Molecular rotation rotational energy levels

Infrared. It can be seen from Fig. 4 that infrared radiation is of an energy suitable for the stimulation of vibrational absorption that occurs within a single electronic level. For gas-phase samples, vibrational bands can be divided into a series of closely spaced absorptions associated with molecular rotational energy levels however, these are not usually observed. Even so, the infrared absorption spectrum for most solid and liquid samples is complex and contains many sharp maxima and minima associated with well-defined vibrational energy levels. 

MICROWAVE molecular spectroscopy involves the observation and analysis of absorption transitions between molecular rotational energy levels of gas-phase molecules. These transitions between rotational levels are associated with a given vibrational state of the ground electronic state, and the transitions fall in the microwave region... 

The progression of sections leads the reader from the principles of quantum mechanics and several model problems which illustrate these principles and relate to chemical phenomena, through atomic and molecular orbitals, N-electron configurations, states, and term symbols, vibrational and rotational energy levels, photon-induced transitions among various levels, and eventually to computational techniques for treating chemical bonding and reactivity. 

Rotational Energy Levels The rotational energy of a molecule depends upon the molecular geometry. For a linear molecule that behaves as a rigid rotator,3... 

When a broadband source of IR energy irradiates a sample, the absorption of IR energy by the sample results from transitions between molecular vibrational and rotational energy levels. A vibrational transition may be approximated by treating two atoms bonded together within a molecule as a harmonic oscillator. 

The result (Equation 4.90) could have been derived more simply. It has been emphasized that the quantum mechanical contribution to the partition function ratio arises from the quantization of vibrational energy levels. For the molecular translations and rotations quantization has been ignored because the spacing of translational and rotational energy levels is so close as to be essentially continuous (As/kT 1). 

It is anticipated that a course dealing with atomic and molecular spectroscopy will follow the student s mastery of the material covered in Sections 1- 4. For this reason, beyond these introductory sections, this text s emphasis is placed on electronic structure applications rather than on vibrational and rotational energy levels, which are traditionally covered in considerable detail in spectroscopy courses. 

The Section on Molecular Rotation and Vibration provides an introduction to how vibrational and rotational energy levels and wavefunctions are expressed for diatomic, linear polyatomic, and non-linear polyatomic molecules whose electronic energies are described by a single potential energy surface. Rotations of "rigid" molecules and harmonic vibrations of uncoupled normal modes constitute the starting point of such treatments. 

In using symmetry to help simplify molecular orbital or vibration/rotation energy level calculations, the following strategy is followed ... 

Energy barriers for internal rotation have been derived, especially during the 1950s, by analyzing (68M12 68M13) microwave spectra of molecules. The method works with molecules with a permanent dipole moment and in the gas phase. Limitations are dictated by the molecular size. The barriers are obtained from rotational energy levels of the molecule as a whole, perturbed by the internal rotor. When different conformers are present in the sample and their interconversion is slower than microwave absorption (barriers smaller than 20 kJ mol can be measured), the spectrum is just a superposition of the lines of the separate species which can be qualitatively and quantitatively determined. 

The analysis of molecular vibration and rotational energy levels in terms of assumed... 

Figure 2.1 Schematic diagram showing possible molecular electronic transitions, and vibrational and rotational energy levels...

To calculate the molecular rotational partition function for an asymmetric, linear molecule, we use Eq. 8.16 for the energy level of rotational state /, and Eq. 8.18 for its degeneracy. As discussed in Section 8.2, rotational energy levels are very closely spaced compared to k/jT unless the molecule s moment of inertia is very small. Therefore, for most molecules, replacing the summation in Eq. 8.50 with an integral introduces little error. Thus the... 

In the calculation of the thermodynamic properties of the ideal gas, the approximation is made that the energies can be separated into independent contributions from the various degrees of freedom. Translational and electronic energy levels are present in the ideal monatomic gas.ww For the molecular gas, rotational and vibrational energy levels are added. For some molecules, internal rotational energy levels are also present. The equations that relate these energy levels to the mass, moments of inertia, and vibrational frequencies are summarized in Appendix 6. 

Although the interpretation of rotational spectra of diatomic molecules is relatively simple, such spectra lie in the far infrared, a region that at present is not as easily accessible to study as are the near infrared, visible, cr ultraviolet. Consequently, most information about rotational energy levels has actually been obtained, not from pure rotation spectra, but from rotation-vibration spectra. Molecules without dipole moments have no rotation spectra, and nonpolar diatomic molecules lack rotation-vibration spectra as well, Thus, II2, N2, 02, and the molecular halogens have no characteristic infrared spectra. Information about the vibrational and rotational energy levels of these molecules must be obtained from the fine structure of their electronic spectra or from Raman spectra. 

In the following sections of this paper, we describe a new model Hamiltonian to study the vibration—inversion—rotation energy levels of ammonia. In this model the inversion motion is removed from the vibrational problem and considered with the rotational problem by allowing the molecular reference configuration to be a function of the large amplitude motion coordinate. The resulting Hamiltonian then takes a form which is very close to the standard Hamiltonian used in the study of rigid molecules and allows for a treatment of the inversion motion in a way which is very similar to the formalism developed for the study of molecules with internal rotation [see for example ]. 

A schematic representation of two molecular electronic states with the accompanying vibrational and rotational energy levels is given in Fig. 14.52. Note that each electronic energy level has a set of quantized vibrational states and that each vibrational state has a set of rotational states. 

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