33 - 15.73 Lecture #33updated September 19,L-S Terms via L2, S2 and ProjectionLAST TIME:* method of ML, MS boxes. [For 3L states, cross out boxes starting fromboth (ML=L,MS=1) and (ML=L,MS=0).]complete (2L + 1)(2S + 1) dimension for each L-S term [# of boxes]*nl2 pattern*(nl)2 n′l′* method of ladders plus orthogonalityTODAY:L2, S2 method to obtain |LMLSMS〉, especially for ML,MS boxes in which thewhere method of ladders plus orthogonality is most inconvenient: ML = 0,MS = 0other, strong spin-orbit basis setsModern calculations use projection operators: designed to project away allunwanted parts of ψ yet preserve normalization.* L2 → L+L– only for ML = 0 block. Every L–S term is representedin this most evil block.* set up and diagonalize S2 — easy — by forming ± linearcombinations(singlet and triplet) αβ – βα αβ + βα* transform L2 to singlet, triplet basis (block diagonalization), thendiagonalize L2 by knowing (from crossing out boxes method)eigenvalues: L(L + 1)33 - 25.73 Lecture #33updated September 19,Look at the ML = 0, MS = 0 block of f2 and construct all L – S basis states. Allextant L-S terms of f2 are present once in the ML = MS = 0 block. Never try toget to this block by ladders and orthogonality!Cute trick that works especially well in ML = 0 and MS = 0 blocksbecause many otherwise awful terms vanish.diagonal butvanishes inML = 0nondiagonalψαβψβαψαβψβαψαβψβαψαβ123456733332222111100=−=−=−=−=−=−=Do d2 in lecture L L LL LL L LL LL L LLL LLL LLL LL LLL22 222 212122=+ +()=+ + −[]()[]==− + =+ ++− −+ +− +− + −+−+− −+zzzzz zz,,()hhhsame as 2So for ML = 0 block only, can replace L2 by L+L– (or L–L+) and, forMS = 0 only, replace S2 by S+S–.33 - 35.73 Lecture #33updated September 19,ψαβ ψψψ ψψψψβα ψψψ ψψψψαβ ψψψ ψψψψψβα ψψψ ψ12112211322212222432334231 35424342433 6 633 6 622 6 16 1022=− =+ =+=− =+ = +=− =+ =+ +=− =+ =SLSLSLSL6616101 1 10 22 121 1 10 22 1200 0 12 12 242465255625357626562646772727567ψψψψαβ ψψψ ψ ψ ψ ψψβα ψψψ ψ ψ ψψψαβ ψ ψψψψ++=− =+ = + +=− =+ = + −=− = = − +SLSLSLall easy require a bit more worknow we know, for 2e–, S2 can only have 2h2 and 0h2 eigenvalues (tripletand singlet)diagonalize S2 by inspectionψψψ ψψψψψψ ψψψψψψ ψψψψψ11212 1121221234 21234312563125647222222tststss=+()=−()=+()=−()=+()=−()=−−−−−−//////ts : : αβ βααβ βα+−← This alsohas αβ − β α formignorefactorsof h2SSSSLLL L22121212 1233 33 33 33 3333 33 6236 12 6 3 3 12 6 2 263 3 2 2αβ αβ ββ αβ βααβ αβ αβαβ αβαβ αβ−= −= −=−+−−= −= −=−[]−+−[]−[]=−+−[]+++− +–////etc.For f2:33 - 45.73 Lecture #33updated September 19,Confirm that these functions diagonalize S2 and give correctdiagonal elements.00What does L2 look like in basis set that diagonalizes S2? LL2112 2132 412122266661266 6ψψψψψψψttt=+++[]=+[]=− /hhhNONLECTURE S2222200001231234=htttssssa diagonal element:an off - diagonal element:ψ ψ ψψ ψψψψ ψ ψψψ ψψ ψψψψψψψψ12112212212 1 222121122122121211212221222 21212tttsSSSS=+()+()=+()+()=+()==+()−()=+()+−−hhhh2212100()==also ψψssS ψψψ ψ ψψ ψψψψψψψψψψ22122222234234234135246221266 6121261610616101216 16 16ttttLL=+[]==++[]= + +++++=+()=hhhLhhh33 - 55.73 Lecture #33updated September 19, L22121266 0616100102266 0 0616 10 0010 22 2420 0 24 2 241231234=⋅⋅−−h//tttssss00These 2 matrices are easier to diagonalize than the full 7 × 7 matrix, especiallybecause we know the eigenvalues in advance!Our goal is actually the eigenvectors not the eigenvalues66 06161001022302=()abcabcL eigenvector equation660302464461610300102230810521abc a b a a ababc babc c b c c b++= →= = =++=++= →= =122212=++[]abc/abc==()=()−4282125 42121212/////TRIPLETS L23 2 30 0 30 00HM M HLS ,===h33 - 65.73 Lecture #33updated September 19, 6642 636 66 16 10 42 6 10 26 10 255224 2 24 42 24 2 181032125450967712 12121212ab a b a ababc b ac bc bcbcd d cd d bb+= = ⇒ =++= += = =⋅+= ⋅= =⋅++ +=()−−//////normalization: 1= b136Similarly,Note that each ψnt basis state gets completely “used up” and all eigenvectors arenormalized and mutually orthogonal. You check both “used up” and orthogonality.Nonlecture: Singlets 312112212300 42 8 21 25 42Htt t=+()+()− /////ψψ ψ312123312112212300 3009142714FPtttttt=+−()=−+−−−−////ψψψψψψ L21 2 1121200 42 0066 0 0616 10 0010222420 0 24 2 2442IIabcdabcd=⋅⋅=−−h//112112212312411211221230016677677526771033770097749771771877IGss s ssss=+++=++−// / /// /ψψ ψ ψψψ ψA lot of algebra skipped here:112411212123124112112212312400254209428420027272717////// //ψψψ ψ ψψψψψsss s sssssDS=−++−=−+−+33 - 75.73 Lecture #33updated September 19,Again note that each ψns is used up. Check for orthogonality!Two opposite strategies:1. ladder down from extreme ML, MS2. L2 + S2 matrices are large but easy to write out for ML = 0 and MS = 0ONLY — could then ladder up from any L2, S2 eigenfunction (no needto use orthogonality).Before going to Projection Operators, look at the problems associatedwith getting 2 other kinds of basis states.JM LSJ“coupled” orbitals — important for strong spin-orbitlimit with HEAVY ATOMS.(HSO is diagonal in jω and in JMLS)ζnl >> energy separations between L-S terms(all ζnl are ≥ 0)coupled many-electron L-S-J states.Again — useful in strong spin-orbit limitJM LSJLM SMLS3- j or ladders←→3- j or ladders←→ nmsms11 111ll()()…J2(ladders arealmostuseless)ladders andL2 and S2 and3-j inmultiplesteps jsωlmj nj s11 1 11ωl()…to get here, must go long way around oruse projection operators.33 - 85.73 Lecture #33updated September 19,#()()()()()()()()MJ0 7 77772 6 7775 77553 5 7773 7753 75556 4 7771 7751 7573 7553 7355