optimized multiple quantum mas lineshape simulations in

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

−


+(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

−


+(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

−


+(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

−


+(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

−


+(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

−


+(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

−


+(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

−


+(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

−


+(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

−


+(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

−


+(x−µx)2

σ2x

2 (1−ρ2)

2 π√

1 − ρ2 σx σy

478

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