Quantum Chemistry. Lecture 1. Disposition. Sources. Matti Hotokka Department of Physical Chemistry Åbo Akademi University

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1 Lere Q hesry M Hookk epre of Physl hesry Åbo Akde Uversy oes Irodo o hs orse The HrreeFok eqos sposo Sores ) The Hükel ehod ) The HrreeFok eqos ) Bss ses d oher prles 4) Wh be lled 5) orrelo 6) The FT eqos ) Prl FT llos Lere oes he web ( Swedsh) > Tehg Aks & Fred, Molelr Q Mehs, Oxford U Press Leve, Q hesry, Pree Hll Hehre & l., Ab Io Molelr Orbl Theory, Wley Koh & Holhse, A hes s Gde o esy Fol Theory, Wley 4

2 Ao s Ao s Bss e = e = h = 4 0 = All eqos old be wre SI Q hess ss o sg o s bese hey splfy he eqos The vles of qes hge Legh.. = bohr = 5.9x0 (=0.5Å) Mss.. = e = 9.x0 kg Eergy.. = hrree = 4.6x0 8 J hrge.. = e =.60x0 9 The eqos re splfed Ke eergy SI h 8π e.. olob ero QQ ZZ SI.. 4πε r r Shrödger eqo The Hlo Sory ses HΨ = EΨ Tedepede Ψ se = HΨ The orelvs Hlo operor H = T V The relvs Hlo operor H = α p β 8

3 ( ) # # & (, ) $, ( ) & & & (, ) $, The elero Hlo I oleles, oly he elero lod s osdered H = h( ) g(, ) el N =, = A, B < A< B N Z A h( r ) = A= RA r g( r, r ) = r r H Ψ = E Ψ el el el el 9 Z AZ R AB B Tke he Shrödger eqo HelΨel = EelΨel Igore he eleroelero ero (Hükel oly) Hel Heff = h'( ) = oseqely, he vrbles be sepred. The wf s prod d he eergy s s (Hükel oly) (,,, ) = ψ ( ) ψ ( ) ψ ( ) Ψ el < Ψ H Ψ >= < ψ h '( ) ψ > = ε el eff el = = 0 For sgle elero (or elero pr) h'( ) ψ ( ) = εψ ( ) Irode MOLAO pproxo ψ = φ p p Iser hs o he oeelero Srödger eqo h'( ) φ ( ) = ε φ ( ) p p p p Mlply he eqo wh 0 d egre!! < φ ( ) h" '( ) φ ( ) > = ε < φ φ > p p p p The lply wh 0, 0,..., 0. Ths gves syse of eqos h h h $ $% $ h h h h h h S S S S S S = ε $ $'% $ S S S h =< φ ( ) h'( ) φ ( ) > S =< φ φ > rs r s rs r s

4 = ) Oly oged ; eleros re osdered ) No eleroelero ero (oeelero odel) ) MOLAO (oly oe p AO per rbo o) 4) The egrls re S rs = δ rs α f r = s h = rs β f r d sreeghbors 0 oherwse, exple α β β α β 0 0,Bdee 4 β α β 0 0 β 0 0 α 4 = ε 4 4 Molelr orbls MO 4 < =.6> = 0.6> = 0.6> =.6> AO AO AO AO = > = > = > Vslze he MO s < = > 5 6

5 ? A Hükels ehod The vrol prple Orbl eerges Iozo eerges Elero ffes UVvs exo eerges Tol eergy elolzo eerges => ro hrer Wvefo hrge dsrbos Bod orders Sperosop rso probbles hel reves The beer he wvefo he lower he eergy Therefore vry he wvefo E el l he lowes possble eergy s obed E El ΨEl HEl ΨEl = = M! Ψ Ψ El El 8 Sp orbls Molelr orbls The ol wvefo wh N oordes shold be opzed. Leve o he ler oordes (BorOppeheer pproxo). A desol proble s sll oo lrge. The wvefo s spl o oeelero fos lled sp orbls ( pproxo) ϕ = ϕ( x, y, z, s) Eh elero hs poso x,y,z he hreedesol spe, d sp oorde s Uslly elero prs d olelr orbls F re osdered 9 Sepre he sp fro he spl oordes F s olelr orbl d G sp fo, eher H or I. The sp pr be reed lylly. A olelr orbl hold elero pr. α( s) ϕ ( x, y, z, s) = ψ ( x, y, β( S) The resred HrreeFok (RHF) ehod s obed whe he se F s sed for boh he H d I elero. The resred HrreeFok (UHF) ehod s obed whe dffere F s sed for he H d I eleros; he ses y be sp oed. 0

6 J N P P O Q R Z U Y ^ b Sler s deer The HrreeFok eqos The ol wvefo s osred fro he oeelero ses. The sples wy o do hs s o se he Sler deer (pproxo; ore re pproxos exs). Ψ( r ) = P P J K J J J K J ϕ ( x ) ϕ ( x ) ϕ ( x ) ϕ( x) ϕ ( x) ϕ ( x) L L M L! J J K J ϕ ( x ) ϕ ( x ) ϕ ( x ) S S S S The sp orbls d d he orbl eerges e re obed H T T T [ h V V ] H ex ϕ = ε ϕ V V V V VWV V V b b b= Ω b Z V ( r ) ϕ ( x ) = ] dx ϕ ( x ) g( r, r ) ϕ ( x ) ϕ ( x ) ex fro he Fok operor F=hV H V ex =,,..., U V ( r ) ϕ ( x ) = X dx ϕ ( x ) g( r, r ) ϕ ( x ) ϕ ( x ) Hrree poel [ [ [ [ [\[ [ [ b b b= Ω b Exhge poel Me feld Bss ses The er dx` Ω es h elero feels elero oly s e feld, o s expl prle. MOLAO ψ = φ p p The AO s re oo opled fos, se spler oes There re wo oo wys of odelg he AO s f Sler s fos f Gss fos 4

7 g h Sler s fos Gss fos orre Ths Sler s fo s he ex s o orbl of he hydroge o. Iorre orre Iorre 5 6 Bss ses Bss ses MOLAO ψ = φ p p The AO s re oo opled fos, se spler oes There re wo oo wys of odelg he AO s Sler s fos Gss fos MOLAO ψ = φ p p The AO s re oo opled fos, se spler oes There re wo oo wys of odelg he AO s Sler s fos Gss fos Why Gsss? The egrls be lled lylly,.e., fs Ths oweghs he lk of ry 8

8 k GTO ored Gss Type Orbl Ary of he expso Forldehyde Sle oe AO (or STO) by severl Gss fos => STOG e. Propery Eo Ebd r O (p) r H (p) HH (l ) q q O q H () e STOG STOG STO4G STO5G STO6G oblep oblep The MOLAO ow relly s MOLer obo of Model Fos. The ore degrees of freedo (.e., fos) he ore re desrpo of he MO Tke we s y AO s. Yo wll doble he ry of he MO (well, o qe) The sybol o oes fro he Sler orbls Flexbly orbl sze qsr = ζr S ( r, θ, φ; ζ ) = N r e Y ( θ, φ) l l l

9 Splvlee bss ses Polrzo fos The ore orbls o o vry fro olele o olele o o reqre he flexbly of doblev Ms hve re for bese hey o os of he eergy Vlee orbls esrbe he hesry Ms be flexble o o eed o hve errbly re for Ths 6G e Flexbly bod gles Obed by ddg polrso fos (lrger w q ber h orlly reqred,.e., d o rbo, roge, oxyge e.) Noo, e.g., 6G(d) (old oo 6G) xzy = 4 ffse fos AO s h exed frher h orl re eeded f he elero lod s espelly exeded. Ths s very ofe he se os. 5