Harmonic approximation normal modes

Harmonic analysis (normal modes) at given temperature and curvature gives complete time behavior of the system in the harmonic limit [1, 2, 3]. Although the harmonic model may be incomplete because of the contribution of anharmonic terms to the potential energy, it is nevertheless of considerable importance because it serves as a first approximation for which the theory is highly developed. This model is also useful in SISM which uses harmonic analysis. 

In the absence of a current precise knowledge of the potential energy surface, interpolations were used to obtain approximate normal mode frequencies for the conserved modes and methyl radical moments of inertia at intermediate R values, as described in Ref. 1 lc, using the interpolation g(R) specified by Eq. (2.13). The number of quantum states for the conserved modes [Nk in Eq. (2.10)] was obtained at each value of R by a direct count of the approximately harmonic levels. Calculations were made for two values of a in Ref. 1 lc, as reported below. The potential energy and structural parameters that determine V the conserved mode frequencies, and the moments of inertia are given in tables in Ref. 11c. 

Since the very beginning, almost a century ago, spectroscopy has revealed that vibrational dynamics are quantal in nature the energy is quantized and dynamics are described in terms of eigenstates. The quantum theory has developed predominantly within the harmonic approximation, that is the simplest expansion, to quadratic terms, of the potential hypersurface around the equilibrium position. Vibrational dynamics can be thus represented with harmonic oscillators (normal modes) corresponding to coherent oscillations of all degrees... 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

Many thermodynamic quantities can be calculated from the set of normal mode frequencies. In calculating these quantities, one must always be aware that the harmonic approximation may not provide an adequate physical model of a biological molecule under physiological conditions. 

The harmonic normal mode description is quite useful for approximated evaluation of various molecular properties. For example, one can use this description in a convenient way to evaluate the average thermal atomic motion. This is done by using the normal mode vector Ls in eq. (4.12), which can be written as... 

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. 

In this approximation the nuclear wavefunctions are a product of N harmonic oscillator functions, one for each normal mode ... 

How one obtains the three normal mode vibrational frequencies of the water molecule corresponding to the three vibrational degrees of freedom of the water molecule will be the subject of the following section. The H20 molecule has three normal vibrational frequencies which can be determined by vibrational spectroscopy. There are four force constants in the harmonic force field that are not known (see Equation 3.6). The values of four force constants cannot be determined from three observed frequencies. One needs additional information about the potential function in order to determine all four force constants. Here comes one of the first applications of isotope effects. If one has frequencies for both H20 and D20, one knows that these frequencies result from different atomic masses vibrating on the same potential function within the Born-Oppenheimer approximation. Thus, we... 

No first derivative terms appear here because the transition state is a critical point on the energy surface at the transition state all first derivatives are zero. This harmonic approximation to the energy surface can be analyzed as we did in Chapter 5 in terms of normal modes. This involves calculating the mass-weighted Hessian matrix defined by the second derivatives and finding the N eigenvalues of this matrix. 

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... 

Consider the degeneracies of the vibrational levels in the harmonic-oscillator approximation. For the ground level there is only one possible set of vibrational quantum numbers (00- ()) hence the ground vibrational level is always nondegenerate. If none of the normal modes are... 

Since we have abandoned the individual quantum numbers of the degenerate modes Qx and Qy, we will replace the summation over the 3N — 6 normal modes in energy expressions by a summation over the distinct vibrational frequencies. For a linear triatomic molecule there are four normal modes, but only three distinct vibrational frequencies. In the harmonic-oscillator approximation, the energy contribution of the doubly degenerate modes is... 

The direct product enables one to find the symmetry of a wave function when the symmetries of its factors are known. For example, consider In the harmonic-oscillator approximation, the vibrational wave function is the product of 3N—6 harmonic-oscillator functions, one for each normal mode. To find the symmetry of we first examine the symmetries of its factors. Let the distinct vibrational frequencies of the molecule be vx>v2,..., vk,...,vn, and let vk be <4-fold degenerate let the harmonic-oscillator... 

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