First-Principles Nuclear Magnetic Resonance Structural Analysis of

J. Phys. Chem. C 2009, 113, 7917–7929

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First-Principles Nuclear Magnetic Resonance Structural Analysis of Vitreous Silica Thibault Charpentier,*,† Peter Kroll,‡ and Francesco Mauri§ CEA, IRAMIS, SerVice Interdisciplinaire sur les Systèmes Mole´culaires et Mate´riaux, LSDRM F-91191 Gif-sur-YVette cedex, France, Department of Chemistry and Biochemistry, The UniVersity of Texas at Arlington, 700 Planetarium Place, Arlington, Texas 76019-0065, and IMPMC, CNRS-IPGP-UniVersite´ Paris 6 et 7, 140 rue de Lourmel, Paris, France ReceiVed: January 12, 2009; ReVised Manuscript ReceiVed: March 16, 2009

Gauge including projector augmented wave (GIPAW) NMR calculations combined with hybrid Monte Carlo/ molecular dynamics simulations are carried out in order to investigate the relationships between the oxygen17 and silicon-29 NMR spectra of vitreous silica and its local structure in terms of the Si-O-Si bond angle and Si-O distance distributions. Special attention is paid to the structure and NMR parameters of three- and four-membered rings, and the effect of their concentration on glass density is studied. It is shown that our simulations provide a new insight into the features of the 17O NMR parameters distribution. Accordingly, a new analytical model is presented and applied for the reconstruction of the Si-O-Si angle from the NMR spectrum. The reliability of the procedure is demonstrated conclusively through the excellent consistency of the analysis of the oxygen-17 and silicon-29 NMR experimental data of vitreous silica. Si-O-Si angle distribution mean values of 147.1° and 148.4°, respectively, and standard deviations of 11.2° and 10.8°, respectively, are obtained from the oxygen-17 and silicon-29 NMR experimental spectrum (Clark et al., ref 4) of the same sample. Introduction Vitreous silica is one of the most important glass formers and, because of its significant scientific and technological importance it has been extensively studied. There is general agreement that its structure is an infinite three-dimensional disordered network of corner-sharing SiO4 tetrahedra linked through bridging oxygen (BO) atoms forming SiOSi linkages. Structural disorder of the amorphous structure manifests itself predominantly in a variation of the Si-O-Si bond angles. Some uncertainties still remain on the intermediate range order described by the Si-O-Si bond angle distribution and ring size distribution.1-4 Most of the knowledge was obtained experimentally by means of diffraction and nuclear magnetic resonance (NMR) spectroscopy and, theoretically, by means of molecular dynamics (MD) simulations, as recently summarized by Yuan and Cormack.3 Diffraction can well-resolve distances of first-nearest neighbors (providing an accurate measurement of the Si-O, O-O, and Si-Si distance distribution), but the extraction of the Si-O-Si angle distribution is difficult because it requires a precise measurement of distances of second-nearest neighbors.1,3 In this respect, NMR provides a more direct way of measuring the Si-O-Si angle distribution from 29Si NMR2,5,6 or 17O NMR.4,7 The key for such studies is the functional relationship between the investigated NMR parameters and the local structural features such as bond angles, bond lengths, and the distance to other atoms further in the environment. Most of the theoretical investigations of such relationships, generally aiming at the derivation of parametrized analytical forms, were performed on small molecular clusters using ab initio quantum chemical calculations.8-17 Fitting predicted values to experimental data * Corresponding author. E-mail: [email protected]. † CEA/IRAMIS. ‡ The University of Texas at Arlington. § IMPMC/CNRS-IPGP-Universite´ Paris 6 et 7.

for reference crystalline systems of known structures18 provides the underlying parameters. However, crystalline systems generally exhibit a limited diversity of local structures (here, as characterized by Si-O-Si angles and Si-O distances) in contrast to those generally expected and predicted by MD simulations of the vitreous state. This was first pointed out by Mauri et al.2 in their re-examination of the correlation between the 29Si isotropic chemical shift and the Si-O-Si angle in tetrahedral linkages. In order to generate a wide range of Si-O-Si angles, density functional theory (DFT)-relaxed structures of R-cristobalite under compressive or tensile pressure were used as references, and the NMR parameters were computed using a DFT-based method specially devised for solids.19 Since this work, significant progress in DFT computation of NMR parameters in solids has been made with the introduction of the gauge including projector augmented wave (GIPAW)20 method. Its major improvement is the inclusion of an explicit reconstruction of the all electron wave function at the nucleus. The GIPAW method is capable of dealing with large systems (here 108 atoms) and exhibits an outstanding accuracy as demonstrated recently for various crystalline and amorphous silicate systems.21-26 In these respects, GIPAW has now become the method of choice for studying the NMR response of large systems such as those needed to model vitreous solids. In this work, we perform GIPAW-NMR calculations of 29Si and 17O NMR parameters using models of vitreous silicate structures generated by a hybrid Monte Carlo/molecular dynamics (bond switching) algorithm. We first investigate the structure and the NMR response of three- and four-membered rings and the implication of the ring size distribution on the Si-O-Si angle distribution and glass density. A model of the 17O NMR parameter distribution is developed, aiming at providing a more reliable fitting of the experimental 17O NMR spectrum, with respect to the topological structural disorder. Thereafter, we re-

10.1021/jp900297r CCC: $40.75  2009 American Chemical Society Published on Web 04/09/2009

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examine quantitatively the functional dependence of 17O and 29 Si NMR parameters on the Si-O-Si angle and Si-O distance. Finally, merging both of the aforementioned procedures, we map the 17O and 29Si NMR spectra into structural parameter distributions (here Si-O-Si angle and/or Si-O distance)4,7,27 and discuss the results. Computational Approaches. Construction of Network Models of Amorphous SiO2. In the present study, we apply a modified Wooten-Winer-Weaire (WWW)-algorithm28 to generate network models of vitreous silica. For our purpose, this is the best approach because it generates defect-free tetrahedral networks of amorphous SiO2 (a-SiO2) suitable for subsequent NMR calculations. In our algorithm, we start with a 108 atoms structure (36 formula units) of cristobalite SiO2 and randomize the bonding network completely. The WWW-algorithm is coupled to a simulated annealing process that searches effectively for local minima of the structure by employing temperature schemes to decrease the residual potential energy of the network. A high initial annealing temperature guarantees that all memory of the initial topology is lost. As in several preceding studies of vitreous silica and related compounds,29-32 we use a simple Keating-type potential for the cost function. The empirical potential is also used to optimize the geometry of the network simultaneously to the bond switching procedure. The procedure reaches a quasi-glassy state within the MC/MD hybrid algorithm and generates several suitable local minimum structures. A detailed description of the algorithm together with the potential parameters used is presented in ref 33. After generation of the network model structures, we employ density functional theory methods to make a local optimization of the structure. For two selected models, we performed ab initio molecular dynamics simulations up to 40 ps at 2000 °C using a time step of 1 fs. No bond rearrangements were detected during the molecular dynamics simulations, and the structure could be quenched to its initial position. Each model we finally investigated was relaxed to 0 K using a kinetic energy cutoff of 80 Ry for the plane wave expansion and the Perdew-BurkeErnzerhof (PBE)-generalized gradient approximation (GGA) method to approximate the electron exchange-correlation functional. In order to take into account the quasi-systematic error of the GGA functional in the prediction of the bond lengths, the unit cell volume of each model is reduced by 4.5%22 before NMR calculations and structural analysis. Following this procedure, twelve glass model structures were generated and analyzed, each containing 108 atoms confined in a box with densities ranging from 2.167 to 2.378 g/cm3 (the experimental density of vitreous silica is 2.20 g/cm3). Hence, we explore a wide range of local geometries in terms of the Si-O-Si angle and Si-O distance variations; various concentrations of N-fold rings are also generated so we can investigate the 17O and 29Si NMR parameters of such units. NMR Calculations. NMR shielding and electric field gradient tensors are calculated using the gauge including projector augmented wave (GIPAW) method20 with PBE-GGA DFT, as implemented in the PARATEC code34 with pseudopotential descriptions of the core electrons (see ref 22 for details). For plane wave expansion, a kinetic energy cutoff of 80 Ry is used, and integrals over the Brillouin zone are performed using a single Baldereschi point, considering the large size of the unit (super)cell. The PARATEC program outputs the absolute shielding and the electric field gradient tensors, σ and V, respectively. The absolute isotropic shielding σiso ) 1/3Tr{σ},

Charpentier et al. and the component of the electric field gradient largest in magnitude Vzz and the quadrupolar asymmetry parameter ηq are obtained. Additionally, σ is described in terms of its principal components, σ33 g σ22 g σ11 by the shielding span Ωσ ) σ33 σ11 (Ωσ g 0) and skew κσ ) 3(σ22 - σiso)/Ωσ (1 g κσ g -1). The isotropic chemical shift δiso and the quadrupolar coupling constant Cq are determined from the relations δiso ) -(σiso σref) and Cq ) eQVzz/h, respectively, where σref is the isotropic shielding of a reference system and Q is the nuclear quadrupole moment. Here, Q is considered as an adjustable parameter and is obtained by fitting the calculated electric field gradient component Vzz of the coesite, R-quartz, and cristobalite to the experimental quadrupolar coupling constant Cq. This yields Q ) 2.42 × 10-30 m2, a value close to the experimental value Q ) 2.55 × 10-30 m2 (Supporting Information). R-Quartz is used as the reference system for the chemical shift scales, providing σref ) 335.74 ppm for 29Si and σref ) 259.32 ppm for 17O. A specific software was written to perform a correlated NMR and local structure (bond angles, distances with neighboring atoms, and ring size,) analysis of each atomic site. NMR spectra are simulated using a kernel density estimation (KDE)35 of the NMR parameters distribution, as described in the Appendix. KDE is also employed throughout this paper for the best visualization of two-dimensional data sets (xi,yi) by drawing the contour plot of the kernel density estimate of the density probability p(x,y). Results and Discussion NMR Response of Small Rings. The network topology of our model structures is characterized by the ring size distribution using the shortest path analysis,36,37 and the results are given Table 1 and the Supporting Information. In contrast to previous work, both silicon and oxygen atom sites have been considered, yielding six and one closed path per site, respectively. For silicon sites, five-, six-, and seven-membered rings are found most frequently, which is in agreement with previous studies.37 Twomembered rings (edge sharing tetrahedra) are found in none of the model structures. We observe an increase in the fraction of 3- and 4-fold rings as the glass density decreases. A similar trend was already reported for crystalline silicates38,39 in MD studies of the structural response of a-SiO2 to pressure40 and also in other systems such as vitreous boron oxide.41 It is worth noticing that this finding challenges results of vibrational studies,42,43 which infer an increase in the concentration of 3-fold rings in silica glass with increasing pressure on the basis of the Raman signature of 3- and 4-fold rings.44,45 A linear regression analysis of the density of our glass structure F against the fraction of 3- and 4-fold rings, f3 and f4, respectively, for oxygen and silicon sites yields

F ) 1.075 - 0.329f3O - 0.0626f4OR ) 0.736 F0

(1)

F ) 1.079 - 0.162f3Si - 0.0329f4SiR ) 0.759 F0

(2)

where F0 ) 2.20 g/cm3 is the experimental glass density and R is the correlation coefficient. As displayed in Figure 1a, the predicted glass density using eqs 1 and 2 is found to agree within 0.016 g/cm3 (standard deviation) with the theoretical value. No significant improvement is obtained by considering fractions of higher N-fold rings. This result suggests that variation in the network topology might be primarily responsible for the variation in glass density (at least over the investigated range

First-Principles NMR Analysis of a-SiO2

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Figure 1. (a) Predicted vitreous silica density from the number of 3- and 4-fold rings (f3 and f4) per oxygen and silicon sites, respectively, using eqs 1and 2. (b) Contour plot of the theoretical two-dimensional distribution of the Si-O-Si bond angle versus the Si-O bond length of our model structures. Solid circles (b) represent the averaged (Si-O-Si and Si-O) values over all model structures for three-, four-, and other N-membered rings (Table 1). (c) and (d) Variation of the mean Si-O-Si angle and Si-O distance versus each model structure density. Dashed lines are the mean values of the Si-O-Si angle and Si-O distance calculated using all model structure data.

TABLE 1: Ring StatisticsaAccording to the Shortest Path Analysis of Selected Model Structures Investigated ring size atom SiO2.1556, Si O SiO2.1586, Si O SiO2.1642, Si O SiO2.1671, Si O a

F ) 2.328 g/cm

3

4

5

6

7

8

9

10

0.00 0.00

0.56 0.25

1.81 0.54

2.03 0.21

1.39 0.00

0.22 0.00

0.00 0.00

0.00 0.00

0.08 0.04

0.78 0.35

1.83 0.49

1.86 0.11

1.14 0.01

0.25 0.00

0.06 0.00

0.00 0.00

0.25 0.12

0.56 0.25

2.03 0.53

1.33 0.07

1.14 0.03

0.69 0.00

0.00 0.00

0.00 0.00

0.33 0.17

0.89 0.36

0.78 0.18

1.89 0.24

1.94 0.06

0.14 0.00

0.03 0.00

0.00 0.00

3

F ) 2.302 g/cm3 F ) 2.210 g/cm3 F ) 2.167 g/cm3

Number of rings per silicon and oxygen atom, respectively.

of density). It is thus of interest to connect this network topology to the Si-O-Si angle (and Si-O distance) distribution and to 17 O and 29Si NMR parameters. In Tables 2 and 3, we summarize statistics of the relevant structural and NMR parameters of the oxygen and silicon atoms, respectively, for selected model structures, which are averaged over all of the model structures (Supporting Information). The statistics of three- and four-membered rings (denoted 3-ring and 4-ring) are remarkably consistent among the different model structures, especially independent of glass density, as shown in Figure 1c,d. Higher N-fold ring species do not exhibit such a regularity and are therefore analyzed as a single specie (denoted >4-ring). As suggested by Figure 1b, the global trend of an increase in the Si-O distance with the Si-O-Si angle decreasing is more likely to be indicative of the ring size distribution.

The predicted mean of Si-O distances 1.617 Å (3-ring), 1.605 Å (4-ring), and 1.601 Å (>4-ring) are in good agreement with previous theoretical and experimental studies.30,46 Concerning three-membered rings, the small dispersion of the Si-O-Si angle (standard deviation of 4.1°) and its lower mean value (126.7°) are consistent with their structural regularity as already outlined in many theoretical studies.37,44,45,47 The structure of four-membered rings are less regular: a mean Si-O-Si angle of 142.3° and a standard deviation of 10.9°. For higher N-fold rings, the theoretical mean Si-O-Si angle 147°(12.2°) is in remarkable agreement with diffraction experimental data,1,46 which is indicative of the very low concentration of small rings in vitreous silica. Examination of the NMR parameters in Tables 2 and 3 clearly shows that the structural regularity of the small rings translates

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TABLE 2: Structural and 17O NMR Parameters of Selected Model Structures Investigated and Averages over All Model Structuresa sample

17

local structure

ring size SiO2.1556, F ) 2.328 g/cm , f3b ) 0.00, f4c ) 0.25 4 >4 SiO2.1586, F ) 2.302 g/cm3, f3b ) 0.04, f4c ) 0.35 3 4 >4 SiO2.1642, F ) 2.210 g/cm3, f3b ) 0.12, f4c ) 0.25 3 4 >4 SiO2.1671, F ) 2.167 g/cm3, f3b ) 0.17, f4c ) 0.36 3 4 >4 averages over all model structures 3 4 >4

O NMR parameters

SiOSi (deg)

SiO (Å)

Cq (MHz)

η

δiso (ppm)

Ωσ (ppm)

κσ

142.3 (11.2) 148.7 (12.2)

1.606 (0.011) 1.600 (0.010)

5.02 (0.46) 5.21 (0.40)

0.23 (0.14) 0.17 (0.11)

49.25 (5.40) 39.78 (6.66)

69.19 (9.45) 78.80 (10.94)

0.75 (0.17) 0.84 (0.10)

130.2 (0.7) 144.6 (11.3) 145.0 (12.4)

1.613 (0.009) 1.602 (0.010) 1.601 (0.009)

4.63 (0.49) 5.05 (0.44) 5.15 (0.46)

0.30 (0.13) 0.21 (0.13) 0.21 (0.14)

53.29 (9.48) 44.18 (6.44) 40.49 (6.06)

45.98 (2.85) 72.31 (8.10) 76.91 (10.21)

0.51 (0.12) 0.77 (0.13) 0.82 (0.10)

126.4 (6.0) 144.2 (12.4) 145.9 (12.1)

1.617 (0.008) 1.604 (0.009) 1.602 (0.011)

4.55 (0.49) 5.02 (0.33) 5.21 (0.36)

0.39 (0.23) 0.23 (0.11) 0.22 (0.15)

57.43 (6.43) 44.82 (4.10) 40.53 (7.49)

41.71 (5.36) 73.57 (6.15) 79.44 (10.68)

0.69 (0.26) 0.80 (0.07) 0.81 (0.14)

128.4 (3.2) 147.3 (7.9) 146.8 (11.8)

1.612 (0.009) 1.601 (0.007) 1.599 (0.012)

4.60 (0.26) 5.11 (0.35) 5.20 (0.43)

0.37 (0.14) 0.18 (0.10) 0.19 (0.11)

55.65 (9.45) 42.97 (4.29) 37.30 (5.41)

43.42 (9.01) 79.55 (6.68) 81.82 (9.70)

0.35 (0.43) 0.82 (0.10) 0.84 (0.12)

126.7 (4.1) 142.3 (10.9) 147.0 (12.2)

1.617 (0.009) 1.605 (0.011) 1.601 (0.010)

4.61 (0.35) 5.03 (0.39) 5.21 (0.41)

0.38 (0.15) 0.24 (0.13) 0.20 (0.12)

55.92 (7.85) 46.54 (6.42) 39.59 (6.55)

44.43 (7.16) 72.17 (9.52) 79.48 (10.72)

0.44 (0.35) 0.76 (0.15) 0.83 (0.11)

3

a Standard deviations are given in parentheses. four-membered rings per oxygen atom.

b

f3 is the number of three-membered rings per oxygen atom.

c

f4 is the number of

TABLE 3: Structural and 29Si NMR Parameters of Some Representative Model Structures Investigated and Averages over All Model Structuresa sample ring size SiO2.1556, F ) 2.328 g/cm , f3 ) 4 >4 SiO2.1586, F ) 2.302 g/cm3, f3b ) 3 4 >4 SiO2.1642, F ) 2.210 g/cm3, f3b ) 3 4 >4 SiO2.1671, F ) 2.167 g/cm3, f3b ) 3 4 >4 averages over all model structures 3 4 >4 3

29

local structure b

Si NMR parameters

SiOSid (deg)

OSiO (deg)

δiso (ppm)

Ωσ (ppm)

κσ

143.2 (11.6) 150.2 (11.8)

109.5 (2.5) 109.5 (2.4)

-105.41 (6.45) -111.88 (5.32)

24.31 (7.89) 15.19 (6.18)

0.15 (0.32) -0.03 (0.42)

133.8 (6.7) 144.8 (11.8) 145.8 (12.2)

109.5 (3.2) 109.5 (2.6) 109.5 (2.8)

-97.12 (0.57) -106.89 (5.32) -108.11 (5.41)

38.13 (9.38) 22.84 (9.75) 15.07 (4.64)

-0.19 (0.30) 0.05 (0.41) 0.00 (0.40)

135.2 (12.1) 144.9 (11.7) 146.6 (13.3)

109.5 (3.4) 109.5 (2.5) 109.5 (3.1)

-97.84 (5.12) -106.74 (4.22) -108.83 (5.61)

32.92 (8.02) 25.07 (9.03) 19.30 (6.39)

0.13 (0.25) 0.08 (0.37) 0.00 (0.46)

137.4 (12.3) 146.8 (8.9) 148.7 (13.4)

109.5 (3.6) 109.5 (2.8) 109.5 (3.2)

-100.01 (4.50) -109.07 (4.47) -110.60 (6.98)

30.37 (6.42) 20.13 (7.65) 16.96 (7.56)

0.24 (0.29) 0.19 (0.44) -0.04 (0.42)

135.4 (12.4) 144.0 (11.5) 148.3 (12.3)

109.5 (3.2) 109.5 (3.0) 109.5 (2.7)

-97.97 (4.74) -106.09 (5.52) -110.31 (5.53)

31.27 (8.40) 23.66 (8.25) 17.50 (6.83)

0.10 (0.41) 0.02 (0.40) -0.04 (0.41)

0.00, f4 ) 0.56 c

0.08, f4c ) 0.78

0.25, f4c ) 0.56

0.33, f4c ) 0.89

a

Standard deviations are given in parentheses. b f3 is the number of three-membered rings per Si atom. c f4 is the number of four-membered rings per Si atom. d For each silicon atom, the four tetrahedral bonding angles are considered.

into a regularity of the NMR parameters distribution; with respect to their standard deviation values, the mean values are well-separated. This is more pronounced for oxygen-17, as its local structure involves a single Si-O-Si angle but is also visible for silicon-29 despite the dependence upon the four angles of the tetrahedral linkages. It is again worth mentioning that for N-fold ring species N > 4, the mean value of the quadrupolar coupling constant Cq ) 5.21(0.41) MHz, quadrupolar asymmetry parameter ηq ) 0.20(0.12), and isotropic chemical shift δiso ) 39.6(6.6) ppm for oxygen-17 and δiso ) -110.3(5.5) ppm for silicon-29 are very close to previous measurements,4 Cq ) 5.08(0.73) MHz, ηq ) 0.15(0.04), and

δiso ) 36.7(4.4) ppm for oxygen-17 and δiso ) -111.2(5.1) ppm for silicon-29. This is illustrated in Figures 2 and 3, where the theoretical 29Si MAS NMR and 17O DAS NMR spectra of 3-, 4-, and >4-fold ring species are compared to the experimental spectra. In both cases, the simulated theoretical spectrum that takes into account only the >4-fold ring species is found to be in fairly good agreement with the experimental spectrum, despite a slight overestimation of its width. This is again indicative of a very low concentration of small rings in vitreous silica. Notably, the two-dimensional line shape of the 17O DAS NMR spectrum (Figure 3) is also well-reproduced by our simulations. This highlights that the underlying interplay between the 17O


J. Phys. Chem. C, Vol. 113, No. 18, 2009 7921 4). This structurally induced broadening was taken into account by Grandinetti and co-workers only along ν1. Along ν2, it was described with the help of a supplementary Gaussian broadening, thus, neglecting or hiding a significant part of the NMR parameters dispersion. The correlation between Cq and ηq was taken into account as a linear variation of ηq with respect to Cq along the ν1 dimension. As it is here clearly demonstrated by our simulations (Figure 3), a NMR parameter distribution is sufficient to satisfactorily reproduce the DAS spectrum. Here, after close examination of Figure 4, we introduce a new model denoted as pG(Cq,ηq,δiso) as the product

j q ;σC) pG(Cq, ηq, δiso) ) G(Cq - C × G[ηq - fηq(Cq);ση] × G[δiso - fδiso(Cq);σδ]

Figure 2. Comparison between experimental 29Si MAS NMR spectra of vitreous silica with the theoretical 29Si MAS NMR spectra (simulated using all model structures) of three-, four-, and other N-membered rings.

NMR parameters (Cq, ηq, and δiso) are well-described by our model structures as discussed below. In Tables 2 and 3, we also report the shielding span Ωσ and skew parameter κσ. To the best of our knowledge, such experimental data have not been reported in the literature. However, recent advances in NMR experiments could offer some opportunities to measure them.48 Oxygen-17 NMR Parameter Distribution. One- and twodimensional distributions of the 17O NMR parameters are displayed in Figure 4. Strong correlations between each pair of NMR parameters are clearly observed, with trends that are in agreement with previous experimental measurements.4,17 The crucial point is that these correlated variations of the 17O NMR parameters (Cq, ηq, and δiso) govern the line shape of the 17O DAS NMR spectrum (Figure 3), so that they have to be taken into account for fitting the experimental data. Of special importance is the slope of the Cq versus ηq correlation (Figure 4a, dotted dashed line). Our calculations yield a value of 2.5, close to the one predicted by point charge calculations27 (≈ 2.0) and to the value observed for crystalline silicates.13,49 It is, however, much weaker than the value extracted from the analysis of the 17O DAS NMR spectrum of vitreous silica5 (≈ 8.2). This latter discrepancy is clarified below. Similarly correlated variations are also observed for 17O (Cq and Ωσ) and (ηq and κσ) (Supporting Information). At this stage, the main features of the two-dimensional DAS spectrum, denoted S(ν1,ν2), should be briefly reviewed. Each atomic site characterized by the NMR parameters (Cq, ηq, δiso) resonates at the isotropic shift ν1 given by

(

Cq2 ηq2 ν1 ) Rδiso + β 1+ ν0 3

)

(3)

This yields a sharp line along ν1 and gives rise to an anisotropic (i.e., broad) line shape along the ν2 (MAS) dimension, which also depends on (Cq, ηq, and δiso). R and β are numerical constants that depend on the experimental conditions (here, R ) 1 and β ) -2.4.). The structural disorder, which results in a distribution of (Cq, ηq, and δiso), leads to a broadening along both dimensions (ν1, ν2) but with ridge features (Figure 3) that reflect the correlated variations of the NMR parameters (Figure

(4)

where G is chosen as a Gaussian distribution (but others can be considered as well) with a standard deviation denoted as σx (x ) C, η, and δ for Cq, ηq, and δiso, respectively). The constraint of the linear Cq-ηq and Cq-δiso correlated variation is introduced with the help of the functions

fηq(Cq) ) aηCq + bη fδiso(Cq) ) aδCq + bδ

(5)

where aη, bη, aδ, and bδ are adjustable parameters (i.e., determined from the fit of eqs 4 and 5 to the NMR spectrum). In our model, at each value of Cq, the density probabilities of ηq and δiso have a constant width and a mean value that is linearly dependent on Cq. For ηq, the Gaussian distribution is truncated so that 0 e ηq e 1. To assess eqs 4 and 5, the theoretical 17O DAS NMR spectra of each of our model structures (for which the NMR parameter distribution is known) is fitted. The mean and standard deviation values of Cq, ηq, and δiso for this yield agree within 0.1 MHz and 0.01 and 0.5 ppm, respectively, with the theoretical values. Figure 5 shows an example. A good level of agreement, defined b/∫(IEXP)2dν b), of about 1.0% is obtained. Using as ∫(IFIT - IEXP)2dν the method of Grandinetti and co-workers,4 we achieved the same level of agreement, and mean values are retrieved at the same level of accuracy as with our method. However, the slope of the Cq versus ηq correlation is not recovered ( Figure 5d, dashed line), and the standard deviation of ηq is found to be underestimated by about 50%. Structural Interpretation of Oxygen-17 NMR Data. The relationship between the 17O quadrupolar NMR parameters and the Si-O-Si angle has been extensively studied in silicates.7-11,13-17,21,49 The variations of 17O Cq, ηq, and δiso versus the Si-O-Si angle Ω and the Si-O distance d (here averaged over the two O-Si bonds) observed in our model structures are displayed in Figure 6 in terms of their respective density probabilities. The variations with respect to Ω are in very good agreement with previous experimental measurements,4 notably for δiso, for which no general trend with local structural features has been established so far (see discussion in ref 4). In the present case, our calculations indicate that δiso is decreasing with the Si-O-Si angle. The strong dependence of each of the NMR parameters upon the Si-O-Si angle Ω easily explains their respective correlations displayed in Figure 4. Addressing the dependence upon the Si-O distance d, we find it difficult to extract reliable trends from the simple visual examination of the NMR/Si-O correlation (Figure 6). The influence of the Si-O distance on the quadrupolar parameters in Si-O-Si linkages has been only recently investigated. Grandinetti and co-workers18 used ab initio calculations for (OH)3Si-O-Si(OH)3 clusters and showed that

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Figure 3. (a) Experimental4 and theoretical 17O DAS NMR spectra (calculated using all model structures of vitreous silica) of (f) three-, (e) four-, and (d) other N-membered rings. Comparison between experimental and theoretical (b) isotropic and (c) MAS dimension projections.

the quadrupolar coupling constant Cq is a function of both the Si-O-Si angle Ω and the Si-O distance d, whereas the quadrupolar asymmetry parameter ηq depends almost only on Ω. Cq was predicted to be a linear increasing function of d. On the contrary, using point charge calculations of large model structures as generated by classical MD, Sen et al.27 found that Cq decreases with increasing Si-O distance. The weakness of their approach was the use of a point charge method for the calculation of the 17O quadrupolar parameters, quantities which are known to be highly dependent on the covalent nature of the Si-O bonding in Si-O-Si linkages. To investigate the pure geometrical dependence of the 17O NMR parameters on Ω and d, i.e., in such a manner that the effect of the correlation between Ω and d is removed, we rescale the atomic coordinates of one model structure isotropically by multiplying the unit cell volume by a factor ranging from 0.94 to 1.08 around its PBE-GGA DFT relaxed value. This procedure generates a set of atomic configurations covering a large range of Si-O bond lengths but with fixed angles Ω around each oxygen atomic site (data shown in Supporting Information). A linear variation of Cq with d is observed with a positiVe slope

that is found to be only slightly dependent upon Ω. This corroborates the small cluster calculations of Grandinetti and co-workers, while it contradicts the point charge calculations of Sen and co-workers. ηq is found to be almost exclusively dependent on the Si-O-Si angle Ω, whereas δiso exhibits a more complex behavior but increases as d increases as recently observed in MgSiO phases.26 For a quantitative investigation, we use the following analytical forms4,18 for the quadrupolar parameters R

( 21 + coscosΩΩ- 1 ) + m (d - d ) 1 cos Ω η (Ω) ) B( 2 cos Ω - 1 )

Cq(Ω, d) ) A

d

β

q

0

(6) (7)

Other analytical forms such as cos Ω polynomials were also investigated but did not yield any significant improvement for predicting the NMR parameters. The contribution of the local structure around the bridging oxygen atoms to the 17O isotropic chemical shift δiso using eq 6 was also investigated. But this quantity was found to be predicted poorly using the short-range



Figure 4. Contour plots of theoretical two-dimensional distributions of (a) the quadrupolar coupling constant Cq versus the asymmetry parameter ηq, (b) Cq versus the isotropic chemical shift δiso, and (c) ηq versus δiso for the oxygen atoms in the model structures of vitreous silica. Solid circles (b) represent the averaged values over all model structures for three-, four-, and other N-membered rings. In panel a, the dotted-dashed line represents the linear regression analysis of the Cq versus ηq correlation. Theoretical one-dimensional distribution of (d) Cq, (e) ηq, and (f) δiso.

information provided by the Si-O-Si angle and Si-O distance. As suggested in previous studies, δiso seems to be sensitive to variation of structure beyond the first coordination sphere.14 A least squares fit of eqs 6 and 7 to the GIPAW values generated with our model structures yields A ) 6.72 MHz, R ) 1.81, md ) 25.92 MHz/Å, d0 ) 1.638, B ) 4.19, and β ) 1.10. Predicted Cq and ηq values with eqs 6 and 7 are found to agree with GIPAW values within 0.15 MHz and 0.08 (standard deviations), respectively. Omitting the d dependence in eq 5 (md ) 0) yields a precision of 0.30 MHz for Cq, supporting the significant contribution of the d dependence. Parameters A, R, B, and β are close to those obtained by Grandinetti and co-workers, whereas our value of md is much higher. Applying eqs 6 and 7 to several crystalline polymorphs of SiO2 (coesite, R-quartz, and cristobalite), Cq and ηq are found to agree within 0.20 MHz and 0.03 (standard deviations), respectively.

As proposed in ref 18, Cq and ηq can hereafter be used to estimate the Si-O-Si angle Ω and Si-O distance d, so that the distribution p(Cq,ηq) can be mapped to the distribution p(Ω,d). In practice, ηq(Ω) in eq 7 is first inverted to determine Ω, and d is obtained from Cq(Ω,d) in eq 6. In order to assess the accuracy of this transformation, we have performed this mapping to our theoretical model structures data, i.e., recovering the Si-O-Si angle and Si-O distance at each oxygen site from the GIPAW NMR parameters. As shown in Figure 7, the global shape of the distribution p(Ω,d) is well-reproduced. The obtained mean and standard deviation values of Ω and d, and the correlation coefficient FΩ,d are Ω ) 140.6°(10.8)°, d ) 1.607(0.012) Å, and FΩ,d ) -0.388, close to those of the model structures, Ω ) 143.0°(12.7)°, d ) 1.604(0.011) Å, and FΩ,d ) -0.492. This confirms the robustness of our approach. Additionally, applied to coesite, R-quartz, and cristobalite, Ω and d are found to agree within 3.0°and 0.007 Å (standard

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Charpentier et al.

Figure 5. (a) Theoretical 17O DAS NMR spectrum of a glass model structure of vitreous silica (F ) 2.302 g/cm3). (b) Comparison between the simulated 17O DAS NMR spectra using parameters obtained from the fit of (a) employing eq 4 (solid lines) and the method of Grandinetti and co-workers4 (dashed lines, a Gaussian broadening of 3.8 ppm along the MAS dimension was used). (c) Theoretical p(Cq,ηq) distribution of the model structure in panel a; Cq ) 5.07(0.52) MHz, ηq ) 0.22(0.14), and δiso ) 42.6(7.6) ppm. (d) p(Cq,ηq) distribution as obtained from the fit of the spectrum in panel a to eq 4; Cq ) 5.14(0.49) MHz, ηq ) 0.21(0.14), and δiso ) 41.3(8.0) ppm. The dashed line represents the Cq versus ηq correlation using the method of Grandinetti and co-workers,4 yielding Cq ) 5.10(0.50) MHz, ηq ) 0.19(0.11), and δiso ) 42.7(7.0) ppm.

deviations). Such a level of agreement is very to close the one obtained in ref 18, despite the significant difference in the md coefficient. Structural Inversion of Oxygen-17 NMR Spectrum. The procedures described above can be merged to yield a method of structural inVersion of the 17O DAS NMR spectrum. We first determine the accuracy of this method, applying it to the 17 O DAS NMR spectrum generated with all of our model structures. This gives mean (and standard deviation) values of Ω ) 141.8°(13.7°), d ) 1.607(0.012) Å, and a correlation coefficient FΩ,d ) -0.329, which are very close to those of the model structures, Ω ) 143.0°(12.7)°, d ) 1.604(0.011) Å, and FΩ,d ) -0.492. The results of the inversion of experimental 17O DAS NMR4 is displayed in Figure 8. Concerning the fitting of the spectrum, our approach and the approach in ref 4, provide very similar levels of agreement, 3.5% and 3.7%, respectively. However, the slopes of the Cq versus ηq correlation strongly differ as displayed in Figure 8c. It is worth noticing that our method predicts a slope that is not only better in agreement with our theoretical data, but also with the experimental data of the crystalline silica polymorphs. The parameters of the NMR j q ) 5.12, σC distribution p(Cq, ηq, and δiso) in eqs 4 and 5 are C ) 0.48, aη ) -0.197, bη ) 1.145, ση ) 0.06, and aδ ) -2.301, bδ ) 48.199, and σδ ) 7.48. Statistics of the NMR parameters calculated using this distribution are summarized in Table 4. As expected from the discussion of the 17O NMR parameter distributions, we obtain mean values that are very close to those of Grandinetti and co-workers, but their respective standard deviations are different, especially for the quadrupolar asym-

metry parameter ηq and the isotropic chemical shift δiso. In this respect, it would be of great interest to process data from ref 7 following our approach. The result of the mapping of p(Cq,ηq) (Figure 8c) into the distribution of the Si-O-Si angle and Si-O distance p(Ω,d) using eqs 6 and 7 is shown in Figure 8d, and statistics are given in Table 5. Obtained values for both the Si-O-Si angle and Si-O distance distribution are in very good agreement with previous diffraction studies.1,46 The major difference from results of Grandinetti and co-workers mainly concerns the Si-O-Si angle distribution width. In addition, we do not observe a decrease in the trend of the Si-O distance with a decrease in the Si-O-Si angle; the value of the correlation coefficient FΩ,d ) 0.1 is rather small. Comparison of Table 5 with Table 2 clearly suggests that most of the N-fold rings in vitreous silica are of size greater than 4. At this stage, it is of great interest to investigate the consistency of the 17O NMR predicted angle distribution and that derived from the 29Si NMR data. Structural Inversion of Silicon-29 NMR Spectrum. To describe the correlation of the 29Si isotropic chemical shift δiso with four Si-O-Si angles of the tetrahedral linkages, we follow the method of Mauri et al.2 We consider that the contribution of each of the four angles Ωn is independent so that 4

δiso )

∑

∑

1 F(Ωn) with F(Ω) ) ck cos(kΩ) 4 n)1 k)0,1,2

(8)

The least squares fit of our 29Si NMR data to eq 8 yields c0 ) -102.36, c1 ) -6.95, and c2 ) -33.82, and the predicted 29Si



Figure 6. Contour plots of theoretical two-dimensional distributions (a) p(Cq,Ω), (b) p(ηq,Ω), (c) p(δiso,Ω), (d) p(Cq,d), (e) p(ηq,d), and (f) p(δiso,d) for the oxygen atoms in our model structures of vitreous silica. Ω is the Si-O-Si bond angle, and d is the Si-O bond distance. Solid circles (b) represent the averaged values over all model structures for three-, four-, and other N-membered rings.

Figure 7. Comparison between (a) the two-dimensional distribution p(Si-O, Si-O-Si) of the model structures of vitreous silica and (b) the predicted distribution using eqs 6 and 7 and the GIPAW 17O NMR quadrupolar parameters of the model structures.

δiso is found to agree within 1.0 ppm (standard deviation) with the GIPAW values (Figure 9a). For the crystalline silica polymorphs, an agreement of 1.5 ppm is obtained. F(Ω) is

displayed in Figure 9b and compared with the function which was derived in ref 2. The observed discrepancy is ascribed to the fact that the reconstruction of all electron wave functions is

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Charpentier et al.

Figure 8. Comparison between the fitted (solid lines) and experimental (dashed lines; Clark et al., ref 4) 17O DAS NMR spectrum using (a) their method and (b) our method (see text). (c) Comparison between the distribution p(Cq,ηq) obtained from the fit (a) (dotted-dashed lines) and fit (b) (solid lines). Experimental values of (Cq,ηq) for crystalline silica polymorphs are also shown. (d) The theoretical distribution of the Si-O-Si bond angle and the Si-O bond length obtained from the distribution of (Cq,ηq) displayed in (c) (solid lines). The mean value of (Si-O-Si, Si-O) is also shown (solid ().

TABLE 4: Statistics of 17O NMR Parameters Calculated with Distribution p(Cq,ηq,δiso) Extracted from the Fit of the Experimental 17O DAS NMR Spectrum (Figure 8) to Eq 4 Clark et al.4

this work mean

standard deviation

5.07 MHz Cq ηq 0.157 δiso 36.5 ppm

0.453 MHz 0.095a 7.55 ppma

a

mean

standard deviation

5.08 MHz 0.150 36.7 ppm

0.372 MHzb 0.0414b 4.30 ppmb

a Correlation coefficients FCq,ηq ) -0.798, FCq,δiso ) -0.138, and Fηq,δiso ) 0.110. b The fit of the spectrum using the method of Clark et al.4 yields the relationships Cq ) -8.62ηq + 6.29 and a Gaussian broadening of 4.2 ppm.

fully considered in the present case with the GIPAW method, which was not used in the previous work. Present results are therefore expected to be much more accurate. Equation 8 can be used to compute the mean and standard deviation of the Si-O-Si angle distribution p(Ω) from the NMR spectrum. Assuming a Gaussian distribution

p(Ω) ) k exp

( )

j 2 Ω-Ω sin(Ω) 2σΩ2

where k is a normalization constant, the spectrum is then calculated according to

G(δiso) )

4

(

4

29

(9)

Si MAS NMR

)

1 dΩnp(Ωn) × δ δiso - ∑ F(Ωn) ∫∏ 4 x)1 n)1

(10)

assuming the statistical independence of each of the bond angles Ωn. Indeed, a calculation performed over all of the quadruplets

Ωn)1,4 gives a small correlation coefficient FΩ ) 0.1. Fitting G(δiso) to the 29Si NMR spectrum of interest yields the j and σΩ, then using eq 9, the mean and standard parameters Ω deviation of the Si-O-Si angle Ω can be determined. Applied to our model structures for which Ω ) 143.0°(12.7)°, our inversion procedure yields Ω ) 142.8°(14.7)°. The small overestimation of the standard deviation σΩ indicates a weak correlation between the four bond angles Ωn. Such a correlation could be assigned to the structural regularity of three-membered rings. Nevertheless, our model sizes are presently too small to properly address this question. Panels c and d of Figure 9 show the inversion of the 29Si MAS NMR spectrum of the vitreous silica studied in ref 4 (fitted to a Gaussian line shape centered at -111.2 ppm with a full width at half-maximum (fwhm) of 12 ppm) with eq 10. The j ) 153.3° and σΩ ) 12.0°, yielding obtained parameters are Ω Ω ) 148.4°(10.8)°. These values are in agreement with the 17O NMR predicted values (Table 5). Consequently, both the silicon29 model (eqs 9 and 10) and oxygen-17 model (eqs 4 and 5) are consistent and applied here to 29Si and 17O NMR data acquired on the same sample. In another study,50 a much narrower 29Si MAS NMR described by a Weibull function with the maximum at -111.8 ppm and a fwhm of 9.5 ppm was j ) reported. Our inversion scheme yields Ω ) 149.0°(8.3)° (Ω 151.4° and σΩ ) 8.7°), values which are close to results of Neuefind and Liss,1 Ω ) 146.7°(7.3)°. The difference in width of the two spectra may be due either to a real structural difference between the two samples or to a residual NMR broadening resulting from 17O isotopic enrichment. A discussion of this issue is out of the scope of the present work. Both results are consistent with a recent review3 concluding that, after



Figure 9. (a) Comparison between the predicted 29Si isotropic chemical shift δiso from Si-O-Si angles (eqs 8-10) and GIPAW values of our model structures of vitreous silica. (b) Predicted variation of 29Si δiso as a function of the Si-O-Si bond angle Ω [F(Ω) eq 8] using parameters determined by Mauri et al.2 and in this work. (c) Experimental (solid lines) and simulated (dashed lines) 29Si MAS NMR spectra of vitreous silica. (d) Si-O-Si bond angle distributions as obtained from diffraction (Neuefind and Liss, ref 1) and inversion of experimental 29Si MAS NMR spectra plotted in panel c, employing eqs 9 and 10.

TABLE 5: Statistics of the Structural Parameters Calculated with Distribution p(Si-O-Si,Si-O) Extracted from the Fit of the Experimental 17O DAS NMR spectrum (Figure 8) to eq 4a Clark et al.4

this work mean Si-O-Si Si-O a

147.1° 1.597Å

standard deviation 11.17° 0.0106Å

mean 146.6° 1.583Å

diffraction1,46


mean 148.3° 1.605Å


Correlation coefficient FSi-O-Si,Si-O ) 0.100.

surveying the literature, the most probable bond angle should be close to 147° and the standard deviation around 9.8°-12.8°. Our result points out, that despite the 29Si δiso dependence upon the four angles of the tetrahedral linkage, it may serve as valuable data to derive reliable information on the Si-O-Si bond angle distribution. Conclusion In this paper, we have combined first-principles calculations of NMR parameters with molecular dynamics simulations to investigate the structure of vitreous silica. The 17O and 29Si NMR parameters of N-fold rings, correlation between the ring size distribution, and glass density as well as the Si-O-Si angle and Si-O distance distributions have been studied. Within the range of our model structures, we observe the increasing trend of the fraction of the small rings with a decreasing glass density. Well-resolved NMR parameters of three- and four-membered rings with respect to higher N-membered rings are predicted. Accordingly, it is expected that they could be better identified in experimental spectra. We present an analytical model of 17O NMR distribution consistent with our simulations, which provides a more reliable extraction of the NMR parameters

distribution from the experimental spectra. With the help of the relationships between 17O and 29Si NMR parameters with the Si-O-Si bond angle and Si-O bond length, we develop a structural inVersion method to reconstruct the Si-O-Si bond angle and Si-O bond length distributions from the 17O and 29Si NMR spectrum. When applied to published 17O and 29Si NMR data from the same sample, our method yields consistent information about the Si-O-Si angle distribution; mean values of 147.1° and 148.4°, respectively, and standard deviation values of 11.2° and 10.8°, respectively. The consistency of both approaches indicates the robustness of our method. Our results of the angular distribution function are in very good agreement with diffraction data. Additionnally, the comparison of these experimental values to our MD data, clearly suggests that most of the N-fold rings in vitreous silica are of size greater than 4. Consequently, an experimental approach combining diffraction and NMR will significantly improve the accuracy of structure determination (including torsion angles), mostly because of the high sensitivity of NMR to the bond angles distribution. An open question remains concerning the correlation between Si-O-Si angles and associated Si-O bond lengths.

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This work clearly demonstrates that the GIPAW method combined with MD simulations offer appealing novel opportunities to explore the structural content of NMR data. While we have presented analysis of 17O and 29Si NMR spectra, it can be expected that the state-of-the-art NMR methods such as 17 O-29Si correlation spectroscopy will further refine our structural analysis. Such work is in progress. Appendix Kernel Density Estimation. Given a set of d-dimensional data, denoted (xi), of size N, the density probability function p(x) can be approximated using a kernel estimate defined as N

p(x) )

∑

1 K (x - xi) N i)1 H

(11)

where the kernel K is (a Gaussian kernel have been employed in the present work)

KH(x) )

1 1 exp - xTH-1x 2 (2π)d⁄2det1⁄2 H

(

)

(12)

H is the so called bandwidth matrix. For sake of simplicity, it has been chosen proportional to the covariance matrix σ of the data (xi)35 H ) N-[1/(4+d)]σ. Simulation of NMR Spectra. The NMR spectra are simulated as follows. We consider, for example, a set of 17O NMR parameters, denoted {Cq,ηq,δiso}i, and the corresponding set of i ), where b ν represents a one or two NMR spectra I(ν b;Cqi ,ηqi ,δiso component frequency vector for a one- or two-dimensional NMR spectra, respectively. The theoretical NMR spectrum could be simply obtained as the sum

.I(νb ) )

∑ I(νb ;Ciq, ηiq, δisoi )

(13)

i

However, because of limited statistics, NMR spectra (especially two-dimensional) obtained with eq 13 generally exhibits too many undesirable discontinuities. Generally, smoothing in the frequency space b ν is performed so as to obtain a more realistic line shape. Here, we employ a new method consisting of building first a kernel density estimation of the density probability function p(Cq,ηq,δiso), from the initial set of NMR parameters {Cq,ηq,δiso}i (Kernel Density Estimation). This corresponds to a smoothing in the NMR parameter space, which is then discretized on a uniform three-dimensional grid, denoted k ). The NMR spectra at each node of the grid (Cqk ,ηqk ,δiso k ) are generated only once. The NMR spectrum of I(ν b;Cqk ,ηqk ,δiso the model structure is obtained as

∫ dCqdηqdδisoI(Vb;Cq, ηq, δiso) × p(Cq, ηq, δiso) k k ) ∆Cq∆ηq∆δiso∑ I(V b ;Ckq, ηkq, δiso ) × p(Ckq, ηkq, δiso )

I(V b) )

(14)

k

where ∆x denotes the discretization step length of parameter x. In the case of a two-dimensional DAS (and MQMAS) experiment, the integration over the three NMR dimensional space can be simplified into a two-dimensional integration over (Cq,ηq) only.52 Acknowledgment. We thank P.J. Grandinetti for providing the 17O DAS NMR data. The NMR calculations were performed at the Supercomputer facility (CCRT) of the CEA under Grant p556. Structural models were generated using the computer facility of the FZ Ju¨lich. This research is supported in part by the Agence Nationale de la Recherche, Grant RMNSOLIDE-

HR-HC (2005-2008). P.K. acknowledges a Heisenberg award of the German Science Foundation (KR 1805/9). Supporting Information Available: GIPAW calculated 29Si and 17O NMR parameters of the crystalline silica polymorphs, coesite, R-quartz, and cristobalite; ring, structural, and NMR parameter statistics for each of the studied model structures; plots of theoretical two-dimensional distributions p(Cq,Ωσ) and p(ηq,κσ) for oxygen-17; and variation of 17O NMR parameters under isotropic rescaling of the unit cell volume for one of the model structure. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Neuefind, J.; Liss, K. D. Ber. Bunsen-Ges. 1996, 100, 1341–1349. (2) Mauri, F.; Pasquarello, A.; Pfrommer, B.; Yoon, Y.-G.; Louie, S. Phys. ReV. B 2000, 62, R4786–R4789. (3) Yuan, X.; Cormack, A. J. Non-Cryst. Solids 2003, 319, 31–43. (4) Clark, T. M.; Grandinetti, P. J.; Florian, P.; Stebbins, J. Phys. ReV. B 2004, 70, 064202. (5) Gladden, L. F.; Carpenter, T. A.; Elliot, S. R. Philos. Mag. B 1986, 53, L81–L87. (6) Pettifer, R. F.; Dupree, R.; Farnan, I.; Sternberg, U. J. Non-Cryst. Solids 1988, 106, 408–412. (7) Farnan, I.; Grandinetti, P. J.; Baltisberger, J. H.; Stebbins, J. F.; Werner, U.; Eastman, M. A.; Pines, A Nature (London) 1992, 358, 31–35. (8) Tossell, J. A.; Lazzaretti, P. Chem. Phys. 1987, 112, 205–212. (9) Tossell, J. A.; Lazzaretti, P. Phys. Chem. Miner. 1988, 15, 564– 569. (10) Tossell, J. J. Non-Cryst. Solids 1990, 120, 13–19. (11) Lindsay, C. G.; Tossell, J. A. Phys. Chem. Miner. 1991, 18, 191– 198. (12) Bussemer, B.; Schro¨der, K.-P.; Sauer, J. Solid State Nucl. Magn. Reson. 1997, 9, 155–164. (13) Vermillion, K. E.; Florian, P.; Grandinetti, P. J. Chem. Phys. 1998, 108, 7274–7285. (14) Xue, X.; Kanzaki, M. Phys. Chem. Miner. 1998, 26, 14–30. (15) Xue, X.; Kanzaki, M. J. Phys. Chem. B 1999, 103, 10816–10830. (16) Xue, X.; Kanzaki, M. Solid State Nucl. Magn. Reson. 2000, 16, 245–259. (17) Clark, T. M.; Grandinetti, P. J.; Florian, P.; Stebbins, J. F. J. Phys. Chem. 2001, 105, 12257–12265. (18) Clark, T. M.; Grandinetti, P. J. J. Phys.: Condens. Matter 2003, 15, S2387–S2395. (19) Mauri, F.; Pfrommer, B. G.; Louie, S. G. Phys. ReV. Lett. 1996, 77, 5300–5303. (20) Pickard, C. J.; Mauri, F. Phys. ReV. B 2001, 63, 245101. (21) Profeta, M.; Mauri, F.; Pickard, C. J. J. Am. Chem. Soc. 2003, 125, 541–548. (22) Charpentier, T.; Ispas, S.; Profeta, M.; Mauri, F.; Pickard, C. J. J. Phys. Chem. B 2004, 108, 4147–4161. (23) Profeta, M.; Benoit, M.; Mauri, F.; Pickard, C. J. J. Am. Chem. Soc. 2004, 126, 12628–12635. (24) Tielens, F.; Gervais, C.; Lambert, J. F.; Mauri, F.; Costa, D. Chem. Mater. 2008, 20, 3336–3344. (25) Ashbrook, S. E.; Polles, L. L.; Pickard, C. J.; Berry, J.; Wimperis, S.; Farnan, I. Phys. Chem. Chem. Phys. 2007, 9, 1587–1598. (26) Ashbrook, S. E.; Berry, A. J.; Frost, D. J.; Gregorovic, A.; Pickard, C. J.; Readman, J. E.; Wimperis, S. J. Am. Chem. Soc. 2007, 129, 13213– 13224. (27) Sen, S.; Russell, C. A.; Murkerji, T. Phys. ReV. B 2005, 72, 174205. (28) Wooten, F.; Winer, K.; Weaire, D. Phys. ReV. Lett. 1985, 54, 1392– 1395. (29) Tu, Y.; Tersoff, J.; Grinstein, G.; Vanderbilt, D. Phys. ReV. Lett. 1998, 81, 4899–4902. (30) Vink, R. L. C.; Barkema, G. T. Phys. ReV. B 2003, 67, 245201. (31) Kroll, P. J. Mater. Chem. 2003, 13, 1657–1668. (32) Kroll, P. J. Non-Cryst. Solids 2005, 351, 1127–1132. (33) Kroll, P. Modelling Amorphous Ceramic Structures. In Ceramics Science and Technology; Riedel, R., Chen, I.-W., Eds., University of Chicago Press: New York, 2007. (34) Pfrommer, B.; Raczkowski, D.; Canning, A.; Louie, S.; Mauri, F.; Cote, M.; Yoon, Y.; Pickard, C.; Haynes, P. PARAllel Total Energy Code; National Energy Research Scientific Computing Center (NERSC): Berkeley, CA, 1990, http://www.nersc.gov/projects/paratec/. (35) Scott, D. W. MultiVariate Density Estimation: Theory, Practice and Visualization; Wiley-Interscience: New York, 1992. (36) King, S. V. Nature (London) 1967, 213, 1112–1113.

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J. Phys. Chem. C, Vol. 113, No. 18, 2009 7929 (46) Grimley, D. I.; Wright, A. C.; Sinclair, R. N.; Martin, S. W. J. Non-Cryst. Solids 1990, 129, 49–64. (47) Uchino, T.; Kitagawa, Y.; Yoko, T. Phys. ReV. B 2000, 61, 234– 240. (48) Sakellariou, D.; Jacquinot, J.-F.; Charpentier, T. Chem. Phys. Lett. 2005, 411, 171–174. (49) Grandinetti, P. J.; Baltisberger, J.; Farnan, I.; Stebbins, J.; Werner, U.; Pines, A. J. Phys. Chem. 1995, 99, 12341–12348. (50) Mahler, J.; Sebald, A. Solid State Nucl. Magn. Reson. 1995, 5, 63– 78. (51) The full width at half-maximum height (fwhm) is transformed to a standard deviation, assuming a Gaussian distribution, i.e., fwhm ) 2(2 ln 2σ)1/2. (52) Angeli, F.; Charpentier, T.; Faucon, P.; Petit, J.-C. J. Phys. Chem. B 1999, 103, 10356–10364.

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First-Principles Nuclear Magnetic Resonance Structural Analysis of

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