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Code and scripts for wavefunction projections of heterogenous systems.

IMPORTANT: I haven't gotten around to learning the github markup language yet for my documentation files, so some of the following might not render properly in your browswer. The gnuplot command below, for instance, is correctly written in this text file, but is incorrectly displayed through the website!

Frequently in nano and material science, we wish to understand how an environment affects the electronic structure of a molecule (or vice-versa). Suppose you know the ionization energy and electron affinity of a molecule in the gas phase. When this molecule sticks to a surface, how do those numbers change? Do the orbitals of the molecule look the same in this new environment, or are they different? You might approach this problem by doing a calculation using a method like DFT and visualizing the wavefunctions and corresponding energies. The problem with this approach is that the orbitals you will get will be mixtures of the molecule states and surface states. While these are the eigenstates of your Hamilitonian, they don't answer the basic question of what changed. To do this, we need to look at the Hamiltonian of the combined, mixed system in basis of the original orbitals. This is analogous to a molecular orbital diagram seen in 1st year chemistry textbooks. KIB is a code which takes wavefunctions from planewave, periodic DFT codes and represents the Hamilitonian of a mixed geometry in terms of the individual components.

The code works with three sets of wavefunctions, each corresponding to a different Hamiltonian.

iota = the isolated molecule states
beta = the bare surface states
kappa = the combined molecule and surface states

In practice, one starts with a geometry of interest (e.g. a molecule on a surface). The corresponding wavefunctions are the kappa states. To generate the iota and beta wavefunctions, create two new geometries and do an electronic structure calculation on each: one with the molecule removed (this will give the beta wavefunctions) and one with the surface removed (this will give the iota)

Once you've completed this step, you are basically done. Everything after this is linear algebra and a bit of plotting/analysis.


N.B. This code is currently built around reading and working with wavefunctions from the ancient but beautiful paratec code. I intend to add support for Quantum Espresso, but have not yet had the chance. You're probably wondering whether it does or will work with vasp. The short answer is no. The internal workings of the WAVECAR file seem to be a bit of a mystery. If you have a straightforward way to get the planewave-coefficients out of this file, let me know and I would be happy to add support!

With that said, here's how to use the code. Again, everything is based on wavefunctions generated with paratec.

Compile the code on your machine using the makefile.

Then go into the iota directory. Here you will find the real and imaginary part of a wavefunction on a real space grid. The actual electronic structure problem was solved with planewaves, but has since been backtransformed into real space.

To run the executable, from this iota directory type

../exe.osx WR.01.001.00001.dx WI.01.001.00001.dx

(Paratec stores real and imaginary part of the wavefunction was actually stored in two different files. Really we are feeding it one complex wfn)

You should see the following output:

 # The norm of the isolated state is   3455999.9821567237     
 # I think there should be     3456000 i.e.    99.99 %
 1 -0.46954201505784937       0.87873213954149121     
 2 -7.04689635864237055E-003  1.31880306517494130E-002

The first two lines are the header, and are a basic sanity check. If you don't see a value close to 100%, it means there is probably a mismatch between the gridpoints between your iota and kappa wavefunctions, or the number of gridpoints in the file has not been understood by the code. Having that number be close to 100% is a necessary though not sufficient condition for normal operation.

What follows is the main output of the code. Each line is the projection of the iota you specified in the command line argument with the kappa states in the other directory.

In this example, there is a significant weight on the 1st kappa state, and none on the second.

Due to file sizes, only two kappa states and one iota state have been included in this example. Normally, one would include all occupied kappa states and some number of unoccupied states in the kappa directory. The exact number of unoccupied states to include is system specific, and is essientially a controlled approximation. The more unoccupied states you include, the more flexible your basis will be and the better you will span the vector space. Fortunately, one can check if a sufficient number of kappas have been included in the expansion by evaluating the inner product of an iota expressed in the basis of kappa. In the iota directory, there is an additional directory called precomputed. Here are the results of using the code on several iota states (iota001 corresponds to the dx files we just used), where the projection was done using 1244 kappa states (rather than just two as per the example).

From this directory, the script ../../ can be called. This will compute the inner product and gives the following output:

iota001  0.999863
iota002  0.999826
iota003  0.999909
iota004  0.999999
iota005  0.999874
iota006  0.999991
iota007  0.999881
iota009  0.999928
iota010  0.999857
iota011  0.999884
iota012  0.999977
iota013  0.999984
iota014  0.999953
iota015  0.999185
iota016  0.999856
iota017  0.9999
iota018  0.997309
iota019  0.994961
iota020  0.999993
iota021  0.995398
iota022  0.999823
iota023  0.981272

For all but iota023, more than 99% of the wavefunction is accounted for, meaning we used enough kappa states in our expansion. If you find that you are not getting a norm close to 1.0, you probably need to include more kappas.

Since we have a decent expansion, we can now used these states to look at how strongly coupled our system is.

One thing to look at is < iota | kappa_i > < i_ kappa |

Using gnuplot (for example), we can plot the projection:

plot "iota021" u 1:($2**2 + $3**2)

Using that command will give square modulus.

To get a matrix element of the form < iota_i | H(\kappa) | iota_j >

use the command

../../ iota021 iota022 ../../kappa/eigenvalues

This will print the real and imaginary part of the matrix element. Essientially this tells you what the coupling between iota021 and iota022 is when they are in the presence of the combined Hamiltonian

This is how bonds form.

Cool right?

NOTE, files contained in the wavetrans directory were not written by me. They are the very nice code written by R. M. Feenstra and M. Widom, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213. I am working to make a vasp friendly version of KIB, and happily it looks like this will do the trick. Their page can be found here:

*** I will be adding vasp compatiblity shortly (or at least I plan to, thanks to this: