Understanding Chemical Changes Across the Α‑Cristobalite to Stishovite Transition Path in Silica † † ‡ † † Miguel A

Article

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Understanding Chemical Changes across the α‑Cristobalite to Stishovite Transition Path in Silica † † ‡ † † Miguel A. Salvado,́*, Pilar Pertierra, A. Morales-García, J. M. Menendez,́ and J. M. Recio † Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain, and MALTA-Consolider Team ‡ Departamento de Química Física I, Universidad Complutense de Madrid, Madrid, Spain, and MALTA-Consolider Team

ABSTRACT: The interplay of structural, energetic, and chemical bonding features across the pressure-induced α- → cristobalite stishovite phase transition in SiO2 is fully characterized by ﬁrst principles calculations and topological analysis of the electron localization function. Under a martensitic approach, the transition mechanism is theoretically modeled at the thermodynamic transition pressure using the α -cristobalite P41212 space group. A soft and symmetric transition path is determined with an activation barrier lower than 100 kJ/mol. No bond breaking is found, but a synchronous bonding formation process leading to Si 6-fold and O 3-fold coordinations. Only when the third coordination sphere of oxygens approaches a given Si at a distance close to 2.0 Å do new Si−O bonding basins appear. This situation occurs at a transformation stage well beyond the maximum in the energetic proﬁle of the transition path.

■ INTRODUCTION tetrahedral to octahedral silicon coordination change. α- cristobalite structure can be related to the C9 cubic structure Understanding chemical changes induced by thermodynamic 17 variables in crystals is of great interest in solid-state chemistry. through a rotation of SiO4 tetrahedra around their C2 axis. Further rotation leads to a rutile packing of oxygen atoms. On It covers structural, energetic, and chemical bonding character- ’ ff ization of local environments of the atomic constituents of the basis of this idea, O Kee e and Hyde proposed a fi mechanism for the transformation mixing tetraedra rotations solids, and involves the modi cations within the same 17 crystalline structure or between different polymorphs if a and silicon displacements. Later, this mechanism was − successfully modeled using a periodic Hartree−Fock ap- solid solid phase transition occurs. When the transformations 15 are induced by pressure, the outstanding capability of this proach, although the limited basis sets and geometrical variable to promote high metal coordinations in crystalline constraints used in that work do not allow for a global solids leads to densification processes of fundamental interest in quantitative description of the transition path. Huang et al.11 suggested from ab initio and MD calculations areas ranging from planetary sciences to materials engineering. α fi A paradigmatic example of such a capability that merits that -cristobalite under pressure rst transforms into a hp- cristobalite phase with the same tetragonal space group and investigation in detail is the emergence of Si hexacoordinated at ff high pressures from the tetrahedral environment displayed at Wicko positions as the original cristobalite, and associated this new phase with the X-I phase found in experiments. On the normal conditions in silica polymorphs. In fact, thermodynamic 7 and kinetic aspects have been the subject of a number of studies basis of three-dimensional single-crystal data, Dera et al. show dealing with silica transformations under pressure, both that the unit cell for this new phase is not tetragonal but − − experimentally1 7 and theoretically.8 14 However, only few monoclinic, although a structural model remains elusive. efforts have been put forward to the understanding of how the Nevertheless, it is noteworthy to emphasize that an easy transformation between cristobalite and rutile forms of GeO2 chemical bonding network evolves from low to high Si 18 coordination,15,16 an issue that needs to be addressed if a has been also described. global characterization of the densification process is desired. Besides energy and geometry, chemical bonding constitutes Cristobalite has been one of the most widely studied phases the third leg of the triad that fully characterizes the changes α involved in a reconstructive phase transition. The electron of SiO2.Thetetragonal -phase is metastable at room conditions (α-quartz being the stable phase) and is usually localization function (ELF) may be considered, among obtained quenching the cubic β-cristobalite, a stable high quantum chemical topological formalisms, as one of the most ff temperature polymorph. At low temperature and pressures appropriate tools to o er a clear view of those regions within higher than 7.5 GPa, stishovite (octahedral silicon) is the stable phase. Stishovite is also tetragonal, and its space group is a Received: February 23, 2013 supergroup of the α-cristobalite one. For this reason, the Revised: April 2, 2013 transformation between them has been taken as a model for Published: April 5, 2013

© 2013 American Chemical Society 8950 dx.doi.org/10.1021/jp401901k | J. Phys. Chem. C 2013, 117, 8950−8958 The Journal of Physical Chemistry C Article the unit cell that are especially sensitive in terms of chemical unit cell can be chosen to describe the transformation with the activity under pressure.19 The topology induced by the ELF requirement of belonging simultaneously to a common field provides a partition of the space into basins or regions that subgroup of the initial and final structures. Once the unit cell follow the language of Lewis’ theory: atomic-like core shells is defined, a transformation coordinate and a transition path can (C), lone pairs (LP), and bonds (B).20 Using ELF analysis, it is be proposed. Under this view, the whole process is modeled in possible to track how these basins are modified and when new a way very similar to a chemical reaction. α α bonds are formed as we move forward along the -cristobalite -Cristobalite belongs to the P41212 space group. Its to stishovite transition path. conventional unit cell contains 4 Si and 8 O atoms with Si at In this work, our contribution is aimed at a full under- 4a(x,x,0) and O at general 8b(x,y,z) positions. Stishovite standing of the interplay between unit cell structural changes, belongs to the P42/mnm space group. Its conventional unit cell energetic profiles, and chemical bonding reorganization of silica contains 2 Si and 4 O atoms with Si at 2a(0,0,0) and O at from α-cristobalite to stishovite. To this end, we will carry out: 4f(x,x,0). Both structures have been described in detail (i) electronic structure first principles calculations of a number elsewhere (see, for example, refs 17,29). of periodic systems following a transformation coordinate that In the present microscopic study of the α-cristobalite to α links -cristobalite and stishovite structures under a common stishovite transformation, a P41212 common unit cell with four space group, and (ii) ELF topological analysis of all electron formula units can be chosen to describe the mechanism because solutions of the optimized structures across the transition path. the space group of α-cristobalite is a subgroup of that of The main outcome of these calculations consists of atomic stishovite. In this space group, the stishovite cell volume trajectories, cell shapes, energy barriers, and the quantitative doubles its value, and Si and O atoms occupy the same Wyckoff fi α identi cation of chemical bonding indexes related to the positions as in -cristobalite with xSi = 0.5, yO = xO, and zO = emergence of the new Si−O bonds. 0.25. fl Three more sections complete this Article. Next, we brie y The phase transition mechanism using the P41212 common present the computational parameters used in our electronic unit cell can be monitored through a normalized transformation structure calculations, along with the main ideas of our coordinate, ξ, evolving from 0 (α-cristobalite) to 1 (stishovite) fi ξ − cr st − cr cr st martensitic approach for the transition path and some basic and de ned as follows: =[xSi xSi]/[xSi xSi], xSi and xSi concepts of ELF. Section 3 provides the discussion of the being the x coordinate of silicon in α-cristobalite and stishovite, calculated structural, energetic, and chemical bonding reorgan- respectively, and xSi the x coordinate of Si at each stage of the ization results across the α-cristobalite → stishovite transition transition path. To determine the transition path, we start with path. A brief summary and the main conclusions will be the equilibrium structure of α-cristobalite at 6 GPa, close to the presented in the last section. calculated thermodynamic transition pressure. At each step, we update the xSi value and optimize the cell parameters and the ■ COMPUTATIONAL DETAILS AND MODELING oxygen coordinates (a total of five parameters), keeping the Total Energy. Static total energy (E) calculations at selected silicon coordinate frozen. volumes (V) in the α-cristobalite and stishovite unit cells were Basic Concepts of ELF. The electron localization function 30 carried out under the generalized gradient approximation (ELF), first introduced by Becke and Edgecombe, provides a (GGA) of the density-functional theory as implemented in the measure of the localization of electron pairs in atomic and Vienna ab initio simulation package (VASP).21 The projector- molecular systems that is able to recover the chemical augmented wave (PAW) all-electron description of the representation of a molecule consistent with Lewis’ valence electron−ion−core interaction22,23 and the exchange and picture. The ELF has been defined to have values between 0 correlation functional proposed by Perdew−Burke−Ernzerhof and 1 (1 corresponding to perfect localization) to ease the (PBE)24 were used. Brillouin-zone integrals were approximated visualization of its isosurfaces. The topological analysis of ELF using Γ-centered Monkhorst−Pack meshes25 where the surfaces provides a partition of the three-dimensional space into ⃗ numbers of subdivisions along each reciprocal lattice vector bi nonoverlapping basins, which can be thought of as electronic ×| ⃗| ff were given by Ni = max(1.15 bi + 0.5). An energy cuto of basins corresponding to bonds (B), lone pairs (LP), and atomic 520 eV was used to ensure convergence of the total energy core shells (C). Hence, the integration of the density over their within 10−3 eV. The four internal coordinates and the c/a ratio volumes assigns a population to the basins. These populations were optimized in α-cristobalite, while the only one internal are well-known31 to follow the expected values and tendencies coordinate and the c/a ratio were optimized for stishovite. from the Aufbau principle and the valence shell electronic pair Numerical and analytical (Vinet)26 equations of state were repulsion (VSEPR) theory. used to describe (E,V) points of the two structures and, The partition of the space into atomic cores, lone pairs, and therefore, to provide pressure (p)−V data, equation of state bonds comes after the identification of critical points in the 3D- fl (EOS) parameters (bulk modulus, B0, and its pressure space where the ELF gradient vanishes, and of the zero ux ′ derivative, B0 , both evaluated at zero pressure), and enthalpy gradient surfaces that surround each of these types of basins. (H)−p curves. H is the appropriate thermodynamic potential to To this end, an in-house developed computational code, determine phase stability at static conditions (zero temperature CRITIC,19 has been used. and zero point vibrational contributions neglected). These All electron wave functions are preferable for a quantitative calculations have been performed with the GIBBS2 code.27,28 analysis of the ELF topology in a crystalline phase. Thus, the Martensitic Approach. The static or martensitic approach optimized structures obtained with VASP were recalculated is based on the assumption that the crystal acts as a concrete with the CRYSTAL98 code32 to obtain the required all electron block. Unlike the nucleation and growth mechanisms, domains wave functions for the two polymorphs and all of the transition are not expected to form, but the crystal as a whole goes structures at 6 GPa. Si and O basis sets were chosen from through a phase transition. That is, the atoms move previous calculations.16 Thanks to our computational interface simultaneously maintaining some translational symmetry: a to the CRYSTAL98 package,19 the following features of the

8951 dx.doi.org/10.1021/jp401901k | J. Phys. Chem. C 2013, 117, 8950−8958 The Journal of Physical Chemistry C Article

ELF topology were rigorously evaluated: (i) ELF profiles for between coesite and stishovite35 or between metastable α- selected Si−O bonds, (ii) all of the critical points of the ELF quartz and stishovite.36 surface, in particular attractors and first-order saddle points, and We also tried to construct the E−V curves for the tetragonal α (iii) plots illustrating the basins and the localization of the P41212 (the same group as -cristobalite) X-I phase proposed 11 critical points. by Huang et al. and also for the orthorhombic C2221 (a A careful topological analysis of the ELF function was subgroup of the previous one) intermediate structure proposed performed to ensure the consistency of our calculations. for the same authors. In both cases, the optimized structures Accordingly, we tried to follow as much as possible the same were identical to the α-cristobalite at all of the studied volumes; type of calculation and equivalent computational parameters in that is, we did not find new E−V curves for these phases. the all electron CRYSTAL98 calculation as in the VASP Furthermore, a close look at Figure 3 of the cited paper reveals calculation. This strategy has been successfully carried out in that the common tangent to the X-I and α-cristobalite phase previous topological analysis of ELF in a variety of high has higher slope than the common tangent to the stishovite and pressure studies.33,34 The ELF topology and the properties of α-cristobalite. This is not compatible with X-I being stable as the basins of the α-cristobalite and stishovite structures have compared to α-cristobalite at intermediate pressures. We have been studied at 6 GPa. Similar analysis was carried out for the also compared the proposed X-I structure given at 20 GPa, with calculated transition structures across the α-cristobalite → that of our calculated α-cristobalite at the same pressure. In the stishovite P41212 transition pathway. proposed structure, the SiO4 tetrahedra is tilted (see discussion below) relative to the α-cristobalite at 0 GPa and is also ■ RESULTS AND DISCUSSION distorted with an angle open to 130°. In our E−V calculations, Structure, Stability, and Equation of State. Figure 1 we also observe tetrahedra tilting at high pressures, but the shows calculated total energies as a function of volume per tetrahedra remains regular. It must be noted that in our conception of the transition mechanism, we model the α-cristobalite to stishovite path at constant pressure using an appropriate transformation coordinate, mimicking the methodology used to study mechanisms of molecular reactions. Thus, the meaning of mechanism in our approximation is equivalent to that used in chemical reactions, whereas in other previous studies of this transformation,10,11 the focus is on how the structure changes as pressure is gradually increased. Transition Path: Energy Profile and Structure. In Table 2, we collect the calculated static GGA values of the cell

Table 2. Calculated Transition Pressure Structural Properties (Cell Parameters and Atomic Positions) of α- Cristobalite and Stishovite According to the P4 2 2 a 1 1 Figure 1. Calculated energy−volume curves. Mechanism α-cristobalite TS stishovite a (Å) 4.743 4.186 4.196 c (Å) 6.402 6.134 5.366 formula unit for α-cristobalite and stishovite. Our calculated x 0.3300 0.4200 0.5000 lattice parameters, atomic coordinates, and EOS parameters at Si xO 0.2260 0.2753 0.3059 zero pressure are collected in Table 1 along with other y 0.1625 0.2653 0.3059 representative experimental and theoretical data. The equili- O zO 0.2080 0.2329 0.2500 brium pressure between the two phases was estimated as 5.6 a GPa. This compares well with the pressure of the transition Properties of the transition state (TS) are also included.

Table 1. Summary of Zero-Pressure Structural and Cohesive α Properties of α-Cristobalite and Stishovite parameters of -cristobalite and stishovite in the P41212 common cell at 6 GPa, close to the transition pressure (5.6 α-cristobalite stishovite GPa). The enthalpy energy proﬁle of the mechanism is shown exp.a GGAb exp.c GGAb in Figure 2, where H is represented versus the normalized ξ α a (Å) 4.9717(4) 5.111 4.17755(16) 4.234 transformation coordinate evolving from 0 ( -cristobalite) to ξ c (Å) 6.9223(3) 7.152 2.66518(34) 2.693 1 (stishovite). Eighteen values have been considered ξ ﬁ xSi 0.30028(9) 0.2906 including the ending structures. The H( ) pro le is notably

xO 0.2392(2) 0.2425 0.3067(3) 0.3069 symmetric and soft. This is a good indication of an appropriate y 0.1044(2) 0.0863 choice for the transformation coordinate. The transition state O ξ zO 0.1787(1) 0.1705 appears at around = 0.55. The computed activation barrier is 3 V0/SiO2 (Å ) 42.77 46.75 23.26 24.15 0.8 eV (lower than 100 kJ/mol). The calculated volume

B0 (GPa) 11.5(7) 12.2 309.9(1.1) 260.8 collapse in this reconstructive transformation is around 34%. At ′ B0 9.0(2) 5.7 4.59(0.23) 5.7 the transition state (TS), the volume reduction is already close to 25%, indicating that the unit cell at TS (see Table 2) is more aReference 37. bPresent work. cReferences 38,39. similar to stishovite than to α-cristobalite.

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chain. Parallel chains are linked through the two other oxygen atoms forming a three-dimensional network. In the structure of stishovite, these helices are transformed into chains of edge- sharing octahedra (see ξ = 1 structure in Figure 4). The oxygen atoms labeled Oe and Of are the two that complete the − − octahedra at the end of the transformation. The Ob Si Oc angle will become 180°. The two other octahedral axes will − − − − − − fi be:Of Si Od and Oe Si Oa. The Oe Si Of nal angle will ° − − ° be 98 , and its complementary one, Od Si Oe,is81. This deviation from the ideal octahedral symmetry is also observed in the experimental stishovite structure. In this process, the coordination of Si atoms changes from 4-fold to 6-fold, whereas that of O atoms changes from 2-fold to 3-fold. At each step of the transformation coordinate, Si and O displacements (relative to the atomic positions in the cell at the Figure 2. Calculated enthalpy profile along the transition path. previous step) were calculated to be less than 0.08 Å. The total Si and O displacements relative to the initial α-cristobalite cell α The -cristobalite structure consists of SiO4 corner-sharing were 1.14 and 0.82 Å, respectively. For the final stishovite cell, tetrahedra. The structure can be described as being built up the degree of lattice distortion, defined as the square root of the from (−O−Si−O−Si−) helices running along the c axis (see sum of the squared eigenvalues of the strain tensor η divided by Figure 3). Each tetrahedra uses two oxygen atoms to build this 3 (see ref 40), referred to the initial α-cristobalite cell was S = 0.07131, whereas the lattice distortions at each step were less than 0.011. All of these internal and cell changes are compatible with the reconstructive character of this transformation. In Figure 5, we show how the most relevant Si−O distances progress along the studied transition path. Initially (ξ =0,α-

Figure 5. Evolution of relevant Si−O distances along the transition path. Each distance is 2-fold degenerated and corresponds to two ξ α oxygen atoms: at =0(-cristobalite), d1 and d2 result in 4-fold Figure 3. A representation of α-cristobalite structure showing (black ξ coordination; at = 1 (stishovite), d1, d2, and d4 result in 6-fold lines) preﬁgured stishovite Si−O new bonds. − coordination. d3 corresponds to Si Og distance.

Figure 4. Sequence of structural changes from α-cristobalite to stishovite along the transformation coordinate.

8953 dx.doi.org/10.1021/jp401901k | J. Phys. Chem. C 2013, 117, 8950−8958 The Journal of Physical Chemistry C Article cristobalite), the shortest Si−O distance is 4-fold degenerated, two new bonds. In this context, two tilt angles are needed to fi − − δ but it splits into two double degenerated (2 + 2) distances, d1 de ne the rotation of the plane Oa Si Od ( 1) and the ξ fi − − δ and d2,as departures from 0. In the rst part of the transition rotation of the plane Ob Si Oc ( 2). These planes are − − α path, the distance corresponding to Si Oa and Si Od (d2) perpendicular in a regular tetrahedra. The tilt angles in - − − ° ° increases faster than that corresponding to Si Ob and Si Oc cristobalite at 0 GPa are 20.2 and 19.4 (with the experimental (d ). In the second part, the trend inverts, and at the transition value37 being 23.3°). When α-cristobalite is compressed to 6 1 ° ° state the four distances become equal. Beyond this point, d1 GPa, these angles increase to 35.9 and 33.3 . Along the remains longer than d . We now focus on the key distance of transition path at constant pressure (6 GPa), both tilt angles 2 ° δ the transformation, the one that experiences the largest continue to increase up to 45 (stishovite phase), but 1 δ δ ° reduction. It has the label d4 in Figure 5. It is also doubly approaches faster than 2 to this limit. 1 =45 implies xO = − − δ ° degenerated (Si Oe and Si Of, see Figure 3) and has an initial yO, and 2 =45 implies zO = 0.25. value as large as 3.7 Å. At the end of the transformation (ξ =1, The sum of three factors, translation of original tetrahedra, stishovite), d4 becomes equal to d2 with a value close to 1.8 Å, their rotation, and the cell deformation, can be analyzed in thus belonging to the first coordination sphere of Si in this terms of three steps in the transformation path. First, an δ phase, and suggesting the creation of two new Si−O bonds. increase of 1 implies a decrease of the x coordinate and an − increase of the y coordinate of oxygen atom O ,but Another doubly degenerated Si O distance (d3) that is worth e exploring involves two oxygens that belong to the second simultaneously Oe displaces along the square diagonal following coordination sphere of Si in α-cristobalite (one of them, O ,is the movement of its attached silicon atom, increasing both x g ff displayed in Figure 5; the other one is not represented for and y coordinates. As illustrated in Figure 6, the global e ect is clarity). The d value is initially 3.2 Å and decreases in the first a nearly constant x coordinate and a continuous increasing of y 3 ξ part of the transition path, but recovers the original distance at coordinate from = 0 to approximately 0.3 when Oe x and y coordinates have taken a similar value and then the oxygen the end of transformation. In summary, it is remarkable to fi conclude that not the next nearest oxygen neighbors to Si, but atom is close to the square diagonal. This rst stage of the the two oxygen atoms at the third coordination sphere are the transformation is accompanied by a shortening of a and b axes ones involved in the creation of the new Si−O bonds. (see Figure 7), whereas the c axis is held nearly constant. The O’Keeffe and Hyde17 proposed a relation between α- cristobalite and stishovite based on the change of a tilt angle δ fi ( ), de ned as the angle of rotation around the C2 axis of the tetrahedra with respect to the orientation in C9 structure (Fd3m, ideal β-cristobalite). In the stishovite, δ reaches its limit value of 45°. Along with this tilt angle, one must take into account additional displacements of the silicon and oxygen atoms to really achieve octahedral symmetry around cations. The authors supposed that likely during the transformation the cation and anion shifts would occur simultaneously. In our model, the silicon atoms move in the ab plane toward the center (Si with z = 0 and z = 1) and corners (Si with z = 0.25 and z = 0.75) of the cell along the square diagonals (x = y and y =1− x, respectively) with change in x proportional to the transformation coordinate ξ. During the whole process, the silicon displacement toward its stishovite position is accom- Figure 7. Lattice change along transition path. panied by displacements of oxygens as illustrated by the change in the coordinates of Oe displayed in Figure 6. − α translation movement leads to the opening of the angle Ob Actually, tetrahedra in -cristobalite are not strictly regular, − − − and along the transition path each SiO unit deforms to accept Si Oc, whereas the distances Si Ob and Si Oc (d2) are 4 constrained to moderate values by the cell contraction. δ Rotation around 2 can be followed in z(O) coordinate that increase from approximately ξ = 0.4 to 0.7 until a value close to 0.25 is reached. The displacement of Oe along the square diagonal continues in this stage. Now the c axis begins to decrease faster than a = b leading to a maximum in the c/a ratio slightly before the maximum of enthalpy. Finally, in a third stage, the Oe nearly stops moving, whereas Si continues its displacement to its final position as imposed by the choice of the transformation coordinate. In this final phase, the a = b axis does not change, whereas the c axis continues decreasing down to its final value. Transition Path: Chemical Bonding. In quantitative terms, Tables 3 and 4 gather the information on ELF attractors for α-cristobalite and stishovite, respectively. These critical points can be related to five chemical entities: Si and O cores − (Ci), O lone pairs (LPi), Si outer cores (Li), and Si O bonds Figure 6. Oxygen atom coordinates along the transition path. (Bi). We can also characterize the valence basins by the number

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Table 3. ELF Attractors or Maxima ((3,−3) Critical Points) a Found for α-Cristobalite

xyzMη CHM 0.3300 0.3300 0.0000 4 0.9999 C(Si) 0.2260 0.1625 0.2080 8 0.9999 C(O) 0.7458 0.3041 0.0223 8 0.8886 LP(O) − 0.2910 0.2521 0.1564 8 0.8456 B1(Si O) − 0.2089 0.0383 0.2305 8 0.8523 B2(Si O) 0.3388 0.2727 0.9927 8 0.8687 L1(Si) Figure 8. Tetrahedral (α-cristobalite) and octahedral (stishovite) 0.3462 0.3612 0.9650 8 0.8687 L2(Si) coordinations. White spheres are critic points (CP): LP (lone pairs), B aPositions (x,y,z), multiplicities (M), ELF value at the critical point (bonds), and L (outer core). Banana-shaped isosurfaces (yellow) (η), and chemical meaning (CHM) are collected. represent lone pairs.

Table 4. ELF Attractors or Maxima ((3,−3) Critical Points) extending VSEPR (valence shell electron pair repulsion) to just a Found for Stishovite an EPR theory because (outer) core electrons also obey the same rules as valence ones. In the case of α-cristobalite, the xyzMη CHM outer core attractors are tetrahedrally situated along directions 0.5000 0.5000 0.0000 4 0.9999 C(Si) crossing the centers of the faces of the O4 tetrahedron 0.3059 0.3059 0.2500 8 0.9999 C(O) surrounded Si, whereas in the stishovite phase these attractors − 0.1979 0.1979 0.2500 8 0.8802 B1(Si O) form a cube, which is the dual polyhedron of the SiO6 − 0.3628 0.3628 0.1599 8 0.8484 B2(Si O) octahedron. This new arrangement of the Li(Si) attractors is − 0.6372 0.6372 0.8401 8 0.8485 B3(Si O) only detected at the final stage of the transition (ξ > 0.88) 0.1040 0.7062 0.5000 8 0.8752 LP1(O) probably due to the similar values they have in both structures. 0.3960 0.2062 0.2500 8 0.8752 LP2(O) Concerning the oxygen lone pair attractor in α-cristobalite 0.5325 0.4675 0.9625 8 0.8673 L1(Si) (see also Figure 8), it splits progressively as we displace across 0.0325 0.0325 0.2875 8 0.8673 L2(Si) the transtition path, but it is not until ξ = 0.65 when it evolves 0.0131 0.9349 0.7500 8 0.8674 L3(Si) into two lone pair and one bonding attractors. Thus, in the 0.9349 0.0131 0.7500 8 0.8674 L4(Si) stishovite structure, each oxygen is bonded to three silicon aPositions (x,y,z), multiplicities (M), ELF value at the critical point atoms (planar-triangular geometry) with two lone pairs located (η), and chemical meaning (CHM) are collected. above and below this bonding plane, as depicted in Figure 8. Finally, we directly analyze the emergence of the new Si−O bonds. By means of ELF profiles plotted in Figures 9 and 10, it of core basins with which they share a boundary. This number is called the synaptic order. Monosynaptic basins correspond to the lone pairs (LPi), and disynaptic basins correspond to two- 41 center bonds (Bi). Thus, there are one monosynaptic and two disynaptics basins in α-cristobalite, whereas in stishovite the number of basins around oxygen increases to two mono- synaptics and three disynaptics. The ordering of ELF values (remember that η = 1 denotes the highest probability of finding localized electron pairs) shows that the separation of these basins follows the same sequence as in the list above. Notice that η differences (except for core basins) are small, which ffi makes more di cult the separation between LPi,Li, and Bi basins. Along with the increasing number of bonds from 4 to 6 (multiplicity has to be taken into account when Tables 3 and 4 are analyzed), there is a splitting of the O lone pair and the Si outer core in stishovite with respect to α-cristobalite. These three topological features have been evaluated across the transition path. We have also integrated the electron density fi α − within the volume space of the oxygen basins of the low and Figure 9. ELF pro le for -cristobalite. ELF values versus a Si O reduced distance (0 for Si, 1 for O) are drawn. high pressure phases. We find that the core remains untouched (2.16 electrons in both structures), whereas the electron population of the lone pairs change from 6.87 in α-cristobalite to 4.46 (2.21 + 2.25) in stishovite, and that of the bonding is possible to distinguish the main differences between the basins from 2.41 (1.33 + 1.08) in α-cristobalite to 3.13 (1.47 + initial and final structures as regards the three Si−O profiles 0.83 + 0.83) in stishovite. It is apparent that in the formation of involved in the oxygen coordination spheres of Si that new Si−O bonds a conversion of lone pair basins into bonding participate in the transformation (see Figure 5). These − − − ones occurs. distances are denoted by d1 (Si Oa or Si Od), d2 (Si Ob or fi − − − We would like to bring the attention rst to the geometrical Si Oc), and d4 (Si Oe or Si Of) (Figure 3 shows O labels). arrangement of the Si outer core attractors (see Table 3, Table To plot the curves for the three distances in the same figure, we 4, and Figure 8). As previously found in other crystals,42 the use a reduced coordinate with a value equal to 0 corresponding position of these attractors minimizes their mutual repulsion to the Si atom and a value of 1 for the O atom position. This

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Figure 10. ELF profile for stishovite. ELF values versus a Si−O fi Figure 11. ELF pro les along the d4 reduced coordinate for values of reduced distance (0 for Si, 1 for O) are drawn. the transition coordinate ranging from ξ =0toξ = 1. Arrows indicate progression of the transformation. reduced coordinate is not to be confused with the transformation coordinate ξ. Just to introduce the main information from these profiles to bond breaking but the creation of two new primary Si−O those not familiar with the ELF analysis, it is illustrative to bonds. associate the regions between two minima with chemical Lewis entities. Thus, d and d curves for α-cristobalite, and d , d , and ■ CONCLUSIONS 1 2 1 2 α d4 for stishovite show the same features separated by three -Cristobalite and stishovite polymorphs of silica manifest very minima along the reduced coordinate: Si core, Si outer core, different responses to hydrostatic pressure. At normal Si−O bond, and O core. It is apparent how the curve conditions, the unit cell volume of α-cristobalite is almost − α corresponding to the largest Si O separation (d4)in - twice that of stishovite, whereas at 6 GPa the ratio has reduced cristobalite shows a different shape with two minima at around to ca. 1.5. This is a consequence of the 6-fold coordination of Si 0.55 and 0.95 that enclose a lone pair basin for the O atom. in stishovite and its high bulk modulus (around 300 GPa) as Other minor features of this curve, as the shoulder around 0.3 compared to the 4-fold coordination of Si in α-cristobalite and fi in Figure 9, have to do with projections on this line of ELF its low B0 value (around 10 GPa). Our rst-principles values related to other atoms, and are not discussed now. simulations are able to quantitatively account for this different Because each curve has a 2-fold multiplicity, we can conclude pressure behavior of the two silica polymorphs. Extensive α that the ELF profile between the two minima around 0.4 and optimizations within the P41212 space group of -cristobalite − 0.9 is a clear signature of the Si O chemical bond in SiO2. do not reveal previously reported X-I-like or orthorhombic 11 ff Now, we can monitor the bond reorganization that C2221 phases, but the same structure with obviously di erent accompanies the phase transition along the P41212 path. This cell parameters and atomic coordinates. can be done analyzing the changes in the ELF profile associated By following a martensitic (static) approximation, the α- − − → with the d4 (Si Oe or Si Of) curve as the transformation cristobalite stishovite phase transition can be smoothly coordinate moves forward from α-cristobalite (ξ =0)to modeled using the x = y fractional coordinate of Si as the stishovite (ξ = 1). Figure 11 displays such information at five transformation coordinate under the common P41212 space stages of the transition path denoted by an increasing of the ξ group. This is a reconstructive transformation with a volume coordinate from 0 to 1. Although there is not an unequivocal collapse around 34% and an energetic barrier of less than 100 indicator of the change in the nature of the profile, it can be kJ/mol at ξ = 0.55. Along this transition path, two oxygen established that the formation of the new bonds is achieved atoms from the third coordination sphere of Si create new Si− between ξ = 0.53 and ξ = 0.71. The mean value within this O chemical bonds when they approach a Si atom at a distance interval corresponds to d(Si−O) around 2.0 Å and a H(ξ) value close to 2.0 Å. No bond breaking is found. As a result, the relative to the initial α-cristobalite structure of approximately 50 original bond network changes toward a more efficient atomic kJ/mol. Some features of the d4 curve help to suggest the packing with greater coordinations for Si (4 to 6) and O (2 to emergence of two Si−O bonds in this interval: (i) 3). The ELF topological analysis reveals that Si outer core disappearance of the shoulder, (ii) nonzero values for ELF attractors change from tetrahedral to cubic symmetry, whereas around 0.5 in the reduced coordinate plot, and (iii) a the O lone pair splits into two lone pairs and one bonding progressive transformation of the O lone pair region into a attractor. The most apparent feature along this transition path, − more symmetric shape close to the two existing ones in the d1 the emergence of two new Si O chemical bonds, is located and d2 curves. We recall that the split of the O lone pair once the energetic barrier has been overcome, as was already discussed above produces two new Si−O bonds at ξ around detected in other reconstructive phase transformations. 0.65. It is also to be emphasized that equivalent curves for d1 fi ■ AUTHOR INFORMATION and d2 present negligible modi cations along the transition path with respect to the similar curves found for α-cristobalite Corresponding Author and stishovite. Thus, this transition path does not involve any *E-mail: [email protected].

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