Electronic states homonuclear

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. 

H3 (and its isotopomers) and the alkali metal triiners (denoted generally for the homonuclears by X3, where X is an atom) are typical Jahn-Teller systems where the two lowest adiabatic potential energy surfaces conically intersect. Since such manifolds of electronic states have recently been discussed [60] in some detail, we review in this section only the diabatic representation of such surfaces and their major topographical details. The relevant 2x2 diabatic potential matrix W assumes the fomi... 

For atoms, electronic states may be classified and selection rules specified entirely by use of the quantum numbers L, S and J. In diatomic molecules the quantum numbers A, S and Q are not quite sufficient. We must also use one (for heteronuclear) or two (for homonuclear) symmetry properties of the electronic wave function ij/. ... 

In Figure 7.25 are shown stacks of rotational levels associated with two electronic states between which a transition is allowed by the -F -F and, if it is a homonuclear diatomic, g u selection rules of Equations (7.70) and (7.71). The sets of levels would be similar if both were states or if the upper state were g and the lower state u The rotational term values for any X state are given by the expression encountered first in Equation (5.23), namely... 

It is important to realize that electronic spectroscopy provides the fifth method, for heteronuclear diatomic molecules, of obtaining the intemuclear distance in the ground electronic state. The other four arise through the techniques of rotational spectroscopy (microwave, millimetre wave or far-infrared, and Raman) and vibration-rotation spectroscopy (infrared and Raman). In homonuclear diatomics, only the Raman techniques may be used. However, if the molecule is short-lived, as is the case, for example, with CuH and C2, electronic spectroscopy, because of its high sensitivity, is often the only means of determining the ground state intemuclear distance. 

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. 

Within the approximation that the valence electronic states can be described adequately as combinations of the above valence CSFs, the three JE, JE, and CSFs must be combined to form the three lowest energy valence electronic states of E symmetry. For the homonuclear case, the E CSF does not couple with the other two because it is of ungerade symmetry, while the other CSFs JE and1E have gerade symmetry and do combine. 

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,... 

Interchanging the nuclear coordinates does not affect R, but it does affect the electronic spatial coordinates since they are defined with respect to the molecule-fixed xyz axes, which are rigidly attached to the nuclei. To find the effect on el of interchanging the nuclear coordinates, we will first invert the space-fixed coordinates of the nuclei and the electrons, and then carry out a second inversion of the space-fixed electronic coordinates only the net effect will be the interchange of the space-fixed coordinates of the two nuclei. We found in the last section that inversion of the space-fixed coordinates of all particles left //e, unchanged for 2+,n+,... electronic states, but multiplied it by —1 for 2, II ,... states. Consider now the effect of the second step, reinversion of the electronic space-fixed coordinates. Since the nuclei are unaffected by this step, the molecule-fixed axes remain fixed for this inversion, so that inversion of the space-fixed coordinates of the electrons also inverts their molecule-fixed coordinates. But we noted in Section 1.19 that the electronic wave functions of homonuclear diatomics could be classified as g or m, according to whether inversion of molecule-fixed electronic coordinates multiplies ptl by + 1 or -1. We conclude that for 2+,2,7,11, IV,... electronic states, i//el is symmetric with respect to interchange of nuclear coordinates, whereas for... 

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.)... 

Consider first chemical bonding, i.e., the formation of molecules from atoms. In the formation of a homonuclear molecule, the two electronic states of the two identical atoms at large interatomic distances R have the same energy, and hence... 

Good examples are the core hole excited states of homonuclear molecules. When one electron is removed from a core orbital, the original Dooh symmetry is lowered to C v The D h group can be decomposed into two CooV components related by a C, or Cs operation, so it is fair to consider that the core-hole excited states are described by resonance between the two structures. The adiabatic subsystems have, by definition, zero overlap in the real space. Their interaction is defined only in complex space through the explicit overlap between the many-electron states. 

In this section we have concentrated on calculations for H-T only, which have particular relevance to the fine and hyperfine constants determined from Jefferts experiments. Many other papers deal with calculations of the vibration-rotation level energies, for which there is much less experimental data. There are also many papers dealing with the heteronuclear molecule, HD+, which is really a special case because the Bom Oppenheimer approximation collapses, particularly for the highest vibrational levels of the ground electronic state. Even the homonuclear species H and D exhibit some fascinating and unusual effects in their near-dissociation vibration rotation levels. Finally we note that in order to match the accuracy of the experimental measurements for all the hydrogen molecular ion isotopomers, it is necessary to include radiative and relativistic effects. 

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 ... 

Fig. AIII.2. Potential energy curves, electronic orbitals, and vibrational levels are schematically depicted for the electronic ground state and an excited electronic state of a homonuclear diatomic molecule such as Hg. The molecular c-axis is assumed to be perpendicular to the...

As was shown in the preceding discussion (see also Sections VIII 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 electronic states antisymmetric for odd J values in and electronic states symmetric for odd J values in S7 and electronic... 

---