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Electronic and structural properties of CaH2: an ab initio Hartree–Fock study
Chemical Physics 252 Ž2000. 1–8 www.elsevier.nlrlocaterchemphys
Electronic and structural properties of CaH 2 : an ab initio Hartree–Fock study Abderrahman El Gridani ) , Mohamed El Mouhtadi ´ Laboratoire de Chimie Physique, Equipe Chimie Theorique appliquee, ´ ´ UniÕersite´ Ibn Zohr, Faculte´ des Sciences, 80 000 Agadir, Morocco Received 22 June 1999
Abstract The Hartree–Fock ab initio method has been used to evaluate some structural and electronic properties of the CaH 2 crystal. The calculated quantities include the crystalline parameters, binding energy, elastic constants, band structure, density of states and electronic charge distribution. Taking into account the correlation effects, the experimental binding energy is well reproduced. The band energies, density of states and charge density maps are analyzed, and the nature of chemical bonding is discussed, showing significant deviations from ionicity Ž z ca s 1.867 < e <.. The values of elastic constants C11, C12 , C13 , C22 , C23 , C33 , C44 , C55 , C66 and bulk modulus, determined for the first time, are 14.4, 110.3, 67.9, 75.1, 143.1, 36.6, 111.1, 68.7, 81.6 and 202.8 GPa, respectively. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Metal hydrides have received great attention due to their utilization in technological applications and their scientific interest they present w1–3x. First of all, they constitute the most convenient porters of hydrogen considered as being the energy source which is the most promising one in the future. In addition, interesting superconductivities have been observed in some metal hydrides such as PdH x w4x, of which critical temperature increases with concentration of hydrogen. Finally, they constitute reliable a means for testing theoretical models. The nature of bonds in metal hydrides is usually enough complex w5–7x and it is explored by means
)
Corresponding author. Fax: q212-81-22-01-00.
of various theoretical and experimental techniques. Historically, three theoretical models w8x generally have been employed to understand experimental data from such diverse fields as photoelectron spectroscopy with X-rays ŽXPS. or synchrotron radiation ŽSRPES. Ženergy of crystal, Compton profile, electronic specific heat and magnetic susceptibility. w9,10x. It concerns anionic, protonic and covalent models corresponding to the three extremes of typical bonding which can exist in a crystal. The anionic model supposes that an electron of the metallic ion is transferred to the hydrogen atom; the protonic model, contrarily to the precedent one, supposes that an electron is yielded by hydrogen to join the conduction metallic band partially occupied. Finally, the covalent model supposes that hydrogen atoms are linked to calcium ones in a covalent manner. The anionic model is more adequate to study the alkali-metal hydrides, which are typically ionic. For
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A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
a transition hydride such as TiH x w10x, the protonic model seems to be well adapted. For some alkalineearth-metal hydrides, different properties of the system cannot be fit with a single model. This is the case for the BeH 2 compound, which was considered covalent. MgH 2 was considered intermediate between ionic and covalent w11x, but recently the calculation of band structures has shown that this compound is largely ionic with a weak covalent character w12x. Concerning CaH 2 , known as the most useful of the three light-metal hydrides ŽLiH, MgH 2 , CaH 2 ., although its ionic character resulting from the large difference between hydrogen and calcium electronegativities and sustained by XPR study w7x, the description of its Compton profile has failed using the IPC models w6x. The failure of these models, in the study of these systems, can be attributed to two main reasons. Firstly, structural effects and solid state bonds are largely omitted in these models since they normally assumed a single 1s Slater-type orbital Ž1s STO. to describe the electron of hydrogen and adopted the rigid-band approximation for the metal. However, Kunz w13x and Pattison et al. w14x have shown that, because of the solid-state effects, the radial behaviour of the hydrogen wavefunction must differ appreciably from 1s STO. Secondly, these models suppose the existence of bonds with extremely bonding character, while the supposition of the existence of those bonds with mixed character would be more reasonable for this type of compounds. The best way allowing to understand completely the electronic properties of metal hydrides is proved to be by means of band-structure calculations. However, because of the low orthorhombic symmetry, no calculation of band structure for CaH 2 has been undertaken in its real structure. Indeed, only its bands have been determined in a hypothetical structure of the fcc CaF2 type w15x. Thereby, structural effects, in this structure, are neglected and, therefore, deserve to be re-examined by a computation based on the real structure. In this paper, we have used the all-electron ab initio method at the self-consistent field Hartree– Fock ŽSCF–HF. level to compute the equilibrium structure, band structure, density of states, charge densities, Mulliken’s population and elastic constants of the CaH 2 system. These magnitudes are compared
to the corresponding available experimental and calculated ones.
2. Computational method The all-electron ab initio method, allowing the calculation of wavefunctions of crystalline systems, is implanted in the CRYSTAL95 program w16x. It is described in numerous previous works Že.g., w17,18x. and has been recently applied to simple ionic combined systems such as CaO w19x and CaF2 w20x, in order to determine their electronic, dynamic and structural properties. In the current study, an error source, due to the intrinsic limit of the Hartree–Fock approximation, is possible Žcorrelation error.. According to the molecular experience w21x, the HF approach underestimates Žin covalence systems. energies of formation Žsometimes by 30%., while bond lengths are overestimated Žby 0.5–1%.. Similar results have been obtained in a systematic study of III–IV and III–V semiconducting compounds w22x. Concerning the ionic systems, a few studies are realized and seem to confirm this tendency. As it will be shown in this work, the energy of formation can be easily corrected in order to take into account the correlation contribution. Under ambient conditions, CaH 2 crystallises in an orthorhombic phase of complicated PbCl 2 type, having Pnma as space group w23x. Each elementary cell contains four CaH 2 molecules with two different sites of non-equivalent hydrogen. This structure possesses three crystalline parameters Ž a, b, c ., six fractional coordinates Ž x ŽCa., x ŽH1., x ŽH2., z ŽCa., z ŽH1., z ŽH2.. and nine components of the elastic tensor. The atomic basic set, used in this work ŽTable 1., is an all-electron one taken from Refs. w12x and w20x for the hydrogen atom and calcium atom, respectively. Six and nine atomic orbitals are employed for hydrogen and calcium, respectively, and are function combinations of gaussian type. These two basis sets can be written as 5-111G for hydrogen and 8-65113G for calcium Žnotation of Ref. w21x., where the numbers refer to the level of contraction. Note that the 5sp orbital exponent has been re-optimized on the CaH 2 system in order to adapt the calcium basis on the system in question Ž j Ž5sp. s 0.2757 bohry2 ..
A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
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Table 1 Exponents Žbohry2 . and coefficients of the Gaussian functions adopted for the present study Shell type
Calcium exponents
Shell type coefficients s
1s
2sp
3sp
4sp 5sp 3d
1.913Žq5. 2.697Žq4. 5.696Žq3. 1.489Žq3. 4.483Žq2. 1.546Žq2. 6.037Žq1. 2.509Žq2. 4.486Žq2. 1.057Žq2. 3.469Žq1. 1.35Žq1. 5.82 1.819 2.075Žq1. 8.4 3.597 1.408 7.26Žy1. 4.53Žy1. 2.757Žy1. 3.191 8.683Žy1. 3.191Žy1.
2.204Žy4. 1.925Žy3. 1.109Žy2. 4.995Žy2. 1.701Žy1. 3.685Žy1. 4.034Žy1. 1.452Žy1. y5.75Žy3. y7.67Žy2. y1.122Žy1. 2.537Žy1. 6.88Žy1. 3.49Žy1. y2.00Žy3. y1.255Žy1. y6.96Žy1. 1.029Žq0. 9.44Žy1. 1.0 1.0
Hydrogen exponents
p or d
coefficients s
1s
2s 3s 2p
1.2Žq2.2.67Žy4. 4.0Žq1.2.249Žy3. 1.2Žq1.6.386Žy3. 4.0Žq0.3.291Žy2. 1.2Žq0.9.551Žy2. 5.0Žy1. 1.0 1.3Žy1. 1.0 2.0Žy1. 1.0
p
1.0
8.47Žy3. 6.027Žy2. 2.124Žy1. 3.771Žy1. 4.01Žy1. 1.98Žy1. y3.65Žy2. y6.85Žy2. 1.75Žy1. 1.482Žq0. 1.025Žq0. 1.0 1.0 1.6Žy1. 3.13Žy1. 4.06Žy1.
The contraction coefficients multiply individually normalized Gaussians. y Ž"z . stands for y = 10 " z.
Hereafter, the notations nx ŽH. and nx ŽCa., where n is the number of the orbital x, designate the orbital x of hydrogen atom and calcium atom, respectively.
3. Equilibrium structure and binding energy Crystalline parameters and optimized fractional coordinates are given in Table 2 compared to their corresponding experimental ones. The calculation of the equilibrium structure is obtained, for the first time, by the optimization of the crystal’s total energy according to each crystalline parameter by fixing the two others and the fractional coordinates to their experimental values. Then, from optimized parameters, this energy is re-optimized according to the different fractional coordinates. A polynomial interpolation leads to a minimal value of energy corresponding to the HF energy y2711.73977 au. These
results are in good agreement with the measured data knowing that the HF calculation overestimates the unit-cell volume w22x. For comparison, the relative errors to volumes in CaF2 w20x and VO w19x crystals case are in the order of 5.2% and 8.9%, respectively. The total crystal energy is calculated from optimized crystalline parameters at the HF level. Total energies of isolated atoms are computed using the crystalline basis sets essentially designated to describe ionic states, and furthermore, by addition of two diffuse orbitals s and p on each center and by optimization of the most diffuse orbital exponent in order to provide additional variational freedom accounting for the tails of atomic wavefunction. According to the HF approximation, the binding energy is equal to the difference between the total energy and the sum of atomic energies, i.e. 21.39 eV per unit cell. Data calculated by the present method ignore all vibrational contributions to the energy,
A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
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Table 2 Calculated and experimental lattice parameters and binding energy for CaH 2
a b c x ŽCa. x ŽH1. x ŽH2. z ŽCa. z ŽH1. z ŽH2. V BEŽHF. BEŽCS. BEŽP.
Calculated
Experimental ŽRef. w23x.
D Ž%.
6.137 3.694 7.098 0.2405 0.2405 1.0078 0.1128 0.4319 0.7688 160.9 21.39 24.73 24.07
5.936 3.600 6.838 0.2600 0.2600 0.9960 0.1100 0.4300 0.7580 146.1 24.81a
3.4 2.6 3.8 7.5 7.5 1.2 2.5 0.4 1.4 10.1 13.8 0.3 3.0
˚ Žcell volume Ž V . in For lattice parameters, values are given in A ˚ 3 .. For binding energy ŽBE., values are given in eV. CS means A Colle Salvetti’s functional. P means Perdew’s functional. a From Ref. w24x.
which is lower Žby 13.8%. than its corresponding experimental one Ž24.81 eV. evaluated from the heat of formation of CaH 2 , H 2 and Ca using the Born– Haber cycle w24x. Concerning the correction due to the electronic correlation of the formation energy, it is equal to the difference between the correlation energies of crystal and atoms, and evaluated by means of the density functional proposed by Colle and Salvetti w25,26x, and Perdew w27,28x, applied to the HF charge density. Note that ŽTable 2. the absolute correction of BE calculated by means of the density functional proposed by Colle and Salvetti Ž E s 3.34 eV. is higher Žby 19.8%. to the one calculated using Perdew’s formalism Ž E s 2.68 eV.. The difference between our correlated energies Ž24.73 and 24.07 eV. and those experimentally determined is in the order of 0.08–0.74 eV per cell. The data obtained by our method do not take into account the vibrational contribution to the energy, and are compared to the experimental value corrected to 0 K.
Fig. 1. Electron energy bands of CaH 2 along two symmetry lines in the first Brillouin zone.
Born type w29x. In this method the central force potentials used to represent the ground state energy of the solid as a function of the nuclear positions require a set of empirical parameters to specify their functional form. These parameters are generally determined by fitting available experimental parameters such as bond energy, lattice parameters and dielectric constants. The Hartree–Fock approach, as implemented in CRYSTAL, permits a parameter-free ab initio determination of such properties even for Table 3 Effects of isotropic compression of crystalline parameters a, b ˚ . on the charges Ž< e <. of Ca and H, which are evaluated and c ŽA according to the Mulliken partition scheme a b c Ca
H1
H2
4. Elastic properties Elastic constants of ionic and partially ionic compounds are generally calculated using classic approaches based on a semi-empirical formula of the
1s 2sp 3sp 4sp 5sp d total 1s 2s 3s p total 1s 2s 3s p total
6.137 3.694 7.098 y2.000 y8.094 y4.615 y2.177 y1.125 y0.123 y18.133 y0.176 y0.374 y1.363 y0.032 y1.945 y0.168 y0.312 y1.426 y0.016 y1.922
5.830 3.509 6.743 y2.000 y8.094 y4.611 y2.184 y1.101 y0.156 y18.146 y0.184 y0.399 y1.331 y0.026 y1.939 y0.173 y0.335 y1.397 y0.010 y1.915
Shell symbols refer to Table 1.
5.539 3.334 6.406 y2.000 y8.093 y4.609 y2.185 y1.073 y0.199 y18.159 y0.193 y0.422 y1.298 y0.020 y1.933 y0.181 y0.355 y1.371 y0.002 y1.909
5.262 3.167 6.086 y2.000 y8.091 y4.605 y2.188 y1.035 y0.254 y18.173 y0.204 y0.443 y1.126 y0.010 y1.921 y0.190 y0.371 y1.352 y0.006 y1.907
A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
materials where experimental data are unavailable w30,31x. In addition, the HF method, as it has become known, reproduced reasonably experimental elastic constants of ionic and semi-ionic systems. In this section, we present the nine elastic constants, calculated for the first time, and which have not yet been determined experimentally, to our knowledge. Using the atomic orbital basis sets of Table 1, the total energy of CaH 2 was calculated for a number of different deformations of the unit cell. The energy
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values against strain components hi were fitted to polynomial functions up the fourth order. Thus the second derivatives of energy Žat the energy minimum. could be calculated and yielded the elastic constants Ci j according to the relation: 1 E2 E Ci j s , V Ehihj 0
where V is the volume of the cell and E is the energy per cell. The found values for these elastic
Fig. 2. Projected and total densities of states ŽDOS. of CaH 2 . A Mulliken partition scheme was used to obtain atomic orbital contributions.
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A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
constants C11 , C12 , C13 , C22 , C23 , C33 , C44 , C55 , C66 and bulk modulus B are 14.4, 110.3, 67.9, 75.1, 143.1, 36.6, 111.1, 68.7, 81.6 and 202.8 GPa, respectively.
5. Electron band structure, density of states, Mulliken’s population and charge density The band structure of CaH 2 is represented in Fig. 1. As one would have to expect, bonds are flat and are separated by a large energy gap Ž Eg .. This gap, between the valence band and the conduction one, is around 10 eV, almost twice larger than the experimental one Žin the order of 5 eV w15x.. This difference between theory and experience is due to the fact that the HF hamiltonian overestimates the energy gap. For comparison, the gap value, calculated for the MgH 2 system w12x, is in the order of 13 eV while the measured one is 5.16 eV. Note that our value is also different from that calculated by ChuanYun Xia et al. w32x using the cluster method which underestimates the insulator and semiconductor gaps Ž3.32 eV.. Analysis of the Mulliken electronic population confirms the ionic nature of bonding in CaH 2 ŽTable 3.. Indeed, the net charges on the calcium and the two non-equivalent hydrogens are 1.867 < e <, y0.945 < e < and y0.922 < e <, respectively, and are close to formal ionic values Žq2 < e <, y1 < e < and y1 < e <,
respectively.. Furthermore, Chuan-Yun Xiao et al. w32x have calculated the charge of the calcium atom using their model, and have shown that CaH 2 is less ionic than in our case Ž q ŽCa. s 1.61 < e <.. Note that the ab initio calculations of the MgH 2 w12x, MgF2 w31x and CaF2 w20x systems show that the charge values of heavy atoms are 1.886 < e <, 1.803 < e < and 1.868 < e <, respectively. It results, therefore, that CaH 2 is strongly ionic with a weak covalent character. In addition, the bonding indices Ca–H1 and Ca–H2 are in the order of y0.012 < e < and 0.003 < e <, respectively. It is, therefore, interesting to analyze the distribution of charges on different ions, especially their modification induced by isotropic compression of the crystal ŽTable 3.. From the analysis of this table, we note that a decrease by 5% of each crystalline parameter increases the total charge of calcium by 0.013 < e <. This indicates a tendency to a less ionic Ca–H bonding. The supplementary negative charge is entirely situated on the dŽCa. orbital y0.076 < e < and the electronic cloud spŽCa. is quite contracted. Indeed, we notice that y0.015 < e < are transferred from the 2spŽCa. and 5spŽCa. orbitals towards the inner orbital 3spŽCa. and the dŽCa. orbital, respectively. A similar transfer of 3sŽH. and 2pŽH. electrons, towards the 1sŽH. and 2sŽH. orbitals, respectively, is observed for hydrogen H1 and hydrogen H2. Also, we note that the charge transferred to calcium per each hydrogen is in the order of y0.013 < e < of which a part is transferred to the dŽCa. orbital.
Fig. 3. Ža. Total electron density map on the Ž100. plane. The separation between isodensity curves is 0.01 erbohr 3 . Žb. Difference Žcrystal minus ionic superposition. electron density map on the Ž100. plane. Continuous, dashed and dotted lines indicate positive, zero and negative valus, respectively. The separation between isodensity curves is 0.01 erbohr 3.
A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
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Fig. 4. Ža. Total electron density map on the Ž110. plane. The separation between isodensity curves is 0.01 erbohr 3 . Žb. Difference Žcrystal minus ionic superposition. electron density map on the Ž110. plane. Continuous, dashed and dotted lines indicate positive, zero and negative valus, respectively. The separation between isodensity curves is 0.01 erbohr 3.
Fig. 2, which reports on total and partial density of states, shows that the valence band is almost purely atomic without any mixed contribution, except perhaps a weak participation of 2sŽH. Ž q2s s 0.399 < e <. and dŽCa. Ž qd s 0.123 < e <. orbitals with the 3sŽH. orbital. However, Chuan-Yun Xia et al. w32x have shown that the valence band is formed essentially by the 1sŽH. orbital with a weak contribution of the 1sŽH. and dŽCa. ones. The conduction band is formed primarily by 2pŽH. and dŽCa. orbitals with a minor role of the 5spŽCa. one. Total and difference Žs total crystal minus ionic superposition. electron densities are calculated on the Ž100. and Ž110. planes limited to a portion containing eight calcium atoms and accompanying hydrogen ones, and are reported in Fig. 3a, b and Fig. 4a, b, respectively. A contraction of the electronic cloud in the crystal, by comparison to free ions, appears clearly and is due both to the repulsion induced by the electronic exchange ŽPauli’s exclusion principle. and to the compression effect of the crystal due to the electrostatic field of the anions.
6. Conclusions The present work represents the first attempt for the calculation of the nine elastic constants for CaH 2 in its orthorhombic structure by means of the LCAO HF ab initio method.
The computation of the band structure and the analysis of Mulliken’s population show that CaH 2 is an ionic system with a low covalence character. The crystal formation is accompanied by contraction of both the anion and cation. Under compression, a further contraction of ions is apparent, and also a charge transfer from 3sŽH. and 2pŽH. orbitals towards dŽCa. and 3spŽCa. ones, indicating an important contribution of d orbitals to the crystal stability. The analysis of the total density of states and partial densities of states shows that the valence band is almost purely atomic without any mixed contribution, except perhaps a weak participation of 2sŽH. and dŽCa. orbitals with the 3sŽH. orbital. Furthermore, the conduction band is formed primarily by 2pŽH. and dŽCa. orbitals with a minor role of the 5spŽCa. one.
References w1x G.G. Libowitz ŽEd.., Solid-state Chemistry of Binary Metal Hydrides, Benjamin, New York, 1965. w2x J.J. Reilly, G.D. Sandrock, Sci. Am. 242 Ž1980. 98. w3x P. Jean, C.H. Satterthwaite ŽEds.., Electronic Structure and Properties of Hydrogen in Metals, Plenum, New York, 1983. w4x T. Skiskiewicz, Phys. Status Solidi A 11 Ž1972. K123. w5x J. Felsteiner, M. Heilper, I. Gertner, A.C. Tanner, R. Opher, K.F. Berggren, Phys. Rev. B 23 Ž1981. 5156.
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A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
w6x N.G. Alexandroppoulous, T. Chatzigeogiou, I. Theororidou, Philos. Mag. B 57 Ž1988. 191. w7x H.F. Franzen, J. Merrick, M. Umana, A.S. Khan, J.R. MacCreary, R.J. Rhon, J. Electron Spectrosc. Relat. Phenom. 11 Ž1977. 439. w8x J.D. Fast, Cases in Metals, Macmillan, London, 1976. w9x R. Lasser, B. Lengeler, Phys. Rev. B 18 Ž1978. 673. w10x N.G. Alexandroppoulous, I. Theororidou, M.J. Cooper, J. Phys. C: Solid State Phys. 20 Ž1987. 1201. w11x C.M. Stander, A. Paaacey, J. Phys. Chem. Solids 39 Ž1978. 829. w12x I. Baraille, C. Pouchan, M. Causa, C. Pisani, Chem. Phys. 179 Ž1994. 39. w13x A.B. Kunz, Phys. Status Solidi 36 Ž1969. 301. w14x P. Pattison, W. Weyrich, J. Phys. Chem. Solids 40 Ž1979. 213. w15x J.H. Weaver, M. Gupta, D.T. Peterson, Solid State Commun. 51 Ž1984. 805. w16x R. Dovesi, V.R. Saunders, C. Roetti, M. Causa, N.M. Harrison, R. Orlando, E. Epra, CRYSTAL95 User’s manual, Theoretical Chemistry Group, University of Turin, TurinrCCLRC Daresbury Lab., Warrington, 1996. w17x J.M. Andre, ´ L. Gouverneur, G. Leroy, Int. J. Quantum Chem., 1 Ž1967. 427, 451. w18x J.M. Andre, ´ J. Chem. Phys. 50 Ž1969. 1536. w19x W.C. Mackrodt, N.M. Harrison, V.R. Saunders, N.L. Allan,
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M.D. Towler, E. Epra, R. Dovesi, Philos. Mag. 68 Ž1993. 653. M. Catti, R. Dovesi, A. Pavese, V.R. Saunders, J. Phys.: Condens. Matter 3 Ž1991. 4151. W.H. Here, L. Radom, P.R. Schleyer, J.A. Pople, Ab Initio Molecular Orbital Theory, Wiley, New York, 1986. R. Orlando, R. Dovesi, C. Roetti, V.R. Saunders, J. Phys.: Condens. Matter 2 Ž1990. 7769. E. Zintl, A. Harder, Z. Elecktrochem. 41 Ž1935. 33. R.C. Weast, M.J. Astle, W.H. Beyer ŽEds.., Handbook of Chemstry and Physics, CRC Press, Boca Raton, FL, 1985– 1986. R. Colle, O. Salvetti, Theor. Chim. Acta 37 Ž1975. 329. R. Colle, A. Fortunelli, O. Salvetti, J. Chem. Phys. 80 Ž1984. 2654. J. Perdew, Phys. Rev. B 33 Ž1986. 882. J. Perdew, Phys. Rev. B 34 Ž1986. 7406. C.R.A. Catlow, M. Dixon, W.C. Mackrodt, in: C.R.A. Catlow, W.C. Mackrodt ŽEds.., Comptur Simulation of Solids, Springer, Berlin, 1982. R. Dovesi, C. Roetti, C. Freyria-Fava, M. Prencipe, V.R. Saunders, Chem. Phys. 156 Ž1991. 11. M. Catti, A. Pavese, R. Dovesi, C. Roetti, M. Causa, Phys. Rev. B 44 Ž1991. 3509. C.Y. Xiao, J.L. Yang, K.M. Deng, Z.H. Bian, K.U. Wang, J. Phys: Condens. Matter 6 Ž1994. 8539.
Electronic and structural properties of CaH 2 : an ab initio Hartree–Fock study Abderrahman El Gridani ) , Mohamed El Mouhtadi ´ Laboratoire de Chimie Physique, Equipe Chimie Theorique appliquee, ´ ´ UniÕersite´ Ibn Zohr, Faculte´ des Sciences, 80 000 Agadir, Morocco Received 22 June 1999
Abstract The Hartree–Fock ab initio method has been used to evaluate some structural and electronic properties of the CaH 2 crystal. The calculated quantities include the crystalline parameters, binding energy, elastic constants, band structure, density of states and electronic charge distribution. Taking into account the correlation effects, the experimental binding energy is well reproduced. The band energies, density of states and charge density maps are analyzed, and the nature of chemical bonding is discussed, showing significant deviations from ionicity Ž z ca s 1.867 < e <.. The values of elastic constants C11, C12 , C13 , C22 , C23 , C33 , C44 , C55 , C66 and bulk modulus, determined for the first time, are 14.4, 110.3, 67.9, 75.1, 143.1, 36.6, 111.1, 68.7, 81.6 and 202.8 GPa, respectively. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Metal hydrides have received great attention due to their utilization in technological applications and their scientific interest they present w1–3x. First of all, they constitute the most convenient porters of hydrogen considered as being the energy source which is the most promising one in the future. In addition, interesting superconductivities have been observed in some metal hydrides such as PdH x w4x, of which critical temperature increases with concentration of hydrogen. Finally, they constitute reliable a means for testing theoretical models. The nature of bonds in metal hydrides is usually enough complex w5–7x and it is explored by means
)
Corresponding author. Fax: q212-81-22-01-00.
of various theoretical and experimental techniques. Historically, three theoretical models w8x generally have been employed to understand experimental data from such diverse fields as photoelectron spectroscopy with X-rays ŽXPS. or synchrotron radiation ŽSRPES. Ženergy of crystal, Compton profile, electronic specific heat and magnetic susceptibility. w9,10x. It concerns anionic, protonic and covalent models corresponding to the three extremes of typical bonding which can exist in a crystal. The anionic model supposes that an electron of the metallic ion is transferred to the hydrogen atom; the protonic model, contrarily to the precedent one, supposes that an electron is yielded by hydrogen to join the conduction metallic band partially occupied. Finally, the covalent model supposes that hydrogen atoms are linked to calcium ones in a covalent manner. The anionic model is more adequate to study the alkali-metal hydrides, which are typically ionic. For
0301-0104r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 9 . 0 0 3 3 3 - X
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A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
a transition hydride such as TiH x w10x, the protonic model seems to be well adapted. For some alkalineearth-metal hydrides, different properties of the system cannot be fit with a single model. This is the case for the BeH 2 compound, which was considered covalent. MgH 2 was considered intermediate between ionic and covalent w11x, but recently the calculation of band structures has shown that this compound is largely ionic with a weak covalent character w12x. Concerning CaH 2 , known as the most useful of the three light-metal hydrides ŽLiH, MgH 2 , CaH 2 ., although its ionic character resulting from the large difference between hydrogen and calcium electronegativities and sustained by XPR study w7x, the description of its Compton profile has failed using the IPC models w6x. The failure of these models, in the study of these systems, can be attributed to two main reasons. Firstly, structural effects and solid state bonds are largely omitted in these models since they normally assumed a single 1s Slater-type orbital Ž1s STO. to describe the electron of hydrogen and adopted the rigid-band approximation for the metal. However, Kunz w13x and Pattison et al. w14x have shown that, because of the solid-state effects, the radial behaviour of the hydrogen wavefunction must differ appreciably from 1s STO. Secondly, these models suppose the existence of bonds with extremely bonding character, while the supposition of the existence of those bonds with mixed character would be more reasonable for this type of compounds. The best way allowing to understand completely the electronic properties of metal hydrides is proved to be by means of band-structure calculations. However, because of the low orthorhombic symmetry, no calculation of band structure for CaH 2 has been undertaken in its real structure. Indeed, only its bands have been determined in a hypothetical structure of the fcc CaF2 type w15x. Thereby, structural effects, in this structure, are neglected and, therefore, deserve to be re-examined by a computation based on the real structure. In this paper, we have used the all-electron ab initio method at the self-consistent field Hartree– Fock ŽSCF–HF. level to compute the equilibrium structure, band structure, density of states, charge densities, Mulliken’s population and elastic constants of the CaH 2 system. These magnitudes are compared
to the corresponding available experimental and calculated ones.
2. Computational method The all-electron ab initio method, allowing the calculation of wavefunctions of crystalline systems, is implanted in the CRYSTAL95 program w16x. It is described in numerous previous works Že.g., w17,18x. and has been recently applied to simple ionic combined systems such as CaO w19x and CaF2 w20x, in order to determine their electronic, dynamic and structural properties. In the current study, an error source, due to the intrinsic limit of the Hartree–Fock approximation, is possible Žcorrelation error.. According to the molecular experience w21x, the HF approach underestimates Žin covalence systems. energies of formation Žsometimes by 30%., while bond lengths are overestimated Žby 0.5–1%.. Similar results have been obtained in a systematic study of III–IV and III–V semiconducting compounds w22x. Concerning the ionic systems, a few studies are realized and seem to confirm this tendency. As it will be shown in this work, the energy of formation can be easily corrected in order to take into account the correlation contribution. Under ambient conditions, CaH 2 crystallises in an orthorhombic phase of complicated PbCl 2 type, having Pnma as space group w23x. Each elementary cell contains four CaH 2 molecules with two different sites of non-equivalent hydrogen. This structure possesses three crystalline parameters Ž a, b, c ., six fractional coordinates Ž x ŽCa., x ŽH1., x ŽH2., z ŽCa., z ŽH1., z ŽH2.. and nine components of the elastic tensor. The atomic basic set, used in this work ŽTable 1., is an all-electron one taken from Refs. w12x and w20x for the hydrogen atom and calcium atom, respectively. Six and nine atomic orbitals are employed for hydrogen and calcium, respectively, and are function combinations of gaussian type. These two basis sets can be written as 5-111G for hydrogen and 8-65113G for calcium Žnotation of Ref. w21x., where the numbers refer to the level of contraction. Note that the 5sp orbital exponent has been re-optimized on the CaH 2 system in order to adapt the calcium basis on the system in question Ž j Ž5sp. s 0.2757 bohry2 ..
A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
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Table 1 Exponents Žbohry2 . and coefficients of the Gaussian functions adopted for the present study Shell type
Calcium exponents
Shell type coefficients s
1s
2sp
3sp
4sp 5sp 3d
1.913Žq5. 2.697Žq4. 5.696Žq3. 1.489Žq3. 4.483Žq2. 1.546Žq2. 6.037Žq1. 2.509Žq2. 4.486Žq2. 1.057Žq2. 3.469Žq1. 1.35Žq1. 5.82 1.819 2.075Žq1. 8.4 3.597 1.408 7.26Žy1. 4.53Žy1. 2.757Žy1. 3.191 8.683Žy1. 3.191Žy1.
2.204Žy4. 1.925Žy3. 1.109Žy2. 4.995Žy2. 1.701Žy1. 3.685Žy1. 4.034Žy1. 1.452Žy1. y5.75Žy3. y7.67Žy2. y1.122Žy1. 2.537Žy1. 6.88Žy1. 3.49Žy1. y2.00Žy3. y1.255Žy1. y6.96Žy1. 1.029Žq0. 9.44Žy1. 1.0 1.0
Hydrogen exponents
p or d
coefficients s
1s
2s 3s 2p
1.2Žq2.2.67Žy4. 4.0Žq1.2.249Žy3. 1.2Žq1.6.386Žy3. 4.0Žq0.3.291Žy2. 1.2Žq0.9.551Žy2. 5.0Žy1. 1.0 1.3Žy1. 1.0 2.0Žy1. 1.0
p
1.0
8.47Žy3. 6.027Žy2. 2.124Žy1. 3.771Žy1. 4.01Žy1. 1.98Žy1. y3.65Žy2. y6.85Žy2. 1.75Žy1. 1.482Žq0. 1.025Žq0. 1.0 1.0 1.6Žy1. 3.13Žy1. 4.06Žy1.
The contraction coefficients multiply individually normalized Gaussians. y Ž"z . stands for y = 10 " z.
Hereafter, the notations nx ŽH. and nx ŽCa., where n is the number of the orbital x, designate the orbital x of hydrogen atom and calcium atom, respectively.
3. Equilibrium structure and binding energy Crystalline parameters and optimized fractional coordinates are given in Table 2 compared to their corresponding experimental ones. The calculation of the equilibrium structure is obtained, for the first time, by the optimization of the crystal’s total energy according to each crystalline parameter by fixing the two others and the fractional coordinates to their experimental values. Then, from optimized parameters, this energy is re-optimized according to the different fractional coordinates. A polynomial interpolation leads to a minimal value of energy corresponding to the HF energy y2711.73977 au. These
results are in good agreement with the measured data knowing that the HF calculation overestimates the unit-cell volume w22x. For comparison, the relative errors to volumes in CaF2 w20x and VO w19x crystals case are in the order of 5.2% and 8.9%, respectively. The total crystal energy is calculated from optimized crystalline parameters at the HF level. Total energies of isolated atoms are computed using the crystalline basis sets essentially designated to describe ionic states, and furthermore, by addition of two diffuse orbitals s and p on each center and by optimization of the most diffuse orbital exponent in order to provide additional variational freedom accounting for the tails of atomic wavefunction. According to the HF approximation, the binding energy is equal to the difference between the total energy and the sum of atomic energies, i.e. 21.39 eV per unit cell. Data calculated by the present method ignore all vibrational contributions to the energy,
A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
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Table 2 Calculated and experimental lattice parameters and binding energy for CaH 2
a b c x ŽCa. x ŽH1. x ŽH2. z ŽCa. z ŽH1. z ŽH2. V BEŽHF. BEŽCS. BEŽP.
Calculated
Experimental ŽRef. w23x.
D Ž%.
6.137 3.694 7.098 0.2405 0.2405 1.0078 0.1128 0.4319 0.7688 160.9 21.39 24.73 24.07
5.936 3.600 6.838 0.2600 0.2600 0.9960 0.1100 0.4300 0.7580 146.1 24.81a
3.4 2.6 3.8 7.5 7.5 1.2 2.5 0.4 1.4 10.1 13.8 0.3 3.0
˚ Žcell volume Ž V . in For lattice parameters, values are given in A ˚ 3 .. For binding energy ŽBE., values are given in eV. CS means A Colle Salvetti’s functional. P means Perdew’s functional. a From Ref. w24x.
which is lower Žby 13.8%. than its corresponding experimental one Ž24.81 eV. evaluated from the heat of formation of CaH 2 , H 2 and Ca using the Born– Haber cycle w24x. Concerning the correction due to the electronic correlation of the formation energy, it is equal to the difference between the correlation energies of crystal and atoms, and evaluated by means of the density functional proposed by Colle and Salvetti w25,26x, and Perdew w27,28x, applied to the HF charge density. Note that ŽTable 2. the absolute correction of BE calculated by means of the density functional proposed by Colle and Salvetti Ž E s 3.34 eV. is higher Žby 19.8%. to the one calculated using Perdew’s formalism Ž E s 2.68 eV.. The difference between our correlated energies Ž24.73 and 24.07 eV. and those experimentally determined is in the order of 0.08–0.74 eV per cell. The data obtained by our method do not take into account the vibrational contribution to the energy, and are compared to the experimental value corrected to 0 K.
Fig. 1. Electron energy bands of CaH 2 along two symmetry lines in the first Brillouin zone.
Born type w29x. In this method the central force potentials used to represent the ground state energy of the solid as a function of the nuclear positions require a set of empirical parameters to specify their functional form. These parameters are generally determined by fitting available experimental parameters such as bond energy, lattice parameters and dielectric constants. The Hartree–Fock approach, as implemented in CRYSTAL, permits a parameter-free ab initio determination of such properties even for Table 3 Effects of isotropic compression of crystalline parameters a, b ˚ . on the charges Ž< e <. of Ca and H, which are evaluated and c ŽA according to the Mulliken partition scheme a b c Ca
H1
H2
4. Elastic properties Elastic constants of ionic and partially ionic compounds are generally calculated using classic approaches based on a semi-empirical formula of the
1s 2sp 3sp 4sp 5sp d total 1s 2s 3s p total 1s 2s 3s p total
6.137 3.694 7.098 y2.000 y8.094 y4.615 y2.177 y1.125 y0.123 y18.133 y0.176 y0.374 y1.363 y0.032 y1.945 y0.168 y0.312 y1.426 y0.016 y1.922
5.830 3.509 6.743 y2.000 y8.094 y4.611 y2.184 y1.101 y0.156 y18.146 y0.184 y0.399 y1.331 y0.026 y1.939 y0.173 y0.335 y1.397 y0.010 y1.915
Shell symbols refer to Table 1.
5.539 3.334 6.406 y2.000 y8.093 y4.609 y2.185 y1.073 y0.199 y18.159 y0.193 y0.422 y1.298 y0.020 y1.933 y0.181 y0.355 y1.371 y0.002 y1.909
5.262 3.167 6.086 y2.000 y8.091 y4.605 y2.188 y1.035 y0.254 y18.173 y0.204 y0.443 y1.126 y0.010 y1.921 y0.190 y0.371 y1.352 y0.006 y1.907
A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
materials where experimental data are unavailable w30,31x. In addition, the HF method, as it has become known, reproduced reasonably experimental elastic constants of ionic and semi-ionic systems. In this section, we present the nine elastic constants, calculated for the first time, and which have not yet been determined experimentally, to our knowledge. Using the atomic orbital basis sets of Table 1, the total energy of CaH 2 was calculated for a number of different deformations of the unit cell. The energy
5
values against strain components hi were fitted to polynomial functions up the fourth order. Thus the second derivatives of energy Žat the energy minimum. could be calculated and yielded the elastic constants Ci j according to the relation: 1 E2 E Ci j s , V Ehihj 0
where V is the volume of the cell and E is the energy per cell. The found values for these elastic
Fig. 2. Projected and total densities of states ŽDOS. of CaH 2 . A Mulliken partition scheme was used to obtain atomic orbital contributions.
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A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
constants C11 , C12 , C13 , C22 , C23 , C33 , C44 , C55 , C66 and bulk modulus B are 14.4, 110.3, 67.9, 75.1, 143.1, 36.6, 111.1, 68.7, 81.6 and 202.8 GPa, respectively.
5. Electron band structure, density of states, Mulliken’s population and charge density The band structure of CaH 2 is represented in Fig. 1. As one would have to expect, bonds are flat and are separated by a large energy gap Ž Eg .. This gap, between the valence band and the conduction one, is around 10 eV, almost twice larger than the experimental one Žin the order of 5 eV w15x.. This difference between theory and experience is due to the fact that the HF hamiltonian overestimates the energy gap. For comparison, the gap value, calculated for the MgH 2 system w12x, is in the order of 13 eV while the measured one is 5.16 eV. Note that our value is also different from that calculated by ChuanYun Xia et al. w32x using the cluster method which underestimates the insulator and semiconductor gaps Ž3.32 eV.. Analysis of the Mulliken electronic population confirms the ionic nature of bonding in CaH 2 ŽTable 3.. Indeed, the net charges on the calcium and the two non-equivalent hydrogens are 1.867 < e <, y0.945 < e < and y0.922 < e <, respectively, and are close to formal ionic values Žq2 < e <, y1 < e < and y1 < e <,
respectively.. Furthermore, Chuan-Yun Xiao et al. w32x have calculated the charge of the calcium atom using their model, and have shown that CaH 2 is less ionic than in our case Ž q ŽCa. s 1.61 < e <.. Note that the ab initio calculations of the MgH 2 w12x, MgF2 w31x and CaF2 w20x systems show that the charge values of heavy atoms are 1.886 < e <, 1.803 < e < and 1.868 < e <, respectively. It results, therefore, that CaH 2 is strongly ionic with a weak covalent character. In addition, the bonding indices Ca–H1 and Ca–H2 are in the order of y0.012 < e < and 0.003 < e <, respectively. It is, therefore, interesting to analyze the distribution of charges on different ions, especially their modification induced by isotropic compression of the crystal ŽTable 3.. From the analysis of this table, we note that a decrease by 5% of each crystalline parameter increases the total charge of calcium by 0.013 < e <. This indicates a tendency to a less ionic Ca–H bonding. The supplementary negative charge is entirely situated on the dŽCa. orbital y0.076 < e < and the electronic cloud spŽCa. is quite contracted. Indeed, we notice that y0.015 < e < are transferred from the 2spŽCa. and 5spŽCa. orbitals towards the inner orbital 3spŽCa. and the dŽCa. orbital, respectively. A similar transfer of 3sŽH. and 2pŽH. electrons, towards the 1sŽH. and 2sŽH. orbitals, respectively, is observed for hydrogen H1 and hydrogen H2. Also, we note that the charge transferred to calcium per each hydrogen is in the order of y0.013 < e < of which a part is transferred to the dŽCa. orbital.
Fig. 3. Ža. Total electron density map on the Ž100. plane. The separation between isodensity curves is 0.01 erbohr 3 . Žb. Difference Žcrystal minus ionic superposition. electron density map on the Ž100. plane. Continuous, dashed and dotted lines indicate positive, zero and negative valus, respectively. The separation between isodensity curves is 0.01 erbohr 3.
A. El Gridani, M. El Mouhtadir Chemical Physics 252 (2000) 1–8
7
Fig. 4. Ža. Total electron density map on the Ž110. plane. The separation between isodensity curves is 0.01 erbohr 3 . Žb. Difference Žcrystal minus ionic superposition. electron density map on the Ž110. plane. Continuous, dashed and dotted lines indicate positive, zero and negative valus, respectively. The separation between isodensity curves is 0.01 erbohr 3.
Fig. 2, which reports on total and partial density of states, shows that the valence band is almost purely atomic without any mixed contribution, except perhaps a weak participation of 2sŽH. Ž q2s s 0.399 < e <. and dŽCa. Ž qd s 0.123 < e <. orbitals with the 3sŽH. orbital. However, Chuan-Yun Xia et al. w32x have shown that the valence band is formed essentially by the 1sŽH. orbital with a weak contribution of the 1sŽH. and dŽCa. ones. The conduction band is formed primarily by 2pŽH. and dŽCa. orbitals with a minor role of the 5spŽCa. one. Total and difference Žs total crystal minus ionic superposition. electron densities are calculated on the Ž100. and Ž110. planes limited to a portion containing eight calcium atoms and accompanying hydrogen ones, and are reported in Fig. 3a, b and Fig. 4a, b, respectively. A contraction of the electronic cloud in the crystal, by comparison to free ions, appears clearly and is due both to the repulsion induced by the electronic exchange ŽPauli’s exclusion principle. and to the compression effect of the crystal due to the electrostatic field of the anions.
6. Conclusions The present work represents the first attempt for the calculation of the nine elastic constants for CaH 2 in its orthorhombic structure by means of the LCAO HF ab initio method.
The computation of the band structure and the analysis of Mulliken’s population show that CaH 2 is an ionic system with a low covalence character. The crystal formation is accompanied by contraction of both the anion and cation. Under compression, a further contraction of ions is apparent, and also a charge transfer from 3sŽH. and 2pŽH. orbitals towards dŽCa. and 3spŽCa. ones, indicating an important contribution of d orbitals to the crystal stability. The analysis of the total density of states and partial densities of states shows that the valence band is almost purely atomic without any mixed contribution, except perhaps a weak participation of 2sŽH. and dŽCa. orbitals with the 3sŽH. orbital. Furthermore, the conduction band is formed primarily by 2pŽH. and dŽCa. orbitals with a minor role of the 5spŽCa. one.
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