1

CHAPTER 1

Introduction

1.1 A brief introduction to metal cluster

Clusters are the materials consisting of a few to hundreds of atoms, which provide a link

between atoms, molecules and bulk material. Finite structure materials such as metal clusters

[1], carbon nanotubes [2], graphenes [3], fullerenes [4] and biological structures [5] are not

traditionally discussed in the solid-state context. However, such finite structure systems are

important, particularly in the field of nanotechnology [6-8], where the size of functional

material is in nanometer range.

As clusters, both molecules and nanoparticles are also aggregates of atoms. The main

difference between molecule and cluster is their size and composition. The size and composition

of a molecule is fixed whereas the size and composition of clusters can be varied. Clusters

interact strongly with each other and have a strong tendency to coalesce and form larger

clusters, but molecules interact with each other very weakly. Cluster often exhibits large

numbers of isomers in which atoms are arranged in different geometrical configurations. On the

other hand, atoms have definite geometrical position in molecules, and molecules hardly exhibit

isomerism.

To differentiate between clusters and nanomaterials, one can say that clusters are the

ultimate nanoparticles where the size and composition can be controlled with atomic precision.

To be more elaborate, number of atoms in nanomaterials is not specified, but the size and

composition of cluster is known exactly. In small clusters, properties change nonmonotically,

and sometimes drastically with size. One of the most significant features of atomic clusters is

that their structures may undergo drastic changes even with the addition of a single atom. As the

size of the cluster grows, the variation of properties becomes less drastic, and on further

increase in size, the properties smoothly approach the bulk value. Both the fields of cluster and

nanoparticles are growing rapidly in recent years and distinction between them has been

narrowed down. As clusters becoming larger and nanoparticles becoming smaller, the

Chapter 1

2

demarcation between the two cannot be specified, and clusters are sometimes referred to as

nanoclusters.

The first report of cluster synthesis in laboratory came from mass spectrometer ion

sources experiments [9]. In this experiment, intense molecular beams were used to produce

clusters by supersonic expansion [9]. Knight and co-workers [10] first produced metallic

clusters. The clusters were formed by supersonic expansion of gas composing of sodium atoms.

Nowadays, clusters of most of the elements in the periodic table and also compound clusters

(consisting binary and ternary elements) can be produced using laser vaporization techniques

[11-14].

The study of cluster science has evolved through a number of reincarnations. Earlier,

clusters were regarded as annoyance causing contaminations in the mass spectrometry peaks,

leading to difficulties in resolving and identifying species. Later, it was realized that their study

can provide deeper insight on nucleation phenomena, interactions and evolution of condensed

matter. The study of connections between cluster science and nano-science is of particular

significance because it forms the basis for understanding nanosystems. One can look at clusters

as embryonic solids, in which the evolution from atom to macroscopic state was captured at

very early stage. Looking at the way in which these systems are being studied by an ever-

increasing number of physicists, chemists and material scientists, they are expected to bring

revolutionary changes in the present-day technology. Clusters composed of metallic or covalent

elements are being considered to be building blocks of new types of materials with intriguing

properties [15]. It is suggested that clusters can be designed and synthesized in such a way that

they mimic the properties of atoms. Such clusters are referred to as unified atoms or superatoms

[16]. These superatoms can be used as building blocks of materials instead of atoms. Even they

can form the basis of a new three-dimensional periodic table with superatoms constituting the

third dimension. They may lead to a new era in material science, where there are unlimited

possibilities of creating new materials. Such materials will allow their electronic, optical,

magnetic, mechanical and chemical properties to be tailored for a specific purpose. On the

other hand, since the miniaturization of conventional silicon-based electronics approaches

fundamental performance limits, finding new materials that can overcome these limits has

become important; and cluster emerges as a promising candidate in this context [17]. Crystals

made of clusters have been synthesized, which show very interesting properties, as for example,

Chapter 1

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crystals of C60 clusters become high-temperature superconductors on doping with metallic

elements. Scientists have been trying to identify other magic clusters that could be used as

building blocks for new materials. Si4 clusters deposited on a graphite substrate can survive as

individuals, and thus have the potential to form a new, cluster-based silicon material [18]. Small

molybdenum clusters of tiny cubes are capable of self-assembling into larger cubes [19].

Nanostructures assembled from cobalt clusters exhibit giant magnetic response of more than

500% at low temperatures with strong field dependence [20]. This cluster-assembled material

has potential applications in the field of spintronics. Furthermore, clusters can be used as

models to study defects in crystals [21] since the properties of defects are primarily governed by

interactions with a few neighboring atoms. The finite size clusters in which atoms occupy the

positions of crystal structure provide us a good model for defect study.

Compared to their bulk counterparts, cluster shows distinct physical and chemical

properties, which arise due to the quantum size effects and coulomb-charging effects [22]. High

surface to volume ratio and significant change in electronic structures are the important features

of the clusters. Metal clusters having only a few to tens of atoms whose sizes are comparable

with the Fermi wavelength of electrons are a fascinating area, since electrons are confined to

molecular dimensions. Due to this confinement, they exhibit unique properties such as molecule

like energy gaps [23-26], strong photoluminescence [27-29] and strong catalytic properties [30-

32]. The fluorescent emission of clusters can be tuned by controlling the cluster size [33]. For

the catalytic properties, gold clusters containing ten Au atoms supported on iron oxide exhibit

strong catalytic activity for CO oxidation [34]. The small clusters not only open the possibility

of tuning a catalytic process by changing cluster size, but also can be used to catalyze chemical

reactions at low temperatures [35]. Thus the shape and the microstructure play a major role in

controlling the catalyst‟s activity. The study of the mechanical, chemical and electromagnetic

properties of the clusters and their relationships to size and shape will enhance our

understanding of these small clusters and also their applications in the chemical and material

industry.

Metal clusters have been successfully synthesized with controlled structure and

properties [36-38]. An important feature of clusters thus formed is the existence of shell-

structure similar to the electronic shell in atoms. It paves a way to design and synthesize clusters

with closed electronic shells, which are not only stable but also inert. In addition to the

Chapter 1

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electronic shell, geometrical shape also plays an important role in the stability of clusters. In

many cases, clusters with closed geometric shells such as FCC, BCC or icoshedron are more

stable than their neighbors.

Unlike bulk system, it is very difficult to assign cluster structures. Many low-lying

isomers with small differences in their binding energies often exist in the ground state. Thus it is

possible that many isomers are populated under typical experimental conditions. From the

theoretical view-point, determination of ground state is difficult due to the existence of many

low-lying isomers. On the other hand, the present experimental techniques do not allow

unambiguous determination of cluster structure. Many of the clusters are too small for

diffraction techniques. Electron tomography requires a series of images taken for many different

orientations, and this approach is suitable for stable and stationary structures. But small clusters

are intrinsically and structurally unstable and may interact with the incident electron beam [39].

Theory and experiment should complement each other; however, the cooperation is not an easy

task. The easiest task for the theorists is sometimes the most difficult for the experimentalists

and vice versa. Experimental studies are performed for investigating many of the cluster

properties like ionization potential, electron affinity, reactivity and magnetic behavior [40-42].

Clusters of particular stability, i.e. magic number clusters can be observed with the help of mass

spectrometroscopy measurements [43]. Some of the common experimental techniques used for

comparison with calculated results are photoelectron spectroscopy (PES), trapped ion electron

diffraction (TIED), ion mobility and infrared spectroscopy. As the experimental determination

of structure and specific electronic characterizations of these nanoscale systems are quite

difficult; one has to rely on the predictive power of computational methods. In addition to

determining the equilibrium states, theoretical calculations also complement other experimental

results. Accurate prediction of vibrational frequency, IR intensity and Raman spectra facilitates

in identifying the true cluster geometries, gives a kind of assurance that the theoretically

determined structures are true. Advancement of computational techniques has enabled to predict

the structure and properties of clusters with a high degree of accuracy nowadays.

Chapter 1

5

1.1.1 Electronic structures of clusters

The energy levels in atoms are discrete and the stability, reactivity and electronic

properties of atoms are greatly dependent upon the difference between the highest occupied and

lowest unoccupied molecular orbitals (HOMO-LUMO gap). In clusters, aggregation of atoms

leads to overlapping of energy levels. This overlapping changes the highest occupied and lowest

unoccupied energy levels. In crystals of metallic atoms, the overlapping between energy levels

becomes very high resulting energy bands, and the HOMO-LUMO gap disappears. In

semiconducting and insulating crystals, the energy bands exhibit a gap at the Fermi level. There

is always a gap between the highest occupied and lowest unoccupied molecular orbitals in small

clusters.

One of the most popular models to describe electronic structure of clusters is the jellium

model [10, 44]. This model is based on the assumption that valence electrons are distributed in

electronic shells and the positive charge of the ions is distributed uniformly in a sphere of radius

R. Jellium model is a simple, very popular and powerful model to describe the electronic

structure of simple metal clusters. Pedersen et al. [45] directly observed and demonstrated the

quantum shells in clusters of sodium atoms consisting up to 3000 atoms. Their experiment was

based on measurements of the size distribution of metal droplets produced by expansion of

metal vapor through a fine nozzle. The shortcoming of Jellium model is that it cannot be used to

determine the electronic structure of non-metallic clusters, and very small clusters do not have

metallic properties irrespective of whether clusters consist of metal or non-metallic elements.

Hence, the simplified jellium model is not sufficient to properly understand the electronic

structure of clusters.

1.1.2 Properties of clusters

Depending on the size and composition, clusters have exciting properties. Some of the

peculiar properties of clusters are:

1.1.2.1 Reactive properties

Due to their large surface-to-volume ratio and low coordination number clusters exhibit

unique reactivity properties. For example, while bulk gold is chemically inert, gold cluster is

Chapter 1

6

reactive and can act as good catalysts [46]. Au8 cluster grown on an oxygen vacancy in a MgO

film converts CO to CO2 in the presence of oxygen at low temperature [47]. Vajda and

coworkers reported that the catalytic activity of Pt clusters (consisting of 8 to 10 atoms) for the

oxidative dehydrogenation of propane is 40–100 times higher than that of more conventional

platinum catalysts [48]. Again, though bulk iron does not bind with hydrogen, but iron clusters

can form stable hydrides [49]. One can expect good catalysts materials based on clusters of

inexpensive and more abundant elements. Clusters also provide a fertile ground for

understanding catalysis mechanism since one can manipulate the size of clusters as well as

number of constituent particles. Electron affinity is one of the properties which govern the

reactivity of materials. Clusters can have much higher electron affinity values than a single

atom. Superhalogens, a kind of cluster consisting of metal atoms surrounded by halogen atoms

can have electron affinity of 10 eV [50]. Here it can be noted that electron affinity of chlorine,

highest among the elements, is only 3.61 eV. Another class of materials referred to as

hyperhalogens, consisting of a metal atom at the core surrounded by superhalogen moieties

have higher electron affinity than superhalogens [50]. Interaction of molecules with clusters can

be used to study their geometry. As an example, Ni7 cluster when exposed to N2 gas molecules

first absorbs one N2 molecule, which is followed by absorption of six more N2 molecules [51].

This suggests that a capped octahedron is the most likely structure of Ni7 cluster, in which one

N2 molecule first binds to the Ni atom at the capping site, followed by six more N2 molecules

binding to six Ni atoms in the octahedral site.

1.1.2.2 Optical properties

The optical properties of materials are governed by the electronic structure and optical

band gap. The energy gap in a bulk system is fixed, but the energy gap for clusters varies with

size and composition. Since the electrons of clusters are confined in molecular dimensions and

discrete energy levels, the electronic and optical properties of clusters show unique properties

like, molecule-like energy gaps [52] and strong photoluminescence [53]. According to Mie

theory, the position and intensity of the absorption bands of ultraviolet-visible spectra are

strongly influenced by the particle size, shape, concentration, composition and dielectric

properties of clusters and their local environment [54]. Thus one can design clusters with

tunable optical properties by simply changing their size and composition. For example, the

Chapter 1

7

optical absorption spectra of even-numbered gold clusters are sharper and intense than odd-

numbered clusters [55]. The fluorescent emission of clusters can be tuned from UV to near IR

regions by changing their size [56]. Gold clusters with sizes Au5, Au8, Au13, Au23 and Au31

exhibit UV, blue, green, red and near IR emission respectively [56]. Noble metal clusters

(mainly silver and gold) also exhibit significant promise for the direct conversion of solar to

chemical energy using photocatalysts [57].

1.1.2.3 Magnetic properties

Magnetism plays a key role in technology. Understanding magnetism, i.e., its origin and

its magnitude are essential to design new magnetic materials. In magnetic material, the atoms

carry magnetic moments, which come from unpaired electrons. In solids, the magnetic moment

per atom is usually smaller than those of constituent individual atoms. Magnetic moments of

materials are affected by interatomic distance, dimensionality and coordination [58]. As size of

the system grows, increasing overlap between electron spins at neighboring sites resulted in

reduction of magnetic moments. Thus atoms in the linear chain are more magnetic than surface

atoms, which in turn are more magnetic than atoms in bulk. As most of the atoms in clusters are

at the surface, lower dimension, lower coordination and increased interatomic distance causes

clusters to have higher magnetic moment per atom than bulk. The magnetic moment per atom of

clusters are observed to change nonmonotically with increasing cluster size, and their value lies

in between those of individual atom and atoms in bulk system. Magnetism in clusters also

exhibits peculiar properties, for example, paramagnetic materials in bulk form (V and Rh)

become ferromagnetic [59, 60], antiferromagnetic material (Mn) becomes ferromagnetic [61].

Magnetism in clusters is also closely related to their geometry. As an example, Li4 tetrahedron

configuration has a magnetic moment of 2 μB while Li4 planar is non-magnetic [62]. Thus, if we

can develop a mechanism to change their structures, clusters can be used as nano-magnetic

switch.

Chapter 1

8

1.1.2.4 Melting properties

In material science, it is essential to understand the transition between solid and liquid

phases of materials. An accurate picture of material‟s performance under extreme conditions

depends on this knowledge. At the melting temperature, a macroscopic crystal exhibits a spike

in the heat capacity due to the latent heat. If the size of the system is reduced to nanometer

regime, the surface energy makes an important contribution to the total energy (because of

increased surface-to-volume ratio). This leads to depression of the melting point, which has

been observed and confirmed in many studies [63-65]. For gold (bulk melting point=1337K),

3.8 nm particles melt at ~1000K and 2.5 nm particles melt at ~500K [63]. The melting point

depression is expected to scale smoothly with particle size. In fact, Pawlow [66] suggested that

the depression should show a 1/r dependency (r, being the radius of the particle). However, if

we further decrease the particle size up to the cluster size regime, the melting properties start

showing size dependent fluctuations [67]. In this size regime, adding a single atom may have

significant effect on the meting temperature. As most of the atoms in clusters are at the surface

and hence low coordination among atoms, the clusters are expected to have lower meting

temperature than bulk. But the melting behavior of clusters is different from one element to

another. The melting point of some clusters (e.g. sodium clusters) is lower than their

corresponding bulk [68]. However, small tin and gallium clusters have considerably higher

melting temperature than the bulk value [69, 70]. The increase of melting temperature has been

attributed to the clusters having different bonding than the bulk [71].

1.1.3 Transition metals clusters

In alkali metal cluster, it is observed that small clusters are planar and get transformed

into three-dimensional structure on increasing cluster size, as for example, Na clusters become

three-dimensional when containing more than five atoms [72]. Due to the unfilled ns1 outer

shell configurations, the alkali atoms of the cluster interact strongly with each other. However,

for alkaline-earth metal clusters, there is weak interaction between the atoms due to closed ns2

outer shell configuration. As the cluster size increases, the s and p orbitals begin to overlap

causing a gradual increase in interaction between the clusters. The properties of transition metal

clusters are mainly governed by the unfilled d orbitals with strong spin-dependent correlations

and are expected to be more complex than those of alkali and alkaline-earth metals. The

Chapter 1

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geometry and electronic structure of transition metal clusters are very much different from those

of alkali and alkaline-earth metal clusters. The relative stability of many transition metal

clusters does not follow any specific rule with the variation of size. Among the transition metal

clusters, coinage group of metals Cu, Ag and Au clusters have similar relative stability pattern

as that of alkali metal clusters. This similarity is attributed to the monovalent nature of coinage

and alkali metals. Interests in the transition metal clusters are due to fundamental as well as

practical reasons [73]. The transition metal clusters have unusual electronic properties [74],

which may lead to their application in the development of new technologies, like miniaturized

electronic devices [75]. The transition metal clusters are shown to have many novel catalytic

properties [76, 77], finding their applications in areas of photocatalysis [78] and electrocatalysis

[79].

1.1.4 Heteroatomic metal clusters

Clusters containing more than one kind of atoms are known as heteroatomic clusters.

Depending on the number of different kinds of metal atoms present, clusters may be bimetallic

(or binary), trimetallic (or ternary) and so on. Heteroatomic metal clusters have evoked a lot of

interest in recent years due to their potential applications in optical [80], electronic [81],

magnetic [82] and catalytic [83] uses. Monometallic (or pure elemental clusters) have been

extensively studied, and in comparison to monometallic clusters, studies on heteroatomic metal

clusters are relatively few. Heteroatomic metal clusters are reported to have higher catalytic

activity and selectivity than the monometallic clusters [84, 85]. However, the study of

heteroatomic metal clusters is much more complex than usual monometallic clusters due to the

introduction of new metallic components. As for example, Wu et al. [86] have reported that the

structures and properties of trimetallic clusters are different from those of monometallic and

bimetallic clusters. The properties of heteroatomic clusters can be tuned by altering size,

composition and atomic orderings. For example, the occurrence of order-disorder transition and

surface segregation of Ni-Cu-Rh trimetallic cluster depends on the cluster size, shape,

composition and atomic ordering [87].

In general, heteroatomic clusters can be considered to be similar with alloys. However,

heteroatomic clusters may be formed by the combination of any elements, even if, the

corresponding elements cannot be alloyed in the bulk form. Heteroatomic clusters having two or

Chapter 1

10

more different elements may be used to study how different atoms interact with each other.

Information on their structure and properties will be useful guidance for the development of

new types of materials.

1.2 Quantum chemical approach

First-principles theoretical approaches can describe with good accuracy the complex

chemical bonds in metallic clusters, but the involved high computational cost involved with

increasing system size limits their applications. Among the first-principle methods, density

functional theory (DFT) makes the best compromise between accuracy and computational cost,

and it is the most commonly used method to describe cluster systems. DFT has made

remarkable progress over the years, becoming as a mainstay for computational chemistry,

physics, material science, atmospheric science and life sciences. It has crossed new frontiers and

its applications extend to structure optimization, reactivity, dynamics, excited states, and

different types of spectroscopic properties.

The starting point of quantum chemical approach to find the electronic structure of

matter is the Schrödinger equation. One has to find the solution of time-independent non-

relativistic Schrödinger equation

Ĥ𝑖(𝑟 1, 𝑟 2, …… , 𝑟 𝑁 , 𝑅 1, 𝑅 2, …… , 𝑅 𝑀) = 𝐸𝑖𝑖

(𝑟 1, 𝑟 2, …… , 𝑟 𝑁 , 𝑅 1, 𝑅 2, …… , 𝑅 𝑀 (1-1)

here Ĥ is the Hamiltonian operator for a system of M nuclei and N electrons. The Hamiltonian

operator Ĥ is represented as

Ĥ = − ћ

2

2𝑚𝑒𝛻𝑟 𝑖

2𝑁𝑖=1 −

ћ2

2𝑀𝐼𝛻

𝑅 𝐼

2𝑀𝐼=1 −

𝑍𝐼𝑒2

𝑅 𝐼−𝑟 𝑖 𝑀𝐼=1

𝑁𝑖=1 +

1

2

𝑒2

𝑟 𝑖−𝑟 𝑗

𝑁𝑗 >𝑖

𝑁𝑖=1 +

1

2

𝑍𝐼𝑍𝐽 𝑒2

𝑅 𝐼−𝑅 𝐽 𝑀𝐽>𝐼

𝑀𝐼=1 (1-2)

where i and j run over N electrons while I and J run over M nuclei in the system. me is the mass

of an electron and 𝑀𝐼 is the mass of the nucleus I. The first two terms represent kinetic energies

of an N electrons and M nuclei respectively. The last three terms correspond to the attractive

electrostatic interaction between the nuclei and electrons, the repulsive interaction between the

electron-electron and nucleus-nucleus respectively. The wave function

𝑖(𝑟 1, 𝑟 2, …… , 𝑟 𝑁 , 𝑅 1, 𝑅 2, …… , 𝑅 𝑀) corresponds to the i‟th state of the system. This wave

function contains all information about the system and depends on 3N spatial coordinates and N

Chapter 1

11

spin coordinates of the electrons, collectively denoted as 𝑟 𝑖 and 3M spatial coordinates of

nuclei 𝑅 𝐼 .

The mass of the nuclei is very large as compared to that of electron. Even the lightest of

all nuclei, i.e, the proton (1H) weighs roughly about 1800 times the mass of an electron. As a

consequence, the nuclei move much slower than the electrons. We may then assume that during

the short time in which electrons change their state, the nuclei are fixed. In other words, the

electrons can be considered to move in the field of fixed nuclei. This assumption which

simplifies the Schrödinger equation to great extent is the famous Born-Oppenheimer or

clamped-nuclei approximation. With this approximation, the kinetic energy term of nuclei

becomes zero and the potential energy due to nucleus-nucleus interaction can be represented as

a constant term. Thus the Hamiltonian reduces to

Ĥ = − ћ

2

2𝑚𝑒𝛻𝑟 𝑖

2𝑁𝑖=1 −

𝑍𝐼𝑒2

𝑅 𝐼−𝑟 𝑖 𝑀𝐼=1

𝑁𝑖=1 +

1

2

𝑒2

𝑟 𝑖−𝑟 𝑗

𝑁𝑗>𝑖

𝑁𝑖=1 . (1-3)

Equation (1-3) is still very difficult to solve for getting due to the complexity of

electron. Because of the antisymmetric wave function nature of fermions, when two electrons of

the same spin interchange position, must change sign i.e. 𝑟 1, 𝑟 2, …𝑟 𝑖 , 𝑟 𝑗 … , 𝑟 𝑁 =

− 𝑟 1, 𝑟 2, … , 𝑟 𝑗 , 𝑟 𝑖 , … , 𝑟 𝑁 . This property, due to antisymmetry is known as exchange property.

Each electron is also affected by the motion of other electrons in the system, a property known

as correlation property. In order to solve equation (1-3), many methods have been suggested,

which tries to simplify the inherent complex nature of the equation.

The Hamiltonian in equation (1-3) can be written as

Ĥ = 𝑇 + 𝑉𝑒𝑒 + 𝑉𝑒𝑥𝑡 (1-4)

where 𝑇 = − ћ

2

2𝑚𝑒𝛻𝑟 𝑖

2𝑁𝑖=1 ; 𝑉𝑒𝑒 =

1

2

𝑒2

𝑟 𝑖−𝑟 𝑗

𝑁𝑗 >𝑖

𝑁𝑖=1 ; 𝑉𝑒𝑥𝑡 =

𝑍𝐼𝑒2

𝑅 𝐼−𝑟 𝑖 𝑀𝐼=1

𝑁𝑖=1

1.2.1Hartree-fock approximations

The difficulty with the solution of many-body problem lies in the interaction between

the individual atoms. However, the many-body problem will decouple into one-body problem in

the absence of electron–electron interaction. Hartree formulated the independent-particle model,

according to which each electron moves in an effective potential. The attraction of the nucleus

and the average repulsive interactions due to other electrons are taken into account in the

Chapter 1

12

effective potential. Each electron is then described by its own wave function. And many-body

wave function can be written as a product of electron wave functions

𝐻 𝑟 1, 𝑟 2, . . . … , 𝑟 𝑁 = ∅1(𝑟 1)∅2(𝑟 2) ……∅𝑁(𝑟 𝑁) (1-5)

here the wavefunctions ∅𝑖(𝑟 𝑖) are the normalized states of the individual electrons. This is

known as Hartree approximation. As electrons are fermions, the antisymmetric nature of the

electron wave function has to be incorporated. The Hartree total wave function given in

equation (1-5) is not antisymmetric in the electron coordinates. The generalization of the

Hartree method which takes into account the antisymmetry requirement can be most simply

done by approximating the N-electron wave function to be slater determinant

𝐻𝐹 𝑟 1, 𝑟 2, . . . … , 𝑟 𝑁 =

∅1 𝑟 1 ∅1 𝑟 2 … ∅1 𝑟 𝑁

∅2 𝑟 1 ∅2 𝑟 2 … ∅2 𝑟 𝑁 … … ……………………………………………………………

∅𝑁 𝑟 1 ∅𝑁 𝑟 2 … ∅𝑁(𝑟 𝑁)

(1-6)

This approximation is known as the Hartree-Fock approximation. It has the desired

effect since interchanging positions of two electrons changes its sign. Thus, in Hartree-Fock

approach, the N-electron wave function is the antisymmetric product of individual electron

spin-orbitals. Using the variational method, the optimal slater determinant with „best‟ electron

spin-orbitals can be determined.

1.2.2 Density functional theory

Another approach of looking into the problem of many-electron system is density

functional theory (popularly known as DFT). DFT has its origin in a landmark paper by

Hohenberg and Kohn which appeared in 1964 [88]. The basic ideas of DFT are contained in the

two papers of Hohenberg, Kohn and Sham [88, 89]. The idea is that instead of dealing with the

many-body Schroedinger equation involving the many-body wavefunction 𝑟 1, 𝑟 2, . . . … , 𝑟 𝑁 ,

one deal with a formulation involving only the total electron density 𝑛(𝑟 ). Thus instead of

dealing with 3N degrees of freedom, one has to deal only with electron density with 3 degrees

of freedom; this is a huge simplification.

The approach of DFT has its foundations in the two famous theorems known as

Hohenberg-Kohn theorems:

Chapter 1

13

1.2.2.1 The first Hohenberg-Kohn theorem

The first theorem (quoting from the original paper) states that „the external potential is

𝑉𝑒𝑥𝑡 (𝑟 ) is (to within an additive constant) a unique functional of 𝜌(𝑟 ); since, in turn 𝑉𝑒𝑥𝑡 (𝑟 )

fixes H we see that the full many particle ground state is a unique functional of 𝜌(𝑟 ).‟ Hence the

ground state density uniquely specifies the external potential, which in turn specifies the

Hamiltonian and all other properties of the system. In can be summarized as

𝜌𝑜𝑉𝑒𝑥𝑡 H o𝐸o .

Since the ground state energy is a functional of the ground state electron density, the

individual components of ground state energy must also be functional of ground state density

𝐸o 𝜌𝑜 = T 𝜌𝑜 + 𝐸ee 𝜌𝑜 + 𝐸ext [𝜌𝑜] (1-7)

T denotes the kinetic energy term, Eee denotes term due to electron-electron interaction and Eext

is for the nuclei-electron interaction. The energy expression can be separated into two parts: the

part independent of the system along with external perturbation if any, i.e. T+Eee ; and part

which depend on the system, i.e., due to nuclei-electron attraction and perturbation Eext ρo =

ρo

(r ) Vext dr .

The system independent part is collectively referred to as Hohenberg-Kohn functional

𝐹𝐻𝐾 [𝜌𝑜], also called universal functional because their form is independent of the system

𝐹𝐻𝐾 𝜌𝑜 = T 𝜌𝑜 + 𝐸ee 𝜌𝑜 .

Thus equation (1-11) can be written as

𝐸o 𝜌𝑜 = 𝐹𝐻𝐾 𝜌𝑜 + ρo

(r ) Vext d𝑟 (1-8)

1.2.2.2 The second Hohenberg-Kohn theorem: variational principle

The second Hohenberg-Kohn theorem comes from the energy variational principle. It

states that for any trial density 𝜌 (𝑟 ), which satisfies necessary boundary conditions such as

𝜌 𝑟 ≥ 0, 𝜌 𝑟 𝑑𝑟 = 𝑁 and which is associated with some external potential, the energy

obtained is upper bound to the true ground state energy E0, i.e.,

𝐸o ≤ 𝐸[𝜌 ]= 𝐹𝐻𝐾 𝜌 + 𝜌 (r ) Vext d𝑟 .

E0 results if and only if ground state density is used in equation (1-8). In plain words, the

ground state energy is obtained only if the input density is the true ground state density 𝜌𝑜 .

Chapter 1

14

1.2.2.3 Kohn-Sham approach

Kohn and Sham in 1965 suggested an approach to determine the unknown universal

functional FHK. They introduced the concept of a non-interacting reference system (in which the

electrons behave as uncharged fermions which do not interact with each other via coulomb

repulsion) built from a set of orbitals so that a major part of the kinetic energy can be computed.

The remaining part of kinetic energy is merged with the non-classical portion of the electron-

electron interaction, which are also unknown, but usually small. Together, the two merged

quantity is known as exchange-correlation functional. The search for an accurate exchange-

correlation functional is the greatest challenge in the density functional theory. The exchange-

correlation energy contains: (i) kinetic correlation energy, which comes from the difference in

the kinetic energies of real interacting and non-interacting systems; (ii) coulombic correlation

energy, which arises from the inter-electronic repulsion; (iii) the exchange energy, which comes

from the antisymmetric requirement of the wavefunction and (iv) self interaction correction.

The ground state energy of many-electron system is given by

𝐸 𝜌 = T 𝜌 + 𝐸ee 𝜌 + 𝜌(r ) Vext d𝑟 (1-9)

here, Eee contains the classical coulomb interaction term, 𝐽[𝜌] and non-classical portion, 𝐸𝑛𝑐𝑙 [𝜌]

due to self-interaction correction, exchange and electron correlation effects. The explicit form of

𝐽[𝜌] is

𝐽 𝜌 =1

2

𝜌(r )𝜌(r′ )

r − r′ 𝑑r 𝑑r′ .

Equation (1-9) can be rewritten as

𝐸 𝜌 = T 𝜌 +1

2

𝜌(r )𝜌(r′ )

r −r′ 𝑑r 𝑑r′ + 𝐸𝑛𝑐𝑙 [𝜌] + 𝜌(r ) Vext d𝑟 . (1-10)

A major technical problem is the accurate determination of the kinetic energy functional

T 𝜌 . By introducing orbitals of a non-interacting reference system, Kohn and Sham avoided

this problem. The corresponding Kohn Sham kinetic energy is

𝑇𝐾𝑆 𝜌 = |T →ρo

min (1-11)

here is the Slater determinant composed of Kohn Sham orbitals ∅𝑖(r ) . The Kohn Sham

kinetic energy can be calculated from the Kohn Sham orbitals as

Chapter 1

15

𝑇𝐾𝑆 𝜌 = ∅𝑖 −1

2∇2 ∅𝑖

N

i=1

.

The Kohn Sham orbitals are functional of the density. Zhao and Parr [90] have shown

that they can be derived by imposing the following constraint

𝜌𝑜 r = ∅𝑖 r 2

N

i=1

to the variation of equation (1-11) as

−1

2∇2 + 𝑉𝐾𝑆 𝑟 ∅𝑖 r = εi∅𝑖 r . (1-12)

These are the Kohn Sham equations and 𝑉𝐾𝑆 𝑟 determines the external potential of the

Kohn Sham system.

Equation (1-9) can be rewritten as

𝐸 𝜌 = 𝑇𝐾𝑆 𝜌 +1

2

𝜌 r 𝜌 r′

r −r′ 𝑑r 𝑑r′ + 𝜌 r Vext d𝑟 + 𝐸𝑥𝑐 𝜌 (1-13)

where 𝐸𝑥𝑐 𝜌 = T 𝜌 − 𝑇𝐾𝑆 𝜌 + 𝐸𝑛𝑐𝑙 𝜌 . The quantity 𝐸𝑥𝑐 𝜌 is called the exchange-

correlation energy functional and consists of all nonclassical interactions between the electrons

and the difference of the kinetic energies of the interacting and non-interacting systems. This

quantity is the most important part in density functional theory and accuracy of the Kohn Sham

method is determined by the quality of approximation used for 𝐸𝑥𝑐 𝜌 .

The functional derivative of the energy functional with respect to the electron density, in

equation (1-13) gives

𝛿𝐸 𝜌

𝛿𝜌 r =

𝛿𝑇𝐾𝑆 𝜌

𝛿𝜌 r + Vext +

𝜌 r′

r −r′ 𝑑r′ +

𝛿𝐸𝑥𝑐 𝜌

𝛿𝜌 r =

𝛿𝑇𝐾𝑆 𝜌

𝛿𝜌 r + Vext +

𝜌 r′

r −r′ 𝑑r′ + 𝑉𝑥𝑐 𝜌 (1-14)

where the exchange-correlation potential is defined as 𝑉𝑥𝑐 𝜌 =𝛿𝐸𝑥𝑐 𝜌

𝛿𝜌 r .

Now equation (1-14) can be written in terms of Kohn Sham potential as

𝛿𝐸 𝜌

𝛿𝜌 r =

𝛿𝑇𝐾𝑆 𝜌

𝛿𝜌 r + 𝑉𝐾𝑆 r

where the Kohn Sham potential has the explicit form

𝑉𝐾𝑆 r = Vext + 𝜌 r′

r −r′ 𝑑r′ + 𝑉𝑥𝑐 𝜌 . (1-15)

Using equation (1-15) in equation (1-12), we get

Chapter 1

16

−1

2∇2 + Vext +

𝜌 r′

r −r′ 𝑑r′ + 𝑉𝑥𝑐 𝜌 ∅𝑖 r = εi∅𝑖 r . (1-16)

Equation (1-16) is the canonical Kohn Sham orbital equation, which have to be solved

iteratively.

1.2.2.4 Local density approximation (LDA) and Generalized Gradient

approximation (GGA)

As we have discussed earlier, the exchange-correlation term is an unknown quantity and

the search for an accurate 𝐸𝑥𝑐 𝜌 is the greatest challenge in DFT method. In the local density

approximation (LDA), electrons are assumed to be distributed uniformly, and they move in a

positive background charge distribution. Hohenberg and Kohn showed that if the density 𝜌 r

varies extremely slowly with r , then 𝐸𝑥𝑐 𝜌 can be written as

𝐸𝑥𝑐𝐿𝐷𝐴 𝜌 = 𝜌 r 휀𝑥𝑐

𝐿𝐷𝐴[𝜌(r )]𝑑r (1-17)

where 휀𝑥𝑐𝐿𝐷𝐴 𝜌(r ) is the exchange-correlation energy per electron of a homogenous electron gas

with electron density 𝜌 r . However, most of the real problems (systems) which we have to deal

in physics and chemistry do not consist of uniformly distributed electrons. In order to account

for the non-homogeneity of electrons, the generalized gradient approximation (GGA) is

suggested. In this approach, the charge density 𝜌 r at a particular point 𝑟 is supplemented with

the gradient of density, ∇𝜌 r . In the generalized gradient approximation (GGA), the functional

depends not only on the charge density at a particular point but also on the gradient of density

𝐸𝑥𝑐𝐺𝐺𝐴 𝜌 = 𝜌 r 휀𝑥𝑐

𝐺𝐺𝐴 𝜌 r , ∇𝜌 r 𝑑r (1-18)

1.2.2. 5 Chemically relevant concepts from DFT

From the second Hohenberg-Kohn theorem, the ground state density must satisfy the

equation

𝛿(𝐸 − 𝜇 𝜌 r 𝑑r ) = 0 (1-19)

where μ is a Lagrange multiplier associated with the constraint of constant number of electrons.

Parr et al. [91] identified this Lagrange multiplier, μ as chemical potential of the system. From

the above equation (1-23), the chemical potential can be written as

Chapter 1

17

𝜇 = 𝑣 r +𝛿𝐹𝐻𝐾

𝛿𝜌 (r ) (1-20)

where 𝑣(r ) is the external potential and FHK is the Hohenberg-Kohn universal functional. Parr

and co-workers [91] realized that the µ can be written as the partial derivative of the system‟s

energy (E) with respect to the number of electrons (N) at fixed external potential 𝑣(r ) as

𝜇 = 𝜕𝐸

𝜕𝑁 𝑣(r )

. (1-21)

In a landmark paper, Kohn and his colleagues [92] stated that “µ characterizes the

escaping tendency of electrons from the equilibrium system. Systems (e.g. atoms or molecules)

coming together must attain at equilibrium a common chemical potential. This chemical

potential is none other than the negative of the electronegativity concept of classical structural

chemistry.”

The electronegativity (χ) which can be defined as the power of attracting electrons

towards itself can be expressed as the negative of the chemical potential [91, 92].

χ = −𝜇 = − 𝜕𝐸

𝜕𝑁

𝑣(r ). (1-22)

This identification of χ is an important step, as there was no systematic way of

evaluating electronegativity values [93]. Equation (1-22) provided a way to calculate

electronegativity values for atoms, molecules, functional groups and clusters.

Parr and Pearson [94] defined chemical hardness (η) as the second derivative of the

energy with respect to the number of electrons at fixed external potential

𝜂 = 𝜕2𝐸

𝜕𝑁2 𝑣(r )

. (1-23)

This definition provided a direct way to calculate chemical hardness values and compare

with experimental values obtained via ionization and electron affinity values [95, 96]. Equation

(1-23) can also be written as

𝜂 = 𝜕𝜇

𝜕𝑁 𝑣(r )

. (1-24)

Equation (1-24) shows that η is the resistance of the chemical potential to changes in the

number of electrons.

Chapter 1

18

Using a finite difference method, working equations for χ and η can be expressed in

terms of ionization potential (IP) and electron affinity (EA) as [97]

χ =𝐼𝑃+𝐸𝐴

2 (1-25)

𝜂 = 𝐼𝑃 − 𝐸𝐴 (1-26)

IP and EA can be obtained from the total electronic energy calculations on the N-1, N and

N+1electron systems as

𝐼𝑃 = 𝐸 𝑁 − 1 − 𝐸(𝑁) (1-27)

𝐸𝐴 = 𝐸 𝑁 − 𝐸 𝑁 + 1 . (1-28)

An alternative method is replacing IP and EA by the energies of the highest occupied

and lowest unoccupied molecular orbitals, respectively, using Koopmans‟ theorem [98]

𝐼𝑃 = −휀𝐻𝑂𝑀𝑂 (1-29)

𝐸𝐴 = −휀𝐿𝑈𝑀𝑂 (1-30)

where 휀𝐻𝑂𝑀𝑂 and 휀𝐿𝑈𝑀𝑂 are the energies of the highest occupied and lowest unoccupied

molecular orbitals, respectively. Then, χ and η can be rewritten as

χ = −휀𝐻𝑂𝑀𝑂 +휀𝐿𝑈𝑀𝑂

2 , (1-31)

𝜂 = 휀𝐿𝑈𝑀𝑂 − 휀𝐻𝑂𝑀𝑂 . (1-32)

Maynard and co-workers [99, 100] have shown that there exists a strong correlation

between reaction rates and the square of the electronegativity divided by its chemical hardness.

Their observation was based on the fluorescent decay of the human immunodeficiency virus

type 1 (HIV-1) interacting with several electrophilic agents [99]. Maynard and his colleagues

considered [99, 100] that the quantity χ2/η is related to the capacity of an electrophile to promote

a covalent reaction. Based on this finding, Parr et al. [101] defined electrophilicity index (ω) as

𝜔 =𝜇2

2𝜂=

𝜒2

2𝜂 . (1-33)

Parr et al. [101] have shown that the quantity ω is a measure of the stabilization in

energy when the system acquires an additional electronic charge. They suggested that EA gives

a measure of the electron accepting capacity of a system. However, it does not give any idea

about how the electron transfer process contributes to lowering of the total energy. This

lowering of energy can be measured using ω [101].

The quantities μ, χ, η, and ω measure the response of the system when the number of

electrons, N varies at fixed external potential, 𝑣(r ). The behavior of the system when 𝑣(r )

Chapter 1

19

changes at constant N is given by the linear density response function [97]. The static electric

dipole polarizabilty (α) is a measure of the linear response of the electronic cloud of a chemical

species to a weak external electric field. The linear response of the electron density to an electric

field F is measured in terms of electric dipole polarizability. It represents a second-order

variation in energy

𝛼𝑎 ,𝑏 = − 𝜕2𝐸

𝜕𝐹𝑎𝜕𝐹𝑏 ; 𝑎, 𝑏 = 𝑥, 𝑦, 𝑧 ; (1-34)

The polarizability (α) is then given by

< α >=1

3(𝛼𝑥𝑥 + 𝛼𝑦𝑦 + 𝛼𝑧𝑧 ). (1-35)

The above mentioned reactivity parameters can be properly understood in terms of their

related electronic structure principles like the electronegativity equalization principle [102,

103], the maximum hardness principle [104, 105], the minimum polarizability principle [106],

the minimum electrophilicity principle [107-109], etc. The electronegativity equalization

principle (EEP) [102, 103] states that, “During an electron-transfer process in a chemical

reaction, electrons flow from a species of lower electronegativity (higher chemical potential) to

one with higher electronegativity (lower chemical potential) until the electronegativities get

equalized to a value approximately equal to the geometric mean of the electronegativities of the

isolated species.” The maximum hardness principle (MHP) [104, 105] is stated as, “There

seems to be a rule of nature that molecules arrange themselves so as to be hard as possible.” The

minimum polarizability principle (MPP) [106] may be stated as, “The natural direction of

evolution of any system is towards a state of minimum polarizability.” The minimum

electrophilicity principle (MEP) [107-109] states that, “Electrophilicity will be a minimum

(maximum) when both chemical potential and hardness are maxima (minima).”

In this thesis, we attempt to understand the structure, electronic, chemical, magnetic and

optical properties of small metal clusters using DFT calculation. Several reactivity descriptors

have been used to describe the reactivity properties of metal clusters. Chapter 2 presents the

scenario of earlier research works on metal clusters and some important works are briefly

reviewed. In chapter 3, structure, reactivity, thermodynamic and magnetic properties of

palladium (4d-transition metal) clusters are presented. Chapter 4 deals with 5d-transition metal

cluster consisting of platinum atoms. The structures, vibrational, chemical and optical properties

of platinum clusters are analyzed. In chapter 5, we have studied gold-palladium bimetallic

Chapter 1

20

clusters. Chapter 6 analyzes interaction between NO molecule and nickel-manganese clusters

using GGA approach of DFT. In chapter 7, the possibility of using nickel cluster decorated

carbon nanotube for detecting CO molecule has been investigated.

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