Multi-electron atoms (11) 2010 update Extend the H-atom picture to more than 1 electron: H-atom sol'n use for N-elect., assume product wavefct.

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1 Mult-electron atoms (11) 2010 update Extend the H-atom pcture to more than 1 electron: VII 33 H-atom sol'n use for -elect., assume product wavefct. n ψ = φn l m where: ψ mult electron w/fct φ n l m one electron w/fct we mght expect the lowest energy state lke H-atom wth n for all electrons --> but ths s not an allowed multelectron wavefuncton Electon Spn changes how thngs work for electrons: Paul Prncple: a. Every wavefuncton for fermon (spn 1/2 partcle) must be ant-symmetrc wth respect to exchange of dentcal partcles b. For electrons n atoms ths turns out to mean each electron has dfferent set of quantum numbers But there s also spn quantum number, ±½, so for each n l m l 2 electrons maxmum Spn ntrnsc magnetc moment or angular momentum no physcal pcture, no functonal form - represent spn w/f as (note no coordnates!): α, β S z α = m s hα = ½ hα S z β = m s hα = -½ hβ where S z spn ang. mom. op. S 2 α = s(s+1) h 2 α = ¾ h 2 α and S 2 sq. tot. ang. mom. 33

2 Mult-electron Atoms -- Smplest dea f H-atom descrbes electrons around nucleus use solutons to descrbe mult-electron atom VII 34 Problem potental has electron-electron repulson V(r) = -Ze 2 /r + e 2 /r j added repulson term j = 1 r j = [(x x j ) 2 + (y y j ) 2 + (z z j ) 2 ] 1/2 dstance between electrons (r e - nucl) H = T + V = -h 2 /2m 2 + -Ze 2 /r + ½ j e 2 /r j K.E. sum over e - attracton repulson assume C of M f gnore 3rd term H 0 ~ h (r ) - separable each h (r ) s H-atom problem wth soluton that we know E 0 = ε ψ 0 = φ (r ) sum of orbtal E product H-atom soluton Whch orbtals to fll? (approxmaton: confguraton/shell) a) Could put all e - n 1s lowest energy, but Paul prevent that b) Put 2e - each orbtal (opposte spn), fll n order of ncreasng energy 34

3 Order of fllng Aufbau (buld up) n order of ncreasng n (dea: low to hgh energy) and ncreasng l (skp one n for d and agan for f) VII 35 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f Mult-electron atoms, need to account for electron repulson, radal node structure (# nodes = n-l-1) leads to splt of s,p,d,f energes for mult-electron atoms, Cross-over of dashed lnes e.g. see 3d go above 4s,4p so skp fllng n order - same for other d,f orbtals Cross over Atomc Orbtal Energy (ε ) vs. Atomc umber (Z) 35

4 Why ths order? relates back to the d and f orbtals beng smaller because fewer nodes but complex, n+1 s,p fll frst nsde node, feel nucleus Added electrons sheld outer electron from attracton to nucleus (3rd term left out).e. as Z ncreases 1s has more negatve energy VII 36 Same for n=2 etc. but each shelded by 2e - n 1s and 2e - n 2s, 6 n 2p, etc. Dfferent s and p rad cause: n-level splt wth l: E E nl But d, f abnormal do not fll untl fll (s + p) hgher n Seems counter-ntutve, but goes lke nodes more nodes e - get sucked n close to nucleus Aufbau mnemonc for rememberng fllng order: Key: use spn and Paul Prncple 2 e - per orbtal Atomc w/f buld up (Aufbau) or fll n the order of: ~ (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 6 (4s) 2 (3d) 10 (4p) 6 etc. 36

5 For atoms, number of electrons equals atomc #, neutral Ths s confguraton - Represents ψ 0 = φ n l m (r ) Soluton to H 0 = Σh (r ) -- Product w/f sum E: E = ε = 1 VII 37 Recall: h(r ι ) φ ι (r ι ) = ε ι φ ι (r ι ) Varaton get extra stablty by half-flled shell balanced by splt of s, p effect bgger for d,f electrons Transton (3d, 4d, 5d seres, and 4f, 5f) get ns 1 (n-1)d 5 Transton metal ons a bt dfferent yet Ion pattern mples that fll 3d last but lose 4s frst nner orbtals (d) stablze by ncreasng effectve charge 37

6 Orbtal Product confguraton Rep. as: ψ 0 = φn l m (r ) Sum Hamltonan Product w/f sum Energy E = = 1 ε VII 38 Approxmaton -- Where does ths come from? these orbtals for multelectron atoms must adjust H-atom lke soluton to account for e 2 /r j,j Warnng: some books use SI unts: e 2 /4πε 0 r j Central feld approxmaton: V(r) = [-Ze 2 /r + V(r )] +,j [e 2 /r j V(r )] ot separable - pull out of repulson - part depends on r Solve problem for just the left hand term Thnk of t as: average potental for electron a) attracted to nucleus b) repelled by all other electron j (average) Result: Stll a problem wth central force, now separable and nclude average repulson Msses out on correlaton nstantaneous e e moton/nteracton (keep apart) Soluton ψ(r 1, r 2, r 3, ) = get product wavefuncton get summed energy (orbtals): E = φ r l m (r, θ, φ ) snce only change potental, angular part same: Y LM (θ,φ) 38 ε

7 Method underlyng ths approach: Varaton Prncple (go for gettng the dea, f not the detals) f use exact H, approxmate (guess) a w/fct: ψ a then compute expectaton value of energy n ψ a H = ψ a *Hψ a dτ / ψ a *ψ a dτ E 0 {where E 0 s true ground state energy -- H > E exp } VII 39 guess w/fct ψ(λ) wth a parameter λ chooses form then H / λ = 0 wll gve best value λ (mnmum E) mprovement n ψ(λ,μ) alter form, add parameter μ Example: He-atom 2 electrons ψ ~ φ 1s (r 1 ) φ 1s (r 2 ) f e - sheld then Z Z' (less attracton to nucleus) ψ a ~ e -Z'r 1/a0 e -Z'r 2/a0 -- here, Z s varaton parameter E 0 = 2 4 E H = 8 E H ~ ev but E exp ~ ev bg error! for Z = 2 solve H / Z' = 0 Z' = Z 5/16 = 27/16 for best fct varaton mprovement: E' = ev To get better add more varaton e.g. ψ'' = (1 + br 12 ) e -Z'r1/a0 e -Z'r 2/a0 get: Z' ~ 1.85 E'' ~ ev b ~ 0.364/a 0 error ~ 0.5% could go on and get E calc more precse than E exp!! 39

8 For atoms represent orbtal as sum of functons φ nl (r ) = ckƒ k (r ) ƒ k could be varous exponentals k or other forms (lke Gaussans) Varaton: do optmzaton: H / c k = 0 fnd best c k lnear combnaton solve problem VII 40 Actual modern research uses Hartree-Fock method underlyng Varaton Prncple s same but optmze V(r ) to calculate average repulson then solve for mproved orbtals untl self-consstent Hartree-Fock conventonal method Self-consstent approach, means cycle Approxmate set of φ 0 orbtals compute V(r ) then nsert V(r ) nto H solve for mproved φ' then, use φ' - average potental V (r ) - all electrons cycle through: V'(r) φ'' V'' untl no change (Approxmate ψ HF + Approxmate H SCF ) msses Confguraton Interacton (CI) agan Other approaches called CI, perturbaton methods, excted confguratons (empty orbtals) also ncluded Alternatve s Densty Functonal Theory (DFT) expresson for electron densty s optmzed to gve lowest energy, then best wave functon these methods have parameters but otherwse are comparable, but more accurate, than Hartree Fock 40

9 Perodcty and the buldup gradual fllng orbtals apprecate ts orgns n quantum mechancs look at t VII 41 Shape of table reflects the fllng sequence, Aufbau concept number of electrons n each type of orbtal and the skppng of 3d and 4f n sequence 41

10 Atomc Radus - Increase down col., dec. across perod Ln (4f) contract - cause 5d transton metals very dense VII 42 Ionzaton potental Perodc table orentaton Left sde ns and np outsde of rare gas core hgh sheldng gves easy onzaton Rght sde fllng p-orbtal less shelded, attracts e - Halfflled - gves sngularty: O : (2p) 3 (2p) 4 Transton seres - favor (3d) m (4s) 1 for m=5 (Cr, Mo, W) 42

11 Shells - f n these models: H 0 = product then ψ 0 = φ (r ) and E = ε h (r ) sum of orbtal energes summed Orbtal 1 electron wavefuncton fcton - not molec. State mult electron w/f descrbe atom or molecule VII 43 Each orbtal s a soluton to H-atom exact potental all have Y lm (θ,φ) egenfuncton L 2, L z or l 2 and l z for electron Potental stll central Angular Momentum conserved - total w/f also egenfuncton angular momentum L = l vector sum, need be careful L z = lz scalar sum easer Closed shells maxmum number electrons n orbtal M = m = 0 (snce for each m there s -m up to ±l) shel snce only possble M = 0, then L = 0 (tot. ang. mom.) Called 1 S state (total L=0, also: all spns pared 2S+1=1) Open shells ex. (2p) 2, (3d) 4, (4f) 9, These can rearrange many ways dfferent ang. mom. M L = m l scalar sum over electrons n open shell M S = m s same over spn, shel 43

12 called Russell-Saunders pcture VII 44 L more complex Vector sum 2e - : L max = l 1 + l 2 snce angular mom, steps of 1 L mn = l 1 - l 2 (S = 1, 0) hgher use Paul and M to work out Rules: stll egenfuncton: L 2 2 ψ LM = L (L + 1) h ψ LM L z ψ LM = M h ψ LM Term symbol ψ LM ~ 2S+1 L J J = L+S, L+S-1,, L-S 2S + 1 multplcty (number M S values) also ndcates number of J values (f L > S) J total angular moment spn and orbt combne Paul s crtcal on what allowed gets trcky But to determne ground state.e. lowest energy state there are smple rules for open shells, lowest term Hund s Rules: Maxmum S lowest E LS Maxmum L of these s lowest Mnmum J less half / or Maxmum J more than half-flled shell Total number of states n term: (2S + 1)(2L + 1) n shell: L,S ground state easy: eg (2p) 2 e.g. C atom vs (2S + 1)(2L + 1) Max S = 1 Max L = 1 Mn J = 0 3 P 0 44

13 (4d) 3 e.g. b +2 on or V + (4s) 1 (3d) 3 Max S = 3/2 Max L = F 3/2 VII 45 (3d) 8 e.g. +2 on or Co + (3d) Max S = 1 Max L = 3 Max J = 4 3 F 4 Pcture: same spn dfferent orbtals keeps electron further apart on average maxmum orbtal angular moment more spatal varaton 45