Diatomic molecules Born-Oppenheimer approximation

Perhaps the first evidence for the breakdown of the Born-Oppenheimer approximation for adsorbates at metal surfaces arose from the study of infrared reflection-absorption line-widths of adsorbates on metals, a topic that has been reviewed by Hoffmann.17 In the simplest case, one considers the mechanism of vibrational relaxation operative for a diatomic molecule that has absorbed an infrared photon exciting it to its first vibrationally-excited state. Although the interpretation of spectral line-broadening experiments is always fraught with problems associated with distinguishing... 

Obviously, there is an isotope effect on the vibrational frequency v . For het-eroatomic molecules (e.g. HC1 and DC1), infrared spectroscopy permits the experimental observation of the molecular frequencies for two isotopomers. What does one learn from the experimental observation of the diatomic molecule frequencies of HC1 and DC1 To the extent that the theoretical consequences of the Born-Oppenheimer Approximation have been correctly developed here, one can deduce the diatomic molecule force constant f from either observation and the force constant will be independent of whether HC1 or DC1 was employed and, for that matter, which isotope of chlorine corresponded to the measurement as long as the masses of the relevant isotopes are known. Thus, from the point of view of isotope effects, the study of vibrational frequencies of isotopic isomers of diatomic molecules is a study involving the confirmation of the Born-Oppenheimer Approximation. 

At this stage we are at the very beginning of development, implementation, and application of methods for quantum-mechanical calculations of molecular systems without assuming the Born-Oppenheimer approximation. So far we have done several calculations of ground and excited states of small diatomic molecules, extending them beyond two-electron systems and some preliminary calculations on triatomic systems. In the non-BO works, we have used three different correlated Gaussian basis sets. The simplest one without r,y premultipliers (4)j = exp[—r (A t (8> Is) "]) was used in atomic calculations the basis with premultipliers in the form of powers of rj exp[—r (Aj (8> /sjr])... 

The methods described above are all based on the Born-Oppenheimer approximation. Therefore, they can be used to calculate polarizabilities of diatomic molecules for a given internuclear distance R. However, if one is interested in values of the polarizability tensors, and C", for a particular vibrational state /i )), one has to average the polarizability radial functions a(R) and C(R) with the vibrational wavefunction i.e., one has to... 

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... 

In the Born-Oppenheimer approximation, the molecular wave function is the product of electronic and nuclear wave functions see (4.90). We now examine the behavior of if with respect to inversion. We must, however, exercise some care. In finding the nuclear wave functions fa we have used a set of axes fixed in space (except for translation with the molecule). However, in dealing with if el (Sections 1.19 and 1.20) we defined the electronic coordinates with respect to a set of axes fixed in the molecule, with the z axis being the internuclear axis. To find the effect on if of inversion of all nuclear and electronic coordinates, we must use the set of space-fixed axes for both fa and if el. We shall call the space-fixed axes X, Y, and Z, and the molecule-fixed axes x, y, and z. The nuclear wave function of a diatomic molecule has the (approximate) form (4.28) for 2 electronic states, where q=R-Re, and where the angles are defined with respect to space-fixed axes. When we replace each nuclear coordinate in fa by its negative, the internuclear distance R is unaffected, so that the vibrational wave function has even parity. The parity of the spherical harmonic Yj1 is even or odd according to whether J is even or odd (Section 1.17). Thus the parity eigenvalue of fa is (- Yf. 

The p and He2+ are thus regarded as two atomic centers in a diatomic molecule. Because of the dual character as an exotic atom and an exotic molecule Antiprotonic Helium is often called antiprotonic helium atom-molecule, or for short, atomcule. Since the Is electron motion, coupled to a large-(n, l) p orbital, is faster by a factor of 40 than the p motion, the three-body system pHe+ is solved by using the Born-Oppenheimer approximation, as fully discussed by Shimamura [6]. 

In chapter 2 we discussed at length the separation of nuclear and electronic coordinates in the solution of the Schrodinger equation. We described the Born-Oppenheimer approximation which allows us to solve the Schrodinger equation for the motion of the electrons in the electrostatic field produced by fixed nuclear charges. There are certain situations, particularly with polyatomic molecules, when the separation of nuclear and electronic motions cannot be made satisfactorily, but with most diatomic molecules the Born-Oppenheimer separation is acceptable. The discussion of molecular electronic wave functions presented in this chapter is therefore based upon the Born-Oppenheimer approximation. 

The Born-Oppenheimer approximation states the vibrational and rotational energies of a molecule can be separated and the individual terms added. The overall energy E can thus be considered as a function of the vibrational quantum number v (taking values 0, 1, 2,. ..) and the rotational quantum number J (which independently takes values 0, 1, 2,...). For a diatomic molecule it can be approximated by the function... 

The potential constants k, a, and h are independent of isotopic substitution (within the framework of the Born-Oppenheimer approximation). The isotopic mass dependence is completely situated in the reduced mass /X. It is physically reasonable to assume for the diatomic molecule-oscillator that a and h are sufficiently small so that V can be regarded as a perturbation to Equation 3 and that it is necessary to consider no terms in the perturbation energy higher than the second power in a and higher than the first power in b. The term only yields a non-vanishing contribution in second order while the term yields a first-order contribution to the energy. One obtains by standard methods... 

When extending the molecular orbital concept developed for the monoelec-tronic species H2 to polyelectronic diatomic molecules, we start by acknowledging the role of two fundamental approximations (a) one associated with the existence of two nuclei as attractive centres, namely the Born-Oppenheimer approximation, as already encountered in H2" and (b) the other related to the concept of the orbital when two or more electrons are present, that is the neglect of the electron coulomb correlation, as already discussed on going from mono- to polyelectronic atoms. Within the orbital approach, an additional feature when comparing to H2" is the exchange energy directly associated with the Pauli principle. 

Consider a diatomic molecule such as H Cl. Within the Born-Oppenheimer approximation, we focus attention on the electronic wavefunction and calculate enough data points to give a potential energy curve. Such a curve shows the variation of the electronic energy with intemuclear separation. The nuclei vibrate in this potential. 

For homonuclear molecules, the g or u symmetry is almost always conserved. Only external electric fields, hyperfine effects (Pique, et al., 1984), and collisions can induce perturbations between g and u states. See Reinhold, et al., (1998) who discuss how several terms that are neglected in the Born-Oppenheimer approximation can give rise to interactions between g and u states in hetero-isotopomers, as in the HD molecule. An additional symmetry will be discussed in Section 3.2.2 parity or, more usefully, the e and / symmetry character of the rotational levels remains well defined for both hetero- and homonuclear diatomic molecules. The matrix elements of Table 3.2 describe direct interactions between basis states. Indirect interactions can also occur and are discussed in Sections 4.2, 4.4.2 and 4.5.1. Even for indirect interactions the A J = 0 and e / perturbation selection rules remain valid (see Section 3.2.2). 

How does the Born-Oppenheimer approximation allow the discussion of a diatomic molecule s energy as a function of only the internuclear separation (and not the coordinates of electrons) ... 

Bunker, P.R., Moss, R.E. Breakdown of Born-Oppenheimer approximation—effective vibration-rotation Hamiltonian for a diatomic molecule. Mol. Phys. 1977,33,417-24 ... 

Within the Born-Oppenheimer approximation, the ground-state energy E = E[p], where the functional is unknown, can be reduced formally, for a homonuclear diatomic molecule at internuclear separation R, to... 

In the adiabatic and the Born-Oppenheimer approximations, the total wave function is taken as a product I = i/k(r R)fk(R) of the function fk(R), which describes the motion of the nuclei (vibrations and rotations) and the function k(f, R) that pertains to the motion of elections (and depmds parametrically on the configuration of the nuclei here, we give the formulas for a diatomic molecule). This aj ioximation relies on the fact that the nuclei are thousands of times heavier than the elections. 

In a diatomic molecule isotope effects appear in the vibrational, rotational, and electronic spectra. Isotope effects on vibrational spectra can be interpreted within the framework of the Born-Oppenheimer approximation, which is the cornerstone of most theories dealing with the effects of isotopic substitution on molecular properties. In this approximation, the potential energy surface for the vibrational-rotational motions of a molecular system is taken as being independent of the masses of the nuclei. Thus, the nuclei of different masses move on the same potential surface because to good approximation the electronic structure is independent of isotopic substitution. Then, the harmonic vibrational frequencies of the two isotopic variants of a diatomic molecule (prime denotes the lighter molecule) can be given as... 

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