ANTISYMMETRI WAVEFUNTIONS: SLATER DETERMINANTS (08/4/4) Wavefuctos that descrbe more tha oe electro must have two characterstc propertes. Frst, scll electros are detcal partcles, the electros coordates must appear wavefuctos such that the electros are dstgushable. Ths meas that the coordates of electros a atom or molecule must eter to the wavefucto so that the may-electro probablty dstrbuto, Ψ = Ψ*Ψ, every electro s detcal. The secod requremet, ad ths s a more completd rgorous statemet of the Paul excluso prcple, s that the wavefucto for a system of two or more electros must chage sg ay tme we permute the coordates of ay two electros, Ψ(,,,, j, N ) = Ψ(,, j,,, N ). Ths s a property of fermos (amog whch are electros, protos, ad other halftegral sp partcles); systems wth more tha oe detcal fermo, oly probablty dstrbutos correspodg to atsymmetrc wavefuctos are observed. Let us revew the -electro case. If wttempt to costruct a two-electro wavefucto as a product of dvdual electro orbtals, ad, the ether () () or () () alore satsfactory sce we requre that the electros be dstgushable. The combatos () () ± () () do meet the requremet of dstgushablty, but these fuctos just descrbe the spatal dstrbuto of the electros; we must also cosder ther sp. If the two electros have dfferet sp egefuctos, dstgushablty meas that ether α β or α β s satsfactory, but α β ± β α arcceptable as are α α ad β β, of course. As we ve oted, the overall wavefucto for two electros must be atsymmetrc wth respect to terchage of the electros labels. Ths admts four possbltes, as log as both ad are sgly occuped (ormalzato costats cluded): 3 Ψ = ##### Symmetrc "##### $ Atsymmetrc ##" ## $ Ψ = ( ϕ a ()ϕ b () +ϕ b ()ϕ a () ) (α β β α ) ##### Atsymmetrc "##### $ ### Symmetrc "### $ α α ( ϕ a ()ϕ b () ϕ b ()ϕ a () ) ( α β + β α ) β β The left superscrpts o Ψ ad 3 Ψ are the sp multplctes (S + ); the trplet wavefuctos arll egefuctos of Ŝ wth egevalue S(S + ) = ad they are degeerats log as we cosder sp-depedet cotrbutos to the eergy (.e., there s o appled magetc feld ad sp-orbt couplg s eglected). The values of M S (= M S 0 α 0 β
m s + m s, the z-compoets of S) for each wavefucto are gve, ad the umber of values M S takes, S +, for a gve S s the sp multplcty for the state. Slater poted out that f we wrte may-electro wavefuctos as (Slater) determats, the atsymmetry requremet s fulflled. Slater determats are costructed usg sporbtals whch the spatal orbtals are combed wth sp fuctos from the outset. We use the otato () ϕ a ()α ad add a bar over the top to dcate spdow, () ϕ a ()β. Slater determats are costructed by arragg sporbtals colums ad electro labels rows ad are ormalzed by dvdg by N, where N s the umber of occuped sporbtals. Ths arragemet s uversally uderstood so the otato for Slater determats ca be made very compact; four Slater determats ca be costructed usg,,, ad :,,, ad What does ths otato mea? To see, let s expad out, step-by-step: φ a () () ϕ = a ()α ϕ b ()α () () ϕ a ()α ϕ b ()α = ϕ a ()α ϕ b ()α ϕ b ()α ϕ a ()α Thus, = ( ϕ a ()ϕ b () ϕ b ()ϕ a () )α α = 3 Ψ( M S = ) Notce that the otato assumes the determat s ormalzed ad that we havdopted the covetos metoed: rug over sporbtals colums ad over electro labels rows. Proceedg the same way for, = () () () () = ϕ a ()β ϕ b ()β ϕ a ()β ϕ b ()β = ϕ a ()β ϕ b ()β ϕ b ()β ϕ a ()β Thus, = ( ϕ a ()ϕ b () ϕ b ()ϕ a () )β β = 3 Ψ( M S = ) ombatos of ad yeld the two wavefuctos wth M S = 0: = = () () () () = ϕ a ()β ϕ b ()α ϕ a ()β ϕ b ()α = ϕ a ()ϕ b () β α ϕ b ()ϕ a () α β () () () () = ϕ a ()α ϕ b ()β ϕ a ()α ϕ b ()β = 3 Ψ( M S = 0) = Ψ( M S = 0) = ϕ a ()ϕ b () α β ϕ b ()ϕ a () β α ( + ) = ( ϕ a ()ϕ b () ϕ b ()ϕ a () ) β α + α β β α α β = ϕ a ()ϕ b () +ϕ b ()ϕ a ()
Take partcular ote of the fact that the spatal parts of all three trplet wavefuctos are detcal ad are dfferet from the sglet wavefucto. To summarze, terms of determats the sglet ad trplet wavefuctos are 3 Ψ( M S = ) = ; 3 Ψ( M S = 0) = ( φ a + ) ; 3 Ψ( M S = ) = Ψ = ( φ a ) Determats ca be represeted dagramatcally usg up- ad dow-arrows orbtals a maer famlar to chemsts. However, the dagrams ow take o more precse meags. Whle the M S = ± compoets of the trplet statre represeted as sgle determats, the sglet wavefucto ad the M S = 0 compoet of the trplet state should be wrtte as combatos of the determats: The geeral form of a Slater determat comports wth ths dscusso. Whe expaded, the determat for N electros N sporbtals yelds N terms, geerated by the N possble permutatos of electro labels amog the sporbtals ad dfferg by a multplcatve factor of for terms related by oe parwse permutato. To be explct, wrtte out determatal form we have Ψ(,,,, j, N ) = N Dfferet Sp Orbtals olums #%%%%%%%%%%% $ %%%%%%%%%%% & () () φ p () φ q () φ z () () () φ p () φ q () φ z () " " " " " () () φ p () φ q () φ z () " " " " " ( j) ( j) φ p ( j) φ q ( j) φ z ( j) " " " " " (N ) (N ) φ p (N ) φ q (N ) φ z (N ) Sce electroc wavefuctos for two or more electros should be wrtte as determats, our goal s to determe the symmetry characterstcs of determats, or more specfcally, how the determats form bases for rreducble represetatos. learly, we wat to avod expadg the determat out to exhbt N terms, f possble. To do that, three propertes of determats ca be used (I these expressos, the sporbtals ca carry ether sp, φ() = ϕ() [α or β ]: = electro label): Swap ay two rows (or colums) of a determat, ad the sg chages, Dfferet electros rows 3
φ A φ P φ Q = φ A φ Q φ P Therefore, f ay two rows (or colums) are detcal, the determat s zero. Ths guaratees that we ca t volate the Paul prcply usg the same sporbtal twce. olums (or rows) ca be factored, φ A φ P + φ Q φ R = φ A φ P φ R + φ A φ Q φ R Ay costat (cludg ) ca be factored out, φ A c φ p φ R = c φ A φ p φ R The latter two rules wll be useful whe evaluatg the results of symmetry operatos. Let s see how these rules apply to a closed-shell molecule, HO, for whch we wll exam Slater determat costructed from the valece MOs. The four valece MOs for water are depcted herd the trasformato propertes of the MOs are summarzed as follows a b R +a for all symmetry operatos, R. R + for R = E,σ v ad for R =,σ v. R +b for R = E,σ v ad b for R =,σ v. If we combe the orbtal trasformato propertes wth the rules gve above for determats, we ca fd the symmetry of the groud electroc state wavefucto. Each symmetry operato operates o all the the determat ad the rules gve above wll be used to evaluate the rreducble represetatos to whch that groud state determat belogs: a a b b a a b a a ( b )( b )a a ( ) σ a a b b a a b v a a ( b )( b )a a = + a a b b a a a a b b a a σ v a a b b a a = + a a b b a a = + a a b b a a Ths sglet, closed-shell electroc state wavefucto (a Slater determat) belogs the totally symmetrc represetato, A. Sce electros are pared orbtals closedshell molecules, f the doubly-occuped orbtals all belog to oe-dmesoal 4
represetatos, the wavefucto wll always belog to the totally symmetrc represetato. Although t s ot as trasparetly true, ths apples to closed-shell molecules wth degeerate orbtals as well. To cosdeust the 3 operato actg upo the groud state determat for NH3, we frst recall how the orbtals trasform: a 3 +a e 3 x e + 3 d x y e y 3 3 e e x y Trasformato of the determats s a bt laborous, but straghtforward ad we ca gore the odegeerate sporbtals: e x e x e y e y 3 ( e + 3 e )( e + 3 e )( 3 e e )( 3 e e ) x y x y x y x y We expad out to four determats obtaed by multplyg through the frst two parethetcal factors,... = 4 e x e x ( 3 e x e y )( 3 e x e y ) + 3 4 e y e y ( 3 e x e y )( 3 e x e y ) 3 4 e y e x ( 3 e x e y )( 3 e x e y ) 3 4 e x e y ( 3 e x e y )( 3 e x e y )... expadg ths out further, we recall that ay determat wth two detcal colums s zero, whch elmates all but oe term for each of these four determats,... = 6 e x e x e y e y + 9 6 e y e y e x e x 3 6 e y e x e x e y 3 6 e x e y e y e x... fally we perform two colum swaps the secod determat ad oe colum swap each of the thrd ad fourth determats, leavg the sg of the secod uchaged ad swtchg the sg of the thrd ad fourth,... = 6 e x e x e y e y + 9 6 e x e x e y e y + 3 6 e x e x e y e y + 3 6 e x e x e y e y So we fally coclude that e x e x e y e y 3 + e x e x e y e y. All the other 3v operatos yeld the same result. The coeffcets work out the ed because the symmetry operatos are orthogoal trasformatos (utary trasformatos, the complex case) - readers are ecouraged to covce themselves that ths s the case. I the smplest ope-shell case, a state s represeted by a sgle determat wth oe upared electro a odegeerate orbtal ad the state symmetry s the sams the symmetry of the half-occuped MO. If two odegeerate orbtals are half occuped, the symmetry of the state s determed by takg the drect product of the two orbtals. 5
rreducble represetatos. For the trplet state of methylee (:H), the methylee valece orbtal symmetres are the sams those for water (above) ad the trplet state electroc cofgurato s (a ) (b ) (a ) ( ). The the M S = ± compoets of the trplet state work out qute smply to trasform as, e.g, a a b b a b a = a a b b a a a b b a σ v a = + a a b b a a a b b a σ v a where the closed shells are ot wrtte out. Let s also cofrm that the M S = 0 compoet also behaves as a bass fucto: Let s exame the electroc states of cyclobutadee (), for whch there s a halfoccuped degeerate set of orbtals. The four π orbtals are depcted below; the lowest-eergy cofgurato s (a u ) (e g ). s D4h, but oly the D4 subgroup eed be cosdered becausll the states that ca arse from ths cofgurato are gerade. We focus etrely o the partally occuped e g orbtals, whch trasform as follows, Therre sx determats of terest:,,,,, ad. The frst two clearly belog to a trplet ad trasform as follows: = a a b b a a + a b a ( ) + a ( ) = a + a σ a + a b v + a + a σ a + a b v a e 4 a +eb ; e 4 b ea e 4 a ; e 4 b e x a eb ; e x b ea (x= y) (x= y) ; e 4 b eb ( ) = + e 4 b ( )( ) = + + a ( ) = a + a 6
Ths demostrates that ad belog to the A represetato (A g D4h). As the reader ca readly verfy, the combato + also belogs to A (A g ). It s straghtforward to show that has ( g ) symmetry: e 4 b eb ( ) ( ) = + = e b e 4 b ( )( ) ( )( ) = + e b e x b ( eb )( ) ( )( ) = = + e b e (x= y) b ( ) ( ) = e b Fally, ad form bass for a reducble represetato that yelds the A (A g g ) represetatos, e 4 a eb ; e 4 b ( ea )( ) = e 4 a ( )( ) = ; e 4 b ( )( ) = x ( eb )( ) = ; ( ea )( ) = (x= y) (x= y) x ; ( )( ) = The reader ca demostrate that + has A g symmetry ad has g symmetry. (A g ad g projecto operators appled to or wll also geerate thpproprate combatos.) I summary, the (a u ) (e g ) cofgurato gves rse to 3 A g, g, A g, ad g states ad we ve establshed the determatal wavefuctos for each of these states: M S M S g : ( ) 0 3 A g e ( a + ) 0 A g : ( + ) 0 g : e ( a ) 0 7
ackgroud: Eerges of Determatal Wavefuctos Several texts quatum chemstry offer rgorous ad complete dervatos for eergy expressos of determatal wavefuctos. I ths documet, we ll provd graphcal method arrvg at the results after provdg a physcal motvato for the method. To accomplsh the latter purpose, let s reexame the determats from whch the sglet ad trplet two-electo wavefuctos were costructed. (A smple example: a helum atom a excted s s cofgurato; = s ad = s): I thbsece of explctly sp-depedet terms the Hamltoa (lk appled magetc feld or sp-orbt couplg), the eerges of these wavefuctos are oly affected by the spatal dstrbuto of the electros specfed by these expressos, so let s exame just the spatal factors: ; Ψ space = ϕ a ()ϕ b () + ϕ b ()ϕ a () 3 Ψ space = ( ϕ ()ϕ () ϕ ()ϕ () a b b a ) Let s evaluate the eerges by takg the expectato values of the Hamltoa (the + sgs apply to the sglet ad the sgs apply to the trplet):,3 E = ϕ ( a ()ϕ b () ±ϕ b ()ϕ a ())H ( ϕ a ()ϕ b () ±ϕ b ()ϕ a ())dτ dτ,3 E = ϕ a ()ϕ b () H ϕ a ()ϕ b ()dτ dτ + ϕ b ()ϕ a () H ϕ b ()ϕ a ()dτ dτ ± ϕ a ()ϕ b () H ϕ b ()ϕ a ()dτ dτ ± ϕ b ()ϕ a () H ϕ b ()ϕ a ()dτ dτ The frst two tegrals have the same valud are evaluated a straghtforward way, e ϕ a ()ϕ b () H ϕ a ()ϕ b () dτ dτ = ϕ a ()ϕ b () ĥ + ĥ + ϕ r a ()ϕ b () dτ dτ = ϕ ()ĥ a ϕ a ()dτ + ϕ ()ĥ b ϕ b ()dτ + e ϕ a ()ϕ b () dτ r dτ = h a + h b + J ab. ( where the ormalzato has bee used: ϕ a dτ = ϕ b dτ = ) The operators ĥ ad ĥ would clude, for the helum s s case, the ketc eergy operators ad electro-uclear oulombc attracto terms for each of the electros geeral they cludll the ketc ad potetal eergy terms that deped oly o each electro s dvdual coordates. h a ad h b are hece referred to as oe electro 8
eerges. J ab s called a oulomb tegral ad has a semclasscal terpretato that t ca be vewed as the electro-electro repulso eergy betwee oe electro charge cloud ϕ a ad a secod electro charge cloud ϕ b. The last two tegrals are equal to each other as well, e ± ϕ a ()ϕ b () H ϕ b ()ϕ a () dτ dτ = ± ϕ a ()ϕ b () ĥ + ĥ + ϕ r b ()ϕ a () dτ dτ e = ± ϕ a ()ϕ b () ϕ r b ()ϕ a () dτ dτ = ±e ϕ a ϕ b () ϕ a ϕ b () dτ r dτ = ±K ab E = h a + h b + J ab + K 3 ab E = h a + h b + J ab K ab K ab s called a exchage tegral ad K ab = E 3 E, the sglet-trplet eergy gap. Note that f we had foud the expectato values of or (p. ), the cross-terms that gve the exchage tegrals do t survve due to orthogoalty of the sp fuctos, ad ther eerges are h a + h b + J ab. Exchage tegrals are varably postve sce () ad () wll ted to have same sg whe the tegrad has ts greatest magtude (whe r 0). K ab s largest whe ad exted over the same rego of space. Thtsymmetrc ature of the trplet spatal wavefucto guaratees that the trplet state the electros ad are ever at the same locato ( 3 Ψ = 0 f the two electros have the same coordates),.e., the trplet state les lower eergy because there s less electro-electro repulso. The Rules: Thbovackgroud wll serve to ratoalze the followg rules for evaluatg eerges of determats, whch cota the followg terms: () a oeelectro orbtal eergy, ε, for each electro. ε wll geerally clude two-electro terms volvg e-e repulsos wth thtomc core electros (screeg) whch dstgushes ε from the symbol h used above, () for each parwse e - e repulso, a oulomb term (a Jj cotrbuto), ad (3) a exchage stablzato (a Kj cotrbuto) for each lke-sp e - e teracto. As a example, cosder the fve determats llustrated here. Assocated wth each of these determats are sx oulomb tegral cotrbutos sce there must be sx uque parwse repulsos wth four electros. The two determats wth MS = ± are Proof: Slater, J.. Quatum Theory of Atomc Structure, Vol. I, McGraw-Hll: New York, 960, p. 486. 9
assocated wth three exchage stablzatos whle those wth MS = 0 arssocated wth two exchage stablzatos. Ths s algebracally summarzed as E gr = ε a + ε b + J aa + J bb + 4J ab K ab E (3) ex = ε a + ε b + ε c + J aa + J ab + J ac + J bc K ab + K ac Mxg of the two MS = 0 determats, Ψex(A) ad Ψex(), yelds a trplet ad a sglet wavefucto. The trplet eergy must be equal to the eerges for the MS = ± trplet wavefuctos that are represetabls sgle determats. K bc E A ex = E ex = ε a + ε b + ε c + J aa + J ab + J ac + J bc K ab + K ac E A+ ex + E A ex = E A ex + E ex ; but E A+ (3) ex = E ex E () ex = ε a + ε b + ε c + J aa + J ab + J ac + J bc K ab + K ac E () ex = E A ex = E A ex + E (3) ex E ex + K bc ; E () ex E ex States arsg from degeerate orbtals wth two or more electros (3) = +K bc Molecules ad os wth ope shell electroc cofguratos are qute commo trasto metal chemstry. efore proceedg further wth applcatos, however, let s derve some formulas that allow us to work wth characters dervg states for multelectro cofguratos degeerate orbtal sets. Whe -fold degeerate orbtals {ϕ,,ϕ } belogg to rreducble represetato Γ are flled wth, say, two electros or two holes, oe caot smply evaluat drect product to determe the states that derve from such cofguratos. The -dmesoal drect squared represetato (Γ Γ) wll have the parwse products of these orbtals, {ϕ,,ϕ,ϕ ϕ j, j}, as bass fuctos. The characters, χ Γ Γ (R), are just χ Γ (R). Now, f wre costructg permssble sglet/trplet state wavefuctos, the spatal part of the wavefuctos are symmetrc/atsymmetrc wth respect to permutato of the electro labels whle the sp fucto s atsymmetrc/symmetrc: " $$$$$$$ Symmetrc # $$$$$$$ % Ψ = ϕ ()ϕ () ϕ ()ϕ (), < j ϕ ()ϕ j () +ϕ j ()ϕ () " Atsymmetrc $$ # $$ % (α β β α ) 0
3 Ψ = ###### Atsymmetrc "###### $ ( ϕ ()ϕ j () ϕ j ()ϕ ()), < j The set, {ϕ,,ϕ }, s thass for a rreducble represetato so the character for each operato wth a class wth respect to ths bass wll be the same, ad depedet of ay choce of orthogoal lear combatos of these orbtals we make. Suppose that we ve sgled out a partcular operato R from each class ad assume that we have chose a lear combato of the orbtals such that the matrx for each R s dagoal: ϕ R r ϕ =,, ; R = The combatos of bass fuctos that dagoalze R wll geerally be dfferet for each operato, but the characters are, as always, the same for every member of a class. Operatg o the spatal parts of the wavefuctos for both the sglets ad the trplets, r 0 0 0 0 0 0 r ### Symmetrc "### $ α α ( α β + β α ) β β ; χ(r) = r +"+ r R ϕ ()ϕ () r ϕ () r ϕ () = r ϕ ()ϕ () R ϕ ()ϕ () r ϕ ()ϕ () R ( ϕ ()ϕ j () + ϕ j ()ϕ ()) r ϕ ()ϕ j () + ϕ j ()ϕ () R ( ϕ ()ϕ j () ϕ j ()ϕ ()) r ϕ ()ϕ j () ϕ j ()ϕ () < j < j So the characters for the operatos thass spaed by all the symmetrc sglet ( χ + ) ad atsymmetrc trplet ( χ ) wavefuctos are χ + (R) = r + r ; χ (R) = r = As oted above, the characters for the ormal drect product bass are just χ (R) = r = < j ad sce thass fuctos are chose so our class represetatve operatos have dagoal matrces, the characters for the squares of the operatos are dagoal as well, < j = r + r = r + r = j = < j For all the umbers r,..., r, r = ; geeral, r could b complex umber, r = e α.
r 0 0 R ϕ r ϕ =,, ; R = 0 0 ; χ(r ) = r 0 0 r = If we take the sum ad dfferece of the expressos for χ (R) ad χ(r ) ad dvde each by two, we obta formulas for thtsymmetrc ad symmetrc drect products, where the cotext of the dervato gve makes t clear that these two formulas are oly defed whe takg a drect product of a degeerate rreducble represetato wth tself ad they re used to hadle two-electro (or two-hole) cases. It s easy to show that these formulae recover our results for cyclobutadee. The D4 subgroup s aga suffcet, sce the (eg) cofgurato wll geerate oly gerade states: [E E] + 3 3 = A Formulas for three electros (or three holes) a 3-fold- or hgher-degeerate set of orbtals ca be derved usg permutato group theory 3 ad are ad for four electros (or four holes) a 4-fold- or hgher-degeerate set of orbtals: Applcatos to Lgad Feld Theory χ + (R) = χ S =0 (R) = χ (R) + χ(r ) χ (R) = χ S = (R) = χ (R) χ(r ) D 4 E 4 ( 4 ) [E E] = A χ S = (R) = 3 χ 3 (R) χ(r 3 ) χ S = 3 (R) = 6 χ 3 (R) 3χ(R)χ(R ) + χ(r 3 ) χ S =0 (R) = χ 4 (R) 4χ(R)χ(R 3 ) + 3χ (R ) χ S = (R) = 8 χ 4 (R) χ (R)χ(R ) + χ(r 4 ) χ (R ) χ S = (R) = 4 χ 4 (R) 6χ (R)χ(R ) + 8χ(R)χ(R 3 ) 6χ(R 4 ) + 3χ (R ) A uderstadg of bodg trasto-metal complexes, partcularly classcal Werer complexes, demads that wccout for electro-electro repulso o a equal footg wth the quas-depedet electro terms mplct our focus o molecular orbtals ad ther respectve orbtal eerges. I lgad-feld theory, oe seeks to correlate atomc (o) state eerges (.e., Russell-Sauders terms) wth molecular states bult up from molecular orbtal cofguratos. I the lgad-feld approach, the ope-shell wavefuctos arssumed to reta ther d-lke character ad the lgad cotrbuto to See D. I. Ford, J. hem. Ed., 49, 336-40 (97); Appedx below gves proof of the three electro 3 formulae.
the partally-flled orbtals s accouted for through ther effect o orbtal eergy splttg ad by treatg the d-d repulso eerges as adjustable parameters (maly the Racah parameter ) that wll geerally be smaller tha free os because whe the electros are delocalzed oto lgads, repulsos are lesseed by the relef of ther crowdg to relatvely cotracted d orbtals. Thesre matters we wll refer to oly tagetally - our focus wll rema o the symmetry-cotrolled characterstcs of lgad feld theory. Let s tur our atteto to the electroc states of octahedrally-coordated d os, for whch therre three possble cofguratos: (tg), (tg) (eg), ad (eg). The sglyexcted (tg) (eg) cofgurato s hadled easly: therre sx assgmets of the tg electro (three orbtals, sp-up or sp-dow) ad four assgmets of the eg electro, so the state wavefuctos from ths cofgurato are wrtte terms of 4 determats. The symmetres of the states are easly determed by takg the drect product, tg eg =,3 T,3 T where the left superscrpts dcate that both sglets ad trplets ca be formed for each symmetry (wth oe electro each tg ad eg, o Paul Prcple volatos occur). We ll retur to the ssue of fdg wavefuctos for these states below. O E 8 3 3 (= 4 ) 6 4 6 A x + y + z A E 0 0 (z x y,x y ) T 3 0 (R x, R y, R z ); (x, y, z) T 3 0 (xy, xz, yz) T E 6 0 0 0 T T + T T E E + E E 3 0,3 T,3 T To fd the states arsg from the (tg) ad (eg) cofguratos, the formulas derved the precedg secto arppled, the results of whch are show thugmeted character tablbove for the O group (the g symmetry of the d orbtals s uderstood). If we correlate the d o atomc states wth these molecular states, the dagram show here emerges. Ths s a modfed verso of Fgure 9.4 gve otto s text. Atomc states art left, molecular orbtal cofguratos are show at far rght. The dagram also shows some eergy splttgs that are ot gve the correspodg dagram foud otto s text. I the otto s Fgure 9.4, the cofguratos are labeled as Strog teracto referrg the the stregth of the lgad-metal teracto. Note that eve the strog lgad feld lmt, the electro-electro repulso that causes the state splttg 3 T 6 0 0 A E T 3 0 3 3 A A E 3
would stll be preset. The fgure gve o the followg paglso cludes eergy splttgs betwee some of the states o both sdes of the dagram the orgs of whch shall be explaed below. Now let s fd wavefuctos for some of these states. For the (tg) cofgurato, Hud s rule predcts that the lowest eergy state wll be the trplet, 3 Tg. The MS = determats are smple to llustrate graphcally ad are show below the correlato dagram: To verfy that these determats do deed belog to the Tg represetato, we frst apply the symmetry operatos to the orbtals... 4
3 xy yz ; yz 3 xz ; xz 3 xy ; xy 4z xy ; xz yz ; yz xz 4z 4z...the apply these to the determats, whch do deed behavs Tg bass fuctos: 4z 4z xy xy ; xz xz ; yz yz (x= y) (x= y) (x= y) xy xy ; xz yz ; yz xz 4z xy xz 3 yz xy = xy yz ; xy yz 3 yz xz = yz xz ; xy xz 4z xz yz 3 xy xz χ(3 ) = 0 4z xy yz = xy yz ; xy yz xy xz = xy xz ; xz yz 4z yz xz = xz yz χ(4 ) = xy xz 4z xy xz xy xz = xy xz ; xy yz xy yz = xy yz ; xz yz 4z xz yz = xz yz (x= y) xy yz = xy yz ; xy yz xz yz The determats xyxz, xz yz, ad xy yz are lkewse the wavefuctos wth MS = for the 3 Tg state. The MS = 0 wavefuctos belogg to 3 Tg are combatos of determats: ( xy xz + xy xz ), ( xz yz + xz yz ) ad ( xy yz + xy yz ). The reader may verfy by drect operato that the Tg wavefuctos are the orthogoal combatos of the same determats: ( xy xz xy xz ), ( xz yz xz yz ), 4z χ( 4 ) = (x= y) xy xz = xy xz ; (x= y) yz xz = xz yz χ( ) = ad ( xy yz xy yz ). Fally, we ca costruct a reducble represetato spaed by determats correspodg to two electros each oe of the tg orbtals. Reducto of that represetato shows that these determats form the Ag ad Eg states. 5
O E 8 3 3 ( 4 ) 6 4 6 xy xy, xz xz, yz yz 3 0 3 A E The combatos of these determats for these two states, dervato of whch s left as a exercse, ars follows: Ψ( A g ) = xy xy + xz xz + yz yz 3 Ψ a ( E g ) = xy xy xz xz yz yz 6 ; Ψ b ( E g ) = xz xz yz yz. Descet symmetry as a tool for dervg electroc wavefuctos Let s cosder the states we derved for the (tg) (eg) cofgurato:,3 Tg,3 Tg. Perhaps the smplest way to fd determatal wavefuctos for each of these s to proceed by lowerg the symmetry ( ths case, from Oh to D4h) ad explotg the symmetry correlatos that apply to both the orbtals ad the states. Whe Oh symmetry s lowered to D4h symmetry, the correlato of d orbtals goes as llustrated here: The paret (tg) (eg) cofgurato ca yeld four descedet cofguratos D4h: (eg) (ag), (eg) (bg), (bg) (ag), ad (bg) (bg), whch respectvely gve rse to,3 Eg,,3 Eg,,3 g, ad,3 Ag states, as determed by evaluatg drect products usg the D4h orbtals. However, the D4h descedet states must also correlate drectly wth ther Oh paret states,,3 Tg (Oh),3 Eg,,3 Ag (D4h),,3 Tg (Oh),3 Eg,,3 g (D4h). We ca coclude that (bg) (ag) ad (bg) (bg) cofguratos ad ther correspodg determats D4h must respectvely derve from,3 Tg ad,3 Tg Oh. We therefore kow some represetatve wavefuctos for each of these two states, M S M S xy z xy x y 3 T g : xy z + xy z 0 3 T g : xy x y + xy x y 0 xy z xy x y 6
M S T g : xy z xy z 0 M S T g : xy x y xy x y 0 Relatve State Eerges Now we wll depart from a purely symmetry-based aalyss ad evaluate the eerges of the trplet states ad several of the sglet states for a octahedral d system. Just as for the qualtatve correlato dagram, the two lmtg cases are thtomc os ad the strog lgad feld lmt. Let s beg wth the strog-feld lmt: = ε t g + J xy,xz K xy,xz = ε t g + (A + ) (3 + ) = ε t g + A 5 = ε t g + ε eg + J xy,z K xy,z = ε t g + ε eg + (A 4 + ) (4 + ) = ε t g + ε eg + A 8 = ε t g + ε eg + J xy,x K y xy,x = ε y t g + ε eg + (A + 4 + ) () = ε t g + ε eg + A + 4 = ε eg + J z K,x y z = ε,x y eg + (A 4 + ) (4 + ) = ε eg + A 8 = ε t g + J xy,xz + K xy,xz = ε t g + (A + ) + (3 + ) = ε t g + A + + = ε t g + ε eg + J xy,z + K xy,z = ε t g + ε eg + (A 4 + ) + (4 + ) = ε t g + ε eg + A + = ε t g + ε eg + J xy,x + K y xy,x = ε y t g + ε eg + (A + 4 + ) + () = ε t g + ε eg + A + 4 + E 3 T g,t g E 3 T g,t g e g E 3 T g,t g e g E 3 A g,e g E T g,t g E T g,t g e g E T g,t g e g I practce, wre terested the relatve eerges of these states, so let s take eergy dffereces to get the excted state eerges relatve to the groud state, E( 3 Tg, tg ): E 3 ( T g,t g ) = ε eg ε t g 3 = Δ o 3 3 E 3 ( T g,t g ) = ε eg ε t g 3 = Δ o 3 3 E 3 T g,t ( g e g ) E 3 ( T g,t g ) = Δ o + 9 9 E ( T g,t g ) E 3 ( T g,t g ) = 6 + o 0 6 + E T g,t ( g e g ) E 3 ( T g,t g ) = Δ o + 5 + 5 + E T g,t ( g e g ) E 3 ( T g,t g ) = Δ o + 9 + 9 + E 3 T g,t g e g E 3 A g,e g At ths pot, we eed to recogze that these expressos arased o thssumpto that each statrses from a sgle (strog feld) cofgurato. As o 0, the eerges of the 3 Tg, 3 Ag, ad the lowest 3 Tg states must be equal scll three states correlate back to the samtomc state ( 3 F). I thbove expressos, however, we ca see that the eergy of the frst two states s 3 relatve to the the lowest 3 Tg stats o 0. The resoluto of ths dffculty les cofgurato teracto (I). Therre two 3 Tg states, for whch we ve wrtte lgad-feld sgle-determat wavefuctos the precedg dscusso, ad they must teract wth each other sce they are of the same symmetry. The wavefuctos wrtte abovre very good approxmatos for large o; the extet of I s small whe the eergy dfferecetwee these lke-symmetry determats from two dfferet cofguratos s large. As o 0, however, these 7
wavefuctos mx to yeld the 3 P atomc statd a compoet of the 3 F atomc state. Sce the 3 Tg state s the oly trplet state of that symmetry, t correlates back to the 3 F atomc state wthout ay mxg wth other cofguratos. We therefore choose the eergy of the 3 Tg statt o = 0 as the zero of eergy, so that at o = 0, E( 3 Tg, tg ) = 3 ad E( 3 Tg, tg eg ) =. Now, we kow that f the mxg betwee the two 3 Tg states s accouted for, the lower of the two states must have E = 0 whe o = 0. We ca therefore wrt secular equato that accouts for the I ad t must be of the form, E x x 3 E = 0 E (5)E + 36 x = 0 where x s the matrx elemet due to the teracto of the two 3 Tg states. 4 The lower eergy soluto s E = 0, whch s satsfed f x = 36 ; the hgher eergy root s therefore E = 5. The teracto betwee the two states s due to electro-electro repulso ad s therefore depedet of the lgad-feld splttg,.e., eve for ozero o, the offdagoal etry the secular equato s 6: E ± ( 3 T g ) = o + E 6 6 3 E = 0 o + 5 ± Sce the lower of the two 3 Tg states s the groud statd we wsh to express all the other states eerges relatve to the groud state, we must subtract E ( 3 Tg) from each. I partcular, for the trplet states we obta E 3 T g,t g e g E + ( 3 T g ) E ( 3 T g ) E ( 3 T g ) E 3 ( A g,e g ) E ( 3 T g ) = = o The Tg sglet determats also mx va I, ad we kow that as o 0, the eerges of the hgher Tg statd the Tg statoth correlate to the G atomc state wth eergy εd + A + 4 +. We ca therefore wrt secular equato (applcabls o 0) that accouts for the I, o o 5 + = 3 o 5 + + 8 o + 5 + 8( o ) + 5 o o + 8 o + 5 + 8 o + 5 A symmetry argumet s used here to deduce the off-dagoal matrx elemet. A drect calculato s gve 4 allhause,. J. Molecular Electroc Structures of Trasto Metal omplexes, McGraw-Hll: New York, 980, p. 58. 8
E x x + E = 0 x = f E + = 4 +. As before, x = for all values of o. The geeral I secular equato ad s roots: E ± E ± ( + ) E ( o + ) E = 0 = + o E ( 3 T g ) = 7 + E T g,t g e g o + ± The reader may ow compare these results wth those publshed by Taabd Sugao the epoymously-amed dagram for a d o: o o + 49 + 8 o + 5 ± E ( 3 T g ) = + 9 + o + o o o + 49 + 8 o + 5 Left: a reproducto of the Taabe-Sugao dagram for a d o; Rght: Plots of the state eerges for selected states of a d o as derved ths documet for the parameter choce = 4.5. 9
Appedx : Relatos Ivolvg oulomb ad Exchage Itegrals p orbtals deftos J 0,0 = J z,z = J x,x = J y, y J, = J, = J, = ( )(J x,x + J x, y ) J J,0 = J,0 = J x, y = J x,z = J y,z, j = ϕ * ()ϕ * j ()( r )ϕ ()ϕ j ()dτ dτ ; K K, = K, = K x, y = J x,x J, j = ϕ * ()ϕ * j ()( r )ϕ ()ϕ j ()dτ dτ x, y K,0 = K,0 = K x, y = K x,z = K y,z d orbtals J 0,0 = J z,z J, = J, = J, = ( )(J xy,xy + J x y ),xy J, = J, = J, = J, = J xy,xz J,0 = J,0 = J xy,z J, = J, = J, = ( )(J xz,xz + J xz, yz ) J,0 = J,0 = J xz,z K, = K xz, yz = J xz,xz J xz, yz K, = K xy,x y = J xy,xy J xy,x y K, = K, = K xy,xz ϕ xz ()ϕ xy ()( r )ϕ yz ()ϕ x y ()dτ dτ K, = K, = K xy,xz + ϕ xz ()ϕ xy ()( r )ϕ yz ()ϕ x y ()dτ dτ K,0 = K,0 = K xy,z K,0 = K,0 = K xz,z Eerges of real d orbtal tegrals terms of Racah parameters J xy,xy = J xz,xz = J yz, yz = J z,z = J x y,x y A + 4 + 3 J xz, yz = J xy, yz = J xy,xz = J x y = J, yz x y,xz A + J xy,z = J x y,z A 4 + J yz,z = J xz,z A + + J x y,xy A + 4 + K xy, yz = K xz, yz = K xy,xz = K x y = K, yz x y,xz 3 + K xy,z = K x y,z 4 + K yz,z = K xz,z + K x y,xy ϕ xz ()ϕ xy ()( r )ϕ yz ()ϕ x y ()dτ dτ 3 0
Eerges of complex d orbtal tegrals terms of Racah parameters J 0,0 A + 4 + 3 J, = J, = J, A + 4 + J, = J, = J, = J, A + J,0 = J,0 J, = J, = J, J,0 = J,0 K, K, K, = K, K, = K, K,0 = K,0 K,0 = K,0 A 4 + A + + A + + 6 + 6 + 4 + + For st -row trasto metal os, Racah parameters ad have typcal rages: 650 00 cm, 3800 5500 cm. (State eergy dffereces do t volve A.) We ve chose to express the oulomb ad Exchage tegrals terms of Racah parameters, but the Slater-odo parameters arlso commoly ecoutered. They are defed by F k e k r r 0 r < 0 k + r R (r ) R l l (r ) k dr > dr ; r < k + r = > r k r k + f r > r r k r k + f r > r ad ( the d-shell): F 0 F 0, F = F 49, F = F 4 4 44 The "Racah Parameters" are related to the Slater-odo parameters by A = F 0 49F 4 = F 5F 4 = 35F 4
Appedx : States from three electros three or more degeerate orbtals We ca exted the method used to dervtsymmetrc drect formulae to hadle 3 electro cofguratos 3 or more degeerate orbtals. If wre costructg permssble quartet state (S = 3 /) wavefuctos, the wavefucto ca be factored to separate spatal ad sp parts, ad just as was the case for the trplet wavefuctos for the two-electro case, spatal part of the wavefuctos artsymmetrc wth respect to permutato of the electro labels whle the sp fucto s symmetrc: 4 Ψ = ########## Atsymmetrc "########## $ ( ϕ ()ϕ ()ϕ (3) ϕ ()ϕ ()ϕ (3) 6 j k j k ϕ ()ϕ (3)ϕ () ϕ (3)ϕ ()ϕ (), < j < k j k j k + ϕ ()ϕ (3)ϕ () + ϕ (3)ϕ ()ϕ () j k j k ) ###### Symmetrc "###### $ α α α 3 α α β + α β α + β α α ( 3 3 3 3) ( α β β + β α β + β β α 3 3 3 3) β β β 3 whe operated o by operato R, each of the terms the spatal wavefucto yeld the same egevalue: r r k. The set {ϕ,,ϕ } s thass for a rreducble represetato so the character for each operato wth a class wth respect to ths bass wll be the same, ad depedet of ay choce of orthogoal lear combatos of these orbtals we make. As before (p. ), we suppose that we ve sgled out a partcular operato R from each class ad have chose a lear combato of the orbtals such that the matrx for each R s dagoal: χ (R) = χ 3 (R) = Let s show that the formula gve o p., χ S = 3 (R) = 6 χ 3 (R) 3χ(R)χ(R ) + χ(r 3 ), yelds just ths result: < j <k r r k χ 3 (R) = 3χ(R)χ(R ) = 3 r r 3 3 = r + 3 r + 6 r r k j r = 3 r 3 3 r + χ(r 3 ) = + r 3 j j k k j Whch, whe we take the sum ad dvdy 6, yelds the desred result for χ 3/ (R). Usg the frst ad last of these equatos, we obta = r χ 3 3 (R) χ(r 3 ) + r r k j j k k j
The reader may verfy that wth a set degeerate orbtals oe ca costruct ( )( )/ dstct determats wth MS = / (or /) whch three dfferet orbtals (φ, φ j, φ k ) are occuped ad ( ) dstct determats wth MS = / (or /) whch oe orbtal (φ ) s doubly-occuped ad oe orbtal (φ j ) s sgly-occuped. The characters for each of the determat of the frst typre r r k ad those of the secod type have characters r. The formula χ / (R) = 3 ( χ 3 (R) χ(r 3 )) follows from takg the sum over all thass determats of both types. 3
Appedx 3: osos As t happes, vbratoal exctatos behave lkosos, amog whch are deuteros, alpha partcles, ad other teger-sp partcles. I solds, vbratoal exctatos are customarly gve a partcle-lke ame: phoos. The wavefucto for a system of two or morosos must reta the same sg ay tme we permute the coordates of ay two partcles, Ψ(,,,, j, N ) = +Ψ(,, j,,, N ). osder, for example, ad overtoe or combato eergy level whch a molecule has two quata of vbratoal exctato. I the harmoc approxmato, the vbratoal Hamltoa for a molecule ca be expressed as a sum of separate cotrbutos from each ormal mode, 3N 6 H vb = H k ; H k = Q + k λ Q k k ad therefore the vbratoal wavefuctos ar product of dvdual harmoc oscllator wavefuctos for each modd the vbratoal eergy s sum of the eerges of dvdual modes. osder, for example, a overtoe or combato eergy level whch a molecule has two quata of vbratoal exctato, 3N 6 Ψ vb = ψ ( 0) ψ j ψ k ; E = 3N 6 hν + 3 j,k j,k h(ν +ν ) j k where mode quatum umbers,, are dcated paretheses. Recall that the groud state ( = 0) vbratoal wavefuctos all trasform as the totally symmetrc represetato, so the symmetry of a combato level volvg odegeerate modes s gve by the drect product, Γ. If the j th ad k th j Γ k modes are members of a degeerate represetato, Γp, three wavefuctos are possble, ψ j (), ψ k (), ad ψ j ()ψ k (). There s, of course, o physcal dstcto betwee the fuctos ψ j ()ψ k () ad ψ k ()ψ j (). Thus, the wavefuctos for a degeerate overtoe vbratoal states form thass for the symmetrc drect product represetato [ Γ ] + p Γ p, the characters of whch are determed usg the formula derved o p., For three quata, the formula s k= χ + (R) = χ (R) + χ(r ). χ + (R) = 6 χ 3 (R) + 3χ(R)χ(R ) + χ(r 3 ). 4