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All-electron method refers to electronic structure methods that consider all electrons in the system explicitly in solving for the wave functions (or electron densities). This is in contrast to the pseudopotential method where the influence of inner-shell electrons is taken into account through an effective potential (i.e., pseudopotential). All-electron methods are, in general, more accurate than pseudopotential methods. An additional merit is that all-electron methods are more suited for simulation of X-ray spectroscopy which involve inner-shell electrons. On the other hand, the calculation cost is higher than the pseudopotential method due to the larger number of electrons under consideration. Software codes employ various measures for efficiently handling steep variations in the inner-shell orbitals and gradual changes in interstitial regions at the same time.

Many calculation codes expand the wave function using functions localized around atoms. Usually, numerical solutions to the Kohn-Sham equation for the atom or atom-localized Gaussians are used. Since the wave function in molecules and solids usually resemble a linear combination of atomic wave functions, atom-localized basis sets can describe the wave function accurately using a much smaller number of basis functions compared to the plane wave basis set. However, since atom-localized basis functions are not orthogonal in general, the improvement with increasing basis size is not always monotonic. Moreover, atom-localized basis sets are usually unsuitable for describing systems where electrons exist at positions away from atoms (such as in electrides or floating electron states).

The Bayesian optimization is one of the machine learning methods to find the optimal value (maximum value or minimum value) of a function whose shape is unknown (black box function). From the function values at a finite number of points, the Bayesian optimization predicts the functional form by the Gaussian process from the function values at a limited number of points. Then, a point, which is expected to have the optimum value or that with large uncertainty is examined preferentially. By sequentially adding search points and gradually increasing the accuracy of prediction, this method can find the optimum point even in a relatively high-dimensional search space.

In the context of condensed matter physics, methods of Markov-chain Monte Carlo simulation for many-body systems obeying classical statistical mechanics are called classical Monte Carlo methods. The representative example is the one by the Metropolis-Hastings method. The Markov-chain Monte Carlo method for simulated annealing is also a classical Monte Carlo method. In particular, a local update method is the method in which a degree of freedom is selected at each step for update. A local update generally suffers from the slow convergence near the critical point (critical slowing-down). To solve this problem, global update methods are proposed, in which a group of elements are updated at each time. However, it is necessary to design the algorithm depending on each specific system, and the applicability of the algorithm is limited compared to the local update methods.

An approximation often used for disordered systems. In general, it is difficult to describe the disordered systems, including alloys, within the framework of the band theory that usually assumes periodic structures. CPA successfully describes realistic alloys quantitatively, by confining the effect of the disorder within each site in a self-consistent fashion. As a part of KKR method, CPA is used in computing the electronic state calculations of disordered systems.

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 to solving a strongly correlated quantum lattice model. This treats correlations along imaginary time (dynamical correlations) with accuracy but ignores spatial ones. This can exactly solve infinite dimensional models such as models on the Bethe lattice. In this method, one first reduces the original model to an impurity model (Anderson model) by dividing the original lattice model into a center site (impurity) and the surrounding sites (effective medium) . Second, one solve this impurity problem under the self-consistent condition that a Green function and a self energy of the medium are equal to those of the original lattice model. Exact diagonalization, numerical renormalization, and path-integral Monte Carlo method are used for an impurity solver. Including spatial correlations has been studied.

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.