### 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.

### 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.

### 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 1994, A. 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.