Many-electron wave functions#

Hilbert space of state vectors#

State vectors for a system with \(N\) electrons are elements of a Hilbert space of the form

\[ \mathcal{V} = \mathcal{V}^1 \otimes \mathcal{V}^2 \otimes \cdots \otimes \mathcal{V}^N \]

where the separate particle spaces are described under the section discussing orbitals.

Pauli principle#

Electrons are fermions and obey the Pauli principle so \(\mathcal{V}\) is limited to include only anti-symmetrized state vectors

\[ | \Psi \rangle = \frac{1}{\sqrt{N!}} \sum \mathcal{P}_{1,2,\ldots,N} \Big[ | \psi_{1} \rangle \otimes | \psi_{2} \rangle \otimes \cdots \otimes | \psi_{N} \rangle \Big] \]

where the sum includes all \(N!\) permutations of the \(N\) orthonormal spin orbitals, \(| \psi_{i} \rangle\), in the \(N\) particle spaces.

Slater determinants#

Expressed in coordinate space, the anti-symmetrized state vectors are known as Slater determinants

\[\begin{split} | \Phi \rangle = |\psi_{1}, \ldots, \psi_{N} \rangle \leftrightarrow \frac{1}{\sqrt{N!}} \begin{vmatrix} \psi_{1}(\mathbf{r}_1) & \cdots & \psi_{N}(\mathbf{r}_1) \\ \vdots & \ddots & \vdots \\ \psi_{1}(\mathbf{r}_N) & \cdots & \psi_{N}(\mathbf{r}_N) \\ \end{vmatrix} \end{split}\]

With a set of spin orbitals that spans the one-electron space \(\mathcal{V}^1\), the set of all distinct Slater determinants span the \(N\)-electron space \(\mathcal{V}\).

Unitary orbital transformations#

Let \(\mathbf{A}\) be a matrix with orbital values as elements

\[\begin{split} \mathbf{A} = \begin{pmatrix} \psi_{1}(\mathbf{r}_1) & \cdots & \psi_{N}(\mathbf{r}_1) \\ \vdots & \ddots & \vdots \\ \psi_{1}(\mathbf{r}_N) & \cdots & \psi_{N}(\mathbf{r}_N) \\ \end{pmatrix} \end{split}\]

such that the Slater determinant is equal to

\[ \Phi(\mathbf{r}_1, \ldots, \mathbf{r}_N) = \frac{1}{\sqrt{N!}} \mathrm{det}\big(\mathbf{A}\big) \]

Let matrix \(B\) relate to \(A\) by means of a unitary tranformation

\[ \mathbf{B} = \mathbf{A U} \]

For deteminants, we have

\[ \mathrm{det}\big(\mathbf{B}\big) = \mathrm{det}\big(\mathbf{A U}\big) = \mathrm{det}\big(\mathbf{A}\big) \mathrm{det}\big(\mathbf{U}\big) = \mathrm{det}\big(\mathbf{A}\big) e^{i\phi} \]

This shows that a unitary transformation of occupied orbitals in a Slater deteminant does not change the many-electron wave function with more than a trivial overall phase factor.

Normalization#

A general multi-electron wave function is expanded in the basis of Slater determinants

\[ | \Psi \rangle = \sum_i c_i | \Phi_i \rangle \]

and it is normalized according to

\[ \langle \Psi | \Psi \rangle = \int \cdots \int \Psi^\dagger(\mathbf{r}_1, \ldots, \mathbf{r}_N) \Psi(\mathbf{r}_1, \ldots, \mathbf{r}_N) \, \mathrm{d}^3 \mathbf{r}_1 \cdots \mathrm{d}^3\mathbf{r}_N = 1 \]

Probabilistic interpretation#

The probability of simultaneously finding electron 1 in the infinitesimal volume element \(\mathrm{d}^3\mathbf{r}_1\), electron 2 in \(\mathrm{d}^3\mathbf{r}_2\), etc. is equal to

\[ | \Psi(\mathbf{r}_1, \ldots, \mathbf{r}_N) |^2 \mathrm{d}^3 \mathbf{r}_1 \cdots \mathrm{d}^3\mathbf{r}_N \]

as illustrated below

../../_images/wf_interpretation.svg

\(N\)-particle density#

The quantity

\[ n(\mathbf{r}_1, \ldots, \mathbf{r}_N) = | \Psi(\mathbf{r}_1, \ldots, \mathbf{r}_N) |^2 \]

is referred to as the \(N\)-particle density.

Reduced particle densities#

One-particle density#

The probability density of finding any one electron in an infinitesimal volume element at position \(\mathbf{r}\) regardless of the positions of other electrons is equal to

\[ n(\mathbf{r}) = N \int \cdots \int \Psi^\dagger(\mathbf{r}, \mathbf{r}_2, \ldots, \mathbf{r}_N) \Psi(\mathbf{r}, \mathbf{r}_2, \ldots, \mathbf{r}_N) \, \mathrm{d}^3 \mathbf{r}_2 \cdots \mathrm{d}^3\mathbf{r}_N \]

Two-particle density#

The probability density of finding any two electrons in separate infinitesimal volume elements at positions \(\mathbf{r}_1\) and \(\mathbf{r}_2\) regardless of the positions of other electrons is equal to

\[ n(\mathbf{r}_1, \mathbf{r}_2) = N (N-1) \int \cdots \int \Psi^\dagger(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3, \ldots, \mathbf{r}_N) \Psi(\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3, \ldots, \mathbf{r}_N) \, \mathrm{d}^3 \mathbf{r}_3 \cdots \mathrm{d}^3\mathbf{r}_N \]