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Calculations of Optical Properties of the Tetraphenyl-X Family ofIsomorphous Crystals (X=C, Si, Ge, Sn, Pb)

Kacey Claborn, Bart Kahr* and Werner Kaminsky†

Department of Chemistry, Box 351700, University of Washington, Seattle, WA 98105, USA* kahr@chem.washington.edu, † kaminsky@wintensor.com

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As part of a program to determine how small structural changes become manifest in the optical properties of crystals we used classicaldipole-dipole interaction calculations to estimate the linear birefringence and optical rotatory power of the crystals Ph4X where X=C,Si, Ge, Sn, and Pb. Field induced effects including second harmonic generation, the electro-optic response and electrogyration werecalculated using the dipole electron shifting model (DES) model. The calculated induced effects are larger than those in standardmaterials such as KH2PO4. All of the properties tend to increase in magnitude with increasing polarizability except for optical rotation,which is largest for Ph4C. We propose an interpretation for the unusual behaviour of the optical rotation in terms of competing helicalcircuits of closely bonded atoms.

IntroductionTetraphenylmethane crystals belong to a family includingthe isomorphous tetraphenyl derivatives of silicon,germanium, tin and lead. This is a rare family ofmolecular crystals whereby a main group element in the2nd row can be replaced by any other atom in thecorresponding family, group 14 in this case, withoutchanging the essential features of the crystal structure. Ascrystal engineers aspire to understand the effects of smallprescribed structural changes on the physical properties ofcrystals, the isomorphous family, Ph4X where X = C, Si,Ge, Sn and Pb, provides unparalleled opportunities forcomparison. Here, we present classical calculations of theoptical properties of the Ph4X family including linearbirefringence, optical rotation, as well as second harmonicgeneration (SHG), the electro-optic effect and inducedoptical rotation (electrogyration), non-linear opticaleffects that are of great interest to many scientists in thecrystal engineering community. The parameterization ofthese calculations is based on redetermined experimentalvalues of refractive indices that are at great variance withthose previously reported.

Computation Methodology

Dipole-Dipole Interaction

Optical rotation may be calculated using the dipole-dipoleinteraction theory and has been applied with success toionic crystals.1 This theory based on electronicpolarizabilities is described elsewhere.2 Here, we outlinethe basic ideas behind the calculations. Polarizabilityvolumes were selected so that the dipole-dipole modelcalculations were close to the refractive indices derivedexperimentally.

The cumulative effect of the dipole-dipoleinteractions between all atoms in the crystal lattice actingon an atom s in unit cell l at a position rs

l is described byan electrical potential, V:2

where Z is the Hertz vector potential. The term in braceshas the periodicity of the lattice. The overall Hertzpotential is built from terms describing the frequencydependent incident wave, the interaction between allatoms in the unit cell and the interaction between all unitcells in the lattice, respectively.

The electric field E(rsl) at atom rs

l originatingfrom the dipole waves emanating from all other atoms(point dipoles) in the structure is described by (Figure 1):3

where psl is the electronic polarization at position rs

l andαs is the electronic polarizability volume of atom s. Ass´ isa matrix of complex numbers.

The imaginary part contained therein describesthe phase shifts that result in radiative interference andoptical rotation. When the sum of the contributions to theelectric field is then taken the series converges onlyconditionally. As a remedy, the function is decomposedinto a Fourier series that can be separated into two

Figure 1. Illustration of the dipole-dipole interaction theory of Devarajanand Glazer (1986). When an initial wave with polarization E passes thefirst atom positioned at (x,0,0) in a Cartesian reference system (z//k, k isthe wave vector), a dipole field is created which induces in a secondatom at (0,y,z) a secondary dipole field. This field oscillates in adirection different to E for y, z ≠ 0. The interference of all inducedwaves with the initial wave (calculated via an Ewald sum) on passingthrough the crystal leads to a rotation of E when at least four atoms adopta chiral arrangement. The result is a conversion of the atomicpolarizabilities from an isotropic to an anisotropic state.

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absolutely convergent parts, one in real space and theother in reciprocal space according to the Ewald theorem.4

A new matrix Css´ is defined from variablesdescribing the electric field E(rs

l):

The imaginary part of this matrix is related to opticalrotation. For the calculation of Qss´ see Appendix 1. Theoptical rotation ρ(k) and optical dielectric constants εij arederived as follows:1

(erij = Levi-Civita symbol, k = wave vector, n = averagerefractive indices, v = unit volume and δij = Kroneckerdelta). The Levi-Civita operation erij accomplishes thecross product between the spatial coordinates of Css′ andthe wave vector.The dipole-dipole interaction distorts thepolarizabilities of the atoms resulting in an anisotropydescribed by the effective polarizabilities αs

eff:

This theoretical model reliably calculates optical rotationon the basis of the interacting forces in inorganicstructures. It was further applied successfully tomolecular crystals where the interacting fields within amolecule are more significant than those between themolecules.5,6 Here, we are trying to evaluate theapplicability of the classical theory to aromatic molecularcrystals, even though we have explicitly neglected theelectronic structure of the molecules in our calculations.While progress has been made in the application ofquantum mechanics to the calculation of optical rotationin molecules7, in crystals we chose a theory that embodieslong range interactions and accommodates the periodicity.

Dipole Electron Shifting (DES)

To calculate the higher order optical properties, theclassical polarizability theory was employed according tothe procedure of Kaminsky and Glazer, whereby a virtualexternal electric field displaces the nuclei from the centersof the surrounding electron clouds in proportion to theatomic electronic polarizabilities.5

In the DES model, the external electric fieldshifts the atomic nuclei by distance x in proportion to thepolarizability of the kth atom (Figure 2): 8

e is the charge of an electron, εo is the permittivity of freespace and Ej

loc is the local electric field connected with theexternal field Eext via the effective relative dielectricconstant ε' in the direction of the external field:

where a spherical depolarization field (Lorentz-depolarization) is assumed.

The largest approximation in the DES model isthe substitution of the ionic polarizability, which shouldbe used to calculate the shifts x for static electric fields, bythe electronic polarizability volumes. The ionicpolarizabilities are not well known, in contrast to theelectronic polarizabilities, which are derived fromrefractive indices and which are tabulated elsewhere.9Naturally, atoms with small electronic polarizabilities areonly slightly affected by a static electric field.10

Furthermore, if the sum of the molar polarizabilities isincreased only by about a factor of three, the dielectricconstant tends already to infinity, as is easily seen fromthe Clausius-Mosotti relation. Thus we can expect only arelatively small error by this approximation as long as thedielectric constant remains small. This is not the casewhen a crystal is ferroelectric, or has field dependentinternal dynamics. It is of particular interest, therefore, tosee if this crude approximation can be justified in thePh4X family.

The dipole-dipole theory was used to calculatethe rotatory power ρij (o/mm) and the optical relativedielectric constant εij with and without the applied field.The polarization tensor aij is the inverse of εij: aij=εij

-1,11 nis the average refractive index and λ is the wavelength.The dispersion of αij is roughly approximated by a λ-1/2

dependence for the SHG calculations. These quantities aswell as the electrogyration gijk and linear electro-opticeffect at constant strain rijk were calculated from thefollowing equations:

ρij(Ekexternal) - ρij(0) = gijkEk

external/nλ (electrogyration atconstant strain)

aij(Ekexternal) - aij(0) = rijk

Ekexternal (electro-optic effect

at constant strain)εij(Ek

light) - εij(0) = 2dijkEklight (second harmonic

generation).

The rotatory power ρij is defined as the clockwise rotationof linear polarized light passing through a non-birefringentsample of 1mm thickness as observed when lookingtowards the light source. The tensor symmetry (dijk = dikj)was applied artificially to the calculated d-coefficients.Since εij=εji, the resultant dijk was totally symmetric, asexpected for transparent crystals (Kleinmann’s rule).12 All

Figure 2 The shift of the nuclei relative to the electron clouds by anexternal electric field E. The radius of the clouds indicates the size of thepolarizability. The external field induces a small change in the value ofthe electronic polarizability volumes, but this effect is neglected in theDES model. Furthermore, in the hard-spheres approximation, theelectronic clouds of different atoms are not free to move relative to eachother. In a classical picture at optical frequencies, the atom’s nucleus isthe center of the light-wave induced vibration of the shell.

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other tensors were calculated assuming no symmetry atall. Therefore, it was satisfying to discover that theresulting tensors conform to Neumann’s law.

Experimental

Crystal Growth

The tetraphenyl derivatives of carbon, germanium, tin andlead were obtained from Aldrich and crystals were grownby slow evaporation from saturated solutions ofchloroform in a water bath at 30°C. Crystals oftetraphenylsilane grown from the melt were donated byProf. S. Haussühl (Cologne, Germany).X-Ray Difraction Analysis

For internal consistency x-ray crystal structures of all ofthe members of the Ph4X family were redetermined13 witha Nonius Kappa CCD diffractometer. Data were refinedto R=2-5%. All measurements were carried out at roomtemperature. The hydrogen atoms on Ph4Pb were placedusing a riding model. Lloyd and Brock recently presenteda detailed comparative analysis of these crystalstructures.14 The tetraphenyl family crystallizes in thetetragonal space group P421c (D2d) and the molecules siton the improper fourfold axes. In this point group there isonly one independent gyration tensor element, g11 = -g22just as for KH2PO4 and its isomorphs.†

Refractive Index Determination

Refractive indices were determined (Table 1) using amicron scale microscope and the three height methodderived from Snell’s law of refraction (Figure 3). This'old-fashioned' method was more reliable than the prismmethod because Ph4X crystals polish poorly. Thebirefringence measurement was refined using standardmethods of polarized light microscopy confirming

negative optical character throughout the series.15

The experimentally determined linear birefringenceshowed a strong variation that is correlated with thepolarizability of the central atom.

Curiously, Newkirk reported that the refractive indices forall the crystals in the Ph4X family were essentially thesame at 1.58±0.05, and that the crystals were all uniaxialpositive.16 Both statements are incorrect. Our results arereliable within 0.01 of the absolute values and thedifference between largest and smallest refractive index is0.39.

Results of Calculations and DiscussionFor all calculations, one needs the x-ray structurecoordinates as well as the absolute refractive indices asreference. The polarizability volumes αH for hydrogenatoms and αC for carbon atoms were chosen to best matchthe experimental optical character. The values of αX werechosen to best match the experimental average refractiveindices. In Ph4Si and Ph4Ge we could not simultaneouslymatch the absolute value of the refractive indices and theexperimental optical character. We then sacrificed theoptical character, which is less important than the absolutevalues. The resulting estimation of optical properties isshown in Table 2 and the corresponding tensor surfaces inFigure 4.

Table 2 Calculated optical properties at 670nm (d-coefficients at1064nm) and structural details. αC (phenyl) = 0.580Å3, αH = 0.058Å3. β:angle between adjacent phenyl groups (see Figure 6); γ: dihedral anglebetween phenyl rings.

X= C Si Ge Sn Pb

αX(Å3) 0.62 4.50 5.00 8.80 9.50ne 1.425 1.755 1.649 1.824 1.747no 1.455 1.667 1.608 1.833 1.767

ρ11 (°/mm) 252 32 -33 -75 -40d123 (10-12m/V) -0.030 -0.094 -0.094 -2.57 -2.58d312 (10-12m/V) -0.034 -0.89 -0.88 -2.47 -2.39

ε-1r123 (10-12m/V) -0.273 5.8 3.5 9.2 9.2ε-1r231 (10-12m/V) 1.42 7.7 4.7 11.0 9.5ε-1g231 (10-5 º/V) -0.14 -0.92 -0.30 -1.75 -1.87

β (º) 106.8 108.0 108.4 110.9 111.1γ (º) 74.8 80.6 83.6 88.9 87.6

β-γ (º) 32.0 28.4 24.8 22.0 22.5X-C2 (Å) 1.5491 1.8730 1.9517 2.1430 2.2225

a (Å) 10.9050 11.4448 11.6160 12.0680 12.1110c (Å) 7.2850 7.0640 6.9020 6.5570 6.5430

Figure 3 Experimental determination of absolute refractive indices usingthe 3-height method:

where nair is assumed to be one.

Table 1 Experimental refractive indices in Ph4X at 650nm. Reliabilityof absolute refractive indices is ca. ±0.01.

X= C Si Ge Sn Pb

ne 1.446 1.706 1.602 1.757 1.730no 1.476 1.742 1.647 1.840 1.789∆n 0.030 0.036 0.045 0.083 0.050

Figure 4 Representation surfaces in which the radius of the surface isproportional to the size of the effect. Red indicates negative signs.

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Optical Rotation

When employing the theory as outlined above we found avariation of the optical rotation contrary to intuition basedsolely on atomic polarizabilities. The calculated values ofoptical rotation for Ph4C, -Si, -Ge, -Sn and -Pb are 252,32, -33, -75 and -40°/mm, respectively (Figure 5).Unexpectedly, the optical rotation rose sharply on goingfrom the polarizable tin derivative to the least polarizabletetraphenylmethane.

We attempted to confirm the expectation forlarge optical rotations in the Ph4X family with pointsymmetries42m(D2d). Generally, however, measurementsof chiro-optical properties such as optical rotation arethwarted by the much larger birefringence. We thereforehad to resort to special polarimetric techniques.17 In thisway we confirmed that Ph4Sn was levorotatory along theabsolute a direction determined on the basis of anomalousX-ray diffraction according to the coordinates given inAppendix 2. The error remains large because of scatteringfrom the very soft, hard-to-polish surfaces. Smallcrystallites of Ph4C only extinguished with the polarizerand analyzer nearly parallel. This indicates a huge opticalrotation in Ph4C, which is predicted in our calculation.

For optical rotation, we venture a structuralinterpretation. In the propeller shaped Ph4X moleculesone can identify at least two different helical circuits ofclosely bonded atoms when looking toward the a axis(Figure 6). The sense of rotation opposes the helix if themaximum of the anisotropic polarizability volumes pointtowards the helix axis whereas the sense of rotationfollows the helix when the polarizability volumes aretangential.18

It appears that there is a competition between thetwo helices, 1-2-3-4-5 and A-B-C-D. The helix 1-2-3-4-5that defines the general disposition of the phenyl ringsproduces dextrorotation. However, this “aromatic” opticalrotation is counterbalanced by the increasing levorotationof A-B-C-D, which increases with αX. In all compounds,except Ph4C, the polarizability volume of the phenylcarbon coordinated to the central atom (C2) was orientedradially, or towards the helix axis (Figure 7). Thus forPh4C, A-B-C-D cannot counterbalance 1-2-3-4-5.

As shown in Figure 5, the optical rotation reachesa minimum for Ph4Sn and begins to rise again for Ph4Pb.Nevertheless, the X-C2 bond increases throughout theseries reaching a maximum length at Ph4Pb, and therebymitigates the opposing levorotatory influence of helix A-B-C-D. Thus, the optical rotation begins again to rise.

We also observed that the dihedral angle betweenphenyl groups γ (Figure 6) was correlated with opticalrotation. The larger the value of γ, the stronger thepositive contribution of that helix. In fact, the torsionangle C1-C2-C4-C5 (approximated by β-γ) followed thetrend of optical rotation (Table 2, Figure 5).

Induced Optical Properties

The induced effects in each group increase by and largewith increasing polarizability in accordance with the DESmodel, which predicts larger effects for largerpolarizability differences.

It is interesting to note how the calculatedelectro-optic phenomena compare with commonly usedmaterials. The values estimated for electrogyration inPh4Pb and Ph4Sn reach far above the largest value reliablyobserved in a solid crystalline material, 0.1×10-5 o/V(normalized by the dielectric constant) in Pb(NO3)2 along[111].19 The d-coefficients are characterized by the figureof merit (d2n-3). Ph4Sn gives here a value of 1.07(pm/V)2.This is to be compared with KH2PO4,20 0.045(pm/V)2 andthe best known material, β-bariumborate, 56(pm/V)2.Newkirk cites a personal communication with Bölger andJerphagnon concerning the d-coefficients, which are in therange of KH2PO4.16 The calculated values for the electro-optic effect are remarkable. With expected values of therelative dielectric constants for the Ph4X series of 5-15,the calculated electro-optic effect exceeds that in KH2PO4by at least a factor of 10. Newkirk also reported a larger123 of about 17±1pm/V with a relative dielectric constantof about 10; neither the particular compounds measured

Figure 5 Calculated optical rotation plotted against the polarizability ofX in Ph4X.

Figure 7 Calculated anisotropic electronic polarizabilities in Ph4X. Threehydrogen polarizabilities diverged, an unavoidable consequence offitting the experimental refractive indices.

Figure 6 Helical circuits of closely bonded atoms in Ph4X, 1-2-3-4-5(atoms polarized tangentially) and A-B-C-D (atoms polarized radially,except for Ph4C), and angles. Projection is normal to (100).

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nor the variation within the series was specified.16 Yet,our calculated values were 2 to 5 times larger (Table 2).

Alternative Determination of Electronic Polarizability

If the estimated polarizabilities chosen to best matchexperimentally determined indices of refraction haverealistic values, is there a way to determine them based onexperiment? In general, the electronic polarizabilityshould be in proportion to the atomic electron densities.The same is true for the x-ray scattering factors. By usingthe peak heights in the Fourier maps derived for structuresolutions, which represent the electron densities at atoms

we hoped to find a correlation between these peak heightsand the polarizabilities that were found by comparing thecalculated and experimental refractive indices. The peakheights are not absolute but can be normalized to acommon atom, i.e. the phenyl carbon atoms. Theselection of electronic polarizabilities αX, fractions of peakheights, and values obtained by multiplying the fractionswith the polarizability of carbon (αC=0.58Å3) are shownin Table 3.

For previous attempts to calculate electronicpolarizabilities from X-ray structure amplitudes seeIvanov – Smolenskii et al.21

Surprisingly, we found a relatively largepolarizability for aromatic hydrogen atoms derived fromboth estimated refractive indices and the Fourier maps.

Appendix 1

Indices α, β and γ refer to a Cartesian reference systemand R is a parameter chosen so as to ensure convergence.The term corresponding to unit cell index l’=0 with S=S’is ignored in the summation over l’. In the summation

over the face-normal vector h, the term h = (000) shouldbe omitted.1

Appendix 2

AcknowledgementsWe are grateful to Prof. S. Haussühl for donatingtetraphenylsilane crystals and for financial support fromthe Petroleum Research Fund of the American ChemicalSociety (35706-AC06) and from the National ScienceFoundation (CHE-0092617).

References† Supplementary information on the crystal structures has been depositedand can be obtained free of charge from the publisher.1 V. Devarajan and A. M. Glazer, Acta Crystallogr. Sect A., 1986, 42,

560.2 G. Beurskens-Kerssen, J. Kroon, H. J. Endemann, J. van Laar and J.

M. Bijvoet, in Crystallography and Crystal Perfection, ed. G.Ramachandran, Academic Press, London, 1963, p. 225; J. M.Bijvoet, A. F. Peerdeman and A. J. Van Bomel, Nature, 1951, 168,271; M. Born and M. Goeppert-Mayer, in Handbuch der Physik,1933, 24, 623; G. Bruhat and P. Grivet, J. Phys. Radium, 1935, 6,12; J. van Laar, H. J. Endemann and J. M. Bijvoet, Acta Crystallogr.Sect. A., 1968, 24, 52; R. Reijnhart, PhD thesis, Delft, 1970.

3 J. D. Jackson, Classical Electrodynamics, Second Edition, JohnWiley & Sons, New York, 1975, p. 220.

4 P. P. Ewald, Ann. Phys., 1921, 64, 253.5 W. Kaminsky and A. M. Glazer, Z. Kristallogr., 1997, 212, 283.6 W. Kaminsky and A. M. Glazer, Phase Transitions, 1997, 66, 1.7 R. K. Kondru, P. Wipf, and D. N. Beratan, Science, 1998, 282, 2247.8 C. Kittel, Introduction to Solid State Physics, Fifth Edition, John

Wiley & Sons, Inc., New York, 1976, p. 399.9 J. R. Tessman, A. H. Kahn and W. Shokley, Phys. Rev., 1953. 92,

891.10 N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt,

Rinehart and Winston, London, 1976, p. 546.11 L. Bohatý, Z. Kristallogr, 1984, 166, 97.12 D. A. Kleinmann, Phys. Rev., 1962, 128, 133.13 A. Robbins, G. A. Jeffrey, J. P. Chesick, J. Donohue, F. A. Cotton,

B. A. Frenz and C. A. Murillo, Acta Crystallor. Sect. B., 1975, B31,2395; V. Gruhnert, A. Kirfel, G. Will, F. Wallrafen and K. Recker,Z. Kristallogr., 1983, 163, 53; A. Karipides and D. A. Haller, ActaCrystallogr. Sect. B., 1972, B28, 2889; P. C. Chieh and J. Trotter, J.Chem. Soc., 1970, 911; V. Busetti, M. Mammi, A. Signor and A.Del Pra, Inorg. Chim. Acta., 1967, 1, 424.

14 M. A. Lloyd and C. P. Brock, Acta Crystallogr., Sect. B, 1997, B53,780.

15 N. H. Hartshorne and A. Stuart, Practical Optical Crystallography,Edward Arnold Publishers, London, 1964, p. 230.

16 H. W. Newkirk, J. Organomet. Chem., 1972, 44, 269.17 W. Kaminsky and A. M. Glazer, Ferroelectrics, 1996, 183, 133; J.

Kobayashi and Y. Uesu, J. Appl. Crystallogr., 1983, 16, 204.18 A. M. Glazer and K. Stadnicka, J. Appl. Crystallogr., 1986, 19, 108.19 H. J. Weber and S. Haussühl, Z. Kristallogr., 1977, 146, 3030; W.

Kaminsky, A. Fahnenstich and S. Haussuehl, Ann. Phys., 1992, 1,92.

20 J. Jerphagnon and S. K. Kurtz, Phys. Rev., 1970, B1, 1749.21 G. A. Ivanov – Smolenskii, V. G. Tsirelson, and R. P. Ozerov, Acta

Crystallogr., 1983, A39, 411.

Structure Coordinates of Ph4Sn

Atom x/a y/b z/c

Sn 0 0 0C2 -0.0189(2) 0.1451(3) 0.1853(4)C3 0.0539(3) 0.2333(3) 0.1674(5)C4 0.0424(4) 0.3265(3) 0.2876(6)C5 -0.0419(4) 0.3342(3) 0.4257(6)C6 -0.1159(4) 0.2479(3) 0.4454(6)C7 -0.1043(4) 0.1540(4) 0.3264(7)

Table 3 Electron density and electronic polarizability. PX/PC is the ratioof Fourier peak heights of atom X and the aromatic carbon, and αX is thepolarizability of atom X.

X= H C Si Ge Sn Pb

PX/PC 0.12 1.1 4.0 11 16 18αX=αC PX/PC 0.070 0.61 2.3 6.4 9.3 10

α X (Å3, Table 2) 0.060 0.62 4.5 5.0 8.8 9.5

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