Chapter12.docx

Electronic Structure of Ligand-Passivated Metal Nanoclusters

Ph.D. In Physics Engineering University of Gaziantep

Supervisor Prof. Dr. Zihni Öztürk

by Mikail Aslan January 2013

ABSTRACT ELECTRONIC STRUCTURE OF LİGAND-PASSIVATED METAL NANOCLUSTERS

ASLAN, Mikail Ph.D. in Physics Engineering Supervisor: Prof. Dr. Zihni ÖZTÜRK Co-supervisor: Assoc. Prof. Dr. Ali SEBETCİ January 2013, pages

The geometric, electronic, and magnetic properties of neutral and/or ionic ligandpassivated metal nanoclusters has been studied. The adsorption of a number of organic molecules such as thiols, alcohols, alkane, alkene, or other carbon-containing groups of atoms on Pt, Co, Au, and/or Ag clusters is investigated by means of density functional theory calculations. Various interaction patterns between the ligands and the metal clusters are constructed and screened with extensive structure searches for the low-lying isomers of the ligand-protected complexes. The optimal adsorption sites are identified and the corresponding electronic and magnetic states are analysed. Keywords: Nanoclusters, Transition Metals, DFT, Ligands

ÖZET LİGANDLARLA

PASİFLEŞTİRİLMİŞ

METAL

NANOTOPAKLARIN

ELEKTRONİK YAPILARI ASLAN, Mikail Doktora Tezi, Fizik Müh. Bölümü Tez Yöneticisi: Prof. Dr. Zihni ÖZTÜRK Yardimci Tez Yöneticisi: Assoc. Prof. Dr. Ali SEBETCİ Ocak 2013, sayfa Nötr ve / veya iyonik ligandlar ile pasifize edilmiş metal nanotopakların geometrik, elektronik ve manyetik özellikleri çalışıldı . Yoğunluk Foksiyonel Teorisi kullanılarak tiyoller, alkoller, alkan, alken veya başka bir karbon-içeren bir takım organik moleküllerin

Pt, Co, Au, ve / ya da Ag topakları üzerindeki adsorpsiyon etkileri

incelenmiştir. Ligandlar ve metal topakları arasındaki çeşitli etkileşim şekilleri inşa edildi ve ligandlarla çevrilmiş basit yapılı kompleks izomerlerin

için kapsamlı bir

şekilde taramalar yapılmıştır. En kuvvetli adsorpsiyon için atomların geometrik konumları tespit edildi ve geometrik konumları belirlenen yapıların elektronik ve manyetik durumları analiz edildi.

Anahtar

kelimeler:

Nanotopaklar,

Geciş

Metalleri,

DFT,

Ligandlar

TABLE OF CONTENTS ÖZET............................................................................................................................. i LIST OF TABLES ..................................................................................................... iv LIST OF FIGURES .................................................................................................... 5 INTRODUCTION ....................................................................................................... 6 1.1 Nanoscience Background ............................................................................................. 6 1.2 Metal Clusters .................................................................. Error! Bookmark not defined. 1.3 Computational Simulation ......................................................................................... 11

THEORY OF DENSITY FUNCTIONAL METHOD ........................................... 12 2.1 Introduction ................................................................................................................ 12 2.2 The Schrödinger Equation ......................................................................................... 14 2.3 The Variational Principle .......................................................................................... 17 2.4 The Hatree-Fock (HF) Method................................................................................ 18 2.5 Avoiding the Solution of the Schrödinger Equation ................................................ 21 2.6 The Hohenburg and Kohn Theorems ....................................................................... 22 2.7 The Energy Functional ............................................................................................... 24 2.8 The Exchange Correlation Functionals .................................................................... 28

THE INFLUENCE OF LIGANDS ON STABILIZATION OF PURE CLUSTERS................................................................................................................ 39 2.1 Introduction ................................................................................................................ 39 2.2 Computational Details ................................................................................................ 41 2.3 Results and Discussion ............................................................................................... 42 2.3.1 Geometric Optimization ........................................................................................ 42 2.3.2 Stability and Energetics ......................................................................................... 47 2.3.3 Electronic Properties ............................................................................................. 52 2.3.4 DFT Chemical Reactivity Descriptors ................................................................. 54 2.4. Conclusion .................................................................................................................. 56

THE INTERACTION OF C2H WITH BIMETALLIC Co-Pt CLUSTERS ....... 58 3.1 Introduction ......................................................................................................... 58 3.2 Computational Details ........................................................................................ 60 ii

3.3 Result and Discussion ......................................................................................... 60 3.3.1 Diatomic [ ComPtnC2H ]-1 ( m +n =2) nanoalloys ................................................. 61 3.3.2 [ ComPtnC2H ]-1 ( m +n =3) ................................................................................... 62 3.3.2 [ ComPtnC2H ]-1 ( m +n =4) ................................................................................... 65 3.3.3 [ ComPtnC2H ]-1 ( m +n =5) ................................................................................... 68 3.4 Energy and Electronic Structure............................................................................... 71 3.3.5 Magnetic Properties............................................................................................... 79 4 Conclusion.................................................................................................................. 80

LIST OF REFERENCES ......................................................................................... 81

iii

LIST OF TABLES Table 2.1 The hierarchy of exchange correlation functionals [taslak referance] ............................. 36 Table 2.1 Electronic Properties of Pt4 (CH)n (n=1 to 11) clusters .................................................. 48 Table 2.2 Energetic Reactions of Pt4 (CH)n (n=1 to 11) cluster ...................................................... 52 Table 2.3 The Chemical Descriptors Indexes of Pt4 (CH)n (n=1 to 11) cluster ................................. 55

iv

LIST OF FIGURES Page Figure 2.1 The lowest energy Structure of Pt4(CH)n (n = 1 to 6) clusters ............................................ 44 Figure 2.2 The lowest energy Structure of Pt4(CH)n (n = 7 to 11) clusters........................................... 46 Figure 2.3 The Binding Energy per atom of Pt4(CH)n (n = 1 to 11) cluster ........................................... 49 Figure 2.4 The second finite difference energies of Pt4(CH)n (n =2 to 10) cluster ............................... 50 Figure 3.3.1 The optimized structure of some isomers of [ Co mPtn ]-1 (2  m +n 3) .......................... 64 Figure 3.3.2 The optimized structure of some isomers of [ Co mPtnC2H ]-1 (m +n =4) .......................... 67 Figure 3.3.3 The optimized structure of some isomers of [ Co mPtnC2H ]-1 (m +n =5) .......................... 70

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CHAPTER 1 INTRODUCTION 1.1 Nanoscience Background

For the past few decades, many scientists and engineers who work on material science has been willing to do research on materials that are sized 100 nm or smaller scale to find amazing inventions or implement their discoveries to new applications. Small nanosized particle shows unique properties that are different from extended bulk states or from atomic states, which is due to the some fundamental reasons: the ratio surface / volume and the quantum size confinement effect [1]. In order to make electronic devices very smaller size, quantum concept become more important key than the conventional understanding of electronic devices. For example, the lithographic techniques are a kind of fabricating of nanometer scale structure is not even applicable for the structures are in atomic scale. Moreover, for the production of regular structures, self-assembly is reliable technique but it is very sensentive to fluctuations. Thus, the quantum size confinement effect should be taken into consideration so as to comprehend deeply the behavior of nano-devices in microscopic world and to control the nano-structure precisely when building it [2].

Naturally, physical systems is inclined to find their most decisive states after doing some process like thermal annealing which is a kind of process for altering material or changing in its properties such as electronegativity and ductility by heating. The most decisive state is the lowest total energy the structure of a physical system possesses when compared to all other possible structures. Finding the lowest total energy the structure ownes in theoretical way can provide to determine the properties of the systems. Although determining the structure of the complex system is challenging work and finding this kind of structure would take cost a lot for the time efficiency, theoretical studies for structural determination plays important roles, which represents the candidate structures manually relying on human intuition [2].

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1.2 Clusters

Clusters, intermediate between single atoms and molecules and bulk matter, may range their size from 2 atoms to tens of thousands. Metal clusters have been gaining importance extensively due to their novel properties for the few decades [3]. One of the main differences between cluster and bulk material is the surface area / volume. However for the case of macroscopic systems, this ratio may be less significant, for small clusters, this ratio keeps its importance, as this ratio is large. Thus, the physical properties of the clusters like electronegativity, which is intrinsically different from the bulk material, can be changed via the properties of the surface. For instance, the tiny clusters of metals may show the properties of insulators or semiconducting with remarkable electronic properties [4]. Moreover, They have probabilities to show oddeven oscillations of their properties such as ionization potential, electron affinities etc so the structures of clusters are not only cut outs of solid materials [5]. The clusters generally modify exciting shapes because of the large number of surface atoms with unsaturated coordination. The geometries and electronic structures of the metal cluster have been deduced in terms of models such as jellium model or nuclear liquid drop model [3].

Due to the development of nano-science and nano technology, the study about electronic properties of nanoclusters has attained considerably significance. Computer simulations play an important part for comprehending the evolution of structure, physical and chemical properties of clusters. This kind of cluster properties at the quantum mechanical level can be understood via computer simulation. When the cluster grows in size, it is fascinating to know at what size of cluster the energy bands form and it behaves like a solid material [4].

1.2.1 Metal Clusters

In particular, metal clusters (MCs) play an important role in catalysis, medical science, and nanostructured electronic devices. For the medical science field, to defeat some

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diseases or treatment of the diseases such as, filariasis, malaria, brain fever, dengue that is due to the existence of mosquitoes, Plant obtained with the synthesis of metal nanoparticles may be prior to control the population of mosquitoes so as to reduce the effect of such kind of diseases via appropriate control methods [6]. For instance, Selvaraj and co-workers used mesocarp layer extract of Cocosnucifera and managed to synthesize silver clusters due to their biocompatibility, low toxicity, green approach and environmental friendly nature and antimicrobial properties that they show by treating silver nitrate solution with aqueous extract of C. nucifera coir at 60 Celsius [7; 8] They checked their method reliability by analyzing the excitation of surface plasmon resonance (SPR) using UV-vis spectrophometer at 433 nm [9]. For the nanostructured electronic devices, MCs has possible application in the microelectronics and sensor technology subfields [9; 10]. For example, Nicola Cioffi and co-workers achieved gold/surfactant core/shell colloidal nanoparticles with a controlled morphology and chemical composition using sacrificial anode technique with galvanostatic mode for the surface-modification of gate electrodes implemented in field effect capacitor sensors for NOx sensing [11]. They preferred Au-MCs based gas sensor due to the new reactivity properties, increased surface area-volume ratio, high sensing performance level and adjust of sensor properties [11-13].

Among Mono-Metallic Cluster, platinum is used as an essential electrode material due to its relatively high catalytic activity and stability so platinum metal cluster offer a wide range of remarkable properties as a variety of technology applications. For small platinum clusters, the determination of the ground state structure by experiment is challenging. Thus, there exists only a small minority of studies using a wide range of spectroscopic techniques, viz. resonance two-photon ionization, dissociation methods, lasers induced fluorescence and stark spectra [14-20]. Theoretical studies done by ab initio methods [21-35] and empirical potential methods [36-41] have been reported.

Co clusters are one of the important metal because of their magnetic properties [42-47] and significance in magnetic storage devices. The significance of studying small Co clusters may be listed as its possible contribution to the solution of the technologically

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important question of how magnetic characteristic varies in reduced dimensions [48]. The ground state structure of the metal nanoparticles is generally determined from photoelectron spectroscopy and chemical probe experiments. A study was conducted [49] that 50–800 numbers of atoms of Nin and Con nanoparticles have icosahedral structures but the structures of small Con (n 50) mono metal clusters were not well classified. Relatedly, for both pure and ammoniated nanoparticles within the size range of n = 50–100, the Con reactions with water and ammonia [50] reveal icosahedral structures. Nevertheless, it was reported that small, monometallic clusters could adopt different type of structures [51]. Yoshida and coworkers [52] accomplished to find the geometry and electronic structures of Con anions (n = 3-6) by using photoelectron spectra. The magnetic features of monometallic Con particles within the range of n=20300 were studied by Stern–Gerlach experiments by various scientist groups [42-44; 47] and concluded that the magnetic moment per atom for small Con particles are considerably larger than the value of bulk (1.7 µB [53]).

Pd is one of the essential materials as catalyst in numerous applications as well. Reduction of very active NO by CO happens in the existence of very dispersed Pd clusters supported on alumina [54]

Additionally, ultradispersed supported Pd

nanoparticles of up to 2 nm in size behaves highly active catalysts in hydrogenation processes [55] owing to much higher selectivity in the conversion of triple to double bonds than that of bulk Pd [56] Furthermore, current observation of Pd nanoparticles in terms of ferromagnetism provides the probability of potential use of these particles in magnetic storage materials [57]. Wide-ranging experimental studies have been conducted on Pd particles over many years [58-62]. Early Stern-Gerlach experiments indicated the absence of magnetic moments in Pd nanoparticles among these experimental studies [59; 61; 62] Photoemission experiments 9 proposed a Ni-like spin distribution in Pdn nanoparticles at n=6 and a nonmagnetic behavior at n =15. Despite the encouraging experimental marks, they could not give satisfactory info on the electronic and geometrical structures of very minute Pd nanoparticles. These properties are required to comprehend the size dependence of chemisorption and catalytic properties of nanosized particles [63].

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The case of nanoalloy substances is of great concern in the chemical industry. Such constituents are motivating since one of the metals may adjust the catalytic properties of the other substances due to structural and electronic and effects. It has been reported that in many particles at nanoscale, the low lying geometry is one of the fundamental part that particles are predominantly occupied by one of the constituents and its surface holds most of the second element. Smaller volumes of element might be adequate to get similar effects as those achieved by single element catalysts if the active element is the one that isolates from the surface [64]. Thus, it is significant to study CoPd nanoalloy particles at the nanoscale. In Fisher-Tropsch reactions [65-67], CoPd nanoparticles displayed better selectivity over pure Co particles. The artificial fabrication of fuels by changing carbonmonoxide and hydrogen is appealing many researchers in this area due to the recent huge fluctuations in the prices of hydrocarbons [64].

The study of Co–Pt nanoalloy particles is one of the major aims of the low-temperature polymer electrolyte membrane fuel-cell technology in order to develop and reduce Pt loading as the oxygen-reduction catalyst [68]. Numerous analyses [69-72] such as the series of binary Pt–X alloys (X =Mn, Cr, Co, Ni) supported on carbon [73], have been conducted to solve the role of alloying in the electrocatalytic activity of Pt for the oxygen-reduction reaction. Those analyses revealed that the intrinsic activity of nanoparticles depends on particle size, shape and composition. By decreasing the size of the clusters, their catalytic activities are inclined to upturn due to the bigger surface areas of smaller nanoparticles and the structural sensitivity to some reactions.

Using ligands with functional groups attached to the surface metal atoms allows for the construction of functional nano assemblies and binding the clusters to surfaces of various substrates. Moreover, the ligand environment could significantly affect the electronic properties of the clusters themselves. There is a vast amount of information available for small bare or nanoalloy metal clusters, however much less theoretical emphasis has been given to the ligation of these clusters with stabilizing molecules. The purpose of most of the computer studies is to determine the structure-property

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relationships. The knowledge of the property variation with geometry also allows one to carry out computer experiments to design new stable clusters with desired properties. Thus, this computational study would be a contribution to the understanding of the influence of ligands on the stabilization of metal clusters to design new stable clusters with desired properties

1.3 Computational Simulation

Computational method is widely used tool for the field of chemistry, physics, and material engineering. The increasing speed and availability of computational resources in grouping with newly, developed and increasingly accurate models provides scientists to solve and rationalize chemical phenomenon. Additionally, Computer simulation allows us to investigate probabilities to check whether the exciting properties of clusters are preserved in different environments. For instance, whether the high magnetic moments of gas-phase clusters will be retained when they are deposited on a substrate. It also allows one to check whether highly stable clusters can be assembled to form solids. This type of approach can lead us to the development of novel cluster based materials.

Another point is that the experiments on clusters especially on gas-phase clusters yield little information about their structures. In this regards simulations complement the experiments by providing structural input. From the application point of view, the interaction between the cluster and the substrate matrix is of paramount importance. The properties of such systems are governed by many body electron ion interactions.

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CHAPTER 2 THEORY OF DENSITY FUNCTIONAL METHOD 2.1 Introduction

For the quantum mechanical simulation of cluster, molecular, periodic and solid systems in the scope of physics, material science, chemistry and multiple branches of engineering, the density functional theory (DFT) has been the dominant method over past 30 years. DFT is prevailing implement for computations of the quantum state of atoms, molecules, crystals, surfaces and their interactions, and of ab-initio molecular dynamics.

The introduction of the DFT began after the foundation of quantum mechanics. In 1927, ın order to determine approximate wavefunctions and energies for atoms and ions, Hatree developed a function called Hartree function. A few years later, Fock and Slater, the students of Hatree, proposed a self consistent function in regard to Pauli Exclusion Principle and multi-electron wavefunction in the form of Slater-determinant (the determinant of one particle orbitals) separately so as to cope with the entirely noconsideration of anti-symetric properties of electrons. Although Hatree Fock Method was not prevalent until 1950s due to the complexity of calculation of Hartree-Fock model, the model is a good approach for the approximation to the real result.

In the same year that Hatree derived Hatree Function, by approximating the distribution of electrons in an atom, Thomas and Fermi suggested a statistical model to calculate the energy of atoms. In this model, the kinetic energy of an atom is expressed by the functional of electron density and add two terms of electron-electron and nuclearelectron interactions in order to compute the atom energy. In the early stage of this model, it did not include the exchange energy of an atom that results from Pauli

12

exclusion principle but it is identified in Harree Fock Theory. In 1928, Dirac enlarged the Hartree Function by adding an exchange energy functional part. It is accepted that the Thomas-Fermi model is a significant first step of DFT but its uses are very restricted presently due to the largest source of error that originates from the statement of the kinetic energy functional term that is only approximation. This approximation leads to inaccuracy in the exchange energy and this is due to the whole disregard of electron correlation. Despite this, it is known as an antecedent of the DFT.

In 1964, Kohn and Hohenberg wrote an important article that provides the basis of DFT. In this paper, they used electron density functional that includes merely 3 variables instead of the complicated many electron wavefunction that includes 3N variables, where N is electron numbers and each electron has 3 spatial coordinates. Thus, the huge amount of variables become out of the concern by using electron density functional that has only 3 variables. According to their first theorem, the ground state energy merely depends on the electron density. Also, Hohenberg and Kohn proposed the second theorem that verified the ground state energy can be found via optimizing the energy of the system in terms of electron density.

Hohenberg-Kohn

(H-K) theorem postulates the relations between electron density

functional and system properties. However, it does not precisely describe what the relations between them are. Kohn-Sham approach is the most common used method instead of optimizing the energy of the system that is implemented at the H-K theorem. Only one year later than their significant article published in 1964, Kohn-Sham (KS) solved the multi electron issue by simplifying it into non-interaction electron problem inside an effective potential involving the external potential and the influence on the Coulomb for the exchange and correlation interactions between electrons in the article they wrote. It is challenging work to deal with the correlation and exchange energy in the scope of the KS-DFT. Up to this point in time, to find out the exchange and correlation energy, there are not still an exact method. Nevertheless, the Local Density Approximation (LDA), the simplest approximation, is related to the uniform electron gas model to obtain exchange energy that exact value can be obtained from Thomas-

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Fermi model and to obtain the correlation energy from fits to the uniform electron gas. In an effective potential, wavefunction can be effortlessly exemplified by a Slater determinant of orbitals and also the kinetic energy functional of this system is unerringly known while the total energy functional still has unknown exchange correlation (XC) part.

By using the Local Density Approximation, the results are excellent with experimental data when comparing to other methods related to the solution of many body problem in quantum mechanics. However, Density Functional Theory was still inaccurate till 1990s in quantum chemistry field. Now, DFT is widely used method to determine electronic structure of the structure in many areas after refining approximation methods to better model XC interaction.

2.2 The Schrödinger Equation

The most chemical approaches are based on the approximate solution of the timeindependent Schrödinger equation that is given as the following:

Hy k (x1, x2, x3,.........xA, R1, R2,.....RB ) = Ek Yk ((x1, x2, x3,.........xA, R1, R2,.....RB )

(2.1)

Here, H is a differential Hamiltonian operator of a molecular system with a number of electrons and B number of nuclei, which denotes the total energy:

1 é A ¶2 ¶2 ¶2 1 B ¶2 ¶2 ¶2 ù A B Zq A A 1 B B Z q Z m H = - êå 2 + 2 + 2 + + + ú - å å + å å +å å å 2 êë k=1 ¶xk ¶yk ¶zk M q q=1 ¶xq2 ¶yq2 ¶zq2 úû k=1 q=1 rkq k=1 l>k rkl q=1 m>q Rqm (2.2)

14

Where, in the equation, k and l are for A electrons while q and m show the B nuclei. The first two expressions depict, in order, the kinetic energy of electrons and nuclei where Mq is the mass of nucleus q. The rest three expressions describe, in order, the potential part of the Hamiltonian that shows electrostatic interaction between nuclei and electrons and the potential formed by the electron and electron and nucleus and nucleus interactions. Rab (and correspondingly rab) is the distance between particles a and b. y k (x1, x2 , x3,.........xA, R1, R2,.....RB ) describes the electronic wave function of the k’ th state of the system and the electronic wave function is a function of 3A spatial coordinates rk and the A spin coordinates sk, shown as the expression xk and the 3B spatial coordinates of the nuclei Rl as well. k covers all possible information of the quantum state. Lastly, Ek is numerical value of the energy state defined by k.

The Shrödinger equation can be simplified if we assume nuclei do not move. Even for the Hidrogen atom having 1 proton particle, the weigth of the proton is roughly 1800 times more than the weight of the electron. In short, the movement of the nuclei is much slower when comparing to the movement of the electron so we can accept that the electrons move in the area of moveless nuclei. This is known as Born-Oppenheimer approach. As coming to the potential energy due to the repulsion between nucleus and nucleus, the potential energy becomes only constant. Thus, the related terms of the Hamiltonian equation can be omitted as

1 A 2 A B Zq A A 1 H = - å Ñk - å å + å å = T + VNe + Vee 2 k=1 k=1 q=1 riq k=1 l>k rkl

(2.3)

The solution of the Schrödinger equation with Hamiltonian operator is the electronic wave function and the electronic energy varies with the electron coordinates. On the other hand, nuclear coordinates do not explicitly appear in the electronic wave function. The total energy Etot has two terms, which are electronic energy Eelec and the constant nuclear repulsion term

15

B

B

Enuc = å å q=1 m>q

Zq Zm Rqm

, i.e.,

H l = EY for the electron particle

(2.4)

and

Etot = Eelec + Enuc

(2.5)

As the electronic wave function is not observable, the physical explanation can only be done by taking square of the electronic wave function in that

Y(x1, x 2 ,.......xA dx1dx2......dxA 2

(2.6)

shows the probability that all electrons are found at the same time in volume elements. There must be no difference in the probability on the condition that any two electrons’ coordinate are interchanged, viz.,

Y(x1, x 2 ,..xk , xl ...xA ) = Y(x1, x 2 ,..xl , xk ...xA 2

2

(2.7)

The second wave function formed by the interchange is same as the first one for bosons particle having integer spin while for fermions having half-integral spin, the interchange causes the sign change between two electronic wave function. Electrons have half spin so that they are fermions. Thus, the wave function must change its sign in the case of the interchange :

Y(x1, x 2 ,..xk , xl ...xA ) = -Y(x1, x 2 ,..xl , xk ...xA )

16

(2.8)

This is called antisymmetry principle. The result of this principle is the generalization of Pauli exclusion principle that is two electrons cannot occupy the same state. Thus, the probability of finding A number of electrons anywhere in space must be exactly unity,

ò ....ò Y(x , x 1

,.......xA dx1 dx2 ......dx A = 1 2

2

(2.9)

A wave function is said to be normalized if it satisfies the equation 2.9.

2.3 The Variatiional Principle E0 is the lowest energy and y 0 is the probability density of finding an electron any set 2

of coordinates. The average total energy is the expectation value of H operator, i.e;

ˆ dx º Y Hˆ Y E[Y] = ò Y* HY

(2.10)

The expression E[Y] refers that the total energy is a functional of the wave function. The variational theorem states that the energy is higher than the ground state energy if not the wave function Y matches to Y 0 :

Etrial [Y] ³ E0 = Y 0 Hˆ Y 0

(2.11)

The method in regard to the variation theorem to determine the lowest energy and its wave function is to optimize the functional E[Y] by searching through all suitable Aelectron wave functions. If the electron wave function gives the lowest energy, then, it will be Y 0 and the energy will be E0 . This can be shortly expressed by

E0 = min Y®A E[Y] = minl ®A Y Tˆ + VNe + Vee Y

17

(2.12)

Here, Y ® A shows that Y is an allowed A-electron wave function. the variational principle can be applied as well to subsets of all possible functions. The aim of the use of these subsets to optimize the equation given above within some algebraic scheme. The calculation will be the best approximation to the real electronic wave function using this particular subset.

2.4 The Hatree-Fock (HF) Method

The Hatree Fock Method is the milestone of nearly all wave function based quantum mechanical methods. Comprehending of logic behind this approximation will be better help for the analysis of the density functional theory.

As mentioned before, It is not possible to solve the equation 2.12 by examining through all acceptable A-electron electronic wave function so a suitable subset needs to be described. The subset should offer a physically sensible approximation to the exact wave function as well. HF theory includes ansatz for the structure of electronic wave functionit is an antisymmetric product of functions z k (xk ) each of which depends on in the coordinates of a single electron, that is ;

z1 (x1 ) z 2 (x1 )

. . .

z 2 (x2 ) z 2 (x2 ) Y HF = f =

1 N!

z A (x1 ) z A (x2 )

. . .

z1 (x A ) z 2 (x A )

(2.13)

. . .

z A (x A )

or using a convenient short-hand notation :

f=

1 det{z1 (x1 ) z 2 (x2 )....z A (x A )} N!

(2.14)

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The functions z k (xk ) are spin orbitals, which consists of one of two spin functions a (s) or b (s) and spatial orbital q k (r ) .

z (x) = q (r )s (s),

s = a, b

(2.15)

The spin functions also show the orthonormal property, that is, < a a > =< b b > = 1 and < a b > =< b a > = 0. It is vey important for computational calculation.

Substitution of the equation 2.14 for electronic wave function into the equation 2.1 produces the following expression for the Hartree Fock energy: A

A

A

1 EHF = f Hˆ f = å (i hˆ i) + å å (ii jj) - (ij ji) 2 i j i

(2.16)

where B ìï 1 * Z üï (i hˆ i) = ò z (x1 ) í- Ñ2 - å q ýz i (xi )dx1 i q r1q ï ïî 2 þ

(2.17)

describes the term because of the interaction between electron and nucleus and kinetic energy and

(ii jj) = òò z i (x1)

2

2 1 z j (x2 ) dx1dx2 r12

(ij ji) = òò z i (x1 )z *j (x1 )

(2.18)

1 z j (x2 )z *i (x2 )dx1dx2 r12

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(2.19)

are the so-called Coulomb and exchange integrals, respectively. They describe the electron interaction for two electrons. EHF is the functional of the spin orbitals,

EHF = E [{z i }]. Using variation theorem to the equation 2.16 for the choice of the orbitals, we can find the best orbitals that make EHF the lowest value with the restriction that the orbitals are orthonormal and thus this changes the Hatree-Fock expression:

ìï 1 2 B Zq r (x ' ) üï dx ýz i (x) + ò n x (x, x ' )z i (x ' )dx ' = e iz i (x) í- Ñ - å + ò ' r -r q r1q ïî 2 ïþ

(2.20)

Here, e i is the ground state orbital energies and n x , the non local exchange potential, is such that

B

' ' ' ò n x (x, x )z i (x )dx = -å ò j

z j (x)z *j (x ' ) z i (x ' )dx ' ' r -r

(2.21)

The HF equations explain non-interacting electrons under the Coulomb potential and a non-local exchange potential.

Although from this starting point, correlated methods (best approximations) for electronic structure and ground state energy are achieved, it costs very high in terms of computational calculation and the restriction of the number of the electrons treated [ref 4]. Moreover, to obtain accurate solutions, a large basis set that means very flexible description of spatial variation of the electronic wave function should be used. For the molecular calculation, Many correlated methods were developed such as, MP2, MP3, MP4, CISD, CCSD, CCSD(T). The calculation done by the last one is accurate enough for predicting the chemical properties of the structure, such as reaction rates, stability… etc. On the other hand, despite rapid advances in computer technology, the application of these methods to realistic model of the systems of interest is not feasible due to computational expense [ref 5].

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In the condensed matter science, the discussion above refers the direct solution of the Schrödinger equation is not currently possible. For the development and use of DFT, this is a major motivation since to find ground state energy, we may not need to solve Schrödinger equation exactly.

2.5 Avoiding the Solution of the Schrödinger Equation

It is not necessary to know 3N dimensional wave function so as to calculate the total energy. Knowledge of probability of density, i.e, the density of electrons at a particular position in space is adequate.

The pair density matrix for the energy expression is defined as

P2 (r1', r2' ;r1, r2 ) =

A(A -1) Y* (x1' , x 2' ,...x 'A )Y(x1, x2 ,...x A )dx3 dx4...dx A ò 2

(2.22)

The diagonal elements of P2 , often referred to as the two-particle density matrix, is as

P2 (r1, r2 ) = P2 (r1, r2 ;r1, r2 )

(2.23)

This equation is for two-electron probability function and entirely defines all twoparticle operators. The first order density matrix can be written with respect to P2:

P1 (r1' , r2 ) =

2 P2 (r1' , r2 ;r1, r2 )dr2 ò A -1

(2.24)

The total energy is defined in terms of P1 and P2 : A éæ 1 Zq E = Hˆ Pˆ = ò êç - Ñ12 - å êëè 2 q r1 - Rq

(

)

ù ö 1 ' P (r , r ) P2 (r1, r2 )dr1dr2 ÷ 1 1 1 ú dr1 + ò r1 - r2 úû ' ø r1 =r1

21

(2.25)

Hence, the total energy can be computed by P1 and P2. This makes the calculation easy since the solution of the Schrödinger equation reduces from 3N spatial coordinates to 6 dimensional space. Shortly, It is enough to find the first and second order density matrixes rather than solving the Schrödinger equation exactly in order to determine the energy.

Methods related to the direct optimization of E(P1, P2) has the particular problem of assuring that the density matrices are legal, that is, the density matrices should be constructible from an antisymmetric . İmplementing this restriction is important and is presently an unresolved issue [6,7]. Thus, without using electronic wave function, to calculate the total energy, the equation 2.25 is not consistent approach.

2.6 The Hohenburg and Kohn Theorems They proved the two theorems in 1964 [8]. The first theorem states:

The electron density determines the external potential as an additive constant.

According to this statement, the electron density individually resolves the equation 2.3. This continues as the equation (Hamiltonian operator) is indicated by the potential due the electrons and nucleus and N number of electrons. Then, the equation can be calculated from the density basically by integration over all space. Hence, in fact, the operator can be individually defined by the given charge density so the all state of electronic wave functions  and all properties of the systems can be determined.

They did an explicit proof of the given theorem that became general to contain systems with degenerate states in proof given by Levy [9]. The theoretical spectroscopist E. B. Wilson came up with a very straightforward proof of this theorem in a conference in 1965 at which it was being presented. Wilson’s comment is that the electron density 22

uniquely describes the charges of the nuclei and the positions and thus unimportantly determines the Hamiltonian. In the content of this proof, the electron density possesses a pinnacle at the nucleus:

Za =

-1 é ¶r (ra ) ù ê ú 2 r (0) ë ¶ra û r =0

(2.26)

a

Here, r (r) is the spherical average of the charge density, . We can uniquely determine Hamiltonian by examining charge density carefully. The Wilson observation causes to start the interest for the interaction between electrons and nucleus although Levy proof is more general. Shortly, The first theorem states that the energy is a functional of the density.

The second theorem forms a variational principle:

For any positive trial density, t, such that

ò r (r)dr = N then E[ r ] ³ E t

t

0

The second theorem’ proof is very clear since it is known from the first theorem that a unique trial Hamiltonian, Ht is found by the trial density. Hence, the trial wave function,

y t ; E[ rt ] = y t Hˆ y t ³ E0 obeys the variational theorem for the Schrodinger equation (see the equation 2.15). By the second theorem, DFT is restricted to the studies for finding ground state.

With respect to these two theorems, the following fundamental statement of DFT can be done:

d éë E[ r ] - m (ò r (r)dr - N)ùû = 0

(2.27)

The ground state energy and density are result of the minimum of some functional E[p]

23

obsessing the limitation that density consists of the correct number of electrons. The Lagrange multiplier of this limitation is chemical potential (µ).

The universal functional E[p], independent of the external potential, could be used for finding the ground state density and energy by minimizing the equation 2.27 with respect to this functional if its form is known.

2.7 The Energy Functional

It can be conculeded from the equation 2.3 that the energy functional consists of three parts, which are the external potential, the attraction between electrons and the kinetic energy so the energy functional can be written as:

E[ r ] = T[ r ] +Vext [ r ] + Vee [ r ]

(2.28)

The external potential is insignificant:

Vext [ r ] = ò Vˆext r (r)dr

(2.29)

The remaining two terms are unknown. Direct minimization of the energy would be achievable on the condition that good approximations to those functionals could be done. This issue is the subject of the current research. [Ref 10].

There was an approach made by Kohn and Sham who propositioned the following method to predict the kinetic energy and electron-electron functionals [ref 11]. They presented fictitious systems of N number of non-interacting electrons to be designated by a single determinant wave function in N number of orbitals. The electron density and kinetic energy are identified from the orbitals in this system:

24

Ts [ r ] = -

1 N z i Ñ2 z i å 2 i

(2.30)

The kinetic energy is the energy of non-interacting electrons in the system rather than true kinetic energy, which produce the ground state density: N

r (r) = å z i

2

(2.31)

i

A set of orbitals forming explicitly the density confirms that it can be constructed from an asymmetric wave function.

The electron-electron interaction, the classical Coulomb interaction, can be written in terms of the density:

VH [ r ] =

1 r (r1 )r (r2 ) dr1 dr2 2 ò r1 - r2

(2.34)

The energy functional may be modified as:

E[ r ] = Ts [ r ] + Vext [ r ]+VH [ r ] + Exc [ r ]

(2.35)

Here, the last term is the exchange-correlation functional, which is shown as:

Exc [ r ] = (T[ r ]- Ts [ r ]) + (Vee[ r ]-VH [ r ])

(2.36)

The exchange correlation functional is basically the total of the error made in using a non-interacting kinetic energy and treating the electron-electron interaction classically. After writing the functional explicitly in terms of the density, apply it to the equation 2.27. Then, we determine the orbitals minimizing the energy, which satisfies the given equations:

25

é 1 2 ù r (r ') dr '+ vxc (r)úz i (r) = e iz i (r) ê - Ñ + vext (r) + ò r - r' ë 2 û

(2.37)

Here, vxc (r) , local multiplicative potential, is the functional derivative of the exchange correlation energy.

vxc (r) =

d Exc [ r ] dr

(2.38)

The conduct of non-interacting electrons in an effective local potential is described by this set of non-linear equations (the Kohn-Sham equations). The orbitals provide to find exact local potential and exact ground state energy for the exact local potential by using the equation 2.31 and 2.36 respectively.

These Kohn-Sham equations and the HF equations have, see the equation 2.20, the identical structure with the non-local exchange potential replaced by the local exchangecorrelation potential, νxc. At this point, it is noted that the terminology in general use and reproduced here is very deceiving. As mentioned before for the exchange correlation energy, it includes an element of the kinetic energy and is not the sum of the exchange and correlation energies as they are understood in Hartree-Fock and correlated wave function theories.

The Kohn-Sham methodology manages to an exact correspondence of the density and ground state energy of a system comprising of non-interacting Fermions and the real many body systems described by the Schrödinger equation. A cartoon that describe this relationship simply is given in Figure 1

26

Figure 2.1 A cartoon shows the relationship between the real many body system (left side) and the non-interacting system of Kohn Sham density functional theory (right side) [ref]

Only if the exact functional is known, the correspondence of the charge density and energy of the many-body and non-interacting system can be exact. In this point, KohnSham density functional theory is an empirical methodology that we do not know the exact functional but the functional is independent of the materials that are studied. Basically, we could solve the Schrödinger equation exactly and find the energy functional and its associated potential for any particular system. However, this costs greater struggle than a direct solution of the energy. Nonetheless, the capability of determining precise properties of the functional in the systems provides exceptional predictions to the functional to be improved and utilized in predictive studies of a wideranging of materials. This property is generally associated with an ab initio theory so the approximations to density functional theory are often referred to as ab initio or first principle methods.

27

The computational cost to solve the equation 2.37 (the Kohn-Sham equation) scales conventionally as N3 owing to the requirement to carry on the orthogonality of N orbitals but in current practice is reducing as N1 through the exploitation of the locality of the orbitals. [ref]

In order to calculate global minimum of the structures, DFT represents a useful and potential highly accurate option instead of the wave function methods mentioned before. The effectiveness of the DFT, in practice, depends on the approximation used for the exchange correlation energy.

2.8 The Exchange Correlation Functionals

The achievement of the Kohn-Sham method is due to the opinion of the parting as the kinetic energy of independent particle and Hartree terms. The remaining part is the exchange correlation energy, which can be approximated by a functional of local density. There are some exchange correlational functionals, which can be classified as two main categories that are empirical and non-empirical functionals. The empirical functionals can be derived from the experimental results for particular materials. For the non-empirical functionals are obtained from the results of first-principles calculations. In this paper, the non-empirical functionals will be reviewed.

The total exchange-correlation energy can be given as:

Exc [ r ] = ò r (r)e xc ([ r (r)])dr

(2.39)

Here, e xc ([ r (r)]) is the exchange correlation energy per electron of a homogenous electron gas with density, r (r). An exchange-correlation hole encloses every electron, which in turn reduces its potential energy. The relationship between e xc ([ r (r)]) and exchange correlation hole that is the alteration in electron density induced by the 28

existence of an electron at a position r is achieved through the coupling constant integration formula by changing the charge from 0 to 1 and making the density constant. The exchange-correlation energy can be modified as:

Exc [ p] =

1 hxc (r, r ') r (r)dr dr ' 2 ò ò r - r'

(2.40)

Where, hxc (r, r') is the coupling constant averaged exchange-correlation hole. By regarding the equation 2.39 and 2.40, the exchange correlation potential per electron can be modified as:

e xc ( r[r]) =

1 hxc (r, r ') dr ' 2 ò r - r'

(2.41)

This may be considered as the potential formed by the interaction between an electron and its exchange-correlation hole. The exchange correlation potential ( Vxc (r) ) can be derived from the derivative of the exchange correlation energy.

The potential energy can be written as:

Vxc (r) = e xc ([ r (r)]) + r (r)

de xc ([ r (r)]) dr (r)

(2.42)

The last term of the above equation presents the variation in exchange correlation hole with respect to the density.

The exchange expression originates from the Pauli principle, which refers that two identical electrons cannot occuppy the same quantum state simultaneously. The correlation expression comes from the motion of the electrons, which we accept independent of each other. The exchange energy term is precisely known from the Hartree Fock method [ref out-93] whereas the correlation expression is identified merely 29

in parameterized forms, and can be determined by using the quantum Monte Carlo (QMC) method with some limiting cases[ref].The next subsection is related to the summary of the exchange and correlation schemes.

2.8.1 The Local Density Functional By utilising the local approximations, The easiest and the most regularly used exchangecorrelation functional were presented. For this reason, Spin-polarized Local Density Approximation (LSDA) and Local Density Approximation (LDA) [ref73] are known methods. The LDA exchange-correlation energy is defined as:

ExcLDA = ò r (r)e xcLDA [r ]dr

(2.43)

Here, e xcLDA [ p] is the exchange correlation energy per electron. In this approach, e xcLDA [ p] is accepted as a functional of the local density and is identified as:

e xcLDA [ r ] = e xchorn [ r ]

(2.44)

Here, e xchorn [ r ] states the energy density of a homogenous electron gas per particle of the density r . Hence, in this method, the exchange-correlation potential and the exchange correlation energy are substituted by the related terminologies for the homogenous electron gas. The homogenous electron gas exists a system having a uniform density in its ground state. Thus, this density describes it. For this reason, LDA exchange correlational function is good choice where the density of the systems changes gradually.

After regarding all above-mentioned, The LDA per particle is approximated as:

e

LDA xc

1 hxcLDA (r, r ') [r] = ò dr ' 2 r - r'

(2.55)

30

The exchange correlation energy is non-local as this energy for an electron is not independent of other particles around through the exchange correlation hole. This approach defines the exchange correlation potential as:

VxcLDA (r) = e xcLDA ([ r (r)]) + r (r)

de xcLDA ([ r (r)]) dr (r)

(2.56)

The exchange energy of the homogenous electron gas can be determined analytically whereas the correlation energy is obtained with quantum mechanic calculation [ref 74]. The form of the exchange energy density modified from Dirac [ref 74] is shown as:

3 3

e x [ r ] = - ( )1/3 r1/3 4 p

(2.57)

Here, 1/ r = 4p rz3 and rz is the radius of the sphere including an electron. Perdew and Zunger derived the most known approximation for the correlation. Ref75They used the output of quantum mechanical calculation of Ceperley and Alder [ref 76]. The correlation is expressed as the following:

ì AInrz + B + Crz Inrz + Drz , ï e x[r] = í ïg / (1+ b1 rz + b 2 rz ), î

rz £ 1 rz ³ 1

ü ï ý ï þ

(2.58)

Local Density Approximation was preferred for those systems having quite smooth density such as bulk materials. However, There exist some weaknesses for this approximation, which is listed below. 

Dielectric constant is over estimated for the structure making ionic, covalent and metallic bond.

31



Calculating binding energies with this approximation gives huge error by it may go %30.



The binding is too strong for small structures having weak bond type like Hydrogen atom.



LDA flops for atomic structure having inhomogeneous density.



Generally, the calculation for magnetic properties of the system is not adequate.



Lattice constant for solids is usually reported smaller than that of the experiments, which in turn bulk modulus or bulk compressibility are calculated too large.

2.8.2 The Generalized Gradient Functional

In this approximation, in order to allow for the influence on the inhomogeneity in charge density of the physical system, LDA needs to be modified by not only the information of the density but also the dependency of the density gradient Ñr . Moreover, the exchange correlation energy is constructed according to the some guidelines, i.e. the sum rule of the exchange correlation hole [reff 8]. Functionals includes the gradient of the density and makes certain of the hole constrains. These kind of functionals are known as Generalized Gradient Approximation (GGA), which can be shown by Taylor series expansion of the exchange correlational functional:

ExcGGA [ ra , rb ] = ò r (r)e xc ( ra , rb )dr + å ò Cxcs ,s ' ( ra , rb ) s ,s '

Ñra Ñrb

ra2/3 rs2/3'

dr +...

Here, ra , rb is electronic density for spin up and spin down respectively.

(2.59)

The

s ,s ' coefficient Cxc was found to be proportional to 1/ r 4/3 . However, GGA could not

32

contribute a systematic improvement on the LDA approximation since the exchange correlation interaction was not determined physically meaningful in this approach. Furthermore, the calculation of the higher order corrections of the density gradient is extremely challenging. On the other hand, Perdew and coworkers offered a more sophisticated approach for the GGA, which can be written as general form:

ExcGGA [ ra , rb ] = ò f ( ra ,rb , Ñra , Ñrb )dr

(2.60)

As usual, it is accepted that ExcGGA has two parts, which are exchange and correlation

ExcGGA = ExGGA + EcGGA

(2.61)

and the approximation of the functional is generally made separately for both of them.

Perdew and Wang exchange correlation functional include no experimental parameter. This functional is obtained from the constant gas approximations with exact limitations in 1991 [ref 77- 79]. Perdew, Burke, and Ernzerhof, PBE exchange correlation functional did a correction on Perdew and Wang functional [ref 80]. Lee, Yang and Parr (LYP) is another popular functional for the correlation [ref 81]. It is not based upon constant electron gas distribution and determined the correlation energy explicitly as the gradient of density functional. The LYP functional holds one empirical parameter. This correlation functional is usually associated with Becke’s exchange functional and is known as BLYP as well.

Importantly, GGA does not offer a complete non-local functional. Only advantage of GGA to be successful is that it contains the local variation of the densities. Furthermore, for both the energy and potentials, GGA in its original formula does not produce the simultaneous asymptotic behavior. In current developed functional, to obtain satisfactory results, a cut-off procedure on density is implemented. On the other hand, in quantum chemistry and condensed matter physics, the except for in the long range weakly

33

bonding system, such as in van der Waals interaction, GGA exchange correlation functional indicates corrections over LDA.

2.8.3 The Meta-GGA Functionals

Lately, for exchange correlation functionals, a rather elegant formalism which is dependent of the semi-local information in the Laplacian of the spin density or of the local kinetic energy density have been developed [ref 32-34]. These kinds of functionals are generally referred to as meta-GGA functionals.

The general form of the functionals can be written as:

ExcGGA = ò r (r)e xc ( r, Ñr , Ñ2 r, t )dr

(2.62)

t is he kinetic energy density, which is shown as: t=

1 2 ÑYi å 2 i

(2.63)

The performance of these functional will be discussed later.

2.8.4 The Hybrid Functionals

The non-interacting density functional system and the fully interacting many body system have direct connection between each other by means of the integration of the work done in gradually turning on the electron-electron interactions. According to this adiabatic connection approach [ref35], the functional can be generally written as:

1 l e2 Exc [ r ] = ò ò ò [< r (r)r (r ') > r,l - r (r)d (r - r ')] dr dr 'd l 2 l =0 r - r' 1

34

(2.64)

Here, < r(r)r(r') > r,l - r(r)d (r - r') > is the density-density correlation function, which is calculated at density for a system described by the effective potential:

Veff = Ven +

1 l e2 å 2 i¹ j ri - rj

(2.65)

Hence, on the condition that the variation of the density-density correlation function with the coupling constant ( l ) was known, the exact energy could be determined. By replacing the pair correlation function with that for the homogeneous electron gas, the LDA is improved.

With the adiabatic integration approach, a different approximation for the exchangecorrelation functional is proposed. The non-interacting system accounts for identically to the Hartree Fock ansatz at l = 0 whereas the LDA and GGA exchange correlation functions are formed to be perfect approximations for the fully interacting homogenous electron gas, i.e. a system with l = 1. Thus, it make sense to approximate the integral over the coupling constant as a weighted sum of the end points, i.e. we may set:

Exc » aEFock + bExcGGA

(2.66)

Coefficients are computed from the exact results in the system that is known. Becke modified this method by defining a new functional with coefficients found by a fit to the observed energies such as atomization energies, proton affinities, ionization potentials and total atomic energies for numerous small molecules [ref 36]. The resultant energy functional is as following:

Exc = ExcLDA + 0.2(ExFock - ExLDA ) + 0.72DExB88 + 0.81DEcPW 91

(2.67)

Where, DExB88 and DEcPW 91 are extensively GGA improvements [ref 37, 38] to the LDA exchange and correlation energy respectively. 35

This kind of Hybrid functionals are now very commonly applied in chemical applications, The accuracy of the calculations, such as binding energies, geometries and frequencies is better than the best GGA functionals.

2.8.5 The Performance of the Exchange Correlation Functionals

There is a natural hierarchy among the exchange correlational types. The studies for the corrections are going on the underlying functional form and the description of ground state properties. The most remarkable latest advance is the non-local nature of the exchange potential that is presented in one form or another. A brief hierarchy is given in Table 1.

Table 2.1 The hierarchy of exchange correlation functionals [taslak referance] Dependencies

Family

Exact exchange, Ñr , 

Hybrid

Ñ2 r ,

Meta-GGA

Ñr

GGA



LDA

For generating functionals, there are two main idea, which may be listed: 

Obtain a suitable functional form and establish parameters found by experimental data or data taken from explicitly correlated calculations. This is fundamentally empirical method.



To find both its structure and the parameters in its functional form, use the exact properties of the functional.

36

Obviously, there exist a number of functionals that cross the boundary between the two main whereas the distinction is frequently valuable in considering the expected precision of a specific functional in a new application.

The further detail that indicates the exchange correlation functions originate from which family and the number of experimentally found parameters is listed in Table 2 so as to examine a variety of functionals

Table 2.1 The characteristics of exchange correlation functionals [taslak referance] Functionals

Family

Parameters

LDA

Local

-

BLYP

GGA

Light

PBE

GGA

-

HCTH

GGA

18

VS98

Meta-GGA

21

PKZB

Meta-GGA

1

Hybrid

Hybrid-exchange

3

Kurth and coworkers and Adamo and coworkers have recently been studied the performance with these functionals in calculations of a number of molecular and material properties [ ref 42-43]. Some of the key data from these studies are listed in Table 3.

The error of the LDA is by 20-30 %. For the GGA functionals, there are important corrections with relative errors in the range 3-7 % and an average absolute error of 17 kcal/mol. The performance of the BLYP and PBE functionals are less accurate than the highly parameterised HCTH functional’s performance.

37

Table 2.1 The Atomization Energies in terms of the mean relative error (M.R.E) for a collection 20 molecules [42] and the mean absolute error (M.A.E) in kcal/mol for a collection of 148 molecules [43] with the maximum absolute error given in brackets. Atomization Energies M.R.E %

M.A.E (max) kcal/mol

LDA

22 %

-

BLYP

5%

-

PBE

7%

17(51)

HCTH

3%

-

VS98

2%

3(12)

PKZB

3%

5(38)

Hybrid

-

3(20)

The meta-GGA functionals result in relative errors of 2-3 % and average absolute errors of 3-5 kcal/mol. The other parameterized functional, VS98 achieves somewhat better result than PKZB. The meta-GGA functionals perform an apparent correction over the GGA functionals. The relatively lightly parameterized hybrid functional presents accurate results as well. An error of 1-2 kcal/mol is adequate for many chemical reactions. For this accuracy, the meta-GGA and hybrid functionals are suitable. However, there is no systematic performance at this level. Maximum errors can be in the range of 12-38 kcal/mol for difficult system. Furthermore, The accuracy for each functional can be very different for different system.

2.9 Conclusion

38

CHAPTER 3 THE INFLUENCE OF LIGANDS ON STABILIZATION OF PURE CLUSTERS 2.1 Introduction

Due to the development of nano-science and nanotechnology, the study about energetic, geometric structure and electronic properties of small metal nanoclusters has attained considerably significance because small metal nanoparticles exhibit chemical, magnetic, and optical properties that are quite different from the properties of their counterparts. Thus, many scientists and engineers who work on material science has been willing to do research on materials that are sized 100 nm or smaller scale to find new materials with tailored properties or implement their discoveries to new applications. In particular, metal clusters (MCs) play an important role in catalysis, medical science, and nanostructured electronic devices. For the medical science field, to defeat some diseases or treatment of the diseases such as, filariasis, malaria, brain fever, dengue that is due to the existence of mosquitoes, Plant obtained with the synthesis of metal nanoparticles may be prior to control the population of mosquitoes so as to reduce the effect of such kind of diseases via appropriate control methods [7]. For instance, Selvaraj and co-workers used mesocarp layer extract of Cocosnucifera and managed to synthesize silver clusters due to their biocompatibility, low toxicity, green approach and environmental friendly nature and antimicrobial properties that they show by treating silver nitrate solution with aqueous extract of C. nucifera coir at 60 Celsius [8, 9] They checked their method reliability by analyzing the excitation of surface plasmon resonance (SPR) using UV-vis spectrophometer at 433 nm [9]. For the nanostructured electronic devices, MCs has possible application in the microelectronics and sensor technology subfields [10,11]. For example, Nicola Cioffi and co-workers achieved gold/surfactant core/shell colloidal nanoparticles with a controlled morphology and

39

chemical composition using sacrificial anode technique with galvanostatic mode for the surface-modification of gate electrodes implemented in field effect capacitor sensors for NOx sensing [12]. They preferred Au-MCs based gas sensor due to the new reactivity properties, increased surface area-volume ratio, high sensing performance level and adjust of sensor properties [12-14].

Among application areas, particularly, colloidal metal particles are cardinal for comprehension of catalytic process by formulating model catalytic systems. Using industrial catalysts supported small MCs with large variations in size and shape generally leads to face with the problems related to control the distribution of the active sites of different reaction products on the catalyst surfaces [15,16]. For this reason, in the last few decades, the study of colloidal metal particles in solution, in the nanoscale regime, with tailored properties have been carried out as the high surface volume ratio of MCs for obtaining high amounts of active sites per metal mass and most catalytic reaction changed according to the surface characteristics for given metals are important key to take under control catalytic reaction by controlling over the nanoparticle size and shape, which makes this field popular to be studied under the nanoscale regime. Thus, colloidal of metal particles of technological application due to the stabilization of metallic particles efficiently in solution have recently attracted more attention to the systems related to the synthesis of colloidal metallic particles starting from single atoms, ions, or small clusters in the model catalytic systems [17]. There exists different ways to make colloidal metal nanoparticles

by means of the

advance in wet chemical synthesis strategies that are under the three main categories: chemical reduction of metal salt precursors, controlled decomposition of organometallic compounds and metal-surfactant complexes and electrochemical synthesis. In order to get under control the growth of produced metallic particles using ligands with functional groups attached to the surface metal atoms like donor ligands, polymers and surfactants, which can be called stabilizer allow for the construction of functional nanoassemblies and binding the clusters to surfaces of various substrates, which is widely used. Moreover, to keep the growing units from coalescence into thermodynamic equilibrium phase, the ligands could significantly affect the growing units. Briefly, the upper

40

methods are implemented with stabilizing agent

for the preparation of nanometallic

particles including Pt, Co, Cu, Ag, Ni, Au, Fe, etc. Pt Metal Clusters are one of the ingredients of colloidal suspensions in the field of catalysis [14-15]. Approximately 1 nanometer sized Pt particles on chlorinated for industrial catalysts were introduced in the 1960s. Lewis and co-workers contributed for the addition of Pt particles in catalytic hydrosylilation reactions. After them, Colloidal metal nano particles started to use many homogenous catalytic reactions in solution, from hydrogen peroxide decomposition to cross-coupling and supported metal nanoparticles on substrate-heterogeneous catalytic reactions. Barcaro and Fortunelli studied Pt metalorganic complexes via changing the number of metal atoms and the quantity of organic ligands. They preferred small Pt atoms since via the metal vapor deposition subtechnique of the upper methods mentioned, very minute metallic clusters like Pt clusters usually containing nore more than 10 atoms, coated by solvent molecules can be produced but the lack of this technique for obtaining this particle is low thermal stability of the solvates, despite at low temperatures, decompose slowly construction of insoluble aggregates. They tried to find possible solution for the stabilization of solvents by using ligands, which are used for inhibiting the coalescence of aggregates. One of our aims is to continue the Barcaro and Fortunelli’ work. They investigated Ptn (from n =1 to n =3) with different number of organic ligands but in this article we have investigated Pt4 by varying the number of hydrocarbon ligands within the framework of density functional theory (DFT) due to the interesting catalytic properties of tetrahedral Pt. Furthermore, the geometric, energetic, magnetic, electronic features, vibrational frequency and global reactivity of tetramer Pt with hydrocarbon ligands were studied. We hope it is possible to find some useful information by deriving from such studies in order to deduce from experimental observations. 2.2 Computational Details

For DFT calculation, NWChem 6.0 package has been used to perform geometry optimizations, and to find vibrational frequencies, the highest occupied and the lowest unoccupied molecular-orbital (HOMO–LUMO) gaps, and the total energies. CRENBL basis set and relativistic effective core potential (ECP) have been chosen for Pt where 41

the outer most 18-electrons (5s2 5p6 5d9 6s1) are treated as valence to reduce the number of electrons explicitly considered in the calculations. For C and H atoms the split valence 6-31G* basis set has been employed. The reliability of the CRENBL basis set and ECP were determined by comparing atomic excitation energies with accurate allelectron calculations where maximum errors were found to be less than 0.12 eV for Pt. The default convergence criteria have been employed during the calculations which are 1x10-6 Hartree for energy and 5x10-4 Hartree / a0 for energy gradient. The hybrid B3LYP exchange-correlation functional is chosen for the transition metal cluster with hydrocarbon ligand studied in the present work. Also, the most stable structures after the calculation have been examined again by employing the BPW91 and SVWN exchange correlation functional. The comparison between these three functionals will be made in the later sections. By building reasonable initial configurations, the structural search without any geometric restrictions with applying various electron spin multiplicity, which is kept fixed during optimization was carried out and minimized their energies at the lowest DF level via the program.

2.3 Results and Discussion 2.3.1 Geometric Optimization The most stable structure of Pt4(CH)n

( 1  n 11 ) hydrocarbon nanoparticles are

presented in the Figure 1. It can be concluded from the figures that, CH is adsorbed on the tetrahedron platinum clusters in a molecular form with carbon instead of hydrogen bonding to platinum for every case studied in the present work. Although for all cluster structures, each addition of the hydrocarbon to the cluster tends to break the tetrahedral Platinum bond type, tetrahedral Platinum almost preserves their structure until the clusters Pt4(CH)4. Furthermore, we re-examined the most stable structures determined by using B3LYP with different exchange functions (BPW91 and SVWN). Similar trend is observed for geometric optimization of the given study. The lowest energy structure of Pt4 is equilateral triangular pyramid, which has C3V point group symmetry. When the CH absorbed on tetramer Platinum as a hollow type (see figure 1) at the optimized structure, the point group symmetry is reduced to CS.

42

43

Figure 2.1 The lowest energy Structure of Pt4(CH)n (n = 1 to 6) clusters

44

The Platinum bond distances of the optimized structure of Pt4CH do not differ much from that of pure tetramer platinum, which is 2.7 A (reference). The average platinum bond distance of this nanoparticle is 2.63 A (see Figure 1). The average Pt-C bond length is 1.95 A and the C-H bond distance is 1.1 A. As understood from the Figure 1, the addition of the CH leads Pt4 to get close to the base formed by other ones. The second CH ligand adsorbed on the tetrahedral platinum as hollow type. The average PtC bond distance does not change importantly after the addition of the second CH on the particle. C2V symmetry is observed in this structure as well and the Platinum bonds get weaken slightly in the particle since the average Pt-Pt distance is 2.93 A. After the addition of the third CH, Pt –Pt distances continue to increase slightly. The third carbon binds the Pt and C as bridge type that cause to break to bond between Pt1- C6. The C-C bond distance is 1.42 A. The structure of the Pt4(CH)4 has CS symmetry. The last added Carbon atom leads C6 to adsorb on the Carbon surfaces as bridge type. The C8 and C7 bind surfaces hollow type and the average C–C bond distances is 1.43 A, which seems not to be affected after the forth CH adsorption. Also, there is no significant change between C5 and Pt atoms structure and the average Pt-C bond distances is 2 A. The fifth C added increased the distance as 1 A between Pt4-Pt1 that cause Platinum atoms not to preserve its tetrahedral structure but the average Pt- Pt distance is approximately 3.3 A, which is not differ much from previous one. In this particle, all Carbon atoms except for C5 form isosceles trapezoid structure. The average C-C bond distance is 1.44 A. Although the bond distance between C6-C8 is slightly decreasing, the double bond between carbon atoms was seen firstly in this cluster. The C9, last added, adsorbed on triangular surface formed by Carbon and two platinum atoms as hollow type. As shown in the figure 1, for the Pt4(CH)6, the carbons keep their isosceles trapezoid structure but C5 converted their adsorption type from hollow to bridge. In this structure, CS symmetry is observed. Although Pt4-Pt1 gets close to the each other, the distance between Pt3-Pt2 goes up 1.8 A, which shows the effect of the adsorption of C10 bonded hollow type to the surface. The last absorbed C11 bound the triangular platinum surface as hollow type and the average bond distance C-Pt is 2A. All carbons almost preserved their structure and the average C-C bond distance is 1.4 A.

45

Figure 2.2 The lowest energy Structure of Pt4(CH)n (n = 7 to 11) clusters 46

After the adsorption of the C12, the structure of the cluster is affected very much since C11 bound the triangular surface formed by C and Pt atoms as hollow type and C5 changed its bonding type. Thus, 5 carbon atoms and 3 carbon atoms bound between them separately. For the Pt4(CH)9, Pt atoms except for Pt4 bond together and the average bond distance between Pt atoms is 2.55 A. The last adsorbed Carbon (C13) bound hollow type and the average C-C bond distances is 1.44 A. Furthermore, the Pt-C bond distance did not changed significantly and the average bond distance is 2 A. In the structure for n = 10, 6 Carbon and 4 Carbon atoms bound together each other respectively. These two kinds of formation are connected via the C11-C7 bond. As seen in the figure 2, 10 carbon atoms push Pt atoms towards to the sides of the cluster. The C14 adsorbed on the surface formed by 2 Platinum atoms and Carbon atom as hollow type and Pt atoms except Pt4 kept bonding type but the Pt1- Pt4 distance increased 0.34 A. For n = 11, Platinum atoms except for Pt4 were affected by the last absorbed CH ligand (see figure 2) and the average Pt-Pt bond distance is 2.6 A.

2.3.2 Stability and Energetics

In order to predict the relative stabilities of the Pt4(CH)n

( 1  n 11 ), the absolute

value of the binding energy per atom and the second finite different energies of the Pt4(CH)n with different size are calculated. The binding energies per atom and the second finite different energies for the lowest structure of this cluster with different sizes (from n= 1 to n = 11) are plotted in Figure 3 and Figure 4 respectively. The absolute value of the binding energy per atom (BE/n) has been obtained in the fallowing way:

BE / n =

nE[C] + nE[H ] + 4E[Pt] - E[Pt 4 (CH)n ] n

from n= 1 to n = 11

(2.1)

where E[*] is the total energy of the neutral Carbon, Hydrogen, Platinum and the cluster respectively.

47

Table 2.1 Electronic Properties of Pt4 (CH)n (n=1 to 11) clusters Cluster

Electron

I.E

B.E per atom HOMO-

Affinity (eV)

(eV)

(eV/atom)

Lowest and highest

LUMO gap vibrational (eV)

frequencies

Pt4CH

1,5

6,7

3,7

1,4

60-2208

Pt4(CH)2

1,8

7,0

4,2

2,0

45-3017

Pt4(CH)3

0

6,4

4,5

2,3

57-3098

Pt4(CH)4

0,8

9,4

4,8

2,4

59-3174

Pt4(CH)5

1,8

6,1

4,9

2,0

37-3185

Pt4(CH)6

2,0

6,0

5,0

2,3

31-3185

Pt4(CH)7

1,3

5,5

5,0

2,5

46-3178

Pt4(CH)8

1,1

6,5

5,2

2,9

45-3190

Pt4(CH)9

1,5

7,1

5,3

2,4

51- 3173

Pt4(CH)10

1,6

6,1

5,3

2,2

56-3169

Pt4(CH)11

1.0

6,0

5,4

1,8

25-3167

48

(cm1)

It is clear that the binding energies per atom of the species continuously increase with increasing CH atoms (see Figure 3 and Table 1) indicating that the species can constantly gain energy during the growth process whereas the

gain energy rate is

slowing down during the growth process (see Figure 3). The binding energy per atom of the pure platinum tetramer is 2.4 eV/atom. After the absorption of the one CH molecule, this binding energy increased rapidly as seen in Table 1,

6 5

Ev/atom

4 3 2 1 0 0

2

4

6

8

10

12

#CH Figure 2.3 The Binding Energy per atom of Pt4(CH)n (n = 1 to 11) cluster

To further illustrate the stability of the species and their size dependent behaviors, we have considered the second finite difference in energies that is a sensitive quantity frequently used as a measure of the relative stability of the complexes and is often compared directly with the relative abundances determined in mass spectroscopy 49

experiments. Moreover, Clusters are especially abundant magic number sizes in mass spectra as they are most stable ones. The second finite different energies (Dn,m) can be calculated as

Dn,m = E n+1,m-1 + E n-1,m+1 -2En,m

(2..2)

where En,m is the total energy of the cluster Pt4(CH)n. The second finite difference (D) in energies versus cluster size are given in Figure y.

0.8 0.6 0.4

Dn,m (eV)

0.2 0 -0.2

0

2

4

6

8

10

12

-0.4

-0.6 -0.8 -1 -1.2

# CH atoms

Figure 2.4 The second finite difference energies of Pt4(CH)n (n =2 to 10) cluster Due to definition of the second finite difference in energies, we examined the Pt4(CH)n from n = 2 to n = 10. The relative stability of the species in first three order within the size range studied is Pt4(CH)9 > Pt4(CH)10 > Pt4(CH)4 and Pt4(CH)8 has the smallest D energy among the size range studied, see figure y, which all show the inverse ordering of the CH adsorption ability around their neighborhoods. Commonly, the more stable the 50

species, the lower reactivity of the species to absorb CH. Thus, two species (Pt4(CH)4 and Pt4(CH)9 ) are expected to be more abundant in mass spectra when comparing to the other clusters. The formation of the chemical bond for pure tetramer platinum releases a significant amount of energy, 9.6 eV. The total variation of energy in the reaction

Pt 4 + CH ® Pt 4CH

DE= -6.1 eV

(2..3)

corresponds to 6.1 eV, see table, which shows the energetic analysis of the given study. For all the reactions, we used exchange correlations becke88 perdew91 as well to compare our result but there has not been experimental data yet to check the results of the table. Maybe, these parts give inspiration the experimenter to study the energetic of the given clusters. The reaction using B3LYP and becke88 perdew91 xc functional is respectively as the fallowing:

Pt 4CH + CH ® Pt 4 (CH)2

DE= -6.1 eV (2..4)

Pt 4CH + CH ® Pt 4 (CH)2

DE= -6.5 eV

The gain energies obtained both two functionals are slightly different to the each other. The growth of the cluster with CH ligands is energetically favorable, see Table 2, but the less energetically favorable structure is calculated at the bond formation of the Pt4(CH)6.

51

Table 2.2 Energetic Reactions of Pt4 (CH)n (n=1 to 11) cluster

Complex

E (eV)

Pt4

+

CH  Pt4CH

-7,1

Pt4CH

+

CH  Pt4(CH)2

-6,1

Pt4(CH)2

+

CH  Pt4(CH)3

-6,1

Pt4(CH)3

+

CH  Pt4(CH)4

-6,0

Pt4(CH)4

+

CH  Pt4(CH)5

-5,9

Pt4(CH)5

+

CH  Pt4(CH)6

-6,1

Pt4(CH)6

+

CH  Pt4(CH)7

-6,1

Pt4(CH)7

+

CH  Pt4(CH)8

-6,1

Pt4(CH)8

+

CH  Pt4(CH)9

-7,1

Pt4(CH)9

+

CH  Pt4(CH)10

-6,4

Pt4(CH)10 +

CH  Pt4(CH)11

-5,9

2.3.3 Electronic Properties

Vertical ionization potentials (IP) and electron affinities (EA) are calculated as the total energy difference of neutral, anionic and cationic species in the fallowing way: IP[Pt4(CH)n ] = E[Pt4(CH)n+] - E[Pt4(CH)n]

(2..5)

EA [Pt4(CH)n ] = E[Pt4(CH)n] - E[Pt4(CH)n -]

(2.6)

Here, E[Pt4(CH)n], E[Pt4(CH)n -] and E[Pt4(CH)n+] refer to total energy of the neutral, anionic and cationic cluster respectively. The highest occupied (HOMO) indicates the ability to give an electron, the lowest occupied molecular orbital energies (LUMO) as an electron acceptor. A large HOMO-LUMO gap has been considered as significant requirement for chemical stability.

52

Figure 2.5 The 3D HOMO-LUMO density plot of the structure of Pt4(CH)4 and Pt4(CH)9 clusters respectively.

53

Calculated ionization potentials (IP), electron affinities (EA), HOMO-LUMO gap and lowest and highest vibrational frequencies are given in the Table and also, the 3D plots of the frontier orbitals, HOMO and LUMO for Pt4(CH)4 and Pt4(CH)9 are given in the Figure 5.

Pt4CH has the lowest HOMO-LUMO gap with the value 1.4 eV in the given study. It was found that the HOMO-LUMO gap energy of Platinum tetramer with B3LYP xc functional is 0,96 eV. İt can be pointed out that the adsorption of the CH ligand cause to increase the gap energy that makes species more stable. The EA and IP energy are calculated as 1.5 eV and 6.7 eV respectively (see Table 1) and also, in this structure, the first excited transition state is observed during the frequency calculation and the lowest and highest vibrational frequencies are 60 cm-1 and 2208 cm-1. The lowest electron affinity is seen in the Pt4(CH)3 cluster and the HOMO-LUMO gap energy of this cluster is little larger than n  3 clusters. The Pt4(CH)4 cluster is the highest ionization energy with the value of 9.4 eV and low EA energy is calculated for this structure. The HOMO-LUMO gap is 2.4 eV and the lowest and highest vibrational frequencies for this cluster are 59 cm-1 and 3174 cm-1 respectively. The highest electron affinity in the given study was seen at n=6 ( 2 eV ) and its IP energy is 6 eV. The Pt4(CH)9 possesses high stability due to its large HOMO-LUMO gap ( 2.4 eV) when comparing with other clusters. the lowest and highest vibrational frequencies for this cluster are 3173 cm-1 respectively. The Pt4(CH)11

51 cm -1 and

has the second smallest HOMO-LUMO gap

within the range studied, which indicates high chemical activity like Pt4CH.

2.3.4 DFT Chemical Reactivity Descriptors

In this section, we center our attention on the characterization of the species we have studied in terms of reactivity descriptors such as chemical potential (), chemical hardness () and electrophilicity index (w). Those quantities are displayed in Table 3. We calculated electronic chemical potential and chemical hardness according to the finite difference approximation:

54

1 2

(2.7)

1 2

(2.8)

m = - (IP + EA)

h = - (IP - EA)

where  determines the escaping tendency of electrons from an equilibrium system and  can be defined as a resistance to the charge transfer [89]. The global electrophilicity index is derived from the chemical potential and hardness by the fallowing:

m2 w= 2h

(2.9)

 measures the stabilization in energy as the environment gives systems extra electronic charges.

Table 2.3 The Chemical Descriptors Indexes of Pt4 (CH)n (n=1 to 11) cluster Cluster

Chemical

Chemical

Electrophilicity

Potential ()

Hardness ()

Index (w )

Pt4CH

-4,1

2,60

3,3

Pt4(CH)2

-4,4

2,60

3,8

Pt4(CH)3

-3,2

3,21

1,6

Pt4(CH)4

-5,1

4,30

3,0

Pt4(CH)5

-4,0

2,16

3,7

Pt4(CH)6

-4,0

1,99

4,0

Pt4(CH)7

-3,4

2,13

2,7

Pt4(CH)8

-3,8

2,71

2,7

Pt4(CH)9

-4,3

2,81

3,4

Pt4(CH)10

-3,8

2,25

3,3

Pt4(CH)11

-3,5

2,48

2,5

55

Global indicators constitute good indexes to describe the reactivity of system, intrinsic electronic properties of the system, shows the feasibility of the chemical change. It is harder to lose an electron but easier to take another one when  becomes more negative . It is observed in table that Pt4(CH)4 has the biggest chemical potential with the value  generally shows parallel tendency with chemical potential descriptors.

5.1 eV.

Pt4(CH)4 is the hardest (4.3 eV) and the smallest one is Pt4(CH)6 with the value of 1.99 eV (see table). It might be concluded that results for chemical hardness indicate oscillating behavior during the growth process within the range of this study. According to the  values in the table, the Pt4(CH)6 is the most susceptible from the external environment. The electrophilicity index, which can be one of the essential parameter for the selection of a catalyzer, shows oscillating behavior in the table results.

2.4. Conclusion

In the present study, we have performed spin-polarized density functional theory calculations to study the interaction of the Pt4(CH)n ( n = 1 to 11 ) metalorganic complexes by varying the number of the CH ligand. The geometric properties, stability and energetics, electronic properties and chemical reactivity indexes have been discussed. From the analyses of the results, several conclusions can be drawn. First of all, during the growh process with CH on platinium tetramer, the atop site adsorption side of CH to the species is not energetically favorable. CH is adsorbed on the tetrahedron platinum clusters in a molecular form with carbon instead of hydrogen bonding to platinum. Secondly, For each case in the present work, Carbon atoms form stronger bond each other than with Pt atoms but after n = 4, Pt-Pt bond gets weaker than between C-Pt atoms since after n =4, platinum atoms in the cluster cannot preserve their distorted tetrahedron structure and the average Pt-Pt distance is little much differ from previous ones.

`

56

Thirdly, In order to prevent the coalescence of the Platinum atoms during the growth of the freshly formed Platinum Particles, it may be concluded from the Figure and Table that C6H6 and C7H7 ligands are good model. Fourthly, regarding the relative stability criteria, energertics, chemical reactivity descriptor indexes and electronic properties of the studied clusters, Pt4(CH)4 and Pt4(CH)9 metal hydrocarbon complexes are found most stable structure.

57

CHAPTER 3 THE INTERACTION OF C2H WITH BIMETALLIC Co-Pt CLUSTERS 3.1 Introduction Transition metal clusters have many potential applications in different areas of nanotechnology such as medicine and biochemistry [1,25], heterogeneous catalysis [31], magnetic recording media [44]. In recent years, a great deal of effort, in particular, has been devoted to studying magnetic CoPt nanoalloy clusters. [5-31] due to the usage of ultra-high density magnetic storage applications [32]. The characterization of the magnetic anisotropy energy distribution of a diluted assembly of CoPt nanoparticles with a mean diameter of 3 nm by using superconducting quantum interference device magnetometry was reported by Tournus and coworkers. They found experimental evidence of a meaning anisotropy constant dispersion with a comparison of unselected CoPt clusters and size-selected Co clusters [33]. Tizitzios and coworkers explained the synthesis of a 3D ferromagnetic CoPt polypod–like nanostructure [15]. On the theoretical side, Sebetci studied small bimetallic CoPt clusters in terms of the structural, energetic, electronic and magnetic properties by using density functional theory method within the generalized gradient approximation. As a general trend, he found the average binding energies of Co-Pt metallic clusters increase with Pt doping [34]. Another theoritical study conducted by Feng and coworkers. They examined magnetic and electronic properties of CoPt nanoparticles, which has equal number of Pt and Co atoms [35]. The ethynyl radical (C2H) is a significant reactive intermediate in hydrocarbon combustion process [36-37]. İt is also a widespread interstellar molecule including a variety of sources [38-43]. In past decades, C2H radical has been attracted much attention [43-48]. Linebeger at al determined electron affinity of C2H as a value of 2.97 eV [49]. Neumark at al. studied C2H and C2D radicals by slow electron velocity-map imaging of the related anions [50]. At the same time, investigating the interactions between metal and organic molecules is significant in both heterogeneous and

58

homogenous catalytic systems. Adsorption of a small molecule or atom may modify the electronic and magnetic properties and stability of the transition metal clusters [51]. Experimental and theoretical works were performed to study the interaction complexes formed between metal atoms and ethynyl radical [52-57]. The linear geometric structure of CrC2H in ground and excited states and the vibronic spectrum of CrC2H at the 11 100 and 13 300 cm-1 region were reported by Brugh and coworkers [58]. By combination of resonance-enhanced

two-photon

ionization,

laser

induced

fluorescence,

and

photoionization efficiency spectroscopy experiments with DFT calculations, Loock at al. determined the frequencies of different modes of YbC2H [59]. FeC2H-1 and PdC2H-1 were studied experimentally [60-61]. Da-Zhi Li and coworkers investigated the interaction of C2H radical with neutral and anionic small gold clusters [62]. Yuan at al. investigated the small anionic CoC2H complexes by mass spectrometry, the photoelectron spectra and DFT calculations [63]. However, to the best our knowledge, there has been no studies reported to date on the interaction between C2H radical and bimetallic clusters. Thus, it is worthy to investigate anionic bimetal CoPtC2H nanoalloy complexes systematically in order to understand the mechanism and elucidate more details on the formation of small CoPtC2H nanoalloy complexes

In this work, We have investigated the structural, energetic, electronic and magnetic properties of [ConPtmC2H ]-1 (2  n+m  5) bimetallic clusters within the framework of the density functional theory and addressed the following important questions: How does C2H adsorption on change with the composition at a given cluster size and structure? How does it change the magnetic properties of bimetallic Co-Pt clusters? What is the influence of cluster size and chemical component on the reactivity of Co-Pt clusters to absorb C2H? Locally stable isomers are distinguished from transition states by vibrational frequency analysis as well. We present the obtained results and discuss the interaction complexes formed between CoPt bimetalic atoms and ethynyl radical in the following sections.

59

3.2 Computational Details

NWChem 6.0 package [64] has been used to perform geometry optimizations, and to find the total energies, the vibrational frequencies, and the highest occupied and the lowest unoccupied molecular-orbital (HOMO–LUMO) gaps by DFT calculations. CRENBL [65] basis set and relativistic effective core potential (ECP) have been chosen for Pt where the outer most 18-electrons (5s2 5p6 5d9 6s1) and Co where the outer most 17-electrons (3s23p63d74s2) are treated as valence to reduce the number of electrons explicitly considered in the calculations. The corresponding Gaussian basis functions of Pt and Co are (5s5p4d) and (7s6p6d) respectively. For C and H atoms the split valence 6-31G* basis set has been employed. The reliability of the CRENBL basis set and ECP were determined by comparing atomic excitation energies with accurate all-electron calculations where maximum errors were found to be less than 0.12 eV for Pt and 0.05 eV for Co [65]. The default convergence criteria of the code have been employed during the calculations which are 1x10-6 Hartree for energy and 5x10-4 Hartree / a0 for energy gradient. The generalized gradient approximation (GGA) of Becke’s exchange functional [Ali hoca] and Lee-Yang-Parr correlational functional [Ali hoca] is chosen for the nanoalloy cluster with ethnyl radical ligand studied in the calculations. The geometry optimizations without any symmetry constraints in various electronic spin multiplicities were carried out.

3.3 Result and Discussion In the following, The optimized geometries of the low-lying some isomers of anionic ComPtnC2H ( 2  m +n  5) clusters obtained with DFT calculation are given in Figure 1, Figure 2 and Figure 3 , where the most stable structures are on the left. We have considered many spin multiplicities and various initial structures. Also, Table 1 represents spin moment, binding energy per atom, homo-lumo gap, vertical detachment energy (VDE) and highest and lowest vibrational frequencies of the studied clusters.

60

3.3.1 Diatomic [ ComPtnC2H ]-1 ( m +n =2) nanoalloys The low lying some anionic isomers of diatomic CoPt ethynyl clusters are displayed in Figure 1. In the ground state structures,The C-C bond distance upon ethynl radical seems to be less affected (see Figure 1) since the C-C distance of the free ethynl radical is calculated as 1.25 A and the ratio Pt/Co has slight effect on the bond length of C-C in C2H radical as well. The ground state geometry of anionic Co2C2H nanoparticles has been identified as a linear structure with point group symmetry Cs. When C2H radical approaches to diatomic Co atoms and is molecularly absorbed to be anionic Co2C2H cluster, only one C atom prefers to bond with Co atom and Co-C bond distance is 1.92 A The Co-Co bond distance is elongated as 2.30 A in the quartet magnetic state ( 4 μB spin moment) while Co-Co bond distance in pure diatomic cobalt particles in literature is 1.99, 2.01 and 2.13 A. The B.E per atom and VDE are calculated as 4.46 and 1.25 eV (Table 1). The theoritical VDA and bond length we found are very close to the experimental value (VDA; 1.50 eV ) and theoretical values (the Co-Co bond length; 2.43 A and the Co-C bond length; 1.93 A ). The second isomer 1B having Cs symmetry is , see Figure 1, 1.27 eV higher in energy than the ground state configuration. The magnetic moment of 1B is 6 μB while 1A and 1D are 4 μB and 1D with Cs symmetry has 0.91 eV relative energy as well. 1C is the lowest energetically favorable structure with 8 μB spin moment. The ground state configuration of anionic CoPtC2H is linear structure with Cs group symmetry. The C2H is adsorbed onto the Co atom rather than the Pt atom where the CPt bond distance is 1.94 A. The elongation between Co-Pt distance upon ethynyl radical absorption is by 0.04 A while the Co-Pt bond length in pure bimetallic CoPt nanoalloy is 2.24 A. The second low-lying minima of anionic CoPtC2H is a linear structure which has 0.59 eV higher energy than the lowest one. The magnetic moments of 2A and 2B are 3 and 1 μB respectively and the third isomer 2E has 3 μB , which has 0.78 eV higher energy than the ground state. The ground state configuration of anionic ethnyl bimetal Pt cluster with Cs symmetry is nonlinear planer structure in doublet state. The angle between Pt-Pt-C atoms is 160. The Pt-Pt distance of the pure bimetal Pt cluster is 2.21 A. The adsorption of ethnyl radical leads to a stretch in the Pt-Pt bond length where the Pt-Pt bond distance becomes 2.46 A. In addition, Bridge site adsorptions have higher

61

energy than atop structures, which continues for the trend that atop structures have lower energy than bridge site adsorptions. The elongation for Pt-Pt and C-C distance is by 0.24 A and 0.05 A respectively in bridge site adsorption whereas there is no significant change for the typical Pt-C distance in both adsorption type. 3.3.2 [ ComPtnC2H ]-1 ( m +n =3) The low lying some anionic isomers of triatomic CoPt ethynyl clusters are displayed in Figure 1. The VDE of the particles for m +n =3 ranges from 4.29 to 5.06 eV( see Table 1). The doping of the Pt atom in these structures leads to increase in VDE. Futhermore, The C-C bond distance is ethynyl radical in the lowest ground state isomers is between 1.24 and 1.44 A where the CC bond of acetylene (1.20 A)[referance] and the CC bond of ethene (1.33 A) [referance]. The lowest energy structure of bare cobalt trimer is computed as a triangular structure [] or linear structure[] by different research groups. Yoshida and coworkers reported CoCo distances of the anionic bare Co trimer possessing linear structure are between 2.252.50 according to their experimental and theoretical results. The anionic Co3 ethynl nanoparticle is planar Y-like structure with a BE of 4.29 eV and it has high point symmetry C2v with 7 μB spin moment as well. The average Co-Co distance of the ground state 5A having a perfect isosceles triangular Co3 unit is 2.4 A. In addition, The doping of one Co atom on [Co2C2H]- leads to increase in VDE energy where the VDE in the structure 5A is 1.52 eV that is consistent with the results in Ref (see Table 1). The second isomer 5B and third isomer 5C has apprixomate relative energies but in different magnetic states. The forth isomer 5D is formed by adding a ethenyl radical to the cobalt trimer cluster with two C atoms attached to the terminal C atom of C2H forming four membering ring in septet magnetic state. The Co-C bond lengths are 2.04 and 1.96 A in this structure. As one of the Co atoms is replaced by a Pt atom, the lowest energy structure of anionic Co2Pt ethnyl nanoparticles becomes the structure possessing triangular bimetallic unit including a Pt atom at apex and 2 Co atoms at base with average 2.44 A Co-Pt and 2.38 A Co-Co bond lengths. While the distances between Co and Pt atoms are very little stretched as the amounts of 0.06 and 0.11 A, the speraration between Co atoms is shrunk as much as 0.15 A after the adsorbtion of ethynyl radical. Its BE is 4.71 eV/atom where 62

the magnetic state is the same with the bare cluster Co2Pt. In the second isomer B having atop ethynyl adsorbtion site, the Co-C bond distance 1.92 A with a relative energy of 0.49 eV while the third C and forth D isomers have average 1.91 and 1.98 A of the Co-C bond length respectively. As Pt doping increases, the CoPt2 ethynyl cluster has also triangular unit with increasing B.E as 4.89 eV. Ethynl radical prefers energitically to bind Pt atom rather than Co atoms which is different adsorption type from other ground bimetallic nanoalloy structures. The C-Pt distance is 1.94 A and the Pt-Pt distance is 2.81 A while these bond lenghts of 3A is 1.95 and 2.46 A. The magnetic moment of all three isomers possessing Cs symmetry in this cluster are 3 μB, 1 μB and 7 μB respectively. It can be seen in Figure 1 that the second isomer B and third isomer C have close ground state energies with A. In addition, in these two isomers B and C, two Carbon atoms bind the bare metals as distrinct from A. The B.E of Pt3 etynyl cluster is calculated as 5.06 eV which is the highest value at m+n = 3. It can be noted that the doping of Pt atom to the cluster leads to increase in B.E value. However, the difference of B.E between CoPt2 and Pt3 ( 0.17 eV) is close to the that of the Co2Pt-CoPt2 ( 0.18 eV) but it is smaller than that of the Co3-Co2Pt (0.42 eV). In this structure, Four coordination of Carbon that indicates sp3 hybridization resulting from delocalized electrons in ethynl radical was seen firstly

among all

nanoparticles we have mentioned. This triggers to expand the the C-C bond length in C2H radical importantly. In addition, the magnetic state of the second isomer B is doublet while the third isomer C has same magnetic moment as the case of the isomer A ( see Figure 1).

63

1A Cs 4

0.0 eV

1B Cs 6 1.27 eV

1C 8 4.00 eV

D Cs 4 0.91 eV

2A Cs 3 0.00 eV

2B Cs 1 0.59 eV

2ECs 3 0.78 eV

3A

3B C1

2 0.96 eV

A Cs 4 0.00 eV

B C1 6 0.49 eV

C

E Cs

4 0.73 eV

A C2V 7 0.00 eV

B C1 7 0.22 eV

DC C1 5 0.27 eV

F D C1 7 0.52 eV

A Cs

D Cs 1 0.19 eV

G Cs 3 0.02eV

A C1 0.00 eV

B Cs 2 0.50 eV

3

0.0 eV

C1 0

Cs 2 0.00 eV

C1

2 0.49 eV

0.44 eV

Figure 3.3.1 The optimized structure of some isomers of [ ComPtn ]-1 (2  m +n 3) 64

3.3.2 [ ComPtnC2H ]-1 ( m +n =4) The low lying some anionic isomers of triatomic CoPt ethynyl clusters are displayed in Figure 2. In these structure, the VDE continues for the trend that doping of Pt atom to the cluster leads to increase VDE that ranges from 1.53 eV to 3.09 eV. The lowest energy structure of anionic Co4 ethynyl cluster adsorbed on bridge site with a B.E of 4.09 eV. The ground state structure of bare Co4 in perivious study was found as out of plane rhombus structure [] while the ground state structure of bare Co4 by theoretical calculation varies in the literature. It has been reported to be a planar rhombus structure or an out of plane rhombus structure and a distorted tetrahedral structure. The lowest geometrical structure of Co4C2H- simulated by Yuan at al is consistent with our lowest lying isomer Co4 ethynyl cluster we optimized. Upon ethynyl adsorption, the magnetic moment in the structure A is reduced to 8 μB which is same as second isomer B (see Figure x) while the third isomer C has 10 μB spin moment with a relative energy of 0.44 eV. The energy separation of the second isomer from the first isomer is relatively low. The Co-Co and Co-Pt edge bond lengths of the bare Co3Pt having rhombus structure are 2.25 and 2.38 A respectively. The adsorption of ethynyl radical leads to expand the Co-Co bond lengths. It can be noted that the doping of one Pt atom to anionic Co3 ethynyl nanoparticle cause to weaken the bond between cobalts. In the second isomer having relative energy of 0.71 eV, the terminal of C atom in C2H radical bound only the Pt side of the cluster in triplet magnetic state with a Pt-Co bond length of 1.96 A. The third and forth isomers are not energetically favorable due to the great relative energies. As one of the Co atoms replaced by Pt atom, the BE and VDE of the lowest energy structure of Co2Pt2C2H- becomes 4.70 and 1.87 eV respectively. The BE and VDE of this structure is slightly lower and little higher than that of CoPt2 ethynyl structure respectively whereas they are lower than that of the Co2Pt ethynyl structure (see Table 1). The C-C distance of isomer A is 1.30 A, longer than that of the CC bond of acetylene (1.20 A)[referance] and shorter than that of the CC bond of ethene (1.33 A). When the spin magnetic moments are considered, it is seen that there is initial abrupt decrease from 7 μB to 3 μB the transition from Co3 to Co2Pt ethynyl cluster. The 65

magnetic state of the second isomer B with Cs symmetry is 6 μB while third and forth isomers have 2 μB and 8 μB magnetic states respectively. Compared with bare CoPt3 cluster [referance], the CoPt3 in [CoPt3C2H]- is distorted obviously while the bare CoPt3 structures in other isomers are perturbed and distorted slightly. This indicates the adsorption gives rise to considerable structural change for the lowest energy structure A. Besides obvious lengthening of Pt-C bond lenght where two Carbon atoms prefers to bond with same one Pt atom, the Pt-Pt bond distance in this structure is longer than that in corresponding pure CoPt nanoalloy cluster [].On the other hand, the magnetic moment of bare CoPt3 is 5 μB. The fist isomer and second isomer of anionic CoPt3C2H have 3 μB magnetic moment whereas the third isomer has 3 μB magnetic moment with Cs symmetry. The total energies of second and third isomers are 0.37 eV and 0.71 eV higher than the first one, respectively. The ground state structure of Pt tetramer is a distorted tetrahedron in doublet magnetic state with Pt-Pt bond distance of 2.60 A. In our previous study [], The Pt-C and the Pt-Pt distances in Pt4methylidyne were calculated as 1.95 A and 2.63 A respectively and the two different bond lengths of Pt-C in Pt4ethyne are 2.1 A and 1.9 A. In addition, The B.E of Pt4methylidyne, Pt4ethyne, Pt4benzene and Pt4(benzene)2 is 3.71 eV, 4.20 eV, 5.13 and 5.52 eV respectively. On the other hand, , due to the adsorption of ethynyl radical, the Pt units in the structure A does not preserve its tetrahedral structurel type in doublet magnetic state. The Pt-Pt bond lengths in this particle are 2.58 A and 2.74 A. The B.E of this structure is 4.95 eV as well. As one Co atom replaced by Pt atom, The C-C bond distance in anionic Pt4C2H becomes longer than that of the CoPt3C2H structure. The lenghtening of the C-C bond distance may be due to losing of some sharing valence electrons between carbon atoms over new Platinum-Carbon interactions. In the second isomer possessing 0.31 eV relative energies, the tetrahedral unit becomes a nonplanar, rhombus like structure. While the magnetic moment of the first and second isomers are 2 μB, the third isomers having distorted tetrahedral unit has 4 μB magnetic moment that is not energetically favorable with a 1.32 eV relative energy.

66

A C1

4

C

Cs

A C1 7 0.00 eV

C

A

B

C1 8 0.00 eV

B C1 3 0.37 eV

C

C1

6

E

C1

C1 3 0.71eV

D

C1

C1 8 0.09eV

C

D Cs 5 0.71eV

2

7

F C1

1.28 eV G C1 9 2.04 eV

C1 10 0.44 eV

A 2

8

0.00 eV

A C1 3 0.00 eV

B C1 2 0.31 eV

4 1.32 eV

Figure 3.3.2 The optimized structure of some isomers of [ ComPtnC2H ]-1 (m +n =4)

67

3.3.3 [ ComPtnC2H ]-1 ( m +n =5)

The low lying some anionic isomers of diatomic CoPt ethynyl clusters are displayed in Figure 3. The B.E values in this section vary 3.94 eV to 4.79 eV. The increase rate of B.E from Co5 to Pt5 ethenyl cluster is slowing down as seen in Table 1.

The ground state structure of bare Co5 in perivious study was calculated as planar W-like structure[] while It has been reported to be a C4v square pyramid, a C2v rhombus pyramid and a D3h trigonal pyramid in the literature. The lowest geometrical structure of Co5C2H- we optimized contain nearly planar W-like structure,which is consistent with the result found by Yuan at al. that use B3YLP xc functional in their optimizations. In this structure, the ground state structure of anionic Co5 ethynyl cluster adsorbed carbon atoms on the shortest side of its structure with a B.E of 3.94 eV. We made VDE calculation that is close to experimental value as well (see Table 1). In all isomers except for the last one, the magnetic moment is 9 μB but the bare Co pentamer and the fifth isomer have high magnetic moment: 11 μB. In the anionic Co2-3ethenyl structure, The ethenyl radical binds only one Co atom with the distances of 1.24 A, slightly longer than the acetylene possesing three bond Carbon structure with the length of 1.20 A, but much shorter than the ethene possessing two bond Carbon structure with the length of 1.33 A. It can be inferred that the ethnyl radical having three bond carbon is slightly affected by the Co2-3. In the anionic Co4-5ethenyl structure, the C-C bond lengths are calculated as 1.29 A and 1.30 A ,which are between the CC and CC bond distances. Therefore, The Co4-5 nanoparticles have more consequence on the CC bond of ethnyl radical those of Co2-3 nanoparticles, which might be valuable for the CC bond activation.

As a single Co atom is replaced by a Pt, the lowest energy morphology does not change (see Figure 3) where two atoms are slightly out of the plane. The Co-Pt bond lengths of anionic Co4Pt ethynyl cluster are 2.41 A and 2.39 A while the Co-Co distances are between 2.21 A and 2.61 A. The second isomer of the Co4PtC2H- is constructed by atop adsorption of C2H radical to one of the most coordinated Co atoms with an B.E of 4.76 68

eV. In the second isomer, which has capped tetrahedron structure, the magnetic moment is 10 μB with a relative energy of 0.69 eV. The third and forth isomers have similar structure with the first one in octet magnetic state.

The Co3Pt2 ethnyl cluster has biyramidal unit where Cobalt atoms with the average CoCo bond length as 2.63 A are on the base of pyramids and Platinium atoms with average Pt-Co as 2.51 are on the apexes of pyramids. A similar structure with 5 μB magnetic moment yields the second isomer but it is not energitically favorable with huge relative energy (see Figure 3). The third and forth isomers have same magnetic moment with the first one and their energies are 0.34 eV and 0.78 eV higher than the lowest isomer.

When a single Co atom is replaced by a Pt atom, the lowest energy morphology does not change. In the ground state of the Co2Pt3 ethenyl structure with a BE of 4.64 eV, ethenyl radical adsorbed on the Pt side instead of Co in sextet magnetic state that is same with the bare Co2Pt3 cluster. The Co-Co bond length of this cluster is 2.35 A , which is shorter than that of the Co3Pt2 ethenyl cluster. In all isomer except for the second one that has 0.55 eV relative energy, the C2H radical prefers to adsorbed on the atop side. The second isomer has 4 μB magnetic moment with the C-C bond length as 1.36 eV while the magnetic moment of the first isomer having Cs symmetry is 4 μB magnetic moment that has a C-C bond length of 1.24 A.

The CoPt4 ethenyl structure is distorted pyramid structure where 4 Pt atoms with an average Pt-Pt bond length of 2.85 A are on the nearly base plane and the Co atom with an average Co-Pt bond length of 2.46 A is at the apex of pyramid. In the second isomer B possessing 7 μB magnetic moment, The adsorption of the ethnyl radical adsorbed on this cluster on atop site. In this case, The terminal Carbon atom in C2H radical prefers to bind Co atom instead of one of the Pt atoms but its energy is higher (0.87 eV) than the lowest energy one. The second isomer has C4 point group symmetry as well. Two Carbon atoms of C2H in the last isomer of this structure prefers to bind Pt-Co side of the cluster on the bridge site in quintet magnetic state.

69

A

Cs 6 0.00 eV

B C1

A

7 0.0 eV

B

5 1.20 eV

D

7 0.34 eV

E 7 0.78 eV

A 8 0.0 eV

B

10 0.69 eV

D

8 0.76 eV

E 8 1.12 eV

A 9

B 9 0.51 eV

G 9 0.25 eV

ı

J 11 0.38 eV

A 5

0.00 eV

B C4

C

D 5 0.75 eV

A

0.00

B

0.0 eV

2

4 0.55 eV

E C1

eV

4

6 0.70 eV

7 0.87 eV

0.16 eV

F C1 8 0.65 eV

Cs 9 0.26 eV

5 0.0 eV

C 2

1.02 eV

Figure 3.3.3 The optimized structure of some isomers of [ ComPtnC2H ]-1 (m +n =5) 70

The B.E of Pt pentamer ethenyl nanoparticle is slightly higher than that of the CoPt4 nanoparticle (see Table 1). Among Pt2-5 ethenyl cluster, the B.E of Pt2C2H- cluster is the highest value as 5.26 eV. As depicted Table 1, the B.E value of the PtnC2H- cluster is decreasing when n varies from n=2 to 5. On the other hand, the VDE values in Table 1 indicates wawe like manner among Pt2-5 ethenyl cluster.

As coming to the Carbon bond lengths, The C-C bond length in the Pt5C2H- is calculated as 1.46 A. It can be inferred that in the case of Carbon atoms that bind the core cluster with a highly coordinated number, such as sp3 hybridization, the C-C bond length in this particle is highly affected. 3.4 Energy and Electronic Structure The binding energy generally can be defined as a measurement of the given clusters’ thermodynamic stability. Thus, to predict the relative stabilities of the ConPtm ethenyl structures, the binding energies is calculated in Table 1 and plotted in Figure 4. The binding energy (B.E ) has been obtained in the following way:

B.E =

2E[C] + nE[H ] + nE[Co] + mE[Pt] - E[ConPtmC2H - ] for 2  n+m  5 n+m+3

(1)

where E[*] is the total energy of the Carbon, Hydrogen, Platinum atoms, Cobalt atoms and the cluster, respectively. The B.E value of lowest ground state structure we have studied changes from 3.81 eV to 5.06 eV. From Table 1 and Figure 4 ,The highest B.E value belongs to the Pt2 ethenyl structure while the lowest one belongs to the structure Co5 ethenyl structure. The B.E values in Figure 4 fluctuate in a zigzag wave-like manner. It can be noted that the binding energies of the species increase with doping Pt atom to the species or removal of Co atoms from the species as the gain energy rate slow down. In other words, provided that the size of the species keep constant, this can be well understood as increasing Pt composition means more bonds involving Pt atoms, that is, the Pt-Pt or Pt-Co and Pt-C bonds, which are stronger than those of the Co-Co and Co-C. However, increasing the size of the cluster leads to decrease the B.E values of 71

the clusters gradually. This indicates that the species cannot constantly gain energy during the growth process.

6.00 5.00

e.V

4.00 3.00 2.00 1.00 0.00

Figure 3.4.1 The B.E of [ ComPtnC2H ]-1

for 2  n+m  5

To further illustrate the stability of the species and their size dependent behaviors, we have considered the second finite difference in energies that is a sensitive quantity frequently used as a measure of the relative stability of the complexes and is often compared directly with the relative abundances determined in mass spectroscopy experiments. Moreover, clusters are especially abundant magic number sizes in mass spectra as they are most stable ones. The second finite different energies (Dn) can be calculated as

Dn,m = En+1,m-1 + En-1,m+1 - 2En,m

(2)

72

where En,m is the total energy of the cluster ConPtmC2H. The second finite differences in energies of the some of the studied cluster is given in Figure 5. Due to the definition of the second finite difference in energies, we examined only 2, 3 and 4 species consisting of 3, 4 and 5 metal atoms respectively. Noticeable peaks at the size of ConPtm ( m +n = 4) indicates that the Co2Pt2C2H cluster is more stable than the neighboring clusters. The Co3Pt2 ethenyl structure can be considered as the least stable structure as it corresponds to a dip in the plot.

In order to assess the adsorption strength of Co-Pt nanoalloy to anionic C2H radical, we obtained the computed adsorption energies of anionic ethenly radical on the bimetallic ConPtm ( 2  n+m  5 ) which is shown in Figure 5. The adsorption energies are given as

Eads = E[Con Ptm ] + E[C2 H -1 ] - E[Con PtmC2 H -1 ] ( 2n+m5 )

(3)

where E[*] is the total energy of given species in the equation. The adsorption energies reveal a growing tendency with the ratio Pt/Co increasing in the studied species. This trend can be obviously seen for the species having dimers and tetramers metal atoms in Figure 5. The global peak of adsorption energy at the cluster Pt4C2H is seen in Figure 5 while the Co2C2H corresponds to a dip in the plot. These two clusters have adsorption energy values of 6.09 eV and 3.59 eV respectively as well. In addition, among Co1-5 ethenyl cluster, increasing size of the cluster leads to go up the adsorption energies sligthly.

Another sensitive quantity to reflect the relative stability is the dissociation energy D. For the anionic ConPtmC2H species, the dissociation channels ConPtmC2H-  ConxPtm-yC2H-

+ CoxPty and ConPtmC2H-  Con-xPtm-y + CoxPtyC2H- are investigated

within the range of the study and the corresponding dissociation energies are computed as respectively :

Dx,y = E[Con-x Ptm-yC2 H - ] + E[Cox Pt y ] - E[Con PtmC2 H - ]

73

()

Dx,y = E[Con-x Ptm-y ] + E[Cox Pt yC2 H - ] - E[Con PtmC2 H - ]

()

7 2nd Finite Difference 6

5

4.89

5.28

4.89

5.71 5.58

5.27

4.71

5.05

4.77

4.59

4.34

Energy (eV)

6.06

Adsorption Energy

4.59

4.45

4.27

5.00

4 3.59

3.68

3

2 1.44 1.02

1

0

0.23

0.09

0.80 0.28

0.72 0.58 0.07

Figure 3.4.2 The B.E of [ ConPtmC2H ]-1

The selected dissociation channels and the corresponding dissociation energy are given in Table 2. The most favorable dissociation channels related to the minimum dissociation energies. From our DFT result, When n +m and x+ y are odd numbers, the clusters prefer to dissociate a cobalt monomer. This is consistent with experimental results on cationic and anionic noble metal clusters [46,48]. According to this experiments, the small even numbered clusters in the systems evaporate a neutral

74

monomer. On the other hand, when n+m and x+y are even numbers, the clusters tend to dissociate a CoPt bimetal molecule.

Cluster

Co2C2H-

Present

Spin

BE per

HOMO-

VDE

Lowest and Highest

moment

atom

LUMO

(eV)

Vibrational

(B )

(eV)

Gap (eV)

4

4.66

0.33

Frequencies (cm-1) 1.25

55-3360

Work Literature

1.53

Experiment

1.50

CoPtC2H-

3

5.08

0.63

2.53

37 -3369

Pt2C2H-

2

5.26

0.28

2.66

43-3384

7

4.29

0.03

1.52

64-3368

Co3C2H-

Present Work Literature

1.59

Experiment

1.81

Co2PtC2H-

4

4.71

0.73

1.47

68-3154

CoPt2C2H-

3

4.89

0.72

2.26

56-3387

Pt3C2H-

0

5.06

1.05

1.64

41- 2968

8

4.09

0.73

1.53

171-3191

Co4C2H-

Present Work Literature

1.82

Experiment

1.63

Co3PtC2H-

7

4.41

0.71

1.70

58-3249

4

4.70

0.74

1.87

42- 3175

CoPt3C2H-

3

4.85

0.45

2.01

39-3216

Pt4C2H-

2

4.95

0.16

3.09

28-3055

9

3.94

0.51

1.96

65-3183

Co2Pt2C2H

-

Co5C2H

Present Work Literature

2.15

Experiment

1.88

Co4PtC2H-

8

4.24

0.63

2.82

35- 3195

Co3Pt2C2H-

7

4.45

0.53

2.22

16-3224

Co2Pt3C2H-

6

4.64

0.76

2.63

37 -3390

75

CoPt4C2H

5

4.76

0.33

2.44

25-3101

Pt5C2H

2

4.79

1.21

2.16

23-2975

In pure Platinum ethenly clusters, as depicted in Table 1, the dissociation of dimers is a most favorable reaction while among Co3-5 ethenyl clusters, the dissociation of monomers is generally a most favorable one. From the data of Table 2, ıt can be seen that the dissociation energy of Platinum atoms are larger than those of cobalt atoms. This is consistent with the trend of binding energies. Futhermore, the most favorable reactions are Co3C2H-  Co2C2H- + Co

and Co3Pt2C2H-  Co2PtC2H- + Co

since the reaction energy for them are the least values in the Table I. On the other hand the least favorable reactions are Co2Pt3C2H-  Co2C2H- + Pt3

and Co3Pt2C2H-  Co3C2H- + Pt2

since they both, see Table 2, have the highest reaction energies. These results are in consistent with second finite difference in total energies for m +n = 3 and 4 but not for n+m =5.

Finally, it is worth pointing out that the dissociation energy of the ConPtmC2H( 2n+m5) clusters is independent of the cluster size. This is reflected by the fact that similar dissociation energies are seen within the size range studied without a noticeable trend of growing or lessening with altering the cluster size. This is also consistent with binding energy results we found.

The highest occupied (HOMO) indicates the ability to give an electron, the lowest occupied molecular orbital energies (LUMO) as an electron acceptor. The energy gap between HOMO and LUMO reveals the ability of electrons to bounce from HOMO to

76

LUMO and determines a molecule to involve in chemical reactions to some degree, that is, a large HOMO-LUMO gap has been considered as significant requirement for chemical stability.

n,m , x and y

Dissociation Channel

D (eV)

n=3, m= 0, x=1 and y = 0

Co2C2H-

2.44

n=2, m= 1, x=0 and y = 1

Co2C2H- + Pt

4.97

n=1, m=2, x=0 and y= 1

CoPtC2H- + Pt

3.96

n=0, m= 3 , x=0 and y = 1

Pt2C2H-

4.07

n=4, m= 0 , x=2 and y = 0

Co2C2H- + Co2

2.68

n=4, m= 0 , x=1 and y = 0

Co3C2H- + Co

2.88

n=3, m= 1 , x=0 and y = 1

Co3C2H- + Pt

5.14

n=3, m= 1 , x=1 and y = 0

Co2PtC2H- + Co

2.61

n=2, m= 2 , x=1 and y = 1

CoPtC2H- + CoPt

3.53

n=2, m= 2 , x=1 and y = 1

Pt2C2H-

3.99

n=2, m= 2 , x=0 and y = 2

Co2C2H- + Pt2

5.29

n=0, m= 4 , x=0 and y = 1

Pt3C2H-

+ Pt

4.29

n=0, m= 4 , x=0 and y = 2

Pt2C2H-

+ Pt2

4.05

n=5, m= 0 , x=1 and y = 0

Co4C2H- + Co

2.90

n=5, m= 0 , x=2 and y = 0

Co3C2H- + Co2

3.14

n=5, m= 0 , x=3 and y = 0

Co2C2H- + Co3

3.24

n=4, m= 1 , x=1 and y = 0

Co3PtC2H- + Co

3.06

n=4, m= 1 , x=2 and y = 0

Co2PtC2H- + Co2

3.03

n=4, m=1, x=1 and y= 1

Co3C2H-

4.22

n=4, m= 1 , x=3 and y = 0

CoPtC2H- + Co3

3.56

n=3, m= 2 , x=1 and y = 0

Co2Pt2C2H- + Co

2.65

n=3, m= 2 , x= 2 and y = 0

CoPt2C2H- + Co2

3.56

n=3, m= 2 , x=1 and y = 1

Co2PtC2H- + CoPt

3.31

n=3, m= 2 , x= 0 and y = 2

Co3C2H-

+ Pt2

5.50

n=2, m= 3 , x=0 and y = 3

Co2C2H-

+ Pt3

5.81

77

+ Co

+ Pt

+ Co2

+ CoPt

n=0, m= 5 , x=0 and y = 4

Pt4C2H- + Pt

3.67

n=0, m= 5 , x=0 and y = 2

Pt3C2H-

+ Pt2

3.65

n=0, m= 5 , x=0 and y = 3

Pt2C2H-

+ Pt3

4.04

The calculated HOMO-LUMO gap

(HLG) and lowest and highest vibrational

frequencies are presented in the Table 1. We have also given HLG and the Vertical Detachment Energies of the ConPtmC2H- species in the Figure 5.

3.50 3.00

HOMO -LUMO GAP VDE

Energy (eV)

2.50 2.00 1.50 1.00 0.50 0.00

Figure 3.4.2 The B.E of [ ConPtmC2H ]-1

Co3C2H- has the lowest HOMO-LUMO gap with the value 0.03 eV in the given study. It was found that the HOMO-LUMO gap energy of pure Co3 is 1.04 eV (sebetci). With a comparison to the Sebetci study which is used same xc functional and basis functions with this study (See referance), ıt can be pointed out that the adsorption of the C2H radical tends to decrease the energy gap among Pt2-5C2H- , indicating that the species

78

become more conductive while among Co1-5C2H, it tends to increase the energy gap, indicating that the species become more stable. According to the Table 1, The Pt3C2Hand Pt5C2H-cluster possess high stability due to its large HOMO-LUMO gaps ( 1.05 eV and 1.21 eV respectively) when comparing with other clusters. This is consistent with the B.E calculation. In addition, As depicted in Figure, There is no obivious trend in terms of size dependency and the Pt/Co composition effect.

VDE is expressed as the energy difference between neutral clusters at optimized anion geometry clusters and optimized anion cluster. The VDE results we have found for Co25C2H- species as well as the corresponding experimental and theoretical results are recorded in Table 1. As seen in Table 1, some of the results can be compared with available experimental results in the literature []. These results are good agreement with experimental values. According to the Figure, The peak is seen at the anionic cluster of Pt4C2H with a value of 3.09 eV while the dip is seen at the anionic cluster of Co2C2H ethenyl cluster. There is no obvious trend except for m + n = 4 that the VDE values goes up by increasing the ratio of Pt/Co.

The stability of the species can be invoked with vibrational frequencies. The highest and lowest vibrational frequencies are given in Table 1. Locally stable isomers are distinguished from transition states by vibrational frequency analysis. In addition, our frequency results may capable of comparison with future experiments.

3.3.5 Magnetic Properties

The total magnetic moment of lowest lying structures of ConPtm and ConPtmC2Hspecies within the range study is given in Figure. For the ethenyl clusters, the total magnetic moment increases with doping of Co atoms in the species. From Figure, the magnetic moments of lowest energy structure of the studied cluster are close to those of bare ConPtm clusters. That indicates that the magnetic moments of bare clusters we mentioned are slightly affected upon adsorption of ethenly radical. Contrary, for bare Co clusters, according to the Knickelbein, the electronic structures of the cobalt species are 79

affected importantly upon chemiadsorption of benzene molecules, which leads to quench their magnetic moments []. This effect may originate from the interaction between the  electrons of benzene molecules and cobalt d orbitals is very strong due to the formation of the cobalt-benzene species as sandwich or rice-ball structures whereas ethenyl radical react with only one or two cobalt atoms through its terminal carbon atom.

12

Total Magnetic Moment (μB)

10

Bare Clusters Ethynly Clusters

8 6 4 2 0

Figure 3.4.2 Total magnetic moments of the ground state, bare and ethenly bimetallic clusters ( 2n+m5 )

4 Conclusion

80

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