# Basic density functional theory

• kttsbnj
In summary, the process of solving for a wave function in DFT involves assuming a density, using a Hamiltonian to find a wave function, and then using that wave function to find another density. However, to calculate the wave function, one must first know the energy eigenvalue, which can be obtained through an eigenvalue problem. This problem can be solved by expanding the wave function in a finite basis and using a diagonalization routine to find both the eigenvalue and eigenvectors.
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.

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?

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.)

## What is density functional theory (DFT) and what is its basic principle?

Density functional theory (DFT) is a computational method used to study the electronic structure of atoms, molecules, and solids. It is based on the principle that the total energy of a system can be determined by the electronic density distribution, rather than the wave function as in traditional quantum mechanical methods.

## What are the advantages of using DFT over other methods for studying electronic structure?

DFT is advantageous because it is relatively fast and accurate, making it useful for studying large systems. It also does not require any empirical parameters, unlike other methods such as Hartree-Fock theory, and it can be applied to a wide range of systems including molecules, solids, and surfaces.

## What are the limitations of DFT?

One limitation of DFT is that it is based on the local density approximation (LDA) or the generalized gradient approximation (GGA), which are approximate methods for describing the exchange-correlation energy. This can lead to errors in predicting properties such as bond lengths and energies. Additionally, DFT does not take into account dispersion interactions, which can be important in some systems.

## What are some applications of DFT in scientific research?

DFT has a wide range of applications in various fields of science, including chemistry, materials science, and physics. It is used to study the electronic structure and properties of molecules, solids, and surfaces, as well as to predict chemical reactions and catalytic processes. DFT is also used in the design and development of new materials, such as for batteries and solar cells.

## How does one perform a DFT calculation?

To perform a DFT calculation, one must first choose an appropriate functional and basis set. The functional describes the exchange-correlation energy, while the basis set describes the wave function of the electrons. The calculation is then carried out using specialized software, which solves the equations of DFT to determine the electronic density and total energy of the system.

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