Linear molecules rotational energy

In the first approximation we replace the centrifugal potential [J(J+l) 2ft ] fe /2nR by a constant eJ, which for collinear transition states is the usual linear-molecule rotational energy at the transition state configuration. With this replacement eq. (25) becomes... 

In addition to vibrational motions, the molecule can undergo rotational motion perpendicular to the bond axis. For linear molecules, the energy associated with rotational transitions is approximated by the rigid-rotor model. 

Whereas monatomic molecules can only possess translational thermal energy, two additional kinds of motions become possible in polyatomic molecules. A linear molecule has an axis that defines two perpendicular directions in which rotations can occur each represents an additional degree of freedom, so the two together contribute a total of 1/2 R to the heat capacity. For a non-linear molecule, rotations are possible along all three directions of space, so these molecules have a rotational heat capacity of 3/2 R. Finally, the individual atoms within a molecule can move relative to each other, producing a vibrational motion. A molecule consisting of N atoms can vibrate in 3N-6 different ways or modes1. For mechanical reasons that we cannot go into here, each vibrational mode contributes R (rather than 1/2 R) to the total heat capacity. 

We stated earlier (Section 9.1) that for the rotational motion of a linear molecule the energy levels are given by... 

For diatomic and other linear molecules, the energies of rotational levels are given (to a good degree of approximation) in terms of a rotadmial quantum number (AT) and the moment of inertia (/ab) l>y the expression ... 

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

The simplest case is a transition in a linear molecule. In this case there is no orbital or spin angular momentum. The total angular momentum, represented by tire quantum number J, is entirely rotational angular momentum. The rotational energy levels of each state approximately fit a simple fomuila ... 

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

An N-atom molecular system may he described by dX Cartesian coordinates. Six independent coordinates (five for linear molecules, three fora single atom) describe translation and rotation of the system as a whole. The remaining coordinates describe the nioleciiUir configuration and the internal structure. Whether you use molecular mechanics, quantum mechanics, or a specific computational method (AMBER, CXDO. etc.), yon can ask for the energy of the system at a specified configuration. This is called a single poin t calculation. ... 

The rotational energy levels for a prolate and an oblate symmetric rotor are shown schematically in Figure 5.6. Although these present a much more complex picture than those for a linear molecule the fact that the selection mles... 

For a symmetric rotor the modification Eg to the rotational energy levels in an electric field S is larger than in a linear molecule and is given, approximately, 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. 

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... 

The envelope of the Stark structure of the rotator in a constant orienting field, calculated quantum-mechanically in [17], roughly reproduces the shape of the triplet (Fig. 0.5(c)). The appearance of the Q-branch in the linear rotator spectrum indicates that the axis is partially fixed, i.e. some molecules perform librations of small amplitude around the field. Only molecules with high enough rotational energy overcome the barrier created by the field. They rotate with the frequencies observed in the... 

FIGURE 6.17 The translational and rotational modes of atoms and molecules and the corresponding average energies of each mode at a temperature T. (a) An atom or molecule can undergo translational motion in three dimensions, (b) A linear molecule can also rotate about two axes perpendicular to the line of atoms, (c) A nonlinear molecule can rotate about three perpendicular axes. 

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 ... 

The molar heat capacities of gases composed of molecules (as distinct from atoms) are Higher than those of monatomic gases because the molecules can store energy as rotational kinetic energy as well as translational kinetic energy. We saw in Section 6.7 that the rotational motion of linear molecules contributes another RT to the molar internal energy ... 

See also the theoretical description of a micro reactor for optical photocatalytic dissociation of non-linear molecules in [140]. Here, a mathematical model for a novel type of micro reactor is given. Rotating non-linear molecules at excitation of valent vibrations are considered, having a magnetic moment. Resonance decay of molecules can be utilized with comparatively weak external energy sources only. 

Then, for a linear molecule, as-j c = fiK = 0, the rigid-rotator energy is given simply by... 

It was shown in the previous chapter that the Schr dinger equation for molecular rotation depends on the type of rotator, as defined in Section 9.2.2. For linear molecules and, hence, diatomics, the energy is given by Eq. (9-40),... 

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