Phases, of atomic orbitals

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 phases and the nodes (the points of change of the phase) of atomic orbitals in molecular orbitals were, however, kept. 

Figure 7.30 (a) In-phase and (b) out-of-phase Is atomic orbitals on the hydrogen atoms of linear... 

It has been pointed out that a different array of atomic orbitals might be conceived of in large conjugated rings. The array, called a Mobius twist, results in there being one point in the ring at which the atomic orbitals would show a phase discontinuity. ... 

Figure 2.14. The molecular orbitals of gas phase carbon monoxide, (a) Energy diagram indicating how the molecular orbitals arise from the combination of atomic orbitals of carbon (C) and oxygen (O). Conventional arrows are used to indicate the spin orientations of electrons in the occupied orbitals. Asterisks denote antibonding molecular orbitals, (b) Spatial distributions of key orbitals involved in the chemisorption of carbon monoxide. Barring indicates empty orbitals.5 (c) Electronic configurations of CO and NO in vacuum as compared to the density of states of a Pt(lll) cluster.11 Reprinted from ref. 11 with permission from Elsevier Science.

Molecular orbitals are obtained by the linear combination of atomic orbitals, and the question of phase will, of course, arise with them too. Thus we can write the two MOs (it and it, cf. p. 12) arising from the two p atomic orbitals in ethene,... 

In order to eliminate the problems with the invariance, we proposed some time ago a topological approximation based on the so-called overlap determinant method [43]. This approximation is based on the transformation matrix T that describes the mutual phase relations of atomic orbitals centred on molecules R and P, and thus plays in this approach the same role as the so-called assigning tables in the overlap determinant method (Eq. 4)... 

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. 

ODENDOR ODLTS ODMR OICTS OLCAO OMVPE OSC optically detected electron nuclear double resonance optical deep level transient spectroscopy optically detected magnetic resonance optical isothermal capacitance transient spectroscopy orthogonalised linear combination of atomic orbitals organo-metallic vapour phase epitaxy on-surface-cracking... 

Fig. 31 Bonding and antibonding molecular orbitals formed by in- and out-of-phase combinations of atomic orbitals...

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. 

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]. 

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