Chemistry Reference
In-Depth Information
converges and (ii) that the unperturbed eigenfunction
0
satisfies exactly
the zeroth-order equation with eigenvalue E
0
. While E
1
, the first-order
correction to the energy, is the average value of the perturbation over
the unperturbed eigenfunction
c
c
0
(a diagonal term), the second-order
term E
2
is given as a transition (nondiagonal) integral in which state
c
0
is changed into state
c
1
under the action of the perturbation V. Further
details are left to Chapter 10.
Møller-Plesset theory (Møller and Plesset, 1934) starts from E(HF)
considered as the result in first order of perturbation theory, E
ð
HF
Þ¼
E
0
þ
E
1
, assuming as unperturbed
c
0
the single determinant HF wave-
function, and as first-order perturbation the difference between the
instantaneous electron repulsion and its average value calculated at the
HF level. Therefore, it gives directly a second-order approximation to
the correlation energy, since by definition
E
ð
correlation
Þ¼
E
ð
true
Þ
E
ð
HF
Þ
ð
8
:
8
Þ
when E(true) is replaced by its second-order approximation
E
ð
true
Þ
E
0
þ
E
1
þ
E
2
¼
E
ð
HF
Þþ
E
2
ð
MP
Þ
ð
8
:
9
Þ
Hence:
E
ð
true
Þ
E
ð
HF
Þ
E
2
ð
MP
Þ¼
E
ð
MP2
Þ
¼
second-order approximation to the correlation energy
ð
8
:
10
Þ
It is seen that only biexcitations can contribute to E
2
, since mono-
excitations give a zero contribution for HF
0
(Brillouin's theorem).
Comparison of SCF and MP2 results for the
1
A
1
ground state of the
H
2
O molecule (Rosenberg et al., 1976; Bartlett et al., 1979) shows that
MP2 improves greatly the properties (molecular geometry, force con-
stants, electric dipole moment) but gives no more than 76% of the
estimated correlation energy.
Y
8.4 THE MP2-R12 METHOD
This is a Møller-Plesset second-order theory, devised by Kutzelnigg and
coworkers (Klopper and Kutzelnigg, 1991),
3
which incorporates the
3
Presented at the VIIth International Symposium on Quantum Chemistry, Menton, France,
2-5 July 1991.
