Diatomic molecules normal mode

Even for molecules in the ground electronic state, our knowledge about cross sections is largely limited to the room-temperature condition, in which vibrational and rotational states are populated in a thermal distribution. Then, for a diatomic molecule, the ground vibrational state is predominantly populated. However, for a polyatomic molecule, normal modes with small quanta must be appreciably excited. For the full understanding of kinetics in plasma chemistry, it is important to assess the role of the internal energy of reactant molecules. 

The infrared spectra of polyatomic molecules contain one fundamental band for each normal mode whose motion modulates the dipole moment of the molecule. Normal modes that do not modulate the dipole moment of the molecule are not seen in the infrared spectrum. Overtone bands occur as with diatomic molecules, along with combination bands, which are produced when two normal modes make simultaneous transitions. 

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,... 

In diatomic molecules, T2 = 0, and thus the expectation value of C vanishes. This is the reason why this operator was not considered in Chapter 2. However, for linear triatomic molecules, t2 = / / 0, and the expectation value of C does not vanish. We note, however, that D J is a pseudoscalar operator. Since the Hamiltonian is a scalar, one must take either the absolute value of C [i.e., IC(0(4 2))I or its square IC(0(412))I2. We consider here its square, and add to either the local or the normal Hamiltonians (4.51) or (4.56) a term /412IC(0(412))I2. We thus consider, for the local-mode limit,... 

Consider the possible normal modes of a diatomic molecule oriented with the axis normal to the surface. If we neglect the surface structure, we get starting from the lowest energy as schematically shown in Fig. 6 ... 

Fig. 6. Schematic representation of the normal modes of an adsorbed diatomic molecule neglecting the surface structure, after Richardson and Bradshaw . In parentheses the experimentally measured values for CO in the ontop position on Pt(lll). (a) A frustrated translation (60 cm (b) A frustrated rotation (not yet detected), (c) The metal-molecule stretch (460cm ) . (d) The intramolecular stretch model (2100cm" ) . ...

For this case, the primary change that is observable in the IR spectrum is due to changes in the vibrahonal frequencies of the probe molecule due to modificahons in bond energies. This can lead to changes in bond force constants and the normal mode frequencies of the probe molecule. In some cases, where the symmetry of the molecule is perturbed, un-allowed vibrational modes in the unperturbed molecule can be come allowed and therefore observed. A good example of this effect is with the adsorption of homonuclear diatomic molecules, such as N2 and H2 (see Section 4.5.6.8). 

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. 

The representation as a two-dimensional potential energy diagram is simple for diatomic molecules. But for polyatomic molecules, vibrational motion is more complex. If the vibrations are assumed to be simple harmonic, the net vibrational motion of TV-atomic molecule can be resolved into 3TV-6 components termed normal modes of ibrations (3TV-5 for... 

As a simple example, we shall outline the normal-mode treatment of a diatomic molecule, leaving the reader to fill in the details (Problem 6.5). Let the nuclei be numbered 1 and 2. The a principal axis passes through the two nuclei the b and c principal axes are perpendicular to the internuclear axis. The origin of the abc axes is at the center of mass. The equilibrium coordinates of the nuclei are (tfi,e>0,0) and (tf2,e 0>0)- The b and c coordinates of each nucleus are always zero, since the axes translate and rotate with the molecule. From (6.3) and (6.1), the mass-weighted... 

Verify all the equations given in the normal-mode treatment of a diatomic molecule in Section 6.2. 

Up to this point, the molecule has been considered to be a rigid rotor, but the work in Chapter 4 on diatomics shows that we must add corrections for rotation-vibration interaction and centrifugal distortion. For a polyatomic molecule, there are several normal modes of vibration, each with its own vibrational quantum number (see Chapter 6). By analogy to (4.75), we write for polyatomic molecules... 

One of the earliest models is the quasi-diatomic model (10-13). This model is based on the assumption that the normal modes describing the state(s) of the photofragments are also the normal modes of the precursor molecule. This means, for example, that in the photodissociation of a linear triatomic molecule ABC A + BC (e.g., photodissociation of ICN - I + CN), the diatomic oscillator BC is- assumed to be a normal mode vibration in the description of the initial state of the triatomic molecule ABC. This means that the force constant matrix describing the vibrational motion of the molecule ABC can be written in the form (ignoring the bending motion) ... 

Figure 2.6 Potential energy of a diatomic molecule as a function of internuclear separation r. The equilibrium separation is re. A normal mode in a polyatomic molecule would have a similar potential curve, with a parameter characterizing the phase of the motion replacing r.

An understanding of what is implied by a Morse curve is necessary here, in order to understand the following point. A diatomic molecule has one normal mode of vibration. If the vibration behaves as a harmonic oscillator, the PE is proportional to the square of the displacement from the equilibrium internuclear distance. The... 

Consider a diatomic molecule A-B the normal-mode frequency (there is only one for a diatomic, of course) is given by [16] ... 

The IR activity of small molecules can be determined by inspection of the mode of a normal vibration (normal mode). Obviously, the vibration of a homopolar diatomic molecule is not IR-active, whereas that of a heteropolar diatomic molecule is IR-active. As shown in Fig. 1-13, the dipole moment of the H2O molecule is changed during each normal vibration. Thus, all these vibrations are IR-active. From inspection of Fig. 1-11, one can readily see that V2 and v3 of the CO2 molecule are IR-active, whereas v is not IR-active. 

The analysis given thus far applies to diatomic molecules. For simple polyatomic molecules, several characteristic vibrational motions, called normal modes, are possible. As an example, the normal modes of vibration for the SOi molecule are shown in Fig. 14.60. 

---