# Photocontrollable Mixed-Valent States in Platinum-Halide Ladder Compounds
## 2 Modeling  
To describe _MX_ ladder compounds (Figure 1), we employ the two-orbital extended Peierls-Hubbard Hamiltonian  
\[\mathcal{H}=\mathcal{H}_{\text{el}}+\mathcal{H}_{\text{lat}}+\mathcal{H}_{ \text{el-lat}}: \tag{1}\]  
\[\mathcal{H}_{\text{el}} =-\sum\limits_{n,l,s}\bigg{(}t_{\text{MX}}^{\parallel}c_{n+1:lM \text{s}}^{\dagger}c_{n:lX\text{s}}+t_{\text{MX}}^{\dagger}c_{n:lX\text{s}}^{ \dagger}c_{n:lM\text{s}}+\frac{t_{\text{MM}}^{\perp}}{2}c_{n:l+1M\text{s}}^{ \dagger}c_{n:lM\text{s}}+\text{H.c.}\bigg{)} \tag{2}\] \[+\sum\limits_{n,l,s}\big{(}\varepsilon_{M}n_{n:lM\text{s}}+ \varepsilon_{X}n_{n:lX\text{s}}\big{)}+\sum\limits_{n,l,s,s^{\prime}}\bigg{\{} \sum\limits_{A=M,X}\frac{U_{A}}{2}n_{n:lAs}n_{n:lAs}\delta_{n,-s^{\prime}}\] \[+V_{\text{MX}}^{\parallel}(n_{n+1:lM\text{s}}+n_{n:lM\text{s}}) n_{n:lX\text{s}^{\prime}}+V_{\text{MX}}^{\times}(n_{n+1:lM\text{s}}+n_{n:lM \text{s}})n_{n:3-lX\text{s}^{\prime}}\] \[+\sum\limits_{A=M,X}\Bigl{(}V_{A}^{\parallel}n_{n+1:lAs}n_{n:lAs ^{\prime}}+\frac{V_{A\perp}^{\perp}}{2}n_{n:lAs}n_{n:3-lAs^{\prime}}+V_{AA}^{ \times}n_{n+1:lAs}n_{n:3-lAs^{\prime}}\Bigr{)}\bigg{\}},\]  
\[\mathcal{H}_{\text{lat}} =\sum\limits_{n,l}\bigg{\{}\frac{K_{\text{MX}}}{2}\left[(u_{n:lX \text{X}}-u_{n:lM})^{2}+(u_{n+1:lM}-u_{n:lX})^{2}\right]+\frac{M_{\text{M}}}{ 2}\dot{u}_{n:lM}^{2}+\frac{M_{X}}{2}\dot{u}_{n:lX}^{2}\bigg{\}}, \tag{3}\]  
\[\mathcal{H}_{\text{el-lat}} =-\sum\limits_{n,l,s}\big{[}\alpha(u_{n:lX\text{X}}-u_{n+1:lM})c_{ n+1:lM\text{s}}^{\dagger}c_{n:lX\text{s}}-\alpha(u_{n:lX\text{s}}-u_{n:lM})c_{ n:lX\text{s}}^{\dagger}c_{n:lM\text{s}}+\text{H.c.}\big{]} \tag{4}\] \[-\sum\limits_{n,l,s}\big{[}\beta_{M}(u_{n:lX}-u_{n-1:lX})n_{n:lM \text{s}}+\beta_{X}(u_{n+1:lM}-u_{n:lM})n_{n:lX\text{s}}\big{]}\,.\]  
Figure 1: Modeling of _MX_ ladder compounds. Metal, halogen, and interchain-ligand sites are denoted by \(M\), \(X\), and \(L\), respectively. \(a\) is a lattice constant. \(\mathcal{H}_{\text{el}}\) denotes the electronic part of the total Hamiltonian in the second-quantization representation, where \(n_{n:lAs}\)\((=c_{n:lAs}^{\dagger}c_{n:lAs})\) is a number operator of an \(s\)\((=\uparrow,\downarrow=+,-)\)-spin electron at site \(A\)\((=M,X)\) on leg \(l\)\((=1,2)\) and unit \(n\)\((=1,\cdots,N)\). \(N\) is a total number of units. We consider half-filled \(d_{z^{2}}\) orbitals on metal (\(M\)) ions and fully-filled \(p_{z}\) orbitals on halogen (\(X\)) ions, where \(\varepsilon_{A}\) (\(A=M,X\)) denotes the orbital energy of the isolated ion. The intrachain and interchain transfer integrals are denoted by \(t_{MX}^{\parallel}\) and \(t_{MM}^{\perp}\), respectively. The on-site and intersite Coulomb repulsions are modeled by \(U_{A}\) (\(A=M,X\)) and \(V_{AA^{\prime}}^{\parallel,\perp,\times}\) (\(A,A^{\prime}=M,X\)), respectively. \(\mathcal{H}_{\text{lat}}\) describes the elastic and kinetic energies of a lattice system, where leg-direction displacement and the mass of the metal (halogen) ion are denoted by \(u_{n:lM(X)}\) and \(M_{M(X)}\), respectively. \(K_{MX}\) is a spring constant. In the present paper, the lattice displacement \(u_{n:lM(X)}\) is treated as a classical value [32; 33; 34]. \(\mathcal{H}_{\text{el-lat}}\) represents electron-lattice interactions, which contain Peierls (\(\alpha\)) and Holstein (\(\beta_{M}\), \(\beta_{X}\)) type couplings. The intrachain transfer integral and the spring constant are taken as \(t_{MX}^{\parallel}=1.54\) (eV) and \(K_{MX}^{\parallel}=6.00\) (eVA\({}^{-2}\)), respectively [1; 2]. The values of the Coulomb parameters and the electron-lattice couplings are as follows, \(U_{M}/t_{MX}^{\parallel}=U_{X}/t_{MX}^{\parallel}=1.00\), \(V_{MX}^{\parallel}/t_{MX}^{\parallel}=0.35\), \(V_{MX}^{\times}/t_{MX}^{\parallel}=0.13\), \(V_{MM}^{\parallel}/t_{MX}^{\parallel}=0.15\), \(V_{XX}^{\parallel}/t_{MX}^{\parallel}=V_{XX}^{\perp}/t_{MX}^{\parallel}=0.10\), and \(\beta_{M}/\sqrt{K_{MX}t_{MX}^{\parallel}}=\beta_{X}/\sqrt{K_{MX}t_{MX}^{ \parallel}}=3\alpha/\sqrt{K_{MX}t_{MX}^{\parallel}}=0.75\).