8.2.3.2 (2) Complete Active Space Calculations (CAS)
The plethora of stationary points found by the GVB (for 1,3-propanediyl) and CAS
(for 1,4-butanediyl) methods cannot be rivalled by ordinary model chemistry
methods. We now look at complete active space (CAS) methods, which are the
standard techniques for treating singlet diradicals; CAS was briefly mentioned in
Section 5.4.3, as a type of multiconfiguration CI (MCSCF) calculation. In CASSCF,
the coefficients of the determinants in the (limited) CI expansion of the molecular
wavefunction,andthe coefficients of the basis functions in the expansion of each
molecular orbital within the determinants of the expansion, are optimized. The
model chemistries are unable to reliably handle singlet diradicals because they
formulate the wavefunction as a single determinant which places the electrons of an
even-electron molecule pairwise in orbitals (Section 5.2.3.1). This is the Hartree–
Fock wavefunction, written as a Slater determinant. More than one determinant
is really needed because a single-determinant wavefunction presupposes the
absence of degenerate (or nearly degenerate) orbitals: if such orbitals are present,
the algorithm will simply fill one of them with a pair of electrons. Treating these
diradicals within the ab initio framework requires configuration interaction (CI,
Section 5.4.3). Here the molecular wavefunction is represented as a weighted sum
of determinants, rather than simply as one determinant. A full CI calculation would
include all the determinants derived from the Hartree–Fock one, including a
determinant in which one of the degenerate orbitals is doubly occupied, and one
in which the other is doubly occupied, as well as determinants corresponding to all
other possible excitations of electrons from occupied into formally empty (virtual)
orbitals (the number of virtual orbitals depending on the number of electrons and
the number of basis functions (Fig. 5.5). Such a full CI calculation, if done with an
infinitely big basis set, would exactly solve the Schr€odinger equation. This is out
of the question, and even with a large finite basis set full CI is applicable only
to very small molecules. The standard method for computations on singlet diradi-
cals is a limited form of CI, in which the molecular wavefunction is represented
by a weighted sum of the Hartree–Fock determinant and a set of determinants
corresponding to shuffling electrons among a carefully chosen set of molecular
orbitals. The chosen set of MOs is theactive space, and method is thecomplete
active space method(CAS). To refine the coefficients that, with the basis functions
comprise the MOs, we use the iterative SCF method (Sections 5.2.2 and 5.2.3.6.2),
so the full appellation of the technique iscomplete active space SCForCASSCF.
This gives a limited-CI wavefunction with corresponding geometry and energy, and
if needed the other usual properties that can be obtained from a wavefunction.
To do a CASSCF calculation, one must first choose the active space, that is, the
relevant MOs. Which MOs are relevant depends on the purpose of the calculation,
and on how “complete” one wants the active space to be. The unattainable limit of
course would be full CI. This will be illustrated with a few examples. Consider the
diradicals 1,3-propanediyl and 1,4-butanediyl. Intuitively, it seems that we should
consider at least these two MOs: the MO that resembles a bonding linear combina-
tion (Section 5.2.3.6) of the two p-type atomic orbitals on the end carbons and the
8.2 Singlet Diradicals 539