Page 73 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

58 C. VALDEMORO
2.2.3. The Holes Replacement Operators
An important operator, complementary to the RO, is the Holes Replacement
Operator (HRO). The general form of this operator is:

The HRO’s are holes density operators and operate by first filling orbitals with
electrons (i.e. they annihilate holes) and then removing electrons from orbitals (i.e.
they create holes). These operators generate the Holes Reduced Density Matrix
(HRDM) which in our notation takes the form:

I wish to stress that the meaning of the word Hole here is different and far more
general than in Many Body Perturbation Theory. Indeed, no specific reference state
is required in this definition and the difference between the RO’s and the HRO’s
follows exclusively from the different order of the creator operators with respect to
the annihilator operators in E and in respectively.

3.The Contraction Mapping in Matrix Form (MCM)

The MCM is at the basis of the two formalisms which are described in this work.
Its general form for is:

where and are N-electron states. The symbols denote q-electron states
which may correspond to simple Slater determinants, to eigenstates of the spin op-
erator or to any kind of q-electron correlated states.
It must be underlined that, in (13), while the resulting RDM, may be represented
either in an orbital basis or in a spin-orbital one, as in (Ref.17), the symbols
stand for uniquely defined states depending on both space and spin variables.
Relation (13) allows us to contract any q – RDM and what is more, it also allows us
to contract any q – HRDM by replacing the number N by the number (2K – N).
The derivation of the MCM is based on the important and well known relation

which must be used q – p times in order to obtain relation (13).
In many cases a simpler form of this mapping may be used. Thus, the RDM by
itself, when it is not involved in matrix operations it can be contracted by using