Apéndice I A.1.- El Modelo de Bandas para Metales de ...tesis.uson.mx/digital/tesis/docs/20581/Apendice.pdf · y la densidad de estados i n d son representados por la función de

Apéndice

94

Apéndice I

A.1.- El Modelo de Bandas para Metales de Transición

En el modelo de bandas las propiedades de cohesión de los metales de transición

y la densidad de estados idn son representados por la función de paso rectangular de

anchura W y altura 10 W , donde el factor 10 es asociado al número máximo de

electrones de la banda d . La energía de Fermi FE para metales con Z electrones en la

capa d está relacionada a la anchura de la banda W por la relación FE =W(Z-5) 10 . La

energía para el caso Z=0 está en el centro de la banda y coincide en este modelo con la

posición de los niveles atómicos d . La energía de cohesión mV debida a los electrones d

es dada por26

:

m W

V = Z(10-Z)20

. A.1

La tendencia parabólica en la energía de cohesión en la serie de metales de

transición predicha por el modelo es generalmente observada experimentalmente87

y la

anchura de la banda puede ser calculada utilizando la aproximación del segundo

momento para la densidad de estados a través de la relación87

e

e

p

p n

n

μ = ε , A.2

donde pμ

es el p-ésimo momento y es función de la distancia de separación entre

electrones, en n dε =E -E

es la energía de eigenvalor nE con respecto al nivel de energía

dE de los electrones de la capa d, en corre sobre todos los N estados electrónicos

ocupados o desocupados por electrones. Siendo H la matriz hamiltoniana diagonal con

respecto a la base de eigen funciones n

Ψ se puede escribir87

Apéndice

95

e e en m n n mH =ε δ . A.3

La energía de enlace de la capa d depende de la magnitud de las integrales de

enlace ddζ , ddπ y ddδ que determina la densidad de estados TB. Se puede relacionar la

anchura de la banda con la integral de enlace considerando el segundo momento de la

densidad de estados asociada al átomo i pudiendo escribir87

ii 2

2 d iα,jβ jβ,iα-

j α,β

μ = ε n ε dε= H H

, A.4

donde dε=E-E y idn es la densidad de estados de la banda d dada por TB. Realizando

los saltos del átomo i al átomo j, la matriz iα,jβH es diagonal con elementos ddζ , ddπ ,

ddπ , ddδ y ddδ . Si el eje z es colocado a lo largo de ijr se obtiene para la red con

coordinación zc la expresión26, 87

2

i 2

2 c

Wμ = =z 5h

12, A.5

donde 2 2 2 2h =1 5 ddζ +2ddπ +2ddδ . Considerando solamente interacciones a primeros

vecinos se puede escribir i

2μ como26

:

i 2

2 ij

i j

μ = ε r

, A.6

donde sustituyendo la ec. A.6 en la ec. A.5, despejando W de la ec. A.5 y sustituyendo

en la ec. A.1 tenemos26

1 2

m 2

ij ij

i j

3V r = Z 10-Z ε r

10

. A.7

Escribiendo αβαβ 0ξ = 3 10 Z 10-Z ε , donde 0ε es el valor de ijε r en 0r , siendo

0r la separación interatómica de bulto se puede escribir

Apéndice

96

αβ

1 2

αβm 2

ij ij

i j0

ξV r = ε r

ε

. A.8

Citando a Friedel y Ducastelle (88) (89) (90) se puede asumir que la integral de

trasferencia varía exponencialmente en las vecindades de 0r26

αβ ij 0

αβ

-κ r -r

ij 0ε r =ε e . A.9

Multiplicando y dividiendo en el factor exponencial por 0 0r r y haciendo

αβ 0 αβκ r =q tenemos

ij

αβ0

αβ

r-q -1

r

ij 0ε r =ε e

. A.10

Sustituyendo la ec. A.10 en la ec. A.8 y haciendo 2

αβ αβξ =ζ se obtiene

ij

αβ0

1 2r

-2q -1rm

ij αβ

i j

V r = ζ e

, A.11

donde ijr representa la distancia entre los átomos i y j , 0r es la distancia de primeros

vecinos en la red αβ , αβξ es una integral de traslape y

αβq describe la dependencia de la

distancia interatómica relativa26

.

La ec. A.11 proporciona sólo la contribución de la energía de cohesión de la

banda d aumentando conforme la separación decrece. Por otro lado la hibridación s - d es

más difícil de tratar. Sin embargo citando a Gelatt, Ehrenreich y Watson (GEW)91

que

asumían que la hibridación s - d tiene cualitativamente el mismo comportamiento que la

contribución de la banda d , los elementos de la matriz de hibridación incrementan con la

anchura de la banda d y la contribución de la hibridación a la energía de cohesión es casi

la mitad de la contribución de la banda d , excepto que ésta no se desvanece para la banda

Apéndice

97

d llena como se puede ver en la ec. 1 para Z=10 . La información detallada del

comportamiento de la contribución s - d a la energía de cohesión como una función de

la distancia interatómica no es fácil de obtener por eso se asumirá que su contribución

puede estar incluida en la ec. A.11 vía una adecuada modificación de los parámetros. La

interacción atractiva entre la capa d en sitios vecinos resulta en una reducción de la

separación interatómica y por lo tanto en una compresión del gas de electrones libres. La

estabilidad de la red requiere que se incluya una contribución repulsiva de corto

alcance26

.

La contribución repulsiva a la energía de cohesión en metales de transición surge

de los electrones s y es descrita por el término rV . En la vecindad de la separación

interatómica de equilibrio se puede asumir que esta varíe exponencialmente de la forma

ijαβ

0

r-p -1

re

ya que rV es una integral interatómica de traslape y es de esperarse que varíe

aproximadamente en la forma dada26

. Así, el potencial repulsivo es:

ijαβ

0

r-p -1

rr

αβ

i j

V = A e

, A.12

donde A depende la integral de salto. Se asume normalmente que el potencial es aditivo

por pares, descrito por una suma de repulsión ion-ion de Born-Mayer originando el

incremento en la energía cinética la conducción de electrones constreñidos dentro de dos

iones aproximados; así, el parámetro αβp depende solo de las especies interactuantes y

está relacionado a la compresibilidad del metal en bulto30

. La ec. A.12 debería contener la

interacción electrostática más los términos de correlación e intercambio que, de hecho,

pueden ser aproximadamente representados por formas aditivas a pares. Sin embargo se

ha sugerido que el término de Born-Mayer pueda acumular la suma de los términos del

Apéndice

98

campo cristalino que son sustraídos en la construcción de la integral de traslape. De este

modo la energía potencial del sistema puede ser expresada en la forma30

:

N

r m

i=1

V= V -V . A.13

La forma analítica del potencial de Gupta mostrado en la ec. A.13 fue obtenido

para modelar materiales cristalinos que mostraban contracción entre planos vecinos en

superficies metálicas. Sin embargo, debido a que fue diseñado para modelar las

interacciones interatómicas entre los átomos metálicos, su rango de aplicación se ha

extendido satisfactoriamente como una buena función semiempírica de energía potencial

en la descripción de los enlaces interatómicos en metales nobles y de transición. En lo

sucesivo se mencionarán sólo algunas de las técnicas de simulación computacional en las

que se implementará el potencial de Gupta con los parámetros de Massen et al.76

para

modelar las interacciones interatómicas entre los átomos metálicos.

Apéndice II

A.2.- Funciones de Distribución de Átomos

Desde el punto de vista teórico, la cantidad fundamental para la adecuada

descripción de muchos sistemas químicos y su dinámica es la energía como función de

los grados de libertad iónicos y electrónicos23

. Los grados de libertad electrónicos

contribuyen a la energía con un valor fijo (0)

elecE (r) , además se asume que la energía de

vibración del punto cero no contribuye a la energía total del sistema, así la cantidad de

interés es la energía como función de los grados de libertad iónicos que pueden ser

Apéndice

99

tratados esencialmente como partículas clásicas asociando a cada estado una energía que

consiste de la suma de dos términos: la energía cinética KE y potencial VE 35,43

.

Siendo N

1 NR = r ,...,r y N

1 NP = p ,...,p las posiciones y los momentos de las

partículas en el ensamble canónico, escribimos el hamiltoniano del sistema como74

:

N N

N N N N

N K VH (P ,R )=E (P )+E (R ) , A.14

donde N

N

KE (P ) es la energía cinética y N

N

VE (R ) es la energía potencial debida a la

interacción entre partículas. Analizando al sistema desde el marco teórico del formalismo

del ensamble canónico, se puede encontrar la densidad de probabilidad de equilibrio de

un sistema de N partículas esféricas idénticas74

:

N N

NN N N -3N

0

N

exp -βH (P ,R )1f (P ,R )= h

N! Z (V,T)

, A.15

donde h es la constante de Planck, N! es el factor asociado a la indistinguibilidad de las

partículas y el factor de normalización NZ (V,T) es la función de partición del ensamble

canónico

N N N N

N NZ (V,T)= ... exp -βH (P ,R ) dP dR . A.16

Para propósitos prácticos la descripción completa de la densidad de probabilidad

proporciona información innecesariamente detallada. Si se está solamente interesado en

el comportamiento de un subconjunto de n partículas, se puede eliminar la información

irrelevante integrando sobre las coordenadas y momentos restantes de las N-n partículas.

Lo anterior conduce a definir funciones de distribución reducidas en el espacio fase

n n n

0f (P ,R ) como74

:

Apéndice

100

n n n (N) N N (N-n) (N-n)

0 0

N!f (P ,R )= … f (P ,R )dP dR

(N-n)! , A.17

donde se ha utilizado la notación (N-n)

n+1 NdP = p ,...,p y (N-n)

n+1 NdR = r ,...,r . El factor

N!/(N-n)! nos proporciona las diferentes combinaciones que se pueden hacer al tomar n

partículas de N . Para un sistema en equilibrio, la integración de las funciones de

distribución reducidas sobre los momentos restantes conduce a las funciones de densidad

de equilibrio de partículas (n) nρ (R ) 74

. Estas funciones son tales que podemos escribir

(n) n nρ (R )dR , que es N!/(N-n)!, la probabilidad de encontrar n partículas en el elemento

de volumen mdR , independientemente de las posiciones de las partículas restantes y de

los momentos. Las funciones de densidad de partículas y las funciones de distribución de

partículas dan una completa y compacta descripción de la estructura de sólidos, líquidos

y, en nuestro caso, cúmulos.

Dada la forma funcional de n

0f , se puede escribir la función de densidad de n

partículas como74

:

N N N (N-n)

N(n) n

N

N

… exp -βH (P ,R ) dP dRN!ρ (R )=

(N-n)! Z (V,T)

. A.18

Integrando los términos dependientes de NP en N N

NH P ,R y NZ V,T

obtenemos

N

N (N-n)

V(n) n

N

N

… exp -βE (R ) dRN!ρ (R )=

(N-n)! Q (V,T)

, A.19

donde definimos la integral de configuración NQ (V,T) como

( , ) exp ( )N

N N

N VQ V T E R dR . A.20

Apéndice

101

La función de distribución radial de partículas o átomos (1)

N 1n(r)=ρ (r ) es la más

sencilla, mas no por eso carece de importancia pues nos proporciona una firma

característica del cúmulo en cuanto al ordenamiento espacial mostrado por los átomos y

su integral en el espacio es proporcional al número de partículas o átomos en el cúmulo74

(1)

N 1 1ρ (r )dr =N . A.21

Por otro lado, la función de distribución de n partículas (n) N

Ng (R ) es definida en

términos de la correspondiente densidad de partículas como:

n

(n) N (n) (i)

N N 1 n N i

i=1

g (R )=ρ (r ,...,r ) ρ (r ) . A.22

La función de distribución mide el exceso de lo que se desvía la estructura de la

fase sólida, liquida o gaseosa siendo la naturaleza de las interacciones aproximadas por

potenciales dependientes únicamente de la distancia de separación átomica 12 2 1r = r -r .

Apéndice III

A.3.- Dispersión de Electrones y Rayos X

A.3.1.- Dispersión de Electrones por Átomos

Para la adecuada descripción de estos fenómenos es necesaria una expresión

analítica que describa los efectos de interferencia entre los electrones dispersados por

varios átomos en una molécula. Para el caso de dispersión elástica es necesario analizar el

efecto de dispersión de un solo átomo y subsecuentemente grupos de átomos pudiendo

así hacer la descripción de los patrones de dispersión de moléculas o cúmulos

considerando su posición relativa fija72

. El sistema bajo estudio consiste de un haz de

Apéndice

102

electrones o rayos X viajando libremente a velocidad uniforme en una dirección dada en

el espacio. El haz es dispersado por un átomo que es representado por una pequeña región

dentro de un campo de fuerza central, y esta distribución del haz dispersado es observado

a gran distancia del átomo. Si r es la distancia del punto de observación del átomo, θ es

el ángulo de la dirección del haz dispersado con la dirección del haz incidente y N es el

número de electrones en el haz incidente cruzando por unidad de área por unidad de

tiempo, así el número de electrones cayendo sobre una pequeña área dS por unidad de

tiempo es dado por72

:

2

dSNI(θ) NI(θ)dω

r

, A.23

donde I(θ) es la función a ser calculada y dω es el diferencial de ángulo solido. El

problema es tratado empleando la ecuación de Schrödinger escrita de la forma:

2

2

V2

8π mΨ x,y,z + W-E x,y,z Ψ x,y,z =0

h . A.24

La solución de para el caso estacionario es una función de las coordenadas x ,

y y z con el origen ubicado en el centro del átomo y 2

puede representar la

distribución de electrones en el haz incidente y dispersado. W es la energía cinética de

los electrones en el haz y permanece constante antes y después de la colisión, y es

determinada por el potencial de aceleración utilizado para producir el haz. VE es la

energía potencial de las interacciones de los electrones con la partícula cargada

Apéndice

103

Fig.A1.- Coordenadas en la dispersión atómica72

.

compuesta de átomos y que se asume es una función de r, es decir que el átomo es tratado

como una partícula esféricamente simétrica72

.

La solución de la ecuación A.24 para r grandes debe ser de la forma

ik×r

ik×z eΨ r ~e + f θ

r

, A.25

el primer término representa la onda plana incidente viajando a través del eje z y es una

solución del electrón libre

2 2Ψ r +k Ψ r =0 . A.26

La constante 2k es función de W y está relacionada con la longitud de onda por

Apéndice

104

2 2

2

2 2

8π m 4πk = =

h λ. A.27

Expresando la energía cinética en términos del momento se obtiene la relación de

de Broglie λ=h/p . El segundo término en A.25 es asociado a la onda dispersada de la

siguiente manera72

:

2

I θ = f θ. A.28

Escribiendo la ec. A.24 en la forma

2 2 2 2

VΨ r +k Ψ r = 8π m h E Ψ r , A.29

entonces la solución más general puede ser de la forma72

2

0 V2

exp ik× r-r1 8π mΨ r =Ψ r - E r Ψ r dη

4π r-r h

, A.30

siendo 0Ψ r la solución general de la ec. A.26. El vector r se extiende al elemento de

volumen dη que contiene el material dispersor. Se puede decir que la integral representa

la amplitud y fase en un punto r de la onda dispersada en el punto dη y de amplitud

2 2

V8π m h E r Ψ r dη . La integral es la suma de todas las ondas producidas por

toda la materia dispersante. Las coordenadas primadas se refieren al átomo y las no

primadas a la onda dispersada. La fig. A1 muestra la relación del sistema coordenado72

.

El primer término en la ec. A.30 representa la onda incidente. Lo anterior es

compatible con la definición de 0Ψ r y escogiendo una onda plana infinita moviéndose

en dirección del eje z tal que:

ik×z

0Ψ z =e . A.31

Apéndice

105

La condición sobre la cual la ec. A.30 es una solución aceptable es que la integral

tenga la forma asintótica propia como es mostrada por el segundo término de la ec. A.25.

Considerando que la distancia del átomo al punto de observación es muy grande, r»r , lo

que permite escribir r-r ~r- r r ×r y la integral toma la forma72

:

ikr

V2

2πm e- exp ik× r r ×r E r Ψ r dη

h r . A.32

De acuerdo con la forma asintótica de la ec. A.30 para grandes r toma la forma

de la ec. A.25 con:

V2

2πmf θ = exp ik× r r ×r E r Ψ r dη

h , A.33

la integral es sólo función de θ ya que la coordenada primada aparece en la forma r r ,

un vector unitario a un ángulo θ en la dirección del haz incidente. Para evaluar la ec.

A.33 se requiere de la aproximación de Born, Ψ r es compuesta de dos partes, 0Ψ r

y la integral correspondiente a la integral en la ec. A.30. La aproximación consiste en

despreciar la expresión de la integral de donde se obtiene72

:

ikz

0Ψ r -Ψ r ~e , A.34

lo que es equivalente a decir que dentro del átomo la amplitud de la onda incidente es

mucho mayor que la onda dispersada, o que la onda dispersada por una parte del átomo

puede no ser dispersada de nuevo por otra parte del átomo. Además no hay cambio de

fase durante el proceso de dispersión. Por otro lado si se escoge 0z = n ×r donde 0n es

el vector unitario a lo largo del eje z y n representa el vector unitario r r , la sustitución

de la ec. A.34 en la ec. A.33 da72

:

Apéndice

106

0 V2

2πmf θ = exp ik× n -n ×r E r dη

h . A.35

Para realizar la integración se debe rescribir la exponencial ya que θ es el ángulo

entre 0n y n , el valor absoluto de 0n -n es igual a 2senθ 2 lo que permite escribir:

0k× n -n ×r =2ksen θ 2 r cosα , A.36

donde α es el ángulo entre r y el vector 0n -n . Tomando este vector como el eje polar

y dejando las primas se integra la ec. A.35 para obtener72

:

2π π i2ksen θ 2 rcos α 2

V2 0 0 0

2πmf θ = dβ senαdα e E r r dr

h

,

2

2

V2 0

sen sr8π mf θ = E r r dr

h sr

, A.37

en el que s=2ksen θ 2 =4πsen θ 2 λ . Ya que VE r es el potencial debido al campo

atómico, este puede ser expresado en términos de la densidad de carga del átomo72

22

2

V

Φ rZεE r =- +ε dη

r r-r

, A.38

el primer término es debido al núcleo con carga Zε y el segundo es debido a la

distribución de electrones en el átomo; Φ r es la solución de la ecuación de

Schrödinger para un átomo con una distribución de carga esférica72

.

La ec. A.38 puede sustituirse en la ec. A.37 y el resultado se obtiene integrando

por partes:

2 2

2 2

2 2 0

sen sr8π mε 1f θ = Z-4π Φ r r dr

h s sr

, A.39

Apéndice

107

Fig. A2.- Sistema coordenado que ilustra la dispersión de rayos X y electrones por moléculas. Mitad

superior del cúmulo icosaédrico de Pd12Pt1.

2 2

2 2

Z-F θ8π mεf θ =

h s,

donde

2 2

0

sen srF θ =4π Φ r r dr

sr

. A.40

La solución completa al problema se puede escribir de la forma72

2 2 ikr

ikz

2 2

Z-F θ8π mε eΨ r =e +

h r s

, A.41

F θ tiene la misma expresión para el factor de dispersión atómica de rayos X. Nótese

que F θ tiende a cero cuando incrementa s , al igual que para longitudes de onda

pequeñas72

.

Apéndice

108

A.3.2.- Dispersión de Electrones por Moléculas

La difracción por moléculas en un gas es observada bajo condiciones

experimentales tal que el haz incidente cae sobre un gran número de moléculas que tienen

orientación completamente aleatoria unas con respecto a otras. La concentración del gas

de moléculas es suficientemente pequeña para que cada molécula disperse

independientemente de las demás. La dispersión es tratada calculando los efectos para

una sola molécula en orientación fija y promediando el resultado sobre todas las posibles

orientaciones. El movimiento de las moléculas durante el proceso de dispersión puede ser

despreciado72

.

Aplicando lo anterior al caso de cúmulos de dimensiones manométricas en estado

gaseoso podemos tratar el problema de cada cúmulo de m átomos, cada uno de los

cuales dispersa electrones de acuerdo con la ec. A.41. Lo anterior requiere la suposición

de que el potencial VE r es independiente para cada átomo sin mostrar contribuciones

entre ellos y se mantienen esféricamente simétricos72

.

De acuerdo con la ec. A.41, la amplitud y fase de la onda dispersada es dada por

ikre r f θ . Tomando el origen del sistema coordenado en el centro de la molécula

como se muestra en la fig. A2, que en general puede no estar en el centro de un átomo, y

denotando la coordenada del i-ésimo átomo con el subíndice i , entonces la onda

dispersada por este átomo es72

ii ikz

i i

i

exp ik r-rΨ r = e f θ

r-r

, A.42

donde r es el vector del punto de observación y ir es el centro del i-ésimo átomo. Ya que

la fase de la onda dispersada depende solamente de la fase de la onda incidente, es

Apéndice

109

requerido el factor iikze y utilizando de nueva cuenta que r»r y la aproximación

i ir-r ~r-n×r se escribe iΨ r de la forma72

:

ikr

i i i i iΨ r = e r exp ik z -n×r f θ, A.43

ikr

i i 0 i iΨ r = e r exp ik n -n ×r f θ.

La suma de las ondas dispersadas debidas a los átomos puede representar la onda

dispersada por la molécula:

ikr

i i 0 i i

i i

Ψ r = Ψ r = e r exp ik n -n ×r f θ . A.44

Existen relaciones de fase aleatorias entre las ondas dispersadas por diferentes

moléculas de manera que tomando el promedio sobre todas las posibles orientaciones y

obteniendo el cuadrado de la amplitud por una sola molécula se obtiene la expresión para

Ψ r como72

:

2

*

0 i i2i

1I=Ψ r Ψ r = exp ik n -n ×r f θ

r

,

A.45

i j 0 ij2i j

1I= f f exp ik n -n ×r

r ,

donde ij i jr =r -r es la separación entre el i-ésimo y j-ésimo átomo en el cúmulo. La

orientación del cúmulo puede ser designada por solo uno de los vectores ijr y el promedio

es obtenido integrando cada uno de los miembros de la doble suma sobre las variables

angulares referidas al vector 0n -n como el eje polar72

2π π

i j ij ij ij ij2 0 0i j

1 1I θ = f f dβ exp isr cosα dα

r 4π , A.46

Apéndice

110

ij

i j2i j ij

sen sr1I θ = f f

r sr , A.47

if es dada por la ec. A39 asociada a la onda dispersada por el átomo en el que ésta incide.

Esta fórmula con excepción del factor if fue derivada por Debye e independientemente

por Ehrenfest en 1915 para dispersión de rayos X por moléculas. De manera indistinta

podemos utilizarla para el cálculo del patrón de dispersión de rayos X por cúmulos

bimétalicos72

.

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Physics of Metals.

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Copyright © 2010 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol. 7, 1–6, 2010

Energetic and Structural Analysis of 102-Atom Pd–PtNanoparticles: A Composition-Dependent Study

Rafael Pacheco-Contreras1, Alvaro Arteaga-Guerrero1, Dora J. Borbón-González2,Alvaro Posada-Amarillas3�∗, J. Christian Schön4, and Roy L. Johnston5

1Programa de Posgrado en Ciencias (Física), Departamento de Investigación en Física, División de Ciencias Exactas y Naturales,Universidad de Sonora, 83190 Hermosillo, Sonora, México

2Departamento de Matemáticas, Universidad de Sonora, Hermosillo, Sonora, México3Departamento de Investigación en Física, Universidad de Sonora, Apdo. Postal 5-088, 83190 Hermosillo, Sonora, México

4Max Planck Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany5School of Chemistry, University of Birmingham, Edgbaston, B15 2TT, Birmingham, UK

We present an extensive study of the structural and energetic changes of 102-atom PdmPt102−m

nanoparticles as a function of composition m, where the interatomic interactions are modeled withthe many-body Gupta potential. The minimum energy structures are obtained through a geneticalgorithm. The excess energy is calculated, as well as the radial distribution of atoms, n�r �, whichis computed for each composition. The results indicate a multi-layer segregation for some composi-tions, with a shell growth sequence as follows: a core with a small number of Pd atoms is followedby an intermediate shell of Pt atoms and the external shell consists of Pd atoms. A region where Pdand Pt atoms are mixed is observed between the outermost and intermediate shells. Furthermore,the pure Pd102 and Pt102 nanoparticles have the capped Marks symmetry, while several differentlowest energy structures are observed for the bimetallic clusters in the range of compositions stud-ied here.

Keywords: Bimetallic Clusters, Structural Properties, Genetic Algorithms.

1. INTRODUCTION

Nanoalloys have recently become the subject of intenseexperimental and theoretical studies1 motivated mainlyby the possibility of creating new materials that willdisplay enhanced physical or chemical properties, withpotential applications in medicine, biology, catalysis andelectronics. The possibility to extend the range of thenanoparticle properties—compared to those of the purecomponents—by mixing two or more chemical elementsto form heteroatomic nanoparticles, constitutes one of themain reasons which have increased scientific interest onthis new type of nanomaterials.2–5 In particular, transitionand noble metal nanoparticles have received much atten-tion because of the possible applications in fields such asoptics,6 catalysis7 and hydrogen storage.8 The ability tocontrol size, shape and chemical ordering in the synthe-sis process, drives experimental work towards fabricatingtailor-made materials designed for specific purposes suchas biomedical applications.9

∗Author to whom correspondence should be addressed.

Many chemical and physical methods have been usedin the preparation of bimetallic nanomaterials such asco-reduction, successive reduction of two metal salts,colloidal solutions and vapor deposition.10�11 Successivereduction is an important method as it is currently usedto prepare core–shell bimetallic nanoparticles, such atomicordering (i.e., “chemical ordering”) is an essential char-acteristic of catalyst nanoparticles (along with a highsurface/volume ratio) as plenty of the chemical reac-tions take place at nanoparticle surface sites.12 In par-ticular, considerable efforts have been devoted to thepreparation of Pd/Pt nanoparticles, mainly due to recentexperimental studies indicating high catalytic activity, long-term stability and good quality as catalysts for severalmajor chemical reactions such as hydro-dechlorination ofdichlorodifluromethane and combustion of methane.13 Fur-thermore, the use of Pd/Pt nanoalloys in catalytic convertersfor the reduction of CO, NO, and aromatic hydrocarbonsfor pollution control in exhaust gases14 is of great impor-tance. These nanoalloys have been demonstrated to workbetter as catalysts in the reactions mentioned above thanmonometallic catalysts (e.g., Pd).15

J. Comput. Theor. Nanosci. 2010, Vol. 7, No. 1 1546-1955/2010/7/001/006 doi:10.1166/jctn.2010.1345 1

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Energetic and Structural Analysis of 102-Atom Pd–Pt Nanoparticles: A Composition-Dependent Study Pacheco-Contreras et al.

Experimental studies of binary transition metal (TM)nanoparticles have shown fascinating morphological char-acteristics. They exhibit a variety of structural motifs,ranging from non-crystalline icosahedral or decahedralpacking, to motifs typical of crystalline structures (e.g., fccand hcp dense packings).16 Exhaustive experimental stud-ies on Pd/Pt nanoalloys have been carried out in order toreveal their intrinsic structural details. Through a combi-nation of experimental techniques, it has been elucidatedthat 1–5 nm diameter Pd/Pt nanoparticles are truncatedoctahedra (TO) with a Pt-rich core enclosed by a Pd-richshell.14�17 This type of segregation was also found byBazin et al., by combining X-ray absorption spectroscopywith TEM and volumetric H2–O2 titration to characterizesmall Pd/Pt nanoparticles deposited on �-alumina. Theirproduction method provided “cherry-like” nanoparticleswith Pd-atoms segregated to the surface.14

On the theoretical side, structural characterization ofnanoparticles has been studied by both first principlesmethods and semiempirical potentials. The former has thedisadvantage of requiring large computational resources,and it has been only feasible to study nanoalloys in therange of 2 to 40 atoms using these approaches combinedwith optimization strategies that employ semiempiricalpotentials.18 The latter approach requires phenomenolog-ical potentials to describe the interatomic interactions, inparticular the many-body nature of metal atom interac-tions, for the structural analysis of nanoalloys. One of themain reasons for using empirical potentials (EP) to modelinteratomic interactions during global explorations of theenergy landscape of clusters, is the ability to study largecluster sizes, since empirical potentials are not as compu-tational expensive as first principles approaches. SeveralEPs can be found in the literature, such as the embed-ded atom method of Foiles, Baskes and Daw,19 EPs pro-posed by Voter and Chen,20 Sutton and Chen,21 Murrelland Mottram,22 and the one used in our study, the Guptapotential.23

In our study, we perform an exhaustive analysis ofthe structures and energies of large Pd–Pt 102-atom sizeclusters, considering all possible compositions. We usea genetic algorithm (GA), which incorporates the Guptapotential as formulated by Cleri and Rosato, to explorethe potential energy surface (PES) of 102-atom Pd/Ptnanoalloys.24�25 The cluster size N = 102 is especiallyinteresting because it has been determined as a verystable structure for monatomic bare or ligand-protectedclusters.26�27 Performing exhaustive GA searches will pro-vide more reliable candidates for global minima (GM) thatcan serve as initial structures for further high level firstprinciples calculations. The unbiased excess energy28 isused to find the most stable clusters and the structural evo-lution of atomic ordering as a function of the number of Pdatoms. We also analyze radial distributions of the atoms,n�r�, for selected compositions. We perform a comparison

r0 1 2 3 4 5 6 7

n (r

)

0

2

4

6

8

10

12

56 Pd atoms42 Pt atoms

Fig. 1. Radial distribution of atoms, n�r�, of the Leary tetrahedronobtained previously for the Pd56Pt42 98-atom cluster. Radial distance, r , isin angstroms.

between different structures found across the compositionrange, as well as a comparison with the Leary Tetrahedron(LT) structure found for 98-atom Pd–Pt nanoparticles.5�29

It is worth mentioning that the LT structure resemblesthe “cherry-like” symmetry structures found previously byBazin et al.14 in a series of experiments on Pd/Pt nanoal-loys (sizes around ∼1–10 nm).

2. COMPUTATIONAL METHODOLOGY

In this work we have used the Gupta potential with theparameterization of Massen et al.30 to model the inter-atomic interaction between metal atoms. It is based onthe tight binding method and employs the second momentapproximation in order to obtain an analytic expression interms of atomic coordinates:

Vclus =N∑

i

�V r�i�−V m�i�� (1)

In Eq. (1), the repulsive part is given by:

V r�i�=N∑

j �=i

A�a� b� exp(

−p�a�b�

(rij

r0�a� b�−1

))

(2)

Table I. Gupta potential parameters used in this study.

Parameter Pt–Pt Pd—Pd Pt–Pd

A/eV 0�2975 0�1746 0�23� /eV 2�695 1�718 2�2p 10�612 10�867 10�74q 4�004 3�742 3�87r0/Å 2�7747 2�7485 2�76

Source: Reprinted with permission from [30], C. Massen et al., J. Chem. Soc.,Dalton Trans. 4375 (2002). © 2002.

2 J. Comput. Theor. Nanosci. 7, 1–6, 2010

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Pacheco-Contreras et al. Energetic and Structural Analysis of 102-Atom Pd–Pt Nanoparticles: A Composition-Dependent Study

Fig. 2. Excess energy (Gupta102 � plot for the 102-atom clusters. White cir-

cles refer to GA optimized configurations and black triangles are assignedto the homotops with lower excess energy. The composition of the lowestenergy cluster is Pd58Pt44. Figures on the extremes are Pt102 and Pd102capped Marks decahedra.

and the attractive many-body term is:

V m�i�=[

N∑

j �=i

�2�a� b� exp(

−2q�a�b�(

rij

r0�a� b�−1

))]

(3)

In the expressions (2) and (3), a and b are the atomlabels for atoms i and j , respectively, rij is the distancebetween atoms i and j and the set of parameters A�a�b�,r0�a� b�, p�a�b�, q�a�b�, ��a� b� are fitted to experimentalvalues of the cohesive energy, lattice parameters and inde-pendent elastic constants for the reference crystal struc-tures at 0 K. For the Pd/Pt nanoalloys, the parameters

Fig. 3. Pd58Pt44 bimetallic nanoparticle with waist-capped Marks deca-hedron structure. The inner 5 Pd atoms are shown as white spheres andthe 53 Pd atoms forming the external shell are represented by dark bluespheres. Pt atoms have been omitted.

assume different values according to the interaction type,e.g., Pd–Pd, Pt–Pt, and Pd–Pt. In Table I, we give theparameter values used in this work.30

The genetic algorithm used here is a global searchmethod which finds lowest energy configurations after aseries of operators are applied to an initial population ofindividuals, namely the clusters. All cluster coordinates arerandomly generated initially and are subsequently relaxedby using a local minimization procedure. The cluster totalpotential energy is used to assign a fitness value, such thatlow-energy clusters have high fitness and vice versa. Thegenetic operators for mating, mutation and selection arerepeated over a specified number of generations, obtain-ing at the end of the process the lowest energy cluster.The Birmingham Cluster Genetic Algorithm code31 wasused in the optimization process with a number of initial

Table II. Main energetical and structural characteristics for Pd/Ptnanoparticles Dark blue spheres represent Pd atoms; while gray spherescorrespond to Pt atoms.

MorphologyExcess

Composition energy (eV) Top view Side view

Pd49Pt53 −14.1268

Pd52Pt50 −14.0358

Pd53Pt49 −14.2370

Pd56Pt46 −14.8407

Pd58Pt44 −15.2134

Pd60Pt42 −15.0185

Pd61Pt41 −15.4503

Pd62Pt40 −14.3150

Pd64Pt38 −14.9134

J. Comput. Theor. Nanosci. 7, 1–6, 2010 3

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random configurations. Typically we carried out hundredsof optimizations for each composition in order to performa reliable exploration of the potential energy hypersurface.This algorithm has been described in detail

elsewhere,5�24�30�31 therefore we give here only the param-eters employed in this study: population size = 40;crossover rate = 0�8; crossover type = 1-point weighted;selection= roulette; mutation rate= 0�1; mutation type=mutate_move; number of generations = 500; number ofGA runs for each composition = 100. This high numberof GA runs is necessary due to the relatively large sizeof the clusters and the fact that the number of homotopsdepends combinatorially on the composition,32 reachingthe maximum number of homotops for Pd51Pt51. Thus, avast computational effort for locating minima is carriedout with a rapidly increasing cost in computational timeas the cluster size is increased. Here, one should note that

r0 1 2 3 4 5 6 7

n (r

)

0

2

4

6

8

10

12


r0 1 2 3 4 5 6 7

n (r

)

0

2

4

6

8

10

12

14


r0 1 2 3 4 5 6 7 8

n (r

)

0

2

4

6

8

10

12

14

16


r0 1 2 3 4 5 6 7

n (r

)

0

2

4

6

8

10

12

14


r0 1 2 3 4 5 6 7

n (r

)

0

2

4

6

8

10

12


Fig. 4. Continued.

not only the cost of the energy calculations increases butthat the total number of minima grows exponentially.For the relevant low energy structures for each com-

position of the PdmPt102−m nanoalloys, we computed theexcess energy (Gupta

102 ) as a function of the number of Pdatoms, m:

Gupta102 = E

GuptaTotal �PdmPtn�−m

EGuptaTotal �Pd102�

102

−nE

GuptaTotal �Pt102�

102(4)

This quantity represents the energy gain (or loss) fora mixed cluster with respect to pure 102-atom Pd and Ptclusters, as defined by Ferrando et al.28 Having found themost stable structures, according to Eq. (4), we calculatedthe radial distribution of the atoms, n�r�, with the aim


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r0 1 2 3 4 5 6 7 8

n (r

)

0

2

4

6

8

10

12

14

16

18


r0 1 2 3 4 5 6 7 8

n (r

)

0

2

4

6

8

10

12


r0 1 2 3 4 5 6 7 8

n (r

)

0

2

4

6

8

10


r0 1 2 3 4 5 6 7

n (r

)

0

2

4

6

8

10

12


Fig. 4. Plots of radial distributions of atoms for selected compositions of the lower energy PdxPt102−x nanoparticles. Pd–Pt–Pd layered segregation isapparent, as well as a small subsurface alloying region. Radial distance, r , is in angstroms.

of discerning the internal atomic ordering of the lowestexcess energy structures obtained in our searches.

3. RESULTS AND DISCUSSION

Figure 2 shows the excess energy (Gupta102 ) plotted against

Pd concentration. The observed Gupta102 behavior is corre-

lated with the energy difference between neighboring com-positions, most noticeably for compositions where a highdegree of mixing between Pd and Pt (bottom of the curve)is possible, close to the 1:1 composition that maximizesthe possible number of homotops (i.e., Pd51Pt51 clusters).Gray circles correspond to

Gupta102 values of structures opti-

mized by the GA code. An atom-exchange operation wasapplied to the final GA structures at each composition inorder to search for low-energy homotops, which mighthave not been found during the GA searches. Black tri-angles denote

Gupta102 values for these configurations. In

most the cases, the atom-exchange procedure finds lowerenergy homotops (same structure, different chemical order-ing), especially for those compositions found at the bottomof the

Gupta102 curve. From Figure 2 one can identify the

composition with the lowest energy, which correspondsto Pd58Pt44, the most energetically favorable configuration.One Pd atom is waist-capping the complete Marks deca-hedron and the segregation of Pd atoms is evident sinceonly 5 Pd atoms remain inside the nanoparticle, that is to

say, ∼95% of Pd atoms are on the bimetallic nanoparti-cle’s surface (See Fig. 3). In Table II we show the energiesand morphological features of several of the lowest Gupta

102

configurations for a wide composition range.The morphological analysis of our simulations shows

that the surface cluster sites are occupied mainly byPd atoms. A similar behavior is found experimentally,where Pd atoms tend to segregate to the surface of Pd/Ptnanoalloys14�17 in a PdshellPtcore-type atomic arrangement.Large Pd/Pt nanoparticles consisting of 147 and 309 atoms(icosahedral and decahedral motifs, respectively) havebeen studied by Cheng et al. using a Monte Carlo method33

at three different temperatures: 100, 300 and 500 K. Theyreported detailed structural information, revealing the exis-tence of onion-ring structures. The scarcity of experimen-tal data and the difficulty of observing internal atomicordering of small nanoparticles experimentally, make com-putational simulations an essential tool in order to explorethe cluster core structures. In this sense, we have per-formed a combined global optimization combined withhomotop local minimization on 102-atom Pd/Pt nanopar-ticles, in order to determine the atomic ordering of thelowest energy configurations.Figure 4 shows the radial distribution function, n�r�

plotted against r for several high-symmetry compositions(see Table II), for which layered structure are found. Thelowest energy configuration corresponding to the compo-sition Pd58Pt44 has an inner core consisting of a Pd dimer


RESEARCH

ARTIC

LE


surrounded by an intermediate shell of Pt atoms and anexternal shell of Pd atoms. It is convenient to highlight theexistence of a region where Pd and Pt atoms are mixed(alloyed), which is observed between the outermost andintermediate shells. Our computational results are in agree-ment with XAS experiments which concluded that a fewPd atoms are located in the center of the nanoparticle andthat the Pd–Pt alloying prefers the core region,34 and aresimilar to the icosahedral and decahedral clusters investi-gated by Cheng et al.33 Identical behavior is presented forthe neighboring low energy structures.

4. CONCLUSIONS

We have obtained the structures and energies of the fullrange of 102-atom Pd/Pt nanoparticles as a function ofcomposition, where we have employed a global searchmethod to find the lowest energy structures for every pos-sible composition. The excess energy criterion indicatesthat the most stable composition corresponds to a Pd58Pt44cluster, although a number of nanoparticles of neighbor-ing compositions differ by only a small amount of excessenergyGupta

102 . The onion-ring growth sequence is exhibitedfor the ground state of this particular kind of cluster, inagreement with the results of finite temperature studies.Further investigations, in order to find out the essentialfeatures that enhance their stability, are still needed; manyof the lowest energy structures obtained here are poten-tial candidates to serve as initial configurations in a firstprinciples calculation. The existence of homotops havinglow

Gupta102 values is an indication of the complexity of the

potential energy hypersurface; this complexity might beclarified by an energy landscape analysis.

Acknowledgment: Alvaro Posada-Amarillas acknowl-edges CONACyT for financial support through project24060. Rafael Pacheco-Contreras is grateful also to CONA-CyT for the award of a Ph.D. scholarship. Special thanksare due to Lauro Oliver Paz-Borbón for technical help.

References

1. R. Ferrando, J. Jellinek, and R. L. Johnston, Chem. Rev. 108, 845(2008).

2. J. Jellinek, Faraday Discuss. 138, 11 (2008).3. L. O. Paz-Borbón, R. L. Johnston, G. Barcaro, and A. Fortunelli,

J. Phys. Chem. C 111, 2936 (2007).4. R. A. Guirado-López and F. Aguilera-Granja, J. Phys. Chem. C

112, 6729 (2008).

5. L. O. Paz-Borbón, T. V. Mortimer-Jones, R. L. Johnston, A. Posada-Amarillas, G. Barcaro, and A. Fortunelli, Phys. Chem. Chem. Phys.9, 5202 (2007).

6. J. Joswig, G. Seifert, T. A. Niehaus, and M. Springborg, J. Phys.Chem. B 107, 2897 (2003); P. Rooney, A. Rezaee, S. Xu, T. Manifar,A. Hassanzadeh, G. Podoprygorina, V. Böhmer, C. Rangan, andS. Mittler, Phys. Rev. B 77, 235446 (2008).

7. J. Gavnholt and J. Schiøtz, Phys. Rev. B 77, 035404 (2008).8. H. Kobayashi, M. Yamauchi, H. Kitagawa, Y. Kubota, K. Kato, and

M. Takata, J. Am. Chem. Soc. 130, 1818 (2008).9. N. Sounderya and Y. Zhang, Recent Patents on Biomedical Engi-

neering 1, 34 (2008).10. C. Burda, X. Chen, R. Narayanan, and M. A. El-Sayed, Chem. Rev.

105, 1025 (2005).11. N. Toshima and T. Yonezawa, New J. Chem. 22, 1179 (1998).12. F. Calvo, E. Cottancin, and M. Boyer, Phys. Rev. B 77, 121406(R)

(2008).13. D. Cheng, S. Huang, and W. Wang, Chem. Phys. 330, 423 (2006).14. D. Bazin, D. Guillaume, Ch. Pichon, D. Uzio, and S. Lopez, Oil Gas

Sci. Tech. 60, 801 (2005).15. A. Stanislaus and B. H. Cooper, Catal. Rev.–Sci. Eng. 36, 75 (1994).16. F. Balleto and R. Ferrando, Rev. Mod. Phys. 77, 371 (2005).17. A. J. Renouprez, J. L. Rousset, A. M. Cadrot, Y. Soldo, and

L. Stievano, J. Alloys Compd. 328, 50 (2001); F. J. Cadete-Santos-Aires, C. Geantet, A. J. Renouprez, and M. Pellarin, J. Catal. 22, 163(2001).

18. L. O. Paz-Borbón, R. L. Johnston, G. Barcaro, and A. Fortunelli,J. Chem. Phys. 128, 134517 (2008).

19. S. M. Foiles, M. I. Baskes, and M. S. Saw, Phys. Rev. B 33, 7983(1986).

20. A. F. Voter and S. Chen, Characterization of Defects in Materials,MRS Symposia Proceedings No. 82, edited by R. W. Siegel, J. R.Weertman, and R. Sinclair, Materials Research Society (1987),p. 175.

21. A. P. Sutton and J. Chen, Philos. Mag. Lett. 61, 139 (1990).22. J. N. Murrell and R. E. Mottram, Mol. Phys. 69, 571 (1990).23. R. P. Gupta, Phys. Rev. B 23, 6265 (1985).24. R. L. Johnston and C. Roberts, Cluster Geometry Optimization

Genetic Algorithm Program, University of Birmingham (1999);C. Roberts, Genetic algorithms for cluster optimization, PhD Thesis,University of Birmingham (2001).

25. F. Cleri and V. Rosato, Phys. Rev. B 48, 22 (1993).26. P. D. Jadzinsky, G. Calero, C. J. Ackerson, D. A. Bushnell, and R. D.

Kornberg, Science 318, 430 (2007).27. K. Manninen, J. Akola, and M. Manninen, Phys. Rev. B 68, 235412

(2003).28. R. Ferrando, A. Fortunelli, and G. Rossi, Phys. Rev. B 72, 085449

(2005).29. R. H. Leary and J. P. K. Doye, Phys. Rev. E 60, R6320 (1999).30. C. Massen, T. V. Mortimer-Jones, and R. L. Johnston, J. Chem. Soc.,

Dalton Trans. 4375 (2002).31. R. L. Johnston, J. Chem. Soc.., Dalton Trans. 4193 (2003).32. J. Jellinek and E. B. Krissinel, Theory of Atomic and Molecular

Clusters, edited by J. Jellinek, Springer-Verlag, Berlin (1999), p. 277.33. D. Cheng, W. Wang, and S. Huang, J. Phys. Chem. B 110, 16193

(2006).34. C. Chen, B. Hwang, G. Wang, L. S. Sarma, M. Tang, D. Liu, and

J. Lee, J. Phys. Chem. B 109, 21566 (2005).

Received: 19 March 2009. Accepted: 27 April 2009.


Structural Insights into 19-Atom Pd/Pt Nanoparticles: A Computational Perspective

Dora J. Borbon-Gonzalez,† Rafael Pacheco-Contreras,‡ Alvaro Posada-Amarillas,*,§

J. Christian Schon,| Roy L. Johnston,⊥ and Juan Martın Montejano-Carrizales#

Departamento de Matematicas, Programa de Posgrado en Ciencias (Física), and Departamento deInVestigacion en Fısica, UniVersidad de Sonora, Hermosillo, Sonora, Mexico, Max Planck Institut furFestkorperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany, School of Chemistry, UniVersity ofBirmingham, Edgbaston, B15 2TT, Birmingham, United Kingdom, and Instituto de Fısica, UniVersidadAutonoma de San Luis Potosı

ReceiVed: May 14, 2009; ReVised Manuscript ReceiVed: July 15, 2009

We present a systematic study of the structural changes of 19-atom PdnPt19-n nanoparticles as a function ofcomposition, modeling the interatomic interactions with the many-body Gupta potential and using a geneticalgorithm to obtain the lowest energy structures for all possible compositions. Topological analysis revealsthat most of the structures are based on icosahedral packings and are strongly composition dependent. Thepure Pd19 nanoparticle exhibits a double icosahedral geometry, while the Ino decahedron is the basis of thePt19 cluster structure, which has a lower symmetry. Several structural motifs of the predicted lowest energyconfigurations are observed for bimetallic clusters in the range of compositions studied here. Six ideal structuralfamilies have been identified. Our results show that, for Pt-rich clusters, Pt atoms segregate into the core andthe number of Pd-Pt bonds increases, while for Pd-rich clusters, the surface-segregated Pd atoms tend notto be nearest-neighbors. X-ray diffraction structure factors are simulated for all the predicted structures.

I. Introduction

Nanoalloys have recently become the subject of intenseexperimental and theoretical studies, mainly due to the potentialthat nanoparticles have for applications in fields such aschemistry, biology, and electronics.1 The addition of a secondmetallic component increases the complexity of nanoparticlesand also affords the opportunity to improve desired properties.2

Thus, tailor-made nanoalloys have become a reality, resultingin the production of extraordinary new nanomaterials forapplications in, for example, heterogeneous catalysis andbiological sensing.3-5 In particular, transition and noble metalnanoparticles have received considerable attention as a conse-quence of their possible applications in fields such as optics,heterogeneous catalysis, hydrogen storage, and biomedicine.3,6-9

Wet chemical synthesis methods are widely used in thepreparation of bimetallic nanomaterials, usually starting fromthe reduction of metal ions to metal atoms, such as successivereduction of two metal salts.10,11 This is an important methodbecause it is currently used to prepare core-shell bimetallicnanoparticles, as in the catalyst nanoparticles used, for example,in hydrogen fuel-cell devices for selective CO oxidation.12

Chemical ordering and a high surface/volume ratio are importantcharacteristics of these types of catalysts, and, in particular, ofthose having Pd atoms on the nanoparticle’s external shell. Ithas been shown that Pd catalysts are highly active and selectivefor the production of methanol in the CO hydrogenationreaction.4 It has also been found that the size, morphology and

atomic ordering of nanoalloys are dependent on the preparationconditions, regardless of the synthesis method. In particular, ithas been shown that the calcination temperature plays a keyrole in the formation of Pd-Pt nanoalloys and that heattreatment and hydrogenation can change the surface compositionof (for example) Pt-Ru and Pt-Rh nanoalloys.13

Exceptional attention has been paid to the preparation of Pd/Pt nanoparticles, mainly due to experimental studies indicatinghigh catalytic activity, long-term stability, and good quality ascatalysts for several major chemical reactions, such as hydro-dechlorination of dichlorodifluoromethane and combustion ofmethane.14 Bimetallic Pd/Pt nanoalloys are used in catalyticconverters for the removal of CO, NOx, and aromatic hydro-carbons for pollution control in exhaust gases,15 essentiallybecause these nanoalloys have proven to be better catalysts inthe above-mentioned chemical reactions than monometalliccatalysts, e.g., pure Pd nanoparticles.16

Experimental and computer simulation studies of binarytransition metal (TM) nanoparticles have revealed intriguingmorphological characteristics, with structural motifs rangingfrom noncrystalline (amorphous, icosahedral, or decahedral) tocrystalline (fcc, hcp) structures.17 Theoretical computationalstudies on structural characterization of nanoparticles have beenperformed using both first principles methods and semiempiricalpotentials. First principles calculations involve large computa-tional resources, and only limited studies have been performedon nanoalloys because of the high number of permutationalisomers.18 Approaches using semiempirical potentials requirephenomenological models to describe the interatomic interac-tions, in particular, the many-body nature of metal atominteractions. Several model potentials have been proposed andused in a number of previous investigations.19-22 In this study,we use the semiempirical many-body Gupta potential to mimicmetallic bonding.23

* To whom correspondence should be addressed. E-mail: [email protected] (Alvaro Posada-Amarillas), Tel. +52-662-2592156.

† Departamento de Matematicas, Universidad de Sonora.‡ Programa de Posgrado en Ciencias (Física), and Departamento de

Investigacion en Fısica, Universidad de Sonora.§ Departamento de Investigacion en Fısica, Universidad de Sonora.| Max Planck Institut fur Festkorperforschung.⊥ University of Birmingham.# Universidad Autonoma de San Luis Potosı.

J. Phys. Chem. C 2009, 113, 15904–1590815904

10.1021/jp904518e CCC: $40.75 2009 American Chemical SocietyPublished on Web 08/18/2009

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In this paper, we give a theoretical and computer simulationperspective, performing a comprehensive analysis of the struc-tural characteristics of 19-atom Pd/Pt clusters, considering allpossible compositions. We use a genetic algorithm (GA) whichincorporates the Gupta potential, as formulated by Cleri andRosato, to explore the potential energy surface (PES) of thesenanoalloys.24,25 The 19-atom cluster is particularly interestingbecause the double icosahedron structure is very stable for singleelement clusters26,27 and the appearance of diverse structuresacross the range of compositions makes it suitable for theanalysis of the relationship between structure and composition.We also present a structural fingerprint for each of the predictedstructures, through the simulated intensity of the X-ray diffrac-tion pattern (XRD).28 Clearly, a comparison of simulations andexperiment is crucial in furthering our understanding of bimetal-lic nanoalloys, as well as for improving theoretical methods. Inthe future, it is expected that a more precise structural deter-mination will be obtained by combining HRTEM and X-rayabsorption spectroscopy techniques.29

II. Methodology

Cluster configurations were obtained by global optimizationof the atom positions using the Gupta potential,23 a semiem-pirical potential energy function which has proven to satisfac-torily describe the interatomic bonding in late transition andnoble metal clusters. This potential function is obtained from asecond moment approximation to the tight-binding density ofstates for the d electrons, with parameters fitted from experi-mental data, such as the cohesive energy, lattice parameters,and independent elastic constants for the reference crystalstructure at 0 K.25 The Birmingham cluster genetic algorithm(BCGA) code30 was used in the optimization process, for manyinitial random configurations. Typically, we carried out hundredsof optimizations for each composition in order to perform areliable exploration of the potential energy hypersurface, focus-ing on PdnPt19-n (n ) 0, · · · , 19) bimetallic clusters.

The BCGA code has been described previously;19,24,30,31 thus,we give here only the parameters used in this study: populationsize ) 40; crossover rate ) 0.8; crossover type ) 1-pointweighted; selection ) roulette; mutation rate ) 0.1; mutationtype ) mutate_move; number of generations ) 500; numberof GA runs for each composition ) 100. This large number ofGA runs is necessary since the number of homotops dependscombinatorially on the composition.32 This vast computationaleffort for locating minima guarantees a high probability offinding the global minima for our 19-atom bimetallic nanopar-ticles. After finding the lowest energy structures, we calculatedseveral important structural properties, including simulated XRDpatterns, with the aim of quantifying structural similarities andof providing useful insights into the structural analysis of thesebimetallic clusters.

III. Results and Discussion

In previous studies on Pd/Pt nanoalloys,19 the strong depen-dence of the structure on composition has been demonstrated.For the 19-atom Pd/Pt nanoalloy, it was found that the doubleicosahedron (DI) structure appears for Pd19, Pd18Pt1, Pd17Pt2,Pd16Pt3, and Pd15Pt4 (DI type I). It was also found that the lowersymmetry Pt19 structure (Dh-Ino) is displayed by the clustersPd8Pt11, Pd7Pt12, and for Pd5Pt14 to Pd1Pt18. The Pd6Pt13 clusterexhibits an hcp-type packing, corresponding to a Pt13-centeredanticuboctahedron capped on the six square faces by Pd atoms(hcp-type). The remaining compositions resemble the icosahe-dron or the DI structure, with examples including capped and

waist-capped structures. The present study reproduces the formerstructural details, emphasizing the existence of several structuralfamilies and computing their XRD patterns. We hope that ourresults will provide tools for the analysis of complex energylandscapes, describing the structural patterns for these types ofnanoalloys using straightforward calculations.

We have classified the nanoparticles into six structuralfamilies; for each, a representative is shown in Figure 1. It

Figure 1. The ideal structures of the Pd/Pt nanoparticles obtained inthis work. (a) Double icosahedron type I (DI_I). (b) Double icosahedrontype II (DI_II). Icosahedron with (c) 6 adjacent equatorial faces capped(Ih-w) and (d) 4 adjacent faces capped plus 2 atoms (Ih-v). (e) Inodecahedron with equatorial faces capped (Dh-Ino). (f) The structurebased on a fragment of the infinite hcp-structure.

Figure 2. The structural families obtained in this study for the 19-atom PdnPt19-n nanoparticles as a function of composition n. (a) DI_Istructures for n ) 19, 18, 17, 16, and 15. (b) Ih-v structure for n ) 14.(c) DI_II structures for n ) 13, 12, and 11. (d) Ih-w structures for n )10 and 9. (e) Dh-Ino structures for n ) 8, 7, 5, 4, 3, 2, 1, and 0. (f)The hcp-based structure for n ) 6.

Structural Insights into 19-Atom Pd/Pt NPs J. Phys. Chem. C, Vol. 113, No. 36, 2009 15905

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should be noted that, apart from the hcp-based structures, allthe structures have fivefold pseudosymmetry. In Figure 1a, theregular double icosahedron (DI) is labeled DI_I (DI type I). Ithas two internal atoms, two apex atoms, five equatorial, andtwo sets of five waist atoms, with 68 M-M bonds. In Figure1b, we show the DI type II (DI_II) structure. This structure isbased on the DI, but one of the apex atoms is moved to theequator, capping the two faces formed by two equatorial andtwo waist atoms, with 66 M-M bonds. In Figure 1c and d, wepresent two structures based on the regular icosahedron (Ih),one with six atoms capping six adjacent equatorial faces (Ih-

w), dark atoms in Figure 1c, and another that can be describedas an Ih with four atoms capping four adjacent faces, formingan irregular tetrahedron (dark atoms in Figure 1d) and two morecapping faces formed by three of these and one of the originalIh (Ih-v) (Figure 1d). There are 65 M-M bonds in both clusters.The structure based on the Ino decahedron (Dh-Ino) is shownin Figure 1e, which is a Dh-Ino with five atoms capping thefive equatorial faces and one more on the equator capping thefaces formed by two of these and two of the original Dh-Ino,with 60 M-M bonds. Finally, in Figure 1f we show the 19-atom hcp structure, with 60 bonds. Figure 2a-f shows for n )

Figure 3. Simulated XRD structure factors for the structures found in this study for bimetallic 19-atom PdnPt19-n nanoalloys. Wave vector (k) unitsare Å-1.

TABLE 1: Geometrical Characteristics of the Structures Obtained in This Study

Pdatoms

Ptatoms structure type

No. ofPd-Pd bonds

No. ofPt-Pt bonds

No. ofPd-Pt bonds

Pd-Pdbond length

Pt-Ptbond length

Pd-Ptbond length

averageinteratomic distance

19 0 DI_I 68 0 0 2.673 2.67318 1 DI_I 56 0 12 2.698 2.583 2.67717 2 DI_I 45 1 22 2.731 2.556 2.590 2.68216 3 DI_I 39 3 26 2.741 2.614 2.608 2.68515 4 DI_I 33 5 30 2.750 2.624 2.625 2.68614 5 Ih-v 25 9 31 2.741 2.700 2.609 2.67213 6 DI_II 21 12 33 2.776 2.693 2.618 2.68212 7 DI_II 17 16 33 2.806 2.699 2.612 2.68311 8 DI_II 13 19 34 2.840 2.696 2.619 2.68510 9 Ih-w 8 21 36 2.763 2.691 2.644 2.6759 10 Ih-w 6 25 34 2.777 2.695 2.641 2.6748 11 Dh-Ino 2 25 34 2.658 2.659 2.635 2.6467 12 Dh-Ino 1 29 31 2.706 2.657 2.637 2.6486 13 hcp 0 36 24 2.666 2.609 2.6435 14 Dh-Ino 0 38 24 2.666 2.659 2.6634 15 Dh-Ino 0 44 18 2.672 2.644 2.6493 16 Dh-Ino 0 50 12 2.677 2.614 2.6652 17 Dh-Ino 0 54 8 2.673 2.614 2.6651 18 Dh-Ino 0 58 4 2.670 2.617 2.6660 19 Dh-Ino 0 62 0 2.668 2.668

15906 J. Phys. Chem. C, Vol. 113, No. 36, 2009 Borbon-Gonzalez et al.

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0, · · · , 19 the ground-state structure of the PtnPd19-n clustersgrouped into these six structural families.

Computational work is useful for analyzing the underlyingstructural details of this type of nanoparticles. Figure 3a-f showsthe simulated XRD patterns for each of the predicted ground-state structures. There are no significant differences betweenthose structures where the icosahedron is the basic motif (Figure3a-d) except for k values of around 2.5 Å-1, where peaksplitting is apparent as the number of Pt atoms rises. Figure 3fcorresponds to the hcp-type structure, and the plot shows a firstsmooth peak while a shoulder appears in the second peak. Athird behavior is clearly seen in Figure 3e, corresponding tothe Ino decahedron symmetry. In this figure, major differencesare observed, especially in the high k value region, suggestingthe existence of structural discrepancies as the compositionvaries. The differences in the curves of Figure 3e can beunderstood by analyzing Figure 2. There are two kinds ofstructures: one where the Ino decahedron is well-defined (Figure2a,b), and the other where a combination of Ino decahedronand icosahedral motifs are present in the structure of the

nanoalloys (Figure 2c-f). If we analyze our results from thePt19 cluster structure (n ) 0 in Figure 2e), we note that, as thenumber of Pd atoms rises, the structural symmetry also increasesuntil we have a well-defined Ino decahedron (n ) 8 in Figure2e) as the basic structural motif.

The atomic ordering of the lowest energy configurations forthe structural families found here was explored systematically,along with the radial distances of the atoms in the bimetallicnanoparticles (see Table 1 and Figure 4a-f). The structuraltrends of the predicted structures are similar to the idealstructures for the DI_I and hcp-type families (Figure 4a,e). ICOtypes I and II show an oscillatory behavior, which is morepronounced for the ICO type I structure (Figure 4b,d). The DI_IIand the Ino decahedra are the structures that have evidentdiscrepancies with respect to the ideal structures. For the DI_IIstructure, there is a very steep step at concentrations where thenumber of Pt atoms ranges from 13 to 19 (Figure 4c) and theIno decahedron structures differ considerably from the idealstructure for practically all the concentrations where this

Figure 4. Radial distances (Å) of the atoms in the families of structures obtained in this study. (a) DI_I, (b) Ih-v, (c) DI_II, (d) Ih-w, (e) hcp, and(f) Dh-Ino.

Structural Insights into 19-Atom Pd/Pt NPs J. Phys. Chem. C, Vol. 113, No. 36, 2009 15907

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structure is observed, in particular, for concentrations having14-19 Pt atoms (Figure 4f).

In Table 1, we list the number of different bonds (Pd-Pd,Pt-Pt, and Pd-Pt) in the clusters, the bond lengths, and alsotheir average length, as well as the average interatomic distance.It can be seen that, in the PdnPt19-n nanoparticles for 0 < n < 9,the number of Pt-Pt bonds increases as the Pt concentrationincreases, but simultaneously the number of Pd-Pd bondsdecreases and the number of Pt-Pd bonds increases. This meansthat the Pt atoms tend to aggregate together, i.e., they tend tosegregate in the cluster core. For n > 13, there are no Pd-Pdbonds, which means that the Pd atoms tend to occupy sites thatare not nearest neighbors. On the other hand, for n ) 1-2 thePd-Pd distance is lower than the corresponding value of theGupta potential (2.7485 Å), although the number of Pd-Pdbonds is high. This is most likely due to the effectivemultiparticle interactions mediated by the Pt atoms enforcedby the embedding of the Pd atoms in the Pt-rich structure. ThePt-Pt and Pt-Pd distances are also lower than the respectiveGupta potential parameters (2.7747 Å and 2.76 Å, respectively)for all values of n.

IV. Conclusions

We have obtained a composition-dependent structural de-scription of 19-atom Pd/Pt nanoparticles using a GA globalsearch method to find the lowest energy structures, for allpossible compositions. A comprehensive analysis of the mor-phology is presented, along with structural information givingdetails on the atomic ordering and the interatomic distances forboth the ideal structural motifs and the predicted structures. Wefound that for Pd-rich compositions the Pt atoms tend tosegregate, increasing the number of Pd-Pt bonds. However,for Pd-rich clusters, the Pd atoms tend to be separated. We alsoobserve that from composition Pd9Pt10 to Pd19 the icosahedralgeometry is dominant over the decahedron-type structures. Thesimulated XRD patterns present characteristic behavior andstructural discrepancies for the Dh-Ino nanoalloys, revealing thatan increase in the number of Pd atoms in the range n ) 0-8produces higher symmetry structures. The interatomic distancesin the predicted structures are generally smaller than those ofthe corresponding ideal structure (Table 1 and Figure 4), exceptfor the Dh-Ino structure. Recent theoretical studies have shownthat the energetic distribution of minima may change whenhigher-level theoretical tools are used,33 which suggests thatfuture work must be carried out using quantum chemistrymethods, such as DFT, in order to verify the predicted ground-state structures and the structural families obtained usingsemiempirical Gupta (or other) potentials.

Acknowledgment. A.P.A. acknowledges CONACyT forfinancial support through project 24060. R.P.C. is also gratefulto CONACyT for the award of a PhD scholarship. J.M.M.C.acknowledges financial support from CONACyT through grantno. 50650 and partial financial support from PIFI (Mexico)through grant 2007-24-21.

References and Notes

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(2) Rousset, J. L.; Cadrot, A. M.; Cadete Santos Aires, F. J.; Renouprez,A.; Melinon, P.; Perez, A.; Pellarin, M.; Vialle, J. L.; Boyer, M. J. Chem.Phys. 1995, 102, 8574.

(3) Ferrando, R.; Jellinek, J.; Johnston, R. L. Chem. ReV. 2008, 108,845.

(4) Coq, B.; Figueras, F. J. Mol. Catal. A 2001, 173, 117.(5) Ching, S.-H.; Hoffman, A.; Guslienko, K.; Bader, S. D.; Liu, C.;

Kay, B.; Makowski, L.; Chen, L. J. Appl. Phys. 2005, 97, 10R101.(6) Joswig, J.; Seifert, G.; Niehaus, T. A.; Springborg, M. J. Phys.

Chem. B 2003, 107, 2897. Rooney, P.; Rezaee, A.; Xu, S.; Manifar, T.;Hassanzadeh, A.; Podoprygorina, G.; Bohmer, V.; Rangan, C.; Mittler, S.Phys. ReV. B 2008, 77, 235446.

(7) J. Gavnholt, J.; Schiøtz, J. Phys. ReV. B 2008, 77, 035404. Toshima,N.; Yonezawa, T. New J. Chem. 1998, 21, 1179.

(8) Kobayashi, H.; Yamauchi, M.; Kitagawa, H.; Kubota, Y.; Kato,K.; Takata, M. J. Am. Chem. Soc. 2008, 130, 1818.

(9) Sounderya, N.; Zhang, Y. Recent Patents on Biomedical Engineer-ing 2008, 1, 34.

(10) Burda, C.; Chen, X.; Narayanan, R.; El-Sayed, M. A. Chem. ReV.2005, 105, 1025.

(11) Toshima, N.; Yonezawa, T. New J. Chem. 1998, 22, 1179.(12) Alayoglu, S.; Nilekar, A. U.; Mavrikakis, M.; Eichhorn, B. Nat.

Mater. 2008, 7, 333.(13) Bazin, D.; Triconnet, A.; Moureaux, P. Nucl. Instrum. Methods B

1995, 97, 41. Bazin, D.; Mottet, C.; Treglia, G. Appl. Catal., A 2000, 200,47. Wu, M.-L.; Chen, D.-H.; Huang, T.-Ch. Langmuir 2001, 17, 3877.

(14) Calvo, F. Faraday Discuss. 2008, 138, 75. Cheng, D.; Huang, S.;Wang, W. Chem. Phys. 2006, 330, 423.

(15) Bazin, D.; Guillaume, D.; Pichon, Ch.; Uzio, D.; Lopez, S. OilGas Sci. Technol. 2005, 60, 801.

(16) Stanislaus, A.; Cooper, B. H. Catal. ReV.-Sci. Eng. 1994, 36, 75.(17) Balleto, F.; Ferrando, R. ReV. Mod. Phys. 2005, 77, 371, and

references therein.(18) Paz-Borbon, L. O.; Johnston, R. L.; Barcaro, G.; Fortunelli, A.

J. Chem. Phys. 2008, 128, 134517. Paz-Borbon, L. O.; Johnston, R. L.;Barcaro, G.; Fortunelli, A. J. Phys. Chem. C 2007, 111, 2936. Dıaz-Ortiz,A.; Aguilera-Granja, F.; Michaelian, K.; Berlanga-Ramırez, E. O.; Mon-tejano-Carrizales, J. M.; Vega, A. Physica B 2005, 370, 200.

(19) Lloyd, L. D.; Johnston, R. L.; Salhi, S.; Wilson, N. T. J. Mater.Chem. 2004, 14, 1691.

(20) Michaelian, K.; Garzon, I. L. Eur. Phys. J. D 2005, 34, 183.(21) Paz-Borbon, L. O.; Mortimer-Jones, T. V.; Johnston, R. L.; Posada-

Amarillas, A.; Barcaro, G.; Fortunelli, A. Phys. Chem. Chem. Phys. 2007,9, 5202.

(22) Cheng, D.; Huang, S.; Wang, W. Chem. Phys. 2006, 330, 423.(23) Gupta, R. P. Phys. ReV. B 1981, 23, 6265.(24) Johnston, R. L.; Roberts, C. Cluster Geometry Optimization Genetic

Algorithm Program, University of Birmingham, 1999; Roberts, C. GeneticAlgorithms for Cluster Optimization, PhD Thesis, University of Birming-ham, 2001; Johnston, R. L. J. Chem. Soc., Dalton Trans. 2003, 4193.

(25) Cleri, F.; Rosato, V. Phys. ReV. B 1993, 48, 22.(26) Jadzinsky, P. D.; Calero, G.; Ackerson, C. J.; Bushnell, D. A.;

Kornberg, R. D. Science 2007, 318, 430.(27) Manninen, K.; Akola, J.; Manninen, M. Phys. ReV. B 2003, 68,

235412.(28) Koga, K.; Takeo, H.; Ikeda, T.; Ohshima, K. Phys. ReV. B 1998,

57, 4053.(29) Park, J.; Joo, J.; Kwon, S. G.; Jang, Y.; Hyeon, T. Angew. Chem.,

Int. Ed. 2007, 46, 4630.(30) Johnston, R. L. Dalton Trans. 2003, 4193.(31) Massen, C.; Mortimer-Jones, T. V.; Johnston, R. L. J. Chem. Soc.,

Dalton Trans. 2002, 4375.(32) Jellinek J.; Krissinel, E. B. Theory of Atomic and Molecular

Clusters; Springer: Berlin, 1999.(33) Ferrando, R.; Fortunelli, A.; Johnston, R. L. Phys. Chem. Chem.

Phys. 2008, 10, 640.

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RESEARCHARTICLE

Copyright © 2010 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol. 7, 1–4, 2010

A Theoretical Kohn-Sham Density Functional TheoryBased Study of Pt@Pd12

Maribel Dessens-Félix1, Rafael Pacheco-Contreras2, Catalina Cruz-Vázquez3,Alvaro Posada-Amarillas4�∗, and Andreas M. Köster5

1Programa de Posgrado en Ciencias de Materiales, Departamento de Investigación en Polímeros y Materiales,Universidad de Sonora, 83000 Hermosillo, Sonora, México

2Programa de Posgrado en Ciencias (Física), Departamento de Investigación en Física, División de Ciencias Exactas y Naturales,Universidad de Sonora, 83000 Hermosillo, Sonora, México

3Departamento de Investigación en Polímeros y Materiales, Universidad de Sonora, 83000 Hermosillo, Sonora, México4Departamento de Investigación en Física, Universidad de Sonora, Apdo. Postal 5-088, 83190 Hermosillo, Sonora, México

5Departamento de Química, CINVESTAV, Avenida Instituto Politécnico Nacional, 2508 A. P., 14-740 México, D.F. 07000, México

We report a theoretical calculation of the structural properties of neutral Pt@Pd12 cluster, usingdensity functional theory (DFT) to search for the minimum energy structure and a genetic algorithm(GA) to obtain the initial configuration. The structural reoptimization and frequency analysis wereperformed with the generalized gradient approximation employing the PBE exchange-correlationfunctional. Besides the radial distribution of the atoms, n�r �, we calculated the pair distributionfunction, g�r �, and X-ray scattering intensity spectrum, I�k�, of both initial and reoptimized configu-rations. Our results indicate that a slightly distorted minimum energy structure is obtained after theab-initio optimization procedure. We also found that the average interatomic distance is larger in theDFT optimized structure than in the GA-predicted ground state. Jahn-Teller and electron correlationeffects might be responsible of this behavior.

Keywords: Bimetallic Clusters, Density Functional Theory, Structural Properties.

1. INTRODUCTION

Nanoscience and nanotechnology have promoted researchon the nanoscale systems mainly due to the promis-ing expectations that nanoparticles have in fields such asmedicine, biology, and electronics.1 Transition metal (TM)nanoparticles occupy a special place in this new worldof discoveries being important for a variety of applica-tions, from catalysts to hydrogen storage devices.2 A vari-ety of techniques have been implemented in order toprepare nanomateriales with specific properties, includingchemical, physical or biological processes. The additionof a second metallic component increases complexity ofnanoparticles and has demonstrated to improve the desiredproperties.3 Thus, tailor-made nanoalloys have become areality with the production of extraordinary new nano-materials for applications in, for example, heterogeneouscatalysis.4 But before such applications can be industrial-ized, the physical and chemical properties must be revealedboth experimentally and theoretically.

∗Author to whom correspondence should be addressed.

The understanding of the structural and electronic prop-erties of nanoalloys has become one of the most activeresearch fields in nanoscience; shape, size and compositiondetermine their properties as well as their electronic distri-bution. Theoretical methods based on semiempirical modelpotentials have shown to be outstanding in the descriptionof structural properties with the condition that interatomicbonding is well-described by the potential energy function.Monte Carlo, Molecular Dynamics or Genetic Algorithmsare well established methodologies to perform explorationsof the potential energy surface.5 But all of these methodsfail when the nature of the chemical bond is poorly repre-sented by a semi-empirical model potential. In these cases,it is necessary to utilize more advanced levels of theory,such as ab initio methods.

Quantum chemical methods have been applied to a num-ber of metallic nanoparticles, frequently incorporating sev-eral doping agents. It has been observed that substitutingone or several metal atoms into metallic clusters alters notonly the shape but also the properties of the host nanopar-ticles. A number of examples can be found in the liter-ature, for instance the large magnetic anisotropy energies

J. Comput. Theor. Nanosci. 2010, Vol. 7, No. 8 1546-1955/2010/7/001/004 doi:10.1166/jctn.2010.1501 1

RESEARCHARTICLE

A Theoretical Kohn-Sham Density Functional Theory Based Study of Pt@Pd12 Félix et al.

induced in Co and Fe nanoparticles by the addition of Pdand Pt atoms,6 the improvement of specific catalytic reac-tions by adding Ag, Au, Cu, Co, Cr, Fe atoms7 to Pd cata-lysts, the increasing in the melting temperature induced bythe doping of Ag icosahedral clusters with Ni or Cu singleimpurities,8 similar effect is observed in Al clusters dopedwith a single Cu atom,9 although in the latter case no con-siderable change in the electronic or geometric structuresof the host cluster is produced.

Structural trends of small late-transition-metal clustershave been carefully studied by density functional theory(DFT) methods, and in a number of studies Pd13 clus-ter has been found to prefer the Ih structure.10�11 How-ever, alternative structure motives for the ground state havebeen suggested, too.12�13 Thus, it is important to study the13-atom cluster because it is the first magic number asstated by the geometric shell model, which predicts highsymmetric structures. The aim of this paper is to studythe geometric ground state structure of Pt@Pd12 cluster.We have chosen optimized icosahedral symmetry initialconfiguration of PtPd12 obtained from a genetic algorithm(GA) and then we reoptimize by using the Kohn-ShamDFT method.

This combined approach is reliable and computerresources are used in a more efficient way. Besides, recenttheoretical results14 have shown that energetic distributionof minima may change when quantum chemistry toolsare used under this scheme. Therefore, it is convenient tocheck for the ground state structures obtained using semi-empirical potentials with higher level theoretical tools,where the quantum and correlation effects may entail clus-ter restructuration through a reordering of atoms whichmay lead to a different stable structure.

This paper is organized as follows. In Section 2 webriefly describe the computational methodology utilized inthis work, Section 3 contains our results and discussion,and in Section 4 we summarize our main conclusions.

2. COMPUTATIONAL DETAILS

Initial configuration was obtained optimizing the Guptapotential,15 a semi-empirical potential energy functionwhich has proven to describe appropriately the interatomicbonding in transition and noble metal clusters. This poten-tial function is obtained from a Tight-Binding SecondMoment Approximation of the Density of States for thed electrons with parameters fitted from experimental infor-mation of the cohesive energy, lattice parameters and inde-pendent elastic constants for the reference crystal structureat 0 K.16 We used the Birmingham Cluster Genetic Algo-rithm code17 in the optimization process with several initialrandom configurations. Typically we performed hundredsof optimizations in order to perform a reliable explorationof the potential energy hypersurface.

The Kohn-Sham DFT calculations were carried out uti-lizing the software package deMon2k18 The deMon2k pro-gram solves the Kohn-Sham equations using the linearcombination of Gaussian-type orbital approximation. TheCoulomb energy was calculated by the variational fit-ting procedure proposed by Dunlap, Connoly and Sabin19

employing a GEN-A2 auxiliary function set, which con-tains s, p and d auxiliary functions.20 The approximateddensity was used for the calculation of the exchange-correlation potential, too. Thus, all Kohn-Sham calcula-tions were performed in the framework of auxiliary densityfunctional theory.21 Scalar relativistic effects were incor-porated by 18 valence electron quasi-relativistic effec-tive core potentials22 for Pd and Pt. We used the PBEfunctional23 for exchange and correlation, based on previ-ous validation calculations on the Pd and Pt dimers, whichshowed good agreement with experimental and other the-oretical results. The equilibrium geometries were obtainedby means of an unconstrained structure optimization usinga quasi-Newton restricted step algorithm and the BFGSupdate scheme for the Hessian matrix. The convergence ofthe structure optimization was based on the analytic energygradient and displacement vectors with thresholds of 10−4

and 10−3 a.u., respectively. After convergence the vibra-tional frequencies were calculated in order to ensure thatno imaginary frequency was present, i.e., that a true min-imum structure was reached. This procedure was repeatedon several potential energy surfaces with different odd-numbered multiplicities.

3. RESULTS AND DISCUSSION

Figure 1 shows the atom distribution in the cluster. TheDFT structure refers to the minimum on the nonet poten-tial energy surface which yielded the lowest energy inour local optimizations of the GA start structure. Thecore/shell structure is apparent for both the GA (S1) andDFT (S2) predicted structures, being the Pt atom at thecenter and the 12 Pd atoms in the external shell. Weobserve that the external shell is located around distancevalues of 2.6 Å for the S2 structure while the S1 struc-ture has the external shell located around 2.5 Å, both withrespect to the central Pt atom. The structural analysis indi-cates that, for the S1 configuration four different inter-atomic distance values are found, 12 corresponding to Pt–Pd bonds of 2.560 Å while the rest are Pd–Pd radial dis-tances, 30 of 2.692 Å, 30 of 4.356 Å and 6 of 5.120 Å. TheS1 structure symmetry is Ih, the icosahedral point group.All searches consistently give the same configuration. Inthe case of the S2 structure, there is a wider distributionof distances. This is an indication of the structural distor-tion produced after taking into consideration the inherentelectronic effects.

Interatomic interactions are reflected in the structuraldetails of the reoptimized structure, exhibiting quantitative


RESEARCHARTICLE

Félix et al. A Theoretical Kohn-Sham Density Functional Theory Based Study of Pt@Pd12

r (Å)0.0 0.5 1.0 1.5 2.0 2.5 3.0

n(r)

0

2

4

6

8

10

12

14

Ga structureDft reoptimization

S1 S2

Fig. 1. Radial distribution of the atoms �n�r for both the GA opti-mized PtPd12 nanoparticle (S1) and the DFT reoptimized structure (S2).Central sphere represent the Pt atom (core) and surface spheres corre-spond to 12 Pd atoms (shell). For the S1 structure the shortest bonddistance is 2.5601 Å (Pt–Pd bond) while the longest one is 5.1203 Å,corresponding to a Pd–Pd bond. S2 structure has a shortest bond distanceof 2.6536 Å and the longest one is 5.3229 Å (Pd–Pd bond).

differences with regard to the GA result. A wider distancesdistribution emerges and the point group symmetry is low-ered. Significant differences are noticed in the g�r, theinteratomic distances are shorter in the S1 structure anda first-peak splitting is appreciated below 3 Å (Fig. 2).This aspect deserves to be pointed out because the short-range order is disrupted, reflecting the appearance of struc-tural distortions after using quantum chemistry tools. Theresulting structural distortion might be related to a possi-ble Jahn-Teller (JT) effect, which is known to be presentin small nanoparticles.24 The high symmetry of the start-ing configuration (Ih might induce orbital degeneracy ofthe highest occupied electronic states, which is removedvia a structural distortion that creates a vibrational degen-

r (Å)

0 1 2 3 4 5 6

g(r)

0.0

0.2

0.4

0.6

0.8

Fig. 2. Pair distribution function comparison for both the S1 and S2structures. The S2 structure distortion is clearly observed through the firstpeak splitting (black lines) between 2.6–2.8 Å. Gray lines correspond toS1 structure pair distribution function.

k (Å–1)

1 2 3 4 5

I(k)

(A

rb. u

nits

)

k (Å–1

)

3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

I(k)

(A

rb. U

nits

)

Fig. 3. XRS intensity calculated for S1 (gray line) and S2 (black line)structures. Short-range order distortion is confirmed in the large k valuesregion.

eracy but at the same time lowers the total energy of thesystem.25 In this way, the icosahedral structure might notbe stable for the PtPd12 and would have to undergo a JTdistortion to a lower symmetry structural ordering.

In order to confirm this result we computed the X-rayscattering (XRS) intensity to obtain more evidence on thestructural distortion of the S2 structure. In Figure 3 wedepict the calculated pattern for both S1 and S2 structures.For large k values, the structural distortion is evident. Theinset focuses on this region where a clear difference isobserved in the XRS intensity, demonstrating the existenceof structural differences between the similar S1 and S2structures through straightforward calculations.

4. CONCLUSIONS

In the present study a theoretical analysis of the struc-tural properties of Pt@Pd12 cluster has been carried out.Semi-empirical potential functions yield an icosahedralcore/shell ordering of atoms (Ih symmetry). A differentresult is observed when quantum chemistry DFT method isused to reoptimize the cluster structure. Whereas the semi-empirical potential approach generates structures with highsymmetry, ab initio reoptimization gives rise to a dis-torted icosahedral structure. Cluster distortion is attributedto a possible Jahn-Teller effect. Additional studies arein progress in order to check for structural stability of13-atom clusters in the range of possible compositionsof PtxPd13−x clusters, for x = 0�1� � � � �13, using a com-bined approach based on semi-empiric potential and DFTcalculations.

Acknowledgments: Maribel Dessens-Félix acknowl-edges Laboratorio de Simulación y Cálculo Computacionaldel DIFUS for providing computational resources for this


RESEARCHARTICLE

A Theoretical Kohn-Sham Density Functional Theory Based Study of Pt@Pd12 Félix et al.

research. Alvaro Posada-Amarillas acknowledges CONA-CYT for funds received through project 24060 and to Pro-fessor Roy L. Johnston for giving access to the BCGAcode. Andreas M. Köster gratefully acknowledges fundingfrom CONACYT (60117), ICYTDF (PIFUTP08-87) andCIAM (107310).

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6. G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans, Phys.Rev. B 44, 12054 (1991); D. Weller and A. Moser, IEEE Trans.Magn. 35, 4423 (1999).

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12. C. M. Chang and M. Y. Chou, Phys. Rev. Lett. 93, 133401 (2004).13. L.-L. Wang and D. D. Johnson, Phys. Rev. B 75, 235405 (2007).14. R. Ferrando, A. Fortunelli, and R. L. Johnston, Phys. Chem. Chem.

Phys. 10, 640 (2008).15. R. P. Gupta, Phys. Rev. B 23, 6265 (1981).16. L. O. Paz-Borbón, T. V. Mortimer-Jones, R. L. Johnston, A. Posada-

Amarillas, G. Barcaro, and A. Fortunelli, Phys. Chem. Chem. Phys.9, 5202 (2007).

17. R. L. Johnston, Dalton Trans. 4193 (2003).18. A. M. Köster, P. Calaminici, M. E. Casida, R. Flores-Moreno,

G. Geudtner, A. Goursot, T. Heine, A. Ipatov, F. Janetzko, J. M. delCampo, S. Patchkovskii, J. U. Reveles, D. R. Salahub, and A. Vela,deMon2k, Release 2.4.2, Cinvestav, Mexico-City, Mexico (2006),http://www.demon-software.com.

19. B. I. Dunlap, J. W. D. Connoly, and J. R. Sabin, J. Chem. Phys.71, 4993 (1979).

20. P. Calaminici, F. Janetzko, A. M. Köster, R. Mejia-Olvera, and B. A.Zuniga-Gutierrez, J. Chem. Phys. 126, 044108 (2007).

21. A. M. Köster, J. U. Reveles, and J. M. del Campo, J. Chem. Phys.121, 3417 (2004).

22. http://www.theochem.uni-stuttgart.de/pseudopotentials (2003).23. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865

(1996).24. J. J. Castro, E. Yépez, and J. R. Soto, Rev. Mex. Fís. 50, 123

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M. E. McHenry, J. Non-Cryst. Sol. 75, 97 (1985).

Received: 12 September 2009. Accepted: 1 October 2009.


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