Bonding orbital, defined

In the sp d hybridization scheme of an octahedral complex, the three t2g non-bonding orbitals defined with respect to the C3 axis are ... 

The concept of natural orbitals may be used for distributing electrons into atomic and molecular orbitals, and thereby for deriving atomic charges and molecular bonds. The idea in the Natural Atomic Orbital (NAO) and Natural Bond Orbital (NBO) analysis developed by F. Weinholt and co-workers " is to use the one-electron density matrix for defining the shape of the atomic orbitals in the molecular environment, and derive molecular bonds from electron density between atoms. 

The determination of atomic charges has been a controversial subject because there has been disagreement on how the atoms in a molecule should be defined or, in other words, how the electronic charge should be apportioned between the atoms. Proposed orbital methods for determining atomic charges include the natural bond orbital (NBO) method (Reed et al., 1985, 1988) and the Mulliken method (1985). However, these orbital methods have some unsatisfactory features. There is a certain arbitrary character associated with them because, as we saw in Chapter 3, the choice of orbitals to describe a given molecule is not unique and... 

The concept of a bond has precise meaning only in terms of a given model or theory. In the Lewis model a bond is defined as a shared electron pair. In the valence bond model it is defined as a bonding orbital formed by the overlap of two atomic orbitals. In the AIM theory a bonding interaction is one in which the atoms are connected by a bond path and share an interatomic surface. 

Chemical bonds are defined by their frontier orbitals. That is, by the highest molecular orbital that is occupied by electrons (HOMO), and the lowest unoccupied molecular orbital (LUMO). These are analogous with the top of the valence band and the bottom of the conduction band in electron band theory. However, since kinks are localized and non-periodic, band theory is not appropriate for this discussion. 

Figure 16.1 The chemical hardness of an atom, molecule, or ion is defined to be half. The value of the energy gap between the bonding orbitals (HOMO—highest orbitals occupied by electrons), and the anti-bonding orbitals (LUMO—lowest orbitals unoccupied by electrons). The zero level is the vacumn level, so I is the ionization energy, and A is the electron affinity, (a) For hard molecules the gap is large (b) it is small for soft molecules. The solid circles represent valence electrons. Adapted from Atkins (1991).

FIGURE 17. On the left is shown the pair of localized banana bond orbitals Ab>Up and Ab.down obtained according to step 2 of Section II.E.2. Their out-of-phase linear combination n = (Ab,Up - Ab.downl/v defined in equation 5 yields the n basis orbital shown on the right... 

A defining feature for carbenes is the existence of two non-bonding orbitals on one carbon atom. There are two electrons to distribute among these two orbitals and their placement defines the electronic state of the molecule. A simple representation showing the electron occupancy of the non-bonding orbitals is displayed in Fig. 1. The orbital perpendicular to the... 

The geometry about a ligand nitrogen coordinated to a metal ion is assumed to be planar with bond directions defined in Fig. 28. The hybridizations of the nitrogen orbitals for different angles between the bonds in the plane can be written as ... 

Group theory was used to define the orbitals of the dihydrogen molecule. This led to the molecular orbital diagram for H2 and the relative energies of atomic, bonding and anti-bonding orbitals. 

At any rate, in the two electron system we adore so much, we can identify the two occupied spin orbitals Xi (4 X2 as the spin up and spin down halves of the single lowest lying spatial orbital, a Is in helium or the a bonding orbital in H2 for example. These can be more precisely defined as... 

Figure D.l to create so-called banana-bond orbitals. We define these as...

In contrast to o -orbitals and o-bonds, a tt-orbital or w-bond is defined as one which has one nodal surfaoe or plane containing the bond axis. (The reader might note that, though we will not deal with them, d-orbitals and d-bonds have two nodal surfaces which intersect on the bond axis.) If a w-bond is to be formed in MO fashion by combining two orbitals, one on each of the two bonded atoms, it is obviously necessary that each orbital have 7r-character with respect to the bond axis and that their two nodal planes coincide. The formation of a w-type bonding MO from two ir-type AOs is shown in Fig. 11-4.1, where the plus and minus signs refer to the sign of the wavefunction. 

In the above example, we have not given attention to the shieldings of the carbon atoms directly participating in the twisting bond C6-C7. To understand the conformation dependence of the C6 and C7 shieldings, a special treatment is required. For simplicity, we consider a four 71-electron system such as HEX. It is now assumed that the x, y and z axes coincide with the direction of the principal axes for o33, a22 and on, respectively. Then, the rotation of the C2-C3 bond modifies the pz orbitals of these carbons, resulting in a distortion of the 7C-orbitals, which is represented by the mixing of the pz orbital with px y orbital defined as... 

The electrons in the 5a orbitals have now been allowed to pair up in the 5aA + 5aB bonding orbital. The energy of the electron pair bond is now estimated from (defined as) AEpb = AEpb - AE°. Figure 8, which displays the various... 

The bond order in a diatomic molecule is defined as one-half the difference between the number of electrons in bonding orbitals and the number of antibonding orbitals. The factor one-half preserves the concept of the electron pair and makes the bond order correspond to the multiplicity in the valence-bond formulation one for a single bond, two for a double bond, and three for a triple bond. Fractional bond orders are allowed, but are not within the scope of this discussion. 

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