Rotation linear molecules

What are the consequences of these considerations for depolarized light scattering In a dilute gas where reorientation is predominantly inertial, we expect the spectrum to be what is normally called the pure rotational Raman spectrum of the molecule. As higher densities are approached, the discrete spectral lines broaden and overlap to form a continuous band. We show how the band shape can be computed for freely rotating linear molecules and spherical top molecules and then indicate the assumptions that have been used by several authors to include collisions in the theory. 

Non-linear molecules have 3n — 6 non-zero vibrational modes. The six zero or trivial vibrational modes correspond to three modes of translation, and three modes of rotation. Linear molecules have 3n — 5 non-zero vibrational modes. Three of the five trivial modes correspond to translation, but one rotational mode is missing. This missing mode corresponds to rotation about the molecular axis. 

Another common system in quantum mechanics is the so-called rigid rotator. The energy levels of a rotating linear molecule are given to good approximation by the following expression. 

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 ... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

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. ... 

To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles (usually termed 0 and ([)). For any non-linear molecule, three angles (usually a, P, and y) are needed. Hence the rotational Schrodinger equation for a nonlinear molecule is a differential equation in three-dimensions. 

Linear molecules belong to the axial rotation group. Their symmetry is intermediate in complexity between nonlinear molecules and atoms. 

As before, when pf i(Rg) (or dpfj/dRa) lies along the molecular axis of a linear molecule, the transition is denoted a and k = 0 applies when this vector lies perpendicular to the axis it is called n and k = 1 pertains. The resultant linear-molecule rotational selection rules are the same as in the vibration-rotation case ... 

For linear molecules, the coulombic potential is unchanged (because the set of all inter-particle distances are unchanged) by rotations about the molecular axis (the z axis) ... 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

Figure 5.5 The rotational angular momentum vector P for (a) a linear molecule and (b) the prolate symmetric rotor CH3I where is the component along the a axis...

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... 

Figure 5.12 shows the J= — 0 transition of the linear molecule cyanodiacetylene (H—C=C—C=C—C=N) observed in emission in Sagittarius B2 (Figure 5.4 shows part of the absorption spectrum in the laboratory). The three hyperfine components into which the transition is split are due to interaction between the rotational angular momentum and the nuclear spin of the nucleus for which 1= 1 (see Table 1.3). The vertical scale is a measure of the change of the temperature of the antenna due to the received signal. 

Measurement and assignment of the rotational spectmm of a diatomic or other linear molecule result in a value of the rotational constant. In general, this will be Bq, which relates... 

Such bands also obey the rotational selection rules in Equation (6.82) and appear similar to a U-I band of a linear molecule. 

Each of the lasing vibrational transitions has associated rotational fine structure, discussed for linear molecules in Section 6.2.4.1. The Sgli transition is — Ig with associated P and R branches, for which AJ = — 1 and +1, respectively, similar to the 3q band of HCN in Figure 6.25. The 3q22 band is, again, with a P and R branch. 

The classical values of each of drese components can be calculated by ascribing a contribution of R/2 for each degree of freedom. Thus tire U ansla-tional and tire rotational components are 3/27 each, for drree spatial components of translational and rotational movement, and (3 — 6)7 for die vibrational contribution in a non-linear polyatomic molecule containing n atoms and (3 — 5)7 for a linear molecule. For a diatomic molecule, the contributions ate 3/27 ti.a s -f 7 [.ot + 7 vib-... 

Let us consider systems which consist of a mixture of spherical atoms and rigid rotators, i.e., linear N2 molecules and spherical Ar atoms. We denote the position (in D dimensions) and momentum of the (point) particles i with mass m (modeling an Ar atom) by r, and p, and the center-of-mass position and momentum of the linear molecule / with mass M and moment of inertia I (modeling the N2 molecule) by R/ and P/, the normalized director of the linear molecule by n/, and the angular momentum by L/. 

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