Thus, Lemma <ref> implies the first inequality. Similarly, by deriving $b'=\langle E,\varphi_2\rangle_{\mathcal{H}}$, we can obtain the second inequality. Lastly,

\begin{align}
\tfrac{d}{d\tau}\|P_-\hat u\|_{\mathcal{H}}^2=2\langle \hat u_\tau ,P_-\hat u\rangle_{\mathcal{H}}\leq 2\langle \mathcal{L} P_-\hat u ,P_-\hat u\rangle_{\mathcal{H}} +2\|P_-E\|_{\mathcal{H}}\|P_-\hat u\|_{\mathcal{H}}.
\end{align}

Since $\lambda_k \geq\frac{1}{2}$ for $k\geq 3$, we have

\begin{equation}
\tfrac{3}{2}\langle \mathcal{L} P_-\hat u ,P_-\hat u\rangle_{\mathcal{H}}\leq -\tfrac{3}{4}\|P_-\hat u\|_{\mathcal{H}}^2.
\end{equation}

Also, by integration by parts, we have

\begin{equation}
\tfrac{1}{2}\langle \mathcal{L} P_-\hat u ,P_-\hat u\rangle_{\mathcal{H}}=\tfrac{1}{4}\|P_-\hat u\|_{\mathcal{H}}^2- \tfrac{1}{2}\|(P_-\hat u)_y\|_{\mathcal{H}}^2.
\end{equation}

Therefore, combining with $2\|P_-E\|_{\mathcal{H}}\|P_-\hat u\|_{\mathcal{H}} \leq 4\|P_-E\|_{\mathcal{H}}^2+\frac{1}{4}\|P_-\hat u\|_{\mathcal{H}}^2$ yields

\begin{equation}
\tfrac{d}{d\tau}\|P_-\hat u\|_{\mathcal{H}}^2 \leq -\tfrac{1}{2} \|(P_-\hat u)_y\|^2-\tfrac{1}{4} \|P_-\hat u\|_{\mathcal{H}}^2+4\|P_-E\|_{\mathcal{H}}^2.
\end{equation}

Thus, applying Lemma <ref> completes the proof.

There are some numeric constants $\varepsilon \in (0,10^{-4})$ and $C>0$ such that each $\hat u^i$ satisfies

\begin{equation}
\|\hat u^i(y,\tau)-2^{-\frac{1}{2}}b^i(\tau'-L)y\|_{\mathcal{H}}^2 \leq Cr^{-10}+Ce^{-\frac{1}{4}L}r^{-8}
\end{equation}

in $\tau \in [\tau'-L-4,\tau'-L]$.

We drop the index $i$ for simplicity. Since $\hat u =a+2^{-\frac{1}{2}}yb+P_-\hat u$, we have

\begin{equation}\label{eq:proj.u_y}
\|\hat u_y\|_{\mathcal{H}}^2=\tfrac{1}{2} b^2+    \|(P_-\hat u)_y\|_{\mathcal{H}}^2.
\end{equation}

We consider $\alpha(\tau):=\|P_-\hat u\|_{\mathcal{H}}^2-r^{-2}b^2$. Combining Lemma <ref>, (<ref>), (<ref>),(<ref>),  and (<ref>) yields

\begin{equation}
\alpha'\leq -\tfrac{1}{4}\|P_-\hat u\|_{\mathcal{H}}^2+C\varepsilon r^{-2}b^2+Cr^{-\frac{11}{2}}e^{-\frac{r^2}{8}}\leq -\tfrac{1}{4}\alpha+Cr^{-10}, \label{eqn:3}
\end{equation}

for small enough $\varepsilon$. Since the above inequality implies $\frac{d}{d\tau}e^{\frac{1}{4}\tau}\alpha \leq Cr^{-10}e^{\frac{1}{4}\tau}$,

\begin{equation}
e^{\frac{1}{4}\tau}\alpha(\tau) - e^{\frac{1}{4}(\tau'-4L)}\alpha(\tau'-4L) \leq Cr^{-10}e^{\frac{1}{4}\tau}
\end{equation}

holds for $\tau \in [\tau'-4L,\tau']$. Since (<ref>) guarantees $\alpha(\tau'-4L) \leq Cr^{-8}$, we have

\begin{equation}\label{eq:stable.esti}
\|P_-\hat u\|_{\mathcal{H}}^2 \leq r^{-2}b^2+ C r^{-8}e^{-\frac{1}{4}L}  + Cr^{-10}
\end{equation}

for $\tau \in [\tau'-3L,\tau']$, where $C$ is some numeric constant.

Next, we define $\beta(\tau):=a^2-r^{-4}(b^2+\|P_-\hat u\|_{\mathcal{H}}^2)$. We again combine Lemma <ref>, (<ref>), and (<ref>), and choose small enough $\varepsilon$ so that we obtain

\begin{equation}
\beta' \geq a^2   -C\varepsilon r^{-4}b^2-Cr^{-\frac{11}{2}}e^{-\frac{r^2}{8}}\geq \beta -Cr^{-\frac{11}{2}}e^{-\frac{r^2}{8}}.
\end{equation}

Hence, considering $\frac{d}{d\tau}e^\tau \beta$, we have

\begin{equation}
e^{\tau'}\beta(\tau')-e^{\tau}\beta(\tau) \geq -Ce^{\tau}r^{-\frac{11}{2}}e^{-\frac{r^2}{8}}.
\end{equation}

Thus, if $\tau \in [\tau'-4L,\tau'-L]$ then

\begin{equation}\label{eq:unstable.esti}
a^2\leq r^{-4}(b^2+\|P_-\hat u\|_{\mathcal{H}}^2)+Cr^{-8}e^{-L}+Cr^{-\frac{11}{2}}e^{-\frac{r^2}{8}}.
\end{equation}

Hence, adding (<ref>) and (<ref>) yields

\begin{equation}
\|\hat u-P_0\hat u\|_{\mathcal{H}}^2 \leq 2r^{-2}\|\hat u\|_{\mathcal{H}}^2+Cr^{-8}e^{-\frac{1}{4}L}+Cr^{-10}.
\end{equation}

Therefore, by remembering (<ref>), we have

\begin{equation}\label{eq:minor.mode.esti}
\|\hat u-P_0\hat u\|_{\mathcal{H}}^2 \leq Cr^{-10}+Cr^{-8}e^{-\frac{1}{4}L}
\end{equation}

in $\tau \in [\tau'-3L,\tau'-L]$.

Now, we consider $\gamma(\tau):=b(\tau)-b(\tau'-L)$. Then, for sufficiently small $\varepsilon$, we have

\begin{equation}
|\gamma'| + \tfrac{d}{d\tau}\|P_-\hat u\|_{\mathcal{H}}^2\leq C\varepsilon b^2+Cr^{-\frac{3}{2}}e^{-\frac{r^2}{8}}\leq Cr^{-8}
\end{equation}

by using Lemma <ref>, (<ref>), and (<ref>). Since $\gamma(\tau'-L)=0$,

\begin{equation}
|\gamma(\tau)|\leq \int_{\tau}^{\tau'-L}|\gamma'(s)|ds \leq 4r^{-8}-\|P_-\hat u(\cdot,\tau'-L)\|_{\mathcal{H}}^2+\|P_-\hat u(\cdot,\tau)\|_{\mathcal{H}}^2
\end{equation}

holds for $\tau \in [\tau'-L-4,\tau'-L]$. Thus, (<ref>) implies

\begin{equation}
\|P_0\hat u(y,\tau)-2^{-\frac{1}{2}}b(\tau'-L)y\|_{\mathcal{H}}=|\gamma(\tau)|\leq Cr^{-8}.
\end{equation}

Hence, combining (<ref>) and (<ref>) completes the proof.

There exist some numeric constants $\varepsilon \in (0,10^{-4})$, $C_1> 1$ and $\sigma_0 \in (0,1)$ with the following significance. Suppose that there is  $\tau_1 \leq  \tau_\varepsilon$ such that for each $\tau'\leq \tau_1$, $S_{\tau'}\overline{M}_\tau$ is $\delta$-linear with multiplicity $m$ at each $\tau'$ for some rotation $S_{\tau'}$, where $\delta\leq \sigma_0/L$ for some $L\geq 2$. Then, there exists a small angle $\theta$ with $|\theta |\leq C_1L\delta$ such that the profile bundle $\mathbf{u}$ of $S_{\tau'}\overline{M}_\tau$ is well-defined in $P^{\tau'}_{8,4L}$ and

\begin{equation}
\max_{i=1,\cdots,m}\|u^i(y,\tau)-(\tan \theta) y\|_{L^2([-8,8])}   \leq C_1 (L^{\frac{5}{4}}\delta ^{\frac{1}{4}}+ L^{-1})\delta
\end{equation}

holds for $\tau \in [\tau'-L-4,\tau'-L]$, where  $\mathbf{u}:=(u^1,\cdots,u^m)$.

By Lemma <ref>, $\mathbf{u}$ is well-defined in $P^{\tau'}_{8,4L}$, and we may assume $u^1 > u^i$ for every $i\geq 2$ without loss of generality. Let $\beta_i$ denote $2^{-\frac{1}{2}}b^i(\tau'-L)$ for each $i=1,\cdots,m$. If $\beta_1 \geq \beta_2$, then combining $u^1-u^2>0$ and $-(\beta_1-\beta_2)y\geq 0$ for $y\leq 0$ yields

\begin{equation}
|\beta_1-\beta_2|\leq \|(\beta_1-\beta_2)y\|_{L^2([-2,0])} \leq  \| (u^1-u^2)-(\beta_1-\beta_2)y\|_{L^2([-2,0])}
\end{equation}

Thus, Lemma <ref> implies

\begin{equation}\label{eq:angle.gap}
|\beta_1-\beta_2| \leq Cr^{-10}+Ce^{-\frac{1}{4}L}r^{-8}.
\end{equation}

when $\beta_1 \geq \beta_2$. If $\beta_1\leq \beta_2$, then we can repeat the same process for $L^2([0,2])$ so that we obtain (<ref>). Of course, we can obtain the same estimate for every $|\beta_1-\beta_i|$. Hence, remembering $L\geq 2$, Lemma <ref> leads us to

\begin{equation}
\|\hat u^i - \beta_1 y\|_{\mathcal{H}}^2 \leq Cr^{-10}+Ce^{-\frac{1}{4}L}r^{-8}\leq Cr^{-10}+CL^{-4}r^{-8}.
\end{equation}

Since $4r:=(C_0L\delta)^{-\frac{1}{4}}$, choosing $\theta$ by $\tan \theta:=\beta_1$ with $|\theta| \leq \|\hat u^1(\cdot,\tau'-L)\|_{\mathcal{H}}\ll 1$ completes the proof.

§.§ Fast convergence

In this final subsection, we show that the rescaled flow converges to a unique tangent flow at the backward infinity at an exponential rate.

We first recall the standard rotation matrix to state the following proposition.

\begin{equation}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{pmatrix}.
\end{equation}

There are some numeric constants $C_2>1$ and $\varepsilon_0>0$ with the following significance. Suppose that the graph $\overline{M}_\tau$ of a function $u \in C^{\infty}(P^0_{8,4})$ with $\|u\|_{C^2(P^0_{8,4})}\leq \varepsilon_0$ is a rescaled curve shortening flow. Then, given $\theta \in (-\varepsilon_0,\varepsilon_0)$, the profile $v$ of the rotated flow $R_{-\theta}\overline{M}_\tau$ is well-defined in $P^0_{6,4}$ and the following holds

\begin{equation}
\|v\|_{C^2(P^0_{4,2})}\leq C_2\sup_{ \tau \in [-4,0]}\|u(y,\tau)-(\tan \theta)y\|_{L^2([-8,8])}.
\end{equation}

Choosing $\varepsilon_0$ small enough, $v$ is well-defined in $P^0_{6,4}$. Also, the area between $v$ and $\{x_2=0\}$ in a ball $B_r(0)$ is equivalent to the area between $u$ and $\{x_2=(\tan \theta) x_1\}$ in $B_r(0)$. Since the $L^1$-norm represents the area difference, we have

\begin{equation}
\|v\|_{L^1([-6,6])} \leq \|u-(\tan \theta)y\|_{L^1([-8,8])} \leq 16\|u-(\tan \theta)y\|_{L^2([-8,8])}.
\end{equation}

Note that the second inequality comes from the standard Holder's inequality. On the other hand, the local $L^\infty$ estimate (cf. Lieberman Theorem 6.17) yields

\begin{equation}
\|v\|_{L^\infty(P^0_{5,3})}\leq C \sup_{ \tau \in [-4,0]}\|v\|_{L^2([-6,6])}
\end{equation}

Also, the interior Schauder estimate (cf. Lieberman Theorem 4.9) implies

\begin{equation}
\|v\|_{C^{2,\alpha}(P^0_{4,2})}\leq C \|v\|_{L^\infty(P^0_{5,3})},
\end{equation}

for any $\alpha \in (0,1)$. Combining the inequalities above completes the proof.

There exist some numeric constants $\varepsilon \in (0,10^{-4})$, $L\geq 2$, $C_3>1$, and $\delta_0 \in (0,1)$ with the following significance. Suppose that there is $\tau_1\leq \tau_\varepsilon$ such that  for each $\tau'\leq \tau_1$, $S_{\tau'}\overline{M}_\tau$ is $\delta$-linear, where $\delta\leq \delta_0$, with multiplicity $m$ at $\tau'$ for some rotation $S_{\tau'}$. Then, for each $\tau'\leq \tau_1$, there exists a rotation $\bar S_{\tau'}$ such that $|\bar S_{\tau'}-S_{\tau'}| \leq C_3\delta$ holds and $\bar S_{\tau'}\overline{M}_\tau$ is $\frac{1}{2}\delta$-linear with multiplicity $m$ at $\tau'-L$. Moreover, the profile bundle $\mathbf{u}$ of $S_{\tau'}\overline{M}_\tau$ satisfies $\|\mathbf{u}\|_{C^2(P^{\tau'}_{3,4L})}\leq C_3\delta$.

We recall $\tau_0,\theta$ in Theorem <ref>, and define $\bar S_{\tau'}=R_{-\theta}S_{\tau'}$. Then, we have $|\bar S_{\tau'}-S_{\tau'}| \leq CL\delta$. Also, Proposition <ref> implies $\|\mathbf{u}\|_{C^2(P^{\tau'}_{3,4L})}\leq CL\delta$.

Next, we combine Theorem <ref> and Proposition <ref> so that we have

\begin{equation}
\|v\|_{C^{2}(P^0_{4,2})}\leq C_1C_2(L^{\frac{5}{4}}\delta ^{\frac{1}{4}}+ L^{-1})\delta.
\end{equation}

Now, we fix $L:=4C_1C_2+2$ so that we have

\begin{equation}
\|v\|_{C^{2}(P^0_{4,2})}\leq (C_1C_2L^{\frac{5}{4}}\delta_0^{\frac{1}{4}}+\tfrac{1}{4})\delta.
\end{equation}

Then, we choose $\delta_0 \leq \sigma_0/L$ to satisfy $C_1C_2L^{\frac{5}{4}}\delta_0^{\frac{1}{4}}\leq  4^{-1}$. This completes the proof.

We recall $\delta_0$ and $C_3$ from Lemma <ref>. Then, by the rough convergence theorem (Theorem <ref>), there exists negative enough $\tau_1 \leq \tau_\varepsilon$ such that for each $\tau'\leq \tau_1$, $S^1_{\tau'}\overline{M}_\tau$ is $\delta_0$-linear with multiplicity $m$ at $\tau'$ for some rotation $S^1_{\tau'}$. Then, by Lemma <ref>, each $\tau'\leq \tau_2:=\tau_1-L$ has a rotation $S^2_{\tau'}$ such that  $S^2_{\tau'}\overline{M}_\tau$ is $(\delta_0/2)$-linear  at $\tau'$, and $|S^1_{\tau_1}-S^2_{\tau_2}|\leq C_3\delta_0$.

We repeat this process. Then, each $\tau'\leq \tau_k:=\tau_1-(k-1)L$ has a rotation $S^k_{\tau'}$ such that  $S^k_{\tau'}\overline{M}_\tau$ is $2^{1-k}\delta_0$-linear  at $\tau'$, and $|S^{k-1}_{\tau_{k-1}}-S^k_{\tau_k}|\leq 2^{2-k}C_3\delta_0$. Hence, there is the limit rotation $S_\infty \in SO(2)$ such that $|S_\infty -S^k_{\tau_k}|\leq 2^{2-k}C_3\delta_0$.

We may assume $S_\infty=R_0$ is the identity matrix without loss of generality. Since the profile bundle $\mathbf{u}^k$ of $S_{\tau_k}^k\overline{M}_\tau$ satisfies $\|\mathbf{u}^k\|_{C^2(P^{\tau_k}_{3,4L})}\leq 2^{1-k}C_3\delta_0$ by Lemma <ref>, rememerbing $|S^k_{\tau_k}-R_0|\leq 2^{2-k}C_3\delta_0$, the profile bundle $\mathbf{u}$ of $\overline{M}_\tau$ satisfies $\|\mathbf{u}\|_{C^2(P^{\tau_k}_{3,4L})}\leq 2^{-k}C$ for some $C$. Hence, there exist some $\delta>0$ only depending on $L$ and constant $C$ such that

\begin{equation}
\|\mathbf{u}(\cdot,\tau)\|_{C^2(B_3(0))}\leq Ce^{5\delta\tau}
\end{equation}

holds for $\tau \leq \tau_1$. Hence, for any $\varepsilon>0$, the graphical radius theorem <ref> implies (<ref>) for negative enough time. Also, given $R>0$, Theorem <ref> yields

\begin{equation}
\|\mathbf{u}(\cdot,\tau)\|_{L^\infty(B_{2R}(0))}\leq Ce^{5\delta\tau}
\end{equation}

for some $C$ depending on $R$. Thus, the interior Schauder estimate (cf. <cit.>) guarantees (<ref>). This completes the proof.

§ ESTIMATE OF LOCALIZED GAUSSIAN DENSITY RATIO

There exist $C>0$ and $\delta \in (0, 1)$ small with the following significance.
Let $a,b \in (0, \delta)$ and let $r > \delta^{-1}$. For any connected $C^2$ curve $\Ga$ in $B_r(0)\subset \R^2$ whose $\pa \Ga\subset \pa B_r(0)$ with curvature bound $\lvert \kappa \rvert \leq b r^{-1}$ and $d_0 = \min\ \{ \lvert x \rvert : x\in \Ga\} < \delta$, there hold

\begin{align*}
\frac{1}{\sqrt{4\pi}}\int_{\Ga \cap B_r(0)} \left[1 - a^2 (\lvert x \rvert^2 -2)\right]_+^3
e^{-\frac{\lvert x \rvert^2}{4}}\ ds
\leq 1+b
\end{align*}

\begin{align*}
\frac{1}{\sqrt{4\pi}}\int_{\Ga \cap B_r(0)} \left[1 - a^2 (\lvert x \rvert^2 -2)\right]_+^3
e^{-\frac{\lvert x \rvert^2}{4}}\ ds \geq
\end{align*}

Let $\ga(s)$ be an arclength parametrization of $\Ga$ such that $\ga(0)$ is a closest point to the origin.
Define $d(s) = \lvert \ga(s) \rvert$. We first estimate error of localization function. For $0 <  a < \delta$ is small, there is $C>0$ such that in $B(0, 1/\sqrt{a}) \cap B_r(0)$

\begin{align*}
\left[1 - a^2 (\lvert x \rvert^2 -2)\right]_+^3 \geq 1 - Ca.
\end{align*}

Recall that $\pa_s d^2 = 2\langle \ga, \mathbf{t} \rangle$ and $\pa_{ss} d^2 = 2 \big(1 + \ka \langle \ga, \mathbf{n} \rangle\big)$. Using $|\kappa|\le br^{-1}$ in $B_r(0)$,  we have

\begin{align*}
d_0^2 + (1 - b)s^2 \leq & d^2(s) \leq d_0^2 + (1 + b)s^2.
\end{align*}

\begin{align}
&\frac{1}{\sqrt{4\pi}}\int_{\Ga\cap B_r(0)} \left[1 - a^2 (\lvert x \rvert^2 -2)\right]_+^3 \exp(-\tfrac{d^2}{4})\ ds \notag\\
&\quad\leq  \frac{\exp (-\tfrac{d_0^2}{4})}{\sqrt{4\pi}}\int_{-\infty}^\infty \exp ( -\tfrac{(1 -b)s^2}{4})\ ds = \frac{\exp (-\tfrac{d_0^2}{4})}{\sqrt{1-b}}\leq 1+b. \label{eqn1: appendix}
\end{align}

In addition, we have

\begin{align}
&\frac{1}{\sqrt{4\pi}}\int_{\Ga\cap B_r(0)} \left[1 - a^2 (\lvert x \rvert^2 -2)\right]_+^3 \exp(-\tfrac{d^2}{4})\ ds \notag\\
&\quad\geq (1-Ca)\frac{\exp (-\tfrac{d_0^2}{4})}{\sqrt{4\pi}}\int_{-\min({1/\sqrt{a},r})}^{\min{(1/\sqrt{a},r)}}\exp(-\tfrac{(1+b)s^2}{4})\ ds \notag\\
&\quad\geq (1-Ca)\exp \left(-\frac{d_0^2}{4}\right)\left(1-\frac{\max\left(\sqrt{a}, 1/r\right)}{\sqrt{1+b}}\right).  \label{eqn2: appendix}
\end{align}

In the last inequality, we used the fact that $\exp({-s^2/4})<-1/s^2$ for sufficiently large $s$. Then, (<ref>) and (<ref>) imply the desired conclusion.

§.§ Acknowledgments

KC has been supported by the KIAS Individual Grant MG078902, an Asian Young Scientist Fellowship, and the National Research Foundation(NRF) grant funded by the Korea government(MSIT) (RS-2023-00219980); DS has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIT) (No.2021R1C1C2005144); WS has been supported by the Taiwan MOST postdoctoral research abroad program 111-2917-I-564-002. He would like to thank University of Warwick and NCTS for providing the wonderful research enviroments during the preparation of this paper; KZ would like to thank Nick Edelen for the helpful discussion on GMT.

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ancient solutions to curve shortening flow, Trans. Am. Math. Soc.
376 (2023), no. 12, 8383–8409.


Since three diabatic states were used in the DANN model, and since all three were coupled near the CI, it would be incorrect to plot only two states in Fig. <ref>fig:results(b). We therefore applied a fixed rotation matrix, $\mathbf{U} \neq \mathbf{U}(\vec{R})$, to generate a new diabatic Hamiltonian with only two coupled states. The new diabatic Hamiltonian was given by $\mathbf{H}_d'(\vec{R}) = \mathbf{U}^{\dagger} \hspace{0.05cm} \mathbf{H}_d(\vec{R}) \mathbf{U}$. Note that any position-independent rotation matrix can be brought outside the gradient in Eq. (<ref>eq:hd_nacv), and so $\mathbf{H}_d'$ is still diabatic. The rotation matrix was chosen to diagonalize $\mathbf{H}_d$ at the CI, so that $\mathbf{U}^{\dagger} \hspace*{0.05cm} \mathbf{H}_d (\vec{R}_{\mathrm{CI}}) \hspace*{0.05cm} \mathbf{U} = \mathrm{diag}(\{ E \} )$. This led to $d_{00}' = d_{11}'$ and $d_{01}'=0$ at the CI. Hence the lowest two rotated states were the most important contributors to $E_0$ and $E_1$, and were therefore used in Fig. <ref>fig:results(b).

§ INTENSIVE AND EXTENSIVE QUANTITIES

The DANN model predicts each property by summing over atomic contributions, just as in the PaiNN model. This guarantees size-extensivity. However, as explained below, the off-diagonal elements of $\mathbf{H}_d$ should be intensive, in the sense that atoms not involved in the excitation should not contribute to these values. Nonetheless, we found that summation for these terms gave better results than averaging. This was true even when using a learnable weighted average. Atom-wise summation can still generate accurate $d_{nm}$, because the readout network can simply map unimportant atoms to zero.

To demonstrate extensivity and intensivity, consider two uncoupled subsystems, $A$ and $B$. The total clamped nucleus Hamiltonian is $H(\vec{r}, \vec{R}) = H^A(\vec{r}, \vec{R}) + H^B(\vec{r}, \vec{R}) $. Let the excitation of interest be in subsystem $A$. In this case the adiabatic states involve only excitations in subsystem $A$, so that the $n^{\mathrm{th}}$ diabatic wave function is written as

\begin{align}
\psi_{\mathrm{d}, n}(\vec{r}; \vec{R}) = \psi_{\mathrm{ad}, 0}^{B}(\vec{r}; \vec{R}) \sum_k U_{nk} \psi_{\mathrm{ad}, k}^{A}(\vec{r}; \vec{R}).
\end{align}

The diabatic state is a direct product of the ground state wave function of system $B$, $\psi_{\mathrm{ad}, 0}^{B}(\vec{r}; \vec{R})$, and a rotation of the adiabatic states of system A, $\psi_{\mathrm{ad}, k}^{A}(\vec{r}; \vec{R})$. The matrix elements of $\mathbf{H}_d$ are then given by

(𝐇_d)_nm     = ⟨ψ_d, n |H(r⃗, R⃗) |ψ_d, m ⟩

= ∑_kl U_nk^* U_ml ⟨ψ^B_ad, 0 ψ^A_ad, k   |  H_A(r⃗, R⃗)  + H_B(r⃗, R⃗)   |ψ^B_ad, 0 ψ^A_ad, l  ⟩

= ∑_kl U_nk^* U_ml ( E_k, A + E_B ) δ_kl

= ∑_k U^*_in U_mk  E_k, A + E_B (𝐔 𝐔^†)_mn

= 𝐇^A_d, nm + E_B  δ_nm.

Hence the diagonal elements of $\mathbf{H}_d$ each gain the adiabatic energy of $B$, while the off-diagonal elements remain the same. To understand this result physically, consider that adding a scalar multiplied by the identity yields eigenvalues that are each shifted by the scalar. This means that the eigenvalues of $\mathbf{H}_d$ are each shifted by $E_B$. Therefore the excitation energy is unchanged, which is the expected behavior upon adding an uncoupled system. Note also that extensivity is sometimes described as doubling the energy when the system size is doubled. However, this definition is too narrow, as it applies to only one adiabatic energy at a time. For example, if subsystem B is a copy of subsystem A, then $E_B = E_{0, A}$, meaning that $E_0 = E_{0, A} + E_{B} = 2 E_{0, A}$, and so the ground state energy is indeed doubled. However, $E_1 = E_{1, A} + E_B = 2 E_{1, A} - \mathrm{gap}$, where $\mathrm{gap} = E_{1, A} - E_{0, A}$. Therefore, the excited state energy is not doubled. A more general definition is that the energy of the second subsystem is added to each adiabatic energy.

This analysis shows that the off-diagonal elements of $\mathbf{H}_d$ are intensive. The adiabatic energy gap is also intensive. Intensive here means that adding a subsystem that does not participate in the excitation does not modify the quantity. Such properties can naturally be modeled with an attention mechanism [Bahdanau et al., 2014, Kim et al., 2017, Vaswani et al., 2017, Veličković et al., 2017] that learns the importance of each atom to the excitation. For example, the excitation energy can be modeled as an attention-weighted sum over atomwise quantities. However, as discussed in Supplementary Sec. <ref>, this approach led to worse performance than simply predicting extensive energies for each state.

§ PROOF OF DIABATICITY

Here we prove Eq. (<ref>eq:hd_nacv) in the main text. Using bra-ket notation for the wave functions, we define the diabatic states as a linear combination of adiabatic states,

\begin{align}
& \ket{\psi_{\text{d,}n}} = \sum_{m} \ket{\psi_{\text{ad,}m}} V^*_{nm}, \label{sm_eq:di_def}
\end{align}

where $\mathbf{V}$ is a unitary matrix. Multiplying each side by $V_{nn'}$ and summing over $n$ yields

\begin{align}
& \ket{\psi_{\text{ad,}n}} = \sum_{m} \ket{\psi_{\text{d,}m}} V_{mn},
\end{align}

where we have renamed the dummy indices, $n \to m$ and $n' \to n$. We have also used the fact that $(\mathbf{V}^{\dagger} \mathbf{V})_{nm} = \delta_{nm}$ for any unitary matrix, where $\delta_{nm}$ is the Kronecker delta. Given the connection between diabatic and adiabatic states, we can relate $\mathbf{V}$ to the derivative coupling $\vec{k}$:

\begin{align}
& \vec{k}_{nm} \equiv \bra{ \psi_{\text{ad,}n}}  \nabla_{R} \ket{ \psi_{\text{ad,}m} } \nonumber \\
& = \sum_{ij} V^*_{jn} \left( \nabla_R V_{im} \bra{\psi_{\mathrm{d}, j}}  \ket{\psi_{\mathrm{d, }i}} + V_{im} \bra{\psi_{\mathrm{d}, j}}  \nabla_R \ket{\psi_{\mathrm{d}, i}} \right) \nonumber \\
& = \sum_{i} V_{in}^* \nabla_R V_{im}. \label{sm_eq:deriv_coupling_diabat}
\end{align}

We have used the fact that, by definition, the derivative coupling between any two diabatic states is zero. We have also used the orthonormality of the diabatic states:

\begin{align}
\bra{\psi_{\mathrm{d}, j}} \ket{\psi_{\mathrm{d}, i}} = \sum_{nm} V_{jn} V^*_{im} \bra{\psi_{\mathrm{ad}, n}} \ket{\psi_{\mathrm{ad}, m}} = \sum_{nm} V_{jn} V^*_{im} \delta_{nm} = \left(\mathbf{V} \mathbf{V}^{\dagger} \right)_{ij} = \delta_{ij}, \label{sm_eq:orthonormal}
\end{align}

which follows from the orthonormality of the adiabatic wave functions. Meanwhile, by construction, the Hamiltonian produced by the model has eigenvalues equal to the adiabatic energies:

\begin{align}
\left( \mathbf{H}_{\mathrm{d}} \right)_{nm} & = \left( \mathbf{U} \hspace*{0.05cm} \mathrm{diag} ( \{ E \} ) \hspace*{0.05cm} \mathbf{U}^{\dagger} \right)_{nm} \nonumber  \\
& = \sum_{ij} U_{ni} \hspace*{0.05cm} (E_i \delta_{ij}) \hspace*{0.05cm} U_{mj}^* \nonumber \\
&  = \left(\sum_{i} U_{ni} \bra{\psi_{\mathrm{ad}, i}} \right) \hat{H} \left( \sum_j U_{mj}^* \ket{\psi_{\mathrm{ad} , j}}  \right), \label{sm_eq:h_d_elec}
\end{align}

where $\mathbf{U}$ is the unitary matrix that diagonalizes $\mathbf{H}_d$, $\hat{H}$ is the Hamiltonian operator, and we have used the relation $\bra{\psi_{\mathrm{ad}, j}} \hat{H} \ket{\psi_{\mathrm{ad}, i}} = E_i \bra{\psi_{\mathrm{ad}, j}} \ket{\psi_{\mathrm{ad}, i}} = E_i \delta_{ij}$. Comparing Eqs. (<ref>) and (<ref>), we see that $\mathbf{H}_d$ is the representation of $\hat{H}$ in a different electronic basis. In particular, if we choose $\mathbf{U} = \mathbf{V}$, such that $\mathbf{U}$ satisfies Eq. (<ref>), then $\mathbf{H}_d$ is the representation of $\hat{H}$ in the diabatic basis.

The model could be trained directly with Eq. (<ref>). However, the equation is numerically ill-posed because $\vec{k}_{nm}$ diverges at conical intersections. It is preferable to work instead with the force coupling, $\vec{g}_{nm}$. We now show that Eq. (<ref>eq:hd_nacv), which is defined in terms of $\vec{g}_{nm}$, holds only if (<ref>) is satisfied. The left-hand side of Eq. (<ref>eq:hd_nacv) can be written as

\begin{align}
& \sum_{ij} U^*_{in} \nabla_R \left( \mathbf{U} \hspace*{0.05cm} \mathrm{diag} ( \{ E \} ) \hspace*{0.05cm}  \mathbf{U}^{\dagger} \right)_{ij} U_{jm}\nonumber \\
& = \sum_{ijk} U^*_{in} \left[ (\nabla_R E_k) \hspace*{0.05cm}  U_{ik} U^*_{jk} + E_k \hspace*{0.05cm}  (\nabla_R U_{ik}) U^*_{jk} + E_k \hspace*{0.05cm} U_{ik}  (\nabla_R U^*_{jk}) \right] U_{jm}.
\end{align}

The first term is

\begin{align}
& \sum_{k} (\mathbf{U}^{\dagger} \mathbf{U})_{nk} \hspace*{0.05cm} (\nabla_R E_k) \hspace*{0.05cm} (\mathbf{U}^{\dagger} \mathbf{U})_{km} = (\nabla_R E_n) \delta_{nm},
\end{align}

where we have used the fact that $(\mathbf{U}^{\dagger} \mathbf{U})_{nm} = \delta_{nm}$. Substituting Eq. (<ref>) with $\mathbf{V} = \mathbf{U}$, the second term is

\begin{align}
\sum_{k} \vec{k}_{nk} (\mathbf{U}^{\dagger} \mathbf{U})_{km} E_k = \vec{k}_{nm} E_m.
\end{align}
Performing the same substitution for the third term gives

\begin{align}
\sum_k \vec{k}^*_{mk} (\mathbf{U}^{\dagger}  \mathbf{U})_{nk}  E_k = -\vec{k}_{nm} E_n,
\end{align}

where we have used the anti-Hermitian property $\vec{k}_{mn}^* = -\vec{k}_{nm}$.
Adding the three terms gives

\begin{align}
\left( \mathbf{U}^{\dagger} (\nabla_R H_d) \mathbf{U}\right)_{nm} = \begin{cases} \nabla_R E_n, & \text{ if }n=m, \\ (E_m - E_n) \hspace*{0.05cm} \vec{k}_{nm}, & \text{ if }n \neq m. \end{cases}
\end{align}

Noting that $\vec{f}_n = -\nabla_R E_n$ and $\vec{g}_{nm} = (E_m - E_n) \hspace*{0.05cm} \vec{k}_{nm}$ gives Eq. (<ref>eq:hd_nacv) in the main text.
Hence Eq. (<ref>eq:hd_nacv) can only hold if Eq. (<ref>) is true. Therefore, enforcing Eq. (<ref>eq:hd_nacv) ensures that there is no derivative coupling between any pair of diabatic states. Note that in the main text we used three diabatic states, and trained only on energies and couplings between the first two adiabatic states. This still means that all three diabatic states are properly diabatic: if Eq. (<ref>eq:hd_nacv) is satisfied for even one pair of adiabatic states, then the derivative coupling must be zero between all pairs of diabatic states.

§ TRAINING SPECIES

Here we provide the motifs and substituents used for combinatorial molecule generation, together with a list of the literature species used for dense configurational sampling. Species with a net charge were excluded from training but are given here for completeness. In some cases only the cis or trans isomer was actually investigated experimentally, but in all cases we reference the publication for both isomers.

Many species in both the training and test set had experimental $S_1$ yields in non-polar solution. We did not put all such molecules in the test set, because this would mean losing hundreds of thousands of training geometries. Instead we used the 40 species with the fewest QC calculations.

Motifs used for combinatorial species generation. Examples of literature species for each motif are also given. The species numbers are those in Tables <ref> and <ref>.

Graphs     Literature species

-.45  -.45      1, 2, 25, 26, 63, 64, 125, 126

-.45   -.45      19, 20, 27, 28, 37, 38, 39, 40, 61, 62, 69, 70, 73, 74, 75, 76, 83, 84, 97, 98, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 127, 128, 129, 130, 147, 148, 155, 156, 157, 158, 163, 164

-.45  -.45     67, 68, 87, 88, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108

-.45  -.45      9, 10, 13, 14, 15, 16, 17, 18, 21, 22, 95, 96, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 161, 162

Substituents used for combinatorial species generation. “Azo” denotes the azobenzene attachment site.

SMILES     Graph

[Azo]N(C)Cc1ccccn1     -.45

[Azo]N     -.45

[Azo]N(CC)Cc1ccccc1     -.45

[Azo]C(N)=O     -.45

[Azo]OCc1cc(OCc2ccccc2)cc(OCc2ccccc2)c1     -.45

[Azo]N1CCCCC1     -.45

[Azo]c1ccccc1     -.45

[Azo]NC(=O)CCl     -.45

[Azo]OC(C)=O     -.45

[Azo]N1CCN(C(C)=O)CC1     -.45

[Azo]NC(=O)C[N+](CC)(CC)CC     -.45

[Azo]Cl     -.45

[Azo]CO     -.45

[Azo]C(=O)OCC     -.45

[Azo]CC     -.45

[Azo]C(=O)O     -.45

[Azo]C#C     -.45

[Azo]NC(=O)CN1C(=O)C=CC1=O     -.45

[Azo][N+](=O)[O-]     -.45

[Azo][N+](C)(C)C     -.45

[Azo]O     -.45

[Azo]Oc1c(N)cccc1CC(C)C     -.45

[Azo]F     -.45

[Azo]N1CCN(C(=O)CCl)CC1      -.45

[Azo]Oc1c(N)cccc1C(C)C     -.45

[Azo]NCc1ccccn1     -.45

[Azo]OCc1ccccc1     -.45

[Azo]NC(C)=O     -.45

[Azo]c1cn(-c2ccc3cc4ccccc4cc3c2)nn1     -.45

[Azo]C     -.45

[Azo]C(=O)C(F)(F)F     -.45

[Azo]N1CCN(C)CC1     -.45

[Azo]c1cn(-c2cccc3ccccc23)nn1     -.45

[Azo]c1cn(-c2ccccc2)nn1     -.45

[Azo]C(C)C     -.45

[Azo]C(=O)OC     -.45

[Azo]N1CCOCC1     -.45

[Azo]N(CC)c1ccccc1     -.45

[Azo]N(C)C     -.45

[Azo]N1CCCC1     -.45

[Azo]N1CC2CN(C(C)=O)CC2C1     -.45

[Azo]C(C)(C)C     -.45

[Azo]NC(=O)C=C     -.45

[Azo]N(CCCCCCCC)CCCCCCCC     -.45

[Azo]OCC     -.45

[Azo]N(CC)CC     -.45

[Azo]N(c1ccc2c(c1)C(C)(C)c1ccccc1-2)c1ccc2c(c1)C(C)(C)c1ccccc1-2     -.45

[Azo]NC(=O)CCC[C@H](C[C@H] (N)C(=O)O)C(=O)O     -.45

Test set species and literature quantum yields. For papers that reported the yield at different wavelengths, we chose the wavelength closest to the $n-\pi^*$ ($S_1$) absorption maximum. Quantum yields computed with FS surface hopping using the diabatic model are also shown.

#     SMILES     Graph     Ref.     ($S_1$ yield, solvent)

1     c1ccc(CNc2ccccc2/N=N/ c2ccccc2NCc2ccccn2)nc1     -.45     [Bandara et al., 2010]     Exp: (0, many) Calc:  0 $\pm$ 0

2     c1ccc(CNc2ccccc2/N=N\ c2ccccc2NCc2ccccn2)nc1     -.45     [Bandara et al., 2010]     Exp: None Calc: 0.51 $\pm$ 0.02

3     c1ccc(/N=N/c2cccc3ccccc23)cc1      -.45     [Malkin and Fischer, 1962]     Exp: (0.25, methyl cyclopentane)Calc: 0.43 $\pm$ 0.02

4     c1ccc(/N=N\c2cccc3ccccc23)cc1      -.45     [Malkin and Fischer, 1962]     Exp: (0.49, methyl cyclopentane)Calc: 0.57 $\pm$ 0.02

5     CC1(C)c2ccccc2-c2ccc(N(c3ccc(/N=N/
c4ccccc4)cc3)c3ccc4c(c3)C(C)(C) c3ccccc3-4)cc21     -.45     [Kurita et al., 2014]     Exp: (0.29, toluene)Calc: 0.19 $\pm$ 0.05

6     CC1(C)c2ccccc2-c2ccc(N(c3ccc(/N=N\ c4ccccc4)cc3)c3ccc4c(c3)C(C)(C) c3ccccc3-4)cc21     -.45     [Kurita et al., 2014]     Exp: (0.31, toluene)Calc: 0.55 $\pm$ 0.02

7     Cc1cc(C)c(/N=N/c2ccccc2)c(C)c1     -.45     [Gegiou et al., 1968]     Exp: (0.16, methylcyclohexane and isohexane)Calc: 0.34 $\pm$ 0.03

8     Cc1cc(C)c(/N=N\c2ccccc2)c(C)c1     -.45     [Gegiou et al., 1968]     Exp: (0.38, methylcyclohexane and isohexane)Calc: 0.60 $\pm$ 0.02

9     Cc1cc(C)c(/N=N/c2c(C) cc(C)cc2C)c(C)c1     -.45     [Gegiou et al., 1968, Rau and Yu-Quan, 1988]     Exp: (0.16-0.22, methylcyclohexane and isohexane) [Gegiou et al., 1968], (0.24, n-hexane) [Rau and Yu-Quan, 1988]Calc: 0.41 $\pm$ 0.02

10     Cc1cc(C)c(/N=N\ c2c(C)cc(C)cc2C)c(C)c1     -.45     [Gegiou et al., 1968, Rau and Yu-Quan, 1988]     Exp: (0.44, methylcyclohexane and isohexane) [Gegiou et al., 1968], (0.5, n-hexane) [Rau and Yu-Quan, 1988]Calc: 0.59 $\pm$ 0.02

11     Cc1cc(/N=N/c2ccccc2N (C)Cc2ccccn2)c(N(C)Cc2ccccn2)cc1O     -.45     [Bandara et al., 2011]     Exp: (0[footnote], many)Calc: 0.33 $\pm$ 0.04

12     Cc1cc(/N=N\c2ccccc2N (C)Cc2ccccn2)c(N(C)Cc2ccccn2)cc1O     -.45     [Bandara et al., 2011]     Exp: NoneCalc: 0.53 $\pm$ 0.02

13     CCc1cc(CC)c(/N=N/ c2c(CC)cc(CC)cc2CC)c(CC)c1     -.45     [Rau and Yu-Quan, 1988]     Exp: (0.25, n-hexane)Calc: 0.32 $\pm$ 0.02

14     CCc1cc(CC)c(/N=N\ c2c(CC)cc(CC)cc2CC)c(CC)c1     -.45     [Rau and Yu-Quan, 1988]     Exp: (0.5, n-hexane)Calc: 0.49 $\pm$ 0.02

15     CC(C)c1cc(C(C)C)c(/N=N/c2c(C (C)C)cc(C(C)C)cc2C(C)C)c(C(C)C)c1     -.45     [Rau and Yu-Quan, 1988]     Exp: (0.19, n-hexane)Calc: 0.27 $\pm$ 0.02

16     CC(C)c1cc(C(C)C)c(/N=N\c2c(C (C)C)cc(C(C)C)cc2C(C)C)c(C(C)C)c1     -.45     [Rau and Yu-Quan, 1988]     Exp: (0.55, n-hexane)Calc: 0.53 $\pm$ 0.02

17     CC(C)(C)c1cc(C(C)(C)C)c(/N=N/ c2c(C(C)(C)C)cc(C(C)(C)C)cc2C (C)(C)C)c(C(C)(C)C)c1     -.45     [Rau and Yu-Quan, 1988]     Exp: (0, n-hexane)Calc: 0.01 $\pm$ 0.00

18     CC(C)(C)c1cc(C(C)(C)C)c(/N=N\ c2c(C(C)(C)C)cc(C(C)(C)C)cc2C (C)(C)C)c(C(C)(C)C)c1     -.45     [Rau and Yu-Quan, 1988]     Exp: NoneCalc: 0.78 $\pm$ 0.02

19     CCCCCCCCN(CCCCCCCC)c1ccc (/N=N/c2ccc(C(=O)C(F)(F)F)cc2)cc1     -.45     [Mohr and Grummt, 2004]     Exp: (0.16[footnote+1],  toluene)Calc: 0.08 $\pm$ 0.04

20     CCCCCCCCN(CCCCCCCC)c1ccc (/N=N\c2ccc(C(=O)C(F)(F)F)cc2)cc1     -.45     [Mohr and Grummt, 2004]     Exp: (0.7-1[footnote+1], toluene)Calc: 0.66 $\pm$ 0.02

21     CCOC(=O)c1cc(F)c(/N=N/c2c (F)cc(C(=O)OCC)cc2F)c(F)c1     -.45     [Knie et al., 2014]     Exp: (0.1, hexane)Calc: 0.08 $\pm$ 0.02

22     CCOC(=O)c1cc(F)c(/N=N\c2c (F)cc(C(=O)OCC)cc2F)c(F)c1     -.45     [Knie et al., 2014]     Exp: (0.53, hexane)Calc: 0.44 $\pm$ 0.02

23     CC(=O)Oc1cc(N(C)Cc2ccccn2)c
(/N=N/c2ccccc2N(C)Cc2ccccn2)cc1C     -.45     [Bandara et al., 2011]     Exp: (0.17, C6D6)Calc: 0.30 $\pm$ 0.04

24     CC(=O)Oc1cc(N(C)Cc2ccccn2)c
(/N=N\c2ccccc2N(C)Cc2ccccn2)cc1C     -.45     [Bandara et al., 2011]     Exp: NoneCalc: 0.51 $\pm$ 0.02

25     CN(Cc1ccccn1)c1ccccc1 /N=N/c1ccccc1N(C)Cc1ccccn1     -.45     [Bandara et al., 2010]     Exp: (0.19, CDCl3)Calc: 0.37 $\pm$ 0.03

26     CN(Cc1ccccn1)c1ccccc1 /N=N\c1ccccc1N(C)Cc1ccccn1     -.45     [Bandara et al., 2010]     Exp: NoneCalc: 0.48 $\pm$ 0.02

27     COc1ccc(/N=N/c2ccc ([N+](=O)[O-])cc2)cc1     -.45     [Gegiou et al., 1968]     Exp: (0.17, methylcyclohexane and isohexane)Calc: 0.05 $\pm$ 0.02

28     COc1ccc(/N=N\c2ccc ([N+](=O)[O-])cc2)cc1     -.45     [Gegiou et al., 1968]     Exp: (0.55, methylcyclohexane and isohexane)Calc: 0.46 $\pm$ 0.02

29     COc1cc(N(C)Cc2ccccn2) c(/N=N/c2ccccc2N(C)Cc2ccccn2)cc1C     -.45     [Bandara et al., 2011]     Exp: (0[footnote], many)Calc: 0.35 $\pm$ 0.03

30     COc1cc(N(C)Cc2ccccn2) c(/N=N\c2ccccc2N(C)Cc2ccccn2)cc1C     -.45     [Bandara et al., 2011]     Exp: NoneCalc: 0.54 $\pm$ 0.02

31     Fc1cccc(F)c1/N=N/c1ccccc1     -.45     [Knie et al., 2014]     Exp: (0.32, hexane)Calc: 0.36 $\pm$ 0.02

32     Fc1cccc(F)c1/N=N\c1ccccc1     -.45     [Knie et al., 2014]     Exp: (0.55, hexane)Calc: 0.53 $\pm$ 0.02

33     Fc1cccc(F)c1/N=N/c1c(F)cc(-c2ccccc2)cc1F     -.45     [Moreno et al., 2015]     Exp: (0.15 $\pm$ 0.015, n-hexane)Calc: 0.41 $\pm$ 0.02

34     Fc1cccc(F)c1/N=N\c1c(F)cc(-c2ccccc2)cc1F     -.45     [Moreno et al., 2015]     Exp: (0.28 $\pm$ 0.028, n-hexane)Calc: 0.51 $\pm$ 0.02

35     Nc1ccccc1/N=N/c1ccccc1NCc1ccccn1     -.45     [Bandara et al., 2010]     Exp: (0[footnote], many)Calc: 0.04 $\pm$ 0.04

36     Nc1ccccc1/N=N\c1ccccc1NCc1ccccn1     -.45     [Bandara et al., 2010]     Exp:  NoneCalc: 0.54 $\pm$ 0.02

37     OCc1ccc(/N=N/c2ccc (OCc3ccccc3)cc2)cc1     -.45     [Sierocki et al., 2006]     Exp: (0.4, dichloromethane)Calc: 0.13 $\pm$ 0.03

38     OCc1ccc(/N=N\c2ccc (OCc3ccccc3)cc2)cc1     -.45     [Sierocki et al., 2006]     Exp: (0.61, dichloromethane)Calc: 0.53 $\pm$ 0.02

39     OCc1ccc(/N=N/c2ccc (OCc3cc(OCc4ccccc4)cc (OCc4ccccc4)c3)cc2)cc1     -.45     [Sierocki et al., 2006]     Exp: (0.36, dichloromethane)Calc: 0.13 $\pm$ 0.03

40     OCc1ccc(/N=N\c2ccc (OCc3cc(OCc4ccccc4)cc (OCc4ccccc4)c3)cc2)cc1     -.45     [Sierocki et al., 2006]     Exp: (0.64, dichloromethane)Calc: 0.54 $\pm$ 0.02

footnote-1In Refs. [Bandara et al., 2010, Bandara et al., 2011], the non-reactive azobenzene derivatives were irradiated from 300 to 600 nm. This range covered both $S_0\to S_1$ and $S_0\to S_2$ excitation. Only small changes in absorbance were observed, indicating minimal trans$\to$cis isomerization. However, the $S_2$ transition was highly redshifted, and thus had significant overlap with the $S_1$ transition. The $S_2$ transition also had a much higher oscillator strength. Hence the small absorbance changes could have been due to the $S_1$ transition, and so the $S_1$ yield may not have been precisely 0. The $S_1$ yield should therefore be interpreted as “small”, rather than exactly zero.
footnote0Averaged over excitations at 313, 436, and 546 nm.

Training and validation set literature species used for dense configurational sampling.

#     SMILES     Graph     Ref.

41     c1ccc2c(c1)CCc1ccccc1/N=N/2     -.45     [Siewertsen et al., 2009]

42     c1ccc2c(c1)CCc1ccccc1/N=N\2     -.45     [Siewertsen et al., 2009]

43     c1ccc2c(c1)CNc1ccccc1/N=N/2     -.45     [Lentes et al., 2019]

44     c1ccc2c(c1)CNc1ccccc1/N=N\2     -.45     [Lentes et al., 2019]

45     c1ccc2c(c1)COc1ccccc1/N=N/2     -.45     [Lentes et al., 2019]

46     c1ccc2c(c1)COc1ccccc1/N=N\2     -.45     [Lentes et al., 2019]

47     c1ccc2c(c1)CSc1ccccc1/N=N/2     -.45     [Lentes et al., 2019]

48     c1ccc2c(c1)CSc1ccccc1/N=N\2     -.45     [Lentes et al., 2019]

49     c1ccc2cc(/N=N/c3ccc4ccccc4c3)ccc2c1     -.45     [Malkin and Fischer, 1962]

50     c1ccc2cc(/N=N\c3ccc4ccccc4c3)ccc2c1     -.45     [Malkin and Fischer, 1962]

51     c1ccc(/N=N/c2ccccc2)cc1     -.45     [Moreno et al., 2015, Birnbaum and Style, 1954, Zimmerman et al., 1958, Bortolus and Monti, 1979, Rau and Yu-Quan, 1988, Gegiou et al., 1968, Rau, 1984]

52     c1ccc(/N=N\c2ccccc2)cc1     -.45     [Moreno et al., 2015, Birnbaum and Style, 1954, Zimmerman et al., 1958, Bortolus and Monti, 1979, Rau and Yu-Quan, 1988, Gegiou et al., 1968, Rau, 1984]

53     CC1(C)c2cccc(N)c2Oc2c(/N=N/ c3cccc4c3Oc3c(N)cccc3C4(C)C)cccc21     -.45     [Nagamani et al., 2005]

54     CC1(C)c2cccc(N)c2Oc2c(/N=N\ c3cccc4c3Oc3c(N)cccc3C4(C)C)cccc21     -.45     [Nagamani et al., 2005]

55     Cc1cccc(C)c1/N=N/c1c(C)cccc1C     -.45     [Olmsted III et al., 1983]

56     Cc1cccc(C)c1/N=N\c1c(C)cccc1C     -.45     [Olmsted III et al., 1983]

57     Cc1cccc(C)c1/N=N/c1ccc(C(=O)O)cc1     -.45     [Chen et al., 2012]

58     Cc1cccc(C)c1/N=N\c1ccc(C(=O)O)cc1     -.45     [Chen et al., 2012]

59     Cc1ccc(/N=N/c2ccccc2)cc1     -.45     [Birnbaum and Style, 1954]

60     Cc1ccc(/N=N\c2ccccc2)cc1     -.45     [Birnbaum and Style, 1954]

61     Cc1ccc(/N=N/c2ccc(C)cc2)cc1     -.45     [Birnbaum and Style, 1954]

62     Cc1ccc(/N=N\c2ccc(C)cc2)cc1     -.45     [Birnbaum and Style, 1954]

63     CCc1ccccc1/N=N/c1ccccc1CC     -.45     [Olmsted III et al., 1983]

64     CCc1ccccc1/N=N\c1ccccc1CC     -.45     [Olmsted III et al., 1983]

65     C#Cc1ccc(/N=N/c2ccc (N(CC)CC)cc2)c([N+](=O)[O-])c1     -.45     [Goulet-Hanssens et al., 2014]

66     C#Cc1ccc(/N=N\c2ccc (N(CC)CC)cc2)c([N+](=O)[O-])c1     -.45     [Goulet-Hanssens et al., 2014]

67     CC(C)c1ccc(/N=N/c2ccc (C(C)C)cc2C(C)C)c(C(C)C)c1     -.45     [Rau and Yu-Quan, 1988]

68     CC(C)c1ccc(/N=N\c2ccc (C(C)C)cc2C(C)C)c(C(C)C)c1     -.45     [Rau and Yu-Quan, 1988]

69     C=CC(=O)Nc1ccc(/N=N/c2ccc (NC(=O)C[N+](CC)(CC)CC)cc2)cc1

-.45     [Mourot et al., 2011]

70     C=CC(=O)Nc1ccc(/N=N\c2ccc (NC(=O)C[N+](CC)(CC)CC)cc2)cc1     -.45     [Mourot et al., 2011]

71     CCN1Cc2ccccc2/N=N/c2ccccc21     -.45     [Lentes et al., 2019]

72     CCN1Cc2ccccc2/N=N\c2ccccc21     -.45

[Lentes et al., 2019]

73     CCN(c1ccccc1)c1ccc(/N=N/ c2ccc(NC(=O)C[N+](CC)(CC)CC)cc2)cc1     -.45

[Mourot et al., 2011]

74     CCN(c1ccccc1)c1ccc(/N=N\ c2ccc(NC(=O)C[N+](CC)(CC)CC)cc2)cc1     -.45
[Mourot et al., 2011]

75     CCN(Cc1ccccc1)c1ccc(/N=N/ c2ccc(NC(=O)C[N+](CC)(CC)CC)cc2)cc1     -.45     [Mourot et al., 2011]

76     CCN(Cc1ccccc1)c1ccc(/N=N\ c2ccc(NC(=O)C[N+](CC)(CC)CC)cc2)cc1     -.45     [Mourot et al., 2011]

77     CCN(CC)c1ccc(/N=N/c2ccc (-c3cn(-c4ccc5cc6ccccc6cc5c4)nn3)cc2 [N+](=O)[O-])cc1     -.45     [Goulet-Hanssens et al., 2014]

78     CCN(CC)c1ccc(/N=N\c2ccc (-c3cn(-c4ccc5cc6ccccc6cc5c4)nn3)cc2 [N+](=O)[O-])cc1     -.45     [Goulet-Hanssens et al., 2014]

79     CCN(CC)c1ccc(/N=N/c2ccc(-c3cn (-c4cccc5ccccc45)nn3)cc2[N+](=O)[O-])cc1     -.45     [Goulet-Hanssens et al., 2014]

80     CCN(CC)c1ccc(/N=N\c2ccc(-c3cn (-c4cccc5ccccc45)nn3)cc2[N+](=O)[O-])cc1     -.45     [Goulet-Hanssens et al., 2014]

81     CCN(CC)c1ccc(/N=N/c2ccc (-c3cn(-c4ccccc4)nn3)cc2[N+](=O)[O-])cc1     -.45     [Goulet-Hanssens et al., 2014]

82     CCN(CC)c1ccc(/N=N\c2ccc (-c3cn(-c4ccccc4)nn3)cc2[N+](=O)[O-])cc1     -.45     [Goulet-Hanssens et al., 2014]

83     CCN(CC)c1ccc(/N=N/c2ccc (NC(=O)C[N+](CC)(CC)CC)cc2)cc1     -.45     [Mourot et al., 2011]

84     CCN(CC)c1ccc(/N=N\c2ccc (NC(=O)C[N+](CC)(CC)CC)cc2)cc1     -.45     [Mourot et al., 2011]

85     CCN(CC)c1ccc(/N=N/c2c(F) c(F)c(NC(=O)C[N+](CC)(CC)CC)c(F)c2F)cc1     -.45     [Mourot et al., 2011]

86     CCN(CC)c1ccc(/N=N\c2c(F) c(F)c(NC(=O)C[N+](CC)(CC)CC)c(F)c2F)cc1     -.45     [Mourot et al., 2011]

87     CCN(CC)c1cc(NC(C)=O)ccc1/N=N/ c1ccc(NC(C)=O)cc1N(CC)CC     -.45     [Sadovski et al., 2009]

88     CCN(CC)c1cc(NC(C)=O)ccc1/N=N\ c1ccc(NC(C)=O)cc1N(CC)CC     -.45     [Sadovski et al., 2009]

89     CCOc1ccc(/N=N/c2ccccc2)cc1     -.45     [Birnbaum and Style, 1954]

90     CCOc1ccc(/N=N\c2ccccc2)cc1     -.45     [Birnbaum and Style, 1954]

91     CC(=O)N1Cc2ccccc2/N=N/c2ccccc21     -.45     [Lentes et al., 2019]

92     CC(=O)N1Cc2ccccc2/N=N\c2ccccc21     -.45     [Lentes et al., 2019]

93     CC(=O)Nc1ccc2c(c1)CCc1cc (NC(C)=O)ccc1/N=N/2     -.45     [Samanta et al., 2012]

94     CC(=O)Nc1ccc2c(c1)CCc1cc (NC(C)=O)ccc1/N=N\2     -.45     [Samanta et al., 2012]

95     CC(=O)Nc1cc(Cl)c(/N=N/c2c (Cl)cc(NC(C)=O)cc2Cl)c(Cl)c1     -.45     [Samanta et al., 2013]

96     CC(=O)Nc1cc(Cl)c(/N=N\c2c (Cl)cc(NC(C)=O)cc2Cl)c(Cl)c1     -.45     [Samanta et al., 2013]

97     CC(=O)Nc1ccc(/N=N/c2ccc(NC(C)=O)cc2)cc1     -.45     [Gascón-Moya et al., 2015]

98     CC(=O)Nc1ccc(/N=N\c2ccc(NC(C)=O)cc2)cc1     -.45     [Gascón-Moya et al., 2015]

99     CC(=O)Nc1ccc(/N=N/c2ccc (NC(C)=O)cc2N2CCCC2)c(N2CCCC2)c1     -.45     [Sadovski et al., 2009]

100     CC(=O)Nc1ccc(/N=N\c2ccc (NC(C)=O)cc2N2CCCC2)c(N2CCCC2)c1     -.45     [Sadovski et al., 2009]

101     CC(=O)Nc1ccc(/N=N/c2ccc (NC(C)=O)cc2N2CCCCC2)c(N2CCCCC2)c1     -.45     [Sadovski et al., 2009]

102     CC(=O)Nc1ccc(/N=N\c2ccc (NC(C)=O)cc2N2CCCCC2)c(N2CCCCC2)c1     -.45     [Sadovski et al., 2009]

103     CC(=O)Nc1ccc(/N=N/c2ccc (NC(C)=O)cc2N2CCN(C)CC2) c(N2CCN(C)CC2)c1     -.45     [Sadovski et al., 2009]

104     CC(=O)Nc1ccc(/N=N\c2ccc (NC(C)=O)cc2N2CCN(C)CC2) c(N2CCN(C)CC2)c1     -.45     [Sadovski et al., 2009]

105     CC(=O)Nc1ccc(/N=N/c2ccc (NC(C)=O)cc2N2CCOCC2)c(N2CCOCC2)c1     -.45     [Sadovski et al., 2009]

106     CC(=O)Nc1ccc(/N=N\c2ccc (NC(C)=O)cc2N2CCOCC2)c(N2CCOCC2)c1     -.45     [Sadovski et al., 2009]

107     CC(=O)Nc1ccc(/N=N/c2ccc(NC (C)=O)cc2N(C)C)c(N(C)C)c1
-.45     [Sadovski et al., 2009]

108     CC(=O)Nc1ccc(/N=N\c2ccc(NC (C)=O)cc2N(C)C)c(N(C)C)c1     -.45     [Sadovski et al., 2009]

109     Clc1ccc(/N=N/c2ccccc2)cc1     -.45     [Birnbaum and Style, 1954]

110     Clc1ccc(/N=N\c2ccccc2)cc1     -.45     [Birnbaum and Style, 1954]

111     CN1Cc2ccccc2/N=N/c2ccccc21     -.45     [Lentes et al., 2019]

112     CN1Cc2ccccc2/N=N\c2ccccc21     -.45     [Lentes et al., 2019]

113     CN(C)c1ccc(/N=N/c2ccccc2)cc1     -.45     [Rau et al., 1990, Albini et al., 1983]

114     CN(C)c1ccc(/N=N\c2ccccc2)cc1     -.45     [Rau et al., 1990, Albini et al., 1983]

115     CN(C)c1ccc(/N=N/c2cccc([N+](=O)[O-])c2)cc1     -.45     [Olmsted III et al., 1983]

116     CN(C)c1ccc(/N=N\c2cccc([N+](=O)[O-])c2)cc1     -.45     [Olmsted III et al., 1983]

117     CN(C)c1ccc(/N=N/c2ccc(C(=O)O)cc2)cc1     -.45     [Lee et al., 2009]

118     CN(C)c1ccc(/N=N\c2ccc(C(=O)O)cc2)cc1     -.45     [Lee et al., 2009]

119     CN(C)c1ccc(/N=N/c2ccc([N+](=O)[O-])cc2)cc1     -.45     [Koller et al., 2007]

120     CN(C)c1ccc(/N=N\c2ccc([N+](=O)[O-])cc2)cc1     -.45     [Koller et al., 2007]

121     CN(C)c1ccc(/N=N/c2ccc(S(=O)(=O)O)cc2)cc1     -.45     [Olmsted III et al., 1983]

122     CN(C)c1ccc(/N=N\c2ccc(S(=O)(=O)O)cc2)cc1     -.45     [Olmsted III et al., 1983]

123     C[N+](C)(C)c1ccc(/N=N/c2ccc ([N+](C)(C)C)cc2)cc1     -.45     [Umemoto et al., 2014]

124     C[N+](C)(C)c1ccc(/N=N\c2ccc ([N+](C)(C)C)cc2)cc1     -.45     [Umemoto et al., 2014]

125     COc1ccccc1/N=N/c1ccccc1OC     -.45     [Olmsted III et al., 1983, Tahara et al., 1997]

126     COc1ccccc1/N=N\c1ccccc1OC     -.45     [Olmsted III et al., 1983, Tahara et al., 1997]

127     COc1ccc(/N=N/c2ccc(CO)cc2)cc1     -.45     [Sierocki et al., 2006]

128     COc1ccc(/N=N\c2ccc(CO)cc2)cc1     -.45     [Sierocki et al., 2006]

129     COc1ccc(/N=N/c2ccc(OC)cc2)cc1     -.45     [Olmsted III et al., 1983]

130     COc1ccc(/N=N\c2ccc(OC)cc2)cc1     -.45     [Olmsted III et al., 1983]

131     COc1cc(N2CC3CN(C(C)=O)CC3C2) c(OC)cc1/N=[NH+]/c1cc(OC)c (N2CC3CN(C(C)=O)CC3C2)cc1OC     -.45     [Dong et al., 2015]

132     COc1cc(N2CC3CN(C(C)=O)CC3C2) c(OC)cc1/N=[NH+]\c1cc(OC)c (N2CC3CN(C(C)=O)CC3C2)cc1OC     -.45     [Dong et al., 2015]

133     COc1cc(N2CCN(C(C)=O)CC2)c(OC) cc1/N=[NH+]/c1cc(OC)c(N2CCN(C (C)=O)CC2)cc1OC     -.45     [Dong et al., 2015]

134     COc1cc(N2CCN(C(C)=O)CC2)c(OC) cc1/N=[NH+]\c1cc(OC)c(N2CCN(C (C)=O)CC2)cc1OC     -.45     [Dong et al., 2015]

135     COc1cc(N2CCN(C(=O)CCl)CC2) cc(OC)c1/N=N/c1c(OC)cc(N2CCN (C(=O)CCl)CC2)cc1OC     -.45     [Samanta et al., 2013]

136     COc1cc(N2CCN(C(=O)CCl)CC2) cc(OC)c1/N=N\c1c(OC)cc(N2CCN (C(=O)CCl)CC2)cc1OC     -.45     [Samanta et al., 2013]

137     COc1cc(NC(C)=O)cc(OC)c1/N=N/ c1c(OC)cc(NC(C)=O)cc1OC     -.45     [Samanta et al., 2013]

138     COc1cc(NC(C)=O)cc(OC)c1/N=N\ c1c(OC)cc(NC(C)=O)cc1OC     -.45     [Samanta et al., 2013]

139     COc1cc(NC(=O)CCl)cc(OC)c1 /N=N/c1c(OC)cc(NC(=O)CCl)cc1OC     -.45     [Samanta et al., 2013]

140     COc1cc(NC(=O)CCl)cc(OC)c1 /N=N\c1c(OC)cc(NC(=O)CCl)cc1OC     -.45     [Samanta et al., 2013]

141     COC(=O)c1ccc(/N=N/c2ccccc2)cc1     -.45     [Birnbaum and Style, 1954]

142     COC(=O)c1ccc(/N=N\c2ccccc2)cc1     -.45     [Birnbaum and Style, 1954]

143     Fc1cccc(F)c1/N=N/c1c(F)cccc1F     -.45     [Knie et al., 2014]

144     Fc1cccc(F)c1/N=N\c1c(F)cccc1F     -.45     [Knie et al., 2014]

145     Fc1ccc(/N=N/c2ccccc2)cc1      -.45     [Birnbaum and Style, 1954]

146     Fc1ccc(/N=N\c2ccccc2)cc1      -.45     [Birnbaum and Style, 1954]

147     N[C@@H](C[C@@H](CCCC(=O) Nc1ccc(/N=N/c2ccc(NC(=O)CN3 C(=O)C=CC3=O)cc2)cc1) C(=O)O)C(=O)O     -.45     [Carroll et al., 2015]

148     N[C@@H](C[C@@H](CCCC(=O) Nc1ccc(/N=N\c2ccc(NC(=O)CN3 C(=O)C=CC3=O)cc2)cc1) C(=O)O)C(=O)O     -.45     [Carroll et al., 2015]

149     NC(=O)c1cccc(/N=N/c2ccccc2)c1     -.45     [Archut et al., 1998]

150     NC(=O)c1cccc(/N=N\c2ccccc2)c1     -.45     [Archut et al., 1998]

151     NC(=O)c1ccc(/N=N/c2ccccc2)cc1     -.45     [Archut et al., 1998]

152     NC(=O)c1ccc(/N=N\c2ccccc2)cc1     -.45     [Archut et al., 1998]

153     O=C1c2cccc(c2)/N=N/ c2cccc(c2)C(=O)N2CCOCCOCCN1CCOCCOCC2     -.45     [Rau, 1984, Lednev et al., 1998]

154     O=C1c2cccc(c2)/N=N\ c2cccc(c2)C(=O)N2CCOCCOCCN1CCOCCOCC2     -.45     [Rau, 1984, Lednev et al., 1998]

155     OCc1ccc(/N=N/c2ccc(O)cc2)cc1     -.45     [Sierocki et al., 2006]

156     OCc1ccc(/N=N\c2ccc(O)cc2)cc1     -.45     [Sierocki et al., 2006]

157     O=C(CCl)N1CCN(c2ccc (/N=N/c3ccc(N4CCN(C(=O) CCl)CC4)cc3)cc2)CC1     -.45     [Beharry et al., 2008]

158     O=C(CCl)N1CCN(c2ccc (/N=N\c3ccc(N4CCN(C(=O) CCl)CC4)cc3)cc2)CC1     -.45     [Beharry et al., 2008]

159     O=C(CCl)Nc1ccc2c(c1)CC c1cc(NC(=O)CCl)ccc1/N=N/2    -.45     [Samanta et al., 2012]

160     O=C(CCl)Nc1ccc2c(c1)CC c1cc(NC(=O)CCl)ccc1/N=N\2     -.45     [Samanta et al., 2012]

161     O=C(CCl)Nc1cc(Cl)c(/N=N/ c2c(Cl)cc(NC(=O)CCl)cc2Cl)c(Cl)c1     -.45     [Samanta et al., 2013]

162     O=C(CCl)Nc1cc(Cl)c(/N=N\ c2c(Cl)cc(NC(=O)CCl)cc2Cl)c(Cl)c1     -.45     [Samanta et al., 2013]

163     CCN(CC)c1ccc(/N=N/c2ccc(OC)cc2)cc1     -.45     [Albini et al., 1983]

164     CCN(CC)c1ccc(/N=N\c2ccc(OC)cc2)cc1     -.45     [Albini et al., 1983]

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