Page 71 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

56 C. VALDEMORO
This book also collects the contributions of most specialists in the field.
Two different approaches to this problem will be described in this work. They are
based in quite different philosophies, but both are aimed at determining the RDM
without a previous knowledge of the WF. Another common feature of these two
approaches is that they both employ the discrete Matrix representation of the Con-
traction Mapping (MCM) [17,18]. Applying this MCM is the alternative, in discrete
form, to integrating with respect to a set of electron variables and it is a much simpler
tool to use.
In this work, we will concentrate on describing the ideas leading to the relevant
formulae and only the essential algebraic developments will be described.

2.Notation and Basic Definitions
2.1. CONSTANTS AND STATES

N = number of electrons of the system
K = number of orbitals of the basis set
S = spin quantum number

2.2. OPERATORS AND EXPECTATION VALUES

2.2.1. Replacement Operators and Reduced Density Matrices
Most operators used in this work may be written in terms of the q-order Replacement
Operators (q – RO) [21,27] which, in our notation, take the form:

where and b are the usual fermion operators.
The expectation values of the q-RO’s are the q – RDM’s. Thus, the general definition
of the q – RDM in this formalism is:

When relation (2) defines a transition q – RDM. In what follows, unless it
is necessary, the upper indices which indicate the bra and ket states will be omitted
since that only the case is considered.