Periodic pseudopotential Hartree–Fock study of quartz structure SiO2 and GeO2Bernard Silvi, Philippe D’Arco, and Mauro Causà

Citation: The Journal of Chemical Physics 93, 7225 (1990); doi: 10.1063/1.459445 View online: http://dx.doi.org/10.1063/1.459445 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/93/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Mechanism of dopant-vacancy association in -quartz GeO2 J. Appl. Phys. 113, 083716 (2013); 10.1063/1.4793786 Periodic Hartree–Fock studies of the properties of the Cl2/MgO (001) interface J. Chem. Phys. 96, 6010 (1992); 10.1063/1.462641 The electronic structure of quartz: A periodic Hartree–Fock calculation J. Chem. Phys. 86, 6967 (1987); 10.1063/1.452344 Crystal structure and thermal expansion of quartz SiO2 at low temperatures J. Appl. Phys. 53, 6751 (1982); 10.1063/1.330062 Compression mechanisms in quartz structures—SiO2 and GeO2 J. Appl. Phys. 49, 5473 (1978); 10.1063/1.324517

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Periodic pseudopotential Hartree-Fock study of a-quartz structure Si02 and Ge02

Bernard Silvia) Laboratoir~ de ,Spectrochimie Moleculaire (URA508) Universite Pierre et Marie Curie, 4, Place Jussieu 75230, Pans Cedex 05, France

Philippe D' Areo Laboratoire de Geologie de ['Ecole Normale Superieure (URAJ316), 24 Rue Lhomond 75231, Paris Gedex 05, France

Mauro Causa D~partim.ento di Chimice Inorganica, Chimica Fisica e Chimica dei Materiali, Universita di Torino, via Pietro Gluria, 5 Il0125 Torino, Italy

(Received 11 July 1990; accepted 1 August 1990)

Periodic pseu.d.opotential Hartree-Fock calculations have been performed on the quartz structure of sIllca and germanium dioxide. It is shown that the tilt angle is the most important str~ctural parameter. This parameter emphasizes the role of the oxygen polarization function whlch appear to be necessary to recover the correct geometry oflow quartz. The structure of tJ-quartz is rationalized on the basis of the energy dependence upon the tilt angle and the absence of tJ-structure in Ge02 is explained. Finally the behavior of both oxides under pressure is discussed.

I. INTRODUCTION

Silicates and germanates have very similar properties, in particular, many of the crystalline silicates have germanate analogs. Among these compounds, silicon and germanium dioxides are the simplest and have been the subject of a large body of experimental work. Among the numerous silica po­lymorphs, only the low quartz, low cristobalite, and stisho­vite analogs have been observed for Ge02. Surprisingly, none of the high temperature phases seems to exist.! In gen­eral, under pressure germanates behave like silicates. How­ever, the trends observed at very high pressure for silicates occur at lower pressures for germanates. Therefore, the lat­ter are used instead of the former for experimental conven­ience.

The a-quartz like Ge02 is the only tetra-coordinated polymorph for which extensive structural studies have been carried out under different pressures.2-5 At zero pressure, the GeOGe bond angle is found to be about I S° less than that ofSi02 and consequently the tilt angle, as defined by Grimm and Dorner,6 is larger by 10°. Another striking difference between the Ge02 and Si02 systems is that the Ge04 tetra­hedra are much more distorted than the Si04 ones.

The behavior and the structural relaxation of Si02 and Ge02 under pressure have been investigated by several au­thors. 3,4,7 In a low pressure regime (P<4 GPa), the com­pressibility of a-quartz (Si02 ) is explained by the opening of the tilt angle. Note that this tilt angle corresponds to the r I

soft mode responsible for the a-tJ transition in quartz and cristobalite.8 At higher pressures, the compression is real­ized through angular distortion of the Si04 tetrahedra keep­ing the tilt angle nearly constant at its 4 GPa value. In the case of Ge02' there is.no such low pressure regime and the

a) Present address: Laboratoire de dynamique des interactions moleculaires (ER271) Universite Pierre et Marie Curie, 4 place lussieu 75230 Paris cooex 05, France.

tilt angle does not play any role in the cell relaxation. The purpose of the present paper is to show how quantum chemi­cal calculations can shed light on these differences.

II. CALCULATIONS

The a-quartz phases ofSi02 and Ge02 have been calcu­lated at the periodic Hartree-Fock level9 with the pseudopo­tential lO version of the linear combination of atomic orbi­tals-Hartree-Fock (LCAO-HF) program CRYSTAL. I I Several basis sets have been selected and checked against experimental structural data. The a-quartz system belongs to the P 32 21 space group and is described by four fractional coordinates and two lattice parameters. Alternatively, one can use a set of six internal coordinates which are more close­ly related to the chemical bonding. Since the available ver­sion ofCRYST AL does not perform analytical gradient cal­culations, it is impossible to fully optimize such rather complicated structures. For these reasons, the Si04 /Ge04

t~trahedra have been kept regular and, therefore, only the SlO/GeO bond length and the tilt angle have been varied. The bond length is the stiffest coordinate and the tilt angle (since it is a very soft coordinate) exhibits rather large changes under pressure in the Si02 case. Moreover, this lat­ter coordinate value allows the a and tJ phases to be distin­guished. The OSiO and OGeO bond angles, though they can deviate from 109° 47, do not show important variations with pressure.

The basis set selection criteria are the energy and the ability to predict a realistic structure. The basis set optimiz­ation has only been performed on Si02 for which all electron (AE) results are available. 12,13

A. Minimal basis set

The PS-3G basis set contains the standard STO-3G va­len~e ~hell and is expected to yield results of an overall quali­ty slmtlar to AE STO-3G. Table I reports the optimized SiO

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7226 Silvi, D'Arco, and Causa: a-quartz structure Si02 and Ge02

TABLE I. Optimized bond length, tilt angle, and Mulliken net charges of Si02 • AE and PS minimal basis set calculations.

PS-3G STO-3G Exp.

Si-O CA.) 1.60 1.60 1.614, 1.608 Tilt angle {j n 23 23.0 15.9 Si net charge (a. u. ) 1.499 1.40 o net charge (a.u.) - 0.75 -0.70

bond length, tilt angle, and the Mulliken net charges calcu­lated at the pseudopotential and AE levels. The two struc­tural parameters are in close agreement and the pseudopo­tential and AE charge density are consistent; but one can note that the pseudopotentialleads to a slightly more ionic situation. Moreover, the calculated energy difference between the a and f3 structures is larger at the AE level.

B. Split valence and polarized split valence basis set

The PS-21GlO,14 basis set has been successfully used to calculate bulk silicon and germanium and therefore has been retained as a starting point for the present study. The exter­nal shell exponents have been reoptimized and are 0.12339, 0.11, and 0.37368, respectively, for Si, Ge, and 0. The val­ues chosen for Si and ° are rather different than those used by Nada et al. 13 in their AE calculations for quartz and sti­shovite. The origin of the differences is that they designed their basis set for both tetra and hexacoordinated silicon ox­ides. Standard exponent d shells were added either only on the fourth column elements (PS-21G*) or on every atom (PS-21G**).

The results obtained on Si02 with these three basis sets are listed in Table II. Introduction of the d orbital on silicon lowers the energy by an amount of 0.3 a.u. with respect to the unpolarized split valence basis set. The additional energy gain due to the oxygen d set (PS-21G**) is one order of magnitude less. However, Table II clearly shows that the important effect of these latter polarization functions is to stabilize the a structure with respect to the f3 one. Thus the PS-21G and PS-21G* basis have been discarded. This result indicates that the tilt angle is an important structural param­eter.

III. RESULTS As previously mentioned, only the bond lengths and tilt

angles have been numerically optimized with the assump-

TABLE II. Si02 • Optimized bond length and tilt angle, energy, and Mulli­ken population data (split valence calculations).

Si-O(A) Tilt angle {j n Si net charge (a.u.) ° net charge (a. u. ) Energy (a.u./cell) Binding energy (a.u./ cel\)

PS-2IG

1.62 0.0 2.256

- 1.128 - 105.481269

- 1.2805

PS-2IG* PS-21G**

1.604 1.600 0.0 20.0 1.664 1.808

- 0.832 -0.904 - 105.766318 - 105.817 611

- 1.56568 - 1.616977

TABLE III. Ge02 • Optimized bond length and tilt angle, energy, and Mul­liken net charge population data (PS-21G** calculation).

PS-21G**

Ge-O (A) 1.740 Tilt angle {j n 25.2 ° net charge (a.u.) - 0.827 energy (a. u./ cell) - 105.144785 Binding energy (a. u./ cell) - 1.161 164

tion of regular Si04 and Ge04 tetrahedra. The PS-21G** results of GeOz are given in Table III. The calculated bond­ing energies listed in Tables II and III have not been correct­ed for the basis set superposition error,15 which is minimized by the pseudopotential approach. 16 At the PS-21G** level, the binding energies of SiOz and GeOz are underestimated by 22% and 28%, respectively. The electron correlation which is not accounted for at the self-consistent field (SCF) level, as well as basis set incompleteness, can be invoked to explain these discrepancies.

The optimized bond lengths of both systems are in close agreement with the experimental values. Note that the mini­mal valence basis set PS-3G yields the same results as PS-21 G* *. This result corroborates the molecular calculation experience. Finally the tilt angles are calculated to be 20° and 25° for SiOz and GeOz, respectively. In the case of the for­mer crystal, the tilt angle is larger than the experimental one by about 5°, while for the latter the agreement is almost per­fect. The discrepancy found for SiOz should be partly due to anharmonicity effects since it appears for Fig. 1 that the SiOz potential is much more anharmonic than that ofGeOz . Remember that quantum chemical optimized geometrical parameters correspond to Born-Oppenheimer energy hy-

0.15

LEGEND ..... Sio. 0 GPa r--- .... Sill." CPa f+----+--+----l-----j

i . D. CeO,O CPa \ ~.- GeO," GPa

o ~ •

>- ~~ f:

~ 0.10 +':;t-:\ _-+-_-+-_+-_-+-_+_+_t:-l1f ~ \\ n ~ : \\ II.

,~\\ /' ....... -~ ~-... iff; 0.05 ::. ~\ /If' '\ i~' ;'i

\~i. ~< .;:::: ~:= :~::: .>, .1.# 0.00 +---.::..:. ........ [)(:.. .... !:i ....... :.~ ... .;=:::..--:....~-+----+~""-.::-::: .•. ..:.: .. .:.ii. "jc:;)1~..;. •• .:.... ~

-35 -25 -15 -5 5 15 25 35 tilt angle

FIG. I. Energy (a.u.) vs tilt angle. The zero is taken at the minimum of the curves.

J. Chern. Phys., Vol. 93, No. 10, 15 November 1990

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Silvi, O'Arco, and Causa: a-quartz structure Si02 and Ge02 7227

persurface minima. Per contra, x-ray, and neutron scatter­ing experiments sample the electron and nucleus densities which depend upon the zero point motion. The f3 structures, which will be discussed in more detail in a following para­graph, experimentally characterized by [) = 0, correspond in our calculations to a local maximum of the energy curve connecting the a 1 and a 2 Dauphine twins. For Si02 , as well as for Ge02 , the barrier is dramatically overestimated by the minimal basis set with respect to PS-2IG**. As one can see from Fig. 1, the barrier height found for Ge02 is about three times larger than that ofSi02 • The Mulliken charges calcu­lated with the polarized split valence basis sets indicate a partly covalent character of the bonding, the germanate ap­pearing slightly less ionic than the silicate. The Si02 values agree with the AE results reported by Nada et al. 13

IV. DISCUSSION

A. Charge densities and the role of polarization functions

Figure 2 presents the density difference (crystal super­position of atoms) map of a-SiOz calculated at the PS-21 G level. The main features are the high electron density on the oxygen lone pairs and the low density along the SiO bond which reflects the rather ionic description provided by this basis set. Addition of d polarization functions on silicon (Fig. 3) weakens the lone pair density and simultaneously yields a buildup of electrons along the bonds. This effect is magnified by introducing polarization functions on oxygen (Fig. 4). On both maps, the silicon area is rather flat, the effects being concentrated on the oxygens. It is therefore difficult to interpret the role of the silicon d orbitals in terms of hybridization. These orbitals are used mainly to enlarge the basis set flexibility and thus to spread out the electron density. Without oxygen d orbitals, the basis set remains unbalanced and leads to a wrong prediction of the structure.

The same trends are also observed for Ge02 • Figures 5

a SiD. 21G

contour line interval 0.00250000

FIG. 2. Difference electron density (crystal atoms) of a-quartz (PS-2IG basis set). The electron density is plotted in the SiOSi bonding plane. The electron density spacing is 2.5 X 10 -, e - bohr '. Continuous, dotted, and dashed lines indicate positive, zero, and negative difference values, respec­tively. The basis functions are the same for both atomic and crystal densi­ties.

a. SiO. 21G* - 21G

contour line interval 0.00250000

FIG. 3. Difference electron density (crystal PS-21G*-crystal PS-21G) of a-quartz. Scale and symbols are the same as in Fig. 2.

and 6 show the electron density difference maps calculated at the PS-21 G** levels ofSiOz and GeOz • As expected these two maps are very similar.

B. The role of the tilt angle in u-(3 transitions

Despite the large amount of experimental work on the high-temperature phase transition in quartz, the microscop­ic description of the f3 phase still remains a subject of contro­versy. Two main pictures have been considered. On the one hand is the truly hexagonal structure,I7,18 often reported as idealized f3-quartz, and on the other hand is an average dis­ordered structure over the a l and a z Dauphine twin posi­tions. Most recent experimental investigations l9 have been interpreted within this latter framework. A crude analysis of Fig. I would lead to discard the first picture of f3-quartz, since the corresponding structure is associated with an ener­gy maximum. However, f3-quartz is a high temperature phase for which the nuclear motion cannot be neglected and therefore there is no straightforward correlation between the structure and the minima of the Born-Oppenheimer energy surface. Experimentally, f3-quartz is unquenchable and after annealing the proportion of the twin configurations a 1 and

a SiO. 21G·· - 21G

contour line interval 0.00250000

FIG. 4. Difference electron density (crystal PS-21 G**-crystal PS-21 G) of a-quartz. Scale and symbols are the same as in Fig. 2.

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7228 Silvi, D'Arco, and Causa: a-quartz structure Si02 and Ge02

ex SiO. 21G··

contour line interval 0.00250000

FIG. 5. Difference electron density (crystal atoms) of a-quartz (PS-21 G** basis set). Scale and symbols are the same as in Fig. 2.

a 2 tends to be equal. 17, 19 Finally the transition temperature increases with pressllre.

The two minima of the potential energy curve corre­spond to the twin geometries and are equivalent in energy. The tilt angle coordinate used to plot Fig. 1 has the same value for every cell of a perfect monocrystal. The local poten­tial with respect to 8 for a given cell depends upon the sur­roundings and is therefore asymmetric. At low temperature, the nuclear motion is trapped in a given well which corre­sponds to one of the a 1 and a 2 twin configurations. Increase of temperature allows discrete jumps between configura­tions, which can be favored by defects. This progressively symmetrizes the potential. Near the phase transition tem­perature, the density of Dauphine microtwinning increases; this was recently recognized as in intermediate incommen­surate phase. 20,21 At this point, the potential can be assumed symmetric and tunneling is possible. Consider now an excit­ed state of the r 1 mode lying near the barrier top in order to allow a delocalization of the motion over the two equivalent wells either with or without tunneling. As this level is excit­ed, the wave function has nodes along the normal coordi­nate. For example, the node of the first excited state wave

ex GeO. 21G··

contour line interval 0.00250000

FIG. 6. Difference electron density (crystal atoms) of a-Ge02 (PS-21 G·· basis set). Scale and symbols are the same as in Fig. 2.

and Ge02 0.20 -y----r--,---=--,------r--,..::..----.----,

0.15

t I ~ :\ I'.:J ..:I ~i ~ 0.10 -t:j:~t-___t--+--t_--+--+_-__l-__i~ ~ l\ ~ :1 .,

: ~~ : ~lc : ;,

0.05 +.~--+--+_-7fIf--+--+_:...-+--+_F.-.,."

0.00 +--..::..:...f~~f"L'==+---..:j:...::::!!CiiII.;.;;..,.: ... :..;..:.~ -35 -25 -15 -5 5 15 25 35

tilt angle

FIG. 7. Enthalpy (a.u.) vs tilt angle calculated at - 1.5,0, and 4 GPa for Si02 and at 0 and 4 for Ge02 (PS-2IG·· basis set).

function is located at 8 = O. Therefore, the density vanishes for 8 = 0 and has two symmetrical maxima. At the same time, the expectation value of 8 is equal to zero by symmetry. This picture overcomes the apparent contradiction of the two classical interpretations. It is consistent with x-ray and neutron diffraction which sample electron and nuclear den­sities, respectively. Moreover, this model agrees with two independent experimental facts. First, the a-{3 transition temperature ofSi02 increases with pressure. In Fig. 7, the enthalpy (E + PV) has been plotted at 0 and 4 GPa. The net effect of the pressure is to increase the barrier length and the curvature of the wells. Assuming that entropy is pressure independent, the transition temperature would increase with the barrier height. Second, in the case of Ge02 , the barrier height is about three times larger than that of Si02 • There­fore, one can expect the transition temperature to be notice­ably larger than for Si02. Germanium dioxide melts at 1115 °C, assuming a crude proportionality between transi­tion temperatures and barrier heights, Ge02 a-{3 transition is expected to be above the melting point and therefore does not arise.

This picture must be understood simply as a means to describe the {3 phase from the results of the present quantum chemical calculation. It cannot necessarily be related to any actual physical process. In particular, this model does not account for the mixing of the soft mode with the r 1 symme­try two phonon state22 which plays an important role in the phase transition. Moreover, the effects of temperature are neglected. The thermal expansion can be simulated by a neg­ative pressure. Combining the V( T) expression of Ghiorso et al.23 with the equation of state,3 one finds that the pres­sure equivalent to the transition temperature should be - 1.5 GPa. This lowers the barrier height, but the two mini­

ma of the curve remain.

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Silvi, D'Arco, and Causa: a-quartz structure Si02 and Ge02 7229

C. Remarks on the compression mechanisms

The experimental data obtained under pressure3•4.7 re­

veal two deformation mechanisms. Under pressure, the bridging angles (or the equivalent tilt angle) and the tetrahe­dron bond angles shrink. For Si02, the bridging angles in­volved are below 2 GPa. At higher pressures and at any pressure for Ge02 , the compression is achieved through the second mechanism. In order to check the predictive interest of such calculations, we derived approximate force constants from our data. The OSiO and OGeO force constants have been calculated at the PS-3G level, since it is well known that such force constants are correctly predicted even with mini­mal basis sets. 23 In contrast, the SiOSi and GeOGe force con­stants are strongly dependent upon the basis set quality and therefore the PS-21G** values have been retained. In the case of Si02 the calculated force constants are of similar magnitude 38 and 40 nm - I for the OSiO and SiOSi coordi­nates, respectively, and the uncertainties due to basis set limitations, absence of correlation, etc., do not allow us to decide which of them is the larger. For Ge02, the bridge angle force constant is about two times larger than the tetra­hedron bending force constant. In this latter case, the com­pression is expected to process through tetrahedron angular deformations keeping the bridging angle nearly constant. Our results lead us to expect that the two processes would compete in the case of Si02, as can also be inferred from the experimental values of force constants. 24

v. CONCLUSION

The results presented in this paper demonstrate the in­terpretative power of the periodic Hartree-Fock method. It enabled us to propose a new microscopic quantic description of high quartz consistent with both classical pictures. In ad­dition, the difference between silicon and germanium diox­ide are rationalized on the basis of the tilt angle dependence of energy. The study demonstrates the progress accom­plished since a pioneer calculation on quartz. 12

ACKNOWLEDGMENTS

The authors gratefully acknowledge Professor R. Do­vesi and V. R. Saunders for illuminating discussions and Dr. S. Hillier for critically reading the manuscript. The calcula­tions have been performed on the CRA Y /2 ofCCVR which is thanked.

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