### 1. Introduction

*ab initio*) method is most accurate and relevant for materials studies, without any need for involvement of empirical parts like classical molecular dynamics (MD). In addition, although the rules of the quantum world are bizarre and cannot be derived, they have been justified up to the level of laws by their logical consistency and by the agreement of experiments over the years. They have never been disproved, nor has evidence been found to contradict them.

*ab initio*MD)+hybrid functionals, etc. This review will treat these trends at an introductory level after a brief recapitulation on first principles methods.

### 2. Brief history of First Principles Methods

### 2.1 The beginning

*Ψ*.

^{3)}

*Ĥ*(or energy operator) operates on the wave function

*Ψ*, and yields

*EΨ*. Like any other eigenvalue equation,

*Ψ*remains

*Ψ*, and energy

*E*, which is what we are searching for, comes out as a constant. It has been stated that all information required to define a quantum state is contained in the wave function alone.

*Ĥ*) taking a dove (≡

*E*) out of a supposedly empty hat (≡

*Ψ*). We know that the dove was somewhere in there where we could not see it, and the magician simply moved her from the hat in a tricky way and showed the dove to us. There is no change to the hat due to this act. Similarly, all necessary information about

*E*is contained in

*Ψ*, and it shows up only after the operation of

*Ĥ*. It should be noted that quantum acts are real things, whereas magic simply involves well-designed and manipulated illusions.

*n*-electron problem, just like the

*N*-atom problem in classical mechanics. When many bodies interact at the same time in a classical or quantum way, it is simply beyond our capability to solve exactly an equation for their interaction. The equation is useful in practice only for the simplest systems such as hydrogen-like systems. Scientists naturally engage in the usual routine, approximation. In fact, the development of the first principles method was through a series of approximations, especially for

*Ĥ*and

*Ψ*.

*n*coordinates of

*n*electrons. We need to remember that we are talking about an amount of material of up to several hundred atoms, which could easily contain thousands of electrons. Then, devising and solving calculations for these

*n*-electron systems are completely out of the question. This is the so-called “many-body problem.” Dealing with

*n*electrons that interact with all other electrons at the same time is just too complex to solve, even numerically. Even if we knew how to do such a calculation, no computer that could handle such a massive task is available, and this will be true no matter how fast computers may become in the future.

^{4)}then approximated the

*n*-electron wave function,

*Ψ*, simply by multiplying electron’s wave functions each other, based on the result from the “electrons in a well” model. He further introduced the so-called one-electron model and a self-consistent procedure, both of which have been passed down in all the following methods as parts of the theoretical framework. Here, the one-electron model means that electrons are so near-sighted that they cannot see other electrons; they see only the mean-field created by other electrons. Of course, this method did not consider the quantum effect and yielded unsatisfactory answers, even when calculations were carried out self-consistently.

^{5)}enhanced the Hartree method to a higher degree of perfection. This time, the key move was in the area of the wave function. In the Hartree-Fock (HF) method,

^{5)}the

*n*-electron wave function is approximated as a linear combination of noninteracting one-electron wave functions in the form of a Slater determinant. And, the first quantum effect, the nonlocal exchange energy, came out from the determinant, satisfying Pauli’s exclusion principle at the same time. The HF method improved calculation results while maintaining the method’s parameter-free nature. However, practical application of the HF method is still limited to small systems with several tens of atoms, which are far from the normal regime of materials.

*Ĥ*now consists of four terms for each electron; a kinetic term, a potential term between each electron and fixed nuclei, an electrostatic term between electrons, and a quantum exchange term of short-range repulsion between same-spin electrons. And yet, one more quantum term is missing and, for this reason, the energies calculated using the HF method are higher than those of data from accurately-performed experiments.

### 2.2 Density functional theory (DFT)

^{6)}proved that electron density at spatial points alone is sufficient to completely characterize the ground state of an

*n*-electron system.

^{7)}described how the exact many-body energy can be calculated with a fictitious and non-interacting one-electron system whose energies are functionals of the electron density. This scheme introduced the exchange correlation (XC) energy, which contains all the interacting quantum effects, making the other energy terms free of such effect. The XC energy is formally exact and functional of the electron density. It is, however, unknown.

^{7)}as schematically shown in Fig. 5. The many-body correlation effect comes principally from the short-range repulsion between electrons with antiparallel spins. It can be rather easily approximated from existing many-body calculations based on uniform electron gas (UEG) systems, completing now all terms including the correlation term for the energy operator

*Ĥ*, as shown in Fig. 6. Schematically indicated in Fig. 6 as a background, electron density, the squared form of the wave function, decides everything in any

*n*-electron quantum system, just as the wave function does. The successful debut of this LDA immediately initiated a rush of formulations of other XC functionals in the physics and chemistry communities.

^{8)}And, the electron-density-based DFT (density functional theory) era started covering atoms, molecules, clusters, and materials. The calculated data are very close to data from accurately-performed experiments. Right after the announcement of GGA, people found that the best density functionals outperformed the best wave function methods and that DFT methods were computationally very inexpensive.

*n*-dimensional equation to

*n*separate 3-dimensional equations via the use of electron density. All properties, including electronic, magnetic, optical, and various response properties, can be obtained given that nuclei and electrons are considered in the method. These days, such methods have been so successful in both accuracy and efficiency that scientists are using them routinely in a wide range of disciplines, making computer experiments possible even for materials. One great thing about computational DFT is that absolute energy is not much of a concern. The important thing is the energy difference between two states or systems. This leads to error cancellation and makes the results more accurate.

### 3. Hybrid Exchange Correlation (XC) Functionals

*r*dependence) of the actual potential. The XC potential becomes too shallow near the nucleus and decays too quickly at long distances from the nucleus.

GGA with van der Waals interactions

^{9)}GGA with hybrid functionals

^{10,}^{11)}GW approximation

^{12,}^{13)}based on many-body perturbation of KS eigenvalue, which is especially popular in the solid-state community. It is now the method of choice for calculations of both the ground state and of quasiparticle band structures, as measured in direct and inverse photoemission. It is applicable in practice to molecular systems, open-shell systems, materials, and metals, providing very accurate band gaps, especially for weakly interacting systems such as semiconductors. However, it is still too computationally costly to be practical for general purposes, and is often used only as a reference calculation for materials.Meta-GGAs

B3LYP, which is mostly used in computational chemistry.

Second-order Møller-Plesset theory (MP2), which is exclusively used in computational chemistry and belongs to the HF-based quantum chemistry approach.

Approach based on random-phase approximation (RPA), which deals with non-local correlations in a more systematic and non-empirical way.

^{10)}which has been very successful and is computationally affordable in materials calculations. HSE06 is designed to complement the conventional GGA/PBE functional by bringing in the HF exchange energy.

^{14)}In addition, the HSE06 results for Cu

_{2}ZnSnS

_{4}as a potential photovoltaic material compare very favorably to experimental data for the lattice constant, optical spectrum, and band gap, all of which can be validated using G

_{0}W

_{0}quasiparticle calculations.

^{15)}

^{16)}An example of successful application of a hybrid functional with a mixing parameter other than 0.25 is the case of ZnO.

^{17)}The theoretically derived defect energetics for ZnO always suffer from huge uncertainty between studies, no matter which calculation methods (including LDA, GGA, GGA+U, etc.) are applied. One of the main causes is the inaccurate description of the band structure of ZnO when using local or semi-local XC functionals. By using an HSE (a = 0.375) hybrid functional with finite-size corrections, defect energetics in ZnO consistent with the relevant experimental observations have been reported without resorting to empirical corrections for the valence and conduction-band positions.

*d-*and

*f-*orbitals is still in doubt. Many transition metals and their compounds belong to this category. Recently, there have been various efforts to formulate a combined functional of hybrid and +U considerations together (The DFT+U approach is discussed in more detail in Section 5.).

^{18)}connected both to the on-site correction term of the DFT+U method and indicated that the screening of the onsite electron repulsion is governed by the ratio of the exact exchange in the hybrid functionals. They claimed that the scheme provides a theoretical justification for the combination, and resolves issues caused by overscreening of localized states using tests for chromium impurity in wurtzite AlN and for vanadium impurity in 4H-SiC. There is no doubt that this level of functional will be added to the existing computational packages soon.

### 4. Click Computation

*ab initio*MD), etc.

^{19)}deals with molecules, polymers, mesostructures, biomaterials, and materials. It is loaded with tools to handle quantum, atomistic, mesoscale, statistical, and analytical simulations. The package contains DFT methods (CASTEP

^{20)}), linear scaling DFT methods, QM/MM methods, semiempirical methods, MD, lattice dynamics, Monte Carlo-based methods, and force fields methods.

^{21)}is just about right, with all the essential tools for materials science, such as LAMMPS, Phonon, Surfaces, Interfaces, Nanotubes, Amorphous Materials, Transition State Search, electronics, etc. All inputs can be conveniently delivered via clicking or box filling, and results (band structure, phonon curves and corresponding thermodynamic data, electron density profile, potential profile, barrier energy profile for diffusion, Seebeck coefficient curves, etc.) are plotted or tabulated automatically.

^{22)}

### 5. Big Data

^{23)}led by UC Berkeley’s Prof. Gerbrand Ceder and Dr. Kristin Persson, has been the leader in the sector of computational materials science since they launched the project formally in 2011. Data and analysis tools are open to the public via web-based access requiring only one’s e-mail address. At present, the project presents a database on about 66,000 compounds, about 44,000 band structures, and much more. And of course, data on tens of new compounds are coming in every week.

^{24)}have provided the world’s largest set of data (on about 1,200 compounds) on the elastic properties of inorganic compounds.

^{24)}They claim that data on dozens more compounds are being added every week. This data set is especially valuable for materials scientists working on new structural materials, for which mechanical properties are the prime concern.

### 6. Calculations Beyond Conventional DFT

### 6.1. Addition of dispersion terms

*r*

^{6}dependence of the force with distance. Again, the reason is that the van der Waals force is a non-local correlation effect. The interaction is a seemingly weak one, but the dispersion effect is omnipresent and can add up to a substantial force in large assemblies. It thus can have a very strong influence on systems such as biomolecules, supramolecules, and nanomaterials.

^{25)}is the most popular; this method requires only two new lines (for example, LVDW = .TRUE. and VDW_SCALING = 0.75 for the VASP case). Normally, bulk solids do not require this correction, unless the system has a surface and molecules on it. Fortunately, the treatment of van der Waals interactions in DFT comes at essentially zero cost in terms of calculation time, because pair-wise calculations are rather straightforward.

### 6.2 +U

*d-*and

*f-*bands, and these localized electrons on the same atomic center create on-site Coulomb interaction, which leads to strong repulsion. This situation cannot be properly described with conventional DFT methods, which tend to delocalize electrons over the crystal and to make each electron feel an average of the Coulombic potential. The repulsion between these electrons is largely due to the contracted nature of the multiple charges, which leads to a greater correlation between their motions and to narrow widths for the bands associated with these electrons.

^{26,}

^{27)}is the general practice for this, and uses an additional Hubbard-like term.

^{27)}One study demonstrated the importance of using Hubbard parameters to model reaction energies involving transition metals even when no change in formal oxidation states was occurring. In actual runs, localized electrons are separated from other delocalized electrons (

*s*- and

*p*-electrons) and treated with the parameterized U as an HF-like potential. Because the correlation part is already counted in DFT, some double counting of the correlation part for localized electrons is subtracted from the Hamiltonian using another term.

*ab initio*) calculations, employing a hybrid functional or unrestricted HF method, or are fitted to the experimental binary oxide formation energies.

^{28)}It should be noted that the on-site interaction energies must be constant for any given material.

*d*-bands away from each other due to a repulsive penalty, the GGA+U method yields higher energies and wider band gaps.

### 6.3 *ab initio* MD (AIMD)

^{29)}and AIMD,

^{30)}have enjoyed a rapidly increasing popularity in the past decade, and are now ‘routinely’ applied in many areas of materials science.

^{31)}For transition elements with tightly bound electrons in oxide materials, a combined approach was found to be needed and a highly scalable algorithm for exact exchange was developed and incorporated into this AIMD run. One conclusion is that the lower levels of XC potentials presently used in AIMD simulation cannot be used to reliably predict the properties of many materials. Expanding the length and time scales that are accessible with AIMD will remain a challenge and a topical area of research in years to come.

Designing NiSi

_{2}contact for CMOS to have core-level shift with doping that leads to a lower Schottky barrier height (SBH)^{32)}Thermoelectricity of CuRh

_{1−x}Mg_{x}O_{2}at various temperatures^{33)}Kinetics for the crystal growth of gallium nitride from trimethylgallium and triethylgallium

^{34)}