Scalable properties of metal clusters: A comparative DFT study of ionic-core treatments

Chemical Physics Letters 578 (2013) 92–96

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Scalable properties of metal clusters: A comparative DFT study of ionic-core treatments Remi Marchal a, Ilya V. Yudanov b,c, Alexei V. Matveev c, Notker Rösch c,d,⇑ a

Department Chemie & Wacker-Institut für Siliziumchemie, Technische Universität München, 85747 Garching, Germany Boreskov Institute of Catalysis, Russian Academy of Sciences, 630090 Novosibirsk, Russian Federation c Department Chemie & Catalysis Research Center, Technische Universität München, 85747 Garching, Germany d Institute of High Performance Computing, Agency for Science, Technology and Research, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore b

a r t i c l e

i n f o

Article history: Received 19 March 2013 In final form 28 May 2013 Available online 7 June 2013

a b s t r a c t To assess various ionic-core approximations we carried out density functional calculations on a series of octahedral palladium and gold model clusters Mn with n = 13–147. We compared results for average bond lengths, cohesive energies, vertical ionization potentials, and electron affinities to the corresponding all-electron scalar relativistic results. We used extrapolated bulk values to compare with experiments. The results of the projector-augmented wave method agree best with the all-electron results. When modeling palladium–gold nanoalloy particles, one should keep in mind that current ECP or PAW modeling of palladium moieties appears to be more accurate than modeling of gold species. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction As entities that bridge the molecular regime and extended materials, metal clusters have been of particular interest during the last decades [1,2]. Properties of metal clusters (or nanoparticles) with several dozen atoms and more vary systematically with cluster size. Therefore, properties may be tuned for the needs of various applications, ranging from laser and opto-electronic devices to fluorescence marking and catalysis [1,2]. Electronic structure calculations are an important tool to complement experimental data for nanoparticles, also with regard to scaling of properties [2,3]. Systems studied computationally comprise bare [4–9] and ligand stabilized [1,10,11] metal particles of considerable size. Nanoalloys gained much interest, e.g. as catalysts and for modifying electrodes [12,13], because properties may also be tuned at fixed particle size by varying the composition. Comprehensive computational studies of bimetallic nanoparticles are more demanding than single-metal nanoparticles for two reasons. For one, an enhanced number of calculations is necessary to explore the changes with composition. This calls for an efficient computational method. Yet, any such approximate method has to describe both metals with balanced accuracy, also at extended system size. For theoretical studies of large metal clusters, methods based on Density Functional Theory (DFT) at the GGA level (generalized gradient approximation) became popular because they are sufficiently ⇑ Corresponding author at: Department Chemie & Catalysis Research Center, Technische Universität München, 85747 Garching, Germany. Fax: +49 89 289 13468. E-mail address: [email protected] (N. Rösch). 0009-2614/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2013.05.063

accurate [14], as demonstrated in many investigations [4–9]. Despite the computational efficiency of such procedures, invoking an approximate treatment of ionic cores is advisable when aiming at nanoalloys. Effective core pseudo-potential (ECP) methods [15] and the projector-augmented wave (PAW) method [16,17] come to mind. These approaches also avoid an explicit treatment of relativistic effects that is required for heavier transition metals [18,19]. To assess the performance of these approaches for large transition metal clusters, we report a DFT study where we compared results of these ionic-core treatments to those of a scalar relativistic all-electron (AE) scheme [18,19] to characterize directly the effect of the approximate treatment of core electrons. Focusing on the scaling of properties with cluster size, we compared average bond lengths, cohesive energies, ionization potentials and electron affinities for a set of model clusters Mn (n = 13, 19, 38, 55, 79, 147) of truncated-octahedral or cub-octahedral shape; see Figure S1 of Supporting Material (SM) [8]. As we are interested in probing the accuracy of these methods for large systems, we exploit scaling laws and extrapolate properties of these models to the corresponding bulk limit, for comparison with experimental data. In the present work, we examined results for clusters of palladium and gold; particles of either metal [20,21], but also their alloys [13] are important catalysts.

2. Computational details All geometries of the clusters were optimized under octahedral symmetry constraints (Oh). The Perdew, Burke and Ernzerhof (PBE) [22] variant of GGA was used to describe exchange–correlation

R. Marchal et al. / Chemical Physics Letters 578 (2013) 92–96 Table 1 ECP approaches used in this study.a

Pd

Au

93

3. Results and discussion

ECP

Abbr.

Core

Description

Ref.

CRENBS LANL1DZ CRENBL Stuttgart1 Stuttgart2 LANL2DZ LANL2TZ

CL LDL CS ST1 ST2 LD LT

LC LC SC SC SC SC SC

SCPP Cristiansen et al. (DHF) SCPP Hay et al. (CG) SCPP Cristiansen et al. (DHF) ECPP Andrae et al. (WB) ECPP Andrae et al. (WB) SCPP Hay et al. (CG) SCPP Hay et al. (CG)

[23] [24] [23] [25] [25] [26] [27]

CRENBS LANL1DZ CRENBL Stuttgart1 Stuttgart2 LANL2DZ LANL2TZ

CL LDL CS ST1 ST2 LD LT

LC LC SC SC SC SC SC

SCPP Cristiansen et al. (DHF) SCPP Hay et al. (CG) SCPP Cristiansen et al. (DHF) ECPP Andrae et al. (WB) ECPP Schwerdtfeger et al. (DHF) SCPP Hay et al. (CG) SCPP Hay et al. (CG)

[28] [24] [28] [25] [29] [26] [27]

a Core refers to the core-size of the ECP (SC for Small-Core and LC for Large-Core ECPs). In the pseudo-potential description, EC denotes Energy-Consistent and SC Shape-Consistent ECPs. The relativistic approximations used in the fittting procedure are spin–orbit averaged Dirac-Hartree–Fock (DHF), Wood-Boring (WB) and Cowan-Griffin (CG).

effects. We employed two large-core (LC) and five small-core (SC) ECP variants (Table 1) [23–29], as well as the PAW method [16,17], and a scalar relativistic AE approach [19,30–32]. Details of the ECPs are provided as SM. ECP and AE calculations were carried out with the Linear Combination of Gaussian-Type Orbitals Fitting-Functions Density Functional (LCGTO-FF-DF) approach [30] as implemented in the program ParaGauss (version 3.1.8) [31,32]. Scalar relativistic effects were incorporated in the case of the AE calculations via the second-order Douglas-Kroll-Hess scheme [18,19,33,34]; we represented molecular orbitals with AE Gaussian-type basis sets, contracted in generalized fashion with atomic eigenvectors: Pd – (18s13p9d) [35] contracted to [7s6p4d], Au – (21s17p11d7f) [36] contracted to [8s7p5d3f]. In the LCGTO-FF-DF method, the Hartree interaction is calculated for an approximated electronic density, represented by a set of fitting functions [30]. The s- and r2 exponents of this set were obtained from a subset of s- and p-orbital exponents, scaled by a factor of 2; this fitting basis was augmented by sets of five p- and d-type ‘polarization’ exponents [30], chosen as geometric series with factors of 2.5, starting at 0.1 and 0.2 au, respectively. We employed a fractional occupation number technique to facilitate SCF convergence [30], invoking a Fermi-type broadening (0.15 eV). For this model study, Pd and Au clusters were calculated in spin-unrestricted (SU) fashion. For two largest Pd clusters, Pd79 and Pd147, previous AE calculations were carried out in spin-restricted fashion [8]. Geometry optimizations were terminated when the forces on each atom were less than 105 au. The numerical integration of the exchange–correlation contributions were carried out with a superposition of atom-centered spherical grids, locally accurate up to angular momentum l = 17. For Pd147 we were unable to converge the SCF procedure using the ST2 ECP (Table 1). PAW calculations [16,17] were carried out with the software VASP [37–39]. The plane-wave orbital basis was determined by an energy cutoff of 400 eV. Similar to the calculations with ParaGauss, we employed a fractional occupation number technique (energy broadening of 0.15 eV) [40]; total energies were extrapolated to kT = 0 eV. As all systems studied are finite, only the C point of the reciprocal space was used; cubic unit cells were chosen so that the closest distance between atoms of separate images was at least 7 Å. Geometry optimizations were terminated when the Hellmann–Feynman force on each atom were less than 103 au. Pd and Au bulk systems were calculated using a 999 grid of k-points, for comparison with the extrapolated values.

3.1. Average bond lengths We calculated the average bond length, dav, of a cluster as the arithmetic mean of all nearest-neighbor M–M distances in the Mn cluster (M = Pd, Au). The value of dav is lower than the nearest-neighbor distance of the corresponding bulk metal [41,42] because surface atoms, lower coordinated than atoms in the core region of a cluster, are drawn closer to their nearest-neighbors than atoms in the inner core. The latter atoms exhibit the same coordination as in the bulk. Thus dav is expected to decrease with the increasing surface-to-volume ratio, i.e., with n1/3 [4]:

Dav ¼ K av n1=3 þ dbulk

ð1Þ

as kav is determined negative. Values of dav of individual clusters Pdn and Aun as well as parameters of the corresponding linear fit, Eq. (1), for all theoretical approaches studied are collected in the Tables S1 and S2, respectively, of the SM. All approaches, except the two LC ECPs, CL and LDL, show the well-known increase [4] of dav with cluster size. The scaling, Eq. (1), is confirmed by values r2 of the squared correlation coefficients which are at least 0.97 (Tables S1 and S2 of SM); only the PAW method for Aun yields r2 = 0.956 (Table 2). The LC ECPs CL and LDL yield unphysical trends for Pd and Au, where dav decreases with increasing cluster nuclearity n; see the positive values of kav (Tables S1 and S2 of SM). For Pd, the AE reference result, 279.8 pm, is 5 pm longer than the experimental value dbulk = 275 pm [42], which is overestimated by all (remaining) theoretical models, by 4 pm (ST1) to 6 pm (PAW, LD, LT). The deviations are comparable to those of previous results where this overestimation was attributed to the GGA functional used [8]. The computational results for dbulk from ionic-core treatments may be ordered as follows (Table 2): LDL (-2.0) < ST1 (1.5) < ST2 (0.2) < CS (0.2) < LT (0.9) < PAW (1.1) < LD (1.8) < CL (5.9); the values in parentheses are the deviations (pm) from the AE reference. Table S1 and Figure S2 of SM show that all small-core ECPs exhibit trend lines similar to the AE model; the PAW approach features a steeper slope. The extrapolated PAW value, dbulk = 280.9 pm, agrees satisfactorily with the result of 279.3 pm, calculated directly for the bulk material. For gold, the AE reference overestimates the experimental value, dbulk = 288 pm [41], by 5.4 pm. This GGA induced overestimation of bond distances (except for CL) is comparable to the one determined for Pd (4.8 pm, Table 2) [8]. As for Pd clusters, we order the results for dbulk from ionic-core treatments and list the deviation (pm) from the AE reference: CL (5.5) < CS (0.8) < LT (1.0) < ST1 (1.2) < LD (3.5) < PAW (4.3) < ST2 (4.5) < LDL (7.2) (Table 2). Apparently, the ordering of the results for Pd and Au clusters is rather different. The dbulk results of all ionic-core treatments deviate more from experiment for Au than for Pd; for ST2 and PAW, these deviations, 10 pm, are quite pronounced. Also the extrapolated PAW value dbulk = 297.7 pm differs somewhat from the directly calculated result, 294.9 pm. From the erratic deviations of dbulk obtained with LC ECPs for Pd and Au clusters, it is obvious that a physically valid ECP description is only possible by explicitly treating the sub-valence shells, i.e. 4s and 4p of Pd and 5s and 5p for Au. The corresponding LC results for binding energies Ecoh (Table 2) are even less useful. Therefore, we will refrain from discussing results of LC ECPs in detail. 3.2. Cohesive energy The cohesive energy Ecoh per atom of a cluster Mn is

Ecoh ðMn Þ ¼ EðMÞ  EðMn Þ=n

ð2Þ

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R. Marchal et al. / Chemical Physics Letters 578 (2013) 92–96

Table 2 Deviations of the various ionic-core approximations, compared to our reference AE properties extrapolated to the bulk limit. dbulk in pm, Ecoh in kJ mol1, UVIP, UVEA, DU in eV and DEbulk in eV. Deviations form AE

Pd

dbulk Ebulk

UVIP UVEA DU Au

dbulk Ebulk

UVIP UVEA DU DEbulk a

Absolute values

CL

LDL

CS

ST1

ST2

LD

LT

5.9 198.3 0.18 0.20 0.01

2.0 182.7 0.40 0.08 0.33

0.2 48.3 0.11 0.11 0.00

1.5 20.1 0.09 0.26 0.35

0.2 14.3 0.15 0.01 0.16

1.8 8.2 0.03 0.11 0.07

0.9 23.5 0.07 0.12 0.05

5.5 87.1 0.76 0.57 0.19 111.3

7.2 115.7 0.78 0.24 0.54 67.0

0.8 8.9 0.03 0.06 0.02 57.1

1.2 14.8 0.02 0.04 0.03 34.9

4.5 18.2 0.01 0.03 0.04 32.5

3.5 21.5 0.05 0.08 0.03 29.7

1.0 6.2 0.13 0.16 0.02 29.6

AE

AEa

exp.

1.1 8.8 0.07 0.01 0.07

279.8 377.4 4.95 4.80 0.15

279.7 375.6 5.04 4.90 0.14

275.0 376.0 5.22 5.22 0.00

4.3 19.8 0.06 0.06 0.00 11.0

293.4 325.1 4.72 4.45 0.27 52.3

293.8 330.2 – – – 45.4

288.0 366.0 5.1 5.1 0.0 10.0

PAW

Values from Ref. [8].

where E(Mn) is the total energy of the cluster and E(M) the total energy of a metal atom M. Ecoh(Mn) differs from the bulk cohesive energy Ebulk mainly due to the reduced coordination of the surface atoms [5]. Thus, as the surface-to-volume ratio decreases with the nuclearity n, Ecoh(Mn) increases linearly with decreasing n1/3 to approach the bulk value:

Ecoh ¼ K coh n1=3 þ Ebulk

ð3Þ

Tables S3 and S4 of SM report values of the individual Pd and Au clusters as well as the parameters of the linear fits, Eq. (3). The quality of the scaling, Eq. (3), is again confirmed by adequate values of the squared correlation coefficients r2 of at least 0.98. For Pd clusters, the AE approach yields Ebulk = 377.4 kJ mol1 which agrees very well with the experimental value, 376 kJ mol1[43]. All SC results of Ebulk overestimate the AE result, hence the experiment, in part to a notable extent, by up to 48.3 kJ mol1 (CS), while the PAW approach underestimates the AE result by 8.8 kJ mol1 (Table 2). These results of the ionic-core treatments can be ordered as follows: PAW (9) < LD (8) < ST2 (14) < ST1 (20) < LT (24) < CS (48); deviations from the AE reference (kJ mol1) are given in parentheses (Table 2). The LC approaches, CL and LDL, again do not provide useful results; they strongly underestimate the AE Ebulk, by 198 and 183 kJ mol1, respectively (Table 2). The direct PAW calculation of the bulk yields Ebulk = 359 kJ mol1, even smaller than the extrapolated value, 369 kJ mol1. For gold clusters, all present extrapolated results of Ebulk underestimate, in part notably, the experiment, 366 kJ mol1[44]. The best result, 325 kJ mol1, is again obtained from the AE reference calculations. We order the results for Ebulk and list the deviation (kJmol1) from the AE reference (Table 2): LD (22) < PAW (20) < ST2 (18) < ST1 (15) < CS (9) < LT (6). For Au, the direct bulk calculation with PAW yields Ebulk = 289 kJ mol1 while the extrapolated value is 305 kJ mol1. As mentioned in the Introduction, one motivation for the present study was to assess the accuracy of various ECP DFT approaches for studying PdAu nanoalloys. As experimental data do not seem to be available for the energetics of such nanoparticles, we discuss the difference DEbulk = Ebulk(Pd)Ebulk(Au) between the extrapolated Ebulk values of Pd and Au. This quantity is an important descriptor when studying the surface segregation of alloys [45] and bimetallic nanoparticles [46]. The AE reference, 52 kJ mol1, overestimates the experimental result, by 42 kJ mol1, mainly due to the underestimation of the Au cohesive energy. Unfortunately, none of the various ionic-core treatments offer any favorable error compensation for DEbulk. On the contrary, all ionic-core treatments examined exhibit positive deviations (kJ mol1) from the AE reference: PAW (11) < LD = LT (30) < ST2 (33) < ST1 (35) < CS (57) (Table 2).

With regard to the relative accuracy of describing Pd and Au nanoparticles, only the PAW method, DEbulk = 63 kJ mol1, is close to the AE result. Note that the direct PAW calculations of the bulk materials yield a rather similar difference, 70 kJ mol1. Almost all SC treatments yield larger values for DEbulk, at 85 kJ mol1, notably larger than the experimental difference. For CS ECP one obtains an even larger difference, 110 kJ mol1. Thus, all computational approaches, AE, PAW and SC ECPs, underestimate the stability of Au nanoparticles relative to those of Pd nanoparticles. This is particularly noticeable for the LC ECPs. 3.3. Ionization potential and electron affinity The vertical ionization potential, VIP, and the vertical electron affinity, VEA, of a cluster Mn are defined as

VIP ¼ EðMþn Þ  EðMn Þ; VEA ¼ EðMn Þ  EðMn Þ EðMþ nÞ

ð4Þ

EðM nÞ

and are the total energies of the cluster cation and anion, respectively, at the geometry of the neutral cluster. In the spherical droplet model [47], one derives VIP and VEA of a spherical metallic particle of radius R by applying a correction to the bulk work function that is proportional to 1/R, i.e. to n1/3 [5,6,48]:

VIP ¼ K VIP n1=3 þ UVIP ; VEA ¼ K VEA n1=3 þ UVEA

ð5Þ

The results for all Pd and Au clusters as well as the parameters of the linear fits according to Eq. (5) are reported in Table S5 and S6, respectively, of SM. Leaving aside the results of LC ECPs, then the values of VIP and VEA of Pdn clusters change quite monotonically with the size n of the system; r2 is at least 0.91 (Table S5 of SM). The AE value of UVIP, 4.95 eV, slightly underestimates the experimental work function of bulk Pd, 5.22 eV (Table 2) [44]. We order the results of UVIP according to the deviations (in eV) from the AE reference (Table 2): ST1 (0.09) < PAW (0.07) < LD (0.03) < LT (0.07) < CS = ST2 (0.15). For VEA, the AE result, 4.80 eV, is notably smaller, by 0.42 eV, than the experimental bulk work function (Table 2). The various ioniccore results for VEA deviate similarly from the AE value and thus yield: ST2 = PAW (0.01) < LD = CS (0.11) < LT (0.12) < ST1 (0.26). For large metal clusters, one expects that the difference VIPVEA decreases with n1/3, vanishing in the limit of very large systems:

VIP  VEA ¼ K D n1=3 þ DU

ð6Þ

The residual difference DU thus quantifies the error made when extrapolating the work function from VIP and VEA data. This linear relationship for the energy difference is even better fulfilled than for the individual quantities. The resulting squared correlation coefficients r2 of the difference VIPVEA are 0.99 (with one

R. Marchal et al. / Chemical Physics Letters 578 (2013) 92–96

exception, ST2), and always above the correlation coefficients determined for individually terms, VIP and VEA. The error DU = 0.15 eV of the AE results for the Pdn clusters is notably smaller than the individual extrapolation errors of VIP and VEA from the work function (Table 2), indicating error compensation. The value DU reflects various deficiencies [49]: (1) the incomplete cancelation of the self-interaction error of the exchange–correlation functional, (2) the incomplete, finite basis set, and (3) limitations of the ionic-core approximation, if used. The deviations (in eV) of the various ionic-core approximations from the AE reference yield the following ordering (Table 2): ST1 (0.35) < PAW = LD (0.07) < LT (0.05) < CS (0.00) < ST2 (0.16). For the gold clusters, the linear relationships, Eqs. (5), are notably less well fulfilled. Leaving aside the LC ECPs, then the squared correlation coefficients r2 of the VIP results are 0.75; for the VEA results we note r2 values of only 0.5 (Table S6 of SM). This effect agrees with previous results of our group [5]. Indeed, the spherical droplet predicts scalability of VIP and VEA only for metallic particles. However, some Au clusters examined in this work do not qualify for such scaling in view of large HOMO–LUMO gaps and low densities of states near the Fermi level (cf. Pd clusters of analogous size; Table S7 of SM). Yet, the differences VIPVEA follow the linear relationship, Eq. (6), extremely well, with r2 values very close to 1, including even those for the LC results. As for Pd, the AE results for gold, UVIP = 4.72 eV and UVEA = 4.45 eV (Table 2), slightly underestimate the experimental bulk work function, 5.1 eV [50]. With deviations from AE results given in parentheses (eV), the results for UVIP obtained with ionic-core treatments yield the following series (Table 2): ST2 (0.01) < ST1 (0.02) < CS (0.03) < LD (0.05) < PAW (0.06) < LT (0.13). The corresponding series for UVEA is (Table 2): ST2 (0.03) < ST1 (0.04) < CS (0.06) < PAW (0.06) < LD (0.08) < LT (0.16). These two orderings, including the values of deviations, are very similar. The values of DU for Aun are rather uniform and agree well with the AE reference, 0.24 eV. The deviations (in eV) from the AE reference give rise to the following order: ST2 (0.04) < ST1 = LD (0.03) < LT = CS (0.02) < PAW (0.00).

4. Conclusions The various ionic-core approaches studied yield results of different quality (Table 2), depending on the metal, Pd or Au. We briefly summarize the performance of the various ionic-core approximations examined: - The LC approximations CRENBS and LANL1DZ are unable to reproduce accurately any of the properties of interest. - The PAW method can be placed at the other end of the accuracy spectrum. All properties are adequately described, except for dbulk(Au). Moreover, the difference DEbulk = 63 kJ mol1, of the extrapolated bulk cohesive energies of Pd and Au agrees quite well with the AE reference, 53 kJ mol1. Thus, when aiming at Pd-Au nanoalloys, it appears that the PAW method provides the most accurate ionic-core approach for binding energies, a key quantity for exploring surface segregation. - The accuracy of results of SC ECPs varies with system and property. CRENBL and LANL2TZ are the most accurate small-core ECPs for Au. In contrast, these two approaches are not recommended for Pd clusters in view of the large overestimation of the AE results for Ecoh. For Pd, the two most accurate ECP methods are the Stuttgart2 and LANL2DZ approaches, while both strongly overestimate the AE result for dbulk(Au). Calculated bond distances and binding energies per atom obtained with the SC and the PAW approach follow the anticipated

95

scaling with cluster size. The results for ionization potentials and electron affinities scale less well in the size range examined, especially for Au clusters where the ‘nonmetallic’ behavior of some Au clusters is quite noticeable. None of the ionic-core treatments examined returned ‘optimal’ results for all properties studied, structure, energy and electronic properties of both types of clusters, Pdn and Aun. The large-core approaches CRENBS and LANL1DZ are not recommended in any case. It is noteworthy that the PAW approach overall seems to reproduce best the all-electron results. Acknowledgments We thank Dr. Sven Krüger for very helpful discussions. R.M. is grateful for financial support by the Wacker-Institut für Siliziumchemie at TU München. We thank Leibniz Rechenzentrum München for providing generous computing resources. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.cplett.2013. 05.063. References [1] P. Braunstein, L.A. Oro, P.R. Raithby, Metal Clusters in Chemistry, Wiley, Weinheim, 1999. [2] U. Heiz, U. Landman, Nanocatalysis, Springer, Berlin, Heidelberg, 2007. [3] I.V. Yudanov, A. Genest, S. Schauermann, H.-J. Freund, N. Rösch, Nano Lett. 12 (2012) 2134. [4] S. Krüger, S. Vent, N. Rösch, Ber. Bunsenges. Phys. Chem. 101 (1997) 1640. [5] O.D. Häberlen, S.-C. Chung, M. Stener, N. Rösch, J. Chem. Phys. 106 (1997) 5189. [6] I.V. Yudanov, M. Metzner, A. Genest, N. Rösch, J. Phys. Chem. C 112 (2008) 20269. [7] J. Kleis, J. Greeley, N. Romero, V. Morozov, H. Falsig, A. Larsen, J. Lu, et al., Catal. Lett. 141 (2011) 1067. [8] R. Koitz, T.M. Soini, A. Genest, S.B. Trickey, N. Rösch, J. Chem. Phys. 137 (2012) 034102. [9] P. Nava, M. Sierka, R. Ahlrichs, Phys. Chem. Chem. Phys. 5 (2003) 3372. [10] N. Rösch, L. Ackermann, G. Pacchioni, J. Am. Chem. Soc. 114 (1992) 3549. [11] A. Genest, S. Krüger, A.B. Gordienko, N. Rösch, Z. Naturforsch. 59 (2004) 1585. [12] R. Ferrando, J. Jellinek, R.L. Johnston, Chem. Rev. 108 (2008) 845. [13] F. Gao, D.W. Goodman, Chem. Soc. Rev. 41 (2012) 8009. [14] F. Weigend, R. Ahlrichs, Phil. Trans. R. Soc. A 368 (2010) 1245. [15] M. Dolg, X. Cao, Chem. Rev. 112 (2011) 403. [16] P.E. Blöchl, Phys. Rev. B 50 (1994) 17953. [17] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. [18] M. Reiher, A. Wolf, Relativistic Quantum Chemistry–The Fundamental Theory of Molecular Science, Wiley, Weinheim, 2009. [19] N. Rösch, A.V. Matveev, V.A. Nasluzov, K.M. Neyman, L.V. Moskaleva, S. Krüger, in: P. Schwerdtfeger (Ed.), Relativistic Electronic Structure Theory – Applications, Elsevier, Amsterdam, 2004, p. 656. [20] J. Tsuji, Palladium Reagents and Catalysts: New Perspectives for the 21st Century, Wiley, Weinheim, 2004. [21] M. Stratakis, H. Garcia, Chem. Rev. 112 (2012) 4469. [22] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [23] L.A. Lajohn, P.A. Christiansen, R.B. Ross, T. Atashroo, W.C. Ermler, J. Chem. Phys. 87 (1987) 2812. [24] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 270. [25] D. Andre, U. Häußermann, M. Dolg, H. Stoll, H. Preuß, Theor. Chim. Acta 77 (1990) 123. [26] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 299. [27] L.E. Roy, P.J. Hay, R.L. Martin, J. Chem. Theor. Comp. 4 (2008) 1029. [28] R.B. Ross, J.M. Powers, T. Atashroo, W.C. Ermler, L.A. Lajohn, P.A. Christiansen, J. Chem. Phys. 93 (1990) 6654. [29] P. Schwerdtfeger, M. Dolg, W.H.E. Schwarz, G.A. Bowmaker, P.D.W. Boyd, J. Chem. Phys. 91 (1989) 1762. [30] B.I. Dunlap, N. Rösch, Adv. Quantum Chem. 21 (1990) 317. [31] T. Belling, T. Grauschopf, S. Krüger, M. Mayer, F. Nörtemann, M. Staufer, C. Zenger, N. Rösch, in: H.-J. Bungartz, F. Durst, C. Zenger (Eds.), High Performance Scientific and Engineering Computing, Springer, Berlin Heidelberg, 1999, p. 441. [32] T. Belling, T. Grauschopf, S. Krüger, F. Nörtemann, M. Staufer, M. Mayer, V.A. Nasluzov, U. Birkenheuer, A. Hu, A.V. Matveev, A.V. Shor, M.S.K. Fuchs-Rohr, K.M. Neyman, D.I. Ganyushin, T. Kerdcharoen, A. Woiterski, S. Majumber, N. Rösch, ParaGauss, version 3.1., Theoretische Universität München, 2006. [33] R.J. Buenker, P. Chandra, B.A. Hess, Chem. Phys. 84 (1984) 1.

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