In this section, we train our ML models with an extended data set composed of two small protonated Schiff base, PSB3 and CH2NH+2, to test the flexibility of the ML models across chem- ical space. The CH2NH+2 data were added to the original data set from SSR(2,2) calculations for trajectories of SHXF/MRCI/SA-4-CASSCF(6,4)/6-31G* dynamics [22] in the previous section.
The resulting data set contains 48750 and 14068 points of PSB3 and CH2NH+2 data, respectively.
The data set is split into training, validation, and test set with ratio of 3:1:1 approximately. We train ML models with the extended training set of PSB3 and CH2NH+2 with the same training parameters used in the previous section. We evaluate the ML models on PSB3 test subset, CH2NH+2 test subset, and the entire test set. In other words, we test our ML models, trained for both PSB3 and CH2NH+2, to (i) PSB3 and CH2NH+2 subsets separately, and (ii) the entire set.
PSB3 test subset CH2NH+2 test subset Total test set
Energy Gradient Energy Gradient Energy Gradient
MAE RMSE MAE RMSE MAE RMSE MAE RMSE MAE RMSE MAE RMSE
EP P S 0.21 0.28 0.71 0.98 0.27 0.38 1.29 3.42 0.22 0.30 0.77 1.46
EOSS 0.23 0.31 0.79 1.11 0.66 0.68 1.46 3.38 0.33 0.43 0.86 1.55
∆SA 0.11 0.36 0.41 1.28 0.31 0.47 0.69 1.92 0.15 0.39 0.44 1.36 Total 0.18 0.32 0.64 1.13 0.41 0.53 1.15 2.99 0.23 0.37 0.69 1.46
Table 3: MAE and RMSE of SSR(2,2) diabatic elements (kcal/mol) and their gradients (kcal/mol/Å) for ML models trained with the combined training set evaluated on PSB3, CH2NH+2 test subsets and the total test set.
In Table 3, the overall errors are about 3times larger than the errors in Table 2. Especially, the errors on the CH2NH+2 subset is larger than that on the PSB3 subset with 0.41 (0.53) kcal/mol and 1.15 (2.99) kcal/mol/Å of MAE(RMSE) for energies and gradients, respectively.
The difference of error magnitudes between subsets indicates a biased fitting of the ML models to PSB3 which is originated from the relatively small size of CH2NH+2 subset. Nevertheless, the magnitudes of errors are still very small suggesting that our ML strategy is flexible to train multiple molecules. The model would be enhanced by a data set with larger chemical and nuclear configuration space by adjusting parmeters. In particular, the successful training of the coupling implies the training of phase across chemical compounds may be possible.
chemical space of our ML method has been extended with a combined training set of PSB3 and CH2NH+2 molecules toward universal ML models for ESMD simulations. Without prior ESMD trajectories, one can use efficient sampling procedures for ML-ESMD simulations. The errors would be still small with these sampling schemes since nuclear configurations are well-sampled with important points around minima and CIs. The extended ML models for two different molecules show accurate prediction indicating the transferability of our ML models to untrained protonated Schiff base family, which may be an important subject of studies for following studies in the future.
IV Conclusion
We have introduced ITMQC-XF method and SchNet/SSR(2,2) for efficient correlated electron- nuclear dynamics. ITMQC-XF effectively described the ENC effects, the nuclear wave packet splitting and electronic decoherence for various model system and realistic molecules [22, 23].
Also, various options for the decoherence term calculation method were given and tested. Ac- cording the benchmark test on the one-dimensional two-state model systems, we conclude that the SHXF method in combination with TD width, E-phase, and O-EOM is most promising in terms of accuracy and efficiency. We note that EhXF in combination with the E-phase can be applied to the RT-TDDFT formalism for various applications in large system. In the SchNet/SSR(2,2) scheme, we train SchNet models with SSR(2,2) energies and coupling to con- struct the diabatic Hamiltonian elements [24]. The ML models for energies were well trained without intrinsic singularity in contrast to the BO properties, while we handle the phase freedom in the coupling element by adopting the phase-less loss. The resulting ML models show small errors, well predict a correct MECI and topology of BO PESs around CIs. Furthermore, we were able to reproduce the reference SHXF dynamics of PSB3 using SchNet/SSR(2,2). Finally, the we glimpsed the possibility to extend our model to a wider chemical space from the result of combined training of PSB3 and CH2NH+2 molecules. We believe that, the ITMQC-XF method and SchNet/SSR(2,2) are useful tools for accurated and efficient ESMD simulations, and can serve as starting points for further developments of effective ESMD strategies.
0 0.2 0.4 0.6 0.8 1
population
O-EOM, P-phase
QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.04 0.08 0.12
0 1000 2000 3000
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, P-phase
QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.04 0.08 0.12
0 1000 2000 3000
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
O-EOM, E-phase
0 0.04 0.08 0.12
0 1000 2000 3000
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, E-phase
0 0.04 0.08 0.12
0 1000 2000 3000
coherence
t (a.u.)
Figure 8: BO population (first and third rows) and coherence profiles (second and fourth rows) for the SAC model with the initial momentum k0 = 10.0 a.u. The black solid line represents the QD result. The blue and red lines represent the results of EhXF and SHXF, respectively.
For the EhXF and SHXF dynamics, the solid and dashed lines represent the result for the TD and FG widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8 1
population
O-EOM, P-phase
0 0.08 0.16 0.24
0 500 1000 1500
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
A-EOM, P-phase
0 0.08 0.16 0.24
0 500 1000 1500
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
O-EOM, E-phase
0 0.08 0.16 0.24
0 500 1000 1500
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, E-phase
0 0.08 0.16 0.24
0 500 1000 1500
coherence
t (a.u.)
Figure 9: BO population (first and third rows) and coherence profiles (second and fourth rows) for the SAC model with the initial momentum k0 = 25.0 a.u. The black solid line represents the QD result. The blue and red lines represent the results of EhXF and SHXF, respectively.
For the EhXF and SHXF dynamics, the solid and dashed lines represent the result for the TD and FG widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8
O-EOM, P-phase EhXF-FGQD
EhXF-TD
T1T2 R1R2
0 0.2 0.4 0.6 0.8
10 15 20 25 30
k0 (a.u.) SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8
A-EOM, P-phase EhXF-FGQD
EhXF-TD
T1T2 R1R2
0 0.2 0.4 0.6 0.8
10 15 20 25 30
k0 (a.u.) SHXF-FG SHXF-TD
O-EOM, E-phase
10 15 20 25 30
k0 (a.u.) A-EOM, E-phase
10 15 20 25 30
k0 (a.u.)
Figure 10: Branching ratios for the SAC model system as a function of initial momentum. Blue and red colors represent the fractions of the nuclear wave packets transmitted on the lower and upper BO states (T1, T2), respectively, and green and purple colors represent the fractions of the nuclear wave packets reflected on the lower and upper BO states (R1, R2), respectively. The solid lines represent the QD results. The squares and circles represent the results of EhXF and SHXF, respectively. For the EhXF and SHXF dynamics, the empty and filled symbols represent the results for the FG and TD widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8 1
population
O-EOM, P-phase
QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.05 0.1 0.15
0 800 1600 2400
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, P-phase
QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.05 0.1 0.15
0 800 1600 2400
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
O-EOM, E-phase
0 0.05 0.1 0.15
0 800 1600 2400
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, E-phase
0 0.05 0.1 0.15
0 800 1600 2400
coherence
t (a.u.)
Figure 11: BO population (first and third rows) and coherence profiles (second and fourth rows) for the DAC model with the initial momentum k0 = 16.0 a.u. The black solid line represents the QD result. The blue and red lines represent the results of EhXF and SHXF, respectively.
For the EhXF and SHXF dynamics, the solid and dashed lines represent the result for the TD and FG widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8 1
population
O-EOM, P-phase
0 0.08 0.16 0.24
0 400 800 1200
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
A-EOM, P-phase
0 0.08 0.16 0.24
0 400 800 1200
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
O-EOM, E-phase
0 0.08 0.16 0.24
0 400 800 1200
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, E-phase
0 0.08 0.16 0.24
0 400 800 1200
coherence
t (a.u.)
Figure 12: BO population (first and third rows) and coherence profiles (second and fourth rows) for the DAC model with the initial momentum k0 = 16.0 a.u. The black solid line represents the QD result. The blue and red lines represent the results of EhXF and SHXF, respectively.
For the EhXF and SHXF dynamics, the solid and dashed lines represent the result for the TD and FG widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8
O-EOM, P-phase
EhXF-FGQD EhXF-TD T1 R1T2 R2
0 0.2 0.4 0.6 0.8
15 20 25 30 35
k0 (a.u.) SHXF-FG
SHXF-TD
0 0.2 0.4 0.6 0.8
A-EOM, P-phase
EhXF-FGQD EhXF-TD T1 R1T2 R2
0 0.2 0.4 0.6 0.8
15 20 25 30 35
k0 (a.u.) SHXF-FG
SHXF-TD
O-EOM, E-phase
15 20 25 30 35
k0 (a.u.) A-EOM, E-phase
15 20 25 30 35
k0 (a.u.)
Figure 13: Branching ratios for the DAC model system as a function of initial momentum. Blue and red colors represent the fractions of the nuclear wave packets transmitted on the lower and upper BO states (T1, T2), respectively, and green and purple colors represent the fractions of the nuclear wave packets reflected on the lower and upper BO states (R1, R2), respectively. The solid lines represent the QD results. The squares and circles represent the results of EhXF and SHXF, respectively. For the EhXF and SHXF dynamics, the empty and filled symbols represent the results for the FG and TD widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8 1
population
O-EOM, P-phase
0 0.08 0.16 0.24
0 1000 2000 3000
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
A-EOM, P-phase
0 0.08 0.16 0.24
0 1000 2000 3000
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
O-EOM, E-phase
0 0.08 0.16 0.24
0 1000 2000 3000
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, E-phase
0 0.08 0.16 0.24
0 1000 2000 3000
coherence
t (a.u.)
Figure 14: BO population (first and third rows) and coherence profiles (second and fourth rows) for the ECR model with the initial momentum k0 = 26.0 a.u. The black solid line represents the QD result. The blue and red lines represent the results of EhXF and SHXF, respectively.
For the EhXF and SHXF dynamics, the solid and dashed lines represent the result for the TD and FG widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8 1
population
O-EOM, P-phase
0 0.08 0.16 0.24
0 900 1800 2700
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
A-EOM, P-phase
0 0.08 0.16 0.24
0 900 1800 2700
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
O-EOM, E-phase
0 0.08 0.16 0.24
0 900 1800 2700
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, E-phase
0 0.08 0.16 0.24
0 900 1800 2700
coherence
t (a.u.)
Figure 15: BO population (first and third rows) and coherence profiles (second and fourth rows) for the ECR model with the initial momentum k0 = 32.0 a.u. The black solid line represents the QD result. The blue and red lines represent the results of EhXF and SHXF, respectively.
For the EhXF and SHXF dynamics, the solid and dashed lines represent the result for the TD and FG widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8
O-EOM, P-phase
EhXF-FGQD EhXF-TD
T1T2 R1R2
0 0.2 0.4 0.6 0.8
5 10 15 20 25 30 35 40
k0 (a.u.)
SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8
A-EOM, P-phase
EhXF-FGQD EhXF-TD
T1T2 R1R2
0 0.2 0.4 0.6 0.8
5 10 15 20 25 30 35 40
k0 (a.u.)
SHXF-FG SHXF-TD
O-EOM, E-phase
5 10 15 20 25 30 35 40
k0 (a.u.) A-EOM, E-phase
5 10 15 20 25 30 35 40
k0 (a.u.)
Figure 16: Branching ratios for the ECR model system as a function of initial momentum. Blue and red colors represent the fractions of the nuclear wave packets transmitted on the lower and upper BO states (T1, T2), respectively, and green and purple colors represent the fractions of the nuclear wave packets reflected on the lower and upper BO states (R1, R2), respectively. The solid lines represent the QD results. The squares and circles represent the results of EhXF and SHXF, respectively. For the EhXF and SHXF dynamics, the empty and filled symbols represent the results for the FG and TD widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8 1
population
O-EOM, P-phase
0 0.08 0.16 0.24
0 700 1400 2100
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
A-EOM, P-phase
0 0.08 0.16 0.24
0 700 1400 2100
coherence
t (a.u.) QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8 1
population
O-EOM, E-phase
0 0.08 0.16 0.24
0 700 1400 2100
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, E-phase
0 0.08 0.16 0.24
0 700 1400 2100
coherence
t (a.u.)
Figure 17: BO population (first and third rows) and coherence profiles (second and fourth rows) for the DAG model with the initial momentum k0 = 20.0 a.u. The black solid line represents the QD result. The blue and red lines represent the results of EhXF and SHXF, respectively.
For the EhXF and SHXF dynamics, the solid and dashed lines represent the result for the TD and FG widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8 1
population
O-EOM, P-phase
QD EhXF-FG EhXF-TD SHXF-FG SHXF-TD
0 0.08 0.16 0.24
0 600 1200 1800
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, P-phase
EhXF-FGQD EhXF-TD SHXF-FG SHXF-TD
0 0.08 0.16 0.24
0 600 1200 1800
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
O-EOM, E-phase
0 0.08 0.16 0.24
0 600 1200 1800
coherence
t (a.u.)
0 0.2 0.4 0.6 0.8 1
population
A-EOM, E-phase
0 0.08 0.16 0.24
0 600 1200 1800
coherence
t (a.u.)
Figure 18: BO population (first and third rows) and coherence profiles (second and fourth rows) for the DAG model with the initial momentum k0 = 40.0 a.u. The black solid line represents the QD result. The blue and red lines represent the results of EhXF and SHXF, respectively.
For the EhXF and SHXF dynamics, the solid and dashed lines represent the result for the TD and FG widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8
O-EOM, P-phase
EhXF-FGQD EhXF-TD
T1T2 R1R2
0 0.2 0.4 0.6 0.8
40 42 44 46 48 50
k0 (a.u.) SHXF-FG
SHXF-TD
0 0.2 0.4 0.6 0.8
A-EOM, P-phase
EhXF-FGQD EhXF-TD
T1T2 R1R2
0 0.2 0.4 0.6 0.8
40 42 44 46 48 50
k0 (a.u.) SHXF-FG
SHXF-TD
O-EOM, E-phase
40 42 44 46 48 50
k0 (a.u.) A-EOM, E-phase
40 42 44 46 48 50
k0 (a.u.)
Figure 19: Branching ratios for the DAG model system as a function of initial momentum. Blue and red colors represent the fractions of the nuclear wave packets transmitted on the lower and upper BO states (T1, T2), respectively, and green and purple colors represent the fractions of the nuclear wave packets reflected on the lower and upper BO states (R1, R2), respectively. The solid lines represent the QD results. The squares and circles represent the results of EhXF and SHXF, respectively. For the EhXF and SHXF dynamics, the empty and filled symbols represent the results for the FG and TD widths, respectively. The first and second columns represent the result for the P-phase (equation 50) and the E-phase (equation 53), respectively. The upper and lower four panels represent the results for the O-EOM and the A-EOM, respectively.
0 0.2 0.4 0.6 0.8
O-EOM, P-phase QD EhXF-FG EhXF-TD
0 0.2 0.4 0.6 0.8
40 42 44 46 48 50
k0 (a.u.)
SHXF-FG SHXF-TD
0 0.2 0.4 0.6 0.8
A-EOM, P-phase QD EhXF-FG EhXF-TD
0 0.2 0.4 0.6 0.8
40 42 44 46 48 50
k0 (a.u.)
SHXF-FG SHXF-TD
O-EOM, E-phase
T1 T2 R1 R2
40 42 44 46 48 50
k0 (a.u.) A-EOM, E-phase
T1 T2 R1 R2
40 42 44 46 48 50
k0 (a.u.)
Figure 20: Branching ratios for the DAG model system as a function of initial momentum withσ = 100.0/k0 a.u. Blue and red colors represent the fractions of the nuclear wave packets transmitted on the lower and upper BO states (T1, T2), respectively, and green and purple colors represent the fractions of the nuclear wave packets reflected on the lower and upper BO states (R1, R2), respectively. The solid lines represent the QD results, the empty and filled squares represent the EhXF-FG and EhXF-TD results, respectively, and the empty and filled circles represent the SHXF-FG and SHXF-TD results, respectively. The first column represents the result for the P-phase (equation 50), while the second column represents the result for the E-phase (equation 53). The upper four panels show the results for the O-EOM, and the lower four panels show the results for the A-EOM.
0 0.03 0.06 0.09 0.12 0.15
SAC DAC ECR DAG DAG*
O-EOM, P-phase
EhXF-FG EhXF-TD SHXF-FG SHXF-TD SH
0 0.03 0.06 0.09 0.12 0.15
SAC DAC ECR DAG DAG*
A-EOM, P-phase
EhXF-FG EhXF-TD SHXF-FG SHXF-TD SH
SAC DAC ECR DAG DAG*
O-EOM, E-phase
EhXF-FG EhXF-TD SHXF-FG SHXF-TD SH
SAC DAC ECR DAG DAG*
A-EOM, E-phase
EhXF-FG EhXF-TD SHXF-FG SHXF-TD SH
Figure 21: MAEs of branching ratios for all ITMQC-XF methods and original FSSH. Dark/light blue, dark/light red, and gray colors represent the MAEs of EhXF-TD/-FG, SHXF-TD/-FG, and FSSH, respectively. The first column is the result for the P-phase (equation 50) while the second column is for the E-phase (equation 53), respectively. The first row shows the results for the O-EOM and the second row shows the results for the A-EOM, respectively. "DAG*"
represent the DAG results obtained with σ= 100.0/k0.
-249.5 -249.4 -249.3
-249.5 -249.4 -249.3
ML
SSR (a) EPPS
-249.4 -249.3 -249.2
-249.4 -249.3 -249.2
ML
SSR (b) EOSS
-0.0128 0 0.0128
-0.0128 0 0.0128
ML
SSR (c) ΔSA
Figure 22: Scatter plots for diabatic Hamiltonian elements (in Hartree) predicted by ML models versus SSR(2,2) reference values.
(a) g-vector
(b) h-vector
Figure 23: MECIcen structures and their normalized and scaled branching plane vectors. Red arrows are vectors calculated by SSR(2,2)/ωPBEh/6-31G* and blue arrows are predicted by ML.
-0.018 0 0.018 g/|g| (a.u.) -0.018
0.018 0 h/|h| (a.u.) -0.15-0.1
-0.05 0.05 0.15 0.1 0 E (eV)
E0 E1 (a) ML, E0 and E1
-0.018 0 0.018
g/|g| (a.u.) -0.018
0.018 0 h/|h| (a.u.) -0.15-0.1
-0.05 0.05 0.15 0.1 0 E (eV)
E0 E1 (b) SSR, E0 and E1
-0.018 0 0.018
g/|g| (a.u.) -0.018
0.018 0 h/|h| (a.u.) -0.15-0.1
-0.05 0.05 0.15 0.1 0 E (eV)
EOSS EPPS
(c) ML, EPPS and EOSS
-0.018 0 0.018
g/|g| (a.u.) -0.018
0.018 0 h/|h| (a.u.) -0.15-0.1
-0.05 0.05 0.15 0.1 0 E (eV)
EOSS EPPS
(d) SSR, EPPS and EOSS
-0.018 0 0.018
g/|g| (a.u.) -0.018
0.018 0 h/|h| (a.u.) -0.03
-0.02 -0.01 0 0.01 0.02 0.03 E (eV)
ΔSA (e) ML, ΔSA
-0.018 0 0.018
g/|g| (a.u.) -0.018
0.018 0 h/|h| (a.u.) -0.03
-0.02 -0.01 0 0.01 0.02 0.03 E (eV)
ΔSA (f) SSR, ΔSA
Figure 24: Two-dimensional BO (a, b) and diabatic (c, d) PESs as well as ∆SA (e, f) about MECIcens in branching plane from ML (a, c, e) and SSR(2,2)/ωPBEh/6-31G* (b, d, f)
Figure 25: Eigenvector components for S0 state from the ML models in the branching space, i.e.
a00 (a) anda10 (b).
-100 0 100 200 300 400 500
0 50 100 150 200 250 300
Angle (degree)
Time (fs) SSR
ML
Dihedral angle around central C=C bond
Figure 26: Time evolution of a dihedral angle around central C=C bond of 50 randomly chosen trajectories. Black lines represent SHXF/ML results while gray lines represent the reference SHXF/SSR(2,2) results.
0 0.2 0.4 0.6 0.8 1
0 50 100 150 200 250 300
Population
Time (fs)
MLp1
ML SSRp1
SSR��11�
��11�
Figure 27: Population evolution of S1 state for ML (red) and SSR(2,2) (blue). Solid lines represent S1 population from the running state (equation 60) while dashed lines represent the population from density matrix (equation 58).
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