# Photocontrollable Mixed-Valent States in Platinum-Halide Ladder Compounds
## 4 Photoexcitation and Following Lattice Relaxation
### Real-Time Dynamics
Secondly, we construct the instantaneous Hamiltonian at \(t+\Delta t/2\) as  
\[\mathcal{H}_{\mathrm{HF}}(t+\frac{\Delta t}{2})\equiv\frac{1}{2}\left[ \mathcal{H}_{\mathrm{HF}}(t)+\mathcal{H}_{\mathrm{HF}}(t+\Delta t)\right], \tag{17}\]  
and the force at \(t+\Delta t/2\) as  
\[F_{A}(t+\frac{\Delta t}{2})\equiv\frac{1}{2}\left[F_{A}(t)+F_{A}(t+\Delta t) \right]. \tag{18}\]  
The lattice displacements and velocities at \(t+\Delta t/2\) are estimated as  
\[u_{n:IA}(t+\frac{\Delta t}{2})= u_{n:IA}(t)+\dot{u}_{n:IA}(t)\frac{\Delta t}{2}, \tag{19}\] \[\dot{u}_{n:IA}(t+\frac{\Delta t}{2})= \dot{u}_{n:IA}(t)+\frac{F_{A}(t)}{M_{A}}\frac{\Delta t}{2}, \tag{20}\]  
which are obtained by \(\Delta t\rightarrow\Delta t/2\) in Equations (15) and (16). Finally, we update the electronic wavefunctions and the lattice displacements by using the information at \(t+\Delta t/2\): \(\mathcal{H}_{\mathrm{HF}}(t)\rightarrow\mathcal{H}_{\mathrm{HF}}(t+\Delta t/2)\) in Equation (13), \(\dot{u}_{n:IA}(t)\rightarrow\dot{u}_{n:IA}(t+\Delta t/2)\) in Equation (15), and \(F_{A}(t)\to F_{A}(t+\Delta t/2)\) in Equation (20). Thus, the wavefunctions are represented as  
\[ \begin{array}{ll}|\psi_{\
u,s}(t+\Delta t)\rangle&\simeq\exp\left[-\frac{i \mathcal{H}_{\mathrm{HF}}(t+\Delta t/2)}{\hbar}\Delta t\right]|\psi_{\
u,s}(t )\rangle\\ &=\sum\limits_{\lambda=1}^{44\hbar}\sum\limits_{\tau}\exp\left[-\frac{ie_{ \lambda,\tau}}{\hbar}\Delta t\right]|\phi_{\lambda,\tau}\rangle\langle\phi_{ \lambda,\tau}|\psi_{\
u,s}(t)\rangle,\end{array}  \tag{21}\]  
where \(\epsilon_{\lambda,\tau}\) and \(|\phi_{\lambda,\tau}\rangle\) are the eigenvalue and the eigenfunction of \(\mathcal{H}_{\mathrm{HF}}(t+\Delta t/2)\), respectively. The lattice displacements are given by the following set of equations:  
\[u_{n:IA}(t+\Delta t)= u_{n:IA}(t)+\dot{u}_{n:IA}(t+\frac{\Delta t}{2})\Delta t, \tag{22}\]  
\[\dot{u}_{n:IA}(t+\frac{\Delta t}{2})= \dot{u}_{n:IA}(t)+\frac{F_{A}(t+\Delta t/2)}{M_{A}}\frac{\Delta t }{2}. \tag{23}\]  
The thus obtained solutions Equations (21) and (22) are correct up to \(\mathcal{O}(\Delta t^{2})\)[36; 38; 39]. The optical phonon frequency \(\omega_{\mathrm{ph}}\) is given by \(\omega_{\mathrm{ph}}=\sqrt{2K_{\mathrm{MX}}/M_{\mathrm{eff}}}\) with the effective mass \(M_{\mathrm{eff}}=M_{M}M_{X}/(M_{M}+M_{X})=4.9\times 10^{-26}\) (kg). The time interval is taken as \(\Delta t=5.0\times 10^{-4}\omega_{\mathrm{ph}}^{-1}=7.6\times 10^{-3}\) (fs). We confirmed that the total energy of the system after turnoff of the pumping conserves within \(2\times 10^{-3}\) percent in the numerical simulations. We note that the electronic wavefunction \(|\psi_{\
u,s}(t)\rangle\) deviates from the eigenstates of the instantaneous Hamiltonian, as shown in Equation (21). In the simulations, the kinetic-energy part in Equation (3) serves as a heat bath and therefore the photoexcited electronic system can be relaxed. Thus, the electron system and the classical lattice system are simultaneously updated. The similar methods are applied to electron-spin coupled systems [40] as well, where local spins coupled with conducting electrons are treated classically. We initially add small random values to the lattice displacements as thermal fluctuations, which are three orders of magnitude smaller than the initial lattice displacements [41]. Hereafter, the total number of units \(N\) is set equal to 64, unless otherwise noticed. To observe photoinduced structural changes, we calculate the electron density on the platinum sublattice  
\[d_{n:l}(t)=\sum_{s}\langle c_{n:lMs}^{\dagger}c_{n:lMs}\rangle \tag{24}\]  
and its alternating component defined as  
\[\tilde{d}_{n:l}(t)=\frac{(-1)^{n}}{4}\Big{[}d_{n-1:l}(t)-2d_{n:l}(t)+d_{n+1:l}( t)\Big{]}. \tag{25}\]  
We can distinguish the IP-CDW domain from the OP-CDW one by observing the order parameter  
\[O_{n}(t)=\frac{\tilde{d}_{n:1}(t)\tilde{d}_{n:2}(t)}{\sqrt{|\tilde{d}_{n:1}(t) \tilde{d}_{n:2}(t)|}}. \tag{26}\]  
\(O_{n}(t)\) takes positive and negative values for the IP and OP configurations, respectively.