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Excitation Energy Transfer between Porphyrin Dyes on a Clay Surface: A study employing Multifidelity Machine Learning

October 2024

DOI:10.48550/arXiv.2410.20551

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Natural light-harvesting antenna complexes efficiently capture solar energy using chlorophyll, i.e., magnesium porphyrin pigments, embedded in a protein matrix. Inspired by this natural configuration, artificial clay-porphyrin antenna structures have been experimentally synthesized and have demonstrated remarkable excitation energy transfer properties. The study presents the computational design and simulation of a synthetic light-harvesting system that emulates natural mechanisms by arranging cationic free-base porphyrin molecules on an anionic clay surface. We investigated the transfer of excitation energy among the porphyrin dyes using a multiscale quantum mechanics/molecular mechanics (QM/MM) approach based on the semi-empirical density functional-based tight-binding (DFTB) theory for the ground state dynamics. To improve the accuracy of our results, we incorporated an innovative multifidelity machine learning (MFML) approach, which allows the prediction of excitation energies at the numerically demanding time-dependent density functional theory level with the Def2-SVP basis set. This approach was applied to an extensive dataset of 640K geometries for the 90-atom porphyrin structures, facilitating a thorough analysis of the excitation energy diffusion among the porphyrin molecules adsorbed to the clay surface. The insights gained from this study, inspired by natural light-harvesting complexes, demonstrate the potential of porphyrin-clay systems as effective energy transfer systems.

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Supplementary Information:

Excitation Energy Transfer between Porphyrin

Dyes on a Clay Surface: A study employing

Multiﬁdelity Machine Learning

Dongyu Lyu,†Matthias Holzenkamp,‡Vivin Vinod,‡Yannick Marcel Holtkamp,†

Sayan Maity,†Carlos R. Salazar,†,§Ulrich Kleinekathöfer,∗,†and Peter Zaspel∗,¶

†School of Science, Constructor University, Bremen 28759, Germany

‡School of Mathematics and Natural Science, University of Wuppertal, Wuppertal 42119,

Germany

¶School of Mathematics and Natural Sciences, University of Wuppertal, Wuppertal 42119,

Germany

§Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83

Linköping, Sweden

E-mail: ukleinekathoefer@constructor.university;zaspel@uni-wuppertal.de

arXiv:2410.20551v1 [physics.chem-ph] 27 Oct 2024

Distributions of Excitation energies

1.6 1.8 2.0 2.2 2.4

Site Energy [eV]

Density

DFTB

STO-3G

3-21G

6-31G

def2-SVP

(a) m-type porphyrin molecules.

1.6 1.8 2.0 2.2 2.4

Site Energy [eV]

Density

DFTB

STO-3G

3-21G

6-31G

def2-SVP

(b) p-type porphyrin molecules.

Figure S1: Distributions of excitation energies for diﬀerent levels of quantum chemistry

averaged over all pigments of the same type.

Transition charges from electrostatic potentials

Atom x (Å) y (Å) z (Å) Charge (e)

N 0.421597 -2.132765 -0.040233 0.004221

N -1.983771 -0.368911 -0.035220 0.134251

N -3.605662 -4.502220 -1.031529 0.005530

N 2.050558 0.310653 -0.195799 -0.137842

N -0.340619 2.060808 -0.211058 0.004226

N 5.008518 -2.994448 1.040074 -0.003230

N -4.820361 3.055132 1.091452 -0.000599

N 3.012745 4.886841 0.779343 -0.018144

C -1.936494 -2.908660 0.017158 0.227779

C -0.592297 -3.057533 -0.006731 -0.161439

C -2.572244 -1.533003 0.043459 -0.153517

C 1.689852 -2.671401 -0.041360 0.148796

C -2.985299 0.594980 -0.016338 -0.175660

C 2.910429 -2.092278 -0.071507 -0.218195

C -2.857323 1.947617 -0.081107 0.192928

C -2.821890 -4.088093 0.085790 -0.057952

C 0.120829 -4.341481 -0.001371 0.046537

C -4.043513 -1.414690 0.156894 -0.044387

C 3.016401 -0.576386 -0.173204 0.145679

C -1.602042 2.649152 -0.219007 -0.166818

C 1.442074 -4.115945 -0.008222 -0.046181

C -4.240539 -0.120133 0.091302 0.006181

C 2.053042 2.787755 -0.289399 -0.213202

C 4.144607 -2.906612 -0.087775 0.056347

C -4.077582 2.804998 -0.094049 -0.040800

C 2.645689 1.566310 -0.252764 0.189595

C 0.630520 3.043435 -0.323455 0.172590

C 2.909964 4.009287 -0.347216 0.047661

C 4.364091 0.032361 -0.242907 0.041203

C -1.396375 4.010535 -0.353687 0.082123

C 4.082246 1.325772 -0.268444 -0.007633

C -0.025732 4.244872 -0.440427 -0.076580

C -2.902804 -4.747067 1.263714 0.018754

C 4.443901 -3.576882 -1.230780 -0.019234

C -4.455729 3.341823 -1.283890 0.012739

C -4.441843 -5.605528 -0.861493 0.014622

C -3.595555 -3.841328 -2.329381 0.003907

C -3.767104 -5.892177 1.392982 0.003225

C 3.515863 4.302707 -1.512417 -0.008985

C 6.150667 -3.800456 0.934022 -0.015111

C -5.976699 3.836917 0.977024 0.011786

C -4.550884 -6.306116 0.265665 0.002901

C 4.780040 -2.326283 2.313005 -0.005672

C -4.470466 2.530065 2.401559 0.003310

C 5.621544 -4.404871 -1.287265 -0.002870

C -5.647546 4.157855 -1.347639 0.003229

C 3.806318 6.031995 0.630129 0.002088

C 6.492501 -4.485792 -0.149290 -0.004012

C -6.411432 4.386493 -0.152522 0.000575

C 2.371226 4.668773 2.066005 0.002163

C 4.323421 5.489409 -1.610854 -0.003148

C 4.454485 6.367095 -0.484315 -0.003731

H 0.261394 -1.143165 -0.051560 0.000000

H -0.348384 -5.312816 0.010812 -0.000000

H -4.775179 -2.197865 0.252831 0.000000

H 2.195243 -4.890485 -0.003348 0.000000

H -5.225600 0.348567 0.120893 0.000000

H 5.327980 -0.443695 -0.262616 -0.000000

H -2.149577 4.785825 -0.398553 0.000000

H 4.842302 2.094002 -0.314675 0.000000

H 0.436296 5.225117 -0.552454 0.000000

H -2.344832 -4.471084 2.148587 0.000000

H 3.835005 -3.548910 -2.119916 0.000000

H -3.932434 3.191837 -2.207290 0.000000

H -5.020987 -5.888868 -1.736485 -0.000000

H -2.913668 -2.995132 -2.404093 0.000000

H -4.601480 -3.488041 -2.561983 0.000000

H -3.295689 -4.575277 -3.093514 0.000000

H -3.834607 -6.433284 2.323061 0.000000

H 3.440235 3.703766 -2.403736 0.000000

H 6.780809 -3.835399 1.823010 -0.000000

H -6.528279 3.986771 1.900444 -0.000000

H -5.197358 -7.159778 0.370213 0.000000

H 3.865095 -1.729997 2.344930 -0.000000

H 5.625547 -1.647177 2.515078 0.000000

H 4.727504 -3.063028 3.119384 0.000000

H -3.555538 1.946794 2.429889 -0.000000

H -5.301639 1.897090 2.760437 -0.000000

H -4.346592 3.366288 3.097250 0.000000

H 5.863872 -4.952648 -2.187886 -0.000000

H -5.968453 4.591096 -2.279423 -0.000000

H 3.867598 6.673476 1.512256 -0.000000

H 7.372166 -5.102891 -0.211581 0.000000

H -7.291128 4.991701 -0.210475 0.000000

H 1.781549 3.759567 2.129459 0.000000

H 1.709796 5.517388 2.282484 0.000000

H 3.142764 4.622414 2.844645 -0.000000

H 4.837824 5.728414 -2.536957 -0.000000

H 5.058418 7.248463 -0.568306 0.000000

H -0.130919 1.090401 -0.079390 0.000000

Table S1: Transition charges and Atomic coordinates (x, y, z) for m-TMPyP

Atom x (Å) y (Å) z (Å) Charge (e)

C 5.005311 1.315763 -0.062397 -0.063429

C 6.822858 2.793393 0.644527 0.016456

C 5.452531 2.475338 0.575636 0.031097

C 6.008654 0.519655 -0.631443 0.030054

C 7.373325 0.807816 -0.442659 0.010262

N 7.754891 1.940274 0.188017 0.001928

C 9.188915 2.200493 0.418145 0.013398

C -5.052217 -1.423867 -0.169090 0.051190

C -6.992483 -2.902767 0.223869 -0.011712

C -5.617956 -2.671033 0.111290 -0.041290

C -6.044986 -0.459596 -0.490947 -0.020794

C -7.423356 -0.678102 -0.297683 -0.014965

N -7.865256 -1.882258 0.091848 0.000800

C -9.287730 -2.097999 0.423846 -0.014479

C 2.323920 -6.683852 -1.045090 -0.001502

C 1.939680 -5.337467 -1.137863 -0.008750

C 1.059098 -5.468637 1.087439 -0.009720

C 1.439782 -6.810833 1.128542 -0.000243

N 2.071177 -7.382541 0.074935 -0.005793

C 2.504820 -8.798642 0.187687 0.001520

C 1.313582 -4.709116 -0.054869 0.043909

C -2.118989 6.731555 -1.097417 -0.000216

C -1.701528 5.401750 -1.199121 0.007361

C -1.069160 5.470925 1.112486 -0.000117

C -1.502945 6.798853 1.162907 0.001945

N -2.026010 7.401639 0.074949 0.003539

C -2.514783 8.782573 0.176029 -0.002770

C -1.172951 4.756131 -0.084051 -0.032454

C 0.927242 -3.257775 -0.121570 -0.216674

C 1.900035 -2.332055 -0.022505 0.170788

C 3.298715 -2.654680 0.238283 -0.040914

C 4.015500 -1.521009 0.319571 0.048500

C 3.151940 -0.382099 0.022333 -0.150854

C 3.534724 0.912782 -0.119660 0.235659

C 2.386349 1.918228 -0.278449 -0.139459

C 2.716319 3.313392 -0.590601 -0.041075

C 1.533552 3.914371 -0.474028 0.007256

C 0.477991 2.953611 -0.235422 -0.150672

C -0.839282 3.276963 -0.124360 0.185411

C -1.936775 2.350795 0.048328 -0.178131

C -3.243524 2.666098 0.380580 0.063236

C -3.946086 1.469323 0.554469 -0.072974

C -3.177941 0.416909 0.093080 0.143864

C -3.519603 -1.018215 -0.090718 -0.211386

C -2.502465 -1.941113 -0.125223 0.181780

C -2.618071 -3.364187 -0.155226 0.008662

C -1.444374 -3.979942 -0.285920 0.038749

C -0.506687 -2.849032 -0.194898 0.145077

N 1.895430 -0.958442 -0.083072 0.002560

N 1.090244 1.704229 -0.142735 0.115325

N -1.906131 0.973502 -0.123502 0.004150

N -1.119084 -1.687825 -0.124413 -0.134106

H 7.172348 3.718628 1.101911 0.000000

H 4.761783 3.168529 1.058701 0.000000

H 5.776178 -0.378675 -1.199207 -0.000000

H 8.155051 0.129758 -0.787542 0.000000

H 9.354566 3.286331 0.428975 0.000000

H 9.772077 1.746511 -0.380679 0.000000

H 9.441675 1.762876 1.379397 0.000000

H -7.392522 -3.892071 0.453755 -0.000000

H -5.065105 -3.573468 0.325220 0.000000

H -5.761459 0.517887 -0.874408 -0.000000

H -8.156848 0.098515 -0.433902 0.000000

H -9.481020 -3.168262 0.520690 0.000000

H -9.887691 -1.677071 -0.367777 0.000000

H -9.471451 -1.601390 1.374755 0.000000

H 2.832358 -7.179728 -1.865393 0.000000

H 2.161299 -4.791212 -2.057151 0.000000

H 0.565313 -5.033971 1.948220 0.000000

H 1.240358 -7.440234 1.993114 0.000000

H 3.085954 -9.072683 -0.698984 -0.000000

H 1.607704 -9.411396 0.257808 0.000000

H 3.122131 -8.887079 1.080551 0.000000

H -2.551078 7.256783 -1.950198 0.000000

H -1.822645 4.871201 -2.145350 0.000000

H -0.682918 4.995776 2.013124 0.000000

H -1.440300 7.384339 2.075035 0.000000

H -2.928017 9.113924 -0.769246 0.000000

H -1.673227 9.420067 0.469777 0.000000

H -3.299635 8.802407 0.949666 0.000000

H 3.709373 -3.638351 0.430252 -0.000000

H 5.050082 -1.504249 0.607771 0.000000

H 3.650642 3.770489 -0.840408 0.000000

H 1.401908 4.975229 -0.586045 0.000000

H -3.636689 3.653312 0.563152 0.000000

H -4.965491 1.428800 0.911641 0.000000

H -3.453088 -4.022879 -0.048589 0.000000

H -1.277767 -5.040636 -0.309835 0.000000

H 1.054709 -0.401934 -0.231692 0.000000

H -1.081090 0.425675 -0.356299 0.000000

Table S2: Transition charges and Atomic coordinates (x, y, z) for p-TMPyP

Hyperparameter optimization

In the GPR ML models, we have the set of hyperparamters σ2

n, σ2

f, l2

1, . . . , l2

D, which needs to

be optimized as part of the training process. To this end, we use the marginal log likelihood

(MLL) of the model. GPyTorch1provides the ADAM optimizer, which we run with a

learning rate of lr = 0.1for a total of 200 steps on a separated, randomly chosen validation

set of 1,000 samples. Since optimizers like ADAM are sensible to initial value selection, we

ﬁrst select an improved set of initial values by grid search with 5-times repeated 5-fold cross

validation on the MLL optimization task over the grid given in Table S3. The best initial

values are then used in a ﬁnal optimization procedure over the full validation set.

Table S3: Grid of possible initial values for the ADAM-based hyperparameter optimization.

Hyperparameter Possible values

lengthscales l1, ..., lD{2, 100.75,101.5}

output scale σf{1}

noise σ2

n{10−4,10−6,10−8}

Preliminary Multiﬁdelity Data Analysis

42024

[meV] 1e2

Density

1e 3 Distribution

LC-DFTB

STO-3G

6-31G

SVP

TZVP

LC-DFTB STO-3G 6-31G SVP

ySVP

[meV]

1e1 Fidelity Difference

210123

[meV] 1e2

ySVP

[meV]

1e2 Fidelity Scatter

LC-DFTB

STO-3G

6-31G

SVP

Figure S2: Preliminary multiﬁdelity data analysis as recommended in Ref.2for porphyrin

molecule p9. The STO3G ﬁdelity shows an unfavorable distribution with respect to the

target ﬁdelity of SVP.

210123

[meV] 1e2

Density

1e 3 Distribution

LC-DFTB

6-31G

SVP

TZVP

LC-DFTB 6-31G SVP

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ySVP

[meV]

1e1 Fidelity Difference

21012

[meV] 1e2

ySVP

[meV]

1e2 Fidelity Scatter

LC-DFTB

6-31G

SVP

Figure S3: Same as Fig. S2 but without the STO3G energies.

S10

42024

[meV] 1e2

Density

1e 3 Distribution

LC-DFTB

STO-3G

6-31G

SVP

TZVP

LC-DFTB STO-3G 6-31G SVP

ySVP

[meV]

1e1 Fidelity Difference

2 0 2 4

[meV] 1e2

ySVP

[meV]

1e2 Fidelity Scatter

LC-DFTB

STO-3G

6-31G

SVP

Figure S4: Preliminary multiﬁdelity data analysis as recommended in Ref.2for the concate-

nated trajectories of the p-type porphyrin molecules. The STO3G ﬁdelity shows unfavorable

distribution with respect to the target ﬁdelity of SVP.

42024

[meV] 1e2

Density

1e 3 Distribution

LC-DFTB

6-31G

SVP

TZVP

LC-DFTB 6-31G SVP

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

ySVP

[meV]

1e1 Fidelity Difference

2 0 2 4

[meV] 1e2

ySVP

[meV]

1e2 Fidelity Scatter

LC-DFTB

6-31G

SVP

Figure S5: Same as Fig. S4 but without the STO3G energies.

420246

[meV] 1e2

Density

1e 3 Distribution

LC-DFTB

STO-3G

6-31G

SVP

TZVP

LC-DFTB STO-3G 6-31G SVP

ySVP

[meV]

1e1 Fidelity Difference

2 0 2 4

[meV] 1e2

ySVP

[meV]

1e2 Fidelity Scatter

LC-DFTB

STO-3G

6-31G

SVP

Figure S6: Preliminary multiﬁdelity data analysis as recommended in Ref.2for the concate-

nated trajectories of the m-type porphyrin molecules. The STO3G ﬁdelity shows unfavorable

distribution with respect to the target ﬁdelity of SVP.

S11

2 0 2 4

[meV] 1e2

Density

1e 3 Distribution

LC-DFTB

6-31G

SVP

TZVP

LC-DFTB 6-31G SVP

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

ySVP

[meV]

1e1 Fidelity Difference

210123

[meV] 1e2

ySVP

[meV]

1e2 Fidelity Scatter

LC-DFTB

6-31G

SVP

Figure S7: Same as Fig. S6 but without the STO3G energies.

Additional Machine Learning Results

2 8 32 128 1024

NSVP

train

MAE [meV]

12.3

24.8

2 8 32 128 1024

NSVP

train

MAE [meV]

24.7

29.1

GPR-SVP

6-31G

3-21G

STO-3G

DFTB

2 8 32 128 1024

NSVP

train

MAE [meV]

17.8

22.3

Figure S8: MFML learning curves with STO3G ﬁdelity included.

101102103104

Ttrain data

[hr]

MAE [meV]

(a) p-TMPyP traj-9

101102103104

Ttrain data

[hr]

(b) p-TMPyP concatenated

GPR (SVP)

MFML(LC-DFTB)

(2)

(4)

(8)

(16)

101102103104

Ttrain data

[hr]

Figure S9: Full time-cost plots.

S12

Spectral Densities

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(a) m1

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(b) m2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(d) m4

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(e) p1

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(f) p2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(g) p3

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(h) p4

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(i) p5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(j) p6

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(k) p7

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(l) p8

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(m) p9

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

p10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(n) p10

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

p11

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(o) p11

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

1000

2000

3000

4000

5000

J( ) [cm 1]

p12

0.0

0.1

0.2

0.3

0.4

0.5

0.6

J( ) [eV]

(p) p12

Figure S10: Spectral densities of the individual pigments.

S13

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

500

1000

1500

2000

2500

3000

3500

4000

J( ) [cm 1]

m-type DFTB

m-type MFML

0.0

0.1

0.2

0.3

0.4

0.5

J( ) [eV]

(a) m-type porphyrin

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

500

1000

1500

2000

2500

3000

3500

J( ) [cm 1]

p-type DFTB

p-type MFML

0.0

0.1

0.2

0.3

0.4

J( ) [eV]

(b) p-type porphyrin

Figure S11: Comparison of the average Spectral densities based on MFML and DFTB.

References

(1) Gardner, J.; Pleiss, G.; Weinberger, K. Q.; Bindel, D.; Wilson, A. G. GPyTorch: Black-

box matrix-matrix Gaussian process inference with GPU acceleration. Advances in Neu-

ral Information Processing Systems 2018,31 .

(2) Vinod, V.; Maity, S.; Zaspel, P.; Kleinekathöfer, U. Multiﬁdelity Machine Learning for

Molecular Excitation Energies. J. Chem. Theory Comput. 2023,19, 7658–7670, PMID:

37862054.

S14

Excitation Energy Transfer between Porphyrin

Dyes on a Clay Surface: A study employing

Multiﬁdelity Machine Learning

Dongyu Lyu,†Matthias Holzenkamp,‡Vivin Vinod,¶Yannick Marcel Holtkamp,†

Sayan Maity,†Carlos R. Salazar,†,§Ulrich Kleinekathöfer,∗,†and Peter Zaspel∗,‡

†School of Science, Constructor University, Bremen 28759, Germany

‡School of Mathematics and Natural Science, University of Wuppertal, Wuppertal 42119,

Germany

¶School of Mathematics and Natural Sciences, University of Wuppertal, Wuppertal 42119,

Germany

§Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83

Linköping, Sweden

E-mail: ukleinekathoefer@constructor.university;zaspel@uni-wuppertal.de

Abstract

Natural light-harvesting antenna complexes eﬃciently capture solar energy using

chlorophyll, i.e., magnesium porphyrin pigments, embedded in a protein matrix. In-

spired by this natural conﬁguration, artiﬁcial clay-porphyrin antenna structures have

been experimentally synthesized and have demonstrated remarkable excitation energy

transfer properties. The study presents the computational design and simulation of

a synthetic light-harvesting system that emulates natural mechanisms by arranging

cationic free-base porphyrin molecules on an anionic clay surface. We investigated the

arXiv:2410.20551v1 [physics.chem-ph] 27 Oct 2024

transfer of excitation energy among the porphyrin dyes using a multiscale quantum me-

chanics/molecular mechanics (QM/MM) approach based on the semi-empirical density

functional-based tight-binding (DFTB) theory for the ground state dynamics. To im-

prove the accuracy of our results, we incorporated an innovative multiﬁdelity machine

learning (MFML) approach, which allows the prediction of excitation energies at the nu-

merically demanding time-dependent density functional theory level with the Def2-SVP

basis set. This approach was applied to an extensive dataset of 640K geometries for the

90-atom porphyrin structures, facilitating a thorough analysis of the excitation energy

diﬀusion among the porphyrin molecules adsorbed to the clay surface. The insights

gained from this study, inspired by natural light-harvesting complexes, demonstrate

the potential of porphyrin-clay systems as eﬀective energy transfer systems.

Introduction

In nature, sunlight is collected eﬃciently by pigment molecules embedded in a protein matrix

and subsequently transferred to a reaction center. At the same time, remarkable progress

has been made in recent years in dye chemistry developing functional materials.1To identify

next-generation energy devices, researchers are exploring both experimental and theoretical

biohybrid approaches inspired by biological systems.2Of particular interest are eﬀorts to

translate biological principles directly into synthetic energy systems.3To this end, photo-

chemical systems with a light-harvesting function on inorganic nanosheets are being devel-

oped.4–7In particular, porphyrins and other tetrapyrrole macrocycles have been shown to

have an impressive variety of functional properties that have been and can be exploited in

natural and artiﬁcial systems.8

Experimental results indicate that clay-porphyrin complexes have the potential to be

used in the development of highly eﬃcient artiﬁcial light-harvesting systems.4,9The assem-

bly of porphyrin molecules on clay surfaces can be eﬀectively controlled using electrostatic

interactions.4,5,10 The authors reported an almost 100 % eﬃciency of the energy-transfer re-

action.4In terms of adsorption, the “size-matching eﬀect”, which entails aligning the anionic

charge distribution with the dimensions of the adsorbing molecules, has been posited as a

means of achieving enhanced adsorption. Furthermore, the observation of multiple-step en-

ergy transfer reactions between three adsorbed dyes on the clay surface is noteworthy.5The

combination of a metalloporphyrin as a photocatalyst and a subporphyrin as a photosen-

sitizer in a self-assembling manner has been shown to enable the photochemical conversion

of cyclohexene using a wider range of visible light without any energy loss due to the sup-

pression of unexpected deactivation processes.6Other porphyrin-clay arrangements include,

e.g., the intercalation of a double-decker porphyrin metal complex into clay nanosheets.11

These examples demonstrate potential applications of porphyrin-clay systems. In this study,

we examine a clay-pophyrin system that emulates the experimental setup of Ref. 4. To this

end, we have developed a molecular multiscale approach coupled to multi-ﬁdelity machine

learning in order to simulate the excitation energy transfer.

In several of the experimental studies, the clay saponite is utilized. While the atom-

istic structure has yet to be resolved, numerous individual features have been identiﬁed.12

Given the lack of full atomic-level details for saponite, this study will utilize montmoril-

lonite nanosheets for simulations, as has been done in other studies on clays, including the

adsorption of esters and polymers.13–16 Very recently, ClayCode, a software facilitating the

modeling of clay systems closely resembling experimentally determined structures, has been

made public.17 This will facilitate the simulation of an even greater number of clay types,

as referenced.

During photosynthesis in plants, algae, and some bacteria, the pigment molecules respon-

sible for the absorption of sunlight are (bacterio)chlrophyll molecules. These molecules play

a dual role in the process, serving both as light absorbers and as mediators of excitation en-

ergy transfer to the reaction centers. In the experimental system, which will be emulated in

this study, tetrakis(1-methylpyridinium-3-yl) (m-TMPyP) and tetrakis(1-methylpyridinium-

4-yl) porphyrins (p-TMPyP) adsorbed on a clay surface were selected for the same objective4

(see Fig. 1). Therefore, the present simulation study will focus on the same porphyrin types,

with the understanding that the results may also be applicable to other porphyrin molecules.

Regarding the approach to modeling the excitation energy transfer in the porphyrin-clay

system, we will largely follow a procedure that has been successfully employed to model bi-

ological light-harvesting systems.18–23 In summary, the initial step is to conduct a molecular

dynamics (MD) simulation of the entire system. The equilibrated structure is then used

as the starting point for QM/MM (quantum mechanics/molecular mechanics) simulations

simulating the pigment molecules. These employ the eﬃcient density functional tight bind-

ing (DFTB) approach24 to improve the description of the vibrational motion of the pigment

molecules. Subsequently, the excited states are determined using a machine-learning ap-

proach as detailed below. At the same time, the excitonic couplings are calculated using the

popular TrESP (Transition charges from Electrostatic potential) approach.25,26 Based on this

information, a time-dependent Hamiltonian can be constructed, which in turn is used to es-

timate the exciton dynamics employing the NISE (Numerical Integration of the Schrödinger

equation) formalism,27 also known as Ehrenfest scheme without back-reaction.28

Concerning the calculation of the excitation energies along the QM/MM trajectories, we

want to majorly improve their accuracy by employing the multi-ﬁdelity machine learning

(MFML) to the present problem.29,30 Starting from a training set of pairs of (descriptors

for) molecular geometries and corresponding calculated excitonic properties, we can con-

struct regression models in machine learning (ML) to be used for the determination of the

excitation energies along trajectories. These models are cheap to evaluate and ideally have

low prediction error, relative to unseen molecular geometries that are picked from a neigh-

borhood of the conformation space spanned up by the training data.31 We use ML models

to predict excitation energies at the level of time-dependent density functional theory us-

ing the CAM-B3LYP functional (TD-DFT/CAM-B3LYP) with the basis set def2-SVP. To

get a strong model with low prediction error, we would need a large amount of such train-

ing samples, which would however come at a prohibitive computational cost. Instead, we

use MFML29,30,32,33 models, which are speciﬁcally constructed from training data at diﬀerent

levels of ﬁdelity, here diﬀerent basis set sizes (STO-3G, 3-21G, 6-31G, dev2-SVP) for the TD-

DFT/CAM-B3LYP calculations with a strongly decreasing amount of samples with growing

level of the hierarchy.33 Thereby, as we will show, we clearly reduce the total amount of

computational eﬀort over building “classical” single-ﬁdelity ML models from standard train-

ing data. In addition, we investigate active learning strategies34–37 to reduce the required

amount of training samples, even on a single level. Active learning aims at choosing or

creating new training samples, which are maximally informative to the model and therefore

reduce redundant information in the training data. In our setup, the selection process is

done on a large amount of candidate molecular conﬁgurations (without calculated energies)

from the MD runs via uncertainty sampling.38

The following section outlines the atomistic setup of the porphyrin-clay system, which

will be simulated at the classical MD and QM/MM levels. The MD simulations allow us

to assess the stability and movement of porphyrin molecules on the insulating clay surface.

Furthermore, we examine the nearest-neighbor distances of select porphyrin molecules and

the resulting excitonic couplings between them. Subsequently, we will address the topic

of active learning and multi-ﬁdelity machine learning of excitation energies. Based on these

results, we can determine the spectral densities, which describe the interaction of the primary

modes and those treated as a thermal bath. The excitation energies and excitonic couplings

are used in the NISE approach to determine the exciton dynamics and the diﬀusion of

excitation energy along the clay surface. The contribution concludes with a brief overview

of the key ﬁndings and future directions, while a Methods section gives some more insights

into technical details of the used approaches.

MD and QM/MM simulations

In a ﬁrst step, the clay surface has to be modelled. Here we decided to model a sheet

of montmorillonite since it is widely used in experiments and structural data is available.

The chemical formula of montmorillonite is (K, N a)x[Si4O8][Al(2−x)M gxO2(OH)2]. In the

present study, we use Na and Cl ions as counterions dissolved in the surrounding water. The

parameter x, which can vary between 0 and 0.95, indicates the number of aluminum and

magnesium atoms in the system and can be used to adjust the charge of the surface. Four

diﬀerent values of x, i.e., 0.13, 0.2, 0.45 and 0.94, were tested together with four m-TMPyP

and twelve p-TMPyP molecules (see Methods section for details of the MD and QM/MM

setups). Larger values of xmake the clay surface more anionic, though the cationic porphyrin

molecules have to compete with sodium ion concerning the binding to the surface. It has to be

noted that in the present cationic montmorillonite model, the negative nature of the surface

(a) (b)

Figure 1: (a) The chemical structures of two porphyrins used in experiment4and in the

present study. (b) The montmorillonite surface used in our simulations. Oxygen, silicon,

and hydrogen atoms are shown in red, yellow, and white, respectively, while the aluminum

and magnesium atoms are colored in pink and blue, respectively. Visible in the structure is

the non-regular arrangement of the magnesium defects located in the middle layer.

is due to atom exchange in the so-called octahedral layer while in the experimental setup

with the saponite system the ions were exchanged in the tetrahedral layer4(see Methods

section for details). The latter layer is closer to the surface of the material. Therefore, the

negative charges in the present modeling setup are slightly more screened at the surface,

leading to a somewhat reduced binding of the molecules to the surface. The montmorillonite

surface with a xvalue of 0.45 was chosen for the following calculations, since all porphyrin

molecules stayed on the surface after the MD equilibrium for 30 ns. In simulations with the

other values of x, at least one dye molecule left the surface during the simulations.

Starting from the last frame of the MD simulation, quantum mechanics/molecular me-

chanics dynamics (QM/MM) ground state simulations are performed. For each porphyrin

molecule, 40,000 conformations including all surrounding point charges were saved with a

stride of 1 fs. Subsequently, these conformations were used as the basis of excited state

calculations using TD-LC-DFTB.26,39 In addition, TD-DFT/CAM-B3LYP calculations in

combination with four diﬀerent basis sets, i.e., STO-3G, 3-21G, 6-31G and def2-SVP, were

also used to calculate the excited states of these conformations with a stride of 8, 16, 32 and

64 fs, respectively. Unlike the protein environment in natural light-harvesting systems, the

clay surface is basically rigid and has a high degree of periodicity if one neglects the non-

periodic charge distribution due to the Mg doping inside the material. The conformations

of the porphyrin molecules, however, are ﬂexible and show ﬂuctuations. These changes in

the conformations lead to energy ﬂuctuation. A short piece of such energy ﬂuctuations is

depicted in Fig. 2a. The excitation energies, also termed site energies below, lead to distri-

butions as shown in Fig. 2b. It becomes clear that the m-type porphyrins have site energies

which are, on average, higher in energy than those of the p-type porphyrin dyes. The average

site energies obtained by diﬀerent level of theory show oﬀsets with respect to each other,

but an overall consistent trend as shown in Fig. S1 and the average excitation energies are

shown in Fig. 2c. The same type of pigments have somewhat diﬀerent site energies due to

their diﬀerent placements on the surface and the interactions with other porphyrins on the

0 25 50 75 100 125 150 175 200

Time [fs]

1.7

1.8

1.9

2.0

Site Energy [eV]

(a)

1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2

Site Energy [eV]

Density

p10

p11

p12

(b)

m1 m2 m3 m4 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10p11p12

1.6

1.8

2.0

2.2

Site Energy [eV]

TD-LC-DFTB

STO-3G

3-21G

6-31G

def2-SVP

(c)

Figure 2: (a) An example fragment of the energy ﬂuctuations at TD-LC-DFTB level of

theory for four porphyrins on the surface. (b) Distribution of the excited state energies

using TD-LC-DFTB, which are approximately Gaussian distributed. (c) The average site

energies using diﬀerent levels of quantum chemistry and the predictions of the MFML model.

The numbering of the diﬀerent m-type (m1-m4) and p-type (p1-p12) porphyrin dyes on the

clay surface are discussed below.

surface. These energy values from diﬀerent levels of quantum chemistry are then used as

input for the MFML model, as described in the next section.

Furthermore, an additional 100 ns-long classical MD simulation was performed producing

10,000 snapshots at a stride of 10 ps in order to be able to better evaluate the positioning and

movement of the porphyrin dyes on the clay surface and to determine the excitonic couplings

between them. In Fig. 3, three snapshots at 0, 50 and 100 ns of this additional trajectory give

an impression of the positions and conformational diﬀerence of the porphyrin molecules along

the trajectory. The m-TMPyP molecules (red) are tightly adsorbed, with all four side chain

groups interacting closely with the clay surface for most of the simulation time. It should

be noted that the adsorption positions of the side chains are mostly close to the positions of

the magnesium defects in the montmorillonite. In Refs.4,40 this has been termed the “size-

matching rule”. On the contrary, only two to three side chains of the p-TMPyPs molecules

(a) 0ns (b) 50ns (c) 100ns

Figure 3: Several snapshots of the classical MD trajectory. The montmorillonite surface

is shown in yellow in the background, with gray beads representing the magnesium defects

inside the clay material. The m-TMPyP and p-TMPyP molecules on the surface are colored

in red and blue, respectively.

(a) Single Copy (b) All Copies

Figure 4: Scatter plot of the center of masses of the dyes reported every 100 ps along the

100 ns trajectory in (a) a single unit cell and (b) in a plot also showing the nearest periodic

images to all sides in the plane of the clay surface.

(blue) are adsorbed on the surface, and the rest are “wandering” in the water environment,

making the molecules “stand” with a certain angle, and move more easily on the clay surface.

In order to better visualize the range of movements of the porphyrins along the trajectory,

the center of mass of the pigments along the trajectory was extracted and plotted in Fig. 4.

Compared to the m-type porphyrins, which are approximately ﬁxed on the surface, there are

distinct movements of the p-type ones. These movements also signiﬁcantly modify the now

time-dependent distances between porphyrins. In addition, note in particular that although

Fig. 4a suggests a large range of motion for the porphyrin m3, it only starts moving after

molecule p2 approaches.

In natural light-harvesting systems, the chromophores are embedded in a stable pro-

tein environment without signiﬁcant changes in the distances between them. However, in

this artiﬁcial porphyrin-clay surface system, not only the distance between the porphyrin

0 20 40 60 80 100

Time [ns]

1.6

1.8

2.0

2.2

2.4

2.6

2.8

Nearest Neighbor Distance [nm]

Nearest Neighbor

p10

(a) Nearest neighbors of m1.

0 20 40 60 80 100

Time [ns]

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

Nearest Neighbor Distance [nm]

Nearest Neighbor

p12

(b) Nearest neighbors of m3.

0 20 40 60 80 100

Time [ns]

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

Nearest Neighbor Distance [nm]

Nearest Neighbor

p10

p12

0 20 40 60 80 100

Time [ns]

Strongest Coupling [cm 1]

Coupled To

p10

(d) Strongest coupling to m1.

0 20 40 60 80 100

Time [ns]

Strongest Coupling [cm 1]

Coupled To

p12

(e) Strongest coupling to m3.

0 20 40 60 80 100

Time [ns]

Strongest Coupling [cm 1]

Coupled To

p10

p11

p12

(f) Strongest coupling to p2.

Figure 5: Nearest neighbor distances (top row) and strongest coupling (bottom row) of select

pigments. The colors indicate which neighbor is the nearest (or strongest coupled) at a given

time. The nearest neighbors and pigments with the largest coupling value can be across the

periodic boundary.

molecules, but also which dye is the nearest neighbor can change continuously while the

porphyrins move on the surface. These changes in distance between pigments have a sig-

niﬁcant impact on the excitonic couplings and subsequently the exciton transfer dynamics.

Considering the periodic boundary conditions, the distance between each porphyrin and its

nearest neighbor was extracted from an extended box with 3×3copies (see Fig. 4b) over the

10,000 frames. Here, we choose three porphyrins as examples and show the nearest neighbor

distances in the top row of Fig. 5. The diﬀerent colors indicate the (frequent) change of the

nearest neighbor. The average nearest neighbor distance across all pigments and snapshots

is 2.01 nm, which is of the same magnitude as the value of 2.6 nm reported in the respective

experiment.4However, in analysis of the experiment, a perfect hexagonal packing was as-

sumed for the porphyrins on the surface. The average distance to the six nearest neighbors

is 2.92 nm in our case, suggesting that our distances on average are slightly larger than the

value reported in the experiment.

The couplings were calculated based on the Coulomb interaction of the transition charges

Vmn(t) = f

4πϵ0X

I,J

qm,T

I·qn,T

|rm

I(t)−rn

J(t)|,(1)

where ϵ0denotes the vacuum permittivity, qm,T

Iand qn,T

Jthe transition charges of atom I

and Jfrom pigment mand n, while rm

I(t)and rn

J(t)are the positions and fis the screening

factor. The transition charges were calculated with the “transition charges from electrostatic

potentials” (TrESP) method41,42 based on CAM-B3LYP/6-31G* level of theory and can be

found in Tables S1 and S2. The positions were based on the MD trajectory with a time step

of 10 ps. Furthermore, we chose the pairwise screening factor introduced in Ref. 43 as the

screening factor f. In a ﬁnal step, we usually rescale the coupling to match experimental

values. Since no experimental dipole moments for these types of porphyrin were available,

we used a previous rescaling factor for bacteriochlorophyll (BChl).18 The largest absolute

coupling value was extracted for each porphyrin along the trajectory. The bottom row of

Fig. 5shows the highest coupling values for the same pigments as in the top row. The signs

of the coupling values can vary over time, as the nearest neighbor and the orientation of the

porphyrin molecules with respect to each other change over time. Therefore, the coupling

values indirectly also contain some information about the rotation of the molecules, i.e.,

mainly of the p-type molecules since the m-type molecules are rather immobile. Usually, the

closest pigment is also the one which has the largest excitonic coupling value. In cases of

similar distances to the nearest and the next nearest neighbor, however, the orientation of

the pigments to each other becomes the determining factor. The average highest absolute

coupling is 4.36 cm−1, while the median is only 2.44 cm−1suggesting strong outliers as can

also be seen in Fig. 5. With such strong ﬂuctuations and even change of sign, the usual

practice of taking average couplings is likely very ﬂawed in our case, so that we have to take

the time dependence of the couplings into account for our further investigations.

Machine Learning

Expensive calculations of the excitation energy for the ﬁrst excited state are replaced by

evaluations of MFML models, one model for all 12 p-type molecules and one model for all

m-type molecules. Prior to building these two MFML models on a large, computationally

training dataset, the characteristics of the porphyrin data of the system are being studied

in the context of ML model construction for cheap single-ﬁdelity data, namely excitation

energies calculated using the TD-LC-DFTB approach. A particularly important topic is the

transferability of the ﬁnal models across the pigments. Also, the impact of using Active

Learning is studied.

Transferability Analysis

In Figure 6we analyze the conﬁguration spaces of the 12 diﬀerent p-type molecules, which

are encountered during the MD run, and how they relate to each other. In Fig. 6a a UMAP

(Uniform Manifold Approximation and Projection) plot44 of the Coulomb representations of

the 40,000 conﬁgurations of every p-type trajectory is shown. It can be seen that, in most

cases, the conﬁguration spaces, which are covered by the diﬀerent molecules, are clearly

diﬀerentiated. Only for a few cases, the conﬁguration spaces of two molecules overlap, e.g.,

for p9 and p1. These distinct conﬁgurations spaces are problematic, since they give a ﬁrst

(a)

10 100 1000

Ntrain

100

150

200

MAE [meV]

Test on p9

Test on other trajectories

(b)

10 100 1000

Ntrain

100

MAE [meV]

Test on all trajectories

Test on separate trajectories

(c)

Figure 6: Results of a transferability study of single-ﬁdelity models across p-type pigments:

(a) For an understanding of the variations in the porphyrin conformations, a UMAP (Uniform

Manifold Approximation and Projection) plot is provided based on the MD trajectories for

the 12 p-type porphyrin molecules. (b) Learning curves for a single-ﬁdelity model trained

on p9 and separately evaluated on all p-type pigments. The model does not generalize to

the other pigments. (c) Learning curves for a single-ﬁdelity model trained on the union of

the trajectories of all p-type pigments. The model generalizes much better. Note that for

better visibility, the vertical axis of (b) and (c) have diﬀerent ranges.

indication towards a bad transferability of ML models being trained on a single trajectory

towards model evaluations on the trajectories of the remaining pigments.

To study this transferability further, we train a single-ﬁdelity Gaussian Process Regres-

sion (GPR)45 model only on p9 data, with hyperparameters being optimized on a separated,

randomized validation subset with 1,000 samples of the p9 trajectory. Figure 6b, shows

learning curves, i.e., an analysis of the prediction error for a growing number of training

samples. Here, the model built on p9 data is evaluated on separate, randomized subsets of

size 2,000 for each p-type molecule. Results reﬂect an average over ﬁve GPR models that are

trained using diﬀerent random selections of training data, excluding the test and validation

sets. The model trained exclusively on p9 fails to make reasonable predictions for conﬁgura-

tions from the other trajectories. The only exception is, when testing on p1, where the error

decreases with an increasing number of training samples, although the error is signiﬁcantly

higher than for the test set of p9. In the UMAP, see Figure 6a, the conﬁguration spaces of p1

of p9 are one of the few conﬁguration spaces of diﬀerent trajectories which are overlapping.

This shows that the diﬀerences in conﬁguration spaces among the various trajectories, as

illustrated in the UMAP, are indeed substantial. A model trained solely on samples from

one trajectory cannot be transferred to predict conﬁgurations from other trajectories.

To overcome this challenge, any subsequent model used in this overall study is trained

over the union of all trajectories of all pigments of one type. Fig. 6c depicts learning curves

for a model trained on samples from all trajectories of p-type molecules, using the same

construction approach outlined for the previous set of learning curves. This joint model

shows a prediction error that decreases with an increasing number of training samples across

the test sets of all diﬀerent p-type pigments. We hence obtain a model with largely uniform

prediction errors cross the individual pigments. Still, it needs to be noted that the prediction

error for the joint model is higher than the error of the model trained only on p9, when

predicting the p9 test conﬁgurations. This is however not surprising as we, in average, only

take 1,000/12 samples from trajectories of each pigment, compared to the 1,000 of only p9

in the other case. At this point, we balance transferability with model accuracy.

Assessing Active learning

One important question is, whether we can gain from speciﬁcally choosing training samples

out of a set of unlabeled candidate molecular conﬁgurations, in contrast to randomly or

uniformly sample training data from a given molecular trajectory. This is what is done in

Active Learning (AL). In an exemplary study, we analyze the impact of using uncertainty

sampling with the Gaussian Process Regression standard deviation as uncertainty measure

to select training samples, rather than using a randomized sampling.

200 500 1000 2000 5000

Ntrain

MAE [meV]

p-traj 9

Absolute error

Random

GPR standard deviation

200 500 1000 2000 5000

Ntrain

50 Concatenated

Figure 7: Comparative study between randomized training sample selection and Active

Learning using GPR standard deviation for a model build for a single pigment (left) and for

a model build on all p-type pigments (right).

Figure 7shows, on the left-hand side, learning curves comparing randomized sampling

to active learning for a model that is only built on training samples of p9. Besides of the

selection of the particular training set, the same data and same error evaluation approach as

in the last section is used. We start the study from a model that is build on 200 randomly

selected samples and consecutive add samples form the remaining full trajectory, either with

active learning or in a randomized fashion. As a reference, we altermatively add samples

with the highest actual absolute error, to show the optimum that we could achieve with

a greedy sampling scheme such as AL. For p9, it can be seen that uncertainty sampling

with GPR standard deviation results in a slightly improved prediction error compared to

random sampling. The (red) reference result indicates, that there would be room for further

improvement with a greedy sampling scheme, if a better uncertainty measure than GPR

standard deviation would be available.

In a second study, for which the results are depicted in Figure 7on the right-hand side,

active learning is tested on the union over all trajectories for all p-type pigments, as we need

it for transferability, see last section. In this case, randomized sampling gives better results

than AL, in contrast to the result for p9, only. Based on this mixed picture, we decided to

stick to a randomized training sample selection strategy for the remaining construction of

MFML models.

Multiﬁdelity Machine Learning Models

The ﬁnal models for the prediction of the excitation energies of the p-type and m-type

porphyrin pigments are multiﬁdelity machine learning models (MFML). Such a MFML model

involves the use of quantum chemistry training data at diﬀerent ﬁdelities, which refer to the

accuracy of the data with respect to the actual value.32 The MFML models are built with

respect to some target ﬁdelity and a baseline ﬁdelity. The former, denoted hereon as F, refers

to the most accurate (and thereby costliest) ﬁdelity that one is interested in predicting with

the ML model. The baseline ﬁdelity, fb, is the cheapest (and thereby the least accurate)

ﬁdelity data that is used in the MFML model. This MFML model is denoted by P(F,ηF;fb)

M F ML

where 2ηF=N(F)

train is the number of training samples used at the target ﬁdelity. The number

of training samples at the subsequently cheaper ﬁdelities are determined by the use of a

scaling factor,γwhich is conventionally set to 2 based on Ref. 32. That is, Nf

train =γ·Nf−1

train

for all fb< f ≤F. The detailed theoretical framework of MFML is further presented in

the Section Multiﬁdelity Machine Learning Approach. In this work, the target ﬁdelity is

TD-DFT/CAM-B3LYP with the def2-SVP basis set while the cheapest baseline ﬁdelity is

the LC-DFTB approach. These will be reported using a shorthand notation, that is SVP

and DFTB, respectively. The diﬀerent ML models are assessed based on the MAE using

learning curves which depict the MAE as a function of the number of training samples used

at the target ﬁdelity. In addition, the MAE is studied as a function of the time-cost of

generating training data for a speciﬁc ML model, be it single ﬁdelity GPR or MFML models

with diﬀerent values of fb(see Model Evaluation). These are reported alongside a recently

introduced MFML approach, termed the Γ-curve which ﬁxes the number of training samples

at ﬁdelity Fand varies only the value of γ.46 This is shown to be superior to the conventional

approach of MFML in providing a low-cost high-accuracy ML model for the prediction of

excitation energies of porphyrin.

2 8 32 128 1024

NSVP

train

MAE [meV]

13.9

24.7

GPR-SVP

6-31G

3-21G

DFTB

(a) Based only on a trajectory

of porphyrin pigment p9.

2 8 32 128 1024

NSVP

train

MAE [meV]

24.8

29.1

GPR-SVP

6-31G

3-21G

DFTB

(b) Concatenated trajectories of

the p-type pigments.

2 8 32 128 1024

NSVP

train

MAE [meV]

19.2

22.3

GPR-SVP

6-31G

3-21G

DFTB

the m-type pigments.

Figure 8: MFML learning curves for three cases of predicting excitation energies for por-

phyrin molecules. The prediction error (as MAE) of the single ﬁdelity GPR and of the

MFML model with the DFTB baseline ﬁdelity are explicitly stated for NSV P

train = 1024.

As in the previous sections, single ﬁdelity GPR models and MFML models were built

and compared on the single pigment p9 and on the union over all p-type pigments for trans-

ferability reasons. In addition tests are conducted on the union over all m-type pigments. In

the all-pigment models an even sampling of the training data is used. The models are tested

on a separated holdout test set for which the excitation energies are calculated at the target

ﬁdelity, that is SVP. For p-type pigments, 2,000 test samples are used, while for m-type

pigments 500 test samples are considered, to account for the lower total amount of data.

The resulting MFML learning curves for the diﬀerent cases are shown in Fig. 8. The shown

learning curves are an average over ten learning curves created from shuﬄing the training

data set. The diﬀerent learning curves in a single study are given for a growing number of

utilized ﬁdelity levels starting from the baseline ﬁdelity fb, as indicated in the legend.

The case of training and testing on the same trajectory of p-type porphyrin molecules

is shown in Fig. 8a for diﬀerent fb. With the addition of cheaper baseline ﬁdelities, one

observes that the MFML model predicts with a lower MAE in comparison to the single

ﬁdelity GPR model built with training samples only from the target ﬁdelity. For instance,

with NSV P

train = 1024, the GPR model results in an MAE of 24.7 meV while the MFML model

with the baseline ﬁdelity fbset to DFTB results in an MAE of 13.9 meV. While this is

a promising result, the transferability study from above indicated that a joint model for

all pigments of one type should be constructed. For this reason, the ﬁnal MFML models

that are used to predict the excitation energies for the porphyrin molecules are built with

training data taken from a pool of trajectories for each type of porphyrin molecule. The

MFML learning curves for the p-type porphyrin molecules are delineated in Fig. 8b for

diﬀerent baseline ﬁdelities. While the addition of cheaper baselines does decrease the model

MAE, this drop is not as signiﬁcant as seen in the case for the single trajectory. The drop

in error between single ﬁdelity GPR and MFML with the DFTB baseline ﬁdelity is about

6 meV for NSV P

train = 1024. However, this is anticipated since both the single ﬁdelity GPR

and the MFML models have to cover a wider region of the conformational phase space (see

discussion on the UMAPs from Fig. 6a) as opposed to a smaller region that is to be covered

in the case for the single trajectory models. A similar observation is made for the MFML

learning curves for m-type porphyrin molecules as seen in Fig. 8c with the single ﬁdelity GPR

model reporting an MAE of 22 meV and the MFML model with DFTB baseline reaching

an MAE of 19 meV with 1024 training samples at the SVP ﬁdelity. The slightly overall

lower MAE for the m-type porphyrin dyes can be explained once again by the fact that

the concatenated trajectories of this porphyrin type result in a lower number of geometries

which in turn could span a smaller region of the conformational phase-space as opposed to

the case for the p-type porphyrin molecules. That is, the m-type set has a smaller number of

total geometries when concatenated in comparison to the total geometries of the p-TMPyP

set. The larger number of total geometries for the p-type set implies that the MFML model

with NSV P

train would contain a smaller amount of information about the conformation space of

the molecule, in contrast to the MFML model built for m-type set. This fact is reﬂected in

the learning curves.

101102103104

Ttrain data

[hr]

MAE [meV]

GPR (SVP)

MFML(DFTB)

(8)

(a) Based only on a trajectory

of porphyrin pigment p9.

101102103104

Ttrain data

[hr]

MAE [meV]

GPR (SVP)

MFML(DFTB)

(8)

(b) Concatenated trajectories of

the p-type pigments.

101102103104

Ttrain data

[hr]

MAE [meV]

GPR (SVP)

MFML(DFTB)

(8)

the m-type pigments.

Figure 9: Time-cost of generating training data versus MAE in meV for a single ﬁdelity

KRR contrasted with that for the MFML model built with baseline ﬁndelity fbDFTB. The

Γ(8)-curve is also depicted for increasing values of γas explained in Section .

In order to better assess the computational impact of single ﬁdelity and MFML models

for porphyrin molecules, the model error is studied as a function of the cost of generating

the training data used in the ML model. In this work, the QC calculation times as returned

by the ORCA computing software47 and DFTB+ software24,48 are used. For the GPR, this

cost is directly related to the number of training samples. For the MFML model, this cost

includes not only the training samples used at the top ﬁdelity, but also the cost of the

training samples used at the subsequent lower ﬁdelities. These curves are shown in Fig. 9.

The time required for training the models and predictions over the holdout test set of the

MFML model for the largest training set size used (that is, NS V P

train = 1024) was 12.97 seconds

and 12.45 seconds for the p-type and m-type porphyrin molecules, respectively. Since this is

such a small contribution, only the total time for generating the training data is considered

in the MAE versus time-cost curves.

In addition to the single ﬁdelity GPR and MFML models, a recently introduced MFML

approach, referred to as the Γ-curve,46 is analyzed as well. In conventional MFML theory,

the training samples at the various ﬁdelities are decided by a scaling factor,γ, that is,

train =γ·Nf−1

train for f∈ {2, . . . , F }. Conventionally, a MFML model is built with γ= 2

based on previous studies.32,49–51 However, the use of diﬀerent values of γhas recently been

studied resulting in a reportedly more eﬃcient approach titled the Γ-curve. The Γ-curve is a

plot of MAE versus time-cost of the MFML model with increasing values of γ. The Γ-curve

is built with a ﬁxed number of training samples at the target ﬁdelity, SVP. Figure 9reports

the Γ(8) curve, that is, with NMSV P

train = 8 with varying values of γ. Diﬀerent values of NSV P

train

were considered and are shown in Fig. S9 in the supplementary material.

Figure 9a depicts the MAE versus time-cost of the ML models for the case of the single

trajectory of p-type porphyrin molecules. One observes that for a given time-cost on the

horizontal axis, the curve for the MFML model is always below that for the single ﬁdelity

GPR curve. This implies that for a given time-cost, the MFML model results in a lower

error than the single ﬁdelity GPR model. Furthermore, the Γ(8) curve lies lower than

the conventional MFML curve. Once again, this implies that for a given time-cost, the

multiﬁdelity model built along the Γ(8)-curve results in a lower MAE. A similar observation

is made for the case of concatenated trajectories of the p-type and m-type molecules in

Figs. 9b and 9c, respectively. Although for the m-type porphyrin molecules, the GPR curve

does reach close to the conventional MFML curve, the Γ(8)-curve always lies beneath it. The

ﬁnal multiﬁdelity models that are used in this work for the prediction of excitation energies

correspond to the ﬁnal data point of the Γ(8) curve, which corresponds to γ= 12. The

multiﬁdelity training structure for this model is {8,12·8 = 216,122·8 = 1152,123·8 = 13824}

with decreasing ﬁdelity. For the p-molecules, this model results in an MAE of ∼25 meV

with a time cost of about 1500 hours, while the conventional MFML model reports a similar

error with a time cost of roughly 8000 hours. The single ﬁdelity GPR model only reaches

an MAE of 29 meV with a time-cost of 2000 hours. The use of the multiﬁdelity model along

the Γ(8) curve results in a time-cost beneﬁt of roughly 5 over the conventional MFML model

with γ= 2 and NSV P

train = 1024. For the p-type porphyrin molecules, the corresponding time-

beneﬁt of using the multiﬁdelity model along the Γ(8) curve over the conventional MFML

model is about the same with the former reporting an MAE of about 17 meV for a time-cost

of roughly 1500 hours, while the latter costs as much as 8000 hours for an MAE of about 19

meV.

Exciton dynamics

In this section, we investigate the exciton dynamics in the porphyrin-clay system by con-

structing a time-dependent Frenkel exciton Hamiltonian, employing the Numerical Integra-

tion of the Schrödinger Equation (NISE) method, and analyzing the exciton diﬀusion. Our

aim is to simulate exciton dynamics over 10ns with a 1 fs time step for nine copies of the 16

pigments, totaling in 144 pigments (see Fig. 4).

Building the Frenkel Exciton Hamiltonian

To model the exciton dynamics, we construct a time-dependent Frenkel exciton Hamiltonian

H(t) =

i=1

[Ei+ ∆Ei(t)] |i⟩⟨i|+X

i=j

Vij (t)|i⟩⟨j|,(2)

where Eidenotes the average site energies, ∆Ei(t)the site energy ﬂuctuations, Vij (t)the

time-dependent electronic couplings, and Nthe total number of pigments in the system.

The average site energies Eiand the site energy ﬂuctuations ∆Ei(t)are obtained from the

MFML model, which predicts the excited-state energies based on the atomic positions from

the QM/MM trajectories for each pigment, as described in earlier sections. The electronic

couplings Vij (t)are calculated using the TrESP method41,42 along a 100 ns MD trajectory

with a 10 ps time step, as detailed above. In previous studies, average couplings were of-

ten used to simplify the Hamiltonian.23,28,52,53 As discussed, however, in the Section on the

MD and QM/MM simulations, using average couplings may not be suitable here due to

signiﬁcant coupling ﬂuctuations as well as the canceling of positive and negative couplings.

Therefore, in addition to constructing the Hamiltonian with average couplings, we also con-

struct a Hamiltonian with time-dependent couplings Vij (t), where the couplings between two

snapshots are obtained by linear interpolation.

Our goal is to perform a 10 ns exciton dynamics simulation to investigate long-term exci-

ton transport. However, the QM/MM trajectories used to obtain the site energies and their

ﬂuctuations are only 40 ps long. Generating 10 ns of QM/MM simulations for every pigment

is computationally prohibitive. To overcome this limitation, we employ a noise generation

algorithm based on spectral densities54 to create longer site energy ﬂuctuation trajectories

∆Ei(t). This algorithm allows us to extend the site energy ﬂuctuations to the desired sim-

ulation length while preserving the statistical properties of the original data. As the noise

generation algorithm relies on the spectral densities of the site energy ﬂuctuations, we ﬁrst

compute the spectral densities for each pigment type. As a ﬁrst step, the autocorrelation

function C(t)of the site energy ﬂuctuations ∆Ei(t)is obtained from the MFML model along

the QM/MM trajectories55 using

C(tj) = 1

N−j

i=1

∆E(ti+tj)∆E(ti).(3)

To suppress noise in the autocorrelation function, a Gaussian damping with a timescale of

5 ps is applied. The spectral density J(ω)is then obtained via the Cosine transform of the

autocorrelation function

J(ω) = βω

πZ∞

C(t) cos(ωt)dt , (4)

where β= 1/(kBT)denotes the inverse temperature with kBbeing the Boltzmann constant

and Tthe temperature. Since all pigments of the same type are chemically identical, we

average the spectral densities over all pigments of the same type, i.e., m-type or p-type.

This averaging minimizes the contributions from speciﬁc conﬁgurations or local environ-

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

250

500

750

1000

1250

1500

1750

2000

J( ) [cm 1]

m average

p average

0.00

0.05

0.10

0.15

0.20

0.25

J( ) [eV]

(a) SD of m-type and p-type porphyrin

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

[eV]

250

500

750

1000

1250

1500

1750

2000

J( ) [cm 1]

FMO

p average

0.00

0.05

0.10

0.15

0.20

0.25

J( ) [eV]

(b) Experimental SD of FMO.

Figure 10: Panel (a) shows the average spectral densities of all m-type and p-type porphyrin

molecules, while panel (b) shows an experimental spectral density of FMO as comparison.18,56

ments present in the relatively short 40 ps QM/MM simulations, leading to a more rep-

resentative spectral density for each pigment type. The spectral densities for the m-type

and p-type porphyrin molecules are shown in Fig. 10, together with an experimental spec-

tral density of the Fenna–Matthews–Olson (FMO) complex containing bacteriochlorophyll

molecules (BChl).18,56 Notably, the porphyrins exhibit spectral features up to approximately

0.4eV, which is signiﬁcantly higher than the spectral features of BChl or chlorophyll (Chl)

in plants, typically extending only up to about 0.2eV. These higher frequency components

might originate from the outer rings of the porphyrin molecules, which are absent in BChl

and Chl, or the central nitrogen-hydrogen (N-H) bonds, which have the highest force constant

within porphyrin molecules.57

To generate the noise ∆Ei(t)following the spectral densities, we ﬁrst create white noise

η(t)with zero mean and unit variance. The target noise ∆Ei(t)is then obtained by mul-

tiplying the Fourier transform of the white noise ˜η(ω)with the square root of the power

spectrum ˜

C(ω) = J(ω)2π/(βω)followed by an inverse Fourier transform54

∆Ei(t) = ℜ(IFFT(˜η(ω)˜

C(ω))) .(5)

This process yields noise ∆Ei(t)that follows the desired spectral density and can be used

for the desired 10 ns simulations. Combining all components, the ﬁnal Hamiltonian ˆ

H(t)

is constructed with average site energies Eifrom the MFML model along the QM/MM

trajectories, time-dependent ﬂuctuations ∆Ei(t)generated from the noise algorithm, and

the time-dependent electronic couplings Vij (t)obtained from average or linearly interpolated

TrESP couplings along the MD simulation.

Numerical integration of Schrödinger equation

To enable exciton calculations over 10 ns for 144 pigments, i.e., 9 copies of 16 pigments,

a numerically eﬃcient algorithm is needed. To this end, we choose the Numerical integra-

tion of Schrödinger equation (NISE) as it is fast, easily parallelizable and works with the

time dependent Hamiltonian we just constructed.27,28,54 The NISE approach treats only the

system quantum mechanically and the coupling to the environment classically. This treat-

ment includes an implicit high temperature limit, so that detailed balance is not fulﬁlled,

i.e., the long term distribution is an even distribution rather than the expected Boltzmann

distribution.58 Modiﬁcations to the NISE that address this limitation exist,59,60 but have

shortcomings in systems with low couplings Vij 60 and high frequency contributions in the

noise,54 both of which are present in the porphyrin-clay system. Therefore, we have to keep

this limitation in mind when interpreting the results.

In the NISE method, we solve the time-dependent Schrödinger equation for the excitonic

wave function |ψs(t)⟩

i¯h∂

∂t |ψs(t)⟩=ˆ

H(t)|ψs(t)⟩,(6)

where ˆ

H(t)is the time-dependent Hamiltonian we constructed earlier. The excitonic state

can be expressed in terms of site basis states |m⟩

|ψs(t)⟩=X

cm(t)|m⟩,(7)

with time-dependent coeﬃcients cm(t). These coeﬃcients can be obtained by solving Eq. 6

0246810

time [ns]

0.0

0.2

0.4

0.6

0.8

1.0

Population

p10

p11

p12

(a) Average coupling values.

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time [ns]

0.0

0.2

0.4

0.6

0.8

1.0

Population

p10

p11

p12

(b) Time-dependent coupling values.

Figure 11: Population dynamics of an exciton initially placed on the respective pigment in

the central copy. The populations were calculated using an average over 100 realizations.

Panel (a) shows the simulation for average coupling values, and panel (b) for time-dependent

coupling values.

numerically by assuming H(t)is constant for small time steps ∆t, where we use a time step

of 1 fs. The population of an exciton on site mis then given by

Pm(t) = |cm(t)|2.(8)

Finally this has to be repeated for many realizations to get reasonable results. The software

code TorchNISE which is available on GitHub is used to run the calculations.61 The exciton

dynamics are simulated by initially exciting a single porphyrin molecule in the central copy

and observing the population decay over time as depicted in Fig. 11. Two versions are

shown for the Hamiltonian constructed based on the average and on the time-dependent

coupling values. The results indicate signiﬁcant diﬀerences in the decay of the exciton

population between the two cases. With average couplings, some p-type porphyrins decay

much slower than others. In particular, certain p-type pigments retain higher populations

over longer times, which is not observed with the time-dependent coupling values. We

suspect that averaging the couplings, which can be both positive and negative due to the

relative orientations of the transition dipole moments (see Fig. 5), leads to an underestimation

of the eﬀective coupling strengths as positive and negative values cancel out each other.

Moreover, the transfer from pigment m3 shows notably diﬀerent behavior between the two

cases. In simulations with average couplings, the exciton population on m3 decays much

faster compared to other m-type pigments. This rapid decay is also observed with time-

dependent couplings at short times but is followed by a slowdown in the decay rate. This

behavior can be explained by examining the coupling patterns in the system: pigment m3 is

strongly coupled to p2 during certain periods (as shown in Fig. 5), and p2 becomes strongly

coupled to p1 at other times. With average couplings, these strong interactions are always

present in the Hamiltonian, allowing the exciton to quickly transfer from m3 to p2, then

to p1, and further into the system, resulting in the faster decay of the exciton population

on m3. In contrast, time-dependent couplings capture these strong interactions only during

speciﬁc intervals. The exciton initially equilibrates quickly between m3 and p2 due to their

strong coupling at that time, but transfer to other pigments is slower because they are not

coupled strongly at the same time.

Despite the improved realism provided by time-dependent couplings, our simulations

show that the exciton lifetimes for the m-type porphyrins are much longer than experimen-

tal observations. In the experimental study by Ishida et al.,4the exciton lifetime of m-type

porphyrin was determined to be approximately 0.4 ns, whereas in our simulations, the life-

times are clearly much longer. There are several potential sources for this discrepancy.

Firstly, we have included only the Qxstate in our exciton dynamics simulations. Incor-

porating the Qystate might increase the transfer rates signiﬁcantly when weak couplings

are the result of misaligned transition dipoles. Whenever the transition dipole of the Qx

states of neighboring pigments is not aligned, the dipole of the Qystate would be very well

aligned with the Qxstate, thereby increasing overall transfer. Additionally, a higher time-

resolution of the couplings might impact the transfer rates. While ﬂuctuating couplings are

usually considered less important in exciton calculations in light-harvesting systems, they

clearly have an impact on the present system. It is possible that variations of the couplings

on faster timescales would aﬀect the lifetimes further. Furthermore, in the experiment, a

hexagonal packing of the porphyrins on the clay surface is assumed, with an inter-pigment

distance of approximately 2.6 nm,4whereas in the present simulations, the average distance

to the six nearest neighbors is about 2.9 nm. Since dipole-dipole coupling decreases with

the 6th power of the distance, we might underestimate the couplings about 50% due to the

larger distances, resulting in slower exciton transfer and longer lifetimes in the simulations.

Another possibility is the absence of thermalization eﬀects in the NISE as p-type porphyrin

molecules generally have lower energy levels.

Diﬀusion

To quantify the exciton transport, we analyze the diﬀusion of the exciton over time. The

diﬀusion is deﬁned as the mean squared displacement (MSD) of the exciton and can be

determined using the positions of the porphyrin molecules and the exciton populations

⟨d(t)2⟩=X

i|(xi(t)−x0(0))|2Pi(t),(9)

where Pi(t)denotes the exciton population on site iobtained from NISE, xi(t)the center-

of-mass position of porphyrin i, and x0(0) the initial center-of-mass position of the initially

excited porphyrin. As a comparison, we use a simple classical model. We place the porphyrin

molecules on a hexagonal grid at a distance of 2.6 nm, as reported in Ref. 4. The population

on any given porphyrin molecule will transfer to the six neighboring porphyrins with a

transfer rate tr. In Ref. 4, the transfer rate from m- to p-type porphyrins was reported to be

2.4 ns−1with one m-type porphyrin surrounded by 4.5 p-type porphyrins on average. Hence,

we estimate the pigment-to-pigment transfer rate as 2.4/4.5=0.53 ns−1. We further assume

this transfer rate holds for all types of transfers (m to p, p to m, p to p, m to m), because

no other transfer rates were determined. This is, of course, a signiﬁcant assumption, since

in the present calculations transfer from p-type porphyrins is much faster than from m-type.

The diﬀusion is then calculated from the resulting population using Eq. 9.

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time [ns]

5000

10000

15000

20000

Diffusion [Å2]

NISE average

NISE p-type

NISE m-type

classical hexagonal

(a) Average Coupling

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5000

10000

15000

20000

Diffusion [Å2]

NISE average

NISE p-type

NISE m-type

only molecule motion

classical hexagonal

(b) Time-dependent Coupling

Figure 12: Diﬀusion with average coupling (panel (a)) and time-dependent couplings and

positions (panel (b)) calculated from the same NISE simulation as Fig. 11. In addition, the

diﬀusion based on the classical hexagonal model is shown in both panels and, furthermore,

the diﬀusion based only on the molecular motion is shown in panel (b)

Fig. 12 displays the exciton diﬀusion averaged over the four NISE calculations with the

exciton placed on an m-type porphyrin in the central copy, the 12 calculations with the

exciton placed on a p-type porphyrin, and averaged over all 16 calculations. Additionally,

the classical diﬀusion based on the hexagonal grid is shown. Panel (a) shows the simulation

for average couplings, and panel (b) for time-dependent couplings. Panel (b) also includes the

diﬀusion of the porphyrin molecules themselves, i.e., ⟨di,mol(t)2⟩=|xi(t)−xi(0)|2), averaged

over all porphyrins.

The results show that time-dependent couplings signiﬁcantly enhance the exciton dif-

fusion rate compared to average couplings, which aligns with the generally faster decay of

initial populations in Fig. 11. This also brings the NISE diﬀusion results closer to the classical

diﬀusion. The movement of the pigments themselves contributes negligibly to the diﬀusion,

indicating that exciton transfer is the main driver of exciton diﬀusion. It can also be seen

that, diﬀusion is slower for m-type porphyrins compared to p-type porphyrins, which aligns

again with Fig. 11, where the exciton decay for m-type porphyrins is slower. The diﬀusion

0246810

time [ns]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Diffusion Constant [Å2/ps]

NISE average

NISE p-type

NISE m-type

classical hexagonal

(a) Long time (10ns)

0.0 0.2 0.4 0.6 0.8 1.0

time [ps]

Diffusion Constant [Å2/ps]

NISE no moving average

NISE average

NISE p-type

NISE m-type

classical hexagonal

(b) Short time (1ps)

Figure 13: Diﬀusion “constant” over time, shown as (a) a moving average over 100ps and (b)

compared to instantaneous value for short times. The NISE calculations are taken from the

simulation with the time-dependent couplings.

“constant” D(t)is deﬁned as the time derivative of the diﬀusion

D(t) = d

dt < d(t)2> . (10)

In the classical model, the diﬀusion constant remains constant. However, in the NISE-based

calculations, we observe that the diﬀusion “constant” ﬂuctuates on very short timescales,

comparable to the Rabi frequencies between porphyrin pairs which can be calculated to be on

the order of 30-150fs based on the average energy diﬀerence and coupling (see Fig. 13(b)).62

While these very fast ﬂuctuation are a result of the quantum description of diﬀusion, they

have little impact on the long term behavior. Hence, we compute a moving average of the

diﬀusion constant over 100 ps to avoid the ﬂuctuations dominating the graph and to enable

a better comparison with the classical diﬀusion (see Fig. 13(a)). The moving average reveals

that the diﬀusion constant is initially larger for p-type porphyrins but becomes similar to

that of m-type porphyrins at a timescale comparable to the lifetime of excitons placed on

the m-type porphyrins. This suggests that while p-type porphyrins facilitate faster exciton

transport initially, the diﬀusion rate become equal once the exciton population is distributed

between m- and p-type porphyrins. The initial quick drop in the diﬀusion constant can

be attributed to an initial equilibration of the exciton between strongly coupled nearest

neighbors, as discussed previously for m-3 in Fig. 11. Such strongly coupled nearest neighbors

are absent on an orderly hexagonal grid. The reduction of the diﬀusion constant over longer

times might also be related to some population reaching the edge of our 3×3 copy simulation.

Even small populations at the edges will have a large inﬂuence on diﬀusion as it grows with

the square of the distance.

A property often referred to in experimental papers is the diﬀusion length, which, for

classical diﬀusion, can be calculated from the diﬀusion constant Dand the lifetime of the

exciton τas63

L=√Dτ . (11)

This formula equation is usually used because Dis often calculated directly without calculat-

ing < d(t)2>ﬁrst and Dτ is equal to < d(τ)2>for classical diﬀusion. In our case, we have

calculated < d(t)2>directly. Hence, determining the diﬀusion length as L=p(< d(τ)2>)

is more sensible as it avoids complications with the ﬂuctuations in Dfor the quantum cases.

We chose the experimental lifetime of τ= 5.6ns of the combined porphyrin-clay complex

for all our calculations.4For the time-dependent coupling calculations, we obtain a diﬀu-

sion length of 7.3 nm for excitations starting on an m-type porphyrin molecules, 9.8 nm for

excitations starting on a p-type dye, and 12.3 nm for the classical hexagon-based diﬀusion.

These values are generally in line with experimental values of organic semiconductors, which

are typically between 5 and 20 nm, although longer diﬀusion lengths have also been ob-

served.64 For porphyrin-based systems, diﬀusion lengths between 7.5 and 40 nm have been

observed.65–68 Interestingly, most of these have higher diﬀusion constants, up to 900Å2/ps for

porphyrin-based metal–organic frameworks,68 but shorter diﬀusion times, which ultimately

lead to comparable diﬀusion lengths.

Since the NISE simulations underestimate the transfer from m- to p-type porphyrins,

we expect that our calculations also underestimate the diﬀusion length for the porphyrin-

clay complex. If that underestimation is of similar magnitude as for the population decay

of the m-type, the porphyrin-clay complex might have one of the largest diﬀusion lengths

among comparable systems. This would be an important ﬁnding because a large diﬀusion

lengths is an important property for organic solar cells. To investigate this further, one

could determine the other pigment-pigment transfer rates or directly measure the diﬀusion

length experimentally. Alternatively, the previously mentioned improvements to the exciton

dynamics (inclusion of the Qystate, better-resolved couplings, and closer aligned nearest-

neighbor distances) could align the simulations closer to reality and predict a better diﬀusion

length.

Conclusions and Outlook

Inspired by the energy transfer networks present in the light-harvesting complexes of plants,

bacteria, and algae, a signiﬁcant amount of work has been done over the past years to repli-

cate these natural systems artiﬁcially. Experimental evidence4has shown that porphyrin

molecules attached to inorganic clay surfaces can achieve remarkable energy transfer eﬃ-

ciencies close to 100%. In this study, we tried to emulate the experimental setup in Ref.

4, i.e., cationic free-base porphyrin molecules adsorbed on an anionic inorganic clay sur-

face. The surface model was created using the CHARMM-GUI webserver. Two types of

porphyrins, m-TMPyP and p-TMPyP, were selected due to their positive charge, in a ration

of 1:3 and with a total of 16 porphyrin molecules.

Classical MD simulations were conducted to equilibrate the system, allowing each por-

phyrin molecule to ﬁnd its optimal binding location on the clay surface. The equilibrated

structures were subsequently used as starting structures for QM/MM MD simulations, fol-

lowed by excitation energy calculations using the TC-LC-DFTB method. This workﬂow

established the computational protocol for investigating energy transfer processes among

the pigment molecules. In addition, a novel multiﬁdelity machine learning-based approach,

MFML, was used to compute excitation energies at the computationally demanding TD-

DFT/Def2-SVP level for all porphyrin molecules, each consisting of 90 atoms, across 640,000

geometries. The extensive sampling of high-level quantum excitation energies was then used

to construct a time-dependent exciton Hamiltonian, enabling simulations of exciton dynam-

ics and diﬀusion using the NISE approach in a periodic system. The results demonstrate

the feasibility of simulating energy transfer in artiﬁcial systems by utilizing porphyrin-clay

hybrid materials. The high eﬃciency observed in the energy transfer process underscores the

potential for porphyrin-based materials in designing light-harvesting devices. The compu-

tational protocol in the present study eﬀectively bridges classical and quantum simulations,

enabling an in-depth exploration of energy dynamics at the molecular level. The successful

implementation of the MFML technique for large-scale quantum calculations also highlights

the potential for machine learning to signiﬁcantly accelerate complex computational tasks

in quantum chemistry.

As an outlook, the computational framework developed in this study can be extended

to design more sophisticated artiﬁcial light-harvesting networks using various dye molecules,

paving the way for further innovations in solar energy harvesting. By providing a detailed

theoretical understanding of the mechanisms underlying eﬃcient energy transfer, this work

contributes to the development of sustainable energy solutions. Furthermore, the results

serve as a valuable benchmark for experimental investigations, oﬀering a basis for future

experimental studies to validate and reﬁne the proposed models. Subsequent studies might

investigate more intricate combinations of clay and dye, varying dye proportions, and ex-

ternal inﬂuences like controlling the temperature to enhance the computational design of

synthetic light-harvesting systems to adhere to the “size-matching” principle.

Methods

Simulation and Calculation Details

Here we give some more details for modelling the montmorillonite nanosheet. The aluminum

atoms in the octahedral layer of the system, i.e., the central layer in Fig. 1b, are partially

replaced by magnesium, which leads to a charge diﬀerence and forms a negatively charged

surface. The parameter x, which can vary between 0 and 0.95, indicates the number of

magnesium defects in the system. The cations in the formula imply that the system is in

a water environment with dissolved cations to achieve neutrality. It has to be noted that

in the experimental setup with the saponite system, the silicon atoms in the tetrahedral

layer, which is shown in yellow in Fig. 1b, were replaced by aluminum. Although the layer

of atoms exchanged in our simulations were diﬀerent from the experiment, the distances

between replacement sites were consistent, so the “size-matching rule” still applies. The clay

surface with a size of 10.38 nm ×10.82 nm ×1 nm in vacuum was generated using the

online tool CHARMM-GUI69,70 where the ratio of defects and ion options can be customized

and modiﬁed. Four diﬀerent values of x(0.13, 0.2, 0.45 and 0.94) were selected to generate

the topology and perform individual MD simulations. The size of the initial simulation box

was chosen to ﬁt the material size, with an height of 4 nm. Periodic boundary conditions

(PBC) were ﬁrst only applied for xand yaxis. The structures of the porphyrin molecules

were obtained from ChemSpider (CSID: 133612 and 4086), and the ACPYPE (AnteChamber

PYthon Parser interfacE) tool71 was then employed to generate the topology and coordi-

nates in GROMACS format. After randomly placing four m-TMPyP and twelve p-TMPyP

molecules on top of the clay surface, a short preliminary simulation was performed in vac-

uum for a maximum of 5 ns, allowing the porphyrins to be attracted to the clay surface

solely through electrostatic interactions. The number of the porphyrins was chosen as it was

proved to be one of the ratios with the highest energy transfer eﬃciency.4The last frame of

the simulation was extracted and then used as the initial conﬁguration for the further MD

simulations.

Due to technical reasons, 3D-PBC was required in the subsequent MD simulations. The

height of the simulation box was extended to 25 nm, which is about 2.5 times the width, to

minimize the eﬀect of the mirror structure on the adsorption behavior of porphyrins. After-

ward, the system was solvated using TIP3P water molecules, and neutralized with sodium

and chloride ions, followed by energy minimization, a 5 ns NVT equilibration at 300 K and

a 5 ns NPT equilibration. Thereafter, the classical MD simulations for 30 ns were performed

with a time step of 1 fs employing GROMACS 5.1.472 using the general AMBER force ﬁeld

(GAFF)73 and the interface force ﬁeld (IFF)74 for porphyrins and clay surface, respectively.

The Nose-Hoover thermostat75 and Berendsen barostat76 were employed to control the tem-

perature and pressure, respectively. The cutoﬀ for short-range non-bonded interaction was

set to 1.2 nm and the long-range electrostatics was treated using the Particle Mesh Ewald

(PME) method.77 In addition, the LINCS algorithm was used for bond constrains.78

The montmorillonite surface with a xvalue of 0.45 was chosen for the subsequent quantum

mechanics/molecular mechanics dynamics (QM/MM) since all sixteen porphyrins were still

adsorbed on the surface after the 30 ns MD simulation, while some porphyrins detached in

the simulations using the other xvalues. Each of the porphyrin molecules was assigned to

the QM region and treated separately using the DFTB3 approach with the 3OB-f parameter

set.79 Classical force ﬁelds were employed for the remainder of the system. A 50 ps NPT

equilibration was performed in GROMACS including the DFTB+interface,26 followed by

another 40 ps QM/MM MD simulation with a smaller integrator time step, which was set to

0.5 fs. The atomic coordinates were stored every two steps with 40,000 frames, which were

subjected to excited state calculations employing TD-LC-DFTB39 and TD-DFT with the

CAM-B3LYP functional, as implemented in DFTB+21.124 and the ORCA 5.0.3 package,80

respectively. Four diﬀerent basis sets, STO-3G, 3-21G, 6-31G and def2-SVP, were used for

the TD-DFT calculations according to their quantum chemical hierarchy, with a time stride

of 8, 16, 32 and 64 fs, respectively.

Furthermore, another 100 ns classical MD simulation was performed for 10,000 snap-

shots after the equilibration, in order to evaluate the excitonic couplings between the por-

phyrin molecules the transition charges from electrostatic potentials (TrESP) method was

applied.41,42

Gaussian Process Regression

In GPR,81 a machine learning model is found for a training set T={(xi, yi)}N

i=1, with train-

ing inputs xiand corresponding output measurements yi. Then, starting from a Gaussian

process prior GP(0, k(x,x′)), with covariance (kernel) function k(x,x′), a posterior predic-

tive distribution is found by conditioning on the training set T. The posterior distribution

has a the mean

m(x) = K(x,X)K(X,X) + σ2

nI−1y(12)

that gives predictions for unknown inputs x, i.e. it is the constructed ML model. By X, we

denote the set of training inputs and yis a vector with the corresponding outputs. In the

model, k(x,X)denotes the vector of pair-wise kernel evaluations between the query input

xand the set of training inputs X, whereas K(X,X)is the matrix of all pair-wise kernel

evaluations between all training inputs X. In addition, σ2

nis a hyperparameter that models

the (unknown) variance of the noise that is assumed on the outputs of the training data.

For our tests we chose a Gaussian covariance function

k(x,x′) = σ2

fexp −1

2(x−x′)TM(x−x′).(13)

Here, M=diag 1

1, ..., 1

Dis a diagonal matrix, where Dis the number of features. This

results in a diﬀerent length-scale for every feature, known as automatic relevance determi-

nation.81 Moreover, σ2

fis a hyperparameter, the output-scale. All hyperparamters are found

by ﬁnding by maximising the marginal log-likelihood, with details in the supplementary

material. We used the GPR implementation of GPyTorch.82

In this work, the inputs xare Coulomb matrices,83 representing the molecular conﬁgura-

tions, and the outputs yare the corresponding excitation energies. For Coulomb matrices,

to ensure invariance against atom permutation, often a sorting is carried out, which results

in discontinuities and makes them unfavorable for training ML models.84 In our tests, all we

have ﬁxed atoms given in a ﬁxed order. Therefore, we can safely ignore invariance against

atom permutation and hence use unsorted Coulomb matrices.

Active learning

Active learning (AL) algorithms try to choose or create new training samples, which are

maximally informative to an ML model. In our setup, we have a large amount of unlabeled

molecular conﬁgurations and generating the respective labels (excitation energies) via quan-

tum chemical calculations is expensive. For this setting, uncertainty sampling38 can be a

well suited AL scheme: First, for a small random subset of the data, excitation energies are

calculated. Then, this data is used to train an initial model. In the iterative process to add

favorable new training samples, the standard deviation of the predictive distribution of the

GPR model

σ(x) = qk(x,x)−K(x,X)K(X,X) + σ2

nI−1K(X,x)(14)

is used as uncertainty measure. Those samples are added that have the highest uncertainty.

Once a samples is selected from the remaining pool of unlabeled samples, its label, i.e. the

excitation energy for the given molecule, is calculated. The input-output pair is then added

as additional training sample. The full procedure is summarized in Algorithm 1.

Algorithm 1 Active Learning by Uncertainty Sampling

Require: Set of unlabeled inputs U, initial sample size ninit, number of iterations niter

1: Randomly select ninit samples from Uand store them in set L

2: Obtain labels y(L)for all inputs in L

3: U=U\L

4: Train initial model m, following eq. (12), using labeled data (L, y(L))

5: for i= 1 to niter do

6: Compute uncertainty σ(x), from eq. (14), for each sample in Uusing model m

7: Select sample x∗from Uwith highest uncertainty

8: Obtain label y∗for selected sample x∗

9: L=L∪ {x∗}

10: U=U\{x∗}

11: Retrain model m, following eq. (12), using updated labeled dataset (L, y(L))

12: end for

13: return Trained model m

Multiﬁdelity Machine Learning Approach

Multiﬁdelity machine learning (MFML) systematically combines ML sub-models trained on

multiple ﬁdelities, f, to produce a low-cost high accuracy ML model interested in predicting

the excitation energies for a target ﬁdelity F.32,50 A composite index, s= (f, ηf), is used to

identify the sub-models of MFML. In this index, we have 2ηf=N(f)

train. The MFML model is

built with a target ﬁdelity, F, for a given baseline ﬁdelity fb, which refers to the lowest QC

ﬁdelity used in the model. One can write the MFML model for a query molecular descriptor,

Xqas

P(F,ηF;fb)

MFML (Xq) := X

s∈S(F ,ηF;fb)

βsP(s)

GPR (Xq),(15)

where, the summation is made for the set of selected sub-models of MFML, S(F,ηF;fb). This

selection is decided by the choice of fb, and N(F)

train = 2ηF, the number of training samples

at the target ﬁdelity, as presented in Ref. 50. The βsfrom Eq. (15) are referred to as the

coeﬃcients of the linear combination of these selected sub-models. For MFML, the βsare

chosen to be

βMFML

s=









+1,if f+ηf=F+ηF

−1,otherwise

,(16)

based on previous work from Ref. 32.

In the MFML method, the number of training samples at the subsequent ﬁdelities are

set by a scaling factor,γ. That is, Nf

train =γ·Nf−1

train for fb< f ≤F. The value of γ= 2

is conventionally used in MFML based on previous work.32,49–51 In a recent work, the eﬀect

of diﬀerent values of γin the model error of MFML has been studied.33 Ref. 33 reports

that the use of very little training data at the target ﬁdelity combined with increasing values

of γ, results in a more data eﬃcient model. These models are studied using the model

error and time-cost of generating the training data required for the model. This speciﬁc

curve is referred to as the Γ(NF

train)curve with each data point essentially being the MFML

model built for a speciﬁc value of γ. Herein, the Γ(8) curve is built for γ∈ {2,...,12}.

The supplementary information associated with this work also reports Γ(2),Γ(4), and Γ(16)

curves in Figure S9. The diﬀerence between the Γ(8) and Γ(16) curves were not seen to

be signiﬁcantly diﬀerent. The ﬁnal model used in this work in the prediction of def2-SVP

energies is the MFML model along the Γ(8) curve for γ= 12 as discussed in Figure 9.

Model Evaluation

The diﬀerent ML models that are built in this work, both single ﬁdelity and MFML, are

evaluated using mean absolute error (MAE) over holdout test sets. Given a test set at target

ﬁdelity, F, denoted as QF:= {(Xq, yref

q)}Neval

q=1 , the MAE for an ML model is calculated as

MAE := 1

Neval

q=1 

yML

q−yref

q

1.(17)

The model error is initially reported as a function of number of training samples used at

the target ﬁdelity in the form of learning curves85–87 for both single ﬁdelity and MFML

models since these are a common metric of model evaluation for the ML for QC quantities

workﬂow.88–90

Since the main aim of multiﬁdelity models is to reduce the time-cost of generating training

data, this work also presents a recently introduced analysis of MAE versus time-cost of

generating training data for a certain ML model.49 For single ﬁdelity models, this is simply

the QC calculation cost of the number of training samples used for the ML model. For MFML

models, this cost takes into account the QC computation cost of not just the training samples

used at the target ﬁdelity, but also the cost of training samples used in the entire multiﬁdelity

data structure.

Acknowledgement

The authors are grateful to the developers of the webtool CHARMM-GUI and in particular

Prof. Wonpil Im for assistance in modelling and modifying the montmorillonite nanosheet.

Furthermore, the authors acknowledge support by the DFG through the Priority Program

SPP 2363 on “Utilization and Development of Machine Learning for Molecular Applications

– Molecular Machine Learning” through the project ZA 1175/4-1 and KL 1299/25-1 and

further funding through projects ZA 1175/3-1 and KL 1299/24-1. VV and PZ would also like

to acknowledge the support of the ‘Interdisciplinary Center for Machine Learning and Data

Analytics (IZMD)’ at the University of Wuppertal. Furthermore, part of the simulations

were performed on a compute cluster funded through the DFG project INST 676/7-1 FUGG,

while part of the machine learning training was carried out on the PLEIADES cluster at the

University of Wuppertal, which is supported by the Deutsche Forschungsgemeinschaft (DFG,

grant No. INST 218/78-1 FUGG) and the Bundesministerium für Bildung und Forschung

(BMBF).

Supporting Information Available

Additional ﬁgures, tables, and explanations on excitation energies, transition charges, spec-

tral densities and the MFML approach.

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Modeling Realistic Clay Systems with ClayCode

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Clays are a broad class of ubiquitous layered materials. Their specific chemophysical properties are intimately connected to their molecular structure, featuring repeating patterns broken by substitutions. Molecular dynamics simulations can provide insight into the mechanisms leading to the emergent properties of these layered materials; however, up to now, idealized clay structures have been simulated to make the modeling process tractable. We present ClayCode, a software facilitating the modeling of clay systems closely resembling experimentally determined structures. By comparing a realistic model to a commonly used montmorillonite clay model, we demonstrate that idealized models feature noticeably different ionic adsorption patterns. We then present an application of ClayCode to the study the competitive barium and sodium adsorption on Wyoming montmorillonite, Georgia kaolinite, and Montana Illite, of interest in the context of nuclear waste disposal.

Spectral densities, structured noise and ensemble averaging within open quantum dynamics

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Although recent advances in simulating open quantum systems have led to significant progress, the applicability of numerically exact methods is still restricted to rather small systems. Hence, more approximate methods remain relevant due to their computational efficiency, enabling simulations of larger systems over extended timescales. In this study, we present advances for one such method, namely, the numerical integration of Schrödinger equation (NISE). First, we introduce a modified ensemble-averaging procedure that improves the long-time behavior of the thermalized variant of the NISE scheme, termed thermalized NISE. Second, we demonstrate how to use the NISE in conjunction with (highly) structured spectral densities by utilizing a noise generating algorithm for arbitrary structured noise. This algorithm also serves as a tool for establishing best practices in determining spectral densities from excited state calculations along molecular dynamics or quantum mechanics/molecular mechanics trajectories. Finally, we assess the ability of the NISE approach to calculate absorption spectra and demonstrate the utility of the proposed modifications by determining population dynamics.

Assessing non-nested configurations of multifidelity machine learning for quantum-chemical properties

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Multifidelity machine learning (MFML) for quantum chemical properties has seen strong development in the recent years. The method has been shown to reduce the cost of generating training data for high-accuracy low-cost ML models. In such a set-up, the ML models are trained on molecular geometries and some property of interest computed at various computational chemistry accuracies, or fidelities. These are then combined in training the MFML models. In some multifidelity models, the training data is required to be nested, that is the same molecular geometries are included to calculate the property across all the fidelities. In these multifidelity models, the requirement of a nested configuration restricts the kind of sampling that can be performed while selection training samples at different fidelities. This work assesses the use of non-nested training data for two of these multifidelity methods, namely MFML and optimized MFML (o-MFML). The assessment is carried out for the prediction of ground state energies and first vertical excitation energies of a diverse collection of molecules of the CheMFi dataset. Results indicate that the MFML method still requires a nested structure of training data across the fidelities. However, the o-MFML method shows promising results for non-nested multifidelity training data with model errors comparable to the nested configurations.

The Unique Structure–Activity Relationship of Porphyrins and Clay Mineral Systems in Modern Applications: A Comprehensive Review

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Porphyrins are a group of organic heterocyclic macromolecules, composed of four modified pyrrole subunits interconnected at their α carbon atoms via methine bridges. This unique and significant group of compounds has a special relationship with inorganic clay minerals, which are hydrous aluminium phyllosilicates. Clay minerals, known to be typical layered materials, act as unique hosts for porphyrin molecules or their precursor materials. When the porphyrin compound enters between the layers of a clay, it can orient horizontally, diagonally, or perpendicularly. Cation-exchanged clay can provide the acidity required for the formation of porphyrins from its precursor molecules without the need for other auxiliary factors such as heating. Clays are also capable of facilitating the incorporation of various metals into the porphyrin ring to form a metalloporphyrin. The properties of these metalloporphyrin clay complexes have been studied extensively. Modern applications of such clay-porphyrin materials include their use as heterogeneous catalysts for oxidation and polymerization reactions in the synthesis of new materials. These heterogeneous catalysts have a wide range of uses in vital sectors, such as food manufacturing, pharmaceuticals, and the chemical industry. Additionally, clay-porphyrin systems are employed in constructing various types of self-assembling artificial photosynthesis systems.

Protein Effects on the Excitation Energies and Exciton Dynamics of the CP24 Antenna Complex

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May 2024

In this study, the site energy fluctuations, energy transfer dynamics, and some spectroscopic properties of the minor light-harvesting complex CP24 in a membrane environment were determined. For this purpose, a 3 μs-long classical molecular dynamics simulation was performed for the CP24 complex. Furthermore, using the density functional tight binding/molecular mechanics molecular dynamics (DFTB/MM MD) approach, we performed excited state calculations for the chlorophyll a and chlorophyll b molecules in the complex starting from five different positions of the MD trajectory. During the extended simulations, we observed variations in the site energies of the different sets as a result of the fluctuating protein environment. In particular, a water coordination to Chl-b 608 occurred only after about 1 μs in the simulations, demonstrating dynamic changes in the environment of this pigment. From the classical and the DFTB/MM MD simulations, spectral densities and the (time-dependent) Hamiltonian of the complex were determined. Based on these results, three independent strongly coupled chlorophyll clusters were revealed within the complex. In addition, absorption and fluorescence spectra were determined together with the exciton relaxation dynamics, which reasonably well agrees with experimental time scales. ■ INTRODUCTION Photosynthesis is key to sustaining life on Earth by turning sunlight into chemical energy. The photosynthetic machinery of plants and other photosynthetic organisms orchestrates this energy conversion through a series of intricate processes aided by pigment−protein complexes (PPCs). 1 Within these PPCs, pigment molecules are anchored to a protein matrix in such a way as to form an energy funnel enabling efficient energy transfer to the reaction center where charge separation takes place. 2−5 The pigments with high excitation energies pass the excitations to those with lower excitation energies in a cascaded fashion. The differences in excitation energy of each pigment molecule in the light-harvesting complexes (LHCs) facilitating the energy ladder arise due to different pigment types and the anisotropy in the protein environment. 6,7 Understanding the energy map within PPCs is crucial to comprehend the excitation energy transfer (EET) and associated properties within these complexes. An enhanced understanding of these mechanisms can potentially accelerate the development of efficient artificial light-harvesting systems, bringing sustainable energy closer to reality. Beyond light harvesting and EET, some of the PPCs, e.g., in higher plants, engage in photoprotection activities, shielding the complexes and the pigments therein from excessively high light intensities. 8 In Photosystem II (PSII) of higher plants, the antenna complexes belonging to the Lhc family undergo conformational changes in response to external stimuli such as a high light intensity inducing the transition between the light harvesting and photoprotection states. 9,10 The entirety of the antenna complexes consist of the major light-harvesting complex, i.e., the trimeric LHCII, representing the genes Lhcb1 to Lhcb3 and the minor antenna complexes CP29, CP26, and CP24 representing the genes Lhcb4, Lhbc5, and Lhbc6, respectively. 11,12 The minor antenna complexes are located between the outer LHCII trimers and the PSII core complex. In this study, we focus on the minor antenna complex CP24 of PSII, which functions in both modes, the light-harvesting and the photoprotection mode, similar to the other complexes of the same family. A network of six chlorophyll a (Chl-a) and five chlorophyll b (Chl-b) molecules is mainly responsible for the functional properties of the CP24 complex. Sunlight initiates the excitation of the chlorophyll molecules usually to their lowest excited state, commonly referred to as the Q y state, followed by a movement of the excited state energy within the system and finally toward a reaction center. 13 In addition to the chlorophyll molecules, a β-carotenoid

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Mar 2024

Machine learning (ML) provides access to fast and accurate quantum chemistry (QC) calculations for various properties of interest such as excitation energies. It is often the case that high accuracy in prediction using a ML model, demands a large and costly training set. Various solutions and procedures have been presented to reduce this cost. These include methods such as

\Delta

-ML, hierarchical-ML, and multifidelity machine learning (MFML). MFML combines various

\Delta

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Electrostatic coupling and water bridging in adsorption hierarchy of biomolecules at water-clay interfaces

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Feb 2024
P NATL ACAD SCI USA

Clay minerals are implicated in the retention of biomolecules within organic matter in many soil environments. Spectroscopic studies have proposed several mechanisms for biomolecule adsorption on clays. Here, we employ molecular dynamics simulations to investigate these mechanisms in hydrated adsorbate conformations of montmorillonite, a smectite-type clay, with ten biomolecules of varying chemistry and structure, including sugars related to cellulose and hemicellulose, lignin-related phenolic acid, and amino acids with different functional groups. Our molecular modeling captures biomolecule–clay and biomolecule–biomolecule interactions that dictate selectivity and competition in adsorption retention and interlayer nanopore trapping, which we determine experimentally by NMR and X-ray diffraction, respectively. Specific adsorbate structures are important in facilitating the electrostatic attraction and Van der Waals energies underlying the hierarchy in biomolecule adsorption. Stabilized by a network of direct and water-bridged hydrogen bonds, favorable electrostatic interactions drive this hierarchy whereby amino acids with positively charged side chains are preferentially adsorbed on the negatively charged clay surface compared to the sugars and carboxylate-rich aromatics and amino acids. With divalent metal cations, our model adsorbate conformations illustrate hydrated metal cation bridging of carboxylate-containing biomolecules to the clay surface, thus explaining divalent cation-promoted adsorption from our experimental data. Adsorption experiments with a mixture of biomolecules reveal selective inhibition in biomolecule adsorption, which our molecular modeling attributes to electrostatic biomolecule–biomolecule pairing that is more energetically favorable than the biomolecule–clay complex. In sum, our findings highlight chemical and structural features that can inform hypotheses for predicting biomolecule adsorption at water–clay interfaces.

Investigating Data Hierarchies in Multifidelity Machine Learning for Excitation Energies

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