On the elastic properties of lithium, sodium and potassium oxide. An ab initio study

ChemicalPhysics North-Holland

156 (1991) 11-19

On the elastic properties of lithium, sodium and potassium oxide. An ab initio study R. Dovesi, C. Roetti, C. Freyria-Fava, M. Prencipe Department of Inorganic. Physical and Materials Chemistry. University of Torino, via Giuria 5. I- IO125 Torino, Italy

and V.R. Saunders Daresbury Laboratory, Science and Engineering Research Council, Daresbury, Warrington WA4 4AD. UK

Received 2 April I99 1

The binding energy, equilibrium lattice parameter, elastic constants and central zone phonon frequencies of Li20, Na,O and KZO have been calculated at an ab initio level with CRYSAL, a Hartree-Fock linear combination of atomic orbitals (LCAO) program for periodic compounds. The variational basis set has heen optimized for each compound, and is reported for future reference and work. A quite satisfactory agreement with very recent experimental data is obtained for L&O. With regard to NazO and K20 no experimental data exist, to our knowledge, for the elastic constants and the central zone phonon frequencies, so that the present data represent the first determination of these quantities.

1. Introduction Classical

approaches

based

on Born-type

semi-em-

degrees of sophistication have usually been adopted for the calculation of the elastic properties of ionic or partially ionic compounds [ l-31. The parameters of the (two-body) central force potentials are usually determined by best-fit to experimental data on a subset of the properties of interest (binding energy, lattice parameter, elastic constants, dielectric constants, phonon spectrum ). Due to its simplicity, this approach can be applied to crystals with a complicated structure and hundreds of atoms in the unit cell [ 41. At the opposite extreme quantum mechanical calculations arc becoming feasible, at least for simple highly symmetric structures, which allow an ab initio (that is parameter-free) determination of the static and elastic properties of the system under study also in the cases in which experimental measurements are totally absent. In the present paper the ab initio Hartree-Fock (HF) linear combination of atomic orbitals (LCAO) pirical

formulas

of various

0301-0104/91/$03.50

periodic approach [ $6 ] is applied to the series Li20, Na10 and KzO. This choice has been suggested for many reasons: (i) the systems are sufficiently simple, so that very accurate computational conditions and extended basis sets could be used, (ii) the presence of cations of three different periods allowed the analysis of the influence of size and polarizability on the static and dynamic properties and on the “inner strain” contribution to C_,_,,and (iii) in the case of L&O, recent accurate experimental data exist [8-lo], that represent a good test for the present approach. On the other hand, experimental determinations of the elastic and phonon data for Na20 and K20 are totally absent. This is one of the cases in which the calculation appears to be easier (apparently good NazO and K20 crystals are difficult to prepare and to preserve) and less expensive than the experimental determination. A preliminary study on Liz0 was published more than six years ago [ 111, when however the numerical accuracy of CRYSTAL was much lower than now. Li20, NazO and K20 arc cubic Fm3m, with antifluorite structure.

0 1991 Elsevier Science Publishers B.V. All rights reserved.

I2

R. Dovesi et al. / Elastic properties

2. Method of calculation The all-electron ab initio self-consistent-field (SCF) HF-LCAO computational scheme, as implemented in CRYSTAL [ 61, has been described in previous papers [ 51. We simply remark here that CRYSTAL is a general program for the treatment of crystalline compounds of any space group, which has been applied to the study of semiconductor and molecular crystals, slabs and polymers. As regards ionic crystals,Li,O [ll],LiH [12],Li3N [13]andMgO [ 141 were considered in previous studies, with particular attention to properties such as the charge density, X-ray structure factors, electron momentum density, Compton profiles and equilibrium geometry. Improvements in accuracy and speed of the program allow now the evaluation of elastic constants and central zone phonon frequencies of simple highly symmetric systems by computing numerically the second derivatives of the energy. In order to define the reliability of the present data, we must remember that they are affected by three different kinds of errors. The first two are related to the basis set incompleteness and to the numerical approximations introduced in the implementation of the HF equations. The third source of error is “intrinsic” to the HF method, and is usually called the “correlation energy error” [ 15 1. The basis set problems are discussed in the next section; we can anticipate that the basis set adopted in the present study is large enough to reduce the basis set truncation error to less than 0.5%, 0.2% and 1% for the binding energy (BE), the lattice parameter ( ao) and the elastic constants (Cij), respectively. As regards the numerical error, approximations are introduced in the evaluation of the Coulomb and exchange series which appear in the Fock matrix definition, as well as in the reciprocal space integration performed at each cycle of the self-consistent stage [ 5 1. The total energy error due to such approximations propagates, through the numerical evaluation of the first and second energy derivatives, to the lattice parameter, elastic constants and phonon frequencies. In the present study values have been adopted for the computational parameters which control the accuracy of the calculation, as well as its cost, (see ch. 2 in ref. [ 5 ] ) such that the error bar for

ofLl,O, &‘a20 and k;O

BE, a0 and the C,, is about 0.00 1 hartree, 0.01 A and 1.O GPa, respectively. We now briefly consider the “correlation” error. It is well known [ 15 ] from molecular experience that the Hat-tree-Fock approach underestimates (in covalent systems) the binding energy by about 30%, whereas the bond lengths are overestimated by 0.51W. Similar results were obtained in systematic studies devoted to IV-IV and III-V semiconductors [ 16 1. As regards ionics, things seem a little more complicated. Previous studies [ 1 l- 14 ] on systems containing light atoms (Li, Be Mg, 0, F) indicate a trend similar to the one observed for covalent systems: very small errors for the lattice constants, about + 8% error for the force constants. However, preliminary results [ 171 on the series LiF, NaF, KF and LiCl, NaCl and KC1 indicate that the error is increasing with the size of the cation and/or the anion and decreasing stiffness of the bonds: in this case the lack of correlation effects (not taken into account at the HF level ) produce too large lattice parameters both because the ion clouds are too large (intra-ionic correlation effeet ) and because dispersion effects, which are neglected at the HF level, become more important both in absolute and relative scale (inter-ionic correlation effects). Typically, the errors for the halkali fluorites are +0.8, $1.3 and +3% for u. and -1, -5 and - 16Ohfor BE. We can suppose that the errors for the halkali oxides are slightly smaller, due to the larger electrostatic effects (charge - 2 on oxygen ) . It is worth mentioning here a parallel study devoted to CaF, [ 181; the calculated lattice parameter is 1.7% larger than the experimental results, whereas for C, ,, C, Zand C’,, the error with respect to the most recent experimental data [ 191 is -0.5, -4.7 and + 17.3%, respectively.

3. Basis set problems The basis set adopted for the present calculations is reported in table 1. The exponents of the two most diffuse sp shells of oxygen are slightly different in the three compounds: 0.45 and 0.15 au for Li*O, 0.46 and 0.14 for Na,O, 0.47 and 0.13 for K20, respectively. The relaxation of the anion valence distribution in going from Liz0 to K20 is thus relatively small, despite the larger anion-cation distance, and the con-

I

1.0

I.0

I.0

1.500(-l)

4sP 5sP 3d

9.580( -3) 6.960( -2) 2.065(-l) 3.470( - I )

P

I.0

1.080(-3) 8.040( -3) 5.324( -2) 1.681(-l) 3.581(-l) 3.855(-l) 1.468(-l) 7.280( -2) -8.830( -3) -9.150( -2) -4.020(-Z) 3.790( - I )

8

Coefficients

4.500(-l)

8.020( + 3) 1.338( +3) 2.554( +2) 6.922( + I ) 2.390( + I ) 9.264 3.85 I 1.212 4.943( + I ) 1.047(+1) 3.235 1.217

Exponents

Oxygen ion

3SP

2sP

Is

type

Shell

2.640( -3) 8.500(-3) 3.350( -2) 1.824(-t) 6.379( - 1) 1.000

I.0

0.53

s

Coefficients

8.400( +2) 2.175(+2) 7.230( + 1) 1.966(+1) 5.044 1.5

Exponents

Lithium ion

1.0

P

2.730( -

5.480( -

1)

I)

5.670( +4) 8.060( + 3) 1.704( +3) 4.436( +2) 1.331(+2) 4.580( + I ) 1.775(+1) 7.380 1.190(+2) 2.533( + 1) 7.800 3.000 1.289

Exponent

Sodium ion

1.0

I.0

2.250( -3) 1.910(-3) 1.105( -2) 5.006( -2) 1.691(-l) 3.658(-l) 3.998( - I ) 1.494(-l) -6.730( -3) -7.980( -2) -7.930( -2) 3.056(-l) 5.639(-l)

S

Coefficients

1.0

I.0

8.030( -3) 6.390( -2) 2.074(-I 3.398(-l) 3.726(-l)

P

)

1.725(+5) 2.432( +4) 5.140( +3) 1.344( +3) 4.045( +2) 1.394( +2) 5.439( + 1) 2.271(+ I ) 4.020( +2) 9.350( + I ) 3.075( + I ) 1.191(+1) 5.167 1.582 1.735(+1) 7.550 2.939 1.190 6.740( - I ) 3.890( - 1) 2.160(-l) 3.94 I .072 3.94( - I )

Exponent

P (ord)

2.200( -4) 1.920(-3) 1.109(-2) 4.992( -2) 1.702(-l) 3.679( - 1) 4.036( - 1) 1.459(-l) -6.030( -3) 8.410( -3) 6.020( -2) -8.050( -2) -1.094(-l) 2.117(-l) 2.580( - I ) 3.726( - I ) 6.840( - I ) 4.022( - I ) 3.990( - I ) 1.860(-l) -7.400(-3) -3.210(-2) -1.290(-l) -6.200( -2) -6.834( - I ) 1.691(-l) I .08 1.5 I .03 1.060 1.0 1.0 1.0 I.0 1.60(-l) 3.13(-l) 4.06( - I )

S

Coefftcients

Potassium ion

Exponents (in bohr-r) and coefftcients of the Gaussian-type-functions (GTGs) adopted for the present calculations. The contraction coefficients multiply normalized individual Gaussians. The oxygen basis refers to L&O. For the other two compounds the exponents of the two most diffuse GTCk have been reoptimixed (0.46 and 0.14 for NaaO, 0.47 and 0.13 for KsO). x( ky) stands forxx IO’Y

Table

w

R. Dovesi et ul. /Elastic

14

properties ofLi>O, Na20 and K,O

sequently smaller effects of the Madelung potential at the anion site. In the optimization process we have explored the importance of (a) a second valence shell on the cation, (b) the polarization functions.of the cations and (c) the polarization functions on the anion. The contribution of additional functions has been estimated in terms of energy (table 2 ); for the cases with large energy variation the equilibrium lattice parameter and the elastic constants have been recalculated (table 3 ). Table 2 Total energy differences between calculations performed with and without d orbitals on the cation or on the anion, and with one or two sp single-Gaussian valence shells on the cation. Numbers in parentheses are the exponents of the added single-Gaussian set of d functions (Li, Na and 0) or of the single valence shell that substitute the two valence shells reported in table 1 (Na and K, for Li the two exponents substitute the single one reported in the previous table). In the case of KzO the number in the first line is the total energy difference between calculations with and without the 3G d shell reported in table 1. Energies in hartree, exponents in bohr-* d orbitals

Liz0

Na20

L&O

on X

-0.0001

-0.0018

(0.7)

(0.7)

-0.0095 (3G contraction)

-0.0001

-0.0001

-0.0001

(0.8)

(0.8)

(0.8)

-0.0021 (0.52) (0.25)

-0.0044 (0.502)

-0.0272 (0.303)

on 0

single valence shell on cation

Table 3 Equilibrium lattice parameter and elastic constants obtained by using a single sp shell on Na (exponent as indicated in table 2)

or by dropping d orbitals on K (see table I ). d is the percentage difference between the data of the present table and those obtained with the basis set of table 1, which are reported in table 5. B= (C,, +2C,*)/3. Lattice parameter in A, elastic constants in GPa Na20

a0

C,, C12 C44 B

KzO

single valence

A

no d

A

5.494 127.1 25.1 37.9

+0.2 +0.6 +9.1 +0.3

6.550 71.8 14.2 19.7

+1.3 -3.1 -4.1 +41.7

+2.8

33.4

59.1

-3.5

Let us consider first the influence of the polarization functions when added to the anion. The energy lowering is very small, of the order of low4 hartreel cell. Adding d orbitals to Li produces roughly the same effect; for sodium the energy lowering is one order of magnitude larger (2 x 10m3 hartree/cell); for potassium it is three times larger than for sodium. In this latter case we used also a three Gaussian (3G) contraction, which gave an energy gain about 40°h larger than a single Gaussian d shell. The trend is similar in the case of the addition of a second sp valence shell on the cation (the exponents are reoptimized): the energy gain is 0.002 1 hartree/cell for LiZO, two and ten times larger for Na and K, respectively. In order to check the influence of these modifications with reference to other properties, we computed the elastic constants and the equilibrium lattice parameters with and without d’s on potassium, and using one and two valence sp shells in Na*O. The results are reported in table 3. The d orbitals on K produce, as expected, a contraction of the lattice parameter (0.08 A), and a consequent increase of C,, and C,2 (2.4 and 0.6 GPa, respectively). The effect on C,, is much larger and opposite; this is because d orbitals are essential for the dipolar polarization of potassium, which is by far the most important relaxation mechanism in the C,, deformation. The split of the Na valence shell has no consequences for the lattice parameter, whereas the elastic constants are reduced by a maximum of 2 GPa. From the above discussion it appears that the basis set of table 1 is quite reasonable, and that additional polarization functions on the cations or anions, or an additional sp shell on the cations are expected to have a minor effect on the calculated quantities. Similar conclusions can reasonably be extrapolated to oxygen: however we were unable to perform a direct check of this hypothesis, because the addition of an extra diffuse sp shell to the anion causes numerical instabilities due to basis set linear dependence problems. 4. Results and discussion 4.1. Binding energy The energy data are reported in table 4. The atomic energies of the cations have been obtained by adding

R. Dovesi et al. /Elastic propertiesof Li20, NazOandK,O

15

Table 4 Energy data (in hartree). E( HF) is the Hartree-Fock total energy; BE is the binding energy evaluated at the HF or HF plus correlation level. 6E is the sum of the correlation energy differences between the isolated atoms and ions, as indicated

E( HF) crystal E( HF) atom X E(HF) atom 0 BE (HF) Correlation contributions 2x-+2x+ o-+0*8E (Ionic correlation) BE (HF+ ionic correlation ) BE (Experimental)

Liz0

Na20

f&O

- 89.9643 -7.4313 -74.8012 -0.3005 (-33%)

- 398.6931 - 161.8513 -74.8012 -0.1893 (-43%)

- 1273.1932 -599.1274 -74.8012 -0.1376 (-54%)

+o.oo -0.15 -0.15 -0.45 (+0.3%)

+0.01 -0.15 -0.14 -0.33 (-0.8%)

+0.03 -0.15 -0.12 -0.26 (- 14.2%)

- 0.449 1

to the basis set of table 1 one (two in the Li case) sp shells, and reoptimizing the exponents of the three most diffuse shells (a = 0.5 165,0.0829 and 0.030 for lithium; 0.5288, 0.1721 and 0.0384 for sodium; 0.40 17,0.22 16 and 0.028 1 for potassium). The same scheme has been adopted for oxygen ((~=0.4866, 0.2023,0.0867). The atomic energies are not too far from the near-Hartree-Fock results of Clementi and Roetti [20] (-7.43273, - 161.85890, -599.16453, - 74.80125 for Li, Na, K and 0). The virial coefftcient is 0.99965, 0.99999, 0.99979 and 0.99865 for Li, Na, K, and 0, respectively, whereas the bulk data are 0.99896,0.99973 and 0.99976 for L&O, NazO and KzO, respectively. On the basis of the discussion in section 2, we can argue that the bulk and atomic basis sets are roughly of the same quality. The HF BE is 30 to 50% smaller than the experimental data [ 2 11. The correlation contribution to BE has been estimated by assuming that it is mainly due to the transfer of one electron from each cation of oxygen. According to this simple model, which disregards relaxation effects due to the crystal field and overlap effects between the electronic clouds of the ions, we considered the difference in correlation energy between the isolated species X and X+ (X = Li, Na, K) and 0 and 02-; the data for the X+X+ correlation difference are from Clementi [ 22 1; as regards oxygen, it is well known that the isolated 02- ion is not stable at the HF level; the correlation energy difference 0+02has been estimated by extrapolating the

-0.3321

-0.3002

data for the isoelectronic couples F+-+Fand Ne2+ +Ne [ 221. Table 4 shows that this simple model performs quite well for the present systems (this is also true for other ionic crystals, such as MgFz and CaF, [ 181). In the K20 case the error is larger perhaps because K+ is highly polarizable, and crystal field effects are not negligible. 4.2. Lattice parameter The Liz0 lattice parameter a0 has recently been measured [9,10] at nine temperature values in the range 293-1603 K, using the technique of inelastic neutron scattering on single crystal and polycrystalline samples. In the same experiments the C,,, Cl2 and C,, elastic constants have also been determined. The a0 versus Tcurve shows a typical linear behavior in the 293- 1300 K interval, whereas at higher T values the curve becomes steeper probably because of high defect concentration [ 9 1. Extrapolation to T= 0 K gives 4.573 A, with a decrease of 0.05 A with respect to room temperature data. We were able to find only a very old determination of the equilibrium lattice parameters of NazO and KzO [ 23 1. The X-ray powder diffraction data of Zintl et al. [ 23 1, reported in table 5, were collected at room temperature. In the same paper the Liz0 lattice parameter is reported, which compares quite well with that of ref. [IO] (a0(293 K) =4.611 A, and 4.619 A, respectively) indicating that the Zintl et al. data, although fifty-five

R. Dovesi et al. /Elastic properties oJ‘Lt,O. ,Va20 and K20

16

Table 5 Lattice parameter a, (in 8, ), elastic constants do not include inner relaxation effects

C,, (in GPa)

and central zone phonon

Liz0 talc. 4.573

a0 C,,

Cl2 C‘r‘r B v (LR) v (Raman

)

233.7 22.1 (68.5) 67.5 92.6 479 574

frequencies

Na,O exp.

KzO

talc. 4.573 a)

216k4 25+6

a) a’

67k 1 a’ (425)” (523) ”

‘) Athermal limit from ref. [ 91. b, Room temperature data; the number in parenthesis NaF (for fia20) and KF (for K,O). Ref. [ 231. ‘) Room temperature data from refs. [ 71 and [ 81.

years old, are quite reliable. The problem however remains of the extrapolation to the athermal limit of the ~~(293 K) data for NazO and K20, so they can be compared with the calculated results. Since no thermal expansion coefficients are available, we can simply estimate the shortening of a,, by comparison with similar compounds for which temperature effects are known. For instance the LiF, NaF and K_F lattice parameter shortening is 0.04, 0.05 and 0.05 [ 241, respectively. If we assume that a similar trend is valid for the oxides (Au,, = 0.05,0.06,0.06, respectively), the data reported in square brackets in table 5 are obtained. The calculated aO’s are obtained by fitting a set of energy points as a function of the lattice parameter, by using polynomial functions or the Murnaghan equation of state [ 25 1. Eight points in the interval a,(min) +2.5% were considered. The u,,(min) values obtained with different interpolating functions are very similar, the maximum difference being of the order of 0.003 A. In the case of Liz0 the agreement with the experimental data is excellent; the situation is similar for NazO, whereas in the case of KzO the calculated data is about 1.3% too large. Again, it is useful to compare with LiF, NaF and KF; in the first two cases the difference with respect to experiment is very small (less than 0.5%); in the KF case it is about 2% [ 17 1. Then, apparently, when light and small anions are involved (as is the case of F- and O*- ), going

exp.

5.484

talc.

5.55 b’ (5.49)

6.466

cxp. 6.44 b’ (6.38)

74. I 14.8 (13.9) 12.6 34.6 244 167

126.3 23.0 (37.8) 34.9 57.5 319 262

is the athermal

v (in cm- r ). C,, values in parenthesis

limit value estimated

by using the temperature

dependence

data of

down the first period potassium is the first cation that shows relatively large systematic errors in a0 (see the discussion in section 2 concerning the correlation error). This statement is confirmed by the data concerning MgO (same row as Na; lattice parameter error less than 0.5% [ 141) and CaF, (same row as K; error +1.7% [18]). In summary, the agreement with experiment for a0 seems excellent for Liz0 and Na20, slightly poorer for KzO. 4.3. Elastic constunts The experimental [ 9, lo] C, versus T curves are linear up to about 1350 K, the linearity being a consequence of lattice anharmonicity [ 261. For comparison with our theoretical results, the experimental elastic constants have to be corrected for the temperature effects and the zero-point motion (static limit). This can be accomplished by extrapolating back to 0 K the graph of elastic constants against T from the linear region at higher T [ 26 1. For this extrapolation we used the data reported in table 3 of ref. [ 9 ] and referring to 293, 523, 773, 973, 1123 K. The results are reported in table 5; the indicated error bar is the mean value of the experimental error at the five points used for the extrapolation (table 3 of ref. [ 9 ] ). Let us briefly describe now the technique adopted for obtaining the calculated C,. The total energy of

R. Dovesi et al. /Elastic properties

the three oxides has been calculated for a number of deformations of the unit cell. The energy values against the strain component fti were fitted to polynomial functions up to the fourth order. The “static lattice” elastic constants C’ijare given by the following expression:

c,=

-!- - #E vO [ NiaVj

of Li20, Na10 and K20

17

ergy, as shown in fig. 1a and 1b, respectively. Assuming a second-order expansion of E with respect to q and S, the energy correction due to internal strain can be cast in the form: E(tl, @=E(rl, 0) -El ,

(2)

where

3,

(1)

'

El = Kq2 ,

K= ]E’(tlo) where the derivatives are evaluated at vi = 8 =O. Thus, the second derivative of the energy at the energy minimum yields the elastic constants. In order to obtain C, , and Cl*, the two deformationsDl=[r7,tl,tl,O,O,O]andD2=[&rl,-2q,O,O, 0] have been considered. Dl yields the elastic bulk modulus B that in a cubic system is related to C, , and Cl2 by the relation B= (C,, +2C,*)/3. 02 yields C , , - C,2, so that by linear combination C, , and CIZ are obtained. Nine energy points were used for each tit; in the Dl case the explored interval was ao(calc) ? 2.596, in the 02 cases q ranged from +0.06 to - 0.06 (note that rl is nondimensional); the maximum difference between the a and c parameters of the tetragonally distorted unit cell being about 0.4 A. In order to compute the C,, shear elastic constant, the [ 0, 0, 0, q, q, r,i] deformation was considered, instead of the simplest [0, 0, 0, r~,0, 0] lattice strain. The former maintains a higher symmetry ( 12 point operators, corresponding to a rhombohedral cell, instead of four point operators, corresponding to a monoclinic deformation), with a corresponding saving factor in the cost of the computation. Ten rl values in the interval kO.06 were considered, corresponding to deviations of the (Y,/I, y angles from 90” by about f 3”. The values obtained in this way are shown in parenthesis in table 5. Actually, due to the lower symmetry resulting from the deformation, the cations acquire a positional degree of freedom (along the diagonal in the rhombohedral deformation used here), which can be exploited for reducing the total energy of the system. Then, for each explored 9 value, the total energy of the system must be minimized with respect to the [S, S, 61 motion of the cations. The correction to the energy (see fig. 1c) is such that C.,, decreases by about 8% in the Na20 case. It must be noticed that in the explored rl interval relaxation effects are essentially linear for 6 and parabolic for the en-

(3)

12/14rl31,

(4)

and E’ and S are the first- and second-order coefficients in the expansion E(qo, 8)=E(qo,

0)+E’d+S2.

(5)

Eqs. (2 )- (5 ) show that it is sufficient to explore the 6 coordinate for one r,rovalue, obtaining E’ and S from eq. (5), and using them in eq. (2) for any q value in the parabolic region of the ‘1~0 axis; the q. value used in the present case (see the line and the parabola in fig. la and b) is q= - 0.06. The results are shown in table 5; the correction to C,, is 1.5, 7.8 and 9.3% for the three compounds, respectively. 4.4. Phonon frequencies Farley et al. [ 8,101 measured the phonon spectrum in the [OO<], [O{Q and [&f] directions at 293 K by inelastic neutron scattering experiments. Older central-zone phonon frequencies data collected by Osaka and Shindo [ 71 coincide nearly exactly with Farley’s data. In the present paper the two lowest transverse optical modes at the center of the Brillouin zone (rpoint ) have been evaluated. Due to the high symmetry at r, the dynamical matrix does not require diagonalization; the two (threefold degenerate) normal modes correspond to in-phase and outof-phase motion of the two cation X, and X2 sublattices. The Raman active (symmetric) mode has been studied by displacing the two cations to positions r(X,)=(O.25+6,0.25+6,0.25+6) and r(X,)= (-0.25-6, -0.25-S, -0.25-b). The normal coordinate U= 2( 3) “‘S is the variation of the X,-X2 distance; the reduced mass is fl= m(X) /2. For the IR active mode Ar, =Ar2 = (S, 6, S); the normal coordinate is U= (3) “‘d and corresponds to the variation of the X-O distance; the reduced mass is ~=2m(X)m(O)/(m(O)+2m(X)).Ifatomicunits

R. Dowi et al. / Elastic properties q/LizO, Na@ and KzO

I8

a

b

0.6

Fig. 1.Inner strain contribution (b) to total energy, E,, and corresponding shift 6 of the Na atoms (a) along x, y and z, as a function of the rhombohedml deformation component q in the calculation of the elastic constant C,,. In (c) the full line represents the total energy variation due to deformation; th broken line includes only the “external” contribution; the difference between the two curves is the E, contribution reported in (b). The straight line (a), and the parabola (b) parameters are obtained from the q= -0.06 relaxation data (see text ); E, and E in millihartree, 6 in A, q is nondimensional.

for energy and distance are used, then the vibrational frequency is given by

where bz is the second-order power coefftcient in the polynomial lit of E versus U, and c= 1, c= 154.1079 and c= 5 140.486 for u in au, THz and cm-‘, respectively. The temperature dependence of the two Liz0 frequencies Yis unknown. In the case of alkali halides the increase of ZJgoing from 293 to 0 K ranges from 5 to 10% [ 27 1, the lower limit referring to light cation and anion systems. If a similar correction is applied to Li20, the difference with respect to the calculated data reduces to about +5%. It must be noticed that the inversion between the Raman and the IR frequencies in Na20 and K20 with respect to L&O is a consequence of the mass difference. The force constants K=2b2 show a more regular pattern: 0.0432, 0.0298, 0.0206 and 0.0646, 0.0470 and 0.0290 hartree per bohr’ for the Raman and IR modes, respectively.

4. Conclusions The complete elastic tensor of Li20, Na20 and K20 has been computed by means of the ab initio LCAO Hat-tree-Fock model. Very good agreement has been found with the available experimental data, concerning mainly Li20. The allowance for inner strain effects reduced C.,, by about 1.5,8 and 9% for the three systems, respectively. The d orbitals play a nonnegligible role only when added to K in K20. The effect on a,, C, ,, and Cl2 is, as expected, a contraction of the lattice parameter ( - 1.2%) and a consequent small increase ( < 3%) of the two elastic constants. The effect on C,, is more than one order of magnitude larger, and opposite ( - 4 1%) because d orbitals allow for dipolar deformation of the cation. The electronic structure of the three systems is fully ionic, with net charges quite close to the formal values (Li: +0.97; Na: + 1.04; K: + 1.Ol ). The bond population, as expected, is very small and negative ( -0.0 1, - 0.03, - 0.05 for Li*O, Na,O, K20 respectively), confirming the total lack of covalency. In summary, the HF-LCAO approach, as imple-

R. Dovesi et al. /Elastic properties of Li20, NazO and K,O

mented in CRYSTAL [ 6 1, is a reliable and relatively inexpensive (300,500, and 1500 s, respectively, on a IBM 30901170 scalar machine for each energy point for the three systems) tool for the study of the elastic properties of simple ionics.

Acknowledgement Financial support by CSI Piemonte, by Minister0 della Pubblica Istruzione and by Italian CNR are gratefully acknowledged.

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