Homonuclear diatomics

The state symmetries of heteronuclear diatomic molecules are simpler than those of homonuclear diatomics the g and u states of Doo/i simply coalesce in Coot . Table 3.2 summarizes the correlations of the irreps of Doo/i with those of its subgroup Coov and between the latter and those of C2v  

An X2 molecule has been set up in Fig. 3.9, where - as has been our wont -the X and y directions are fixed with a hypothetical quadrupolar field, so D2/1 labels can be used. 

As the molecule rotates, the displacements are assumed to remain perpendicular to the internuclear axis. Inertia does in fact tend to increase the XX distance, leading to centrifugal distortion and thus to interaction between rotation and vibration this effect can be ignored in the present context. 

Chapter 3. Diatomic Molecules and their Molecular Orbitals 

A no less important point to note is that the six symmetry coordinates comprise a complete set, in terms of which any arbitrary molecular motion can be described. A composite motion like the one in Fig. 3.9 can be constructed by a superposition of symmetry coordinates with suitably chosen phase and amplitude, and is therefore assigned to a reducible representation, the direct sum of its component irreps. It is easy to see that the motion of a single atom also belongs to a reducible representation Displacement of the left-hand X atom parallel to x is clearly a superposition of Tx and the negative phase of Ry, so it belongs to bsu b2g, whereas that of the right-hand atom along z, composed of Tz and transforms as ag 

All the molecules considered so far have had H as one of the participants in the bonding scheme. In this section we will start to consider the MOs produced for interactions between heavier atoms, beginning with homonuclear diatomics, A2, of second-row elements. General MO diagrams for these diatomics will be constructed and then used to discuss the relative stability of those that are observed experimentally N2, O2 and F2 and those that are not found under normal circumstances, Li2, Be2, B2 and C2. In common with H2, these molecules all belong to the Dooh point group and so the 2s valence orbitals will be linked together as 7g+ and SALCs  

The orbitals will be numbered as we construct the MO diagram following the convention that (T-symmetry orbitals are numbered without regard to the g or u labels. This time we will also need to cope with the p-orbitals on the two atoms in the diatomic. We have dealt with the symmetry of p and d-orbitals on the central Au atom in the coh complex Au( CN)2 in Section 5.9. There, it was noted that the pj orbital (aligned with the molecular axis) belongs to a separate irreducible representation to p and p,. For the diatomic molecules we must now take SALCs of p-orbitals on the two equivalent atoms. This means that it is not possible to assign symmetry labels to the individual AOs, only to their linear combinations. 

The reducible representation for the pj orbitals is given in the first line of Table 7.9. The operations which do not swap the two atoms over leave this basis unchanged, and so a character of 2 is found, while for operations that exchange the atoms we find 0. This is identical to the reducible representation for the s-orbitals discussed in Section 7.3, so that the same reducible representations must be present  

The reducible character set for the p and p, orbitals is given on the second line of Table 7.9. We could proceed with the reduction following the elimination method covered in Section 6.2.2, but for brevity the result is given at the bottom of the table and a simple summation of the characters confirms that 

A similar effect can be seen for the 5cTg+ state, but, since this is bonding when formed purely from p -orbitals, the loss of overlap density causes its energy to increase. Finally, the antibonding character of the 6cr + is increased on sp-hybridization. 

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In summary, for a homonuclear diatomic molecule there are generally (2/ + 1) (7+1) symmetric and (27+1)7 antisymmetric nuclear spin functions. For example, from Eqs. (50) and (51), the statistical weights of the symmetric and antisymmetric nuclear spin functions of Li2 will be and respectively. This is also true when one considers Li2 Li and Li2 Li. For the former, the statistical weights of the symmetric and antisymmetiic nuclear spin functions are and, respectively for the latter, they are and in the same order. 

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. 

Here we shall consider a homonuclear diatomic molecule resPicted to a onedimensional x-space (Starzak, 1989) (Eig. 9-5). Although there is only one space coordinate, there are two degrees of freedom. The whole molecule can undergo moPon (Panslation), and it can vibrate. 

Figure 4.11j), for example, and all homonuclear diatomic molecules belong to this point group. 

In general, for a homonuclear diatomic molecule there are (21+ )(/+1) symmetric and (21+ 1)/antisymmetric nuclear spin wave functions therefore... 

All other homonuclear diatomic molecules with / = for each nucleus, such as F2, also have ortho and para forms with odd and even J and nuclear spin statistical weights of 3 and 1, respectively, as shown in Figure 5.18. 

For a homonuclear diatomic molecule with nuclei labelled 1 and 2 the LCAO method gives the MO wave function... 

Hi2 is the resonance integral, usually symbolized by p. In a homonuclear diatomic molecule Hi I = H22 = a, which is known as the Coulomb integral, and the secular determinant becomes... 

Figure 7.14 Molecular orbital energy level diagram for first-row homonuclear diatomic molecules. The 2p, 2py, 2p atomic orbitals are degenerate in an atom and have been separated for convenience. (In O2 and F2 the order of

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... 

Even molecules such as the short-lived SO and PO molecules can be treated, at the present level of approximation, rather like homonuclear diatomics. The reason is that the outer shell... 

The first is the g or m symmetry property which indicates that ij/ is symmetric or antisymmetric respectively to inversion through the centre of the molecule (see Section 4.1.3). Since the molecule must have a centre of inversion for this property to apply, states are labelled g or m for homonuclear diatomics only. The property is indicated by a postsubscript, as in... 

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... 

In a homonuclear diatomic molecule there may be an intensity alternation with J for the same reasons that were discussed in Section 5.3.4 and illustrated in Figure 5.18. 

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. 

The g and m subscripts in Figure 7.28 are appropriate only to a homonuclear diatomic molecule. This is the case also for the x and a labels which may result in intensity alternations for J even or odd in the initial state of the transition. Figure 7.28 would apply equally to a A type of transition. 

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. 

Infrared spectroscopy has broad appHcations for sensitive molecular speciation. Infrared frequencies depend on the masses of the atoms iavolved ia the various vibrational motions, and on the force constants and geometry of the bonds connecting them band shapes are determined by the rotational stmcture and hence by the molecular symmetry and moments of iaertia. The rovibrational spectmm of a gas thus provides direct molecular stmctural information, resulting ia very high specificity. The vibrational spectmm of any molecule is unique, except for those of optical isomers. Every molecule, except homonuclear diatomics such as O2, N2, and the halogens, has at least one vibrational absorption ia the iafrared. Several texts treat iafrared iastmmentation and techniques (22,36—38) and thek appHcations (39—42). 

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. 

Figure 13.18 Bond dissociation energies for gaseous, homonuclear diatomic molecules (from J. A. Kerr in Handbook of Chemistry and Physics, 73rd edn., 1992-3, CRC Press, Boca Raton, Florida), pp. 9.129-9.137.

The Raman spectrum of aqueous mer-cury(I) nitrate has, in addition to lines characteristic of the N03 ion, a strong absorption at 171.7 cm which is not found in the spectra of other metal nitrates and is not active in the infrared it is therefore diagnostic of the Hg-Hg stretching vibration since homonuclear diatomic vibrations are Raman active not infrared active. Similar data have subsequently been produced for a number of other compounds in the solid state and in solution. 

For a homonuclear diatomic, there is no argument that this sharing is equitable, but other authors have produced different sharing schemes for heteronuclear... 

Hurley, A. C., Proc. Roy. Soc. [London) A216, 424, The molecular orbital theory of chemical valency. XIII. Orbital wave functions for excited states of a homonuclear diatomic molecule."... 

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. 

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