## Optical Properties of Metal Clusters from First Principles Calculations

ABSTRACT

Ground State structures of neutral Copper clusters CuN=3􀀀6 were generated and optimized within the framework of the Density Functional Theory (DFT) using Generalized Gradient Approximation (GGA) and ultra-soft pseudo potential. The shapes and binding energies of the clusters obtained showed good agreement with other theoretical works, except for the trimer cluster which has the shape of an equilateral triangle rather than the isosceles triangle as reported in previous works. The optical absorption strength function of the neutral Copper clusters CuN=3􀀀6 in vacuum were then computed within the framework of Time Dependent Density Functional Theory (TDDFPT). The spectrum were computed for the energy range of 1:5eV < ~! < 5:5eV . The clusters that had odd numbers of atoms showed metallic character and could not be implemented in the turboTDDFT package. Whilst the clusters with even numbers of atoms showed semiconductor property with the Highest Occupied Molecular Orbital-Lowest Unoccupied Molecular Orbital (HOMO-LUMO) gap increasing as the number of atoms change from four to six , and full implementation with turboTDDFT was achieved. In both clusters with even number of atoms four absorption peaks were obtained with that of six atoms higher in intensity. The shift in the direction of the peaks as the cluster size increases shows no unique trend. However, there was a slight broadening of the peaks as the cluster size increased. Most of the peaks were observed to be from the electron transiting from the s-orbital in the occupied state to the orbitals in the unoccupied state. Transitions from low-lying orbitals such as d and p orbital were also noticed.

ABSTRACT i
ACKNOWLEDGEMENT ii
DEDICATION iii
List of Figures vi
List of Tables vii
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Current Methods and Challenges . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Previous Work on Copper Clusters . . . . . . . . . . . . . . . . . . . . . 2
1.4 Motivation for Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Aim and Scope of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Structure of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Optical Properties of Materials: Theoretical Framework 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Propagation of Electromagnetic Radiation Through Matter . . . . . . . . 4
2.2.1 Maxwell’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Skin Depth of Conducting Material . . . . . . . . . . . . . . . . . 5
2.2.3 Relation of Complex Dielectric Function to Observables . . . . . . 6
2.3 Interaction of Many atomic Systems with Electromagnetic radiation . . . 7
2.4 Cluster-size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Extrinsic Size Effects of the Optical Properties of single clusters . . . . . 10
2.5.1 Optical material Functions of Bulk Metals . . . . . . . . . . . . . 10
2.6 Intrinsic Size Effects of the Optical Properties of Single Clusters . . . . . 16
2.6.1 Size Dependent Optical Material Functions of Metal Clusters . . . 16
iv
2.7 Classical model for (!;R) . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.8 Quantum Mechanical Models for (!;R) . . . . . . . . . . . . . . . . . . 17
3 Electronic Structure Methods and Calculations 18
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Born Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . 18
3.3 Density Functional Theory (DFT) . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1 Hohenberg and Kohn theorems . . . . . . . . . . . . . . . . . . . 19
3.3.2 Kohn Shams Equations . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.3 Exchange-Correlation Functional . . . . . . . . . . . . . . . . . . 21
3.4 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.1 Norm-conserving pseudopotential . . . . . . . . . . . . . . . . . . 23
3.4.2 Ultra-soft Pseudo-potential . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Kohn-Shams orbitals (Plane wave basis set) . . . . . . . . . . . . . . . . . 24
3.6 Time Dependent Density Functional Theory (TDDFT) . . . . . . . . . . 25
3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6.2 Runge-Gross theorem . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6.3 Time-Dependent Kohn-Shams Equations . . . . . . . . . . . . . . 26
3.6.4 Time-dependent Exchange-Correlation Functional . . . . . . . . . 27
3.7 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Results and Discussions 28
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.1 Optimized Structures of Clusters . . . . . . . . . . . . . . . . . . 28
4.2.2 Density of States of Clusters . . . . . . . . . . . . . . . . . . . . . 29
4.2.3 Binding Energy of Clusters . . . . . . . . . . . . . . . . . . . . . . 29
4.2.4 Absorption Spectrum of Clusters . . . . . . . . . . . . . . . . . . 29
5 Conclusion and Recommendations 32
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Bibliography 36

CHAPTER ONE

1.1 Introduction

Metal clusters are particles composed of a certain number N of atoms with 3 < N < 107 [17]. Early studies on metal clusters are traceable to the work of Faraday on a gold colloidal particle that is responsible for the red stain in glass [8]. Mie in 1908 gave a mathematical and theoretical framework for studying the properties of small particles when he gave the exact solution to Maxwell’s equations for spherical particles with sizes smaller than the wavelength of incident radiation [31, 15]. Since then, the eld of metal clusters (clusters in general) have attracted a lot of attention [10] both at the theoretical and experimental level of investigations. These studies have led to the development of more rened methods and theories for investigating the properties of metal clusters and thus their viability for diverse areas of applications as witnessed today.

1.2 Current Methods and Challenges

In the past, people have paid a lot of attention to metal clusters [10, 22, 15]. This is largely attributed to their chemical and physical properties being size, shape, composition and chemical ordering dependent on their large surface to volume ratio.[2, 22, 10]. The applications of these cut across the search for efficient materials for solar energy conversion in photo-voltaic and photo-catalysis, biomedical nano materials for cancer detection and treatment, and optoelectronic devices [2]. For solar energy conversion, the emphasis has been on the optical properties of the metal clusters. For example, [8, 31, 30] discussed novel methods of enhancing the transport property of haematite using plasmatic nanoparticles such gold and silver for photocatalytic systems;[21] conducted DFT studies on Ag-Cu nano-alloy and found a ferroelectric and ferromagnetic effect that can be applied to non-linear optical devices, [23] conducted TDDFT studies on the dependence of the surface plasmon resonance of gold nanorods on its aspect-ratio and size. All these studies are in the bit to overcome the deficiencies present in most semiconductor materials used for renewable energy generation. These materials as much as they have some good properties that make suitable for energy generation, among other limitations, have the problem of poor charge-transport property and wide band gap (outside the visible spectrum).

1.3 Previous Work on Copper Clusters

A lot has been done on the gold and silver nano-particles with many on-going theoretical and experimental studies because of the availability of data to conrm results obtained.

But for the copper cluster little has been done in comparison with the previous two.[6] investigated the atomic structure of small copper clusters using Density Functional Theory (DFT) and Random Search Algorithm (RSA); and found new unreported geometric structures of copper clusters that are isometric to the ground state structures. [28] found that Copper clusters enhances the photocatalytic property of titania by way of providing states that are above the valence band edge of a semiconductor material such as titania and the formation of mid-gap states. [20] used real space pseudo potential (RSP) within Local Spin Density Approximation (LSDA) to investigate the structural and electronic property of both neutral and anion copper clusters and obtained results that are comparable to experimental values. [14] used tight-binding molecular dynamics scheme to examine the structural and electronic properties of small copper clusters, with results in good agreement with other theoretical work and experimental results. [1] used Density

Functional Theory (DFT) with effective core potential to investigate how properties such as dissociation energy, HOMO-LUMO gap and ionization potential of copper clusters changes as the size of the cluster tends its bulk material, and found that clusters are proper choice to represent a surface than the bulk material. [12] found that through density functional theory reactivity descriptors, a material such as copper clusters can be characterized.

1.4 Motivation for Study

The above cited material show the potential of using copper clusters to enhance the photocatalytic property of semiconductor material like TiO2. A success in determining the best cluster size, shape and composition for applications in photovoltaic and photo-catalytic processes, will greatly lead to the reduction in the over-dependence on gold or silver for those applications. This is in view of the fact that copper is relatively abundant and cheaper than the other two noble metals. Once this specific cluster type is identified, it will further enhance the processes involved in designing a set-up or process for this cluster-type production in the laboratory. Moreover, there are many advances in ab initio tools available for analysis which could be utilized to obtain better results. To the best of our knowledge, only few work if any has been done on the determination of the absorption spectrum of small copper clusters using ab initio methods. It would therefore be expedient to study how the photo-absorption spectrum of small copper clusters depends on their size.

1.5 Aim and Scope of Study

This work is focused on studying how the photo-absorption spectrum of ground state structures of neutral copper clusters CuN=3􀀀6 in a vacuum, obtained from the Density Functional Theory (DFT) using Generalised Gradient Approximation (GGA) functional and ultra-soft pseudo-potential, changes as the size of the cluster is increased. The calculations are conducted within the framework of Time Dependent Density Functional Perturbation Theory (TDDFPT) using the turboTDDFT, and all within the connes of Quantum ESPRESSO package.

1.6 Structure of Work

This work is divided into four chapters. The first chapter gives the general introduction into the work, chapter two deals with the discussion on the theory of optical properties of metals, chapter three is on the description of the electronic structure methods and calculations used for this study and, in chapter four, results, discussion on the results and lastly chapter five, the conclusion and recommendations are presented.