Density Functional Theory DFT is a widely used computational method in quantum chemistry and solid-state physics to investigate the electronic properties of materials. When an external electric field is applied, the electronic properties of the material can change significantly. The changes in the electronic properties can be analyzed using DFT calculations by considering the following factors:1. Polarization: The applied external electric field can induce polarization in the material, which can lead to a redistribution of the electron density. This can affect the electronic properties such as band structure, density of states, and optical properties. In DFT calculations, the change in polarization can be accounted for by including the external electric field in the Kohn-Sham equations.2. Stark effect: The external electric field can cause a shift in the energy levels of the material, known as the Stark effect. This can lead to changes in the band structure and the density of states. In DFT calculations, the Stark effect can be included by adding the interaction between the external electric field and the electron density in the Hamiltonian.3. Field-induced structural changes: The applied external electric field can cause structural changes in the material, such as lattice distortions or phase transitions. These structural changes can affect the electronic properties of the material. In DFT calculations, the structural changes can be accounted for by performing geometry optimizations under the influence of the external electric field.4. Basis set and exchange-correlation functional: The accuracy of the electronic property predictions in DFT calculations depends on the choice of the basis set and the exchange-correlation functional. The applied external electric field can affect the accuracy of these predictions. It is essential to choose an appropriate basis set and exchange-correlation functional that can accurately describe the material's electronic properties under the influence of the external electric field.5. Numerical convergence: The applied external electric field can affect the numerical convergence of the DFT calculations. It is crucial to ensure that the calculations are well-converged with respect to the k-point sampling, plane-wave cutoff energy, and other numerical parameters.6. Finite-size effects: In DFT calculations, the material is often modeled using a periodic supercell. The applied external electric field can cause finite-size effects, such as artificial interactions between the periodic images of the supercell. These effects can be minimized by using a sufficiently large supercell or by employing techniques such as the modern theory of polarization or the Berry phase approach.In summary, the electronic properties of materials can change significantly when an external electric field is applied. To accurately predict these changes using DFT calculations, it is essential to consider factors such as polarization, the Stark effect, field-induced structural changes, the choice of basis set and exchange-correlation functional, numerical convergence, and finite-size effects.