The energy profile of an excited state of a molecule differs from its ground state in terms of the distribution of electrons and the energy levels they occupy. In the ground state, the electrons in a molecule occupy the lowest possible energy levels according to the Pauli Exclusion Principle. In an excited state, one or more electrons have absorbed energy usually in the form of photons and have been promoted to higher energy levels. This change in electron distribution can lead to different molecular geometries, vibrational frequencies, and reactivity.To predict the transitions between the ground state and excited states of a molecule, we can use quantum chemical calculations. These calculations are based on the principles of quantum mechanics, which describe the behavior of electrons in molecules. There are several methods to perform these calculations, such as:1. Time-Dependent Density Functional Theory TD-DFT : This method is an extension of Density Functional Theory DFT , which is widely used for ground state calculations. TD-DFT calculates the excited state energies and transition probabilities by solving the time-dependent Kohn-Sham equations. It is suitable for medium-sized molecules and provides a good balance between accuracy and computational cost.2. Configuration Interaction CI : This method is based on the expansion of the molecular wavefunction in terms of Slater determinants, which are formed by distributing the electrons in different molecular orbitals. CI calculations can provide highly accurate results for excited states, but they are computationally expensive, especially for large molecules.3. Coupled Cluster CC : This method is an improvement over CI, as it uses a more compact representation of the wavefunction in terms of exponential operators. CC can provide accurate results for both ground and excited states, but it is also computationally demanding.4. Many-Body Perturbation Theory MBPT : This method is based on perturbation theory, which involves calculating the corrections to the ground state wavefunction and energy due to the presence of electron-electron interactions. MBPT can be used to calculate excited state energies and transition probabilities, but its accuracy depends on the order of the perturbation expansion.To predict the transitions between the ground state and excited states, quantum chemical calculations provide information about the energies of the states, the oscillator strengths which are related to the probabilities of the transitions , and the wavelengths or frequencies of the absorbed or emitted photons. By analyzing these results, we can gain insights into the electronic structure and properties of the molecule, as well as the possible pathways for photochemical reactions and other excited state processes.