Diatomic molecules terms

A molecular electronic configuration gives rise to one or more electronic terms. Each diatomic-molecule term designation has the form where S is the total... 

In the following, we shall demonstrate techniques for calculating the electronic potential energy terms up to the second order. For simplicity, we shall study the case of H2 molecule, the simplest multi-electron diatomic molecule. 

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. 

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronic-vibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes ... 

Since depends on nuclear coordinates, because of the term, so do and but, in the Bom-Oppenheimer approximation proposed in 1927, it is assumed that vibrating nuclei move so slowly compared with electrons that J/ and involve the nuclear coordinates as parameters only. The result for a diatomic molecule is that a curve (such as that in Figure 1.13, p. 24) of potential energy against intemuclear distance r (or the displacement from equilibrium) can be drawn for a particular electronic state in which and are constant. 

In an approximation which is analogous to that which we have used for a diatomic molecule, each of the vibrations of a polyatomic molecule can be regarded as harmonic. Quantum mechanical treatment in the harmonic oscillator approximation shows that the vibrational term values G(v ) associated with each normal vibration i, all taken to be nondegenerate, are given by... 

As for diatomic molecules, there are stacks of rotational energy levels associated with all vibrational levels of a polyatomic molecule. The resulting term values S are given by the sum of the rotational and vibrational term values... 

In the case of atoms (Section 7.1) a sufficient number of quantum numbers is available for us to be able to express electronic selection rules entirely in terms of these quantum numbers. For diatomic molecules (Section 7.2.3) we require, in addition to the quantum numbers available, one or, for homonuclear diatomics, two symmetry properties (-F, — and g, u) of the electronic wave function to obtain selection rules. 

Our sun is, of course, a star. It is a relatively cool star and, as such, contains a number of diatomic molecules (see Figure 25-3). There are many stars, however, with still lower surface temperatures and these contain chemical species whose presence can be understood in terms of the temperatures and the usual chemical equilibrium principles. For example, as the star temperature drops, the spectral lines attributed to CN and CH become more prominent. At lower temperatures, TiO becomes an important species along with the hydrides MgH, SiH, and A1H, and oxides ZrO, ScO, YO, CrO, AlO, and BO. 

Exercise 1.2. Consider a diatomic molecule that has two orbitals (A and B on the atoms A and B, respectively) and two electrons in the MOs formed by these orbitals. What is the Huckel electronic energy of this system in terms of the parameters aA, aB, and PAB1... 

Let us suppose that the system of interest does not possess a dipole moment as in the case of a homonuclear diatomic molecule. In this case, the leading term in the electric field-molecule interaction involves the polarizability, a, and the Hamiltonian is of the form ... 

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... 

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