The anticommutation relations for two annihilation operators is

and the anticommutation relation for two creation operators is similarly

(tex2html_deferredtex2html_deferred5.tex2html_deferred2 { a_j^, a_i^ } = a_j^ a_i^ + a_i^ a_j^ = 0 ) |

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)

(tex2html_deferredtex2html_deferred5.tex2html_deferred4Ô_1 = _ij^2n i | h | j a_i^ a_j ) | |||

(tex2html_deferredtex2html_deferred5.tex2html_deferred5 Ô_2 = 12 _ijkl^2n ij | kl a_i^ a_j^ a_l a_k ) |

where the sums run over all spin orbitals . Thus the Hamiltonian is

(tex2html_deferredtex2html_deferred5.tex2html_deferred6H = _pq^2n a_p^ a_q [p|h|q] + 12 _pqrs^2n a_p^ a_r^ a_s a_q [pq|rs] ) |

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 2*n* 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

(tex2html_deferredtex2html_deferred5.tex2html_deferred7_a^2n = _a^n + _a^n ) |

The one-electron part of the Hamiltonian becomes

(tex2html_deferredtex2html_deferred5.tex2html_deferred8H_one = _pq^n [p |h| q] a_p ^ a_q + [p |h| q ] a_p ^ a_q + [p |h| q ] a_p ^ a_q + [p|h|q] a_p ^ a_q ) |

After integrating over spin, this becomes

(tex2html_deferredtex2html_deferred5.tex2html_deferred9H_one = _pq^n (p|h|q) { a_p ^ a_q + a_p ^ a_q } ) |

The two-electron term can be expanded similarly to give

(tex2html_deferredtex2html_deferred5.tex2html_deferred10H_two = 12 _pqrs^n (pq|rs) { a_p ^ a_r ^ a_s a_q + a_p ^ a_r ^ a_s a_q + a_p ^ a_r ^ a_s a_q + a_p ^ a_r ^ a_s a_q } ) |

Now we make use of the anticommutation relation (5.1) and we swap the order of

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

= | |||

tex2html_{d}eferredtex2html_{d}eferred5.tex2html_{d}eferred11 |
+ | (tex2html_deferredtex2html_deferred5.tex2html_deferred11 & + & . a_p ^ a_q a_r ^ a_s - _q , r a_p ^ a_s + a_p ^ a_q a_r ^ a_s - _q , r a_p ^ a_s ] ) |

Now we observe that and can both be written , and also that and are both 0. This simplifies our equation to

= | |||

tex2html_{d}eferredtex2html_{d}eferred5.tex2html_{d}eferred12 |
- | (tex2html_deferredtex2html_deferred5.tex2html_deferred12 & - & . _qr a_p ^ a_s - _qr a_p ^ a_s ] ) |

Now we introduce the replacement (or shift) operator

(tex2html_deferredtex2html_deferred5.tex2html_deferred13Ê_ij = a_i ^ a_j + a_i ^ a_j ) |

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.

(tex2html_deferredtex2html_deferred5.tex2html_deferred14 H = _pq^n (p|h|q) Ê_pq + 12 _pqrs^n (pq|rs) ( Ê_pq Ê_rs - _qr Ê_ps ) ) |

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).