Done right means that in comparing the energy of two systems one must utilize
corresponding electron promotions (“excitations”). I’ll illustrate this by compar-
ing the energy of two well-separated beryllium atoms with the twice the energy
of one beryllium atom. I choose the beryllium atom because this 4-electron atom
is the simplest closed-shell species which gives some choice (among 2s and 2p
set) of occupied orbitals, lending a little resemblance in this respect to the
molecular case.
A CASSCF(2,2)/6-31G* calculation was done on one beryllium atom, using a
simplified version of the procedure in Chapter 8 for molecules: an orbital
localization step is pointless for an atom, and in the energy calculation optimi-
zation is meaningless. First an STO-3G wavefunction was obtained and the
atomic orbitals (AOs) were visualized; this showed MO1, 2, 3, 4, and 5 to be,
respectively, 1s, 2s (both occupied), and three energetically degenerate unoccu-
pied 2p orbitals. The active space was chosen to consist of the 2s and a 2p
orbital, and a single-point (no optimization requested) CASSCF(2,2)/6-31G*
calculation was done. The energy was#14.5854725 hartree.
A CASSCF(2,2)/6-31G* calculation was now done on two beryllium atoms
separated by 20 A ̊, where they should be essentially noninteracting; the coordi-
nates of these two atoms were input treating them as one unit, an 8-electron
supermolecule. An STO-3G wavefunction was obtained and visualized. This
showed as expected a set of molecular orbitals (MOs), since this species is
formally a molecule. With five AOs from each atom, we have ten AOs resulting
from plus and minus combinations (bonding and antibonding only in a formal
sense, because of the separation). These were:
MO1, 1sþ1s; MO2, 1s-1s; same energy. These two account for two pairs of
electrons.
MO3, 2sþ2s; MO4, 2s-2s; same energy. These two account for two pairs of
electrons.
MO5, 2pxþ2px; MO6, 2px#2px; ...., MO10, 2pz#2pz, All six of these, 5–10,
same energy, unoccupied.
The critical choice was made of a CASSCF(4,4)/6-31G* calculation; the active
space is thus the degenerate filled 2sþ2s and 2s#2s pair of MOs, and the
degenerate empty 2pxþ2pxand 2px#2pxpair of MOs. CASSCF(4,4) was
chosen because it corresponds to the CASSCF(2,2) calculation on one beryllium
atom in the sense that we are doubling up the number of electrons and orbitals in
our noninteracting system. This calculation gave an energy of#29.1709451
hartree. We can compare this with twice the energy of one beryllium atom, 2%
#14.5854725 hartree¼#29.1709450 hartree.
Let’s compare these CASSCF results with those for a method that is not size-
consistent, CI with no “complete” aspect. We’ll use CISD (configuration inter-
action singles and doubles;Section 5.4.3). Here are the results for CISD/6-31G*:
One beryllium atom,#14.6134355
Two beryllium atoms separated by 20 A ̊,#29.2192481.Answers 649