, the assump-389
tions of a Gaussian statistical model may be inappropriate for the system390
in question. As mentioned earlier, model distributions reflect the underlying391
stochastic nature of bonding in a disordered material. Regardless of the con-392
vention applied in describing the EFG tensor, the sign on the quadrupole393
coupling constant should be single valued and therefore a more appropriate394
distribution may be found in the positive tailed log-normal distribution. Under395
this assumption, random variable y representing the quadrupole coupling con-396
stant is transformed as z = exp(y), ie., y = log(z). The resulting distribution397
would be a bi-variate normal-lognormal distribution in the chemical shielding398
and quadrupole coupling constant respectively.399
4 Conclusions400
Theory has been outlined and an application implemented in the C program-401
ming language that permits the simulation of an MQMAS spectrum, as a func-402
tion of underlying parameter distributions. This simulation relies on the use of403
quasi-Monte Carlo variates to promote convergence and utilizes the OpenMP404
library to permit execution on SMP machines. Owing to the manner in which405
random variates are created in the application, the program is amenable to406
High Throughput Computing (HTC) platforms such as Condor or PBS. Dif-407
ferent nodes within a cluster or grid can be attributed different sections of408
the sample space using a submission script. In addition, an OCTAVE script409
implementing a simulated annealing algorithm is used to optimize the simu-410
lation, providing reliable estimates of NMR parameters. Finally, theory was411
outlined and implemented for providing parameter variance estimates using a412
jackknife approach. As an alternative to the essentially parametric approach413
outlined here, the application may be used to optimize a very large number414
of chemical sites of equal amplitude. In this event, kernel density estimation415
may be applied to parameter estimates to provide a more arbitrary probabil-416
ity distribution model for chemical order. In conjunction with the MQMAS417
experiment, the application described herein enables the characterization of418
materials which may vary greatly in the degree of underlying chemical and419
structural order.420
18
Acknowledgements421
Jeff Nucciarone and the Research Computing and Cyberinfrastructure group422
at Penn State are acknowledged for their generous assistance and use of com-423
putational resources. Marek Pruski kindly provided the MQMAS spinsight424
pulse sequence used for experiments. This work has been funded via National425
Science Foundation grant number CHE 0535656426
Appendix A427
P2(θ) =1
2
(
3 cos2 θ − 1)
428
P4(θ) =1
8
(
35 cos4 θ − 30 cos2 θ + 3)
429
V2,0 =1
2
√6Vzz; V2,1 = −Vxz − iVyz430
V2,−1 = Vxz − iVyz; V2,2 =1
2(Vxx − Vyy) + iVxy431
V2,−2 =1
2(Vxx − Vyy) − iVxy; K
(2,0) =1√6[3I2
z − I(I + 1)]432
K(2,1) = −1
2I+(2Iz + 1); K(2,−1) =
1
2I−(2Iz − 1)433
K(2,2) =1
2I−I−434
A(4)(I, r, c) = 18I(I + 1) − 34(r2 + rc + c2) − 5435
A(2)(I, r, c) = 8I(I + 1) − 12(r2 + rc + c2) − 3436
A(0)(I, r, c) = I(I + 1) − 3(r2 + rc + c2)437
Appendix B438
Referring to equation 31, the kernal of the integrand is:439
I =440
19
e−
(f2−f2m)2
2 λ22
−(f1−f1m)2
2 λ21 ǫ
2 π λ1 λ2+ λ1 λ2 (1−ǫ)
(λ21+(f1−f1m)2) (λ2
2+(f2−f2m)2)
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2) A
2 π√
1 − ρ2 σx σy
441
where it is understood that the total intensity is computed via summation of I442
over all L chemical sites, as well as powder angles α, β. Each partial derivative443
listed is performed independently for each chemical site and the total variance444
matrix calculated in the manner described previously. Random variates for445
chemical shift and quadrupole coupling constant are x and y respectively, with446
corresponding mean µ and standard deviation σ labeled with the appropriate447
subscript. Frequency coefficients:448
clb0 = −I(I + 1) + 3/4449
clb1 = −18I(I + 1) + 34/4 + 5450
clb2 = (r − c)(I(I + 1) − 3(r2 + rc + c2))451
clb3 = (r − c)(18I(I + 1) − 34(r2 + rc + c2) − 5)452
Derivatives:453
∂I∂η
=∂I
∂f1m
df1m
dη+
∂I∂f2m
df2m
dη454
df2m
dη=455
clb1y2
11520f0I2(2I − 1)2
cos2 β (140 cos(4.0α)η + 60.0η − 480 cos(2α))+456
cos4 β (−70.0 cos (4 α) η − 70.0 η + 420 cos (2 α)) − 70.0 cos(4.0α)η − 6.0η + 60.0 cos(2α)
457
− clb0y2η
5f0I2(2I − 1)2458
df1m
dη=459
−k · clb1y2
11520f0I2(2I − 1)2
cos2 β (140 cos(4.0α)η + 60.0η − 480 cos(2α))+460
cos4 β (−70.0 cos (4 α) η − 70.0 η + 420 cos (2 α)) − 70.0 cos(4.0α)η − 6.0η + 60.0 cos(2α)
461
20
+k · clb0y
2η
5f0I2(2I − 1)2− clb2y
2
11520f0I2(2I − 1)2
cos2 β (140 cos (4.0α) η + 60.0η − 480 cos(2α))+462
cos4 β (−70.0 cos (4 α) η − 70.0 η + 420 cos (2 α)) − 70.0 cos (4.0 α) η − 6.0 η + 60.0 cos(2α)
463
+clb3y
2η
5f0I2(2I − 1)2464
where k is the shear factor.465
∂I∂f2m
=466
(f2−f2m) e−
(f2−f2m)2
2 λ22
−(f1−f1m)2
2 λ21 ǫ
2 π λ1 λ32
+ 2 (f2−f2m) λ1 λ2 (1−ǫ)
(λ21+(f1−f1m)2) (λ2
2+(f2−f2m)2)2
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2) A
2 π√
1 − ρ2 σx σy
∂I∂f1m
=467
(f1−f1m) e−
(f1−f1m)2
2 λ21
−(f2−f2m)2
2 λ22 ǫ
2 π λ2 λ31
+ 2 (f1−f1m) λ2 λ1 (1−ǫ)
(λ22+(f2−f2m)2) (λ2
1+(f1−f1m)2)2
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2) A
2 π√
1 − ρ2 σx σy
∂I∂λ2
=468
A2 π
√1 − ρ2 σx σy
−e− (f2−f2m)2
2 λ22
− (f1−f1m)2
2 λ21 ǫ
2 π λ1 λ22
+(f2 − f2m)2 e
− (f2−f2m)2
2 λ22
− (f1−f1m)2
2 λ21 ǫ
2 π λ1 λ42
+
λ1 (1 − ǫ)(
λ21 + (f1 − f1m)2
) (
λ22 + (f2 − f2m)2
) − 2 λ1 λ22 (1 − ǫ)
(
λ21 + (f1 − f1m)2
) (
λ22 + (f2 − f2m)2
)2
×
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2)
21
∂I∂λ1
=469
A2 π
√1 − ρ2 σx σy
−e− (f2−f2m)2
2 λ22
− (f1−f1m)2
2 λ21 ǫ
2 π λ21 λ2
+(f1 − f1m)2 e
− (f2−f2m)2
2 λ22
− (f1−f1m)2
2 λ21 ǫ
2 π λ41 λ2
+
λ2 (1 − ǫ)(
λ21 + (f1 − f1m)2
) (
λ22 + (f2 − f2m)2
) − 2 λ21 λ2 (1 − ǫ)
(
λ21 + (f1 − f1m)2
)2 (
λ22 + (f2 − f2m)2
)
×
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2)
∂I∂ρ
=470
e−
(f2−f2m)2
2 λ22
−(f1−f1m)2
2 λ21 ǫ
2 π λ1 λ2+ λ1 λ2 (1−ǫ)
(λ21+(f1−f1m)2) (λ2
2+(f2−f2m)2)
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2) A
2 π√
1 − ρ2 σx σy
471
×(
ρ
(1 − ρ2)− ρ(x − µx)
2
(1 − ρ2)2σx+
(x − µx)(y − µy)
(1 − ρ2)σxσy+
2ρ2(x − µx)
(1 − ρ2)σxσy− ρ(y − µy)
2
(1 − ρ2)2σy
)
472
∂I∂σy
=473
− A4 π (1 − ρ2)
32 σx σy
e− (f2−f2m)2
2 λ22
− (f1−f1m)2
2 λ21 ǫ
2 π λ1 λ2+
λ1 λ2 (1 − ǫ)(
λ21 + (f1 − f1m)2
) (
λ22 + (f2 − f2m)2
)
×
(
2 ρ (x − µx) (y − µy)
σx σ2y
− 2 (y − µy)2
σ3y
)
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2)
−
e−
(f2−f2m)2
2 λ22
−(f1−f1m)2
2 λ21 ǫ
2 π λ1 λ2+ λ1 λ2 (1−ǫ)
(λ21+(f1−f1m)2) (λ2
2+(f2−f2m)2)
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2) A
2 π√
1 − ρ2 σx σ2y
22
∂I∂µy
=474
−
e− (f2−f2m)2
2 λ22
− (f1−f1m)2
2 λ21 ǫ
2 π λ1 λ2+
λ1 λ2 (1 − ǫ)(
λ21 + (f1 − f1m)2
) (
λ22 + (f2 − f2m)2
)
×
(
2 ρ (x−µx)σx σy
− 2 (y−µy)σ2
y
)
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2) A
4 π (1 − ρ2)32 σx σy
∂I∂σx
=475
− A4 π (1 − ρ2)
32 σy σx
e− (f2−f2m)2
2 λ22
− (f1−f1m)2
2 λ21 ǫ
2 π λ1 λ2
+λ1 λ2 (1 − ǫ)
(
λ21 + (f1 − f1m)2
) (
λ22 + (f2 − f2m)2
)
×
(
2 ρ (y − µy) (x − µx)
σy σ2x
− 2 (x − µx)2
σ3x
)
e−
(x−µx)2
σ2x
−
2 ρ (y−µy) (x−µx)σy σx
+(y−µy)2
σ2y
2 (1−ρ2)
−
e−
(f2−f2m)2
2 λ22
−(f1−f1m)2
2 λ21 ǫ
2 π λ1 λ2+ λ1 λ2 (1−ǫ)
(λ21+(f1−f1m)2) (λ2
2+(f2−f2m)2)
e−
(x−µx)2
σ2x
−
2 ρ (y−µy) (x−µx)σy σx
+(y−µy)2
σ2y
2 (1−ρ2) A
2 π√
1 − ρ2 σy σ2x
∂I∂µx
=476
−
e− (f2−f2m)2
2 λ22
− (f1−f1m)2
2 λ21 ǫ
2 π λ1 λ2+
λ1 λ2 (1 − ǫ)(
λ21 + (f1 − f1m)2
) (
λ22 + (f2 − f2m)2
)
×
(
2 ρ (y−µy)σy σx
− 2 (x−µx)σ2
x
)
e−
(x−µx)2
σ2x
−
2 ρ (y−µy) (x−µx)σy σx
+(y−µy)2
σ2y
2 (1−ρ2) A4 π (1 − ρ2)
32 σy σx
23
∂I∂A =477
e−
(f2−f2m)2
2 λ22
−(f1−f1m)2
2 λ21 ǫ
2 π λ1 λ2+ λ1 λ2 (1−ǫ)
(λ21+(f1−f1m)2) (λ2
2+(f2−f2m)2)
e−
(y−µy)2
σ2y
−
2 ρ (x−µx) (y−µy)σx σy
+(x−µx)2
σ2x
2 (1−ρ2)
2 π√
1 − ρ2 σx σy
478
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