Analysis of O2 Adsorption Stability and Strength Over Gold Clusters Using DFT and Logistic...

ORI GIN AL PA PER

Analysis of O2 Adsorption Stability and Strength OverGold Clusters Using DFT and Logistic Regression

M. Erdem Gunay • Tugba Davran-Candan •

Ramazan Yıldırım

Received: 7 October 2011 / Published online: 1 December 2011

� Springer Science+Business Media, LLC 2011

Abstract In this work, the stability and strength of O2 adsorption over Au2–10

clusters were studied. The density functional theory (DFT) computed adsorption

data were classified using multiple logistic regression. The effects of user defined

descriptors were analyzed and it was found that stable O2 adsorption requires the

presence of an unpaired electron while its strength depends on the size and the

charge of the cluster. As the size of the cluster increases, the probability of strong

adsorption decreases, and the odds of finding strong adsorption is higher for the

anionic clusters compared to neutral and cationic clusters. The effects of the elec-

tronic properties were also studied and HOMO–LUMO gap was found to be the

most significant property determining the stability of O2 adsorption; as its value

increases, the probability of stable adsorption decreases. The strength of the

adsorption, on the other hand, was found to be mostly dependent on the ionization

potential, which has a negative effect.

Keywords CO oxidation � Structure activity relationship � DFT � Multiple logistic

regression � O2 adsorption stability and strength

Introduction

The field of catalysis is very important in chemical industry, and developing a

catalyst with high activity, selectivity and stability for a specific reaction usually

requires long, laborious and expensive experiments. To ease these efforts, quantum

mechanical modeling of catalysts has been studied extensively to understand the

physical/chemical properties of the catalysts and the catalytic mechanisms on the

surface. Nevertheless, the quantum mechanical calculations are also quite time

M. E. Gunay � T. Davran-Candan � R. Yıldırım (&)

Department of Chemical Engineering, Bogazici University, 34342 Istanbul, Turkey

e-mail: [email protected]

123

J Clust Sci (2012) 23:221–235

DOI 10.1007/s10876-011-0422-2

consuming and expensive although the progress in this field has been remarkable

due to the developments in hardware and software in recent years.

The success of the quantum mechanical modeling of catalysis is highly

dependent on the initial set of parameters and assumptions because there are

numerous possibilities for the structures and properties of the catalyst and catalyst–

reactant complex. Only a small fraction of modeling attempts produces significant

information that can be used to understand the catalytic phenomena. Hence, there is

a need for new tools and approaches that can be used to predict the outcome of

quantum mechanical calculations at least in some extent, and help us to concentrate

our time and resources on the promising set of initial conditions.

The quantum chemical modeling of catalysis starts with setting some input

variables (descriptors). If the cluster model, which represents the catalyst by a group

of finite number of atoms arranged similar to the actual catalyst, is used, the user

defined descriptors are usually cluster size, cluster charge, the presence of unpaired

electron, the coordination number of the atom on which the adsorption takes place

and so on. One can determine the optimum energy and structure of the cluster, the

adsorption energies, and the activation energies of possible mechanisms (output

variables) depending on these descriptors. One can also calculate some electronic

properties such as HOMO–LUMO gap, binding energy per atom, ionization

potential and electron affinity to explain or interpret the output variables and their

dependency to descriptors. Some studies in the literature suggest that the relations

among these three types of variables [user defined descriptors, density functional

theory (DFT) computed electronic properties and DFT computed adsorption

energies] can be generalized to predict the possible outcomes of the DFT

simulations for the catalytic system [1–4] as well as for some other areas [5–10].

This allows the researcher to select the values of the user defined descriptors in a

way that the chance for the desired results will be higher.

CO oxidation over gold clusters have been studied extensively due to its

significance for selective CO oxidation over Au based catalysts as a part of ongoing

efforts to purify hydrogen for the fuel cell applications [11]. Gold was both

experimentally and theoretically shown to be active when it is in the form of nano-

sized clusters or finely dispersed on metal-oxides [2, 12, 13]. For example, the

experimental studies indicate that Au particles with diameters less than 5 nm

dispersed on a support like Al2O3, ZrO2 and TiO2 in the absence or presence of a

promoter such as MgO, MnOx, and FexOx are highly active for CO oxidation or

selective CO oxidation in hydrogen rich streams [14–16]. Similarly, there are

numerous computational works focusing on adsorption of CO [17, 18] or O2

[19, 20] and their reaction [21–24]. DFT calculations are usually performed using

cluster model because it physically suits well to the CO oxidation over Au based

catalysts even though these clusters are much larger than those used in DFT

calculations.

It is often suggested in the literature that CO oxidation over Au catalysts

proceeds through Langmuir–Hinshelwood mechanisms, which involves oxygen

adsorption. The O2 adsorption takes place through an electron transfer from Au to

the antibonding p* orbital of O2, and the clusters having odd number of electrons

adsorb oxygen while the clusters with even number of electrons do not [20]. For this

222 M. E. Gunay et al.

123

reason, O2 adsorption is not stable or strong in all possible adsorption sites making

this system a good candidate to test the possibility to predict the outcome

(adsorption strength in this case) of the quantum mechanical modeling using the

techniques like logistic regression.

Multiple logistic regression is a statistical method of nonlinear regression that

models the relation between the predictor variables and a categorical response

variable [25–27]. In the last decade, logistic regression has been applied in fields

such as microbiology [28, 29], epidemiological studies [30, 31], environmental

issues [32, 33], and catalysis [34]. The use of logistic regression seems to be quite

suitable for the cases with the outcome variable having only two possible values

such as stable–unstable and weak–strong adsorption as in the present work.

In this work, we analyzed the stability and strength of O2 adsorption over Au2–

Au10 clusters using multiple logistic regression to show that the role of user defined

descriptors and DFT computed electronic properties on the stability and strength of

O2 adsorption can be predicted. This may be extended to the other catalysis systems

and help the researcher to identify the promising structures before starting

comprehensive DFT studies.

Computational

DFT Calculations

Gaussian 03W [35] was used for DFT calculations. Becke’s three-parameter hybrid

functional (B3LYP) [36] was employed together with the Los Alamos National

Laboratory second Double-Zeta (LANL2DZ) basis set [37] and the corresponding

Los Alamos relativistic effective core potential [37] for the heavy gold atom. This

functional and the basis set combination was reported to be reliable for the study of

O2 adsorption over gold clusters [38, 39] which was also verified by the comparison

of our computed values with the available experimental results [4]. Optimizations

were performed without any symmetry constraints. No corrections for inter and

intramolecular basis set superposition error (BSSE) were done. The structures were

fully relaxed during geometry optimizations.

The descriptors, which were used to generate adsorption data were taken as

cluster size (from 2 to 10), cluster charge (anionic, neutral and cationic), the

unpaired electron (presence or absence) and coordination number of the Au atom on

which the O2 molecule was adsorbed. 92 different adsorption sites over 27 clusters

were identified and O2 adsorption energies were computed for each site. The

binding energy per atom, HOMO–LUMO gap, ionization potential and electron

affinity were also calculated for the clusters as the electronic properties. The

adsorption energies of O2 molecules and the binding energies of clusters were

calculated by Eqs. 1 and 2, respectively:

Eads;O2¼ E Aun

v�O2ð Þ � E Aunvð Þ � E O2ð Þ ð1Þ

Analysis of O2 Adsorption Stability and Strength 223

123

Eb Aunvð Þ ¼ n� 1ð ÞE Au1ð Þ þ E Au1

vð Þ � E Aunvð Þj j

nð2Þ

where n denotes the number of Au atoms involved in the cluster and m denotes the

charge of the cluster. E(Aunm–O2) is the total energy of the cluster–O2 complex,

E(Aunm) corresponds to the total energy of the cluster and E(O2) denotes the total

energy of the O2 molecule.

Since the ionization potential and electron affinity are not linearly independent,

they were not used together in the same model as input variables. Only the models

generated using ionization potential alone will be presented here because the fitness

of the models constructed replacing the ionization potential with the electron

affinity was almost the same. The full data set for the 27 cluster and 92 adsorption

sites is presented in Tables 1, 2, and 3.

Multiple Logistic Regression Modeling

While multiple linear regression modeling approximates the relationship between

predictor variables and a continuous response (output) variable, multiple logistic

regression makes an approximation for the relation between predictor variables and

a categorical response [25–27]. The general form of the multiple logistic regression

for ‘‘m’’ independent variables is shown in Eq. 3.

YðXÞ ¼ exp b � Xð Þ

1þ exp b � Xð Þ ð3ÞQ

(X) is the predicted response variable while (b � X) represents

b0 ? b1x1 ? b2x2 ? … bmxm, where the xs are the predictor variables and the bs

are the corresponding regression coefficients. SinceQ

(X) approaches to one when the

exponential term approaches to infinity and to zero if the exponential term approaches

to zero, the numerical value ofQ

(X) indicates the probability of response factor falling

into one of the two possible categories (the value of one is 100% probability for one

category while the value of zero is 100% probability of the other) [25]. The multiple

logistic regression coefficients (b) were calculated by the maximum likelihood

estimation of the log likelihood function, which is shown in Eq. 4 [25–27].

ln L bð Þ½ � ¼Xn

j¼1

Yj ln P Xj

� �� þ 1� Yj

� �ln 1�

YXj

� �h in oð4Þ

where n is the number of data points, Yj and Xj represent the jth experimental

response and the predictor variables, respectively.

The significance of the model was determined by inspecting the G value

(corresponding F in multiple linear regression):

G ¼ �2 lnL bð ÞRL bð ÞF

� �ð5Þ

where (L(b)F) is the likelihood function with all the predictor variables while

(L(b)R) is the same function calculated in the absence of these.


123

G follows a Chi-square (v2) distribution with ‘‘m’’ degrees of freedom. For a 95%

confidence interval (a = 0.05), if G B v2(1 - a, m), the null hypothesis (all the

coefficients of multiple logistic regression are zero) is true, meaning that the model

Table 1 Lowest-energy structures of Aunm (n = 2–10, m = -1, 0, ?1)

Cluster Structure Cluster Structure Cluster Structure

Au2- Au5

- Au8-

Au2 Au5 Au8

Au2? Au5

? Au8?

Au3- Au6

- Au9-

Au3 Au6 Au9

Au3? Au6

? Au9?

Au4- Au7

- Au10-

Au4 Au7 Au10

Au4? Au7

? Au10?


123

is not significant. On the other hand, if G [ v2(1 - a, m), the coefficient of at least

one predictor is nonzero i.e. the model is accepted to be significant [28].

The significance of each predictor variable was checked using the Wald test.

ZWald is calculated by dividing the regression coefficients with their standard errors

SE(b), and the two tailed p value of ZWald has to be less than 0.05 for 95%

confidence for a variable to be accepted as significant [25–27]. The upper and lower

limits of the coefficients (for the 95% confidence) are also used as the indicators of

significance since they should both have the same sign to make sure that the

regression coefficient cannot be zero.

The extrapolation ability of the logistic regression was tested by constructing a

model using the data belonging Au2–Au8 clusters to predict the results of Au9–Au10

clusters (data unseen by the model). Logistic regression models were created using

computer codes written in MATLAB 7.9 (R2009b) environment.

Results and Discussion

The results are presented and discussed in two parts. In part one, the data set was

divided into two categories, namely ‘‘stable’’ and ‘‘unstable’’ adsorption depending

on the sign of adsorption energies, where the computed positive adsorption energies

indicated instability. In part two, the stable adsorption values were analyzed in terms

of strength, dividing the data set as ‘‘strong’’ (if the value of the adsorption energy

was smaller than -0.5 eV) and ‘‘weak’’ (if the value of the adsorption energy was

higher than -0.5 eV). It should be noted that the -0.5 eV border separating strong

and weak adsorption has no physical significance. It was selected by trial and error

such that the data points can be classified effectively based on the values of the

descriptors and the electronic properties, which is the main purpose of the work.

Table 2 Electronic properties for Aunm (n = 2–9, m = -1, 0, ?1)

Cluster BE

(eV/

atom)

H–L

(eV)

IP

(eV)

Cluster BE

(eV/

atom)

H–L

(eV)

IP

(eV)

Cluster BE

(eV/

atom)

H–L

(eV)

IP

(eV)

Au2- 0.87 1.44 2.03 Au5

- 1.48 2.20 3.18 Au8- 1.64 0.90 2.78

Au2 0.94 3.27 10.59 Au5 1.28 1.25 7.16 Au8 1.56 2.78 7.95

Au2? 0.35 0.57 15.30 Au5

? 1.73 3.16 13.30 Au8? 1.75 1.01 11.21

Au3- 1.42 3.10 3.68 Au6

- 1.50 1.03 2.15 Au9- 1.69 2.07 3.81

Au3 0.92 1.33 7.26 Au6 1.50 3.48 8.45 Au9 1.51 1.01 6.24

Au3? 1.64 3.95 15.39 Au6

? 1.66 1.17 12.90 Au9? 1.86 3.35 11.94

Au4? 1.38 1.17 3.02 Au7

- 1.63 2.18 3.51 Au10- 1.67 0.95 3.16

Au4 1.17 1.93 7.94 Au7 1.44 1.12 6.99 Au10 1.57 2.26 7.34

Au4? 1.54 1.31 12.89 Au7

? 1.79 2.97 12.48 Au10? 1.78 1.03 10.08

BE binding energy per atom, H–L HOMO–LUMO gap, IP ionization potential


123

Analysis of Adsorption Stability: Modeling Descriptors Versus O2 Adsorption

Stability

The stability analysis was performed by using both the descriptors and the electronic

properties. However, the ‘‘stable’’ or ‘‘unstable’’ classification based on the

descriptors can be treated as the demonstration and verification of our approach,

otherwise one needs no special tool to make this classification. It is very clear from

the visual inspection of the data that the stability of adsorption depends on the

Table 3 O2 adsorption energies over available adsorption sites

Cluster a.sa (c.n) Eads (eV) Cluster a.sa (c.n) Eads (eV) Cluster a.sa (c.n) Eads (eV)

Au2- 1 (1) -1.37 Au7

- 4 (3) 0.66 Au9 5 (4) -0.12

Au2 1 (1) 1.01 Au7- 5 (2) 0.64 Au9 6 (3) -0.17

Au2? 1 (1) -1.58 Au7

- 6 (1) 0.65 Au9 9 (2) -0.27

Au3- 1 (1) 0.56 Au7

- 7 (4) 0.65 Au9 7 (6) -0.11

Au3- 2 (2) 0.69 Au7 1 (2) -0.16 Au9

? 3 (3) 1.30

Au3 1 (1) -0.52 Au7 2 (4) -0.12 Au9? 4 (3) 1.30

Au3 2 (2) -0.52 Au7 3 (5) -0.41 Au9? 5 (5) 1.41

Au3? 1 (2) 1.03 Au7 4 (2) -0.26 Au9

? 6 (5) 1.26

Au4? 2 (2) -1.01 Au7 5 (4) -0.33 Au9

? 7 (6) 1.35

Au4? 3 (1) -1.02 Au7 6 (3) -0.33 Au9

? 9 (2) 1.26

Au4 1 (3) 0.61 Au7 7 (2) -0.41 Au10- 1 (4) -0.50

Au4 2 (2) 0.69 Au7? 1 (3) 1.21 Au10

- 3 (4) -0.27

Au4? 1 (3) -0.33 Au7

? 2 (3) 1.21 Au10- 5 (4) -0.27

Au4? 2 (2) -0.33 Au7

? 4 (3) 1.21 Au10- 7 (6) -0.11

Au5- 1 (3) 0.39 Au7

? 7 (6) 1.27 Au10- 8 (2) -0.33

Au5- 3 (2) 0.35 Au8

- 4 (4) -0.84 Au10- 10 (2) -0.32

Au5- 4 (4) 0.72 Au8

- 8 (2) -0.85 Au10 3 (3) 0.92

Au5 1 (3) -0.39 Au8 3 (4) 1.20 Au10 4 (6) 0.92

Au5 3 (2) -0.28 Au8 8 (2) 1.09 Au10 5 (3) 0.89

Au5 4 (4) -0.20 Au8? 1 (4) -0.17 Au10 6 (4) 0.86

Au5? 1 (2) 1.08 Au8

? 3 (3) -0.24 Au10? 1 (3) -0.29

Au5? 4 (4) 0.70 Au8

? 5 (3) 0.16 Au10? 2 (6) -0.16

Au6- 1 (2) -0.95 Au8

? 7 (6) -0.11 Au10? 3 (6) 0.32

Au6- 2 (4) -0.63 Au8

? 8 (2) -0.27 Au10? 4 (3) -0.29

Au6 1 (2) 1.14 Au9- 1 (4) 0.98 Au10

? 5 (6) -0.83

Au6 2 (4) 1.25 Au9- 2 (4) 0.90 Au10

? 6 (3) -0.29

Au6? 1 (2) -0.31 Au9

- 3 (3) 0.55 Au10? 7 (6) -0.16

Au6? 2 (4) -0.26 Au9

- 7 (6) 1.29 Au10? 8 (6) -0.47

Au6? 4 (2) -0.32 Au9

- 8 (2) 0.68 Au10? 9 (6) -0.82

Au6? 5 (4) -0.26 Au9 3 (5) 0.22 Au10

? 10 (3) -0.27

Au7- 3 (4) 0.88 Au9 4 (3) -0.44

a.s adsorption site, c.n coordination number


123

absence or the presence of the unpaired electron. If an unpaired electron was present

in the cluster, the adsorption was found to be stable except for a few cases.

However, the implementation of the logistic regression for this ‘‘known’’ problem

seemed to form a good demonstration. Hence, we first discussed this model and then

continued with the analysis of stability through the electronic properties.

The descriptors of the clusters were used as the predictor variables for the

probability of stable adsorption, and the results of the logistic regression model

developed are presented in Table 4. Since the G value (106.4) of the model is much

greater than the critical v2 value of 9.49 (for four degrees of freedom), the null

hypothesis that all the coefficients are zero (meaning no significant pattern or

correlation) was rejected and the model was concluded as being significant. As

indicated from the p values in the table, the stability of O2 adsorption depends only

on the presence of unpaired electron. This is also evident from the upper and lower

bounds of the regression coefficients (b). Except unpaired electron, the lower and

upper limits of the regression coefficients for all the descriptors have values with

opposite signs, which indicate that their coefficients can have the value of zero

within 95% confidence interval. Hence, these variables cannot be deterministic for

the stable–unstable classification. On the other hand, lower (1.89) and upper (7.04)

limits of the regression coefficient for unpaired electron have the same sign (no

possibility to have the value of zero), indicating that it is a significant variable. The

positive value of the regression coefficient also indicates that the presence of

unpaired electron affects the stability positively, i.e. clusters having odd number of

electrons adsorb O2, while clusters with even number of electrons show small or

zero reactivity towards O2, parallel to the previous findings [4, 20, 40–42]. This

result clearly demonstrates that logistic regression modeling can be easily and

effectively employed in this subject.

The logistic curve obtained for the unpaired electron versus the probability of

having stable adsorption is shown in Fig. 1. The open symbols indicate the status of

stability for the quantum mechanically computed values; they have 100 or 0%

probabilities, because O2 either adsorb or do not adsorb on a given cluster. The full

symbols, on the other hand, show the corresponding model predictions, which are

quite close to the computed data. Since x axis has only two values (two categories),

the separation between the data points in the same category has no physical

Table 4 Statistics of the model for descriptors versus O2 adsorption stability

Descriptors b (regression

coefficient)

SE(b) (standard

error of regression

coefficient)

ZWald

(b/SE(b))

p Value

of ZWald

blower bupper

Constant term 0.65 2.47 0.26 0.793 -4.19 5.49

Size -0.13 0.36 -0.36 0.722 -0.83 0.57

Charge -0.57 0.85 -0.67 0.503 -2.24 1.10

Coordination number -0.30 0.45 -0.66 0.510 -1.18 0.59

Unpaired electron 4.46 1.31 3.40 0.001 1.89 7.04

G = 106.4, v2 (0.95, 4) = 9.49


123

meaning; they were just created to prevent overlapping and make the visual

inspection easier.

The test of extrapolation ability of the model was not necessary in this section

because the size was found to be not significant as a variable. The model can predict

the adsorption stability of any cluster if the absence or presence of unpaired

electron, which is the only variable determining the stability, is known.

Analysis of Adsorption Stability: Modeling Electronic Properties Versus O2

Adsorption

Table 5 shows the results of the logistic regression when the electronic properties

were used as the predictor variables against the stability of adsorption. Since the

ionization potential and the electron affinity were not linearly independent, only the

ionization potential was used here as one of the electronic properties. On the other

hand, the coordination number was also added as a predictor variable to identify the

location of the adsorption, although it is not an actual structural property of the

cluster. Like the previous case, the G value (97.4) of the model is much greater than

the critical v2 (9.49) indicating that the model is significant. The Wald test

demonstrates that all the electronic properties had p values quite higher than 0.05,

except the HOMO–LUMO gap. This means that among the electronic properties

considered, HOMO–LUMO gap is the only structural property effective on the

stability of O2 adsorption.

Next, a new reduced model was constructed based only on HOMO–LUMO gap

so that the results could be further verified and visually presented in a two

dimensional plot. As it is shown in Table 6, the G value (96.2) of the new model is

very close to the G value (97.4) of the full model verifying the significance of

HOMO–LUMO gap. This effect seems to be through the presence of unpaired

electron, which is the most significant variable for HOMO–LUMO gap [4].

0

20

40

60

80

100

O2

adso

rpti

onpr

obab

ility

(%)

predicted calculated

Absence ofunpaired electron

Presence ofunpaired electron

Fig. 1 Effects of absence orpresence of unpaired electron onO2 adsorption probability


123

The negative value of the coefficient of HOMO–LUMO gap indicates that the

probability of stable O2 adsorption is negatively affected by the increase of HOMO–

LUMO gap, the cut off value being 1.44 eV. This makes sense when the even–odd

pattern observed in the change of HOMO–LUMO gap with size is considered. The

lower values of the gap correspond to the clusters with odd number of electrons,

which were shown to adsorb O2 [4, 20, 40–42].

The plot of HOMO–LUMO gap versus the probability of O2 adsorption stability

shows a logistic curve supporting these findings (Fig. 2). Except three data points,

all the predicted O2 adsorption probabilities lie quite close to the actual situations.

In order to test the generalization accuracy of the reduced model, 4-fold cross

validation was applied. First, the data were divided into four subsets, and three

subsets were used for the logistic regression model while one subset was used for

the validation. After repeating these four times, the G value of 93.9 was calculated

by combining the validation subsets covering the entire data. This value is almost

the same as that of the model (96.2) shown in Table 6, indicating the high

generalization accuracy of the model [43, 44].

Analysis of Adsorption Strength: Descriptors Versus O2 Adsorption Strength

In the second part of the work, the stable adsorption data (50 data points) were

divided into two categories as explained at the beginning of ‘‘Results and

discussion’’: strong adsorption if Eads \ -0.50 eV and weak adsorption if Eads [-0.50 eV. The effects of descriptors and electronic properties on the adsorption

strength are presented below.

Table 5 Statistics of the model for electronic properties versus O2 adsorption stability


coefficient)

SE(b) (standard

error of regression

coefficient)

ZWald (b/SE(b)) p Value

of ZWald

blower bupper

Constant term 6.26 2.91 2.15 0.032 0.55 11.97

HOMO–LUMO gap -4.55 0.99 -4.59 0.0001 -6.49 -2.60

Binding energy 1.08 2.01 0.54 0.592 -2.86 5.01

Ionization potential 0.10 0.14 0.73 0.466 -0.17 0.38

Coordination number -0.40 0.35 -1.15 0.252 -1.09 0.29

G = 97.4, v2 (0.95, 4) = 9.49

Table 6 Statistics of the reduced model for electronic properties versus O2 adsorption stability

Electronic

properties

b (regression

coefficient)

SE(b) (standard error of

regression coefficient)

ZWald

(b/SE(b))

p Value

of ZWald

blower bupper

Constant term 7.06 1.30 5.44 0.0001 4.52 9.60

HOMO–LUMO gap -4.42 0.90 -4.94 0.0001 -6.18 -2.67

G = 96.2, v2 (0.95, 1) = 3.84


123

Since all the clusters with unpaired electron already had the ability to adsorb O2,

the clusters without unpaired electron were excluded; therefore, this descriptor was

not treated as a predictor variable anymore. Hence; size, charge and coordination

number were used as the descriptors of this model. The results are summarized in

Table 7. Although the G value of 16.4 is not as high as the G values of the models

discussed above for the adsorption stability, it is still well above the critical v2 value

(7.81). Thus, this model should also be accepted as significant.

According to the findings summarized in Table 7, the strength of O2 adsorption

depends on the size and the charge of the cluster (p values are 0.018 and 0.010,

respectively). The coordination number seems to have no significant effect.

The coefficient of size and charge are both negative as shown in Table 7,

indicating that O2 adsorption becomes weaker as the cluster size increases or the

charge changes in the order of anionic–neutral–cationic (Fig. 3a). Indeed, the effect

of charge on the adsorption strength of O2 over Au clusters was shown by previous

studies. Despite of the early claims that cationic and neutral clusters show no

reactivity towards O2 except for Au10?, the cationic and neutral clusters were also

0

20

40

60

80

100

0 1 2 3 4

O2

adso

rpti

on p

roba

bilit

y(%

)

HOMO-LUMO gap (eV)

predicted calculatedFig. 2 Effects of HOMO–LUMO gap on O2 adsorptionprobability

Table 7 Statistics of the model for descriptors versus O2 adsorption strength


coefficient)

SE(b) (standard

error of regression

coefficient)

ZWald

(b/SE(b))

p Value

of ZWald

blower bupper

Constant term 0.82 68.5 0.01 0.99 -133.4 135.1

Size -0.52 0.22 -2.36 0.018 -0.96 -0.09

Charge -1.53 0.59 -2.59 0.010 -2.68 -0.37

Coordination number 0.27 0.35 0.78 0.44 -0.41 0.96

G = 16.4, v2 (0.95, 3) = 7.81


123

shown to adsorb O2 if they possess unpaired electrons [4, 19, 20, 41, 42, 45, 46].

Nonetheless, the strength of O2 adsorption on anionic Au clusters was always

demonstrated to be greater than the adsorption strength on cationic and neutral

clusters [4, 19, 20, 45, 46].

The decrease of strong adsorption probability with increasing size is also clear

from Fig. 3a. It should be noted that, the figure indicates the mean probabilities for

strong adsorption, since the percent probability values shown in the figure were

calculated by averaging the values over all the possible adsorption sites for that

cluster, rather than considering only the maximum adsorption energies. Some

cluster sizes do not appear in the figure (for example the anionic cluster with the size

of three), because they do not result any stable adsorption for that charge; hence

they were eliminated from the data set in this section.

The decrease of the mean probabilities for strong adsorption with the increasing

size can be attributed to the increase in the number of inner atoms (atoms located on

the flat surfaces of the clusters) with high coordination number parallel to the

arguments in the literature that the strength of adsorption decreases with increasing

coordination number [1, 2]. The reason why coordination number was found to be

insignificant in this work may be related to its mild correlation with the size

(Pearson correlation coefficient of 0.57). On the other hand, it was also argued that

the underlying factor determining the adsorption ability of a cluster was the shape of

the frontier orbitals (HOMO, in this case) rather than the coordination number of the

adsorption site [47, 48]. In other words, O2, being an electron acceptor, was argued

to adsorb on the sites where HOMO protruded furthest in the vacuum [48]. Since the

lobes of HOMO over the inner atoms are generally smaller than those on the

corners/edges, and the number of inner atoms increases with increasing the cluster

size, the orbital shape seems to reflect its effect through the cluster size.

The extrapolation ability of logistic regression was tested by constructing a

model from the data belonging to Au2–Au8 clusters, and this model was used to

predict the probability of strong adsorption over Au9–Au10 clusters (data unseen by

0

20

40

60

80

100

2 3 4 5 6 7 8 910 2 3 4 5 6 7 8 910 2 3 4 5 6 7 8 910

% P

roba

bilit

y of

str

ong

O2

adso

rpti

on

Size

anionic neutral cationic(a)

0

20

40

60

80

100

2 3 4 5 6 7 8 910 2 3 4 5 6 7 8 910 2 3 4 5 6 7 8 910

% P

roba

bilit

y of

str

ong

O2

adso

rpti

on

Size

anionic neutral cationic(b)

Fig. 3 Dependency of strong adsorption probability on cluster size and charge for: a full model, b usingAu2–Au8 clusters to predict the results of Au9–Au10


123

the model). The new model was found to be also significant with its G value of 27.7,

which is significantly higher than the critical v2 value (7.81). Figure 3a, b compares

the probabilities of models generated from Au2–Au10 and Au2–Au8, respectively;

the numerical values of predicted probabilities for Au9–Au10 were also given on the

top of the corresponding bars. The predictions of both models for Au10- clusters are

quite close with each other (26.8–26.3%) as well as with the DFT computed value

(16.7%). The probability of strong adsorption for Au9 is predicted better by the new

model (2.3% compared to 11.9% of the original model) since the DFT computed

value is 0%. The probability of the Au10? cluster changes from 2.4% to about 0%,

both being lower than the actual probability of 20% found by DFT calculations;

however the model still successfully captures the low probability of strong

adsorption for this cluster.

Analysis of Adsorption Strength: Electronic Properties Versus O2 Adsorption

Strength

Finally, the electronic properties were used to model the strength of O2 adsorption

(Table 8). The G value (14.3) of the model is above the critical v2 value (9.49)

indicating that the model is significant. The only significant variable for the

adsorption strength, which has a p value below 0.05, is the ionization potential.

The ionization potential has a negative coefficient meaning that the strength of

adsorption is negatively affected by this variable. This is something expectable for

the anionic and neutral clusters since the adsorption of O2 on these clusters are

known to be accompanied by an electron transfer from Au to the p* orbital of O2

[20, 46, 47]. Hence, the lower the value of ionization potential, which is defined as

the amount of energy required to pull off an electron, the easier to make the electron

transfer required for the adsorption. On the other hand, an electron transfer in the

opposite direction (from O2 to Au) is observed for the cationic clusters during O2

adsorption [4]. This means that for the cationic clusters, adsorption of O2 need not

be stronger as ionization potential decreases, which is contradictory to the

suggestion of the model. This explains the relatively lower G value (lower

prediction ability) for this model compared to the previous models above.

Table 8 Statistics of the model for electronic properties versus O2 adsorption strength

Electronic

Properties

b (regression

coefficient)

SE(b) (standard

error of regression

coefficient)

ZWald

(b/SE(b))

p Value

of ZWald

blower bupper

Constant term 3.84 3.65 1.05 0.294 -3.31 11.0

HOMO–LUMO gap 0.37 2.36 0.15 0.877 -4.27 5.00

Binding energy -2.50 1.55 -1.61 0.107 -5.54 0.54

Ionization potential -0.25 0.12 -2.11 0.035 -0.48 -0.02

Coordination number 0.03 0.30 0.10 0.924 -0.56 0.61

G = 14.3, v2 (0.95, 4) = 9.49


123

Conclusions

The effects of user defined descriptors and electronic properties of the Au clusters

on the stability and strength of the O2 adsorption were analyzed by classifying the

DFT computed adsorption data using multiple logistic regression. It was found that,

among the user defined descriptors, unpaired electron is the most significant variable

for the stability of O2 adsorption, whereas the size and charge of the cluster

determine the adsorption strength. The probability of finding strong adsorption

decreases with increasing cluster size and changing the charge of the cluster in the

order of anionic–neutral–cationic. For the electronic properties, the stability of the

O2 adsorption depends on HOMO–LUMO gap; probability of stable adsorption

is smaller for the clusters with high HOMO–LUMO gaps. The strength of the

adsorption, on the other hand, is mostly determined by the ionization potential. It

was also concluded that multiple logistic regression modeling can successfully

classify the DFT computed data, and help to improve the effectiveness of the

computational studies.

Acknowledgments The financial support provided by TUBITAK Project 109M207 and Bogazici

University Research Fund Project 09A503D are gratefully acknowledged.

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