Simple harmonic motion displacement

Vibrational energy, which is associated with the alternate extension and compression of die chemical bonds. For small displacements from the low-temperature equilibrium distance, the vibrational properties are those of simple harmonic motion, but at higher levels of vibrational energy, an anharmonic effect appears which plays an important role in the way in which atoms separate from tire molecule. The vibrational energy of a molecule is described in tire quantum theory by the equation... 

For small displacements molecular vibrations obey Hooke s law for simple harmonic motion of a system that vibrates about an equilibrium configuration. In this case the restoring force on a particle of mass m is proportional to the displacement x of the particle from its equilibrium position, and acts in the opposite direction. In terms of Newton s second law ... 

Imagine a Maxwell liquid placed between two parallel plates and sheared by moving the upper plate in its own plane. However, instead of moving the plate at a constant velocity as discussed in Chapter 1, let the displacement of the plate vary sinusoidally with time, ie the plate undergoes simple harmonic motion. If the maximum displacement of the upper plate is X and the distance between the plates is h, then the amplitude A of the shear strain in the liquid is given by... 

Figure 2.4 Simple harmonic motion of the mass m caused by an initial displacement of a0 from its equilibrium position...

Consider the situation shown in Figure 2.4 where a mass m is caused to oscillate by an initial displacement up to an amount oq at t = 0. The amplitude a would have to be smaller than shown for simple harmonic motion as a real spring would only obey Hooke s law over a limited strain amplitude. However the assumption is that Hooke s law is obeyed and the restoring force from both spring displacements is — IJcoq where k is the force constant or elastic modulus of the spring. So we may write the force at any position as... 

This can be seen from elementary considerations. The distortion of the surrounding medium that leads to the intermediate state with energy WH can be described by simple harmonic motion, with a parameter x for the displacement, a potential energy px2 and a wave function of the form r=const xexp(—ax2). The probability P 2 of a configuration with potential energy WH is thus... 

Fig. 3.4 Simple harmonic motion, (a) Displacement xand velocity vas a function of time t. (b) Variation of potential and kinetic energies during the motion.

FIGURE 3.4 Two systems that undergo simple harmonic motion if displacements are small (left, pendulum right, mass suspended by a spring). 

The Vibration of Diatomic Molecules.—In addition to their rotation, we have seen that diatomic molecules can vibrate with simple harmonic motion if the amplitude is small enough. We shall use only this approximation of small amplitude, and our first stop will be to calculate the frequency of vibration. To do this, we must first find the linear restoring force when the interatomic distance is displaced slightly from its equilibrium value / ,. We can get this from Eq. (1.2) by expanding the force in Taylor s series in (r — rt). We have... 

In a diatomic molecule, the masses mv and m2 vibrate back and forth relative to their centre of mass in opposite directions, as shown in the following figure. The two masses reach the extremes of their respective motions at the same time. The diatomic molecule has only one vibrational degree of freedom, i.e., it has only one frequency, called the fundamental vibrational frequency. During vibrational motion, the bond of the molecule behave like a spring and the molecule exhibits a simple harmonic motion provided the displacement of the nuclei from the equilibrium configuration is not too much. At the two extremes of motion which correspond to extension and compression of the chemical bond between the two atoms, the potential energy is maximum. On... 

Succession of vibrational states For a simple harmonic motion the acceleration is related to the displacement from the equhibiium position by the equation... 

If the displacements are x, and x respectively we shall have, from the equations of simple harmonic motion,... 

As the molecule vibrates (undergoes atom displacements)) the electronic charge distribution and, hence, the polarizability (a) varies in time. The polarizability is related to the electron density of the molecule and is often visualized in three dimensions as an ellipsoid and represented mathematically as a symmetric second-rank tensor. The time-dependent amplitude (Q ) of a normal vibrational mode executing simple harmonic motion is written in terms of the equilibrium amplitude Q , the normal mode frequency o), and time t). 

Simple harmonic motion n. Periodic oscillatory motion in a straight Une in which the restoring force is proportional to the displacement. If a point moves uniformly in a circle, the motion of its projection on the diameter (or any straight hne in the same plane) is simple harmonic motion. If r is the radius of the reference circle, (o the angular velocity of the point in the circle, 0 the angular displacement at the time t after the particle passes the mid-point of its path, the linear displacement... 

Properties. It is of considerable importance to examine the nature of die solutions obtained above. It is evident from Eq. (9), Sec. 2-2, that each atom is oscillating about its equilibrium position with a simple harmonic motion of amplitude Aik — Kkhk, frequency x /27t, and phase e. Ihirthermore, corresponding to a given solution X of the secular equation, i he frequency and phase of the motion of each coordinate is the same, but I lie amplitudes may be, and usually are, different for each coordinate. On account of the equality of phase and frequency, each atom reaches its position of maximum displacement at the same time, and each atom pa.sscs through its equilibrium position at the same time. A mode of ibration having all these characteristics is called a normal mode of vibra-iion, and its frequency is known as a normal, or fundamental, frequency of (he molecule. 

Each mode involves approximately harmonic displacements of the atoms from their equilibrium positions for each mode, i, all the atoms vibrate at a certain characteristic frequency, V/. The potential energy, V(r), of a harmonic oscillator is shown by the dashed line in Figure 1.1 as a function of the distance between the atoms, r. For any mode in which the atoms vibrate with simple harmonic motion (i.e., obeying Hooke s law), the vibrational energy states, can be described... 

To a reasonable approximation, at least for small displacements, the vibrations in a polyatomic molecule can be described as a kind of simple harmonic motion. This is essentially equivalent to considering a chemical bond between two atoms as a weightless spring that obeys Hooke s law (i.e., the force is proportional to the displacement). The simplest vibration is that in a diatomic molecule, and we can refer back to the potential energy curve for He (O Fig. 10-1) as an example. At the minimum, near the bottom of the weU, the potential energy curve is indeed parabolic to a very good approximation. 

The quadrupolar field of the ion trap exerts a restoring force on ions near its center that is linear with respect to displacement from the trap s center. This results in simple harmonic motion. 

To separate the effects of static and dynamic disorder, and to obtain an assessment of the height of the potential barrier that is involved in a particular mean-square displacement (here abbreviated (x )), it is necessary to find a parameter whose variation is sensitive to these quantities. Temperature is the obvious choice. A static disorder will be temperature independent, whereas a dynamic disorder will have a temperature dependence related to the shape of the potential well in which the atom moves, and to the height of any barriers it must cross (Frauenfelder et ai, 1979). Simple harmonic thermal vibration decreases linearly with temperature until the Debye temperature Td below To the mean-square displacement due to vibration is temperature independent and has a value characteristic of the zero-point vibrational (x ). The high-temperature portion of a curve of (x ) vs T will therefore extrapolate smoothly to 0 at T = 0 K if the sole or dominant contribution to the measured (x ) is simple harmonic vibration ((x )y). In such a plot the low-temperature limb is expected to have values of (x ) equal to about 0.01 A (Willis and Pryor, 1975). Departures from this behavior indicate more complex motion or static disorder. 

K for myoglobin (Parak et al., 1981). Thus, measurements of (x ) at temperatures below this value should show a much less steep temperature dependence than measurements above, if nonharmonic or collective motions (whose mean-square displacement is denoted (x )c) are a significant component of the total (x ). Figure 21 illustrates the expected behavior of (x )v, x, and their sum for a simple model system in which a small number of substates are separated by relatively large barriers. In practice, the relative contributions of simple harmonic vibrations and coUective modes will vary from residue to residue within a given protein. 

The moment of inertia of a ring of particles, mrki was used as the criterion for stability to define a closed orbit that combines circular motion with simple harmonic displacements. A more general discussion that substantiates the derivation is given by Goldstein [11]. The frequency of revolution is obtained in the form of a square root, defined by a set of integers,... 

Displacements may arise not only from thermal motion but also from static disorder when corresponding atoms in different unit cells take up slightly different mean positions. Certain side chains, especially those exposed, may take up a few radically different conformations in different molecules so that separate images of them can be seen with reduced occupancy in electron density maps. The mean square displacement will also include contributions from lattice disorders but these are usually small in protein crystals that diffract well to high resolution [191]. In principle, the thermal vibrations can be distinguished from static disorder by varying temperature. Simple harmonic vibrations are expected to decrease linearly with temperature. 

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