Basic concepts for DFT calculations
All the material properties can be described by the seemly simple many-body Schrodinger’s equation (for relativistic effects, we need to use the Dirac’s equation):
However, numerically, to describe the many body wavefunction Y [Y is Psi] we need M^N data points (where M is the number of grid points for the space, and N is the number of electron). Thus direct description for such wavefunction is impractical for large systems (the scaling is N^N, i.e., exponential), except for some statistical methods (quantum Monte-Carlo methods). In the field of continue-function description of Y, there has been a long history of attempts to approximate Y using the single electron wavefunctions y(x). This is based on good physical intuitions: Slater determinate for molecules and quasiparticle wavefunctions for solids. This can also be based on the natural orbitals of the charge density matrix. Note that, there will be N occupied and mutually orthogonal single particle wavefunctiuons y(x) [y is psi] Thus, the numerical data needs to describe them is M*N. The scaling now is N^2, acceptable for thousand electron systems. For even larger systems, modern schemes try to take the advantage of the locality of the density matrix r(r1,r2), thus make the whole calculation scale as O(N).
Formally, the attempt to change from Y to y has
been helped significantly by the density functional theory (DFT) [P.
Hohenberg
and W. Kohn, Phys. Rev. 136, B864 (1964)]. From the above many body
equation,
it is easy to see that the ground state many body wavefunction Y is a
functional of the external potential 
.
The DFT says that the V® can actually be thought as a functional of the ground state charge density r. As a result, every electronic property of the system is a functional of r. Why this is a big deal? It gives people basis and confidence to approximate everything (e.g., the exchange-correlation energy) based on r (density functional theory). The result is a surprising success of the DFT to describe the energetics of materials. Now, back to the Y to y transition. It turns out, all the other energies (Coulomb energy and exchange-correlation energy) can be directly represented or accurately approximated by the charge density r, except the kinetic energy. To represent the kinetic energy accurately, Kohn and Sham went back to use the kinetic energy of the single particle wavefunctions y, replacing the true many body wavefunction kinetic energy, and shifting the difference to other terms (the exchange-correlation terms). This results in the Kohn-Sham algorithm [W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965)], and the Kohn-Sham single particle equation:
Here, V(r)[rho] is a functional of the charge density r, and
and the local density approximation (LDA) for V(r)[rho] is:
here mu_xc(r) is a fixed, pre-fitted (e.g, from quantum Monte-Carlo simulation for homogeneous electron gas) function. This is at the heart of the local density approximation, because the exchange-correlation energy density at point r only depends on the charge density at that point. So, to do a calculation based on LDA [or similarly LSDA (include spin), GGA (include some gradient information in the function mu_xc)] is to solve the Kohn-Sham equations for a given atomic position set {R}, and to solve these equations self-consistently (i.e., the input potential V(r) in the Kohn-Sham equation equals the output potential V(r)[rho] generated from the output charge density r. These solutions are also the variational minimum energy solutions in a Kohn-Sham total energy expression. As a result, the force on the atoms can be calculated based on the charge density using the Hellmann-Feynman theory. This gives us the basis to move the atoms.
For a review of the planewave pseudopotential algorithms used to solve the Kohn-Sham equation, please see: M.C. Payne, M.P. Teter, D. C. Allan, T.A. Arias, and J.D. Joannopoulos, Review of Modern Physics, Vol.64, 1045 (1992).
