Why Don't Atom Forces Follow Inverse Square Law? Help Needed!

[tasnim rahman]

Briefly,the forces between two atoms arise from nuclear and electron-cloud repulsions, and nuclear to electron-cloud attraction. But why does the repulsion vary with respect to distance, without following the inverse square law? Should not electrostatic charges follow the inverse square law? Someone help quick.

 

[Andrew Mason]

Briefly,the forces between two atoms arise from nuclear and electron-cloud repulsions, and nuclear to electron-cloud attraction. But why does the repulsion vary with respect to distance, without following the inverse square law? Should not electrostatic charges follow the inverse square law? Someone help quick.

The electrostatic repulsion between two nuclei will follow the inverse square law. Although the repulsion between individual electrons will obey the inverse square law, one has to take into account the fact that these are electron clouds so distances between electrons are not all the same.

But the problem is that the atoms have both electrons and protons. The repulsion between electron clouds is countered by the electrons of one atom being attracted to its positive nucleus and to the positive nucleus of another nearby atom - this is how bonds are made between atoms. So as the electron clouds approach and repel each other, the repulsion is offset by the attraction to each other's nuclei.

AM

 

[alxm]

There's no net charge, so why would it follow Coulomb's law at distance?

 

[Andrew Mason]

There's no net charge, so why would it follow Coulomb's law at distance?

With the atoms separated by many atomic diameters, they will each appear neutral. But as the atoms approach, the distribution of electrons changes. The negative electrons repel each other and results in asymmetrical distribution of negative charge. The water molecule is a good example. The water molecule is a polar molecule. One side is slightly positive and the other slightly negative. Overall it is neutral, but as another water molecule gets close to it one bonds are formed between these slightly negative and positive sides. It is very important. In fact, your life depends on it.

AM

 

[tasnim rahman]

Thanks everyone. But I was thinking whether it had anything to do with thisView attachment inversecubelaw.pdf. And despite the atoms being themselves neutral this may only cause overall zero net charge, but the laws for attraction and repulsion, between nuclei and electron cloud still persist, governing the bond radius.

 

[alxm]

AM: I was addressing the original poster.

But since you repeated your explanation to me, I might as well correct them:
The effective electron-electron potential in atoms does not follow an inverse-square law at any range because of exchange and correlation effects. The long range potential is entirely correlation (dispersion forces). Dispersion accounts for essentially the entire interatomic potential at a range of a single vdw diameter, not 'many atomic diameters'.

The repulsion between electron clouds is countered by the electrons of one atom being attracted to its positive nucleus and to the positive nucleus of another nearby atom - this is how bonds are made between atoms.

No, it is not. It's wrong to imply that chemical bonding could be rationalized in purely electrostatic terms, or anything short of an quantum-mechanical model. Even then, a model of an atom based on the electrostatic interactions of a simple 'electron gas' will fail to reproduce bonding, per Teller's no-bonding theorem.

 

[alxm]

Thanks everyone. But I was thinking whether it had anything to do with thisView attachment 29067. And despite the atoms being themselves neutral this may only cause overall zero net charge, but the laws for attraction and repulsion, between nuclei and electron cloud still persist, governing the bond radius.

As I said above, chemical bonding can't be described in terms of classical electrostatics in any form.

 

[Andrew Mason]

Thanks everyone. But I was thinking whether it had anything to do with thisView attachment 29067. And despite the atoms being themselves neutral this may only cause overall zero net charge, but the laws for attraction and repulsion, between nuclei and electron cloud still persist, governing the bond radius.

Matter is generally neutral over macroscopic distances. All bonds between atoms are electrical. So chemical bonds have everything to do with the creation of asymmetrical charge distributions, or electric dipoles, at atomic distances.

AM

 

[Andrew Mason]

AM: I was addressing the original poster.

But since you repeated your explanation to me, I might as well correct them:
The effective electron-electron potential in atoms does not follow an inverse-square law at any range because of exchange and correlation effects. The long range potential is entirely correlation (dispersion forces). Dispersion accounts for essentially the entire interatomic potential at a range of a single vdw diameter, not 'many atomic diameters'.

We have to speak about probability distributions rather than actual positions of electrons around a nucleus. So we cannot really apply the inverse square law within an atom.

But that is very different than saying that the coulomb force inverse square law does not operate at that scale.

The inverse square law (for electrical force or the inverse law for electrical potential) still applies to proton, alpha particle or electron scattering, for example.

No, it is not. It's wrong to imply that chemical bonding could be rationalized in purely electrostatic terms, or anything short of an quantum-mechanical model. Even then, a model of an atom based on the electrostatic interactions of a simple 'electron gas' will fail to reproduce bonding, per Teller's no-bonding theorem.

One may have to apply quantum mechanics - a bond between two H atoms is formed not from the electron being always between the two nuclei but from the greater probability of the electrons being between the nuclei than being elsewhere, for example. But the potential is electrical. So, surely chemical bonds are electrical in nature.

AM

 

[alxm]

But that is very different than saying that the coulomb force inverse square law does not operate at that scale.

That's not what I said. The Coulomb potential takes the exact same form in quantum mechanics. What I wrote was that the effective potential between two interacting atoms does not do so. Which was my main point - the interaction between electrons in an atom is not limited to or described by the coulomb potential alone, and you most certainly cannot describe chemical bonding in terms of the Coulomb potential of a "charge cloud".

One may have to apply quantum mechanics - a bond between two H atoms is formed not from the electron being always between the two nuclei but from the greater probability of the electrons being between the nuclei than being elsewhere, for example.

No, for the second time, it is not. Quantum mechanics means more than the fact that the electrons form a charge density cloud! If it was as simple as that, you could describe it with the Poisson equation and classical electrostatics. Such a model (absent any effective 'quantum potential') can not describe atoms, much less chemical bonds.

Chemical bonding is not due to electrostatics alone. You must take into account dynamics. Electrons move. Their kinetic energy, equal to half the total potential energy, is what stops that 'density cloud' from collapsing in on the nucleus. The complicated dynamics of electron-electron interactions, as well as the purely quantum-mechanical Pauli principle must be taken into account to accurately chemical bonding. Quantum mechanics does not mean that you can just substitute a model of the electron as a point-charge for a model as a static density cloud and expect that to work. It does not work, and using that as a rationale for chemical bonding is extremely misleading, since it does not reflect the fact that electrons are moving - even if their charge densities are not.

But the potential is electrical. So, surely chemical bonds are electrical in nature.

So what's the exchange and correlation energy then?

 

[tasnim rahman]

Thanks again, people. This link gives an http://www.chemistry.mcmaster.ca/esam/Chapter_6/section_1.html" , can anyone verify this.

 

[alxm]

Thanks again, people. This link gives an http://www.chemistry.mcmaster.ca/esam/Chapter_6/section_1.html" , can anyone verify this.

That's by Richard Bader, a quite well-known quantum chemist, who created 'Bader analysis', which is indeed an interpretation of chemical bonding in terms of the electronic charge density, eliminating the need for working explicitly with quantum wave functions.

This doesn't contradict anything I was saying; on the contrary, his introduction points out that classically: "no model of the atom which invokes some stationary arrangement of the electrons around the nucleus is possible".

You can analyze and explain chemical bonding if you know the electronic density. You can calculate the electronic energy as well (in theory) if you know the density - this is the basis for Density Functional Theory, the development of approximate methods to do just that.

Now, what Bader is saying there, is that, for a chemical bond at equilibrium the electrostatic forces cancel out. (This comes from the Hellmann-Feynman theorem in QM) I can see how you might think this is at odds with me saying that you can't explain chemical bonding in electrostatic terms, but it's actually why I was saying that. Because this is still only part of the picture. You should not conclude from that, that the electrostatic forces alone govern chemical bonding. What he's saying is that the forces on the nuclei can be understood completely in terms of the electrostatic interaction with the electron 'cloud', and from that, you can arrive at facts about chemical bonding since two bonding atoms at equilibrium should have no net force on the nuclei.

This is all interesting, but it's also side-stepping the big issue, which is how you determine the density itself. And that is not purely electrostatic. The interactions between the electronic density and itself is not governed by Coulomb repulsion alone. You have to take into account the kinetic energy of the electrons and also, the exchange energy, which is purely quantum-mechanical and comes from the Pauli principle. Determining the electronic energy in terms of the density is very difficult (the Coulomb part is of course very simple, a 1/r potential), after 50 years of work, we still don't know how to determine this very well. A simple model that includes all electrostatic interactions, but only makes a crude approximation of the kinetic energy, will not lead to chemical bonding (the Teller theorem i mentioned).

In short: Once you know the density, you can find out quite a lot of interesting things about chemical properties from looking at it, and from electrostatic analysis. But the big issue is still how you determine the density - and that is not something you can arrive at from electrostatics alone (which Bader also makes quite clear from the outset)

 

[calhoun137]

To answer the question posed by the OP, the interaction between otherwise nuetral molecules at a distance is called "van der wall's" interaction, see Landau vol 5, section 76.

This is sometimes known as the "three six law", because for a non-ideal gas we have the following equation of state:

P = NT/(V-Nb) - N^2a/V^2

It's clear from this expression that the pressure (force/area) depends on volume through one term of order V^-1, and another term of order V^-2.

 

[tasnim rahman]

That's by Richard Bader, a quite well-known quantum chemist, who created 'Bader analysis', which is indeed an interpretation of chemical bonding in terms of the electronic charge density, eliminating the need for working explicitly with quantum wave functions.

This doesn't contradict anything I was saying; on the contrary, his introduction points out that classically: "no model of the atom which invokes some stationary arrangement of the electrons around the nucleus is possible".

You can analyze and explain chemical bonding if you know the electronic density. You can calculate the electronic energy as well (in theory) if you know the density - this is the basis for Density Functional Theory, the development of approximate methods to do just that.

Now, what Bader is saying there, is that, for a chemical bond at equilibrium the electrostatic forces cancel out. (This comes from the Hellmann-Feynman theorem in QM) I can see how you might think this is at odds with me saying that you can't explain chemical bonding in electrostatic terms, but it's actually why I was saying that. Because this is still only part of the picture. You should not conclude from that, that the electrostatic forces alone govern chemical bonding. What he's saying is that the forces on the nuclei can be understood completely in terms of the electrostatic interaction with the electron 'cloud', and from that, you can arrive at facts about chemical bonding since two bonding atoms at equilibrium should have no net force on the nuclei.

This is all interesting, but it's also side-stepping the big issue, which is how you determine the density itself. And that is not purely electrostatic. The interactions between the electronic density and itself is not governed by Coulomb repulsion alone. You have to take into account the kinetic energy of the electrons and also, the exchange energy, which is purely quantum-mechanical and comes from the Pauli principle. Determining the electronic energy in terms of the density is very difficult (the Coulomb part is of course very simple, a 1/r potential), after 50 years of work, we still don't know how to determine this very well. A simple model that includes all electrostatic interactions, but only makes a crude approximation of the kinetic energy, will not lead to chemical bonding (the Teller theorem i mentioned).

In short: Once you know the density, you can find out quite a lot of interesting things about chemical properties from looking at it, and from electrostatic analysis. But the big issue is still how you determine the density - and that is not something you can arrive at from electrostatics alone (which Bader also makes quite clear from the outset)

Thanks for giving your long time to clarify this to me alxm, i think i get it now. Thnx a lot again.