Diatomic molecule rotational energy levels

Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

As for diatomic molecules, there are stacks of rotational energy levels associated with all vibrational levels of a polyatomic molecule. The resulting term values S are given by the sum of the rotational and vibrational term values... 

Quantum mechanics predicts that a diatomic molecule has a set of rotational energy levels ej given by... 

Equation (4.18) applies only to a diatomic or linear polyatomic molecule. Similar kinds of rotational energy levels are present in more complicated molecules. We will describe the various kinds in more detail in Chapter 10. 

The rotational energy levels of (a) a heavy diatomic molecule and (b) a light diatomic molecule. Note that the energy levels are closer together for the heavy diatomic molecule. Microwaves arc absorbed when transitions take place between neighboring energy levels. 

Rotational energy levels in a diatomic molecule are described well by the expression ... 

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. 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

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. 

Fig. VI1-3.—Rotational energy levels for a diatomic molecule. The first four rotational states are 2 represented. The lowest state,...

Finding the quantum mechanical rotational energy levels for a molecule gets complicated very quickly. However, the results for a linear molecule are simple and illustrative. The rotational energy levels for a heteronuclear diatomic (e.g., HC1) or asymmetric linear... 

The rotational energy levels for a homonuclear diatomic molecule follow Eq. 8.16, but the allowed possibilities for j are different. (The rules for a symmetric linear molecule with more than two atoms are even more complicated, and beyond the scope of this discussion.) If both nuclei of the atoms in a homonuclear diatomic have an odd number of nuclear particles (protons plus neutrons), the nuclei are termed fermions if the nuclei have an even number of nuclear particles, they are called bosons. For a homonuclear diatomic molecule composed of fermions (e.g., H— H or 35C1—35C1), only even-j rotational states are allowed. (This is due to the Pauli exclusion principle.) A homonuclear diatomic molecule composed of bosons (e.g., 2D—2D or 14N—14N) can only have odd- j rotational levels. 

I 2.1 Rotational Energy Levels of Diatomic Molecules, K I 2.2 Vibrational Energy Levels of Diatomic Molecules, 10 I 2.3 Electronic Stales of Diatomic Molecules, 11 I 2.4 Coupling of Rotation and Electronic Motion in Diatomic Molecules Hund s Coupling Cases, 12 1-3 Quantum States of Polyatomic Molecules, 14... 

For diatomic molecules, corrections can be made for the assumption used in the derivation of the rotational partition function that the rotational energy levels are so closely spaced that they can be considered to be continuous. The equations to be used in making these corrections are given in Appendix 6. Also given are the equations to use in correcting for vibrational anharmonicity and nonrigid rotator effects. These corrections are usually small.22... 

Hougen, J.T. (1970). The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules (National Bureau of Standards (U. S.), Monograph 115, Washington D.C.). 

Spectral studies of rotational energy levels have proved most profitable for linear molecules having dipole moments, particularly diatomic molecules (for example, CO, NO, and the hydrogen halides). The moment of inertia of a linear molecule may be readily obtained from its rotation spectrum and for diatomic molecules, interatomic distances may he calculated directly from moments of inertia (Exercise 14d). For a mole-... 

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 practice values of B are also often quoted in cm-1. For the simple rigid rotor the rotational quantum number J takes integral values, J = 0, 1, 2, etc. The rotational energy levels therefore have energies 0, 2B, 6B, 12B, etc. Elsewhere in this book we will describe the theory of electric dipole transition probabilities and will show that for a diatomic molecule possessing a permanent electric dipole moment, transitions between the rotational levels obey the simple selection rule A J = 1. The rotational spectrum of the simple rigid rotor therefore consists of a series of equidistant absorption lines with frequencies 2B, 4B, 6B, etc. 

Rotational energy levels for the ground vibrational state v" = 0) and the first excited vibrational state (v = 1) in a diatomic molecule. The vertical arrows indicate allowed transitions in the i and Pbranches numbers in parentheses index the value J of the lower state. Transitions in the (3t>ranch (Ay = 0) are not shown since they are not infrared active. 

Rotational Levels and Transitions. The vibrational-rotational energy levels for a linear molecule are similar to those for a diatomic molecule and to a good approximation are given in cm units by the sum G u V2...) + where ... 

Ah diatomic molecnles for which the constituent nnclei have spin exhibit the phenomenon of spin isomerism. The nnclear spins can be parallel (the ortho isomer) or opposed (the para isomer). For most diatomic molecules, which might be expected to exhibit, spin isomerism the energy separation of the rotational states is small compared to kT, even at low temperatures. However, in the case of hydrogen molecules, which have the smallest moment of inertia of any diatomic molecnles, the energy difference between the rotational energy levels is relatively large and only the lowest states are popnlated at room temperature. 

This method was introduced in 1965 by Aslund9 and has been extensively used by the Swedish workers and others in analyzing the spectra of diatomic molecules. The initial objective is to reduce a system of assignments and measurements to a set of term values by least squares methods. The term values for the upper and lower states are then fitted independently to rotational energy level expressions. It has always been assumed that upper and lower state energy levels could be separated in this way recently, however, Albritton and co-workers10 have shown that the upper and lower state term values are correlated. A correlation matrix is given by these authors for two upper and one lower state of... 

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