Is my methodology for learning density functional theory correct?

[RedX]

I'm trying to learn density functional theory by myself, but I'm a bit confused as to how to use it. Is the following methodology correct (I think it'd take forever to use LaTex to write the equations, so I have a link to small webpage that already has the equations laid out and numbered)?

http://docserver.ub.rug.nl/eldoc/dis/science/f.kootstra/c2.pdf

Once you have an expression for the exchange correlation energy as a functional of the density:

1) Guess at a density to plug into 2.31
2) Solve 2.3.1
3) Using the energy eigenfunctions obtained from 2), calculate 2.22 and 2.33
4) Plug 3) into 2.24 to get the energy

Now using the density obtained in 3), go through the whole process again, until you get no change in 4). Then you're at the ground state energy?

I understand that if you want to find the ground state energy of any problem, you can pick arbitrary wave functions and evaluate:

<Energy>=<wave function | Hamiltonian | wave function>

until you find the wave function which minimizes <energy>.

Now

<wave function | Hamiltonian | wave function>
=<wave function | T | wave function>
+<wave function | Vexternal | wave function>
+<wave function | Vinternal | wave function>

Now the last two terms, once given the density, is an electrostatics problem. The problem is calculating the average value of the kinetic energy given the density. Using the auxilliary non-interacting system with the Kohn-Sham equations allows you to calculate a wavefunction which is a function of just 4 variables (including spin), and you can get a kinetic energy from the wavefunctions. Am I correct in this assesment?

 

[santoshroy]

Yes your interpretation is correct. This way of getting a minimised energy eigen value is actually variational principle where variation in E should be a extremum w.r.t the assumed density functional.

 

[R0man]

It appears that your methodology for learning density functional theory is on the right track. However, it is always recommended to consult with a teacher or textbook to ensure that you fully understand the concepts and equations involved.

In general, the steps you have outlined seem to follow the basic principles of density functional theory. By guessing a density and then solving for the energy, you are essentially using the variational principle to find the density that minimizes the energy. This is a common approach in DFT calculations.

Furthermore, your understanding of the role of the Kohn-Sham equations in calculating the kinetic energy is correct. The auxiliary non-interacting system allows for a simpler calculation of the kinetic energy, and the resulting wavefunctions can then be used to calculate the total energy.

However, it is important to note that DFT is a complex and constantly evolving field, and there may be additional considerations and nuances to take into account in your methodology. It is always best to consult with experts in the field or consult reputable sources to ensure that your approach is accurate and comprehensive.