ˆ 2. Chapter 4 The structure of diatomic molecules. 1 Treatment of variation method for the H 2+ ion 1. Shroedinger equation of H 2. e - r b.

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Percival Norman
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1 Chpte 4 The stutue of tom moleules Tetment of vton metho fo the on. hoenge equton of Bon-Oppenheme Appoxmton The eletons e muh lghte thn the nule. Nule moton s slow eltve to the eleton moton. A θ e - R B The hmltonn opeto e hoenge equton of R Whee n R e elte R osθ y : Moleul Otl Theoy We oul now solve:,r,r R,R e Ths s possle ut teous n e solve extly usng onfol ellptl oontes: z R ξ /R η - /R s otton oun z π; ξ ; - η

2 ele Fξ,η π -/ e m whee m, ±, ±, ±3, The ssote quntum nume s λ. otl ngul momentum λ m h eleton level wth m s ouly egenete,.e. m,- m toms: l,,,... n the tom otls e lle: s,p,, et. toms: λ,,,... n the moleul otls e: σ, π, δ, et. λ lette σ π δ 3 4 γ ele Fξ,η π -/ e m σ λ lette σ π π δ 3 4 γ λ m δ

3 . The Vton Theoem Fo ny well-ehve wvefunton, the vege enegy fom the mltonn of the system s lwys gete o lose to the ext goun stte enegy fo tht mltonn, > < xmple: Devse tl vton funton fo the ptle n one-mensonl ox of length l. A smple funton tht hs the popetes of the goun stte s the pol funton: x l x h m fo x l x l h lx x lx x x 5 l x l x x l 3 5h h > < 4 ml 8ml π l 3 l 6m

4 δ The genfuntons of n emtn opeto onsst of omplete set.,, 3 onsst of n othogonl nomlze set of wvefuntons ε stte goun > < ts Poof > < ε mke pmete, the ustng ε Bse on ths pnple, the pmetes e egulte y the mnmzton outne so s to otn the wvefunton tht oespons to the mnmum enegy. Ths s tken to e the wvefunton tht losely ppoxmtes the goun stte. > < < > ε 3. Lne Vton Funtons n n n f f f f... A lne vton funton s lne omnton of n lnely nepenent funtons f, f, f n.

5 xmple ε s sme ε x y x x x y y x mke ε ε x y let ε

6 sequl equton me equton get get, esolve, get The lge equton hs oots, n.... n n n.. n n.. n n n nn.. n n nn The lge equton hs n oots, whh n e shown to e el. Angng these oots n oe of nesng vlue: n. Fom the vton theoem, we know tht the lowest vlue of oot W s the uppe oun fo the system s goun-stte enegy. W Moeove, t n e pove tht the lne vton metho poves uppe ouns to the eneges of the lowest n sttes of the system. W, 3 W 3,, n W n, We use the oots s ppoxmtons to the eneges of the lowest sttes. If ppoxmton to the eneges of moe sttes e wnte, we moe funtons f k to the tl funton. The ton of moe funtons f k n e shown to nese the uy of the pevously lulte eneges.

7 3. The soluton of Fo tht s : e s A π A θ e - R B e s B π Note : we hve s mny lne omntons s we hve tom otls Tl funtons sequl equton Q hs the sme fom s, ± f f α β α β

8 equton sequl to susttutng / / / / ] [ onton nomlzton equton sequl to susttutng

9 β α β α - Cn we eue the n fom the moleul symmety? Ovelp ntegl α J J J R R R R 5.5% oulom ntegl

10 β K R R R R esonne ntegl K J K J K J K J β α

11 The lulte n expementl moleul potentl enegy uves fo hyogen moleule-on. J K J - K - Moleul Otl Theoy A epesentton of the onstutve ntefeene tht ous when two s otls ovelp n fom onng σ otl. σ s

12 Moleul Otl Theoy The eleton ensty lulte y fomng the sque of the wvefunton. Note the umulton of eleton ensty n the ntenule egon. ρ Moleul Otl Theoy A epesentton of the estutve ntefeene tht ous when two s otls ovelp n fom n ntonng σ otl. - σ s

ρ Moleul Otl Theoy A moleul otl enegy level gm fo otls onstute fom the

13 Moleul Otl Theoy The eleton ensty lulte y fomng the sque of the Wvefunton. Note the elmnton of eleton ensty fom the ntenule egon. ρ Moleul Otl Theoy A moleul otl enegy level gm fo otls onstute fom the ovelp of s otls; the septon of the levels oespons to tht foun t the equlum on length.

14 Moleul otl theoy n tom moleules. Moleul otl theoy. vey eleton n moleule s n n vege potentl fel of the nule n the othe eletons. The stte of eleton s ese y the wvefunton the one-eleton wvefunton ϕ,,... n... n n. The fomton of moleul otl MO. The MO my e ppoxmte fom the lne omnton of tom otls LCAO. Thee s equements n the fomton of MO: The AOs shoul hve omple enegy, hve omptle symmety, e le to hve mxmum ovelp.

15 Why shoul the AOs shoul hve omple enegy? sequl equton f [ [,, β, 4β ] 4β ] - >> β >, > - β, β. The fomton of moleul otl MO. The MO my e ppoxmte fom the lne omnton of tom otls LCAO. Thee s equements n the fomton of MO: The AOs shoul hve omple enegy, hve omptle symmety n e le to hve mxmum ovelp. The ulng-up pnple n moleules: Pul exluson pnple, the mnmum enegy pple n un s ule.

16 . The htest stuton n lssfton of moleul otl. σ-otl n σ-on σ s σ s σ p σ p s p z - - σ ps - - σ ps

17 . The htest stuton n lssfton of moleul otl. π-otl n π-on. The htest stuton n lssfton of moleul otl. δ-otl n δ-on

18 3. The stutue of homonule tom moleules. The goun stte eleton onfgutons F F σ u F Fo oxygen n fluone, p n s e well septe. π g negy p π u p p x,p y p z O : KKσ s σ s σ p π p 4 π p σ g σ u F : KKσ s σ s σ p π p 4 π p 4 s s σ g

19 negy Moleul Otl Theoy p B LUMO OMO B σ u σ g π g π u B p p x,p y p z The effet of ntetons etween s n p. At the stt of the seon ow L-N, we hve mxng of s n p. The esult s tht σ u s pushe own n enegy whees σ g s se. B : KKσ g σ u π u s σ u s N : KKσ g σ u π u 4 σ g σ g leton onfgutons σ g e 3 σ gs σ us L 6 KKσ g B KKσ g σ u π u C KKσ g σ u π u 4 N 3 KKσ g σ u π u 4 σ g N 4 KKσ g σ u π u 4 σ g O 5 KKσ gs σ us σ gp π up 4 π gp O 6 KKσ gs σ us σ gp π up 4 π gp F 8 KKσ gs σ us σ gp π up 4 π gp 4 s-p z mxng

20 3. The stutue of homonule tom moleules. The on oe Bon oes : n n n: letons n onng otls n:letons n ntonng otls Dtoms Bon oes : n n s-p z mxng Moleule L Be B C N O F Ne Bon Oe 3 Bon Length Å.67 n/ n/ Bon negy kj/mol 5 n/ n/ Dmgnet / Pmgnet p n/ p p n/

21 3. The stutue of homonule tom moleules. The moleul spetosopy - tem Moleul Otl Theoy Dtoms Tem symols Moleule Confguton Tem symol σ g Σ g pn multplty L Tz : Σ Π Δ T YML z Refleton Pty

22 Moleul Otl Theoy Dtoms Tem symols Moleule Confguton Tem symol σ g Σ g σ g σ u Σ u s s σ u e σ g σ u Σ g L σ g σ u σ g Σg s s s s σ g σ u Be σ g σ u σ g σ u Σ g pn multplty L Tz : Σ Π Δ s s T Refleton YML z Pty σ g Moleul Otl Theoy Dtoms Tem symols Moleule Confguton Tem symol B π u 3 Σg Δg Σg C π u 4 N 3σ g π u 4 Σg Σg 3σ g N 3σ g π u 4 Σ g π u pn multplty L Tz : Σ Π Δ T YML z Refleton Pty

23 Moleul Otl Theoy Dtoms Tem symols Moleule Confguton Tem symol N 3σg π u 4 π g Π g Πg 3σ u πg O 3σ g π u 4 π g 3 Σg Δg Σg F 3σ g π u 4 π g 4 Σ g π u pn multplty L Tz : Σ Π Δ T YML z Refleton Pty 3σ g 4. The stutue of hetonule tom moleules

MO Theoy fo eteonule Dtoms MO s wll no longe ontn equl ontutons fom eh AO. AO s ntet f symmetes e omptle. AO s ntet f eneges e lose. No nteton wll ou f eneges e too f pt. A nononng otl wll fom.

24 MO Theoy fo eteonule Dtoms MO s wll no longe ontn equl ontutons fom eh AO. AO s ntet f symmetes e omptle. AO s ntet f eneges e lose. No nteton wll ou f eneges e too f pt. A nononng otl wll fom. Ψ X mkes gete ontuton to the Ψ MO Ψ Y mkes gete ontuton to the Ψ MO xmple: F The F s s muh lowe n enegy thn the s so they o not mx. The F s otl mkes non-onng MO. We etnly on t hve to woy out the F s euse s MUC lowe n enegy. The s n F p s e lose n enegy n o ntet. The px n py on t hve the ppopte symmety though n theefoe fom nononng MO Only the pz n s mx. Kσ 3σ π 4

25 Isoeleton ule: The MO s on fomton n eleton onfgutons e sml mong the soeleton tom moleules. CO s soeleton wth N. KK3σ 4σ π 4 5σ eleton onfgutons of NO Moleule eletons eleton onfguton tem L 4 Kσ Σ Be 5 Kσ 3σ Σ C 7 Kσ 3σ π Π N 8 Kσ 3σ π 3 Σ O 9 Kσ 3σ π 3 Π F Kσ 3σ π 4 Σ BeO, BN KK3σ 4σ π 4 Σ CN, 3 KK3σ 4σ π 4 5σ Σ BeF CO 4 KK3σ 4σ π 4 5σ Σ NO 5 KK3σ 4σ π 4 5σ π Π

26 3 Vlene onvb theoy fo the hyogen moleule n the ompson VB theoy wth Moleul Otl theoymo In vlene onvb theoy we lolze one eleton to eh. A B o A B e e e e The etle-lonon tetment: f AB f AB The tl vton funton: Ψ f f AB AB We hve the vlene on wvefunton Ψ, VB N[A B A B] [ α β β α] In moleul otl MO theoy eh eleton moves ove the whole moleule. A B n A B e e Both eletons n e on the sme nule The unnomlze LCAO-MO wve funton fo the goun stte s: Ψ, MO N[A B][A B] [ α β β α] AA B B A B A B - - Covlent tems

27 Compson of MO n VB theoes VB Theoy epte toms e ought togethe to fom moleules. The eletons n the moleule p to umulte ensty n the ntenule egon. The umulte eleton ensty hols the moleule togethe. letons e lolze elong to spef ons. yzton of tom otls Bss of Lews stutues, esonne, n hyzton. Goo theoy fo petng moleul stutue. Moleul otl theoy Moleul otl theoy Moleul otls e fome y the ovelp n nteton of tom otls. letons then fll the moleul otls ong to the ufu pnple. letons e elolze on t elong to ptul ons, ut e spe thoughout the moleule. Cn gve ute on ssoton eneges f the moel omnes enough tom otls to fom moleul otls.

8. Two Ion Interactions

8. Two Ion Interactions 8. Two on ntetons The moels of mgnet oe hve een se on mltonns of the fom on J J zw. Wht s the physl ogn of ths two on ntetons 8. Dpol nteton The ogn of lge moleul fels nnot e the wek mgnet pol nteton CD