α 2 α 1 β 1 ANTISYMMETRIC WAVEFUNCTIONS: SLATER DETERMINANTS (08/24/14) - PDF

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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

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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 ) 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 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 e x e x e y e y e x e x e y e y 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 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 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] = 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, (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
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