Page 74 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

REDUCED DENSITY MATRIX VERSUS WAVE FUNCTION 59
(14) directly, without any reference to the intermediate matrix The
algorithm in this case is:

This direct sum with respect to an index which is common to the creator and the
corresponding annihilator cannot always be carried out as will be seen in the following
sections, hence the usefulness of the general MCM (relation (13)).

4.The Spin adapted Reduced Hamiltonian (SRH)
The Spin adapted Reduced Hamiltonian (SRH) is the contraction to a p-electron
space of the matrix representation of the Hamiltonian Operator, in the N-
electron space for a given Spin Symmetry [17,18,25,28]. The basis for the matrix
representation are the eigenfunctions of the operator. The block matrix which is
contracted is that which corresponds to the spin symmetry selected; In this way, the
spin adaptation of the contracted matrix is insured.
Let us consider the spectral resolution of

Now, in order to obtain the p – SRH, let us apply the MCM to both sides of this
relation,

According to relation (17), the p–SRH matrix is a sum of terms and each term is the
product of the energy and the RDM corresponding to an eigen-state of the system.
Therefore for the p – SRH matrix has all the relevant information about the
eigen-states of our system and is represented in a reduced space which renders it easy
to handle.
Note that by generalizing the concept of N-representability it can be said that the
p – SRH are both N-representable and S-representable.

4.1. CONSTRUCTION OF THE p–SRH MATRIX

A very convenient feature is that, in order to construct the p – SRH matrix it is not
necessary to evaluate the matrix and then apply the MCM since both operations
can be combined and carried out simultaneously. Thus, by replacing in relation (17)