Gold Clusters
CHAPTER 6. G OLD CL USTERS 1 1
al. [234] . In a study comprising Gaussian-based and plane-wave DFT, MP2 and CCSD (T)
calculations, it is found t hat the neutral gold octamer, optimized at t he DFT level, adopts a 3D structure in t he framework of t he two latter methods and a 2D at t he DFT level [235] .
This result is in contradict ion with t he studies of Ran and Kim et al. , who both also employed DFT, MP2 and CCSD (T) methods and report a planar st ructure of Aus for all t hree methods to be most stable [236, 237] . R emacle et al. [238] and Walker [239]
re-established t hat the size t hreshold for the 2D-3D coexistence is lower for cationic gold clusters t han for neutral ones in t heir DFT st udy of charged and neutral gold clusters, where Walker calculated structures and energetics of neutral and positively charged gold clusters up to t he nonamer employing t hree different DFT functionals and t hree basis sets. Xiao et al. calculated various electronic properties and magnetic moments of gold clusters AUn
(
n = 2 - 1 4 , 20, 55) , also including linear and zig-zag chains. They reportt he t ransition from 2D to 3D at AU13 and that cluster stabilities calculated with the Sutton-Chen potential disagree with those yielded from DFT calculations [240] . Dong
et al. studied the vibrational breathing modes of clusters up to 1 6 atoms employing all-electron DFT calculations, and they predict t he AU13 cluster to be more st able in a 3D arrangement [241,242]. A global-minimum search for AUn (n = 1 5 - 1 9 ) by means
of a basin-hopping method coupled with DFT, was undertaken by Zeng e t al. [243] . By
photo-dissociat ion experiments, Vogel et al. determined the unimolecular decay rates and monomer-dimer branching ratios of Au; (n = 7 - 27) clusters and observed only
direct fragments AU�_l for all even- ized cluster and t ho e containing more t han 16
atoms. For the remaining odd-sized clusters, dimeric fragmentation is found to be in general preferred [244] . vVang e t al. calculated structural and electronic properties of gold clusters up to 14 atoms from DFT LANL2DZ theory and found the turnover point from 2D to 3D to occur at AU12 . They also report the stat ic electric dipole p olarizabilities per atom and find t hem to oscillate with cluster size, giving smaller values for even cluster sizes [245] . J ell inek and coworkers investigated the dipole polarizabilities and optical absorption spectra of low energy structures of AUn clusters, n = 2 - 14 and 20, obtained
from static (GG A and LDA) and t ime-dependent D FT (LDA) calculations [246] . T hey employed a valence double-zeta basis set including d polarization functions and a large core pseudo potential, leaving 1 1 valence electrons. Their obtained polarizabilities per atom computed with t he finite-field met hod exhibit odd-even oscillation and increase with cluster size for the planar clusters (AU4 - AU1 3) . Those obtained from t he TDLDA
CHA P TER 6. G OL D CL USTERS 1 14
excit at ion energies and oscillator strengt hs show t he same trend but give absolute values
that are smaller by at least 10 %.
Further photoemission, -ionization and other spectroscopic studies on small gold clu - ters can be found in refs. [247-252]. A variety of sophisticated t heoretica l calculations on atomic, dimeric and trimeric gold are reported in [2 1 1,2 1 7,253-2 70]. Finally, ext remely stable gold coated nano-clusters of the form J'vi@Aun (M = various transit ion met a l ,
17, = 6, 1 2 ) were found recent ly and are reported in refs. [271-273].
The motivation for this work was t o y temat ically search for t he lowest-lying isomers
of small gold clusters up to 20 atoms, a DFT based genetic algorithm and to
accurately calculate the tat ic electric dipole polarizability of the clusters among ot her elect ronic and struct ural propert ies. To my best knowledge, ther are only four reports
on calculated polarizabilities of gold clusters [230, 232, 245, 246] and no experimental mea
surements are available. In all t hese t heoret ical approaches, however, no systemat ic search
for the lowest-lying isomers was undertaken. Furthermore, Hou et al. only invest igated
the polarizabilities of planar struct ures and Wang et al. employed solely t he rather small
L A N L2 0z basis set and pscudopotent ial, which were used in this work for preliminary
calculat ions only.
6 . 2 Methods
The predicted low spin global minima struct ures of gold clusters consist ing from four up to twenty atoms were obtained utilizing the genetic algorithm code B E L P H E G O R as described
in detail in section (3. 1 ) . The init ial populations of respect ive cluster sizes consisted of t he Lennard-Jones global minima, from at least ten randomly generated structures and from published energet ically lowest-lying structure [2 1 9 , 220, 222,225, 230 , 23 1 , 233, 239, 240, 243] .
The minimum energy difference dVi was set to 0.005 eV, dmin and dmax parameters were fixed between 2 . 4 A and 4.5 A , respectively. The t erminat ion criteria was 150 mating and local minimization steps for clusters up to ten atoms and 1 00 steps for t he remainder . The mutation probability was varied between 10 % and 20 %.
Typically, ten of t he t hus obtained energetically lowest-lying isomers were t hen furt her relaxed from L A N L2 0 z basis set and pseudopotential calculations to t heir local minima. Depending on t he relative energy d istribution, t wo to four of t he energetically lowest lying true minima isomers obtained by t hese means were t hen furt her optimized using
CHA.PTER 6. G OLD CL USTERS 1 1·5
t he extensive S T U T T G A RT valence basis set t ogeth r with t he energy-consistent scalar
relat ivistic pseudopotential ( labelled large ba i ) for gold [274] . Fin811y, in p-R.ch case, t he nature of the stat ionary point was determined by calculating and diagonalizing t he ma t rix of energy second derivat ive (Hessian) . For t he exchange-correlation potential, t he hybrid functional b3pw91 , according to t he parameterization suggested by Becke [ 196] and Perdew and Wang [275] , was employed in a self-consistent fashion as implemented in the G AUSS I A N 0 3 program package. No symmetry constraints were applied dur ing t he optimizat ion procedure. The large basis set , which was used for the accurate calculations,
cont ains 58 basis functions, whereas t he L A N L 2 0 z basis set which was used for prelim
inary calculations contains only 2 2 . Since DFT scales with N?, where N denotes t he
number of basis functions, calculat ions from t he large basis are extremely computer-time demanding and could not be finished by t h complet ion o f t his t hesis, as t he computer resources at Massey University were rather limited . More accurate results from large b asis set calculations are only reported up to t he decamer, and LANL2Dz calculations are report ed for t he whole range of gold clu ters st udied in t his work. It will, however, become apparent t hat t he deviations for most electronic properties and also for structural
propert ies from LA
L2DZ
and large ba i et calculat ions are less t han 5 o/c. It should alsobe noted t hat many publications on gold clusters only employ t he LANL20z basis set and pseudopotential. It is, however, found t hat t he relat ive energy differences between respec
tive isomers obtained at t he LA L20z level can be quite different from t hose obt ained
from t he extensive basis set .
The static electric d ipole polarizability of t he ground state gold atom was calculated utilizing several DFT functionals comprising different exchange and correlation functions such as b3p86, b3lyp, b3pw9 1 , blyp, bp86, pw9 1 pw9 1 , svwn, svwn5, pbepbe, mpw 1 pw9 1 , b 1 b95 and pbe 1 pbe. As depicted in figure ( 6 . 2 ) , where calculated polarizabilities o f atomic gold are plotted against various D FT functionals, t he b3pw9 1 functional yields t he small est deviation from t he benchmark all-electron, relativistic CCSD (T) c alculat ion in t he framework of t he Douglas-Kroll-Hess transformation, and was t hus employed for all gold cluster calculations.
CHAP TER 6. GOLD CL USTERS 1 16 5.5 5.458 • 5.426 5.4 ·5.370 5.3 • 5 . 2 76 • 5.265 5.24 1 ·5.250 · 5 . 1 72 • 5. 1 69 5 . 1 5 • 5.0 1 2 • This work • 4.955 - - R , PP, QCI S O(T) - R, OKH, CCSO(T)
F igure 6 . 2 : Static electric dipole polarizability of the ground state gold atom in A 3 as a function of different DFT exchange-correlation functionals and compared to benchmark calculations. 5.20 , A3 R , DKH, CCSD (T) [276] and 5 .35 A3 ,
R , PP, QCISD(T) [277]
6 . 3 Results and D iscussion
6 . 3 . 1 Structur al Data
F igures (6.3-6.7) depict the predicted low-spin global minima of neutral gold clusters up
to 20 atoms. In table (6. 1 ) , the average neighbor distance, t5E, VIPs, YEAs and cohesive
energies of these clusters are presented.
For clusters up to ten atoms, planar struct ures are found to represent t he energetically lowest-lying isomers. Essenti ally the same structures are reported as global minima by
Wang et al. [245] , Xiao et al. [240] , vValker2 [239] , Jellinek et al. [246] and Fernandez et
al [23 1] ' thus confirming the efficiency of B EL PHEGOR and the employed methods in this
CHA PTER 6. GOLD CL US TER S 1 1 7
Table 6 . 1 : Average neighbor distances, r lat ive energies, vertical ionization potentials, vertical electron affinit ies and cohesive energies
(
not corrected for zero-point vibrat ional energy)
of t he lowest-energy AUn cluster i omers 2 :s: n :s: 20 . The geometry notat ion is that of figure (6.3) . The notationIn
implie t he value is given per atom. Values in parenthesis are obtained from LANL20z and those without from large basis set calculations.d 6.E VIP VEA ECoh d 6.E vrp VEA ECoh
Cluster (A) (eV) (eV) (eV) (eV In ) Cluster (A) (eV) (eV) (eV) (eV/n)
LO
-
-
9.353 2.093-
( I Ll ) (2.766) (0.267) (7.092) (3.656) ( 1 .687) ( LO)-
-
(9.340) (2. 101 )-
( 1 L2) (2.836) (0.389) (6.814) (3. 134) ( 1 . 676) 2_0 2.519-
9.273 1.785 1 . 02 1 ( 1 2_0) (2.768) (0.0) (7.449) (3. 278) ( 1 . 746) (2_0) (2.547)-
(9.450) ( 1 .887) (0.951 ) ( 1 2_1 ) (2.858) (0.018) (7.565) (2.836) ( 1 . 745) 3_0 2.564 0.0 8. 195 3. 1 36 1 . 029 ( 1 2_2) (2.804) (0.098) (7.325) (2.76 ) ( 1 . 737) (3_0) (2.61 1 ) (0.0) (8.46 1 ) (3.382) (0.942) ( 1 2_3) (2.8 19) (0.205) (7.32 1 ) (2.393) ( 1 .72 ) 4_0 2.675 0.0 7. 743 2.347 1 .337 ( 1 3_0) (2.80-1) (0.0) (6.594) (3. 405 ) ( 1 . 74 1 ) (4-0) (2.719) (0.0) (7.915) (2.4 72) ( 1.235) ( 1 3_1 ) (2.820) (0.053) (6. 525) (3.399) ( 1 737) 5_0 2.687 0.0 7.355 2.825 1 . 478 ( 1 3_2) (2. 768) (0. 1 60) (6. 700) (3. 5 1 2 ) (1 . 728) (5_0) (2.737) (0.0) (7.517) (2.994) ( 1 .363) ( KO) (2. 39) (0.0) (7. 452) (2.428) ( 1 .804) 6_0 2.685 0.0 8.219 1.897 1 . 7 1 3 ( 1 4-1 ) (2.839) (0. 1 63) (7. 232) (2.783) ( 1 793) (6_0) (2 .733) (0.0) (8.482) (2.000) ( 1 .596) ( 1 5_0) (2.824) (0.0) (6.38 1 ) (3. 1 50 ) ( 1 . 794) 7_0 2.696 0.0 7.046 2.993 1 . 662 ( 15_1 ) (2.849) (0.02 1 ) (6.753) (3.410) ( 1 . 793) (U) (2.748) (0.0) (7.238) (3. 1 79) ( 1 .538) ( lU) (2.852) (0.032) (6.971) (3.657) ( 1 792) 8_0 2.670 0.0 8.012 2.535 1 . 786 ( I U) (2. 774) (0.266) (6.956) (3.928) ( 1 . 777) (8_0) (2.717) (0.0) (8. 1 39) (2.636) ( 1 .663) ( 1 6_0) (2.83 1 ) (0.0) (7. 1 94) (2.883) ( 1 .846) 9_0 2.712 0.0 6. 904 3. 188 1 . 757 ( 16_1 ) (2.845) (0.2 1 8) (6.889) (2.784) ( 1 .832) (9_0) (2.766) (0.0) (7. 1 06) (3.403) ( 1 .62 1 ) ( 1 6_2) (2.852) (0.242) (7.032) (2.954) ( 1 . 8310 9_1 2.710 0. 1 35 7.42 1 3.589 1 . 742 ( I U) (2.826) (0.0) (6.947) (3. 708) ( 1 .850) (9_1) (2.766) (0. 1 56) ( 7.553) (3.826) ( 1 .604) ( I U ) (2.86 1 ) (0.037) (6.327) (3. 124) ( 1 .848) 1 0_0 2.714 0.0 1 .838 ( 1 8_0) (2.827) (0. 0) (7. 1 39) (2.944) ( 1 .8 9) ( 10_0) (2.766) (0.0) (7.626) (2.770) ( 1 . 695) ( 1 8_1 ) (2.830) (0.034) (6.999) (2.809) ( 1 .887) 10_1 2.718 0.259 1.812 ( 1 9_0) (2.864) (0.0) (6.584) (3.497) ( 1 .920) ( 1 0_1 ) (2.767) (0.049) (7.798) (2. 1 05) ( 1 .690) ( 1 9_1 ) (2.869) (0.471 ) (6.379) (3.356) ( 1 .895) ( 1 0_2) (2.774) (0.063) (7.05 1 ) (2.48 1 ) ( 1 . 6 ) (20_0) (2.875) (0.0) (7.438) (2.419) ( 1 .969) ( I LO ) (2.762) (0.0) (6. 138) (2.888) ( 1 . 7 1 2)CHA PTER 6. G OLD CL USTERS 1 1
2 0 3 0
1 0 1 1 0 2
Figure 6.3: Predicted global minima and lowest-energy i o mers of AU2 - 10, ordered ( from left t o right and top to bot t om) by increased size and energy. The cluster
CHA PTER 6. GOL D CL USTERS 1 19
1 2 0 1 2 1 1 2 2
1 3 0 1 3 1 1 3 2
Figure 6.4: Predicted global mIDlma and lowest-energy Isomers of AUl l- 13 , ordered
(
from left to right and top to bottom)
by increased size and energy. TheCHAP TER 6. G OLD CL USTERS 1 4 0 1 5 0 1 5 2 1 20 1 4 1 1 5 1
Figure 6.5: Predicted global minima and lowest-energy isomers of A U 14- 15 , ordered ( from left to right and top to bottom) by increased size and energy. T he cluster n_m is t he mth energetic isomer wit h 17, atoms.
CHAP TER 6. GOLD CL USTERS 1 2 1
1 6 0 1 6 1
1 7 0 1 7 1
1 7 2
F igure 6.6: Predicted global minima and lowest-energy isomers of AU 16- 1 7 , ordered
(from left to right and top to bot tom) by increased size and energy. The cluster run is t he m.th energetic isomer wit h n atoms.
CHA PTER 6. GOLD CL USTERS 1 22
1 8 0 1 8 1
1 9 0 1 9 1
2 0 0
Figure 6.7: Predicted global minima and lowest-energy isomers of AU18-20 , ordered
( from left to right and top to bottom) by increased size and energy. The cluster n _m is t he m t h energetic isomer wit h n atoms.