Rotation-Vibration-Electronic Spectra of Diatomic Molecules

1 Rotation-Vibration-Electronic Spectra of Diatomic Molecules 

Even systems as seemingly simple as diatomic molecules often act as complex, many-body systems. Mechanistic understanding and insight, as opposed to mere empirical description, are based on the existence and discovery of patterns that owe their existence to approximate constants of motion. An approximate constant of motion is the eigenvalue of an operator that commutes with most, but not all, terms in the exact molecular Hamiltonian. Nonconservation of this quantity results in subtle rather than catastrophic corruption of the simple patterns on which spectroscopic assignments and mechanistic interpretations are based. 

The simple spectra described in this chapter and the ideas presented in this book provide the language, concepts, and intuitive framework required for the design and interpretation of even the most elaborate and innovative modern molecular structure and dynamics experiments. 

Although few bona fide examples of textbook spectra of gas phase diatomic molecules exist, we begin this book with an elementary description of a spectrum devoid of the esoteric details that encode the most interesting diatomic molecular dynamics. The approach in this chapter is unashamedly phenomenological the physical principles, derivations of formulas, and theoretical context will be presented in Chapters 2-9. 

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The above three sources are a classic and comprehensive treatment of rotation, vibration, and electronic spectra of diatomic and polyatomic molecules. 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

To investigate the spectra of diatomic molecules, we need the selection rules for radiative transitions. We now investigate the electric-dipole selection rules for transitions between vibration-rotation levels belonging to the same 2 electronic state. (Transitions in which the electronic state changes will be considered in Chapter 7.)... 

We now consider radiative transitions foi which both v and J change, but the electronic state does not these transitions give the vibration-rotation spectra of diatomic molecules. The selection rules for these transitions were found in Section 4.4 to be ( 2 states only)... 

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. 

For electronic states of diatomic molecules in which the electrons have a resultant angular momentum around the internuclear line (see section 25) J may also stay unaltered. Since the vibration-rotation spectra are all obtained in absorption and hence refer to the ground state, and since NO is the only diatomic molecule in the ground state of which the electrons have an angular momentum around the internuclear line, we disregard this special case in the subsequent investigation. 

Gas phase electronic spectra of polyatomic molecules are more complicated than the spectra of diatomics. The number of vibrational modes and the possibility of combination bands usually lead to numerous vibrational bands, and these may be overlapping. Also, the rotational fine structure tends to be more complicated, as we might expect from the differences between diatomic and polyatomic infrar (IR) spectra. Conventional absorption spectra can prove to be a difficult means of measuring and assigning transitions, and so numerous experimental methods have been devised to select molecules in specific initial states and to probe the absorption or the emission spectrum with narrow frequency range lasers. 

The information contained in a diatomic molecule rotation-vibration-electronic wavefunction is enormous. But this is dwarfed by the information content of a time-evolving wavefunction that originates from a non-eigenstate pluck. A simplified, reduced-dimension representation, rather than an exact numerical description, is prerequisite to visualization and understanding. The concepts and techniques presented in this book, developed explicitly for diatomic molecule spectra and dynamics, are applicable to larger molecules. Indeed, any attempt... 

Until now we have only considered the rotation and vibration-rotation spectra of polyatomic molecules. The information to be gained from electronic hands is here much more limited than in the case of diatomic molecules. As mentioned already in section 31, the reason for this is a twofold one on the experimental side the diffuse character of the electronic spectra in most cases, due to causes which we shall discuss in the next chapter on the theoretical side the greater difficulty of finding suitable means to classify the electronic states. 

For molecules with more than two vibrational degrees of freedom recourse must be had to a geometrical representation in a space with more than three dimensions. The reasoning is the same as before, but one may expect that the differences with the case of diatomic molecules become still more accentuated. Hence, it is quite comprehensible why discrete electronic band spectra should occur only as an exception for polyatomic molecules (see sections 31 and 46), the excited electronic states being apt to suffer from predissociation over the whole or almost the whole range of their vibration-rotation levels. 

When two atoms are combined to form a diatomic molecule, isotope effects appear in the vibrational, rotational, and electronic spectra. To the extent that the Bom-Oppenheimer approximation is valid, the potential function for a given electronic state is independent of the masses of the nuclei, hamonic vibration frequencies of two isotopic variants of a diatomic molecule in the same electronic state are then related by the equation... 

The electronic spectra of isotopically varied diatomic molecules reflect the effects of changes in the vibrational and rotational energy levels of both the ground and the excited electronic states. This subject is of great historical importance since the isotopes 0, " 0, and were all identified in... 

The electronic spectra of molecules, even the smallest diatomic molecules, are more complicated than those of atoms because more than one nucleus is present. However, now we can take advantage of molecular symmetry. Just as with vibrational spectroscopy, electronic spectroscopy of molecules uses group-theoretical ideas for simplification. Because all diatomic molecules have either Coov or symmetry, for the present those two point groups will be important. There are also some similarities in electronic spectra and rotational spectra for diatomic molecules, so a review of rotational motions might be useful. 

The spectra of polyatomic molecules are more complicated than those of atoms or diatomic molecules. As with diatomic molecules, rotational transitions can occur without vibrational or electronic transitions, vibrational transitions can occur without electronic transitions but are generally accompanied by rotational transitions, and electronic transitions are accompanied by both vibrational and rotational transitions. 

The molecular constants o , B, Xe, D, and ae for any diatomic molecule may be determined with great accuracy from an analysis of the molecule s vibrational and rotational spectra." Thus, it is not necessary in practice to solve the electronic Schrodinger equation (10.28b) to obtain the ground-state energy o(R). 

Kratzer and Loomis as well as Haas (1921) also discussed the isotope effect on the rotational energy levels of a diatomic molecule resulting from the isotope effect on the moment of inertia, which for a diatomic molecule, again depends on the reduced mass. They noted that isotope effects should be seen in pure rotational spectra, as well as in vibrational spectra with rotational fine structure, and in electronic spectra with fine structure. They pointed out the lack of experimental data then available for making comparison. 

Aside from vibration and rotation constants, an important piece of information available from electronic spectra is the dissociation energies of the states involved. In electronic absorption spectroscopy, most of the diatomic molecules will originate from the c"=0 level of the ground electronic state. The vibrational structure of the transition to a given excited electronic state will consist of a series of bands (called a progression) representing changes of 0—>0, 0—>1, 0- 2,..., 0— t nax, where... 

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