Thus we write the electronic wavefunction as
.
Why have we been able to avoid including
spin until now? Because the non-relativistic Hamiltonian does not
include spin. Nevertheless, spin must be included so that the
electronic wavefunction can satisfy a very important requirement,
which is the *antisymmetry principle* (see Postulate 6 in Section
4). This principle states that for a system of fermions, the
wavefunction must be antisymmetric with respect to the interchange of
all (space *and* spin) coordinates of one fermion with those of
another. That is,

(156) |

The Pauli exclusion principle is a direct consequence of the antisymmetry principle.

A very important step in simplifying
is to expand it
in terms of a set of one-electron functions, or ``orbitals.'' This
makes the electronic Schrödinger equation considerably easier to
deal with.^{3} A *spin orbital* is a function of
the space and spin coordinates of a single electron, while a *spatial* orbital is a function of a single electron's spatial
coordinates only. We can write a spin orbital as a product of a
spatial orbital one of the two spin functions

(157) |

or

(158) |

Note that for a given spatial orbital , we can form

Where do we get the one-particle spatial orbitals ? That is beyond the scope of the current section, but we briefly itemize some of the more common possibilities:

- Orbitals centered on each atom (atomic orbitals).
- Orbitals
centered on each atom but also symmetry-adapted to have the correct
point-group symmetry species (symmetry orbitals).
- Molecular orbitals obtained from a Hartree-Fock procedure.

We now explain how an *N*-electron function
can be
constructed from spin orbitals, following the arguments of Szabo and
Ostlund [4] (p. 60). Assume we have a complete
set of functions of a single variable
.
Then any
function of a single variable can be expanded exactly as

(159) |

How can we expand a function of

(160) |

Now note that each expansion coefficient

(161) |

Substituting this expression into the one for , we now have

(162) |

a process which can obviously be extended for .

We can extend these arguments to the case of having a complete set of
functions of the variable
(recall
represents x, y,
and z and also ). In that case, we obtain an analogous result,

(163) |

Now we must make sure that the antisymmetry principle is obeyed. For the two-particle case, the requirement

(164) |

implies that

= | |||

= | (165) |

where we have used the symbol to represent a

(166) |

We can extend the reasoning applied here to the case of *N* electrons;
any *N*-electron wavefunction can be expressed exactly as a linear
combination of all possible *N*-electron Slater determinants formed
from a complete set of spin orbitals
.