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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Semi-empirical electronic state calculation

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.

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Tensor network method

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.

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Thermal Pure Quantum State

In textbooks of statistical physics, it is shown that finite-temperature properties (for example, internal energies or specific heat) can be calculated by the ensemble averages, which take the averages of physical properties with respect to the Boltzmann weight. To take the ensemble average numerically, it is necessary to perform the full diagonalization. The numerical cost of the full diagonalization is very high and it is almost impossible to calculate the finite-temperature properties for realistic systems.

However, recent progress of the quantum statistical physics shows that the finite-temperature properties can be calculated by the expectation values of the single wave function in principle.  In the pioneering work by Imada and Takahashi, it was pointed out that finite-temperature properties can be calculated by the expectation values of the wave functions. After that, several people rediscovered or proved that ensemble average can be replaced by the expectation value of the wave function [J.Jaklic and P.Prelovsek PRB 1994A. Hams and H. de Raedt PRE 2000, S.Lloyd 1988].

In the paper of 2012, Sugiura and Shimizu prove that the finite-temperature properties can be exactly calculated by the single wave functions in the thermodynamic limit and show the simple way to construct the single wave function. They call the wave function “Thermal Pure Quantum (TPQ) State”. The TPQ method is implemented in  HΦ and it is easy to perform the finite-temperature calculation for a wide range of the quantum lattice model such as the Hubbard model and the Heisenberg model. Recent applications of the TPQ state using HΦ to the frustrated Hubbard model is found in the paper.

Time-dependent density functional theory (TDDFT)

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.

Universality class

Singular behaviors of physical properties around the phase transition point are governed by a few parameters (these parameters are called critical exponents) and the combination of the critical exponents is called universality class. Symmetries and spatial dimensions of the system determine the universality class and details of the system such as strength of interactions and the lattice structures do not change universality class. To determine the universality class of the phase transitions, it is necessary to perform the high-accuracy calculations such as the unbiased Monte Carlo calculations. Unbiased Monte Carlo calculations can be done with ALPS or DSQSS and the estimation of the critical exponents from the numerical data can be done with BSA.

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