Page 203 - Academic Press Encyclopedia of Physical Science and Technology 3rd Organic Chemistry

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450 Organic Chemical Systems, Theory

on the position of the transition state are given by the is possible to obtain useful approximate solutions. These
Hammond postulate: The transition structure moves away becomegraduallylessaccurateasthemoleculeinquestion
from the corner that has been stabilized. The shape of the increases in size.
surface along the other diagonal is just the opposite: The Essentiallyalloftheapproximatemethodsforthecalcu-
reaction path is of low energy relative to the two corners. lation of electronic wave functions for an organic molecule
For this reason a change in the relative stability of the start with a basis set of AOs mentioned above in the qual-
geometries represented by the two corners has an effect itative discussion. In a so-called minimum basis set de-
opposite to that discussed in the Hammond postulate. A scription, the AOs assigned to an atom in a molecule are
stabilization of one of the corners will move the transition those that would be occupied in a neutral isolated atom
point closer to that corner. of that element plus the remainder of its valence shell.
While it can be generally assumed that the molecules in The AOs can be used in two ways for the construction of
the initial catchment basin are almost exactly at thermal molecular electronic wave functions.
equilibrium, which is being perturbed quite insignificantly
bytheescapeofthemostenergeticmoleculesoverasaddle
1. The Valence–Bond Method
point or saddle points, this is not always true. In hot ground
state reactions a molecular assembly is initially generated In the conceptually simple but computationally very diffi-
with a vibrational energy content much higher than would cult valence–bond (VB) method one first assigns all avail-
correspond to the temperature of the surrounding medium, able electrons to individual AOs in a way that satisfies the
perhaps as a result of being born in a photochemical pro- Pauli principle; that is, placing no more than two elec-
cess or in a very highly exothermic thermal process. Such trons in any one AO and assigning opposite spins to elec-
an assembly can then react at much higher rates. trons that share the same AO. The method of doing this
A study of the reaction rates of molecules starting in is not described in detail here, but it is possible to ensure
individual vibronic levels provides much more detailed that the resulting many-electron wave function is properly
information than the study of molecules that are initially spin-adapted and represents a singlet, a doublet, a triplet,
at thermal equilibrium but has so far remained a domain and so on, as appropriate. An example is the H H VB
+
−
of small-molecule chemical physicists and has had little structure for the H 2 molecule, in which the two available
impact on the theory of organic reactions. electrons are both assigned to the 1s orbital of the hy-
An important point to consider is the possibility that a drogen on the right. Since they are then necessarily spin-
molecule has a choice of more than one saddle over which paired this represents a singlet structure.
it can escape from an original catchment basin. The nature It is generally possible to write a very large number of
of products isolated from such competing reactions can such VB structures for any given molecule. For instance,
depend on the choice of reaction conditions. Under kinetic for the H 2 molecule we could equally well have written
−
control the molecules of the two products are not given H H or four additional structures in which one electron
+
an opportunity to travel back over their respective saddles was assigned to each of the available 1s AOs and which
into the initial catchment basin. The preferred product will differed only in the assignment of electron spin. In order
be that which is being formed faster. On the other hand, to obtain the desired final electronic wave function ψ it is
if the molecules are provided with an opportunity to return necessary to allow these VB structures to mix.
to the initial catchment basin so that an equilibrium is Mathematically,thiscorrespondstotakingalinearcom-
eventually approached, the reaction is said to be run under bination of all structures of the same spin multiplicity. In
thermodynamic control and the thermodynamically more order to determine the coefficients with which each of the
stable product will be isolated. structures enters into the final mixture, one uses the vari-
ational theorem, which states that the best approximation
to the ground wave function ψ(S 0 ) is given by those co-
B. Electronic Wave Functions
efficients that lead to a wave function of lowest energy.
Once a wave function of a quantum mechanical state is Determining the coefficients requires the diagonalization
available, it is possible to calculate its observable prop- of a matrix the dimension of which is equal to the num-
erties in a straightforward manner. Unfortunately, the ber of VB structures of a given multiplicity possible for
Schr¨odinger equation is very difficult to solve, and this the molecule. This increases very rapidly with the size of
has not been achieved exactly for any but the simplest the molecule, making a full calculation impractical even
molecular systems. The fundamental mathematical prob- for quite small organic molecules. Another important dif-
lem can be traced to the fact that even for fixed nuclear ficulty in the calculation is the complicated nature of the
positions the molecule still represents a many-body sys- computationofeachoftheelementsofthematrix,dueulti-
tem if it contains more than one electron. Nevertheless, it mately to the fact that the AOs are not orthogonal.Because