Energy simple harmonic motion

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... 

The simple harmonic motion of a diatomic molecule was treated in Chapter 1, and will not be repeated here. Instead, emphasis is placed on polyatomic molecules whose electronic energy s dependence on the 3N Cartesian coordinates of its N atoms can be written (approximately) in terms of a Taylor series expansion about a stable local minimum. We therefore assume that the molecule of interest exists in an electronic state for which the geometry being considered is stable (i.e., not subject to spontaneous geometrical distortion). 

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... 

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. 

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.

A diatomic molecule has only one mode of vibration, represented by the coordinate Ar, the change in the internuclear distance r. The potential energy of the molecule U r), whose vibration is approximated as simple harmonic motion, is given by ... 

There is an equivalence between the differential equations describing a mechanical system which oscillates with damped simple harmonic motion and driven by a sinusoidal force, and the series L, C, R arm of the circuit driven by a sinusoidal e.m.f. The inductance Li is equivalent to the mass (inertia) of the mechanical system, the capacitance C to the mechanical stiffness and the resistance Ri accounts for the energy losses Cc is the electrical capacitance of the specimen. Fig. 6.3(b) is the equivalent series circuit representing the impedance of the parallel circuit. 

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... 

The principle of classical equation solving, which is at the heart of MM, can be appreciated by imagining two objects, connected by a spring, executing simple harmonic motion collinear with the spring. If the force constant k of the spring is known, it is possible to calculate the potential energy V of the system at any separation x of the objects as... 

The energy levels of a molecule vibrating with simple harmonic motion are given by sv-lb — hv(v + ), where v is the vibrational frequency and v the vibrational quantum number. To obtain the vibrational partition function we sum the energy levels measuring the energy from the lowest available level, e0 %hv (the zero-point energy), to obtain... 

Figure 1 illustrates the simplest theoretical treatment for the energy absorption behavior of covalent bonds. Hooke s law curve corresponds to bonds vibrating in simple harmonic motion. In practice, covalent bond vibrations are anharmonic and follow the potential energies depicted in the Morse curve, also illustrated in Figure 1. It is this phenomenon of anharmonicity that allows the overtones or harmonics to occur in the NIR region. Thus, the frequencies of the overtones are slightly less than whole number...

Vibrations are considered in terms of the classical expressions governing motion of nuclei vibrating about their equilibrium positions with a simple harmonic motion (40). The potential and kinetic potential energies of molecules are defined in terms of the coordinates most appropriate to the molecular structures. All relative motions of atoms about the center of mass (vibrations) are linear combinations of a set of coordinates, known as normal coordinates. For every normal mode of vibration, all coordinates vary periodically with the same frequency and pass through equilibrium simultaneously. 

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. 

Equation (3.3.2) can be solved for the vibration energy levels accessible to any diatomic molecule undergoing simple harmonic motion. The solution proceeds by trying wavefunctions of the form... 

Fig. 3.3.2 Energy levels of a simple harmonic oscillator. The potential energy for harmonic motion is parabolic. The internuclear distance, r, is the equilibrium separation.

Consider a 5 kg mass attached to a spring with a spring constant / = 400Nm undergoing simple harmonic motion with amplitude A = 10 cm. Assume that the energy this mass can attain is quantized according to Eq. 3.97. What is the quantum number nl The potential energy is... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

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