H 2+ : A Model System for Understanding Chemical Bonds

Transcription

1 : Modl Sytm fo Undtanding Chmical ond a - b R Th fit iu w hav to dal with i th multipl nucli; now w can hav nucla vibation and otation fo th fit tim. Impotant digion: on-oppnhim appoximation. V NN E l Rationalization fo bonding Ovall potntial can b ationalizd in tm of comptition btwn ) intnucla pulion, which blow up a R dca, and ) th puly lctonic ngy, which i mo favoabl a R dca. R0 Rbig

2 Gnal Molcula amiltonian ˆ Tˆ Tˆ V V V N NN N ˆ T N Tˆ Kintic h α h m m α i i α N nucla ; lctonic i,j indic ov lcton α,β indic ov nucli Z nucla chag V NN V N V Potntial α β > α Z α α i i j> i Z αβ Z ij β α iα

3 on-oppnhim ppoximation ˆ Tˆ Tˆ V V V N NN N Thi tm i toublom bcau it coupl lcton motion to nucla motion. Without it, w could olv fo nucla ignfunction and lctonic ignfuction indpndntly. Th on-oppnhim appoximation invok an adiabatic paation of th nucla and lctonic motion, on th bai of th lag diffnc in ma btwn nucli and lcton. Claically you can nviion that th nucli mov much low than th lcton, uch that th lctonic dg of fdom intantanouly adjut to th nucla poition. Opationally, w mak th lctonic amiltonian paamtically dpndnt on nucla poition.

4 on-oppnhim ppoximation, cont d h Z ˆ ff m i α i i α ( ) α α i VNN i j ij Thi appoximation undli th nti concpt of potntial ngy ufac, and by xtnion, foc fild (any tim you aociat a ingl ngy with a givn nucla configuation). Oth adiabatic appoximation paat vibation fom otation. Impotant bakdown of th O appoximation includ intnal convion and intytm coing (adiationl tanition impotant in photochmity). Thi can b addd in bcau it a contant fo fixd nucla poition.

5 ack to a - b R ˆ h m It tun out that th a analytical olution, obtaind in confocal lliptic coodinat: z z a a R R b b ( z ) ( z ) you might imagin, th divation i complx, and in any ca, thi i not a gnal mthod. Intad, lt appoach thi a a vaiational poblm. a b φ angl aound intnucla axi

6 Vaiational Tatmnt of a tial wavfunction, lt u omthing vy impl, pcifically a um of obital cntd aound ach nuclu (thi fit with out intuition about bonding aiing fom obital ovlap): Th cula dtminant i ES ES E ES c c obital cntd on nuclu obital cntd on nuclu c cofficint ES ES ES E to b dtmind 0 S ˆ ˆ S S S S ˆ ˆ

7 Clo Look at th Intgal J E E m m b b b a b a ˆ ˆ h h Th typ of intgal a fd to a Coulomb intgal, and togth with xchang intgal (nxt lid), play a vy impotant ol in lctonic tuctu calculation. Thi i jut th amiltonian fo a atom cntd on, and thu i an ignfunction of it.

8 Clo Look at th Intgal () K SE E m m a a a b b a ˆ ˆ h h Thi i an xchang intgal. Thi i jut th amiltonian fo a atom cntd on, and thu i an ignfunction of it.

9 W Can Now Solv Scula Dtminant E ± ± ± S E 0 ± ± J ± ± K S ( ) ( ) ± S Th tm ally giv i to bonding. W v xpd th igntat in tm of th typ of intgal: ovlap, Coulomb, and xchang. Th implicitly dpnd on R, th intnucla ditanc. Fo, th tun out to b analytical: J K S R R R R ( R) R R 3

10 Eignfunction σ g σ no nod along φ angl g/u ymmty wt invion * antibonding * σ u R R ( ) ( ) S ( ) ( ) S onding molcula obital Enhancd lcton dnity in intnucla aa, hlping to cn poton fom ach oth. nti-onding molcula obital Dpltd lcton dnity in intnucla aa, dhilding th poton fom ach oth.

11 Eignngi (Potntial Engy Cuv) E - (anti-bonding) E (bonding) ~.7 Å

12 Wa thi all a wat of tim? Notic that th wavfunction do not involv J o K, th Coulomb and xchang intgal: ± ± ( ) ( ) ± S In fact, th fom of th wavfunction i dictatd by th ymmty of th poblm (mmb th ymmtic doubl wll potntial?). So w didn t ally hav to invok th vaiational mthod at all to olv fo th ignfunction. owv, th xpion w got fo th ignngi a non-tivial, and giv a fling fo how lctonic tuctu poblm a olvd in gnal, a w will.

13 Excitd Stat Exampl: Mixing p z obital Exampl: Mixing p x/y obital bonding Th a th π obital. anti-bonding Th molcula obital a familia, but thy can all b calculatd quantitativly uing th am vaiational tatmnt a fo th gound tat.

14 Molcula Obital Diagam ally povid th foundation fo th molcula obital appoach to lctonic tuctu.

15 Multi-Elcton Sytm No multi-lcton ytm can b olvd analytically. W will build up ou intuition with th modl ytm: : building up to multi-lcton atom : fit al molcul Int-lcton pulion i th biggt challng. lo nd to dal with lcton pin: Pauli pincipl.

16 Pauli Pincipl Idntical paticl a inditinguihabl in quantum mchanic. Exchanging paticl chang th wavfunction, at mot, by a ign chang (i.., applying pmutation twic mut tun you to oiginal wavfunction). Pˆ ˆ ˆ (, ) P ± (, ) ( ) P, Sub-atomic paticl hav a popty calld pin, which i an intinic angula momntum which can hav ith intg o half-intg valu. TERE IS NO CLSSICL NLOG OF SPIN; it i a puly QM popty. Popl omtim imagin th paticl lik littl top, pinning about an axi, but thi i a highly impfct analogy. Paticl with intg pin a calld boon; intchanging th giv no ign chang of wavfunction. Paticl with half-intg pin (lcton a impotant xampl) a fmion; intchanging th do chang ign of wavfunction.

17 Pauli Pincipl and Elcton Sytm ( ) ( ) ˆ,, P If ( ) (,, ) Thi can only b th ca if th wavfunction i zo. Concluion: Th i zo pobability that lcton can b imultanouly found in th am plac. Thi i on of th mo common dfinition of th Pauli pincipl; howv, a w v n, it mo gnal (and alo mo abtact) in ality.

18 Pauli Pincipl and Elcton Sytm (cont d) Elcton hav pin angula momntum of J/. Thi angula momntum i imila to ou tatmnt of angula momntum of th atom (i.., lcton otating aound th nuclu), xcpt that th total angula momntum can tak half-intg valu fo pin. y analogy, th a poibl valu of th m quantum numb, m±/. 4 poibl pin tat: α () α( ) α ( ) β ( ) β ( ) α( ) β ( ) β ( ) Thi nomalization contant nu that th qua of th wavfunction intgat to (h th intgation i not ov phyical pac, but ov th pin coodinat ). Th do not oby inditinguihability; ymmtiz! [ α () β ( ) ± β () α( ) ] ± α: m/ ( pin up ) β: m-/ ( pin down )

19 Now lt apply th pmutation opato to th... P P P ˆ ˆ P ˆ ˆ () α( ) α( ) α( ) α () β ( ) β ( ) β ( ) β [ α() β ( ) β () α( ) ] [ α() β ( ) β () α( ) ] [ α( ) β ( ) β ( ) α( ) ] [ α( ) β ( ) β ( ) α( ) ] Th 3 pin tat a ymmtic with pct to lcton xchang. Thi ingl pin tat i anti-ymmtic with pct to lcton xchang.

20 Pauli Pincipl pplid to Th gound tat of ha both lcton in th obital: ( ) ( ) Thi i ymmtic with pct to xchang ( ) ( ) ( ) ( ) P ˆ and thu, by itlf, do not oby th Pauli pincipl. owv, th complt wavfunction conit of both th obital (patial) wavfunction and th pin wavfunction. To obtain a wavfunction that oby th Pauli pincipl, th i only on pin wavfunction poibl (th anti-ymmtic on): gound () ( ) [ α( ) β ( ) β ( ) α( ) ] Pˆ gound gound

21 Pauli Pincipl pplid to (cont d) What about an xcitd tat of wh on lcton i pomotd to th obital: ( ) ( ) To oby inditinguihability, mut ymmtiz [ () ( ) ± () ( ) ] Th combination mut combin with th anti-ymmtic pin tat, and th - combo with th ymmtic pin tat, to oby Pauli pincipl: [ () ( ) () ( ) ] [ α() β ( ) β () α( ) ] nti-ymmtic combination: Singlt [ ( ) ( ) ( ) ( ) ][ α () β ( ) β () α( ) ] [ ( ) ( ) ( ) ( ) ][ α() α( ) ] [ () ( ) () ( ) ][ β () β ( ) ] Symmtic combination: Tiplt

22 Pauli Pincipl pplid to 3 Elcton Sytm (Li) Naivly, w might xpct th gound tat of Li to b ( ) ( ) ( 3) Of cou, w know fom intoductoy chmity that thi i not th ca. Th aon i that it i not poibl to atify th Pauli pincipl fo uch a wavfunction. Sinc th obital potion of th wavfunction i ymmtic with pct to xchang of any lcton, w would hav to find a pin function that i antiymmtic with pct to xchang of all pai. Thi i not poibl! Mo gnally, Pauli pincipl impli that lcton that occupy th am obital mut hav diffnt pin, i.., [ and a indic fo diffnt ( ) ( ) α( ) α( ) Pˆ obital.] ( ) ( ) [ ( ) ( ) ( ) ( ) α β β α ] [ () ( ) () ( ) ] α() α( ) Violation of Pauli pincipl! ut th a jut fin

23 Pauli Pincipl Symmtization fo ny Numb of Elcton Not that w can wit,.g., th gound tat atom wavfunction () ( ) () ( ) () ( ) [ ] α β β α a a dtminant ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) β α β α Thi i uggtiv. In fact, any anti-ymmtic wavfunction (with pct to pmutation of any pai of lcton) can b wittn a ( ) ( ) ( ) ( ) ( ) ( ) () () () O M M M L L L 3 3 3! f f f f f f f f f n Thi i calld a Slat dtminant

24 Why do thi wok? It bad on impotant, wll-known popti of dtminant:. Exchanging any ow in a dtminant multipli it by -. [Thi i quivalnt to paticl xchang.]. Exchanging any column in a dtminant multipli it by -. [Thi i quivalnt to obital xchang.] 3. If ow o column a idntical, th dtminant i zo. [Thi i a way of tating that no lcton may ha th am obital ND am pin.] Now, all of a uddn it ay to wit out th poply ymmtizd gound tat wavfunction fo Li: 6 ( ) α( ) ( ) β ( ) ( ) α( ) ( ) α( ) ( ) β ( ) ( ) α( ) () α() 3 () 3 β () 3 () 3 α() 3 compact notation fo witing thi Slat dtminant i

25 Gnal multi-lcton atomic amiltonian nucla chag ˆ ff m i i i h ( ) α i Z i j> i ij Ky nw tm, lativ to atom: int-lcton pulion; lad to hilding ffct, plitting of /p/d/f obital fo am valu of n, tc.

26 tomic Unit Mot lctonic tuctu calculation a pfomd uing an xtmly convnint t of unit calld atomic unit, in which both Planck contant and th lcton ma and chag hav valu of xactly : Ma: m (ma of lcton) Chag: (chag of lcton) ngula momntum: ħ (Planck contant) Lngth: unit of oh (0.53 Å; adiu of claical atom) Engy: unit of at (4.36 x 0-8 J; potntial ngy of claical atom) Tim: I don t know what th unit a calld (.4 x 0-7 ) Spd of light: 37 In th unit, th amiltonian implifi nicly,.g., fo multilcton atom: ˆ ff i ( ) α i i Z i i j> i ij

27 ˆ ˆ ˆ h h ff Ptubation thoy tatmnt of gound tat: U ()() wavfunction a th zo-od olution to b ptubd by th lcton-lcton pulion. () ( ) () ( ) 8 5 Z E E E E Z n Z n Z E E Expimntal valu i

28 - - R In atomic unit, and invoking th on-oppnhim pincipl, th amiltonian i ˆ ff R ( ) ( ) α

29 Valnc ond Tatmnt It i poibl to go though a vy imila vaiational tatmnt a w applid pviouly fo. I m not going to go though it in dtail. On ky diffnc i that w hav to dal with lcton inditinguihability and th Pauli pincipl. ut th ally ky diffnc i th lcton-lcton pulion. Th baic tatgy i to patition th amiltonian into a zo-od pat and th ptubation. Th a quivalnt way to do thi in th valnc bond appoach: ˆ ff ( ) α hˆ hˆ Coupling tm -atom amiltonian fo lcton intacting with nuclu -atom amiltonian fo lcton intacting with nuclu

30 O quivalntly... ˆ ff ( ) α hˆ hˆ Coupling tm -atom amitonian fo lcton intacting with nuclu -atom amitonian fo lcton intacting with nuclu Uing a bai t of -atom obital on ach atomic cnt, w wind up with intgal in th cula dtminant that a vy imila to th coulomb, xchang, and ovlap intgal that w divd fo, xcpt that now thy involv lcton, which mak thm mo complicatd. [It tun out that all th pin tuff baically dop out in th nd; it mo impotant fo 3 and mo lcton. So th intgal involv only patial (obital) intgation.]

31 ( ) ( ) 0 ' ' ' ' S SK K J J E E J J ES K SK S ES K SK S E J J ± ± () () () () () () () ( ) () ( ) () ( ) () ( ) K J S K J Th a idntical to th intgal divd fo, and a fd to a -cnt intgal bcau thy involv intgation ov th poition of jut on lcton. Th a nw lativ to and ai fom lcton-lcton pulion. Thy a -cnt intgal. [Can gt up to 4- cnt intgal fo lag ytm.]

32 lthough th ignngy xpion a much mo complicatd, th wavfunction wind up looking vitually idntical (again, ymmty of th amiltonian dictat thi): ± ( ± S ) [ ( ) ( ) ± ( ) ( ) ] To gt a complt wavfunction, w hav to add in th pop pin wavfunction to atify th Pauli pincipl, and w wind up with a inglt gound tat and a tiplt xcitd tat. numb of impovmnt a poibl to thi valnc bond tatmnt:. U mo ophiticatd bai t (a oppod to th impl atom function) that tak into account lcton cning of th nucli (impl tatgy: cal th chag on th nucli down a bit).. dd in p-obital, tc. Thi i fd to in th litatu a Gnalizd Valnc ond. 3. dd in ionic tuctu. [W ll com back to thi in a cond.]

33 Molcula Obital Tatmnt Th baic ida: Intad of going back to th lina combination of atomic obital bai t that undli th valnc bond appoach, why not u th ult that w obtaind fo ( molcula obital ) a th bai t fo. In oth wod, th bai would b ( ) ( ) σ g σ g o mo pcily, a poply anti-ymmtizd Slat dtminant σ σ g g ( ) α( ) σ g ( ) β ( ) ( ) α( ) σ ( ) β ( ) g Thi tim, w goup tm in th amiltonian diffntly:

34 ˆ ff hˆ hˆ Coupling tm amiltonian fo lcton amiltonian fo lcton Th concptual advantag i cla; w know th olution of th zo-od ingl-lcton amiltonian, and o w only nd to wok on th pky lcton-lcton pulion. Still, w know that th molcula obital can b xpd imply in tm of impl atomic obital, a in th valnc bond appoach, o th mut b om ptty dp connction btwn th appoach. Fo xampl, σ g ± ( ) [ ] ± S

35 So lt fom a lcton molcula obital (fogtting about pin fo th momnt): σ g () σ g ( ) [ ( ) ( ) ][ ( ) ( ) ] () ( ) () ( ) () ( ) () ( ) Thi i jut th valnc bond ignfunction but w alo hav th xta tm. Phyically, th xta tm copond to placing both lcton on on atom, i.., thy pnt ionic tat: -. Th i a nic connction with Lwi dot tuctu: vu : : So th molcula obital tatmnt includ ionic tat, but valnc bond do not. Do thi man that MO thoy i btt than V? Y and no. Th i om pobability of finding both lcton on th am atom, UT th pobability i lativly mall (du to pulion). In thi naiv MO appoach, th i a 50/50 chanc of finding th lcton on th am atom! V appoach i fixd by adding ionic tat into th bai t. MO appoach i fixd by adding in configuation intaction (includ xcitd, antibonding tat in th bai t)...

36 ut what xcitd tat do w mix in?? * σ u σ g Total ymmty: g g u u u u Th total amiltonian itlf i ymmtic with pct to invion (i.., g ), o intgal of th typ found in th cula dtminant a zo, i.., σ ˆ σ g 0 Thu, th fit xcitd configuation that can mix with th gound tat i * σ u u * () σ u ( ) [ ( ) ( ) ][ ( ) ( ) ] () ( ) () ( ) () ( ) () ( ) Not that th ionic and covalnt bai function com in with oppoit ign fom th gound tat bai function, allowing thm to b dcoupld.

37 Concluion of Invtigation Valnc bond ionic tm Molcula obital configuation intaction W v now achd th pinnacl of puity in thi cou. Th vaiational pincipl allow highly accuat quantum igntat, but at a cot. Evything fom h on in will b a i of incaingly datic appoximation, anging fom at-fock and DFT, to molcula mchanic with xplicit and thn implicit olvnt, and finihing with docking. It a long way down!