Aru P. Hill

Carmelo Sgarlata

Daniel L. Wellman

Kenneth N. Raymond

Comptes Rendus Chimie

C. R. Chimie 13 (2010) 849–852Contents lists available at ScienceDirectComptes Rendus Chimiewww.sciencedi rec t .comFull paper/MémoireCurvature of the lanthanide contraction: An explanationKenneth N. Raymond *, Daniel L. Wellman, Carmelo Sgarlata, Aru P. HillDepartment of Chemistry, University of California, Berkeley and Chemical Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720-1460,United StatesA R T I C L E I N F OArticle history:Received 22 December 2009Accepted after revision 16 March 2010Available online 31 May 2010Keywords:LanthanidesX-ray diffractionStructure elucidationElectrostatic interactionsBond theoryA B S T R A C TA number of studies have shown that for isostructural series of the lanthanides (elementsLa through Lu), a plot of equivalent metal-ligand bond lengths versus atomic numberdiffers significantly from linearity and can be better fit as a quadratic equation. However,for hydrogen type wave functions, it is the inverse of the average distance of the electronfrom the nucleus (an estimate of size) that varies linearly with effective nuclear charge.This generates an apparent quadratic dependence of radius with atomic number. Plottingthe inverse of lanthanide ion radii (the observed distance minus the ligand size) as afunction of effective nuclear charge gives very good linear fits for a variety of lanthanidecomplexes and materials. Parameters obtained from this fit are in excellent agreementwith the calculated Slater shielding constant, k. 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionThe lanthanide contraction, the change in ion size withatomic number, was one of the first structural features ofthe lanthanides and papers referring to it date back to atleast 1926 [1]. Because early diffraction studies were time-consuming, the entire lanthanide series (here defined as Lathrough Lu, excepting the radioactive Pm) was typicallynot examined and the decrease in ionic radius was usuallyshown using representative members of the series [2] andthe contraction was assumed to be linear with atomicnumber [3]. Modern advances in structure analysis haveyielded a number of whole or near-whole series ofstructures for isostructural compounds [4–8], generatingmuch more data for a plot of the lanthanide contractionand showing that the contraction is significantly curved, afeature that we and others in the field have generallyaccepted but not explained.The early X-ray and neutron diffraction characteriza-tion of the series [Ln(H2O)9][CF3SO3]3 was especiallysignificant [6,9], because of its relevance to the solution* Corresponding author.E-mail address: raymond@socrates.berkeley.edu (K.N. Raymond).1631-0748/$ – see front matter  2010 Académie des sciences. Published by Edoi:10.1016/j.crci.2010.03.034chemistry of the lanthanide aqua ions. This series isisostructural and the cations have C3h crystallographic sitesymmetry. Thus, each vertex and each cap H2O isequivalent within its class. In an ionic model, the cappingligands have more ligand–ligand repulsion and so havelonger Ln–O distances. This result is mirrored in moderncalculations [10]. The authors of the [Ln(H2O)9][CF3SO3]3papers struggled to explain both the curvature in the Ln–Odistances across the series as well as the differentcurvature seen for the two different water molecules [6].Subsequently, Quadrelli examined this and the isostruc-tural series [Ln(H2O)9][EtOSO3]3 [8], and found both to bemore closely fit by a quadratic expression [11].More recently, Ibers et al. determined the structure of aseries of compounds with the general formula RbLnSe2[4]. The lanthanide ions lie in octahedral coordinationenvironments, and each coordinating selenium atom isidentical. The Ln–Se internuclear distances of the com-plexes were found to be more closely fit by a quadraticexpression, as were the Ln–Te distances in the seriesCsLnZnTe3 [5].We earlier presented an explanation of the apparentquadratic nature of the contraction based on the nuclearshielding of the f orbitals [12] and Slater’s model for atomicshielding [13–15]. That paper described a structural serieslsevier Masson SAS. All rights reserved.K.N. Raymond et al. / C. R. Chimie 13 (2010) 849–852850for a multidentate ligand. Here we expand upon that modeland show its effective explanation of Ln3+ ionic radii for anumber of published series of isostructural complexes.2. Modeling and plotting ionic radius from crystal-structure dataFor hydrogen atom wave functions, it is not the averagedistance of the electron from the nucleus, <r>, that varieslinearly with effective nuclear charge but rather theaverage inverse, <1/r>. This suggests that 1/r(x), theinverse of the lanthanide ionic radius (where x is thenumber of f electrons in the lanthanide ion of interest,treated as a continuous variable), can be represented by alinear equation:1rðxÞ ¼ aþ bx: (1)As will be shown, this is mathematically equivalent tothe Slater shielding formula.Here the intercepta ¼ 1r0(2)is the inverse of the ionic radius of lanthanum and theslopeb ¼ @ð1=rðxÞÞ@x(3)is the incremental change in the inverse of the ionicradius from one lanthanide to the next. Thus, theinternuclear distance between the lanthanide and thecoordinating atom of the ligand, d(Ln-L), is given bydðLn LÞ ¼ 1aþ bxþ rðLÞ (4)where r(L) is the radius of the coordinating ligand.Eq. (4) can be expanded through a Maclaurin series [16]to give an equation of the formf ðxÞ ¼ f ð0Þ þ f 0ð0Þxþ f 00ð0Þx2=2þ ::: (5)Table 1Internuclear distances (Å) in lanthanide complexes with various ligands.Lanthanide Number of f electrons [Ln(H2O)9][CF3SO3]3Vertex CappingLa 0 2.513 2.611Ce 1 2.489 2.594Pr 2 2.469 2.579Nd 3 2.451 2.571Sm 5 2.418 2.548Eu 6 2.408 2.536Gd 7 2.395 2.536Tb 8 2.378 2.527Dy 9 2.363 2.520Ho 10 – –Er 11 – –Tm 12 – –Yb 13 2.302 2.532aLu 14 2.287 2.519aa Values omitted from the refinement due to steric crowding.which, if terminated at the third term, gives thepolynomial described by QuadrellidðLn LÞ ¼ A0  A1xþ A2x2 (6)whereA0 ¼ 1=aþ rðLÞA1 ¼ b=a2A2 ¼ b2=a3Equations (1) to (6) suggest that a plot of 1/r(x) as afunction of the number of f electrons will be linear.The ionic packing model, with derived ionic radii, waspioneered by Goldschmidt [17] and Pauling [18], culmi-nating in tabulations such as the widely used values ofShannon [19]. The relevant Ln–L internuclear distances areshown in Table 1. By subtracting the known ionic radii ofthe ligating atoms from the Ln–L internuclear distances insimple structures, the ionic radii of the Ln ions can bedetermined for a series of experimentally determineddistances.A least-squares fit of the crystallographic data to obtainthe parameters a, b, and r(L) of Eq. (4) for each ligand wasperformed by minimizing:Xni¼1ðdiðxÞobs  diðxÞcalcÞ2(7)where di(x)obs is the observed internuclear distance anddi(x)calc is the corresponding predicted values d(Ln-L) forthe ith crystallographic data point. In order to account fordifferences in lanthanide ionic radius that were due to thedifferent coordination environments, Eq. (8) [20] was usedto normalize the values of both a and b for coordinationnumber 6 and 9:irCNðqÞirCNðpÞ¼ CNðqÞCNð pÞ 1=10(8)where irCN(q) and irCN(p) are the ionic radii of the lanthanideions having coordination number (CN) q and p, respective-ly. Values for the radius of the vertex water oxygen atomsof [Ln(H2O)9][CF3SO3]3 and [Ln(H2O)9][EtOSO3]3, theCsLnZnTe3 RbLnSe2 [Ln(H2O)9][EtOSO3]3Axial Equatorial Vertex Vertex Capping3.204 3.174 3.030 2.517 2.616– – 2.999 2.491 2.6003.162 3.139 2.978 2.470 2.5833.147 3.125 2.959 2.457 2.5703.119 3.103 2.935 2.430 2.55– – – 2.415 2.5423.103 3.086 2.914 2.401 2.5363.087 3.078 2.893 2.382 2.5273.073 3.067 – 2.371 2.5173.065 3.058 2.874 2.362 2.5113.053 3.046 2.85 2.352 2.5033.041 3.043 – 2.340 2.504a3.033 3.042 – 2.324 2.503a– – 2.837 2.318 2.497aFig. 1. Plot of the inverse of the ionic radius vs. the number of f electronsfor the lanthanide complex RbLnSe2.Fig. 3. Plot of the inverse of the ionic radius vs. the number of f electronsfor the lanthanide complex [Ln(H2O)9][CF3SO3]3.K.N. Raymond et al. / C. R. Chimie 13 (2010) 849–852 851selenium atoms of Rb[Ln]Se2, and equatorial telluriumatoms of Cs[Ln]ZnTe3 were fixed according to literaturevalues [19] in order to decrease the number of parametersto be refined. With the refined values of r(L) (supportinginformation), the inverse of the ionic radius is graphed as afunction of the number of f electrons for the four chosenseries.The series RbLnSe2 (Ln = La, Ce, Pr, Nd, Sm, Gd, Tb, Ho, Er,Lu) [4] crystallizes with lanthanide ion site symmetry D3d;the lanthanide ions are nearly octahedral and eachcoordinating selenium atom is identical. A coordinationnumber of 3 for the selenium atoms was used to fix theionic radii of the selenium atoms to previously publishedvalues [19] for the purposes of the data refinement. A plotof 1/r(x) as a function of the number of f electrons is shownin Fig. 1.The series CsLnZnTe3 (Ln = La, Pr, Nd, Sm, Gd, Tb, Dy, Ho,Er, Tm, Yb) [5] crystallizes in the layered KZrCuS3 structuretype in the orthorhombic space group Cmcm. The structureis composed of LnTe6 octahedra, with axial and equatorialtellurium atoms in the unit cell. The equatorial atom ionicradii were fixed to the three-coordinate literature value[19], and the axial tellurium ionic radii, due to theirdifferent coordination environment, were refined. Fig. 2shows our results for these complexes.In the lanthanum nonaaqua triflate and ethyl sulfatesalts ([Ln(H2O)9][CF3SO3]3, Ln = La, Ce, Pr, Nd, Sm, Eu, Gd,Tb, Dy, Yb, Lu and[Ln(H2O)9][EtOSO3]3, Ln = La, Ce, Pr, Nd,Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu) [6,8] the cations areFig. 2. Plot of the inverse of the ionic radius vs. the number of f electronsfor the lanthanide complex CsLnZnTe3.also tricapped-trigonal prismatic in geometry. The vertexionic radii were fixed to a literature value for water [21]and the capping radii were refined, constrained to a singlevalue (the values for Tm, Yb, and Lu are anomalous, whichwe attribute to steric crowding, and were omitted). Thelinear plots are shown in Figs. 3 and 4.3. Calculating the screening constant for 4f electronsUsing Slater’s revised model [14], the ionic radius of acomplexed lanthanide ion can be written in the form:rðxÞ ¼ r0Z0Z0 þ xð1 kÞ(9)where r(x) is the ionic radius of the lanthanide, Z0* is itseffective nuclear charge, r0 is its ionic radius, x is thenumber of f electrons and k is the atomic screeningconstant [12]. Therefore:1rðxÞ ¼Z0 þ xð1 kÞr0Z0(10)and, after taking the first derivative of 1/r(x) with respect tox,@ð1=rðxÞÞ@x¼ 1 kr0Z0(11)Fig. 4. Plot of the inverse of the ionic radius vs. the number of f electronsfor the lanthanide complex [Ln(H2O)9][EtOSO3]3.K.N. Raymond et al. / C. R. Chimie 13 (2010) 849–852852we can combine Eq. (12) with Eq. (3) to getk ¼ 1 br0Z0 (12)Substituting Eq. (2) gives:k ¼ 1 baZ0 (13)that is, the apparent curvature of the ionic radii can be usedto calculate the Slater screening constant. Calculating thescreening constant using our refined values for a, b and anextrapolated value for the effective nuclear charge oflanthanum, Z*0 in Eq. (13), results in a value for k of 0.74, ingood agreement with the previously reported value of 0.69obtained from the ionization energies of the lanthanides[22].4. ConclusionsThe lanthanide contraction is a topic of considerablehistory and practical as well as theoretical significance. Theincreasing number of structures and the more completesets of data for the entire series of lanthanides in a givenstructural type show that the change of lanthanide toligand bond distance does not change linearly with atomicnumber. The apparent curvature of the ionic radii withatomic number is because it is 1/r that varies linearly withcharge. This simple model gives good fits to the data and auseful value of the Slater screening constant.AcknowledgementsThis article is dedicated to Professor Buenzli on theoccasion of his retirement from the EPFL. Research wassupported by the Director, Office of Science, Office of BasicEnergy Sciences, and the Division of Chemical Sciences,Geosciences, and Biosciences of the U.S. Department ofEnergy at LBNL under Contract No. DE-AC02-05CH11231.We thank Dr. Michael Seitz for helpful discussions.Appendix A. Supplementary dataSupplementary data associated with this article can befound, in the online version, at doi:10.1016/j.crci.2010.03.034.References[1] T. Barth, G. Lunde, Z. Phys. Chem. 121 (1926) 78.[2] K.S. Krasnov, N.I. Giricheva, G.V. Girichev, J. Struct. Chem. 17 (1976)575.[3] D.A. Vander Griend, S. Malo, T.K. Wang, K.R. Poeppelmeier, J. Am. Chem.Soc. 122 (2000) 7308.[4] B. Deng, D.E. Ellis, J.A. Ibers, Inorg. Chem. 41 (2002) 5716.[5] J.Y. Yao, B. Deng, L.J. Sherry, A.D. McFarland, D.E. Ellis, R.P. Van Duyne,J.A. Ibers, Inorg. Chem. 43 (2004) 7735.[6] A. Chatterjee, E.N. Maslen, K.J. Watson, Acta Cryst. B44 (1988) 381.[7] N. Matsumura, T. Takeuchi, A. Ouchi, Bull. Chem. Soc. Jpn. 63 (1990)620.[8] R.E. Gerkin, W.J. Reppart, Acta Cryst. C40 (1984) 781.[9] J.M. Harrowfield, D.L. Kepert, J.M. Patrick, A.H. White, Aust. J. Chem. 36(1983) 483.[10] E. Furet, K. Costuas, P. Rabiller, O. Maury, J. Am. Chem. Soc. 130 (2008)2180.[11] E.A. Quadrelli, Inorg. Chem. 41 (2002) 167.[12] M. Seitz, A.G. Oliver, K.N. Raymond, J. Am. Chem. Soc. 129 (2007) 11153.[13] E. Clementi, D.L. Raimondi, J. Chem. Phys. 38 (1963) 2686.[14] E. Clementi, D.L. Raimondi, W.P. Reinhardt, J. Chem. Phys. 47 (1967)1300.[15] J.C. Slater, Phys. Rev. 36 (1930) 57.[16] W.H. Beyer (Ed.), CRC Standard Mathematical Tables, CRC Press, BocaRaton, FL, 1987, p. 299.[17] V.M. Goldschmidt, M. Barth, T. Lunde, W.H. Zachariasen, Skr. NorskeVidensk. Akad. MN 1 (1926).[18] L. Pauling, J. Am. Chem. Soc. 49 (1927) 765.[19] R.D. Shannon, Acta Cryst. A32 (1976) 751.[20] L. Pauling, The Nature of the Chemical Bond and the Structure ofMolecules and Crystals: An Introduction to Modern StructuralChemistry, 3rd Ed., Cornell University, Ithaca, NY, 1960.[21] R. Heyrovska, Chem. Phys. Lett. 429 (2006) 600.[22] K.A. Waldron, E.M. Fehringer, A.E. Streeb, J.E. Trosky, J.J. Pearson, J.Chem. Ed. 78 (2001) 635.

Curvature of the lanthanide contraction: An explanation

https://comptes-rendus.academie-sciences.fr/chimie/item/10.1016/j.crci.2010.03.034.pdf

Curvature of the lanthanide contraction: An explanation

Abstract

Similar works

Full text

Available Versions

Comptes Rendus Chimie