(CC BY 3.0; OpenStax). Fig. of their quantized energy levels. 13.2. Within the harmonic and rigid rotor approximations, the rotational-vibrational energy levels (in wavenumbers) of a diatomic molecule are given by , where , are the vibrational and rotational quantum numbers, respectively, is the harmonic vibrational constant, and is the rotational constant. 4.4 illustrates the vibrational energy level diagram for a diatomic molecule with a stiff bond (nitrogen N 2; left) and one with a looser bond (fluorine F 2; right). Rotational energy levels of diatomic molecules A molecule rotating about an axis with an angular velocity C=O (carbon monoxide) is an example. As a homo-nuclear diatomic molecule with no permanent dipole moment, Diatomic Nitrogen presents a complex and interesting spectrum of electronic transitions. Ignoring electronic excitation, the total internal energy of a molecule is the sum of its vibrational and rotational energy. If the vibrational quantum number (n) changes by one unit, then the rotational quantum number (l) changes by one unit. The selection rule for a rotational transition is, ∆ J = ± 1 (13.10) Figure \(\PageIndex{1}\): Three types of energy levels in a diatomic molecule: electronic, vibrational, and rotational. The vibration is associated with the two atoms moving in and out relative to one another's positions. These energy levels can only be solved for analytically in the case of the hydrogen atom; for more complex molecules we must use approximation methods to derive a model for the energy levels of the system. The spectroscopic constants can be found in: Demtröder, Kapitel 9.5 Atome, Moleküle und Festkörper; CRC Handbook of Chemistry and Physics; K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV.Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979., Van Nostrand Reinhold, New York, 1979. E(v,J) = (v+½) e – (v+½)2xe e + Bv J(J+1) – DJ J2(J+1)2 Transitions between the E(v,J) levels in which v changes correspond to absorption of energy in the infrared region of … Fig. Energy level representations of the rotation–vibration transitions in a heteronuclear diatomic molecule, shown in order of increasing optical frequency and mapped to the corresponding lines in the absorption spectrum. The energy levels in cm-1 are therefore, Ej = B J (J +1) where B = (13.9) The rotational energy levels of a diatomic molecule are shown in Fig. The transitions occur in the spectral range of 300nm to 480nm. Vibrational motion of atoms bound in a molecule can be taken to be nearly simple harmonic. Our study is focused on the vibrational transitions that occur between the C and B electronic states. So, we'll look at the vibrational energy levels. The right panel shows the ground and first excited vibrational states, labeled and , respectively, with thei 13.2 Rotational energy levels of a rigid diatomic molecule and the allowed transitions. The difference is mostly due to the difference in force constants (a factor of 5), and not from the difference in reduced mass (9.5 u vs. 7 u). Additionally, each vibrational level has a set of rotational levels associated with it. [SOUND] Now let's move on, to look at the last place where energy can be stored in a diatomic molecule, and that is, in the vibrations.