Kohn Sham Equations

There is an analogy that I’d like to create between the many body quantum electronic hamiltonian and the Kohn-Sham equations.

Like all analogies it would be woefully skimpy on details but when I first started out I thought it was a useful way of looking at things. That and the fact that I like to recast equations into stories.

Consider the complete many-body electronic hamiltonian:

\[ Many body hamiltonian \]

I like to think of this way. We have a society of electrons that we would like to describe. The society might occupy these buildings which do not move.

Of course buildings do move, say in response to earthquakes but most are stationary.

To explain that society of electrons in toto we need to find out about each and every member of that society.

This is akin to doing a thorough census. Or say a novel with a third person omniscient view in which the author, writes a gigantic Tolstoyan work, going into the motivations of every character.

That would be a detailed work and it would tell us what the society is fundamentally like, its ground state energy.

But say you are only interested in this fundamental aspect of society, not in the story of every single electron. Well what can you do?

Exactly what Kohn and Sham did.

To describe what the common denominator of that society is, you can move to a first person point of view.

You find out how that one member of society relates to other members of the society (taking society in a smeared out, abstracted sense). You find out how that one person interacts with all the buildings.

But since you are now not taking all the members your results would have an uncertainty. You would have to make presumptions to take into account the unique quirks of that individual.

You lump all that in the \(E_XC\) term. But if you’re making assumptions about what that unique person is then society’s interaction with that individual electron are mediated through his quirky behavior. So you have to explore his character, reducing his unique traits to a minimum.

When you’ve removed all those contributions, in a sensible way you will have all the different mental states and moods of the central character of your novel.

Kohn-Sham are essentially saying that if you pin down an individual’s unique traits and are able to negate them, you will have the same picture of what the society is fundamentally like; that you would have had if you’d taken the sum over all the individual members.