ElectronLocalizationandDelocalization inOpen-ShellMolecules XAVIERFRADERA,MIQUELSOLA` InstituteofComputationalChemistryandDepartmentofChemistry,UniversityofGirona, 17071Girona,Catalonia Received11February2002;Accepted16May2002 PublishedonlineXXXXXX2002inWileyInterScience(www.interscience.wiley.com).DOI10.1002/jcc.10141 Abstract:LocalizationanddelocalizationindicesderivedintheframeworkofthequantumAtomsinMolecules theoryhaverecentlybeenusedtoanalyzetheelectron-pairstructureofclosed-shellmolecules.Herewereport calculationsoflocalizationanddelocalizationindicesforopen-shellmoleculesattheHartree-Fock(HF)level.Several simpledoubletandtripletradicalmoleculesarestudied.Ingeneral,interatomicdelocalizationbetweenbondedatoms isheavilydependentontheorderandpolarityofthebond.Unpairedelectronsalsohaveasignificanteffectonthe interatomicdelocalizationindices.Indeed,formanyradicals,theanalysisofthespincomponentsrevealsthat theinteratomicdelocalizationisverydifferentfor ␣ and ␤ spin electrons in many cases. In general, at the HF level, the resultscanberationalizedintermsoforbitalcontributions.However,thedefinitionoflocalizationanddelocalization indicesiscompletelygeneral,andtheycouldbecalculatedatanyleveloftheory,providedthattheone-andtwo-electron densitiesareavailable. ©2002WileyPeriodicals,Inc. JComputChem23:1347–1356,2002 Keywords:electronlocalization;electrondelocalization;open-shell;topologicalanalysis;electron-pairdensity;non- nuclearattractor;AtomsinMoleculestheory Introduction electron-pairdensityusedinthesecalculationsisobtainedby usingtheHFapproximationwiththeKohn-Shamorbitals.Thus, TheAtomsinMoleculesTheory(AIM)1 definestheatomsina theDFTresultsreportedinrefs.14–20considerelectroncorrela- moleculefromthetopologicalanalysisofthemolecularelectron tionbeyondtheHFlevelonlyindirectly.21 Localizationandde- density, ␳(r). Briefly, an atomic basin is a region in real space localizationindiceshavealsobeenappliedsuccessfullytothe boundedbyzero-fluxsurfacesintheelectrondensityorbyinfinity. analysisofseveralreactions4 andtothedescriptionofsolvent Propertiesofanatom,suchastheelectronpopulation,N(A),can effectsinseveralmoleculesandintheMenshutkinreaction.5 becalculatedbyintegrationthroughthecorrespondingbasin.By Sofar,calculationsof ␭(A) and ␦(A,B) or related quantities for ⌫ open-shellmoleculeshavebeenveryscarce.MacDougalland usingtheelectron-pairdensity, (r1,r2),onecandefinealocaliza- 22 tionindex, ␭(A), which corresponds to the number of electrons Bader havecomputedatomicpropertiesofcarbenesandsil- localizedinanatomA,andadelocalizationindex, ␦(A,B), which lylenesintripletstates,includingpercentagesoflocalization[100 ␭ 23 isthenumberofelectronsdelocalizedbetweentwoatoms,Aand N(A)/ (A)] for selected atoms. Recently, Chesnut and Bartolotti B.2 Theconceptsoflocalizedanddelocalizedelectronscanbe havecalculateddelocalizationindicesforthetripletstateofa relatedtotheconceptsofsharedandnonsharedelectrons,respec- chargeresonancecomplex.Theaimofthepresentcontributionis tively,intheLewismodel.3 However,theanalysisintermsof ␭(A) ␦ ␳ ⌫ 2 and (A,B) is based in the physical observables (r) and (r1,r2). Correspondenceto:X.Fradera;e-mail:[email protected] Recently,localizationanddelocalizationindiceshavebeen Contract/grantsponsor:SpanishDGES;contract/grantnumber:PB98- usedextensivelytoanalyzeanumberofclosed-shellmoleculesat 0457-C02-01 theHartree-Fock(HF)andconfigurationinteraction(CI)levelsof Contract/grantsponsor:Departamentd’Universitats,RecercaiSocietat 2,4–13 theory. Ingeneral,theHFmethodexaggeratestheinter- delaInformacio´oftheGeneralitatdeCatalunya(M.S.) atomicdelocalizationbetweencovalentlybondedatoms,withre- ThisarticleincludesSupplementaryMaterialavailablefromtheauthors specttoCI.Localizationanddelocalizationindiceshavebeen uponrequestorviatheInternetatftp://ftp.wiley.com/public/journals/jcc/ calculatedalsofromKohn-Shamorbitalsobtainedintheframe- suppmat/23/1347orhttp://www.interscience.wiley.com/jpages/ workofthedensityfunctionaltheory(DFT).14–20 However,the 0192-8651/suppmat/v23.1347.html ©2002WileyPeriodicals,Inc. 1348 Fradera and Sola` • Vol. 23, No. 14 • Journal of Computational Chemistry to calculate localization and delocalization indices for a number of theory. Then, localization and delocalization indices were calcu- representative radical molecules in doublet and triplet spin states, lated using eqs. (3) and (4). Furthermore, the ␣ and ␤ spin and to analyze in detail the contributions of the ␣ and ␤ spin contributions were also calculated, as well as the contributions of electrons to ␭(A) and ␦(A,B). different symmetry groups. The precision of all the calculations was ensured by checking that both the sum of the atomic popula- tions and the sum of all the localization and delocalization indices Methods yielded the number of electrons in each molecule with good accuracy. Errors in the summations are less than 0.002 for all the Restricted open shell HF (ROHF) and unrestricted HF (UHF) molecules. au are used throughout the article. wave functions were calculated for a number of diatomic and triatomic neutral radical molecules using the GAUSSIAN98 pack- age.24 The spin multiplicity considered was 2 (doublet) for BeH, Results and Discussion BO, CH, CN, NO2, CNO, OH, and C2H, and 3 (triplet) for CH2, ϩϩ 25 HNC, NF, NH, O2, and PH. The 6-311 G(2d,2p) basis set Table 1 shows the ROHF results for all the molecules, including was used for all the calculations, and the molecular geometries the contributions of the ␣ and ␤ spin components The contribu- were completely optimized in all cases. The topological analysis of tions of the orbitals in each symmetry group are also detailed in the electron densities and the atomic integrations were performed Table 1. All the linear molecules are considered with Cϱv sym- using the AIMPAC package.26 In the CH, OH, and OCN systems, metry, and their orbitals are divided into ␴ and ␲ sets, with the ␲ ␲ ␲ ␲ the electron density loses the linear symmetry because the x and set including the x and y orbitals. The nonlinear molecules ␲ y orbitals have different occupations at the ROHF and UHF exhibit C2v or Cs symmetry; however, only the Cs subgroup can be levels. These symmetry problems could be solved by means of used for the orbital analysis (the partition of the localization and CASSCF calculations. However, the breaking of symmetry does delocalization indices into orbital contributions requires that all the not have a major influence on the electron-pairing patterns of these overlaps between orbitals belonging to different groups are zero two molecules, and we have preferred to keep the ROHF and UHF within each atomic basin2). Thus, orbitals in all the nonlinear levels of theory throughout. molecules are separated into those that are symmetric (aЈ) and At any level of theory, localization and delocalization indices antisymmetric (aЉ) with respect to the molecular plane. Similar can be defined in terms of the one- and two-electron densities: results are obtained using the UHF method, except for molecules that suffer from heavy spin contamination at the UHF level (CN, HNC, and C2H). Thus, results at the UHF level are reported only ␭͑A͒ ϭ Ϫ͵ ͑2⌫͑r , r ͒ Ϫ ␳͑r ͒␳͑r ͒͒dr dr (1) 1 2 1 2 1 2 for these three molecules. UHF results for the rest of molecules are A included as supplementary material. Hydrides ␦͑A, B͒ ϭ Ϫ2 ͵ ͵ ͑2⌫͑r , r ͒ Ϫ ␳͑r ͒␳͑r ͒͒dr dr (2) 1 2 1 2 1 2 A B The series of hydride molecules BeH, CH, NH, OH, and PH can be used to show how the polarity of the XOH bond is reflected in the respectively. The integrations in the equations above were carried atomic populations and localization and delocalization indices ϭ out through one or two atomic basins, as indicated. At the HF level (X Be, C, N, O, P). For instance, interatomic charge transfer is ␦ of theory, these equations reduce to the following:2 very low for the CH radical, and the (C,H) value is about 1.0. Thus, the COH bond in this molecule can be considered as apolar covalent. Similar considerations apply to CH . In turn, the XOH ␭͑A͒ ϭ ͸ S ͑A͒2 (3) 2 ij bonds in the rest of hydride molecules appear to be polarized to some extent. For the BeH and PH radicals, there is a transfer of ␦͑A, B͒ ϭ 2 ͸ S ͑A͒S ͑B͒ (4) ij ij charge from the heavy atom to the H, whereas the NH and OH molecules exhibit positive charges on the H atom. The NOH bond where the summations run over all the pairs of molecular spin- in NH appears to be only slightly polar, with a charge of ϩ0.33 on ␦ orbitals (MSOs) of the molecule, and {Sij(A)} are overlaps of pairs the H atom and a (N,H) value close to 1 (0.94). In contrast, BeH of MSOs within the basin of atom A. Eqs. (3) and (4) are appli- and OH exhibit a considerable degree of charge transfer and cable to restricted HF (RHF), ROHF, and UHF wave functions. relatively low interatomic delocalization. Finally, the PH radical, Taking into account that overlaps between ␣ and ␤ MSOs are zero, although having a charge of Ϫ0.58 on H, exhibits a ␦(P,H) value the separation of ␣ and ␤ contributions to the ␭(A) and ␦(A,B) of 1.13. The fact that ␦(P,H) is larger than 1 means that the indices is straightforward. Note that the ␣ and ␤ components are no-bonding valence electrons in the P atom are somewhat delo- exactly equivalent at the RHF level, but not at the ROHF and UHF calized to the H atom (vide infra). levels. Furthermore, for linear or planar molecules, the contribu- The comparison of the ␣ and ␤ components of the atomic tions of groups of orbitals belonging to different symmetry groups populations and localization and delocalization indices (see