
==== Front
J Mol Model
J Mol Model
Journal of Molecular Modeling
1610-2940
0948-5023
Springer Berlin Heidelberg Berlin/Heidelberg

39153076
6099
10.1007/s00894-024-06099-5
Original Paper
New global minimum conformers for the Pt19 and Pt20 clusters: low symmetric species featuring different active sites
Guevara-Vela José Manuel 1
Gallegos Miguel 2
Rocha-Rinza Tomás 3
Muñoz-Castro Álvaro 4
Kessler Peter L. Rodríguez plkessler@cio.mx

5
Martín Pendás Ángel ampendas@uniovi.es

2
1 https://ror.org/01cby8j38 grid.5515.4 0000 0001 1957 8126 Departamento de Química Física Aplicada, Universidad Autónoma de Madrid, C. Francisco Tomás y Valiente, 7, Madrid, 28049 Spain
2 https://ror.org/006gksa02 grid.10863.3c 0000 0001 2164 6351 Departamento de Química Física y Analítica, Universidad de Oviedo, Av. Julián Clavería, 8, Oviedo, 33006 Asturias Spain
3 https://ror.org/01tmp8f25 grid.9486.3 0000 0001 2159 0001 Instituto de Química, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, Delegación Coyoacán, 04510 Mexico City Mexico
4 https://ror.org/04jrwm652 grid.442215.4 0000 0001 2227 4297 Facultad de Ingeniería, Arquitectura y Diseño, Universidad San Sebastián, Bellavista 7, Santiago, 8420524 RM Chile
5 https://ror.org/00q8h8k29 grid.466579.f 0000 0004 1776 8315 Centro de Investigaciones en Óptica A.C., Loma del Bosque 115, Col. Lomas del Campestre, León, 37150 Guanajuato Mexico
17 8 2024
17 8 2024
2024
30 9 31016 7 2024
30 7 2024
© The Author(s) 2024
2024
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Context

The study of platinum (Pt) clusters and nanoparticles is essential due to their extensive range of potential technological applications, particularly in catalysis. The electronic properties that yield optimal catalytic performance at the nanoscale are significantly influenced by the size and structure of Pt clusters. This research aimed to identify the lowest-energy conformers for Pt18, Pt19, and Pt20 species using Density Functional Theory (DFT). We discovered new low-symmetry conformers for Pt19 and Pt20, which are 3.0 and 1.0 kcal/mol more stable, respectively, than previously reported structures. Our study highlights the importance of using density functional approximations that incorporate moderate levels of exact Hartree-Fock exchange, alongside basis sets of at least quadruple-zeta quality. The resulting structures are asymmetric with varying active sites, as evidenced by sigma hole analysis on the electrostatic potential surface. This suggests a potential correlation between electronic structure and catalytic properties, warranting further investigation.

Methods

An equivariant graph neural network interatomic potential (NequIP) within the Atomic Simulation Environment suite (ASE) was used to provide initial geometries of the aggregates under study. DFT calculations were performed with the ORCA 5 package, using functional approximations that included Generalized Gradient Approximation (PBE), meta-GGA (TPSS, M06-L), hybrid (PBE0, PBEh), meta-GGA hybrid (TPSSh), and range-separated hybrid (ωB97x) functionals. Def2-TZVP and Def2-QZVP as well as members of the cc-pwCVXZ-PP family to check basis set convergence were used. QTAIM calculations were performed using the AIMAll suite. Structures were visualized with the AVOGADRO code.

Supplementary Information

The online version contains supplementary material available at 10.1007/s00894-024-06099-5.

Keywords

Metal clusters
Platinum clusters
DFT
Hybrid functionals
http://dx.doi.org/10.13039/501100004837 Ministerio de Ciencia e Innovación PID2021-122763NB-I00 Dirección General de Cómputo y de Tecnologías de Información y ComunicaciónLANCAD-UNAM-DGTIC 250 http://dx.doi.org/10.13039/501100002850 Fondo Nacional de Desarrollo Científico y Tecnológico 1221676 http://dx.doi.org/10.13039/501100020702 Instituto Potosino de Investigación Científica y Tecnológica TKII-E-0424-I-080424-4/PR-6 issue-copyright-statement© Springer-Verlag GmbH Germany, part of Springer Nature 2024
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pmcIntroduction

Platinum nanoparticles are highly versatile species that find application in a wide range of fields due to their electrochemical [1, 2] and optical properties  [3, 4]. Their importance is even more pronounced in the field of catalysis [5, 6] and especially for the photocatalytic production of H2  [7–10]. Indeed, the importance of hydrogen gas production can hardly be overemphasized, being central to various industrial sectors, including oil refining and steel processing. Furthermore, achieving a global increase in the production of H2 is essential for the decarbonisation of heavy industry, long-haul transportation, and seasonal energy storage  [11]. In this context, the catalytic activity of small Pt clusters outperforms that of larger nanoparticles and bulkier metals due to their unique structures and electronic properties  [12, 13]. Understanding the reasons behind these outstanding characteristics could potentially lead to the development of cheaper and more abundant catalysts, thereby advancing sustainable hydrogen production and helping to reduce our dependence on fossil fuels in the fight against climate change  [14].

This background has given rise to a wide interest in acquiring a comprehensive understanding of the electronic properties and structure of small Pt nanoparticles, whose catalytic performance is strongly influenced by their shape and size  [15]. Consequently, significant effort has been invested in identifying the most thermodynamically stable structures of different small platinum clusters. Despite the high computational cost associated with Density Functional Theory (DFT) calculations in comparison to other methodologies such as the Gupta [16] or Sutton-Chen [17] model potentials, they have been the preferred approach for this endeavor. For instance, Kumar and Kawazoe  [18], on one hand, and Wei and Liu  [19], on the other, conducted systematic investigations using the PBE functional and projected augmented wave (PAW) pseudopotentials to identify isomers of up to 44 and 46 atoms, respectively. These studies have indicated that the addition of single atoms to existing clusters may prove an effective method for identifying novel minimum structures [20]. The search for the most stable structures of these moieties has employed a wide range of techniques, including biased searches  [21], prescreening using empirical potentials  [22], genetic algorithms  [22, 23], simulated annealing  [24], or a combination of them. Moreover, specific studies have focused on finding the global minimum (GM) of specific individual clusters, such as Pt13  [21, 25–28], Pt15  [22, 24], and Pt55  [26, 29, 30]. The use of different functionals or methodologies may result in varying conformer orderings, and this has occasionally given rise to controversies. This underscores the necessity for highly reliable results, which can be achieved by considering a number factors, including the use of large and flexible enough basis sets [31] or of appropriate aproximate exchange-correlation functionals that incorporate a suitable amount of Hartree-Fock exchange  [32].

With this in mind, we employed a simulated annealing procedure based on molecular dynamics simulations, using a machine learning potential trained from DFT calculations, to investigate the structures of Pt18–Pt20. This approach led to the discovery of new and asymmetric stable minima for Pt19 and Pt20. While the structure of the new minimum for Pt19 is similar to the previously reported ones, the new minimum for Pt20 differs significantly from previously accounted structures. Furthermore, our analysis incorporates different DFT approaches and emphasizes the importance of incorporating moderate amounts of Hartree-Fock exchange and the use of large basis sets to accurately determine the energetic ordering of platinum clusters. Finally, employing the Quantum Theory of Atoms in Molecules (QTAIM), we have reveiled the differences in the electronic distribution of the two lowest isomers of the Pt20 cluster. We believe that the insights presented in this article will (i) provide important guidelines for the design of future simulations of these systems and (ii) contribute to the accurate energetic ordering of isomers leading to the identification of bona fide GM of metallic clusters.

Results and discussion

Figure 1 depicts the structures of the lowest-lying energy isomers of Pt18, Pt19, and Pt20. The xyz files corresponding to each structure at the TPSSh level of theory are available as Supporting Information. For Pt18, we confirm that the GM, 18.1, is the three-layer trigonal prism structure with D3h symmetry proposed by Kumar and Kawazoe [18]. Furthermore, an analogous structure, slightly higher in energy, was identified wherein a Pt atom is displaced in an upward direction from the base, creating a rectangular opening in one of the sides of the pyramid. This results in Cs symmetry. The third Pt18 isomer depicted in Fig. 1 exhibits a distorted three-layer trigonal prism structure with C2 symmetry. It is noteworthy that this isomer is identified as the GM when described with the M06-L and ωB97x functionals. As will be discussed in greater detail below, meta-GGA functionals tend to disfavor the three-layer trigonal prism structure. Despite repeated attemps, it was not possible to optimize the 18.1 isomer using the ωB97x functional.Fig. 1 Lowest-lying energy isomers of Pt18, Pt19, and Pt20 along with their labels and relative energy in kcal/mol at the TPSS/Def2-QZVPP//TPSS/Def2-TZVP level of theory

It is notable that the use of functional approximations that partially incorporate exact Hartree-Fock exchange results in a reduction in the energy differences between the 18.1 and 18.2 isomers. Indeed, the aforementioned discrepancy is 4.4 kcal/mol for the TPSS functional, while it is only 0.4 kcal/mol for the TPSSh, which includes 10% HF exchange. Furthermore, going from the PBE to the PBE0 functional reverses the energetic order of the isomers. With PBE, 18.1 is the lowest energy isomer, while 18.2 is the GM for PBE0, a hybrid functional with 25% exact HF exchange. Indeed, it is established that the inclusion of HF exchange can be pivotal in ensuring the correct energetic ordering [33]. This is because it mitigates the many-electron self-interaction error. This error arises from the inability of approximate exchange-correlation functionals to exclude the interaction of an electron with itself. Consequently, the inclusion of exact exchange facilitates a more precise characterization of electronic systems  [34, 35].

In the case of Pt19, a novel GM, designated as the 19.1 structure, was identified that exhibits subtle differences from the previously reported minima, 19.2. Both structures are derived from the addition of a Pt atom to one of the triangular sides of the Pt18 three-layer trigonal prism (18.1). In the case of 19.2, the atom is added to one of the interstitial spaces between three other Pt atoms, resulting in a structure with Cs symmetry. With regard to the 19.1 isomer, the additional atom is located at an equal distance from the three sides of the triangular face of the three-layer trigonal prism, thereby resulting in C3v symmetry. The third reported isomer, 19.3, exhibits no symmetry elements beyond the identity element. Regarding the energetic order, in this case, there is a more uniform assessment from the different functionals: PBE, PBE0, TPSS, and TPSSh all favor the 19.1 structure as the GM. In contrast, and similarly to what happens with Pt18, the M06-L and ωB97x functionals predict disordered structures over those containing a three-layer trigonal prism, with a difference in energy of over 10 kcal/mol. As was the case with 18.1, it was not possible to optimize 19.1 using the ωB97x functional.

For Pt20, the minimum energy structure proposed in previous works [18, 19, 22] (20.2) is a three-layer trigonal prism with two platinum atoms appended to one of its sides. In contrast, our proposed GM (20.1) possesses C1 symmetry and can be better described as a square base formed by nine Pt atoms, with a second and third levels formed by six and five Pt atoms, respectively. It appears that this unstructured arrangement favors the number of connecting atoms over the directionality of connections, which may indicate the initial stages of the transition from ordered nanoclusters to a metallic arrangement. The third isomer of Pt20 put forward in Fig. 1, 20.3, is formed by a rhomboidal-like base with nine atoms with two additional levels on top containing seven and four atoms, respectively. At first glance, its symmetry appears to be C2-like; however, the presence of minor discrepancies between the edges destroys this pseudo-symmetry, leaving 20.3 as an additional C1 isomer. Energetically, the TPSS, TPSSh, and ωB97x functionals render 20.1 as the GM. The PBE0 functional yields quasi-degenerate energies for the 20.1 and 20.2 isomers, and PBE favors the latter by 3.1 kcal/mol. Ultimately, optimization of the 20.2 isomer was not feasible when using the M06-L functional. Regarding the energetic ordering of the other two isomers, 20.3 sits 5.4 kcal/mol lower in energy. Notice, however, that the M06-L 20.3 structure has C2 symmetry (Table 1).Table 1 Relative energies (kcal/mol) of the three lowest-lying isomers of Pt18, Pt19, and Pt20 computed with different exchange-correlation functionals as well as the corresponding multiplicity

Label	M	PBE	PBEh	PBE0	TPSS	TPSSh	TPSS0	M06-L	ωB97x	
18.1a	9	0.0	0.0	5.4	0.0	0.0	7.1	12.2	—	
18.2	5	5.1	4.9	0.0	4.4	0.4	0.0	12.9	8.2	
18.3	9	13.2	13.2	7.4	7.9	4.6	3.6	0.0	0.0	
19.1	9	0.0	0.0	0.0	0.0	0.0	0.0	13.5	—	
19.2a	9	2.3	2.5	4.7	1.5	3.0	2.8	12.8	13.6	
19.3	3	7.0	7.5	6.5	4.3	4.3	0.6	0.0	0.0	
20.1	5	3.1	3.3	0.0	0.0	0.0	1.0	5.4	0.0	
20.2a	5	0.0	0.0	0.0	0.7	1.0	0.0	—	9.1	
20.3	3	7.3	7.9	2.7	2.4	3.4	9.0	0.0	23.1	
a Ref. [18]

Fig. 2 Relative energy of the 20.1 cluster with respect to 20.2 as a function of the number of basis set functions. All calculation were done using the TPSSh functional

On the role of the basis set

Besides the importance of selecting the correct functional approximation, the selection of basis sets is of paramount importance for the accurate assignment of the energy ordering in these systems. Figure 2 illustrates the relative energy of the 20.1 cluster as compared to 20.2 for different basis sets taken from the cc-pwCVXZ-PP family  [36]. We observe that the smallest basis set, cc-pwCVDZ-PP with 1080 functions, places 20.1 higher in energy than 20.2. However, when the cc-pwCVTZ-PP basis set is considered, 20.1 is correctly identified as the true GM. Notably, this behavior is not oscillatory, and the selection of 20.1 as the GM is confirmed by the larger cc-pwCVQZ-PP and cc-pwCV5Z-PP basis sets, which have 2880 and 4120 functions, respectively. These results suggest that the use of basis sets of at least QZ quality is essential for the accurate differentiation of conformers with similar energies.

Table 2 illustrates the relative energies of the 20.1 and 20.2 isomers using different functionals and basis sets of triple and quadruple-zeta quality. In particular, we selected the functionals mentioned above (PBE, PBE0, TPSS, TPSSh, and ωB97x) and included also the popular BP86 [37, 38] and B3PW91 [39, 40] approximations in combination with the Def2-TZVP (TZ) and Def2-QZVPP (QZ) basis sets. Remarkably, older GGA functionals (BP86 and PBE) exhibited a pronounced preference for 20.2 over 20.1, with energy differences of 9.8 and 7.9 kcal/mol, respectively. Moreover, the incorporation of exact exchange has a clear stabilizing effect on 20.1 relative to 20.2: while PBE/TZ favored 20.2 by 7.9 kcal/mol, PBE0/TZ favored it only by 5.2 kcal/mol, a difference of 2.7 kcal/mol. In contrast, the newer ωB97x, a range-separated hybrid functional, favored the 20.1 isomer by 4.2 kcal/mol.Table 2 Relative energies (kcal/mol) of the two lowest-lying isomers of Pt20 computed with different exchange-correlation functionals and basis sets. In the last column, the indicated quantities are ΔE(QZ) - ΔE(TZ) where ΔE(QZ)= ΔE20.2(QZ) - E20.1(QZ). Ditto for ΔE(TZ)

Functional	20.1	20.2a	ΔE(QZ) - ΔE(TZ)	
ωB97x/TZ	0.0	4.1		
ωB97x/QZ	0.0	9.2	5.1	
TPSSh/TZ	4.4	0.0		
TPSSh/QZ	0.0	1.1	5.5	
TPSS/TZ	4.4	0.0		
TPSS/QZ	0.0	0.7	5.1	
PBE0/TZ	5.2	0.0		
PBE0/QZ	0.0	0.0	5.2	
PBE/TZ	7.9	0.0		
PBE/QZ	3.1	0.0	4.8	
B3PW91/TZ	7.8	0.0		
B3PW91/QZ	3.0	0.0	4.8	
BP86/TZ	9.8	0.0		
BP86/QZ	5.2	0.0	4.6	
a Ref. [18]

Fig. 3 IR spectra for the lowest energy structures of Ptn (n=18, 19, 20) clusters at the TPSSh/Def2TZVP level of theory

It is striking that the transition from TZ to QZ-quality basis sets has significant implications for all functionals, irrespective of whether they are GGA, meta-GGA, or hybrid. Using the PBE functional and the TZ basis sets, 20.2 is 7.9 kcal/mol more stable than 20.1, while using the QZ basis reduces this difference to only 3.1 kcal/mol. Even more importantly, the use of QZ basis sets is cabable of reversing the energetic ordering, as observed with the TPSS and TPSSh functionals. For TPSS/TZ, 20.2 is the GM, while it is 20.1 in the case of TPSS/QZ. Indeed, going from the TZ to the QZ basis sets favors 20.1 over 20.2 by approximately 5 kcal/mol, regardless of its reference value. This is observed in the last column of Table 2. From these observations, we can infer that basis sets incompleteness has a selective impact on the relative stability of different isomers. Furthermore, these findings confirm the importance of using high-quality basis sets to obtain meaningful results for relative energies and other properties  [31]. Our results emphasize the importance of selecting an appropriate functional approximation and a sufficiently large basis set to ascertain the relative stability of metal clusters in general and Ptn isomers in particular.Fig. 4 Electrostatic potential (VS) mapped onto the van der Waals envelope for the minimum energy structures of Pt18, Pt19, and Pt20. σ-holes, identified as maximally negative VS sites, are highlighted by yellow arrows. See the text for further details

Fig. 5 Bond paths and atomic charges for the minima structures of the 20.1 and 20.2 clusters. Two-dimensional relieve map of the Laplacian of the electronic density for the 20.1 cluster plotted in the plane formed by the Pt8, Pt14, and Pt18 atoms

IR fingerprints

In order to provide fingerprints for structural identification, we have calculated the vibrational modes of the most stable clusters. The characteristic peaks for Ptn (n=18, 19, 20) clusters were found at 194.3, 207.6, and 200.8 cm-1, respectively. These correspond to the antisymmetric stretching of the central layers. The lowest vibrational frequencies, observed at 27.1, 25.7, and 16.8 cm-1, are twisting vibrations, while the vibrations at the highest frequencies are located at 214.0, 213.7, and 225.6 cm-1, are antisymmetric. The IR spectra of these clusters were generated using the orca_mapspc utility within the ORCA program and are shown in Fig. 3. These spectra could serve as benchmarks for future experimental characterization of Pt clusters.

Electrostatic potentials and QTAIM analyses

Theoretical studies on the lowest energy structures of Pt clusters provide insight into the underlying mechanisms that contribute to their excellent catalytic performance, as observed in experiments [24]. The overall catalytic efficiency of a given aggregate can be related to the existence of specific catalytically active sites, which are determined by the location of low-coordinated atoms within the cluster structure. As observed in gold and other noble metals, such sites lead to electron-deficient regions, the so-called σ-holes, that can be evaluated via the electrostatic potential mapped onto the van der Waals envelope (conveniently approximated as an electron density isosurface with isovalue 0.001 a.u.) [41, 42].

As illustrated in Fig. 4, the resulting electrostatic potential surface indicates the formation of six equivalent low-coordinated edges in Pt18, which give rise to six analogous σ-holes. These pinpoint reactive sites, as evidenced by the negative values of the electrostatic potential (VS). The addition of a Pt atom to form Pt19 results in three equivalent σ-holes that retain some characteristics of the Pt18 parent, together with the appearance of a new deep hole around the capping atom. We thus expect different reactive sites in different regions of the cluster. This is exacerbated in Pt20, where the marked decrease in symmetry gives rise to several different σ-hole sites. Since the number of catalytic active regions in this cluster is greater than that observed in Pt18 and Pt19, we conclude that this cluster will be particularly reactive. Hence, the controlled growth of these types of aggregates may prove an effective strategy for modifying the number of catalytic sites within a narrow size range.

The Quantum Theory of Atoms in Molecules (QTAIM) is a powerful tool for studying chemical interactions within a molecule. It allows for an orbital invariant analysis of electron density, thereby providing insights into the nature and strength of interactions between atoms. The QTAIM has been successfully employed in the study of intermetallic interactions [43–45] and metallic clusters [46–51]. Figure 5 provides such an analysis for the 20.1 and 20.2 clusters, illustrating both the bond paths [52, 53] between the platinum atoms and their atomic charges  [54]. The electron densities of these two clusters are strikingly similar, exhibiting concentration of charge at the edge atoms coupled with a depletion of density in the interior atoms, which have a larger number of immediate neighbors. This is in contrast with previous findings in AlnSc clusters, where the electron density was found to be larger for the endohedral atom [51]. Furthermore, Fig. 5 also shows a two-dimensional relief map of the Laplacian of the electron density. This dissection reveals charge depletion zones that coincide with the catalytic active areas depicted in Fig. 4. This observation suggests a potential link between the distribution of the electron density near the nucleus and the catalytic properties of these Pt clusters, which merits further investigation.

Methods

We employed a machine-learning interatomic potential constructed from DFT energies and forces calculated at the TPSS/Def2-TZVP level of theory in order to ease the mapping of the complex potential energy surface of the Pt20 clusters. This potential was developed using E(3)-equivariant graph neural networks within the NequIP program by Batzner and coworkers  [55], and it was used in conjunction with the Atomic Simulation Environment suite  [56]. We used this interatomic potential to perform simulated annealing minimisations based on molecular dynamics simulations. The resulting structures were reoptimised using different exchange-correlation functionals, namely BP86  [37, 38], PBE  [57], PBEh  [58], PBE0  [59], B3PW91  [39, 40], TPSS  [60], TPSSh [61], M06-L  [62], and ωB97x  [63], in combination with the Def2-TZVP (TZ) and Def2-QZVP (QZ)  [64–66] basis sets which include a relativistic pseudopotential replacing 60 core electrons of the platinum atom. With the aim of calibrating the performance of DFT methods, in our previous work, we conducted a benchmark calculation for Pt2 dimers described by different types of functionals [23]. By comparing the ionization potential and dissociation energies of these clusters, the results indicated that the hybrid-GGA M06-L, B3PW91, and meta-GGA TPSS functionals were good choices to evaluate the structures of the clusters. Interestingly, the TPSSh functional provided a bond length of 2.34 Å, which is slightly smaller than that predicted by TPSS (with 2.35 Å), but in better agreement with the experiment of Airola and Morse [67] (2.33 Å). Thus, the TPSSh functional seems realiable in predicting molecular geometries and vibrational frequencies for these systems. All DFT calculations were conducted with the aid of the Orca 5 software [68]. For the QTAIM [54] analyses, densities were obtained using the Zeroth-Order Regular Approximation [69–71], and the density partitions themselves were carried out using the AIMAll program  [72]. The resulting structures were visualized using the Avogadro code  [73].

Conclusion

In this study, we employed Density Functional Theory (DFT) to investigate the lowest energy structures and electronic properties of Pt18, Pt19, and Pt20 clusters. Our findings have revealed the existence of novel, more stable isomers for the Pt19 and Pt20 systems, 3.0 and 1.0 kcal/mol more stable, respectively, than the previously reported minimum energy structures. Furthermore, the role of different DFT approximations, including GGA (PBE), meta-GGA (TPSS, M06-L), hybrid (PBE0), meta-GGA hybrid (TPSSh), and range-separated hybrid (ωB97x) functionals, was investigated in relation to the relative energies of the studied clusters. Our findings indicate that the energy ordering of different isomers is highly sensitive to the use of density functional approximations that include some exact Hartree-Fock exchange. Furthermore, it is evident that the use of basis sets of at least quadruple-zeta quality is necessary. A QTAIM analysis highlights significant distinctions in both the electron distribution and the nature of interatomic contacts among isomers, even those with comparable total energies. The MEP of the novel Pt20 minimum indicates the existence of potential catalytic sites. The insights presented in this article are expected to facilitate the design of future simulations and contribute to the accurate energetic ordering of isomers of metallic clusters, ultimately enabling the identification of true global minima in noble metal clusters as prototypical catalytic reactive species.

Supplementary information

A supplementary file with the structures of the studied clusters is available.

Supplementary Information

Below is the link to the electronic supplementary material.Supplementary file 1 (pdf 121 KB)

Acknowledgements

This work was supported by FONDECYT ANID Regular 1221676. Powered@NLHPC: This research was partially supported by the supercomputing infrastructure of the NLHPC (ECM-02). IPICyT’s National Supercomputing Center supported this research with the computational time grant TKII-E-0424-I-080424-4/PR-6. T.R.-R. is thankful to DGTIC/UNAM for computer time (project LANCAD-UNAM-DGTIC 250). A.M.P. and M.G. thank MCIN/AEI/10.13039/501100011033 and ERDF A way of Making Europe, grants PID2021-122763NB-I00. M.G. also thanks the Spanish MICIU for a predoctoral FPU grant, FPU19/02903.

Author Contributions

J.M.G.-V. and T.R.-R. wrote the initial draft of the manuscript text. M.G. and P.L.R-K. and A.M-C. performed the theoretical calculations. Á.M.-C., P.L.R.-K., and A.M.P. reviewed and edited the manuscript. All authors reviewed the final manuscript.

Funding

Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.

Data Availability

No datasets were generated or analyzed during the current study.

Declarations

Conflict of Interest

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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References

1. Hrapovic S Liu Y Male KB Electrochemical biosensing platforms using platinum nanoparticles and carbon nanotubes Anal Chem 2003 76 4 1083 1088 10.1021/ac035143t
Hrapovic S, Liu Y, Male KB et al (2003) Electrochemical biosensing platforms using platinum nanoparticles and carbon nanotubes. Anal Chem 76(4):1083–1088. 10.1021/ac035143t10.1021/ac035143t
2. Claussen JC Kumar A Jaroch DB Nanostructuring platinum nanoparticles on multilayered graphene petal nanosheets for electrochemical biosensing Adv Funct Mater 2012 22 16 3399 3405 10.1002/adfm.201200551
Claussen JC, Kumar A, Jaroch DB et al (2012) Nanostructuring platinum nanoparticles on multilayered graphene petal nanosheets for electrochemical biosensing. Adv Funct Mater 22(16):3399–3405. 10.1002/adfm.20120055110.1002/adfm.201200551
3. Zhang N Han C Xu YJ Near-field dielectric scattering promotes optical absorption by platinum nanoparticles Nat Photonics 2016 10 7 473 482 10.1038/nphoton.2016.76
Zhang N, Han C, Xu YJ et al (2016) Near-field dielectric scattering promotes optical absorption by platinum nanoparticles. Nat Photonics 10(7):473–482. 10.1038/nphoton.2016.7610.1038/nphoton.2016.76
4. Samadi A Bendix PM Oddershede LB Optical manipulation of individual strongly absorbing platinum nanoparticles Nanoscale 2017 9 46 18449 18455 10.1039/c7nr07374g 29159358
Samadi A, Bendix PM, Oddershede LB (2017) Optical manipulation of individual strongly absorbing platinum nanoparticles. Nanoscale 9(46):18449–18455. 10.1039/c7nr07374g29159358 10.1039/c7nr07374g
5. Cheng N Banis MN Liu J Extremely stable platinum nanoparticles encapsulated in a zirconia nanocage by area-selective atomic layer deposition for the oxygen reduction reaction Adv Mater 2014 27 2 277 281 10.1002/adma.201404314 25405600
Cheng N, Banis MN, Liu J et al (2014) Extremely stable platinum nanoparticles encapsulated in a zirconia nanocage by area-selective atomic layer deposition for the oxygen reduction reaction. Adv Mater 27(2):277–281. 10.1002/adma.20140431425405600 10.1002/adma.201404314
6. Choi KM Na K Somorjai GA Chemical environment control and enhanced catalytic performance of platinum nanoparticles embedded in nanocrystalline metal–organic frameworks J Am Chem Soc 2015 137 24 7810 7816 10.1021/jacs.5b03540 26023888
Choi KM, Na K, Somorjai GA et al (2015) Chemical environment control and enhanced catalytic performance of platinum nanoparticles embedded in nanocrystalline metal–organic frameworks. J Am Chem Soc 137(24):7810–7816. 10.1021/jacs.5b0354026023888 10.1021/jacs.5b03540
7. Cheng N, Stambula S, Wang D et al (2016) Platinum single-atom and cluster catalysis of the hydrogen evolution reaction. Nat Commun 7(1). 10.1038/ncomms13638
8. Zhang J Zhao Y Guo X Single platinum atoms immobilized on an MXene as an efficient catalyst for the hydrogen evolution reaction Nat Catal 2018 1 12 985 992 10.1038/s41929-018-0195-1
Zhang J, Zhao Y, Guo X et al (2018) Single platinum atoms immobilized on an MXene as an efficient catalyst for the hydrogen evolution reaction. Nat Catal 1(12):985–992. 10.1038/s41929-018-0195-110.1038/s41929-018-0195-1
9. Mei L Gao X Gao Z Size-selective synthesis of platinum nanoparticles on transition-metal dichalcogenides for the hydrogen evolution reaction Chem Commun 2021 57 23 2879 2882 10.1039/d0cc08091h
Mei L, Gao X, Gao Z et al (2021) Size-selective synthesis of platinum nanoparticles on transition-metal dichalcogenides for the hydrogen evolution reaction. Chem Commun 57(23):2879–2882. 10.1039/d0cc08091h10.1039/d0cc08091h
10. Tang P Lee HJ Hurlbutt K Elucidating the formation and structural evolution of platinum single-site catalysts for the hydrogen evolution reaction ACS Catal 2022 12 5 3173 3180 10.1021/acscatal.1c05958 35558899
Tang P, Lee HJ, Hurlbutt K et al (2022) Elucidating the formation and structural evolution of platinum single-site catalysts for the hydrogen evolution reaction. ACS Catal 12(5):3173–3180. 10.1021/acscatal.1c0595835558899 10.1021/acscatal.1c05958
11. Griffiths S Sovacool BK Kim J Industrial decarbonization via hydrogen: a critical and systematic review of developments, socio-technical systems and policy options Energy Res Soc Sci 2021 80 102208 10.1016/j.erss.2021.102208
Griffiths S, Sovacool BK, Kim J et al (2021) Industrial decarbonization via hydrogen: a critical and systematic review of developments, socio-technical systems and policy options. Energy Res Soc Sci 80:102208. 10.1016/j.erss.2021.10220810.1016/j.erss.2021.102208
12. Imaoka T Kitazawa H Chun WJ Magic number Pt13 and misshapen Pt12 clusters: which one is the better catalyst? J Am Chem Soc 2013 135 35 13089 13095 10.1021/ja405922m 23902457
Imaoka T, Kitazawa H, Chun WJ et al (2013) Magic number Pt13 and misshapen Pt12 clusters: which one is the better catalyst? J Am Chem Soc 135(35):13089–13095. 10.1021/ja405922m23902457 10.1021/ja405922m
13. Schweinberger FF Berr MJ Döblinger M Cluster size effects in the photocatalytic hydrogen evolution reaction J Am Chem Soc 2013 135 36 13262 13265 10.1021/ja406070q 23961721
Schweinberger FF, Berr MJ, Döblinger M et al (2013) Cluster size effects in the photocatalytic hydrogen evolution reaction. J Am Chem Soc 135(36):13262–13265. 10.1021/ja406070q23961721 10.1021/ja406070q
14. Acar C Dincer I Review and evaluation of hydrogen production options for better environment J Clean Prod 2019 218 835 849 10.1016/j.jclepro.2019.02.046
Acar C, Dincer I (2019) Review and evaluation of hydrogen production options for better environment. J Clean Prod 218:835–849. 10.1016/j.jclepro.2019.02.04610.1016/j.jclepro.2019.02.046
15. Ahmadi TS Wang ZL Green TC Shape-controlled synthesis of colloidal platinum nanoparticles Sci 1996 272 5270 1924 1925 10.1126/science.272.5270.1924
Ahmadi TS, Wang ZL, Green TC et al (1996) Shape-controlled synthesis of colloidal platinum nanoparticles. Sci 272(5270):1924–1925. 10.1126/science.272.5270.192410.1126/science.272.5270.1924
16. Gupta RP Lattice relaxation at a metal surface Phys Rev B 1981 23 12 6265 6270 10.1103/physrevb.23.6265
Gupta RP (1981) Lattice relaxation at a metal surface. Phys Rev B 23(12):6265–6270. 10.1103/physrevb.23.626510.1103/physrevb.23.6265
17. Sutton AP Chen J Long-range Finnis–Sinclair potentials Philos Mag Lett 1990 61 3 139 146 10.1080/09500839008206493
Sutton AP, Chen J (1990) Long-range Finnis–Sinclair potentials. Philos Mag Lett 61(3):139–146. 10.1080/0950083900820649310.1080/09500839008206493
18. Kumar V, Kawazoe Y (2008) Evolution of atomic and electronic structure of Pt clusters: planar, layered, pyramidal, cage, cubic, and octahedral growth. Phys Rev B 77(20). 10.1103/physrevb.77.205418
19. Wei GF Liu ZP Subnano Pt particles from a first-principles stochastic surface walking global search J Chem Theory Comput 2016 12 9 4698 4706 10.1021/acs.jctc.6b00556 27482921
Wei GF, Liu ZP (2016) Subnano Pt particles from a first-principles stochastic surface walking global search. J Chem Theory Comput 12(9):4698–4706. 10.1021/acs.jctc.6b0055627482921 10.1021/acs.jctc.6b00556
20. Rodríguez-Kessler PL Rodríguez-Domínguez AR Muñoz-Castro A Systematic cluster growth: a structure search method for transition metal clusters Phys Chem Chem Phys 2021 23 8 4935 4943 10.1039/d0cp06179d 33621288
Rodríguez-Kessler PL, Rodríguez-Domínguez AR, Muñoz-Castro A (2021) Systematic cluster growth: a structure search method for transition metal clusters. Phys Chem Chem Phys 23(8):4935–4943. 10.1039/d0cp06179d33621288 10.1039/d0cp06179d
21. Rodríguez-Kessler PL, Rodríguez-Domínguez AR (2015) Size and structure effects of PtN (N = 12 - 13) clusters for the oxygen reduction reaction: first-principles calculations. J Chem Phys 143(18). 10.1063/1.4935566
22. Wang X Tian D Structures and structural evolution of PtN (N=15–24) clusters with combined density functional and genetic algorithm methods Comput Mater Sci 2009 46 1 239 244 10.1016/j.commatsci.2009.02.031
Wang X, Tian D (2009) Structures and structural evolution of PtN (N=15–24) clusters with combined density functional and genetic algorithm methods. Comput Mater Sci 46(1):239–244. 10.1016/j.commatsci.2009.02.03110.1016/j.commatsci.2009.02.031
23. Guevara-Vela JM Rocha-Rinza T Rodríguez-Kessler PL On the structure and electronic properties of PtN clusters: new most stable structures for N = 16–17 Phys Chem Chem Phys 2023 25 42 28835 28840 10.1039/d3cp04455f 37853760
Guevara-Vela JM, Rocha-Rinza T, Rodríguez-Kessler PL et al (2023) On the structure and electronic properties of PtN clusters: new most stable structures for N = 16–17. Phys Chem Chem Phys 25(42):28835–28840. 10.1039/d3cp04455f37853760 10.1039/d3cp04455f
24. Rodríguez-Kessler PL Muñoz-Castro A Rodríguez-Domínguez AR Structure effects of Pt15 clusters for the oxygen reduction reaction: first-principles calculations Phys Chem Chem Phys 2023 25 6 4764 4772 10.1039/d2cp05188e 36692089
Rodríguez-Kessler PL, Muñoz-Castro A, Rodríguez-Domínguez AR et al (2023) Structure effects of Pt15 clusters for the oxygen reduction reaction: first-principles calculations. Phys Chem Chem Phys 25(6):4764–4772. 10.1039/d2cp05188e36692089 10.1039/d2cp05188e
25. Watari N Ohnishi S Atomic and electronic structures of Pd13 and Pt13 clusters Phys Rev B 1998 58 3 1665 1677 10.1103/physrevb.58.1665
Watari N, Ohnishi S (1998) Atomic and electronic structures of Pd13 and Pt13 clusters. Phys Rev B 58(3):1665–1677. 10.1103/physrevb.58.166510.1103/physrevb.58.1665
26. Aprá E Fortunelli A Density functional calculations on platinum nanoclusters: Pt13, Pt38, and Pt55 J Phys Chem A 2003 107 16 2934 2942 10.1021/jp0275793
Aprá E, Fortunelli A (2003) Density functional calculations on platinum nanoclusters: Pt13, Pt38, and Pt55. J Phys Chem A 107(16):2934–2942. 10.1021/jp027579310.1021/jp0275793
27. Chang CM, Chou MY (2004) Alternative low-symmetry structure for 13-atom metal clusters. Phys Rev Lett 93(13). 10.1103/physrevlett.93.133401
28. Piotrowski MJ, Piquini P, Da Silva JLF (2010) Density functional theory investigation of 3, 4, an 5 13-atom metal clusters. Phys Rev B 81(15). 10.1103/physrevb.81.155446
29. Da Silva JLF, Kim HG, Piotrowski MJ, et al (2010) Reconstruction of core and surface nanoparticles: the example of Pt55 and Au55. Phys Rev B 82(20). 10.1103/physrevb.82.205424
30. Ewing CS Veser G McCarthy JJ Effect of support preparation and nanoparticle size on catalyst–support interactions between Pt and amorphous silica J Phys Chem C 2015 119 34 19934 19940 10.1021/acs.jpcc.5b05763
Ewing CS, Veser G, McCarthy JJ et al (2015) Effect of support preparation and nanoparticle size on catalyst–support interactions between Pt and amorphous silica. J Phys Chem C 119(34):19934–19940. 10.1021/acs.jpcc.5b0576310.1021/acs.jpcc.5b05763
31. Neese F Prediction of molecular properties and molecular spectroscopy with density functional theory: from fundamental theory to exchange-coupling Coord Chem Rev 2009 253 5–6 526 563 10.1016/j.ccr.2008.05.014
Neese F (2009) Prediction of molecular properties and molecular spectroscopy with density functional theory: from fundamental theory to exchange-coupling. Coord Chem Rev 253(5–6):526–563. 10.1016/j.ccr.2008.05.01410.1016/j.ccr.2008.05.014
32. Tekarli SM Drummond ML Williams TG Performance of density functional theory for 3d transition metal-containing complexes: utilization of the correlation consistent basis sets J Phys Chem A 2009 113 30 8607 8614 10.1021/jp811503v 19572689
Tekarli SM, Drummond ML, Williams TG et al (2009) Performance of density functional theory for 3d transition metal-containing complexes: utilization of the correlation consistent basis sets. J Phys Chem A 113(30):8607–8614. 10.1021/jp811503v19572689 10.1021/jp811503v
33. Hostaš J, Pérez-Becerra KO, Calaminici P et al (2023) How important is the amount of exact exchange for spin-state energy ordering in DFT? Case study of molybdenum carbide cluster, Mo4C2. J Chem Phys 159(18). 10.1063/5.0169409
34. Mori-Sánchez P, Cohen AJ, Yang W (2006) Many-electron self-interaction error in approximate density functionals. J Chem Phys 125(20). 10.1063/1.2403848
35. Cohen AJ Mori-Sánchez P Yang W Challenges for density functional theory Chem Rev 2011 112 1 289 320 10.1021/cr200107z 22191548
Cohen AJ, Mori-Sánchez P, Yang W (2011) Challenges for density functional theory. Chem Rev 112(1):289–320. 10.1021/cr200107z22191548 10.1021/cr200107z
36. Figgen D, Peterson KA, Dolg M et al (2009) Energy-consistent pseudopotentials and correlation consistent basis sets for the 5d elements Hf–Pt. J Chem Phys 130(16). 10.1063/1.3119665
37. Perdew JP Density-functional approximation for the correlation energy of the inhomogeneous electron gas Phys Rev B 1986 33 12 8822 8824 10.1103/physrevb.33.8822
Perdew JP (1986) Density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys Rev B 33(12):8822–8824. 10.1103/physrevb.33.882210.1103/physrevb.33.8822
38. Becke AD Density-functional exchange-energy approximation with correct asymptotic behavior Phys Rev A 1988 38 6 3098 3100 10.1103/physreva.38.3098
Becke AD (1988) Density-functional exchange-energy approximation with correct asymptotic behavior. Phys Rev A 38(6):3098–3100. 10.1103/physreva.38.309810.1103/physreva.38.3098
39. Perdew JP Chevary JA Vosko SH Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation Phys Rev B 1992 46 11 6671 6687 10.1103/physrevb.46.6671
Perdew JP, Chevary JA, Vosko SH et al (1992) Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation. Phys Rev B 46(11):6671–6687. 10.1103/physrevb.46.667110.1103/physrevb.46.6671
40. Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98(7):5648–5652. 10.1063/1.464913
41. Stenlid JH Brinck T Extending the σ-hole concept to metals: an electrostatic interpretation of the effects of nanostructure in gold and platinum catalysis J Am Chem Soc 2017 139 32 11012 11015 10.1021/jacs.7b05987 28770602
Stenlid JH, Brinck T (2017) Extending the -hole concept to metals: an electrostatic interpretation of the effects of nanostructure in gold and platinum catalysis. J Am Chem Soc 139(32):11012–11015. 10.1021/jacs.7b0598728770602 10.1021/jacs.7b05987
42. Stenlid J Johansson A Brinck T σ-holes on transition metal nanoclusters and their influence on the local Lewis acidity Curr Comput-Aided Drug Des 2017 7 7 222 10.3390/cryst7070222
Stenlid J, Johansson A, Brinck T (2017) -holes on transition metal nanoclusters and their influence on the local Lewis acidity. Curr Comput-Aided Drug Des 7(7):222. 10.3390/cryst707022210.3390/cryst7070222
43. Puyo M Lebon E Vendier L Topological analysis of Ag–Ag and Ag–N interactions in silver amidinate precursor complexes of silver nanoparticles Inorg Chem 2020 59 7 4328 4339 10.1021/acs.inorgchem.9b03166 32157877
Puyo M, Lebon E, Vendier L et al (2020) Topological analysis of Ag–Ag and Ag–N interactions in silver amidinate precursor complexes of silver nanoparticles. Inorg Chem 59(7):4328–4339. 10.1021/acs.inorgchem.9b0316632157877 10.1021/acs.inorgchem.9b03166
44. Guevara-Vela JM Hess K Rocha-Rinza T Stronger-together: the cooperativity of aurophilic interactions Chem Commun 2022 58 9 1398 1401 10.1039/d1cc05241a
Guevara-Vela JM, Hess K, Rocha-Rinza T et al (2022) Stronger-together: the cooperativity of aurophilic interactions. Chem Commun 58(9):1398–1401. 10.1039/d1cc05241a10.1039/d1cc05241a
45. Burguera S Bauzá A Frontera A A novel approach for estimating the strength of argentophilic and aurophilic interactions using QTAIM parameters Phys Chem Chem Phys 2024 26 23 16550 16560 10.1039/d4cp00410h 38829286
Burguera S, Bauzá A, Frontera A (2024) A novel approach for estimating the strength of argentophilic and aurophilic interactions using QTAIM parameters. Phys Chem Chem Phys 26(23):16550–16560. 10.1039/d4cp00410h38829286 10.1039/d4cp00410h
46. Mandado M Krishtal A Alsenoy CV Bonding study in all-metal clusters containing Al4 units J Phys Chem A 2007 111 46 11885 11893 10.1021/jp074973y 17966993
Mandado M, Krishtal A, Alsenoy CV et al (2007) Bonding study in all-metal clusters containing Al4 units. J Phys Chem A 111(46):11885–11893. 10.1021/jp074973y17966993 10.1021/jp074973y
47. Foroutan-Nejad C Al42-; the anion-π interactions and aromaticity in the presence of counter ions Phys Chem Chem Phys 2012 14 27 9738 10.1039/c2cp40511c 22684037
Foroutan-Nejad C (2012) Al; the anion- interactions and aromaticity in the presence of counter ions. Phys Chem Chem Phys 14(27):9738. 10.1039/c2cp40511c22684037 10.1039/c2cp40511c
48. Badri Z Pathak S Fliegl H All-metal aromaticity: revisiting the ring current model among transition metal clusters J Chem Theory Comput 2013 9 11 4789 4796 10.1021/ct4007184 26583397
Badri Z, Pathak S, Fliegl H et al (2013) All-metal aromaticity: revisiting the ring current model among transition metal clusters. J Chem Theory Comput 9(11):4789–4796. 10.1021/ct400718426583397 10.1021/ct4007184
49. Foroutan-Nejad C Bonding and aromaticity in electron-rich boron and aluminum clusters J Phys Chem A 2021 125 6 1367 1373 10.1021/acs.jpca.0c11474 33538582
Foroutan-Nejad C (2021) Bonding and aromaticity in electron-rich boron and aluminum clusters. J Phys Chem A 125(6):1367–1373. 10.1021/acs.jpca.0c1147433538582 10.1021/acs.jpca.0c11474
50. Lacaze-Dufaure C Bulteau Y Tarrat N Coordination of ethylamine on small silver clusters: structural and topological (ELF, QTAIM) analyses Inorg Chem 2022 61 19 7274 7285 10.1021/acs.inorgchem.1c03870 35485936
Lacaze-Dufaure C, Bulteau Y, Tarrat N et al (2022) Coordination of ethylamine on small silver clusters: structural and topological (ELF, QTAIM) analyses. Inorg Chem 61(19):7274–7285. 10.1021/acs.inorgchem.1c0387035485936 10.1021/acs.inorgchem.1c03870
51. Guevara-Vela JM de la Vega AS Gallegos M Wave function analyses of scandium-doped aluminium clusters, AlnSc (n= 1–24), and their CO2 fixation abilities Phys Chem Chem Phys 2023 25 28 18854 18865 10.1039/d3cp01730c 37403600
Guevara-Vela JM, de la Vega AS, Gallegos M et al (2023) Wave function analyses of scandium-doped aluminium clusters, AlnSc (n= 1–24), and their CO fixation abilities. Phys Chem Chem Phys 25(28):18854–18865. 10.1039/d3cp01730c37403600 10.1039/d3cp01730c
52. Bader RFW A bond path: a universal indicator of bonded interactions J Phys Chem A 1998 102 37 7314 7323 10.1021/jp981794v
Bader RFW (1998) A bond path: a universal indicator of bonded interactions. J Phys Chem A 102(37):7314–7323. 10.1021/jp981794v10.1021/jp981794v
53. Martín Pendás Á Francisco E Blanco M Bond paths as privileged exchange channels Chem Eur J 2007 13 33 9362 9371 10.1002/chem.200700408 17674344
Martín Pendás Á, Francisco E, Blanco M et al (2007) Bond paths as privileged exchange channels. Chem Eur J 13(33):9362–9371. 10.1002/chem.20070040817674344 10.1002/chem.200700408
54. Bader RFW (1990) Atoms in molecules: a quantum theory. Oxford University Press
55. Batzner S, Musaelian A, Sun L et al (2022) E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nat Commun 13(1). 10.1038/s41467-022-29939-5
56. Hjorth Larsen A Jørgen Mortensen J Blomqvist J The atomic simulation environment—a python library for working with atoms J Phys: Condens Matter 2017 29 27 273002 10.1088/1361-648x/aa680e 28323250
Hjorth Larsen A, Jørgen Mortensen J, Blomqvist J et al (2017) The atomic simulation environment—a python library for working with atoms. J Phys: Condens Matter 29(27):273002. 10.1088/1361-648x/aa680e28323250 10.1088/1361-648x/aa680e
57. Perdew JP Burke K Wang Y Generalized gradient approximation for the exchange-correlation hole of a many-electron system Phys Rev B 1996 54 23 16533 16539 10.1103/physrevb.54.16533
Perdew JP, Burke K, Wang Y (1996) Generalized gradient approximation for the exchange-correlation hole of a many-electron system. Phys Rev B 54(23):16533–16539. 10.1103/physrevb.54.1653310.1103/physrevb.54.16533
58. Heyd J Scuseria GE Ernzerhof M Hybrid functionals based on a screened coulomb potential J Chem Phys 2003 118 18 8207 8215 10.1063/1.1564060
Heyd J, Scuseria GE, Ernzerhof M (2003) Hybrid functionals based on a screened coulomb potential. J Chem Phys 118(18):8207–8215. 10.1063/1.156406010.1063/1.1564060
59. Adamo C, Barone V (1999) Toward reliable density functional methods without adjustable parameters: the PBE0 model. J Chem Phys 110(13):6158–6170. 10.1063/1.478522
60. Tao J, Perdew JP, Staroverov VN et al (2003) Climbing the density functional ladder: nonempirical meta–generalized gradient approximation designed for molecules and solids. Phys Rev Lett 91(14). 10.1103/physrevlett.91.146401
61. Staroverov VN Scuseria GE Tao J Comparative assessment of a new nonempirical density functional: molecules and hydrogen-bonded complexes J Chem Phys 2003 119 23 12129 12137 10.1063/1.1626543
Staroverov VN, Scuseria GE, Tao J et al (2003) Comparative assessment of a new nonempirical density functional: molecules and hydrogen-bonded complexes. J Chem Phys 119(23):12129–12137. 10.1063/1.162654310.1063/1.1626543
62. Zhao Y, Truhlar DG (2006) A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. J Chem Phys 125(19). 10.1063/1.2370993
63. Chai JD, Head-Gordon M (2008) Systematic optimization of long-range corrected hybrid density functionals. J Chem Phys 128(8). 10.1063/1.2834918
64. Weigend F Ahlrichs R Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: design and assessment of accuracy Phys Chem Chem Phys 2005 7 18 3297 3305 10.1039/b508541a 16240044
Weigend F, Ahlrichs R (2005) Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: design and assessment of accuracy. Phys Chem Chem Phys 7(18):3297–3305. 10.1039/b508541a16240044 10.1039/b508541a
65. Weigend F Accurate coulomb-fitting basis sets for H to Rn Phys Chem Chem Phys 2006 8 9 1057 10.1039/b515623h 16633586
Weigend F (2006) Accurate coulomb-fitting basis sets for H to Rn. Phys Chem Chem Phys 8(9):1057. 10.1039/b515623h16633586 10.1039/b515623h
66. Hellweg A Hättig C Höfener S Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn Theor Chem Acc 2007 117 4 587 597 10.1007/s00214-007-0250-5
Hellweg A, Hättig C, Höfener S et al (2007) Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn. Theor Chem Acc 117(4):587–597. 10.1007/s00214-007-0250-510.1007/s00214-007-0250-5
67. Airola MB Morse MD Rotationally resolved spectroscopy of Pt2 J Chem Phys 2002 116 4 1313 1317 10.1063/1.1428753
Airola MB, Morse MD (2002) Rotationally resolved spectroscopy of Pt2. J Chem Phys 116(4):1313–1317. 10.1063/1.142875310.1063/1.1428753
68. Neese F (2017) Software update: the ORCA program system, version 4.0. Wiley Interdiscip Rev Comput Mol Sci 8(1):e1327. 10.1002/wcms.1327
69. van Lenthe E Baerends EJ Snijders JG Relativistic regular two-component Hamiltonians J Chem Phys 1993 99 6 4597 4610 10.1063/1.466059
van Lenthe E, Baerends EJ, Snijders JG (1993) Relativistic regular two-component Hamiltonians. J Chem Phys 99(6):4597–4610. 10.1063/1.46605910.1063/1.466059
70. van Leeuwen R van Lenthe E Baerends EJ Exact solutions of regular approximate relativistic wave equations for hydrogen-like atoms J Chem Phys 1994 101 2 1272 1281 10.1063/1.467819
van Leeuwen R, van Lenthe E, Baerends EJ et al (1994) Exact solutions of regular approximate relativistic wave equations for hydrogen-like atoms. J Chem Phys 101(2):1272–1281. 10.1063/1.46781910.1063/1.467819
71. van Lenthe E Baerends EJ Snijders JG Relativistic total energy using regular approximations J Chem Phys 1994 101 11 9783 9792 10.1063/1.467943
van Lenthe E, Baerends EJ, Snijders JG (1994) Relativistic total energy using regular approximations. J Chem Phys 101(11):9783–9792. 10.1063/1.46794310.1063/1.467943
72. Keith TA (2019) Aimall (version 19.02.13). TK Gristmill Software, Overland Park KS, USA, 2019 (aim.tkgristmill.com)
73. Hanwell MD Curtis DE Lonie DC Avogadro: an advanced semantic chemical editor, visualization, and analysis platform J Cheminform 2012 4 1 2946 4 10.1186/1758-2946-4-17
Hanwell MD, Curtis DE, Lonie DC et al (2012) Avogadro: an advanced semantic chemical editor, visualization, and analysis platform. J Cheminform 4(1):2946–4. 10.1186/1758-2946-4-1710.1186/1758-2946-4-17
