Molecular orbitals in AH molecules

In the same way that the study of Ha and Ha served as a basis for the construction of m.o.s in other diatomic molecules, the discussion of the simplest triatomic species H3+ will greatly help the study of more complex molecules of the general formula AH . 

This imstable species was first detected in the gaseous phase by mass spectrometry when Ha was subjected to an electric discharge. It is known from spectroscopic studies that it has an equilateral geometry with bond lengths aroimd 100 pm. 

Let ns begin, however, by considering a linear arrangement of three H atoms, each contributing a Is orbital 

There will be three a m.o.s. That of lowest energy is equally bonding between A and B and between B and C and is given by  

The normalization factor Ni is included and /z is a numerical coefficient which accounts for the fact that B is in a different situation when compared to A and C. The shapes and signs of the contributing atomic orbitals are shown below Eq. (7.4). 

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In the previous section, we discussed the construction of the a molecular orbitals in AH systems. In this section, we confine our treatment to the jz molecular orbitals in cyclic conjugated polyenes. In most molecules, there are a bonds as well as n bonds, and these systems can be treated by the methods introduced in these two sections. 

Figure 1.6. Molecular orbitals for AH molecules (with the electronic occupation appropriate for molecules with four valence electrons, such as BH or AIH in their lowest singlet state). 0-0 bonding (Ta-h MO...

If we now consider a planar molecule like BF3 (D3f, symmetry), the z-axis is defined as the C3 axis. One of the B-F bonds lies along the x-axis as shown in Figure 5.9. The symmetry elements present for this molecule include the C3 axis, three C2 axes (coincident with the B-F bonds and perpendicular to the C3 axis), three mirror planes each containing a C2 axis and the C3 axis, and the identity. Thus, there are 12 symmetry operations that can be performed with this molecule. It can be shown that the px and py orbitals both transform as E and the pz orbital transforms as A, ". The s orbital is A/ (the prime indicating symmetry with respect to ah). Similarly, we could find that the fluorine pz orbitals are Av Ev and E1. The qualitative molecular orbital diagram can then be constructed as shown in Figure 5.10. 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

Bent AH2 Molecules.—A bent AH molecule belongs to the symmetry class C2r. The definitions of the symbols appropriate to the non-localized orbitals of such a molecule are given below. The z axis bisects the HAH angle and lies in the molecular plane. The y axis also lies in the molecular plane and is parallel with the H H line. C2(z) means a rotation by 180° about the z axis. wave function does not or does change sign when one of the symmetry- operations C2(z) or av(y) is carried out. 

Another way of rationalizing these results is to treatx, y, z as px, py, p- orbitals of central atom A in an AH molecule. A molecule with C2v symmetry is H2S. A convention to set up the Cartesian coordinates for this molecule is as follows. Take the principle axis (C2 in this case) as the z axis. Since H2S is planar, we take the x axis to be perpendicular to the molecular plane. Finally, the y axis is taken so as to form a right-handed system. Following this convention, the px, py, p- orbitals on sulfur in a H2S molecule are shown in Fig. 6.3.2. When the orientations of these orbitals are examined, it is obvious that the p- orbital is symmetric with respect to all four operations of the C2v point group E, C2, other hand, the px orbital is symmetric with respect to E and arv(xz), but antisymmetric with respect to C2 and [Pg.182]

The electronic structures of Group lA and IB metal clusters have been determined using two theoretical methods ah initio molecular orbital theory and the semi-empirical diatomics-in-molecules (DIM)... 

Fig. 6.16 Molecuiar orfaita] picuires and qnalilalive energies of linear and beni AH molecules. Open and shaded areas represent differences in sign (+ or -) df the wave functions. Changes in shape which increase in-phase overlap lower the molecular orbital energy- Frum Cunarc.

Generally, as we shall see in this Chapter, there is only one conformation of a molecule in any one crystal structure. One of the most common questions asked by solution chemists about the results of a crystal structure analysis is how can one be sure that the solid-state conformation is the same as that observed in solution The conformation found for a flexible molecule in the crystalline state is that of one of the various conformers found in solution. This has been verified by other physical methods such as nuclear magnetic resonance. If, however, a molecule is found to have the same conformation in several different crystal structures, it is reasonable to assume that this conformation has a low (although not necessarily the lowest) energy. This assumption can often be tested by calculation (by ah initio molecular orbital calculations, for example) of the appropriate theoretical potential energy curve. 

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