Independent internal coordinates

As V is a unitary matrix, Y = VTX is just an equivalent set of Cartesian coordinates, and = UTZ is just an equivalent set of internal coordinates, simply linear combinations of the Zn. The i, , N-6, change independently, in proportion to changes in linear combinations of the Cartesian coordinates. So, locally, we have defined 3N — 6 independent internal coordinates. Every different configuration of the molecule, X, will have a different B matrix, and hence a different definition of local internal coordinates, defined automatically. 

Vectors and matrices are given as boldface symbols throughout.) E is the unit matrix G is a matrix which depends, only although not in a simple fashion, on the geometry and the atomic masses of the molecule. F is the matrix of force constants expressed in the 3 N — 6 independent internal coordinates. Its elements are ... 

We have studied the potential hypersuifaces for the Van der Waals dimers ( 2 (CjH by varying all the independent internal coordinates (in the rigid molecule cq>proximation, 3 angles for (Nj), 5 angles for and the distance R in both... 

A diatomic molecule has only the internuclear distance Q as an internal coordinate (F = I). Unless //, vanishes because d>, and are of different symmetry, it is in general impossible to find a value of Q that would satisfy simultaneously both conditions. The energies , and E, therefore are different, and in one dimension, two states of the same electronic symmetry cannot cross. In a system with two independent internal coordinates Q, and Q2 (F = 2), for... 

Fig. 1. The structure of polymethineimine and the five independent internal coordinates rN=C> rC-N> rC-H> and p...

A nonlinear molecule of N atoms has 32V — 6 internal vibrational degrees of freedom, and therefore 3A — 6 normal modes of vibration (the three translational and three rotational degrees of freedom are not of vibrational spectroscopic relevance). Thus, there are 32V — 6 independent internal coordinates, each of which can be expressed in terms of Cartesian coordinates. To first order, we can write any internal displacement coordinate ry in the form... 

Figure 6.7 Five of the seven independent internal coordinates for acetylene. The remaining two internal coordinates are bending motions normal to the plane of the paper, and are not considered in our treatment.

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],... 

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