Dear friend Yp Hu

Ah, the intricate world of band gaps! Now, let's dive into the difference between the Kohn-Sham band gap and the band gap calculated by the Materials Project system, as perceived by me:

1. **Kohn-Sham Band Gap:**

- The Kohn-Sham band gap is a concept derived from density functional theory (DFT), a quantum mechanical modeling method.

- It's part of the Kohn-Sham equations, which are used to describe the electronic structure of a material.

- In DFT, the electron-electron interaction problem is mapped onto a set of non-interacting electrons in an effective potential, known as the Kohn-Sham potential.

- The band gap from these non-interacting electrons is called the Kohn-Sham band gap.

- It's worth noting that the Kohn-Sham band gap might not always perfectly match experimental values, as DFT has certain approximations and limitations.

2. **Materials Project System's Band Gap:**

- The Materials Project is a platform that provides open access to computed information on known and predicted materials.

- The band gap provided by the Materials Project is typically calculated using high-throughput DFT calculations.

- This band gap represents the energy difference between the top of the valence band and the bottom of the conduction band in the electronic band structure of a material.

- The Materials Project uses DFT calculations and materials databases to predict and explore the properties of a wide range of materials.

In essence, both the Kohn-Sham band gap and the band gap calculated by the Materials Project are results of electronic structure calculations, but they may differ due to variations in computational methods, approximations, and the underlying models used. Each serves as a tool for understanding the electronic properties of materials, but the specifics can vary based on the context and methodology.

Remember, my opinionated take is all about embracing the complexities of quantum mechanics and materials science!

The Kohn-Sham band gap and the material project system's band gap represent two different approaches to determining the electronic band gap of a material.

The Kohn-Sham band gap is calculated using density functional theory (DFT), which is a widely used quantum mechanical method for studying the electronic structure of materials. In the Kohn-Sham approach, the electronic structure of a material is approximated by a set of non-interacting electrons moving in an effective potential. The band gap is then obtained as the energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO).

On the other hand, the material project system's band gap refers to the band gap value provided by the Material Project database. The Material Project is a computational project that aims to calculate and provide various material properties, including electronic band gaps, using high-throughput calculations and machine learning techniques. The band gap values in the Material Project database are obtained through a combination of first-principles calculations, empirical models, and experimental data.

While both the Kohn-Sham band gap and the material project system's band gap aim to determine the energy difference between electronic energy levels in a material, they differ in the method and approach used for the calculations. The Kohn-Sham band gap is based on a theoretical quantum mechanical framework, while the material project system's band gap is derived from a combination of computational and empirical methods. Consequently, the band gap values obtained from these approaches may have slight differences due to variations in the theoretical models, computational methods, and empirical corrections used in the calculations.

Density Functional Theory

In principle, density functional theory (DFT) is an exact formulation of quantum mechanics that uses the electron probability density, rather than the wavefunction, to calculate observables efficiently. In practice, the form of the density functional for each many-body observable, such as the total energy, is unknown and must be approximated.

Kohn-Sham Theory

The Kohn-Sham system is a fictitious system comprised of completely non-interacting (single-particle) electrons subject to an effective potential, known as the Kohn-Sham potential. The role of the Kohn-Sham potential is to ensure that the non-interacting electrons have the same electron probability density as the many-body system of interacting electrons.

The rational behind the use of this auxiliary Kohn-Sham system is that obtaining its electron density is relatively computationally efficient and, because, in principle, it is that of the many-body system, it can be used to calculate real-world, many-body observables via the appropriate density functional. For practical calculations, the Kohn-Sham potential must be approximated, as well as the density functional, both of which can be a source of errors.

The Band Gap

There are no reliable approximate density functionals that yield the band gap (also known as the fundamental gap) of a material. Without a more rigorous and reliable alternative, the Kohn-Sham single-particle energies are used to obtain the many-body gap. In this case, the gap is assumed to be the difference between the energy of the lowest unoccupied Kohn-Sham orbital and the energy of the height occupied Kohn-Sham orbital. This is known as the Kohn-Sham band gap.

The Kohn-Sham band gap is generally a poor approximation to the many-body band gap. This can be due to the approximation to the Kohn-Sham potential. However, it's important to note that even if the exact Kohn-Sham potential is used, the Kohn-Sham band gap will, in general, differ from the many-body band gap significantly, because the Kohn-Sham system is a system of non-interacting electrons that is designed to yield the interacting density only, not any other many-body quantities.

The Materials Project

The Materials Project uses Kohn-Sham DFT calculations to obtain band gaps for a variety of materials. As such, the computed gap can differ from the many-body gap by up to 40%. The project uses methods designed to improve the band gap obtained from a Kohn-Sham DFT calculation. However, these methods can be unreliable for some materials and hence computed gaps should be treated with caution.

In short, the Materials Project uses Kohn-Sham DFT to obtain band gaps. Hence, generally speaking there is no difference between the two, and, owing to the challenge in calculating the many-body band gap, these calculations can be unreliable.

Same