CHM 501 Lecture 13 Ligand Field Theory

Occupation of the t2g orbitals gives a little extra stabilization of the complex. Ligand Field Stabilization Energy (LFSE). d electron configurationOh Field ... LigandFieldTheory CrystalFieldTheory Anionicapproachtounderstandingbondingintransitionmetalcomplexes Assumptions Metalionsaretreatedaspointcharges Ligandsaretreatedaspointchargesorpointdipoles dorbitalsonthemetalareconsideredbutligandorbitalsareignored Theelectrostaticsofthemetalandligandscreatesanelectricfield,denotedV,thatistreatedasaperturbationintheHamiltonian describingtheenergiesofthedelectrons. Inthecaseofagasphase,octahedralcomplexV=Vsphere+Voct Vsphereraisestheenergiesofallthedorbitalsthesameamount,essentiallyresettingthe"zero"ofenergy Themathismessy,butthesolutionoftheperturbationequationgivestwofactorsthatarethesameforallofthedelectrons: D=35Ze2/4(4πεo)a4 q=2/105 sothatDq=[1/6][Ze2/4πεoa5] whereZisthechargeonthemetalion(+1,+2,etc) eisthechargeontheelectron=1.602×10–19C risthedistanceofthedelectronfromthemetalnucleus;indicatestheaverageofthedistancetothefourthpower aisthedistancebetweenthemetalandtheligand εoisthepermittivityoffreespace;4πεo=1.113×10–10 J–1C2m–1 Solvingforeachindividualdorbitalgives: E(dz2)=E(x2–y2)=+6Dq E(dxy)=E(xz)=E(yz)=–4Dq Graphically: Δo=10Dq=CrystalFieldSplitting Orderofmagnitudeof10Dq: ForZ=+2,r=1Å,anda=2Å,10Dq~145kJ/mol,i.e.thisisontheorderofweakcovalentbondenergies Occupationofthet2gorbitalsgivesalittleextrastabilizationofthecomplex LigandFieldStabilizationEnergy(LFSE) delectronconfigurationOhFieldconfiguration LFSEunpairedspins  d1t2g1 4Dq1  d2t2g2 8Dq2  d3t2g3 12Dq3  d4t2g4 16Dq–P2(lowspin)  d4t2g3eg1 6Dq4(highspin)  d5t2g5 20Dq–2P1(ls)  d5t2g3eg2 0Dq5(hs)  d6t2g6 24Dq–2P0(ls)  d6t2g4eg2 4Dq4(hs)  d7t2g6eg1 18Dq–P1(ls)  d7t2g5eg2 8Dq3(hs)  d8t2g6eg2 12Dq2  d9t2g6eg3 6Dq1  d10t2g6eg4 0Dq0  P=spinpairingenergy:thisenergyisnotincludedforanyrequiredspinpairing hs=highspin ls=lowspin UsingGroupTheory Thenumberofenergylevelsanddegeneraciesforanygivengeometrycanbereadilyobtainedusinggrouptheory. Thetransformationpropertiesofthedorbitalsgivestheirreduciblerepresentations,whichwilleachbeofadifferentenergy. InthecaseofanOhcomplex,thedorbitalstransformast2g+eg,whichareeasilyidentifiedbylooking atthequadraticbasisfunctionsonthefarrightofthecharactertable. OhCharacterTable Oh E 8C3 6C2 6C4 3C2(=C42) i 6S4 8S6 3σh 3σd     a1g 1 1 1 1 1 1 1 1 1 1   x2+y2+z2 a2g 1 1 –1 –1 1 1 –1 1 1 –1     eg 2 –1 0 0 2 2 0 –1 2 0   (2z2–x2–y2,x2–y2) t1g 3 0 –1 1 –1 3 1 0 –1 –1 (Rx,Ry,Rz)   t2g 3 0 1 –1 –1 3 –1 0 –1 1   (xy,yz,xz) a1u 1 1 1 1 1 1– –1 –1 –1 –1     a2u 1 1 –1 –1 1 –1 1 –1 –1 1     eu 2 –1 0 0 2 –2 0 1 –2 0     t1u 3 0 –1 1 –1 –3 –1 0 1 1 (x,y,z)   t2u 3 0 1 –1 –1 –3 1 0 1 –1     Thismakesiteasytoidentifyorbitaldegeneracyforanygeometry.Grouptheorydoesnotindicatetheenergyorder. NonoctahedralComplexes Labelingtheorbitals:whenthesymmetrydropsbelowOhlabelingthedorbitalsast2gandeg isnolongerappropriateorcorrect.Identifyingthecorrectpointgroupandthenusingthecorrespondingcharactertable quicklygivesthecorrectirreduciblerepresentationstolabeltheorbitals. AsecondwaytodothisistouseaCorrelationTable,whichshowstheconnectionbetweenthelabelsofvariouspointgroups. PointGroup: Oh D4h C4v D2d D3 Td IrreducibleRepresentation: eg a1g+b1g a1+b1 a1+b1 e e IrreducibleRepresentation: t2g b2g+eg b2+e b2+e a1+e t1 Grouptheorydoesnotgiveustherelativeenergiesoftheorbitals,however. Consideratetragonalcase: Needtointroduceadditionalparameters,δ1andδ2 ThesituationissuchthatE(dx2–y2)–E(dxy)=10Dq (movingtheligandalongthezaxisshouldhavenoeffectontherelativeenergiesoftheorbitalsinthexyplane). Forcompressioncase: [E(dx2–y2)+δ2]–[E(dxy)+2δ1]=10Dq [E(dx2–y2)–E(dxy)]+δ2–2δ1=10Dq 10Dq+δ2–2δ1=10Dq δ2=2δ1 Canwepredictwhenthiswillhappen?Yes,usingtheJahn-Tellertheorem Jahn-TellerTheorem:Inanonlinearmoleculeadegenerateelectronicstatewilldistorttoremove thedegeneracyandtoincreasethestability Considerd1 InanOhgeometry,theelectronicstateistriplydegenerate(thesingleelectroncanbeinoneofthreeorbitalsofidenticalenergy). Axialelongationgivesastatethatisstilldegenerate(doubly)sowouldneedtofurtherdistort. Axialcompressionleadstoasinglydegeneratestateandincreasedstability. LFSE=–4Dq–2δ1 Thisshouldoccurevenifalltheligandsarethesame! WhichconfigurationsshouldbeJ-Tactive? configuration active? distortiongeometry d1 yes compression d2 yes elongation d3 no   d4(hs) yes either d4(ls) yes compression d5(hs) no   d5(ls) yes elongation d6(hs) yes compression d6(ls) no   d7(hs) yes elongation d7(ls) yes either d8 no   d9 yes either,(nearlyalwaysiselongation,oftentoCN=4) d10 no   TetrahedralComplexes TetrahedralsymmetryisfairlycommonbutcannotbetreatedasadistortionfromOh Ligandsbetweenaxesaredestabilized,ligandsalongaxesarestabilized. ThesplittinginTdcomplexesisalwayslessthanthesplittinginOhcomplexeswiththesameligands (ΔtZn2+ Spectrochemicalseries ligandsorderedbyrelativesizeofDqforanymetalionligandsorderedbyligandfieldstrength I–