Polyatomic molecules vibrational motion

The representation as a two-dimensional potential energy diagram is simple for diatomic molecules. But for polyatomic molecules, vibrational motion is more complex. If the vibrations are assumed to be simple harmonic, the net vibrational motion of TV-atomic molecule can be resolved into 3TV-6 components termed normal modes of ibrations (3TV-5 for... 

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

We will consider a simple diatomic molecule first, then generalize our final equations for a polyatomic molecule that has 3N — 6 (or 3N — 5 for linear molecules) vibrational motions, where N is the number of atoms in the molecule. If we make the assumption that the single vibration of a diatomic molecule is an ideal harmonic oscillator, then quantum mechanics gives us an equation for the quantized energy of that harmonic oscillator ... 

The goal of this book is to present in a coherent way the problems of the laser control of matter at the atomic-molecular level, namely, control of the velocity distribution of atoms and molecules (saturation Doppler-free spectroscopy) control of the absolute velocity of atoms (laser cooling) control of the orientation, position, and direction of motion of atoms (laser trapping of atoms, and atom optics) control of the coherent behavior of ultracold (quantum) gases laser-induced photoassociation of cold atoms, photoselective ionization of atoms photoselective multiphoton dissociation of simple and polyatomic molecules (vibrationally or electronically excited) multiphoton photoionization and mass spectrometry of molecules and femtosecond coherent control of the photoionization of atoms and photodissociation of molecules. 

In order to illustrate the approximations involved when trying to mix quantum and classical mechanics we consider a simple system with just two degrees of freedom r and R. The R coordinate is the candidate for a classical treatment - it is, e.g., the translational motion of an atom relative to the center of mass of a diatomic or polyatomic molecule. This motion is slow compared to the vibrational motion of the diatom - here described by the r coordinate. Thus if we treat the latter quantum mechanically we could introduce a wavefunction t) and expand this in eigenstates of the molecule, i.e., eigenstates to the hamiltonian operator Hq for the isolated molecule. Thus we have... 

In many cases the dynamical system consists of fast degrees of freedom, labeled x, and slow degrees of freedom, labeled y. An example is that of a fluid containing polyatomic molecules. The internal vibrations of the molecules are often very fast compared to their translational and orientational motions. Although this and other systems, like proteins, have already been treated using RESPA,[17, 34, 22, 23, 24, 25, 26] another example, and the one we focus on here, is that of a system of very light particles (of mass m) dissolved in a bath of very heavy particles (mass M).[14] The positions of the heavy particles are denoted y and the positions of the light particles rire denoted by X. In this case the total Liouvillian of the system is ... 

Treating the full internal nuclear-motion dynamics of a polyatomic molecule is complicated. It is conventional to examine the rotational movement of a hypothetical "rigid" molecule as well as the vibrational motion of a non-rotating molecule, and to then treat the rotation-vibration couplings using perturbation theory. 

For a polyatomic molecule, the complex vibrational motion of the atoms can be resolved into a set of fundamental vibrations. Each fundamental vibration, called a normal mode, describes how the atoms move relative to each other. Every normal mode has its own set of energy levels that can be represented by equation (10.11). A linear molecule has (hr) - 5) such fundamental vibrations, where r) is the number of atoms in the molecule. For a nonlinear molecule, the number of fundamental vibrations is (3-q — 6). 

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

The harmonic oscillator energies and wavefunctions comprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomic molecule are often characterized in terms of individual bond-stretching and angle-bending motions each of which is, in turn, approximated harmonically. This results in a total vibrational wavefunction that is written as a product of functions one for each of the vibrational coordinates. 

For a non-linear polyatomic molecule, again with the centrifugal couplings to the vibrations evaluated at the equilibrium geometry, the following terms form the rotational part of the nuclear-motion kinetic energy ... 

Unlike the case of simple diatomic molecules, the reaction coordinate in polyatomic molecules does not simply correspond to the change of a particular chemical bond. Therefore, it is not yet clear for polyatomic molecules how the observed wavepacket motion is related to the reaction coordinate. Study of such a coherent vibration in ultrafast reacting system is expected to give us a clue to reveal its significance in chemical reactions. In this study, we employed two-color pump-probe spectroscopy with ultrashort pulses in the 10-fs regime, and investigated the coherent nuclear motion of solution-phase molecules that undergo photodissociation and intramolecular proton transfer in the excited state. 

The vibration-rotation interaction term makes the Hamiltonian for nuclear motion of a polyatomic molecule difficult to deal with. Frequently, this term is small compared to the other terms. We shall make the initial approximation of omitting Tvib rot. The rotational kinetic energy TTOt involves the moments of inertia of the molecule, which in turn depend on the instantaneous nuclear configuration. However, the vibrational motions are much faster than the rotational motions, so that we can make the approximation of calculating the moments of inertia averaged over the vibrational motions. 

Whereas for diatomic molecules the vibration-rotation interaction added only a small correction to the energy, for a number of polyatomic molecules the vibration-rotation interaction leads to relatively large corrections. Similarly, although the Born-Oppenheimer separation of electronic and nuclear motions holds extremely well for diatomic molecules, it occasionally breaks down for polyatomic molecules, leading to substantial interactions between electronic and nuclear motions. 

To consider the quantum mechanics of rotation of a polyatomic molecule, we first need the classical-mechanical expression for the rotational energy. We are considering the molecule to be a rigid rotor, with dimensions obtained by averaging over the vibrational motions. The classical mechanics of rotation of a rigid body in three dimensions is involved, and we shall simply summarize the results.2... 

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

I 2.1 Rotational Energy Levels of Diatomic Molecules, K I 2.2 Vibrational Energy Levels of Diatomic Molecules, 10 I 2.3 Electronic Stales of Diatomic Molecules, 11 I 2.4 Coupling of Rotation and Electronic Motion in Diatomic Molecules Hund s Coupling Cases, 12 1-3 Quantum States of Polyatomic Molecules, 14... 

The potential energy of such an oscillator can be plotted as a function of the separation r, or, for a normal mode in a polyatomic molecule, as a function of a parameter characterizing the phase of the oscillation. For a simple harmonic oscillator, the potential energy function is parabolic, but for a molecule its shape is that indicated in Figure 2.6. The true curve is close to a parabola at the bottom, and it is for this reason that the assumption of simple harmonic motion is justified for vibrations of low amplitude. 

---