Chapter 9 Many Electron Atoms

Transcription

1 Chem 356: Introductory Quntum Mechnics Chpter 9 Mny Electron Atoms MnyElectron Atoms A: HrtreeFock: Minimize the Energy of Single Slter Determinnt HrtreeFock Itertion Scheme Chpter 9 Mny Electron Atoms MnyElectron Atoms Hmiltonin for He Z = Ĥ =! m 1! e m! e m α Ze Ze + α r 1 r r 1, = T ˆ + Vˆ e + Vˆ ee 1) Simplifiction: neglect kinetic energy of hevy nucleus. Assume nucleus is fixed in origin (exct: seprte center of mss motion) T ˆ = Tˆ el + T ; neglect Tˆ, only consider electronic prolem ) Simplify units. We will use tomic units e Atomic Units Mss 1 e m =.u kg 19 Chrge e = 1.u C Length 0 = 1.u. 0 =! m e e = m Angulr Momentum! = 1.u.! = Js Energy (Hrtree) E h = 1.u. E h = In tomic units (.u.) e 0 = J e = m e =! = 0 = = 1.u. This simplifies equtions, s ll the constnts re gone. Chpter 9 Mny Electron Atoms 11

2 Chem 356: Introductory Quntum Mechnics In electronic structure theory, everyone uses.u. nd converts to physicl units s the need rises ow we wish to solve Ĥ = r 1 r r 1 Ĥψ (! r 1,! r ) = Eψ (! r 1,! r ) (For the moment we neglect spin) Let us e more generl nd write down the Hmiltonin for n electron tom Ĥ = 1 Z 1 i!ri + where r = r! i r! j i=1 i=1 i=1 j<i r The wvefunction ψ (! r 1,! r,...! r ) hs 3 coordintes This is very complicted prolem. However, if we could neglect the e simple. Ĥ = ĥ(1) + h() +...ĥ( ) el V, 1 r terms, the prolem would Try product wvefunction ĥ(i) = 1 Z i! ri ψ = ϕ (! r 1 )ϕ (! r )...ϕ z (! r ) Solve h(! r 1 )ϕ ) = ε ϕ (! r 1 ) oneelectron oritls, oritl energies Moreover, ech prolem for ech electron is the sme: only the nme of the vrile is chnged. This would simply e Hydrogen like solutions. ψ (! r 1...! r M ) = ϕ (! r 1 )ϕ (! r )...ϕ z (! r M ) E = ε + ε + ε c +...ε z Trivil! Question: wht would the lowest energy stte e? Chpter 9 Mny Electron Atoms 1

3 Chem 356: Introductory Quntum Mechnics Answer: It is tempting to fill up these oritl levels with electrons, of either spin. Tht is not the lowest energy solution to Ĥψ = Eψ! Wht is? Put ll electrons in the sme lowest energy oritl E = ε + ε + ε... ε = ε We ll know this is physiclly incorrect, nd need to invoke the Puli principle. Let us try to formulte it, nd understnd where it origintes. Let us go ck to sics: If n opertor  commutes with the Hmiltonin eigenstte of Ĥ then: ( ˆψ) = ˆ ˆψ = ˆ ψ = ( ˆψ) Hˆ A AH AE E A Hence Âψ is lso n eigenstte with the sme energy. H ˆ, A ˆ = 0 nd ψ is n Specil types of opertors tht commute with the Hmiltonin re the permuttions of coordintes. Eg. ˆP1 h(1) + h() + 1 r 1 ψ (1,) 1 = h() + h(1) + ψ (,1) r1 = HP ˆ ψ (1, ) systems. 1 PH ˆ ˆ = HP ˆˆ This is true for ny permuttion, true for mny electron Or 1 1 Specil permuttions re trnspositions T, which interchnge coordintes of prticles i nd j. These opertors re Hermitin, nd TT = 1, interchnging twice does nothing. Ĥ nd T hve set of common eigenfunctions, since they commute Tˆ ψ(1,... i, j, M) = λψ Chpter 9 Mny Electron Atoms 13

4 Chem 356: Introductory Quntum Mechnics Tˆ Tˆ ˆ ( ψ ) = Tλψ = λ ψ = ψ Hence λ = 1 λ = 1, λ = 1 From these considertions (plus it more solid resoning) we cn deduce there re types of prticles: 1) Bosons λ = 1: The wve function is symmetric under interchnge of coordintes (ny trnsposition) ψ (1,...i,... j,... ) =ψ (1,... j,...i,... ) ) Fermions λ = 1: The wvefunction chnges sign under trnsposition T ψ =ψ (1,... j,...i,... ) = ψ (1,...i,... j,... ) All spin 1 prticles, like protons nd electrons re Fermions. I do not hve simple explntion for this (spinsttistics theorem in quntum field theory). All permuttions cn e written s product of trnspositions p = T T kl T mj... either odd or even. Hence ction of ll T determines ction of P for ny permuttion. For electrons wvefunction is ntisymmetric under interchnge of electron coordintes (This includes the spin degree of freedom) If we return to our oritl product wvefunction ψ ϕ (1) ϕ ()... ϕ ( ) to mke ntisymmetric wvefunctions = one cn design clever trick z Put the oritls in determinnt: ϕ (1) ϕ (1) ϕ () ϕ () = ϕ () ϕ (1)ϕ () Or c 1 ϕ (1) ϕ (1) ϕ c (1) ϕ () ϕ () ϕ c () ϕ ϕ (3) ϕ (3) 3 (3) Interchnging the in determinnt. utomtic minus sign. Also: interchnge oritls nd introduces minus sign c coordintes of electrons = interchnge rows We cn simplify nottion nd write ψ = ϕ ()ϕ c (3) indicting the digonl terms only, Then ϕ ()ϕ (1)ϕ c (3) = ϕ ()ϕ c (3) Chpter 9 Mny Electron Atoms 14

5 Chem 356: Introductory Quntum Mechnics = ϕ (1)ϕ ()ϕ c (3) Slter determinnts esily fold in ntisymmetry. If we consider ϕ (1) s spinoritl, then in the determinnt we flip spinoritls (i.e. lels,, etc.) y permuttions, or we should flip spce nd spin coordintes upon permuttions. Finlly : ( h(1) + h() +...h( )) ϕ ()...ϕ z ( ) ( ) ( ) = ε + ε +... ε ϕ (1)... ϕ ( ) z z If h(1) ϕ (1) = εϕ (1) i i i ow we hve the Puli principle too: If two oritls in determinnt re equl then ψ = 0 ϕ ()...ϕ (k)... = φ (1)...φ (k)... = 0 ( equl columns ) For this reson we put one electron in ech spinoritl A simple Slter determinnt is the pproprite ntisymmetric generliztion of product of oritls. These would e eigenfunctions if worry out Vˆ el 1 = r. i j Hˆ = hˆ () i, sum of oneelectron Hmiltonins. It is not, we hve to i To del with V ˆ el one pplies the vritionl procedure in steps. ) Minimize the energy of single determinnt wvefunction. This is the HrtreeFock self- consistent field procedure. It leds to n optiml set of oritls (oneelectron wvefunctions) nd oritl energies. ) Apply liner vritionl principle nd minimize energy of c λ Φ λ in which Φλ re Slter λ determinnts. This is clled Configurtion Interction. It is importnt to go ehind Hrtree Fock (single determinnt) to get ccurte results. Moreover for (nerly) degenerte sttes one hs to use CI to get splitting (see tomic term symols lter on.). There re mny other pproches to improve results eyond the sic HrtreeFock result. Chpter 9 Mny Electron Atoms 15

6 Chem 356: Introductory Quntum Mechnics A: HrtreeFock: Minimize the Energy of Single Slter Determinnt. If we lel oritls in the determinnt σ, to indicte sptil index, nd spin index σ = α, β, then the following integrls re relevnt: The σσ ' J = J σσ ' k h σ = d r! 1 ϕ σ *( r! 1 )h( r! 1 ) ϕ σ ( r! 1 )d r! 1 σ σ = d r! 1 d r! ϕ σ *( r! σ 1 ) ϕ ' )* 1 ϕ σ σ r )ϕ ' ) σ σ 1 σ ' σ ' = d r! 1 d r! ϕ σ σ 1 )* ϕ ' σ )* ϕ ' r )ϕ σ ) σ σ ' σ ' σ 1 J integrl descries the Coulom interction etween ρ ) = ϕ ) And ρ ) = ϕ ) It is independent of spin. σσ ' The k integrl rises due to the ntisymmetry in Φ, This integrl is the Coulom interction etween distriutions ρ ( r) = ϕ ( r)* ϕ ( r ) 1 1 ρ ( r ) = ϕ *( r ) ϕ ( r ) And It is clled exchnge interction. It only rises for spinoritls tht hve equl spin. In the energy it enters with minus sign 1 ' 1 EHF = h σ + J σσ k σσ σ σ,, σ σ ' If one hs closed shell with equl α nd β oritls, the expression reduces to E = h + ( J k ) HF, The exchnge interction is the reson tht if oritls re degenerte, it is preferle to lign spins prllel. It mximizes the numer of k likespin contriutions. See Hund s rules for mnyelectron toms. We now hve n expression (did not derive) for the HrtreeFock energy. This expression cn e minimized to find n optiml set of oritls. If one expnds the oritls in sis set X i (! r ), one otins mtrix eigenvlue eqution. F c jλ = ε λ c iλ (orthonoml sis) j The mtrix F depends on the occupied oritls. For this reson the equtions hve to e solved self- consistently Chpter 9 Mny Electron Atoms 16

7 Chem 356: Introductory Quntum Mechnics HrtreeFock Itertion Scheme Guess strting oritls ϕ ( old ) Construct F ϕ old! new Digonlize Fc = cε pick up the lowest energy oritls F ϕ new old Check if F ϕ = F ϕ If not: updte oritls to get new set of improved oritls If new old F ϕ F ϕ convergence reched This is clled the self consistent field (SCF) itertion scheme. It is powerful numericl scheme to minimize the energy of single determinnt. Some otes: The oritls in HF re not unique. The only thing tht mtters is the determinnt. In determinnt I cn mke liner comintions of rows nd columns. Sme is true for oritls 1 ( ) 1 ( ϕ ϕ ) = ϕ ϕ ϕ +ϕ The energy E HF is not equl to ε One needs to use the full expression, given efore. HF is surprisingly ccurte in mny cses. For molecules it often yields ~ 99% of the exct energy. For purposes in chemistry this is not ccurte enough. 1% of energy > 100 kcl mol 1 typiclly! HF SCF equtions re tricky. It is esy to get trpped in higher energy solution. Pitfll of non- liner optimiztion. In prctice one uses sis set. The ccurcy of HF clcultion is limited y the size of the sis set. HrtreeFock results cn e qulittively wrong when you expect degenercies etween ground nd excited sttes; or when certin oritls in degenerte mnifold re occupied, nd others not in the HF determinnt. Usully one needs to go eyond HF to get qulittively correct results then. One lwys needs to go eyond HF to get quntittively ccurte results Chpter 9 Mny Electron Atoms 17