Density functional method refers to methods for calculation of the total energy and various physical properties of physical systems based on density functional theory (DFT). DFT has its foundations in Hohenberg-Kohn theorems, which basically states that the energy and physical properties of many-electron systems can be calculated from a universal functional of the electron density. This theory shows that in order to describe an N-electron system, there is no need to solve the Schroedinger equation for a wave function with 3N variables, but that it suffices to handle the electron density with 3 spatial variables. Thus, it is a very powerful theory for describing many-electron systems, but there is a caveat; it does not give a prescription for the form of the density functional. Good approximations for the density functional is still an active area of research.
DMRG is a method for computing expectation values of various quantities at the ground state of a given Hamiltonian that describes a quantum many-body system in discrete space. It can be viewed as a variational method using the matrix product state (MPS) as a variational wave function. Therefore, it provides a precise approximation in the cases where the matrix product state is a good description of the target state. One-dimensional systems is a typical example. The precision of the approximation can be controlled by the dimension of the tensor indices (the bond dimension). The quality of the result can therefore be evaluated by altering the bond dimension. Since the MPS is a special case of the tensor network state (TNP), DMRG can be regarded as one of the tensor network methods.
One of the quantum Monte Carlo methods. The method is based on the observation that the evolution in the imaginary-time can be used as a projection operator onto the ground state. In the DMC, one introduces many walkers, each carrying a weight, and let them do random walks in the space of the basis states in such a way that the evolution of the ensemble of the walkers stochastically satisfies the imaginary-time dependent Schrodinger equation. DMC is often used with other variational, e.g., for generating the initial distribution of walkers.
A method for first-principles calculation in treating charged systems and electronic structures of materials under electric fields for slab models (= models with periodic boundary conditions along two spatial coordinates in three dimension). In this method, one first imposes a periodic boundary condition for two directions, and an appropriate boundary condition for the other direction, and then solves the Poisson equation with Green’s function method. By this method, charged states and effect of applied electronic fields are appropriately taken into account. A number of first-principles calculation packages supports this method.
A method for studying quantum many-body systems by diagonalizing the Hamiltonian matrix. Due to its exactness, this method is widely used in spite of the strong size limitation. By switching from the full diagonalization to partial diagonalization in which only a small number of eigenvalues and eigenvectors are computed, one can relax the size limitation. As program packages based on the Lanczos method, TITPACK, KobePack, and SpinPack are available. A full diagonalization can be found in ALPS package. HΦ provides codes suitable for massively parallel computers for a broad range of models.