To determine how the calculated band gap of a zinc oxide ZnO crystal changes when the lattice constant is varied using density functional theory DFT calculations, you would need to perform a series of calculations with different lattice constants and analyze the resulting band structures.Here's a step-by-step approach to solving this problem:1. Choose a range of lattice constants: Select a range of lattice constants around the experimental or theoretical equilibrium lattice constant for ZnO. For example, you could choose lattice constants from 3.20 to 3.30 with a step of 0.01 .2. Perform DFT calculations: For each lattice constant in the chosen range, perform a DFT calculation using a suitable software package e.g., VASP, Quantum Espresso, or Gaussian . Make sure to use the same exchange-correlation functional e.g., PBE, LDA, or hybrid functionals like HSE06 and other computational settings e.g., plane-wave cutoff energy, k-point sampling for all calculations to ensure consistency.3. Extract the band structure: After completing the DFT calculations for each lattice constant, extract the band structure information, specifically the energies of the valence band maximum VBM and conduction band minimum CBM .4. Calculate the band gap: For each lattice constant, calculate the band gap as the difference between the CBM and VBM energies.5. Analyze the results: Plot the calculated band gap as a function of the lattice constant. This will allow you to visualize how the band gap changes with the variation in the lattice constant.6. Interpret the findings: Analyze the plot to determine if there is a clear trend in the band gap as the lattice constant changes. For example, you might observe that the band gap increases or decreases with increasing lattice constant, or that it exhibits a minimum or maximum value at a specific lattice constant.Keep in mind that DFT calculations, especially with the commonly used exchange-correlation functionals like LDA and GGA, tend to underestimate the band gap of semiconductors. To obtain more accurate band gap values, you may need to use more advanced methods such as hybrid functionals, GW approximation, or the Bethe-Salpeter equation.