5.7 Lectue #4 e / ij and Sate Sum Rue Method 4 - LAST TIME:. L,S method fo setting up NLM L SM S many-eecton basis states in tems of inea combination of Sate deteminants * M L = 0, M S = 0 boc: L L+ L S S S + * diagonaize S (singets and tipets) * diagonaize L in same basis that diagonaizes S [Reca: to get matix eements of L, fist evauate L and then eft mutipy by ψ j. couped epesentations njωs and NJLSM J. Pojection opeatos: automatic pojection of L eigenfunctions * emove unwanted L pat * peseve nomaization of wanted L pat * emove oveap facto ψ i TODAY:. Sate Sum Rue Tic (tace invaiance): MAIN IDEA OF LECTURE.. evauate e ij matix eements (tedious, but good fo you) i> j [-e opeato, spatia coodinates ony, scaa wt J,L,S] * mutipoe expansion of chage distibution due to othe eectons * matix eement seection ues fo e / ij in both Sate deteminanta and many-e basis sets * Gaunt Coefficients (c ) (tabuated) and Sate-Condon (F,G ) Couomb and Exchange paametes. Because of sum ue, can evauate mosty ab ab and ab ba type matix eements and neve ab cd ij ij ij type matix eements.. Appy Sum Rue Method 4. Hund s st and nd Rues updated Septembe 9,
5.7 Lectue #4 4 -. Sate s Sum Rue Method It is amost aways possibe to evauate e / ij matix eements without soving fo a LM L SM S basis states. * tace of any Hemitian matix, expessed in ANY epesentation, is the sum of the eigenvaues of that matix (thus invaiant to unitay tansfomation) * e / ij and evey scaa opeato with espect to J ˆ (o L,S) ˆ ˆ has i> j nonzeo matix eements diagona in J and M J (o L and M L ) and independent of M J o (M L,M S ) [W-E Theoem: J is a GENERIC ANGULAR MOMENTUM with espect to which e / ij is cassified] Reca fom definition of, that e / ij is a scaa opeato with espect to ˆ ˆ ˆ J, L, S but not with espect to j i o i. Inteeectonic Repusion: e ij i> j * destoys obita appoximation $$ fo eectonic stuctue cacuations * coeation enegy, shieding e e e e at (, θ, φ ) at (, θ, φ ) scaa with espect to J, L, S, s i but not j i, i = = + cos, [ ] / = + ( ) updated Septembe 9,
5.7 Lectue #4 expand as powe seies in < > whee < is smae of, ( integas evauated in egions : <, < ) 4 - engthy ageba see Eying, Wate, and Kimba Quantum Chemisty pages 69-7 and, fo eationship between Legende poynomias and Y m ( θφ, ), pages 5-59. mutipoe expansion n 4 = π n+ ij n= 0 m=n n < m m Y Y n n i i n j j (, ) (, + ) > [ ] θ φ θ φ not pincipa q.n.! convegent seies n -poe moment (n=0 monopoe, n= dipoe, ) angua momenta magnitude n, pojection m scaa poduct of angua momenta, one fo i-th patice, one fo j-th * * convets m to m n-poe chage distibution n-th an tenso n+ components No dependence on s, so / ij is scaa with espect to S, si, sj. m [ Yn ( θi, φi)= θi, φi i = n, m = m i ] updated Septembe 9,
5.7 Lectue #4 4-4 The eason fo this athe compicated ooing expansion is that it is we suited fo integas ove atomic obitas which ae expessed in tems of i, θ i, φ i, which ae coodinates of the i-th e with espect to the cente of symmety (nuceus) athe than the othe e. It enabes use of AO basis states. Othewise / ij integas woud be nightmaes. Seection ues fo matix eements: not pincipa q. n. i n, m = m, m = i s 0 i obitas j n, m = m, m = j s 0 j tiange ue, n + i i i i (steps of because of paity) tem in mutipoe expansion ovea: L = 0, S = 0, M L = 0, M S = 0, and indep. of M L, M S Can use any M L, M S fom box diagam. It is aso cea how to evauate the angua factos of the atomic obita matix eements using -j coefficients. Specia tabes of Gaunt Coefficients (aso C&S pages 78-79, Goding, page 4, see handout). updated Septembe 9,
5.7 Lectue #4 4-5 genea / matix eement ( so = 0,, and ae possibe) e ± ab cd = ab cd ab dc e ± ab cd = m m m m m m m m (, ) (, ) +, + { = 0 tenso an fo poduct of AOs occupied by e ± # must be same as fo # fo scaa poduct of n-th an tensos δ δ δ ( ) s s s s a c b d a b c d / does not opeate ** ** on spin coodinates GAUNT COEFFICIENTS ANGULAR FACTOR OF INTEGRAL ( ) R n n n n a a b b c c d d adia facto / scaa with espect to ˆ = ˆ + ˆ L ( a a c ) ( c b b d ) d e e c m, m c m, m / c ( m, m ) + A A 000 + tabuated m m, m, m Cebsch-Godan coefficients that esut fom intega ove poduct of spheica hamonics one fom opeato, two fom obitas tiange ue:- + + + = even ( fom popeties of A 000 ) ( paity) updated Septembe 9,
5.7 Lectue #4 estictions on and m: e intega m + m = m m n m Y n m tiange ue 4-6 fo intaconfiguation matix eements, R (abcd) taes on especiay simpe fom (because the same one o two obitas appea in the ba and in the et). ( ) ( ) ( ) ( ) R ab, ab F a, b R abba, G ab, Sate Condon paametes (these ae educed matix eements dependent ony on a, b, c, d and not on any of the m quantum numbes.) A L-S states fom one configuation ae expessed in tems of the same set of F, G paametes. ab e ab J ab, δ m, m K ab, = ( ) ( ) ( ) DIRECT s a spins must match o K tem vanishes s b EXCHANGE ( ) = ( ) ( ) ( ) J ab ab e ab c m m c m m F,,, n, n, ( ) = δ( sa s ) b [ ( a a b ) b ] ( a a b b) = 0 b ( am, bm a b ) K ab ab e ba m m c m m G,,, n, n fo specia cases, such as nd, n = n and F = G Now we ae eady to use tabes of c (o, moe convenienty, a and b to set up e = 0 ij matix a a a a b b a m, m a a b b b ( a a b ) b b a a b b [ a*( ) a( ) Opb ˆ *( ) b( ) d τ dτ] chage distibutions [ a*( ) b( ) Opa ˆ ( ) b*( ) d τ dτ] something ese! ) updated Septembe 9,
5.7 Lectue #4 4-7 Easy exampe: nf ( eca I, H, G, F, D, P, S) I 60 = αβ H 5 = αα I e = ( ) ( ) ( ) ( ) ( ) = 046,,, = 046,,, [ c, ] F ( nf, nf ) [ ] ( ) I c, c, F nf, nf δαβ, c, G nf, nf = ( ) these ae the ony L-S states epesented by a singe Sate deteminant extemes of M L,M S box diagam since e / ij is a scaa opeato with espect to Lˆ, Sˆ, Jˆ, matix eements ae M L, M S, M J independent so we can use any M L,M S component to evauate the matix eement whicheve is most convenient! H e = 046,,, e m e = 0 one spin α othe spin β {[,, ] (, ) [ (, ) ] (, )} both spins α e e H c c F nf nf c G nf nf = ( ) ( ) (a,a) (b,b) (a,b) Hee is whee eveyone maes mistaes! F ( nf ) F (nf ) Use tabe of c in Goding/C&S handout (C&S page 79). Note that [/76 64] / is impicit afte the fist enty fo f, = 6. convenient facto = 0 4 6 c (,) / / [/76 64] / c (,) 0 7/ [6/76 64] / c (,) 0 +/ 0 / / [7/76 64] / D 5 089 = 76 64 updated Septembe 9,
5.7 Lectue #4 4-8 D is a facto that simpifies the expessions. Each tem has the fom F /D. Ca this atio F. Get simpe ooing expessions when you epace F by D F (D appeas in denominatos of c as [ /D ] / ) I e I = F + F F F + 9 + 76 64 0 4 6 = F + 5F + 9F + F 0 4 6 Aways have two factos of c. Thus F gets divided by D to yied F. H e H = F + ( ) ( ) F + F F 0 4 6 0 7 0 6 7 + 76 64 5 = 9 ( ) 76 64 = F 5F 5 F F 0 4 6 F F F F 0 4 6 A ot of boo eeping, but easy to ean how to use tabes of c, a, b, D. But it is much moe wo fo f than fo f. SUM RULE METHOD: Basic idea is that the sum of diagona eements in the singe Sate deteminant basis set within an M L, M S box is equa to the sum of the eigenvaues! Loo at ML =, MS = box: α0 α and αα. This box geneates H and F, but tace is E( H) + E( F) and we aeady now E( H)! So E E I H E F α0α αα E H E G αβ βα αβ E I E H E ( ) = ( ) = ( ) = + + + D αβ αα ( ) = + ( ) ( )= + + ( ) ( ) α β β α α0β β0α αβ E I E G) ( ) E H E( F) + ( ) ( ( ) = α α + α α + α0α ( ) ( ) ( )= S+ ( ) E P E H E F E S sum of seven sum of six E L updated Septembe 9,
5.7 Lectue #4 4-9 This seems athe aboious, but it is much easie than: * geneating each LM L = L SM S = S as an expicit inea combination of Sate deteminants * then cacuating matix eements of e / ij, because thee ae many nonzeo offdiagona matix eements between Sate deteminants in the same M L,M S box. Hee is the fina esut fo the enegies of a (nf) S+ L tems: 0 E = E + E + E ( 0) ( 0) Z R E = sum of obita enegies fom h = = n ( ) E = e SO + H ij next ectue ( ) E intaconfiguationa spin - obit inteconfiguationa e = ( )+ ( ij) Fo nf ( ) ( ) ( ) eady now shieded by a fied subshes shieded by same subshe ε CI n Bae nuceus hydogenic obita enegy o paty shieded by fied shes. I ε nf 0( ) F nf F nf ( ) F( nf ) F( nf ) + + 5 + 9 + 4 6 H nf G nf F nf D nf P nf S nf ε + F 0 5 F 5 F 4 F 6 ε + F 0 0 F + 97 F 4 + 78 F 6 ε + F 0 0 F F 4 86 F 6 ε + F 0 + 9 F 99 F 4 + 75 F 6 ε + F 0 + 45 F + F 4 87 F 6 ε + F 0 + 60 F + 98 F 4 + 76 F 6 shieded-coe configuationa enegy intaconfiguation L-S tem spittings (thee is NO cente of Gavity Rue fo degeneacy weighted L-S tems) updated Septembe 9,
5.7 Lectue #4 4-0 Now it is easy to show that a F s ae > 0 and F >> F + etc. (oughy facto of 0 pe step in ) Fom this we get an empiica ue Lowest E of a L S tems is the one with * MAXIMUM S * of those with Maximum S, owest is the one with MAXIMUM L These ae Hund s fist and second (of thee) ues. Note aso that Hund s ues do nothing about pedicting the enegy ode of L-S tems except fo the identity of the singe, owest enegy L-S tem. updated Septembe 9,
5.7 Lectue #4 Nonectue 4 - Thee ae sevea inteesting pobems aso soved by this e / ij fomaism.. Enegy spittings between and Sate deteminanta chaactes of two o moe L,S tems of the same L and S that beong to the same L,S configuation e.g. d two D tems see pages 47-50 of Goding fo secua deteminant fo D of d. matix eements of e / ij between same L,S tems that beong to two diffeent configuations e.g. nd ndn d S, P, D, F, G S, P, D, F, G no Paui estictions SPDFG,,,, so thee wi be S~ S P~ P D~ D F~ F G~ G inteconfiguationa CI s, and each of these 5 inteaction matix eements wi NOT be of the same magnitude. updated Septembe 9,