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Supplementary Material

Scalable Properties of Metal Clusters:

A Comparative DFT Study of Ionic-Core Treatments

Remi Marchal,1 Ilya V. Yudanov,2,3 Alexei V. Matveev,3 and Notker Rösch3,4*

1. Department Chemie & Wacker-Institut für Siliziumchemie, Technische Universität München, 85747 Garching, Germany

2. Boreskov Institute of Catalysis, Russian Academy of Sciences, 630090 Novosibirsk, Russian Federation

3. Department Chemie & Catalysis Research Center, Technische Universität München,

85747 Garching, Germany

4. Institute of High Performance Computing, Agency for Science, Technology and Research,

1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore

* To whom correspondence should be addressed. E-mail: roesch@mytum.de

1. ECP formalism

The general form of an ECP [15] is

[pic] (7)

where Vlocal is an attractive local part and Vnl corresponds to a non-local and generally repulsive part. The local part is:

[pic] (8)

where Q = Z – Nc is the total charge of the core of an atom, Z being the nuclear charge and Nc the number of core electrons represented by the ECP. For the large-core ECPs CRENBS (CS) and LANL1DZ (LDL), Nc(Pd) = 36 and Nc(Au) = 68, while for all (small-core) ECPs Nc(Pd) = 28 and Nc(Au) = 60. The non-local part Vnl of the potential is given as the sum of products of Gaussian functions and projection operators [pic] onto the subspace of spherical harmonics [pic][15]:

[pic] (9)

[pic] (10)

Blk and [pic]are adjustable parameters, [pic] is an integer. The projectors [pic] apply only to angular momentum up to Lmax. To avoid this drawback, some of ECPs are constructed by adding a further local part to the potential [15]:

[pic] (11)

Here the first term is a local term that acts on all angular momentum components of the wave-function. The ECPs CRENBS (CS), CRENBL (CL), Stuttgart1 (ST1), LANL2DZ (LD), LANL2TZ (LZ) and LANL1DZ (LDL) are built in this latter way, while the Stuttgart2 (ST2) ECP used in this study does not include such an additional local part. This could explain why we were not able to converge the SCF procedure for Pd147 when we applied the ST2 ECP.

2. Tables

Table S1. Average bound lengths dav of Pdn clusters, optimized with different ionic-core approximations, and parameters of the linear fit as a function of n-1/3. Distances in pm. For the abbreviation, refers to Table 1.

| |CL |LDL |CS |ST1 |ST2 |LD |

|Pd13 |0.10 |13 |Au13 |0.62 |4 | |

|Pd19 |0.21 |14 |Au19 |0.09 |3 | |

|Pd38 |0.07 |26 |Au38 |0.12 |5 | |

|Pd55 |0.00 |40 |Au55 |0.29 |6 | |

|Pd79 |0.03 |60 |Au79 |0.06 |9 | |

|Pd147 |0.03 |104 |Au147 |0.01 |19 | |

3. Comparison of Spin-Restricted and Spin-Unrestricted Results for Pd Clusters

For Pd clusters, a detailed comparison between results of spin-restricted and spin-unrestricted calculations shows that spin polarization leaves all extrapolated properties unchanged (Table S1, S3, and S5). Indeed, for Pd79, the largest (absolute) deviations, in the case are 0.1 pm for dav, 1.6 kJ·mol-1 for Ecoh, 0.15 eV for VIP, and 0.14 eV for VEA. These minor deviations would change extrapolated values only insignificantly: 0.1 pm for dbulk, 1.8 kJ·mol-1 for Ebulk, 0.09 eV for ΦVIP, and 0.10 eV for ΦVEA.

4. Figures

[pic][pic][pic]

M13 M19 M38

[pic][pic]

M55 M79

[pic]

M147

Figure S1. Sketches of octahedral clusters of nuclearities 13, 19, 38, 55, 79 and 149.

[pic]

Figure S2. Functions linear in n–1/3 fitted to average bond lengths dav (in pm) of the Pdn cluster studied. The experimental value dbulk = 275 pm is shown as dashed horizontal line.

[pic]

Figure S3. Functions linear in n–1/3 fitted to average bond lengths dav (in pm) of the Aun cluster studied. The experimental value dbulk = 288 pm is shown as dashed horizontal line.

[pic]

Figure S4. Functions linear in n–1/3 fitted to cohesive energies Ecoh (in kJ·mol–1) of the Pdn cluster studied. The experimental value Ecoh = 376 kJ·mol–1 is shown as dashed horizontal line.

[pic]

Figure S5. Functions linear in n–1/3 fitted to cohesive energies Ecoh (in kJ·mol–1) of the Aun cluster studied. The experimental value Ecoh = 366 kJ·mol–1 is shown as dashed horizontal line.

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