On the Effects of All-Electron Basis Sets and the Scalar Relativistic

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On the Effects of All-Electron Basis Sets and the Scalar Relativistic Corrections in the Structure and Electronic Properties of Niobium Clusters Fernando Néspoli Nassar Pansini, António C. Neto, Miguel Gustavo de Campos Batista, and Rafaela Moraes de Aquino J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b04600 • Publication Date (Web): 07 Jul 2017 Downloaded from http://pubs.acs.org on July 14, 2017

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The Journal of Physical Chemistry

On the Effects of All-Electron Basis Sets and the Scalar Relativistic Corrections in the Structure and Electronic Properties of Niobium Clusters F. N. N. Pansini,∗,† A. C. Neto,∗,† M. de Campos,∗,‡ and R. M. de Aquino‡ Departamento de F´ısica, Universidade Federal do Esp´ırito Santo, 29075-910 Vitória, Brazil, and Departamento de F´ısica, Universidade Federal de Roraima, 69310-000 Boa Vista, Brazil E-mail: [email protected]; [email protected]; [email protected]

∗ To

whom correspondence should be addressed de F´ısica, Universidade Federal do Esp´ırito Santo, 29075-910 Vitoria, ´ Brazil ‡ Departamento de F´ısica, Universidade Federal de Roraima, 69310-000 Boa Vista, Brazil † Departamento

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Abstract In this paper, an augmented all-electron double zeta basis set is used in calculations of the structure and electronic properties of small niobium clusters. The B3PW91 and M06 DFT functionals with and without second order Douglas-Kroll-Hess (DKH) scalar relativistic corrections are also utilized. Furthermore, an additional d Gaussian type function is introduced in the standard basis sets in order to improve the description of the clusters orbitals in the valence band. Our findings show that the extra d function is important to yield accurate results of electronic properties and, in addition, the DKH corrections can be relevant when the all-electron basis sets are used in the calculations. Our best results are obtained with the M06 functional together with the DKH second order corrections and with the extra d function added to the all-electron basis set.

Introduction Various theoretical studies of niobium clusters have been performed by many authors, 1–13 in the most cases, the density functional theory (DFT) together with the effective core potential (ECP), that is synonymous with pseudopotential (PP), are utilized. For ECP approach, only the valence electrons are explicitly treated and a few suitable functions are used to modeling the core. Kumar and Kawazoe 5 have studied Nbn clusters for 2 ≤ n ≤ 23 by using the ultrasoft PP and plane wave expansion with the generalized gradient approximation 14 exchange-correlation functional. Nhat et al. 7–9 performed the calculations with the cc-pVXZ-PP basis sets (X = D, T), where PP denotes the pseudopotential. Specifically, the authors have used the BPW91 and M06 DFT functionals with ccpVTZ-PP basis set for Nbn clusters with 2 ≤ n ≤ 6, 7 and cc-pVDZ-PP with 7 ≤ n ≤ 20. 8,9 In most cases, the ECP can provide good values of geometrical parameters and reproduce many experimental results. In this approach (ECP), the relativistic effects can be previously included in the core when the pseudo-potential is calibrated, while in all2


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electron calculations, the relativistic effects are in fact obtained for the molecule. Thus, when an all-electron basis sets are used in the relativistic calculations a better description of the molecular orbitals is expected. Note that relativistic effects can be relevant for small clusters 15 and even for molecules with first-row elements of the Periodic Table. 16 The niobium clusters are difficult target for theoretical predictions, due to the presence of various 4d (and one 5s) electrons. 7 The accurate description of these electrons is important to correctly describe the chemical bonding of the clusters. Therefore, the main goals of our work, all related to Nb clusters, are the following: to understand the influence of the scalar relativistic effects; to investigate the performance of the standard all-electron basis sets in the structure and electronic properties calculations; to improve the description of the valence band by adding one d Gaussian type function (GTF) in the basis sets; to yield accurate results for structure and electronic properties. The computational and theoretical details are shown in the next section, then the results and discussions are presented, and finally, the concluding remarks.

Computational and Theoretical Details In the DFT conceptual framework, the electron chemical potential (µ) and the chemical hardness (η ) were exactly defined by Pearson, Parr, and Yang 17,18 as

µ=

∂E ∂N

(1) ν (r )

and η=

∂2 E ∂N 2

= ν (r )

3

∂µ ∂N

, ν (r )


(2)


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where the ν(r) is the external potential. The ionization potential (I) and electron affinity (A) can be defined as I = E ( N − 1 ) − E ( N ),

and

(3)

A = E ( N ) − E ( N + 1 ). Where E( N − 1), E( N ), and E( N + 1) are the energies of the system containing, respectively, ( N − 1), N, and ( N + 1) electrons. Taking into account the definitions given by Equations (1) and (2), and from the numerical differentiation, we can write 1 −µ = χ M = ( I + A), and η = ( I − A). 2

(4)

Where the chemical potential can be associated with the Mulliken electronegativity (χ M =

−µ), that provide the capacity of the system in to attract electrons. The Maximum Hardness Principle (MHP) 17–19 says to be a rule of nature that molecules arrange themselves to be as hard as possible. So, the hardness provides important informations about the stability of the system. Now, we can approximate I and A with the energies of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), i.e., I ≈ −ǫ HOMO and A ≈ −ǫLUMO . 17–19 Thus, from the Equations (4) we finally found 1 −χ M = µ = (ǫLUMO + ǫ HOMO ), 2

and (5)

η = (ǫLUMO − ǫ HOMO ). From the MHP, the most stable structure is the one that provides the larger value of η, in other words, that posses the highest gap between ǫLUMO and ǫ HOMO . 17–19 Of course, Equations (5), and (4) are approximate, but one can use them to calculate the chemical potential (electronegativity) and chemical hardness with relatively good results, at least qualitatively. 19,20 4


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From the η and µ, we can define the electrophilicity index as 21,22

ω=

µ2 , 2η

(6)

that measures the stabilization of the system when its acquires an additional electron from the neighborhood. So, for the most stable systems, are expected low values of the electrophilicity, i.e., the system is less prone to accept one additional electron to become more stable. In this work, the ionization potential (I) and electroaffinity (A) were calculated with the vertical detachment energies in the geometries of the neutral clusters, i.e., the E( N + 1) is the energy of the Nbn cluster with one additional electron in the system and the E( N − 1) is the Nbn cluster with one less electron. In all cases, the doublet spin state was adopted for the calculations of the E( N + 1) and E( N − 1) energies at the restrict openshell DFT framework. The hardness, chemical potential, and electrophilicity were then calculated from the I and A values. Note that, in addition to the approximations due to finite difference method used in the Equations (4), there are approaches produced by the relaxation of the orbitals when the E( N + 1) and E( N − 1) energies are calculated and used in Equation (3). Dealing specifically with our object of study, the Nbn (n = 4, 6, 8, 10) neutral clusters are investigated by using the Density Functional Theory (DFT) with the B3PW91 23,24 and M06 25 functionals. The basis sets considered are: the augmented all-electron (ae) contracted double-ζ 26 (ADZP), and the ADZP with Douglas-Kroll-Hess corrections, 27–29 ADZP-DKH, all of them, of Jorge and coworkers. 30 The ADZP-DKH basis used here, is the standard DZP-DKH 30 but with the augmented functions taken from the ADZP 26 basis set. Therefore, we have performed our calculations with (ADZP-DKH) and without (ADZP) second order Douglas-Kroll-Hess 27–29 (DKH) relativistic corrections. The ADZP and ADZP-DKH basis sets have shown good performance in the ab initio and DFT calcu-

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lations when transition metal atoms are present 15,26,31–35 in the molecule. Furthermore, the B3PW91 and M06 functionals were used in previous works with a good performance when applied in the electronic structure calculations of the metal clusters. 7–11,15,36,37 In order to better describe the clusters orbitals in the valence band, we have added one d function to the basis sets ADZP and ADZP-DKH for the Nb atom. All molecular calculations were performed with computational chemistry programs GAMESS 38 and MOLPRO. 39

Results and Discussion To find the most stable structure of the niobium clusters (Nbn , n = 4, 6, 8, 10), the optimization at all-electron B3PW91/ADZP level of theory is performed by starting from the low lying isomers obtained in the references 7 and 8, which no constraint was imposed during this process. Additionally, the frequencies were also analyzed in order to verify if the obtained structure is a true minimum. Note that Nb6 cluster has different multiplicity, 7,8 but the spin-state splitting between the most stable singlet and triplet structures is only of 0.08 eV. 7 Consequently, the coexistence of different spin-states is perfectly possible. Therefore, in the present work, only the singlet spin-state is analyzed. Due the importance of the relativistic corrections observed for some small clusters, 15 we made the geometry re-optimization with the second order DKH 27–29 relativistic correction by taking as the starting point the most stable geometries obtained at ae-B3PW91/ADZP level. The all-electron basis set ADZP-DKH together with the B3PW91 functional was adopted in the DKH calculations. In order to improve the description of the valence band we added one uncontracted d GTF in the ADZP basis set. The new d GTF was the one that provided the lower total MP2 (5 correlated electrons) energy for the Nb atom, that is the methodology also used by Jorge and coworkers 40 to generate the standard ADZP basis set. Thus, the exponent of the additional function assume the value of 1.5220. For the relativistic basis set ADZP-

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DKH we have added the same d GTF. The new basis sets will be labeled as ADZP(+d) and ADZP-DKH(+d) , respectively. The Figure 1 shows the most stable structure of the clusters obtained at ae-B3PW91/ADZP level of theory, and the smallest and largest bond lengths for the relativistic (DKH) and non-relativistic (NR) levels. The x, y, z coordinates for each level of theory are available in the Supporting Information. For the ADZP basis sets, the DKH distances presented in the Figure 1 are larger than the NR ones (except for the Nb10 largest bonding), maximum ˚ respectively. Considering all bond lengths, and minimum deviation of 0.051 and 0.001 A, ˚ for the Nb6 . By comparthe average differences assume the maximum value of 0.005 A, ing the bond distances obtained here with the others theoretical results, 5,7,8 we can note that the ae-B3PW91/ADZP values overestimate the previous ECP theoretical data 5,7,8 for the most cases. For example, our results for the Nb4 bonding distances are 0.07 and 0.08 ˚ (respectively for NR and DKH levels) larger than those obtained by Nhat et al. 7 at A BPW91/aug-cc-pVTZ-PP level of theory. This behavior is also observed for the Nb6 and Nb8 clusters, and when the results are compared with the Kumar et al. 5 values. For the ˚ than those calculated by Nhat et al., 8 and smaller (0.2 Nb10 , our results are larger (0.06 A) ˚ than the ones calculated by Kumar et al. 5 The ADZP(+d) bonding lengths, in all of A) cases (DKH and NR), are smaller than those obtained from ADZP or ADZP-DKH basis, and closer to the Nhat et al. 7,8 results. Comparing the ADZP(+d) and ADZP-DKH(+d) , the relativistic distances are larger than the NR if we take only the lowest and largest bonding lengths. In the average (considering all distances), the Nb4 and Nb6 clusters present larger bond distances with DKH corrections than the non-relativistic results, differences of 0.015 ˚ respectively. For the Nb8 and Nb10 the opposite behavior is observed, difand 0.002 A, ˚ respectively for DKH and NR. Note that for the geometric ferences of 0.043 and 0.002 A parameters, the relativistic corrections can have a significant effect for some systems. Table 1 gathers the binding energy (BE), HOMO-LUMO gap, hardness (η), electronegativity (χ M = −µ), and the electrophilicity (ω) for the Nbn clusters. Additionally, the

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Figure 1: Ground state neutral clusters structures calculated at ae-B3PW91/ADZP and aeB3PW91/ADZP(+d) levels of theory, with (DKH) and without (NR) relativistic correction. theoretical 5,7,8 and experimental 41 reference data are also shown. Notable is that aeB3PW91/ADZP and ae-B3PW91/ADZP-DKH binding energy values are larger than all reference data, theoretical and experimental. The ADZP and ADZP-DKH results can be 1.6 eV larger than the experimental ones. The mean percentage deviations with respect the experimental values are 28% and 36% for NR and DKH levels, respectively. The BE results obtained from the basis sets with the additional d function (see Table 1) together with B3PW91 functional have presented a significant improvement when compared with the ae-B3PW91/ADZP and ae-B3PW91/DKH-ADZP levels of theory. Mean percentage

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deviations of 4% (NR) and 3% (DKH) were obtained when the BPW91 values of Nhat et al. 7,8 are taken as reference. For the M06/cc-pVXZ-PP data, 7,8 the average deviations obtained with (+d) basis sets are 15% and 8% for the NR and DKH levels, respectively. When the Kumar et al. 5 data are taken as reference, the deviations are 10% and 3% for the NR and DKH levels, in this order. For the experimental data, our results shown relatively good agreement, mostly for the ae-B3PW91/ADZP-DKH(+d) level of theory, where the mean deviation is of 8% (0.4 eV). For the ae-B3PW91/ADZP(+d) level, 15% of average deviation was obtained. Additionally, the DKH values are larger than the NR ones, average deviation of 0.3 eV (see Figure 2). Table 1 also shows the results obtained with the M06/ADZP(+d) and M06/ADZPDKH(+d) levels of theory, where all geometries were re-optimized with the respective approaches. For the BE, the M06 functional provides better results than the B3PW91 when the experimental values are taken as reference, deviations of 11% and 3% for the NR and DKH, in this order. Thus, by following the same behavior observed for the B3PW91 results, the M06/ADZP-DKH(+d) binding energy values are closer to the experimental than those calculated without scalar relativistic corrections. As shown in Figure 2, the M06/ADZP-DKH(+d) level of theory is the only one that provides results within experimental uncertainties. As expected, the BE increases with the cluster size, 5,7–9 which is observed for all levels of theory used in this work. Summarizing, the ADZP(+d) and ADZP-DKH(+d) basis sets are most suitable for BE calculation than the standard ADZP and ADZP-DKH basis. The HOMO-LUMO gaps (see Tab. 1) calculated with ADZP and ADZP-DKH basis sets have presented smaller values than those obtained with the (+d) basis sets. When compared with the Nhat et al. 7,8 results, the best agreement was found with the (+d) sets. Furthermore, the Kumar data 5 are significantly lower than the our results and those calculated by Nhat et al. 7,8 . In general, the DKH corrections decrease the HOMO-LUMO gaps when compared with the NR calculations.

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Table 1: Binding Energy (eV/atom), HOMO-LUMO gap (eV), ionization potential (eV), electron affinity (eV), hardness (eV), electronegativity (χ M = −µ) (eV), and electrophilicity for the Nbn clusters, calculated from M06 and B3PW91 functionals with the all-electron basis sets ADZP, ADZP-DKH, ADZP(+d) , and ADZP-DKH(+d). The theoretical and experimental reference values are also shown. Functional Cluster

BE a

( ǫL − ǫH ) b

I

A

η

−µ = χ M

ω

BE a ( ǫL − ǫH ) b

ADZP

I

A

η

−µ = χ M

ω

ADZP-DKH

B3PW91 Nb4

4.95

1.95

5.70

1.05

4.65

3.38

1.23

5.29

1.89

5.66 1.04 4.62

3.35

1.22

Nb6

5.75

1.21

4.95

1.20

3.75

3.07

1.26

6.07

1.16

4.86 1.05 3.81

2.96

1.15

Nb8

6.25

1.15

5.11

1.67

3.45

3.39

1.67

6.57

1.07

5.10 1.46 3.63

3.28

1.48

Nb10

6.61

1.82

6.12

1.54

4.59

3.83

1.60

6.96

1.73

5.60 1.62 3.98

3.61

1.64

ADZP(+d)

ADZP-DKH(+d)

B3PW91 Nb4

3.30

2.13

5.57

0.67

4.90

3.12

0.99

3.54

2.08

5.53 0.72 4.81

3.13

1.02

Nb6

3.69

1.14

5.05

1.38

3.67

3.21

1.41

4.04

1.22

5.10 1.29 3.81

3.37

1.39

Nb8

4.22

1.61

5.61

1.37

4.24

3.49

1.44

4.53

1.66

5.41 1.33 4.08

3.37

1.39

Nb10

4.42

2.35

5.53

0.86

4.67

3.19

1.09

4.73

2.05

5.43 1.03 4.40

3.23

1.19

Nb4

3.52

2.29

5.80

0.81

4.99

3.31

1.10

3.80

2.20

5.72 0.86 4.86

3.29

1.12

M06 Nb6

3.85

1.21

5.32

1.70

3.61

3.51

1.71

4.28

1.16

5.30 1.46 3.84

3.38

1.49

Nb8

4.41

1.69

5.65

1.52

4.13

3.59

1.56

4.75

1.79

5.43 1.47 3.96

3.45

1.50

Nb10

4.61

2.45

5.55

1.06

4.49

3.30

1.21

4.94

2.19

5.46 1.20 4.25

3.33

1.30

BE a

(ǫ L − ǫ H )b

I

Reference Nb4

3.90±0.16c

2.25d 1.20 f 5.64g 5.66h

A

η

1.10i 0.86h

4.54 j 4.80k

−µ = χm ω 3.37 j 1.25 j 3.26k 1.11k

3.49d 3.99e 3.74 f Nb6 4.44±0.20c 1.44d 0.31 f 5.38g 1.48i 3.80 j 3.48 j 1.59 j 3.97d 4.42e 5.22h 1.48h 3.74k 3.35k 1.50k 4.18 f Nb8 4.90±0.24c 1.86d 0.78 f 5.53g 1.45i 4.08 j 3.49 j 1.49 j 4.28d 4.87e 5.38h 1.33h 4.05k 3.36k 1.39k f 4.65 Nb10 5.09±0.27c 2.25d 0.99d 5.48g 1.45i 4.03 j 3.47 j 1.51 j 4.57d 5.06e 5.41h 1.48h 3.93k 3.45k 1.49k 4.85 f a Binding Energy. b Hardness calculated from the HOMO-LUMO gap. c Experimental values taken from the references 5 and 41. d Theoretical data 7,8 at BPW91/cc-pVXZ-PP level (X = T for n = 4, 6 and X = D for n = 8, 10). e Theoretical data 7,8 at M06/cc-pVXZ-PP level (X = T for n = 4, 6 and X = D for n = 8, 10). f Theoretical data from the reference 5. g Experimental data from the reference 42 (uncertainty ±0.05 eV). h Theoretical data from the reference 13 at BPW91/cc-pVXZ-PP level (X = T for n = 4, 6 and X = D for n = 8, 10). i Experimental data from the reference 43 (uncertainty ±0.05 eV). j Calculated here from the experimental data 42,43 with the Eq. (4). k Calculated here from the theoretical data 13 with the Eq. (4).

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6.00

binding energy (eV/atom)

BE B3PW91/ADZP(+d) B3PW91/ADZP−DKH(+d) Experimental

5.50

M06/ADZP(+d) M06/ADZP−DKH(+d)

5.00

4.50

4.00

3.50

3.00 4.00

6.00

8.00

10.00

number of Nb atoms

Figure 2: Binding energy versus the numbers of Nb atoms. 8.00

M06

7.00

(+d)

ADZP

η

χM ω

6.00

η, χM, ω (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ADZP−DKH η χM ω

(+d)

Exp. η χM ω

5.00 4.00 3.00 2.00 1.00 0.00 4.00

6.00

8.00

10.00

number of Nb atoms

Figure 3: Hardness (η), electronegativity (χ M = −µ), and electrophilicity (ω) calculated at ae-M06/ADZP(+d) and ae-M06/ADZP-DKH(+d) levels of theory, and experimental data (Exp.). All levels of theory used here are appropriated to describe the ionization potential (I). The maximum deviation was obtained by the B3PW91/ADZP (7%), and the best results, are those calculated by the M06/ADZP-DKH(+d) (1%). The Nhat et al. 13 values are also in good agreement with the experimental, deviation of 2%. In average, the DKH corrections and the (+d) bases have yielded improvements when compared to the NR I results for all levels of theory. Note that for the B3PW91/ADZP-DKH we have 5% of deviation from the experimental values, and for the B3PW91/ADZP-DKH(+d) level, the error is only of 3% (0.1 eV).

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8.00 7.00

ADZP−DKH

(+d) B3PW91

M06

Exp. η χM ω

η χM ω

6.00

η, χM, ω (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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η χM ω

5.00 4.00 3.00 2.00 1.00 0.00 4

6

8

10

number of Nb atoms

Figure 4: Hardness (η), electronegativity (χ M ), and electrophilicity (ω) calculated at aeB3PW91/ADZP-DKH(+d) and ae-M06/ADZP-DKH(+d) levels of theory, and experimental data (Exp.). The results obtained here for the electroaffinity (A) shown relatively large deviations when compared with the I deviations values (see Tab. 1). The maximum average error was obtained by the B3PW91/ADZP(+d) (24%), and the minimum, for the M06/ADZPDKH(+d) level of theory (12%). Notable is the Nhat et al. 13 results also present large deviations from experimental values calculated by vertical (10%) and adiabatic (15%) detachment energies procedures. Specifically, to find A with the vertical detachment 13 (VDE), the E( N + 1) energy was calculated at the anion geometry, and the E( N ) value was determined by removing one electron of this system, while for adiabatic detachment energy (ADE) procedure 13 , the E( N ) and E( N + 1) energies were calculated at the respective optimized geometries. As commented above, when the E( N + 1) and E( N − 1) energy values are calculate, the orbitals relaxation are inevitable by bringing approximations in the η, µ, and ω properties. The Nhat et al. 13 approaches, used to calculate the I and A properties, increases the error in the finite difference approximation, Equations (4), when compared with our approach. However, we have used the I and A values of Nhat et al. 13 to calculate the η, µ, and ω properties. For hardness, the DKH corrections have improved the results for all cases. The average deviations, when the experimental data are taken as reference, were respectively of 12


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4% and 8% for the DKH and NR calculations. Notable is that the (+d) and standard basis sets provide the same deviations for the NR and DKH levels when the B3PW91 and the M06 functionals are used. The chemical potential calculated from the B3PW91 functional presents the same deviation, 6%, in all cases analyzed here. For the M06 with the (+d) bases, the DKH and NR calculations also show the same error, 3%. For the electrophilicity, the B3PW91/ADZP and B3PW91/ADZP-DKH levels of theory yielded practically the same deviation (∼ 11%) when the experimental data are used as reference. Furthermore, the (+d) bases do not improve the ADZP and ADZP-DKH values, deviations of 15% were found from the B3PW91/ADZP(+d) and B3PW91/ADZP-DKH(+d) values. Again, the best results were obtained with the M06/ADZP-DKH(+d) level of theory, 8% of deviation. The M06/ADZP(+d) calculations have yielded an error of 11% from the experimental data (see Figures 3 and 4). By considering the VDE values of Nhat et al. 13 , the deviations with respect to the experimental values are of 3%, 3%, and 6%, respectively for η, χ M = −µ, and ω. When the ADE values 13 of A and I are used to calculate the η, χ M = −µ, and ω properties, the deviations are 3%, 6%, and 11%, respectively. As shows in Figures 3 and 4, the levels of theory used here were able to describe the behavior of the η, χ M = −µ, and ω. For Nb6 , the hardness and electrophilicity have opposite trends, i.e., when the η is minimum the ω is maximum. This behavior is expected, since that the most stable systems tend to present low values of electrophilicity, because they are less prone to accept one additional electron to become more stable. Conversely, we expected that η assume large values for the most stable molecules (maximum hardness principle). Considering all the studied clusters, the electronegativities (χ M = −µ) shown a slighter linear behavior for all levels of theory. From the Equations (4) and (5), it is clear that we can calculate the η, χ M = −µ, and ω properties from two different ways with completely different results. Note that the gap between HOMO and LUMO shown in Table 1 is the hardness calculated from Equation (5). Where the large differences observed between the both approximations are expected, 20

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mainly because the LUMO energies tend to produce rather poor approximations for electron affinity in DFT calculations. However, the qualitative trends of the hardness were correctly described by the values obtained from Equations (4) and (5). The low accuracy obtained with the ADZP and ADZP-DKH basis sets, for some cases, can be understood. Nhat et al. 7–9 have shown that the overlap of the atomic 4d and 5s orbitals mainly contribute to form the valence band in the clusters orbitals. The basis set used by Nhat et al. 7 for systems with n = 4 and 6 was the cc-pVTZ-PP with [5s5p4d2 f 1g] GTFs to describe the valence electron. For clusters with n = 8 and n = 10, the basis set was 8 the cc-pVDZ-PP with [4s4p3d1 f ] GTF for the valence. Although the basis sets ADZP and ADZP-DKH have more GTF [8s7p4d2 f ] than the cc-pVXZ-PP (X = D, T), the four d functions were not enough to accurately describe the valence band of the Nbn clusters. Thus, the additional d function is fundamental to yield accurate results of the structure and electronic properties. We highlight that our methodology, ae-DFT/ADZPDKH(+d) , best describes, from theoretical viewpoint, the systems studied here when compared with the previous ECP calculations. 5,7,8 It is because our methodology includes a suitable all-electron basis sets and take into account the scalar relativist effects directly in the molecule.

Conclusions In this work, we have analyzed the performance of the all-electron basis sets in the structure and electronic properties of small niobium clusters. Additionally, the calculations with second order scalar DKH relativistic corrections were also taken into account. For the bonding lengths, the standard bases predictions overestimates the previous theoretical results and our best results. The standard basis sets, ADZP and ADZP-DKH, have presented low accuracy in the binding energy predictions when the experimental and theoretical values are taken as reference, even when the DKH corrections is used in the

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calculations. In order to better understanding the all-electron basis sets effects, and to improve the valence band description of Nbn clusters, an additional GTF was introduced in the standard bases, ADZP and ADZP-DKH. The ADZP-DKH(+d) results are closer to the experimental data than the ones obtained with the ADZP(+d) , and our best values were calculated at the M06/ADZP-DKH(+d) level of theory. Thus, along with the all-electron basis sets used here, the DKH corrections can produce a significant effect in the geometric parameters for some systems. For the ionization potential, the DKH and (+d) basis sets have improved the results when compared with the standard bases and with the nonrelativistic results. However, the DKH and (+d) basis sets did not improve the values in comparison with the stand basis and NR results in the electroaffinity calculation. For the hardness, the DKH corrections have improved the results, but the additional d function do not change the deviations for the M06 and B3PW91 functionals. For the chemical potential, the improvement was only obtained by the functional change, wherein the best results were calculated with the M06 together with (+d) basis sets. For the electrophilicity, the B3PW91/ADZP and B3PW91/ADZP-DKH levels of theory yielded practically the same deviation from the experimental data, and the (+d) basis sets have not improved the results calculated with the B3PW91 functional. When the M06 data are analyzed, one can say that the DKH corrections are important for the η, and the M06/ADZP-DKH(+d) level of theory was that yielded the better results. In abstract, the additional d function is of primary importance for the geometrical parameters, gap between the HOMO LUMO, and for the binding energy. For the others properties analyzed hare, the (+d) bases do not represent an important change in the deviations from the experimental data. The DKH corrections are significant for the most properties studied here and our best results were obtained at the M06/ADZP-DKH(+d) level of theory.

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Acknowledgement To FAPES (Fundaça˜ o de Amparo a` Pesquisa e Inovaça˜ o do Esp´ırito Santo), CAPES (Coordenaça˜ o de Aperfeiçoamento de Pessoal de N´ıvel Superior) for the financial support, and to the Laboratory of Atomic and Molecular Physics of Department of Physics at the Federal University of Esp´ırito Santo, for the computational support.

Supporting Information Available Tables of geometrical parameters for each level of theory. This material is available free of charge via the Internet at http://pubs.acs.org/.

References (1) Goodwin, L.; Salahub, D. R. Density-Functional Study of Niobium Clusters. Phys. Rev. A 1993, 47, R774. (2) Gronbeck, ¨ H.; Rosén, A. Investigation of Niobium Clusters: Bare and CO-adsorption. Phys. Rev. B 1996, 54, 1549. (3) Fournier, R.; Pang, T.; Chen, C. Structural Characterization of Niobium-Cluster Anions from Density-Functional Calculations. Phys. Rev. A 1998, 57, 3683. (4) Fowler, J. E.; Garc´ıa, A.; Ugalde, J. M. Many Low-Lying Isomers of the Cationic and Neutral Niobium Trimer and Tetramer. Phys. Rev. A 1999, 60, 3058. (5) Kumar, V.; Kawazoe, Y. Atomic and Electronic Structure of Niobium Clusters. Phys. Rev. B 2002, 65, 125403. (6) Majumdar, D.; Balasubramanian, K. Theoretical Study of the Electronic States of Niobium Trimer (Nb3 ) and its Anion (Nb3− ). J. Chem. Phys. 2003, 119, 12866–12877.

16


Page 17 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(7) Nhat, P. V.; Ngan, V. T.; Nguyen, M. T. A New Look at the Structure and Vibrational Spectra of Small Niobium Clusters and Their Ions. J. Phys. Chem. C 2010, 114, 13210. (8) Nhat, P. V.; Ngan, V. T.; Tai, T. B.; Nguyen, M. T. Electronic Structure, Vibrational and Thermochemical Properties of Neutral and Charged Cluster Nbn , n = 7 − 12. J. Phys. Chem. A 2011, 115, 3523. (9) Nhat, P. V.; Nguyen, M. T. Structures, Spectra, and Energies of Niobium Clusters from Nb13 to Nb20 . J. Phys. Chem. A 2012, 116, 7405. (10) Wang, H.-Q.; Li, H.-F.; Wang, J.-X.; Kuang, X.-Y. Structural, Electronic and Magnetic Effects of Al-doped Niobium Clusters: A Density Functional Theory tudy. J. Mol. Model. 2012, 18, 2993–3001. (11) Wang, H.-Q.; Li, H.-F. Probing the Structural and Electronic Properties of Al-Doped Small Niobium Clusters. Chem. Phys. Lett. 2012, 554, 231–235. (12) Li, H.-H.; Kuang, X.-Y.; Ding, L.-P.; Shao, P.; Han, L.-L.; Lu, T.-T. Evolution of Geometrical Structures, Stabilities and Electronic Properties of Neutral and Anionic Nbn−1 Co (n = 2 − 9) Clusters: Comparison with Pure Niobium Clusters. Comput. Mat. Sci. 2014, 95, 600–609. (13) Nhat, P. V.; Majumdar, D.; Leszczynski, J.; Nguyen, M. T. Clusters; Springer, 2017; pp 87–135. (14) Perdew, J. P.; Ziesche, P.; Eschrig, H. Electronic Structure of Solids 91; Akademie Verlag, Berlin, 1991; Vol. 11. (15) Jorge, F. E.; Ferreira, I. B.; Soprani, D. D.; Gomes, T. Estimating the Impact of an All- Electron Basis Set ans Scalar Relativistic Effects on the Structure, Stability, and Reactivty of Small Clusters. J. Braz. Chem. Soc. 2016, 27, 127.

17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 22

(16) Feller, D.; Peterson, K. A.; De Jong, W. A.; Dixon, D. A. Performance of Coupled Cluster Theory in Thermochemical Calculations of Small Halogenated Compounds. J. Chem. Phys. 2003, 118, 3510–3522. (17) Parr, R. G.; Yang, W. Density Functional Theory of Atons and Molecules; Oxford University Press: New York, 1989. (18) Pearson, R. G. Chemical Hardness: Aplications from Molecules to Solids; Wiley-VCH: Oxford, 1997. (19) Pearson, R. G. Chemical Hardness and Density Functional Theory. J. Chem. Sci. 2005, 117, 369. (20) Torrent-Sucarrat, M.; Duran, M.; Solà, M. Global Hardness Evaluation Using Simplified Models for the Hardness Kernel. J. Phys. Chem. A 2002, 106, 4632–4638. (21) Parr, R. G.; Szentpaly, L. v.; Liu, S. Electrophilicity Index. J. Am. Chem. Soc. 1999, 121, 1922–1924. (22) Chattaraj, P. K.; Roy, D. R. Update 1 of: Electrophilicity Index. Chem. Rev. 2007, 107, PR46–PR74. (23) Perdew, J. P.; Wang, Y. Accurate and Simple Analytic Representation of the ElectronGas Correlation Energy. Phys. Rev. B 1992, 45, 13244. (24) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648–5652. (25) Zhao, Y.; Truhlar, D. G. The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06Class Functionals and 12 Other Functionals. Theo. Chem. Acc. 2008, 120, 215–241.

18


Page 19 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(26) de Oliveira, P. J. P.; Barros, C. L.; Jorge, F. E.; Neto, A. C.; Campos, M. Augmented Gaussian Basis Set of Double Zeta Valence Quality for the Atoms Rb and Y − Xe: Application in DFT Calculations of Molecular Electric Properties. J. Mol. Struc.: {THEOCHEM} 2010, 948, 43–46. (27) Douglas, M.; Kroll, N. M. Quantum Electrodynamical Corrections to the Fine Structure of Helium. Ann. Phys. 1974, 82, 89. (28) Hess, B. A. Applicability of the No-Pair Equation with Free-Particle Projection Operators to Atomic and Molecular Structure Calculations. Phys. Rev. A 1985, 32, 756. (29) Hess, B. A. Relativistic Electronic-Structure Calculations Employing a TwoComponent No-Pair Formalism with External-Field Projection Operators. Phys. Rev. A 1986, 33, 3742. (30) Barros, C. L.; de Oliveira, P. J. P.; Jorge, F. E.; Neto, A. C.; Campos, M. Gaussian Basis Set of Double Zeta Quality for Atoms Rb Through Xe: Application in NonRelativistic and Relativistic Calculations of Atomic and Molecular Properties. Mol. Phys. 2010, 108, 1965–1972. (31) Camiletti, G. G.; Neto, A. C.; Jorge, F. E.; Machado, S. F. Augmented Gaussian Basis Sets of Double and Triple Zeta Valence Qualities for the Atoms K and Sc − Kr: Applications in HF, MP2, and DFT Calculations of Molecular Electric Properties. J. Mol. Struc.: {THEOCHEM} 2009, 910, 122–125. (32) de Berrêdo, R. C.; Jorge, F. E.; Jorge, S. S.; Centoducatte, R. An Augmented Gaussian Basis Set for Calculations of Molecular Polarizabilities on Platinum Compounds. Comput. Theor. Chem. 2011, 965, 236–239. (33) Neto, A. C.; Jorge, F. E. All-Electron Double Zeta Basis Sets for the Most Fifth-Row Atoms: Application in DFT Spectroscopic Constant Calculations. Chem. Phys. Lett. 2013, 582, 158–162. 19



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(34) Sousa, F. A. L.; Jorge, F. E. Basis Set Convergence on Static Electric Dipole Polarizability Calculations of Alkali-Metal Clusters. J. Braz. Chem. Soc. 2013, 24, 1357. (35) Pansini, F. N. N.; de Souza, F. A. L. Trends in the Spin States and Mean Static Dipole Polarizability of the Group VIIIA Metallocenes. J. Phys. Chem. A 2016, 120, 2771–2778. (36) Belyaev, S. N.; Panteleev, S. V.; Ignatov, S. K.; Razuvaev, A. G. Structural, Electronic, Thermodynamic and Spectral Properties of Mgn (n = 2 − 31) Clusters. A DFT Study. Comput. Theor. Chem. 2016, 1079, 34–46. (37) Li, C.-G.; Zhang, J.; Yuan, Y.-Q.; Tang, Y.-N.; Ren, B.-Z.; Chen, W.-G. Geometries, Stabilities and Electronic Properties of Copper and Selenium Doped Copper Clusters: Density Functional Theory Study. Physica E 2017, 86, 303–310. (38) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. J. et al. General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 1347–1363. (39) Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schutz, ¨ M. Molpro: A GeneralPurpose Quantum Chemistry Program Package. WIREs Comput. Mol. Sci. 2012, 2, 242–253. (40) Neto, A. C.; Muniz, E. P.; Centoducatte, R.; Jorge, F. E. Gaussian Basis Sets for Correlated Wave Functions. Hydrogen, Helium, First- and Second-Row Atoms. J. Mol. Struct.:{THEOCHEM} 2005, 718, 219. (41) Hales, D. A.; Lian, L.; Armentrout, P. B. Collision-Induced Dissociation of Nb+ (n = 2 − 11): Bond Energies and Dissociation Pathways. Int. J. Mass Spectrom. Ion Processes 1990, 102, 269. (42) Knickelbein, M. B.; Yang, S. Photoionization Studies of Niobium Clusters: Ionization Potentials for Nb2 − Nb76 . J. Chem. Phys. 1990, 93, 5760–5767. 20


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(43) Kietzmann, H.; Morenzin, J.; Bechthold, P. S.; Gantefor, ¨ G.; Eberhardt, W. Photoelectron Spectra of Nbn −Clusters: Correlation Between Electronic Structure and Hydrogen Chemisorption. J. Chem. Phys. 1998, 109, 2275–2278.

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On the Effects of All-Electron Basis Sets and the Scalar Relativistic

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