Equivalent bond functions

In tetrahedral symmetry the o--bond functions induce a representation AI -j- Tj. The partners px, Py, Pz in T, and the s orbital of Ai type can therefore be used to form the four equivalent bond functions of sp type. The results for other common molecular symmetries are obtained in the same way. Whenever the symmetry of the molecule is lowered, say by bending, then the symmetry restrictions on the content of hybrid orbitals are relaxed and other types of orbitals that have the proper symmetry in the point-group of the bent molecule, may participate. For example, if NOi" becomes nonplanar, the symmetry group becomes Czv, ( 3 ) correlates with A i(Cjr) and since p, transforms like A1, the pz orbital can enter into hybrid bond function can be obtained from symmetry. Let be the ith bond function. Then if Tr is a... 

Each bond orbital in a molecule may be described by a function that is localized in the bond region. If the bond is of rotations about the bond axis, and reflections in planes containing the bond axis. The x bond functions are antisymmetric to 180 rotations about the bond axis, and antisymmetric to reflections in the molecular plane or in the bond plane. A set of equivalent orbitals of this type give a reducible representation of the molecxilar point group since operations that interchange identical nuclei also interchange equivalent bond functions. Some aspects of this procedure were already outlined, leading to Eq. 7.119. 

On the other hand, the results of an MO analysis might seem at odds with the obvious fact that in molecules such as CH4 or SFb all of the a bonds are equivalent. Actually, there is no inconsistency. If the electron density in the molecule is computed from the wave functions for all of the filled MOs (e.g., the A i and T2 MOs in CH4) the equivalence of all the bonds will be evident. At the same time, the fact that these equivalent bonds are a result of the presence of electrons in nonequivalent MOs is also experimentally verifiable by the technique of photoelectron spectroscopy. 

The final molecule of this series is methane, the tetrahedral structure of which follows if a fourth unit positive charge is removed from the nucleus in the ammonia lone-pair direction. There are now four equivalent bonding orbitals, which may be represented approximately as linear combinations of carbon s-p hybrid and hydrogen Is functions. The transformation from molecular orbitals into equivalent orbitals or vice versa is exactly the same as for the neon atom. 

The photodissociation of symmetric triatomic molecules of the type ABA is particularly interesting because they can break apart into two identical ways ABA — AB + A and ABA — A + BA. Figure 7.18(a) shows a typical PES as a function of the two equivalent bond distances. It represents qualitatively the system IHI which we will discuss in some detail below. We consider only the case of a collinear molecule as illustrated in Figure 2.1. The potential is symmetric with respect to the C -symmetry line 7 IH = i HI and has a comparatively low barrier at short distances. The minimum energy path smoothly connects the two product channels via the saddle point. A trajectory that starts somewhere in the inner region can exit in either of the two product channels. However, the branching ratio ctih+i/cti+hi obtained by averaging over many trajectories or from the quantum mechanical wavepacket must be exactly unity. 

We have extended this model to the naphthalene triplet 2D lattice, where each site has four nonequivalent bonds and two equivalent bonds. Therefore, we have considered the two types of bond separately, with two bond potentials se (S ) and two interactions we (w ) for the effective equivalent (nonequivalent) bonds. These four functions are determined by the vanishing of the mean scattering on each bond separately see Fig. 4.17. The calculation is quite analogous to the preceding one. The system to be solved involves four equations ... 

For the production of three bonds the carbon atom will have to be brought into the excited configuration already mentioned. In the formation of three equivalent bonds with the electrons of the hydrogen atoms one electron naturally remains over unpaired. The three electron pairs will occupy the functions with the lowest energy 2s + 2 X 2p, for example 2s + 2px + sp, thus sp2 hybridization and bond angles of 120° the lone electron is then a pz electron. The radical will then have a plane structure with the dumb-bell-shaped wave function of... 

The most recent study, using UHF, MP2, BLYP and B3LYP methodologies, reached somewhat different conclusions. Density functional theory calculations showed one imaginary frequency for the rhombic structure (four equivalent bonds, 157.3 pm), suggesting that it is a (very low-lying) transition structure between two parallelograms (two pairs of equivalent bonds, 149.5 and 169.5 pm, respectively)... 

Finally, it should be mentioned that the structures outlined above are only used for convenience. In the total antisymmetrized wave-function (a Slater determinant, section III) the distinction between different states of hybridization disappeare. Thus by proper transformation (keeping the total wavefunction unchanged) we can replace the two or-two ttd molecular orbitals of N3 by four equivalent bananashaped orbitals (see section III.F.l). (In the latter picture the central nitrogen atom may be regarded as being in a hybridization state which resembles rp .) This transformation is similar to that employed for the nitrogen and acetylene molecules, where the three ott bonds can be replaced by three equivalent bonds . 

A pair-function calculation, where each electron pair is associated to a quasi-localized bond function built from Lowdin orthogonalized Is orbitals. This is equivalent to the treatment of ethane mentioned above 14) ... 

The alkyne-Co2(CO)6 complexes 1 are the binuclear cluster complexes of the acetylenic derivatives with the hexacarbonyldicobalt moiety. These complexes can be readily prepared by treatment of alkynes with commercially available octacarbonyldicobalt [Co2(CO)g] and can regenerate the parent triple bond functionality under some mild oxidation conditions. Two synthetically very useful reactions have so far been developed by taking advantage of the characteristic properties of the alkyne-Co2(CO)6 complexes 1 one is so-called Nicholas reaction" and the other is so-called Pauson-Khand reaction (Scheme 1). The alkyne-Co2(CO)6 complexes 1 possessing a hydroxyl group or its equivalent at carbon p- to alkyne moiety (propargyl alcohol derivatives) could easily... 

As X goes to zero, Eqs. 8.63 are recovered. Clearly, X is related to the angle between the bonds, at 120° X is zero and at 90° it should be infinite (no s character remains). On the assumption that the bonds are straight (that is, that the maximum overlap density occurs on the line joining the nuclei) we can compute the angle between the bonds as a function of the parameter X. The cosine of the angle between equivalent hybrids is obtained from the inner product of the directional portions of two bond functions. Thus, if we write 2 and 3 in the slightly modified way... 

We assume that a molecule, such as NH3, belongs to the point group Csv, and that it has three equivalent bonds represented b3rfunctions 1, 4 2, and Si s, as well as a lone-pair orbital 4 which is not equivalent to the bond orbitals. The z axis is the principal axis. If we act on the i j with the projection operator for the Xth rep, the result will be a linear combination of the functions in 4, that transform like the Xth rep. In this manner we can project new functions d> each of which is a linear combination of the "iTj. The inverse of the transformation that takes the 4 y into the is the transformation that gives us the in terms of the To keep the computation down to a minimum we will treat the projected functions 4> not as linear combinations of s, p, p , and p but as the combinations of the base functions of the axial rotation group so, pi, po, and p i. Thus, there are only three functions involved, namely, 00, 01, and 0 i. These functions transform according to the A(0o) and E [Pg.317]

The basis functions of such an EBO model are orthonormal two-centre equivalent bond orbitals (EBO) which we designate by = Their self-energies and interaction crossterms are defined as follows ... 

In order to pass to this limit, it is noted that IF,// is the interaction energy between two equivalent bonds per unit length squared. The interaction energy for two equivalent bonds of length As and As is then approximately Wfj ASf AsJP. Again, as in (3.12), this approximation becomes exact in the limit (6.7). Thus for the discrete chain the end-to-end vector Green s function is... 

Under what conditions are the molecular-orbital and valence-bond descriptions of the c bonding in an octahedral complex equivalent Derive the valence-bond functions shown in Fig. 9-7 from the general molecular-orbital functions. 

In the previous section, we showed how one could transform from a basis set of STOs to a basis set of symmetry orbitals. Since these two sets are related through a unitary transformation, they are equivalent and must lead to the same MOs when we do a linear variation calculation. However, there are an inhnite number of unitary transformations available, and so the set of symmetry orbitals is only one of an infinite number of possible equivalent bases. Of course, this set has the unique advantage of being a set of bases for representations of the symmetry group, which makes it easy to work with. Another set of equivalent basis functions are the hybrid orbitals. These have the distinction of being the functions that are concentrated along the directions of bonds in the system. Consider, for example, methane, which was discussed in detail in Chapter 10. The minimal basis set of valence STOs on carbon can be transformed to form four tetrahedrally directed hybrids ... 

Table 72. Bond lengths of the two crystallographically-independent molecules in pm (standard deviation). Below the mean bond lengths (pm) of functionally equivalent bonds from both molecules, as well as their averaged bond angles (degrees).

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