0
Much of the literature in CI theory makes use of the notation of
second-quantization. Szabo and Ostlund [1]
give a good introduction
to second-quantized operators. Here we will only summarize the
anticommutation relations between creation and annihilation operators, and
then proceed to express the
Hamiltonian in second quantized form for spatial orbitals, rather
than for spin orbitals. Then we will use these results to derive the
Hamiltonian in terms of the unitary group generators.
The anticommutation relations for two annihilation operators is
and the anticommutation relation for two creation operators is similarly
The anticommutation relation between a creation and an annihilation operator
is
Now we will find an expression for the Hamiltonian in terms of creation and
annihilation operators over spatial orbitals. We begin with the
second-quantized form of the one- and two-electron operators (see Szabo and
Ostlund [1] p. 95)
where the sums run over all spin orbitals 
. Thus the
Hamiltonian is
From the previous equation we can see that the second-quantized form
of the Hamiltonian is independent of the number of electrons in
the system.
Now integrate over spin, assuming that spatial orbitals are constrained
to be identical for 
and 
spins. A sum over all 2n spin
orbitals can be split up into two sums, one over n orbitals with
spin, and one over n orbitals with spin. Symbolically, this is
The one-electron part of the Hamiltonian becomes
After integrating over spin, this becomes
The two-electron term can be expanded similarly to give
Now we make use of the anticommutation relation (5.1)
and we swap the order of 
and 
,
introducing a minus sign. This yields
Now we use the anticommutation relation between a creation and an annihilation
operator, which is given by (5.3).
This relation allows us to swap the and 
in each term,
to give
Now we observe that 
and
can both be written 
, and also that
and 
are both 0.
This simplifies our equation to
Now we introduce the replacement (or shift) operator
which Paldus has shown [5] to be isomorphic to the
generators of the unitary group. This replacement
operator is commonly referred to as a unitary group generator, but
as Duch has pointed out [27], such usage is somewhat
dubious in papers where no unitary group theory is employed.
This is the Hamiltonian in terms of replacement operators.
Contemporary papers on CI theory often express
the Hamiltonian in the form of equation
(5.15).