Homonuclear diatomic molecules electronic states

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

The molecular orbital energy-level diagrams of heteronuclear diatomic molecules are much harder to predict qualitatitvely and we have to calculate each one explicitly because the atomic orbitals contribute differently to each one. Figure 3.35 shows the calculated scheme typically found for CO and NO. We can use this diagram to state the electron configuration by using the same procedure as for homonuclear diatomic molecules. 

Next, we address some simple cases, begining with homonuclear diatomic molecules in 1S electronic states. The rotational wave functions are in this case the well-known spherical harmonics for even J values, yr(R) is symmetric under permutation of the identical nuclei for odd J values, y,.(R) is antisymmetric under the same permutation. A similar statement applies for any D.yjh type molecule. 

Suppose the spin of each nucleus of the homonuclear diatomic molecule is zero. The nuclei are then bosons and p must be symmetric with respect to nuclear exchange. For example, the C12 nucleus has 7=0, and we will consider the rotational levels of the ground electronic state of C2, which is a 2 state (Levine, Section 13.6). The rotational levels with 7 = 0,2,4,... have ipf symmetric (Fig. 4.14) and require a symmetric pns. With 7 = 0,... 

Recall that homonuclear diatomic molecules have no vibration-rotation or pure-rotation spectra due to the vanishing of the permanent electric dipole moment. For electronic transitions, the transition-moment integral (7.4) does not involve the dipole moment d hence electric-dipole electronic transitions are allowed for homonuclear diatomic molecules, subject to the above selection rules, of course. [The electric dipole moment d is given by (1.289), and should be distinguished from the electric dipole-moment operator d, which is given by (1.286).] Analysis of the vibrational and rotational structure of an electronic transition in a homonuclear diatomic molecule allows the determination of the vibrational and rotational constants of the electronic states involved, which is information that cannot be provided by IR or microwave spectroscopy. (Raman spectroscopy can also furnish information on the constants of the ground electronic state of a homonuclear diatomic molecule.)... 

Interatomic distance is calculated by mathematical modelling of the electron exchange that constitutes a covalent bond. Such a calculation was first performed by Heitler and London using Is atomic wave functions to simulate the bonding in H2. To model the more general case of homonuclear diatomic molecules the interacting atoms in their valence states are described by monopositive atomic cores and two valence electrons with constant wave functions (3.36). 

Table 6.3. Ground state electron configurations and states for homonuclear diatomic molecules in the first row of the periodic table...

D N2) was determined as 9 79, 7 90, 7 42, 6 23, or 5-76 eV according to the assumed states of excitation of the nitrogen ion and the nitrogen atom produced. Spectroscopically obtained values for Z)(N2) are 9 76 or 7 38 eV, depending on the assumptions made. The retarding potential and appearance potential measurement alone is satisfactory for the interpretation of electron impact processes in homonuclear diatomic molecules, where there can be no doubt about the mass number of the ions. Possible confusion for heteronuclear diatomic molecules is not likely to be very great, but the method by itself is clearly inapplicable to dissociative ionization processes in polyatomic molecules, where the number of possible products is large. 

The electronic states of homonuclear diatomic molecules may now be built up by feeding the electrons into the various orbitals, provided that the relative order of molecular orbital energies is known. This has been determined by Mullikan from molecular spectra data and is generally found to be ... 

Hoffmann-Ostenhof and Morgan (1981) were able to prove that the ground-state charge distribution of a one-electron homonuclear diatomic molecule can exhibit maxima in p only at the positions of the nuclei. In this proof an important inequality is used (Hoffmann-Ostenhof and Hoffman-Ostenhof 1977),... 

Curves for the negative-ion states of H2 and L are chosen to illustrate the procedures for the homonuclear diatomic molecules. Curves for benzene and naphthalene are examples of excited states for larger molecular negative ions. These illustrate the relationship between gas phase acidities and thermal electron attachment reactions. Such correlation procedures can be applied to systematic predictions for many different problems. 

The electron affinities of the main group homonuclear diatomic anions have been measured by PES. A few experimental values for the transition metal dimers are also available. The electron affinities of all the 3d homonuclear diatomic molecules have been calculated using density functional methods [1-4], Only the AEa of I2, 2.524 eV C2, 3.27 Si2, 2.2o S2, 1.67 F2, 3.0g Cl2, 2.4s Br2, 2.5, and 02, 1.07 have been measured by more than one method [1-3]. CURES-EC calculations confirm these to within 0.1 eV. Positive excited states Ea have been measured for 02, C2, and I2 and are inferred for other X2 [5-8]. Just as in the case of the atomic Ea, the trends in the Periodic Table can support the assignments of AEa for the other elements. 

Fig. 5 This figure shows ground-state energies divided by Z2 for the 2-electron isoelec-tronic series for homonuclear diatomic molecules, Z being the nuclear charges. The energies in Hartrees are shown as functions of the interatomic distance R, measured in Bohrs. The dotted curves are electronic energies alone, while the solid curves also include intemuclear repulsion. For both the solid and dotted curves, the lowest curve corresponds to the smallest value of Z. As in Fig. 4, a very restricted basis set was used for the calculation...

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