# The quantum chemical theory Atoms in Molecules

### Background

'A bond is a bond is a bond !' This definition gave Richard Bader after the defense of one of his PhD students, and I heard it again several times in discussions with him, what a chemical bond is.

Bader uses quantum mechanics to answer the question, what (and where) a bond is. He describes in his book 'Atoms in Molecules - A Quantum Theory' [Bader 1990] all the basic concepts in chemistry - atoms, molecules, chemical bonds, functional groups - using quantum mechanics.

To do this, he divides molecules in quantum mechanic subsystems, in atomic areas. The topology of the electron density is used to define where one atom ends and the next begins.

Each subsystem (or atom) is surrounded by a surface, through which the gradient vector field of the electron density has no flux.

Don't worry if you don't have a good quantum mechanic background. Here is a simple phenomenological picture what the formula means:

Take two atoms, for example two hydrogen atoms. You have around each of the atoms a sperical distribution of electron density which is highest at the position of the nuclei and falls down exponential with the distance from the nuclei.

The gradient vector field could be described by lines following the steepest decent. Like you staying on a hill and walk down in the direction of steepest decent.

The gradient vector field is zero at any point, where you have no decent. In the case of our two hydrogens this is exactly in the middle of them, describing a valley between two hills.

If you now take an oxygen instead of one of the hydrogen atoms, then the picture still hold true, just that the oxygen compared to the tiny hydrogen would be better described as a mountain next to our hill.

Using now 3 dimensions instead of two and the topology of the electron density, Bader describes 4 so called critical points.

• (3, -3) critical point: The electron density falls down in all three perpendicular directions of space. This is a local maximum of electron density.
• (3, -1) critical point: The electron density falls down in two perpendicular directions of space and rises in the third direction. This is a saddle point of electron density with a maximum of electron density in 2 directions of space and a minimum in the third one.
• (3, +1) critical point: The electron density falls in one direction of space and rises in the two other perpendicular directions of space. Again this is a saddle point with a maximum in one and a minimum in two directions of space.
• (3, +3) critical point: This is a local minimum with electron density rising in all 3 directions of space.

So how are this critical points related to chemistry? Here is some explanation about the chemical meaning:

• (3, -3) critical point: This is the position of an atom. For all atoms except hydrogen it is also the position of the nuclei. The point therefore is also called an atomic critical point.
• (3, -1) critical point: This points are between two neighbouring atoms defining a bond between them. This point is therefore also called a bond critical point.
• (3, +1) critical point: This point is to be found in the middle of several bonds forming a ring. It is also called a ring critical points.
• (3, +3) critical point: This point is found when several rings form a cage and is therefore called a cage critical point.

Connecting the critical points in space give again a space filling partitioning of space in atomic subunits. So in principle this model yields an exact quantum mechanic description where in a molecule bonds are. The topology works equally for coulomb bonds as for weak bonds.

There is only one disadvantage: You need the electron density first to identify the critical points in space. The model was originally intended to describe molecules in vacuum. But an example in the next chapter will show, that it may be used for solids too, even when the example shown is far, far away from the size of a protein.

Last Updated: 26 October 1996