Page 88 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

REDUCED DENSITY MATRIX VERSUS WAVE FUNCTION 73
These symmetry relations impose that the following corrections should also be intro-
duced:

• must also be made equal to zero.
• The value should be subtracted from
Under these conditions, the 3-RDM of the three lower states of the Beryllium atom
and the two lower ones of the Water molecule were determined [48] by taking as initial

data the 2 -RDM obtained in a Full Configuration Interaction. In Table 4 some of
these results are given and as can be seen they are very satisfactory.
The results for approximating the 4-RDM in terms of the lower order RDM’s are
slightly inferior but still very good. In consequence, I expect that the iterative pro-
cedure proposed in the previous section may prove to be a realistic one.
In spite of the good results obtained we continue our search for simple auxiliary
conditions directed at ensuring that the approximated matrix is positive and that its
trace has the correct value. This search is mainly focused at improving the quality
of the 2-RDM obtained in terms of the 1-RDM,which at the moment is the less
precise procedure [46]. When this latter aim is fulfilled we expect that the iterative
solution of the 1-order CSchE will also be successful although in this CSchE the
information carried by the Hamiltonian only influences the result in an average way
which probably will retard the convergence.

6.Conclusion
The two previous sections outline the main formal and applicative results obtained
in our search for a theoretical framework where the number of variables which are
explicitly taken into account would be as small as the observables allow. This frame-
work should permit the use of different levels of approximation for the Hamiltonian
operator and its orbital representation. That is, the size of the basis set and the
kind of approximation used for the integrals should not be predetermined by the
formalism.
Both lines of research are far from being closed and we are confident that their
development will contribute useful results. However, without considering future per-
formances I think that it can already be said that it is a good strategy to project the
future Quantum Chemical methodology in such a way that the WF is by-passed and
the 2-RDM or (better still but more difficult) the 1-RDM are directly determined.

Acknowledgements

Investigación Científíca y Técnica del Ministerio de Educación y Ciencia under project
PB90-0092.

References
1. P. A. M. Dirac, Proc. Cambridge Phil. Soc. 27, 240 (1931).
2. K. Husimi, Proc. Phys. Soc. Japan 22, 264 (1940).