Internal displacement coordinate polyatomic molecule

It is now fundamental to define the normal coordinates of this vihrational system - that is to say, the nuclear displacements in a polyatomic molecule. Again in the limit of small amplitudes of vibration, the normal coordinates in the form of the vector Q, are related to the internal coordinates by a linear transformation, viz. 

The generalization of a force constant, k, and reduced mass, fi, from a one dimensional harmonic oscillator to the normal mode oscillators of a polyatomic molecule is accomplished by the F, G matrix methods of Wilson, et al., (1955). For the present discussion it is sufficient to know that a force constant and a reduced mass may be uniquely defined for each of 3N — 6 linearly independent sets of internal coordinate displacements in a polyatomic molecule (see also Section 9.4.12).] The harmonic oscillator Hamiltonian... 

For the diatomic molecule the same results apply except that the X displacement coordinate is replaced by the internal coordinate r — rj. For polyatomic molecules, each nonlinear molecule acts as though it consisted of 3N - 6 separate harmonic oscillators in each of which the X displacement coordinate is replaced by the appropriate normal coordinate Q. 

Polyatomic molecules have more than one vibrational frequency. The number can be calculated from the following. One atom in the molecule can move independently in three directions, the x, y, and z directions in a Cartesian coordinate system. Therefore, in a molecule with n atoms, the n atoms have 3n independent ways they can move. The center of mass of the molecule can move in three independent directions, x, y, and z. A nonlinear molecule can rotate in three independent ways about the x, y, and z axes, which pass through the center of mass. A linear molecule has one less degree of rotational freedom since rotation about its own axis does not displace any atoms. These translations of the center of mass and rotations can be performed with a rigid molecule and do not change its shape or size. Substracting these motions, there remain 3n — 6 degrees of freedom of internal motion for nonlinear molecules and 3n —5 for linear molecules. These... 

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