Interactions of More Than Two Orbitals

The theory of two-orbital interactions has been described in the preceding sections. The elements of the chemical orbital theory also include the theories of of three-orbital interactions and cyclic interactions of more than two orbitals (Scheme 1). 

Another theory as an important element of the chemical orbital theory is an orbital phase theory for cyclic interactions of more than two orbitals. The cyclic orbital interactions are controlled by the continuity-discontmuity of orbital phase [21-23]. 

When more than two orbitals are involved, the energy change must take into account all important orbital interactions. This will be illustrated for the formaldehyde-ethylene case following the method of Herndon and Giles/131 If we assume the bonds are half-formed in the transition state, then the exchange integral y is just equal to iP- Since p has a value of about 40 kcal/mole, then... 

Use two-orbital interaction diagrams to explain the observed features of the following systems. Note A clear orbital interaction diagram includes pictures of the orbitals before and after the interaction and shows the disposition of the electrons. A brief verbal explanation of the diagram is also desirable. If more than two orbitals seem to be involved, use your judgment to choose the two most important orbitals. 

The hydrogen bond is an electrostatic interaction between a hydrogen atom bound to an electronegative atom such as oxygen, fluorine or nitrogen, and the free electrons of other atoms. An actual covalent bond is not possible, because that would result in the presence of more than two electrons in the orbital around the hydrogen atom. 

To describe atoms with several electrons, one has to consider the interaction between the electrons, adding to the Hamiltonian a term of the form Ei< . Despite this complication it is common to use an approximate wave function which is a product of hydrogen-like atomic orbitals. This is done by taking the orbitals in order of increasing energy and assigning no more than two electrons per orbital. 

Radicals and excited states have an orbital occupied by one electron. The interaction of the singly occupied orbital with a vacant orbital (Scheme 15) and with a singly occupied orbital (Scheme 16) leads to the stabilization. The stabilized orbitals occupy one and two electrons, respectively. There are no electrons in the destabilized orbital. For the interaction with a doubly occupied orbital there are two electrons in the stabilized orbital and one electron in the destabilized orbital (Scheme 17). Although the destabilization of the out-of-phase combined orbital is greater than the stabilization of the in-phase combination, there is one more electron in the stabilized orbital. Net stabilization is then expected. 

The strengths of the couplings between pairs of CSFs whose energies cross are evaluated through the SC rules. CSFs that differ by more than two spin-orbital occupancies do not couple the SC rules give vanishing Hamiltonian matrix elements for such pairs. Pairs that differ by two spin-orbitals (e.g. I.. (f>a... (f>b...l vs I.. (f>a. .. ) have interaction... 

There are a number of examples of pericyclic reactions for which the interaction diagram is not simply connected. We define a non-simply connected pericyclic system as one in which in the interaction diagram at least one basis orbital is connected to more than two others. An example is shown with its interaction diagram in Equations 11.37-11.38. 

Another way to describe delocalized bonding uses die MO approach. The same principles of overlap of AOs can be applied to systems where more than two p AOs overlap to form n systems. First, die number of MOs produced by die overlap will be die same as die number of atomic p orbitals which interact. Thus for the allyl system where three contiguous p orbitals interact, there will be three MOs produced from die interaction of three 2p AOs. For the butadienyl system where there are four contiguous p orbitals interacting, four MOs will result, and so on. 

The effectiveness of NSO s in reducing the expansion size in systems with more than two electrons is not as great and, in fact, for larger systems, their use is not practical. The loss in practicality is immediately obvious when one realizes that in order to obtain them, one must diagonalize the first-order density matrix of the exact wavefunction, i.e. a full configuration interaction must first be performed. Two methods have been introduced in order to regain the initial usefulness of natural orbitals the pseudonatural orbital method and the approximate or iterative natural orbital method. 

---