Vibrational modes and force constants

In a classical mechanics approach, the description of motion requires a dynamic trace of the displacement of an object in time, a trajectory that collects the variation in time of some coordinates q. But which coordinates For molecules, a first choiee could be cartesian coordinates for each nucleus in a laboratory reference frame the 

In general, molecular motion is a combination of translation, rotation, and vibration. If one is interested only in vibration, the translational and rotational components of motion must be deconvoluted out, and that requires a change in coordinates. Translation could be screened out by using a reference system with its origin at the center of mass, but that would still leave a mixture of rotation and vibration. What is needed in order to have pure vibrations is a set of coordinates that vary in time in such a way that there is no net angular velocity around the three inertial axes. 

For an N-atom molecule there are 3N - 6 internal degrees of freedom, so there must be 3N - 6 independent vibrations and one needs as many independent coordinates for their description. Chemical intuition works in terms of chemical bonds, so that the most intuitive way of representing molecular vibrations is in terms of the variations in bond lengths, bond angles, and torsion angles. These are internal coordinates, [ i]. Cartesian coordinates are independent by definition, but finding a set of truly independent internal coordinates for a large polyatomic molecule is all but a trivial matter [1], 

a rule of thumb says that valid normal coordinates must be either entirely symmetric or entirely antisymmetric over the most common symmetry operations if m and n label any two internal coordinates related by symmetry, then either Ukm = cikn or akm = —akn- This rule of thumb holds also in the construction of the coefficients of atomic orbitals in molecular orbitals (see Section 3.5). The bending mode of water straddles the mirror plane and is already symmetry-adapted if the displacements of the two hydrogen atoms are equal and in opposite directions (if the motions were in the same direction, that would be a molecular rotation, not a vibration). 

The vibrational potential energy, V, can be written as a function of any of the coordinate sets, V = V(qc) or V = V(qi) or V = V(( n), and coordinate sets transform one into another using linear combinations that involve geometrical relationships. The potential energy can be written as a Taylor series the first derivatives are null at equilibrium, and if the vibrations are as a first approximation considered harmonic, for any set of coordinates [g] the second derivatives are the force constants,/(i,j), and higher-order derivatives are neglected  

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