The conjugation length of a polymer refers to the number of alternating single and double bonds in a continuous chain within the polymer. This conjugation allows for the delocalization of -electrons, which in turn affects the optical properties of the polymer, such as absorption and emission spectra.The relationship between the conjugation length and the optical properties can be understood through the following points:1. Bandgap: As the conjugation length increases, the energy gap between the highest occupied molecular orbital HOMO and the lowest unoccupied molecular orbital LUMO decreases. This is because the energy levels of the molecular orbitals become closer together as the conjugation length increases. A smaller bandgap results in lower energy absorption and emission transitions.2. Absorption and emission spectra: With an increase in conjugation length, the absorption and emission spectra of the polymer shift to longer wavelengths lower energies . This is known as the redshift or bathochromic shift. This shift occurs because the energy difference between the electronic states decreases as the conjugation length increases.3. Intensity: The intensity of the absorption and emission spectra also depends on the conjugation length. Generally, an increase in conjugation length leads to an increase in the intensity of the absorption and emission spectra due to the increased probability of electronic transitions.To calculate the excited states of conjugated polymers using quantum mechanical methods, several approaches can be employed, such as:1. Time-dependent density functional theory TD-DFT : This is a widely used method for calculating the excited states of molecules and polymers. It is an extension of the ground-state density functional theory DFT and provides a good balance between computational cost and accuracy.2. Configuration interaction CI : This method involves the calculation of the wavefunction as a linear combination of different electronic configurations. It can provide highly accurate results but can be computationally expensive, especially for large systems like polymers.3. Many-body perturbation theory MBPT : This approach is based on the perturbation of the ground-state wavefunction to obtain the excited states. It can be more accurate than TD-DFT but is also more computationally demanding.4. Quantum Monte Carlo QMC methods: These are a family of stochastic methods that can provide highly accurate results for excited states. However, they are computationally expensive and may not be suitable for large systems like polymers.The choice of method depends on the size of the system, the desired accuracy, and the available computational resources.