The polarisability of the sulphide ion in MgS

Volume 149, number 3

THE POLARISABILITY

CHEMICAL PHYSICS LETTERS

19 August 1988

OF THE SULPHIDE ION IN MgS

P.W. FOWLER and P. TOLE Department of Chemistry, University ofExeter,

Stocker Road, Exeter EX4 4QD, UK

Received 16 May 1988; in tinal form 6 June 1988

Ab initio self-consistent field and coupled Hartree-Fock calculations on a cluster of ions embedded in a point-charge lattice predict a polarisability ~~~29.3 ai (4.33 A’) for the sulphide ion in MgS. Correlation corrections, as estimated by second-order Moller-Plesset theory are small (less than 2%) and bring (Yto within 5% of the value deduced from the refractive index of MgS (3 I .O aA (4.60 A3) ). (Y(S*- ) shows a marked dependence on environment, as expected for an ion that is unbound in vacua. However, in an idealised (001 > surface of MgS the sulphide ion is only 12% more polarisable than the bulk species.

1. Introduction Isolated, doubly charged, monatomic anions are unstable systems. The first and second electron affinities of the sulphur atom are +200 and - 590 kJ mol- I, respectively [ 11, and so a free sulphide ion S2- would lose an electron to produce the mono anion, S- . An oxide ion would behave similarly. In spite of this, stable and apparently ionic sulphides and oxides exist as crystalline solids. Clearly, the crystal has a crucial effect on the stability of these anions. More generally, the electrical properties of anions are known to be affected by the crystalline environment, and drastic effects might be expected for an ion that has no existence outside the crystal. For example, the polarisability of a free sulphide ion is plausibly regarded as infinite, and yet the refractive index of MgS [2] can be interpreted in terms of a sulphide polarisability of only 31.02 au [ 31. In previous work [ 4,5 ] it has been shown that ab initio calculations which simulate the crystalline environment can reproduce the changes in properties which occur when stable anions are enclosed by the crystal. These changes are accounted for by electrostatic and overlap compression of the anion by its neighbours in the crystal [ 4,6]. The same theoretical methods have also been applied to the oxide ion in MgO [ 71. Here we extended this work to a secondrow, unbound ion: S*- in MgS. Such a calculation has at least two points of in-

terest. First, S2- is predicted from empirical models [ 31 to have a large polarisability, even in MgS, and to show greater variation with environment than other ions studied so far. Calculation of its polarisability, and comparison with the predictions from the Clausius-Mossotti equation, are correspondingly severe tests of the method and the underlying physical ideas. Secondly, it is desirable to establish the relative importance of electron correlation in the S2- system. It is already known that on descending the periodic table from fluoride to chloride there is a drop in the fractional contribution of electron correlation to the ionic polarisability [ 5 1. In view of the small correlation polarisability found for the in-crystal oxide ion (~1 aiinatotalof %12ai for02-inMg0 [7]), the uncorrelated description of a for its second-row analogue, S*-, could be quite accurate. If the downward trend were confirmed, it would be an encouraging sign for calculations on even larger systems, where explicit evaluation of correlation polarisabilities rapidly becomes too expensive.

2. Method A brief summary of the method established in a series of earlier calculations [ 4,5,7 ] is given here. The calculation has three stages: first we treat the isolated anion, then the ion in a point-charge lattice that sim-

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ulates the Madelung potential of the crystal, and finally a cluster comprising one anion and its nearestneighbouring cations, the whole embedded in the point-charge lattice. The second and third stages add the effects on the central ion of electrostatic and overlap compression, respectively. This sequence is well-defined for a “normal” anion such as F- or Cl- and leads to a step-by-step calculation of the factors affecting an in-crystal anion [ 41. However, for the present case, the concept of a calculation on an unbound anion requires some explanation. Although the isolated sulphide ion is in reality unbound, it is possible to obtain a HartreeFock single-determinant wavefunction corresponding to the 1s22s22p63s23p6closed-shell configuration. This solution to the SCF equations exists, both in a finite basis and at the numerical Hartree-Fock limit, because the constraints of symmetry equivalence and double occupancy imposed on the 3p orbitals force the system to “choose” between this metastable Szconfiguration and the even less stable S4+. The positive value of the 3p orbital energy (0.076 Et, in our basis) is a symptom of the instability of the spherically symmetric closed shell, which would eject an electron if the constraints were relaxed. Because a Hartree-Fock ground-state solution exists, it is possible to define a coupled Hartree-Fock polarisability for isolated S2- under the same constraints. This is the quantity calculated here as a “free ion” polarisability. It is very large (720 ai in our basis) and highly basis dependent and can be seen as a finitebasis simulation of the true polarisability, which would be infinite for a system with a free electron. It is interesting to compare the energetic5 of the oxide and sulphide ions. Experimentally these elements are similar. The neutral atom has a positive electron affinity, so that the process ASe-+Ais exothermic for A = 0 or S. The dianion is unstable with respect to both A- and Aa since the process A- +e--+A’is strongly endothermic as would be anticipated from considerations of Coulomb repulsion. For oxygen the first electron affinity (EA) is 142.6kO.2 kJ mol-‘, the second EA is -75Ok30 kJ mol-’ [l]. At the Hat-tree-Fock level the energies of sulphur species 274

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follow the experimental sequence S2~ > S > S #’[ 8 ] whereas the electron affinity of the oxygen atom is wrongly predicted to be negative [ 9, lo], A correlated calculation is necessary to give the correct order of energies: O2> O> O-. In a crystal the stabilities of A- and A*- are reversed, with the larger Madelung stabilisation of the dianion predominating, so much that singly charged ions are only encountered in the gas phase or in irradiated solids. A large and flexible basis is needed to describe the effects of the crystalline perturbation on this diffuse anion. In the present work a ( 12~10~) contraction from a ( 15s12p4d) basis of Gaussian-type orbitals was used for S, constructed from the smaller ( 12~9~) Huzinaga set [ 111 by following the prescription of Reinsch and Meyer [ 121. This leads to four sets of d functions with exponents in the pattern (325; 95; 3[, &!)where { is a diffuse p exponent [=0.020317 a;*. The energy-optimised core of a basis constructed in this way gives an accurate description of the occupied orbitals, whilst the diffuse d polarisation functions describe their response to an electric field. The isolated ion has an SCF energy of -397.3532 E,, in this basis. In the cluster calculations a special highly contracted basis ( 1Os8p) + [ 2sl p ] was used for Mg2+. This minimal set gives a near Hartree-Fock energy of - 198.8236 & for free Mg’+, reproducing exactly the full (10~8~) energy, but with a contraction that is compact enough to be used in a calculation on a seven-ion (P) (Mgz’)6 cluster. A similar basis was used in previous work on MgO [ 71. A minimal cation basis can be used here because the role of magnesium ions in the calculation is simply to provide compression of the sulphide ion. Their intrinsic polarisability is very small (0.486 ai [ 31) and insensitive to environment; it may be calculated separately. MgS has the rocksalt structure, with a shortest anion-cation distance of 2.595 8, [2]. A cubic fragment of 5 x 5 x 5 charges, with scaling on the faces of the cube, was used to represent the infinite lattice. Within the cube each lattice point carries its full #’The near Hartree-Fock results of Clementi and Roetti are E(S)= -397.5049 Eh and E(S-I) = -397.5382 E,,, and we find E(S2-)= -397.3532 E,, in the present basis. Clementi andRoettigiveE(O)=-74.8094E,,and E(O-)= -74.7895 E,,, and E( 0 2- ) = - 74.4648 E,, is the finite-basis result in ref. [71.

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nominal charge; on the faces of the cube the charges are halved, on the edges divided by four and on the comers divided by eight. This relatively small number of charges is sufficient to give a converged Madelung potential and polarisability for an ion in the electrostatic crystal field [ 41.

3. Results and discussion Table 1 shows the results of uncorrelated calculations for the free S*- ion, the ion in point-charge lattices appropriate to the compounds BaS, SrS, CaS and MgS and finally for the ion in a cage of magnesium ions embedded in a lattice of charges. The lattice parameters for the sulphides are taken from ref. [ 2 1. Trends in the polarisability can be understood by following the changes in ionic size, as measured by the radial expectation value

which is equal to the average second moment of the charge density (apart from sign), and in the binding of the outer electrons, represented by the orbital energy for the 3p shell, tjp. The entries in the table represent the ion in a series of successively more compressive environments. The free ion has a very large polarisability, a large radial extent and a positive HOMO eigenvalue. Imposition of the Madelung potential places the electrons of the anion in a potential well, and the orbital eigenvalues reflect this increased stability. The radial extent of Table 1 Hartree-Fock calculations of properties of the sulphide ion in various environments. 01is the polarisability, 0 is the unsigned second moment and c(3p) is the 3p orbital eigenvalue, all in atomic units Environment

(Y

e

t(3p)

free S

720

140

+0.076

point-charge lattice BaS SrS CaS MgS

83.84 67.99 61.79 47.18

58.82 56.14 55.02 51.91

- 0.406 -0.439 -0.453 -0.491

full cluster+electrostatics MB

29.25

46.32

-0.418

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the electron cloud is reduced and the polarisability drops even more rapidly. As a rough guide, Q!would be expected to fall by twice the fractional change in @ for a one-electron atom [ 131, and by rather more for a many-electron system. The final row in table 1 refers to the in-crystal ion: S2- in MgS. Anion properties may be extracted from those of the cluster in several slightly different ways. 0, as an expectation value, is obtained by subtraction, a(anion) =&cluster) -&cage), of the expectation value for an “empty” cage from the cluster value. The polarisability, as a second-order property, is obtained from a CHF calculation in which the 30 occupied orbitals that are heavily dominated by Mg 2+ functions are frozen, and only the filled S2orbitals are allowed to polarise. Such a calculation should be essentially free of the artefacts of basis set superposition error (BSSE), and also does not need correction for physical dipole-induced-dipole (DID) enhancement of the polarisability. Further details on these points are given in refs. [ 4,7]. The technique of freezing cation orbitals depends for its success on the assumption that the orbitals separate cleanly into “cage” and “anion” MOs. This assumption was tested by performing a finite-field calculation on the full cluster. The resulting polarisability is 29.43 a& and this increase of 0.18 ai over the CHF result is well within the expected BSSE for the finite-field calculation, so freezing the cation orbitals does not miss out any important contribution to the sulphide polarisability. It seems that MgS and MgO are both well described as ionic solids, at least insofar as their refractive indices are concerned. Even when compressed by its nearest neighbours, the sulphide ion has a large electron cloud. Although second-shell effects are thought to be insignificant for fluoride and chloride ions, it is conceivable that sulphide-sulphide interactions could contribute to the in-crystal value of (Y(S2-). A full treatment of these effects is not feasible with present computational resources, but the sensitivity of cy( S2- ) can be estimated very roughly by comparing the results of two calculations, one for the cluster ( S2- ) (Mg*+ )6 in the point-charge lattice and one for the same cluster in free space. In the former case the secondneighbour sulphide ions are present as point charges, in the latter they are not present at all. Total removal 275

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of the lattice leads to a decrease in LY(S*- ) of 0.072 a: ( -c 0.3%). The sulphide ion is therefore not unduly sensitive to second-shell electrostatic effects and it seems unlikely that the influence of second-shell overlap would be much greater. The dominance of first-shell effects shown in the present ab initio calculations provides the key to the failure of empirical approaches that ignore first-shell overlap. For example, Yousif and Kassim [ 141 use a Kirkwood approximation to calculate the polarisability of oxide and sulphide ions in rocksalt structures, taking into account only the Madelung potential at the ion site, The results are incompatible with known refractive indices, overestimating the anionic polarisabilities by factors of two or more, as might be expected from the purely electrostatic results in table 1. Polarisabilities of many-electron systems in the Kirkwood approximation are also prone to overestimation [ 15 1. The checks that we have carried out suggest that the value of a( S*- ) = 29.25 a: is a reliable estimate of the CHF polarisability for an in-crystal sulphide ion. However, in view of the known importance of electron correlation for the polarisabilities of free anions [ 41, it is also necessary to check whether correlation would affect this value. An economical way of including electron correlation in a property calculation is via Moller-Plesset perturbation theory. Its application to correlation corrections for multipole moments [ 16 ] and polarisabilities [ 4 ] has been discussed. Different treatments of the perturbation series lead to two inequivalent second-order MnrllerPlesset corrections to the CHF polarisability [ 41. It should be borne in mind that only when these corrections are small and nearly equal is Moller-Plesset theory an appropriate way to deal with correlation. If the computed values fail to satisfy these criteria then the method is a posteriori unsuitable. The correlation polarisability was evaluated for the in-crystal sulphide ion by finite-field differentiation of the second-order Msller-Plesset energy and dipole moment. The same basis, cluster geometry, point-charge lattice and set of active orbitals were used as in the CHF calculation. The MP2 corrections to (w(S*- ) turn out to be remarkably small: +0.41 ad from the energy series and -0.07 ai from the dipole series. These corrections are smaller than found for the first-row oxide ion [ 7 1, and suggest that 276

19 August 1988

correlation is almost entirely quenched for S2- in the crystalline environment. Electron correlation can affect the polarisability of an atom or ion in one of two ways, depending on whether radial (in/out) or angular correlations predominate. Reinsch and Meyer [ 12 ] rationalised the trends in correlation polarisability for neutral atoms across first and second rows of the periodic table in terms of configurations mixed into the SCF wavefunction. Atoms at the left of the table have empty, low-lying p orbitals and angular correlation predominates. This causes shrinkage of the valence electron distribution and thus a lowering of the polarisability. For atoms at the right of the row excitation to lowlying (n+ 1)s orbitals becomes easier and the valence electrons can spread radially outwards, leading to a higher polarisability. In the second row the core is larger and more polarisable, and consequently angular correlation increases in importance [ 12 1. Similar rationalisations have been applied to ionic polarisabilities. Species with nominally identical electronic configurations show strongly charge-dependent correlation contributions. For example, in the three lo-electron ls22s22p6 systems F-, Ne and Na+ the correlation contribution to cy varies from large and positive for F- to almost negligible for Nat [ 17 1. F- expands greatly when the electrons are correlated, whereas in Na+ the radial and angular changes are more subtle. As the size of the cation increases the sign of the correlation polarisability changes to negative (e.g. -0.01 ai for Rb+ [ 181). For the second-row halide, Cl-, the correlation polarisability is still positive but it is a much smaller percentage of the CHF value ( 19% in Diercksen and Sadlej’s calculation [ 191). Some unpublished calculations on Br- show that this trend continues down the group. In the present investigation we have found that the correlation polarisability is smaller for in-crystal S*than for in-crystal 02-, both in absolute magnitude and as a fraction of the total. This trend could be interpreted either as a more effective quenching of radial correlation for the larger ion or (which seems more likely) as an increase in angular core-valence correlation for the second row ion with its accessible empty d orbitals. The MP2 calculations are suggestive, in that both perturbation series give a small correlation polaris-

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ability, but not definitive in that they do not determine its absolute magnitude or even its sign. More extensive correlated studies would be necessary to establish these points and to recover the last few percent of the total polarisability. Finally, the calculated value of c~u( S2- ) can be compared with experiment. The refractive index of MgS extrapolated to infinite wavelength is 2.1935 [ 21. From the Clausius-Mossotti and LorentzLorenz equations

Table 2 Some properties of the sulphidc ion in the (001) face of an MgS crystal. p= and e,, are the components of the dipole and quadrupole moments along the outward normal to the face. Other symbols and units are as in table I The origin for all properties is the anion lattice site Value

/& %

- 0.258 0.137 15.860 15.905 15.768 32.69 32.48 33.10

;,= Q”” Q,, d (Y, =(Y,=(Y, (Y,,=c&;

E-1 n*-1 --24nc-w --t+2 - n2+2 - 3 V, the polarisability per MgS unit is 4.6688 A’ (3 1SO7 a:). Subtracting a constant 0.486 ai for Mg*+ leaves a(S’- in MgS)=31.02 ai. Ifwe take cyfrom the ab initio calculations to be the sum of CHF and MPE contributions, the theoretical value underestimates the experimental one by some 4%. Errors of this order could easily be caused by residual incompleteness in the basis, higher-order correlation contributions or neglect of S 2- -S2- interactions. We conclude therefore that an ionic model of MgS is successful in accounting for the refractive index of the crystal, and that charge-transfer contributions, if present at all, are limited to less than 5% of a~,,,.Ab initio calculations within this approach will be reported for a range of oxides and sulphides in a later paper. The approach used here for a perfect crystal of magnesium sulphide can also be used to probe the sensitivity of the sulphide anion to another environment: the surface of the crystal. As recent empirical models [ 141 predict unphysically large results, for the reasons discussed above, it may be useful to give a brief account of ab initio calculations on the surface S2- ion. A cluster consisting of (Mg2’),(S2-) embedded in a 7 x 7 x 4 slab of point charges was used to simulate the anion site on a (001) face of an MgS crystal. The results are summarised in table 2. Analysis of the cluster properties by methods described previously [20] shows that the surface anion has a mean polarisability 12% greater than the in-crystal value, with a small anisotropy (Y,,-(Y, =0.6 1 ai, a small dipole moment and a small quadrupole. These changes are slightly larger than those found for surface oxide ions in MgO, but still suggest a picture in

Property

which surface effects can be viewed as a perturbation of the bulk ion. A detailed discussion of the effects of coordination number and surface relaxation and rumpling in MgO is given in a recent paper [ 201.

References [ I] J.E. Huheey, Inorganic chemistry: principles of structure and reactivity, 2nd Ed. (Harper and Row, New York, 1978). [2] I.M. Boswarva, Phys. Rev. B 1 (1970) 1698. [ 31 P.W. Fowler and N.C. Pyper, Proc. Roy. Sot. A 398 ( 1985) 377. [ 41 P.W. Fowler and P.A. Madden, Mol. Phys. 49 ( 1983) 9 13. [ 5] P.W. Fowler and P.A. Madden, Phys. Rev. B29 ( 1984) 1035. [ 61 G.D. Mahan, Solid State Ionics 1 ( 1980) 29. [ 71 P.W. Fowler and P.A. Madden, J. Phys. Chem. 89 ( 1985) 2581. [ 81 E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14 (1974) 177. [9] D.B. Cook, J. Chem. Sot. Chem. Commun. ( 1980) 623. [ IO] D.B. Cook, Mol. Phys. 42 ( I98 I ) 235. [ 111 S. Huzinaga, Technical Report: Approximate Atomic Functions, Division of Theoretical Chemistry, University of Alberta (1970). [ 121 E.A. Reinsch and W. Meyer, Phys. Rev. A 14 (1976) 915. [ 131 J.G. Kirkwood, Physik. Z. 33 (1932) 57. [ 141 S.Y. Yousifand H.A. Kassim, Surface Sci. 197 (1988) 509. [ 151 D.M. Bishop and A. Cartier, Mol. Phys. 48 (1983) 175. [ 161 R.D. Amos, Chem. Phys. Letters 73 (1960) 602. [ 171 G.H.F. Diercksen and A.J. Sadlej, Theoret. Chim. Acta 61 (1982) 485. [ 18 ] P.W. Fowler, N.C. Pyper and P.J. Knowles, Mol. Phys. 56 (1985) 83. [ 191 G.H.F. Diercksen and A.J. Sadlej, Chem. Phys. Letters 84 (1981) 390. 1201 P.W. Fowler and P. Tole, Surface Sci. 197 (1988) 457.

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