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## The Nature of Many-Electron Wavefunctions

Let us consider the nature of the electronic wavefunctions . Since the electronic wavefunction depends only parametrically on , we will suppress in our notation from now on. What do we require of ? Recall that represents the set of all electronic coordinates, i.e., . So far we have left out one important item--we need to include the spin of each electron. We can define a new variable which represents the set of all four coordinates associated with an electron: three spatial coordinates , and one spin coordinate , i.e., .

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,

 (184)

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

 (185)

or
 (186)

Note that for a given spatial orbital , we can form two spin orbitals, one with spin, and one with spin. The spatial orbital will be doubly occupied. It is possible (although sometimes frowned upon) to use one set of spatial orbitals for spin orbitals with spin and another set for spin orbitals with spin.4

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 -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

 (187)

How can we expand a function of two variables, e.g. ?

If we hold fixed, then

 (188)

Now note that each expansion coefficient is a function of a single variable, which can be expanded as
 (189)

Substituting this expression into the one for , we now have
 (190)

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,

 (191)

Now we must make sure that the antisymmetry principle is obeyed. For the two-particle case, the requirement
 (192)

implies that and , or
 (193)

where we have used the symbol to represent a Slater determinant, which in the genreral case is written
 (194)

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

Next: Matrix Mechanics Up: Solving the Electronic Eigenvalue Previous: Solving the Electronic Eigenvalue   Contents
David Sherrill 2006-08-15