Diatomic molecule homonuclear

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

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. 

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. 

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.

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. 

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. 

Hiroshima, 721 histidine, 443, 774 hole, 195 homeostasis, 386 HOMO, 126, 580 homogeneous alloy, 202 homogeneous catalyst, 565 homogeneous equilibria, 362 homogeneous mixture, F53 homolytic dissociation, 80 homonuclear diatomic molecule, 103 Hooke s law, 92 hormone, 670 horsepower, A4, 791 hour, A4 HPLC, 354 HRF products, 723 HTSC, 192 Humphreys series, 51 Hund, F 35 Hund s rule, 35, 37 Hurricane Rita, 144 hyaluronic acid, 344 hybrid orbital, 109 hybridization bond angle, 131 molecular shape, 111 hydrangea color, 463 hydrate, F32 hydrate isomer, 676 hydration, 178 hydrazine, 627... 

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 ... 

Figure 6.4. Schematic energy diagram of a homonuclear diatomic molecule. Note that the splitting 2/ is P proportional to the overlap of the...

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... 

Hertwig, R. H., Koch, W., 1995, On the Accuracy of Density Functionals and Their Basis Set Dependence An Extensive Study on the Main Group Element Homonuclear Diatomic Molecules Li2 to Br2 , J. Comput. Chem, 16, 576. 

The limitation of the above analysis to the case of homonuclear diatomic molecules was made by imposing the relation Haa = Hbb> as in this case the two nuclei are identical. More generally, Haa and for heteronuclear diatomic molecules Eq. (134) cannot be simplified (see problem 25). However, the polarity of the bond can be estimated in this case. The reader is referred to specialized texts on molecular orbital theory for a development of this application. 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

Show that the variational energies of a homonuclear diatomic molecule are given in the LCAO approximation by Eq. (137) and that the corresponding wavefiine-tions are as indicated in Eqs. (141) and (142). 

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). 

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