# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 7 Applicability of the Frozen-Density Approximation
The corresponding relation taking into account the evolution of \(\Psi(t)\) can be derived for NRXS from the general relation in Equation (15) resulting in  
\[\frac{dP^{\text{NRXS}}(\mathbf{Q})}{d\Omega}= \mathrm{P}_{0}^{\text{NRXS}}\sum_{F}\int d^{3}r_{1}\int d^{3}r_{2 }e^{i\mathbf{Q}\cdot(\mathbf{r}_{1}-\mathbf{r}_{2})} \tag{52}\] \[\qquad\times\sum_{I,K}\langle\Psi_{K}(t_{p})|\hat{G}_{F}^{\text{ NRXS}}(\mathbf{r}_{1},\mathbf{r}_{2})|\Psi_{I}(t_{p})\rangle e^{-\Omega_{FK}^{2}\tau_{p}^{2}/(8\ln 2 )}e^{-\Omega_{Fl}^{2}\tau_{p}^{2}/(8\ln 2)},\]  
Figure 8: Time-resolved photoelectron spectra in the case of the probe-pulse duration being comparable with the shortest beating period of the electron density of indole (**a**) computed with Equation (51) accurately treating electron dynamics during the probe and (**b**) computed with Equation (42) neglecting electron dynamics during the probe. Figure is reproduced with permission from Reference [32]. Copyright American Physical Society, 2016.  
where \(\Omega_{F\Gamma(FK)}=\omega_{\rm in}-\omega_{\bf k_{s}}-E_{F}+E_{I(K)}\). In addition, such a relation for RXS results from the general relation in Equation (28), leading to  
\[\frac{dP^{\rm RXS}(\mathbf{Q})}{d\Omega}= \mathbb{P}_{0}^{\rm RXS}\sum_{C_{1},C_{2}}e^{i\mathbf{Q}\cdot( \mathbf{R}_{C_{1}}-\mathbf{R}_{C_{2}})}\sum_{F}\sum_{I,K}\langle\Psi_{K}(t_{p} )|\hat{G}_{F,C_{1},C_{2}}^{\rm RXS}|\Psi_{I}(t_{p})\rangle e^{-\Omega_{F}^{2} \tau_{p}^{2}/(8\ln 2)}e^{-\Omega_{I}^{2}\tau_{p}^{2}/(8\ln 2)}. \tag{53}\]  
In both cases, the relations are not connected to the electronic state at the time of measurement \(\Psi(t_{p})\) that cannot be singled out. Instead of that, all three relations in Equations (42), (52) and (53) include the electronic state \(\Psi(t_{p})\) convoluted with the amplitude of the electric field [cf. Equation (17)]. Thus, an assumption that a time-resolved signal obtained by a probe pulse not satisfying the frozen-density approximation depends on the electronic state simply averaged over the pulse duration, \(\int_{t_{p}-\tau_{p}/2}^{t_{p}+\tau_{p}/2}dt\Psi(t)/\tau_{p}\), is incorrect. Actually, Equations (18), (30) and (42) assuming the frozen-density approximation can be obtained from corresponding Equations (51)-(53) by substituting the mean energy \(\langle E\rangle\) of the superposition \(\Psi(t_{p})\) for \(E_{I(K)}\) in factors \(\Omega_{F\Gamma(FK)}^{\rm el}\) and \(\Omega_{F\Gamma(FK)}\), correspondingly. This means that the frozen-density approximation is valid as long as the maximum energy difference of the eigenstates involved in the dynamics is negligible compared to the probe-pulse bandwidth. Since the dependence of Equations (52) and (53) on the probe-pulse duration \(\tau_{p}\) encoded in the factors \(e^{-\Omega_{F\Gamma(FK)}^{2}\tau_{p}^{2}/(8\ln 2)}\) is analogous to this dependence in the case of TRAPES in Equation (51), it is possible to transfer the conclusion of Reference [32] to the case of time-resolved X-ray scattering. Thus, if the duration of an X-ray probe-pulse is about ten times shorter than the shortest beating period of the electron density, then time-resolved scattering patterns are connected to the electronic state \(\Psi(t_{p})\) and can be described by corresponding Equations (18) and (30). Whereas, if the duration of the X-ray probe pulse is comparable with the shortest beating period, then this connection is lost, and scattering patterns are determined by \(\Psi(t_{p})\) convoluted with the amplitude of the probe-pulse electric field as described by corresponding Equations (52) and (53).