Friday, June 17, 2005

Walter Kohn and Density-Functional Theory

I have just attempted, and failed, to read Walter Kohn's Nobel Lecture: "Electronic Structure of Matter -- Wave Functions and Density Functionals" (pdf). I didn't learn the chemistry lesson, but I did learn another lesson in humility.

Well . . . I did also manage to uncover an error in the lecture. Page 221 is duplicated. I've contacted the Nobel Prize organization to alert them to this egregious flaw, and I expect soon to be awarded my own Nobel Prize for Proofreading.

But I still don't understand much about Density-Functional Theory, which I promised to get back to you on. Let me supply some weblinks that might be of use to those of you with greater intellectual capacity than I have (which is likely a large number).

Here's a nice site by Wilfried Gerhard Aulbur, professor of physics at Ohio State University. By 'nice,' I mean that it has some colorful pictures and color-coded graphs and tables. For those who can understand such things, this nice site also provides an "overview over the basic principles of DFT and some neat applications of DFT to real life problems." Or so says Professor Aulbur on the mother site, where he also tells us:

"Density functional theory is an extremely successful approach for the description of ground state properties of metals, semiconductors, and insulators. The success of density functional theory (DFT) not only encompasses standard bulk materials but also complex materials such as proteins and carbon nanotubes."

Well, it sounds very practical, as well as successful. And all of this is based on a main idea:

"The main idea of DFT is to describe an interacting system of fermions via its density and not via its many-body wave function. For N electrons in a solid, which obey the Pauli principle and repulse each other via the Coulomb potential, this means that the basic variable of the system depends only on three -- the spatial coordinates x, y, and z -- rather than 3*N degrees of freedom."

As Zippy the Pinhead would say, "Are we there yet?"

Well, this pinhead isn't. First, I need to know what a "ground state" is in order to understand what "ground state properties" means. I don't think that it refers to the state of coffee beans when I finish pulverizing them for brewing each morning. I think that it must refer to physics or chemistry. So . . . let's see Wikipedia's entry:

"In physics, the ground state of a quantum mechanical system is its lowest-energy state."

Unfortunately, this doesn't say much to me, for I would imagine that the lowest-energy state of a system is at absolute zero, but that doesn't seem to fit this context. Proteins? Carbon nanotubules? Do proteins even exist near absolute zero? Do carbon nanotubules?

Well, this hasn't gotten me very far. But at least, I know what fermions are. That is, I've seen the word before and know that it refers to a sort of subatomic particle. That's not really knowing, though, is it? Back to Wikipedia, this time on fermions:

"Fermions . . . are particles which form totally-antisymmetric composite quantum states. As a result, they are subject to the Pauli exclusion principle and obey Fermi-Dirac statistics. The spin-statistics theorem states that fermions have half-integer spin. One possible way of visualizing spin is that particles with a 1/2 spin, i.e. fermions have to be rotated by two full rotations to return them to their initial state."

This is about to spin out of control. Following up these multiple, branching links could lead to infinity -- and beyond! But then comes some information to bring us back to our ground state:

"All elementary particles are either fermions or bosons. Composite particles composed of fermions may be either bosons (such as mesons) or fermions (such as baryons) depending on their total spin. The elementary particles which make up matter are fermions, belonging to either the quarks (which form protons and neutrons) or the leptons (such as electrons). The Pauli exclusion of fermions is responsible for the stability of the electron shells of atoms, making complex chemistry possible. It also allows the stability of degenerate matter under extreme pressures."

Uh . . . wait a minute. What did that say? "Composite particles composed of fermions may be either bosons (such as mesons) or fermions (such as baryons)." This says that fermions make up bosons and fermions. Assuming that this is no error, then the term "fermion" is ambiguous between elementary particle and composite particle.

Look, I'm going to leave the rest of this to you folks out there in cyberland. If anyone with expertise wants to post a comment clarifying this fermion matter and exposing me for the density-dysfunctional thickhead that I am, please do so.

Oh, and please explain density-functional theory while you're at it.

Meanwhile, I intend to look into this "degenerate matter." It sounds almost Gnostic . . .


At 2:48 AM, Blogger Dennis Mangan said...

Couldn't tell you about DFT, but I do know that ground state refers to the electrons in their orbits (or shells) around the nucleus. These electrons will have the same energy regardless of the temperature: the temperature refers to the atoms or molecules in conjunction, their energy level as they bounce around. Proteins certainly can exist at absolute zero, where their electrons will still be merrily spinning away.

At 5:25 AM, Blogger Horace Jeffery Hodges said...

Thanks Dennis, that bit about "ground state" makes sense to me.

I'm still surprised about proteins at absolute zero -- I thought that the Bose-Einstein Condensation effect would alter proteins, forcing them to collapse into a superatom.

But what do I know?


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