October 2005

Most algorithms in *ab initio* electronic structure theory compute
quantities in terms of one- and two-electron integrals. Let us consider
the form of these integrals and their permutational symmetries. Here we
find it helpful to employ the notation of Szabo and Ostlund, *Modern
Quantum Chemistry*.

Let us start with molecular spin orbitals,
, which
describe the motion of a single electron as a function of spatial coordinates
*and* a spin coordinate, denoted collectively by , where
or
, with being a formal ``spin coordinate'' used by
Szabo and Ostlund. Typically, a spin orbital is written as a product of a
spatial part times a spin function (usually just or ), i.e.,
or
, where
is a spatial
orbital that depends only on the spatial coordinates such as
or
.

There are two standard notations for integrals in terms of molecular spin orbitals, denoted ``physicists' notation'' and ``chemists' notation.'' The physicists' notation lists all complex-conjugate functions to the left, and then non-complex-conjugate functions to the right. For two-electron integrals, within a pair of complex-conjugate functions (or non-complex-conjugate functions), the orbital for electron 1 would be listed first, followed by the orbital for electron 2. In chemists' notation, by contrast, one lists the functions for electron 1 on the left, followed by functions for electron 2 on the right. Within each pair, one lists the complex-conjugate functions first, followed by the non-complex-conjugate functions.

The one-electron integrals are the easiest. One-electron integrals over spin orbitals in physicist's notation are defined as

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

Permutational symmetries in the two-electron integrals are somewhat
more interesting. The two-electron integral in physicists' notation
is

(11) |

(12) |

Clearly the integral is unchanged if the dummy indices of integration are
permuted. This leads to the symmetry

(13) |

(14) |

(15) |

(16) |

(17) |

For the case of real orbitals, we can clearly remove the complex
conjugations in the equations above, leading to a four-fold permutational
symmetry in the two-electron integrals. However, an additional symmetry
arises if the orbitals are real: in that case, the same integral
is obtained if and (or and )
are swapped in
. It is trivial to verify that this leads to an
overall eightfold permutational symmetry,

(18) | |||

or

(19) | |||

Finally, it is worthwhile to consider the permutational symmetries
in the antisymmetrized two-electron integral,
,
defined as

(20) | |||

(21) |

In the general case, the permutational symmetries are

(22) | |||

One consequence of these relationships is that

(23) |

If we integrate out spin, we are left again with integrals over spatial
orbitals
. Most frequently, two-electron integrals over
spatial orbitals are written in chemists' notation as,

(24) |

(25) |

(26) |