The so-called time-displacement autocorrelation \( \phi(t) \) for a quantity \( \mathbf{M} \) is given by

$$ \phi(t) = \int dt' \left[\mathbf{M}(t')-\langle \mathbf{M} \rangle\right]\left[\mathbf{M}(t'+t)-\langle \mathbf{M} \rangle\right], $$

which can be rewritten as

$$ \phi(t) = \int dt' \left[\mathbf{M}(t')\mathbf{M}(t'+t)-\langle \mathbf{M} \rangle^2\right], $$

where \( \langle \mathbf{M} \rangle \) is the average value and \( \mathbf{M}(t) \) its instantaneous value. We can discretize this function as follows, where we used our set of computed values \( \mathbf{M}(t) \) for a set of discretized times (our Monte Carlo cycles corresponding to moving all electrons?)

$$ \phi(t) = \frac{1}{t_{\mathrm{max}}-t}\sum_{t'=0}^{t_{\mathrm{max}}-t}\mathbf{M}(t')\mathbf{M}(t'+t) -\frac{1}{t_{\mathrm{max}}-t}\sum_{t'=0}^{t_{\mathrm{max}}-t}\mathbf{M}(t')\times \frac{1}{t_{\mathrm{max}}-t}\sum_{t'=0}^{t_{\mathrm{max}}-t}\mathbf{M}(t'+t). \tag{20} $$