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The only basic definition of improper torsion/improper dihedral I have is - improper dihedral is formed when we have a central atom bonded to the three atoms. Can anyone help me visualize/help to understand how these improper dihedrals help to maintain the planar structure of let's say - benzene?

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    $\begingroup$ well, it helps keep them all in the same plane, which is usually the best thing for benzene. I was going to post a hand made image, but vpn won't let me... but if you look at any carbon, it is bonded to two carbons and one hydrogen. It would be odd if that carbon you chose was out of plane of the other atoms $\endgroup$
    – Wesley
    Nov 29, 2021 at 13:15

2 Answers 2

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The key thing to remember is that while a "proper" torsion includes atoms 1-2-3-4 with bonds between 1-2, then 2-3, and then 3-4, an improper torsion is just defined by any set of 4 atoms.

So here's a proper torsion C-C-C-H in benzene:

proper torsion 1-2-3-4 in benzene

If I want the ring to be flat, I'd define this torsion to be, e.g. ±180°

But I don't need to have four atoms bonded in row. I can define an improper torsion like this - notice atoms 1-2 and 2-3 aren't even bonded. I just picked the three atoms around number 4.

improper torsion in benzene

Here if I wanted the ring to be flat, I'd set the torsion to be 0° as depicted. (It's going to be the angle between vector 1-2 and 3-4, which should naturally be zero for a planar structure.)

The nice thing about improper torsions, opposed to other kinds of out-of-plane potential functions, is that they can take the same form as a proper torsion, and use the same type of angle calculation. No need to define a plane, etc.

Bezene isn't a great example, because I can keep it planar with many proper torsions. Something like formaldehyde is a classic case because there's no way to define a 1-2-3-4 bonded set, but it should be planar:

formadehyde

Naturally, adding an improper torsion to a force field will help ensure planarity of formaldehyde and similar compounds. It's also commonly used to define inversion potentials for N and P pyramidal geometries.

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A plane can be defined with 3 atoms and two planes can be defined with 4 atoms. Thus it's natural to define an angle between these two planes which is the dihedral (or torsion) angle.

You can think of this dihedral angle as an opened book and place two atoms on the book spine. For improper dihedral, the other two atoms connect to the same atom on the spine

| \ / |
|  x  |
|  |  |
|  x  |
| / \ |
x     x

For proper dihedral, the other two atoms connect to different atoms on the spine

x
| \ / |
|  x  |
|  |  |
|  x  |
| / \ |
      x

In either case, we have an angle between the book pages

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