Essentials of Computational Chemistry

7.2 MULTICONFIGURATION SELF-CONSISTENT FIELD THEORY 207
squares of all CSF coefficients is unity and the percent contribution of any CSF to the wave
function is simply its expansion coefficient squared.
MCSCF calculations in practice require much more technical expertise than do singleconfiguration
HF analogs. One particularly difficult problem is that spurious minima in
coefficient space can often be found, instead of the variational minimum. Thus, convergence
criteria are met for the self-consistent field, but the wave function is not really optimized. It
usually requires a careful inspection of the orbital shapes and, where available, some data
on relative energetics between related species or along a reaction coordinate to ascertain if
this has happened.
A different issue requiring careful attention is how to go about selecting the orbitals that
should be allowed to be partially occupied, and how to specify the ‘flexibility’ of the CSF
expansion. We turn to this issue next.
7.2.2 Active Space Specification
Selection of orbitals to include in an MCSCF requires first and foremost a consideration of
the chemistry being examined. For instance, in the TMM example above, a two-configuration
wave function is probably not a very good choice in this system. When the orbitals being
considered belong to a π system, it is typically a good idea to include all of them, because
as a rule they are all fairly close to one another in energy. Thus, a more complete active
space for TMM would consider all four π orbitals and the possible ways to distribute the
four π electrons within them. MCSCF active space choices are often abbreviated as ‘(m,n)’
where m is the number of electrons and n is the number of orbitals, so this would be a (4,4)
calculation.
Sometimes reaction coordinates are studied that involve substantial changes in bonding.
In such an instance, it is critical that a consistent choice of orbitals be made. For instance,
consider the electrocyclization of 1,3-butadiene to cyclobutene (Figure 7.2). The frontier
orbitals of butadiene are those associated with the π system, so, as just discussed, a (4,4)
approach seems logical. However, the electrocyclization reaction transforms the two π bonds
into one different π bond and one new σ bond. Thus, a consistent (4,4) choice in cyclobutene
would involve the π and π* orbitals and the σ and σ * orbitals of the new single bond.
frontier orbitals
p1 , p2 , p3 , p4 diabatic correlation
active orbitals
p, p*, s, s*
Figure 7.2 The frontier orbitals of s-cis-1,3-butadiene are the four π orbitals (π2 is the specific
example shown). If these orbitals are followed in a diabatic sense along the electrocyclization reaction
coordinate, they correlate with the indicated orbitals of cyclobutadiene