Page 72 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

REDUCED DENSITY MATRIX VERSUS WAVE FUNCTION 57

Since

where the symbol s denotes the set of spin variables, the replacement operators can
be further generalised. Thus, in the operator

the sum of annihilators destroy q electrons which are in the state J and the sum of
creators create the state I.
Therefore, in this basis one has:

2.2.2. The Hamiltonian Operator and the Energy
The spin free many-body Hamiltonian Operator can be written in compact form by
employing the 2-RO

where

In this latter formula, the two electron repulsion integral is written following Mulliken
convention and the one electron integrals are grouped in the matrix In this way,
the one-electron terms of the Hamiltonian are grouped together with the two electron
ones into a two electron matrix. Here, the matrix is used only in order to render
a more compact formalism.
An element of the matrix representation of the Hamiltonian in the. N-electron space
is therefore:

or equivalently

A particular case of this is the well known expression of the energy of a normalized
state