Hybrid orbitals, wavefunctions

FIGURE 3.13 These contours indicate the amplitude of the sp hybrid orbital wavefunction in a plane that bisects it and passes through the nucleus. Each sp hvbrid orbital points toward the corner of a tetrahedron. 

To improve our model we note that s- and /7-orbitals are waves of electron density centered on the nucleus of an atom. We imagine that the four orbitals interfere with one another and produce new patterns where they intersect, like waves in water. Where the wavefunctions are all positive or all negative, the amplitudes are increased by this interference where the wavefunctions have opposite signs, the overall amplitude is reduced and might even be canceled completely. As a result, the interference between the atomic orbitals results in new patterns. These new patterns are called hybrid orbitals. Each of the four hybrid orbitals, designated bn, is formed from a linear combinations of the four atomic orbitals ... 

FIGURE 3.16 Three common hybridization schemes shown as outlines of the amplitude of the wavefunction and in terms of the orientations of the hybrid orbitals, (a) An s-orbital and a p-orbital hybridize into two sp hybrid orbitals that >oint in opposite direc tions, forming a linear molecular shape, (b) An s-orbital and two p-orbitals can blend together to give three ip hybrid orbitals that point to the corners of an equilateral triangle, (c) An s-orbital and three p-orbitals can blend together to give four sp hybrid orbitals that point to the corners of a tetrahedron. 

We now use a Pauling-like approach to show how hybrid orbitals for a variety of combinations of s, p, and d orbitals may be formulated.10 We assume that the radial dependences of the s, p, d orbitals are similar so that they can be neglected. The angular parts of the orbital wavefunctions are given by the following expressions (in the usual spherical coordinates 9, ) ... 

There are two ways in which the state of hybridization at the carbon atoms of [1.1.1]-propellane has been deduced. The results disagree dramatically (Figure 1), and at least one is clearly wrong. The hybridization at the carbon atoms can be derived from (i) the empirical correlations between the s orbital character of a carbon hybrid orbital and the, 3C- H and 13C-13C NMR coupling constants across the bonds it forms, (ii) the analysis of computed wavefunctions, using one of several possible schemes. 

Hybrid Orbitals. Orbitals, as one-electron energy levels, and corresponding wavefunctions are mathematical concepts only states are physically observable. Nevertheless, the simple picture of orbitals as the rungs of an energy ladder is very helpful, and is in many cases sufficient to account for the photophysical and photochemical properties of molecules. In more accurate pictures of orbitals it is necessary to consider their interactions, as they are not really totally independent. In this respect the concept of hydrid orbitals is important such hybrid orbitals are formed from a combination of elementary orbitals defined by their quantum numbers n, /, and m. The best... 

Recalling from Section 3.4.1, we use the s orbital and the pz orbital to form two equivalent hybrid orbitals, one pointing in the +z direction and the other in the —z direction. These two orbitals are called sp hybrids, since they are formed by one s and one p orbital. The wavefunctions of the sp hybrid orbitals are given by eqs. (3.4.6) and (3.4.7). In matrix form the wavefunction are... 

Relationship between the coefficient matrices for the hybrid and molecular orbital wavefunctions... 

One way to generate surfaces is by explicit QM calculation of species as they are followed through some mechanism. SCF-CI calculations have proven of considerable value in the author s research. The philosophy here has been to include as basis orbitals only those atomic and hybrid orbitals which are part of chromo-phores or make up bonds which are altered, broken, formed or modified, during the photochemical transformation. Additionally, basis orbitals aimed along the directions of bonds are used, since then the SCF wavefunctions are linear combinations of recognizable orbitals of bonds rather then arbitrary vertically and horizontally oriented atomic orbitals. 

By overlapping each H Is orbital with each one of these hybrid orbitals of C, in a maimer similar to H2, four equivalent functions are defined each localized in each formal CH bond four localized functions., as graphically illustrated in Fig. 8.6. After eventual mixing of ionic contributions and the inclusion of the spin-functions, the global electronic wavefunction can be constructed. 

Although we have introduced references to bond formation with hybrid orbitals, we have not yet really tackled how to describe molecules using orbitals. The starting point for molecules, as for atoms, is the Schrodinger equation, and we can solve this to obtain electron wavefunctions or molecular orbitals. However, for molecules the electrons are attracted to all the nuclei in the molecule, not just one, and we have to include the repulsion between the nuclei in the energy. There are several methods and many programs available to calculate molecular orbitals. Nearly all employ two approximations. 

One possible combination of a 2s atomic orbital and 2p atomic orbital is shown in Figure 4.2a. In the figure, the colour of the orbital lobe corresponds to a particular phase (see Section 1.6) and the addition of the 2s component reinforces one lobe of the 2p, atomic orbital but diminishes the other. Equation 4.1 represents the combination mathematically. The wavefunction V sp hybrid describes a normalized (see Section 1.12) sp hybrid orbital which possesses 50% 5 and 50% p character. Although equation 4.1 and Figure 4.2a refer to the combination of 2s and 2p atomic orbitals, this could just as weU be 2s with 2py or 2p, or 3 with hp, and so on. 

Equations 4.1 and 4.2 represent two wavefunctions which are equivalent in every respect except for their directionalities with respect to the x axis. Although the orbital energies of the initial 2s and 2p atomic orbitals were different, mixing leads to two hybrid orbitals of equal energy. 

A similar scheme to those described above can be derived to generate four sp hybrid orbitals from one 2s and three 2p atomic orbitals. The sp hybrid orbitals are described by the normalized wavefunctions in equations 4.6-4.9 and are shown pictorially in Figure 4.6a. Each sp hybrid orbital possesses 25% character and 75% p character, and the set of four equivalent orbitals defines a tetrahedral framework. 

Figure 4.4 shows the formation of three sp hybrid orbitals (see equations 4.3-4.5). (a) Confirm that the directionalities of the three hybrids are as specified in the figure, (b) Show that equations 4.3 and 4.5 correspond to normalized wavefunctions. 

What do the four hybrid orbitals look like Each sp orbital takes three-quarters of its character from a p orbital and one-quarter from an s orbital. It has a planar node through the nucleus like a p orbital but one lobe is larger than the other because of the extra contribution of the 2s orbital the symmetry of the 2s orbital means that adding it to a 2p orbital will increase the size of the wavefunction in one lobe, but decrease it in the other. 

Figure 5.4 gives a pictorial representation of the way in which the three sp hybrid orbitals are constructed. Remember that a change in sign for the atomic wavefunction means a change in phase. The resultant directions of the lower two hybrid orbitals in Figure 5.4 are determined by resolving the vectors associated with the 2p and 2py atomic orbitals. 

Natural bond orbital (NBO) analysis The NBO analysis transforms the canonical delocalized Hartree-Fock (HF) MOs and non-orthogonal atomic orbitals (AOs) into the sets of localized natural atomic orbitals (NAOs), hybrid orbitals (NHOs), and bond orbital (NBOs). Each of these localized basis sets is complete, orthonormal, and describes the wavefunction with the minimal amount of filled orbitals in the most rapidly convergent fashion. Filled NBOs describe the hypothetical, strictly localized Lewis structure. NPA charge assignments based on NBO analysis correlate well with empirical charge measures. ... 

As long as one works within the atomic and molecular orbital approximations, two wavefunctions that yield the same total electron density can be shown to be mathematically equivalent. Whether we describe an excited Be atom as having the electron configuration l5225 2p or as having the one valence electron in each of the two spz hybrid orbitals, we are really talking about the same state ... 

---