## 2 Adiabatic Nature of Exciton Self-Localization  
As shown by [PERSON] [9], the PISPT phenomenon is closely related to the aforementioned self-localization of an exciton in an insulating crystal. It can be simply described by the following model Hamiltonian \((\equiv\mathrm{H_{F}},\hbar\ =\ 1)\) for an exciton,  
\[\mathrm{H_{F}}\ =\ -\mathrm{T_{F}}\ \sum_{<l,l^{*}(\
eq l)>}\ \left[\mathrm{F_{I^{*}} ^{+}F_{l}+h.c.}\right]+\sum_{l}(\mathrm{E_{g}}+6\mathrm{T_{F}}-\omega_{0} \mathrm{SQ_{l}})\mathrm{F_{I}^{+}F_{l}}+\frac{\omega_{0}}{2}\sum_{l}\left(- \frac{\partial^{2}}{\partial Q_{l}^{2}}+Q_{l}^{2}\right) \tag{1}\]  
Here, \(\mathrm{T_{F}}\) (?0) is the resonant transfer (energy) of an exciton from a lattice site \(l\) to its nearest neighbouring sites \(l^{*}\) in a simple cubic crystal. The bracket \(<\)\(l\), \(l^{*}\)? in Equation (1) denotes that these two lattice sites \(l\) and \(l^{*}\) are nearest neighbours with each other. \(\mathrm{F_{I}^{+}}\) in Equation (1) is the creation operator of this exciton at the lattice site \(l\). It is not the charge transfer type excitation, but a Frenkel type (intra-atomic, or intra-molecular) one that is well localized only in each lattice site. As schematically shown in Figure 1a, \(\mathrm{E_{g}}\) in Equation (1) denotes the energy gap of this insulator, while S is the dimensionless coupling constant of this exciton to a site localized phonon, of which energy and dimensionless coordinate are \(\omega_{0}\) and \(Q_{l}\), respectively. In this section, the kinetic energy of this phonon is negelected, because of the adiabatic approximation. Usually, \(\mathrm{E_{g}}\), \(6\mathrm{T_{F}}\) and \(\omega_{0}\)S are quantities of the order of eV, while \(\omega_{0}\) is 10 meV or so. Within the adibatic approximation, the eigen-state (\(\equiv|\Psi(Q_{l})>\), \(<\Psi|\Psi>\ =\ 1\)) of this H\({}_{\rm F}\) will be given as a function of \(Q_{l}\). It is unknown at present, but we determine it under the condition that the total number of the exciton is just one, \(\sum_{l}F_{l}^{+}F_{l}\ =\ 1\). After formally taking the average of H\({}_{\rm F}\) with respect to this unknown \(|\Psi>\), we can apply the Hellmann-Feynman theorem to Equation (1), and can get as,  
\[\frac{\partial<\Psi|H_{\rm F}|\Psi>}{\partial Q_{l}}\ =\ 0,<\Psi|F_{l}^{+}F_{l}| \Psi>\ =\ \frac{Q_{l}}{S}. \tag{2}\]  
Substituting this Equation (2) into the original Equation (1), we also get  
\[<H_{\rm F}>\ =\ \left(E_{\rm g}+6T_{\rm F}\right)-T_{\rm F}\ \sum_{<l,l^{\prime}( \
eq l)>}\ \left[<F_{l}^{+}F_{l^{\prime}}>+<F_{l^{\prime}}^{+}F_{l}>\right]-\frac{ \omega_{0}S^{2}}{2}\sum_{l}<F_{l}^{+}F_{l}>^{2}, \tag{3}\]  
where \(\Psi\) is omitted in the averages \(<\cdots>\), for simplicity. We should note that this Equation (3) holds only at local minimum (or extremum) points in the multi-dimentional coordinate space spanned by \(Q_{l}\), since it is obtained by using Equation (2). When the exciton-phonon coupling is sufficiently strong, \(6T_{\rm F}<\left(\omega_{0}S^{2}\right)/2\), according to [PERSON] and [PERSON] [7], we have only two types of minima in the adiabatic potential energy surface of the excited state, as schematically shown in Figure 1a. The first minimum is the global one with \(<F_{l}^{+}F_{l}>\ =\ \delta_{l,0}\), being localized, say, at the origin \(l=0\) with a large lattice displacement, \(Q_{0}\ =\ \rm S\). Its electronic energy \(\left(\left(E_{\rm g}+6T_{\rm F}\right)-\omega_{0}S^{2}\right)\), given by the second term of Equation (1), formally starts from the exciton band center \(\left(E_{\rm g}+6T_{\rm F}\right)\), but goes below the exciton band, as a local lattice displacement \(Q_{0}\) is self-induced \((0\to S)\). It is called the self-trapped, or self-localized exciton (STE) state. The second local minimum is \(<F_{l}^{+}F_{l}>\ =\ 1/\rm N\), where N denotes the total number of the lattice sites in the crystal. This is the plane-wave state of the exciton whose wave-vector (\(\equiv\rm k\)) is zero, \(\rm k=0\), and its energy is just the energy gap \(E_{\rm g}\).