Essentials of Computational Chemistry

186 6 AB INITIO HARTREE–FOCK MO THEORY
r CH
r CCl
H
Cl(a) C Cl(b)
H H
r CCl(b)
∗
H
r CCl(a)
Cl(a) C Cl(b) Cl(a) C Cl(b)
H H
H
∗
H H
Figure 6.8 Reduced-dimensionality PESs for the chloride/methyl chloride system. On the left is the
two-dimensional surface associated with a D3h symmetry constraint. On this surface, the point marked
‡ is a minimum. The simultaneous shortening or lengthening of the C–Cl bonds (simultaneous to
preserve D3h symmetry) while allowing the C–H bond lengths to relax is indicated by the dashed
line on this surface. The same process is indicated by the dashed line on the surface to the right,
whose coordinates are the individual C–Cl bond lengths, and point ‡ again represents the minimum
on this line. However, movement off the dashed line can lower the energy further. Movement along
the solid line, which involves lengthening one C–Cl bond whilst shortening the other, corresponds
to the reaction path for nucleophilic substitution from one equilibrium structure to another (points
marked ∗), and illustrates that the minimum-energy structure under the D3h constraint is actually a TS
structure on the full PES
symmetry, or in an MO having π ∗ NO character that is of a ′ symmetry. These two electronic
states are fundamentally different. The symmetry of a doublet electronic state is simply the
symmetry of the half-filled orbital if all other orbitals are doubly occupied, so we would
refer to the two possible electronic states here as 2 A ′′ and 2 A ′ , respectively. When symmetry
is imposed, we will have a block diagonal Fock matrix and the unpaired electron will appear
in either the a ′ block or the a ′′ block, depending on the initial guess. Once placed there, most
SCF convergence procedures will not provide any means for the electronic state symmetry to
change, i.e., if the initial guess is a 2 A ′ wave function, then the calculation will proceed for
that state, and if the initial guess is a 2 A ′′ wave function, then it will instead be that state that
is optimized. The two states both exist, but one is the ground state and the other an excited
state, and one must take care to ensure that one is not working with the undesired state.
Typically, one can assess the nature of the state (ground vs. excited) after convergence
of the wave function. Continuing with our example, let us say that we have optimized the
2 A ′ state. We can then take that wave function, alter it so that the occupation number of the
highest occupied a ′ orbital is zero instead of one, and the occupation of the lowest formerly