MIT OpenCourseWarehttp://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Andrei Tokmakoff, MIT Department of Chemistry, 3/12/2009 p. 11-1 11.1 VIBRATIONAL RELAXATION* Here we want to address how a quantum mechanical vibration undergoes irreversible energy dissipation as a result of interactions with other intra- and intermolecular degrees of freedom. Why is this process important? It is the fundamental process by which non-equilibrium states thermalize. As chemists, this plays a particularly important role in chemical reactions, where efficient vibrational relaxation of an activated species is important to stabilizing the product and not allowing it to re-cross to the reactant well. We will be looking specifically at vibrational couplings and relaxation, but the principles are the same for spin-lattice relaxation and electronic population relaxation through electron-phonon coupling. For an isolated molecule with few vibrational coordinates, an excited vibrational state must relax by interacting with the remaining internal vibrations or the rotational and translational degrees of freedom. If a lot of energy must be dissipated, radiative relaxation may be more likely. In the condensed phase, relaxation is usually mediated by the interactions with the environment, for instance, the solvent or lattice. The solvent or lattice forms a continuum of intermolecular motions that can absorb the energy of the vibrational relaxation. Quantum mechanically this means that vibrational relaxation (the annihilation of a vibrational quantum) leads to excitation of solvent or lattice motion (creation of an intermolecular vibration that increases the occupation of higher lying states). For polyatomic molecules it is common to think of energy relaxation from high lying vibrational states ( kT << hω0 ) in terms of cascaded redistribution of energy through coupled modes of the molecule and its surroundings leading finally to thermal equilibrium. We seek ways of describing these highly non-equilibrium relaxation processes in quantum systems.Andrei Tokmakoff, MIT Department of Chemistry, 3/12/2009 p. 11-2 Classically vibrational relaxation reflects the surroundings exerting a friction on the vibrational coordinate which damps its amplitude and heats the sample. We have seen that a Langevin equation for an oscillator experiencing a fluctuating force f(t) describes such a process: Qt()+ω0 Q −γQ f t()&&2 2 &= / m (12.1) This equation ascribes a phenomenological damping rate γto the vibrational relaxation; however, we also know in the long time limit, the system must thermalize and the dissipation of energy is related to the fluctuations of the environment through the classical fluctuation-dissipation relationship: () (0)ftf 2 γ Tδ(t) (12.2)= m kWe would also like to understand the correspondence between these classical pictures and quantum relaxation. Let’s treat the problem of a vibrational system HS that relaxes through weak coupling V to a continuum of bath states HBusing perturbation theory. The eigenstates of HSare a and those of HBare α . Although our earlier perturbative treatment didn’t satisfy energy conservation, here we can take care of it by explicitly treating the bath states. HH + (11.1)= 0H0 =HS+ HB (11.2) HS = a E a + b E b (11.3)a b HB =∑α Eαα (11.4) α Haα =(E E)+ aα (11.5)0 a α We will describe transitions from an initial state i = aα with energy E E to a final statea+αf= bβ with energy E E . Since we expect energy conservation to hold, this undoubtedlyb+β requires that a change in the system states will require an equal and opposite change of energy in the bath. Initially, we take pa =1 pb =0 . If the interaction potential is V, Fermi’s Golden Rule says the transition from i to f is given by VAndrei Tokmakoff, MIT Department of Chemistry, 3/12/2009 p. 11-3 2k fi = 2π∑ pi i V f δ(E − E ) (11.6)f i ,hif 2aVα bβ δ (( Eb + Eβ)−( Ea + Eα )) (11.7)= 2π∑ pa,αha,,α b,β 1 +∞ −iE −E + E −E t h(( ba )(β α))= h2 ∫−∞ dt∑ pa,α aα V bβ bβ V aα e (11.8) a,α b,β Equation (11.8) is just a restatement of the time domain version of (11.6) 1 +∞k fi = h2 ∫−∞ dtV tV 0 (11.9)() ( ) 0 0Vt()= eiH t V e−iH t . (11.10) Now, the matrix element involves both evaluation in both the system and bath states, but if we write this in terms of a matrix element in the system coordinate Vab= aV b : aVα bβ =α Vab β (11.11) Then we can write the rate as +∞ e V e −iE t +iE t αβkba = 12 ∫−∞ dt ∑ pαα β β Vba α e−iωbat (11.12)abhαβ 1 +∞ , e−iωbatkba = 2 ∫−∞ dtV ab tV 0() ba ( ) Bh (11.13) ab () iH t ab −iH t Vt = e BV eB (11.14) Equation (11.13) says that the relaxation rate is determined by a correlation function Ct = Vab t V ba (11.15) which describes the time-dependent changes to the coupling between b and a. The time-dependence of the interaction arises from the interaction with the bath; hence its time-evolution under HB . The subscript Lba () () (0) means an equilibrium thermal average over the bath states B L =∑ pαα L α . (11.16)B α Note also that eq. (11.13) is similar but not quite a Fourier transform. This expression says that the relaxation rate is given by the Fourier transform of the correlation function for the fluctuating coupling evaluated at the energy gap between the initial and final state states.Andrei Tokmakoff, MIT Department of Chemistry, 3/12/2009 p. 11-4 Alternatively we could think of the rate in terms of a vibrational coupling spectral density, and the rate is given by its magnitude at the system energy gap ωba. kba = 12 C% ba (ωab ). (11.17)h where the spectral representation C% ba (ωab ) is defined as the Fourier transform of ba ().Ct Vibration coupled to a harmonic bath To evaluate these expressions, let’s begin by consider the specific case of a system vibration coupled to a harmonic bath, which we will describe by a spectral density. Imagine that we prepare the system in an excited vibrational state in v =1 and we want to describe relaxation 1to v = 0 . 2 2H = hω(P + Q ) (11.18)S 0 α%%2 2 †H = hω p + q = hω aa +1 (11.19)B ∑ α(α α)∑ α(αα 2 ) α % %α We will take the system-bath interaction to be linear in the system and bath coordinates: VHSB =ξα q Q . (11.20)=∑ α α %% Here ξ is a