As a chemist, I can provide you with a qualitative explanation of how the addition of nitrogen atoms affects the electronic and optical properties of a graphene sheet. However, I cannot perform quantum chemical calculations as it requires specialized software and computational resources. Here's the explanation:Graphene is a single layer of carbon atoms arranged in a two-dimensional honeycomb lattice. It is a zero-bandgap semiconductor, which means it has no bandgap between the valence and conduction bands. This property results in high electrical conductivity and unique optical properties, but it limits its application in electronic devices that require a bandgap.Doping graphene with nitrogen atoms can introduce a bandgap and modify its electronic and optical properties. Nitrogen doping can be achieved by substituting some of the carbon atoms in the graphene lattice with nitrogen atoms. There are two common types of nitrogen doping: pyridinic and graphitic.1. Pyridinic nitrogen doping: Nitrogen atoms replace carbon atoms at the edge of the graphene lattice, creating a pentagonal ring with one nitrogen atom and four carbon atoms. This type of doping introduces localized states near the Fermi level, which can open a bandgap and improve the charge carrier mobility.2. Graphitic nitrogen doping: Nitrogen atoms replace carbon atoms within the graphene lattice, forming a hexagonal ring with one nitrogen atom and five carbon atoms. This type of doping introduces impurity states in the band structure, which can also result in a bandgap opening.The bandgap and absorption spectra of nitrogen-doped graphene depend on the concentration of nitrogen atoms and the type of doping. Generally, as the nitrogen concentration increases, the bandgap increases, and the absorption spectra shift towards higher energies shorter wavelengths . This can lead to tunable electronic and optical properties, which can be beneficial for various applications, such as transistors, sensors, and solar cells.To calculate the bandgap and absorption spectra of a graphene sheet with varying nitrogen concentration, you would need to use quantum chemical calculations, such as density functional theory DFT or tight-binding methods, with specialized software like Gaussian, VASP, or Quantum Espresso. These calculations require a deep understanding of computational chemistry and access to high-performance computing resources.