Cog-wheel effect

Non planar pyrocatechin without the cog-wheel effect between the rotors and the wagging atoms. 

Pyramidal Acetone-Like Molecules Without the Cog-Wheel Effect... 

Non-planar acetone-like molecules without any cog-wheel effect. 

The sine x sine symmetric binomial forms are considered to introduce a cog-wheel effect between the rotors into the Hamiltonian operator (30), because each sine function depends on the sense of the rotation [34,35]. 

In this expression, the cross products between cosine and sine functions, which depend on the sense of rotations introduce the cog-wheel effects between... 

Our first simple example of a local rNRG will consider the benzaldehyde molecule, where the two moving parts are assumed to move independently of the torsion sense of each other, i.e., we assure that there is no cog-wheel effect between both movements. This effect is known indeed to be very small. 

From the character table, the symmetry eigenvectors for the double torsion without cog-wheel effect in benzaldehyde are easily derived on the basis of the double free rotor solutions. It is found ... 

Table 10 Character table for the double rotation without the cog-wheel effect in pyrocathechin.

They are the same symmetry eigenvectors as those of pyrocathechin with the cog-wheel effect (34), but the group properties are different. 

When there is no cog-wheel effect between the rotors and the wagging atoms the cross-products between the cosine and sine functions, which appear in the potenticil (41) of the complete Hamiltonian operator may be dropped. So we have ... 

FVom the multiplication properties of the operations the character table for a non-planar pyrocatechine without the cog-wheel effects between the rotors themselves but with the wagging atoms, is easily deduced. This is given in table 11. 

When there are no cog-wheel effect betweai the rotating parts, the sine x sine products may be dropped in (41), (84) or (86). So, the local Hamiltonian operator may be expressed as ... 

The restricted local Hamiltonian operator for the double internal rotation in acetone neglecting the cog-wheel effect between the rotors is similar to that of pyrocatechine (80), except for the periodicity of the rotor. The restricted local NRG is deduced directly from that of pyrocatechin (81), taking into account (42) [21-22] ... 

In this table, there exist four non-degenerate Irreducible representations, one twofold degenerate and four fourfold degenerate. From this table and that of acetone with the cog-wheel effect, the equivalence relations between the irreducible representations of both groups are derivable. These are given in Table 13. 

With cog-wheel effect Without cog-wheel effect... 

The potential energy function for acetone without cog-wheel effect between the rotors is then written in terms of symmetry eigenvectors corresponding to the completely symmetric representation ... 

Dimethylamine in which the methyl groups are rotating in a first approach without a cog-wheel effect between them. The wagging of the N-hydrogen atom, on the contrary, induces an inversion of the whole molecule, that cannot be easily neglected. 

This expansion contains only seven terms. This expression means that only seven conformational energy values have to be determined for retaining all the main features of potential energy surface of pyrocatechin. These seven conformations have to be conveniently chosen in order to avoid linear dependency. In particular, the 90 , 90 and 90 , -90 conformations have to be considered in order to take accurately into account the cog wheel effect. This particularity is often forgotten by the spectroscopists when they determine the V/i term, i.e, the Aff of the potential energy function (111). 

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