The calculation of vertical excitation energies and oscillator strengths of a molecule can vary significantly when different levels of theory are employed, such as time-dependent density functional theory TDDFT and coupled-cluster CC theory. These two methods have different approaches to solving the electronic structure problem, which can lead to differences in the calculated properties.1. Time-Dependent Density Functional Theory TDDFT :TDDFT is a widely used method for calculating excited state properties of molecules due to its relatively low computational cost and good accuracy for many systems. It is an extension of ground-state density functional theory DFT and involves solving the time-dependent Kohn-Sham equations. In TDDFT, the vertical excitation energies and oscillator strengths are obtained from the linear response of the electron density to an external perturbation, such as an electric field.The accuracy of TDDFT calculations depends on the choice of the exchange-correlation functional, which approximates the unknown exact functional. There are many functionals available, and their performance can vary depending on the system being studied. Some functionals may provide better results for certain types of excitations or molecular systems, while others may struggle with charge-transfer excitations or systems with strong electron correlation.2. Coupled-Cluster CC Theory:Coupled-cluster theory is a highly accurate ab initio method based on the exponential ansatz for the wavefunction. It systematically includes electron correlation effects by considering all possible excitations from a reference determinant, such as a Hartree-Fock or a complete active space self-consistent field CASSCF wavefunction. The accuracy of CC calculations can be systematically improved by including higher-order excitations, such as singles and doubles CCSD , or singles, doubles, and triples CCSDT .For excited state calculations, the equation-of-motion coupled-cluster EOM-CC approach is commonly used. EOM-CC can provide very accurate vertical excitation energies and oscillator strengths, especially for systems with strong electron correlation effects. However, the computational cost of EOM-CC calculations is significantly higher than TDDFT, which can limit its applicability to larger molecular systems.In summary, the choice between TDDFT and coupled-cluster theory for calculating vertical excitation energies and oscillator strengths depends on the balance between computational cost and desired accuracy. TDDFT is generally more affordable and can provide good results for many systems, but its accuracy depends on the choice of exchange-correlation functional. Coupled-cluster theory, particularly EOM-CC, can provide highly accurate results but at a higher computational cost.