Normal modes of harmonic

Step 0 Perform the normal modes of the harmonic part of Hg to get the vibrational frequencies v and normal mode vectors which compose the transformational matrix V. 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

A normal mode of vibration is one in which all the nuclei undergo harmonic motion, have the same frequency of oscillation and move in phase but generally with different amplitudes. Examples of such normal modes are Vj to V3 of H2O, shown in Figure 4.15, and Vj to V41, of NH3 shown in Figure 4.17. The arrows attached to the nuclei are vectors representing the relative amplitudes and directions of motion. 

If the displacements of the atoms are given in terms of the harmonic normal modes of vibration for the crystal, the coherent one-phonon inelastic neutron scattering cross section can be analytically expressed in terms of the eigenvectors and eigenvalues of the hannonic analysis, as described in Ref. 1. 

There are many different solutions for X1 and X2 to this pair of coupled equations, but it proves possible to find two particularly simple ones called normal modes of vibration. These have the property that both particles execute simple harmonic motion at the same angular frequency. Not only that, every possible vibrational motion of the two particles can be described in terms of the normal modes, so they are obviously very important. 

We have seen this expression before in Chapter 5, where it was the starting point for describing vibrational modes. We found in Chapter 5 that a natural way to think about this harmonic potential energy surface is to define the normal modes of the system, which have vibrational frequencies v, (/... 

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... 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

For such molecules, all of the vibrations are active in both the infrared and Raman spectra. Usually, certain of the vibrations give very weak bands or lines, others overlap, and some are difficult to measure, as they occur at very low wavenumber values.40 Because the vibrations cannot always be observed, a model of the molecule is needed, in order to describe the normal modes. In this model, the nuclei are considered to be point masses, and the forces between them, springs that obey Hooke s law. Furthermore, the harmonic approximation is applied, in which any motion of the molecule is resolved in a sum of displacements parallel to the Cartesian coordinates, and these are called fundamental, normal modes of vibration. If the bond between two atoms having masses M, and M2 obeys Hooke s law, with a stiffness / of the spring, the frequency of vibration v is given by... 

An understanding of what is implied by a Morse curve is necessary here, in order to understand the following point. A diatomic molecule has one normal mode of vibration. If the vibration behaves as a harmonic oscillator, the PE is proportional to the square of the displacement from the equilibrium internuclear distance. The... 

The statistical mechanical form of transition theory makes considerable use of the concepts of translational, rotational and vibrational degrees of freedom. For a system made up of N atoms, 3N coordinates are required to completely specify the positions of all the atoms. Three coordinates are required to specify translational motion in space. For a linear molecule, two coordinates are required to specify rotation of the molecule as a whole, while for a non-linear molecule three coordinates are required. These correspond to two and three rotational modes respectively. This leaves 3N — 5 (linear) or 3N — 6 (non-linear) coordinates to specify vibrations in the molecule. If the vibrations can be approximated to harmonic oscillators, these will correspond to normal modes of vibration. 

We expand the potential energy surface at the saddle point to second order in the coordinates at the top of the barrier and determine the normal modes of the activated complex one of them is the reaction coordinate y identified as the mode with an imaginary frequency. Since the other normal modes of the activated complex are not coupled to the reaction coordinate in the harmonic approximation, we do not consider them here because they are irrelevant. For the harmonic solvent, we may likewise find the normal modes S. We use these normal modes to write down the Hamiltonian, and then add a linear coupling term representing the coupling between the reaction... 

A molecule composed of A atoms has in general 3N degrees of freedom, which include three each for translational and rotational motions, and (3N — 6) for the normal vibrations. During a normal vibration, all atoms execute simple harmonic motion at a characteristic frequency about their equilibrium positions. For a linear molecule, there are only two rotational degrees of freedom, and hence (3N — 5) vibrations. Note that normal vibrations that have the same symmetry and frequency constitute the equivalent components of a degenerate normal mode hence the number of normal modes is always equal to or less than the number of normal vibrations. In the following discussion, we shall demonstrate how to determine the symmetries and activities of the normal modes of a molecule, using NH3 as an example. 

Similarly, during their effort to understand the thermal energy of solids, Einstein and Debye quantized the lattice waves and the resulting quantum was named phonon. Consequently, it is possible to consider the lattice waves as a gas of noninteracting quasiparticles named phonons, which carries energy, E=U co, and momentum, p = Uk. That is, each normal mode of oscillation, which is a one-dimensional harmonic oscillator, can be considered as a one-phonon state. 

The normal modes of polyatomic molecules in the harmonic approximation can be calculated with the help of computational methods. 

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