DOI:10.1002/chem.201602949 Full Paper & Elemental Radii Atomic and Ionic Radii of Elements 1–96 MartinRahm,*[a] Roald Hoffmann,*[a] and N. W. Ashcroft[b] Abstract: Atomic and cationic radii have been calculated for tive measureofthe sizes of non-interacting atoms, common- the first 96 elements, together with selected anionicradii. ly invoked in the rationalization of chemicalbonding, struc- The metric adopted is the average distance from the nucleus ture, and different properties. Remarkably,the atomic radii where the electron density falls to 0.001 electrons per bohr3, as defined in this way correlate well with van der Waals radii following earlier work by Boyd. Our radii are derived using derived from crystal structures. Arationalizationfor trends relativistic all-electron density functional theory calculations, and exceptionsinthose correlations is provided. close to the basis set limit. They offer asystematic quantita- Introduction cule,[2] but we prefer to follow through with aconsistent pic- ture, one of gauging the density in the atomic groundstate. What is the size of an atom or an ion?This question has been The attractivenessofdefining radii from the electron density anatural one to ask over the centurythat we have had good is that a) the electron density is, at least in principle, an experi- experimental metricinformation on atoms in every form of mental observable,and b) it is the electron density at the out- matter,and (more recently) reliable theory for thesesame ermost regionsofasystem that determines Pauli/exchange/ atoms. And the momentone asks this question one knows same-spinrepulsions, or attractive bondinginteractions, with that there is no unique answer.Anatom or ion coursing down achemical surrounding. We are also interested in aconsistent amolecular beam is different from the “same” atom or ion in density-dependent set of radii, for the systematic introduction amolecule, or amolecular crystal,oranionic salt, or ametal. of pressure on small systemsbythe XP-PCMmethod,[3] which So long as we are aware of this inherent ambiguity in the we will report on in the nearfuture. Our aim there is to pro- concept of “size” of an atom, we can proceed. In doing so we vide aconsistentaccount of electronegativity undercompres- use amethod (and associate numerical techniques) as de- sion. scribed in the summary at the end of the article. In this work we present atomic radii calculatedfor elements 1–96. The radii History we compute are clearly specified in terms of the electronden- sity.Wedefine (as bestasour calculations allow,and quite ar- As we implied, the size of individual atoms or ions is along- bitrarily,even as we offer arationale) aradius as that average standingissue, with many approaches to their estimation. distance from the nucleus wherethe electron density falls to EugenSchwarz reminded us that Loschmidtwas the first to 3 3 [4] 0.001 electrons per bohr (0.00675 eÀ ). The criterion is not calculate the diameter of amolecule, Experimental estimates originalwith us;the presented data is arevisiting of earlier of atomic radii beganwith Lothar Meyer’s periodic curve of work by Boyd,[1] as will be detailed below. atomic volumes,[5] followed by early X-ray structurework by The definition and estimate of radii whichwewill use here Bragg[6] and Pauling,[7] the atomicvolumes derived by Biltz,[8] focusesonthe electron density of isolated atoms andions. We the Wigner–Seitz radius, the Bondi radii (a refinement of Paul- are wellaware that the orbitalconfiguration of afree atom ing’s values),[9] the extensive compilation of Batsanov,[10] empiri- may not reflect well the configurationittakes up in amole- cally function-fitted radii,[11] and more recently,van derWaals radii obtained from aremarkable statistical analysisofthe Cambridge StructuralDatabase (CSD) by Alvarez.[12] Islam and [a] Dr.M.Rahm, Prof. Dr.R.Hoffmann Ghoshhave also calculated radii based on spectroscopicevalu- Department of Chemistry and Chemical Biology [13] CornellUniversity,Ithaca, New York, 14853 (USA) ation of ionization potentials. E-mail:[email protected] Theoreticalestimates of atomicradii began, to the best of [email protected] our knowledge,with Slater,[14] who, followedby many others, [b] Prof. Dr.N.W.Ashcroft devised sets of radii calculated from the maximum radial densi- LaboratoryofAtomic and Solid State Physics ty of the outermost single-particle wavefunction, or from the CornellUniversity,Ithaca, New York, 14853 (USA) radialexpectation values of individual orbitals.[15–17] Other ap- Supporting information and the ORCID identification number(s) for the au- thor(s) of this article can be found under http://dx.doi.org/10.1002/ proachesuse radii defined using potentialminima(orbital chem.201602949. nodes),[18] radii calculated from the gradients of the Thomas– Chem. Eur.J.2016, 22,14625 –14632 14625 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Full Paper Fermi kinetic energy functional andthe exchange correlation energy,[19] and aDFT–electronegativity-based formulation.[20] Calculations invoking asimulated noblegas probe have been used to refine the Bondi radii of main-group elements,[21] and to calculate atomic and ionic radii for elements 1–18.[22] Calcu- lations of the electronic second moment,[23] and periodic trends[11] are other ways for the estimation of the size of atoms and molecules. It maybenoted that our estimates of radiiare very different from average values of powers of r (such as those tabulated for atoms in the Desclaux tables),[16] or from maxima in the radial density distribution. The latter are understandably small- er than those defined by adensity cutoff. Each indicator has good reasons for its use, and the various radii complement each other.The ones we discussclearly address the (in away, primitive) questionofthe size of an atom or ion. In general there are multiple uses for radii in chemistry and physics.Van der Waalssurfaces (based on radii)are commonly invoked,together with weak intra- and supra-molecular inter- actions,for explaining crystallinepacking and structure.[24,25] Vander Waals surfaces and radii can also be useful descriptors when rationalizing materialproperties, such as melting points,[26] porosity,[27] electrical conductivity,[28] and catalysis.[29] We also have covalent and ionic radii, sometimes differentiated by their coordinationnumber.[30,31] Figure 1. Radial decayofthe natural logarithm of the electrondensity in se- lectedatoms and singly charged ions. The dashed horizontal line marks 3 adensity of 0.001 e bohrÀ Adensity metric for atomic radii So many radii. Clearlyradii for atoms, however defined, are The radii useful.And just as clearly,all such definitions are human con- structs, as noted above. We present aset of computed radii for Our calculated radii for the ground state atoms (Z=1–96) are isolated atoms based on adensity metric. Because of steady shownintwo ways. In Figure 2they are displayed on the peri- advancement in electronic structure theory over the last odic table, and in Figure 3asaplot versus atomic number. 39 years, our radii are markedly different from those originally Figure 3also shows the cation radii, which we will discuss in calculated by Boyd in 1977.[1] Afurtheradvantage of this set of due course. radii is that they are available for 96 elements;wealso comple- If we are concerned with quantifying the sizes of non-inter- ment the radii for neutralatoms with cationic radii and aselec- acting atoms, one of few relevant experimental comparisons tion of anionic radii. availableiswith the noble-gas elements. Figure 4shows how Bader et al. were first to propose an electron density cutoff our radii agree with Alvarez’s experimental radii obtained from 3 of 0.002 ebohrÀ as the van der Waals boundary of mole- astatistical analysisof1925 noble-gas containing structures re- [32] 3 cules, Alimiting electron density value of 0.001 ebohrÀ was portedinthe Cambridge Structural (CSD), Inorganic Crystal later motivated by Boyd, who showedthat the relative radii of Structure (ICSD)and the molecular gas phase documentation differentatoms are nearly invariant with further reduction of (MOGADOC) databases.[37] The experimental van der Waals the cutoffdensity.[1] Figure 1shows the motivation clearlyin radii for the noble gaselements are afurther refinement of Al- another way—it illustrates the computed density (actually its varez’s original set.[12] natural logarithm) for some strictly spherical atoms as afunc- The dashed line in Figure 4, of slope 1and 0intercept is tion of distance. There are variations to be avoided(see the Li aprettygood fit.[38] Thus the density cutoff criterion choice 3 curve) until the density settles down to its expected exponen- (0.001 ebohrÀ )isareasonable one for asituation of little inter- tial falloff. The Boyd density cutoff has been used as apractical action. Wemight have expected deviations from this line as outer boundary for quantum chemical topology domains.[33, 34] the atomic numbersofthe noble gas elements increase(and Other limiting values have been proposed,[35] and variable iso- the strength of dispersioninteractions grow),but only asmall density surfaces have been used for the construction of solva- deviation—in the expected direction—is seen for Ar,Kr, Xe tion spheres in implicitsolvation models.[36] and Rn. The radii we present are based on free atom densities. In contrast, most experimentally motivated radii in the literature are based on atoms in amolecular or extended structure, Chem. Eur.J.2016, 22,14625 –14632 www.chemeurj.org 14626 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Full Paper Figure 2. Calculated radiiofelements1–96. The ground state configurations of the atoms are shown in small print. Figure 3. Atomic and mono-cationic radii versus element number. often in acrystal. Broughtuptoanother atom, even one as in- be explored. Thus, in achemical environment, for most other nocent of interacting as He or Ne, our reference atom will ex- situations aside from the noblegas elements, we expect small- perience attractive interactions, at the very least those due to er interatomic distances than those shown in Figure 2, due dispersion.