Machine learning (ML) has shown to advance the research field of quantum chemistry in almost any possible direction and has also recently been applied to investigate the multifaceted photochemistry of molecules. In this paper, we pursue two goals: (i) We show how ML can be used to model permanent dipole moments for excited states and transition dipole moments by adapting the charge model of Gastegger et al. [Chem. Sci. 8, 6924–6935 (2017)], which was originally proposed for the permanent dipole moment vector of the electronic ground state. (ii) We investigate the transferability of our excited-state ML models in chemical space, i.e., whether an ML model can predict the properties of molecules that it has never been trained on and whether it can learn the different excited states of two molecules simultaneously. To this aim, we employ and extend our previously reported SchNarc approach for excited-state ML. We calculate UV absorption spectra from excited-state energies and transition dipole moments as well as electrostatic potentials from latent charges inferred by the ML model of the permanent dipole moment vectors. We train our ML models on CH2NH2+ and C2H4, while predictions are carried out for these molecules and additionally for CHNH2, CH2NH, and C2H5+. The results indicate that transferability is possible for the excited states.

Photosynthesis,1,2 the ability of beings to see, and photorelaxation of, e.g., DNA and proteins to prevent them from photodamage3–5 are fascinating examples of the importance of light–matter interactions for our daily lives. Another marvelous aspect are the colors of every thing and every being, which are related to the absorption of a part of the incident light spectrum. In order to get a deeper understanding of these phenomena and to find out about the possibility of a molecule to be excited by light, answers to the following questions have to be provided: At which wavelengths can a molecule absorb electromagnetic radiation? How much of these wavelengths is absorbed? Can this absorption be used to identify a molecule?

In order to answer such questions, experiments or quantum chemical calculations are usually carried out. Assuming the resonance condition, i.e., the equivalence of the energy of one or more photons of the incident light with the energy gap between two electronic states, single- or multiphoton excitations can take place to one or more excited states if an oscillating dipole is induced.6–8 The oscillator strength, fijosc, between electronic states i and j, is related to the transition dipole moment, μij, of the respective electronic states as well as the energy difference, ΔEij, between them9 and is given in a.u.: fijosc=23ΔEijμij2. The larger the oscillation strength, the more likely a transition takes place.

The corresponding experiments such as UV/visible spectroscopy often lack the possibility to distinguish and characterize the different electronic states and rely on theoretical simulations to identify the states and provide detailed insights into their characters. However, the required calculations of the excited states involved in photo-induced processes are limited by the high costs for solving the underlying quantum chemical equations. Especially, the excited states necessitate highly accurate quantum chemical methods whose computational costs scale unfavorably with the number of electronic states and atoms considered in the calculations.10,11 Furthermore, sampling of many different molecular configurations followed by statistical averaging is often required in order to accurately reproduce the shape of the experimentally obtained spectra. Many calculations are required to obtain accurate results, which seriously limits this nuclear ensemble approach.

A solution to the aforementioned problems can be obtained with (atomistic) machine learning (ML) models, which have shown to be extremely powerful for the electronic ground state to provide ML potentials for energies or dipole moments (see, e.g., Refs. 12–31). ML force fields exist,15,18,19,28,32–39 and the transferability of properties has also been demonstrated.32,40–45 The main advantage of ML models is that they can sample a huge number of molecular configurations with the accuracy of the underlying quantum chemical calculations at only a fraction of the original costs.37,46

Recently, the interest to advance the research field of photochemistry and to tackle the excited states with ML has also increased.47–50 The fitting of molecule-specific potential energy surfaces (PESs) and coupling values51–66 or dipole moments as single values57,58,64 has been demonstrated up to date, and energy gaps, HOMO–LUMO gaps, and oscillator strengths have been fitted.45,67–71 The novel proposed ML models are mostly based on many configurations of a single molecule.47,53–55,57,58,63–66,72–75 Only a few ML models treat different molecules in their energetic equilibrium structure, which is mapped to a single output, e.g., the oscillator strength.76 Yet it is unclear, whether such a universal ML force field as it exists for the electronic ground state is feasible also for the excited states. The description of many molecular systems with one ML model further requires the construction of excited-state properties from atomistic contributions, but most ML models targeting the excited states employ molecule-wise descriptors, and some studies suggest molecular descriptors to be superior to atom-wise descriptors for the excited states.47,48,69

Another limitation of many existing ML models for the computation of absorption spectra is that they fit the oscillator strengths rather than the excited-state energies and transition dipole moments. The fitting of the latter properties is beneficial as they can be used, e.g., for the computation of photodynamics, ML/MM (ML/molecular mechanics) schemes77 similar to QM/MM (quantum mechanics/MM) schemes,78 or the investigation of explicit light interaction,79,80 to name only a few applications.

Transition dipole moments and permanent dipole moments can be computed by applying the dipole moment operator as implemented in many electronic structure programs. Permanent dipole moments can also be constructed from atomic charges using the point charge model [Eq. (5)]. By having access to the atomic charges of a molecule, not only can the dipole moment vector be computed but also the charge fluctuations within dynamics or different reaction coordinates can be investigated and electrostatic potentials can be computed.81,82 The atomic charges of the excited states can further be used to construct the approximated excited-state force fields83 or can be used to investigate how the charge distribution changes due to light excitation. Although atomic charges are considered one of the most intuitive chemical concepts, they cannot be obtained directly by solving the Schrödinger equation.84 Subsequent analysis of the charge distribution in a molecule is highly dependent on the underlying partitioning scheme applied.85 

Dipole models based on ML17,33,39,86–93 can provide access to the density or latent partial charges while being based on the underlying electronic structure theory. For instance, the latent charges obtained from the dipole moment ML model reported in Ref. 17, which never learn atomic charges directly, show good agreement with common charge models (CHELPG94 and Hirshfeld95), which are considered to be more reliable than, for example, Mulliken96 charges.82 They have been used to plot electrostatic potentials and assess the changes of atomic charges with respect to molecular geometries for the electronic ground state in Refs. 82 and 97. Electrostatic potentials are further interesting81 to interpret noncovalent interactions,98 for quantitative structure–activity relationship,99 or for force fields.100 

Unfortunately, especially, the fitting of transition dipole moments is challenging, as the sign of properties resulting from two different electronic states is arbitrary due to the arbitrary phase of the wave function,57,101 and because rotational covariance has to be preserved for vectorial properties. To the best of our knowledge, only one study exists72 in which phase corrected transition dipole moments were treated in a rotationally covariant way and a single-state fashion64 with ML. The trained ML models were used to fit model Hamiltonians for subsequent prediction of UV spectra. Yet an ML model that can describe many different PESs, forces, and dipole moment vectors simultaneously for the prediction of UV spectra does not exist.

In this work, we adapt the aforementioned ground-state charge model to describe the permanent dipole moments of the excited states, and in addition, we extend it to model the transition dipole moments in a rotationally covariant way. To this aim, we use the SchNarc deep learning approach,65 originally developed for photodynamics simulations, to additionally enable the computation of UV spectra. By doing so, we extend the SchNarc approach enabling a simultaneous modeling of permanent and transition dipole moment vectors of a given number of electronic states in addition to a manifold of excited-state potentials, forces, and couplings thereof. The methylenimmonium cation, CH2NH2+, and the isoelectronic molecule ethylene, C2H4, are used as model systems to assess the accuracy of ML-fitted transition dipole moments and latent partial charges by computation of UV spectra and electrostatic potentials.

In addition, we aim to evaluate the possibility of training one ML model on a set of molecular conformations of CH2NH2+ and C2H4, i.e., a multi-molecule model. The performance of this model is assessed by comparison to the single-molecule ML models. As SchNarc constructs energies and dipole moments from atomic contributions, the transferability of this model toward other molecules not included in the training set is evaluated. Thus, in addition to CH2NH2+ and C2H4, the molecules CH2NH, CHNH2, and C2H5+ are described, which are not included in the training set and have never been seen by the ML model.

Recently, we reported the SchNarc approach for efficient photodynamics simulations with ML-fitted PESs, derivatives, and couplings.65 In this work, we extend this deep learning model to fit permanent and transition dipole moments in a rotationally covariant way of a given number of states and pairs of states, respectively. The number of electronic states, which can be predicted, depends on the number of electronic states described in the reference calculations but is not a hard-coded parameter, so the training set can contain information on any desired number of states of different spin multiplicity.

In order to train an ML model on the different excited-state PESs and properties simultaneously, a training set has to be provided that consists of molecular conformations (atomic numbers and positions) on the one hand and the corresponding PESs, forces, and excited-state properties on the other hand. The molecular geometries are automatically transformed into molecular descriptors by SchNet33 and are intrinsic to the network architecture, which further relates these tailored molecular descriptors to the excited-state properties in an end-to-end fashion.65,91 The number of atoms is not fixed; hence, the molecules of different sizes and compositions can be treated. Similarly, the number of electrons is neither a hard-coded parameter nor an input to the neural network in the current implementation. Therefore, we limit our study to isoelectronic molecules. If, e.g., CH2NH2+ and CH2NH2 need to be described with the same machine learning model, an additional input parameter will have to be included so that the number of electrons or the electronic charge of the species can be distinguished.

The loss function, LSchNarc, which is used to monitor the error on the different properties during training, includes permanent and transition dipole moments, summarized in the term μ, in addition to energies, forces, and different types of couplings,

LSchNarc=tELE+tFLF+tSOCLSOC+tNACLNAC+tμLμ.
(1)

The trade-offs between the errors of the different properties are labeled with the letter, t, and the error of each property with L. The subscripts E, F, μ, SOC, and NAC denote energies, forces, permanent and transition dipole moments, spin–orbit couplings (SOCs), and nonadiabatic couplings (NACs), respectively. It is favorable to choose the trade-offs such that the error of each property is equally weighted in the loss function. The couplings are not accounted for in this work, as the SOCs arise between states of different spin multiplicity102 and NACs can be approximated from the Hessians of the PESs.65 

While the energies and forces can be monitored using the mean squared error (MSE) between predicted properties by the ML model (denoted by the superscript “ML”) and the quantum chemical reference values (denoted as “QC”),

LE=EQCEML2 and LF=FQCFML2,
(2)

a phase-less loss function has to be applied for coupling values and transition dipole moments unless they are phase corrected.57 Different variants of such a phase-free training algorithm have been proposed by us, which depend on the type of calculation and can be found in detail in Ref. 65. SchNarc automatically determines the most suitable phase-free training process, which is in its simplest form the minimum function of the MSEs assuming once a negative and once a positive sign of a coupling value and dipole moment, respectively. The minimum function can be used when only one excited-state property with arbitrary signs is treated, which is the case here,

Lμ=minεμ+,εμ,
(3)

with

εμ±=μQC±μML2.
(4)

The dipole moments are treated as vectorial properties, and thus, the signs within a vector are conserved. As permanent dipole moment vectors are described together with transition dipole moment vectors, they are also trained in a phase-free manner. As a consequence, they are only defined up to an arbitrary sign, which can lead to permanent dipole moment vectors pointing into the wrong direction. Hence, they have to be adjusted for a reference molecular geometry when making predictions. A more detailed discussion can be found in Sec. S4 of the supplementary material.65 

The model for permanent and transition dipole moments used here is based on the charge model of Ref. 17: Since the ML model is an atomistic one, atomic contributions to the molecular dipole moment can be automatically obtained. These atomic contributions are taken as latent atomic charges of a given state i, qi,a, i.e., they have to be multiplied by the distance, raCM, of the atoms, a, to the center of mass of the molecule and are then summed up for all atoms, Na, of the molecule, before feeding the resulting dipole moment into the loss function. In the same way as the permanent dipole moment of the electronic ground state, SchNarc fits the permanent dipole moment of a given number of states, μi, and transition dipole moments, μij, between different electronic states according to Eqs. (5) and (6), respectively,

μi=aNaqi,araCM,
(5)
μij=aNaqij,araCM.
(6)

Note also that here the atomic charges are latent variables and the “atomic transition charges” between two different states, qij,a, used to obtain transition dipole moments do not have a direct physical meaning. However, these charges are the quantities that are used in the predictions. They are then multiplied with raCM and, in this way, allow for rotational covariance of the transition dipole moment vectors.

The training sets and reference computations of all molecules are based on the multi-reference configuration interaction method accounting for single and double excitations (MR-CISD) out of the active space of six electrons in four orbitals with the double-zeta basis set aug-cc-pVDZ (augmented correlation consistent polarized valence double zeta) as implemented in Columbus.103 The molecules investigated in this study are the methylenimmonium cation (CH2NH2+), ethylene (C2H4), aminomethylene (CHNH2), methylenimine (CH2NH), and ethenium (C2H5+).

1. Training sets

The training set of the methylenimmonium cation, CH2NH2+, forms the basis, as this training set already exists and can be taken from Ref. 57. It consists of 4000 data points of three singlet states, which has been shown to cover the relevant configurational space visited after photo-excitation to the second excited singlet state, S2.

In order to compute an ample training set for ethylene in the most efficient way, the molecular geometries of the available CH2NH2+ training set are used and the nitrogen atom is replaced by a carbon atom. 3969 MR-CISD/aug-cc-pVDZ calculations are converged with which the training set for ethylene is built. No optimizations of state minima or crossing points are carried out as this would lead to considerably higher computational effort. Thus, it is not guaranteed that the S2/S1 and S1/S0 minimum energy conical intersections of ethylene are covered comprehensively, and additional sampling, such as adaptive sampling, might be necessary to improve the density of data in critical regions of the PESs of ethylene. The high diversity of the previously sampled structures of CH2NH2+,57 which are included in the training set, nevertheless makes us confident that this training set is sufficient to compute UV/visible absorption spectra and electrostatic potentials. The same reference method as for the training set of CH2NH2+ is used in order to allow for merging of the two training sets. Hence, the Rydberg states of ethylene are not described explicitly, and no special care was taken to comprehensively include Rydberg orbitals in the active space. Rydberg states have also been neglected in some previous studies104–110 and are considered to be less relevant in two-state photodynamics.105,111 The inactive, doubly occupied molecular orbitals and the active molecular orbitals of the equilibrium structures of CH2NH2+ and C2H4 are shown in Fig. S1 of the supplementary material.

As CH2NH2+ is considered to be a three-state problem with a bright second excited singlet state and C2H4 is referred to as a two-state problem with a bright first excited singlet state, S1,112–115 these two molecules and their distinct photodynamics are considered to be a perfect testbed for the purpose of this study.

Statistically significant results for the computation of UV/visible absorption spectra can be obtained by sampling a lot of different molecular conformations. Here, the reference UV/visible absorption spectra are obtained from excited-state calculations of 500 molecular conformations sampled from a Wigner distribution.116,117 The same method as for the training set generation is used for every molecule. Except for the equilibrium structure of CH2NH2+ and C2H4, these 500 data points are not included in the training set. Alternatively, sampling could also be carried out with Born–Oppenheimer MD simulations, but Wigner sampling is considered to be superior for small molecules118 and is the standard procedure in SHARC.119 The calculated vertical excitations from every sampled conformation in combination with the corresponding oscillator strengths and a Gaussian broadening yield the UV/visible spectra. The width of the Gaussians is specified in Table S1 of the supplementary material. In addition to the molecules, on which the ML models are trained on, the UV/visible spectra of CH2NH, CHNH2, and C2H5+ are computed from 500, 500, and 100 Wigner-sampled conformations, respectively. C2H5+ is used as a negative example; hence, 100 sampled conformations are sufficient to show that the ML model fails in this case. The molecular structures of these molecules are optimized at the MP2/TZVP level of theory using the program ORCA.120 

The electrostatic potentials are plotted with Jmol121 and correspond to the energetically lowest lying conformation of each molecule. The Hirshfeld charges are obtained from MP2/TZVP calculations, while the Mulliken charges are available in Columbus; hence, they are obtained from respective calculations with MR-CISD/aug-cc-pVDZ.

As a deep learning model, SchNarc is used, which combines the continuous-filter convolutional-layer neural network SchNet33,91 for excited states and the MD program SHARC (Surface Hopping including ARbitrary Couplings).79,119,122 As the SchNarc model, originally developed for photodynamics, is described in detail elsewhere,65 we only shortly describe the technical details and timings of the computations.

As ML is computationally efficient compared to quantum chemistry, more conformations can be sampled and more trajectories can be initiated, while still being computationally less expensive. To this aim, 20 000 initial conditions are sampled from a Wigner distribution from which the UV/visible absorption spectra are computed using the oscillator strengths obtained from ML energy gaps and transition dipole moments in combination with Gaussian broadening. The computation of the three potential energies at 500 and 20 000 initially sampled molecular conformations takes about 7 s (39 s) and 4.5 min (26 min), respectively, on a Tesla-V100 graphics processing unit (GPU) [Xeon E5-2650 v3 central processing unit (CPU)] using the largest trained ML model. In contrast, 500 computations of three PESs with MR-CISD/aug-cc-pVDZ take about 17 h on an Intel Xeon E5-2650 v3 CPU.

SchNarc models are trained on 3000 data points of CH2NH2+ and C2H4 separately using 200 additional data points for validation during training and the remaining points for the test set. The network architecture of the single-molecule models is the same, although hyperparameters were evaluated for each molecule separately. By monitoring the loss function of the validation set, overfitting can be controlled. Five hidden layers and 256 features to describe the atoms within a cut-off region of 5 Å are used to generate the molecular descriptors. The model that is trained on both molecules takes 7000 randomly shuffled data points out of a training set consisting of 3969 data points of C2H4 and 4000 data points of CH2NH2+. By including similar amounts of molecules in the training set, no molecule is favored over each other. 500 data points are used for validation, and the rest is held back as a test set. The network architecture is enlarged and comprises seven hidden layers and 512 features with a cut-off region of 6 Å as more information has to be learned by the model. This larger architecture is avoided for the single-molecule models in order to avoid overfitting. The training of the single-molecule SchNarc model takes about 5 h and of the model trained on both molecules with the larger network architecture about 8 h on the aforementioned GPU. A more detailed discussion on the choice of network parameters and training can be found in Sec. S2 of the supplementary material.

Independent of the training set, the atomic charges and positions are input to the ML models. The number of atoms is not a hard-coded parameter; hence, in principle, molecules of different sizes can be treated, and training sets can be combined as wished. As the number of electronic states, which can be described, depends on the training set, the number of electronic states described for each molecule has to be consistent. For predictions, it is only necessary to provide the ML model with the atomic positions and atomic charges similar to an xyz-file.

The trade-offs for each trained property along with the mean absolute error (MAE) obtained from all three states or all possible pairs of states on the test set of each model are given in Table I. For determining the trade-offs, the following procedure is employed: The training is started with equal trade-offs and stopped after a few epochs. Then, the contribution of the different losses of energies, forces, and dipoles is assessed, and the training is restarted with trade-offs such that each property is weighted equally large in the overall loss function [Eq. (1)]. If a trade-off is set to 0, then the corresponding property does not have any influence on the training process and the ML model is not trained on this property. The scatter plots of the models are shown in Fig. 1. The largest errors can be estimated from the scatter plots. Especially in critical regions of the PESs, quantum chemical calculations are difficult to converge and can show artifacts and energy jumps in PESs,57,64 which is discussed for the methylenimmonium cation in the supplementary material in Sec. S1 (Fig. S2) together with state orderings close to conical intersections and optimizations thereof (Fig. S3). Thus, the scatter plots should be taken with care. The predicted dipole moments obtained with SchNarc are about a factor of 5 more accurate than our previously reported kernel ridge regression models64 and multi-layer feed-forward neural networks,57,64 which fit dipole moments in a direct way—as single values with kernel ridge regression and as single elements put together in one vector with neural networks.

TABLE I.

Trade-offs used to train energies, forces, and dipole moments along with the mean absolute error (MAE) and root mean squared error (RMSE) on the test set for each property. Permanent and transition dipole moments are shown together as they are processed together with SchNarc. The mean over all states and pairs of states are shown. The respective scatter plots are given in Fig. 1.

ModelMAE (RMSE) energy (eV)tE
CH2NH2+ 0.047 (0.13) 1.0 
C2H4 0.11 (0.23) 1.0 
Combined 0.060 (0.15) 1.0 
ModelMAE (RMSE) energy (eV)tE
CH2NH2+ 0.047 (0.13) 1.0 
C2H4 0.11 (0.23) 1.0 
Combined 0.060 (0.15) 1.0 
MAE (RMSE) forces (eV/Å)tF
CH2NH2+ 0.21 (0.49) 1.0 
C2H4 0.32 (0.63) 1.0 
Combined 0.23 (0.52) 1.0 
MAE (RMSE) forces (eV/Å)tF
CH2NH2+ 0.21 (0.49) 1.0 
C2H4 0.32 (0.63) 1.0 
Combined 0.23 (0.52) 1.0 
MAE (RMSE) dipoles (D)tμ
CH2NH2+ 0.14 (0.44) 0.001 
C2H4 0.19 (0.39) 0.1 
Combined 0.13 (0.31) 0.2 
MAE (RMSE) dipoles (D)tμ
CH2NH2+ 0.14 (0.44) 0.001 
C2H4 0.19 (0.39) 0.1 
Combined 0.13 (0.31) 0.2 
FIG. 1.

Scatter plots showing the reference energies, forces, permanent, and transition dipole moments plotted against the ML predictions of the model trained on CH2NH2+ and C2H4 simultaneously.

FIG. 1.

Scatter plots showing the reference energies, forces, permanent, and transition dipole moments plotted against the ML predictions of the model trained on CH2NH2+ and C2H4 simultaneously.

Close modal

The computed UV/visible absorption spectra are shown in Fig. 2 with the reference method on the left and the ML predictions on the right. Panels (a) and (b) illustrate the spectra of CH2NH2+ and C2H4, which are both included in the training set. The filled spectrum with solid lines is obtained from SchNarc models trained solely on CH2NH2+ or C2H4 (i.e., a single-molecule model), and the dotted lines are obtained from the SchNarc model trained on the combined training set, which includes both molecules (i.e., a multi-molecule model). As it is visible, both models can be used to accurately predict the UV/visible absorption spectra. Remarkably, the S2 state is correctly predicted to be bright for CH2NH2+ in panel (a), while the S1 state is dark, while the inverse relation is predicted correctly in panel (b) for C2H4. The results indicate that although the transition dipole moments are completely different for the different electronic states in both molecules, SchNarc can accurately capture the absorption behavior of both molecules. Interestingly, the model trained on both molecules is even slightly more accurate than the ML model trained solely on C2H4 for the prediction of the UV/visible absorption spectrum in panel (b). We did not expect such an outcome because force fields with increasing generality become usually less accurate for specific examples. The results suggest that it should be possible to construct transferable ML models for the excited states, at least when similar molecules are described. As suggested in a previous study by us in which we computed learning curves using different types of ML models trained on the methylenimmonium cation,64 the learning capacity of the ML models might not be fully exploited. This could be the case for the single-molecule ML models as well as for the transferable ML models. Thus, it is likely that the single-molecule ML models become comparably accurate using larger training set sizes.

FIG. 2.

UV/visible absorption spectra computed from 500 Wigner sampled conformations with MR-CISD/aug-cc-pVDZ on the left and from 20 000 Wigner sampled conformations with SchNarc on the right. The molecules (a) CH2NH2+ and (b) C2H4 are included in the training set and the performance of the ML model trained on one (solid lines) and both molecules (dashed lines) is compared, while (c) CH2NH and (d) CHNH2 are not included in the training set and the ML model trained on both molecules is used for the prediction. In panel (b), the experimental UV/visible absorption spectrum (abbreviated as “Expt.”) is plotted, where the data are taken from Ref. 123.

FIG. 2.

UV/visible absorption spectra computed from 500 Wigner sampled conformations with MR-CISD/aug-cc-pVDZ on the left and from 20 000 Wigner sampled conformations with SchNarc on the right. The molecules (a) CH2NH2+ and (b) C2H4 are included in the training set and the performance of the ML model trained on one (solid lines) and both molecules (dashed lines) is compared, while (c) CH2NH and (d) CHNH2 are not included in the training set and the ML model trained on both molecules is used for the prediction. In panel (b), the experimental UV/visible absorption spectrum (abbreviated as “Expt.”) is plotted, where the data are taken from Ref. 123.

Close modal

Furthermore, we plot the experimental UV/visible spectrum of ethylene in the gas phase, which is available in Ref. 123, next to the MR-CISD/aug-cc-pVDZ spectrum of ethylene in panel (b) on the left. As expected, the quantum chemical spectrum is in worse agreement with the experimental spectrum than the ML model is to the quantum chemistry reference method.

Due to the advantage of the atom-wise molecular descriptor, which enables a description of different molecules of different sizes, the transferability capabilities of SchNarc for the prediction of a manifold of PESs and transition dipole moments throughout chemical compound space are evaluated. To this aim, the UV/visible spectra of CHNH2 and CH2NH are additionally computed, which are shown in panels (c) and (d), respectively. In order to ensure that these molecules are not included in the training set, an analysis of the maximum bond distances in the training set is carried out. According to the unrelaxed dissociation scans of CH2NH2+, the hydrogen atoms can be considered as dissociated at a bond length of about 2.5 Å. No geometry inside of the training set has an N–H bond length larger than 2 Å, and eight geometries have a C–H bond length larger than 2 Å, where only one is larger than 2.5 Å. The same is true for the training set of C2H4 with regard to the C–H bond length. Thus, it can be safely said that the assessment of the performance of SchNarc is not biased by an unusual large amount of dissociated configurations in the training set.

As is clearly visible in panels (c) and (d), the energies of the S1 and S2 states are lower compared to the energies of CH2NH2+ and C2H4 in panels (a) and (b). This trend is predicted correctly with SchNarc for both CH2NH and CHNH2. In addition, the bright and dark states are predicted qualitatively correct. In panel (d), the S1 state is much darker than the S2 state, whereas the S1 state is brighter in panel (c). Although the spectra of the SchNarc models of the unknown molecules are broadened compared to the quantum chemical spectra, they can be used to obtain a qualitatively correct picture of the UV/visible light absorption at almost no additional costs.

CH2NH and CHNH2 both contain one atom less than the molecules described in the training set. Thus, one might also assume that the ML model trained solely on CH2NH2+ can be used to predict qualitatively correct UV/visible absorption spectra, as only atoms have to be removed. However, the evaluation of the single-molecule models shows that this model cannot be used to capture the correct absorption behavior and energy range of the two molecules not included in the training set. The performance of the ML model trained solely on CH2NH2+ is even comparable to the ML model trained solely on C2H4, which would be expected to be at least worse.

As already indicated, the molecular structures of the tested molecules, CH2NH and CHNH2, are similar to CH2NH2+ and C2H4. In order to assess the performance of SchNarc for the computation of the UV/visible absorption spectra of molecules with a different structure, the isoelectronic molecule C2H5+, which contains one atom more, is additionally chosen.

Figure 3 shows the reference spectrum on the left and the ML-predicted spectrum on the right. The trained SchNarc models cannot be used to predict the UV/visible absorption spectrum of C2H5+. While the S1 state is predicted to be dark and the S2 state to be bright, which is in accordance with the reference spectrum, the energy range is off. A reason for this discrepancy can be the larger system size, the different shapes of the molecules, or both cases. As three hydrogen atoms are bound to a carbon atom in C2H5+, the structure of this molecule is completely different from the structures inside of the training set.

FIG. 3.

UV/visible absorption spectrum of C2H5+ computed with MR-CISD/aug-cc-pVDZ for two excited singlet states from 100 Wigner sampled conformations. 20 000 Wigner sampled geometries are used to obtain the spectrum on the left computed with SchNarc trained on CH2NH2+ and C2H4.

FIG. 3.

UV/visible absorption spectrum of C2H5+ computed with MR-CISD/aug-cc-pVDZ for two excited singlet states from 100 Wigner sampled conformations. 20 000 Wigner sampled geometries are used to obtain the spectrum on the left computed with SchNarc trained on CH2NH2+ and C2H4.

Close modal

The results shown here leave us to conclude that isoelectronic molecules with a similar molecular structure can be predicted and that our ML models are to a certain extent transferable throughout chemical compound space also for excited-state PESs and properties thereof. It would be interesting to assess the transferability capacity of ML for the excited states when treating a larger number of molecules. Unfortunately, the high expenses and complexity of multi-reference quantum chemistry methods remain a clear bottleneck in this regard.

The transition dipole moments and energies provide a measure of the quality of the molecular properties that are constructed from atomic contributions with SchNarc. As mentioned above, SchNarc also provides direct access to latent ground-state and excited-state partial charges based solely on the dipole moment data of the underlying electronic structure method. In order to assess, whether the ML model provides meaningful partial charges, the electrostatic potentials obtained from SchNarc are compared to those obtained from Mulliken and Hirshfeld charges. Note that the latter are rarely implemented in quantum chemistry programs for excited states. The results are thus shown only for the electronic ground state in Fig. 4(a).

FIG. 4.

Electrostatic potentials of C2H4, CH2NH2+, CHNH2 and CH2NH obtained from Mulliken charges, Hirshfeld charges computed at the MP2/TZVP level of theory, and latent charges of the ML models trained on the training set containing both C2H4 and CH2NH2+ molecules for the (a) electronic ground state and (b) excited states—the latter being only predicted with ML. Red indicates the regions of negative charge, while blue refers to positive charges.

FIG. 4.

Electrostatic potentials of C2H4, CH2NH2+, CHNH2 and CH2NH obtained from Mulliken charges, Hirshfeld charges computed at the MP2/TZVP level of theory, and latent charges of the ML models trained on the training set containing both C2H4 and CH2NH2+ molecules for the (a) electronic ground state and (b) excited states—the latter being only predicted with ML. Red indicates the regions of negative charge, while blue refers to positive charges.

Close modal

The first and second columns show the molecules, which are included in the training set. Red indicates negative charges, while blue indicates positive charges. The electrostatic potentials in the first line are obtained from Mulliken charges. As is visible, the Mulliken scheme shows that negative charges are located at hydrogen atoms and positive charges at the carbon atoms, which is in contrast to the Hirshfeld scheme given in the second line and also in contrast to chemical intuition. The electrostatic potentials obtained from the ML model trained on both molecules are shown in the third line. A similar charge distribution is obtained for ML models trained on a single molecule (see Fig. S2 of the supplementary material). The partial charges obtained from SchNarc are in good agreement with the Hirshfeld charges. Similar agreement, at least qualitatively, can be obtained for CH2NH2+.

As the charge distribution of the electronic ground state is in qualitatively good agreement with the Hirshfeld partitioning scheme, the redistribution for the excited states can be analyzed. In the case of C2H4 in the first column of panel (b), the negative and positive charges do not redistribute considerably in the case of the S1 state, but the distribution is inverted for the S2 state. The positive charge is then located between the carbon atoms. For CH2NH2+, the positive charge is located at the far end of the nitrogen side of the molecule for the ground state. In the S1 state, the positive charge is still located at the nitrogen but closer to the center of the molecule. In the S2 state, the distribution is similar to the ground state. These distributions give rise to dipole moments, which perfectly agree with the reference calculations [QC/Ml: 1.5 a.u. (S0), 1.2 a.u. (S1), and 1.5 a.u. (S2); the vectors all point from C toward N].

In addition to the molecules included in the training set, the transferability of SchNarc to predict electrostatic potentials is tested too. Although the ML model has never been trained on CHNH2 or CH2NH, the ground-state electrostatic potentials agree arguably better with the Hirshfeld distributions than the Mulliken ones. This is especially true for CHNH2. Comparing the S0 distribution with the one from S2, an inversion of the charge locations is visible, which is also present in C2H4, but not in CH2NH2+.

The last column illustrates the electrostatic potentials of CH2NH, where the negative charge is located at the nitrogen atom according to the Hirshfeld partitioning, but rather at the adjacent hydrogen according to ML.

All these results indicate that the charge distributions obtained with SchNarc can be used to obtain the electrostatic potentials of molecules included in the training set and that transferability is also possible for latent partial charges, at least for isoelectronic molecules.

In this work, the SchNarc deep learning approach for photodynamics is extended to describe permanent and transition dipole moments in a rotationally covariant manner and for any given number of electronic states. The dipole moment vectors can be trained in one ML model in addition to the ground-state energies and forces as well as a manifold of excited-state energies and forces. SchNarc can be used to accurately predict UV/visible absorption spectra, and the latent partial charges can be used to assess the charge distribution via the electrostatic potentials of molecules. As SchNarc is trained not only on the ground state but also on the excited states, the charge distribution for the excited states can be assessed. As the partial charges for the ground state are in qualitatively good agreement with the Hirshfeld charges and also the excited-state molecular dipole moments agree between ML and the reference, we consider the charges to be equally accurate also for the excited states. The latent partial charges are based on highly accurate quantum chemistry and provide direct access to the charge distribution after light excitation.

Due to the atom-wise tailored descriptor, many different molecules can be described in one model, which contain different numbers of atoms. At least when isoelectronic, similarly structured molecules are treated, transferability is confirmed for UV/visible absorption spectra and partial charges. These properties can be computed with our ML approach at least qualitatively at almost no additional costs. Remarkably, the ML model can treat charged species on the same footing as neutral species.

Especially interesting would be to assess the improvement one can achieve by including many more molecules than just two isoelectronic ones. At the current stage of research, the high complexity and costs of accurate multi-reference quantum chemical methods hamper an ample assessment of the transferability in the excited states. Nevertheless, the trend clearly shows that ML models trained on more molecules are superior to ML models trained on single molecules, even if these molecules exhibit a completely different photochemistry and overall charge.

See the supplementary material for concerning molecular orbitals of the methylenimmonium cation and ethylene, conical intersections, UV/visible spectra, and electrostatic potentials.

This work was financially supported by the AustrianScience Fund [Grant No. W 1232 (MolTag)] and the uni:docs program of the University of Vienna (J.W.). The computational results presented have been achieved, in part, using the Vienna Scientific Cluster (VSC). P.M. thanks the University of Vienna for continuous support, also in the frame of the research platform ViRAPID. J.W. and P.M. are grateful for an NVIDIA Hardware Grant. The authors thank Michael Gastegger for helpful discussions regarding the extension of the charge model of Ref. 17 for SchNarc and concerning the latent partial charges.

The training set for CH2NH2+ is published in Ref. 57 and the training set of C2H4 will be made available as the supplementary material in the same format as the previous training set, i.e., the one used by SHARC.119 The SchNarc model is updated and freely available at https://github.com/schnarc/schnarc.

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