Lctur 6 Titl: Fundmntls of th Quntum Thory of molcul formtion Pg- In th lst modul, w hv discussd out th tomic structur nd tomic physics to undrstnd th spctrum of toms. Howvr, mny toms cn comin to form prticulr molculs,.g. Chlorin (Cl) nd Sodium (N) toms form NCl molculs. Bonding twn ions, s in th ngtiv chrgd chlorin ion nd th positivly chrgd sodium ion, could undrstood in th light of coulom intrction (ttrction) twn oppositly chrgd odis. But toms of th sm typ cn lso form onds, s for xmpl in th cs of H. It rmind, howvr, inxplicl tht two similr toms, which r lctriclly nutrl, could form ound stt. In this lctur, w will undrstnd th formtion of molcul from tom in th quntum mchnicl frmwork.
Pg-1 It only cm possil with th id of quntum mchnics to ttin fundmntl undrstnding for th formtion of molcul. Evn in th cs of ionic onding, sic nw insights hv n otind through quntum thory. First, for xmpl, it must undrstood why th ions form in th first plc, nd why th lctron which is trnsfrrd from sodium to chlorin thus finds n nrgticlly mor fvorl stt. In th following, w will dvlop som importnt sic ids for th quntum thory of chmicl onding. Howvr, Physics & Chmistry r still fr wy from complt solution to ths prolms. To undrstnd chmicl onding, th intrctions of svrl prticls must tkn into ccount: givn n tomic nucli nd m lctrons, on would hv to find th complt wvfunction nd th corrsponding nrgis of th totl systm. It is usful to kp in mind tht th nuclr msss r much grtr thn thos of th lctrons. Thus th lctronic motions r much fstr thn nuclus. Thn w my ignor th motion of th nucli nd trt thm s fixd. In tomic physics, w wr l to otin much informtion from spctroscopic osrvtions nd could dirct our ttntion to oth th ground stts nd th xcitd stts. In th study of chmicl onding, th dtrmintion of th wvfunction of th ground stt of th prticulr molcul plys mor importnt rol.
Pg- Th Hydrogn Molcul Ion + H Crtinly th simplst cs of chmicl onding occurs in th hydrogn molcul + ion H. This spcis is osrvd s ound stt in gs dischrgs in hydrogn tmosphr, in such gs dischrg, th hydrogn molcul loss on lctron. Th onding nrgy, quivlnt to th dissocition nrgy, hs n dtrmind to.65 V. W r dling with two nucli ( nd in figur-6.1) nd on lctron. If th nucli r fr rmovd from on nothr, w cn imgin tht th lctron is loclizd on on nuclus or othr. Th wvfunctions r thos of th hydrogn tomic ground stt. So th Hmiltonin for nuclus, = m r H ( r ) E ( r ).. (6.1) r - r And corrspondingly for nuclus, ( ) ( ) H r = E r.. (6.) R Figur-6.1 So, E = E = E If w lt th nucli pproch on nothr, th lctron, which ws, for xmpl, t first ttchd to nuclus, will xprinc th ttrctiv Coulom forc of nuclus. Convrsly, n lctron which ws t first ound to nuclus will xprinc th ttrctiv Coulom forc of nuclus. W must thrfor st up Schrodingr Eqution which contins th Coulom potntil for oth. Furthr in ordr to clcult th totl nrgy of th systm, w must tk into ccount th Coulom rpulsion twn nucli. Th dditionl nrgy R is not dirctly rltd to th nrgy of th lctron, it will only produc constnt shift of ll th nrgy ign vlus. W will introduc it t th nd.
Pg-3 Rfrring to th figur 6.1, lt us dfin, R = R R r = r R r = r R Thus, Schrödingr qution, thn ψ = Eψ m r r... (6.3) Hr E nd ψ r nrgy ign vlu nd th wvfunction for th whol systm rspctivly nd r yt to clcultd. W will us th prturtion mthod. In principl, th lctron could in th nighorhood of nuclus or of nuclus, with th sm nrgy. Ths two stts nd r thus dgnrt. Now howvr, th othr nuclus, from which th lctron is y chnc mor distnt, cts, s prturtion to th lctronic stt. W thus xpct tht th dgnrcy will liftd y this prturtion. In th prsnc of th dgnrcy, w tk th totl wvfunction is th linr comintion of th hydrogn wvfunctions dfind in qution 6.1 nd 6. nd dfind s, ψ = c + c 1 Whr c 1 nd c r th cofficints. This is gnrlly known s linr comintion of th tomic oritl (LCAO) nd ψ is known s th molculr oritl. So, putting this, in qution 6.3, nd rrrnging w gt, H c1 + H c = E c1 + c r r ( ).. (6.5) Whr H = m r nd H = m r
Pg-4 Sustituting th vlus from qutions 6.1 nd 6. in 6.5, w gt, E c E c Ec Ec 1 + = 1 + r r E E c + E E c = 1 E r E r.. (6.6) Whil nd r functions of position, th cofficints r indpndnt of positions. W ssum tht th functions nd r rl, s in th cs of hydrogn tom ground stt wvfunctions, nd th functions nd r not orthogonl sinc th lctron is ssocitd with two diffrnt nucli nd. So, dv = S nd dv = dv = 1 Now, Eqution 6.6 is multiplid y.. (6.7) E c1 dv + c E dv r = r E c c r r dv + c E dv c r r dv = ( ) ( ) ( ) ( ) ( ) 1 1 r r ( Ec1 c1c) ( Ec S c) ( E C) c ( E S ) c + = + = Whr, ( ) ( ) r ( ) ( ) r 1 + r r dv = C + r r dv =.(6.8)
Pg-5 Eqution 6.6 is multiplid y nd following th sm procdur w gt, ( ) ( ) E S c + E C c =.. (6.9) 1 So, to gt th solution, following dtrminnt from qutions 6.8 nd 6.9 should vnish. Th dtrminnt: E+ C E S + c1 = E S + E+ C c So, ( E C) ( E S ) + + = E + C + E C E S E S = E 1 S C S E C ( ) ( ) ( ) + + = ( C S ) ( C S ) ( S )( C ) 1 ( S ) ± 4 41 E = ( ) C S ± C + S C S C + + C S S = ( C S ) ( CS ) ± = ( 1 S ) ( C S ) ( CS ) ± = Two solutions: ( 1 S ) ( 1 S ) ( i) ( ii) C+ S + CS C+ = 1 S 1+ S C + S + CS C = 1 S 1 S.(6.1) C± So, E = 1 ± S
Pg-6 If w tk, solution (i) in qution 6.1 nd thn, Sustituting C+ E = 1+ S ( E+ C) c1+ ( E S + ) c = + ( C+ ) in Eqn.(6.8) C S + C c 1+ + c = 1+ S 1+ S C + C + CS CS S + + S c1 + 1+ S 1+ S CS CS c1+ c = 1+ S 1+ S CS CS c1 c = 1+ S 1+ S c = So, c1 c c = = nd thus, ψ c( ) = + + E E = E C+ E = E = E (6.11) + 1+ S Now if w tk, solution (ii) in qution 6.1 nd C Sustituting E = in Eqution.(6.9) 1 S ( E S+ ) c1+ ( E+ C) c = ( C ) S C + c1+ + C c = 1 S 1 S CS + S + S C + + C CS c1 + c = 1 S 1 S CS CS c1+ c = 1 S 1 S So, c1 c c = = nd thus ψ c( ) C = = 1 S E E E =
Pg-7 Lt us undrstnd th mning of this wvfunctions ψ + nd ψ. Figur-6. shows th plotting of wvfunctions nd s wll s ( ) +. ( ) ψ = + c + X nsity distriution of lctron Figur -6. Th following osrvtions cn md. () Symmtric wvfunction ψ + is formd y +. Bcus of th ovrlp, th occuption proility for ψ + twn two nucli is incrsd. () Thus th dnsity distriution of lctron in th ψ + stt shown in th lowr prt of figur 6. incrss s th distnc twn nd dcrss.
Pg-8 Antisymmtric wvfunction ψ is formd from. Th occuption proility is clrly zro in th pln of symmtry. ( ) ψ = c X nsity distriution of lctron Figur-6.3 As cn sn from figurs-6. nd 6.3 tht th distnc twn nd is n importnt fctor to hv finit ovrlp twn th nd. Now w will concntrt on th nrgy vlus E + nd E. To do tht w hv to undrstnd th mning of th quntitis S, C nd. W will try to vlut ths quntitis through th digrms.
Pg-9 Th quntity dv = S From figur 6.4 it is clr tht this quntity S is considrl whn R is sufficintly clos nd S incrss whn R dcrss. Th quntity S is known s ovrlp intgrl. (i) X (ii) 1. 1..8.6 (iii).4. X. 1 3 4 rdil distnc Figur-6.4 Figur-6.4(i) rprsnts th 1s tomic oritl of hydrogn for two nucli nd. Figur-6.4(ii) shows tht whn th nucli r clos nough th ovrlp incrss. Figur-6.4(iii) plots th vlu of ovrlp intgrl S with rspct to th intrnuclr distnc R. Whn oth th nucli r coming closr th ovrlp incrss nd th vlu of S incrss. Whn thy fll on ch othr, th ovrlp intgrl coms 1. Howvr, this sitution dos not ris du to th nuclr-nuclr rpulsion trm. W will visuliz this pictur ltr.
Pg-1 C = + r r dv ( ) ( ) r = + r r dv ( ) ( ) r This trm is known s th dirct coulom trm. Figur-6.5(i) shows th oritl th coulom ttrction potntil. As th two nucli pprochs to ch othr, th ovrlp twn thm incrss. Figur-6.5(ii) plots th trm C with rspct to th intrnuclr distnc R. Hr lso w s tht, this trm is pprcil whn th oth th nucli r clos nough. nd (i) X 1..8.6 (ii) r 4πε.4.. 1 3 4 rdil distnc Figur-6.5
Pg-11 = + r r dv ( ) ( ) r = + r r dv ( ) ( ) r (i) 1..8 (ii) X.6.4 r 4πε.. 1 3 4 rdil distnc Figur-6.6 Th trm is known s cross intgrl trm. Th mning of this is tht thr should ovrlp twn, nd coulom ttrction trm ( ) s shown in figur 4πε r 6.6(i). Th vlu of this intgrl is plottd in figur 6.6(ii). It is clr from this figurs tht whn thr is no ovrlp twn nd, th vlu of gos to zro.
Pg-1 In figur 6.7(i), th quntity SC,, r plottd togthr to hv comprison. It cn sn tht ll ths quntitis dpnd on th R. 1. 1..8.6.4.. (i) S C curv 1 curv curv 3 1 3 4 rdil distnc Figur-6.7 1..5..5 (ii) C 1 S 1. 1 3 4 rdil distnc C+ 1 + S C+ C In figur 6.7(ii), th quntitis nd r plottd with rspct to th 1 + S 1 S intrnuclr distnc R. As cn sn, from this plot tht ths quntitis r dgnrt whn th two nucli r fr wy from ch othr. If two nucli pproch on nothr th quntitis split up in wy dpnding on whthr symmtric or ntisymmtric / onding or ntionding. In Symmtric cs nrgy rducs whrs in Antisymmtric cs nrgy incrss.
Pg-13 Now, if w dd nuclr nrgy, th nrgy digrm is shown in figur-6.8 symmtric cs thr is minimum, so it forms ound stt. Antionding is not stl stt. Equilirium intrnuclr distnc 1..5 Anti-onding Bonding E..5 1 3 4 rdil distnc Figur-6.8 Physicl mning: in th ound stt th proility dnsity of lctron is quit lrg in th middl, profit Coulom ttrction from oth th nucli, nrgy gos down. Whrs in th ntionding cs, only lctron xprincs ttrctiv forc of on nuclus t tim. Th intrnuclr distnc whr th nrgy is hving minimum is known s quilirium intrnuclr distnc. Thus, this molcul H + - ion is stl with th onding nrgy lowr thn th individul hydrogn tom.
Pg-14 Rcp From this lctur w hv dvlopd th sic id of th quntum thory of chmicl onding. To undrstnd chmicl onding, th intrctions of svrl prticls should tkn into ccount: givn n tomic nucli nd m lctrons, on would hv to find th complt wvfunction nd th corrsponding nrgis of th totl systm. It is usful to kp in mind tht th nuclr msss r much grtr thn thos of th lctrons. Thus th lctronic motions r much fstr thn nuclus. This is known s Born-Oppnhimr pproximtion. In th study of chmicl onding, th dtrmintion of th wvfunction of th ground stt of th prticulr molcul plys mor importnt rol. Th dgnrt tomic nrgy lvls split du to th intrction twn th tomic oritls. Th no. of molculr oritl is sm th no. of tomic oritl. Th molcul will stl only if minimum is chivd during th clcultion of ground stt nrgy of th molcul. Th minimum riss cus th coulom ttrctions ring th nucli closr on th othr hnd th nuclr-nuclr rpulsion rstrict thm to com too clos. An optimum distnc, th two nrgis form minimum for th molcul nd thus quilirium is chivd. Th intr-nuclr distnc t quilirium is th ond lngth of th molcul.