All the deductions from correlation diagrams are distilled in the general Woodward-Hoffmann Rule (F 40 - see page 44 for the rule for excited states):

A ground-state pericyclic change is symmetry-allowed when the total number of (4q + 2)_s and (4r)_a components is odd.

2 Carbons come from an alkene and 1 carbon from a carbene or carbenoid.
Dihalocarbene addition (F 47-48, V&S 962.

Simmons-Smith reaction (V&S 848):-

Syn,syn (suprafacial,suprafacial) concerted cycloaddition is forbidden in the ground state, so reactions usually have to occur stepwise (with one bond formed at a time).
Zwitterion intermediates i.e. C+-C-C-C- requires suitable substitutents to stabilise the charges (F 26), and this mode of reaction is favoured by polar solvents (2nd year lab experiment).

A diradical, i.e. C.-C-C-C., is also possible, again favoured by suitable stabilising groups (F 27).
Supra,supra-concerted is allowed photochemically (F 17, C&S B310).
Alkene + alkene ® cyclobutane.
Alkene + carbonyl compound ® oxetane (Paterno-Buchi reaction).
It is also found that alkene + ketene (R2C=C=O) additions occur efficiently in thermal reactions. This may involve both double-bonds of the ketene (F 27, 45-46).

In principle:- alkene (2p ) + allyl anion (4p ) ® cyclopentyl carbanion. In practice this is an important method of 5-membered heterocycle synthesis known as 1,3-dipolar cycloaddition involving heteroatomic systems which are isoelectronic with allyl anion (F 10-12, C&S B300-307). These reactants are known as dipolarophiles, and they always have to be represented by zwitterionic (or dipolar) structures (make sure you can draw them!).

1,3-Dipolar cycloadditions always involve syn (suprafacial)-addition to both alkene and dipolarophile, and may be regioselective (but the regiochemistry is often complicated and will not be discussed).

The effects of orbital symmetry are seen in the stereochemistry of electrocyclic reactions, in particular whether the process is conrotatory or disrotatory.

Cyclobutene ring opening is conrotatory(V&S 607-610):-

Hexatriene ring closures are disrotatory (V&S 610-612):-

The rules also apply to various reaction of anions and cations (F 58, 64-67). Cyclopropyl halides solvolyse with disrotatory ring opening to produce allyl cations. (F 64-65, C&S A607). In the developing cation, overlap occurs under the ring, which means that in the reaction shown below the product should be the least stable 1,3-dimethylallyl cation - and it is!

The simplest sigmatropic reaction is a 1,2-shift, and it can be shown that these can only be concerted and suprafacial in carbocations (F 76, V&S 335-339). In the 1,2-methyl shift shown below, the three carbons are held together by a three-centre two-electron bond at the transition state. Any extra electrons, as would be present in the radical or anion, would have to go into anti-bonding orbitals.

Other sigmatropic shifts can be analysed by the general Woodward-Hoffmann rule, but a simple and reliable way to analyse the bonding in the transition state is to formally break the rearranging bond and look at the interaction of the two radicals. Thus in a suprafacial 1,3-hydrogen shift, the s orbital of the hydrogen atom cannot bond simultaneously to both termini of the allyl radical, since the odd electron there is in an antisymmetric orbital. Therefore these 1,3-shifts are forbidden, and reactions like keto-enol tautomerism always occur stepwise, e.g.:

On the other hand, suprafacial 1,5-shifts are allowed and lots of examples are known (F 72-3). 1,7-Shifts should occur antarafacially, and this is possible geometrically as shown below (F 72-3). Finally, the transition state for 3,3-shifts can be seen as composed of two allyl radicals.

An interesting example of a 1,7-hydrogen shift occurs with vitamin D(F 5):-

It is found that a chair-form transition state is usually preferred for 3,3-sigmatropic reactions, as in the following Cope rearrangement (F 78):

Important preparative examples of 3,3-shifts include oxy-Cope rearrangements (C&S B319-321). These are strongly catalysed by converting the alcohol to an alkoxide, and the conversion of this (by rearrangement) to an enolate provides the driving force to overcome the increase strain energy on going from a 6- to a 10-membered ring.:

Claisen rearrangements (V&S 1006) are also preparatively useful:

The ene reaction (F 84-6, C&S B332) is another type of reaction which can have an aromatic transition state. The McLafferty rearrangement (V&S 916) is a retro (reverse)-ene reaction and so is ester pyrolysis and similar reactions (F 86):