# 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

### $$N$$-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

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