Phase sign atomic orbitals

Now comes an important general rule if we begin with n atomic orbitals, we must end up with n orbitals after hybridization. Figure 4.2b and equation 4.2 show the second possibihty for the combination of a 2s and a 2p atomic orbital. The sign change for the combination changes the phase of the 2p orbital and so the resultant hybrid points in the opposite direction to the one shown in Figure 4.2a. (Remember thatp atomic orbitals have vector properties.)... 

The in-phase and out-of-phase relations mean the same and opposite signs of the atomic orbitals in (j) and nearest to the charge. 

Figure 1.8 The overlapping of two hydrogen Is atomic orbitals with the same phase sign (indicated by their identical color) to form a bonding molecular orbital.

In order to understand it let us consider the interference of waves. Now if the crest of one wave overlaps with the crest of the other, the two waves interact in a constructive interference and therefore the new resulting wave is reinforced, i.e. add up. In other words, there is in phase overlap or addition overlap. In the similar way, addition overlap of atomic orbitals with same signs leads to the formation of bonding molecular orbital. 

All four carbon atoms of buta-1,3-diene are sp2 hybridized, and (in the planar conformation) they all have overlapping p orbitals. Let s review how we constructed the pi molecular orbitals (MOs) of ethylene from the p atomic orbitals of the two carbon atoms (Figurel5-3). Each p orbital consists of two lobes, with opposite phases of the wave function in the two lobes. The plus and minus signs used in drawing these orbitals indicate the phase of the wave function, not electrical charges. To minimize confusion, we will... 

It can be seen how the phase of a Bloch sum changes in a periodic lattice by considering a simple one-dimensional lattice of (tr-bonded) p atomic orbitals, with a repeat distance d. Figure 4.4 shows such a chain and the sign combinations of the atomic... 

The combination of hydrogen Is atomic orbitals to form MOs. The phases of the orbitals are shown by signs inside the boundary surfaces. When the orbitals are added, the matching phases produce constructive interference, which give enhanced electron probability between the nuclei. This results in a bonding molecular orbital. When one orbital is subtracted from the other, destructive interference occurs between the opposite phases, leading to a node between the nuclei. This is an antibonding MO. 

Chapter 14 has been modified significantly. Material has been added on the phases of atomic orbitals and the orbital art has been modified to include signs in the lobes. This approach makes it easier for students to understand how bonding and antibonding molecular orbitals result from the linear combination of atomic orbitals. Also, the treatment of spectroscopy in Chapter 14 has been greatly expanded in response to requests by users. There are new sections on electronic, vibrational, and rotational spectroscopy. A new section on magnetic resonance spectroscopy has been added. 

The actual sign ("phase") of the molecular orbital at any given point r of the 3D space has no direct physical significance in fact, any unitary transformation of the MO s of an LCAO (linear combination of atomic orbitals) wavefunction leads to an equivalent description. Consequently, in order to provide a valid basis for comparisons, additonal constraints and conventions are often used when comparing MO s. The orbitals are often selected according to some extremum condition, for example, by taking the most localized [256-260] or the most delocalized [259,260] orbitals. Localized orbitals are often used for the interpretation of local molecular properties and processes [256-260]. The shapes of contour surfaces of localized orbitals are often correlated with local molecular shape properties. On the other hand, the shapes of the contour surfaces of the most delocalized orbitals may provide information on reactivity and on various decomposition reaction channels of molecules [259,260]. 

An approximate description of the MOs in H2 can be obtained by considering them as linear combinations of atomic orbitals (LCAOs). Each of the H atoms has one 1 atomic orbital let the two associated wavefunctions be ipi and In Section 1.6, we mentioned the importance of the signs of the wavefunctions with respect to their overlap during bond formation. The sign of the wavefunction associated with the I5 atomic orbital may be either + or —. Just as transverse waves interfere in a constructive (in-phase) or destructive (out-of-phase) manner, so too do orbitals. Mathematically, we represent the possible combinations of the two I5 atomic orbitals by equations 1.27 and 1.28, where N and N are the normalization factors. Whereas V mo is nn in-phase (bonding) interaction, ipuQ is an out-of-phase (antibonding) interaction. 

Figure 4.4 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 wavefunc-tion means a change in phase. The resultant directions of the lower two hybrid orbitals in Figure 4.4 are determined by resolving the vectors associated with the and 2py atomic orbitals. 

---