properties, like geometry, spectra, and dipole moment. There are various require-
ments that can be enforced to produce different kinds of localized orbitals [ 63 ]; the
most widely used MO localization schemes in CAS calculations are probably NBO
(natural bond orbitals) and Boys localization. Boys localization [ 64 ] generates MOs
that are as compact as possible, and NBO localization [ 65 ] creates MOs each of
which is essentially composed of basis functions on just two atoms; both might thus
be expected to resemble Lewis structures. We visualize the localized orbitals and
inspect them, in search of which ones to assign to theactive space.
The active space is the set of MOs among which the electrons will be distributed:
electrons will be promoted from the formally occupied orbitals into the formally
unoccupied ones in a CI calculation limited to the chosen orbitals. The orbitals
are chosen according to the purpose of the calculation. If we simply wish to obtain
the geometry of a diradical like 1,4-butanediyl, then we look for the troublesome
orbitals, the ones now (we hope) localized on the end carbons. An orbital corres-
ponding to this occupied and one corresponding to this vacant orbital constitute the
minimum active space for our calculation. Since two electrons and two orbitals are
involved, this is called a CASSCF(2,2) calculation; withaelectrons andborbitals
we have a CASSCF(a,b) calculation. Figure8.9clarifies this: the algorithm will
recognize this active space as consisting of the two frontier orbitals (HOMO and
LUMO); we want these to be the two MOs that are localized on the end carbons.
If we had decided to use a (6,6) active space, by including in addition the two
proximate C–Csbonds and their antibonding counterparts, the active space would
be recognized as the HOMO, HOMO-1, HOMO-2, and LUMO, LUMOþ1, and
LUMOþ2. If an orbital that should be in the active space (as shown by its
.
.
..
.
.active space for a CAS(2, 2)
calculation on .CH 2 CH 2 CH 2 CH 2.EnergyMO 16MO 17Fig. 8.9 The active space for a CASSCF(2,2) calculation on 1,4-butanediyl. There are two
relevant MOs: the highest occupied and lowest unoccupied MO, and two electrons to be
distributed among these. The relevant MOs must be determined by inspection (preferably visual)
to be the right ones for the purpose of the calculation: see Fig. 8.10
8.2 Singlet Diradicals 541