Theory

Basis expansion of the Schrödinger equation

We are going to solve the Schrödinger equation

\[\hat{H}|\psi\rangle = \epsilon|\psi\rangle,\]

where $\hat{H}$ is the Hamiltonian of a quantum few-body system, $|\psi\rangle$ and $\epsilon$ are the eigenfunction and the eigenvalue to be found.

We shall expand the wave-function $|\psi\rangle$ in terms of a set of basis functions $|i\rangle$ for $i = 1 \ldots n$,

\[|\psi\rangle = \sum_{i=1}^{n} c_i |i\rangle.\]

Inserting the expansion into the Schrödinger equation and multiplying from the left with $\langle k|$ for $1 \leq k \leq n$ gives

\[\sum_{i=1}^{n} \langle k|\hat{H}|i\rangle c_i = \epsilon \sum_{i=1}^{n} \langle k|i\rangle c_i.\]

Or, in the matrix notation

\[Hc = \epsilon Nc,\]

where $H$ and $N$ are correspondingly the Hamiltonian and the overlap matrices with the matrix elements

\[H_{ki} = \langle k|\hat{H}|i\rangle, \quad N_{ki}\]

Gaussians as basis functions

We shall use the so-called Correlated Gaussians (or Explicitly Correlated Gaussians) as the basis functions. For a system of $N$ particles with coordinates $\vec{r}_i$, $i = 1 \ldots N$, the Correlated Gaussian is defined as

\[g(\vec{r}_1, \ldots, \vec{r}_N) = \exp \left( - \sum_{i,j=1}^{N} A_{ij}\vec{r}_i \cdot \vec{r}_j - \sum_{i=1}^{N} \vec{s}_i \cdot \vec{r}_i \right),\]

where $\vec{r}_i \cdot \vec{r}_j$ denotes the dot-product of the two vectors; and where $A$, a symmetric positive-defined matrix, and $\vec{s}_i$, $i=1,\ldots,N$, the shift-vectors, are (cleverly chosen) parameters of the Gaussian.

In matrix notation,

\[g(\vec{r}) = \exp \left( -\vec{r}^T A \vec{r} + \vec{s}^T \vec{r} \right),\]

where $\vec{r}$ is the column of the coordinates $\vec{r}_i$ and $\vec{s}$ is the column of the shift-vectors $\vec{s}_i$,

\[\vec{r} = \begin{pmatrix} \vec{r}_1 \\ \vdots \\ \vec{r}_N \end{pmatrix}, \quad \vec{s} = \begin{pmatrix} \vec{s}_1 \\ \vdots \\ \vec{s}_N \end{pmatrix},\]

and

\[\vec{r}^T A \vec{r} + \vec{s}^T \vec{r} = \sum_{i,j} \vec{r}_i \cdot A_{ij}\vec{r}_j + \sum_i \vec{s}_i \cdot \vec{r}_i.\]