Computational Chemistry

LCAO expansion, Eq.5.164¼5.52. A widely-used version of the MCSCF method
is the CASSCF method, in which one carefully chooses the orbitals to be used in
forming the various CI determinants. Theseactive orbitals, which constitute the
active space, are the MOs that one considers to be most important for the process
under study. Thus for a Diels–Alder reaction, the twopand twop MOs of the
diene and thepandp MO of the alkene (the dienophile) would be a reasonable
minimum [ 98 ] as candidates for the active space of the reactants; the six electrons in
these MOs would be theactive electrons, and with the 6–31G basis this would be a
(specifying electrons, MOs) CASSCF (6,6)/6–31G calculation. CASSCF calcula-
tions are used to study chemical reactions and to calculate electronic spectra. They
require judgement in the proper choice of the active space and are not essentially
algorithmic like other methods [ 99 ]. An extension of the MCSCF method is multi-
reference CI (MRCI), in which the determinants (the CSFs) from an MCSCF
calculation are used to generate more determinants, by promoting electrons in
them into virtual orbitals (multireference, since the final wavefunction “refers
back” to several, not just one, determinant). Just as HF geometries can be subjected
to MPn (commonly MP2) single-point calculations to account for dynamic correla-
tion and obtain better relative energies, geometries from CASSCF calculations,
which are commonly used to take static correlation into account, can be subjected to
(usually single-point) perturbational calculations to account fordynamiccorrela-
tion. The most reliable and widely-used of these “post-CAS” methods is the
CASPT2N (complete active space perturbational theory second order, nondiagonal
one-particle operator, a kind of analogue of MP2) [ 100 ]. CASSCF calculations are
illustrated in some detail in Section 8.2.
Thecoupled cluster(CC) method is actually related to both the perturbation
(Section 5.4.2)andtheCIapproaches(Section 5.4.3). Like perturbation theory,
CC theory is connected to the linked cluster theorem (linked diagram theorem)
[ 101 ], which proves that MP calculations aresize-consistent(seebelow).Like
standard CI it expresses the correlated wavefunction as a sum of the HF ground
state determinant and determinants representing the promotion of electrons from
this into virtual MOs. As with the Møller–Plesset equations, the derivation of the
CC equations is complicated. The basic idea is to express the correlated wave-
functionCas a sum of determinants by allowing a series of operatorsT^ 1 ,T^ 2 ,...to
act on the HF wavefunction:

C¼ 1 þT^þ

T^^2

2!

þ

T^^3

3!

þ(((

!

CHF¼e

T^

CHF ð 5 : 170 Þ

whereT^¼T^ 1 þT^ 2 þ(((.TheoperatorsT^ 1 ,T^ 2 ,...areexcitation operatorsand
have the effect of promoting 1, 2, etc., respectively, electrons into virtual spin
orbitals. Depending on how many terms are actually included in the summation
forT^,oneobtainsthecoupled cluster doubles(CCD),coupled cluster singles and

274 5 Ab initio Calculations