# Keywords

## Atom-localized basis/Gaussian basis

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).

## Coherent Potential Approximation (CPA)

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.

### Related keywords

KKR method

## Density functional method

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.

## Density matrix renormalization group (DMRG)

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.

### Related keywords

tensor network method

### Related applications

ALPS, iTensor

## Diffusion Monte Carlo (DMC)

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.

### Related applications: CASINO, QWalk

## Dynamical Mean Field Theory (DMFT)

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.

### Related applications

ALPS, TRIQS

## Effective screening medium method (ESM)

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.

## Exact Diagonalization (ED)

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.

### Related keywords

Lanczos method, Krylov subspace method, Heisenberg model, Hubbard model

## Fragment molecular orbital method (FMO)

A method used in quantum chemistry calculation for large molecules. In this method the whole molecule is split into multiple clusters (fragments). For each fragment molecular orbital calculation is carried out with the static electric potential caused by the other fragments.

### Related applications

GAMESS, ABINIT-MP, PAICS

## GW approximation

A method for calculating quasiparticle energies of many-electron systems using perturbation theory. The self energy of the many-electron system is approximated by the single-particle Green's function G and screened coulomb interaction W. This makes possible much more quantitative calculations of quantities such as photoemission spectra compared to the Kohn-Sham method with semilocal functional approximations. Nowadays, the accuracy is, in many cases, comparable to experiment. The GW approximation is usually coded as a perturbation on Kohn-Sham wave functions.

## Heisenberg model

One of the most basic models that describes the spin degrees of freedom in solid, which is a reasonable description of many materials in low-energy (low-temperature) scale. There are many variants according to the types and ranges of the interactions, and lattice structures. In particular, the antiferromagnetic Heisenberg model can be obtained from the half-filled Hubbard model through the perturbation with respect to the hopping constant, which makes the model relevant in the study of high-temperature cuprate superconductivity. The case where frustration exists, as in the case of the antiferromagnetic model on a kagome lattice, is a target of active research with the expectation of novel quantum states. The cases without frustration can be studied by quantum Monte Carlo and software packages such as ALPS and DSQSS are available whereas more general cases can be dealt with by exact diagonalization using for exacmple TITPACK, SpinPack, KobePack, and Hphi. Variational Monte Carlo is also an option in the latter case, for which mVMC is available.

### Related applications

ALPS, DSQSS, TITPACK, SpinPack, KobePack, HPhi, mVMC

## Hubbard model

A theoretical model proposed by Hubbard, Kanaori and Gutzwiller for describing the electronic states of transition metal oxides. It consists of the transfer term and on-site Coulomb interaction. Despite its simplicity, its mathematical treatment is rather difficult, allowing exact solution only for very restricted cases, e.g., the one-dimensional system. The model is extensively studied since it exhibits many phases of interest: ferromagnetic, antiferromagnetic, Mott insulator, high-temperature superconducting, etc. Exact diagonalization of small clusters of the model can be done with ALPS, SpinPack, Hphi, etc. By Hphi, calculation of dynamic properties, and finite-temperature calculation is also possible. Calculation by dynamical mean-field theory can be done by ALPS and pyDMFT. Variational Monte Carlo calculation can be done with mVMC.

### Related applications

ALPS, SpinPack, HPhi, pyDMFT, mVMC

## Ising Model

One of the most fundamental models in statistical physics that is first introduced as an effective model of the ferromagnetic-paramagnetic phase transition. This model consists of "spins" that take value of +/- 1 on a lattice. When each spin tends to take the same value as that of neighbor spins (ferromagnetic interaction), all spins take the same value at absolute zero (ferromagnetic phase or ordered phase). At the high temperature limit, on the other hand, all spins takes independent values from each other (paramagnetic phase or disordered phase). Finite-temperature phase transition is a phenomena that the system changes from one phase to the other phase at some temperature. When randomness of interactions is introduced, this model becomes difficult to solve even numerically. For example, the ground state cannot be found in polynomial time and the thermal-relaxation time becomes very long (spin glass). Ising model is a classical model originally, but one can obtain quantum version of Ising model by introducing transverse magnetic field flipping a spin as a source of quantum fluctuation. In this model (transverse field Ising model,) ferromagnetic-paramagnetic phase transition manifests oneself even at absolute zero by controlling the strength of a field (quantum phase transition). Since Ising model itself is very simple but supports many phenomena, it has been well studied and used in not only classical / quantum statistical physics but also classical / quantum information theory and machine learning.

## Job management

For use of a massively large computer system, an user requires a machine resource (the number of CPU cores and wall-time) and runs one's calculation on assigned resources from job manager of the supercomputer. Some tools and libraries implement useful functions that automatically require resources, effectively use assigned resources (load balancing), save a snapshot of the simulation and restart from the snapshot (checkpoint), and store pairs of input and output of each simulation. MateriApps project refers these functions as "job management."

### Related applications

OASIC, ALPS, ALPScore

## KKR method

One of the first-principles calculation methods. The name of the method comes from the capital letters of the developers, Korringa, Kohn, and Rostoker. In this method, the Green's function which describes multiple electron scattering due to effective potentials is numerically solved. By combining with density functional theory (DFT), electronic structure calculation can be performed efficiently. Furthermore, by combining with CPA (coherent potential approximation), this method can be applied to electronic state calculation for impurity systems and random alloys. Although various methods related to DFT can, in principle, be used, open packages usually employs pseudo-potential methods and muffin-tin potentials.

### Related keywords

CPA

### Related applications

SPRKKR, AkaiKKR

## Kohn-Sham method

The most majorly-used calculation methodology based on density functional theory (DFT). The Kohn-Sham method was developed to provide a good approximation for the kinetic energy of many-electron systems, which was found to be very difficult when trying to parametrize as a pure (orbital-free) density functional. The kinetic energy, which holds the largest portion of the total energy, is approximated by the kinetic energy of a reference noninteracting system with the same electron density. The rest of the total energy was assigned to the exchange-correlation energy functional, and as a result, the many-body problem was reformulated as a system of single-electron Schroedinger-like equations (Kohn-Sham equations) in an effective potential.

## Markov-chain Monte Carlo mothod (MCMC)

Efficient way of computing statistical average of physical quantities at equilibrium by replacing the statistical summation over all microscopic states by stochastic sampling. It is often called just "Monte Carlo method". For example, in the case of Ising model, the total number of microscopic states increases as a function of the number of spins. Therefore, it is practically impossible to computer the expectation values strictly following the definition. In Markov-chain Monte Carlo method, a stochastic process is defined so that it satisfies the ergodic condition and the balance condition. Temporal averages over the microscopic states generated in this way should equal the thermal averages at equilibrium. Slow relaxation is often problematic for systems near the criticality or with frustration. There are a number of techniques designed for dealing with this problem, such as extended ensemble methods and variational Monte Carlo method.

### Related keywords

quantum Monte Carlo, variational Monte Carlo

## Molecular Dynamics (MD)

Methods of simulating many particle systems by solving equations of motion such as the Newton equation. Mathematically, in molecular dynamics simulation one solves a system of simultaneous ordinary differential equations by using, e.g., the Runge-Kutta method and the velocity Verlet method. While a simple implementation would allow simulation with fixed energy and fixed volume, introducing the Nose-Hoover thermostat makes it possible to simulate with fixed temperature possible as well. Similarly, simulation with fixed pressure or with chemical potential is possible. There are various choices of the force field, the interaction energy between particles, ranging from the simple short-ranged force such as the hard sphere potential and the Lennard-Jones potential to the long-ranged one such as the Coulomb potential or more realistic and more complicated ones, depending on the purpose of the simulation.

## Monte Carlo

A method of simulation is called a Monte Carlo method if sampling with pseudo random numbers is used. The simplest example is the random sampling with the weight that is uniform in the configuration space. An important category is that of importance sampling methods, e.g., Markov-chain Monte Carlo. The method is also used for solving optimization problems via simulated annealing.

### Related keywords

Markov-chain Monte Carlo, quantum Monte Carlo, path-integral Monte Carlo, variational Monte Carlo, diffusion Monte Carlo

## Nonequilibrium Green's function method

A method for calculating quantum transport properties of a nanostructure coupled to two or more leads under bias. The electron density and conductance of the system under bias can be obtained by calculating the Green's function of the nanostructure using self energies that account for the effect of the leads.