Path-integral Monte Carlo

A Markov-chain Monte Carlo method for quantum many body systems. Simulation by this method is based on a Markov process in the space of d+1 dimensional configurations obtained via path-integral representation of d dimensional quantum many-body systems. The method is also called the world-line Monte Carlo. While the weight of a given state can be easily computed in classical systems, its computational cost for quantum systems is exponentially high. Therefore, the mapping into the d+1 dimensional classical problem through Suzuki-Trotter transformation or high-temperature series expansion is necessary for reducing the cost to a manageable level. The method is applied to various strongly-correlated lattice models such as transverse Ising model, Heisenberg model, and Hubbard model, as well as bosonic systems such as He4. However, in application to frustrated Heisenberg models and fermionic systems, the weight can be negative in general, making the method unpractical. This problem is called the negative sign problem, the most severe limitation of the method in studying quantum many-body systems.

Phase field method

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.

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Plane wave basis

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.

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Pseudopotential method

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

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Quantum chemistry methods

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 Monte Carlo method (QMC)

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.

Real space basis

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).

Recursive Green’s function method

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.

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Rietveld analysis

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.

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