Tag Archives: Density Functional Theory

​Density Functional Theory: A Philosophical Explanation

Density Functional Theory was proposed in 1970s. The idea behind the Density Functional Theory (DFT) can be understood from a well known and widely popular story of Archimedes. It was 200 B.C. when Archimedes proposed his theory on ‘Density’, which is proportional to the displacement of water of a particular mass by that substance. Knowing density, we can find various other properties of that substance, like volume, mass depending on its value we can predict whether a substance will float or sink in a particular liquid, whether an aeroplane will fly or not etc. Not only macroscopic substance, but also we can predict the properties of the molecules depending on its electron densities. This is the basis of this theory.

DFT is different from the Hartree-Fock theory in the sense that the later completely depends on the total molecular wave function of the molecule on the other hand, the former depends on the density of electron cloud. So we need to call it- Density ‘Function’ Theory (cause it calculates energy and all related things as a function of density), but why the term ‘Functional’ is used in place of ‘Function’. The suffix ‘-al’ is related more to the mathematics than the philosophy.

According to mathematics,

if y= f(x) (Read: y is a function of x)

and z= f(y),

then z= [x] (Read: z is a functional of x)

In mathematics, function of a function is known as functional. This is how the term ‘Functional’ came in Density Functional Theory.

Let me explain it technically,

E.g, E = f(density) = f(radius),

Therefore, energy and radius are in functional relationship.

We know,

Total Energy of a molecule,

E = E(ke)+E (n,n)+E (e,n)+E(xc)

Where,

E(ke)= Kinetic energy of electrons

E (n,n)= Nuclear-Nuclear repulsion energy

E (e,n)= Electron-Nucleus attraction energy

E(xc)= Exchange correlation energy

The value of exchange correlation energy, E(xc), is very less in the order of large values of negative power of 10. But it produces a huge impact on the total energy of the molecule. If we don’t consider the value of E(xc), and calculate the ionisation energy with the remaining equation the value will come out in negative, that’s an error, as we know the ionisation energy can not have negative values.
DFT is named that way because the theory is contributed by tens of scientist. So we can’t give it a name unlike Schrödinger theory or Hartree-Fock theory. But we can not ignore the pioneering work of Kohn and Sham, who gave the fundamental equation of DFT, that is known as Kohn-Sham equation.

For calculating energy from the Kohn-Sham equation, we need to find density, rho.
Density can be calculated as follows.

The above calculation is done computationally using Gaussian software. For this, we need to feed it with different basis set followed by the name of method (can be selected by keeping in mind what percentage of accuracy we want in our result as compared to experimental result). Basis set is almost similar to atomic orbitals.
Example of basis set: 6-31+G* (Six minus three one plus G star)

Where, 6 gives 6Gaussians for the core electrons, 3 gives 3Gaussians, 1 gives 1Gaussians respectively for the valence electrons- which gives a total of 10Gaussians for a single electron system, i.e, H atom. Higher the number of Gaussian, higher is the accuracy of the result with experimental values for a particular atom or molecule. For higher accuracy in result, we can use the term CBS (means: complete basis set). In other words, ‘Gaussian’ is the unit of accuracy.
For Hydrogen atom (1s1), if we use 6-31G basis set, we will get 10G. For Lithium (1s2 2s1), we will get 30G.

For DFT, the total number of integration is given by, N^3, for Hartree-Fock Theory, the number of integration is N^4 and for MP2 theory the number of calculation is N^5 (where N is the number of Gaussians). Higher the value of integration, higher is the accuracy and higher is the time it consumes. Here is a latest theory that gives the theoretical result with so accuracy that even let us doubt the value of experimental result. This is called CCSD(T), i.e, Couple Cluster Singlet Doublet with perturbation Triplet. Now imagine the calculation of energy with the help of CCSD(T)/CBS. This takes a total of N^7 integration. It may take 2 years approximately to calculate the energy of Hydrogen molecule with our laptops.

Now the fact is that, DFT gives a poor value of exchange energy but a good value of correlation energy. On the other hand, Hartree Fock theory gives a good value of exchange energy and poor value of correlation energy. The human nature is to experiment with things that they do not understand and Scientists are no different. So they made the hybrid of HF and DFT to get a better result of exchange and correlation energy. In the language of Gaussian software, it is called B3LYP (Becke’s 3 parameter hybrid functional with correlation from Lee, Yang and Parr).

B3LYP: DFT+HF

To get a pure result based on DFT, one can use BP86 (Becke and Perdew exchange correlation functional invented in the year 1986).

BP86: DFT