Homonuclear diatomic electronic

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. 

Some heteronuclear diatomic molecules, such as nitric oxide (NO), carbon monoxide (CO) and the short-lived CN molecule, contain atoms which are sufficiently similar that the MOs resemble quite closely those of homonuclear diatomics. In nitric oxide the 15 electrons can be fed into MOs, in the order relevant to O2 and F2, to give the ground configuration... 

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. 

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. 

Despite its very simple electronic configuration (Is ) hydrogen can, paradoxically, exist in over 50 different forms most of which have been well characterized. This multiplicity of forms arises firstly from the existence of atomic, molecular and ionized species in the gas phase H, H2, H+, H , H2" ", H3+. .., H11 + secondly, from the existence of three isotopes, jH, jH(D) and jH(T), and correspondingly of D, D2, HD, DT, etc. and, finally, from the existence of nuclear spin isomers for the homonuclear diatomic species. 

Ishiguro, E., Kayama, K., Kotani, M., and Mizuno, Y., J. Phys. Soc. Japan 12, 1355, Electronic structure of simple homonuclear diatomic molecules. II. Lithium molecule. ... 

In Section 2.12, we saw that a polar covalent bond in which electrons are not evenly distributed has a nonzero dipole moment. A polar molecule is a molecule with a nonzero dipole moment. All diatomic molecules are polar if their bonds are polar. An HC1 molecule, with its polar covalent bond (8+H—Clfi ), is a polar molecule. Its dipole moment of 1.1 D is typical of polar diatomic molecules (Table 3.1). All diatomic molecules that are composed of atoms of different elements are at least slightly polar. A nonpolar molecule is a molecule that has no electric dipole moment. All homonuclear diatomic molecules, diatomic molecules containing atoms of only one element, such as 02, N2, and Cl2, are nonpolar, because their bonds are nonpolar. 

In the molecular orbital description of homonuclear diatomic molecules, we first build all possible molecular orbitals from the available valence-shell atomic orbitals. Then we accommodate the valence electrons in molecular orbitals by using the same procedure we used in the building-up principle for atoms (Section 1.13). That is,... 

FIGURE 3.31 Atypical molecular orbital energy-level diagram for the homonuclear diatomic molecules Li2 through N2. Each box represents one molecular orbital and can accommodate up to two electrons. 

HOWTO DETERMINE THE ELECTRON CONFIGURATION AND BOND ORDER OF A HOMONUCLEAR DIATOMIC SPECIES... 

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. 

In Eq. (44), gei(T ) is the ratio of transition state and reactant electronic partition functions [31] and the rotational degeneracy factor = (2ji + l)(2/2 + 1) for heteronuclear diatomics, and will also include nuclear spin considerations in the case of homonuclear diatomics. 

To describe the band structure of metals, we use the approach employed above to describe the bonding in molecules. First, we consider a chain of two atoms. The result is the same as that obtained for a homonuclear diatomic molecule we find two energy levels, the lower one bonding and the upper one antibonding. Upon adding additional atoms, we obtain an additional energy level per added electron, until a continuous band arises (Fig. 6.9). To describe the electron band of a metal in a... 

For a homonuclear diatomic molecule such as Cl2 the interatomic surface is clearly a plane passing through the midpoint between the two nuclei—in other words, the point of minimum density. The plane cuts the surface of the electron density relief map in a line that follows the two valleys leading up to the saddle at the midpoint of the ridge between the two peaks of density at the nuclei. This is a line of steepest ascent in the density on the two-dimensional contour map for the Cl2 molecule (Fig. 9). 

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. 

Atoms do not all have the same ability to attract electrons. When two different types of atoms form a covalent bond by sharing a pair of electrons, the shared pair of electrons will spend more time in the vicinity of the atom that has the greater ability to attract them. In other words, the electron pair is shared, but it is not shared equally. The ability of an atom in a molecule to attract electrons to it is expressed as the electronegativity of the atom. Earlier, for a homonuclear diatomic molecule we wrote the combination of two atomic wave functions as... 

A homonuclear diatomic molecule contains a nonpolar bond, since the electron pair between the two atoms is shared equally. Cl2 is an example of a homonuclear diatomic molecule. 

Textbook discussions of homonuclear diatomic molecules are commonly based on the familiar type of MO energy diagram shown in Fig. 3.28, which underlies the standard MO Aufbau procedure for constructing many-electron molecular configurations (which is analogous to the well-known procedure for atoms). Figure 3.28 purports to represent the energies and compositions of available MOs, which are... 

Figure 3.28 A schematic energy diagram for valence MOs of homonuclear diatomic molecules (following Mulliken). The order of 02P and 7t2p MOs should be reversed for 14 or fewer electrons.

MO wave functions in the above form give equal importance to covalent and ionic structures, which is unrealistic in homonuclear diatomic molecules like H2. This should be contrasted with (/>Vb> which in its simple form neglects the ionic contributions. Both and i//MO are inadequate in their simplest forms while in the VB theory the electron correlation is overemphasized, simple MO theory totally neglects it giving equal importance to covalent and ionic structures. Therefore neither of them is able to predict binding energies closer to experiment. The MO theory could be... 

There exists no uniformity as regards the relation between localized orbitals and canonical orbitals. For example, if one considers an atom with two electrons in a (Is) atomic orbital and two electrons in a (2s) atomic orbital, then one finds that the localized atomic orbitals are rather close to the canonical atomic orbitals, which indicates that the canonical orbitals themselves are already highly, though not maximally, localized.18) (In this case, localization essentially diminishes the (Is) character of the (2s) orbital.) The opposite situation is found, on the other hand, if one considers the two inner shells in a homonuclear diatomic molecule. Here, the canonical orbitals are the molecular orbitals (lo ) and (1 ou), i.e. the bonding and the antibonding combinations of the (Is) orbitals from the two atoms, which are completely delocalized. In contrast, the localization procedure yields two localized orbitals which are essentially the inner shell orbital on the first atom and that on the second atom.19 It is thus apparent that the canonical orbitals may be identical with the localized orbitals, that they may be close to the localized orbitals, that they may be identical with the completely delocalized orbitals, or that they may be intermediate in character. 

Usually the electronic structure of diatomic molecules is discussed in terms of the canonical molecular orbitals. In the case of homonuclear diatomics formed from atoms of the second period, these are the symmetry orbitals 1 og, 1 ou, 2ag,... 

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