Vibrational Normal Modes and Wavefunctions

Excitations of molecules to higher vibrational states typically occur in the mid-infrared region of the spectrum, between 200 and 5,000 cm ( 1 = 2.5-50 pm). In this chapter we consider the main factors that determine the energies and strengths of vibrational excitations and describe several applications of infrared spectroscopy to macromolecules. Chapter 12 discusses Raman spectroscopy, in which vibrational transitions accompany the scattering of light at higher frequencies. 

As we discussed in Chap. 2, the solutions to the Schrodinger equation for a quadratic potential energy function of coordinate x are the harmonic-oscillator wavefunctions, 

Within the limits of the harmonic approximation, a vibrational wavefimction for a nonlinear molecule is simply a product of the wavefunctions of the 3N—6 or 3N — 5 independent harmonic oscillators  

And to the same approximation, the vibrational energy of a molecule is the sum of the energies of the individual normal modes  

Consider a classical system with N particles and 3N Cartesian coordinates x, X2,. . XjN- Let s define the coordinates of each atom relative to the equilibrium position of that atom, so that X, X2 and x are stated relative to the equilibrium position of atom 1 X4, X5 x, relative to the equUibiium position of atom 2 and so forth. The total kinetic energy of the system then is 

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