Basic density functional theory

[kttsbnj]

I try to learn DFT by myself(Kohn-Sham Equations), but the concept is still not so clear for me.

So far, if I start with assuming any density, and then I would be able to find V(KS)

Then I use this hamiltonian and solve for a wave function. And I use this wave function to find another density to repeat all steps.

The question is How can I solve for a wave function if I don't know the energy eigenfunction, or how can I get the energy eigen function.

 

[kttsbnj]

following the web ... worldwideweb.tcm.phy.cam.ac.uk/~ajw29/thesis/node11.html

equation 1.34

how can I calculate wavefunction. Basically one must know the energy eigen value first, right?

 

[kanato]

Equation 1.34 is an eigenvalue problem. What is generally done is to expand $$\psi$$ in some finite basis, then you have a matrix eigenvalue problem for the coefficients of the basis function and the energy eigenvalue. Once you have this matrix you can pass it along to any diagonalization routine that can handle Hermitian matrices with complex entries, and that will produce both the eigenvalue and the eigenvectors. There are a variety of such methods which produce both the eigenvalues and eigenvectors at once, without prior knowledge of either.

(As a practical matter, if you have an extremely large basis set, such as with plane waves, then you need to use tricks based on the fact that the Hamiltonian matrix will be sparse and you are only interested in some number of the lowest eigenstates.)