MO that resembles an antibonding linear combination of these atomic orbitals. We
want these to be our HOMO and LUMO. The CAS wavefunction would then be
composed of the HF determinant plus all determinants in which the two formally
unpaired electrons are distributed (cf.Fig. 5.2.2) among the HOMO and LUMO.
This is the minimum active space for a CAS calculation on these species, and is
called a CASSCF(2,2) calculation (2 electrons, 2 MOs). This means that two
electrons are being distributed in all possible ways among two MOs.
A CASSCF Calculation on 1,4-Butanediyl
The procedure will be described first for 1,4-butanediyl, which failed all our simple
model chemistries tests. We first choose a starting geometry. This will depend
somewhat on the purpose of our study. If we wish to compute the reaction profile
for ring opening of cyclobutane to the proximate diradical, i.e. to the immediate
relative minimum following ring opening (the concept of a well-defined transition
state stationary point seems inapplicable here [ 62 ]), we might select a starting
geometry that resembles cyclobutane with a stretched C–C bond. If we wish to
explore the whole 1,4-butanediyl potential energy surface, we would perform
geometry optimizations starting with all reasonably distinct conformations, created
randomly or by systematically altering the torsion angles of a beginning structure.
Here we consider a CASSCF calculation starting with the C2hconformation of
1,4-butanediyl (Fig.8.8). The exact keywords for each step depend on the program,
and are not given specifically here.
Step 1 obtains a wavefunction for our starting “guess” geometry. For speed
and to limit the number of MOs (which appear in step 2), an STO-3G basis set
(Sections 5.3.2 and 5.3.3) is usually used. A single point calculation with the
specified basis is requested and the wavefunction is stored in a file (Gaussian [ 49 ]
calls this acheckpoint file) to be recalled in subsequent steps.
Step 2 uses the wavefunction from step 1 tolocalizethe MOs. To recapitulate
(Section 5.2.3.1): normally the Hartree–Fock wavefunction is represented straight-
forwardly as a Slater determinant in which the chosen basis set {f} is used to
expand the occupied MOscas linear combinations of theffunctions. The Fock
matrix derived from this determinant is called thecanonical Fock matrix, and when
repeatedly diagonalized and refined in the SCF process it yields a set of MOs, the
canonical MOs. These MOs commonly do not resemble the bonding (or inferred
antibonding) orbitals of Lewis structures: for example, visualizing the canonical
MOs of H 2 O, one does not see one MO corresponding to one of the O–H bonds, and
one corresponding to the other O–H bond. Canonical MOs tend to be delocalized
over the whole molecule, eluding correspondence with conventional Lewis bonds.
However, it is possible to combine the canonical MOs so as to get localized orbitals
corresponding to bonds and lone pairs. This is done (in outline) by manipulating the
canonical Hartree–Fock wavefunction determinant by adding multiples of rows or
columns to other rows or columns. The wavefunction is unaltered mathematically
(Section 4.3.3.7,Determinants, property number 6): it will give the same observable
540 8 Some “Special” Topics