structure is a minimum and the bridged nonclassical structure is a transition state,
but with the 6–31G basis the bridged ion has become a minimum and the classical
one, although the global minimum, is not securely ensconced as such, being only
3.4 kJ mol#^1 lower than the bridged ion. At the post-Hartree–Fock (Section 5.4)
MP2 level with the 6–31G basis the bridged ion is a minimum and the classical
one has lost the dignity of being even a stationary point. The ethyl cation and
several other systems have been reviewed [ 75 ].
In summary, in many cases [ 39 ] the 3–21G (i.e. 3–21G()) or 6–31G basis sets,
or for that matter even the much faster molecular mechanics (Chapter 3) or
semiempirical (Chapter 6) methods, are entirely satisfactory, but thereareproblems
that require quite high levels of attack.
5.4 Post-Hartree–Fock Calculations: Electron Correlation.................
5.4.1 Electron Correlation..............................................
Electron correlation is the phenomenon of the motion of pairs of electrons in atoms
or molecules being connected (“correlated”) [ 76 ]. The purpose ofpost-Hartree–
Fock calculations (correlated calculations) is to treat such correlated motion
better than does the Hartree–Fock method. In the Hartree–Fock treatment, electro-
n–electron repulsion is handled by having each electron move in a smeared-out,
average electrostatic field due to all the other electrons (Sections 5.2.3.2and “Using
the Roothaan–Hall Equations to do Ab initio Calculations – the SCF Procedure”),
and the probability that an electron will have a particular set of spatial coordinates
at some moment is independent of the coordinates of the other electrons at that
moment. In reality, however, each electron at any moment moves under the
influence of the repulsion, not of an average electron cloud, but rather ofindividual
electrons (in fact current physics regards electrons as point particles – with wave
properties of course). The consequence of this is that the motion of an electron in a
real atom or molecule is more complicated than that for an electron moving in a
smeared-out field [ 77 ] and the electrons are thus better able to avoid one another.
Because of this enhanced (compared to the Hartree–Fock treatment) standoffish-
ness, electron–electron repulsion is really smaller than predicted by a Hartree–Fock
calculation, i.e. the electronic energy is in reality lower (more negative). If you walk
through a crowd, regarding it as a smeared-out collection of people, you will
experience collisions that could be avoided by looking at individual motions and
correlating yours accordingly. The Hartree–Fock method overestimates electron–
electron repulsion and so gives higher electronic energies than the correct ones, even
with the biggest basis sets, because it does not treat electron correlation properly.
Hartree–Fock calculations are sometimes said to ignore, or at least to neglect,
electron correlation. Actually, the Hartree–Fock method allows forsomeelectron
correlation: according to our current understanding, two electrons of the same spin
5.4 Post-Hartree–Fock Calculations: Electron Correlation 255