What is the Role of Wavefunction in DFT for Molecular Systems?

[Karrar]

DFT used electron density instead wavefunction, and in same time DFT use Schrödinger equation that use wavefunction. !??

so it still depend on wavefunction of Kohn-Sham and not depend on density only non interaction system wavefunction of still stand

please I have many Paradox to interpretation of DFT

 

[alxm]

DFT used electron density instead wavefunction, and in same time DFT use Schrödinger equation that use wavefunction. !??

No, DFT doesn't necessarily use the Schrödinger equation. The http://cmt.dur.ac.uk/sjc/thesis_ppr/node12.html" showed (from the S.E.) that the ground-state energy can be determined by a functional of the density, (i.e. the exact density functional or EDF) and that the variational principle also applies to the density (for the EDF! Note that the variational method is still used with the approximate functionals used in DFT, although this is mathematically not justified. It's just generally assumed that since the EDF is variational, a good approximation will be as well).

This proves the EDF exists, but it doesn't give us any information at all on what it is (and we still don't know). But forget anything you might know about Kohn-Sham theory for a second and consider the basic problem of finding out the density functional. You know you can split the energy into two parts, the kinetic and potential energy. What can we say about these? The potential energy in terms of the electronic density is very straightforward: You have the Coulomb attraction between the density and the (assumed-to-be-stationary) nuclei, so that's simply:
$$V_{ne}[\rho] = -\sum_{A}\int\frac{Z_A\rho(\mathbf{r})}{|\mathbf{r} - \mathbf{R_A}|} d\mathbf{r}$$
And the Coulomb repulsion between the electrons is simply:
$$V_{ee}[\rho] = \frac{1}{2}\int\int\frac{\rho(\mathbf{r})\rho(\mathbf{r'})}{|\mathbf{r} - \mathbf{r'}|} d\mathbf{r'}d\mathbf{r}$$
With a factor 1/2 to avoid double-counting.

So now we're missing two things: The kinetic energy, and the exchange energy (i.e. the difference in energy due to the Pauli principle). How would you go about determining that from the density? Well, there's no good way to do this. The simplest approach would be to just take some non-interacting electrons in a box (homogeneous electron gas) and figure out their energy in terms of this homogeneous density. You can also (from the S.E.) calculate the exchange energy of this homogeneous electron gas, and include that.

This leads to the Thomas-Fermi and Thomas-Fermi-Dirac density functionals, which were the first to be developed (well before the HK-theorems even). So far, no orbitals and no Schrödinger equation at all. But the TF/TFD models are very bad models for molecules, which have anything but a homogeneous electron density! (Although in light of how bad the underlying assumptions are, you could also say it's surprisingly good.) They have had some use in solid-state physics, however. Work continues, but it's been very difficult to improve on this approach (so-called 'orbital-free' or 'non-KS' DFT), especially compared to Kohn-Sham DFT methods.

Now, what the Kohn-Sham scheme does, is to (seemingly) take one step backwards in order to move forward. We know from the Schrödinger equation that:
$$\sum_i^N<\psi_i|-\nabla^2|\psi_i>$$
Is the exact ground-state kinetic energy for a system of $$N$$ non-interacting electrons. We also know from Hartree-Fock theory that this accounts for up to 98% of the true kinetic energy.

The Kohn-Sham idea is to introduce this non-interacting reference system of orbitals for non-interacting electrons, and build our density out of them. The density functional becomes an effective potential that acts on the non-interacting electrons for the purpose of calculating the kinetic energy. We're solving the Schrödinger equation again, but not the Schrödinger equation of the actual, interacting system, but a much simpler S.E. for non-interacting electrons in a potential.

Hence, you have the all the potential energy and most of the kinetic energy accounted for. What remains is the exchange and correlation energies, since we've not taken into account either the Pauli principle, or the change in kinetic energy due to electron-electron interactions. In everyday DFT terminology this is the 'functional', since the rest is implied. Now you can solve this problem using a variational/SCF approach and in a similar fashion to the Hartree-Fock approach leads to the Roothaan–Hall equations, you get the analogous (and very similar) Kohn-Sham equations. (In fact, if you solved the KS equations without any functional at all, it'd essentially amount to the Hartree method, i.e. HF without exchange*)

The difference here is that you're not solving the Schrödinger equation for the real system, but for the non-interacting reference system. The Kohn-Sham orbitals do not correspond to real orbitals, strictly speaking. You'll get different opinions on what their physical significance is (if any) when you ask different quantum chemists.

(* If you've managed to follow my explanation, you might get the idea: Why not calculate the Hartree-Fock exchange for the non-interacting reference system and use that as well? Well, we do that with modern DFT methods. The famous B3LYP functional does so, using a mixture of this 'exact' exchange with the LSDA and B88 exchange functionals, and LYP/VWN for correlation. The first two parts are analytical, the other parts and the mixing ratios are semi-empirical though. Given that DFT methods are usually referred to as 'semi-empirical', it's worth underlining that only the exchange-correlation functional is, and not all functionals, either. DFT is not inherently semi-empirical)

 

[Karrar]

that great, I so happy to your reply it is so clear explanation , thanks alxm