Vibrational motions

Consider first of all a very simple classical model for vibrational motion. We have a particle of mass m attached to a spring, which is anchored to a wall. The particle is initially at rest, with an equilibrium position xe along the x-axis. If we displace the particle in the +x direction, then experience teaches us that there is a restoring force exerted by the spring. Likewise, if we displace the particle in the — x direction and so compress the spring, then there is also a restoring force. In either case the force acts so as to restore the particle to its rest position xe. 

For very many springs, the restoring force turns out to be directly proportional to the displacement x - xe  

According to Newton s second law, force is mass times acceleration 

1 explained the connection between force and potential energy in Chapter 0. For a one-dimensional problem 

This equation has exactly the same form as the equation of motion for the single particle, except that the reduced mass jx replaces the mass of the single particle, where 

The total energy, kinetic plus potential, of the system is easily shown to be 

For our classical model of a molecule (two balls connected by a spring), the potential energy is given by JJ(r) = k(r — re)2/2, where k is the force constant and re is the position of the potential minimum (the rest length of the spring), and oscillation occurred 

The potential energy function for a chemical bond is far more complex than a harmonic potential at high energies, as discussed in Chapter 3. However, near the bottom of the well, the potential does not look much different from the potential for a harmonic oscillator we can then define an effective force constant for the chemical bond. This turns out to be another problem that can be solved exactly by Schrodinger s equation. Vibrational energy is also quantized the correct formula for the allowed energies of a harmonic oscillator turns out to be  

For example, molecules of 12C160 can be excited from v = 0tov = lbya photon with energy 4.257 x 10-20 J (co = E/Ti = 4.037 x 1014 radians per second coe = 2140 cm-1). This energy difference corresponds to the infrared region of the spectrum. This means the force constant for the C = O bond is k = 1855 N m-1. All of the force constants in Table 3.2 were found from the experimentally measured vibrational frequency. 

The lowest lying level (v = 0) has E = hcv/2 = Tico/2, not zero. In Problem 6.11 we showed that the wavefunction for the lowest state is 

FIGURE 8.6 High -resolution infrared spectrum of HC1 gas. All the transitions here have Au = +1, but the lines divide into two branches, corresponding to Ay = +1 or Ay = —1. A few lines are labeled. Note that each line is actually slightly split, because the separation between rotational states is difference for H35C1 (75% natural abundance) and H37C1 (24% natural abundance). The force constant and the bond length can both be extracted from the line positions. 

The reduced mass is calculated from the masses of the two weights joined by the spring  

From Equation 4.4, it can be seen that the natural frequency of vibration of the harmonic oscillator depends on the force constant of the spring and the masses attached to it but is independent of the amount of energy absorbed. Absorption of energy changes the amplitude of vibration, not the frequency. The frequency v is given in hertz (Hz) or cps. If one divides v in cps by c, the speed of light in cm/s, the result is the number of cycles per cm. This is v, the wavenumber  

N/m is the Systeme International d Unites (SI) unit dyn/cm is the cgs unit. The single-bond force constant in dyn/cm is about 5x10.  

From Equation 4.7, we can deduce that a C-C bond vibrates at a lower wavenumber than a C=C bond, because the force constant for the C-C bond is smaller than that for C=C. For example, C-C vibrates at 1200 cm and C=C vibrates at 1650 cm . In general, force constants for bending vibrations are lower than stretching vibrations. Resonance and hybridization in molecules also affect the force constant for a given bond. 

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There will probably be some similarities, but also some fiindamental differences. We have mainly considered small molecules with relatively rigid structures, in which the vibrational motions, although much different from the low-energy, near-hannonic nonnal modes, are nonedieless of relatively small amplitude and close to an equilibrium stmcture. (An important exception is the isomerization spectroscopy considered earlier, to which we shall return shortly.)... 

In an ideal molecular gas, each molecule typically has translational, rotational and vibrational degrees of freedom. The example of one free particle in a box is appropriate for the translational motion. The next example of oscillators can be used for the vibrational motion of molecules. 

The model of non-mteracting hannonic oscillators has a broad range of applicability. Besides vibrational motion of molecules, it is appropriate for phonons in hannonic crystals and photons in a cavity (black-body radiation). 

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

Vibrational motion is thus an important primary step in a general reaction mechanism and detailed investigation of this motion is of utmost relevance for our understanding of the dynamics of chemical reactions. In classical mechanics, vibrational motion is described by the time evolution and l t) of general internal position and momentum coordinates. These time dependent fiinctions are solutions of the classical equations of motion, e.g. Newton s equations for given initial conditions and I Iq) = Pq. 

In time-dependent quantum mechanics, vibrational motion may be described as the motion of the wave packet... 

Modem photochemistry (IR, UV or VIS) is induced by coherent or incoherent radiative excitation processes [4, 5, 6 and 7]. The first step within a photochemical process is of course a preparation step within our conceptual framework, in which time-dependent states are generated that possibly show IVR. In an ideal scenario, energy from a laser would be deposited in a spatially localized, large amplitude vibrational motion of the reacting molecular system, which would then possibly lead to the cleavage of selected chemical bonds. This is basically the central idea behind the concepts for a mode selective chemistry , introduced in the late 1970s [127], and has continuously received much attention [10, 117. 122. 128. 129. 130. 131. 132. 133. 134... 

Infrared and Raman spectroscopy each probe vibrational motion, but respond to a different manifestation of it. Infrared spectroscopy is sensitive to a change in the dipole moment as a function of the vibrational motion, whereas Raman spectroscopy probes the change in polarizability as the molecule undergoes vibrations. Resonance Raman spectroscopy also couples to excited electronic states, and can yield fiirtlier infomiation regarding the identity of the vibration. Raman and IR spectroscopy are often complementary, both in the type of systems tliat can be studied, as well as the infomiation obtained. 

Both infrared and Raman spectroscopy provide infonnation on the vibrational motion of molecules. The teclmiques employed differ, but the underlying molecular motion is the same. A qualitative description of IR and Raman spectroscopies is first presented. Then a slightly more rigorous development will be described. For both IR and Raman spectroscopy, the fiindamental interaction is between a dipole moment and an electromagnetic field. Ultimately, the two... 

There usually is rotational motion accompanying the vibrational motion, and for a diatomic, the energy as a fiuictioii of the rotational qiiantum iiumber, J, is... 

Ruhman S, Joly A G and Nelson K A 1987 Time-resolved observations of coherent molecular vibrational motion and the general occurrence of impulsive stimulated scattering J. Chem. Phys. 86 6563-5... 

In the reactant channel leading up to the transition region, motion along represents the FI atom approaching the molecule, while motion along / is the vibrational motion of the atom. The initial wavepacket is chosen to represent the desired initial conditions. In Figure 2, the FI2 molecule is initially in the ground... 

The excellent agreement of the results of HCR ab initio studies with the corresponding experimental findings clearly shows that the strongest influence on the numerical accuracy of the vibronic levels have effects outside of the R-T effect, that is, primarly the replacement of the effective bending approaches employed in previous works by a full 3D treatment of the vibrational motions (for an analysis of this matter see, e.g., [17]). Let us note, however, that such a... 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

In this article we describe an extension of SISM to a system of molecules for which it can be assumed that both bond stretching and angle bending describe satisfactorily all vibrational motions of the molecule. The SISM presented here allows the use of an integration time step up to an order of magnitude larger than possible with other methods of the same order and complexity. 

To determine the vibrational motions of the system, the eigenvalues and eigenvectors of a mass-weighted matrix of the second derivatives of potential function has to be calculated. Using the standard normal mode procedure, the secular equation... 

For the model Hamiltonian used in this study it was assumed that the bond stretching and angle i)ending satisfactorily describe all vibrational motions... 

SISM for an Isolated Linear Molecule An efficient symplectic algorithm of second order for an isolated molecule was studied in details in ref. [6]. Assuming that bond stretching satisfactorily describes all vibrational motions for linear molecule, the partitioned parts of the Hamiltonian are... 

The following illustration shows the potential energy surface for vibrational motion along one normal mode ... 

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