where \(H_{ij} = \int \langle i|H|j \rangle \ dx\), (i and j = 0, 1), \(\int \langle 0|0 \rangle \ dx = \int \langle 1|1 \rangle \ dx = 1
\) and \(\int \langle 0|1 \rangle \ dx \ \approx \ 0.
\) Equations (A2) are a set of the secular equation with two variables (\(c_0\) and \(c_1\)). The coefficients are \(H_{00} = E_0\), \(E_{11}=E_1\), and \(H_{01} = H_{10} = - V ( V > 0)\).
Based on an asymmetric double-well system with two distinct \(E_0\) and \(E_1\)(Figures S2(a)), Eq.(A2) can be rewritten as
\(
\begin{bmatrix}
E_0-E \ \ \ \ \ \ \ \ - V \\
- V \ \ \ \ \ \ \ \ E_1 - E \\
\end{bmatrix}
\begin{bmatrix}
c_0 \\
c_1 \\
\end{bmatrix}
= \ 0
\)\(\)
Typically, \(( c_0, c_1 ) \neq (0, 0)
\), the determinant of the 2 x 2 matrix is vanished, and the eigenenergies are \(E\equiv\) \(E_\pm\) \(=\) \(\frac{E_0 + E_1}{2}
\) \(\pm\) \(\sqrt{(\frac{E_0 - E_1}{2})^2 + V^2 }
\), where \(E_-\) and \(E_+\) are the energies for the ground state and the excited state, respectively (Figure S2(b)).
The ratios of population at the ground state and at the excited state are
\((\frac{c_1}{c_0})_{ground} = \frac{E_{av} \ + \ \sqrt{\Delta^2 + V^2} }{V}\) and \((\frac{c_1}{c_0})_{excited} = \frac{E_{av} \ - \ \sqrt{\Delta^2 + V^2} }{V}\), where \(\Delta = \frac{E_0 - E_1}{2}\) and \(E_{av} = \frac{E_0 + E_1}{2}\). Then, the excited state and the ground state are \(\psi_+ = \cos(\frac{\theta}{2}) \ e^{- i \frac{\phi}{2}} \ |0 \rangle + \sin(\frac{\theta}{2}) \ e^{i \frac{\phi}{2}} \ |1 \rangle
\) and \(\psi_- = - \sin(\frac{\theta}{2}) \ e^{- i \frac{\phi}{2}} \ |0 \rangle + \cos(\frac{\theta}{2}) \ e^{i \frac{\phi}{2}} \ |1 \rangle
\), respectively, where \(\tan \ \theta = |V|/\Delta\) and \(V= |V| e^{i \phi}
\).
Furthermore, let us apply this double-well system to the H-bond, i.e. N-H … O, in vacuum, and adopt the atomic orbitals from N and O atoms as \(|0\rangle= \psi_N\) and \(|1\rangle= \psi_O\). Initially, the proton is located at the \(|0\rangle\) state close to the N-atom. We here define the initial state at t = 0 as \(\psi(0) = |0\rangle = e^{i \frac{\phi}{2}} \
[\cos(\frac{\theta}{2}) \ \psi_+
- \sin(\frac{\theta}{2}) \ \psi_-]\) . The time evolution of the state follows