The isotypism of BeH2 and SiO2: an ab initio study

Chemical Physics Letters 378 (2003) 343–348 www.elsevier.com/locate/cplett

The isotypism of BeH2 and SiO2: an ab initio study Ute Hantsch a

a,*

€rn Winkler a, Victor Milman , Bjo

b

Institut f€ ur Mineralogie/Abt. Kristallographie, Senckenberganlage 30, D-60054 Frankfurt am Main, Germany b Accelrys, 334 Cambridge Science Park, Cambridge CB4 0WN, UK Received 10 February 2003; in final form 30 July 2003 Published online: 26 August 2003

Abstract Polymorphs of BeH2 isotypical to SiO2 framework structures have been investigated by density functional theory calculations using a plane wave basis set in conjunction with ultra-soft pseudo-potentials. All calculated structures turned out to be stable with respect to small atomic displacements and are therefore possible metastable BeH2 modifications. The most stable modification at ambient pressure is the already known Ibam structure. We predict a phase transition to a high pressure modification. Linear response calculations have been performed to investigate the connection between the nature of bonding and the lattice dynamics of several BeH2 polymorphs. Ó 2003 Elsevier B.V. All rights reserved.

1. Introduction Molecular BeH2 has attracted a significant amount of interest [1] as due to the small number of electrons it is amenable to high quality quantum mechanical calculations, and therefore may serve as a testing tool for new quantum chemical methods. However, experimental investigations of molecular BeH2 are problematic. It was therefore difficult to measure vibrational modes, and the experimental verification of the intermolecular vibrations has succeeded only recently [1,2]. In addition, crystalline BeH2 has been a subject of numerous discussions, since Overhauser [3] di-

*

Corresponding author. Fax: +496979822101. E-mail address: [email protected] (U. Hantsch).

rected attention to hydrides of light metals in connection with the design of new high-Tc superconductors. However, the effort to study crystalline BeH2 seems to have been limited, and until now there is only one structure determination available [4]. Recently this structure has been shown to be related to that of moganite [5], a SiO2 polymorph which can be described as a quartz structure fulfilling the Brazilian twin law on a unit cell scale [6]. SiO2 is known to crystallize in a large variety of tetrahedral framework structures, ranging from dense structures (coesite, q ¼ 2:92 g/cm3 ) to very open nanoporous structures such as sodalite (q ¼ 1:71 g/cm3 ) [7]. Some families of frameworkconnected polymorphs, such as aluminum phosphates [8] or phosphorous oxynitride [5,9] show similarities to the behaviour of SiO2 , but from a crystallographic point of view differ significantly in

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that there are two distinct cations or anions. Other compounds, such as the group IV metal dioxides GeO2 , PbO2 , and SnO2 , show structural phase transformations similar to those observed in SiO2 at high pressures, e.g. for the transition from the stishovite into a post-stishovite structure [10,11]. The possibility that BeH2 might behave similar to SiO2 and thus allow to extend crystal chemical models explaining phase relations in this large family of structures to more compounds, was therefore intriguing. In this Letter, we present the results of quantum mechanical study of the structure, energetics, and compressibility of hypothetical BeH2 polymorphs.

2. Computational details The quantum mechanical calculations performed here are based on density functional theory, DFT. While DFT itself is exact [12], practical calculations require an approximation for the treatment of the exchange and correlation (xc) effects. The most widely used schemes are the Ôgeneralized gradient approximationÕ, GGA [13], or the local density approximation, LDA. Results based on GGA calculations are generally in better agreement with experiment than those obtained with the LDA [14–17]. In one set of calculations, ultra soft pseudopotentials were used with a maximum cutoff energy of the plane waves of 260 eV. In these calculations the PBE-version of the GGA was employed while academic and commercial versions of the CA S T E P program [18–20] were used. In addition to the cutoff energy, one further parameter determines the quality of the calculations, namely the density of points with which the Brillouin zone is sampled. The wave vectors for the sampling points were chosen according to the scheme proposed by [21]. We use a sampling of reciprocal space such that the distances between

1 . Full geometry grid points are about 0.05 A optimization calculations were performed in which all structural parameters not constrained by the space group symmetry were relaxed. The space groups were Ibam, C12=c1, P 32 2, I  43m, C12=c1, P 42 /mnm, P 41 21 2 and P 21 3 for the BeH2 poly-

morphs with the structure of pseudo-moganite, moganite, a-quartz, sodalite, coesite, stishovite, low- and high-cristobalite, respectively. After the final self-consistency cycle the remaining forces on
, and the rethe atoms were less than 0.003 eV/A maining stress was less than 0.05 GPa. In a second set of calculations in which phonon frequencies were evaluated, the program package AB I N I T [22] was used. Vibrational properties are evaluated analytically in this approach using the linear response formalism. These calculations are based on LDA and use norm-conserving pseudopotentials of the Troullier–Martins type. We will show that the choice of the xc-functional is unproblematic in the present calculations. Both sets of calculations are restricted to the athermal limit, in which temperature effects and zero-point motions are neglected.

3. Results and discussion The results of the geometry optimizations of the known BeH2 polymorph with the Ôpseudo-moganiteÕ-structure and space group symmetry Ibam are listed in Table 1. As expected, the cell parameters obtained with DFT–LDA are in slightly worse agreement with experiment than the DFT–GGA results. The main discrepancy between calculation and experiment is observed in a comparison of the bond lengths, where the difference between the longest and the shortest bond is too small. Similar Table 1 Structural parameters of the known BeH2 polymorph (space group Ibam) Experiment
) a (A
) b (A
) c (A
3 ] V [A
) Be–H min (A
) Be–H max (A H–Be–H min (°) H–Be–H max (°) Be–H–Be (°) Be–H–Be (°)

9.082(4) 4.160(2) 7.707(3) 291.2(2) 1.38(2) 1.44(2) 106.5(5) 113.0(5) 127(1) 130(1)

GGA–PBE

LDA

9.012 4.183 7.645 288.18 1.435 1.439 98.6 113.2 120.0 126.4

8.707 3.959 7.480 257.86 1.428 1.433 94.8 120.3 115.7 116.8

Parameters calculated with DFT–GGA and DFT–LDA are compared with experimental data obtained by Smith et al. [4].

U. Hantsch et al. / Chemical Physics Letters 378 (2003) 343–348

problems have been reported in connection with the calculation of SiO2 quartz [23], where often the hexagonal high quartz was found to be the ground state structure. This problem encountered during

Fig. 1. Isosurface of the electron density difference qself-consistent
qnon-interacting atoms of the BeH2 Ibam structure (Be: dark grey, H: light grey). One can clearly see the covalent character of the Be–H bonds which build the framework structure.

345

the geometry optimization can be explained by the low energies which are required to translate and rotate tetrahedra in rigid unit mode structures [24] and does not affect the basic results presented in this Letter. A population analysis of the Ibam structure clearly indicated the covalent bonding between the Be-atoms and the four hydrogen atoms coordinating them with a bond population of 0.5. This is also evident from the electron density difference surface presented in Fig. 1. As a next step, we computed the ground state structures of BeH2 corresponding to the structure of moganite, a-quartz, high- and low cristobalite and SiO2 -sodalite (Table 2). A comparison of the enthalpies showed that all these structures are less stable than the structure of the known polymorph. However, the small energy differences of 1.06–6.18 kJ/mole imply that they are possible metastable polymorphs. The sequence of densities of BeH2 polymorphs (sodalite least dense, stishovite densest) is equivalent to that of SiO2 polymorphs, whereby the ratio qSiO2 =qBeH2 increases from 2.88 to 4.01 with increasing density (Table 2). The intriguing similarity of BeH2 to ambient pressure phases of SiO2 prompted us to test if the high pressure behaviour is also similar. Quartz is known to undergo phase transitions to coesite and

Table 2 Calculated BeH2 structures at ambient pressure Orig Space group
) a (A
b (A)
) c (A
) b (A
3 /f.u.) V (A
) Be–H min (A
) Be–H max (A H–Be–H min (°) H–Be–H max (°) Be–H–Be (°) Be–H–Be (°) DE (kJ/mole) qBeH2 (g/cm3 ) qSiO2 (g/cm3 ) qSiO2 =qBeH2

Ibam 9.012 4.183 7.645 90 24.02 1.435 1.439 98.6 113.2 120.0 126.4 0 0.763 – –

Mogan C12=c1 11.958 4.388 7.537 129.95 25.39 1.417 1.447 97.1 117.5 129.6 139.1 +5.02 0.721 2.55 3.54

Quartz

Sodalite

Crist-l

Crist-h

Coesite

Stish

P 32 2 4.244 4.244 4.597 90 23.90 1.429 1.434 110.8 114.5 132.9 132.9 +6.18 0.766 2.65 3.46

I 43m 7.183 7.183 7.183 90 30.88 1.434 1.434 106.3 106.3 124.6 124.6 +2.99 0.593 1.71 2.88

P 41 21 2 4.182 4.182 5.622 90 24.59 1.423 1.437 101.8 120.7 127.1 127.1 +1.06 0.745 2.36 3.17

P 21 3 6.005 6.005 6.005 90 27.07 1.414 1.433 104.2 114.2 127.8 180.0 +4.25 0.676 2.17 3.21

C12=c1 6.352 10.882 6.215 119.64 23.34 1.416 1.477 101.1 118.6 128.3 180.0 +13.60 0.785 2.92 3.72

P 42 =mnm 3.863 3.863 2.301 90 17.17 1.570 1.663 85.8 180.0 132.9 132.9 +49.30 1.067 4.28 4.01

Their names correspond to the SiO2 polymorphs moganite (Mogan), quartz, sodalite, high (h) and low (l) cristobalite (Crist), coesite and stishovite (Stish). Orig indicates the original BeH2 Ibam structure.

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stishovite on compression [7]. Both coesite and stishovite as BeH2 trial structures turned out to be stable with respect to small distortions, and are hence possible (metastable) polymorphs. The structural details of these polymorphs at ambient pressure are listed in Table 2. An analysis of the pressure dependence of the lattice energies shows that the initial Ibam structure is always more stable than the coesite structure, while it undergoes a phase transition to the stishovite (rutile) structure at about 33 GPa (Fig. 2). A population analysis of the stishovite structure, in which each Be is coordinated octahedrally by six hydrogen atoms, indicated covalent bonding between the 4 slightly shorter Be–H bonds (the bond population Be–H is again 0.5) while the two longer Be–H ÔbondsÕ are not populated at all. During compression, the distance between the Ônon-bondedÕ Be and H atoms decreases faster than the distance between the 4 bonded Be and H atoms. At 45 GPa, the ratio of length inverts, so that the distances between the Ônon-bondedÕ Be and H atoms become smaller than the Be–H bond lengths (Fig. 3). An electron density isosurface, however, clearly shows that the electrons are distributed around the Be atom but

Fig. 2. Energies of the BeH2 Ibam and BeH2 coesite polymorph relative to the BeH2 stishovite structure. The Ibam structure is most stable up to a pressure of 33 GPa. In contrary to its SiO2 isotype, BeH2 coesite is at no pressures more stable than the ground state structure.

Fig. 3. Be–H distances in BeH2 stishovite. At 45 GPa the calculated distances between the non-bonded Be–H contacts become smaller than the Be–H bond lengths.

not on the bonds themselves, building delocalized ÔresonantÕ bonds (Fig. 4). Further, least squares fitting to third-order Birch–Murnaghan equation of state has been carried out to obtain the bulk moduli of the Ibam and the stishovite structure. The bulk modulus of BeH2 stishovite is 82 GPa, which is about one

Fig. 4. Isosurface of the electron density difference qself-consistent
qnon-interacting atoms of the BeH2 stishovite structure (Be: dark grey, H: light grey). One can see that the electrons are distributed around the Be atom but not on the bonds themselves, building delocalized ‘‘resonant’’ bonds.

U. Hantsch et al. / Chemical Physics Letters 378 (2003) 343–348

third of the bulk modulus of the related SiO2 polymorph (298 GPa, [7]). The bulk modulus for the stable Ibam structure is predicted to be 25 GPa which would be two thirds of the bulk modulus of SiO2 a-quartz. A more detailed comparison is given in Table 3. Linear response calculations of the phonon frequencies have been performed in order to further characterize the interatomic interactions. The accuracy of the calculations was estimated based on the comparison of the calculated and measured vibrational frequencies of an isolated BeH2 molecule. The calculations were performed by placing the molecule in a cubic box with edge length of
). The equilibrium Be–H distance 20 a.u. (10A was obtained from single point energy calculations on sets of BeH2 molecules with varying Be–H distances. The resulting potential curve showed the same curvature for both GGA and LDA which indicates that the choice of the xc-functional should not affect calculated vibrational properties. The dependence of the total energy on the Be–H distance computed in the GGA calculation was used to obtain the force constant and thus the m1 stretching frequency. We used the relationship pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 ðXY2 Þ ¼ k1 =ðmy  4p2 Þ, where k1 is a force constant and my is the mass of atom Y [25]. The obtained value, v1 ¼ 1911 cm
1 , compares well

347

with the result of a LDA-based linear response calculation which gave m1 ¼ 1931 cm
1 . All obtained results are presented in Table 4, where the calculated frequencies are compared to the corresponding experimental values. The results compare very well (within 10 cm
1 ) with those obtained from matrix isolated BeH2 , while they are in slightly worse agreement with recently measured gas phase values, where the disagreement for the stretching vibration is 30 cm
1 . The data obtained here for m2 are in significantly better agreement with the experimental data than those obtained earlier based on Hartree–Fock calculations [26], the present values for m3 are slightly better. No experimental data are available for m1 but both our work and the earlier calculations [26] show that m1 is about 200 cm
1 smaller than m3 . After having established the reliability of the calculations, we compare the phonon frequencies obtained for the isolated molecule to those of solid BeH2 . For the latter, no experimental data are available. We predict that the highest stretching frequency in structures with tetrahedrally coordinated Be occurs at 1990 cm
1 . Hence, we observe a red-shift by 160 cm
1 with respect to the highest frequency of the molecule which is due to the crystal lattice formation. This shift is to be expected, and its magnitude is similar to the

Table 3 Calculated parameters of the third-order Birch–Murnaghan equations of state for the BeH2 -Ibam and BeH2 -stishovite structures
3

V (A /f.u.) B (GPa) B0 (Gpa)

BeH2 -Ibam

BeH2 -Stish

SiO2 -Quartz

SiO2 -Stish

24.02 24.7 (0.9) 3.9 (0.1)

17.17 81.5 (0.1) 3.6 (0.1)

22.71 38.7 (1.0) 4.9 (0.2)

14.04 298 (8) 4.0 (0.5)

The results are compared to the available data for related SiO2 polymorphs [7].

Table 4 Frequencies of a quasi-isolated BeH2 molecule calculated with Abinit (LDA) using linear response approach and with CASTEP from a potential energy curve (GGA)


1

m1 (cm ) m2 (cm
1 ) m3 (cm
1 )
) Be–H (A

IR[2]

IR[1]

LDA

GGA

– 698 2159 1.333

– – 2179 1.334

1931 702 2149 1.338

1911

Experimental data for BeH2 are available from an argon matrix [2] and gas phase measurements [1].

1.330

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observed changes in hydrogen bonded systems, where, for example, the pressure-induced approach of a second oxygen leads to an increase in the hydrogen bond strength, to a decrease in the covalent O–H bond strength and hence to a decrease of the O–H stretching frequency [27]. Calculations of the phonon frequencies along high symmetry directions of the Brillouin zone of the high-pressure phase with the stishovite structure confirmed that at pressures from 0 to 25 GPa BeH2 this polymorph may be (meta-)stable. At higher pressures, phonons at several wave vectors close (0.05 reciprocal lattice units) to the C-point become soft and would induce a phase transition. Lattice energies of the ambient pressure structure and the stishovite structure become equal at about 33 GPa. As the two sets of calculations were performed with different pseudopotentials and different approximations for the xc potential, further calculations are required to establish the exact phase sequence. However, it is clear that SiO2 -like high pressure phases with octahedrally coordinated beryllium may exist metastably and hence the analogy between BeH2 and SiO2 extends to the high pressure regime. This is also evident from the calculation of phonon frequencies. The highest frequency in the high pressure compounds with octahedrally coordinated beryllium is 1750 cm
1 , and therefore about 240 cm
1 less than the highest frequency in the structures with tetrahedrally coordinated beryllium. This shift is similar to that observed for SiO2 phases, where a typical Si–O stretching frequency is 1200 cm
1 in tetrahedral framework structures, while in stishovite the highest frequency is 950 cm
1 [28]. In summary, we have shown that a whole family of BeH2 polymorphs isostructural to the SiO2 polymorphs may be stable. We hope that this will inspire experimental attempts to synthesize and characterize these compounds.

Acknowledgements This study was funded by the German Science Foundation (Grant Wi1232/10).

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