Page 204 - Academic Press Encyclopedia of Physical Science and Technology 3rd Organic Chemistry

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Organic Chemical Systems, Theory 451

of these mathematical difficulties the VB method has not Indeed, in the most commonly used form of the MO
had much numerical use, although it remains important as procedure this final step is omitted altogether, and a single
a conceptual tool. configuration built from optimized molecular orbitals is
used as an acceptable, if poor, approximation of the FCI
electronic wave function ψ. These optimum MOs are
2. The Molecular Orbital Method
known as the self-consistent field (SCF) or Hartree–Fock
The way in which almost all computations of molecular (HF) MOs. They are obtained using the variational prin-
electronic wave functions are performed nowadays is the ciple and demanding that the energy of the one config-
MO method already mentioned. In this method it is rec- uration under consideration be as low as possible. The
ognized from the outset that in a molecule electrons are energy difference between the HF description and the FCI
delocalized over the whole region of space spanned by the description is referred to as the correlation energy. It is
nuclear framework. Accordingly, AOs are first combined important to note that weak intermolecular interactions
into MOs. Unlike the VB structures mentioned above, (van der Waals interactions), important in processes such
these are still one-electron wave functions. as molecular recognition and complexation, cannot be cal-
Mathematically, the MOs are written as linear combi- culated at the SCF level. In calculations of these effects,
nations of atomic orbitals. From n AOs it is possible to inclusion of correlation effects is essential.
construct up to n linearly independent MOs. This is more One of the advantages of the MO method is that it is
than is usually needed since in the next step, in which elec- relatively easy to improve the SCF solution partially with-
tron occupancies are assigned to these molecular orbitals, out having to go all the way to the unreachable limit of
each can be occupied by up to two electrons of opposite FCI. Several methods for making such an improvement
spins. The unused MOs are referred to as unoccupied (or are available, such as (1) limited configuration interac-
virtual). Such an assignment of electron occupancies to tion in which the CI expansion is truncated in some syste-
MOs results in a many-electron wave function known as matic fashion well before the FCI limit is reached and (2)
a configuration. The number of configurations possible methods in which correlation effects are viewed as a small
is usually very large and is in fact equal to the number perturbation of the SCF solution and are treated by per-
of VB structures possible for the same spin multiplicity turbation theory. The use of these methods is particularly
(singlet, doublet, etc.). In order to obtain the final elec- important(1)whentheelectronicwavefunctionψ isbeing
tronic wave function ψ it is now necessary to mix all calculated for a geometry far removed from the molecular
possible MO configurations in a fashion analogous to that equilibrium geometry, (2) when the molecule has very low
used to mix the VB structures before. This configuration lying excited electronic states, (3) when the molecule is a
mixing is referred to as configuration interaction (CI). In- biradical, (4) if the calculation is performed for an excited
deed, the final wave function ψ obtained by the VB pro- electronic state, or (5) if intermolecular forces are to be
cedure in which all VB structures are included and that calculated.
obtained by the MO procedure in which all configurations So far we have discussed only the minimum basis set
are included will be one and the same. It is referred to approximation. This might be quite adequate if the AOs
as the full configuration interaction (FCI) wave function. were chosen in a truly optimal manner, but it would require
It cannot be obtained in practice for any but the small- a nonlinear optimization and this is normally not done.
est organic molecules because the number of possible Even so, in order to obtain a truly accurate solution for the
configurations increases astronomically with molecular electronic wave function ψ within the Born–Oppenheimer
size. approximation it would be necessary to increase the num-
The MO path to the FCI wave function ψ involves one ber of basis set AOs used in the calculation (in principle,
more step than the VB path, and this appears to be more to infinity).
complicated at first sight. However, the availability of the The form of the AOs usually adopted in numerical
additional step endows the MO method with more flexibil- work is normally chosen for computational convenience
ity than the VB method, which permits a mathematically (Gaussian-type orbitals) in a way that makes them quite
much simpler formulation. This flexibility arises in the different from the optimum orbitals of an isolated atom.
step of combining AOs in MOs. First, it is easy to ensure Thus, in practice most computations are not performed
that the MOs are mutually orthogonal, and this leads to with a minimum basis set but with extended basis sets,
an immense simplification in the calculation of matrix ele- and several such standard sets are in common use.
ments. Second, it is also possible to optimize the choice of The electronic wave functions resulting from such
the MOs in such a way as to speed up the convergence of large-scale calculations are usually not easy to visualize.
the final summation in which configurations are combined Frequently, it helps to draw the resulting electron densities
to obtain the state wave function. in the form of contour maps. Alternatively, electrostatic