The Core Concept

Density Functional Theory (DFT) simplifies the study of multi-electron systems by replacing the high-dimensional Wavefunction with the 3-Dimensional electron density, n(r), as the fundamental variable. This allows us to calculate ground-state energies and material properties through a self-consistent iteration that treats electrons as non-interacting particles moving within an effective potential.

The mathematical genius of DFT lies in its ability to change a problem with 3N variables to one with only 3 variables.

Example:

 Let us look at the H2 molecule, instead of tracking 2 separate electron we look at the single electron ‘cloud’ they form. In stable H2 molecule it is thickest right   between the 2 molecules.

The Problem

A normal Schrödinger equation for a system of N electrons defines Hamiltonian as

The three terms represent Kinetic Energy, the External Potential (from nuclei), and Electron-Electron Interaction. Solving this directly is computationally impossible for large systems because the wavefunction depends on 3N variables (x, y, z for every electron).

The Solution

DFT reduces the problem to only 3 spatial variables (x,y,z). The Hohenberg-Kohn theorems prove that the total ground-state energy is a unique functional of the electron density, n(r):

Here, FHK(r) is the Universal Functional. Its mathematical form is independent of the external potential Vext(r), making it the theoretical backbone of DFT.

To solve for the density, we use Kohn-Sham mapping. This maps the complex system of interacting electrons onto a hypothetical system of non-interacting electrons that share the same density ,n(r). This density is constructed from single-particle orbitals:

Example:

When we solve for H2 molecule the orbital we get are the MO orbital of H2.As the Kohn-Sham mapping wants the 2 electron in the lowest energy level, the MO has the 2 electron there, creating the high electron density we see at the centre.

Note:

While the math of DFT and MO theory are different , they show that the for H2 molecule the ground-state density is formed by the 2 electron occupying the stable Bonding MO.

This lead to the Kohn-Sham Equation:

another thing we use here is the effective potential Veff, all many-body interactions are condensed into a single effective potential:

1. Hartree Potential (VH): The classical electrostatic repulsion of the electron cloud:

Hatree potential basically calculates the cost of packing the electron together even       though they want to repel each other.

Example:

In the H2 molecule the VH is high at the centre and less near the atoms. i.e. electron density Is high, VH is also high.

2. Exchange-Correlation Potential (Vxc): Captures all remaining quantum effects:

Since Veff depends on the density, and the density is derived from the equations using Veff, we solve the system iteratively:

1.Guess an initial density, n(r).

2.Calculate VH and Vxc using the current n(r)

3.Compute the Kohn-Sham equations to find the orbitals

4.Calculate a new density n(r)

5.Check the difference between the old and new density:

      if it is with set tolerance, use it to find the total energy

      if not, feed the new density into step 2 and repeat

Result:

By doing this process repeatedly with different distances between the H2 molecules we can plot a potential energy curve. This curves tells us how far apart the atoms want to be from each other(Bond length).