Linear molecules vibrational modes

There are (3n—6) modes of vibration in a polyatomic molecule except that for a linear molecule one mode is doubly degenerate so that effectively the number is (3n-5) n is the number of atoms in the molecule. Symmetry rules may restrict the kinds of transitions which can occur during an electronic transition but often the number of possible transitions is very large. 

The description of the vibrations of polyatomic molecules only becomes mathematically tractable by treating the system as a set of coupled harmonic oscillators. Thus a set of 3N - 6 (3N - 5 for linear molecules) normal modes of vibrations can be described in which aU the nuclei in the molecule move in phase in a simple harmonic motion with the same frequency, normal-mode frequencies are solved, the normal coordinates for the vibrations can be determined, and how the nuclei move in each of the normal modes of vibration can be shown. There are two important points that follow from this. First, each normal mode can be classified in terms of the irreducible representations of the point group describing the overall symmetry of the molecule [7, 8]. This symmetry classification of the... 

For chemically bound molecules, it is usual to analyse tlie vibrational energy levels in teniis of normal modes, a non-linear (or linear) molecule witli V atoms has 3 V - 6 (or 3 V - 5) vibrational degrees of freedom. There is a... 

In order to calculate q (Q) all possible quantum states are needed. It is usually assumed that the energy of a molecule can be approximated as a sum of terms involving translational, rotational, vibrational and electronical states. Except for a few cases this is a good approximation. For linear, floppy (soft bending potential), molecules the separation of the rotational and vibrational modes may be problematic. If two energy surfaces come close together (avoided crossing), the separability of the electronic and vibrational modes may be a poor approximation (breakdown of the Bom-Oppenheimer approximation. Section 3.1). 

For a polyatomic molecule, the complex vibrational motion of the atoms can be resolved into a set of fundamental vibrations. Each fundamental vibration, called a normal mode, describes how the atoms move relative to each other. Every normal mode has its own set of energy levels that can be represented by equation (10.11). A linear molecule has (hr) - 5) such fundamental vibrations, where r) is the number of atoms in the molecule. For a nonlinear molecule, the number of fundamental vibrations is (3-q — 6). 

The assignment of (hr) - 5) vibrational modes for a linear molecule and (hr) - 6) vibrational modes for a nonlinear molecule comes from a consideration of the number of degrees of freedom in the molecule. It requires hr) coordinates to completely specify the position of all t) atoms in the molecule, and each coordinate results in a degree of freedom. Three coordinates (x, y, and z) specify the movement of the center of mass of the molecule in space. They set the translational degrees of freedom, since translational motion is associated with movement of the molecule as a whole. Two internal coordinates (angles) are required to specify the orientation of the axis of a linear molecule during rotation, while three angles are required for a nonlinear... 

A nonlinear molecule consisting of N atoms can vibrate in 3N — 6 different ways, and a linear molecule can vibrate in 3N — 5 different ways. The number of ways in which a molecule can vibrate increases rapidly with the number of atoms a water molecule, with N = 3, can vibrate in 3 ways, but a benzene molecule, with N = 12, can vibrate in 30 different ways. Some of the vibrations of benzene correspond to expansion and contraction of the ring, others to its elongation, and still others to flexing and bending. Each way in which a molecule can vibrate is called a normal mode, and so we say that benzene has 30 normal modes of vibration. Each normal mode has a frequency that depends in a complicated way on the masses of the atoms that move during the vibration and the force constants associated with the motions involved (Fig. 2). 

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. 

The results obtained above for a diatomic molecule can be generalized for polyatomic molecules. Each of the 3N-6 normal modes of vibration (or 3N-5 for linear molecules) will contribute an energy given by an expression analogous to Eq. (63), namely,... 

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). 

The internal degrees of freedom are associated with the rotation and vibration of the molecule. A linear molecule has 2 degrees of rotational motion and a non-linear molecule has three. The remaining (3N — 5) or (3N — 6) degrees of freedom describe the motion of the nuclei with respect to each other. For example, the linear CO2 has (3N — 5) = 4 vibrational degrees of freedom and the non-linear SO2 has three. The mode... 

For CS2, it is a linear molecule and should have 3N—5 — 4 vibrational modes. There is a symmetrical stretch, an asymmetrical stretch and a bending in-plane and a bending out-of-plane modes of vibration. 

For linear triatomic molecules 3 N - 5 = 4 and we expect form vibrational modes instead of three as shown in the fig. Out of the four there are two vibrations, one in the plane of the paper and the other in which the oxygen atoms move simultaneously into and out of the plane. The two sorts of motion are termed degenerate and so we have only three vibrations. 

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