Page 206 - Academic Press Encyclopedia of Physical Science and Technology 3rd Organic Chemistry

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Organic Chemical Systems, Theory 453

Most other molecular properties are normally evalu- the probability that an electron in that MO will be found
ated only at the equilibrium geometry, although strictly in that particular AO. This relation is particularly simple
speaking they should be calculated at a large number of in those semiempirical methods that use the zero differ-
geometries and averaged over the vibrational wave func- ential overlap approximation. Then, the AOs are mutually
tion of the state in question. Generally, they are obtained orthogonal and the relation between the square of the co-
by representing the observable by a quantum mechani- efficient and the probability is a simple proportionality.
cal operator and computing the expectation value of this In methods that use nonorthogonal AOs, such as the ab
operator over the wave function. initio ones, it is more difficult to define electron popu-
At the HF level, ab initio or semiempirical, some of lations for AOs and atoms. The procedure usually used
these properties can be obtained in an approximate manner is known as the Mulliken population analysis. This per-
more simply. Thus, the lowest ionization potential of the mits a calculation of electron densities in AOs and of total
molecule is approximately equal to the negative of the en- electron densities on atoms. These in turn can be related
ergy of the highest occupied molecular orbital (HOMO). to molecular dipole moments, infrared spectral intensi-
The electron affinity of the molecule is similarly approx- ties, and, much more approximately, to nuclear magnetic
imated by the negative of the energy of the lowest unoc- resonance (NMR) shielding constants.
cupied molecular orbital (LUMO). The energies of elec- The distributions of unpaired spin obtained in an anal-
tronic excitation are usually more complicated to obtain ogous fashion from the squares of coefficients of a singly
in that the introduction of CI may be quite necessary. occupied orbital are related to the hyperfine coupling con-
Those electron excitations that can be well described stants in electron spin resonance spectroscopy.
as a promotion of an electron from one single occupied There is another class of molecular properties that are
MO to one unoccupied MO are rare but exist in some not related in a simple way to the expectation value of
molecules. An example is the so-called L a band in the an operator: the so-called second-order properties. Some
absorption spectra of aromatic hydrocarbons and the first of the most important of these are molecular polarizabil-
intense band of polyenes. In the SCF approximation the ity, Raman intensities, and chemical shielding in NMR
electronic excitation energy is equal to the energy dif- spectroscopy. They can be computed by introducing an
ference between the orbital out of which the promotion outside perturbation such as an electric or magnetic field
occurs and the orbital into which it occurs, minus the re- explicitly into the calculation of the molecular wave func-
pulsion energy of two electron densities, each provided by tion. These calculations are more difficult and less reliable,
an electron in one of these two MOs. This approximates particularly with respect to magnetic properties, where the
the energy of the triplet excited state. The energy of the incompleteness of the basis sets used tends to make the
singlet excited state is higher by twice the exchange in- results dependent on the choice of origin of coordinates.
tegral between the two MOs involved (the self-repulsion Good progress has been made in the calculation of NMR
energy of a charge density produced by taking a product chemical shielding constants and their anisotropies, while
of the two orbitals). the calculation of intensities in Raman spectra still leaves
Due to the approximate nature of the SCF treatment much to be desired.
and to the additional approximations involved in the state-
ments just made, the results are usually more useful for
an interpretation of trends within a group of compounds IV. REACTION PATHS
rather than the absolute value for any one compound. The
semiempirical methods in particular have had much use
A. Thermal Reactions
in this kind of application.
A property related to this is the formation of intermo- A reaction involving motion through a single transition
lecular complexes characterized by a charge-transfer tran- state is referred to as an elementary reaction step. Most
sition, which frequently occurs in the visible region and reactions of organic compounds involve a sequence of
in which an electron is transferred from one molecule to such reaction steps in which the reacting system passes
another. The energy of this transition is related to the ion- through a series of intermediates and transition states that
ization potential of the donor and the electron affinity of separate them. The reaction intermediates are usually un-
the acceptor moiety and once again can be correlated with observed. An important part of the investigation of organic
the computed MO energies. reactivity is the determination of these individual steps in
Not only the energies but also the coefficients of the an overall sequence, usually accomplished using the tools
MOs computed in the SCF picture are approximately re- of chemical kinetics.
lated to observable properties. Thus, for a given MO, the In terms of transition state theory an understanding
square of the coefficient on a particular AO is related to of the reactivity of organic molecules under conditions