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Adsorption Energies in MOFs

Pre-trained ODAC models are versatile across various MOF-related tasks. To begin, we’ll start with a fundamental application: calculating the adsorption energy for a single CO2 molecule. This serves as an excellent and simple demonstration of what you can achieve with these datasets and models.

For predicting the adsorption energy of a single CO2 molecule within a MOF structure, the adsorption energy (EadsE_{\mathrm{ads}}) is defined as:

Eads=EMOF+CO2EMOFECO2(1)E_{\mathrm{ads}} = E_{\mathrm{MOF+CO2}} - E_{\mathrm{MOF}} - E_{\mathrm{CO2}} \tag{1}

Each term on the right-hand side represents the energy of the relaxed state of the indicated chemical system. For a comprehensive understanding of our methodology for computing these adsorption energies, please refer to our paper.

Loading Pre-trained Models

A pre-trained model can be loaded using FAIRChemCalculator. In this example, we’ll employ UMA to determine the CO2 adsorption energies.

from fairchem.core import FAIRChemCalculator, pretrained_mlip

predictor = pretrained_mlip.get_predict_unit("uma-s-1p1")
calc = FAIRChemCalculator(predictor, task_name="odac")
WARNING:root:device was not explicitly set, using device='cuda'.

Adsorption in rigid MOFs: CO2 Adsorption Energy in Mg-MOF-74

Let’s apply our knowledge to Mg-MOF-74, a widely studied MOF known for its excellent CO2 adsorption properties. Its structure comprises magnesium atomic complexes connected by a carboxylated and oxidized benzene ring, serving as an organic linker. Previous studies consistently report the CO2 adsorption energy for Mg-MOF-74 to be around -0.40 eV [1] [2] [3].

Our goal is to verify if we can achieve a similar value by performing a simple single-point calculation using UMA. In the ODAC23 dataset, all MOF structures are identified by their CSD (Cambridge Structural Database) code. For Mg-MOF-74, this code is OPAGIX. We’ve extracted a specific OPAGIX+CO2 configuration from the dataset, which exhibits the lowest adsorption energy among its counterparts.

import matplotlib.pyplot as plt
from ase.io import read
from ase.visualize.plot import plot_atoms

mof_co2 = read("structures/OPAGIX_w_CO2.cif")
mof = read("structures/OPAGIX.cif")
co2 = read("structures/co2.xyz")

fig, ax = plt.subplots(figsize=(5, 4.5), dpi=250)
plot_atoms(mof_co2, ax)
ax.set_axis_off()
<Figure size 1250x1125 with 1 Axes>

The final step in calculating the adsorption energy involves connecting the FAIRChemCalculator to each relaxed structure: OPAGIX+CO2, OPAGIX, and CO2. The structures used here are already relaxed from ODAC23. For simplicity, we assume here that further relaxations can be neglected. We will show how to go beyond this assumption in the next section.

mof_co2.calc = calc
mof.calc = calc
co2.calc = calc

E_ads = (
    mof_co2.get_potential_energy()
    - mof.get_potential_energy()
    - co2.get_potential_energy()
)

print(f"Adsorption energy of CO2 in Mg-MOF-74: {E_ads:.3f} eV")
Adsorption energy of CO2 in Mg-MOF-74: -0.459 eV

Adsorption in flexible MOFs

The adsorption energy calculation method outlined above is typically performed with rigid MOFs for simplicity. Both experimental and modeling literature have shown, however, that MOF flexibility can be important in accurately capturing the underlying chemistry of adsorption [1] [2] [3]. In particular, uptake can be improved by treating MOFs as flexible. Two types of MOF flexibility can be considered: intrinsic flexibility and deformation induced by guest molecules. In the Open DAC Project, we consider the latter MOF deformation by allowing the atomic positions of the MOF to relax during geometry optimization [4]. The addition of additional degrees of freedoms can complicate the computation of the adsorption energy and necessitates an extra step in the calculation procedure.

The figure below shows water adsorption in the MOF with CSD code WOBHEB with added defects (WOBHEB_0.11_0) from a DFT simulation. A typical adsorption energy calculation would only seek to capture the effects shaded in purple, which include both chemisorption and non-bonded interactions between the host and guest molecule. When allowing the MOF to relax, however, the adsorption energy also includes the energetic effect of the MOF deformation highlighted in green.

To account for this deformation, it is vital to use the most energetically favorable MOF geometry for the empty MOF term in Eqn. 1. Including MOF atomic coordinates as degrees of freedom can result in three possible outcomes:

  1. The MOF does not deform, so the energies of the relaxed empty MOF and the MOF in the adsorbed state are the same

  2. The MOF deforms to a less energetically favorable geometry than its ground state

  3. The MOF locates a new energetically favorable geoemtry relative to the empty MOF relaxation

The first outcome requires no additional computation because the MOF rigidity assumption is valid. The second outcome represents physical and reversible deformation where the MOF returns to its empty ground state upon removal of the guest molecule. The third outcome is often the result of the guest molecule breaking local symmetry. We also found cases in ODAC in which both outcomes 2 and 3 occur within the same MOF.

To ensure the most energetically favorable empty MOF geometry is found, an addition empty MOF relaxation should be performed after MOF + adsorbate relaxation. The guest molecule should be removed, and the MOF should be relaxed starting from its geometry in the adsorbed state. If all deformation is reversible, the MOF will return to its original empty geometry. Otherwise, the lowest energy (most favorable) MOF geometry should be taken as the reference energy, EMOFE_{\mathrm{MOF}}, in Eqn. 1.

H2O Adsorption Energy in Flexible WOBHEB with UMA

The first part of this tutorial demonstrates how to perform a single point adsorption energy calculation using UMA. To treat MOFs as flexible, we perform all calculations on geometries determined by geometry optimization. The following example corresponds to the figure shown above (H2O adsorption in WOBHEB_0.11_0).

In this tutorial, Ex(ry)E_{x}(r_{y}) corresponds to the energy of xx determined from geometry optimization of yy.

First, we obtain the energy of the empty MOF from relaxation of only the MOF: EMOF(rMOF)E_{\mathrm{MOF}}(r_{\mathrm{MOF}})

import ase.io
from ase.optimize import BFGS

mof = ase.io.read("structures/WOBHEB_0.11.cif")
mof.calc = calc
relax = BFGS(mof)
relax.run(fmax=0.05)
E_mof_empty = mof.get_potential_energy()
print(f"Energy of empty MOF: {E_mof_empty:.3f} eV")
      Step     Time          Energy          fmax
BFGS:    0 19:24:56    -1077.274062        0.206406
BFGS:    1 19:24:57    -1077.276780        0.152729
BFGS:    2 19:24:58    -1077.281941        0.169932
BFGS:    3 19:25:00    -1077.284772        0.155752
BFGS:    4 19:25:00    -1077.288839        0.108767
BFGS:    5 19:25:01    -1077.291021        0.086433
BFGS:    6 19:25:02    -1077.293361        0.093433
BFGS:    7 19:25:03    -1077.295409        0.100141
BFGS:    8 19:25:04    -1077.297831        0.102538
BFGS:    9 19:25:04    -1077.300015        0.091599
BFGS:   10 19:25:05    -1077.302008        0.079035
BFGS:   11 19:25:05    -1077.304139        0.105595
BFGS:   12 19:25:06    -1077.306720        0.087897
BFGS:   13 19:25:07    -1077.309520        0.086343
BFGS:   14 19:25:08    -1077.312260        0.086824
BFGS:   15 19:25:09    -1077.314703        0.106300
BFGS:   16 19:25:10    -1077.316991        0.106201
BFGS:   17 19:25:12    -1077.319485        0.085516
BFGS:   18 19:25:13    -1077.322264        0.109632
BFGS:   19 19:25:13    -1077.325134        0.148691
BFGS:   20 19:25:16    -1077.327765        0.125945
BFGS:   21 19:25:17    -1077.329920        0.069098
BFGS:   22 19:25:17    -1077.331955        0.087284
BFGS:   23 19:25:18    -1077.334273        0.125221
BFGS:   24 19:25:18    -1077.336833        0.166687
BFGS:   25 19:25:20    -1077.339544        0.145559
BFGS:   26 19:25:20    -1077.342149        0.087648
BFGS:   27 19:25:21    -1077.344541        0.076175
BFGS:   28 19:25:24    -1077.346890        0.148943
BFGS:   29 19:25:26    -1077.349783        0.170213
BFGS:   30 19:25:29    -1077.352529        0.109285
BFGS:   31 19:25:31    -1077.354746        0.070348
BFGS:   32 19:25:32    -1077.356776        0.089669
BFGS:   33 19:25:33    -1077.358658        0.124289
BFGS:   34 19:25:34    -1077.360604        0.108035
BFGS:   35 19:25:35    -1077.362469        0.068621
BFGS:   36 19:25:35    -1077.364167        0.070223
BFGS:   37 19:25:37    -1077.365715        0.105371
BFGS:   38 19:25:37    -1077.367258        0.104407
BFGS:   39 19:25:38    -1077.368765        0.062612
BFGS:   40 19:25:39    -1077.370120        0.057926
BFGS:   41 19:25:40    -1077.371336        0.064509
BFGS:   42 19:25:41    -1077.372380        0.067290
BFGS:   43 19:25:41    -1077.373369        0.057369
BFGS:   44 19:25:42    -1077.374404        0.042166
Energy of empty MOF: -1077.374 eV

Next, we add the H2O guest molecule and relax the MOF + adsorbate to obtain EMOF+H2O(rMOF+H2O)E_{\mathrm{MOF+H2O}}(r_{\mathrm{MOF+H2O}}).

mof_h2o = ase.io.read("structures/WOBHEB_H2O.cif")
mof_h2o.calc = calc
relax = BFGS(mof_h2o)
relax.run(fmax=0.05)
E_combo = mof_h2o.get_potential_energy()
print(f"Energy of MOF + H2O: {E_combo:.3f} eV")
      Step     Time          Energy          fmax
BFGS:    0 19:25:43    -1091.565588        1.145036
BFGS:    1 19:25:46    -1091.585063        0.314149
BFGS:    2 19:25:47    -1091.590210        0.243429
BFGS:    3 19:25:48    -1091.608170        0.237244
BFGS:    4 19:25:49    -1091.614633        0.227934
BFGS:    5 19:25:50    -1091.625219        0.186790
BFGS:    6 19:25:51    -1091.632357        0.178916
BFGS:    7 19:25:52    -1091.640624        0.175081
BFGS:    8 19:25:53    -1091.648042        0.184522
BFGS:    9 19:25:54    -1091.656144        0.160935
BFGS:   10 19:25:55    -1091.663843        0.178420
BFGS:   11 19:25:57    -1091.672299        0.188697
BFGS:   12 19:25:58    -1091.682088        0.157552
BFGS:   13 19:25:59    -1091.692985        0.177272
BFGS:   14 19:26:00    -1091.704431        0.158116
BFGS:   15 19:26:01    -1091.715508        0.191678
BFGS:   16 19:26:02    -1091.725712        0.197897
BFGS:   17 19:26:04    -1091.735324        0.163753
BFGS:   18 19:26:05    -1091.745537        0.151472
BFGS:   19 19:26:06    -1091.754018        0.170835
BFGS:   20 19:26:08    -1091.761494        0.153898
BFGS:   21 19:26:08    -1091.767888        0.152411
BFGS:   22 19:26:09    -1091.774205        0.165885
BFGS:   23 19:26:10    -1091.780891        0.135476
BFGS:   24 19:26:11    -1091.788350        0.181065
BFGS:   25 19:26:12    -1091.794277        0.204681
BFGS:   26 19:26:12    -1091.800635        0.131379
BFGS:   27 19:26:13    -1091.806516        0.189491
BFGS:   28 19:26:14    -1091.812299        0.199155
BFGS:   29 19:26:15    -1091.817182        0.151652
BFGS:   30 19:26:16    -1091.822210        0.100141
BFGS:   31 19:26:16    -1091.826298        0.125828
BFGS:   32 19:26:17    -1091.832526        0.177421
BFGS:   33 19:26:19    -1091.837110        0.246726
BFGS:   34 19:26:20    -1091.842045        0.112496
BFGS:   35 19:26:21    -1091.845782        0.328688
BFGS:   36 19:26:23    -1091.850879        0.174085
BFGS:   37 19:26:24    -1091.858481        0.159914
BFGS:   38 19:26:24    -1091.865287        0.141291
BFGS:   39 19:26:25    -1091.872005        0.141917
BFGS:   40 19:26:25    -1091.878338        0.247148
BFGS:   41 19:26:27    -1091.880920        0.544926
BFGS:   42 19:26:28    -1091.885629        0.678812
BFGS:   43 19:26:28    -1091.893771        0.216384
BFGS:   44 19:26:29    -1091.899163        0.158099
BFGS:   45 19:26:30    -1091.917105        0.272864
BFGS:   46 19:26:31    -1091.924893        0.362183
BFGS:   47 19:26:33    -1091.941181        0.259165
BFGS:   48 19:26:34    -1091.956498        0.432599
BFGS:   49 19:26:34    -1091.973704        0.775537
BFGS:   50 19:26:36    -1091.966208        1.497817
BFGS:   51 19:26:36    -1092.007345        0.396691
BFGS:   52 19:26:37    -1092.022490        0.285749
BFGS:   53 19:26:38    -1092.072403        0.333420
BFGS:   54 19:26:38    -1092.088434        0.294865
BFGS:   55 19:26:39    -1092.122184        0.319618
BFGS:   56 19:26:40    -1092.127127        0.599146
BFGS:   57 19:26:41    -1092.142228        0.318145
BFGS:   58 19:26:42    -1092.157725        0.268180
BFGS:   59 19:26:43    -1092.169612        0.281630
BFGS:   60 19:26:44    -1092.182070        0.247304
BFGS:   61 19:26:45    -1092.191658        0.349909
BFGS:   62 19:26:45    -1092.203893        0.413967
BFGS:   63 19:26:46    -1092.214866        0.360631
BFGS:   64 19:26:48    -1092.224796        0.242091
BFGS:   65 19:26:51    -1092.232279        0.120391
BFGS:   66 19:26:53    -1092.238301        0.121883
BFGS:   67 19:26:54    -1092.243741        0.127485
BFGS:   68 19:26:55    -1092.249530        0.138036
BFGS:   69 19:26:56    -1092.254531        0.133713
BFGS:   70 19:26:56    -1092.259746        0.114452
BFGS:   71 19:26:57    -1092.264405        0.080882
BFGS:   72 19:26:58    -1092.268434        0.107020
BFGS:   73 19:27:01    -1092.272104        0.135069
BFGS:   74 19:27:02    -1092.275584        0.189900
BFGS:   75 19:27:02    -1092.278848        0.195108
BFGS:   76 19:27:03    -1092.282338        0.145852
BFGS:   77 19:27:04    -1092.285210        0.099664
BFGS:   78 19:27:05    -1092.287629        0.101675
BFGS:   79 19:27:07    -1092.290099        0.113556
BFGS:   80 19:27:07    -1092.292056        0.106041
BFGS:   81 19:27:08    -1092.293817        0.080004
BFGS:   82 19:27:10    -1092.295233        0.088615
BFGS:   83 19:27:11    -1092.296598        0.081745
BFGS:   84 19:27:12    -1092.298022        0.078294
BFGS:   85 19:27:13    -1092.299568        0.077982
BFGS:   86 19:27:14    -1092.301357        0.092660
BFGS:   87 19:27:16    -1092.302951        0.099303
BFGS:   88 19:27:16    -1092.304805        0.093478
BFGS:   89 19:27:17    -1092.306296        0.067367
BFGS:   90 19:27:18    -1092.307679        0.048825
Energy of MOF + H2O: -1092.308 eV

We can now isolate the MOF atoms from the relaxed MOF + H2O geometry and see that the MOF has adopted a geometry that is less energetically favorable than the empty MOF by ~0.2 eV. The energy of the MOF in the adsorbed state corresponds to EMOF(rMOF+H2O)E_{\mathrm{MOF}}(r_{\mathrm{MOF+H2O}}).

mof_adsorbed_state = mof_h2o[:-3]
mof_adsorbed_state.calc = calc
E_mof_adsorbed_state = mof_adsorbed_state.get_potential_energy()
print(f"Energy of MOF in the adsorbed state: {E_mof_adsorbed_state:.3f} eV")
Energy of MOF in the adsorbed state: -1077.091 eV

H2O adsorption in this MOF appears to correspond to Case #2 as outlined above. We can now perform re-relaxation of the empty MOF starting from the rMOF+H2Or_{\mathrm{MOF+H2O}} geometry.

relax = BFGS(mof_adsorbed_state)
relax.run(fmax=0.05)
E_mof_rerelax = mof_adsorbed_state.get_potential_energy()
print(f"Energy of re-relaxed empty MOF: {E_mof_rerelax:.3f} eV")
      Step     Time          Energy          fmax
BFGS:    0 19:27:18    -1077.090553        0.983390
BFGS:    1 19:27:19    -1077.122796        0.871949
BFGS:    2 19:27:20    -1077.172480        0.801385
BFGS:    3 19:27:21    -1077.212450        0.502580
BFGS:    4 19:27:22    -1077.232150        0.439210
BFGS:    5 19:27:23    -1077.250061        0.282549
BFGS:    6 19:27:24    -1077.260795        0.260939
BFGS:    7 19:27:25    -1077.270315        0.246514
BFGS:    8 19:27:26    -1077.279564        0.209651
BFGS:    9 19:27:27    -1077.286616        0.183123
BFGS:   10 19:27:29    -1077.291780        0.151377
BFGS:   11 19:27:30    -1077.295884        0.140513
BFGS:   12 19:27:32    -1077.299531        0.145093
BFGS:   13 19:27:33    -1077.304188        0.184344
BFGS:   14 19:27:34    -1077.308955        0.172553
BFGS:   15 19:27:35    -1077.313551        0.136378
BFGS:   16 19:27:35    -1077.317677        0.164919
BFGS:   17 19:27:36    -1077.321891        0.150550
BFGS:   18 19:27:37    -1077.326254        0.143855
BFGS:   19 19:27:39    -1077.329819        0.113696
BFGS:   20 19:27:39    -1077.332449        0.109350
BFGS:   21 19:27:40    -1077.334689        0.099863
BFGS:   22 19:27:43    -1077.336976        0.121523
BFGS:   23 19:27:44    -1077.339681        0.117428
BFGS:   24 19:27:46    -1077.342297        0.090415
BFGS:   25 19:27:47    -1077.344617        0.090821
BFGS:   26 19:27:49    -1077.346519        0.068008
BFGS:   27 19:27:50    -1077.348068        0.067778
BFGS:   28 19:27:51    -1077.349774        0.091663
BFGS:   29 19:27:52    -1077.351328        0.095574
BFGS:   30 19:27:53    -1077.352936        0.064592
BFGS:   31 19:27:54    -1077.354117        0.050613
BFGS:   32 19:27:55    -1077.355418        0.055644
BFGS:   33 19:27:58    -1077.356790        0.084699
BFGS:   34 19:27:58    -1077.358186        0.079266
BFGS:   35 19:27:59    -1077.359702        0.068661
BFGS:   36 19:28:00    -1077.361108        0.070816
BFGS:   37 19:28:00    -1077.362385        0.074361
BFGS:   38 19:28:01    -1077.363568        0.085450
BFGS:   39 19:28:02    -1077.364887        0.080212
BFGS:   40 19:28:03    -1077.365862        0.066907
BFGS:   41 19:28:04    -1077.366878        0.058635
BFGS:   42 19:28:04    -1077.367970        0.059731
BFGS:   43 19:28:05    -1077.369004        0.069902
BFGS:   44 19:28:08    -1077.370253        0.075623
BFGS:   45 19:28:09    -1077.371406        0.048885
Energy of re-relaxed empty MOF: -1077.371 eV

The MOF returns to its original empty reference energy upon re-relaxation, confirming that this deformation is physically relevant and is induced by the adsorbate molecule. In Case #3, this re-relaxed energy will be more negative (more favorable) than the original empty MOF relaxation. Thus, we take the reference empty MOF energy (EMOFE_{\mathrm{MOF}} in Eqn. 1) to be the minimum of the original empty MOF energy and the re-relaxed MOf energy:

E_mof = min(E_mof_empty, E_mof_rerelax)

# get adsorbate reference energy
h2o = mof_h2o[-3:]
h2o.calc = calc
E_h2o = h2o.get_potential_energy()

# compute adsorption energy
E_ads = E_combo - E_mof - E_h2o
print(f"Adsorption energy of H2O in WOBHEB_0.11_0: {E_ads:.3f} eV")
Adsorption energy of H2O in WOBHEB_0.11_0: -0.685 eV

This adsorption energy closely matches that from DFT (–0.699 eV) [1]. The strong adsorption energy is a consequence of both H2O chemisorption and MOF deformation. We can decompose the adsorption energy into contributions from these two factors. Assuming rigid H2O molecules, we define EintE_{\mathrm{int}} and EMOF,deformE_{\mathrm{MOF,deform}}, respectively, as

Eint=EMOF+H2O(rMOF+H2O)EMOF(rMOF+H2O)EH2O(rMOF+H2O)(2)E_{\mathrm{int}} = E_{\mathrm{MOF+H2O}}(r_{\mathrm{MOF+H2O}}) - E_{\mathrm{MOF}}(r_{\mathrm{MOF+H2O}}) - E_{\mathrm{H2O}}(r_{\mathrm{MOF+H2O}}) \tag{2}
EMOF,deform=EMOF(rMOF+H2O)EMOF(rMOF)(3)E_{\mathrm{MOF,deform}} = E_{\mathrm{MOF}}(r_{\mathrm{MOF+H2O}}) - E_{\mathrm{MOF}}(r_{\mathrm{MOF}}) \tag{3}

EintE_{\mathrm{int}} describes host host–guest interactions for the MOF in the adsorbed state only. EMOF,deformE_{\mathrm{MOF,deform}} quantifies the magnitude of deformation between the MOF in the adsorbed state and the most energetically favorable empty MOF geometry determined from the workflow presented here. It can be shown that

Eads=Eint+EMOF,deform(4)E_{\mathrm{ads}} = E_{\mathrm{int}} + E_{\mathrm{MOF,deform}} \tag{4}

For H2O adsorption in WOBHEB_0.11, we have

E_int = E_combo - E_mof_adsorbed_state - E_h2o
print(f"E_int: {E_int}")
E_int: -0.9691989421842617
E_mof_deform = E_mof_adsorbed_state - E_mof_empty
print(f"E_mof_deform: {E_mof_deform}")
E_mof_deform: 0.28385066986083984
E_ads = E_int + E_mof_deform
print(f"E_ads: {E_ads}")
E_ads: -0.6853482723234219

EintE_{\mathrm{int}} is equivalent to EadsE_{\mathrm{ads}} when the MOF is assumed to be rigid. In this case, failure to consider adsorbate-induced deformation would result in an overestimation of the adsorption energy magnitude.

Acknowledgements & Authors

Logan Brabson and Sihoon Choi (Georgia Tech) and the OpenDAC project.

References
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