Excited states - QWalk/mainline GitHub Wiki

The purpose of the STATES section is to tell the code which orbitals to use in the determinants. The .orb file defines a large list of orbitals taken from the DFT code. They are numbered 1-N, where N is however many orbitals the DFT code output. For an atom, since there is spherical symmetry, each orbital may be identified with an angular momentum quantum number. For example, for carbon without a pseudopotential, it may be as follows:

and so on. (Technically Px,Py, and Pz are not eigenfunctions of the angular momentum operator, but are real linear sums)

So, for the C atom, how do we generate the STATES section for the ground state? Well, we know that the ground state is 1s^2 2s^2 2p^2. Because of Hund's rule, we know that the two electrons in 2p will be of the same spin. By convention, this is spin up. So, in the .sys file, we should have the following line:

NSPIN { 4 2 }

which tells the code that there are 4 spin up electrons and 2 spin down electrons.

Now, in the STATES section, we need to indicate the orbitals that should be included in the determinant. To obtain the ground state, we need to put:

STATES { 
  #first four orbitals go into the up determinant
   1 2 3 4 
  #next two orbitals go into the down determinant
   1 2 
}

Note that the wave function

STATES { 
  1 2 3 5
  1 2 
}

should be exactly the same energy as the above one.

So what if we'd like to evaluate the 2s->2p energy? First I should note that this is not the ground state of the Hamiltonian, which means that there are many caveats to this procedure (read the RMP by Foulkes et al, and references within). None the less, it often works pretty well to do what follows.

There are several states which involve a transfer from 2s->2p in this system, while maintaining the multiplicity 3 total spin. I'll list each, along with the corresponding states section:

excitation within the spin-majority channel

STATES { 
   1 3 4 5 
   1 2
}

excitation within the spin-minority channel into an already-occupied P orbital

STATES { 
  1 2 3 4
  1 3 
}

excitation within the spin-minority channel into an unoccupied P orbital

STATES { 
  1 2 3 4
  1 5 
}

These all may give different energies.

Finally, what if we'd like to evaluate the energy difference between ground states with different multiplicity (total spins)? This is actually much easier. You just perform a normal DFT->QWalk calculation starting with different multiplicities in the DFT code.