Article pubs.acs.org/IC
Fast and Reasonable Geometry Optimization of Lanthanoid Complexes with an Extended Tight Binding Quantum Chemical Method Markus Bursch, Andreas Hansen, and Stefan Grimme* Mulliken Center for Theoretical Chemistry, Institut für Physikalische und Theoretische Chemie, Universität Bonn, 53115 Bonn, Beringstraße 4, Germany S Supporting Information *
ABSTRACT: The recently developed tight binding electronic structure approach GFN-xTB is tested in a comprehensive and diverse lanthanoid geometry optimization benchmark containing 80 lanthanoid complexes. The results are evaluated with reference to high-quality X-ray molecular structures obtained from the Cambridge Structural Database and theoretical DFT-D3(BJ) optimized structures for a few Pm (Z = 61) containing systems. The average structural heavyatom root-mean-square deviation of GFN-xTB (0.65 Å) is smaller compared to its competitors, the Sparkle/PM6 (0.86 Å) and HF-3c (0.68 Å) quantum chemical methods. It is shown that GFN-xTB yields chemically reasonable structures, less outliers, and performs well in terms of overall computational speed compared to other low-cost methods. The good reproduction of large lanthanoid complex structures corroborates the wide applicability of the GFN-xTB approach and its value as an efficient low-cost quantum chemical method. Its main purpose is the search for energetically low-lying complex conformations in the elucidation of reaction mechanisms.
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INTRODUCTION Lanthanoid complexes of various sizes are of interest in a broad field of applications such as photoemitting materials,1 bioimaging,2 luminescent lanthanoid-based metal-organic frameworks,3−5 and homogeneous catalysis,6 to name only a few known applications. Thus, the correct computation of such lanthanoid complex structures is fundamental for theoretical and mechanistic studies in these fields of research. In the last decades, the Sparkle7−9 model extensions of several semiempirical methods like AM1,10−18 PM3,19−25 PM6,26 PM7,27,28 and RM129,30 were developed to optimize the geometries of coordination complexes containing +3 charged lanthanoid ions. Specifically, Sparkle/PM6 still represents the state-of-the-art for semiempirical methods in this field, in which theoretical alternatives are lacking. In this work, we investigate the applicability of the recently developed GFN-xTB31 tight binding electronic structure approach for geometry optimization of a large and diverse set of common lanthanoid complexes. High-quality X-ray molecular structures obtained from the Cambridge Structural Database (CSD) and theoretical DFT-D3(BJ) reference structures for Pm (Z = 61) containing systems are taken as reference. The competitive quantum chemical methods Sparkle/PM626,32 as well as HF-3c33 are applied to the same set of structures in order to put the quality of the new method into some perspective. While GFN-xTB and PM6 formally have the same computational cost (for practical computation times, see the section Computation Time), HF-3c is 1−2 orders of magnitude slower (but an order of magnitude faster than DFT-D3(BJ) with a reasonable AO basis set). Energetic (thermochemical) properties are not considered here, but we note in passing that GFN-xTB has recently been © 2017 American Chemical Society
applied successfully to study high energy processes like the dissociation of transition metal complexes in electron ionization mass spectrometry34 (QCEIMS method35). Furthermore, reasonable atomic charges are in general obtained from GFNxTB which were recently used to improve the widely used D3 dispersion model.36 In a few examples, however, we demonstrate Table 1. Obtained Overall Structural Quality Data for GFN-xTB, Sparkle/PM6, and HF-3c method
RMSD [Å]
MAD (dA−B) [Å]
MAD (CNm)
GFN-xTB Sparkle/PM6 HF-3c
0.65 0.86 0.68
0.23 0.19 0.10
0.95 1.04 0.93
Table 2. RMSD and ΔCNm of the GFN-xTB Optimized Structures to the X-ray Structures (RMSDX‑ray) and the DFT-D3(BJ) Optimized Structures (RMSDcalc.) GFN-xTB entry Pm-1 Pm-2 Pm-3 Ho-5 Pr-5 Sm-5
CSD
287100 812633 963104
RMSDX‑ray
0.15 0.28 0.15
ΔCNm,X‑ray
RMSDcalc.
ΔCNm,calc.
0.47 0.02 1.11
0.28 0.13 0.06 0.09 0.28 0.08
1.22 0.15 0.29 0.00 0.51 0.77
Received: August 3, 2017 Published: October 5, 2017 12485
DOI: 10.1021/acs.inorgchem.7b01950 Inorg. Chem. 2017, 56, 12485−12491
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inorganic ligands (alanates, boranes, metal organic ligands) were also included in the parametrization set. The coordination numbers of the utilized structures range from 3 to 15. For further methodological details on GFN-xTB and its parameters, see ref 31. All Sparkle/PM6 calculations were carried out with the MOPAC201638 program package. HF-3c33 calculations were conducted with the TURBOMOLE 7.0.239,40 program, and the DFT-D3(BJ) reference structures were calculated with the ORCA 4.0.041,42 program package. The GFN-xTB structure optimizations were carried out with the xtb 4.8 stand-alone program, which can be obtained free of charge for academic use (via email to xtb@ thch.uni-bonn.de). GFN-xTB default convergence criteria were used (SCF: 1.0·10−6 Eh; Gradient: 1.0·10−3 Eh). Sparkle/PM6 calculations were conducted with identical convergence criteria. For HF-3c calculations, slightly changed TURBOMOLE convergence criteria were used (SCF: 1.0·10−7 Eh; Gradient: 1.0·10−4 Eh).
the ability of GFN-xTB for the search of energetically low-lying conformations for complexes with flexible ligands.
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COMPUTATIONAL DETAILS
The semiempirical tight binding method GFN-xTB is parametrized for all elements up to Z = 86. The number of parameters is kept small, and element-pair-specific parameters are largely avoided. As the GFN-xTB method does not focus on the description of electronic (spectroscopic) or magnetic properties, the lanthanoids are treated like 4d transition metals in an “f-in-core” approximation.37 Accordingly, the f electrons are not explicitly considered and all lanthanoids contribute three valence electrons to the electronic structure. Note that, in the Sparkle/PM6 approach, no electrons (basis functions) are provided by the lanthanoid atom, meaning that all bonding is basically of electrostatic origin. Furthermore, the parametrization of the lanthanoids was simplified by linear interpolation of the atomic parameters with the nuclear charge Z between the start (Ce, Z = 58) end points (Lu, Z = 71). As reference data, mainly structures including hard bases as ligands (halides, amines, carboxylates, etc.) were employed, as these are predominantly observed in the most common lanthanoid complexes. Nevertheless, several
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RESULTS AND DISCUSSION General Remarks. Our benchmark set consists of 77 disorder-free high-quality crystallographic structures with
Table 3. Data Comparison for Complexes Containing Elements La−Eu. The RMSD Is Given in Å, Wall Time (WT) in Seconds. Structural Correctness (SC): Yes = Y; No = N GFN-xTB
Sparkle/PM6
entry
CSD
RMSD
WT
cycles
ΔCNm
SC
RMSD
WT
cycles
ΔCNm
SC
La-1 La-2 La-3 La-4 La-5 La-6 Ce-1 Ce-2 Ce-3 Ce-4 Ce-5 Ce-6 Ce-7 Ce-8 Ce-9 Ce-10 Pr-1 Pr-2 Pr-3 Pr-4 Pr-5 Pr-6 Nd-1 Nd-2 Nd-3 Nd-4 Nd-5 Pm-1 Pm-2 Pm-3 Sm-1 Sm-2 Sm-3 Sm-4 Sm-5 Eu-1 Eu-2 Eu-3 Eu- 4 Eu-5
1406265 1409125 1410388 1418152_−1 215859 913893 1044879 1409131 636475 774594 797225_+4 804892 836189_−1 904596 220013 862911 211410 222805_+3 229509 789537 812633 815090 1058297 1409126 186711 789536 906651
0.63 1.08 0.49 1.68 0.70 0.50
39 22 10 19 46 19
87 146 48 98 62 51
0.43 0.84 0.02 5.22 1.48 2.42
Y Y Y N Y Y
0.58 0.92 0.90 2.82 0.44 0.45
144 197 29 177 117 80
597 997 270 1699 350 301
1.15 0.65 4.43 2.4 1.07 1.82
Y Y N N Y Y
0.58 0.32 0.89 0.41 1.33 0.77
12 21 229 12 7 101
84 53 179 40 155 100
0.30 0.57 0.39 1.91 1.64 0.24
Y Y N Y N N
0.47 0.34 0.18 0.17 0.65
7 6 9 5 18
33 39 26 32 107
0.15 0.37 0.42 1.00 0.67
Y N Y Y Y
0.28 0.47 1.67 0.55 0.47 0.95 1.02 0.28 0.13 0.06 0.30 0.17 1.23 0.96 0.15 1.43 0.22 0.82 0.17 0.56
13 23 57 23 58 139 142 2 1 0 63 3 58 260 0 144 22 20 4 3
36 75 219 91 71 153 151 27 19 9 77 19 177 196 17 343 55 75 25 98
0.44 0.02 0.27 1.57 0.80 0.47 3.52 0.60 1.22 0.15 0.29 1.01 0.48 0.64 1.95 1.11 0.66 0.42 0.82 0.41
Y Y N Y Y N Y Y Y Y Y Y Y Y Y Y Y Y Y N
0.55 0.43 1.21 1.33 1.99 0.69 1.18 1.67 1.31 0.76 0.34 0.45 1.11 0.43 0.57 1.14 0.50 1.22 0.67 0.97 0.18 0.70 0.31 0.68 0.82 0.27 1.11 0.13 1.45 0.76 2.59 0.20 1.29
58 35 208 45 6 306 752 39 24 52 17 15 230 11 87 124 65 364 265 276 2 7 01 56 7 253 528 0 28 2934 141 13 10
278 195 219 333 103 397 549 127 426 327 323 202 386 78 757 102 291 536 524 424 37 174 66 98 187 209 364 25 71 213 597 198 637
0.95 1.94 1.6 0.5 0.55 2.35 1.38 1.83 0.97 0.19 0.2 0.56 0.81 0.89 0.86 0.09 0.19 0.01 1.08 0.35 2.89 0.14 1.54 1.54 0.71 0.21 0.8 0.62 1.54 0.66 1.01 0.06 2.49
Y Y N Y N Y N N Y Y Y Y N Y Y N Y Y Y Y N Y N N Y Y Y Y N N N Y Y
1016561 209486 228522 914587 963104_−2 1151903 1402204 815914_+2 1158110_+2 1158110_−2
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Inorganic Chemistry R-factors below 0.05 extracted from the Cambridge Structural Database (CSD).43 Three additional promethium containing complex structures were calculated as molecular reference (due to its radioactivity, no crystal structures are available) at the PBE0-D3(BJ)44−48/ZORA-def2-TZVP49 all-electron level of theory applying the SARC2-ZORA-QZV50 basis set for the lanthanoid atoms. The structure numbering and the corresponding CSD identification numbers can be found in Tables 3 and 4. Missing entries indicate not achieved convergence or failed self-consistent field (SCF) convergence occurred during geometry optimization. We observed this in four GFN-xTB, three Sparkle/ PM6, and six HF-3c calculations. For GFN-xTB, this was rarely observed for larger multinuclear lanthanoid clusters with small bridging anionic ligands. In these special cases, an artificial contraction of the lanthanoid framework was observed. The benchmark set contains mononuclear, multinuclear, and lanthanoid-transition metal mixed complexes with versatile
organic and inorganic ligands. Neutral and ionic mono- and multidentate ligands of variable flexibility and size are included. We employ four structural quality criteria for comparison of experimental and theoretical molecular structures: (i) heavyatom (all except H) root-mean-square deviation (RMSD), (ii) absolute mean coordination number (CN) change of the lanthanoid atoms (ΔCNmean), (iii) the mean absolute deviation (MAD) of the Ln−Ligand bond distances (dA−B) and (iv) structural correctness (SC). The latter criteria indicates chemical or coordinative changes that might not be reflected by the previous criteria. Geometry Optimization. Overall, GFN-xTB performs well in all quality criteria compared to the well-established method Sparkle/PM6. It yields a smaller mean heavy-atom RMSD (Figure 1) of 0.65 Å compared to Sparkle/PM6 (0.86 Å) and HF-3c (0.68 Å), indicating that ligand−ligand and electrostatic interactions are well-described within GFN-xTB on a competitive level. The investigation of the lanthanoid−ligand bond distances
Table 4. Data Comparison for Complexes Containing Elements Gd−Lu. The RMSD Is Given in Å, Wall Time (WT) in Seconds. Structural Correctness (SC): Yes = Y; No = N GFN-xTB entry
CSD
Gd-1 Gd-2 Gd-3 Gd-4 Gd-5 Tb-1 Tb-2 Tb-3 Tb-4 Tb-5 Dy-1 Dy-2 Dy-3 Dy-4 Dy-5 Ho-1 Ho-2 Ho-3 Ho-4 Ho-5 Er-1 Er-2 Er-3 Er-4 Er-5 Tm-1 Tm-2 Tm-3 Tm-4 Tm-5 Yb-1 Yb-2 Yb-3 Yb-4 Yb-5 Lu-1 Lu-2 Lu-3 Lu- 4 Lu-5
1145951 191242 809026 854536 908963 1008454 741843 914007 132373 1227115 789527 920636 642445 938031 1209340 1160702 949430 1008527 1107060 287100 1063556_−2 1140170 186712 1276830 1282289 695868 752927 241797 984462 1130736 1016562 136019 1404893 153628 174206 1164391 213481 233097 138662 1236083
Sparkle/PM6
WT
cycles
ΔCNm
SC
RMSD
WT
cycles
ΔCNm
SC
0.16 0.19 0.78 0.10 0.75 0.65 1.51 0.81 0.81 0.60 1.41 0.18 1.91 0.35 0.39 0.28 0.17 0.58 1.80 0.15 1.36 1.14 0.75 0.49 0.04 0.38
2 21 194 6 44 18 96 119 98 24 139 58 10 27 4 2 3 113 108 2 232 30 78 17 1 20
27 31 133 25 70 60 179 184 151 81 196 53 105 42 34 28 38 166 260 19 307 130 121 56 11 100
1.23 0.37 0.36 0.28 0.63 1.69 2.58 0.39 0.31 0.40 1.73 5.59 0.43 1.95 1.08 0.28 2.31 1.37 0.46 0.17 0.47 2.87 0.34 0.42 0.27 0.36
Y Y Y Y N N N Y Y Y N Y N N Y Y Y Y N Y N Y Y Y Y Y
0.80 0.75 0.69 0.43 1.84 0.31 0.89
7 80 506 18 40 20 249
126 126 536 119 65 135 476
0.41 1.28 1.59 1.32 3.25 1.01 1.05
Y Y N Y N Y Y
1.06 0.86 0.96 0.52 0.75 0.65 0.81 0.24 0.41 0.82 1.60 0.50
261 45 65 252 19 134 38 1 3 186 30 5
419 160 137 476 111 382 397 18 60 386 33 141
0.17 1.7 0.83 0.76 2.13 2.5 1.38 3.59 0.26 0.19 0.61 0.53
N Y Y Y N N Y N N Y Y Y
2.06 0.17 0.99 0.21 0.52 0.27 0.61 0.63 0.21 0.53 1.40 0.39 0.39
129 62 36 19 33 25 29 28 10 60 290 26 14
233 52 161 59 53 46 75 91 25 105 241 61 47
0.70 0.51 0.17 0.56 0.74 0.15 1.07 1.22 0.16 1.76 1.85 0.09 0.13
N Y Y Y Y Y Y Y Y Y Y Y Y
2.13 1.28 0.21 0.40 0.52 1.11 1.82 0.61 1.13 1.28 0.49 0.68 0.72 0.76 0.48 0.97 0.36 0.40 0.61
91 790 36 16 30 59 450 264 69 62 1283 914 51 140 243 138 256 117 11
351 1204 90 268 216 226 1359 531 419 213 496 123 173 540 369 326 337 287 31
1.02 0.02 0.09 0.47 0.85 1.63 0.66 0.72 0.6 1.26 0.37 0.97 0.7 0.43 0.93 0.66 0.62 0.1 0.57
Y N Y Y Y Y Y Y Y N Y Y Y Y Y Y Y Y Y
RMSD
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containing cyclopentadienyl ligands, where a partial decoordination of these ligands causes large deviations. For the tabulated data, see the Supporting Information. The mean absolute deviation of the mean coordination number of the lanthanoid atoms (Figure 2) shows comparably good performance with a MAD of 0.95 compared to 1.04 for Sparkle/ PM6 and 0.93 for HF-3c. The few outliers (La-4, Nd-4, and Dy-1) for GFN-xTB refer to hydride bridged lanthanoid cluster structures that show strong reorganization of the inner ligand sphere during the optimization. The individual data as well as the structural correctness criterion are shown in Tables 3 and 4. Overall, the structures obtained with GFN-xTB are of good quality, and some examples are depicted in Figure 3. Remarkably good structures are obtained for structures including conformationally well-defined multidentate ligands like Er-5 (RMSD = 0.04 Å) or Gd-4 (RMSD = 0.10 Å). Even for the chemically rather “exotic” cases such as Sm-1, a thioarsenite bridged trinuclear samarium complex, or Tm-1, including short Tm−Pd contacts, reasonable RMSD values of 0.32 and 0.38 Å, respectively, are obtained with GFN-xTB. The obtained overall structural quality data are summed up in Table 1. At this point, we emphasize that GFN-xTB is less heavily parametrized compared to many other semiempirical methods in common use (e.g., DFTB51−54) and mainly employs a small number of global and element-specific parameters. A direct comparison of six GFN-xTB optimized structures with high-quality DFT-D3(BJ) reference structures (Table 2) further reveals a good and even better accordance of the GFNxTB structures and the high level DFT results for isolated molecules. This indicates that a small part of the residual deviations between theory and experiment are due to crystal “packing” effects, which, however, presently due to technical reasons cannot be resolved by periodic calculations. Particularly, the deviations for spatially expanded structures or such with highly flexible ligand spheres are expected to be significantly smaller in comparison to gas phase reference data, which should be kept in mind when judging the results. Computation Time. In terms of computation (wall) time, GFN-xTB clearly outperforms Sparkle/PM6 in most cases (Figure 4). The generally low computation times (small number of electronic self-consistent field (SCF) cycles) and the small number of required optimization steps emphasizes the high efficiency and robustness of GFN-xTB and the corresponding code (Figure 5). Specifically, the highly efficient optimizer in the
indicates that GFN-xTB reproduces the direct coordination sphere with a MAD of 0.23 Å slightly worse than Sparkle/PM6 with a MAD of 0.19 Å. HF-3c clearly outperforms both, GFN-xTB and Sparkle/PM6, with a MAD of only 0.10 Å. The performance of GFN-xTB for this property can be partly explained by several outliers of sterically overloaded structures
Figure 1. Heavy-atom (all except H) root-mean-square deviation of the optimized structures with respect to the X-ray reference structures for GFN-xTB (blue), Sparkle/PM6 (red), and HF-3c (gray). The deviations are given in Å.
Figure 2. Mean absolute coordination number change of the lanthanoid atoms relative to the reference structures and overall mean absolute deviations for GFN-xTB (blue), Sparkle/PM6 (red), and HF-3c (gray).
Figure 3. Overlays of structures for eight selected lanthanoid complexes. The GFN-xTB optimized structures are depicted in transparent blue, the X-ray structures with color coded atoms. Most hydrogen atoms are omitted for clarity. The heavy-atom RMSD is given in Å. 12488
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was started with the optimized X-ray structure, and the resulting ensemble of energetically low-lying structures was re-evaluated by DFT-D3(BJ) single point energy calculations at the PBE0D3(BJ)/ZORA-def2-TZVP//GFN-xTB level of theory applying the SARC2-ZORA-QZV basis for the lanthanoid atom. The input structures and the DFT-D3(BJ) energetically lowest conformers are depicted in Figure 6. Specifically for Er-2, the maximization of attractive ligand−ligand interactions (π stacking) in the absence of crystal packing becomes apparent in the energetically lowest conformer found. Even though the GFN-xTB conformer energy ranking may vary from that of higher level computations, the generation of an almost complete conformer ensemble of good structural quality is far from being trivial and the structure generation represents in many applications the most time-consuming step. Therefore, the use of GFN-xTB in efficient composite approaches is of high value. An application and details on the automated conformational search algorithm based on GFN-xTB were recently shown in the context of fully automated quantum chemistry based computations of spin−spin coupled nuclear magnetic resonance spectra.55
xtb code in combination with fast SCF calculations lead to the short computation times observed. Note that all timings refer to a single CPU and, hence, parallel runs which are possible further extend the broad applicability of the method to even larger systems at reasonable turnaround times. A Conformational Search Application. For the complexes Er-2 and Ce-9, which have more flexible ligands, we conducted automated conformational searches using a newly developed normal-mode-following algorithm that is implemented in the xtb code and which is published separately.55 The search
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CONCLUSION The new GFN-xTB method proved to be a versatile and fast semiempirical tool for the computation of molecular structures containing elements up to Z = 86 including all lanthanoids (Z = 57−71), the latter specifically considered in this work. Its flexible and robust design qualifies this method to deal with a broad field of chemical applications without requiring large computational resources. The ability to optimize lanthanoid complex structures at state-of-the-art semiempirical or better quality and with very low computational cost was demonstrated in this work for a very versatile and diverse set of 80 reference structures (77 X-ray structures). For 44 out of the 80 cases, the RMSD to the X-ray structure was less than 0.6 Å, meaning that the theoretical structure more or less coincides with the experimental one. Part of the residual deviations are attributed to crystal packing/crystal field effects which are neglected in our study. The robustness of GFN-xTB combined with its excellent cost/accuracy ratio enables the consistent computation of molecular structures up to a few thousand atoms which also allows the investigation of large lanthanoid frameworks. A periodic implementation to enable three-dimensional crystal structure optimization for material design is in progress in our lab. Large scale screening investigations are now possible also for lanthanoid complexes, allowing computational chemists and experimentalists to gain more insight into the structural characteristics of such systems. Furthermore, conformational analysis for systems with flexible ligands can be conducted quite routinely,
Figure 4. Serial computation (wall) time in seconds for GFN-xTB (blue) and Sparkle/PM6 (red).
Figure 5. Number of optimization cycles for GFN-xTB (blue) and Sparkle/PM6 (red).
Figure 6. Exemplary conformational search for Er-2 (left) and Ce-9 (right). GFN-xTB optimized X-ray cutout (left) and the corresponding energetically lowest conformer after re-evaluation obtained by the automated conformational search (right) with their relative DFT-D3(BJ)// GFN-xTB energies in kcal/mol. 12489
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which is important for the elucidation of reaction mechanisms or the computation of spectra.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.7b01950. Tabulated data and Cartesian coordinates of all structures used or generated in this study (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Stefan Grimme: 0000-0002-5844-4371 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the DFG in the framework of the “Gottfried-Wilhelm-Leibniz” prize. The authors thank C. Bannwarth, P. Pracht, J. Pisarek, and C. Bauer for many fruitful discussions and all experimentalists for providing highquality X-ray data to the CSD.
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