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
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
. 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
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.
xmple ε s sme ε x y x x x y y x mke ε ε x y let ε
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.
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
equton sequl to susttutng / / / / ] [ onton nomlzton equton sequl to susttutng
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
β K R R R R esonne ntegl K J K J K J K J β α
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
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
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.
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.
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.
. 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
. 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
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
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
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/.59.4...4 n/ Bon negy kj/mol 5 n/ 89 69 94 494 55 n/ Dmgnet / Pmgnet p n/ p p n/
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
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
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. 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
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
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
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.