Biomedical Engineering Reference
In-Depth Information
excitations alone are included we have CID (
CI doubles
). Such truncations can lead to a
problem called
size consistency
that I can illustrate by considering a very simple problem,
that of two neon atoms. Table 19.1 shows aCISD/6-311G* calculation for dineon at 5000 pm
separation, together with the free atom energy.
Table 19.1
Dineon calculations, 6-311G* basis set
Method
Dineon energy/
E
h
at 5000 pm
Atom energy/
E
h
HF-LCAO
−
257.0451061
−
128.52255305
CISD
−
257.4466147
−
128.7283956
The HF-LCAO energy of a pair of neon atoms at large separation is exactly twice the
free atom value, but this is not the case for the CISD calculation. If we have an ensemble of
n
particles and their energy is
n
times the energy of a single particle, we say that the theory
scales correctly
(or that the method is
size consistent
). Full CI calculations scale correctly,
but truncated CI expansions do not.
After double excitations, quadruple excitations are next in order of importance. If singles,
doubles, triples and quadruples are included the acronym becomes CISDTQ.
19.3 Coupled Cluster Method
The coupled cluster (CC) method was first used by physicists studying nuclear struc-
ture. Bartlett's (1989) review is fairly recent. The fundamental equation relates a HF
wavefunction Ψ
HF
to the best possible wavefunction Ψ by
exp
T
Ψ
HF
Ψ
=
(19.9)
The exponential operator is defined by a Taylor series expansion
exp
T
∞
T
k
k
=
(19.10)
!
k
=
0
and the
cluster operator
is defined as
T
=
T
1
+
T
2
+···+
T
n
(19.11)
where
n
is the number of electrons in themolecule. The operators have the effect of replacing
occupied spin orbitals in Ψ
HF
with virtual spin orbitals.
T
1
performs all singly excited
substitutions,
T
2
performs all doubly excited configurations and so on until all
n
electrons
have been promoted fromfilled to virtual spinorbitals. The effect of the exponential operator
