Symmetry correction

Symmetry corrections for rate or equilibrium constant are usually expressed in a formally different but perfectly equivalent way by means of statistical factors (Leffler and Grunwald, 1963). If the two reagents of (33) have pA and ph equivalent reactive sites, then the rate constant for the forward reaction must be divided by a statistical factor of pA-pB- Since similar considerations apply to the back reaction as well, the equilibrium constant must be divided by a statistical factor ofpApJpc Pv> as n (35). The assign-... 

Thermochemical data from the compilation of Stull et at., 1969. Entropy values are based on a 1 M standard state. The asterisk denotes symmetry-corrected quantities. Symmetry numbers were chosen as follows 18 for the n-alkanes, cis-3-hexene, dibuthyl sulphide, diethyl ether, and diethyl amine 2n for the cycloalkanes and 2 for all of the remaining ring compounds 3 for the alkanols, alkanethiols and alkyl amines 9 for the methyl alkyl sulphides... 

The values of 0(ASD) /2.3O3 R listed in Table 5 are the entropic components of log EM. These are the log EM- alues for ideal strainless cyclisation reactions, i.e. reactions where 0AH° = 0. It is of interest to note that, as far as the entropic component is concerned, symmetry corrected effective molarities on the order of 102 106M are found. This observation leads to the important conclusion that cyclisation reactions of chains up to about 7 skeletal bonds are entropically favoured over reactions between non-connected 1 M end-groups. The intercept of 33 e.u. corresponds to an effective molarity of exp(33/R) or 107 2M, which may be taken as a representative value for the maximum advantage due to proximity of end-groups in intramolecular equilibrium reactions. It compares well with the maximum EM of about 108M estimated by Page and Jencks (1971). 

Fig. 24 Plot of symmetry-corrected 0A5° (e.u.) against number of single bonds for macrocyclisation equilibria of formals [30], calculated from data reported by Yamashita et al. (1980). The curve was calculated from Jacobson-Stockmayer theory. For the meaning of the arrows see text...

Guggenheim S. and Bailey S. W. (1978). Refinement of the margarite structure in subgroup symmetry Correction, further refinement, and comments. Amer. Mineral, 63 186-187. 

Let K,pp and K°, respectively, stand for the experimentally determined equilibrium constant for Eq. (5) and the chemical (i.e., symmetry-corrected) constant ... 

We may now solve the secular equation, using these symmetry-correct MOs, and obtain the MO energies. Thus, for the A orbitals we have the equation... 

Only orbitals of the same symmetry can interact. Formation of Symmetry Correct Orbitals... 

In order to take advantage of the symmetry criterion for orbital interaction, it is necessary to have orbitals that are symmetry correct. The molecular orbitals obtained from atomic orbitals by the methods described in Sections 1.2 and 10.1 will sometimes be symmetry correct and sometimes not. 

It is evident from a comparison of 5 and 4 that the reflection transforms >pi into i/j2. The reader should verify that the a reflection and the C2 rotation also transform molecular orbitals constructed by the LCAO method from hybrid atomic orbitals are subjected to symmetry operations. Each of those orbitals in the set of MO s that is not already symmetry correct will be transformed by a symmetry operation into another orbital of the set. 

How can we obtain symmetry correct orbitals to use in place of i/ and ip2 The procedure is simple, and follows the method we always use in dealing with changes in forms of orbitals. We combine ifi1 and [Pg.546]

These new orbitals are shown in 6 and 7. The reader should verify that these new orbitals are symmetry correct. 

The symmetry correct orbitals can now be classified as to symmetry type. The positive combination (6) is unchanged by reflection in mirror plane a or by rotation C2 we therefore call it symmetric (S). The negative combination (7) changes sign on reflection or on rotation, and is antisymmetric (A). It happens in this instance that each of our new orbitals behaves the same way under... 

Figure 10.6 Energy changes in the interaction of two O—H bonding a orbitals ifi1 and iji2 of the water molecule to form the symmetry correct orbitals Ts and TV...

Let us summarize the steps to be followed in obtaining symmetry correct orbitals. 

Examine each molecular orbital to see whether it is already symmetry correct. 

For orbitals that are not already symmetry correct, group together orbitals that are transformed into each other by the symmetry operations. 

Form sums and differences within these symmetry related groups to obtain symmetry correct orbitals. Remember that the total number of orbitals obtained at the end of the process must always equal the total number at the start. 

It may sometimes happen that not all symmetry elements are needed. Orbitals can be made symmetry correct with respect to selected symmetries, as long as subsequent conclusions are based on these symmetries only. 

An obvious question is, when does one need symmetry correct orbitals The HaO example illustrates that the symmetry correct orbitals will usually extend over a larger region of the molecule than did the symmetry incorrect orbitals from which they were made. The symmetry correct model corresponds to a more highly delocalized picture of electron distribution. We believe that electrons are actually able to move over the whole molecule, and in this sense the delocalized symmetry correct pictures are probably more accurate than their localized counterparts. Nevertheless, for most purposes we are able to use the more easily obtained localized model. The reason the localized model works is illustrated in Figure 10.6. The interaction that produced the delocalized symmetry correct orbitals made one electron pair go down in energy and another go up by an approximately equal amount. Thus the total energy of all the electrons in the molecule is predicted to be about the same by the localized and by the more correct delocalized model. 

Figure 10.7 The three bonding C—H orbitals of the methyl group in CH3CH2 +. 4>c is symmetry correct with respect to mirror plane cr, but

The orbitals [Pg.550]

We expect Ts to be slightly lower in energy than Figure 10.8 shows these new symmetry correct orbitals. 

Now we are ready to assess the interactions of the electrons in the C—Ha and C—H bonds with the vacant p orbital on Gj. First we notice that of our two new symmetry correct orbitals, only has the same symmetry as the vacant orbital. Since Ts has different symmetry, it cannot interact and need not be considered further. With the help of the symmetry we have reduced what looked initially like a moderately complicated problem to a simple one. We have only to consider the interaction of two orbitals. In Figure 10.9 we have placed on the left... 

Find symmetry correct orbitals in each of the following ... 

Symmetric and antisymmetric orbitals The first point is that each symmetry correct molecular orbital may be either unchanged or transformed to... 

There is danger of error only when such elements are used exclusively.) The atomic orbital basis consists of a p function on each of the four carbon atoms Figure 11.15 illustrates these orbitals and the derived symmetry correct orbitals. As a result of the reflection plane that carries one ethylene into the other, these molecular orbitals are delocalized over both molecules. Figure 11.15 also shows the orbitals of the product we consider only the two C—C bonds formed and ignore the other two, which were present from the beginning and did not undergo any change. 

With the permutation-symmetry-corrected results(17) it appears that we cannot successfully fit our QEO data to a two-level model, at least for centrosymmetric structures. We have only one data point (9) to make this judgement, however, and... 

Enough substituted cyclopropanes have now been subjected to careful kinetic studies so that a characteristic pattern of reactivity and stereochemical preferences has emerged. Substituents facilitate stereomutations in proportion to their ability to stabilize 1,3-trimethylene diradical structures. The values for both k 2 and (k + k2) stereomutation rate constants relate linearly with consistent measures of substituent radical stabilization energies with equal sensitivities. Experimentally determined (A , + / 2)- i2 ratios do not vary widely they range from 1.4 to 2.5 over a fair diversity of substituents. Neither do kf.kj ratios vary widely. The majority fall between 1 1 and 2.5 1 the largest yet reported gives 2(CHD) a symmetry corrected kinetic advantage over A i(CDPh) in 1-phenyl-1,2,3-d3-cyclo-propanes of 5 1. 

These symmetry corrections, however, represent only a very first correction. In principle such properties as molecular weight, moment of inertia, and vibration frequency are just as accidental and nonintrinsic from the point of view of molecular interactions. Nevertheless, they can contribute significantly to free energy changes and should be corrected for if a molecular discussion of such changes is to be rigorous, or at least non-speculative. 

Co4(CO) 12 may be described in exactly the same way, except that the site occupied in the iron compound by a face-bridging CO group, and thus by four metal and two ligand electrons, is now to be occupied by (symmetry-correct combinations of) three cobalt lone pairs. The basal cobalt atoms are five-coordinate (if as usual we discount bridged metal-metal bonds) with the sixth site of the coordination octahedron occupied by a lone pair and the 18-electron count achieved by straight, stereochemically inactive cobalt-cobalt bonds. This is exactly the situation required by the straight bond description of Co2(CO)g (H(8) above). 

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