Atomic units The atomic units have been chosen such that the fundamental electron properties are all equal to one atomic unit.

Transcription

1 tomc uts The atomc uts have bee chose such that the fudametal electo popetes ae all equal to oe atomc ut. m e, e, h/, a o, ad the potetal eegy the hydoge atom e /a o. D Cm The use of atomc uts also smplfes Schödge's equato. Fo example the amltoa fo a electo the ydoge atom would be: Othe fudametal costats: h m 4πε 0 e Othe fequetly used eegy uts: a.u. 7.e 67.5Kcal/mol cm - Kcal/mol 4.84KJ/mol Boltzma s costat: k J/K vogado s umbe: mol - Rydbeg costat: R m - Compto wavelegth of electo: λ C m Stefa-Boltzma costat:σ W/m K 4

2 ppoxmatos made atee-fock-roothaa-all theoy

3 Basc cocepts, techques ad otatos of molecula quatum mechacs stuctue of may-electo opeatos e.g. amltoa fom of may-electo wave-fuctos Slate detemats, ad lea combato of them atee-fock F appoxmato moe sophstcated appoaches whch use the F method as a statg pot The electoc poblem The o-elatvstc tme-depedet Schödge equato >E > amltoa opeato fo a system of ucle ad electos R R R R B B RB molecula coodate system B B 443 RB > > T e T - the ato of the mass of ucleus to the mass of a electo the atomc umbe of ucleus T e the opeato fo the ketc eegy of the electos T the opeato fo the ketc eegy of the ucle ee the opeato fo the Coulomb attacto betwee electos ad ucle ee the opeato fo the epulso betwee electos the opeato fo the epulso betwee ucle e ee epesets the geeal poblem to be sepaated two pats: electoc ad uclea poblems

4 Bo-Oppeheme ppoxmato The ucle ae much heave tha electos they move much moe slowly the ucle ca be cosdeed foze a sgle aagemet molecula cofomato the electos ca espod almost stataeously to ay chage the uclea posto the electos a molecule ae movg the feld of fxed ucle -d tem ca be eglected 5-th tem s a costat Electoc amltoa descbes the moto of electos the feld of pot chages elec > Electoc Schödge equato: 4 - s the electoc wave-fucto whch descbes the moto of the electos elec elec >E elec elec > 3 elec elec { I };{R } 4 E elec E elec {R } 5 The total eegy: E tot E elec B> R B B 6 B B 443 RB costat > > T e explctly depeds o the electoc coodates paametcally depeds o the uclea coodates paametc depedece the uclea coodates do ot appea explctly elec. dffeet wave-fucto s defed fo each uclea cofguato Equatos 6 electoc poblem T e ee

5 If the electoc poblem s solved we ca solve fo the moto of the ucle Sce the electos move much faste tha the ucle we ca eplace the electoc coodates by the aveage values aveaged ove the electoc wave-fucto uclea amltoa o descbes the moto of the ucle the aveage feld of the electos ucl { R { R > E E elec pot tot potetal eegy suface PES pot E tot { R } } } B> R B B B> uclea Schödge equato ucl ucl > E ucl > ucl - descbes the vbato, otato ad taslato of a molecule E - total eegy of the molecule the Bo-Oppeheme appoxmato - cludes: - electoc eegy - vbatoal eegy - otatoal eegy - taslatoal eegy R B B Schematc llustato of a potetal eegy suface The equlbum cofomato of the molecule coespods to the mmum of the suface

6 Total wave-fucto Bo-Oppeheme appoxmato: { };{R } elec { };{R } ucl {R } Bo-Oppeheme appoxmato - usually a good appoxmato - bad appoxmato fo: excted states degeeate o cuasdegeeate states The tsymmety o Paul Excluso Pcple electo sp ad sp fuctos complete ad othoomal 0 0 * * * * d d ad d d the electo s descbed by spatal ad sp coodates: x{,} may electo wave-fucto must be atsymmetc wth espect to the techage of the coodate x both space ad sp of ay two electos. x, x,..., x,..., x,...,x -x, x,..., x,..., x,...,x

7 atee ppoxmato atee, 98 P,,...,... sp obtals The fom of P suggests the depedece of Pobablty desty gve by P s equal to the poduct of mooelectoc pobablty destes Ths s tue oly f each electo s completely depedet of the othe electos P - depedet electo model P /3 P /4 P /5P P P s ucoelated depedet wth P. Ucoelated pobabltes Coelated pobabltes I a -electo system of electos the motos of the electos s coelated due to the Coulomb epulso electo-oe wll avod egos of space occuped by electo two. Eε ε ε Electoc amltoa ca be ewtte: E h Whee: h v s the mooelectoc opeato ee v s the mooelectoc tem of the exteal potetal: e v 443 v I ψ P, h wll act oly o the wavefucto coespodg to the -th electo. oweve, ee depeds o pas of electos so that we ca ot sepaate the vaables Schödge equato.

8 atee ppoxmato: the electos do ot teact explctly wth the othes, but each electo teacts wth the medum potetal gve by the othe electos J Usg the vaatoal methods oe obtas the eegy of the system: E whee: J hdτ dτdτ -Coulomba belectoc tegals dτ - coe mooelectoc tegals -epeset the classcal epulso eegy betwee two chage destes descbed by ad Belectoc potetal / felt by the electo, due to the stataeous posto of electo s eplaced by a mooelectoc potetal obtaed by aveagg the teacto betwee the two electos ove the spatal ad sp coodates of electo. Summg ove oe obtas the medum potetal actg o electo ad whch s due to the othe - electos J Coulomb opeato J dτ dτ dτ epesets the local medum potetal felt by electo ad due to the electo descbed by

9 ε dτ Usg the Lagage s multples method atee equatos: J ε - the eeges of molecula obtals - J E ε Total electoc eegy: I ode to fd we eed SCF pocedue SCF pocedue the famewok of atee appoxmato

10 ρ electoc desty coespodg to the -th electo ρ tot ρ total electoc desty Each electo teacts wth a electoc desty obtaed by subtactg ts desty fom the total desty ρ k ρ tot ρ k ee potetal ca be wtte as: ρ k k ee g wth: g ρ ' d' g - teacto eegy of the pot chage the cosdeed dvdual electo wth the othe electos epeseted as a electoc desty amltoa: el v atee equatos: g v g ε

11 Detematal wave-fuctos: atee-fock appoxmato P - does ot satsfy the Paul pcple - gves a o-zeo pobablty fo two electos to be exactly at the same pot space Fock, Slate, 930 SD,,...,! / SD atsmetzed sum of atee poducts wth all the possble dstbutos of the electos the molecula obtals E SD SD,,..., Usg the vaatoal method of Rtz: K SD... - shothad otato J K * * hdτ v dτ * * dd E exchage tegal I atee J appoxmato

12 exchage opeato: K dτ - a o-local opeato because ts esult depeds o the value of o ete space ad ot oly o the value of whee s located the electo J K 0 mzg the eegy by vayg the sp obtals leads to the atee-fock equatos: v ' -' dτ '' * -' dτ ε Defg the Fock opeato: f ε h J K f J K total electoc eeges: E ε E J K ε - J I atee appoxmato molecula obtal eeges: total eegy: E ε T J K B> R B B

13 f we use the spatal obtals: xφ ş xφ SD RF,,...,! / s s s s s s s s s the famewok of RF appoxmato: E / / / J K molecula obtal eeges: ε / J K / E ε / / J K atee-fock equatos alteatve Schödge equato whch the exact amltoa has bee eplaced by a appoxmate Fock opeato - Coulomb opeato has bee eplaced by a opeato whch descbes the teacto of each electo wth the aveage feld due to the othe electos

14 Gve a set of k othoomal spatal obtals O {φ },,...k RF ad UF fomalsms k sp-obtals:,,...,k K x x, estcted O estcted wave-fucto Restcted wave-fucto fo L atom s s s RF But: K ss 0 ad K ss 0 s ad s electos wll expeece dffeet potetals so that t wll be moe coveet to descbe the two kd of electos by dffeet wave-fuctos Uestcted wave-fucto fo L atom s s s UF δ δ S usually, the two sets of spatal obtals use the same bass set

15 UF wave-fuctos ae ot egefuctos of S opeato!!! sp cotamato c c 4 4 c > - exact doublet state 4> - exact quatet state -appoxmately a sglet - appoxmately a doublet 6> - exact sextet state Fo a UF wave-fucto, the expectato value of S s: S S S UF exact δ δ S whee: S exact S UF S exact sp poecto pocedues Gaussa