MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II�Instructor: Prof. Robert Field 5.74 RWF Lecture #1717 – 1 Normal ↔↔↔↔ Local Modes: 6-Parameter Models Reading: Chapter 9.4.12.5, The Spectra and Dynamics of Diatomic Molecules, H. Lefebvre-Brion and R. Field, 2nd Ed., Academic Press, 2004. Last time:ω1, τ, ω2 (measure populations) experiment ω 2 ω1(abcd)←( AB)←g  two polyads. populations in (1234) depend on τ. Eres,kcould use fk = to devise optimal plucks for more complex situations Eres (choice of plucks and probes) * multiple resonances * more than 2 levels in polyad Overtone Spectroscopy nRH single resonance nRH + 1RH double resonance dynamics in frequency domain Today:Classical Mechanics: 2 1 : 1 coupled local harmonic oscillators QM: Morse oscillator 2 Anharmonically Coupled Local Morse Oscillators eff . Antagonism. Local vs. Normal.HLocal Whenever you have two identical subsystems, energy will flow rapidly between them unless something special makes them dynamically different: * anharmonicity * interaction with surroundings spontaneous symmetry - breaking effNext time: HNormal.5.74 RWF Lecture #17 17 – 2 Two coupled identical harmonic oscillators: Classical Mechanics H T= ( )PPR L, V+ ( )Q Q R L, ( )R = Right, L = Left T = ( )    PP P PR L R L , 1 2 G geometry and masses 1 rr( R 2 + PL 2 )+ 2G PP =[ GP rr′ R L ]2 1 RL) F QR V =( QQ  Q2 L force constants 12 rr( R+ Q2 )+ 2F QRQ=[ FQL rr′ RL]2 rr R rr L + 2 2 2 2 () 0() rr R rr L F Q F Q 2 H =  1 GP2 + 12   1 GP2 + 1  H R 0H L rr′ R L + F QQ + GPP rr′ R L kinetic potential (anharmonic)coupling coupling3 5.74 RWF Lecture #17 17 – 3 φ 21 1 1 1 1 +m1 3mGrr =+=+= = Frr k= µ m1 m2 mmm3 m3 13 Frr 1kRL cosφ (projection of velocity of ③ for ① — ③ stretch onto ③ — ② = ′ Grr = ′ m3 direction) kinetic coupling gets small for large m or φ = π /2 Each harmonic oscillator has a natural frequency, ω 0: 12/1 FGrrrr] 12/= 2 1 π c  k µ [πc ω 0 = 2 and the coupling is via 1 : 1 kinetic energy and potential energy coupling terms. Uncouple by going to symmetric and anti-symmetric normal modes. Qs = 2–1/2[QR + QL] Qa = 2–1/2[QR – QL] Ps = 2–1/2[PR + PL] Pa = 2–1/2[PR – PL] plug this into H and do the algebra   1 1 12 2( k + k RL ) QsH + Grr ′ Ps+ =  2 2µ   1 1 12+ 2 ( k − k ) Qa 2 − Grr ′ Pa+   RL2µ no coupling term! 15.74 RWF Lecture #17 17 – 4 1  1 12 ω s = 2πc µ + Grr ′( k + kRL ) / 1  1 12 ω a = 2πc µ − Grr ′( k − kRL ) / simplify to ωs = ω0 + β + λ (algebra, not power series) ωa = ω0 + β –λ RLβ= kG 2 rr ′ (22πc)ω 0 Grr′ can have either sign. It is usually negative because φ > π/2. λ ω β ω = −    +µ[ ]′k k GRL rr2 10 0 Can have either sign. Positive if right bond gets stiffer when left bond is stretched. sign of λ determined by whether potential or kinetic coupling is larger (or by the signs of kRL and Grr′).5.74 RWF Lecture #17 17 – 5 Morse Oscillator The Morse oscillator has a physically appropriate and mathematically convenient form. It turns out to give a vastly more convenient representation of an anharmonic vibration than 1Vr()= 1 frrx 2 + 1 frrrx 3 + frrrrx 4 2 6 24 treated by perturbation theory. 2 r −ar ] 0 ∞VMorse ()= De[1− e (V ()= 0, V ()= De ) rR=−Re 12 213 31 4 4Power series expansion of VMorse ()= (2a D )r −(6a D )r +(14a De )r .r 2 e 6 e 24 ffIf we use frr = 2a2De rrr = –6a3De rrrr = 14a4De in the framework of nondengenerate perturbation theory, we get much better results than we expect or deserve. Why? Because the energy levels of a Morse oscillator have a very simple form: Morse()v hc = E 0 hc + ωm (v +12)+ x (v +12)2/EMorse m / and an exact solution for the energy levels gives E0Morse = 0 2/1 2aDe 12 m m m1 +m ωm = 2πc  µµ= 12 2 2a h xm =− c4πµ Morsewe get the exact same relationship between (De,a) and (E ,ωm , x )by perturbation theory (with a twist)0 m5.74 RWF Lecture #17 17 – 6 H() 10 = frrr 2 + 1 P2 2 2µ quartic term treated() 1 4 to 1st order only!v r vEv 1 = 24 frrrr  v −1r 3 v −v +1r 3 v v −3r 3 v −v +3r 3 v() frrr 21  = +Ev 2  6 ωm 1 3 This works better than we could ever have hoped, and therefore we should never look a gift horse in the mouth. We always use Morse rather than an arbitrary power series representation of V(r). Sometimes we neven use a power series ∑an [1− exp(−ar )]. n Armed with this simplification, consider two anharmonically coupled local stretch oscillators. WHY? What promotes or inhibits energy flow between two identical subsystems? * ubiquitous * Local and Normal Mode Pictures are opposite limiting cases * Heff contains antagonistic terms that preserve and destroy limiting behavior eff * the roles are reversed for HLocal effand HNormal See Section 9.4.12.3 of HLB-RWF Extremely complicated algebra 1. HLocal defined identically to H Local, but with diagonal anharmonicity. 2. Convert to dimensionless P, Q, H and then to a,a†. 3. exploit the convenient V(Q) ↔ E(v) properties of VMorse. 4. van Vleck transformation to account for the effect of out of polyad coupling terms from [GPRPL + kRLQRQL] BUT NOT from VMorse.rr′5. Simplest possible fit model — relationships (constraints) between fit parameters imposed by the identical Morse oscillator model. 6. Next time — transformation from HLocal to HNormal.5.74 RWF Lecture #17 17 – 7 () ()hR 0hR 11. HLocal 21 1 µ PR 2 + 2µ Q2 + V anh (QR )= R 12 +2µ PL 2 + 1 Q +V anh (QL )2µ L PPL +kRLQ Q +Grr ′ R R L ()1HRL ()−1V anh Q()= VMorse Q kQ2 2 This enables us to use Harmonic-Oscillators for basis set but Morse simplification for the separate local oscillators. We are going to expand Vanh(Q) and keep only the Q3 and Q4 terms and treat them, respectively, by second-order and first-order perturbation theory, as we did for the simple Morse oscillator. (0) () () ()1 1 1HLocal = hR + h( L 0)+ hR + hL + HRL ()H 05.74 RWF Lecture #17 17 – 8 ˆ ˆ ˆ …