Chapter One

Theoretically, in order to describe the behavior of a spectrum, a mathematical model which could predict the spectrum characteristics is needed. Since in this study a Two-state system has been used like models which was introduced previously past and could couple with the environment, the former ideas have been extended in this study. we use the second quantized version for writing this Hamiltonian. First, the Hamiltonian of a rotational system is considered in a classic scale, afterwards it is brought to a quantum scale. In the first step, the vibrations and quantum rotation is illustrated for two atom molecules. Then it is devoted to Two-state system and dissipative Twostate system. In the second step, the rotation of a molecular group in a hindering potential is studied in the classic and quantum scales. Finally, at the present of strong coupling constant the Hamiltonian has been applied and a numerical renormalization group approach has been used for numerical solution. Then, by using Hubbard operators, dynamical functions of this oprators are written. The fourier transform of the Greens function is developed, then density of state is calculated.