Normal modes of harmonic systems

As in molecules, the starting point of a study of atomic motions in solid is the potential surface on which the atoms move. This potential is obtained in principle from the Born-Oppenheimer approximation (see Section 2.5). Once given, the 

Such a cell is also called a Wigner Seitz cell. 

A harmonic approximation is obtained by expanding the potential about the minimum energy configuration and neglecting terms above second order. This leads to 

K is a real symmetric matrix, hence its eigenvalues are real. Stability requires that these eigenvalues are positive otherwise small deviations from equilibrium will spontaneously grow in time. We will denote these eigenvalues by ct , that is, 

The individual atomic motions are now obtained from the inverse transformation 

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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, (/... 

In many situations it is convenient and adequately accurate to use the harmonic approximation to TST theory this involves an expansion of the energy over all the normal modes of the system near the equilibrium and the saddle point configuration of the mechanism ... 

The second derivatives with respect to nuclear displacements are crucial for characterizing stationary points on a potential hypersurface. They provide as well the normal modes of the system and can be linked within the harmonic approximation to the vibrational frequencies of the system, which can be measured experimentally by IR or Raman spectroscopy. By taking the derivative of the SCF energy gradient expression (Eq. [152]) with respect to another... 

For many applications, it may be reasonable to assume that the system behaves classically, that is, the trajectories are real particle trajectories. It is then not necessary to use a quantum distribution, and the appropriate ensemble of classical thermodynamics can be taken. A typical approach is to use a rnicrocanonical ensemble to distribute energy into the internal modes of the system. The normal-mode sampling algorithm [142-144], for example, assigns a desired energy to each normal mode, as a harmonic amplitude... 

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. 

While Eq. (9.49) has a well-defined potential energy function, it is quite difficult to solve in the indicated coordinates. However, by a clever transfonnation into a unique set of mass-dependent spatial coordinates q, it is possible to separate the 3 Ai-dirncnsional Eq. (9.49) into 3N one-dimensional Schrodinger equations. These equations are identical to Eq. (9.46) in form, but have force constants and reduced masses that are defined by the action of the transformation process on the original coordinates. Each component of q corresponding to a molecular vibration is referred to as a normal mode for the system, and with each component there is an associated set of harmonic oscillator wave functions and eigenvalues that can be written entirely in terms of square roots of the force constants found in the Hessian matrix and the atomic masses. 

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. 

Lattice vibrations are fundamental for the understanding of several phenomena in solids, such as heat capacity, heat conduction, thermal expansion, and the Debye-Waller factor. To mathematically deal with lattice vibrations, the following procedure will be undertaken [7] the solid will be considered as a crystal lattice of atoms, behaving as a system of coupled harmonic oscillators. Thereafter, the normal oscillations of this system can be found, where the normal modes behave as uncoupled harmonic oscillators, and the number of normal vibration modes will be equal to the degrees of freedom of the crystal, that is, 3nM, where n is the number of atoms in the unit cell and M is the number of units cell in the crystal [8],... 

These are the normal modes of this harmonic system. A motion of this type is a collective motion of all atoms according to (from (4.17))... 

Here Ef is the amplitude, t the duration, and co the frequency of the ith pulse. This scheme has been applied in Ref [46] to a generic two-dimensional HT model which incorporated a H-atom reaction coordinate as well as a low-frequency H-bond mode. In a subsequent work [47] the approach has been specified to a simple model of HT in thioacetylacetone. The Hamiltonian was tailored to the form of Eq. (4.1) based on the information available for the stationary points, that is, the energetics as well as the normal modes of vibration. From these data an effective two-dimensional potential was constructed including the H-atom coordinate as well as a coupled harmonic oscillator, which describes the 0-S H-bond motion. Although perhaps oversimplified, this model allowed the study of some principle aspects of laser-driven H-bond motion in an asymmetric low-barrier system. 

Atoms in a crystal are not at rest. They execute small displacements about their equilibrium positions. The theory of crystal dynamics describes the crystal as a set of coupled harmonic oscillators. Atomic motions are considered a superposition of the normal modes of the crystal, each of which has a characteristic frequency a(q) related to the wave vector of the propagating mode, q, through dispersion relationships. Neutron interaction with crystals proceeds via two possible processes phonon creation or phonon annihilation with, respectively, a simultaneous loss or gain of neutron energy. The scattering function S Q,ai) involves the product of two delta functions. The first guarantees the energy conservation of the neutron phonon system and the other that of the wave vector. Because of the translational symmetry, these processes can occur only if the neutron momentum transfer, Q, is such that... 

In collaboration with experimental groups, we have recently studied some chlorinated-benzene crystals, 1,2,4,5-tetrachlorobenzene (TCB) [59] and 1,4-dichlorobenzene (DCB) [60], as well as solid tetracyanoethene (TCNE) [58]. In these studies we have used empirical atom-atom potentials, of exp-6 type [see Eq. (6)], which we have supplemented with the Coulomb interactions between fractional atomic charges. Lattice dynamics calculations have been performed by the harmonic method, with inclusion of intramolecular vibrations [70], see Eqs. (17) to (24). The normal modes of the free molecules have been calculated from empirical Valence Force Fields, using the standard CF-matrix method [101, 102]. The results of these calculations are used here to illustrate some phenomena occurring in more complex molecular crystals. These phenomena are well known the numerical results show their quantitative importance, in some specific systems. 

The quantum mechanical information that follows from a normal mode analysis must reveal the same mechanical equivalence to a set of disconnected oscillators as the classical analysis. Each such oscillator (normal mode of vibration) can exist in any of the states possible for a one-dimensional harmonic oscillator. Each has its own contribution to the energy of the system, and thus, the Hamiltonian in Equation 7.35 corresponds to a quantum mechanical energy level expression... 

The next step is to perform a normal mode analysis, which is a method for finding the uncoupled orthogonal vibrational modes of the system. Expressed in coordinates, q. jj, from the IS along these D orthogonal modes, a harmonic expansion of the poten-tii in the reactant region can then be established as... 

The description of the vibrations of polyatomic molecules only becomes mathematically tractable by treating the system as a set of coupled harmonic oscillators. Thus a set of 3N - 6 (3N - 5 for linear molecules) normal modes of vibrations can be described in which aU the nuclei in the molecule move in phase in a simple harmonic motion with the same frequency, normal-mode frequencies are solved, the normal coordinates for the vibrations can be determined, and how the nuclei move in each of the normal modes of vibration can be shown. There are two important points that follow from this. First, each normal mode can be classified in terms of the irreducible representations of the point group describing the overall symmetry of the molecule [7, 8]. This symmetry classification of the... 

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