Article pubs.acs.org/JPCB
Cite This: J. Phys. Chem. B 2017, 121, 10726-10732
Oxygen Packing Fraction and the Structure of Silicon and Germanium Oxide Glasses XiangPo Du† and John S. Tse*,†,‡ †
State Key Laboratory for Superhard Materials, Jilin University, Changchun 130012, P. R. China Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E2, Canada
‡
ABSTRACT: The recently proposed relationship between the oxygen volume fraction and topological ordering in solid and liquid oxide glasses at high pressure is examined with Bader’s atoms-in-molecules (AIM) theory using glass structures generated from first principles molecular dynamics calculations. It is shown that the atomic (O/Si and O/Ge) volume ratio derived from AIM theory is not constant with pressure. This finding is due to the continuous change in the electron topology under compression. Unlike crystalline solids, there is no distinctive transition pressure for Si−O and Ge−O coordination in a glass; instead, the changes are gradual and continuous over a broad pressure range. Therefore, relating a unique Si−O or Ge−O coordination number to the properties of the glass at a given pressure is difficult.
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INTRODUCTION The elucidation of the structures of crystalline and disordered solids (glasses) based on the packing of the constituent atoms is a very powerful concept. An understanding of the underlying topical ordering can be used to forecast a variety of properties. The extension of this concept to high-pressure solids is not straightforward as the “size” of an individual atom is influenced by the surroundings and differs significantly from that under ambient conditions. A strategy originally proposed by Goldschmidt et al.1 is often used to estimate atomic radii from observed bond lengths. However, it is already known that ionic radii may not be realistic indicators of the sizes of ions.2 The size of an atom is particularly difficult to define under high pressure when there is substantial rearrangement of the valence electron density and, sometimes, a fundamentally altered nature of the chemical bond.3 Gibbs pioneered the idea of assigning atomic size using information on the electron density,4 obtained either from a high-resolution density map from X-ray crystallography or from first principles calculations.5 This approach is based on the atoms-in-molecules (AIM) quantum theory.6,7 In the AIM theory, atomic regions in a molecule or a solid may be unambiguously defined from the mathematical analysis of the electron topology (the distribution of the electron density), which is governed by a set of rigorous theoretical laws.6 The adaptation of this concept in using the volume defined by the AIM theory removes the arbitrariness and ambiguity of assigning a spherical radius to an otherwise distorted atomic density under extreme environments. This approach has recently been applied and shown to be successful in explaining the atomic mechanisms of high-pressure devitrification8 and the amorphous polymorphism of metallic glasses.9 © 2017 American Chemical Society
A relationship between the structures of crystalline SiO2 and the topological packing of the oxygen atoms under high pressure was first proposed by Binggeli and Chelikowski.10 In a recent paper,11 it was suggested that the properties of a wide variety of solid and liquid oxide glasses under extreme conditions can also be categorized from the oxygen packing fraction calculated from the ionic radii derived from empirical bond length relationships. Because the local geometry surrounding the A atoms (i.e., A−O coordination number, CN) in a typical AO2 (A = Si and Ge) oxide glass changes with pressure, different formulae must be used for different pressure regions. To apply the procedure, one must choose, a priori, the structural type (e.g., 4-, 5-, or 6coordinate) for the oxide glasses, thereby prejudging the structure at a given pressure range. Theoretical studies of highpressure silica and germanium glasses have shown that Si−O12 and Ge−O CNs13 change continuously with pressure and that there is no similar abrupt change in coordination as in the crystalline solids. Furthermore, the CN of Si and Ge in the glass at a given pressure is not unique and is often a mixture of Si−O and Ge−O coordinations. This characteristic calls into question the validity of the assignment of a unique A−O CN in oxide glasses. In addition, there is no sound justification supporting the assumption of a constant atomic radius ratio for a given bond type over a broad pressure range. In view of these problems, here, we reported the results of an analysis of the structure and relevant properties of SiO2 and GeO2 glasses at different pressures using the AIM theory and the theoretical structures obtained from first Received: September 20, 2017 Revised: November 2, 2017 Published: November 3, 2017 10726
DOI: 10.1021/acs.jpcb.7b09357 J. Phys. Chem. B 2017, 121, 10726−10732
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Figure 1. Plot of the Laplacian in a plane defined by the atoms forming the O−Si−O bonds in (a) quartz at 0 GPa, (b) stishovite at 70 GPa, and (c) cotunnite at 150 GPa.
Table 1. Calculated Bader Volume for Si and O, Volume Ratios, and Packing Fraction ηO for Crystalline Phases of SiO2 pressure (GPa) 0
25
70 90 150
quartz coesite cristobalite quartz coesite cristobalite stishovite stishovite α-PbO2 α-PbO2 cotunnite cotunnite
V(O) (Å3)
V(Si) (Å3)
V(O)/V(Si)
ηO = V(O)/V(total)
18.482 16.288 21.355 12.860 13.328 12.859 9.862 8.794 8.846 8.551 8.858 7.997
3.052 3.037 3.140 2.775 2.721 2.825 2.575 2.316 2.290 2.221 2.307 2.121
6.056 5.363 6.801 4.634 4.897 4.552 3.830 3.874 3.864 3.876 3.840 3.770
0.924 0.915 0.931 0.903 0.907 0.901 0.880 0.884 0.885 0.885 0.885 0.883
principles molecular dynamics (FPMD) calculations.13,14 Previously, we have shown that a variety of experimental observations, such as X-ray diffraction (XRD) patterns, densities, sound velocities, and O−K absorption spectra of dense silica, can be correctly reproduced from the structure generated by isobaric−isothermal (NPT) molecular dynamics (MD).11 Here, we used the trajectories obtained in previous work11 for the analysis of SiO2 glass and performed new simulations on GeO2 glass using a similar approach. In addition, calculations of the high-pressure crystalline polymorphs of SiO2 and GeO2 were also performed. The results reinforce the speculations raised above. It is found that although there is a correlation between the oxygen packing fraction and the coordination number, the increase in coordination number in glasses is continuous and does not occur in discrete steps. The mixing of different coordination numbers is always expected in compressed glasses.
used in the Brillouin zone sampling. Calculations with a larger supercell (324 atoms) were also performed at 30 and 90 GPa. The calculated XRD patterns and coordination number distributions are in good agreement with those obtained with the smaller supercell (216 atoms).
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RESULTS AND DISCUSSION We first discuss the results for SiO2. There are several known or proposed high-pressure crystalline forms of SiO2. Here, we optimized the structures of quartz, cristobalite, and coesite with tetrahedral Si−O coordinations (4-coordinated Si), stishovite and α-PbO2 with 6 Si−O nearest neighbor oxygen, and the proposed cotunnite with an even higher Si−O CN. We computed the (Bader) volumes for all of the atoms according to AIM theory. An atom in the crystal is defined as a region of real space bounded by surfaces through which there is no flux in the gradient vector field (the “zero-flux” boundary) defined from the calculation of the Laplacian of the electron density.6 The oxygen occupancy fraction (ηO) is simply the ratio of oxygen Bader volume to the total volume.11 Figure 1 shows the Laplacians plotted in the plane defined by coplanar O···Si···O atoms of representative low and high-pressure polymorphs (quartz at 0 GPa, sitshovite at 70 GPa, and cotunnite at 150 GPa) of SiO2. The significant distortions in the Si and O Bader volumes bound by the zero-flux boundaries at high pressure are evident. The calculated atomic Bader volumes at different pressures are summarized in Table 1 and Figure 2. The oxygen atoms are very compressible, as the “volume” decreases significantly as the pressure is increased. The Bader volumes change from 21.36 Å3 in cristobalite at ambient pressure to 8 Å3 in cotunnite at 150 GPa, a reduction of almost 2/3! In comparison, the Si atom is rather “rigid” and only decreases from 3 Å3 at 0 pressure to 2 Å3 at
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METHODS First principles molecular dynamics calculations implemented in the Vienna ab initio simulation package15 were used to simulate the structural properties of glassy GeO2. An amorphous GeO2 model was prepared by melting the ambient quartz structure solid at 2800 K first with a canonical ensemble (NVT) and then cooling the structure slowly and stepwise to 300 K with the isothermal−isobaric ensemble. A supercell containing 216 atoms was used. Sufficient simulation steps (>60 000, 1 fs) were used to ensure equilibrium at each target pressure. The Perdew−Burke− Ernzerhof16 generalized gradient approximation was chosen as the exchange−correlation function, and augmented plane wave potentials for Ge and O atoms were used to describe the electron−ion interactions. For all calculations, a plane wave basis set with an energy cutoff of 395 eV was used. The Γ point was 10727
DOI: 10.1021/acs.jpcb.7b09357 J. Phys. Chem. B 2017, 121, 10726−10732
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The Journal of Physical Chemistry B
to 12.52 Å3 at 20 GPa. Again, these values are close to those of quartz below 25 GPa. At 30 GPa, the average volume is 10.93 Å3, and this decreases steadily to 8.91 Å3 at 90 GPa, the highest pressure studied. The Bader volume of the Si atoms decreases smoothly and, similar to the crystalline phases, decreases from 3.34 Å3 at ambient pressure to 2.43 Å3 at 90 GPa. A summary of the volumes and oxygen packing fraction in dense silica are given in Table 2. Once again, at the highest pressure, the smallest packing fraction ηO is 0.88. It was clearly shown in the previous FPMD study (Figure 4a of ref 12) that the Si−O CN of silica glass is 4 at 0 pressure and that 5- and 6-coordinate Si−O starts to appear near 15 GPa at the onset of permanent densification.17 Between 15 and 110 GPa, the glass consists of mixed 4-, 5-, and 6-coordinate Si−O at different proportions. The 4-coordinate Si completely vanishes at 60 GPa, and the 6-coordinate Si−O structure becomes dominant above 110 GPa. Therefore, there is no corresponding trend in the volume or Si−O CN in silica glass with the high-pressure crystalline phases held stable at the same pressure regime. A plot of the Si−O CN against the oxygen packing fraction ηO is shown in Figure 4a and shows remarkable similarity with the pattern reported in Figure 2 of ref 11. In previous work,11 an abrupt increase in CN is observed at ηO = 0.53, signaling the change from 4- to 6-coordinate (note that ηO in ref 11 is not related to the Bader volume employed here, as according to the atomic radius defined in ref 11, ηO should increase with pressure, but the opposite is predicted by the Bader analysis). Despite the similarity, the physics behind the two plots are very different, and the present result does not vindicate the use of the empirical bond length relationship in ref 11 because a Si−O CN had to be chosen for the glass to evaluate the oxygen packing fraction. As shown by the MD structures, it is not possible to assign a unique CN between 20 and 60 GPa. Now, we discuss the new MD results for high-pressure GeO2 glass. To validate the structures obtained from MD simulations, the diffraction pattern, the coordination number at different pressures, and the equation of state are compared to the experimental results in Figure 5a−c, respectively. The experimental diffraction patterns show two major bands between Q = 2 and 6 Å−1 with a small shoulder beginning to appear at Q = 4 Å−118 at pressures above 80 GPa. These features are correctly reproduced in the calculated patterns (Figure 5a). In particular, the shoulder at Q = 4 Å−1 becomes progressively more prominent as the pressure increases and emerges as a peak at 150 GPa Q = 4 Å−1. The Ge−O coordination numbers as a function of pressure determined from the analysis of the MD structures are shown in Figure 5b. As shown in the figure, there is a broad distribution of the Ge−O coordination number due to Ge atoms in different environments. Again, it is not possible to assign a unique Ge−O CN at a given pressure, particularly at high pressure. At low pressure, the Ge−O CNs are tightly clustered but at high pressures they become more scattered. However, the theoretical Ge−O CNs reproduced the observed increasing trend with pressure quite closely. Furthermore, the calculated volumes at different pressures are also in good accordance with the experimental equations of state.18 In view of the agreement with several experimental measurements, the glass structures obtained from the FPMD calculations should be reliable. For comparison, we also optimized the structures of several highpressure crystalline forms of GeO2 because the Ge−O coordination number can be unambiguously identified in the crystals.
Figure 2. Variation of the Bader volume for Si and O in the crystalline phases of SiO2.
150 GPa. An abrupt decrease in the oxygen volume is observed at 25 GPa when quartz (Si−O CN = 4) transforms to stishovite (Si−O CN = 6). Further decreases were predicted at 70 and 150 GPa when SiO2 is in the α-PbO2 and cotunnite phases, respectively. The trend in the change in atomic volumes with increasing pressure is exactly as expected. Assuming that SiO2 is an ionic crystal, the normal ionic radius of O2− is already much larger than that of Si4+. After the transformation from 4- to 6- Si− O coordination at approximately 25 GPa, the Si atom becomes more ionic, and therefore, the atomic radius is smaller; further compression leads to an even more ionic (smaller) Si. However, it should be noted that at extreme pressures (>Mbar), there is a possibility that back-transfer of electrons from O to Si may occur. The calculated O/Si volume ratio and the oxygen packing fraction of the different structures under different pressures are presented in Table 2. From 0 to 25 GPa, the ηO values decrease Table 2. Calculated Average Bader Volumes for Si and O, Volume Ratios, and Packing Fractions ηO for SiO2 Glass at Different Pressures pressure (GPa)
V(Si) (Å3)
V(O) (Å3)
V(O)/V(Si)
ηO = V(O)/V(total)
0 10 20 30 50 70 90
3.344 3.265 3.081 2.776 2.606 2.496 2.425
21.250 15.437 12.523 10.925 9.972 9.261 8.911
6.354 4.728 4.064 3.935 3.826 3.710 3.675
0.927 0.904 0.890 0.887 0.884 0.881 0.880
rapidly from 0.924 (quartz) to 0.880 (stishovite). Even though the ηO values for both Si and O atoms decrease more gradually after 25 GPa, there is no apparent change in ηO within the numerical accuracy. Changes in the Si and O Bader volumes of all atoms in the SiO2 glass model with compression are plotted in Figure 3. Because the oxygen atoms are disordered, the volume distribution is quite widespread at 0 pressure but becomes much narrower as the pressure is increased. In the glass, the initial average volume is 21.25 Å3; this volume is composed of almost completely 4coordinate Si−O and is comparable to that of cristobalite. The volume decreases to approximately 15.44 Å3 at 10 GPa and then 10728
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Figure 3. Variation of the Bader volume for Si and O as a function of pressure in a SiO2 glass.
dominant, but there is already a small percentage (5%) of 5coordinate Ge−O. This result is in agreement with a previous MD study using empirical potential.13 The proportion of 4coordinate Ge−O decreases rapidly with pressure and disappears at 20 GPa. Concomitantly, the proportion of 5- and 6-oxygen coordinate Ge increases, with the former maximized at 20−30 GPa. At this pressure range, the 7-coordinate Ge−O starts to appear. At 70 GPa, the 5-coordinate Ge−O disappears completely, and at 80 GPa, the amount of 7-coordinate Ge−O is more than the amount of 6-coordinate Ge−O. This result is consistent with the experimental evidence, suggesting that a structural polymorphism occurs near 80 GPa.18 The change in local structure related to the coordination number is clearly identified in Figure 5c. The changes in the Ge−O coordination from 4 to 5 and 6 has a significant effect on the local structure. In 4-coordinate GeO2 glass, the O−Ge−O angles are close to the ideal tetrahedron value of 109.45°. In a perfectly 6-coordinate Ge−O octahedron, the O−Ge−O angle is expected to be 90°. Therefore, the O−Ge−O angles in 5-coordination should have angles between 90 and 109.45°, and for Ge−O CN > 6, due to the crowding of the nearest oxygen neighbors, the O−Ge−O angles will be smaller than 90°. These expectations are borne from the analysis of the distribution of the O−Ge−O angles as a function of pressure (Figure 9). At 0 GPa, the maximum distribution is indeed close to the tetrahedral angle. At 5 GPa, the distribution becomes bimodal with peaks at 80 and 100°. Above 15 GPa, only one peak is observed, close to 90°. At 50 GPa, the main peak is located at 80° with a smaller peak at 138°. The highangle peak indicates the pressing of more oxygen atoms into the first Ge-coordination sphere. The calculated Bader atomic volume and oxygen packing fractions as a function of pressure are summarized in Table 3. Because of the smaller O/Ge volume ratio, the oxygen packing fraction ηO is 0.810 at 0 pressure and decreases to 0.749 at 120 GPa. A plot of the coordination number vs ηO in Figure 4b shows a break at ηO ∼ 0.77 when the coordination number jumps from 4 to 6. Figure 4b also shows a similar trend as that in Figure 2A of ref 11 and Figure 6 of ref 18, in which the Ge−O coordination changes over a small interval of η. In ref 11, the change in ηO is found to be very small, between 0.67 and 0.7118 for the rutile-tocotunnite transition in GeO2. Here, using AIM formalism, the transition occurs between ηO = 0.78 and 0.76. Once again, without prior knowledge of the Ge−O CN of the underlying stable crystalline phases in the same pressure region, the CN−ηO correlation is not quantitative and may only be used as an empirical guide.
Figure 4. Plot of the variation of the coordination number with the oxygen packing fraction for (a) SiO2 and (b) GeO2 glass.
The computed Bader volumes for Ge and O atoms in crystals with different Ge−O CNs are compared in Figure 6. The Bader volumes of oxygen are larger than those of Ge, and the absolute values are smaller in GeO2 than those in SiO2. In addition, in GeO2, the Ge and O volumes are more comparable. This finding would indicate that GeO2 is not as ionic as SiO2. The lower energy empty Ge d-orbitals participate in bonding at much lower pressures than those in SiO2. In the quartz 4-coordinate Ge−O structure, the oxygen volume is 16 Å3, and this volume decreases abruptly with the 6-coordinate Ge−O rutile structure and continues to decrease in the high-pressure CaCl2, α-PbO2, and pyrite structures. Interestingly, there is an apparent drop in the Ge volume from 8.6 Å3 in the ambient quartz 4-coordinate Ge− O structure to 6 Å3 in the rutile and high-pressure structures with Ge−O CN > 6. The volume reduction becomes much smaller and only decreases to 5 Å3 in the pyrite structure. The volume change in the GeO2 glass follows a similar trend as the silica analogue (Figure 7). The volume reductions are smooth with increasing pressure. The average O volume at 0 GPa is 12.5 Å3, and it decreases to 7.5 Å3 at 120 GPa, a 40% decrease. The change in the average Ge volume from 7.5 to 5.2 Å3 at pressures from 0 to 120 GPa is more noticeable than that in SiO2. Figure 8 shows the variation of the Ge−O CN with increasing pressure in the GeO2 glass. At 0 pressure, 4-coordinate Ge−O is 10729
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Figure 5. Comparison of experimental and calculated (a) X-ray diffraction pattern, (b) coordination number, and (c) equation of state for GeO2 glass.
Figure 6. Variation of the Bader volume for Ge and O in the crystalline phases of GeO2.
Figure 7. Variation of the Bader volume for Ge and O as a function of pressure in GeO2 glass.
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CONCLUSIONS Pressure is a formidable thermodynamic variable that alters the structural, physical, and electronic properties of materials. In elemental solids with low-lying empty orbitals, such as the spatially diffused d-orbitals in Si and alkali and alkaline elements,3
compression can promote hybridization with the filled valence orbitals to alleviate the Pauli repulsion between the electrons in a crowded environment. The concept of chemical bonds and the well-established structural concepts applicable under ambient pressure need to be modified.19 In a theoretical study of silica at 10730
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the covalency in GeO2 at high pressure is significantly more important. Specific to the objective of this investigation, it is shown that the Si/O and Ge/O volume ratios, which are related to the atomic size ratio, are not constant, even in crystalline structures with the same coordination number. The situation is even more uncertain in the oxide glasses. Upon compression, the oxygen packing fraction will reach an asymptotic value. Therefore, it is no surprise that the Si−O and Ge−O CNs are related to ηO. The difficulty lies with the assignment of realistic atomic sizes in a compressed solid and the relationship to the Si− O and Ge−O CNs of the ambient structures. The FPMD calculations performed here show that the electronic structures change with pressure and that there should be no constant scaling rules of atomic sizes to bond lengths regardless of the local coordination. Experimentally, it has already been shown that the average CN varies continuously up to 50 GPa in SiO2 glass.23,24 The proposed oxygen packing model, which relies completely on the choice of oxygen size parameter, is too simplistic. Theoretical calculations support the suggestion of a polyamorphism transition in GeO2 glass with a dominant coordination number >6 above 80 GPa.18
Figure 8. Change and proportion of the coordination number in highpressure GeO2 glass.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
John S. Tse: 0000-0001-8389-7615 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We wish to thank the National Natural Science Foundation of China (No. 11474126) and the Canada Research Chair program for financial support.
Figure 9. Distribution of O−Ge−O angles in the structure of compressed GeO2 glass.
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200 GPa,20 it was found that Si and O ions deformed and became aspherical, leading to the formation of non-close-packed structures.21 The conventional concepts of bonds and bond length may not be applicable, and a correct description of the electronic structure requires a full quantum mechanical treatment.22 As demonstrated in this study, the structural behaviors of SiO2 and GeO2 under pressure may be similar, but the electronic structure cannot be viewed as simple ionic compounds. Clearly
REFERENCES
(1) Goldschmidt, V. M.; Barth, T.; Linde, T.; Zacharisasen, W. H. Skrifer Norske-Videns-kaps, Akademi 1926, 2, 1. (2) Shannon, R. D.; Prewitt, C. T. Effective Ionic Radii in Oxides and Fluorides. Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. 1969, 25, 925−946. (3) Tse, J. S.; Boldyreva, E. V. Electron Density of Crystalline Solids at High Pressure. In Modern Charge Density Analysis; Gatti, C., Macchi, P., Eds.; Springer-Verlag, 2011.
Table 3. Calculated Average Bader Volume for Ge and O, Volume Ratios, and Packing Fraction ηO for GeO2 Glass at Different Pressure pressure (GPa)
V(Ge) (Å3)
V(O) (Å3)
total (Å3)
V(O)/V(Ge)
ηO = V(O)/V(total)
0 5 10 15 20 30 50 70 90 120 150
8.551 7.510 7.112 6.853 6.597 6.324 5.900 5.611 5.386 5.112 4.908
18.207 13.147 11.693 11.298 10.734 10.086 9.271 8.753 8.371 7.763 7.331
44.965 33.804 30.498 29.449 28.065 22.734 24.442 23.117 22.128 20.638 19.570
1.218 1.755 1.644 1.647 1.626 1.595 1.572 1.560 1.555 1.517 1.504
0.810 0.776 0.767 0.767 0.765 0.761 0.759 0.757 0.757 0.752 0.749
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Article
The Journal of Physical Chemistry B (4) Gibbs, G. V.; Spackman, M. A.; Boisen, M. B., Jr. Bonded and Promolecule Radii for Molecules and Crystals. Am. Mineral. 1992, 77, 741−750. (5) Gibbs, G. V.; Wang, D.; Hin, C.; Ross, N. L.; Cox, D. F.; Crawford, T. D.; Spackman, M. A.; Angel, R. J. Properties of Atoms under Pressure: Bonded Interactions of the Atoms in Three Perovskites. J. Chem. Phys. 2012, 137, No. 164313. (6) Bader, R. F. W. A Quantum Theory. In Atoms in Molecules; Oxford University Press: New York, 1990. (7) Bader, R. F. W.; Beddall, P. M.; Cade, P. E. Partitioning and Characterization of Molecular Charge Distributions. J. Am. Chem. Soc. 1971, 93, 3095−3107. (8) Wu, M.; Tse, J. S.; Wang, S.; Wang, C.-Z.; Jiang, J. Z. Origin of Pressure-induced Crystallization of Ce75Al25 Metallic Glass. Nat. Commun. 2015, 6, No. 6493. (9) Wu, M.; Lou, H.; Tse, J. S.; Liu, H.; Pan, Y.; Takahama, K.; Matsuoka, T.; Shimizu, K.; Jiang, J. Z. Pressure-induced Polyamorphism in A Main-group Metallic Glass. Phys. Rev. B 2016, 94, No. 054201. (10) Binggeli, N.; Chelikowski, J. Structural Transformation of Quartz at High Pressures. Nature 1991, 353, 344−346. (11) Zeidler, A.; Salmon, P. S.; Skinner, L. B. Packing and the Structural Transformations in Liquid and Amorphous Oxides from Ambient to Extreme Conditions. Proc. Natl. Acad. Sci. U.S.A. 2014, 111, 10045− 10048. (12) Wu, M.; Liang, Y.; Jiang, J.-Z.; Tse, J. S. Structure and Properties of Dense Silica Glass. Sci. Rep. 2012, 2, No. 398. (13) Guthrie, M.; Tulk, C. A.; Benmore, C. J.; Xu, J.; Yarger, J. L.; Klug, D. D.; Tse, J. S.; Mao, H.-K.; Hemley, R. J. Formation and Structure of a Dense Octahedral Glass. Phys. Rev. Lett. 2004, 93, No. 115502. (14) Tse, J. S. ab initio Molecular Dynamics with Density Functional Theory. Annu. Rev. Phys. Chem. 2002, 53, 249−290. (15) Kresse, G.; Furthmuller, J. Efficient Iterative Schemes for ab initio Total-energy Calculations using A Plane-wave Basis Set. Phys. Rev. B 1996, 54, 11169−11186. (16) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (17) Bridgman, P. W.; Šimon, I. Effects of Very High Pressures on Glass. J. Appl. Phys. 1953, 24, 405−413. (18) Kono, Y.; Kenny-Beson, C.; Iluta, D.; Shibazaki, Y.; Wang, Y.; Shen, G. Ultrahigh-pressure Polyamorphism in GeO2 Glass with Coordination Number >6. Proc. Natl. Acad. Sci. U.S.A. 2016, 113, 3436− 3441. (19) Bader, R. F. W.; Austen, M. A. Properties of Atoms in Molecules: Atoms under Pressure. J. Chem. Phys. 1997, 107, 4271−4285. (20) Oganov, A. R.; Gillan, M. J.; Price, G. D. Structural Stability of Silica at High Pressures and Temperatures. Phys. Rev. B 2005, 71, No. 064104. (21) Gibbs, G. V.; Tamada, O.; Boissen, M. B., Jr. Atomic and Ionic Radii: a Comparison with Radii Derived from Electron Density Distributions. Phys. Chem. Miner. 1997, 24, 432−439. (22) Gibbs, G. V.; Ross, N. L.; Cox, D. F.; Rossoll, K. M.; Iversen, B. B.; Spackman, M. A. Bonded Radii and the Contraction of the Electron Density of the Oxygen Atom by Bonded Interactions. J. Phys. Chem. A 2013, 117, 1632−1640. (23) Sato, T.; Funamori, N. Sixfold-Coordinated Amorphous Polymorph of SiO2 under High Pressure. Phys. Rev. Lett. 2008, 101, No. 255502. (24) Brazhkin, V. V. Comment on “Sixfold-Coordinated Amorphous Polymorph of SiO2 under High Pressure”. Phys. Rev. Lett. 2009, 102, No. 209603.
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