2n + 2 rule

The structures of some biochemically interesting compounds are shown below. Mark the chiral carbons in each and calculate by the 2n rule how many stereoisomers might be expected. Explain why only one pair of enantiomers is known for camphor (ball-and-stick models will be very helpful here). How many different stereoisomers would you expect actually could be prepared of the quinine and codeine structures ... 

At the stationary point defined by the variational conditions (14.1.69), we simplify the reduced Lagrangian according to the 2n + rule, arriving at an expression of the form... 

Arachno Clusters 2n + 6 Systems). In comparison to the number of known closo and nido boranes and heteroboranes, there are relatively fewer arachno species. Partly because of the lack of a large number of stmctures on which to base empirical rules, arachno stmctures appear to be less predictable than their closo and nido counterparts. For example, there are two isomeric forms of one with the arachno [19465-30-6] framework shown... 

The general XT E problem involves a multicomponent system of N constituent species for which the independent variables are T, P, N — 1 liquid-phase mole fractions, and N — 1 vapor-phase mole fractions. (Note that Xi = 1 and y = 1, where x, and y, represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to estabhsh the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by siiTUiltaneous solution of the N equihbrium relations ... 

From this it would appear that the ( - l)th-order wave function is required for calculating the th-order energy. However, by using the turnover rule and the nth and lower-order perturbation equations (4.32), it can be shown that knowledge of the nth-order wave function actually allows a calculation of the (2n-i-l)th-order energy. 

For variationally optimized wave functions (HF or MCSCF) there is a 2n -I- 1 rule, analogous to the perturbational energy expression in Section 4.8 (eq. (4.34)) knowledge of the Hth derivative (also called the response) of the wave function is sufficient for... 

The Lagrange technique may be generalized to other types of non-variational wave functions (MP and CC), and to higher-order derivatives. It is found that the 2n - - 1 rule is recovered, i.e. if the wave function response is known to order n, the (2n + l)th-order property may be calculated for any type of wave function. 

According to the Woodw ard-Hofmann rules the concerted thermal [2n + 2n] cycloaddition reaction of alkenes 1 in a suprafacial manner is symmetry-forbidden, and is observed in special cases only. In contrast the photochemical [2n + 2n cycloaddition is symmetry-allowed, and is a useful method for the synthesis of cyclobutane derivatives 2. 

Figures 6.4 shows some of the variety of possible shapes of P f) for elementary rules shown in the figures are the power spectra for rules Rll, R56, R150 and R200. The plots were generated for lattice size N = 2048, ignoring the first 15 transient steps and averaging a total of 20 runs. Also, since there are only N data points but 2N real Fourier components, half of the components are redundant. Thus, only the first half of the components are shown (see [H89b] or [H87] for a complete set of power spectra).

Molecules like lactic acid, alanine, and glyceraldehyde are relatively simple because each has only one chirality center and only two stereoisomers. The situation becomes more complex, however, with molecules that have more than one chirality center. As a general rule, a molecule with n chirality centers can have up to 2n stereoisomers (although it may have fewer, as we ll see shortly). Take the amino acid threonine (2-amino-3-hydroxybutanoic acid), for example. Since threonine has two chirality centers (C2 and C3), there are four possible stereoisomers, as shown in Figure 9.10. Check for yourself that the R,S configurations are correct. 

The orbital phase theory includes the importance of orbital symmetry in chanical reactions pointed out by Fukui [11] in 1964 and estabhshed by Woodward and Holiimann [12,13] in 1965 as the stereoselection rule of the pericyclic reactions via cyclic transition states, and the 4n + 2n electron rule for the aromaticity by Hueckel. The pericyclic reactions and the cyclic conjugated molecules have a conunon feature or cychc geometries at the transition states and at the equihbrium structures, respectively. 

The c/two-boranes BnH - (5 < n < 12) and the carboranes Bf C2Hf +2 are showpieces for the mentioned Wade rule. Further examples include the B12 icosahedra in elemental boron (Fig. 11.16) and certain borides such as CaB6. In CaB6, B6 octahedra are linked with each other via normal 2c2e bonds (Fig. 13.13). Six electrons per octahedron are required for these bonds together with the 2n + 2 = 14 electrons for the octahedron skeleton this adds up to a total of 20 valence electrons. The boron atoms supply 3 x 6 = 18 of them, and calcium the remaining two. 

The Wade rules can be applied to ligand-free cluster compounds of main-group elements. If we postulate one lone electron pair pointing outwards on each of the n atoms, then g — 2n electrons remain for the polyhedron skeleton (g = total number of valence elec-... 

KT1 does not have the NaTl structure because the K+ ions are too large to fit into the interstices of the diamond-like Tl- framework. It is a cluster compound K6T16 with distorted octahedral Tig- ions. A Tig- ion could be formulated as an electron precise octahedral cluster, with 24 skeleton electrons and four 2c2e bonds per octahedron vertex. The thallium atoms then would have no lone electron pairs, the outside of the octahedron would have nearly no valence electron density, and there would be no reason for the distortion of the octahedron. Taken as a closo cluster with one lone electron pair per T1 atom, it should have two more electrons. If we assume bonding as in the B6Hg- ion (Fig. 13.11), but occupy the t2g orbitals with only four instead of six electrons, we can understand the observed compression of the octahedra as a Jahn-Teller distortion. Clusters of this kind, that have less electrons than expected according to the Wade rules, are known with gallium, indium and thallium. They are called hypoelectronic clusters their skeleton electron numbers often are 2n or 2n — 4. 

B8C18 has a dodecahedral Bg c/o.vo-skclclon with 2n = 16 electrons. In this case, the Wade rule neither can be applied, nor can it be interpreted as an electron precise cluster nor as a cluster with 3c2e bonds. B4(BF2)6 has a tetrahedral B4 skeleton with a radially bonded BF2 ligand at each vertex, but it has two more BF2 groups bonded to two tetrahedron edges. In such cases the simple electron counting rules fail. 

The requirements necessary for the occurrence of aromatic stabilisation, and character, in cyclic polyenes appear to be (a) that the molecule should be flat (to allow of cyclic overlap of p orbitals) and (b) that all the bonding orbitals should be completely filled. This latter condition is fulfilled in cyclic systems with 4n + 2n electrons (HuckeVs rule), and the arrangement that occurs by far the most commonly in aromatic compounds is when n = 1, i.e. that with 6n electrons. IO71 electrons (n = 2) are present in naphthalene [12, stabilisation energy, 255 kJ (61 kcal)mol-1], and I4n electrons (n = 3) in anthracene (13) and phenanthrene (14)—stabilisation energies, 352 and 380 kJ (84 and 91 kcal) mol- respectively ... 

The cyclization of fully conjugated polyenes containing 2n + 2 jr-electrons (equation 1) was termed electrocydie by Woodward and Hoffmann, who showed that the steric course of such reactions was governed by the rules of orbital symmetryI. 3. 

In general, linear 7r-electron systems with Z1r = 2N electrons at the lowest energy levels have closed-shell singlet states while cyclic systems reach closed shell structures only when ZT = 4N + 2. Cyclic 7r-electron systems with Zn 4N + 2 will therefore exhibit multiplet ground states according to Hund s rules, and should be chemically reactive because of the unpaired electrons. Hiickel s rule that predicts pronounced stability for so-called aromatic ring systems with 4jV + 2 7r-electrons is based on this shell structure. The comparison with cyclic systems further predicts that ring closure of linear 7r-electron systems should be exothermic by an amount... 

Balakrishnarajan and Jemmis [32, 33] have very recently extended the Wade-Mingos rules from isolated borane deltahedra to fused borane ("conjuncto ) delta -hedra. They arrive at the requirement of n I m skeletal electron pairs corresponding to 2n + 2m skeletal electrons for such fused deltahedra having n total vertices and m individual deltahedra. Note that for a single deltahedron (i.e., m = 1) the Jemmis 2n + 2m rule reduces to the Wade-Mingos 2n I 2 rule. 

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