Unitary Group Generators

C. David Sherrill

School of Chemistry and Biochemistry

Georgia Institute of Technology

March 1994

We begin with the second-quantized form of the one- and two-electron
operators (see Szabo and Ostlund [1], p. 95),

(1) |

(2) |

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

(3) |

Now integrate over spin, assuming that spatial orbitals are constrained
to be identical for
and
spins. A sum over all 2*K* spin
orbitals can be split up into two sums, one over *K* orbitals with spin, and one over *K* orbitals with
spin. Symbolically, this is

(4) |

The one-electron part of the Hamiltonian becomes

(5) |

After integrating over spin, this becomes

(6) |

The two-electron term can be expanded similarly to give

(7) |

Now we make use of the anticommutation relation

(8) |

and we swap the order of and , introducing a minus sign. This yields

(9) |

Now we use the anticommutation relation between a creation and an annihilation operator, which is

(10) |

This relation allows us to swap the

= | |||

+ | (11) |

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

= | |||

- | (12) |

Now we introduce the unitary group generators, which we write as [2]

(13) |

and the Hamiltonian becomes

(14) |

This is the Hamiltonian in terms of the unitary group generators [3].