Titl: Vibrational structur of lctronic transition Pag- Th band spctrum sn in th Ultra-Violt (UV) and visibl (VIS) rgions of th lctromagntic spctrum can not intrprtd as vibrational and rotational spctrum thos ar discussd in prvious lcturs. Th frquncis in th UV and visibl rgion ar much gratr than th vibrational and rotational frquncis. Thus lin spctra sn in th UV and visibl rgion in th atomic transitions ar du to th lctronic transitions. Thus th band spctra sn in th UV and visibl rgion in th molcular transitions ar possibly du to th lctronic transitions. In th following, w will larn th analysis of th molcular spctra in th UV-VIS rgion.
Pag- W hav larnt from th prvious lctur-8 that th Schrodingr quation of a diatomic molcul can b writtn as n i + V Ψ = EΨ.(.) µ m i= Whr first trm is th kintic nrgy of th nucli, scond trm is th kintic nrgy of th lctrons and V is th potntial and is dfind as th sum of lctron-lctron rpulsion, lctron-nuclar attraction and nuclar-nuclar rpulsion Z Z ZZ V = + i > j= ri j i= ria i= rib R = V V + V n n n A B A B.. (.) n nn Using Born-Oppnhimr approximation w can writ th total wavfunction ( rr, ) φ( rr, ) χ( R) Ψ =..(.) Whr φ ( rr, ) is th lctronic wavfunction dpnds on th lctronic coordinat r and inhrntly dpnds on th nuclar coordinat R and χ ( R) is nuclar wavfunction. Th lctronic nrgy E ( ) n i + V rr rr = E R rr m i= R can b calculatd by fixing th nucli at R from (, ) φ(, ) ( ) φ(, ) (.) As w hav undrstood that this lctronic nrgy with th nuclar-nuclar rpulsion acts as potntial nrgy for th vibrational motion of th molcul. Thrfor, th xcss of th nrgy of a non-rotating molcul ovr th minimum lctronic nrgy must b considrd as th vibrational nrgy. In addition thr is a rotational nrgy of th molcul. Thus th total nrgy E of th molcul is th sum of ths thr nrgis, E = E + E + E.(.4) υ R
Pag- If w writ quation.4 in trms of th wav-numbr unit thn T = T + G( υ) + F( J).(.5) Or, T= T + ω υ+ ωx υ+ + BJ υ ( J+ ) (.6) Whr, ( ) G υ = ω υ+ ωx υ+ + ωy υ+ (.6(a)) And F( J) = BJ( J+ ) υ Th gnral graphical rprsntation of two lctronic stats with thir vibrational and rotational lvls is givn in figur-.. In this figur, vibrational and rotational stats ar also shown for two diffrnt lctronic stats. Hr th singl prim rfrs to th final or xcitd stat and doubl prim is usd for th initial or ground stat. Potntial curv of xcitd lctronic stat J ' = 5 J ' = J ' = 5 J ' = J ' = 5 J ' = υ ' = υ ' = υ ' = Potntial curv of ground lctronic stat Vibrational lvls Rotational lvls Intr-nuclar distanc J " = 5 J " = J " = 5 J " = J " = 5 J " = υ " = υ " = υ " = Figur-.
Pag-4 Th potntial nrgy curv of diffrnt lctronic stats of Li molcul is shown in figur-.. W will larn th nomnclatur of ths lctronic stats in nxt lcturs. Th vrtical arrow dsignats th lctronic transitions btwn th two lctronic stats. Ths transitions consist of vibrational as wll as rotational transitions btwn ths lctronic stats. Thus, a closr look into ths lctronic transitions provids information about th vibrational and rotational structur of th lctronic transitions. This dpnds on th rsolution of th obsrvd spctrum. Now w will focus our attntion to ths structurs. Enrgy Π u Li S Li P ( ) + ( ) Σ + u Σ + u Li S Li S ( ) + ( ) Σ + g Figur-. Intr nuclar distanc Not: Th lctronic transitions ar always drawn by vrtical lin on th potntial curv. Th maning is that th lctronic transitions occur much fastr than th nuclar motion. So during th transition th intr-nuclar distanc rmains sam.
Pag-5 Vibrational structur of lctronic transitions Using quation-.5, th wavnumbrs of th spctral lins btwn two lctronic stats ar givn by, ν = T T = ( T T ) + ( G( υ ) G( υ )) + ( F( J ) F( J )) (.7) Hr, final initial T initial and T final ar th trm valus of th initial and final stats rspctivly. If th final stat is th xcitd lctronic stat and initial stat is th ground stat, thn this is known as lctronic absorption. If th transition is from xcitd lctronic stat to ground or othr lowr lctronic stat thn it is known as mission. Hr th singl prim rfrs to th final stat and doubl prim is usd for th initial stat. So, from quation.7, w can writ that th transition nrgy (in cm - ) is th sum of thr (lctronic, vibrational and rotational) transition nrgis. ν = ν + ν + ν υ r Not that, for an lctronic transition ν is constant and th high rsolution of this spctrum will giv th structur arising from th vibrational and rotational nrgis btwn ths two lctronic stats. For th tim bing, lt us considr that rotational faturs ar much smallr and hnc lt us nglct th rotational part. Thus, th coars structur of th lctronic transition is known as vibrational structur. Substituting th valus of G( υ) from quation-.6(a) in quation-.7, w gt ν = ν + ω υ + ω x υ + + ω y υ + +... ω υ + ω x υ + + ω y υ + +.....(.8)
Pag-6 In actual vibration analysis, howvr, a modifid form of quation-.8 is mployd. Equation-.8 rfrs to th vibrational nrgy trms to th minimum of th potntial nrgy curv whr G( υ ) =. In a modifid form, th vibrational nrgis ar masur with rspct to th ground vibrational lvl i.. vibrational lvl with υ =. Th vibrational trm corrsponding to this υ = lvl can b obtaind from quation-.6(a) as: G x y 4 8 ( ) = ω ω + ω +..(.9) Th nw vibrational nrgy trm valus ar thn masurd with rspct to this G() valu, which, is chosn as nw zro lvl and th corrsponding vibrational trm valus ar rprsntd by th symbol G ( ) υ and may b xprssd as a function of vibrational quantum numbrυ as follows: ( υ) = ωυ ω υ + ω υ G x y Equations (.6(a)),(.9) and (.) ar rlatd by:.(.) ( υ) = ( υ) + ( ) or G ( υ) G( υ) G( ) G G G = (.) Equation.8 may b writtn with (υ ) for th lowr lctronic stat and with (υ ) for th uppr lctronic stat. ( x y ) ( x y ) ν( υ, υ ) = ν + ωυ ω υ + ω υ +... ωυ ω υ + ω υ +... (.) Hr ν is th transition from υ = to υ = and is known as origin of th transition. Whr, ν = ν + ω ω x + ω y +... ω ω x + ω y +... 4 8 4 8..(.) Th transformation is don with th rlations ω = ω ωx + ωy +... 4 ωx = ωx ωy +... ω y = ω y +...
Th cofficintsω y and ω y ar vry small and can b nglctd. Using ( ) for lowr stat and ( ) for uppr stat th abov rlations for two lctronic stats ar: ω = ω ω x ω = ω ω x ω x = ωx ωx = ωx Ths quations giv th rlations btwn two typs of vibrational constants.
Pag-7 Vibrational progrssions : In figur.(a), th group of transitions ar originating from th sam vibrational stat of th uppr lvl υ =, υ =, υ =,. and trminating to th diffrnt vibrational lvls υ of th lowr stat. This is known as υ - progrssion. On th othr hand, in figur.(b), th group of transitions ar originating from th sam vibrational stat of th 4 lowr lvl υ =, υ =, υ υ =,. and trminating to th diffrnt vibrational lvls υ of th uppr stat. This is known as υ -progrssion. (a) υ = υ = υ = υ = υ = 4 4 υ 4 υ (b) 4 υ = υ = υ = υ = υ = 4 υ Figur. Pag-8
υ Somtims, th transitions for a particular vibrational mod is dfind as ν υ. In figur-.4, svral transitions btwn th two vibrational lvls of diffrnt lctronic stats ar shown. ν ν ν ν ν ν ν Figur-.4 ν ν ν υ = υ = υ = υ = υ = υ υ = = υ = Pag-9
Now, th potntial nrgy curv of a particular lctronic stat along with vibrational lvls is shown in figur-.5. If w analyz th vibrational lvls w can gt th information about th bond dissociation nrgy of that molcul. Th dissociation nrgy is th rquird nrgy D for which th vibrational transition will b on th top of th potntial nrgy curv as shown in figur-.5. Th diffrnc btwn th succssiv vibrational lvls will b zro. Lt us say that th vibrational quantum no. isυ. Thn δ G( υ) = ω ωxυd =. δυ D ω So υd =. ω x D = G ( υ ) = ωυ ω xυ D D D ω = 4ω x (.4) Thus if th ω and ωxar known for a particular lctronic stat th bond dissociation nrgy can b calculatd. Enrgy υ D D Potntial nrgy curv υ = D Figur-.5 Intr-nuclar distanc Pag-
Vibrational structur analysis of th Iodin absorption spctrum. If a continuous radiation in th visibl rgion is passd through cll containing Iodin vapor, a band spctrum can b rcordd using a spctromtr arising from absorption by iodin molculs (I ). This is an lctronic transition btwn th ground B Π and th xcitd + u X + g Σ lctronic stat as shown in th figur-.6. Th υ = to υ = is known as th origin transition and will hav th lowst frquncy (highst λ) and can b locatd nar 6 Å. Th othr discrt band xtnds up to about 5 Å in th plac grn rgion byond which a continuum can b obsrvd. Sinc th molculs ar at room tmpratur, th υ = is th most populatd and most of th obsrvd bands ar du to transitions for υ = to υ =,,,,..... Enrgy B Π + u I( P ) + I( P ) D Σ X + g ν I( P ) + I( P ) D Figur-.6 Intr nuclar distanc Pag-
Exprimntal st up : Th xprimntal st up is shown in figur-.7(a) and th obsrvd spctrum is shown in figur -.7 (b). (a) Broad band light sourc collimating lns Iodin sampl cll Light collction fibr Spctromtr (b) transitions from υ = Transitions from υ = Transition from υ = Figur-.7 Pag-
Obsrvations from th spctrum:. Each small hump, or pak such as th (6,) band labld on th spctrum, corrsponds to a transition btwn two vibrational lvls corrsponding to two diffrnt lctronic stats and is calld a band.. Each band is comprisd of svral hundrd lins, ach of which involvs diffrnt uppr and lowr rotational quantum numbrs as mntiond, ths lins ar not rsolvd in th prsnt xprimnt. Th rgion of maximum absorption in ach band is causd by many of ths lins falling togthr; it is calld th band had.. Thr ar thr sris of bands. Th highr nrgy sid (lowr in wavlngth) bands ar from υ = to υ =,,,,.... 4. This sris shows a convrgnc i.. th transitions xtnd up to th vibrational lvl at top of th potntial curv. 5. Sinc th spctrum is takn at th room tmpratur, th highr vibrational lvls of th ground lctronic stat ar also populatd. Th middl sris is from υ = and th highr wavlngth sris is from υ =. CALCULATION OF VIBRATIONAL CONSTANTS: As discussd arlir that in a givn lctronic stat th diffrnc btwn two succssiv vibrational stats (with quantum numbrυ and, and in units of cm-) is givn by: ( ) ( ) ( ) ( ) G = G + G = G + G υ+ υ υ υ υ substituting from quation. and simplifying w gt: ( ) ( ) G = x + = x + υ+ ω ω υ ω ω υ From this, w gt for υ =,,,,... tc: υ υ + G υ+ (.5) G G G ω ω ω ω = Χ = Χ ω 4ω ω ω Χ = Χ = 5 = ω ω Χ = ω 5 6 ω Χ Pag-
For iodin, w will concntrat th only vibrational bands for th xcitd lctronic stat and th transitions originating from υ =. Thn for th uppr lctronic stat, Gυ + = ω ω x = ω ω x..(.6) whr G υ+ is th diffrnc (in cm-) btwn two vibrational lvls withυ = and υ = in th uppr lctronic stat and is calld th first vibrational quantum of th uppr lctronic stat. Furthr, th diffrnc btwn two succssiv vibrational quanta givs approximatly th constant scond diffrnc, G as sn G υ Gυ = Gυ = ω x = ω x = constant.(.7) ( + ) + + + All th obsrvd vibrational bands originating from υ = ar givn in tabl-..
Pag-4 Tabl. Obsrvd vibrational bands in iodin absorption spctrum Srial NO. Quantum Numbr υ Wavlngth nm Wav No. /λ G υ + cm - = G ( υ + ) G ( υ Pag-5 6 57. 75.5 96.7 7 567.87 769.66 97.4 8 564.7 777.89 98. 4 9 56.7 78.46 94.57 5 558.7 7898.5 95.59 6 555.95 7987. 89.8 7 55. 87.7 85.49 8 55.77 856.4 8.67 9 4 548. 84.85 84.45 5 545.8 8.6 8. 6 54.57 896.89 75.84 7 54.4 8476.9 79.9 8 59.7 855.47 74.7 4 9 56.97 86. 7.55 5 54.94 869.69 7.67 6 5.99 876.8 68.9 7 5. 888.4 66.6 8 59. 889.6 64. 9 4 57.5 8956.6 64.46 5 55.79 99 6.7 6 54. 975.96 56.96 7 5.64 9.6 57.67 8 5.7 99.8 57.65 4 9 59.64 944.9 5.8 5 4 58. 996.8 5.7 6 4 56.87 947. 5.4 7 4 55.67 99.5 45. 8 4 54.47 947.48 45. 9 44 5.4 948.7 4.79 45 5.44 954.48 4. 46 5. 9557. 4.74 47 5.4 9594.78 7.55 48 59.7 96.9 7. 4 49 58.54 9664.4.4 5 5 57.7 9695.9.75 6 5 56.97 975. 9.4 7 5 56. 9754.6 9.
From quation.5, w hav Gυ + = ω ωx ωxυ. So w can plot th G υ+ vrsus υ and fit it with a linar function. From th fitting of th graph w gt th vibrational constants ω ωx and ωx from th intrcpt and th slop of th lin as shown in figur -.8 Go (v') in cm- 9 8 7 6 5 4 Equation y = a + Adj. R-Squ.99 Valu Standard E D Intrc..86 D Slop -.9955.8 5 5 5 4 45 5 55 Vibrational Quantum numbr Figur-.8 Slop = ω o x o = -.9955 ω ox o =.998 cm And, Intrcpt = ω ω o x o =. ω =.+.996 =.7 cm Now th dissociation nrgy in th uppr stat ω D = = 44.9 cm 4ω x Pag-6
Rcap In this lctur,. w hav undrstood th lctronic transitions of diatomic molculs. Th vibrational transitions btwn th two lctronic stats giv th vibrational structur of th lctronic transitions. Th vibrational constants of th two lctronic stats can b calculatd by analyzing th vibrational transitions. 4. Th υ = to υ = is known as th origin transition and will hav th lowst frquncy (highst λ) 5. Th group of transitions ar originating from th sam vibrational stat of th lowr lvl υ =, υ =, υ =,. and trminating to th diffrnt vibrational lvls υ of th uppr stat. This is known as υ -progrssion. 6. Th group of transitions ar originating from th sam vibrational stat of th uppr lvl υ =, υ =, υ =,. and trminating to th diffrnt vibrational lvls υ of th lowr stat. This is known as υ -progrssion. 7. Thr ar thr sris of bands in th absorption of iodin molcul. Th highr nrgy sid (lowr in wavlngth) bands ar from υ = to υ =,,,,.... 8. This sris shows a convrgnc i.. th transitions xtnd up to th vibrational lvl at top of th potntial curv. 9. Sinc th spctrum is takn at th room tmpratur, th highr vibrational lvls of th ground lctronic stat ar also populatd. Th middl sris is from υ = and th highr wavlngth sris is from υ =.