# Task 5: Time-dependent expectation values

## Time-dependent overlap

We compute the time-dependent overlap between two Slater determinants (with orthonormal atomic orbital basis functions) by
\begin{align}
    P(t, t_0)
    &= \bigl|\langle \Phi(t) | \Phi(t_0) \rangle \bigr|^2
    = \bigl|\det\bigl(C^{\dagger}_o(t) C_o(t_0)\bigr)\bigr|^2,
\end{align}
where $C_o(t)$ is the coefficient matrix of the time-evolved state with $o$ denoting the occupied columns.

## One-body expectation values

Given a one-body operator $\hat{A}$ we wish to compute the time-dependent expectation values.
```{note}
A one-body operator is an operator that acts on a single particle at a time.
For example, the position operator $\hat{x}$ is a one-body operator, whereas the Coulomb interaction $\hat{u}$ is a two-body operator as it applies to the interaction between two particles at a time.
```
Having already found the matrix elements of $\hat{A}$ in a static (time-independent) atomic basis $\{\psi_{\mu}\}_{\mu = 1}^{l}$, we can compute the expectation value of this operator by
\begin{align}
    \langle A(t) \rangle
    &= \langle \psi_{\mu} | \hat{A} | \psi_{\nu} \rangle D_{\nu \mu}(t),
\end{align}
where $D_{\nu \mu}(t) = \sum_{i = 1}^{n} C^{*}_{\mu i}(t) C_{\nu i}(t)$ is the time-evolved Hartree-Fock density matrix.
Alternatively, by transforming the one-body operator to the time-evolved molecular orbital basis $\{\psi_p(t)\}_{p = 1}^{k}$ we can compute the expectation value from
\begin{align}
    \langle A(t) \rangle
    &=
    \langle \phi_{p}(t) | \hat{A} | \phi_q(t) \rangle
    \rho_{q p},
\end{align}
where $\rho_{q p}$ is the one-body density matrix in the time-evolved Hartree-Fock basis.
This matrix is given by
\begin{align}
    \boldsymbol{\rho}
    &= \begin{pmatrix}
        \boldsymbol{I}_{n \times n} & \boldsymbol{0}_{n \times (k - n)} \\
        \boldsymbol{0}_{(k - n) \times n} & \boldsymbol{0}_{(k - n) \times (k - n)}
    \end{pmatrix},
\end{align}
that is, the matrix is the identity in the occupied block and zero elsehwere.
Both ways of computing the expectation value give the same value.

92e0a8f21aa0c4fa24125eb81c122e43e54bae2c
ipynb
Jupyter Notebook
docs/task-5-td-expectation-values.ipynb
Schoyen/tdhf-project-fys4411
b0231c0d759382c14257cc4572698aa80c1c94d0
MIT
2021-06-03T00:34:57.000Z
2021-06-03T00:34:57.000Z
docs/task-5-td-expectation-values.ipynb
Schoyen/tdhf-project-fys4411
b0231c0d759382c14257cc4572698aa80c1c94d0
MIT
docs/task-5-td-expectation-values.ipynb
Schoyen/tdhf-project-fys4411
b0231c0d759382c14257cc4572698aa80c1c94d0
MIT
Qwen/Qwen-72B
1. YES
2. YES