Supercomputer Research in Chemistry and Chemical Engineering

Chapter 2

Theoretical Approaches to Metal Chemistry 1

1

1

Charles W. Bauschlicher, Jr. , Stephen R. Langhoff , Harry Partridge , Timur Halicioglu , and Peter R. Taylor 2

3

1

Ames Research Center, National Aeronautics and Space Administration, Moffett Field, CA 94035 Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 ELORET Institute, Sunnyvale, CA 94087 2

Downloaded by IOWA STATE UNIV on April 10, 2017 | http://pubs.acs.org Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

3

Recent advances in methodology have made possible accurate ab initio calculations on transition metal diatomic systems as well as small (3-6 atom) simple metal clusters. These accurate calculations can in turn be used to define two- and three-body potentials for use in modelling much larger clusters. Calculations on clusters containing Be, A l and Cu atoms illustrate the accuracy of current work and the diversity of metal bonding. The structure and reactivity of small clusters vary dramatically with size, and very large clusters are required before the cluster structure approaches that of the bulk. For example, even though the bulk structure of Be is hcp, the fcc structure is still considerably more stable than hcp for a 55 atom Be cluster. Comparison of ab initio and model calculations for small Al clusters demonstrates that it is necessary to include three-body terms in the model for quantitative results. The impact of adsorbates on metal-metal bonding is studied for Be X and Al X clusters. The optimal sites for adsorption are often different for small clusters than the bulk, owing to the enhanced ability of small clusters to distort. 13

n

13

n

C o m p u t a t i o n a l chemistry is being applied at N A S A A m e s to numerous problems in chemistry, physics and materials science. One important application is to problems i n re-entry physics that are intractable to experiment, such as the extreme conditions occuring i n the bow shock wave of the aeroassisted orbital transfer vehicle ( A O T V ) (1). N o n e q u i l i b r i u m r a d i a t i o n is a significant component of the heating, owing to the large blunt heat shield of the A O T V and its trajectory through the t h i n upper atmosphere. Accurate knowledge of the chemistry of hot mixtures of nitrogen and oxygen are required for input into c o m p u t a t i o n a l fluid dynamics ( C F D ) codes involved in the heat shield design. A l s o , the chemistry of hydrogen and air mixtures is being studied to aid design of supersonic combustion ramjet engines.

0097-6156/87/0353-0016S06.00/0 © 1987 American Chemical Society

Jensen and Truhlar; Supercomputer Research in Chemistry and Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1987.


2.

BAUSCHLICHER ET AL.

Theoretical

Approaches

to Metal Chemistry

17

In a d d i t i o n to the gas-phase work, we are c o m p u t i n g (2) the vibrational spectra and rotational barriers of polymer fragments to help interpret experi ments. B y achieving a better understanding of polymers and their chemistry, we hope to design longer lifetime and more corrosion resistant polymers. A n o t h e r major c o m p u t a t i o n a l effort is i n the area of metals and their chem istry, w h i c h comprises the subject of this manuscript. T h e studies are directed towards b o t h catalysis and the development of improved materials, such as stronger m a t r i x composites. T h e materials and gas phase work have some over lap. For example, surface recombination affects the heating on the A O T V heat shield a n d on the walls of the scramjet. In a d d i t i o n , desorption of these molecules from the walls of the scramjet could impact the chemistry in the flow. T h e study of molecular systems containing metal atoms, particularly tran sition metal atoms, is more challenging than first-row chemistry from both an experimental and theoretical point of view. Therefore, we have systematically studied (3-5) the c o m p u t a t i o n a l requirements for obtaining accurate spectro scopic constants for diatomic and triatomic systems containing the first- and second-row t r a n s i t i o n metals. O u r goal has been to understand the diversity of mechanisms by w h i c h transition metals b o n d and to a i d i n the interpretation of experimental observations. W h i l e accurate calculations on transition metal compounds are restricted to three or fewer transition metal atoms, it is possible to consider much larger clusters of A l and B e atoms. We have considered (6^9) A l clusters of up to six atoms using correlated wave functions, and A l i , B e i 3 a n d Bess at the S C F level. These calculations give insight into how the b o n d i n g changes w i t h cluster size. Since even for these simple metals the ab initio calculations are time consum ing, we have interfaced (9) our ab initio methods w i t h a parameterized model approach where the potential is expanded i n two- and three-body interaction terms. For single component systems, these potentials can be determined from either b u l k d a t a or calculations. W i t h the parameterized model we can consider larger clusters a n d identify interesting clusters for further ab initio study. The parameterized model can also be used for multi-component materials, although in this case the two- and three-body parameters are not easily deduced from bulk data. T h e m o d e l approach appears well-suited for the study of alloys and m a t r i x composite materials, especially large multi-component systems not d i rectly amenable to ab initio study, but it must rely on ab initio calculations to define the two- and three-body interaction potentials. T h e study of small metal clusters and their chemistry is an active area of experimental research (10). Gas phase experiments have shown ( Π ) a very large variation i n reactivity w i t h cluster size, but have been unable to determine the geometry of the cluster, or the adsorption site if the clusters have been reacted w i t h other molecules. Experiments on supported clusters have determined (12) the average metal-metal b o n d lengths, but only for a d i s t r i b u t i o n of clusters, a n d the effect of the support is u n k n o w n . T h e reactivity and metal-metal bond lengths are often considerably different from the well-studied perfect crystal faces. T h e o r e t i c a l calculations on metal clusters are therefore important for determining o p t i m a l geometries, and to explain the large changes i n reactivity w i t h cluster size. T h e cluster model is also useful for studying the chemistry of perfect crystal faces. W i t h current super computers, it has become possible to model N125X and N125X5 clusters (where X = 0 , F , S, CI), and thereby study changes in bonding w i t h coverage. These calculations (13) explain the experimental observation that oxygen shows a large shift i n v i b r a t i o n a l frequency w i t h coverage, while sulfur does not. T h e theoretical study of perfect crystals, as well as coverage dependence where experimental data is available for comparison, also helps to delineate the accuracy of the small metal cluster work. 3


SUPERCOMPUTER

18

RESEARCH

Section II describes recent improvements in methodology that have signif icantly improved the accuracy of calculations on s m a l l metal clusters. Section III describes the calculation of some accurate dimer and trimer potentials, and the insight they give into the nature of metal chemistry. Section I V reviews the work on s m a l l metal clusters and discusses how the ab initio and parameterized model approaches are interfaced. Section V contains our conclusions.


M e t h o d o l o g i c a l Advances In this section we give a brief overview of recent methodological advances that have significantly improved our capabilities for accurate calculations on molecules containing transition metals as well as on small clusters. Accurate results for transition metals require both large one-particle basis sets including high angular m o m e n t u m functions and a careful treatment of the correlation (or η - p a r t i c l e ) problem. Recently we have carried out (14-20) full configurationinteraction ( F C I ) calculations on molecules to assess the accuracy of correlation methods that truncate the η - p a r t i c l e expansion. T h e most important result from the F C I benchmark calculations is that a carefully designed complete-activespace self-consistent field ( C A S S C F ) calculation to optimize the orbitals, fol lowed by a multi-reference singles plus doubles configuration-interaction ( M R C I ) calculation from the important configurations in the C A S S C F wave function, gives consistently the best agreement w i t h the F C I . Hence this is the method of choice when the resulting configuration expansion is of manageable size (i.e. less than about 1 m i l l i o n configurations). O n e important implication of the F C I studies is that if the η - p a r t i c l e prob lem is treated at the C A S S C F / M R C I level, the l i m i t i n g factor i n the accuracy of the wave function becomes the one-particle basis. However, a recent develop ment by Almlôf and Taylor (21) has greatly increased the size of the gaussian p r i m i t i v e valence and polarization basis sets that can be used i n C I calculations. T h i s is accomplished by using general contractions w i t h coefficients determined from the n a t u r a l orbitals of C I calculations on the atoms. A t o m i c natural orbitals ( A N O s ) define a method of truncating the basis set to equal accuracy in each shell. T h e following prescription has led to extremely accurate results for excitation energies and dissociation energies of diatomic molecules (15,16). In a double-zeta plus polarization A N O basis set we find a C A S S C F / M R C I treatment that reproduces the F C I result for the η - p a r t i c l e problem, and this C A S S C F / M R C I treatment is then taken to near the one-particle l i m i t . A s shown later, this approach (17) gives a definitive prediction for the ground state of AI2, even though the lowest two triplet states are separated by less than 200 c m " . A t present, we are o p t i m i z i n g (22) A N O contractions for the first-row transition metals that are based on the average of the 3 d 4 s and 3 d 4 s occupations to satisfy the extensive basis set requirements for an accurate description of t r a n s i t i o n metal diatomics. A l t h o u g h a properly designed C A S S C F / M R C I treatment i n a large A N O basis set is expected to give quantitative results for molecular systems includ ing transition metals, it can be computationally very intensive. Indeed, this approach quickly becomes intractable for larger clusters, especially when a large number of electrons are correlated. We have, therefore, devoted consid erable effort to calibrating single reference-based correlation methods against C A S S C F / M R C I and F C I calculations. W h e n the molecular system is reason ably well described by a single reference configuration, we have found that the coupled pair functional ( C P F ) approach (23), a size-consistent reformulation of 1

n

2

n + 1

1


2.

BAUSCHLICHER ET AL.

Theoretical

Approaches

to Metal Chemistry

19

S D C I , gives an accurate representation of the molecular state. T h i s is i n con trast to single-reference singles-plus-doubles C I ( S D C I ) , w h i c h is often not very satisfactory for transition-metal diatomics, especially when the molecular state arises from a m i x t u r e of atomic states w i t h different d occupations. T h e C P F approach gives quantitative 'e(eV)

265 265

1.206 1.206

277 277

1.231 1.233 1.401 1.386 1.55±0.15

- 1

u

e

(cm ) 325 325

M R C I (4s3p2d) F C I (4s3p2d) M R C I (6s5p3d2f) M R C I + R e l (6s5p3d2f)

ρ

(cm *)

τ-,

%

and Σ

u

(a ) 5.241 5.240

M R C I (4s3p2d) F C I (4s3p2d) M R C I (6s5p3d2f) M R C I + R e l (6s5p3d2f) EXPT

n

3

U

21

1

Tefcm- ) 252 289 128 158

4.710 4.710

344 343

4.660

350

165 174

a

T h e M R C I calculation is a second-order C I from a l l configurations i n the C A S S C F wave function resulting from a l l arrangements of the 3s and 3p elec trons i n the 3s a n d 3p orbitals. T h e D a r w i n and mass velocity contributions were included using first-order perturbation theory. H u b e r and Herzberg, Ref. 42. c


SUPERCOMPUTER RESEARCH

22

Do of 1.40 e V for AI2 is w i t h i n the error bounds of the experimental value of 1 . 5 5 ± 0 . 1 5 e V determined by Stearns and K o h l (46) using a K n u d s e n cell mass spectrometric m e t h o d and assuming a Σ ~ g r o u n d state.


3

Since accurate m o d e l l i n g of larger clusters requires the inclusion of a threeb o d y interaction function, we have devoted considerable effort to the under standing of t r i a t o m i c systems containing C u , A l and B e . T h i s work is also directed at understanding the nature of b o n d i n g i n alloys and composites. A l though it is not possible to compute the potential energy surface of these t r i atomics to the same accuracy as for the diatomics, a good estimate of the errors in the t r i a t o m i c calculations can be obtained by performing the same level of calculations on the diatomics. L i k e the AI2 molecule, the CU3 molecule is interesting i n its own right, and has been the subject of many experimental and theoretical papers (see for exam ple 26-28 a n d references therein). O u r ab initio study of CU3 gives a B ground state corresponding to a C Jahn-Teller distortion away from a E equilateral triangle geometry. T h e B state was found to lie 59 c m " below the A\ state and 280 c m " below the Dzh equilateral geometry, thus confirming the pseu dorotation barrier and Jahn-Teller s t a b i l i z a t i o n energy deduced by T r u h l a r and T h o m p s o n (27) from an analysis of the fluorescence spectrum of Rohlfing and Valentini (29). Based on our experience (25) w i t h CU2 where b o t h higher exci tations and relativistic effects are important, our ab initio study of CU3 included relativistic effects v i a first-order perturbation theory and correlation effects us ing the C P F f o r m a l i s m . However, this level of correlation treatment required some reduction i n the one-particle basis set and yielded errors of 0.056 a i n r , 0.08 e V i n D and 8 c m " in u for C u compared w i t h the accurate experimen tal values. T h i s provides a good estimate of the errors in the b o n d lengths and b i n d i n g energy of the CU3 cluster. T h i s level of treatment for CU3 is expected to yield an accurate three-body interaction t e r m for use in m o d e l l i n g C u clusters. 2

2

2

f

2v

2

1

2

2

1

0

e

1

e

e

2

T h e existence of two nearly degenerate triplet states w i t h substantially different r values i n A l manifests itself in the A l (9) and C u A l (47) triatomics in terms of low-lying states w i t h considerably different geometries. For example, AI3 has three nearly degenerate states; the A a n d B\ states, which are two Jahn-Teller components of a E state, a n d the A\ state. Experiments yield conflicting d a t a as to the ground state. M a t r i x isolation E S R (48) shows a quartet state w i t h equivalent A l atoms (either an equilateral or pseudorotating triangle), while magnetic deflection experiments have been interpreted (49) as showing a doublet g r o u n d state. L i k e the n and Σ ~ states of AI2, the three states of AI3 have different geometries, w i t h a 0.39 a v a r i a t i o n i n bond length and 15° v a r i a t i o n i n b o n d angle. Since one expects the b o n d i n g i n small clusters to arise from a m i x t u r e of these low-lying states i n AI2 and AI3, we have averaged the results for the low-lying states for evaluation of the two- a n d three-body parameters. In the next section we describe our ab initio a n d parameterized model results for larger A l clusters. O u r theoretical results (47) for the Cu Be systems are summarized i n Table II. For C u B e we find two linear structures, C u - B e - B e where the bonding is very directional owing to the formation of s-p hybrids, and B e - C u - B e where the b o n d i n g is m u c h more delocalized. C u B e also has two low-lying linear structures, one of w h i c h contains delocalized metal b o n d i n g . In b o t h isomers of the C u B e and C u B e 2 linear structures, the very directional b o n d i n g implies a repulsive three-body c o n t r i b u t i o n . A s we show later, this large three-body force explains the apparently strange behavior of B e on a C u ( l l l ) surface (50). A s in the case of C u a n d C u , the ab initio calculations give more t h a n input into e

2

3

2

4

A

2

4

f

2

3

3

u

0

x

y

2

2

2

2

3


2.

BAUSCHLICHER ET AL.

Theoretical Approaches to Metal Chemistry

23

Table II. Spectroscopic constants for selected C u ^ B e ^ systems SDCI

CPF


2

CuBe Σ + (8s6p4d/4s3p) basis 4.098 0.55

r (a ) D (eV) e

0

e

4.090 0.68

(9s7p4d3flg/6s3p3dlf) basis r (a ) D (eV) e

4.022 0.77

0

e

3.999 0.92

C u 2 B e linear s y m m e t r i c * Σ + r (a ) atomization (eV) D (CuBe-Cu)(eV) e

0

4.089

4.017 2.87 2.18

e

linear asymmetric * Σ r (Cu-Cu)(a ) r (Cu-Be)(a ) T (eV) " ' c

a

0

e

0

e

+

4.462 4.402 ...

4.415 4.206 0.62

B e 2 C u linear asymmetric Σ 2

r (Be-Be)(a ) r (Cu-Be)(a ) atomization energy (eV) D (Be-BeCu)(eV) e

0

e

0

4.119 4.055

+

4.169 4.011 1.23 0.56

e

2

linear s y m m e t r i c Σ + r (a ) Te(eV) e

0

4.353 ...

4.312 0.15

I n this basis set the spectroscopic parameters for Cu2 are: r =4.148 a and D =1.77eV. a

c

e


0

24

SUPERCOMPUTER

RESEARCH

the m o d e l i n g approach, they yield insight into the chemistry. T h i s is especially i m p o r t a n t when the complexity of the systems precludes performing calculations on a l l s m a l l systems of interest. B e Clusters C o m p u t a t i o n a l l y the study of s m a l l B e clusters is straightforward, since struc tures are qualitatively correct using s m a l l basis sets and neglecting the effects of electron correlation. For example, at the S C F level using only a D Z basis, the B e - B e b o n d length in B e (of 3.97 a ) is just 0.05 a longer t h a n at the S D C I level using a much larger triple-zeta basis w i t h two sets of polarization func tions ( T Z 2 P ) (6,51). However, the atomization energy per a t o m ( D / a t o m ) is significantly larger (0.70 eV) at the T Z 2 P - S D C I level than at the D Z - S C F level (0.39 e V ) . Therefore, questions of structure can be answered at the S C F level, while correlation must be included to accurately compute the cohesive energy of the cluster. 4

0

0


e

A n i m p o r t a n t question in materials science is how large a cluster must be before its structure and chemistry is the same as that of the b u l k . The smallest strongly b o u n d B e cluster is B e , which has tetrahedral geometry. A tetrahedron can be considered the b u i l d i n g block for both the fee and hep stuctures, the latter being the structure of Be metal (52). A central a t o m i n either the hep or fee structure is surrounded by 12 nearest neighbors, w i t h trigonal symmetry for hep and octahedral symmetry for fee. S C F calculations on B e i 3 clusters, w i t h the constraint that all B e - B e b o n d lengths are equal, yield a D / a t o m that is more t h a n twice as large as B e ; 0.87 e V (hep) and 0.91 e V (fee) (6). However, the fee structure is more stable than the hep structure for the 13 atom B e cluster, whereas the bulk structure is hep. W h i l e all of the bond lengths are equal in the b u l k fee structure, there is always some distortion in the bulk hep structure. If the constraint of equivalent b o n d lengths is eliminated (except that the clusters are still required to have trigonal s y m m e t r y - Osh or D^d), both clusters show modest distortions (up to 0.27 a ). T h e fee structure is stabilized by an a d d i t i o n a l 0.24 e V due to d i s t o r t i o n . Therefore, not only is the lowest energy B e i structure different from the bulk, the 13 atom fee structure differs from the fee bulk structure by undergoing significant distortion. 4

e

4

Q

3

T h e a d d i t i o n of nearest neighbors to the twelve surface B e atoms of Β β χ results i n a 55 a t o m cluster. A t the S C F level, using a slightly smaller basis set t h a n used i n our best treatment of B e 13, the fee structure of Bess is also observed to be more stable t h a n the hep structure (7). However, the relative stability between the two structures decreases to 0.03 e V per atom (favoring fee) in Bess compared to 0.10 e V for B e i 3 . T h e D / a t o m for Be55 is significantly larger than that for B e i 3 (1.33 vs. 0.86 e V / a t o m ) , w h i c h is i n t u r n about twice that found for B e , but it is still significantly less than the bulk value of 3.38 e V / a t o m . A l t h o u g h part of this difference arises from neglect of electron correlation and basis set l i m i t a t i o n s , scaling the Bess b i n d i n g energy by 1.80 (the increase in D between the equivalent and best B e calculations) does not fully account for differences w i t h the b u l k . Note, however, that the D / b o n d increases only slightly (0.07 e V ) between B e and B e , and by even less (0.03 eV) between B e i 3 and B e . If the factor of 1.8 for basis set and correlation errors is applied to the 0.34 e V / b o n d for B e , the resulting value of 0.59 e V is i n good agreement w i t h the 0.56 e V / b o n d deduced from bulk data. T h u s , the D per bond is converging quite quickly w i t h cluster size. T h e structure, however, is probably more influenced by the number of bonds per atom, which is 3.93 for B e 5 , compared to 6 for the bulk. Hence the structure of clusters can be quite different 3

e

4

e

4

e

4

1 3

5 5

5 5

e

5


2.

BAUSCHLICHER ET AL.

Theoretical Approaches to Metal Chemistry

25

from the b u l k , a n d rather large clusters, p r o b a b l y between 100 a n d 300 atoms, are required before the bulk structure is o p t i m a l . A l Clusters A l t h o u g h A l is less important t h a n transition metals as a catalyst, it is a simple metal for which some experimental data is available for comparison w i t h theory (48.49.53). U n l i k e B e clusters, A l clusters are not adequately described at the S C F level i n a s m a l l one-particle basis set. Theoretical b o n d lengths (6,8,9) given i n Table III for the A l a n d A l i clusters indicate that the A l - A l distance decreases w i t h b o t h extensions of the one-particle basis a n d w i t h the inclusion of electron correlation. In the larger basis sets the A l - A l b o n d lengths show an increase w i t h increasing cluster size i n analogy w i t h B e clusters. T h e bond lengths i n b o t h cases approach that of the b u l k from below. T h e inclusion of correlation shortens the A l b o n d length by more than i n B e , but by less than the change w i t h basis set improvement. Since electron correlation increases the b i n d i n g energy by a factor of 1.5, it must be included for a quantitative determination of the cohesive energy. We have considered the larger A l - A l 6 clusters using b o t h ab initio calcula tions a n d the parameterized m o d e l (9). T h e results for A l a n d AI5, summarized in Table I V , show that the parameterized m o d e l a n d ab initio calculations agree well on the relative energetics i f b o t h the two- a n d three-body interactions are included. For A l it is difficult to treat all the structures at the T Z 2 P - C P F level, but for the structures considered, there is reasonable agreement between the ab initio a n d m o d e l results. W h e n the parameters deduced from the calculations o n AI2 a n d AI3 are applied to b u l k A l , the cohesive energy is too s m a l l a n d the b o n d length is too large. T h e s m a l l cohesive energy is expected because our c o m p u t e d AI2 D at the T Z 2 P - C P F level is only 7 1 % of the experimental value (42,46). T h e bulk values are i n much better agreement w i t h experiment if the model is parame terized using the experimental D a n d r values for the Σ ~ state. Hence, the


4

3

4

4

4

4

6

c

3

e

e

Table III. B o n d lengths for metal clusters of A l a n d B e Cluster/method

r (ûo) e

Be DZ S C F B e large basis set C I Beisifcc) D Z S C F Bei (hcp)DZ S C F B u l k * (hep)

3.97 3.92 4.06 4.11 4.26

Al Al Al

5.30 5.10 5.02 5.44 5.44 5.26 5.26 5.41

4

4

3

D Z - S C F rhombus large basis S C F large basis C I

4

4

4

A l i ( f c c ) large basis set S C F bulk (fcc) 3

a

a

R e f . 52 —for B e the average of the two values i n the b u l k , 4.32 and 4.21 is given.


SUPERCOMPUTER

26

RESEARCH

Table I V . C o m p a r i s o n of stability and structure of A l clusters between ab initio and parameterized interaction results w i t h two- and three-body terms (2+3-b) as well as using only the two-body (2-b) interaction. B i n d i n g energies ( D i n eV) per a t o m , and b o n d distances ( r in ao) are given n

e

e

D /atom

r

e

Structure Al Rhombus

e

A b initio

2+3-b

2-b

A b initio

2+3-b

2 ^

1.08

1.13

1.48

5.04

5.10

4.97

1.32 1.27 1.18 1.15 1.05

1.27 1.24 1.23 1.20 1.24

1.67 1.96 1.93 1.44 2.11

4.96 5.18 5.19 5.02/4.94 4.74

5.14 5.25 5.26 5.13/5.07 5.32

4.97 4.96 4.97 4.98/4.95 4.98

3.43°

2.75 3.70

4

Al

5


C2v ^4v

C

8

V2h

bulk

9.97 13.4

6

5.41

5.75 5.41

a

6

6

4.84 4.25

fc

a

E x p e r i m e n t a l value (52). F o r better comparison, these values have been calculated using the two-body potential calibrated w i t h the experimental A l data (42,46) ( D 1.55 e V and r 4.66 a for the Σ " state). 6

2

e

e

3

0

parameters needed to reproduce the bulk properties appear to be closer to those for the Σ ~ excited state, although this could be a consequence of the form of the potential, which is discussed in more detail below. It is interesting to note that if the experimental AI2 data is used, this method has errors i n the lattice constant and cohesive energy that are of the same magnitude as those found i n density functional methods developed to study solids (54). Since the calculated lowest energy structure of B e is fee, whereas the bulk structure is hep, we have carried out S C F calculations on AI13 using a large basis set to see if its structure is also different from that of the bulk. T h i s is i n fact the case since the nearly degenerate icosahedral and hep structures are b o t h about 1 e V more stable t h a n fee, w h i c h is the bulk structure. In a d d i t i o n , neither the hep structure nor the fee structure is significantly distorted from a l l bonds equal. T h i s is also opposite to the situation in the bulk where the hep structure undergoes distortion. A p p l i c a t i o n of the parameterized model (with parameters based u p o n A l and A l ) leads to a planar AI13 being about 1 e V more stable t h a n hep, fee or icosahedral, whereas this structure is 2.6 e V above the most stable structure at the ab initio level. A t present, our modelling approach uses a Lennard-Jones potential for the two-body t e r m 3

1 3

2

3

«(Γ,, ) Γ >

=

ί

( ( ^ )

1

2

- 2 ( ^ )

6

)

(1)

where ro is the e q u i l i b r i u m b o n d distance of the dimer, r y is the distance between atoms i and j and ε is the energy at r = r o . For the three-body interaction we considered the A x i l r o d - T e l l e r form: t

t J


2.

BAUSCHLICHER E T AL.

Approaches

to Metal Chemistry

27

where r , ry*, r^ and e*i, a , c*3 represent the sides and angles, respectively, of the triangle formed by the three particles i , j and k. T h e intensity of the three-body interaction is given by the parameter Z , w h i c h is specific to each combination of different species in the tri-atom interaction. T h e poor results for AI13 using this model may be either due to our choice of potential functions w i t h simple analytical forms or to the neglect of four-body and higher-order interac tions. It might be possible to avoid these higher-order terms by having effective two- and three-body parameters that vary smoothly w i t h cluster size. However, much of the problem may be that the Lennard-Jones potential rises more steeply than the ab initio potential, so that the two-body term underestimates the twob o d y contributions of second nearest neighbors. T h e three-body function is probably incapable of accounting for both the limitations of the two-body t e r m and three-body effects. Alternate forms for the potential are presently under investigation. In addition, we have developed a modified model potential, where the threeb o d y interaction has been reduced slightly to give the ab initio ordering of struc tures for A I 1 3 , i.e. the Ζ value in E q u a t i o n 2 is fitted to the 13-atom ab initio results. T h i s model potential was then used to study (55) the midsized clus ters A l 7 - A l i 5 . T h e o p t i m a l structures are summarized in Table V and shown graphically in Figure 1. T h e b i n d i n g energy per atom increases monotonically w i t h increasing cluster size, but the energy required to remove the an atom varies w i t h a pattern reminiscent of that observed (56) in mass spectroscopic experiments for the ionization potential and abundance. Figure 1 demonstrates that changes in cluster geometry w i t h increasing cluster size can be quite dra m a t i c . For example, the A I 9 and A l n clusters are three-dimensional while A l i o is planar. These studies must be considered qualitative because of unresolved questions about the potential, but such geometrical variations could explain the large changes in cluster properties w i t h size. t J


Theoretical

r

2

M i x e d Clusters of Be and C u Since the relative simplicity of C u - s i m p l e metal systems make them ideal for s t u d y i n g mixed component systems, S C F and model calculations have been car ried out (50) for selected C u ^ - B e x systems. T h e two- and three-body functions in the model are taken from ab initio calculations (26) on C u and CU3, and the 2

Table V . T h e b i n d i n g energy for A I 7 to AI15 based on the modified three-body potential cluster

D /atom

A I 7 planar A l g planar A I 9 3-dimensional Al planar A l n 3-dimensional AI12 planar AI13 3-dimensional Al 3-dimensional A l l 5 3-dimensional 1 0

1 4

a

2.05 1.82 1.94 2.17 2.03 2.20 2.14 2.15 2.26

1.38 1.44 1.49 1.56 1.60 1.65 1.69 1.72 1.76

T h e D has been normalized to that of A l . T h e energy required to remove one A l atom. e

b

a

e

2


SUPERCOMPUTER RESEARCH

28

cross terms are determined from calculations (47) on C u B e and C u 2 B e . We have not used B e and Be3 to define the B e - B e interaction terms, since these systems are weakly b o u n d and not indicative of larger B e clusters. Therefore we have used the parameterized model only for C u B e . T h e C u cluster was previously designed (57,58) to m o d e l chemisorption at a three-fold hollow site on the (111) surface. For this reason the three C u atoms at the adsorption site are treated using an all-electron treatment, while all of their nearest neighbors i n the first layer and the six atoms i n the second layer are modelled using an effective core potential ( E C P ) that explicitly treats only the 4s electrons. T h e geometry of the bare C u cluster was taken from bulk d a t a (52), which along w i t h the details of the C u basis set are described i n earlier studies of Ο a n d NH3 chemisorption (57,58). T h e B e a t o m was described using a (9s4p) /(4s2p) gaussian basis set (59). T h e chemisorption of one B e a t o m into the three-fold hollow is found to be repulsive at the S C F level. T h e inclusion of correlation could lead to a bound system, but it is unlikely that Be w i l l be strongly b o u n d i n the three-fold hollow of C u i g . A n a l y s i s of the parameterized m o d e l results shows that the repulsive three-body interaction overcomes the attractive two-body interaction at these geometries. T h e same three-body forces lead to the directional b o n d i n g noted for C u B e and C u B e . However, the parameterized model does predict that B e is b o u n d for the on-top site. 2


1 8

2

2

T h e interaction of B e w i t h the C u i g cluster was also considered at the S C F level. T h e B e was constrained to have T

Supercomputer Research in Chemistry and Chemical Engineering

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