Page 76 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

REDUCED DENSITY MATRIX VERSUS WAVE FUNCTION 61

4.2. SOME PROPERTIES OF THE p – SRH MATRICES
Let us now derive from the fundamental relation (17), some properties of the p–SRH
matrices. Taking the trace of this matrix we find that:

Since the trace of is an invariant of the system, relation (22) establishes that the
trace of the p – SRH matrix is also an invariant of the system.
The eigen – values of the p – SRH matrices have the form:

where the symbol I denotes the corresponding p – SRH eigen-state. This relation
shows up the average character of the p – SRH eigen-values. However, until now we
have not been able to find the relation linking the values with the energy observables
of the N-electron system.
In what follows we will focus our attention on the p = 2 and p = 1 cases which are the
most useful ones. The eigen-vectors of the p – SRH for these values of p are geminals
and orbitals respectively. In order to simplify the interpretation of the geminals and
to reduce the size of the matrices involved in the calculations it is convenient to apply
to the 2 – SRH a linear transformation which factorizes this matrix into two blocks
according to the representations of the Symmetric Group of Permutations. In this
way the eigen-geminals of both blocks have a clear physical meaning since those of
the symmetric block describe the space part of singlet pair states and those of the
antisymmetric part describe that of triplet states.
Although the physical meaning of the 2- and 1- electron eigen-states of the 2 – SRH
has not been established rigorously we interpret them as describing states of two/one
electrons which in average can be considered independent. This interpretation was
justified [29] through the analysis of the asymptotic form of the 2 – SRH in the
coordinate representation for . . In this analysis, Karwowski et al. showed that
the eigen-geminals of the asymptotic 2 – SRH described isolated pairs of electrons.
Another important feature of the SRH formalism is that it can be generalized by
contracting Taking now the trace of the product of these generalized SRH’s
matrices and the one gets the n + 1 moment of the spectral distribution [30].

4.3. APPLICATIONS OF THE SRH THEORY
An outline of the main applications of the SHR theory is presented in this section. In
6.1 the advantage of using the eigen-vectors the 1-SRH as a basis in CI calculations
is discussed. The main application until now of this theory is summarized in the
following subsection. Then in 6.3 other applications which have been less developed
are mentioned.

4.3.1. Performance of the eigen-vectors of the 1-SRH as a basis in CI calculations
A set of calculations [31] was recently carried out in order to compare the performance
of the 1 –SRH eigen-orbitals with other known and easy to get basis sets when doing