Normal modes in the harmonic approximation

The Xj s are variations from equilibrium position of interatomic distances or angles. In this harmonic approximation we have terms in xj, but also crossed terms in XjXj, that represent harmonic interactions between two different vibrations j and / that are often related to nearly lying atoms. The presence of such terms means that one cannot excite the vibration defined by Xj without also exciting that defined by Xj,. One can, however, eliminate such cross terms in the harmonic approximation by performing a linear transform on the x s, which depends on the various coefficients found in the harmonic development of V, which are called force constants. The newly defined coordinates are those of normal modes of the molecule. They are linear combinations of the x s, which implies that the x s may also be expressed in the form of the reciprocal linear combination of these normal modes. The 

---

The following first section of this appendix describes quantities that are measured when registering spectra obtained using various experimental set-ups and their relations with molecular quantities. These relations form the basis of the interpretations of molecular spectra. The second section describes some general properties of a distribution that are used in various chapters of this book when this distribution is the band of a spectram. The third section deals with such concepts as normal modes in the harmonic approximation, while the fourth section deals with force constants, reduced masses, etc., and offers comparisons of these various quantities. The last section provides a more specific calculation of the first and second moment of a band such as which corresponds to a normal mode characterized by a strong anharmonic coupling with a much slower mode. 

Each of the normal modes in the harmonic approximation is a traveling wave in the crystal and represents a packet of energy by analogy with the wave/photon duality of electromagnetic radiation, the wave packets in crystals are called phonons. The energy of each phonon can be quantified either classically or quantum mechanically. Since the normal modes are independent, the vibrational partition function for the crystal is simply the product of the individual partition functions for each phonon. These are known analytically for harmonic oscillators. The corresponding classical and quantum mechanical partition functions for the crystal are ... 

In quantum chemical calculations, the vibrational problem is normally described in the harmonic approximation. Assuming that the vibrational problem has been solved, potential energy and each internal parameter qn can be expressed as function of Nvib normal mode coordinates Q, [1-6]... 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

The partition functions Z and Za are also easily evaluated in the harmonic approximation from products of the stable normal mode frequencies at the sad-... 

This means that, in the harmonic approximation and to lowest order in h, the classical transition rate is multiplied by a factor depending only on the sums of squares of the normal mode frequencies at the saddle point and minimum ... 

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... 

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

When the molecule is raised to the electronic excited state, the normal mode Q preserves the same decomposition with respect to the nuclear displacements, except for Jahn-Teller effects, which we exclude in this work. In these conditions, still in the harmonic approximation, two parameters, the equilibrium point and the potential curvature, will change in the excited state (in the particular case of non-totally-symmetric vibrations, the equilibrium point does not change). In the excited state e, the nuclear potential becomes... 

Each molecular vibration factor in Equation (3) is a type of molecular time correlation function for the internal vibrational dynamics. In the harmonic approximation, i) and f) would reduce to the harmonic vibrational eigenstates and the qj would be the actual molecular normal modes. Then one has the simplification... 

VA intensity is proportional to the electric dipole strength. In the harmonic approximation, Dlm, the dipole strength for the th normal mode of a fundamental vibrational transition (0 —> 1) can be expressed as... 

To obtain further insight into the meaning of the inelastic neutron spectra, it is necessary to have specific theoretical models with which to compare the experimental results. In the harmonic approximation it is possible to calculate the incoherent inelastic neutron spectrum i.e., the neutron scattering cross section for the absorption or emission of a specific number of phonons can be obtained with the exact formulation of Zemach and Glauber.481 A full multiphonon inelastic spectrum can be evaluated by use of Fourier transform techniques.482 The availability of the normal-mode analysis for the BPTI136 has made possible detailed one-phonon calculations483 for this system the one-phonon spectrum arises from transitions between adjacent vibrational levels and is the dominant contribution to the scattering at low frequencies for typical experimental conditions.483 The calculated one-phonon neutron en-... 

The contribution of each vibrational mode to X, can be obtained by expanding the potential energies of the neutral and cation states in a power series of the normal coordinates (denoted here as 2, and Q2). In the harmonic approximation, the relaxation energy X writes [1-5,17-21] ... 

The classical phase-space averages for bound modes in Eq. (11) are replaced by quantum mechanical sums over states. If one assumes separable rotation and uses an independent normal mode approximation, the potential becomes decoupled, and onedimensional energy levels for the bound modes may be conveniently computed. In this case, the quantized partition function is given by the product of partition functions for each mode. Within the harmonic approximation the independent-mode partition functions are given by an analytical expression, and the vibrational generalized transition state partition function reduces to... 

In the Bom-Oppenheimer approximation the vibronic waveftmction is a product of an electronic waveftmction and a vibrational waveftmction, and its symmetry is the direct product of the symmetries of the two components. We have just discussed the symmetries of the electronic states. We now consider the symmetry of a vibrational state. In the harmonic approximation vibrations are described as independent motions along normal modes Q- and the total vibrational waveftmction is a product of functions, one waveftmction for each normal mode ... 

In the harmonic approximation the problem of small amplitude vibrations (Chapters 6 and 7) reduces to the 3N — 6 normal modes N is the number of atoms in the molecule). Each of the normal modes may be treated as an independent harmonic oscillator. A normal mode moves all the atoms with a certain frequency about their equilibrium positions in a concerted motion (the same phase). The relative deviations (i.e. the ratios of the amplitudes) of the vibrating atoms from equilibrium are characteristic for the mode, while the deviation itself is obtained from them by multiplication by the corresponding normal mode coordinate Q (—oo, oo). The value Q = 0 corresponds to the equilibrium positions of all the atoms, Q and —Q correspond to two opposite deviations of any atom from its equilibrium position. 

In the harmonic approximation, the problem of small amplitude vibrations (discussed in Chapters 6 and 7) reduces to the 3N — 6 normal modes (N is the number of atoms in the molecule). Each of the normal modes may be treated as an independent harmonic oscillator. A normal mode moves all the atoms with a certain frequency about their equilibrium positions... 

The eigenfrequencies are the (internal as well as external) vibrational excitation frequencies of the crystal in the harmonic approximation. The eigenvectors which express the crystal normal modes in terms of the displacement coordinates Ga( ) usually called the polarization vectors. 

Given the potential expansion in Eq. (4) it is relatively easy to calculate the Van der Waals vibrations of the dimers, in the harmonic approximation. Although we realize that the harmonic model will not be appropriate for the larger amplitude Van der Waals vibrations, it is still interesting to consider the harmonic frequencies and the corresponding normal modes, since these give already a clear indication of the extent to which specific monomer rotations will be hindered (see Section 4). First we find, by direct minimisation of the equilibrium structure of the dimer, i.e. the... 

---