Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated.

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1 Rtatinal transitins Vibratinal transitins Electrnic transitins Brn-Oppenheimer Apprximatin is the assumptin that the electrnic mtin and the nuclear mtin in mlecules can be separated. This leads t mlecular wavefunctins that are given in terms f the electrn psitins (r i ) and the nuclear psitins (R j ): " mlecule (ˆ r i, R ˆ j ) =" electrns (ˆ r i, R ˆ j )" nuclei ( R ˆ j ) Invlves the fllwing assumptins: Electrnic wavefunctin depends n nuclear psitins but nt upn their velcities, i.e., the nuclear mtin is s much slwer than electrn mtin that they can be cnsidered t be fixed. The nuclear mtin (e.g., rtatin, vibratin) sees a smeared ut ptential frm the fastmving electrns.

2 Electrnic transitins: UV-visible Vibratinal transitins: IR Rtatinal transitins: Radi E Electrnic Vibratinal Rtatinal Must first cnsider mlecular mment f inertia: " I = m i r i i At right, there are three identical atms bnded t B atm and three different atms attached t C. Generally specified abut three axes: I a, I b, I c. Fr linear mlecules, the mment f inertia abut the internuclear axis is zer. See Physical Chemistry by Atkins.

3 Rtatin f mlecules are cnsidered t be rigid rtrs. Rigid rtrs can be classified int fur types: Spherical rtrs: have equal mments f inertia (e.g., CH 4, SF 6 ). Symmetric rtrs: have tw equal mments f inertial (e.g., NH 3 ). Linear rtrs: have ne mment f inertia equal t zer (e.g., CO, HCl). Asymmetric rtrs: have three different mments f inertia (e.g., H O). The classical expressin fr the energy f a rtating bdy is: E a =1/I a " a where a is the angular velcity in radians/sec. Fr rtatin abut three axes: E =1/I a " a +1/I b " b +1/I c " c In terms f angular mmentum (J = I): We knw frm QM that AM is quantized: E = J a + J b + J c I a I b I c J = J(J +1) Therefre, J(J +1) E J = I, J = 0, 1,,, J = 0, 1,,

4 Last equatin gives a ladder f energy levels. Nrmally expressed in terms f the rtatinal cnstant, which is defined by: hcb = I => B = 4"cI Therefre, in terms f a rtatinal term: F(J) = BJ(J +1) cm -1 The separatin between adjacent levels is therefre F(J) - F(J-1) = BJ As B decreases with increasing I =>large mlecules have clsely spaced energy levels. Transitins are nly allwed accrding t selectin rule fr angular mmentum: "J = ±1 Figure at right shws rtatinal energy levels transitins and the resulting spectrum fr a linear rtr. Nte, the intensity f each line reflects the ppulatins f the initial level in each case.

5 Cnsider simple case f a vibrating diatmic mlecule, where restring frce is prprtinal t displacement (F = -kx). Ptential energy is therefre V = 1/ kx Can write the crrespnding Schrdinger equatin as d " + [E #V]" = 0 µ dx d " µ dx + [E #1/kx ]" = 0 where µ = m m 1 m 1 + m The SE results in allwed energies E v = (v +1/)" # " = k & % ( $ µ ' 1/ v = 0, 1,, The vibratinal terms f a mlecule can therefre be given by G(v) = (v +1/) v v = 1 # k & % ( "c $ µ ' 1/ Nte, the frce cnstant is a measure f the curvature f the ptential energy clse t the equilibrium extensin f the bnd. A strngly cnfining well (ne with steep sides, a stiff bnd) crrespnds t high values f k.

6 The lwest vibratinal transitins f diatmic mlecules apprximate the quantum harmnic scillatr and can be used t imply the bnd frce cnstants fr small scillatins. Transitin ccur fr "v = ±1 This ptential des nt apply t energies clse t dissciatin energy. In fact, parablic ptential des nt allw mlecular dissciatin. Therefre mre cnsider anharmnic scillatr. A mlecular ptential energy curve can be apprximated by a parabla near the bttm f the well. The parablic ptential leads t harmnic scillatins. At high excitatin energies the parablic apprximatin is pr (the true ptential is less cnfining), and des nt apply near the dissciatin limit. Must therefre use a asymmetric ptential. E.g., The Mrse ptential: V = hcd e 1" e "a(r"r e ) ( ) where D e is the depth f the ptential minimum and a = µ" 1/ # & % ( $ hcd e '

7 The Schrödinger equatin can be slved fr the Mrse ptential, giving permitted energy levels: G(v) = (v +1/) v " ( v +1/) x e v where x e is the anharmnicity cnstant: x e = a µ" The secnd term in the expressin fr G increases with v => levels cnverge at high quantum numbers. The number f vibratinal levels fr a Mrse scillatr is finite: v = 0, 1,,, v max Mlecules vibrate and rtate at the same time => S(v,J) = G(v) + F(J) S(v,J) = (v +1/) v + BJ(J +1) Selectin rules btained by cmbining rtatinal selectin rule J = ±1 with vibratinal rule v = ±1. When vibratinal transitins f the frm v + 1 v ccurs, J = ±1. Transitins with J = -1 are called the P branch: v P (J) = S(v +1,J "1) " S(v,J) = v " BJ Transitins with J = +1 are called the R branch: v R (J) = S(v +1,J +1) " S(v,J) = v + B(J +1) Q branch are all transitins with J = 0

8 Mlecular vibratin spectra cnsist f bands f lines in IR regin f EM spectrum ( cm t 0.5 ev). Vibratinal transitins accmpanied by rtatinal transitins. Transitin must prduce a changing electric diple mment (IR spectrscpy). Q branch P branch R branch Electrnic transitins ccur between mlecular rbitals. Must adhere t angular mmentum selectin rules. Mlecular rbitals are labeled, ", #, $, (analgus t S, P, D, fr atms) Fr atms, L = 0 => S, L = 1 => P Fr mlecules, % = 0 => ", % = 1 => # Selectin rules are thus $% = 0, ±1, $S = 0, $"=0, $& = 0, ±1 Where & = % + " is the ttal angular mmentum (rbit and spin).