Calculation of the relative permittivity of Rhodamine 6G using the quantum mechanical method

Abstract

Background: The article describes a method for calculating the permittivity of organic molecules in quantum mechanics using the well-studied Rhodamine 6G molecule as an example. The study of optical properties of large organic molecules requires not only experimental data but also the use of calculations obtained both analytically and numerically. Objectives: Methods for calculating permittivity as phenomenological characteristics of a sample are to be tested on well-studied molecules to be further applied to more complex nonlinear structures. However, the integral changes need to be approximated in the wave functions of large molecules. Material and methods: The numerical simulations in MATLAB were carried out to be compared with the data from Gaussian 09, which are accurate for such small molecules as Rhodamine 6G. MATLAB calculated permittivity values for the frequency domains corresponding to absorption and fluorescence based on the Fermi golden rule. Hence, any molecule can be represented as a composite quantum mechanical system. Meanwhile, Gaussian 09 used the DFT method to determine permittivity. Results: The Fermi golden rule can be applied due to the representation of the molecule as a complex quantum mechanical system. The proposed numerical methods minimize error by using the Dirac delta function. According to our hypothesis, the sum of the wave function of a particle in a potential well and a particle in a ring equals the wave function of the entire system, thus making it possible to study large molecules. As a result of the calculation for two wavelengths of 337 and 573 nm, the permittivity results calculated using the proposed method in MATLAB are 2.98 and 6.27, respectively. Gaussian 09 calculated the same parameters at 2.85 and 6.23. Conclusion: The resulting datasets show a high degree of correlation. Therefore, the research hypothesis has been confirmed. The selected method also proved efficient, hence the enhancement of luminescence can be achieved by changing the relaxation time of the excited state. Plasmonic nanostructures with predetermined properties will controllably enhance the resulting field by the square of the superposition modulus of their near-field. Consequently, conditions for highly coherent radiation with high intensity and polarization can be predicted and calculated before an experiment is carried out.