First principle calculation of lattice structures

[janakiraman]

Hi

Well I have been thinking of this quite sometime, but i really dint know anyone who could answer my question. We all know that may elements exist in the form of crystals e.g. Si, Cu etc. And they have very specific crystal structure like Diamond for Si and face centered for Cu. Can we prove from first principles of quantum mechanics like DFT or some other techniques that only these are the most favorable crystal structure for a particular element? If so can any of you provide me links to the research works that have been done in this field

Thanks
janakiraman

 

[kanato]

This sort of thing is very difficult to prove. For first principles methods, the crystal structure is an input, so the only way to identify which crystal structure is preferred is to compute the total energy for several trial crystal structures and see which is lowest. Of course, this means you can only decide between crystal structures that you test. You can do structural relaxations, but these will only find a local minimum in the energy and they won't break any of the symmetries that you start with, so you have to start close to the right structure to begin with. I seem to recall that some elements have very complex crystal structures (Be or B?) that would be virtually impossible to deduce from scratch.

 

[janakiraman]

This sort of thing is very difficult to prove. For first principles methods, the crystal structure is an input, so the only way to identify which crystal structure is preferred is to compute the total energy for several trial crystal structures and see which is lowest.

.

Sorry but do please correct me if I'm wrong. I think the only exact potential that has been calculated till date is the Learnado Jonnes Potential. I think all the other first principle methods lag way behind in identifying the exact potential of the system. If that's the case, how do you say you can compute the potential for different structures to see which one is the lowest

Of course, this means you can only decide between crystal structures that you test. You can do structural relaxations, but these will only find a local minimum in the energy and they won't break any of the symmetries that you start with, so you have to start close to the right structure to begin with. I seem to recall that some elements have very complex crystal structures (Be or B?) that would be virtually impossible to deduce from scratch.

I also think that almost all crystal structures are within the shapes defined by the Bravia Lattice. I'm wondering if atleast it is possible to show why only these few structures are preferred from a whole lot of possible alternatives for crystal structure?

 

[kanato]

.Sorry but do please correct me if I'm wrong. I think the only exact potential that has been calculated till date is the Learnado Jonnes Potential. I think all the other first principle methods lag way behind in identifying the exact potential of the system. If that's the case, how do you say you can compute the potential for different structures to see which one is the lowest

I said the total energy needs to be calculated, and I mean the total quantum mechanical energy of the electrons (in principle, one should include phonons, but this will only be relevant if comparing two structures with very close electronic energy). You can do this in DFT, with suitable approximations like LDA but you will have to pick trial structures to test.

The Lennard-Jones potential is an effective pair-wise atomic interaction. It isn't exact for any system, but it can be a good approximation for certain species of atoms (mainly noble gases). Since you bring up the LJ potential I'm guessing you're thinking of molecular dynamics (MD). I'm not an expert on MD, but I think it's not very good for getting crystal structures. The main reason is that MD needs to have some finite kinetic energy or temperature to proceed. If you start with very low temperature, the system will stay in the configuration it started in, and if you start at high temperature and lower the temperature then as the crystal forms you usually end up freezing defects into it.

In any case, two body potentials (like LJ) will always lead to a close packed structure like fcc or hcp, and they will not differentiate between them. To do anything more in MD you need 3 or 4 body potentials. These get quite complex and are usually rather specific to the species of atom used. It's are also quite difficult to construct effective many body potentials that are accurate in MD. Also I think there are significant difficulties where magnetism is concerned, IIRC iron prefers one phase when it is magnetic and another when it is not, and magnetic phase transitions may or may not be related to structural transitions.

I also think that almost all crystal structures are within the shapes defined by the Bravia Lattice. I'm wondering if atleast it is possible to show why only these few structures are preferred from a whole lot of possible alternatives for crystal structure?

The Bravais lattices are the different ways you can have discrete translational symmetry in 3D space. Any periodic crystal will be a Bravais lattice, but they do not tell you much about the crystal structure. For instance, diamond is an FCC Bravais lattice with two atoms per unit cell. You can have Bravais lattices with more atoms per unit cell, so this makes the possible configurations that have to be explored quite large. IIRC Gallium has 8 atoms per unit cell.

The most obvious reason for preference of hcp and fcc is that these structures are close packed. Each atom can maximize its number of neighbors and bonds. This is how you would fill space with equals sized classical spheres, or with atoms that interact in an LJ structure. Then the question you have to ask is why do certain atoms deviate from that behavior, or why do certain atoms prefer one stacking of close packed layers (fcc) vs. another (hcp).

 

[Bob S]

Carbon has (up to) eight allotropes or naturally occurring forms. Several are diamond, graphite, lonsdaleite, fullerenes, nanotubes. Which allotrope is the most favored (most likely?)? Do all or most of these allotropes have different total binding energies? What is (are) the heats of combustion of these allotropes? My handbook lists the HHV (higher heating value) as 393.5 kilojoules per mol. Are they all the same?

 

[janakiraman]

Ok i think i got confused. LJ and others are emperical potentials used in MD simulations. I don't have a good idea about Bravais lattice and will get back about it once i learn more details. How about the question above? Which allotrope of C is most preferred and why?

 

[genneth]

I'd imagine it depends strongly on temperature and pressure. I'm fairly sure that at RTP the preferred form is graphite.

 

[janakiraman]

I also think graphite is more preferred at room temperature. Also i would like to know if DFT can predict exactly what crystal structure is preferred at what temperature and pressure. I'm also just curious what happens to the fermi distribution of electrons within an atom with change in temperature and pressure. If it changes can DFT catch this variation?

 

[genneth]

DFT is very good at calculating the ground state electronic energy. However, lattice structure is also dependent on the position and movement of the nuclei. Usually, DFT is used to create the inter-atomic forces; i.e. a Born-Oppenheimer approximation, which is as good as the ratio of electron to nucleus mass. However, it's not a very good idea to simply let the atoms relax into place from some initial arrangement --- whilst being a good simulation of the actual kinetics, it often won't settle into the actual state (remember that for a finite system, the difference between eigenstates goes as exp(-<system size>)).

I believe it is more customary to use quantum monte carlo methods when lattice structures are wanted. I know that the Cambridge TCM group is particularly strong at this (alas, not my field). The idea here is to simply write down the Lagrangian (or Hamiltonian) corresponding to the entire system, and evolve it along the imaginary time direction whilst preserving normalisation of the wavefunction. Each eigenstate will then evolve as exp(-Et), so after a sufficiently long time the ground state will be exponentially dominant. Precise details are, of course, significantly more complicated. If memory serves, this has been used to predict various crystal structure (especially at high pressure) which were then confirmed.