optimization we seek a global minimum onahypersurfaceinanenergyversus
basis function coefficients space defined byC¼f(basis function coefficients).
The wavefunction found may correspond toapointonthehypersurfacethatisnot
even a minimum, but rather a saddle point. Even if it is the global minimum, if we
are using a restricted Hartree–Fock (RHF) wavefunction rather than an unre-
stricted (UHF) one (end of Section 5.2.3.6.5), there are cases in which a lower
energy will be obtained by switching to a UHF function. The RHF function is then
said to show external or triplet instability. If within the type of wavefunction we
are using (RHF or UHF) a better function can be found by moving to another point
on the hypersurface, away from a saddle point or a higher-energy minimum, the
wavefunction is said to show internal instability. There are algorithms that will
test for wavefunction instability and alter coefficients to obtain the best wave-
function from the chosen basis set. SeegerandPoplepioneeredthemathematical
analysis of and some cures for wavefunction instability [ 21 ], and in more chemi-
cal language Dahareng and Dive have examined about 80 molecules for the
phenomenon and offer some generalizations [ 22 ]. Instability can occur also
with post-Hartree–Fock (correlated) (Section 5.4)wavefunctions[ 23 ]. Chemists
do not routinely test for wavefunction stability, and indeed it is rarely a problem
except for unusual molecules, e.g. p-benzyne [ 24 ]. However, when investigating
exotic (as judged by the experienced chemist) molecules, it is good practice to
carry out this check.
The Hartree–Fock SCF method is, of course, in exactly the same spirit as
the procedure described inSection 5.2.2using the Hartree product as our total or
overall wavefunctionC. The main difference between the two methods is that the
Hartree–Fock method representsCas a Slater determinant of component spin MOs
rather than as a simple product of spatial MOs, and a consequence of this is that the
calculation of the average coulombic field in the Hartree method involves only the
coulomb integralJ, but in the Hartree–Fock modification we need the coulomb
integralJ andthe exchange integralK, which arises from Slater determinant terms
that differ in exchange of electrons. BecauseKacts as a kind of “Pauli correction”
to the classical electrostatic repulsion, reminding the electrons that two of them of
the same spin cannot occupy the same spatial orbital, electron–electron repulsion is
less in the Hartree–Fock method than if a simple Hartree product were used. Of
courseKdoes not arise in calculations involving no electrons of like spin, as in H 2
or (Sections 4.4.2 and 5.2.3.6.5) HHe+, which have only two, paired-spin, electrons.
At the end of the iterative procedure we have the MO’sciand their corresponding
energy levelsei, and the total wavefunctionC, the Slater determinant of theci’s.
Theeican be used to calculate the total electronic energy of the molecule, and
the MO’sciare useful heuristic approximations to the electron distribution, while
the total wavefunctionCcan in principle be used to calculate anything about the
molecule, as the expectation value of some operator. Applications of the energy
levels and the MO’s are given inSection 5.4.
196 5 Ab initio Calculations