Real p orbitals

As an example of equivalent representations, we will consider the p-orbital function space. This space may be described by three real p-orbitals pir ps, p8 (commonly written as p p > pg) and we will call this the/basis. Alternatively, we may take three complex p-orbitals Pi Pi, P (commonly written as plF p lf p0) and we will call this the g basis. These two sets of functions are related by the equations ... 

The three real p-orbitals may be written in terms of a set of Cartesian coordinates xlt xt> and xt as... 

In order to demonstrate some of the conclusions of this chapter, we pose the question for the point group, for which irreducible representations do three real p-orbitals and five real d-orbitals or their combinations form a basis of representation ... 

At infinite separation, one arrives at two boron atoms each having a donut-like cylindrical density as indicated in Figure 5-3. However, such a density cannot be obtained from real atomic p-orbitals. In other words, the density that results from the supermolecule is simply inaccessible from calculations on the isolated atoms. Whatever we do, we will never generate the correct charge density (and therefore energy) of the dissociated B2 molecule by calculations of the isolated boron atoms and the requirement of size-consistency is violated. Only if one switches to complex orbitals such as lpx rpyl, are cylindrical atomic densities possible. But even then, we are still in trouble and face a different problem. Just as... 

This corresponds to the component of 2P with ML = 0. Occupying the real px or p orbital results in the same energy - but note that the real p and py orbitals are no longer eigenfunctions of the L2 operator. 

Table 4.2 [E] Angular momentum operations on the real p and d orbitals...

Explain why the number of p orbitals in this imaginary periodic table is less than the number in the real periodic table. 

In Table 6-3.1 we show the matrices for all of the operations of the 8v point group using both real and complex p-orbitals as basis functions. For the operations Ct and Cj we have simply replaced 0 by 27 /3 and 4t /3 respectively in both eqn (6-3.1) and eqn (6-3.2). The matrices for the rejection operations have been obtained in a fashion similar to that used for the rotations. In carrying out these steps it has been assumed that plf p, and p lie along the vectors 6t, e8, and e, respectively (see Fig. 6-3.1). For obvious reasons the matrix representation in the real basis is identical to the one given in 5-3(2) and, further, the reader may verify for himself that the matrices using the complex basis obey the 8v group table (Table 3-4.1). 

It turns out, in fact, that the electron distribution and bonding in ethylene can be equally well described by assuming no hybridization at all. The "bent bond" model depicted at the right requires only that the directions of some of the atomic-p orbitals be distorted sufficiently to provide the overlap needed for bonding. So one could well argue that hybrid orbitals are not real they do turn out to be convenient for understanding the bonding of simple molecules at the elementary level, and this is why we use them. 

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