Development of Exotic Harmonium Model to Investigate Electron-Positively Charged Particle Correlation in Two-Component Quantum Systems

The problem of calculating the electron-positively charged particle correlation energy poses a challenge in the field of quantum chemistry beyond the adiabatic approximation. In this study, a toy model called Exotic Harmonium is developed to enhance our understanding of this type of correlation. Since the analytical methods to solve the eigenvalue equation for Exotic Harmonium lead to the excited rather than the ground state of the system, we employ the variation method in this study to obtain an approximate ground state wave function. By considering the asymptotic behavior of the wave function, we derive a compact but highly accurate variational wave function. Using this wave function, we are able to determine various properties of the system, including energy and its components, as well as single-particle densities. Additionally, we extend key concepts and quantities specifically tailored to study electron correlation in the Harmonium model to the Exotic Harmonium model. For instance, the "correlation hill" concept effectively highlights the distinct nature of electron-positively charged particle correlation compared to electron-electron correlation. Furthermore, we utilize Exotic Harmonium to assess the accuracy of five electron-positively charged particle correlation functionals developed within the context of the two-component density functional theory. Through a regression process, we obtained a compact yet precise analytical form for wave function which explicitly depends on the two crucial variables: oscillator field frequency and positively charged particle mass. This analytical form provides valuable insights into the behavior of the system. Also, the Exotic Harmonium model enables the investigation of electron-positively charged particle correlation in a vast range of particle masses.