Page 79 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

64 C. VALDEMORO

A detailed discussion of these and other variants was given in (Ref.35). Attention
must be called to the fact that these methods are not variational which causes the
energies obtained with them to be lower than those obtained with the FCI method.
The counterpart to this deffect is that excited states, open-shell systems, and radicals,
can be calculated with as much ease as the ground state and closed-shell systems.
Also, the size of the calculation is determined solely by the size of the Hilbert subspace
chosen and does not depend in principle on the number of electrons since all happens
as if only two electrons were considered.
While from the energy point of view, the correlation effects seem to be overestimated,
the RDM’s are particularly satisfactory. Thus, when comparing the 2-RDM’s ob-
tained with these approximations for the ground state of the Beryllium atom with
the corresponding FCI one, the standard deviations are: 0.00208236 and 0.00208338
for the MPS and IP respectively. For this state, which has a dominant four electron
configuration of the type, the more important errors, which nevertheless can
be considered small, are given in table 2.
In table 2, the elements which are equal due to symmetry have been omitted.

It can be seen that, even for this case where no ambiguity exists in applying the IP
approximation, the results are slightly better with the MPS variant which seems to
favor this latter approximation.
The data given in table 2 have been obtained using a double zeta basis [36] and
transforming it to the basis which diagonalizes the 1-SRH matrix which as we saw
in subsection 5.3 is a basis both good and simple to determine.
Analysis of the different terms of the energy
Another interesting analysis of this method can be carried out by applying a par-
titioning of the energy [37] which shows up the role played by the 1–RDM, the
1-HRDM and the 2-HRDM.
Thus, it has been shown that the energy can be partitioned as

where: