Molecular Vibrations: from Harmonic Oscillators to Pendulums *
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“ Chemistry is all about reactions - making and breaking of chemical bonds and the rate at which they do so. Insights into the structure of molecules and their reactivities have been, for many centuries, gained by imagining the molecule to be a tumbling network of springs. ” MOLECULAR VIBRATIONS: FROM HARMONIC OSCILLATORS TO PENDULUMS * Srihari Keshavamurthy Department of Chemistry “No spectrum, no matter how simple, is dynamics-free. Explaining and hence understanding this No dynamical process, no matter how complex, fails to molecular choreography will let us control reveal its essential characteristics in one or a series of molecular reaction dynamics. Well, at least well designed spectroscopic experiments.” that is what physical chemists have been "...patterns are made to be broken, and the breaking fantasizing for many decades now! Parts of of standard patterns is the key to perceiving those the fantasy have been turned into reality in dynamical features that demand explanation.” the last couple of decades but attaining the “holy grail” of control requires an intimate The quotes above are taken from the preface understanding of the nature of molecular to a recent book titled The Spectra and Dynamics vibrations. of Diatomic Molecules (Elsevier, 2004) by H. Lefebvre-Brion and R. W. Field. It elegantly Chemistry is all about reactions - making and sums up the current view in molecular breaking of chemical bonds and the rate at spectroscopy. But why does one want to break which they do so. Insights into the structure the standard patterns? What are those of molecules and their reactivities have been, dynamical features that demand our for many centuries, gained by imagining the explanation? In short, the dynamical features molecule to be a tumbling network of refers to the intricate dance that the atoms in a springs. Such a “ball-and-spring” model, molecule perform upon excitation i.e., sketched in Fig. 1, is inherently a classical intramolecular dynamics. mechanical notion and one cannot help but Figure 1: Morse potential (blue) as compared to the harmonic oscillator (green) as models for vibrations. * This article is dedicated to the fond memory of Lakshmi. 47 wonder about its utility in the present day can only take specific, discrete (quantized) context of single molecule dynamics and energy levels. In fact almost everyday the above spectroscopy. The purpose of this brief article picture of the molecule as a collection of is to argue that studying the classical dynamics harmonic oscillators is confirmed whenever is not only useful from the perspective of someone measures and interprets the infrared gaining insights into the dynamics of the spectrum of a molecule. However, bonds molecules but perhaps such (semi)classical between atoms can and do break! The studies are more relevant than ever. Of course harmonic oscillators with their Hookian Dirac, in his landmark paper on quantum restoring force will never break. Clearly if one mechanics, had already anticipated this. It is wants to describe chemical reactions then it is interesting to note that the ever popular necessary to have a better model for the transition state theory of reaction rates, vibrations. A particularly simple but accurate formulated by Wigner, Eyring, and others in and widely used model is that of the Morse the 1930s, is actually based on classical potential (see Fig. 1 on page 47) mechanics. At the same time it is a sobering thought that quantum mechanics, the correct (2) theory for microscopic systems, is less than a century old but organic and inorganic chemists with D being the dissociation energy. For small have been synthesizing complex molecules for displacements (low energies) the Morse a much longer time! oscillations are harmonic in nature with frequency = 2 a 2 D / m but for large q's the How does one model the molecular vibrations? Morse potential differs substantially from the The simplest model, accurate at low energies harmonic one. In particular V(+∞)=D=const and thus sufficient to characterize the and thus the force dV/dq = 0 and the bond equilibrium structure, is in terms of a harmonic breaks. The frequency of vibrations in a Morse oscillator potential (Fig. 1 on page 47) potential depend on the energy i.e., Ω = Ω(E), in contrast to the harmonic vibrations. Again, (1) for a polyatomic molecule one has a collection o f f r e q u e n c i e s ( Ω 1 , Ω 2 , Ω 3 … … . Ω f ) . between two atoms of the molecule with Several experiments support the Morse being the frequency of oscillation of the oscillator model and thus molecular vibrations diatomic bond, and q being the displacement are examples of nonlinear oscillators. The from equilibrium. A polyatomic molecule is classical and quantum dynamics of the Morse therefore viewed as a collection of harmonic oscillator can also be exactly solved which oscillators with different frequencies and makes it a very useful model. appropriate reduced masses. For a molecule with N atoms the number of such vibrational In a polyatomic molecule the oscillators are modes is f (3N-6) (nonlinear geometry) or coupled - some strongly and some not at all. Why do they couple? Since there is not much (3N-5) (linear) with frequencies( 1, 2,... f). Quantum mechanics tells us that the oscillator space here to go into the details of the theory 48 of molecular structure, and with an inevitable coupling strength λ at the same total energy. risk of sounding cryptic, let me just say that: One can clearly see the drastically different The various vibrational modes couple and that nature of the classical dynamics. Are there is why it is a molecule. It is the coupling which consequences of these features of classical makes life interesting! Coupling of the nonlinear dynamics in the quantum domain? nonlinear oscillators leads to complex classical What does one mean by the term “quantum dynamics and the possibility of chaos as well. It chaos” if that is not a contradiction in itself? is well known, for nearly a century, that even These and many more similar questions have coupling harmonic oscillators with nonlinear been the focus of attention for nearly four coupling results in complicated motion. For decades now. Although the final word on these example, a Hamiltonian that describes two issues is yet to be said, we now know that the uncoupled harmonic oscillators can be nonlinearity of the vibrations is crucial to expressed as understanding the molecular dynamics and its control. In particular it is useful to think of the (3) molecule as a set of harmonic oscillators at low energies whereas at higher energies it is more appropriate to and an example of nonlinear coupling between think in terms of coupled pendulums. the x-oscillator and the y-oscillator is: Imagine that one wants to break a specific bond (4) in a polyatomic molecule. Fig.2: Classical phase space for two nonlinearly coupled harmonic oscillators. For small coupling one sees regular, circular orbits. However, for larger coupling strengths the phase space is a rich mix of regular (red, green, blue, white) and chaotic (violet) regions. In Fig. 2 the ( ) classical phase space section Then, as shown in Fig. 3 on page 51, all that has is shown for very small and relatively large to be done is to excite that bond and dump 49 energy in excess of the bond strength. With ionization rates of atoms and chemical-reaction rates some luck the deposited energy will stay put in of molecules! that bond for a few vibrational time periods (about a few hundred femtoseconds) or This flow of energy within a molecule is called chemically significant timescales and then snap as the phenomenon of Intramolecular the specific bond. Turns out that this viewpoint Vibrational energy Redistribution or IVR for is far too naive mainly due to the fact that short. The questions that we are, as many other molecules excited to such high energies have chemical physicists in the world are, interested complicated intramolecular dynamics. The in: Where does the energy flow? How? Why? excited mode is couled to many other modes How fast? How is this classical notion of ball- and thus the initially localized energy flows and-spring vibrational motion encoded in the rapidly into many other, perhaps undesirable, quantum eigenstates? The significance to modes. In other words the molecular reaction dynamics has to do with the fact that choreography is very complicated. Sometimes IVR is at the heart of one of the best known it is so complicated that it is simple! Underlying and widely applied approach to estimating this complicated song and dance sequence of reaction rates- the Rice-Ramsperger-Kassel- excited molecules is a dynamics of Marcus (RRKM) theory. In essence RRKM, indescribable beauty which, amazingly enough, proposed in the 1950s, make the assumption is the same as one observes while studying that for most molecules the time scale for IVR celestial mechanics. Indeed one could say that is much smaller than typical reaction what was termed nearly a century ago by timescales;this neglects dynamics and renders Poincaré as the fundamental problem of the theory statistical. Does this basic, seemingly dynamics is relevant to the intramolecular reasonable, assumption ring the death bell for motions as well. This might sound very strange mode-specific chemistry? Before answering given the disparity between mass, length, and this question let me point out that around the time scales for molecular and celestial very time that RRKM was proposed a famous phenomena and yet very recently astronomers, computational study was performed by Fermi, mathematicians, chemists, and physicsts have Pasta, and Ulam (FPU). They took a large come together to work on space missions! It is number of nonlinearly coupled oscillators, appropriate to quote from the recent account excited one of them and numerically studied by Porter and Cvitanovic:- In a turn of events that the subsequent flow of energy through the would have astonished anyone but N.