On the physics of exchange, correlationa and coupling phenomena

[askhetan]

I have recently started my phd on ab-initio simulations, and have several confusions about the calculations (lets say with a method like DFT) of solid crystal structures.

1. what exactly are the exchange and correlation phenomena. I have read texts and books which describe its physical effects (in the sense that ferromagnetism occurs due to exchange interaction). However my question actually is - is it some esoteric physics that we still do not know but have only yet observed or is it something which is included in the Hamiltonian of the Schroedinger equation but cannot be solved for exactly for many body problems ?

2. Same for spin-orbit coupling. though the physics of this is much more clear to me because I can see magnetic moments of moving charges interacting, again - are these not included in the Hamiltonian of the Schroedinger equation or is it that they cannot be solved for exactly.

Once I have answers o these I would feel much easy to learn how to use DFT, etc.
Since I do not want to have just an overview, please give me answers as detailed as you'd like.

 

[M Quack]

1) AFAIK it is something well understood in principle, but difficult to treat in many-body theories. It basically concerns the overlap of wave functions of neighboring atoms. If you could write a completely antisymmetric wave function (multi-Hartree-Fock like), then it should pop out all by itself. If DFT it is usually fudged in manually as free parameter (LSDA+U etc).

2) It is, but again for simplicity and computational efficiency often DFT is done for spinless Schroedinger particles, and everything that is spin, LS coupling and so on is added as correction later.

You should be able to find much better answers to all these questions in some basic review text about DFT. Ask your supervisor for some references.

 

[askhetan]

Thanks for the answer, it is also the same that I read in many books - that there is ONE wave function that has to be completely expressed as a function of 3N+1 (3 dimensions+time) degrees of freedom for a N body problem. Also this ONE wave function determines the complete system, so in that way, the whole universe has ONE wave function. (Please correct me if I am wrong).

I guess the problem is that the common methods (Hartree fock, DFT) make use of approximate and decomposed wave functions. So does that mean that if we knew that 'ONE' wavefunction, then the expectation value of energy will automatically account for all such interactions and we will not need separate laws for these phenomena ??

I have also been wondering what are the equations that describe the energy changes due to all kinds of spin interactions (fine, hyperfine, etc). The spin interactions definitely do not seem to be included in the energy hamiltonian. If someone can direct me to a link that has the mathematics of how spin couplings are already included, or if it is otherwise, it would be very kind of them.

P.S. -The problem is that I see my supervisor once in 6 months. I did masters in fluid dynamics and am absolutely a noob in QM. Having a real hard time learning and teaching myself this stuff.

 

[M Quack]

Well, in most cases one looks for eigenstates of energy, so you don't have to deal with time. But otherwise you are correct in principle. In practice in solid state physics one assumes periodic crystals so that you only have to perform your calculations over one part of the Brillouin zone (part because the rest can be obtained by symmetry). If you are dealing with molecules or clusters then this is no longer true. If you are dealing with defects in a perfect crystal, then additional approximations come into play.

Pretty much by definition, if you know the one wavefunction of the complete system without approximations, then all interactions are taken care of and you know exactly what the energy levels are.

"Exchange" is entirely derived from Pauli's exclusion principle, i.e. the fact that Fermion wave functions have to be antisymmetric. This effectively leads to different energies for parallel and antiparallel spin (or total magnetic momentum) states that can be approximated (in the most primitive case) by a dot product of these magnetic moments times an "exchange constant". I have no clue how the value of such a "constant" could be derived from first principle, other than finding hte solutions of the "proper" wave functions for parallel and antiparallel alignment and then using the different in energy.

Wiki gives a surprisingly good introduction to DFT, and the literature list at the bottom is not too shabby either.

http://en.wikipedia.org/wiki/Density_functional_theory

This gives more details and should answer most of your questions.

http://arxiv.org/abs/cond-mat/0211443