THE JOURNAL OF CHEMICAL PHYSICS 141, 214303 (2014)

CCSD(T) calculations of confined systems: In-crystal polarizabilities of F− , Cl− , O2 − , and S2 − F. Holka,1,a) M. Urban,1,2,b) P. Neogrády,2,c) and J. Paldus3,d) 1

Faculty of Materials Science and Technology in Trnava, Institute of Materials Science, Slovak University of Technology in Bratislava, Paulínska 16, 917 24 Trnava, Slovakia 2 Department of Physical and Theoretical Chemistry, Faculty of Natural Sciences, Comenius University, Mlynská dolina, SK–842 15 Bratislava, Slovakia 3 Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

(Received 19 August 2014; accepted 10 November 2014; published online 2 December 2014) We explore dipole polarizabilities of the singly and doubly charged F− , Cl− , O2 − , and S2 −  1 2anions 2 in an external, harmonic oscillator (HO) confining potential i 2 ω ri . We find that in contrast to F− and Cl− those for O2 − and S2 − are unrealistically high due to the instability of the corresponding restricted Hartree-Fock (RHF) solutions. Yet, already a relatively weak HO confining potential stabilizes their RHF solutions and eliminates any possible broken-symmetry solutions. The coupledcluster theory with single, double and noniterative triple excitations (CCSD(T)) then yields considerably reduced polarizabilities for O2 − and S2 − relative to their unconfined values. We showed that polarizabilities of O2 − and S2 − are more sensitive to the strength of a confinement potential than are those for F− and Cl− . This enables us to relate the confining parameter ω with the known experimental polarizabilities for selected crystals (our “training set”) and to find a specific confining parameter ω for which the CCSD(T) polarizability equals the experimental in-crystal polarizability of an anion in the training set. The latter may then be used as an alternative approach for determining the incrystal polarizabilities of anions by exploiting the fact that the characteristic ω values depend linearly on the ionic radius of a cation participating in specific crystals containing these anions. Using this method we then calculate the isotropic dipole polarizabilities for F− , Cl− , O2 − , and S2 − embedded in the LiF, LiCl, NaF, NaCl, KF, KCl, ZnO, ZnS, MgO, MgS, CaO, CaS, SrO, SrS, BaO, BaS, and other crystals containing halogen, oxygen, or sulphur anions. We compare our results with those obtained via alternative models of the in-crystal anionic polarizabilities. [http://dx.doi.org/10.1063/1.4902353] I. INTRODUCTION

While the properties of simple monovalent anions, such as F− , Cl− , and other halogens, are well known both experimentally and theoretically, doubly charged anions, such as O2 − or S2 − , are unknown as isolated species, at least under normal conditions (see, however, Refs. 1–3). Simply stated, the nuclear charge of the free oxygen or sulphur atoms is too small to bind the two extra electrons. Yet, these anions represent textbook examples of the anionic species that constitute many compounds, particularly crystals in which they are stabilized by a cationic environment. They may also be regarded as fragments of polyoxometalates that constitute a huge class of metal-oxygen clusters, usually involving Mo, W, or V, which have many potential applications in catalysis, material science, and medicine (see, e.g., Refs. 4 and 5). Such compounds are frequently described as clathrate-like structures or the so-called Lindqvist anions,6 in which O2 − represents the central oxide ion. A classical example of the stabilization of O2 − or S2 − is represented by ionic crystals, in which the stabilization effect arises due to the field of the surrounding positive ions.7 a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] d) Electronic mail: [email protected]

0021-9606/2014/141(21)/214303/10/$30.00

Properties of atoms, molecules, or ions that are embedded in crystals, solutions, and the like, i.e., when placed in an external confinement, often undergo a significant alteration of their energy spectrum, their relative stability, as well as of their electric, magnetic, optical, and other properties.8–12 Specifically, the knowledge of optical properties, such as the electronic polarizabilities, can be very useful for an understanding and interpretation of macroscopic properties of optical glasses or crystals, since they control the dielectric response of ionic crystals.13 The ionic polarizabilities are also linked with the dipole-dipole dispersion coefficients and thus with van der Waals interactions between ions. Another area of interest are polarizabilities of ions in water, which are particularly useful for models simulating the behavior of salts in water or other solutions. Even here the polarizabilities of participating anionic species are significantly reduced by their environment. For example, the intrinsic polarizabilities of monovalent anions of halogens generally amount to about 35%–55% of their gas-phase values14 as rendered by the B3LYP/DFT calculations employing the Hirshfeld model15 for the partitioning of polarizabilities of cations and anions for specific environments of water clusters.16 The first extensive and systematic study of the electronic polarizabilities of ions in crystals was published about 60 years ago by Tessman, Kahn, and Shockley.7 Their polarizabilities for a series of cations and anions were obtained by

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the least-squares fit of the experimental refraction data while relying on simple additivity rules. An important improvement of the estimates of polarizabilities of various species imbedded in crystals has been achieved by relying on the experimental refractive indices supplemented by ab initio calculations of the ionic polarizabilities.17, 18 A detailed analysis of the empirical electronic polarizabilities of 650 crystals was published by Shannon and Fischer.19 The anionic polarizabilities were then further refined by relying on the variable cation coordination number (CN), specific structural features of different crystals, and other effects. A stepwise introduction of such corrections lead to a very good agreement of the fitted polarizabilities and the experimental crystal polarizabilities with no discrepancies exceeding 4%. From a viewpoint of the electronic structure at the Hartree-Fock (HF) level, the existence of the negatively charged O2 − or S2 − anions as isolated species may be assessed on the basis of HF singlet stability conditions and the existence of related broken symmetry (BS) solutions.20 By continuously varying the nuclear charge Z in ten-electron systems from Z = 10 (Ne) to Z = 9 (F− ) and, finally, to Z = 8, we reach the hypothetical, doubly negative oxide ion O2 − . Stabilities of the O− and O2 − anions was studied by Herrick and Stillinger21 who used analytic continuation methods to extrapolate the nine-to-ten electron energy gaps of Ne and F− as continuous functions of the nuclear charge Z to Z = 8. Testing for the singlet stability20 of the resulting solutions we find22 that while the standard symmetry adapted (SA), i.e., the spherically symmetric, closed-shell, pure singlet solutions for Ne and F− are perfectly stable, thus representing the true minimum on the HF mean-energy hypersurface, the SA solution for O2 − is singlet unstable, the instability onset arising at about Z ≈ 8.6. A detailed analysis of the stability of O2 − or S2 − anions was presented in our more recent work.23 Our main interest in this paper are the optical properties, particularly the static electric dipole polarizabilities of anions in crystals. In a broader sense, the present work builds on our earlier investigations of confined systems.23–25 The first step in this direction represent our calculations of polarizabilities of simple two-electron systems,24 including a quantum dot, the hydrogen anion, the helium atom, and the lithium cation. The polarizability of all these systems considerably decreases when confined by a simple, spherically symmetric, isotropic harmonic oscillator (HO) potential W (r) that is centered at the nucleus and is represented by a sum of one-electron contributions w(ri ), i.e.,  w(ri ), w(ri ) = 12 ω2 ri2 , (1) W (r) = i

where the parameter ω represents the confinement strength. Suggestion of shapes for variation of ion polarizability with confining potential, and the differences between bound and unbound ions in such cases were discussed in works of Fowler.26, 27 The HO potential and some other, more general confinements were also used, e.g., by Góra et al.12 in their study of (hyper)polarizabilities of spatially confined LiH molecule. For example, the static dipole polarizability of the hydrogen anion decreases by two orders of magnitude with a

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relatively small confinement24 of ω = 1.0, although this reduction is less dramatic for He and Li+ that are stabilized by a larger electrostatic field of their nuclei. We note that the extent of this polarizability lowering is essential for an understanding of dielectric properties of metallic hydrides, hydrogen, rare gases, and alkali metals.28 Under extreme conditions, such as a compression or heating, some of these systems may exhibit the phenomenon of a dielectric catastrophe and undergo a metal-insulator phase transition.28 A confinement via the spherical HO potential represents a rather artificial model when exploring the role of confinement on the properties of various systems. More realistic is a modeling of ionic properties by their embedding in the condensed phase. In particular, the anions embedded in crystals are compressed by both the Coulomb (Madelung) forces and by a short-range overlap with the nearest-neighbor cations, thus displaying properties that differ considerably from those in the gas phase. Some of the properties for several anions that are embedded in crystals are known from both the theoretical calculations and the experiment. When compared with their isolated counterparts (assuming they exist) they are smaller, less polarizable, and more strongly bound. A physical picture underlying this effect is one in which the polarizability of a compressible, negative ion is affected by both the number and the distribution of its neighbors. The pseudopotential arising from these neighbors constitutes a “box” confining the anionic electron density, as discussed in detail in Refs. 17, 29, and 30. Thus, the in-crystal anionic polarizabilities clearly depend on their environment and the theoretically computed ionic polarizabilities for ions embedded in the confinement simulating the crystalline environment31 (with “real” nearest neighbors point charges for more distant ions) provide an excellent agreement with experiment. Such models can also explain the variation in properties of the free ions, the ions at a surface, and the in-crystal ions, as well as the difference in properties of ions in different crystals32–35 and can be used to derive consistent empirical ionic polarizabilities and dispersion coefficients that enable a prediction of the crystal refractive indices and energies. In our earlier work, we showed23 that subjecting the O2 − or S2 − ions to a sufficiently strong, external HO confinement all HF instabilities and BS solutions disappear. Consequently, the single determinant HF wave function may serve as a good reference for subsequent coupled-cluster single double triple (CCSD(T)) calculations of the dipole polarizability of these doubly charged anions, precisely in the same way as in calculations of singly charged F− , Cl− anions. Clearly, upon exposing the anions to a confining potential W (r) characterized by the confinement strength parameter ω, Eq. (1), their electron distribution is significantly altered in comparison to the electron distribution of free ions. Considering that the similar effect, i.e., the alteration of the electron distribution is experienced by ions in a specific crystal environment we can try linking ω with a suitable crystal structure parameter. In this work, we employ for this purpose the ionic radius of a cation in a specific crystal. In this regard, we were inspired by Pyper’s considerations of anion polarizabilities as a function of the crystal geometry.18 Another alternative7 for such a correlation with the parametrization by ω is to employ the

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volume corresponding to an anion in a specific crystal that yields the same polarizability as that obtained from the experimental dielectric constant or from the refractive index extrapolated to infinity (in order to obtain the static polarizability), assuming the additivity of polarizability in ionic crystals. An alternative approach that is based on the relationship between the electronic dipole polarizability of an ion and its effective charge has been suggested by Yatsenko.36 The external potential generated by, e.g., a MgO crystal was simulated in our previous work23 by a grid of point charges with altering signs that are allocated at the atomic centers in the crystal. In the following, we assume that an immersion of the anions F− , Cl− , O2 − , or S2 − in a specific confinement potential W (r), Eq. (1), can mimic physical surroundings which an anion experiences in a real physical situation, namely, in a crystal structure. We also assume that polarizabilities calculated for a specific confinement parameter ω can be used to predict the in-crystal polarizabilities by relying on a calibration with respect to the crystal polarizabilities that are available from the literature, mostly from the Clausius-Mossotti equation using the additivity rules for polarizability.7 Finally, we examine the sensitivity of anionic polarizabilities to an external HO confinement. The polarizability of an anion in a spherical HO potential may be accurately calculated by sophisticated ab initio methods upon a simple extension of the existing computer codes that incorporate the one-electron potential arising from the confinement. The polarizability anisotropy and its magnitude can also be accounted for by employing more general confinements, such as prolate or oblate HO confinements used by Sako et al.37 in their investigations of the energy spectra of quantum dots and other confined systems. We should note, finally, that an understanding of a compression of the anionic electron density as an effect resulting due to a confining potential modeling the crystal environment was described by Madden et al.,38 thus elucidating the fact that the polarizabilities of in-crystal anions can considerably differ from their gas-phase values. II. AB INITIO METHODS EMPLOYED

For computation of polarizabilities at the ab initio level, we rely on highly accurate coupled cluster (CC) approaches, namely, on the CCSD(T) method with an iterative determination of singly and doubly excited CC amplitudes and a perturbative estimate of triples.39–42 For open-shell species, we use the Restricted Open-shell Hartree-Fock (ROHF) singledeterminantal reference. In the CCSD step, the singles and doubles are approximately spin adapted using the approach of Ref. 43. With a proper selection of the AO basis this approach provides highly accurate polarizabilities while requiring a reasonable computational effort.24, 44 We used the MOLCAS codes45 supplemented with routines that calculate appropriate one-electron integrals arising from the presence of the symmetric, isotropic, HO confining potential W (r), Eq. (1). The static electric dipole polarizabilities were then calculated via the finite field approach, employing a series of weak external electric fields.46 The polarizability was obtained numerically as the second derivative of the CCSD(T) energy (including the confining potential) with respect to the

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V(r) = -Z/r W(r) = 1/2ω2r2 W(r) + V(r)

4

Energy E / [a.u.]

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2

2p 2p

0 2s 2s

-2

-4 0

2

4

6

8

10

Distance r / [a.u.] FIG. 1. The shape of the one-electron component w(r) of the confining HO potential W (r), Eq. (1), with ω = 0.3, the atomic one-electron potential v(r) = −Z/r, and their superposition v(r) + w(r), as well as the orbital energies associated with the symmetry adapted RHF solutions for the free O2 − anion (thin lines) and for the anion immersed in the potential W (r) (thick lines).

field strength. The basis sets employed in CCSD(T) calculations of polarizabilities of anions exposed to a confining potential are Dunning’s augmented correlation-consistent sets of 5Z quality.47–49 III. ONE-ELECTRON POTENTIALS FOR O2 −

In Fig. 1, we visualize the one-electron component of both the spherically symmetric, isotropic confining HO potential w(r) = 12 ω2 r 2 [cf. Eq. (1)] and the atomic one-electron potential v(r) = −Z/r (Z being the nuclear charge), as well as their superposition v(r) + w(r). The orbital energies of the 2s and 2p orbitals that are associated with the respective RHF solution for the free O2 − ion and that which is immersed in the HO potential with relatively large ω = 0.3 are also indicated in the energy diagram of Fig. 1, illustrating the shift of the reference orbitals due to the confinement. Superimposing the confining HO potential  W (r), Eq. (1), onto the one-electron potential V (r) = − i Z/ri leads to an infinitely high energy barrier at r ≡ |r| → ∞ that compresses the electron density, particularly at larger distances from the nucleus. The energy of the 1s, 2s, and 2p orbitals is shifted upwards relative to those of the free O2 − anion. The alteration in the energy spectrum of atoms when immersed in a confining potential was studied in detail earlier (see, e.g., Ref. 8). The total energy of the O2 − anion increases when immersed into the confining HO potential. However, as mentioned above, all BS solutions are removed leading to the one-electron reference suitable for the subsequent correlated CCSD(T) treatment. This opens a route toward calculations of the in-crystal polarizabilities of O2 − and S2 − , provided that we can link the spherical HO confining potential with a suitable parameter characterizing the crystal in which these ions are embedded. We must of course realize that such a potential represents only a crude model in view of its infinitely high energy barrier. This potential was also employed in our previous work23, 24 and preliminary results of the in-crystal polarizabilities for

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3.0

αF−(CsF)=1.360,(0.119) (a) αF−(RbF)=1.241,(0.137) αF−(KF) =1.201,(0.144) αF−(NaF)=1.030,(0.177) αF−(LiF)=0.887,(0.213)

2.0 1.5 1.0 −

F / CCSD(T) − F / HF−SCF

0.5 0.0

αCl−(CsCl)=3.616,(0.090) αCl−(RbCl)=3.469,(0.097) αCl−(KCl) =3.387,(0.102) αCl−(NaCl)=3.135,(0.118) αCl−(LiCl)=2.877,(0.135)

6.0 5.0 4.0

(b)

3.0 −

15

0

0.05

2−

O / CCSD(T) 2− O / HF−SCF

90

Cl / CCSD(T) − Cl / HF−SCF

2.0 1.0

100

80 10

70

αO2−(BaO)=3.652,(0.174) αO2−(SrO)=2.918,(0.198)

60

αO2−(CaO)=2.505,(0.216)

50

αO2−(MgO)=1.699,(0.274)

5

40 30 0 0.1

20

0.15

0.2

0.25

0.15

0.2

0.3

10 0.1

0.15

0.2

Confinement parameter ω

0.25

0 0.05

0.3

FIG. 2. Electronic polarizabilities of the F− (a) and Cl− (b) anions for different values of the confinement parameter ω as obtained at the HF-SCF and CCSD(T) level of theory using the aug-cc-pV5Z basis set. The arrows on the CCSD(T) curves mark the “experimental” polarizabilities of a respective anion in a specific crystal (see Sec. IV B). Numbers in parenthesis represent the value of ω for which the calculated and the “experimental” anion polarizabilities are equal.

the O2 − and S2 − anions were reported in Ref. 25. Besides, we expect that the calculations of the in-crystal properties of O2 − and S2 − that employ the HO potential W (r) may also serve as a model for studying other properties, in particular the polarizabilities, of species under severe conditions, such as a high pressure. IV. ANION POLARIZABILITIES A. Anion polarizabilities in a confinement

We now consider the electronic polarizabilities as a function of the confinement strength ω for two distinct types of anions embedded in ionic crystals. The first group is represented by the anions like F− or Cl− that are stable under normal conditions as isolated species. These ions, as free species, exhibit no instabilities and no BS solutions at the HF level in contrast to the second group represented by the O2 − and S2 − anions. In accord with the results of the instability analysis, these ions exhibit a very different behavior than do the F− or Cl− ions. The derivatives of the dipole polarizability curves for the F− or Cl− ions with respect to the confining potential strength ω approach zero for the vanishing confinement ω = 0.0 (see Fig. 2), when the polarizability corresponds to that of the stable, free F− or Cl− ions. Note the difference in polarizabilities of F− and Cl− as obtained at the HF and the CCSD(T) level: the effect of electron correlation is particularly significant for free ions and its importance moderately diminishes with the increasing ω. This may have been expected since the effect of electron correlation in isolated systems spreads the electron density further away from the nucleus, whereas the electron cloud is compressed when subjected to the external confining potential. The O2 − and S2 − anions display a different behavior for small values of ω than do F− or Cl− as shown in Figs. 3 and 4. The unrealistically high values of polarizabilities for O2 −

0.1

Confinement parameter ω

0.25

0.3

FIG. 3. Electronic polarizabilities of O2 − for different values of the confinement parameter ω as obtained at the HF-SCF and CCSD(T) level of theory with the aug-cc-pV5Z basis set. See caption to Fig. 2 for further details.

and S2 − in the ω = 0 limit reflect the fact that the second electron is not bound. Due to the HF instabilities in the region of small ω values these polarizabilities correspond in fact to the nonexistent species with a “free” second electron. It is worthwhile to note here that this “free” electron puts extreme demands on the basis set employed when calculating molecular properties24 as was shown in our earlier work23 which clearly demonstrated that the convergence of the total energy toward the complete basis set limit is extremely poor and a divergent behavior may occur. The insets in Figs. 3 and 4 highlight the range of ω values where there are no HF instabilities or BS solutions for O2 − and S2 − anions. We next analyze the relationship between the polarizability of an anion that is immersed in a HO confining potential and its polarizability in a specific crystal structure. As can be seen from Fig. 2, in the studied range of the ω values the electronic polarizability of F− and Cl− decreases to about 50% of its free anion value. For O2 − and S2 − , this lowering of polarizability with the increasing ω values is even more dramatic

100 12

2−

S / CCSD(T) 2− S / HF−SCF

90

Electronic polarizability α / [Å3]

Electronic polarizability α / [Å3]

2.5

Electronic polarizability α / [Å3]

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10

80

αS2−(BaS)=6.405,(0.138) αS2−(SrS)=5.323,(0.161)

70 8

αS2−(CaS)=5.130,(0.166)

60

αS2−(MgS)=4.597,(0.181)

6

50 40

4

30 2 0.1

20

0.15

0.2

0.25

0.3

10 0

0.05

0.1

0.15

0.2

Confinement parameter ω

0.25

0.3

FIG. 4. Electronic polarizabilities of S2 − calculated for different values of the confinement parameter ω as obtained at the HF-SCF and CCSD(T) level of theory with the aug-cc-pV5Z basis set. See caption to Fig. 2 for further details.

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than for the F− and Cl− anions (see the insets in Figs. 3 and 4). Recalling classical considerations of Mayer and Mayer50 (see also Ref. 19), namely, that an anion in a crystal is less polarizable than when it is free, there must clearly exist the value of ω for which the calculated anion polarizability equals the experimentally determined value. Unfortunately, the polarizabilities of individual ions are not accessible to a direct experimental measurement. The conventional approach to the evaluation of ionic polarizabilities exploits the relationship between the refractive index and the molecular polarizability, generally known as the LorentzLorenz equation. The overall molecular polarizability (i.e., the polarizability of a molecular unit in a crystal) as obtained via the Lorentz-Lorenz equation can be further decomposed into its cationic and anionic components by applying the additivity rule. Several procedures have been proposed for this purpose (see, e.g., Refs. 7, 17, 51, and 52). The method we employ is based on a combination of the experimental as well as the ab initio results17 (see Sec. IV B). The basic idea is that having the polarizability for some anions from the “experimental” source, for example, the one based on the ClausiusMossotti equation and the additivity rule) we can find a specific strength of the confining parameter ω that corresponds to this “experimental” anionic polarizability. These cases are indicated by an arrow in the insets in Figs. 2–4, together with the corresponding values of ω. Having established the relationship between the polarizability of an anion in a confinement ω and the “experimental” polarizability, we can then predict the in-crystal polarizability for an anion for which the polarizability in a specific crystal is unknown. This method should work provided that the structural pattern and the coordination number of the crystal for which we predict the anionic polarizability are the same as in the crystal that had been used in fitting the confinement strength ω with the experimental polarizability. The only requirement is to find a model that relates ω with some structural characteristic of the crystal.

B. In-crystal anionic polarizabilities

We now adopt a simple ionic model of essentially cubic crystals with ions represented by spherically symmetrical entities having a closed-shell electronic configuration.53 There are two principal sources10, 29, 38 that cause the electron densities of in-crystal anions to contract (i.e., to be less polarizable) relative to their isolated species. The first one has its origin in the Coulomb (Madelung) interaction of an anion with the surrounding ions. In the ionic model, these ions are treated as point charges. From a simplistic viewpoint, the potential generated by the point charge lattice can then be regarded as a constant attractive potential in a certain region of space which vanishes at the point of closest cation-anion separation. The second effect arises due to the Pauli principle and the spatial overlap of the electronic cloud of the anion with the electronic cloud of the neighboring ions. The mutual action of both effects creates the confining potential that compresses the electron density of the anion and reduces its polarizability.

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There exist several approaches (see, e.g., Refs. 10, 54, and 55) for a calculation of the in-crystal polarizabilities of anions that are based on a simple ionic model. The conditions imposed by such a model can be accomplished by embedding the anion in a finite grid of point charges that mimics the neighboring ions. The size of the crystal grid is adjusted so that it preserves the electrical neutrality of the entire system. This model (also referred to by the acronym CRYS10 ) reproduces the average coulomb potential confining the anionic electron density. Another important factor that leads to a compression of the electron density of an ion in a crystal lattice is the overlap with the electron densities of neighboring ions. The model which takes into account both effects (referred to as CLUS10, 54 ) requires that the nearest neighbors of a cation in the grid are considered with all their electrons as well as the full nuclear charges. The rest of the lattice is then kept in the point charge form. The disadvantage of the CLUS model is that the overall cluster polarizability involves, in addition to the desired anion polarizability, also the cation polarizability, the contribution from the dipole-induced dipole interactions, and the basis set superposition error, since the basis sets of the ions are incomplete. The difficulties may also arise when one tries to describe the ions that are unbound as free species, which is the case of O2 − and S2 − . Moreover, the size of the system containing the central anion and the first level of cations influences the computer time that is required for calculations. Nonetheless, despite these limitations, these methods were successfully applied in calculations of the in-crystal anionic polarizabilities of alkali halides and chalcogenides (see, e.g., Refs. 10, 13, 56, and 57). The model of the crystalline environment which we employ here benefits from our earlier experience with calculations of the static electric polarizabilities of confined systems.23, 24 This model has been implemented in the MOLCAS program package45 and represents a simple alternative to calculations of the in-crystal anionic polarizabilities. The confining potential of Eq. (1) is a spherically symmetric and isotropic HO potential, yet, from the physical viewpoint it has some drawbacks: First, it is artificial in a sense that it does not represent any real physical environment – it can be regarded as a confining box for the anionic electron density. Its strength (i.e., the steepness of the HO well) is controlled by the tuning parameter ω which has to be empirically adjusted by relying on some physical property which defines the surrounding environment; in our case the crystal lattice. Second, the approximation of the Coulomb-overlap potential10, 29 in the region far away from the equilibrium position of the anion in the crystal lattice is also non-physical in view of the infinitely high potential barrier. Thus, the dynamical processes, e.g., the electron tunneling and the ionization or the charge transfer cannot be treated with the HO model potential. Nevertheless, we expect that this simplified model can be used to calculate static properties like the ionic polarizabilities in a crystal. For a calculation of the in-crystal polarizabilities, we have to find parameters of a suitable model function that determines the value of ω for each anion under consideration and is suitable for a particular crystal lattice of a given chemical composition. In our model, we have chosen the ionic radius of a cation participating in a crystal as such a parameter.

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0.30 LiF NaF

0.20

KF

0.15 LiCl

Confinement parameter ω

0.10 0.05

C. Confinement potential and ionic radii of cations in a crystal

ωF−(rM+) ωCl−(rM+)

(a) 0.25

NaCl

KCl

CsF

RbCl

CsCl

ωO2−(rM2+) ωS2−(rM2+)

(b)

0.30

RbF

MgO

0.25

CaO

SrO

0.20

BaO MgS

0.15

CaS

SrS BaS

0.10

ωO2−(rM+)

(c)

0.30 0.25 0.20

Li2O

Na2O

K2O

0.15 0.10

0.6

0.8

1

1.2

1.4 n+

1.6

1.8

Effective ionic radius r(M ) / [Å]

FIG. 5. Dependence of the confining parameter ω on the cationic radius RM n+ for a series of “tuning” crystals. Values of ω in fits for a series of crystals with the F− , Cl− , O2 − , and S2 − anions correspond to the “experimental” polarizabilities as shown in Figs. 2–4.

We recall here that the idea of ionic radii was first proposed in the 1920s by Pauling.58 Considering ions as spheres of a well defined size, the unit cell parameter (obtained from the crystallographic data) is given as a sum of the ionic diameters (for simple unit cells of, e.g., the rocksalt type). Assigning the radius of 1.40 Å to the oxide ion O2 − and comparing the available X-ray data of different compounds, Pauling calculated a set of ionic radii58 (later revised by Shannon30 ) for most of the common elements. The ionic radius of a given ion varies with the coordination number and the spin state, which should be considered when modeling anionic polarizabilities in a particular crystal lattice.

According to the assumption of the additivity of polarizabilities in ionic species the molecular polarizability (i.e., the polarizability of species forming the crystal) can be split into contributions from each of the ions constituting the crystal. The validity of concepts of both the ionic radii and of the additivity of the electronic polarizabilities is limited to strongly ionic and symmetric crystals (e.g., to face-centered cubic lattices). It was reported7 that the additivity rule for polarizability brakes down for O2 − in chalcogenides. Nonetheless, the effect of the environment on polarizabilities of cations was found to be very small54 (see, however, Ref. 19). These facts lead us to the assumption that the ionic radius, as a parameter defining a particular crystal lattice, can serve as a parameter for determining the in-crystal polarizabilities. The key to our approach of predicting the in-crystal polarizabilities is an observation that the confinement strength ω, corresponding to the “experimental” polarizability of an anion in a specific crystal, depends linearly on the ionic radius of the corresponding cation, as shown in Fig. 5 for the F− , Cl− , O2 − , and S2 − ions. Thus, to predict the polarizability of an anion in a specific crystal we associate the cationic radius with the confining parameter ω and the anionic polarizability. For this purpose, we only require to calculate the CCSD(T) static dipole polarizability of F− , Cl− , O2 − , and S2 − as a function of ω. Having in mind the instability of the free O2 − and S2 − anions, we have ascertained that all the polarizabilities that we use in our model are calculated for ω’s at which there exist no BS HF solutions. The next step is then to link the ω values for a set of crystal lattices containing F− , Cl− , O2 − , and S2 − with the known in-crystal anionic polarizabilities. As a reference (or a “training” set), we use fluorides and chlorides of alkali metal cations and oxides (MgO, CaO, SrO, BaO) and sulfides (MgS, CaS, SrS, and BaS) of alkaline earth elements. The values of ω for which the CCSD(T) anionic polarizabilities are the same as those determined from

TABLE I. Electronic polarizabilities of F− in different crystals.a Polarizabilities in Å3 , effective ionic radii in Å.

LiF NaF KF RbF CsF CsF AgF CaF2 SrF2 BaF2 CdF2 PbF2 a

Ref. 17

Ref. 7

CNC

STR

RM n+ b

∞ αm

∞ αm

VI VI VI VI VI VIII VI VIII VIII VIII VIII VIII

NaCl NaCl NaCl NaCl NaCl CsCl NaCl CaF2 CaF2 CaF2 CaF2 CaF2

0.760 1.020 1.380 1.520 1.670 1.740 1.150 1.120 1.260 1.420 1.100 1.290

0.915 1.178 1.992 2.582 3.624

0.909 1.163 2.008 2.572 3.604 3.674

Ref. 17 αM n+

αF −

αF −

ω

2.854 2.5012 3.008 3.9350

0.028 0.148 0.791 1.341±0.022 2.264±0.040 2.264±0.040 1.749 0.473 0.771±0.009 1.497

0.887 1.030 1.201 1.241±0.022 1.360±0.040 1.410±0.040c 1.105 1.014 1.119±0.005 1.219

4.963

2.653

1.155

0.903 1.005 1.180 1.264 1.366 1.420 1.063 1.049 1.116 1.203 1.040 1.132

0.2085 0.1827 0.1471 0.1332 0.1184 0.1114 0.1698 0.1728 0.1590 0.1431 0.1748 0.1560

Crystals above the horizontal line were used to determine tuning parameters in Eq. (2). Effective ionic radii taken from Ref. 30. c ∞ αm employed in calculation of αF − taken from Ref. 7. b

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TABLE II. Electronic polarizabilities of Cl− in different crystalsa . Polarizabilities in Å3 , effective ionic radii in Å. Ref. 17

LiCl NaCl KCl RbCl CsCl CsCl AgCl TlCl SrCl2 CuCl

b

CN

STR

RM n+

VI VI VI VI VIII VI VI VIII VI IV

NaCl NaCl NaCl NaCl CsCl NaCl NaCl CsCl TiO2 d ZnS

0.760 1.020 1.380 1.520 1.740 1.670 1.150 1.590 1.180 0.600

Ref. 7

Ref. 17

This work

∞ αm

∞ αm

D αm

αM n+

αCl −

αCl −

ω

2.905 3.283 4.178 4.810 5.880

2.903 3.263 4.137 4.712 5.829 6.235

2.980 3.360 4.272 4.856 5.984 6.419 5.327 7.727 7.465 4.628

0.028 0.148 0.791 1.341±0.022 2.264±0.040 2.264±0.040 1.941 4.149±0.148 0.771±0.009

2.877 3.135 3.387 3.469±0.022 3.616±0.040 3.971±0.040c 3.089 3.375±0.148 3.347±0.005e

2.917 3.089 3.354 3.468 3.656 3.595 3.182 3.526 3.204 2.820

0.1324 0.1205 0.1040 0.0975 0.0874 0.0906 0.1145 0.0943 0.1131 0.1397

5.030 7.524

a

Crystals above the horizontal line were used to determine tuning parameters in Eq. (2). Effective ionic radii taken from Ref. 30. c ∞ αm employed in calculation of αCl − taken from Ref. 7. d The compound has a deformed rutile structure. e ∞ D αm is not available. αm employed in calculation of αCl − taken from Ref. 7. b

the Clausius-Mossotti equation are plotted as a function of the ionic radius of metallic cations in Fig. 5(a) for F− and Cl− and in Fig. 5(b) for O2 − and S2 − . A similar dependence of ω on the ionic radius in the Li2 O, Na2 O, and K2 O crystals is shown separately in Fig. 5(c). The dependence of ω on the ionic radius RM n+ (n = 1 or 2) of a cation is approximated by the linear function (i = F− , Cl− , O2− , and S2− ) (2) defining our model function that enables us to obtain ω for any ionic crystal lattice as represented by the ionic radius of the participating cation. The parameters ai and bi are then obtained by the least-squares fit of the function (2) using the data shown in Fig. 5. ωi (RM n+ ) = ai RM n+ + bi ,

We note that similar linear dependence of anionic polarizabilities on the ionic radius of a cation can also be constructed employing “experimental polarizabilities” of the training set. One concern regarding the appropriateness of our approach is the assumption that the ions for which we calculate dipole polarizabilities constitute a part of the relevant ionic crystals. Of course, no species are perfectly ionic. In fact, all crystals exhibit, to some extent, the covalent character so that our approach is necessarily approximate. The polarizabilities of F− , Cl− , O2 − , and S2 − in several crystals, as obtained by applying the confining potential W (r) of Eq. (1), as defined by the linear fit of Eq. (2), and by the plots in Fig. 5, are summarized in Tables I–V. The incrystal polarizabilities of F− and Cl− , Tables I and II, are significantly reduced with respect to the polarizabilities of free anions (2.400 and 5.341 Å3 for F− and Cl− , respectively).

TABLE III. Electronic polarizabilities of O2 − in different crystals.a Polarizabilities in Å3 , effective ionic radii in Å.

MgO CaO SrO BaO NiO CoO FeO MnO CdO Li2 O Na2 O K2 O a

Ref. 17

Ref. 52c

Ref. 17

Ref. 52

This work

CNC

STR

RM n+ b

∞ αm

D αm

αM n+

αO 2−

αM n+

αO 2−

αO 2−

ω

VI VI VI VI VI VI VI VI VI IV IV IV

NaCl NaCl NaCl NaCl NaCl NaCl NaCl NaCl NaCl CaF2 CaF2 CaF2

0.720 1.000 1.180 1.350 0.690 0.745 0.780 0.830 0.950 0.590 0.990 1.370

1.7532 2.8583 3.5356 4.6683

1.795 2.973 3.775 5.248 2.472

0.072 0.473 0.771 1.497

1.681 2.385 2.765 3.171

0.094 0.469 0.861 1.595 0.266 0.508

1.699 2.505 2.918 3.652 2.202 2.405

0.544 1.054 0.024

2.303 2.909 2.090

1.741 2.332 2.943 3.774 1.699 1.779 1.837 1.929 2.198 1.580 2.303 3.896

0.2690 0.2252 0.1971 0.1705 0.2737 0.2651 0.2596 0.2518 0.2330 0.2893 0.2268 0.1674

2.129d 2.560d 4.330d

2.846 3.963 2.132

0.028 0.148 0.791

2.073 2.264 2.748

Crystals above the horizontal line were used to determine tuning parameters in Eq. (2). Effective ionic radii taken from Ref. 30. For the transition metal cations, we take the values of rc that corresponds to the high-spin state of the cation. c Calculated from refractive indices and molar volumes reported in Ref. 52. d ∞ D αm not available. αm used in calculation of αO 2− . b

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TABLE IV. Electronic polarizabilities of O2 − in different crystals with the alkali metal cation.a Polarizabilities in Å3 , effective ionic radii in Å. Ref. 17

Li2 O Na2 O K2 O Rb2 O a b

TABLE VI. Parameters ai and bi in Eq. (2) based on the CCSD(T) polarizabilities for F− , Cl− , O2 − , and S2 − in a confinement W (r), Eq. (1). Ion

This work

CNC

STR

RM n+ b

αM n+

αO 2−

αO 2−

ω

IV IV IV IV

CaF2 CaF2 CaF2 CaF2

0.590 0.990 1.370 1.520

0.028 0.148 0.791 1.341

2.073 2.264 2.748

2.043 2.339 2.696 2.860

0.2432 0.2248 0.2073 0.2004

F− Cl− O2 − O2 − S2 −

Li+ –Cs+ Li+ –Cs+ Mg2 + –Ba2 + Li+ –K+ Mg2 + –Ba2 +

ai

bi

−0.098977 −0.045886 −0.156244 −0.046015 −0.063465

0.283641 0.167278 0.381460 0.270314 0.228907

Crystals above the horizontal line were used to determine tuning parameters in Eq. (2). Effective ionic radii taken from Ref. 30. −

Their values for F contained in cubic crystals with alkali metal cations vary between 0.90 Å3 (LiF) and 1.37 Å3 (CsF), which represent about 56% and 86% of their bare ion values, respectively. The polarizabilities for analogous Cl− vary between 2.92 Å3 (LiCl) and 3.66 Å3 (CsCl). These values agree very well with the existing reference data17 or other theoretical calculations (see, e.g., Refs. 31 and 59) for these crystals. Note, however, that our method performs reasonably well even in cases which were not included in our training set that was employed to find the relationship between the confinement parameter ω and the cationic radius for crystals. Further, our value of 1.06 Å3 for αF − in the AgF crystal, while surprisingly low thanks to the small ionic radius for Ag+ , agrees well with the value of 1.11 Å3 taken from Ref. 17. The same holds for the AgCl crystal (Table II). We note that the fitting of parameters that is based on the linear dependence of the confinement strength ω on the cationic radius for alkali metal-halogen cubic crystals (Table VI) works even for crystals with different structural features and for crystals with a different cationic valency. Indeed, in agreement with Ref. 17, the polarizability of F− in the CsF crystal having the CsCl structure differs very little from the polarizability of F− in the cubic crystal. A satisfactory agreement with the literature data is also found for bivalent crystals, such as CaF2 , SrF2 , BaF2 , and PbF2 . The polarizability of F− in CdF2 is not available in the literature, so that the value of 1.04 Å3 represents our prediction. The conclusions concerning the performance of our method for F− also apply to crystals containing Cl− . A similar reduction of polarizabilities (to 47% of their gas-phase value) as in our in-crystal cases was reported14 for F− and Cl− in the fully hydrated condensed-phase limit. The

dipole polarizabilities in aqueous solutions of 1.3 and 3.5 Å3 for F− and Cl− , respectively, obtained by Molina et al.,60 are significantly higher than our in-crystal polarizabilities for crystals with lighter cations, such as LiF or LiCl, but lower than the values for crystals with Cs cations. A similar value of 3.6 Å3 was also reported by Jungwirth and Tobias16 for the polarizability of Cl− in water clusters. The reduced anionic polarizabilities in a water solution and in water clusters relative to bare anions has been interpreted as resulting from a confinement due to the solute molecules, similar to our interpretation of a confinement in crystals. Even larger differences for the in-crystal polarizabilities than for F− and Cl− arise for crystals containing S2 − , and particularly O2 − , as shown in Figs. 3 and 4 and in Tables III–V. In contrast to F− and Cl− , the reference data for the O2 − and S2 − bare anions that are unstable as isolated species, are not available. The polarizabilities of O2 − and S2 − in the MgO, MgS, CaO, CaS, SrO, SrS, BaO, and BaS crystals agree very well with the experimental values as obtained from the Clausius-Mossotti relation assuming the additivity rule. We see that the O2 − and S2 − polarizabilities span a rather wide range of values depending on the model employed. Yet, the fact that for a series of Mg, Ca, Sr, and Ba oxides and sulfides our values agree well with those given by the Clausius-Mossotti relation should not be overestimated, since these crystals have been employed as a “training set” when determining the parameters defining the linear fit of Eq. (2), so that any deviation from the experimental values only indicates a deviation from the linearity of the dependence of ω on the ionic radius R2+ M of the cation. More significant is the agreement of the CCSD(T) polarizability of O2 − with the Clausius-Mossotti value in the BaO crystal. We also note that

TABLE V. Electronic polarizabilities of S2 − in different crystals.a Polarizabilities in Å3 , effective ionic radii in Å.

MgS CaS SrS BaS PbS ZnS CdS a b

b

CNC

STR

RM n+

VI VI VI VI VI IV IV

NaCl NaCl NaCl NaCl NaCl ZnO ZnO

0.720 1.000 1.180 1.350 1.190 0.600 0.780

Ref. 17

Ref. 7

∞ αm

D αm

Ref. 17 αM n+

αS 2−

αM n+

αS 2−

4.6688 5.6034 6.0930 7.9020

5.940 6.447 8.380

0.072 0.473 0.771 1.497

4.5968 5.1304 5.3220 6.4050

1.1 1.6 2.5

4.8 4.7 5.9

0.8 1.8

4.9 5.8

5.729 7.586

Crystals above the horizontal line were used to determine tuning parameters in Eq. (2). Effective ionic radii taken from Ref. 30.

Ref. 7

This work αS 2−

ω

4.529 5.144 5.615 6.128 5.642 4.296 4.652

0.1832 0.1655 0.1540 0.1432 0.1534 0.1908 0.1794

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various literature data often differ by as much as 1.5 Å3 , depending on the model employed. The polarizability of S2 − in BaS is our prediction. A different situation arises in the case of zinc oxide. The polarizability of O2 − in the ZnO crystal differs significantly from the Clausius-Mossotti value in comparison with other crystals, resulting in a largely underestimated value. Here, we must recall that ZnO is a strong ionic compound and its most common crystal structures are of the hexagonal wurtzite and the cubic zincblende types. In both cases, the anion is bound to four cations (i.e., the coordination number is 4). When compared to the rocksalt-type lattice with the coordination number 6 these structures are less spherical. Moreover, there is a relatively more extensive vacant space in the wurtzite or zincblende lattices for the electron density to spread, explaining low calculated values of α in comparison with those obtained from the refractive indices. This example serves as an indication of a deficiency of our model when applied to differently coordinated anions. Its usefulness is also questionable when applied to crystals with a significant covalent character of bonds (but still considered as ionic). We are presently trying to devise a more general empirical function for the parametrization of the confining potential W (r), Eq. (1), that takes into account other properties and can avoid the above mentioned limitations. V. CONCLUSIONS

We present the CCSD(T) results for the static dipole polarizabilities of the F− , Cl− , O2 − , and S2 − anions that are immersed in a confining spherically symmetric, isotropic harmonic oscillator potential W (r). The in-crystal polarizabilities of halogen anions are reduced by half relative to those for their isolated (bare, unconfined) counterparts. The unconfined O2 − and S2 − anions exhibit HF instabilities and implied BS solutions. Their HF and CCSD(T) polarizabilities are unrealistically high exhibiting some features of the singly charged anion and a free, unbound extra electron. Upon exposing O2 − and S2 − anions to an external HO confinement all BS solutions disappear and their polarizabilities are considerably reduced. Their polarizabilities are much more sensitive to the external confinement than are those for halogen anions. We propose a model in which the confining potential W (r), Eq. (1), is related to the ionic radius RM n+ of the metallic cation (n = 1 or 2) that is present in the respective halogen, oxide or sulfide crystals. The polarizabilities of the F− , Cl− , O2 − , and S2 − anions that are immersed in the confining potential W (r) with its strength determined by the linear dependence of the confinement parameter ω on RM n+ – the latter established by relying on the known polarizabilities of a series of “tuning” crystals – have been subsequently calculated by the CCSD(T) method. The polarizabilities of the F− anion range from 0.90 Å3 in the LiF crystal to 1.42 Å3 (CsF) and from 2.92 (LiCl) to 3.66 Å3 (CsCl) for the Cl− anion. For O2 − in different crystals, the polarizabilities vary in a rather large interval of about 2.1 Å3 , namely, from 1.74 Å3 (MgO) to 3.80 and 3.90 Å3 (BaO and K2 O, respectively) and those for S2 − vary from 4.53 Å3 (MgS) to 6.13 (BaS) Å3 .

J. Chem. Phys. 141, 214303 (2014)

Our results provide an alternative demonstration of a considerable dependence of the anion polarizabilities on the environment to which they are exposed. In line with the statement of Mayer and Mayer,50 the negative ions in crystals (and the same holds for solutions) have considerably lower polarizabilities than the same gaseous (or free) ions. In contrast, according to the available literature,19, 50 the cation polarizabilities in crystals are somewhat higher than the free-ion polarizabilities as obtained from the ab initio calculations. Clearly, to generate more uniform data for the in-crystal anionic polarizabilities will require a similar approach as that for the cationic polarizabilities which is based on the ab initio calculations of cations exposed to a realistic confinement. The CCSD(T) treatment of the cationic polarizabilities in crystals will be the subject of our future research. Another problem that has not been discussed so far is related to a possible dependence of the in-crystal polarizability of an anion on the spin state of the participating cations in the transition metal containing crystals. The ionic radii of such cations may depend on the spin state of the cation in a specific crystal. Having values of ω suitable for calculation of in-crystal polarizabilities we may consider more general application of our approach, e.g., calculation of hyperpolarizabilities and other electronic structure related properties. Finally, we believe that our approach may allow the assessment of the sensitivity of ionic polarizabilities in crystals exposed to a high pressure61 employing the derivatives of polarizability with respect to the confinement parameter ω.

ACKNOWLEDGMENTS

The support from the Slovak Grant Agencies, Grant Nos. APVV-0059-10 and VEGA 1/0770/13 is gratefully acknowledged. This work was also supported by the Association EURATOM CU, Contract No. FU07-CT-2006-00441/P1c. The content of the publication is the sole responsibility of its authors and it does not necessarily represent the views of the European Union (EU) Commission or its services. A part of the calculations was performed in the Computing Centre of the Slovak Academy of Sciences and Slovak University of Technology in Bratislava using the supercomputing infrastructure acquired in Project Nos. ITMS 26230120002 and 26210120002 (Slovak infrastructure for high-performance computing) supported by the Research and Development Operational Programme funded by the ERDF. 1 W.

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