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.
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).
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.