Diatomic molecules perturbations

Lefebvre-Brion H and Field R W 1986 Perturbations in the Spectra of Diatomic Molecules (Orlando Academic)... 

In rare gas crystals [77] and liquids [78], diatomic molecule vibrational and vibronic relaxation have been studied. In crystals, VER occurs by multiphonon emission. Everything else held constant, the VER rate should decrease exponentially with the number of emitted phonons (exponential gap law) [79, 80] The number of emitted phonons scales as, and should be close to, the ratio O/mQ, where is the Debye frequency. A possible complication is the perturbation of the local phonon density of states by the diatomic molecule guest [77]. 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

The energy of the diatomic molecule, as given by Bq. (79) does not take into account foe anharmonicity of the vibration. Tte.qjE fect of fop cubic and qoartic terms in Eq. (73) can be evaluated by application of the theory of perturbation (see Chapter 12). 

The interatomic potential function for the diatomic molecule was described in Section 6 5. In the Taylpr-series development of this function (6-72)3 cubic and higher terms were neglected in the harmonic approximation. It is now of interest to evaluate the importance of these so-called anharmonic terms with the aid of the perturbation theory outlined above. If cubic and quartic... 

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

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). 

Btiilding on atomic studies using even-tempered basis sets, universal basis sets and systematic sequences of even-tempered basis sets, recent work has shown that molecular basis sets can be systematically developed until the error associated with basis set truncation is less that some required tolerance. The approach has been applied first to diatomic molecules within the Hartree-Fock formalism[12] [13] [14] [15] [16] [17] where finite difference[18] [19] [20] [21] and finite element[22] [23] [24] [25] calculations provide benchmarks against which the results of finite basis set studies can be measured and then to polyatomic molecules and in calculations which take account of electron correlation effects by means of second order perturbation theory. The basis sets employed in these calculations are even-tempered and distributed, that is they contain functions centred not only on the atomic nuclei but also on the midpoints of the line segments between these nuclei and at other points. Functions centred on the bond centres were found to be very effective in approaching the Hartree-Fock limit but somewhat less effective in recovering correlation effects. 

The selection rule (4.138) differs from previously discussed selection rules in that it holds well for nonradiative transitions, as well as for radiative transitions. In deriving (4.138), we made no reference to the operator d, beyond the statement that it did not involve the nuclear spin coordinates. For any time-dependent perturbation that does not involve nuclear spin, the selection rule (4.138) will hold. Thus molecular collisions will not cause nonradiative transitions between symmetric and antisymmetric rotational levels of a homonuclear diatomic molecule. If we somehow start with all the molecules in symmetric levels, the collisions will not populate the antisymmetric levels. 

The higher-order perturbations also add small corrections to the coefficients in (4.67). For example, the term hve(v+1/2) becomes hve v+ /2)[ + (B /4vj)A], where A is a complicated constant involving ke, Re, U " Re), U,v(Re), Uv(Re), and Uv,(Re), and where ve and Be are defined by (4.68). The coefficients of the third, fourth, fifth, and sixth terms on the right of (4.67) are also affected. Because these corrections involve B /v, they are negligible except for light molecules (hydrides), and are usually omitted. However, they must be included in accurate work on hydrides. The accurate expression for the total energy of a diatomic molecule then has the form... 

F ERMI RESONANCE. In polyatomic molecules. Hvo vibrational levels belonging to different vibrations lor combinations of vibrations) may happen lo have nearly die same energy, and therefore be accidentally degenerate. As was recognized hy Fermi in the case of CO such a "resonance" leads to a perturbation of the energy levels that is very similar to the vibrational perturbations of diatomic molecules. 

For polar molecule perturbers the Bom electron scattering amplitude is quite accurate and Eq. (11.9) is immediately useful. As an example, the squared Born scattering amplitude for / — / - 1 rotational deexcitation of a polar diatomic molecule is given by3... 

If one adopts the correct point of view that the complete wave function of any state of a diatomic molecule has contributions from all other states of that molecule, one can understand that all degrees of perturbation and hence probabilities of crossover may be met in practice. If the perturbation by the repulsive or dissociating state is very small, the mean life of the excited molecule before dissociation may be sufficiently long to permit the absorption spectrum to be truly discrete. Dissociation may nevertheless occur before the mean radiative lifetime has been reached so that fluorescence will not be observed. Predissociation spectra may therefore show all gradations from continua through those with remnants of vibrational transitions to discrete spectra difficult to distinguish from those with no predissociation. In a certain sense photochemical data may contribute markedly to the interpretation of spectra. 

Consider a reaction between two molecules A and B. For simplicity, we assume that each molecule has only one MO 0FA° of energy EA° and T," of energy EB°, respectively). During the reaction, the reagents evolve to produce the supermolecule (A B). As we saw in the previous section, the MOs of (A B) can be calculated by a perturbation approach which is entirely analogous to the Hiickel treatment of a diatomic molecule. In fact, we only need to take the MOs of the diatomic and replace ... 

Moreover, a partial proof was provided for the HSAB principle. Consider the formation of a diatomic molecule AB. Upon neglect of the external potential perturbation, the chemical potential change for the atoms A and B will be ... 

The nondissociative adsorption of diatomic molecules (CO, N2, NO, and H2) on the surface ions of oxides and halides is accompanied by distinct perturbations of the vibrational spectra. This statement is documented in detail for CO in this review. At this stage of the discussion, it is sufficient to mention the following points. 

First, diatomic molecules are usually adsorbed on cations (some exceptions to this empirical rule are mentioned in the case studies). The type of interaction and the resulting vibrational perturbation (purely electrostatic or with some orbital overlap contribution) depend on the charge carried by the cation and by the anions in nearest-neighbor positions and on the electronic structure of the cation (with or without d electrons). As a typical example of the effect of a purely electrostatic perturbation on the stretching mode of a diatomic molecule, we mention the classic case of CO adsorbed on Na+ exposed on NaCl (100) (Fig. 2), in which it is clearly shown that the frequency of adsorbed CO is distinctly blue shifted with respect to that of CO gas (2143 cm-1) (48). More general considerations concerning the role of the electrostatic field in perturbing the adsorbed molecules are discussed elsewhere (12-15, 21-23). 

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