Rotational energy diatomic

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. 

A linear molecule, such as any diatomic molecule, carbon dioxide, and ethyne (acetylene, HC=CH), can rotate about two axes perpendicular to the line of atoms, and so it has two rotational modes of motion. Its average rotational energy is therefore 2 X jkT = kT, and the contribution to the molar internal energy is NA times this value ... 

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. 

In a diatomic molecule with rotation energy and rotation quantum number /, there is an energy proportional to J J + 1), and there are 2/ + 1 configurations (referred to as degeneracy). The probability of finding a molecule in energy state J becomes... 

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 the idealized case for rotation of a diatomic molecule, one assumes the molecule is analogous to a dumbbell with the atoms held at a fixed distance r from each other that is, it is a rigid rotor. The simultaneous vibration of the molecule is ignored, as is the increase of internuclear distance at high rotational energies arising from the centrifugal force on the two atoms. 

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

Solution of the wave equation for rotation of a rigid diatomic molecule leads to the following expression for rotational energy ... 

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. 

We see from (4.104) that, although the vibrational quantum number is not changing, the frequency of a pure-rotational transition depends on the vibrational quantum number of the molecule undergoing the transition. (Recall that vibration changes the effective moment of inertia, and thus affects the rotational energies.) For a collection of diatomic molecules at temperature T, the relative populations of the energy levels are given by the Boltzmann distribution law the ratio of the number of molecules with vibrational quantum number v to the number with vibrational quantum number zero is... 

The presence of degenerate vibrational modes affects the rotational energies. In Section 4.11, we saw the effect of electronic angular momentum on the rotational energies of a diatomic molecule. In this section, we shall assume that there is no electronic angular momentum, only nuclear vibrational and rotational angular momentum. If both electronic and... 

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