Independent normal mode approximation

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... 

To evaluate the turning points we make the independent normal mode approximation, where the potential V ,(s, Q ) in mode m at s along the reaction coordinate is given by Eq. [118]. The turning point for vibrational state in this mode is obtained by solving the equation ... 

Vibrational frequencies may be extracted from the PES by performing a normal mode analysis. This analysis of the normal vibrations of the molecular configurations is a difficult topic and can be pursued efficiently only with the aid of group theory and advanced matrix algebra. In essence, the 3 translational, 3 rotational and 3N-6 vibrational modes (2 rotational and 3N-5 vibrational modes for linear molecules) may be determined by a coordinate transformation such that all the vibrations separate and become independent normal modes, each performing oscillatory motion at a well defined vibrational frequency. As a more concrete illustration, assume harmonic vibrations and separable rotations. The PES can thus be approximated by a quadratic form in the coordinates... 

In contrast to the corresponding coupled equations bijrij = 0 in mass-weighted coordinates, Eq. 6.49 shows that each normal coordinate Q,- oscillates independently with motion which is uncoupled to that in other normal coordinates Qj. This separation of motion into noninteracting normal coordinates is possible only if V contains no cubic or higher-order terms in Eq. 6.4. Anharmonicity will inevitably couple motion between different vibrational modes, and then the concept of normal modes will break down. In the normal mode approximation, no vibrational energy redistribution can take place in an isoFated molecule. 

As before, we make the fundamental assumption of TST that the reaction is determined by the dynamics in a small neighborhood of the saddle, and we accordingly expand the Hamiltonian around the saddle point to lowest order. For the system Hamiltonian, we obtain the second-order Hamiltonian of Eq. (2), which takes the form of Eq. (7) in the complexified normal-mode coordinates, Eq. (6). In the external Hamiltonian, we can disregard terms that are independent of p and q because they have no influence on the dynamics. The leading time-dependent terms will then be of the first order. Using complexified coordinates, we obtain the approximate Hamiltonian... 

To render the KP theory feasible for many-body systems with N particles, we make the approximation of independent instantaneous normal mode (INM) coordinates [qx° 3N for a given configuration xo 3W [12, 13], Hence the multidimensional V effectively reduces to 3N one-dimensional potentials along each normal mode coordinate. Note that INM are naturally decoupled through the 2nd order Taylor expansion. The INM approximation has also been used elsewhere. This approximation is particularly suited for the KP theory because of the exponential decaying property of the Gaussian convolution integrals in Eq. (4-26). The total effective centroid potential for N nuclei can be simplified as ... 

A molecule with N atoms has a total of 37V degrees of freedom for its nuclear motions, since each nucleus can be independently displaced in three perpendicular directions. Three of these degrees of freedom correspond to translational motion of the center of mass. For a nonlinear molecule, three more degrees of freedom determine the orientation of the molecule in space and thus its rotational motion. This leaves 37V - 6 vibrational modes. For a linear molecule, there are just two rotational degrees of freedom, which leaves 3N -5 vibrational modes. For example, the nonlinear molecule H2O has three vibrational modes, while the linear molecule CO2 has four vibrational modes. The vibrations consist of coordinated motions of several atoms in such a way as to keep the center of mass stationary and nonrotating. These are called the normal modes. Each normal mode has a characteristic resonance frequency Vj (expressed in cm ), which is usually determined experimentally. To a reasonable approximation, each normal mode behaves as an independent harmonic oscillator of frequency u . The normal modes of H2O and CO2 are shown in Figs. 14.2 and 14.3. A normal mode will be infrared active only if it involves an oscillation of the dipole moment. All three modes of H2O are... 

The time independent product of exponentials in Eq. (A2.70) becomes the exponential of a sum over all modes. Thus Eq. (A2.57) becomes for the individual normal mode, excited to its n harmonic, in its low temperature approximation. 

Kelvin (the zero point motion). This latter effect is explained by quantum mechanics, and it can in turn explain absorption features of impurities in crystalline matrices. The presentation of the fundamental vibrational modes of crystals is based on the harmonic approximation, where one only considers the interactions between an atom or an ion and its nearest neighbours. Within this approximation, an harmonic crystal made of N ions can be considered as a set of 3N independent oscillators, and their contribution to the total energy of a particular normal mode with pulsation ivs (q) is ... 

If the thermodynamic properties are calculated within the harmonic approximation, in which the normal modes of vibration are assumed to be independent and harmonic, the cell has no thermal expansion. PARAPOCS (Parker and Price, 1989) extends this to the quasi-harmonic approximation. In this method the vibrations are assumed to be harmonic but their frequencies change with volume. This provides an approach for obtaining the extrinsic anharmonicity which leads to the ability to calculate thermal expansion. 

Phonons are normal modes of vibration of a low-temperature solid, where the atomic motions around the equilibrium lattice can be approximated by harmonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled normal modes (phonons) if a harmonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

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-oscillator approximation, the quantum-mechanical energy levels of a polyatomic molecule turn out to be vib 2, (v, + )hv,, where the v s are the frequencies of the normal modes of vibration of the molecule and v, is the vibrational quantum number of the ith normal mode. Each v, takes on the values 0,1,2,... independently of the values of the other vibrational quantum numbers. A linear molecule with n atoms has 3n — 5 normal modes a nonlinear molecule has 3n — 6 normal modes. (See Levine, Molecular Spectroscopy, Chapter 6 for details.)... 

Band and Freed have criticized the quasi-diatomic approximation and emphasized that any complete theory of dissodation must involve the use of the correct sets of normal modes Q and O of the molecule in the initial and final states (> and /> respectively. The two sets are not independent, but are related by a co-ordinate transformation. A detailed, quantum mechanical description has been developed in which the set Q in state ( > are taken to be the normal modes of the unexdted parent molecule for direct photodissodation, or the metastable photo-excited molecule for indirect predissociation, and the set Q ) in the state /> are separated into QUQi, wh e Qi is the reaction co-ordinate on the final repulsive surface and IQi are the normal modes in the photofragments. For a linear, triatomic molecule, Qi is simply the vibrational mode of the diatomic fragment and Q) indudes the symmetric and antisymmetric stretching modes (if collinearity is preserved). The matrix elements for the transition from... 

In the harmonie 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 moleeule). Eaeh 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... 

One practical approach to the inclusion of vibrational anharmoni-city is to neglect the mode-mode coupling of the normal modes and to employ an approximate anharmonic potential curve to describe the motion along each generalized normal mode of the reacting system independently. This is called the INM method.In this approach, the vibrational energy is just the sum of the vibrational energies within each mode,... 

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