Diatomic molecules orbital overlap

Let us first briefly review the construction of molecular orbitals in simple diatomic molecules, AB, using the linear combination of atomic orbitals (LCAO) scheme. The end product for the first long row of the periodic table is the well-known diagram in Fig. 6-1. We focus on two broad principles that are exploited in the construction of this diagram one has to do with symmetry and overlap, the other concerns energies. 

C09-0047. Hydrogen forms diatomic molecules with elements from Group 1 of the periodic table. Describe the bonding in LiH and include a picture of the overlapping orbitals. 

The halogens, the elements from Group 17 of the periodic table, provide an introduction to intermolecular forces. These elements exist as diatomic molecules F2, CI2, Bf2, and I2. The bonding patterns of the four halogens are identical. Each molecule contains two atoms held together by a single covalent bond that can be described by end-on overlap of valence p orbitals. 

The basic principles dealing with the molecular orbital description of the bonding in diatomic molecules have been presented in the previous section. However, somewhat different considerations are involved when second-row elements are involved in the bonding because of the differences between s and p orbitals. When the orbitals being combined are p orbitals, the lobes can combine in such a way that the overlap is symmetric around the intemuclear axis. Overlap in this way gives rise to a a bond. This type of overlap involves p orbitals for which the overlap is essentially "end on" as shown in Figure 3.5. For reasons that will become clear later, it will be assumed that the pz orbital is the one used in this type of combination. 

For diatomic molecules, there is coupling of spin and orbital angular momenta by a coupling scheme that is similar to the Russell-Saunders procedure described for atoms. When the electrons are in a specific molecular orbital, they have the same orbital angular momentum as designated by the m value. As in the case of atoms, the m value depends on the type of orbital. When the internuclear axis is the z-axis, the orbitals that form a bonds (which are symmetric around the internuclear axis) are the s, pz, and dzi orbitals. Those which form 7r bonds are the px, p, dlz, and dyi orbitals. The cip-y2 an(i dxy can overlap in a "sideways" fashion with one stacked above the other, and the bond would be a 8 bond. For these types of molecular orbitals, the corresponding m values are... 

Figure 4.1 The c and n orbital overlaps between 2p atomic orbitals in the formation of some diatomic molecules...

Mulliken derived equations for the variation of overlap integrals with distance between the participating nuclei. For homonuclear diatomic molecules the equations for the overlap of two 2p orbitals in o and n modes are respectively ... 

The concepts which we need for understanding the structural trends within covalently bonded solids are most easily introduced by first considering the much simpler system of diatomic molecules. They are well described within the molecular orbital (MO) framework that is based on the overlapping of atomic wave functions. This picture, therefore, makes direct contact with the properties of the individual free atoms which we discussed in the previous chapter, in particular the atomic energy levels and angular character of the valence orbitals. We will see that ubiquitous quantum mechanical concepts such as the covalent bond, overlap repulsion, hybrid orbitals, and the relative degree of covalency versus ionicity all arise naturally from solutions of the one-electron Schrodinger equation for diatomic molecules such as H2, N2, and LiH. 

This is a simple example of a heteronuclear diatomic molecule which is found in a stable molecular substance. We must first choose the basis set. The only AOs that need to be seriously considered are the hydrogen Is, fluorine 2s and fluorine 2p, written for brevity as ls(H), 2s(F) and 2p(F). The fluorine Is orbital lies very low in energy (700 eV lower than 2p) and is so compact that its overlap with orbitals on other atoms is quite negligible. The fluorine 2p level lies somewhat lower than ls(H), as indicated by the higher ionisation potential and electronegativity of F. Interaction between 2p(F) and 2s(H) is very small and can be neglected for all practical purposes. One is tempted to discard 2s(F), which lies more than 20 eV below 2p(F) the 2s-2p separation increases... 

It is conventional to label the internuclear axis in a diatomic molecule as z. Thus the three 2p(F) orbitals can be labelled 2p, 2py and 2p2 2px and 2py have their lobes directed perpendicular to the internuclear axis, and have nodal surfaces containing that axis, while 2pr clearly overlaps in o fashion with ls(H). (The reader may wonder whether this orientation of 2p, 2py and 2pz is obligatory, or whether it is chosen for convenience. For a spherically-symmetric atom, there are no constraints in choosing a set of three Cartesian axes. Any set of orthogonal p orbitals can be transformed into another equally acceptable set, by a simple rotation which does not change the electron density distribution of the atom. The overlap integral between a hydrogen Is orbital and the set of three 2p(F) orbitals is the... 

First, diatomic molecules are usually adsorbed on cations (some exceptions to this empirical rule are mentioned in the case studies). The type of interaction and the resulting vibrational perturbation (purely electrostatic or with some orbital overlap contribution) depend on the charge carried by the cation and by the anions in nearest-neighbor positions and on the electronic structure of the cation (with or without d electrons). As a typical example of the effect of a purely electrostatic perturbation on the stretching mode of a diatomic molecule, we mention the classic case of CO adsorbed on Na+ exposed on NaCl (100) (Fig. 2), in which it is clearly shown that the frequency of adsorbed CO is distinctly blue shifted with respect to that of CO gas (2143 cm-1) (48). More general considerations concerning the role of the electrostatic field in perturbing the adsorbed molecules are discussed elsewhere (12-15, 21-23). 

In this section we consider homonuclear diatomic molecules (those composed of two identical atoms) formed by elements in Period 2 of the periodic table. The lithium atom has a 1 s22s electron configuration, and from our discussion in the previous section, it would seem logical to use the Li Is and 2s orbitals to form the MOs of the Li2 molecule. However, the Is orbitals on the lithium atoms are much smaller than the 2s orbitals and therefore do not overlap in space to any appreciable extent (see Fig. 14.33). Thus the two electrons... 

It may occur to the reader that it is always possible to bring the orbitals together in such a way that the overlap is positive. For example, in Fig. 5.8g, h if negative overlap is obtained, one need only invert one of the atoms to achieve positive overlap. This is true for diatomic molecules or even for polyatomic linear molecules. However, when we come to cyclic compounds, we no longer have the freedom arbitrarily to invert atoms to obtain proper overlap matches. One example will suffice to illustrate this. 

The Hg atom has a 6s closed electronic shell. It is isoelec-tronic with helium, and is therefore van der Waals bound in the diatomic molecule and in small clusters. For intermediate sized clusters the bands derived from the atomic 6s and 6p orbitals broaden as indicated in fig. 1, but a finite gap A remains until the full 6s band overlaps with the empty 6p band, giving bulk Hg its metallic character. This change in chemical binding has a strong influence, not only on the physical properties of mercury clusters, but also on the properties of expanded Hg, and on Hg layers on solid and liquid surfaces. For a rigid cluster the electronic states are discreet and not continuous as in fig. 1. Also the term band for a bundle of electronic states will be used repeatedly in this paper, although incipient band might be better. As the clusters discussed here are relatively hot, possibly liquid, any discreet structure will be broadened into some form of structured band . 

We will consider first the combination of two s orbitals, as in H2. For convenience, we label the atoms of a diatomic molecule a and b, so the atomic orbital wave functions are i(i( Ir ) and il<( li ). We can visualize the two atoms moving closer to each other until the electron clouds overlap and merge into larger molecular electron clouds. The resulting molecular orbitals are linear combinations of the atomic orbitals, the sum of the two orbitals and the difference between them ... 

The orbital overlaps leading to the formation of the MCO molecular orbitals are shown in Figure 8.10. The Icr and 2cr orbitals have been omitted because they are essentially unperturbed CO molecular orbitals at energies too low to interact with the atomic orbitals of the metal. Similar situations arise in heteronu-clear diatomic molecules (ScO, earlier and HF in Fig. 6.20). Starting at the bottom of the diagram we see that a Itt orbital is formed by the overlap of a metal dy ... 

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