Calculation methods/algorithms

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  • When one irradiates some material with X-ray, the component of X-ray spectrum corresponding to the excitation of the inner shell of an atom contained in the material is absorbed. The fine structure in the spectrum of this X-ray absorption edge is called X-ray absorption fine structure (XAFS), and contains various information on the structure of the material in the atomistic scale. To extract such information, it is necessary to compute the anticipated electronic state of the related atoms and compare it with experiments. Most of applications for X-ray analysis, such as FEFF, Demeter and Missing, come with functions that produce X-ray absorption spectrum, so that the comparison to experiments can be done easily. In addition, by using packages of first-principles calculation with X-ray spectrum calculation capability (e.g., WIEN2k, Exciting, Quantum ESPRESSO, ANINIT, AkaiKKR SPRKKR GPAW) one can do the X-ray analysis in higher precision.
    X-ray spectroscopy analysis
  • 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.
    Diffusion Monte Carlo(DMC)
  • Core electrons play a very small part in chemical bond formation, so that computational load can be decreased with small loss in accuracy by replacing core electrons by a pseudopotential that act on the valence electrons. In this manner, only the relatively slowly-varying valence wave functions need to be considered explicitly, and this allows for decreasing the basis set size when using a plane wave basi
    Pseudopotential method
  • Cluster expansion 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.
    KKR method
  • 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).
    Atom-localized basis/Gaussian basis
  • 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.
    Exact Diagonalization(ED)
  • (Empirical) molecular dynamics
  • A Monte Carlo method is referred to as a quantum Monte Carlo method when it is applied to quantum many-body systems. There is a variety of the methods including: Path-integral Monte Carlo, which is a Markov-chain Monte Carlo in the d+1 dimensional path-integral representation of d dimensional quantum system, diffusion Monte Carlo, in which ground state wave functions are sampled from the random walk of the diffusion process in the imaginary time evolution, and variational Monte Carlo, in which the expectation value of the energy of the trial wave function is evaluated by stochastic sampling.
    Quantum Monte Carlo method(QMC)
  • 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.
    Coherent Potential Approximation(CPA)
  • 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.
    GW method
  • A method for studying the time-dependent electronic state of a given many-electron system within the framework of the density functional theory. While the conventional (static) density functional theory is based on the fact that the ground state energy can be characterized by some functional of the density, the TDDFT is based on its extension to time-dependent density (Runge-Gross theorem). It is possible to obtain the time-dependence of the electronic state of molecules and solids by solving the equation derived from the theorem. Often used in studying the non-linear response to the strong external field. Supported by many standard packages of the first principle calculation and quantum chemistry such as VASP, CASTEP, ABNINT, QUANTUM ESPRESSO, PHASE, Gaussian, GAMESS-US, etc.
    Time-dependent density functional theory(TDDFT)
  • A kind of a base set used in electronic state calculation. The real space is divided into a mesh, and each point is taken as a basis of the wave function. This basis set is suitable for implementation of high-efficiency large-scale calculations using a order-N method, and has flexibility in choosing boundary conditions. In actual calculation, the equation that the wave function follows (such as the Kohn-Sham equation) is expressed as a difference equation, which is solved in combination with an appropriate interpolation method (such as the finite element method or the spline interpolation method).
    Real space basis
  • 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.”
    Job management
  • 3D-RISM theory
  • 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.
    All-electron method
  • First-principles molecular dynamics refers to molecular dynamics calculations using forces calculated from first-principles electronic structure methods.
    First-principles molecular dynamics
  • 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.
    Dynamical Mean Field Theory(DMFT)
  • A method for electronic structure calculation that utilizes parameters obtained from simple approximations or experiments instead of rigorously calculating quantities such as overlap integrals between atomic orbitals. In the field of computational quantum chemistry, the Huckel method (which considers only π orbitals and calculates coulomb and resonance integrals between neighboring atomic orbitals while neglecting overlap integrals) and the extended Huckel method (which considers σ orbitals in addition) are often-used methods. In the field of condensed matter, tight-binding models are sometimes constructed to reproduce experiments.
    Semi-empirical electronic state calculation
  • A method of handling a continuum model in an inhomogeneous field using field variables called phase fields. By introducing a continuous field that describes the state of the phase in addition to the density and temperature fields as field variables, it can be applied to the simulation of many physical phenomena (solidification phenomena, phase transformation, etc.) accompanied by phase transition. The continuous field is described based on the Ginzburg-Landau equation, and the parameters in the model are determined by the phase free energies. Since all physical quantities are written in a continuous field, calculation codes are easy to write, and public or commercially available programs can be used.

    Phase field method
  • 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.
    Fragment molecular orbital method(FMO)
  • In electronic structure calculations, the wave function is often expanded as a linear combination of plane waves. Plane waves comprise an orthonormal basis set, so that increasing the basis size (using many plane waves with different wavelengths) leads to a monotonic improvement in the reproduction of the wave function. However, it is unsuitable for describing steeply-varying wave functions near the core since disproportionately many plane waves are necessary for expanding steep functions. The (L)APW and pseudopotential methods were developed to circumvent this difficulty.
    Plane wave basis
  • A method based on trial wave functions with parameters. By adjusting parameters according to variational principles one obtains optimal wave function. For the trial wave function for fermionic systems, a Slater determinant is often used with Gutzwiller-Jastrow type correlation factor to reflect correlation effects. Monte Carlo sapling is used in computing expectation values. Hence the name of the method. Because of the absence of the negative sign problem, this method is applicable to a broad range of problems including the first principles calculation (e.g., CASINO), quantum chemistry (e.g., QWalk), lattice fermion systems (e.g., mVMC), etc.
    Variational Monte Carlo(VMC)
  • The tensor network method is the method for studying many-body problems based on the tensor network representation (a partially or fully contracted product of many tensors). It is used for computing partition functions of models in classical statistical mechanics as well as ground-state properties of quantum systems in discrete space. In particular, variational principle calculation based with the tensor network state (TNS) is well-known. A TNS is a wave function of which the expansion coefficients in some orthonormal basis are expressed as a product of tensors. Typically, the number of tensors is proportional to the system’s degrees of freedom. When the tensors are of rank 3, the TNS is a matrix product state, and the resulting tensor network method corresponds to DMRG. Similar to DMRG, an arbitrary quantum state can be expressed as a TNS. There are multiple choices for the structure of the network, the method of optimizing the tensors, and the method of contracting network. The tensor network method is a general name referring to all of those.
    Tensor network 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 functional method
  • A method for estimating a crystal structure from powder diffraction patterns of X-rays and neutrons. From measured data, the crystal structure is determined by pattern fitting.
    Rietveld analysis
  • An efficient calculation method for treating scattering of electrons in a potential. Transmission and reflection amplitudes are evaluated by directly calculating the Green’s function of the electrons. It is possible to evaluate electron transport properties of nanoscale devices by combining with Laundauer’s formula. In the recursive Green’s function method, fast computation of the scattering states of electrons are realized by sequentially calculating the Green’s function along the transport direction. The method can also treat magnetic materials and superconducting materials. The method is implemented in Kwant.
    Recursive Green's function method
  • A method for calculating electronic states of mainly isolated systems such as molecules is collectively called quantum chemical calculation. Many applications often implement various calculation methods based on the molecular orbital method. The Hartree-Fock approximation is a method of optimizing the molecular orbital wave functions by approximating the many-body wave function with a single Slater matrix. To incorporate the electron correlation effect ignored in the Hartree-Fock approximation, one can use the perturbation method, the Configuration Interaction (CI) method, the coupled cluster (CC) method, etc. The perturbation theory treats electron correlation within perturbation expansion, and is called the Møller-Plesset (MP) method. Depending on the order of the perturbation, it is expressed as MP2, MP3, MP4, etc. In the CI method, in addition to the ground state obtained by the Hartree-Fock method (a state in which electrons are occupied into the orbits below the Fermi energy), we prepare states in which only one electron is moved from its electron configuration (one electron excited state), states in which two electrons have moved (a two-electron excited state), etc., and minimize energy by considering their linear combination. In the CI method, when the number of prepared slater wave functions is increased, the wave function always approaches an exact wave function. However, since the calculation cost increases exponentially, the expansion is usually terminated at a specific order. Depending on how many sets of the Slater wave functions are prepared, we call the methods as Full-CI, CID (two-electron excitation only), CISD (one- and two-electron excitation only), and QCISD (improved CISD). The CC method improves a wave function by taking a partial sum for a specific configuration. Depending on which configuration is considered, we call the methods as CCD (two-electron excitation only), CCSD (one- and two-electron excitation only), and CCSD (T) (three-electron excitation is considered in perturbation theory). In addition to these Hartree-Fock-based methods, electronic state calculations based on the density functional theory are also widely used.
    Quantum chemistry methods
  • An (artificial) neural network is one of the machine learning methods that imitate the neural structure of the animal brain. A neural network has a structure in which many nodes (neurons) are connected. There are various types of neural networks. Typical examples are feed-forward neural networks (also called perceptrons) used for supervised learning and restricted Boltzmann machines used for unsupervised learning. In recent years, it has become possible to dramatically improve the learning ability by introducing a structure composed of many layers (deep neural networks). Neural networks are widely used in various fields such as image recognition, speech recognition, language analysis, model generation, and class classification. Even in the field of materials science, applications to machine learning force fields, variational wave functions, exploration of new materials (materials informatics), etc., are being advanced.
    Neural network
  • 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.
    Bayesian Optimization
  • Divide and conquer method
  • 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.
    Density matrix renormalization group(DMRG)
  • Machine learning potential

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