Energy Levels and Molecular Spectra

The vibrational structure may be explained as follows For each state of a molecule there is a wave function that depends on time, as well as on the internal space and spin coordinates of all electrons and all nuclei, assuming that the overall translational and rotational motions of the molecule have been separated from internal motion. A set of stationary states exists whose observable properties, such as energy, charge density, etc., do not change in time. These states may be described by the time-independent part of their wave functions alone. Their wave functions are the solutions of the time-independent Schrddinger equation and depend only on the internal coordinates q = 9, Qz,. . . of all electrons and the internal coordinates Q = Q, Qz, of all nuclei. 

Within the Born-Oppenheimer approximation (cf. McWeeny, 1989 Section 1.1) the total wave function of a stationary state is written as 

Hamiltonian which is deflned for a particular electronic state J as 

Due to the product form of the total wave function in Equation (1.12) the energy of a stationary state can be written as 

Similarly, a rotational component jp and a translational component are obtained when all 3N displacement coordinates of the N nuclei are used rather than the internal coordinates, which are obtained by separating the motion of center-of-mass and the rotational motions. 

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