46

CHAPTER 3

Chemical, Magnetic and Thermodynamic Properties of Palladium Clusters

3.1 Introduction

Palladium is used as catalysts materials for many reactions, mainly in hydrogenation and

dehydrogenation reactions [1]. Pd atom has closed-shell 4d10

electronic configuration. In Pd

clusters, the electrons are promoted to 5s and sometimes to 5p orbitals [2]. The electron

promotion shortens the bond lengths and increases binding energy thereby stabilizing small Pd

clusters [2]. This unusual electronic structure makes study of Pd clusters intriguing and

challenging. The small Pd clusters Pdn (n≤30) not only opens the possibility of tuning a catalytic

process by changing cluster size, but also can be used to catalyze chemical reactions at ambient

temperatures [3]. It also plays integral roles in a number of nano-scale devices, for example, Pd

nano-wire-based hydrogen sensors and carbon nanotube field effect transistors [4, 5], etc.

Generally, the catalytic activity is directly proportional to the size of the catalytically active

particles. Moreover, catalyzed reactions exhibit structural sensitivity to the size of the

catalytically active particles. Small metal clusters consisting of few atoms are the most

appropriate catalytic particles. A thorough understanding of their chemical reactivity will be of

immense importance for the catalytic purposes of clusters. In order to understand the physical

and chemical properties of these clusters, the determination of atomic configurations becomes

essential. Hence, our primary aim is to find out the equilibrium configurations of the small Pd

clusters and see how their electronic and chemical properties are related to size. Even though,

there have been a large number of studies, both theoretical and experimental, on the Pd clusters;

a detailed systematic study will always be of great help for understanding its unique properties.

The structural and electronic properties of neutral, anionic and cationic Pd clusters atoms

were studied [6] using density functional theory (DFT) and Pd4 (with 40 valence electrons) was

found to be the magic cluster with relatively high binding energy and stability function (∆𝐸2).

Pd4, Pd6 and Pd8 were observed to be relatively more stable compare to others with relatively

high value of stability function (∆𝐸2) [6]. The second energy calculations of Pd clusters using

DFT have shown that Pdn (n=4, 6,8,10 and 13) clusters are relatively more stable [7]. Using

DFT, Zhang et al. [8] have studied the energetic, electronic, and magnetic properties of Pd

Chapter 3

47

nanoparticles of up to 55 atoms and showed that the stability of the nanoparticles increases with

cluster size and dimensionality.

In bulk form, Pd has no magnetic moment due to its closed electronic configuration of

4d10

5s0. However, in cluster form, the delocalization of electrons and depletion of states because

of sp-d hybridization may give rise to magnetism. The answer to the question, “whether Pd

clusters has magnetic properties or not” is still a controversial issue. Experimental studies either

find non-magnetic or very weak magnetic moments. Pd cluster was found to have no magnetic

moment in Stern-Gerlach deflection experiments [9]. But, Photoemission studies have found Ni-

like magnetic behaviour in Pdn (n=3-6) and nonmagnetic behaviour in n>15 [10]. Theoretical

calculations predict very weak ferromagnetic properties [11], while in some cases, it predict

large magnetic moments [2, 12, 13]. In view of the present circumstances, it seems reasonable to

analyze the magnetic properties of the Pd clusters by using an appropriate exchange-correlation

functional and basis set suitable for describing transition metals.

Thermodynamic study is useful in understanding the behaviour of clusters with change in

temperature. In the realm of small clusters, phase transitions are not sharp but gradual [14], and

the thermodynamic properties are greatly dependent on size [15]. Theoretical study can predict

the structural transitions on increasing temperature for small clusters. For example, theoretical

prediction of structural change from fcc to decahedral and icosahedral structures was made and

observed in many systems [16]. In this chapter, we have studied the dependence of stability,

chemical, magnetic and thermodynamic properties on the structures of small Pdn (n=2-11)

clusters. The stability of the clusters is examined with binding energy [BE] per atom, atom

addition energy change and the stability function. Chemical reactivity parameters like, ionization

potential (IP) [17], electron affinity (EA) [17], chemical potential (μ) [17], electronegativity ()

[18], chemical hardness (μ) [19], electrophilicity (ω) [20], polarizability () [21] are investigated

for these clusters, and their dependence on the cluster size is discussed. The stability of the

clusters is also discussed with the help of these reactivity parameters. The magnetic moments for

each of the clusters are also computed. Thermodynamic functions such as heat capacity at

constant volume, entropy and enthalpy are calculated at different temperatures. The knowledge

of these properties may be useful to understand how these small clusters work as catalysts.

Chapter 3

48

3.2 Calculation methods

We have performed ab-initio molecular dynamics simulation near the melting

temperature of bulk palladium, using SIESTA 3.1 software [22]. During the simulation process,

the cluster goes through various structural changes and we recorded the energy and geometries of

the cluster during simulation and choose some of those structures based on minimum energy

criterion with reasonably different geometry. These minimum energy structures of molecular

dynamics run, are considered as initial geometries (or conformers) of palladium clusters Pdn

(n=2-11). Then geometry optimization is performed on each conformer to determine the

geometry having lowest energy in each of the cluster sizes. We have also considered Pd cluster

geometries reported in earlier studies [6-8]. The optimization is performed within the framework

of density functional theory using the GAUSSIAN 09 suite of program [23]. In this study, we

used B3LYP [24, 25], B3P86 [24, 26] and B3PW91 [24, 27] functionals and LanL2DZ [28],

LanL2MB [28-30] and SSD [31-33] basis sets to see the effects of functionals and basis sets on

the calculated properties. Thermodynamic functions for the lowest energy structures are

calculated at the same level of theory and at different temperatures.

The binding energy per atom 𝐵𝐸

𝑎𝑡𝑜𝑚 of a cluster is defined as the energy gain in

assembling the cluster from its isolated constituent atoms. The binding energy per atom is

calculated as

𝐵𝐸

𝑎𝑡𝑜𝑚 =

nE atom −Ecluster

n (3-1)

where n is the number of atoms in the cluster, Eatom and Ecluster are the energies corresponding to

single atom and the cluster respectively.

The atom addition energy change [ΔE1(Pdn)] which corresponds to the energy change for

cluster growing reaction Pdn-1 + Pd → Pdn is calculated as

𝛥𝐸1(𝑃𝑑𝑛) = 𝐸(𝑃𝑑𝑛) − 𝐸(𝑃𝑑𝑛−1) − 𝐸(𝑃𝑑) (3-2)

A large negative change of ΔE1(Pdn) implies that the cluster Pdn is more stable than the

preceding Pdn-1 structure.

The second finite difference of total energies [ΔE2(Pdn)] (i.e. the stability function) is

calculated to study the relative stability of the clusters. The stability function is defined as

ΔE2(Pdn) = E(Pdn+1) +E(Pdn-1)- 2E(Pd) (3-3)

Chapter 3

49

3.3 Results and Discussion

3.3.1 Structure

In the field of cluster study, locating the minimum in the potential energy surface

depends on the choice of initial structures. One has to search for all the configurations in the

potential energy surface. For small cluster, as there are fewer numbers of possible configurations,

one can easily search the potential energy surface. However, things are not simple for larger

clusters since the number of possible initial configurations become quite large. A number of

structural isomers of Pdn clusters (n=2-11) have been investigated using DFT methods. All the

clusters are fully optimized using three different functionals and three basis sets and the most

stable structure having the lowest energy are determined. The optimized Palladium cluster

geometries with B3LYP functional and LanL2DZ basis set are presented in Figure 3.1. For each

of the cluster sizes, the structure on the left column is the lowest energy structure. Some of the

higher-energy isomers are also shown along with their energy difference from the lowest one.

The number of imaginary frequency for the most stable geometries shown here are zero

signifying that they correspond to minima on the potential energy surface. If not specifically

mentioned otherwise, the reported and discussed results are calculated at B3LYP/LanL2DZ

level.

Most stable structure Higher energy isomers

△E=0.000

△E=0.000

△E=0.031 eV

Chapter 3

50

Most stable structure Higher energy isomers

△E=0.000

△E=0.034 eV

△E=0.039 eV

△E=0.000

△E=0.004eV

△E=0.046 eV

△E=0.000

△E=0.002 eV

△E=0.075eV

△E=0.000

△E=0.004 eV

△E=0.006 eV

△E=0.000

△E=0.001 eV

△E=0.002 eV

△E=0.004 eV

△E=0.000

△E=0.007 eV

△E=0.008 eV

△E=0.009 eV

Chapter 3

51

Most stable structure Higher energy isomers

△E=0.000

△E=0.003 eV

△E=0.004 eV

△E=0.005 eV

△E=0.000

△E=0.003 eV

△E=0.004 eV

△E=0.0044 eV

△E=0.006 eV

Figure 3.1 Some of the low-lying isomers of Pdn clusters. Structures in the leftmost column

correspond to most stable geometries. △E represents energy difference of the given geometry

from the most stable one. The reported energy differences are calculated at B3LYP/LanL2DZ

method.

The spin multiplicities of the small clusters up to seven atoms have triplet state. Pd8, Pd9

and Pd10 have pentet ground state; whereas, Pd11 has septet spin multiplicity. The calculated

inter-atomic Pd-Pd bond length of Pd2 dimer is 2.53Å. Valerio and Toulhout [34] have found the

same (2.53 Å) while Kalita and Deka [6] have found 2.54 Å. The obtained result is slightly

higher than the experimental value of 2.48 Å [35]. The optimized geometry has D∞h point group

symmetry. The Pd3 has optimized structure of an isosceles triangle with the symmetric point

group C2v. The bond lengths in Pd3 are 2.55 and 2.76 Å, in agreement with a previous DFT result

[36]. The lowest energy structure for Pd4 is a tetrahedron with C3v symmetry. Our calculated

average bond length of 2.66 Å agrees with previously reported value of 2.67 Å [6]. Pd5 isomer of

square pyramidal structure with C4V point group symmetry is the most stable one with average

bond length of 2.67 Å. The reported value of average bond length of Pd5 is 2.71 Å [6]. The most

stable structure of Pd6 is a regular octahedron with D4h symmetry. Pd-Pd bond length in the

Chapter 3

52

regular octahedron of Pd6 is 2.72 Å. Another DFT calculation [6] has reported the same average

bond length for Pd6. Pd7 is found to have a pentagonal bipyramid as the ground state with C2

symmetry and an average bond length of 2.75 Å. Bicapped octahedron is found to have the

lowest energy for Pd8 having C2 symmetry with an average bond length of 2.74 Å. The observed

bond length is in good agreement with the previously reported value of 2.75 Å [6]. Pd9 has the

ground-state structure of double trigonal antiprism with C3h symmetry. Two octahedrons sharing

one common edge is found to be the most stable structure for Pd10, which has C2h symmetry. The

average bond lengths for Pd9, Pd10 and Pd11 clusters are 2.74, 2.76 and 2.79 Å respectively. In

general, the average bond length show increasing behaviour as the cluster grows in size.

3.3.2 Binding energy and stability

(a)

Chapter 3

53

(b)

(c)

Figure 3.2 (a) Variation in BE per atom with Pdn cluster size (b) Variation in atom addition

energy with Pdn cluster size (c) Variation in stability function with Pdn cluster size.

The value of binding energy per Pd atom as a function of cluster size is given in Table

3.1, and their variation with cluster size at different levels is plotted in Figure 3.2a. The binding

energy per atom increase as the cluster becomes larger in size. Due to increased number of bonds

in the larger clusters, the atoms are bound more tightly as the number of constituent particles in

the cluster is increased. However, this increment is not homogeneous. For cluster containing 2-4

Chapter 3

54

atoms, the binding energy per atom increases rapidly compared to cluster which contains more

than four atoms. Since the experimental binding energy for the bulk is 3.91 eV/atom [37], much

higher than the binding energy of our studied clusters, we infer that large numbers of atoms are

required in the cluster to obtain the bulk behavior. The experimental dissociation energy for the

Pd2 is reported to be 0.73±0.26 eV [38]. In the present work, the binding energy of Pd2 is

calculated as 0.96 eV at B3LYP/LanL2DZ level, which is close to the experimental value. For

Pd3, the binding energy per atom is calculated to be 0.853 eV/atom. Spin polarized DFT study

[39] shows the binding energy of Pd3 as 0.810 eV/atom. However, in another DFT study, Qui et

al. [40] have calculated the binding energy of Pd3 as 0.774 eV/atom. For Pd4 and Pd5, the present

study shows binding energy per atom are 1.249 and 1.301 eV/atom respectively. These values

are higher than those reported by Qui et al. (1.020 and 1.148 eV/atom for Pd2 and Pd3) [40]. On

the other hand, Kalita et al. [6] reported 1.41 and 1.48 eV/atom as binding energy per atom for

Pd2 and Pd3, which are lower than our calculation. In case of Pd6, Pd7, Pd8, Pd9 and Pd10, we

found binding energy per atom as 1.411, 1.439, 1.467, 1.528 and 1.555 eV/atom respectively,

which are greater than 1.295, 1.321, 1.372, 1.407, 1.449 eV/atom [40] and less than 1.878, 1.903,

1.982, 2.037, 2.095 [41] respectively. Pd11 has a binding energy per atom of 1.745 eV/atom.

Table 3.1 Binding energy per atom, ionization potential, electron affinity, atom addition energy

and stability function of Pdn (n = 2 - 11) clusters.

Cluster

size

BE/atom

(eV/atom)

Ionization

potential

(eV)

Electron

affinity

(eV)

E1(Pdn)

(eV)

E2(Pdn)

(eV)

Pd2 0.481 8.709 0.630 -0.961 -0.638

Pd3 0.853 7.868 0.837 -1.599 -0.839

Pd4 1.249 7.004 0.726 -2.438 0.929

Pd5 1.301 6.674 1.243 -1.509 -0.450

Pd6 1.411 6.703 1.215 -1.959 0.346

Pd7 1.439 6.446 1.754 -1.613 -0.020

Pd8 1.467 6.579 1.865 -1.665 -0.356

Pd9 1.528 6.476 1.869 -2.021 0.223

Pd10 1.555 6.348 2.207 -1.798 -0.101

Pd11 1.745 6.570 2.204 -1.899

Chapter 3

55

The relative stability of clusters can be measured in terms of atom addition energy change

and stability function. The relative stability of clusters is reflected in the mass abundance spectra,

where clusters with higher relative stability are more abundant. In Table 3.1, atom addition

energy change, stability functions of the Pdn clusters are tabulated. The atom addition energy

shows that at n=4, 6 and 9, the change is more negative as compared to their neighbouring

clusters indicating that Pd4, Pd6 and Pd9 are more stable. A more accurate and most commonly

used measure of the relative stability is given by stability function(∆𝐸2). Stability function

(∆𝐸2) has relatively high value for n=4, 6 and 9, which clearly imply that these clusters are

relatively more stable than their neighbouring clusters.

3.3.3 Chemical Reactivity

(a)

Chapter 3

56

(b)

Figure 3.3 (a) Variation in IP with number of atoms in palladium clusters (b) Variation in EA

with number of atoms in palladium clusters.

The ionization potential values are shown in Table 3.1 and their variation as a function of

cluster size is plotted in Figure 3.3a. The variation of ionization potential (with size of the

cluster) is more prominent up to Pd5 as presented in Figure 3.3a. The ionization potential for Pd2

is 8.70 eV, which is higher than the result (7.69eV) of Kalita et al. [6]. In case of Pd3, Pd4, Pd5,

Pd6 and Pd7 we found ionization potentials as 7.86, 7.00, 6.67, 6.70 and 6.44 eV respectively,

which are greater than 7.56, 6.86, 6.60, 6.33, 6.30 eV respectively of Kalita et al. [6]. It may be

noted that B3P86 functional give higher values of ionization potential than B3LYP and B3PW91

functionals.

The electron affinity (EA) values, as given in Table 3.1, for Pd2, Pd3 and Pd4 are observed

to be 0.630, 0.837 and 0.726 eV respectively at B3LYP/LanL2DZ level. Using BLYP functional

and DNP basis set, Kalita et al. [6] found EA values of 1.581, 1.486 and 1.239 eV, respectively.

Their calculated values are lower than our result. Our calculation with B3P86 functional and

Chapter 3

57

SDD basis set shows EA values 1.257, 1.418 and 1.342 eV for Pd2, Pd3 and Pd4, respectively.

Hence, one has to carefully choose the method of calculation as the observed properties are

highly sensitive to the method. We have obtained EA value of 1.243 eV for Pd5, which is in

agreement with the experimental results (1.45±0.10 eV) of Ganteför et al. [42]. Pd6 has EA value

of 1.215 eV, which is lower than the experimental value (1.65±0.10 eV) [42]. Our calculated

value of EA for Pd7 is 1.754 eV, which is in good agreement with the experimental result of

1.70±0.15 eV obtained by Ganteför et al. [42]. For the cluster Pd8 and Pd9, our calculated value

is 1.865 and 1.869 eV respectively, which are also consistent with the experimental value of

(1.85±0.25 eV) and (1.95±0.25 eV) respectively [42]. The EA for Pd10 (2.207 eV) and Pd11

(2.204 eV) are also very close to the experimental results of 2.00±0.25 and 1.95±0.25 eV

respectively [42]. The overall trend of EA with the variation of cluster size, with three

functionals and three basis sets, is depicted in Figure 3.3b. The electron affinity increases with

increasing number of atoms in the cluster. Such close agreement of the observed electron affinity

with the experimental results implies the validity of our calculation methods. Thus, the larger

clusters have higher tendency to gain electrons and become negatively charged species as

compared to the smaller clusters. Pd4 and Pd6 are seen to have lower electron affinity values than

their neighbouring clusters, which support the higher stability of these two clusters as suggested

by atom addition energy change and the stability function. Among the functional and basis sets

considered here, B3P86 functional gives the highest EA, and LanL2DZ basis set gives the lowest

EA.

The chemical hardness values for the Pdn (n=2-11) clusters are given in Table 3.2 and

their variation with respect to number of atoms is plotted in Figure 3.4a. The hardness value is

seen to decrease with increasing cluster size except for a small shoulder at n=6, 8 and 11. The

observed trend is expected since in larger clusters the electrons are more loosely bound, and the

clusters are more prone to change their electronic structure [43]. The chemical potential

measures the escaping tendency of electrons, and electrons tend to flow from a region of high

chemical potential to a region of low chemical potential. The difference in chemical potential

cause electron transfer and the flow of the electrons is linearly proportional to the difference of

chemical potential. Among the clusters in our study, Pd4 has the highest value (-3.865 eV) of

chemical potential, and Pd2 has the lowest chemical potential of -4.670 eV using

B3LYP/LanL2DZ level of calculation. In DFT, chemical potential measures the escaping

Chapter 3

58

tendency of an electron cloud [44]. So, higher chemical potential implies higher escaping

tendency of the electron cloud from the system. Hence when Pd2 or Pd4 clusters interact with any

other cluster, Pd4 shows the highest electron donating power, whereas Pd2 shows the highest

electron accepting power, since the chemical potential of Pd4 is the highest and Pd2 is the lowest

among the other clusters. This result is also in conformity with the EEP [45, 46].

Table 3.2 Chemical hardness, chemical potential, electrophilicity and polarizability of Pdn (n = 2

- 11) clusters.

Cluster size Chemical

hardness (eV)

Chemical

potential (eV)

Electrophilicity

(eV)

Polarizability

(a.u)

Pd2 8.079 -4.670 1.349 84.219

Pd3 7.031 -4.352 1.347 113.165

Pd4 6.278 -3.865 1.189 130.729

Pd5 5.431 -3.958 1.443 161.892

Pd6 5.488 -3.959 1.428 188.602

Pd7 4.692 -4.100 1.791 213.284

Pd8 4.714 -4.221 1.890 274.163

Pd9 4.608 -4.173 1.889 308.018

Pd10 4.140 -4.277 2.266 339.642

Pd11 4.364 -4.387 2.205 388.364

The electronegativity (𝜒) as defined above is the measure of attracting electrons towards

itself and is equal to the negative of chemical potential. In interacting systems, electrons flow

from a region of low electronegativity to a high electronegativity region until the

electronegativity values of the constituent systems equalize. The behaviour of electronegativity

(𝜒) as a function of the cluster size is shown in Figure 3.4b. It is clear from Figure 3.4b that in

general the electronegativity of the cluster increases as the size of the cluster increases albeit

after a sudden decrease from Pd2 to Pd4. In other words, the larger clusters have stronger

tendency to attract electrons. The observed trend is consistent with the behaviour shown by

electron affinity. It is clear from Figure 3.4b that the electronegativity (𝜒) of Pd4, Pd6 and Pd9

clusters are lower compared to their neighbouring clusters. The B3P86 functional shows similar

Chapter 3

59

trend and similar value for chemical hardness with the other two functionals due to the mutual

cancellation of the differences shown by the ionization potential and electron affinity = 𝐼𝑃 −

𝐸𝐴 , however for electronegativity it shows similar trend but a comparatively large value due to

the addition of the differences shown by the ionization potential and electron affinity =

𝐼𝑃+𝐸𝐴

2 . The electrophilicity (𝜔) is a measure of the stabilization in energy when the cluster

acquires an additional electronic charge from the surroundings. It is closely related to the

electron affinity and shows same variation with changing cluster size. In Figure 3.5a, we present

the variation of electrophilicity with respect to the number of atoms in the cluster. In case of

B3P86 functional the electrophilicity values are higher compared with the other two functionals.

We observed that in general 𝜔 increases as the cluster grows in size, i.e. the larger cluster has

stronger tendency to accept electrons. It is consistent with what we observed from electron

affinity and electronegativity. The electrophilicity for Pd10 is highest (4.532 eV) as expected as

Pd10 was found to have the highest value of electron affinity among the clusters in this study. Pd4

has the lowest value of electrophilicity (2.379 eV) and is expected least prone to gain electrons

from its surrounding. The electrophilicity values of Pd4, Pd6 and Pd9 are found to be lower than

their neighbouring clusters, which supports the result of their higher stability as predicted from

high stability function and low atom addition energy. The relative stability of these clusters also

reflects the validity of minimum electrophilicity principle (MEP) since electrophilicity of these

clusters are relatively less than its neighbouring counterparts. The polarizability values are

presented in Table 3.2, and their variation with the number of atoms is plotted in Figure 3.5b. It

is to be noted that the polarizability of the Pdn clusters depend almost linearly on the cluster size,

and it is revealed that smaller clusters have lower polarizability and higher value of chemical

hardness, while larger clusters have higher polarizability and lower chemical hardness value. The

observed trend is expected from the principles of minimum polarizability [47] and maximum

hardness [48, 49].

Chapter 3

60

(a)

(b)

Figure 3.4 (a) Size dependence of chemical hardness in palladium clusters (b) Size dependence

of electronegativity in palladium clusters.

Chapter 3

61

(a)

(b)

Figure 3.5 (a) Change in electrophilicity with number of atoms in Pdn (n = 2–11) clusters (b)

Change in polarizability with number of atoms in Pdn (n = 2–11) clusters.

Chapter 3

62

3.3.4 Magnetic properties

Table 3.3 The variation of magnetic moments (µB per atom) with cluster size calculated in this

work at B3LYP/LanL2DZ level, and by Kumar et al. [2], Granja et al. [12] and Futschek et al.

[13] are presented.

Cluster

size

Magnetic moment

in the present

study

Magnetic moment by

Kumar et al.

Magnetic moment by

Granja et al.

Magnetic moment

by Futschek et al.

Pd2 1.000 1.000 1.000 1.000

Pd3 0.666 0.666 0.666 0.000

Pd4 0.500 0.500 0.500 0.500

Pd5 0.400 0.400 0.400 0.400

Pd6 0.333 0.000 0.333 0.333

Pd7 0.290 0.290 0.290 0.290

Pd8 0.500 0.250 0.500 0.250

Pd9 0.444 0.444 0.444 0.222

Pd10 0.400 0.600 0.600 0.400

Pd11 0.550 0.550 0.550

In this study, we observe that magnetism is associated with Pd clusters. In Table 3.3, we

present the magnetic moment per atom for lowest energy structures of Pd clusters with B3LYP

functional and LanL2DZ basis set and are compared with the results of other DFT calculations.

The other two functionals and basis sets also give the same values. Our calculated magnetic

moments for Pdn clusters up to n=11 are exactly same with the results of Granja et al. [12], with

the only exception of Pd10 for which they have obtained 0.60 µB per atom, greater than our result

of 0.40 µB per atom . The magnetic moments for Pdn clusters up to n=7 are same with Futscheck

et al. [13], with the only exception that Futscheck et al. [13] observed zero magnetic moment for

Pd3. The observed magnetic moments also agree with the results of Kumar et al. [2] up to Pd7,

except for Pd6, which they found to be non-magnetic. In the case of Pd8, our magnetic moment is

similar to Granja et al. [12] but differ from Kumar et al. [2] and Futscheck et al. [13]. In the case

of Pd9 and Pd11, our calculated magnetic moments are in agreement with the results of Kumar et

Chapter 3

63

al. [2] and Granja et al. [12]. For Pd10 our calculated magnetic moment 0.40 µB per atom is in

agreement with the results reported by Futscheck et al. [13]. So in general, our calculated

magnetic moments for Pdn clusters are in good agreement with other DFT study [2, 12, 13]. It is

to be noted that clusters studied here are very floppy and so many isomeric structures with very

close in energy exists. These isomers are not only structural but magnetic also. Researchers have

found very small difference in total energy [50] due to the presence of different magnetic

moments in cluster and the low-lying spin states are basically controlling the electronic and

magnetic properties of the cluster. Due to closed shell electronic configuration, palladium atom

shows no magnetic moment. This closed shell results weak bonding in small cluster. However, in

the cluster, the aggregation of atoms leads to delocalization of electrons. Upon bonding, sp-d

hybridization develops with the depletion of 4d states on each atom. This depletion may be the

reason for magnetism in the small palladium cluster.

Table 3.4 The variation of magnetic moments (µB per atom) with cluster size calculated at

various levels of theory.

Cluster

size

Magnetic moments per atom (µB per atom)

B3LYP B3PW91 B3P86

LanL2M

B

SDD LanL2D

Z

LanL2M

B

SDD LanL2D

Z

LanL2M

B

SDD

Pd2 1.000 1.000 0.999 1.000 1.000 1.000 0.999 1.000

Pd3 0.666 0.666 0.666 0.666 0.666 0.666 0.666 0.666

Pd4 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500

Pd5 0.400 0.400 0.399 0.400 0.399 0.400 0.400 0.400

Pd6 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333

Pd7 0.290 0.290 0.286 0.290 0.286 0.290 0.290 0.290

Pd8 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500

Pd9 0.444 0.444 0.444 0.444 0.444 0.444 0.444 0.444

Pd10 0.400 0.400 0.399 0.400 0.400 0.400 0.400 0.400

Pd11 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550

Chapter 3

64

3.3.5 Thermodynamic properties

Thermodynamic properties, such as heat capacity at constant volume (Cv), entropy (S)

and enthalpy (H), for these Pdn clusters at different temperatures are computed and analyzed. The

vibrational zero point energies for the clusters Pd2, Pd3, Pd4, Pd5, Pd6, Pd7, Pd8, Pd9, Pd10 and Pd11

at B3LYP/LanL2DZ level of calculation are 1.220, 2.790, 4.970, 8.510, 8.130, 9.890, 12.240,

14.400, 16.180 and 17.600 kJ/mol respectively. Increasing number of atoms in the cluster is

associated with a corresponding increase of vibrational mode which in turn resulted to increase

in its vibrational zero-point energy. As the cluster size becomes larger, the heat capacity (Cv) of

these clusters also becomes higher. The variation of the heat capacity (Cv) with temperature for

Pdn is presented in Figure 3.6a. It is clear from Figure 3.6a that (Cv) is monotonically increasing

function of temperature and ultimately levels off for large temperature. Entropy (S) and enthalpy

(H) of Pdn at different temperatures are presented in Figure 3.6b and 3.6c respectively. Both

entropy and enthalpy increases linearly with the increase of cluster size. The same effect is also

observed by increasing the temperature and this is expected since an increase in temperature is

associated with an increase in internal energy and volume of the cluster.

(a)

Chapter 3

65

(b)

(c)

Figure 3.6 (a) Heat capacity (Cv) as function of temperature (at B3LYP/LanL2DZ method) (b)

Entropy as function of temperature and cluster size (at B3LYP/LanL2DZ method) (c) Enthalpy

as function of temperature and cluster size (at B3LYP/LanL2DZ method).

Chapter 3

66

3.4 Conclusions

Palladium clusters, Pdn containing up to 11 atoms are studied using density functional

theory and analyzed in terms of various reactivity parameters like, ionization potential, electron

affinity, chemical potential, electronegativity, electrophilicity, chemical hardness, polarizability,

etc. As the cluster grows in size, the average bond length, binding energy per atom, entropy and

enthalpy are found to increase whereas chemical hardness decreases. Larger clusters have higher

polarizability and lower chemical hardness value, which are consistent with the minimum

polarizability principle (MPP) and maximum hardness principle (MHP). Stability function and

atom addition energy change predict that Pd4, Pd6 and Pd9 are relatively more stable than their

neighbouring clusters. Chemical reactivity parameters like electron affinity, electronegativity and

electrophilicity suggest that larger clusters have stronger tendency to accept electrons. The trend

of reactivity parameters mimics the result of the atom addition energy change and stability

function. All the Pd clusters are found to be magnetic.

In the next chapter, we have studied structure, chemical and optical properties of

platinum clusters.

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