The Nature of Many-Electron Wavefunctions

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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,

$\begin{displaymath}\psi_e({\bf x}_1, \ldots, {\bf x}_a, \ldots, {\bf x}_b, \ldot... ...x}_1, \ldots, {\bf x}_b, \ldots, {\bf x}_a, \ldots, {\bf x}_N) \end{displaymath}$

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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

$\begin{displaymath}\chi({\bf x}) = \psi({\bf r}) \vert \alpha \rangle \end{displaymath}$

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$\begin{displaymath}\chi({\bf x}) = \psi({\bf r}) \vert \beta \rangle \end{displaymath}$

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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

$\begin{displaymath}\Phi(x_1) = \sum_i a_i \chi_i(x_1). \end{displaymath}$

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How can we expand a function of two variables, e.g. $\Phi(x_1, x_2)$ ? If we hold x₂ fixed, then

$\begin{displaymath}\Phi(x_1, x_2) = \sum_i a_i(x_2) \chi_i(x_1). \end{displaymath}$

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Now note that each expansion coefficient a_i(x₂) is a function of a single variable, which can be expanded as

$\begin{displaymath}a_i(x_2) = \sum_j b_{ij} \chi_j(x_2). \end{displaymath}$

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

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

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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,

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

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

$\begin{displaymath}\Phi({\bf x}_1, {\bf x}_2) = -\Phi({\bf x}_2, {\bf x}_1) \end{displaymath}$

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implies that b_ij = -b_ji and b_ii = 0, or

$\displaystyle \Phi({\bf x}_1, {\bf x}_2)$	=	$\displaystyle \sum_{j > i} b_{ij} [ \chi_i({\bf x}_1) \chi_j({\bf x}_2) - \chi_j({\bf x}_1) \chi_i({\bf x}_2) ]$
	=	$\displaystyle \sum_{j > i} b_{ij} \vert\chi_i \chi_j \rangle$	(165)

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

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

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

Next: Matrix Mechanics Up: Solving the Electronic Eigenvalue Previous: Solving the Electronic Eigenvalue