# Photocontrollable Mixed-Valent States in Platinum-Halide Ladder Compounds
## 4 Photoexcitation and Following Lattice Relaxation
### Variational Calculation  
We take a variational approach to lattice dynamics [42; 50] and reveal the stabilization of the OP-CDW configuration in photoexcited states. Any conversion between IP CDW and OP CDW is caused by emergent solitons in pairs. To describe the relaxation channels via soliton pairs, we introduce bond variables,  
\[u_{nl}^{-} \equiv u_{nlX}-u_{nlM}, \tag{31}\] \[u_{nl}^{+} \equiv u_{nl+1:lM}-u_{nlX}, \tag{32}\]  
and define a trial wavefunction,  
\[u_{nl}^{\pm} =\pm(-1)^{n}u_{lSn:l}\ \ \ (l=1,2); \tag{33}\] \[s_{nl} =\left[1+\kappa_{l}\left\{\tanh\frac{|n-N/2|-d_{l}/2}{\xi_{l}}-1 \right\}\right], \tag{34}\]  
where \(u_{l}\) (\(l=1,2\)) is the initial CDW amplitude, \(\kappa_{l}\) and \(\xi_{l}\) are variational parameters to design each soliton defect, and \(d_{l}\) denotes their distance along the legs. We set \(d_{1}\) and \(d_{2}\) equal to \(d\) for simplicity. The \(n\)-photon excited state from the static HF state \(|E_{0}\rangle\) described by Equation (33) is given as  
\[|E_{\text{ex}}\rangle\equiv\prod_{i=1}^{n}\tilde{c}_{i_{\
u_{i}}\tau_{i}}^{ \dagger}\tilde{c}_{i_{\
u_{i}}\tau_{i}}|E_{0}\rangle, \tag{35}\]  
where \(\tilde{c}_{i_{\
u_{i}}\tau_{i}}^{\dagger}\) creates the \(\tau_{i}\)-spin electron of the \(\
u_{i}\)th HF-level energy \(\epsilon_{i_{\
u_{i}}\tau_{i}}\), while the indices satisfy the condition \(\epsilon_{i_{\
u_{i}}\tau_{i}}\leq\epsilon_{\text{F}}\) (Fermi energy) \(<\epsilon_{i_{\
u_{i}}\tau_{i}}\) and \(n\) electron-hole pairs are selected in ascending order of their creation energies \(\epsilon_{i_{\
u_{i}}\tau_{i}}-\epsilon_{i_{\
u_{i}}\tau_{i}}\). The variational parameters are determined to minimize the photoexcited-state energy \(E_{\text{ex}}=\langle E_{\text{ex}}|\mathcal{H}_{\text{HF}}|E_{\text{ex}}\rangle\) at every \(d\), where \(\mathcal{H}_{\text{HF}}\) is the static HF Hamiltonian. An energy of the excited state immediately after photoexcitation (\(\kappa_{1}=\kappa_{2}=d=0\)) is denoted by \(E_{\text{ex}}^{0}\). In the calculation, the lattice displacements are treated adiabatically. In other words, the kinetic-energy part in Equation (3) is discarded. The thus calculated adiabatic potential energy surfaces of the \(n\)-photon excited states are presented in Figure 6a-c. The optimized \(\kappa=(\kappa_{1},\kappa_{2})\) at \(d=0\) and \(16\) are plotted in Figure 6a'-c', where the situation \(\kappa_{1}\geq\kappa_{2}\) is focused on because the energy surfaces are symmetric with respect to the line \(\kappa_{1}=\kappa_{2}\). In the case of \(n=1\) (Figure 6a,a'), the photogenerated CTE begins to be localized with increasing \(\kappa_{l}\) while keeping \(d=0\). In IP CDW, the local deformation appears on each chain (\(\kappa_{1}=\kappa_{2}\
eq 0\) at \(d=0\)). In OP CDW, on the other hand, it appears on either one of the two chains (\(\kappa_{1}\
eq 0,\kappa_{2}=0\) at \(d=0\)). These results are consistent with the \(d\)-\(p\)-\(\pi\) model calculations [31]. With increasing \(d\), the photoexcited-state energies increase with diminishing \(\kappa_{l}\). The local defect at \(d=0\) cannot dissociate into a soliton pair in each CDW. In the case of \(n=4\) (Figure 6b,b'), the situation is almost the same. On both the CDW backgrounds, four CTEs can be merged into a local defect described by the finite \(\kappa_{l}\) at \(d=0\) but global structural changes are never induced. This relaxation dynamics induced by the weak photoexcitation is observed in Figure 3a,b as disappearance of the photoinduced defects. The time-evolutional calculations reveal geminate recombination of solitons (Figure 3a) and background charge oscillations (Figure 3a,b). This dynamics is hard to see within this variational approximation. The situation is drastically changed in the case of \(n=8\) (Figure 6c,c'), where the IP-to-OP phase conversion is possible but the reverse process never occurs. Self-trapping of CTEs is observed on each CDW background as with the one- and four-photon excitation cases.