Linear molecules rotational states

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... 

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]). 

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. 

Once the electronic Schrodinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the PES is known. This can then be used for solving the nuclear part of the Schrodinger equation. If there are N nuclei, there are 3N coordinates that define the geometry. Of these coordinates, three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. Eor a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3N-6(5) coordinates to describe the internal movement of the nuclei, the vibrations, often chosen to be... 

The probability distribution for the n = 2 intermolecular level. Fig. 12c, indicates that this state resembles a bending level of the T-shaped complex with two nodes in the angular coordinate and maximum probability near the linear He I—Cl and He Cl—I ends of the molecule [40]. The measured I C1(B, v = 2f) rotational product state distribution observed following preparation of the He I C1(B, v = 3, m = 2, / = 1) state is plotted in Fig. 12d. The distribution is distinctly bimodal and extends out to the rotational state, / = 21,... 

In general a nonlinear molecule with N atoms has three translational, three rotational, and 3N-6 vibrational degrees of freedom in the gas phase, which reduce to three frustrated vibrational modes, three frustrated rotational modes, and 3N-6 vibrational modes, minus the mode which is the reaction coordinate. For a linear molecule with N atoms there are three translational, two rotational, and 3N-5 vibrational degrees of freedom in the gas phase, and three frustrated vibrational modes, two frustrated rotational modes, and 3N-5 vibrational modes, minus the reaction coordinate, on the surface. Thus, the transition state for direct adsorption of a CO molecule consists of two frustrated translational modes, two frustrated rotational modes, and one vibrational mode. In this case the third frustrated translational mode vanishes since it is the reaction coordinate. More complex molecules may also have internal rotational levels, which further complicate the picture. It is beyond the scope of this book to treat such systems. 

For linear molecules or ions the symbols are usually those derived from the term symbols for the electronic states of diatomic and other linear molecules. A capital Greek letter E, n, A, O,... is used, corresponding to k — 0,1,2,3,..., where A. is the quantum number for rotation about the molecular axis. For E species a superscript + or - is added to indicate the symmetry with respect to a plane that contains the molecular axis. 

The number of fundamental vibrational modes of a molecule is equal to the number of degrees of vibrational freedom. For a nonlinear molecule of N atoms, 3N - 6 degrees of vibrational freedom exist. Hence, 3N - 6 fundamental vibrational modes. Six degrees of freedom are subtracted from a nonlinear molecule since (1) three coordinates are required to locate the molecule in space, and (2) an additional three coordinates are required to describe the orientation of the molecule based upon the three coordinates defining the position of the molecule in space. For a linear molecule, 3N - 5 fundamental vibrational modes are possible since only two degrees of rotational freedom exist. Thus, in a total vibrational analysis of a molecule by complementary IR and Raman techniques, 31V - 6 or 3N - 5 vibrational frequencies should be observed. It must be kept in mind that the fundamental modes of vibration of a molecule are described as transitions from one vibration state (energy level) to another (n = 1 in Eq. (2), Fig. 2). Sometimes, additional vibrational frequencies are detected in an IR and/or Raman spectrum. These additional absorption bands are due to forbidden transitions that occur and are described in the section on near-IR theory. Additionally, not all vibrational bands may be observed since some fundamental vibrations may be too weak to observe or give rise to overtone and/or combination bands (discussed later in the chapter). 

As already noted, in the Born-Oppenheimer approximation, the nuclear motion of the system is subject to a potential which expresses the isotope independent electronic energy as a function of the distortion of the coordinates from the position of the transition state. An analysis of the motions of the N-atom transition state leads to three translations, three rotations (two for a linear molecule), and 3N - 6 (3N- 5 for a linear transition state) vibrations, one which is an imaginary frequency (e.g. v = 400icm 1 where i = V—T), and the others are real vibrational frequencies. The imaginary frequency corresponds to motion along the so-called reaction... 

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. 

To calculate the molecular rotational partition function for an asymmetric, linear molecule, we use Eq. 8.16 for the energy level of rotational state /, and Eq. 8.18 for its degeneracy. As discussed in Section 8.2, rotational energy levels are very closely spaced compared to k/jT unless the molecule s moment of inertia is very small. Therefore, for most molecules, replacing the summation in Eq. 8.50 with an integral introduces little error. Thus the... 

We noted in Section 8.2 that only half the values of j are allowed for homonuclear diatomics or symmetric linear polyatomic molecules (only the even-y states or only the odd- y states, depending on the nuclear symmetries of the atoms). The evaluation of qmt would be the same as above, except that only half of the j s contribute. The result of the integration is exactly half the value in Eq. 8.64. Thus a general formula for the rotational partition function for a linear molecule is... 

---