Page 80 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

REDUCED DENSITY MATRIX VERSUS WAVE FUNCTION 65

The only operation used for obtaining this partitioning is the anticommutation rule
of the fermion operators. Note, that by adding the F and G terms one falls into the
unitarily invariant Absar and Coleman partitioning [32,33] which was obtained by
using a Group theoretical approach.
The interesting point about relation (30) is that each of the terms has a clear physical
interpretation. Thus the term involving F is a sum (for N electrons) of generalised
Hartree-Fock energy levels and clearly is a one-electron term. The term involving
G gives the sum (for N electrons) of the energy of an electron in the average field
of holes. The term is clearly the repulsion energy between the holes and
finally the value shifts the zero of the energy.
In my opinion this partitioning is particularly suitable for analysing electronic corre-
lation effects. To illustrate this point a set of calculations for the three lowest singlet
states of the Beryllium atom are reported in table 3 (in all cases
Hartrees).
Let us start the analysis of the results given in table 3 by commenting on the FCI
one. It is interesting to note that the one-electron term energy becomes lower as
the degree of excitation of the state increases. I find this result rather unexpected,
since in principle, the low energy orbitals will become more empty, At any rate the
stabilization caused by the term is more than counter-balanced by a large
increase of the positive terms of the energy, in particular by
The most stricking features, when comparing the FCI results with the IP and MPS
ones are:
• The and the terms vary for the different states in a very
similar way to the FCI terms. The values obtained with the IP and MPS
approximations for the term for the ground and third state show a
similar behaviour to those of the FCI calculation. However while the
FCI value is higher in the second state (which has a dominant open shell
configuration) than in the other states the opposite happens to the IP and
MPS results.

• The lowering of the energy in the ground state with respect to the FCI result
is due to the term which is much too low in the two approximations.
This error is compensated to a certain extent by errors in the opposite direction
of the two other terms.
• In the second state the two terms depending on the 1-and 2-HRDM compensate
their errors to a large extent but nevertheless the hole –electron positive energy
is too low and a global lowering of this state energy results.
• Finally in the third state the two approximations give very similar energy values,
both with higher energy than the FCI one. In each approximation, the error
of the different terms compensate each other to a certain extent.