Diatomic molecules orientational

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 ... 

The Hamiltonian in this problem contains only the kinetic energy of rotation no potential energy is present because the molecule is undergoing unhindered "free rotation". The angles 0 and (j) describe the orientation of the diatomic molecule s axis relative to a laboratory-fixed coordinate system, and p is the reduced mass of the diatomic molecule p=mim2/(mi+m2). 

Transfer matrix calculations of the adsorbate chemical potential have been done for up to four sites (ontop, bridge, hollow, etc.) or four states per unit cell, and for 2-, 3-, and 4-body interactions up to fifth neighbor on primitive lattices. Here the various states can correspond to quite different physical systems. Thus a 3-state, 1-site system may be a two-component adsorbate, e.g., atoms and their diatomic molecules on the surface, for which the occupations on a site are no particles, an atom, or a molecule. On the other hand, the three states could correspond to a molecular species with two bond orientations, perpendicular and tilted, with respect to the surface. An -state system could also be an ( - 1) layer system with ontop stacking. The construction of the transfer matrices and associated numerical procedures are essentially the same for these systems, and such calculations are done routinely [33]. If there are two or more non-reacting (but interacting) species on the surface then the partial coverages depend on the chemical potentials specified for each species. 

By the last two assumptions the theory, strictly speaking, is only applicable to the monatomic gases A, Kr, Xe, to a somewhat lesser extent to the almost spherical molecules CH4, CF4, SFe, and perhaps to nonpolar diatomic molecules. The rotation of even slightly nonspherical molecules like Q2 and N2 will not be free in the entire cavity when such a molecule comes close to the wall of its cage it will have to orient itself parallel to this wall. Furthermore, some of the cavities are somewhat oblate (cf. Section I.B), and thus the rotation of relatively large, oblong molecules may be seriously... 

Calculate the entropy of a tiny solid made up of four diatomic molecules of a compound such as carbon monoxide, CO, at T = 0 when (a) the four molecules have formed a perfectly ordered crystal in which all molecules are aligned with their C atoms on the left (top-left image in Fig. 7.7) and (b) the four molecules lie in random orientations (but parallel, any of the images in Fig. 7.7). 

If we move the chemisorbed molecule closer to the surface, it will feel a strong repulsion and the energy rises. However, if the molecule can respond by changing its electron structure in the interaction with the surface, it may dissociate into two chemisorbed atoms. Again the potential is much more complicated than drawn in Fig. 6.34, since it depends very much on the orientation of the molecule with respect to the atoms in the surface. For a diatomic molecule, we expect the molecule in the transition state for dissociation to bind parallel to the surface. The barriers between the physisorption, associative and dissociative chemisorption are activation barriers for the reaction from gas phase molecule to dissociated atoms and all subsequent reactions. It is important to be able to determine and predict the behavior of these barriers since they have a key impact on if and how and at what rate the reaction proceeds. 

The intrinsic state (7.126) describes the ground state of a diatomic molecule. The orientation of the axis of the molecule in space can be chosen arbitrarily. It is convenient to choose it along the z direction (Figure 7.3). The coherent state (7.126) depends then only on the magnitude of a, and can be written as... 

For a molecule in a given electronic and vibrational state, it is convenient to define the permanent dipole operator d = (i/r // i/r), where v/) is a product of the electronic and vibrational states. This vector operator depends on the angles that specify the orientation of the molecule with respect to the external field axis. For diatomic molecules, d is directed along the intermolecular axis. The Stark shifts of the molecule in a DC electric field can (almost always) be found by treating the molecule as a rigid rotor and diagonalizing the matrix of the operator... 

Carbon monoxide (CO) is a well-studied species in which the two ends of the diatomic molecule have similar coordinative properties. In the formation of any real crystal of solid CO, a statistical fraction of the molecules are therefore found to enter the lattice in backward orientation ... 

First consider tpn. For a nucleus with spin /, there are 21 +1 possible orientations of the spin Mj = — /. Thus for a diatomic molecule... 

Here R, ru...,rN are respectively the-C.M. position of the diatomic molecule and the position vectors of the fluid atoms, P, PU...,PN are the conjugate momenta, n and Q are the momentum and coordinate characterizing the oscillatory degree of freedom, r is the vector representing the orientation and length of the bond in the diatomic molecule, and L is the angular momentum of the molecule about the C.M. The interaction Hamiltonian has already been linearized in the oscillatory coordinate Q in the last equation. 

First suppose that a spherical surface of unit radius is drawn and the center of this sphere is taken as the origin of a spherical polar coordinate system. Suppose further that p(0), the initial orientation of a diatomic molecule, is represented by the unit vector, k, along the positive Z axis of... 

To understand why reaction rates depend on temperature, we need a picture of how reactions take place. According to the collision theory model, a bimolecu-lar reaction occurs when two properly oriented reactant molecules come together in a sufficiently energetic collision. To be specific, let s consider one of the simplest possible reactions, the reaction of an atom A with a diatomic molecule BC to give a diatomic molecule AB and an atom C ... 

Equation (18) may be derived in a manner similar to that described above for vibration-vibration energy transfer, with the difference that the product of the squared matrix elements of the two vibrations concerned, appears in the expression for transition probability. Since both molecules require suitable orientation for vibrational exchange, for two diatomic molecules the steric factor is usually taken to be ( )2. If AE - 0 (exact resonance), eqn. (18) is no longer valid and the following equation may be employed... 

From the above equations, it is seen that the amplitude factor for a homonuclear diatomic molecule is (2m)-1, which for H2 in atomic mass units would be 0.5. For H20, the amplitude of vibration in the bending mode would be very nearly the same as that of a hypothetical H2 molecule with the same frequency. The amplitude factor is therefore again 0.5. Considering v3 in CH4, in which opposite pairs of H atoms are pinched together, the potential energy is shared between the two pairs. Considering Hooke s law, clearly the squared amplitude is lower by a factor of 2 compared to the hypothetical H2 of the same frequency, and therefore the amplitude factor of CH4(v3) is 0.25. (Note that for CH4, H20 and H2, the product of the amplitude factor and orientation factor is constant.) Stretton33 has compiled a useful list of amplitude factors for substituted methanes. 

It is conventional to label the internuclear axis in a diatomic molecule as z. Thus the three 2p(F) orbitals can be labelled 2p, 2py and 2p2 2px and 2py have their lobes directed perpendicular to the internuclear axis, and have nodal surfaces containing that axis, while 2pr clearly overlaps in o fashion with ls(H). (The reader may wonder whether this orientation of 2p, 2py and 2pz is obligatory, or whether it is chosen for convenience. For a spherically-symmetric atom, there are no constraints in choosing a set of three Cartesian axes. Any set of orthogonal p orbitals can be transformed into another equally acceptable set, by a simple rotation which does not change the electron density distribution of the atom. The overlap integral between a hydrogen Is orbital and the set of three 2p(F) orbitals is the... 

The discovery of confinement resonances in the photoelectron angular distribution parameters from encaged atoms may shed light [36] on the origin of anomalously high values of the nondipole asymmetry parameters observed in diatomic molecules [62]. Following [36], consider photoionization of an inner subshell of the atom A in a diatomic molecule AB in the gas phase, i.e., with random orientation of the molecular axis relative to the polarization vector of the radiation. The atom B remains neutral in this process and is arbitrarily located on the sphere with its center at the nucleus of the atom A with radius equal to the interatomic distance in this molecule. To the lowest order, the effect of the atom B on the photoionization parameters can be approximated by the introduction of a spherically symmetric potential that represents the atom B smeared over... 

The coupling to the electromagnetic field depends on the orientation of the molecule, and this will be reflected in the spatial distribution of the products. Consider, as an example, a diatomic molecule with P12 parallel to the molecular axis. For a spherically symmetric initial state, the angular distribution is given by (cos )2. That... 

This is the correct expression for the rotational partition function of a heteronuclear diatomic molecule. For a homonuclear diatomic molecule, however, it must be taken into account that the total wave function must be either symmetric or antisymmetric under the interchange of the two identical nuclei symmetric if the nuclei have integral spins or antisymmetric if they have half-integral spins. The effect on Qrot is that it should be replaced by Qrot/u, where a is a symmetry number that represents the number of indistinguishable orientations that the molecule can have (i.e., the number of ways the molecule can be rotated into itself ). Thus, Qrot in Eq. (A.19) should be replaced by Qrot/u, where a = 1 for a heteronuclear diatomic molecule and a = 2... 

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