The Nature of Many-Electron Wavefunctions

Let us consider the nature of the electronic wavefunctions $\psi_e({\bf r}; {\bf R})$. Since the electronic wavefunction depends only parametrically on ${\bf R}$, we will suppress ${\bf R}$ in our notation from now on. What do we require of $\psi_e({\bf r})$? Recall that ${\bf r}$ represents the set of all electronic coordinates, i.e., ${\bf r} = \{ {\bf r}_1, {\bf r}_2, \ldots {\bf r}_N \}$. So far we have left out one important item--we need to include the spin of each electron. We can define a new variable ${\bf x}$ which represents the set of all four coordinates associated with an electron: three spatial coordinates ${\bf r}$, and one spin coordinate $\omega$, i.e., ${\bf x} = \{ {\bf r}, \omega \}$.

Thus we write the electronic wavefunction as $\psi_e({\bf x}_1, {\bf x}_2, \ldots, {\bf x}_N)$. 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,

$$\psi_e({\bf x}_1, \ldots, {\bf x}_a, \ldots, {\bf x}_b, \ldo... ...x}_1, \ldots, {\bf x}_b, \ldots, {\bf x}_a, \ldots, {\bf x}_N)$$ (184)

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

A very important step in simplifying $\psi_e({\bf x})$ 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

$$\chi({\bf x}) = \psi({\bf r}) \vert \alpha \rangle$$ (185)

or

$$\chi({\bf x}) = \psi({\bf r}) \vert \beta \rangle$$ (186)

Note that for a given spatial orbital $\psi({\bf r})$, we can form two spin orbitals, one with $\alpha$ spin, and one with $\beta$ 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 $\alpha$ spin and another set for spin orbitals with $\beta$ spin.⁴

Where do we get the one-particle spatial orbitals $\psi({\bf r})$? 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 $\psi_e({\bf x})$ 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 $\{\chi_i(x)\}$. Then any function of a single variable can be expanded exactly as

$$\Phi(x_1) = \sum_i a_i \chi_i(x_1).$$ (187)

How can we expand a function of two variables, e.g. $\Phi(x_1, x_2)$?

If we hold $x_2$ fixed, then

$$\Phi(x_1, x_2) = \sum_i a_i(x_2) \chi_i(x_1).$$ (188)

Now note that each expansion coefficient $a_i(x_2)$ is a function of a single variable, which can be expanded as

$$a_i(x_2) = \sum_j b_{ij} \chi_j(x_2).$$ (189)

Substituting this expression into the one for $\Phi(x_1, x_2)$, we now have

$$\Phi(x_1, x_2) = \sum_{ij} b_{ij} \chi_i(x_1) \chi_j(x_2)$$ (190)

a process which can obviously be extended for $\Phi(x_1, x_2, \ldots, x_N)$.

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

$$\Phi({\bf x}_1, {\bf x}_2) = \sum_{ij} b_{ij} \chi_i({\bf x}_1) \chi_j({\bf x}_2)$$ (191)

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

$$\Phi({\bf x}_1, {\bf x}_2) = -\Phi({\bf x}_2, {\bf x}_1)$$ (192)

implies that $b_{ij} = -b_{ji}$ and $b_{ii} = 0$, or

$$\Phi({\bf x}_1, {\bf x}_2) = \sum_{j > i} b_{ij} [ \chi_i({\bf x}_1) \chi_j({\bf x}_2) - \chi_j({\bf x}_1) \chi_i({\bf x}_2) ]$$

$$= \sum_{j > i} b_{ij} \vert\chi_i \chi_j \rangle$$ (193)

where we have used the symbol $\vert\chi_i \chi_j \rangle$ to represent a Slater determinant, which in the genreral case is written

$$\vert\chi_1 \chi_2 \ldots \chi_N \rangle = \frac{1}{\sqrt{N!... ...\bf x}_N) & \ldots & \chi_N({\bf x}_N) \end{array} \right\vert$$ (194)

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 $\{\chi_i({\bf x})\}$.