Metal Bonding and Interactions in High Temperature Systems

12 Electronic Structure of Small Group IA and IB Metal Clusters STEVEN C. RICHTSMEIER, ROBERT A. EADES, and DAVID A. DIXON University of Minnesota, Department of Chemistry, Minneapolis, MN 55455 Downloaded by UNIV OF LEEDS on January 16, 2018 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch012

JAMES L. G O L E Georgia Institute of Technology, Department of Chemistry, Atlanta, GA 30332

The e l e c t r o n i c s t r u c t u r e s of Group IA and IB metal c l u s t e r s have been determined using two theo retical methods: ab initio molecular o r b i t a l theory and the semi-empirical diatomics-in-molecules (DIM) method. E l e c t r o n affinities f o r Li , Li , Li , Na , and N a are determined and v a r i o u s methods f o r c a l c u lating the electron affinity from ab initio wave -functions are considered. Surfaces for Li and Li are discussed in detail with respect to e l e c t r o n attachment and detachment. S i m i l a r c o n s i d e r a t i o n s are o u t l i n e d f o r Na . The s t r u c t u r e s of the Group IA t r i m e r s , Li , Na , K , Rb and C s have been c a l c u l a t e d u s i n g the DIM method. The trimers are all bound w i t h respect to d i s s o c i a t i o n to atom plus diatomic having C v s t r u c t u r e s , a B (obtuse angle) geometry being the most s t a b l e form. A A s t a t e lies s l i g h t l y higher i n energy. A general d i s c u s s i o n of the p o t e n t i a l energy surface f o r the t r i m e r s i s presented. Further DIM c a l c u l a t i o n s on the Group IB trimers (CU , A g , A u ) are o u t l i n e d and t h e i r similarity to the Group IA trimers is emphasized. V i b r a t i o n a l frequencies f o r the B and A con f i g u r a t i o n s are presented (uncorrected f o r the e f f e c t of higher p o t e n t i a l surfaces) in the hope that they will a i d the search f o r trimer s p e c t r o s c o p i c features in both matrix i s o l a t i o n and gas phase s p e c t r o s c o p i c studies. 2

3

5

2

3

3

3

3

3

3

3

3

3

2

2

2

2

1

3

3

3

2

2

2

1

The i n t e r p l a y between theory and experiment i s of importance i n high-temperature chemistry. The e l e c t r o n i c s t r u c t u r e s of high-temperature molecules are being studied i n d e t a i l with numerous experimental and t h e o r e t i c a l techniques. Molecular 0097-6156/82/0179-0177$12.25/0 © 1982 American Chemical Society

Gole and Stwalley,; Metal Bonding and Interactions in High Temperature Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

Downloaded by UNIV OF LEEDS on January 16, 2018 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch012

178

METAL BONDING AND INTERACTIONS

p r o p e r t i e s i n c l u d i n g ground s t a t e geometries ( 1 ) , p o t e n t i a l energy curves (2) and surfaces ( l a ) , e x c i t a t i o n energies (3), i o n i z a t i o n p o t e n t i a l s (4), e l e c t r o n a f f i n i t i e s (4,5) and v i b r a t i o n a l i n t e n s i t i e s (6) are being determined using t h e o r e t i c a l methods f o r a wide range o f h i g h temperature s p e c i e s . Such t h e o r e t i c a l s t u d i e s are u s e f u l i n p r o v i d i n g a guide f o r the i n t e r p r e t a t i o n of experimental r e s u l t s , p a r t i c u l a r l y i n cases where n o v e l species are encountered. T h i s i s e s p e c i a l l y true f o r metal c l u s t e r s , many o f whose s t r u c t u r e s a r e thought to be f l u x i o n a l and f o r which l i t t l e s t r u c t u r a l chemistry i s yet known, even though they are o f great p r a c t i c a l importance (7). A wide range o f t h e o r e t i c a l methods has been a p p l i e d t o the study o f the s t r u c t u r e o f s m a l l metal c l u s t e r s . The extremes are represented on the one hand by semi-empirical molecular o r b i t a l (Extended Huckel) (8) and valence bond methods ( D i a t o m i c s - i n Molecules) (9) and on the other hand by r i g o r o u s ab i n i t i o c a l c u l a t i o n s w i t h l a r g e b a s i s s e t s and extensive c o n f i g u r a t i o n i n t e r a c t i o n (CI) (10). A number o f approaches l y i n g between these two extremes have been employed i n c l u d i n g the X-a method (11), approximate molecular o r b i t a l methods such as CNDO (12) and PRDDO (13) and Hartree-Fock ab i n i t i o molecular o r b i t a l theory with moderate CI. In a p p l y i n g any of the t h e o r e t i c a l techniques mentioned, i t i s important that one be cognizant of the degree of s o p h i s t i c a t i o n needed t o provide u s e f u l d e s c r i p t i v e i n f o r m a t i o n which w i l l a i d the i n t e r p r e t a t i o n of experiment. Here c e r t a i n c o n s t r a i n t s are o p e r a t i v e ; molecules and/or c l u s t e r s which are the most amenable to experiment may be i n a c c e s s i b l e to r i g o r o u s ab i n i t i o c a l c u l a t i o n s by v i r t u e of the atoms from which they are c o n s t i t u t e d . While r i g o r o u s c a l c u l a t i o n s may be performed on s m a l l l i t h i u m or sodium c l u s t e r s , such e f f o r t s are not yet f e a s i b l e f o r potassium, rubidium, or cesium and one must r e s o r t to a meaningful semie m p i r i c a l theory. In studying the h e a v i e r metal c l u s t e r s using semi-empirical techniques one would hope to employ a s i g n i f i c a n t component of a v a i l a b l e experimental i n f o r m a t i o n as an a i d to the e v a l u a t i o n of q u a n t i t i e s inherent i n the theory (e.g. coulomb and exchange i n t e g r a l s ) . Experimental data should enter i n t o and gauge the c a l c u l a t i o n s . For t h i s reason, we have chosen to apply the d i a t o m i c s - i n molecules method to the study o f the h e a v i e r Group IA t r i m e r s 3 > Rb3, and C S 3 and the Group IB t r i m e r s C U 3 , Ag3, and A U 3 . In t h i s chapter, we w i l l d i s c u s s the a p p l i c a t i o n of both ab i n i t i o and DIM methods t o the study of the s t r u c t u r e of s m a l l metal c l u s t e r s , s p e c i f i c a l l y c l u s t e r s of the Group IA and Group IB atoms. These are, by choice, examples taken from our work but we have endeavored t o comment on the general a p p l i c a b i l i t y of the v a r i o u s techniques. K


12.

RICHTSMEIER E T A L .

Small Group

IA

and IB

179

Clusters


Aspects of T h e o r e t i c a l Methods A b - i n i t i o C a l c u l a t i o n s . A b - i n i t i o molecular o r b i t a l theory has been widely discussed. Konowalow and Rosenkrantz, i n t h i s volume (14), have o u t l i n e d the elegant m u l t i c o n f i g u r a t i o n s e l f c o n s i s t e n t (MCSCF) method (15) while Kurtz and Jordan (16) have presented d e t a i l e d s t u d i e s of metal atom-water i n t e r a c t i o n s and a l k a l i h a l i d e dimers. Here, we o u t l i n e SCF+CI c a l c u l a t i o n s on a number o f small metal c l u s t e r s (4,17,18). Our focus i n t h i s work has been on the determination of e l e c t r o n a f f i n i t i e s f o r the a l k a l i dimers and trimers. A v a r i e t y of ab i n i t i o c a l c u l a t i o n s have been employed i n these s t u d i e s . For example, i f we wish to determine the a d i a b a t i c e l e c t r o n a f f i n i t y (EA) f o r a given molec u l e , we must c a l c u l a t e the energy of the anion a t i t s optimum geometry and the energy o f the appropriate n e u t r a l a t i t s o p t i mum geometry. The d i f f e r e n c e i n these energies y i e l d s a value f o r the EA uncorrected f o r v i b r a t i o n a l e f f e c t s . The c a l c u l a t i o n of e l e c t r o n a f f i n i t i e s a n d , e s p e c i a l l y the molecular s t r u c t u r e of the anion,should be done using extended b a s i s sets i n c l u d i n g a d d i t i o n a l d i f f u s e functions i n order to properly d e s c r i b e the more d i f f u s e nature of the anion charge cloud. Such d i f f u s e functions are a l s o required f o r determining the p o l a r i z a b i l i t y of the molecule and f o r determining the p o t e n t i a l d e s c r i b i n g the a d i a b a t i c i n t e r a c t i o n o f an e l e c t r o n with a molecule (19). An extended b a s i s set may even be required f o r d e s c r i b i n g the p o t e n t i a l energy surface o f the n e u t r a l ( l a , 4 ) . T h i s i s e s p e c i a l l y true f o r h i g h l y f l u x i o n a l molecules where very d i f f e r e n t geometries have s i m i l a r energies (see the d i s c u s s i o n below). The c a l c u l a t i o n s should be extended beyond the SCF l e v e l using methods which account f o r e l e c t r o n c o r r e l a t i o n , e.g. c o n f i g u r a t i o n i n t e r a c t i o n or MCSCF techniques (20). In c e r t a i n instances i t may be appropriate to determine the ground s t a t e wavefunction using the g e n e r a l i z e d valence-bond (GVB) technique (21). T h i s i s e s p e c i a l l y true f o r closed s h e l l anions where the p a i r of e l e c t r o n s i n the highest occupied molecular o r b i t a l (HOMO) may be described by the wavefunction y

GVB

( H 0 M 0 ) = [cJ> Cl)cJ> (2) + c() (2)(t) (l)](a3-3a) 1

2

1

2

(1)

instead of the r e s t r i c t e d Hartree-Fock form ^(HOMO) = [^(1)^(2)

Kae-ga)

(2)

The s p l i t t i n g of the HOMO may provide important f l e x i b i l i t y i f the anion i s best described by the combination of a " n e u t r a l core and a very d i f f u s e e l e c t r o n . There are a number of s l i g h t l y more approximate methods f o r determining the e l e c t r o n a f f i n i t y (EA) based on the r e s t r i c t e d Hartree-Fock (RHF) and s p i n - u n r e s t r i c t e d Hartree-Fock (UHF) methods. F o r c l o s e d s h e l l anions, molecules which d i s s o c i a t e to 11


METAL

180

BONDING A N D INTERACTIONS

an e l e c t r o n plus a r a d i c a l i n a doublet e l e c t r o n i c s t a t e , one can employ Koopmans' theorem (22) to estimate the b i n d i n g energy of the e l e c t r o n . T h i s approach i s most a p p l i c a b l e f o r systems c h a r a c t e r i z e d by moderate to l a r g e e l e c t r o n a f f i n i t i e s . Previous c a l c u l a t i o n s of the e l e c t r o n a f f i n i t i e s of GeH3 and SiH3 (23) using Koopmans theorem have y i e l d e d good agreement with experimental estimates. For doublet open s h e l l anions, which d i s s o c i a t e to an e l e c t r o n p l u s a c l o s e d s h e l l n e u t r a l , the EA can be c a l c u l a t e d u s i n g an SCF energy d i f f e r e n c e obtained from the d i f f e r e n c e between an RHF c a l c u l a t i o n on the n e u t r a l and a UHF c a l c u l a t i o n on the r a d i c a l anion. T h i s approach has p r e v i o u s l y been a p p l i e d to the determination of E A s f o r L i , LiNa and Na with some success (5). The ab i n i t i o c a l c u l a t i o n s described below were performed with the HONDO (Versions 3 and 5) (24) and MOLE programs (25). A l l c a l c u l a t i o n s employed extended b a s i s s e t s constructed from contracted Gaussian type o r b i t a l s . B a s i s sets were, i n g e n e r a l , taken from the work of Huzinaga (26) and Dunning and Hay (27). They are d e s c r i b e d i n d e t a i l elsewhere (4,18). 1

f


2

2

The Diatomics-in-Molecules Approach. The simple v e r s i o n of the DIM method that we employ i s based on the Heitier-London approximation (28). In s p i r i t , i t i s s i m i l a r to the LondonE y r i n g approach except that we use accurate diatomic p o t e n t i a l curves (29a) r a t h e r than an approximate form f o r the diatomic t r i p l e t curve (e.g. the Sato parameter f o r LEPS surfaces) (29b). To exemplify, the energies f o r the doublet s t a t e s of an A3 system, where each A atom i s i n a S s t a t e , are given by the f o l l o w i n g expression (the London equation) 2

E

= J

±

1 2

« 3« 3±21

2

1 / 2

[(K

1 2

-K 3) 1

2 +

(K

1 2

-K 3)

2

2

+

(K 3-K 1

2

2 3

) ]

1 / 2

(3)

(The same energy expression i s a l s o employed i f a l l of the atoms are d i f f e r e n t . ) The terms J . and K ^ are the Coulomb and exchange i n t e g r a l s between atoms i and j . The i n t e g r a l s J . . and K „ are evaluated from the e m p i r i c a l expressions ±

±

1 J

1

1

3

(4)

1

3

(5)

J,,

= 1/2[ E..(R..) + E..(R..)]

K..

= 1/2[ E..(R..) - Z..(R..)]

R

where ^ i j ( ^ - : ) i s the p o t e n t i a l curve f o r the ground e l e c t r o n i c s t a t e of the diatomic at the d i s t a n c e R^ between atoms i and j and ^ i j ( ^ i - j ) i s the lowest t r i p l e t curve; i n g e n e r a l , the t r i p l e t curve i s n e a r l y r e p u l s i v e but may be c h a r a c t e r i z e d by a small a t t r a c t i v e w e l l . These two curves i n the asympototic l i m i t c o r r e l a t e with the ground S s t a t e s of the atoms i and j . T h i s r e p r e s e n t a t i o n of the J . . and K.. i n t e g r a l s f o l l o w s from the 2


12.

RICHTSMEIER

ET AL.

Small Group

IA

and

IB

181

Clusters

simplest form of the Heitier-London approximation of a diatomic molecule formed from two S atoms:

f o r the

energy

2

E(h) 3


E( £)

2

(6)

2

(7)

= (J + K ) / ( l + S ) = (J - K ) / ( l - S )

where S i s the overlap i n t e g r a l between the two S o r b i t a l s . In the approximation we have employed, the value of S i s set equal to zero. For four atoms an energy expression s i m i l a r to that given above (equation (3)) i s employed s i n c e only two Rumer (30) diagrams are needed to d e s c r i b e the molecule. For f i v e and s i x atom molecules there are f i v e Rumer diagrams (30) which must be combined to g i v e the energy of the f i v e appropriate s i n g l e t s t a t e s . The energies of the s i n g l e t s t a t e s f o r s i x atoms are given by the eigenvalues of the 5 x 5 matrix 6 E = I J.. 1 - K (3a) ±>j ~ 1 J

where K i s a matrix whose elements are given by Gelb, Jordan and S i l b e y ( 3 1 ) . T h i s expression, i n s l i g h t l y d i f f e r e n t form, was derived many years ago by E y r i n g et al_. (32) . An important advantage of the DIM method as o u t l i n e d above i s t h a t , f o r a maximum of e i g h t atoms, the method i s computationa l l y e f f i c i e n t ; even f o r ten atoms, a moderate s i z e 42 x 42 matrix need be d i a g o n a l i z e d . The d i a g o n a l i z a t i o n step r e q u i r e s l i t t l e time u s i n g e f f i c i e n t matrix d i a g o n a l i z a t i o n r o u t i n e s such as EIGEN or GIVENS. Thus, extensive s t u d i e s of the p o t e n t i a l energy surface can be made. For c l u s t e r s with more than 10 atoms, the method i s c o n s i d e r a b l y more d i f f i c u l t to apply as a r e s u l t of the valence bond branching diagram. For a twelve atom c l u s t e r , a 132 x 132 matrix must be constructed and then d i a g o n a l i z e d . I f e x c i t e d s t a t e s are i n c l u d e d i n the DIM c a l c u l a t i o n (33) , very l a r g e matrices must be d i a g o n a l i z e d even f o r a three-atom system. Perhaps the most s i g n i f i c a n t advantage of the DIM method i s that the e m p i r i c a l curves used i n e v a l u a t i n g the Coulomb and exchange i n t e g r a l s are "exact" f o r the diatomic. Thus they include both c o r r e l a t i o n and r e l a t i v i s t i c e f f e c t s between a l l atomic p a i r s . T h i s l a t t e r c o r r e c t i o n i s very important f o r heavy atoms (e.g. gold (Au)) (34). Although the s i n g l e t curves are w e l l - e s t a b l i s h e d from experiment (35) or theory, the t r i p l e t curves are, i n general, not w e l l known. T h i s d e f i c i e n c y i s being r a p i d l y remedied by new experimental and t h e o r e t i c a l s t u d i e s on metal diatomics. For example, a combination of t h e o r e t i c a l (36) and experimental work (37) has now provided an e x c e l l e n t represent a t i o n of the £ curve f o r L i . In t h i s chapter, we w i l l focus on the a p p l i c a t i o n of DIM to small c l u s t e r s of the Group IA ( a l k a l i ) and Group IB (coinage) metals. The Group IA atoms are w e l l represented by a S ground 3

+

u

2

2


182

METAL BONDING AND INTERACTIONS

s t a t e . The f i r s t e x c i t e d P s t a t e l i e s at c o n s i d e r a b l y higher energy f o r sodium and potassium decreasing to 1.39 eV f o r cesium (38). Thus, the DIM method should be w e l l - s u i t e d f o r studying c l u s t e r s of sodium and potassium and of moderate u t i l i t y f o r the study of rubidium and cesium. The a p p l i c a t i o n of DIM to c l u s t e r s of Group IB atoms w i l l be v a l i d only i f the ground atomic s t a t e d^s i s well-separated from the f i r s t e x c i t e d state, d s ( f o r A g , d p i s the f i r s t e x c i t e d l e v e l « i s o e n e r g e t i c with d s ) . The e x c i t a t i o n energies (38) are 1.39 eV, 3.75 eV and 1.14 eV f o r Cu, Ag and Au, r e s p e c t i v e l y . Thus, we a n t i c i p a t e that DIM should be q u i t e a p p l i c a b l e to the study of s i l v e r c l u s t e r s and should be of moderate a p p l i c a b i l i t y to the c h a r a c t e r i z a t i o n of copper and gold c l u s t e r s (see f o l l o w i n g d i s c u s s i o n s ) . The parameters f o r the curves employed i n these s t u d i e s are given i n Table I. The s i n g l e t curves are represented by a Morse potential 1

9


1 0

2

1

9

1

Z

ij

=

D {exp[-26(r 1

i j

- r ) - 2exp[-3(r.. - r ) ] } e

e

2

(8)

with the parameters taken from experiment(35). The t r i p l e t curves are represented by Lennard-Jones 6-n p o t e n t i a l s (39) \ ^

= 4D3[(o7r..)

n

6

- (a/r..) ]

(9)

with n = 8 f o r the Group IA metals and n = 12 f o r the Group IB metals. The t r i p l e t curves f o r the Group IB metals were d e r i v e d from the pseudopotential c a l c u l a t i o n s of Ermler et a l . ( 2 ) . The t r i p l e t curve f o r Au2 was f i t to a Lennard-Jones 6-n p o t e n t i a l and the value of n was optimized to 12. The experimental and t h e o r e t i c a l e q u i l i b r i u m d i s t a n c e s f o r the ground s t a t e of Au2 were found to differ slightly. Therefore, the p o s i t i o n of the t r i p l e t minimum was scaled s l i g h t l y using the p o s i t i o n of the experimental s i n g l e t minimum. The d i f f e r e n c e between the s i n g l e t and t r i p l e t minima was assumed to be a constant and the p o s i t i o n of the t r i p l e t minimum was then set w i t h r e s p e c t to the experimental s i n g l e t minimum. The c a l c u l a t e d s i n g l e t curves f o r Cu2 and Ag2 do not agree as w e l l w i t h experiment as does the Au2 s i n g l e t curve. The t h e o r e t i c a l t r i p l e t curves f o r Cu2 and Ag2 were thus scaled with respect to the experimental s i n g l e t curve as described above f o r Au2. The t r i p l e t curves f o r the Group IA diatomics were synthes i z e d from molecular beam s c a t t e r i n g data. The values of O were obtained from the g l o r y s c a t t e r i n g experiments of Helbing and Rothe (40) on a l k a l i - a l k a l i p a i r s i n t e r a c t i n g i n the E s t a t e . We have used t h e i r values f o r O which were obtained with the cons t r a i n t of the van der Waals c o e f f i c i e n t (Set A, Ref. 40). The values of the w e l l depth were taken from the s p i n exchange experiments of P r i t c h a r d et a l . (41). These authors found the w e l l 3


12.


Table 1.

Atom

IA

and

P o t e n t i a l Curve Parameters

IB

183

Clusters

f o r DIM C a l c u l a t i o n s

3

D (eV)

euu" )

r (au)

D (eV)

a(au)

1.0677

0.45706

5.0493

0.02721

6.939

Na

0.7298

0.45312

5.8166

0.02721

7.2633

K

0.5197

0.40426

7.4134

0.02721

7.7573

Rb

0.4935

0.3814

7.8121

0.02721

7.986

Cs

0.3966

0.39029

8.2921

0.02721

8.379

Cu

1.98

0.7616

4.195

0.2349

4.149

Ag

1.63

0.7924

4.724

0.0513

5.507

Au

2.24

0.9087

4.668

0.16245

4.843

Li Downloaded by UNIV OF LEEDS on January 16, 2018 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch012

Small Group

b

1

1

e

3

See text f o r d e t a i l s o f curves and references. ' T r i p l e t curve parameters from Ref. 36 are O = 6.406 au and D~ = 0.0362 eV. The a c t u a l c a l c u l a t i o n s were c a r r i e d out usir a Morse curve w i t h parameters: r = 8.0 au., 3 = 0.45991 au.~ and D = 0.0362 eV. 6


M E T A L BONDING A N D

184

INTERACTIONS

depth to be v i r t u a l l y i n v a r i a n t w i t h respect to s c a t t e r i n g p a r t ner. Thus the w e l l depths f o r the a l k a l i £ s t a t e s were a l l chosen to have the same v a l u e . The only comparison which we can make w i t h accurate ab i n i t i o curves i s with the £ curve c a l c u l a t e d by Konowalow (36) and confirmed by the experimental work of Stwalley et a l . (37). As demonstrated i n Table I, the ab i n i t i o curve y i e l d s a deeper w e l l depth and a smaller value of 0 (the value of £(rij) which the energy i s z e r o ) . The curve constructed from the s c a t t e r i n g r e s u l t s i s not i n s e r i o u s d i s agreement. A d e t a i l e d comparison of the determined s t r u c t u r a l parameters f o r the trimer using both cruves w i l l be presented i n later discussion. 3

3

+

u


3

f

o

r

Results A b - i n i t i o Studies A. Surface f o r LJ3 (Na3). In order to approach the c a l c u l a t i o n of e l e c t r o n a f f i n i t i e s , we f i r s t d i s c u s s the nature of the Li (_la,j4) and Na3 (3) s u r f a c e s . In c o n s i d e r i n g the nature of these s p e c i e s , we emphasize the complications which are e n t a i l e d i n the d e s c r i p t i o n of t h e i r molecular s t r u c t u r e . A l o g i c a l s t r u c t u r e f o r a metal t r i m e r , M3, where M has a S ground s t a t e i s an e q u i l a t e r a l t r i a n g l e . The simplest molecular o r b i t a l theory then y i e l d s a valence e l e c t r o n c o n f i g u r a t i o n . . . a e ' c o r r e s ponding to a E ' s t a t e which must J a h n - T e l l e r d i s t o r t (42). The trimer can d i s t o r t along the bending component of the degenerate "e" v i b r a t i o n (which has a i symmetry f o r a C v p o i n t group) to give e i t h e r an obtuse t r i a n g u l a r (0>6O°) B e l e c t r o n i c s t a t e or an acute t r i a n g u l a r (GK6O ) A i e l e c t r o n i c s t a t e . The presence of a c o n i c a l i n t e r s e c t i o n (43) (the c r o s s i n g of the A i and B surfaces at 6 0 ° ) r e q u i r e s the i n t r o d u c t i o n of a v e c t o r p o t e n t i a l term i n the n u c l e a r Hamiltonian i n order that the Born-Oppenheimer wavefunction be s i n g l e - v a l u e d . T h i s phenomena has been r e f e r r e d to as the molecular Aharonov-Bohm (MAB) e f f e c t (44). The e f f e c t i s s i g n i f i c a n t whenever the n u c l e a r wavefunction i s a p p r e c i a b l e along a closed path about a c o n i c a l i n t e r s e c t i o n . The presence of the MAB e f f e c t r e q u i r e s a pronounced change i n the method of e v a l u a t i o n of p o s s i b l e v i b r a t i o n a l energy l e v e l s e s p e c i a l l y i f the A i and B s t a t e s are connected by a low b a r r i e r . For a f u r t h e r d i s c u s s i o n of the g l o b a l topology of t r i a t o m i c p o t e n t i a l energy s u r f a c e s the reader i s a l s o r e f e r r e d to Davidson (45). The s u r f a c e f o r L i 3 has been explored by a number of workers at the SCF-CI l e v e l ( l a , 4., 46, ^7) (see Table 2 f o r a summary of r e s u l t s ) . These researchers have found that the molecule i s bound w i t h respect to monomer p l u s dimer by 9-10 kcal/mol and with respect to three L i atoms by ~34 kcal/mol. The L i 3 molecule has three low energy forms corresponding to B ( C ) , A i ( C v ) , and E ( D ^ ) e l e c t r o n i c s t a t e s . The E + s t a t e i s simply the l i n e a r form corresponding to the B s t a t e . The B 3

2

2

1

2

2

2

2

0

2

2

2

2

2

2

2

2

2

2

2

2

2 v

2

U

U

2

2

2


2

12.

RICHTSMEIER


Table 2.

Binding energy

R

5

ET AL.

Small Group

IA

185

Clusters

Ab i n i t i o Geometries and Energies f o r L i and N a 3

b e

Q

c e

Binding energy

R a

b

Q

e

c e__

L i (\)

2

Li ( B ) 3

and IB

2

3

34.0

2.77

71

33.3

2.73

54

la

33.6

2.84

73

33.4

2.79

54

44

29.4

2.80

68

29.4

2.70

52

43

28.7

2.96

74

—

—

—

4

51.5

3

Na

2

Na ( B ) 3

26.2

a

2

3.31

73

25.6

3.22

3

Energy i n kcal/mol r e l a t i v e to d i s s o c i a t i o n to three atoms.

b

o E q u i l i b r i u m bond length i n A.

o l a . u . = 0.529177A.

E q u i l i b r i u m bond angle i n degrees.


3

METAL

186

BONDING A N D

INTERACTIONS

2

s t a t e l i e s lowest i n energy; the A i s t a t e has a s l i g h t l y higher but comparable energy. Various c a l c u l a t i o n s i n d i c a t e that the l i n e a r form l i e s between 1 and 6 kcal/mole above the B state. Gole et a l . (4), i n f a c t , f i n d the l i n e a r form to be at a secondary minimum on the p o t e n t i a l energy surface rather than at a saddle p o i n t . A comparison of the d i f f e r e n t c a l c u l a t i o n s demons t r a t e s that v a r i o u s b a s i s s e t s t r e a t d i f f e r e n t geometries at d i f f e r e n t l e v e l s of approximation. Apparently, i t i s d i f f i c u l t to o b t a i n a moderate s i z e d b a s i s set which i s e q u a l l y appropriate f o r both the l i n e a r and h i g h l y bent forms. In h i g h l y f l u x i o n a l molecules such as the a l k a l i t r i m e r s , where small energy d i f ferences between s t r u c t u r e s of v a s t l y d i f f e r e n t geometry are present, s i g n i f i c a n t care must be taken i n choosing the b a s i s set. I t should a l s o be noted that these molecules are v i r t u a l l y unbound at the SCF l e v e l and that b i n d i n g i s obtained only when c o r r e l a t i o n e f f e c t s are considered, e.g., with the use of CI techniques. M a r t i n and Davidson (3) have examined the s t r u c t u r e of the sodium trimer at the SCF-CI l e v e l and f i n d r e s u l t s that are very s i m i l a r to those which have been obtained f o r L i 3 . Na has an optimum C v geometry w i t h B symmetry, a A i c o n f i g u r a t i o n corresponding to a saddle point l y i n g only 0.6 kcal/mol higher i n energy. The l i n e a r form l i e s only 3 kcal/mol above the m i n i mum B geometry. The Na molecule i s bound with respect to the d i s s o c i a t i o n l i m i t Na + Na by 8.5 kcal/mol. 2


2

3

2

2

2

2

2

2

3

2

+

B. P o t e n t i a l Surfaces f o r L i and L i . The p o t e n t i a l energy surfaces f o r the anion and c a t i o n of L i (4) are s i g n i f i c a n t l y simpler i n form than that of the n e u t r a l . At the D geometry, the anion has an a e c o n f i g u r a t i o n , the minimum corresponding to a t r i p l e t coupled s t a t e . The s i n g l e t s t a t e , E , a r i s i n g from the a e c o n f i g u r a t i o n must J a h n - T e l l e r d i s t o r t . Indeed, the d i s t o r t i o n i s q u i t e l a r g e and the anion i s found to have a geometry (^g" ") (see F i g . l a ) . The ^g" s t a t e i s , i n f a c t , the lowest l y i n g s t a t e of L i . In c o n t r a s t , the removal of an e l e c t r o n from L i ( D h ) leads to an a e ° e l e c t r o n c o n f i g u r a t i o n f o r the c a t i o n and the D ^ geometry i s the most s t a b l e s t r u c t u r e (see F i g . l b ) . 3

3

3

3 n

2

2

1

2

f

2

1

1-

3

2

3

3

3

C.

E l e c t r o n A f f i n i t i e s and I o n i z a t i o n P o t e n t i a l s . 1. SCF-CI r e s u l t s f o r l i t h i u m trimer. Various values f o r the e l e c t r o n a f f i n i t y and i o n i z a t i o n p o t e n t i a l are given i n Table 3 and i n F i g . 1. The exact d e f i n i t i o n of v e r t i c a l and a d i a b a t i c energy increments i s complicated by the f l u x i o n a l nature of the L i surface and the presence of a low-lying t r i p l e t surface f o r L i . As noted p r e v i o u s l y , the a d i a b a t i c energy i s defined as the energy d i f f e r e n c e between the minimum energy geometries of the i o n and the n e u t r a l . A secondary a d i a b a t i c quant i t y i s defined w i t h r e l a t i o n to the l i n e a r minimum on the L i surface. A v e r t i c a l energy i s defined as the energy d i f f e r e n c e 3

3

3



3

3

1

1

3

s

+

3

3

2

3h

3

3

+

2

3

3

3

+

3

3

Figure 1. Potential energy surfaces as a function of bond angle for Li , Li ~ and Li obtained from SCF-CI calculations at a bond distance of 5.6 a.u. a: Cut in surfaces for Li and Li ~. Key: A, Li ; O , Li ~ A ; and Li ~ A '(D ). Various vertical electron affinities are represented, b: Cut in surfaces for Li and Li . Key: O , Li and A, Li \ Various vertical ionization potentials are represented. Dissociation to Li + Li at 5.24 eV (4).


METAL


188


Table 3. Summary of E l e c t r o n Binding Energies i n eV f o r L i ^

Quantity

AE(SCF)

AE(C1)

Adiabatic electron a f f i n i t y

0.46

1.10

Secondary a d i a b a t i c e l e c t r o n a f f i n i t y

0.59

1.14

V e r t i c a l detachment

0.66

1.22

Neutral v e r t i c a l electron a f f i n i t y

0.30

0.67

Adiabatic ionization p o t e n t i a l

3.64

3.95

Secondary a d i a b a t i c i o n i z a t i o n potential

3.47

3.91

Vertical ionization potential ( n e u t r a l minimum)

3.74

4.03

4.59

4.39

energy (6.0 a.u.)

Vertical ionization potential (secondary n e u t r a l minimum linear configuration)


12.

RICHTSMEIER ET AL.

Small Group

IA

and

IB

189

Clusters

at a given geometry of the n e u t r a l or the i o n . Table 3 demons t r a t e s that c o r r e l a t i o n e f f e c t s play a s i g n i f i c a n t r o l e i n the c a l c u l a t i o n of the e l e c t r o n a f f i n i t y . At the CI l e v e l , the adiabat i c e l e c t r o n a f f i n i t y i s 1.10 eV while the v e r t i c a l detachment energy i s 1.22 eV (6.0 au.). The v e r t i c a l attachment energy f o r the a d d i t i o n of a t r i p l e t coupled e l e c t r o n i s only 0.67 eV and corresponds to formation of a A2^ anion r a t h e r than a s i n g l e t species. The a d i a b a t i c and v e r t i c a l i o n i z a t i o n p o t e n t i a l s f o r L i 3 are very s i m i l a r , both being approximately 3.95 eV. This results because of the s i m i l a r geometries f o r the B s t a t e of L i 3 ( C v ) and the Ai s t a t e of L i 3 ( D h ) . I t must be noted, however, that the v e r t i c a l i o n i z a t i o n process f o r the removal of an e l e c t r o n from l i n e a r L i to g i v e l i n e a r L i 3 leads to the higher i o n i z a t i o n p o t e n t i a l , 4.39 eV. I f both C and forms are present i n an experiment, a complicated t h r e s h o l d dependence f o r the i o n i z a t i o n process w i l l be observed (4). The r e s u l t s f o r H 3 exemp l i f y that the f l u x i o n a l nature of a s m a l l metal c l u s t e r may complicate the experimental determination of e l e c t r o n a f f i n i t i e s and i o n i z a t i o n p o t e n t i a l s . 3

2

2

2

+

l


3

+

3

2 v

2. F u r t h e r c a l c u l a t i o n of e l e c t r o n a f f i n i t i e s f o r l i t h ium and sodium c l u s t e r s . The e l e c t r o n a f f i n i t i e s of L i , L i 3 , L i 5 , Na and Na3 have been c a l c u l a t e d (16) using the more a p p r o x i mate methods d i s c u s s e d p r e v i o u s l y . Since L i ~ and Na " are open s h e l l anions, the EA's f o r L i and Na are c a l c u l a t e d using the d i f f e r e n c e between the UHF energy of M ~ and the RHF energy of M . Because the anions of L i , L i s , and Na3 are c l o s e d s h e l l s p e c i e s , the EA's f o r t h e i r n e u t r a l c l u s t e r s can be evaluated using Koopmans theorem. Geometries f o r L i " and L i 3 ~ were taken from prev i o u s l y optimized v a l u e s . The s t r u c t u r e determined f o r N a " was geometry optimized u s i n g the g r a d i e n t method (48) i n symmetry f o l l o w i n g the r e s u l t s found f o r L i 3 ~ . The geometry f o r Na _ was estimated u s i n g the r a t i o R ( L i ~ " ) / R e ( L i ) to o b t a i n R (Na ~")/ R (Na ). The L i s geometry was presumed to be Dsh with a bond length of 6.0 au. The v a r i o u s EA's determined u s i n g the above p r e c i p i t i o n are given i n Table 4 where they are compared w i t h other v a l u e s . The present e l e c t r o n a f f i n i t y f o r L i i s the h i g h e s t value yet calculated. T h i s i s probably the r e s u l t of the q u a l i t y of the b a s i s set which, at present, i s the l a r g e s t employed to d e s c r i b e the molecule. There i s good agreement w i t h our values f o r E A ( L i ) and EA(Na ) and those of Shepard et a l . (5) which were determined i n the same f a s h i o n . As a f u r t h e r comparison, we have a l s o given the values of the EA estimated from an a p p l i c a t i o n of Koopmans theorem to UHF wavefunctions f o r L i " and Na ". Koopmans theorem r i g o r o u s l y a p p l i e s only to c l o s e d s h e l l RHF c a l c u l a t i o n s and, as expected, the current r e s u l t demonstrates that the EA i s s e r i o u s l y overestimated. (The a c t u a l EA f o r L i i s e s t i mated to be between 0.50-0.65 eV.) (15). These r e s u l t s should a l s o be compared to the SCF-CI c a l c u l a t i o n of Dixon et a l . (15) 2

2

2

2

2

2

2

2

3

T

2

3

2

e

e

2

2

e

2

2

2

2

2

1

2

2

1

2


190

METAL

Table 4.

Molecule

Approximate E l e c t r o n A f f i n i t i e s f o r A l k a l i Metal C l u s t e r s i n eV

Method

3

EA(ev)

Ref.

KT(UHF)

0.92(V)

T h i s work

Li,

AE(UHF-RHF)

0.49(A)

T h i s work

Li,

AE(UHF-RHF)

0.46(A)

5

Li

2

AE(SCF-CI)

0.45(A)

15

Li

3

KT(RHF)

1.00(V)

4

Li

3

AE(SCF-CI)

1.10(A)

4

KT(RHF)

1.26(V)

4

Li Downloaded by UNIV OF LEEDS on January 16, 2018 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch012


L ±

2

5

Na

2

KT(UHF)

0.85(V)

T h i s work

Na

2

AE(UHF-RHF)

0.43(A)

T h i s work

Na

2

AE(UHF-RHF)

0.42(A)

5

Na

3

KT (RHF)

0.93(V)

T h i s work

^T

= Koopmanns' theorem, UHF = s p i n u n r e s t r i c t e d

Hartree-Fock, RHF = r e s t r i c t e d Hartree-Fock, SCF-CI = S e l f c o n s i s t e n t f i e l d with c o n f i g u r a t i o n i n t e r a c t i o n , AE(UHF-RHF) = E(M "(UHF))-E(M (RHF)), n

n

AE(SCF-CI) = E(M "(SCF-CI))-E(M (SCF-CI)). n

n

Êlectron a f f i n i t y . V = v e r t i c a l , A = adiabatic.


12.


Small Group

IA and IB

191

Clusters

who, i n t h e i r study, employed a b a s i s set of S l a t e r type o r b i t a l s somewhat smaller than that needed to provide s u f f i c i e n t f l e x i b i l i t y i n the valence space d e s c r i b i n g the anion. Consequently, t h e i r determined value f o r the EA was too low. The e l e c t r o n a f f i n i t y has a l s o been determined using the equations-of-motion method (49,50), however the p r e d i c t e d value i s too large because the b a s i s set used i n these c a l c u l a t i o n s i s h e a v i l y biased i n favor o f the anion. The value f o r EA(M ) determined by Koopmans theorem f o r M ~ (closed s h e l l ) corresponds t o a v e r t i c a l detachment energy a t the optimum s t r u c t u r e of Mn~. For the a l k a l i trimers t h i s does not introduce l a r g e e r r o r s s i n c e the energies of the l i n e a r and C v forms are q u i t e s i m i l a r . The value obtained a t the SCF-CI l e v e l (4) i s 1.14 eV f o r L i 3 w h i l e the a p p l i c a t i o n o f Koopmans' theorem y i e l d s 1.00 eV. These values are i n good agreement. The value obtained from Koopmans theorem f o r N a i s 0.93 eV, s i m i l a r to that found f o r L i . We thus expect the a d i a b a t i c EA f o r N a to l i e between 1.0 and 1.1 eV. As discussed p r e v i o u s l y , the i n t e r p r e t a t i o n o f e l e c t r o n a f f i n i t i e s f o r the a l k a l i trimers i s comp l i c a t e d by the f l u x i o n a l nature o f t h e i r p o t e n t i a l energy surfaces. S i g n i f i c a n t l y , more complication a r i s e s f o r s l i g h t l y l a r g e r c l u s t e r s as a r e s u l t o f the l a r g e number of p o s s i b l e geom e t r i c a l isomers. In studying the L i s c l u s t e r and i t s negative i o n , a number of s t r u c t u r e s should be i n v e s t i g a t e d . A few s t u d i e s have been completed. The v e r t i c a l detachment energy f o r the negative i o n c a l c u l a t e d f o r a Dsh geometry i s 1.26 eV. DIM c a l c u l a t i o n s (51) p r e d i c t that the Dsh n e u t r a l s t r u c t u r e , which must J a h n - T e l l e r d i s t o r t , i s s i g n i f i c a n t l y l e s s s t a b l e than the more compact pseudot r i g onal-bipyramidal s t r u c t u r e . Detachment from t h i s t r i g o n a l bipyramidal s t r u c t u r e would y i e l d a somewhat lower a d i a b a t i c EA i f the Dsh geometry i s optimum f o r L i s " . 3. General comments on e l e c t r o n a f f i n i t i e s and i o n i z a t i o n p o t e n t i a l s . From the i n v e s t i g a t i o n s c a r r i e d out thus f a r , i t does appear that the e l e c t r o n a f f i n i t y increases with i n c r e a s i n g odd numbers of metal atoms i n a c l u s t e r . Experimentally, i t i s w e l l known that i o n i z a t i o n p o t e n t i a l s (IP) show an even-odd dependence on c l u s t e r s i z e , where even n c l u s t e r s have a higher IP than odd n c l u s t e r s (52). For e l e c t r o n a f f i n i t i e s the opposite e f f e c t probably e x i s t s w i t h even n c l u s t e r s having lower EA's than odd c l u s t e r s . These r e s u l t s can be explained i n terms o f simple o r b i t a l models. The i o n i z a t i o n p o t e n t i a l i s determined by removing an e l e c t r o n from the highest occupied molecular o r b i t a l (HOMO). F o r an even n c l u s t e r , i n a s i n g l e t s t a t e , the e l e c t r o n which i s r e moved w i l l see a greater nuclear charge than w i l l an e l e c t r o n i n the HOMO o f an odd n c l u s t e r i n a doublet s t a t e . This r e s u l t s because the n u c l e a r charge f o r a c l o s e d s h e l l o r b i t a l i s only p a r t i a l l y screened by the other e l e c t r o n i n that o r b i t a l . I n c o n t r a s t , an odd e l e c t r o n i n what i s probably a more d i f f u s e HOMO 1

n

n


2

1

3

3


3

METAL

192

BONDING

A N D INTERACTIONS


sees an e f f e c t i v e n u c l e a r charge of one s i n c e the other c l o s e d s h e l l e l e c t r o n s screen the remaining nuclear charge. For s i m i l a r reasons, the a d d i t i o n of an e l e c t r o n to form an anion w i l l lead to an opposite e f f e c t . The a d d i t i o n of an e l e c t r o n to an open s h e l l HOMO allows t h i s e l e c t r o n to experience a l a r g e r n u c l e a r charge as a r e s u l t of the p a r t i a l screening e f f e c t of the other e l e c t r o n . T h i s , of course, i s a more s i g n i f i c a n t e f f e c t than the a d d i t i o n of an e l e c t r o n to the LUMO of a c l o s e d s h e l l molecule (an even n s i n g l e t c l u s t e r ) . In t h i s l a t t e r case, the c l o s e d s h e l l e l e c t r o n s screen most of the nuclear charge so that the added e l e c t r o n i s more l o o s e l y bound, r e s u l t i n g i n a lower e l e c tron a f f i n i t y . C a l c u l a t i o n s Based on the Diatomics-in-Molecules Method A. Comparison w i t h a b - i n i t i o r e s u l t s - LJ3 and Na . In order to demonstrate the q u a l i t y of the r e s u l t s which we have obtained using the DIM method, we have compared our r e s u l t s f o r L i 3 and Na with v a r i o u s ab i n i t i o c a l c u l a t i o n s (see Table 2 and Table 5). We have a l s o compared the r e s u l t s obtained with our synthesized t r i p l e t curve f o r L i w i t h those obtained using the more accurate curve determined by Konowalow (36). Our value f o r the atomization energy to three atoms i s i n good agreement w i t h the best ab i n i t i o value obtained by Gerber and Schumacher ( l a ) . The bond angle which we determine using DIM i s too l a r g e and the bond d i s t a n c e i s too long. However, because the surface i s very f l a t , the agreement i s reasonable. The comparison i s important s i n c e i t i s w e l l known that the simple DIM method s e r i o u s l y overestimates the s t a b i l i t y of the s t r o n g l y bent molecule Hi» (53) (Difh symmetry). The r e s u l t s which we o b t a i n u s i n g the more accurate t r i p l e t curve of Konowalow show a b i n d i n g energy 1.7 kcal/mol l a r g e r than the ab i n i t i o r e s u l t s and a bond l e n g t h that i s i n e x c e l l e n t agreement with the ab i n i t i o v a l u e s . The bond angle i n c r e a s e s s l i g h t l y and i s s t i l l too l a r g e when compared w i t h more accurate c a l c u l a t i o n s . The shorter bond l e n g t h found using the ab i n i t i o t r i p l e t curve can be a t t r i b u t e d to the smaller value of the O parameter i n the LennardJones p o t e n t i a l ; the aib i n i t i o t r i p l e t curve i s l e s s r e p u l s i v e i n t h i s r e g i o n than our constructed t r i p l e t curve. The l a r g e r b i n d ing energy found f o r the trimer r e f l e c t s the smaller value of O obtained from the ab i n i t i o t r i p l e t curve. T h i s can e a s i l y be demonstrated u s i n g s i m p l i f i e d expressions d e r i v e d from equation (1). In terms of the p o t e n t i a l energy curves f o r a C v geometry w i t h atom 2 at the apex (29a): 3

3

2

2

2

1

+

E( B ) = 3/2 Z (l-2) 2

2

g

1

+

E ( A ) = 1/2 2: ( l - 2 ) + h 1

3

+ l/2 Z +

+ u

3

(l-2) + E 3

+ u

+

(l-3) = E

( l - 3 ) + 3/2 Z (l-2) u

+

= E_


(10) (11)


8.01

7.94

5.30

6.16

7.51

7.89

8.37

Li

Na

K

Rb

Cs

2

2

a

14.0

17.3

18.1

23.7

35.3

33.0

BE (kcal/mol) 12

8.71

8.25

7.91

6.97

6.26

6.44

R (a.u.)

^ R e s u l t s u s i n g the t r i p l e t curve from Ref. 36.

b i n d i n g energies w i t h r e s p e c t to d i s s o c i a t i o n t o three atoms.

70.6

71.5

9.22

9.67

73.2

85.8

97.0

92.1

0

2

13

8.30

7.81

7.41

5.81

5.09

5.06

R (a.u.)

0

56.9

56.5

55.9

49.3

48.0

46.3

The DIM S t r u c t u r e s of Pure A l k a l i M e t a l Trimers

8.96

8.39

+

E ( B )

5.56

13

R (a.u.)

Li

12

R (a.u.)

Alkali

Table 5.


a

13.8

17.1

17.9

23.4

34.2

33.3

BE (kcal/mol)

194

METAL

BONDING A N D

3

INTERACTIONS

+

An increase i n the w e l l depth of the E curve w i l l lead to an increase i n the b i n d i n g energy o f the B s t a t e s i n c e the E ( l - 2 ) term i s r e p u l s i v e (ra). In a f u r t h e r comparison between the DIM and ab i n i t i o r e s u l t s , we focus on the s t r u c t u r e and energy of the ^ A i s t a t e of L i 3 . The bond angle determined f o r t h i s s t a t e i s s l i g h t l y smaller than that found using ab i n i t i o c a l c u l a t i o n s ; however, the bond length ( R 1 3 ) i s again i n good agreement. The binding energy i s a l s o i n good agreement and DIM p r e d i c t s that the A i s t a t e l i e s only 1.1 kcal/mol above the B s t a t e . This should be compared with a value o f 0.7 kcal/mol determined from the ab i n i t i o r e s u l t s o f Gerber and Schumacher ( l a ) . The DIM method p r e d i c t s that the l i n e a r conformation i s very c l o s e i n energy to the optimum geometry. Although the determined bond d i s t a n c e i s i n good agreement with ab i n i t i o , r e s u l t s , the l i n e a r geometry i s p r e d i c t e d t o be too s t a b l e at the DIM l e v e l . Very s i m i l a r comparisons can be made f o r Na3. The bond length and binding energy o f the B s t a t e a r e i n good agreement with the ab i n i t i o r e s u l t s o f M a r t i n and Davidson (_3) while the bond angle i s again p r e d i c t e d t o be too l a r g e . The s t r u c t u r e and energy of the A i c o n f i g u r a t i o n are i n e x c e l l e n t agreement with the ab i n i t i o r e s u l t s . Again, the energy of the l i n e a r form evaluated at the DIM l e v e l i s too c l o s e t o that o f the B configurat i o n . These comparisons demonstrate that f o r L i 3 and Na3 the DIM method y i e l d s r e s u l t s i n reasonable agreement with ab i n i t i o s t u d i e s . Therefore, we f e e l that i t can be a p p l i e d to the study of the s t r u c t u r e s and energetics of other a l k a l i metal c l u s t e r s and w i l l y i e l d reasonable and u s e f u l r e s u l t s . U

2

2

3

+

3

u

+

u

2

2


2

2

2

2

2

2

B. Parameters Governing P r e d i c t e d S t a b i l i t i e s . The important diatomic terms governing the s t a b i l i t i e s of the B ( E ) and A i ( E _ ) s t a t e s are i n d i c a t e d i n equations 10 and 11. The t r i p l e t curves f o r the a l k a l i dimers have small w e l l s , t h e i r O values being c l o s e to the value of r f o r the s i n g l e t curve. They are not s t r o n g l y r e p u l s i v e i n the region where the a t t r a c t i v e s i n g l e t i n t e r a c t i o n i s l a r g e . The s i n g l e t i n t e r a c t i o n w i l l t h e r e f o r e tend to be the dominant term i n the sum d e s c r i b i n g the binding energies of these molecules. These p r o p e r t i e s w i l l lead to the p r e d i c t i o n o f molecular c l u s t e r s which are s t r o n g l y bound. In c o n t r a s t , the simplest t h r e e - e l e c t r o n t r i a t o m i c , H 3 , has an energ e t i c a l l y unstable s t r u c t u r e with respect to the asymptotic limit, atom plus diatomic. The t r i p l e t curve f o r H i s s t r o n g l y r e p u l s i v e i n the region where the s i n g l e t curve i s s t r o n g l y a t t r a c t i v e . The c a n c e l l a t i o n o f energies i s such that the r e p u l s i v e t r i p l e t e f f e c t s are dominant. T h i s leads to the p r e d i c t i o n of an unstable H 3 s p e c i e s . This argument i s s i m i l a r to that i n i t i a l l y employed by Cashion and Herschbach i n d i s c u s s i n g H (29a) and l a t e r by Dixon, Stevens and Herschbach (54) i n c o n s i d e r i n g H6. 2

2

+

2

e

2

3


12.


Small Group

IA

and IB

195

Clusters

C. General Form o f P o t e n t i a l Energy Surface. The general form of the p o t e n t i a l energy s u r f a c e f o r the a l k a l i trimers i s depicted i n F i g . 2 f o r the L i 3 molecule. There are two low-lying states, B and A i , r e s u l t i n g from a d i s t o r t i o n i n C symmetry, which cross a t a D ^ c o n i c a l i n t e r s e c t i o n ( E ) . These B and A i s t a t e s are l a b e l l e d as E and E_ ( F i g . 2a) r e s p e c t i v e l y since they a r e d e r i v e d from equation (3) g i v i n g equations 10 and 11 r e s p e c t i v e l y . The B ( E . ) s t a t e can e a s i l y d i s t o r t to the l i n e a r form bending past 180 to reach another c o n i c a l i n t e r s e c t i o n . The A\ and B s t a t e s can be connected by an asymmetric motion p l a c i n g the three atoms i n a C symmetry c o n f i g u r a t i o n and allowing the two s t a t e s to mix. T h i s i s a l s o represented i n F i g . 2. An expanded view o f the c o n i c a l i n t e r s e c t i o n i s given i n F i g . 2b. There are two other i d e n t i c a l sets of C d i s t o r t i o n s located at 120° and 240° on the hypersurface r e l a t i v e to the f i r s t C d i s t o r t i o n . These three d i s t o r t i o n s a l l i n t e r s e c t a t t h e i r c o n i c a l i n t e r s e c t i o n s . Because o f the f l u x i o n a l nature of the B s t a t e and the low energy motion r e q u i r e d to generate the l i n e a r form, the hypersurface i s more complicated than that f o r other molecules (trimethylenemethane) which e x h i b i t s i m i l a r behavior (55). We note that our curves f o r E and E_ are very s i m i l a r to those obtained by Gerber and Schumacher ( l a ) i n t h e i r f i t of the Li3 s u r f a c e . The s t r u c t u r e s o f the a l k a l i t r i m e r s are summarized i n Table 5. F o r comparison, experimental r e s u l t s f o r the dimer are given i n Table 2. 2

2

2

2 v

2

f

2

3

2

2

+

2

2

2

2

2


s

2 v

2 V

2

2

+

D. P e r i o d i c i t y o f M Molecular S t r u c t u r e . The general beh a v i o r f o r M3 a l k a l i species i s s i m i l a r to that found p r e v i o u s l y i n our d i s c u s s i o n o f L i 3 and Na3. The lowest energy s t r u c t u r e corresponds to the B s t a t e (C2V, obtuse angle) with the A i conf i g u r a t i o n a t s l i g h t l y higher energy. The £ linear structure i s a l s o of comparable energy to the B s t a t e . The bond angle f o r the B s t a t e decreases w i t h the i n c r e a s i n g s i z e of the a l k a l i atom, approaching a v a l u e o f 60°. The bond lengths, r i 2 and r2 3, ( l a b e l l e d with atom 2 as the unique atom), are s l i g h t l y longer than those found f o r the diatomic. The bond angle f o r the A i c o n f i g u r a t i o n i n c r e a s e s w i t h the i n c r e a s i n g s i z e of the a l k a l i atom a l s o approaching 60°. The trimers a r e a l l bound with r e s pect to the d i s s o c i a t i o n l i m i t , atom p l u s diatomic. We f i n d that the b i n d i n g energy decreases with i n c r e a s i n g atomic number; however, i t i s perhaps more r e l e v a n t to compare t h i s b i n d i n g energy to the b i n d i n g energy of the diatomic. The lowest r a t i o s are found f o r U 3 and N a w h i l e the r a t i o s f o r K 3 , Rb3 and CS3 are approximately 0.5. The s t r u c t u r e o f the A i c o n f i g u r a t i o n i s i n t e r e s t i n g i n that the bond length r i 3 i s v i r t u a l l y that o f the dimer. I t i s apparent that t h i s s t a t e looks very much l i k e an atom bound to a d i a tomic. The b i n d i n g i s , of course, s i g n i f i c a n t l y greater than a t y p i c a l van der Waal's i n t e r a c t i o n . T h i s may r e s u l t , i n part^ from the l a r g e p o l a r i z a b i l i t i e s o f the a l k a l i atom and of the a l k a l i diatomic. In a d d i t i o n , there may a l s o be a p o s s i b l e small admixture of the i o n i c c o n f i g u r a t i o n s M~M and M M ~. 3

2

2

2

2

+

u

2

2

2

2

3

2

+

2

+

2


196


M E T A L BONDING A N D INTERACTIONS

Figure 2.

Scheme of Li potential energy surface obtained from DIM calculations. 3

a: Full surface representing one C distortion as a function of bond angle. The bond distances are taken from the optimized E (r = 5.3) and E. (r = 6.26) structures. The binding energy corresponds to dissociation to three atoms. Two surfaces cross at 60° giving a E state. DIM states E and E_ correspond to the B and A electronic states. The • - • shows the general form of the asymmetric stretch distortion in C symmetry which yields a low energy path between the B and A states. 2v

+

2

2

+

2

2

t

s

2

2

2

±

b: Expanded view of the E' intersection region showing the full hypersurface. The three equivalent C distortions are shown by three sets of curves. At values of 0 60°, the E state ( B ) is dominant. The B states all lead to linear geometries, 6 = 180°. 2

2v

2

t

+

2

2

2

2


12.

RICHTSMEIER

ET AL.

Small Group

IA

and

IB

197

Clusters

E. Comparison with Previous C a l c u l a t i o n s . A number of previous semi-empirical c a l c u l a t i o n s on a l k a l i trimers have been reported and are summarized by Companion. Companion (56) has used the DIM method to study L i 3 and f i n d s r e s u l t s e s s e n t i a l l y i d e n t i c a l to those reported here except that the bond angle f o r the B s t a t e i s s l i g h t l y l a r g e r . Pickup (57), i n studying H 3 , derived an elegant approach to generate the DIM matrices. Emp l o y i n g a t r i p l e t curve with a smaller w e l l depth, Pickup found s l i g h t l y smaller b i n d i n g energies. Whitehead and G r i c e (58) have a l s o examined L i 3 and Na ( a l s o N a L i and L i N a ) with the DIM method. These authors employed a t r i p l e t curve with a very s h a l low w e l l , f i n d i n g the l i n e a r forms to be more s t a b l e than the B C s t r u c t u r e s f o r both L i 3 and Na3. They a l s o s i g n i f i c a n t l y underestimate the b i n d i n g energy of the B s t r u c t u r e . In examining the A i state,formed by a l l o w i n g an atom to approach a d i a tomic, they f i n d a r e l a t i v e energy lower than that of the B s t a t e f o r both L i 3 and Na3, i n c o n t r a s t to the present DIM and ab i n i t i o r e s u l t s . We b e l i e v e the current r e s u l t s (and those of Companion f o r L i 3 ) are more r e l i a b l e s i n c e a more accurate r e p r e s e n t a t i o n of the t r i p l e t curve has been employed. Two pseudop o t e n t i a l c a l c u l a t i o n s have been c a r r i e d out f o r some of the a l k a l i t r i m e r s . Hart and Goodfriend (59) report no binding energies and assume a l i n e a r geometry. The bond lengths determined i n these c a l c u l a t i o n s show moderate agreement with our r e s u l t s . Pickup and Byers-Brown (60), using a d e s c r i p t i o n c l o s e to the SCF l e v e l of approximation with a small " s " b a s i s s e t , found no e v i dence that K and Na3 are bound with respect to monomer plus dimer. T h i s i s not unreasonable i f we consider the previous d i s c u s s i o n of ab i n i t i o c a l c u l a t i o n s on the L i 3 surface where c o r r e l a t i o n e f f e c t s are shown to account f o r most of the binding energy. 2

2

3

2

2

2

2

2 v


2

2

2

2

2

3

F. Extension of DIM t o the Group IB Trimers. The Group IB and IA trimers are q u i t e analogous. P r e v i o u s l y we have reported the r e s u l t s of DIM c a l c u l a t i o n s on the s t r u c t u r e of the Group IB trimers with a focus on the B surface (61). We report here new features of the Group IB surfaces and compare c a l c u l a t e d binding energies with the recent experimental work of H i l p e r t and Ginger i c h (62). In a d d i t i o n , we have a l s o explored the A i surface of the t r i m e r s . The bending p o t e n t i a l s f o r the A i and B s t a t e s of the pure Group IB trimers a r e shown i n F i g u r e 3. These p o t e n t i a l s are determined using optimized B and A i bond lengths obtained at the optimum bond angle. The energy of the A i s t a t e i s comparable to that of the B s t a t e f o r a l l of the trimers (see Table 6). In f a c t , f o r Ag3, the A i geometry i s p r e d i c t e d to be s l i g h t l y more s t a b l e than the B s t r u c t u r e (0.3 kcal/mol). We c a l c u l a t e that the A i s t r u c t u r e s f o r Cu and A U 3 have bond angles c l o s e to 55° and determine a bond angle f o r A g ( A i ) of 51°. T h i s trend i s 2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

3

2

3


Gole and Stwalley,; Metal Bonding and Interactions in High Temperature Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1982. 2

+

2

2

s

2

3

2

s

Figure 3. DIM potential energy surfaces as a function of bond angle for the pure Group IB trimers. The bond distances are taken from the optimized E ( B ) and E. ( A ) structures. The binding energy corresponds to dissociation to three atoms. The state crossing at 60° is due to the presence of a E' state at this geometry. Key: a, Cu ; b, Ag ; and c, Au .


2 H

§ & > §

00


4.99

4.78

Ag

Au

R

5.44

6.19

4.66

+

2

E ( B )

2

13 (au)

69.3

76.6

65.6

6

79.8

52.1

75.2

a

R

5.07

5.57

4.51

12 (au)

R

e

2 a

4.67

4.73

4.19

.(

3

!)

13 (au)

1

54.8

50.8

55.4

6

Êxperimental

r e s u l t s from Ref.62.

79.2

52.4

75.2

BE (kcal/mol) a

Trimers

a

e x

87.7

60.5

70.3

D

h

P( c» )

BE (kcal/mol) '

The DIM S t r u c t u r e s of Pure Group IB Metal

BE (kcal/mol)

6,

B i n d i n g energies w i t h r e s p e c t to d i s s o c i a t i o n to three atoms.

4.30

Cu

a

12 (au)

Metal

R

Table


b

84.8

58.9

68.7

2 v

exp(C )

a

BE (kcal/mol) '

b

200

METAL


reversed from the values of 8 found f o r the B s t r u c t u r e where 6 i s smaller f o r C u and A u than i t i s f o r A g . H i l p e r t and Ginger i c h (62) have generated the Group IB trimers i n a Knudsen c e l l , and, using mass spectrometry, have measured the atomization energ i e s which are given i n Table 6. The c a l c u l a t e d values are i n good agreement w i t h the experimental q u a n t i t i e s i n view of our neglect of v i b r a t i o n a l e f f e c t s . H i l p e r t and G i n g e r i c h (62) have i n t e r p r e t e d t h e i r r e s u l t s assuming a l i n e a r s t r u c t u r e f o r the metal trimer i n c o n t r a s t to our f i n d i n g of an optimum h i g h l y bent s t r u c t u r e . The use of a C v s t r u c t u r e lowers the experimentally determined binding energies by a few kcal/mol which, f o r A g and Au , y i e l d s c l o s e r agreement between theory and experiment. 2

2

3

3

3

2

3


3

G. C a l c u l a t i o n o f V i b r a t i o n a l Frequencies. We have c a l c u l a t e d v i b r a t i o n a l frequencies f o r the i s o l a t e d Group IB A i s t r u c tures; these are tabulated i n Table 7 and compared with the prev i o u s l y determined q u a n t i t i e s f o r the B s t a t e . The symmetric s t r e t c h frequencies are s i m i l a r f o r the B and A i s t r u c t u r e s and are s i g n i f i c a n t l y higher than the s t r e t c h i n g frequency of the pure diatomic. The bending frequencies f o r the A i s t a t e cons i d e r a b l y exceed the values found f o r the B s t a t e . The value of 0)3, the asymmetric s t r e t c h i s even lower i n the A i s t a t e than the B s t a t e . In f a c t , a r e a l value of 0)3 f o r A u cannot be determined, the A i s t r u c t u r e representing a saddle point on the trimer surface. The asymmetric s t r e t c h i s q u a l i t a t i v e l y the mode that connects the A i and B s t a t e s v i a a C s t r u c t u r e . There i s no c r o s s i n g of the p o t e n t i a l curves i n the conversion between the s t a t e s s i n c e both s t a t e s belong to the same r e p r e s e n t a t i o n i n C symmetry. T h i s implies that our treatment of the f u l l v i b r a t i o n a l motion i s only q u a l i t a t i v e because we have not included the e f f e c t s of the c o n i c a l i n t e r s e c t i o n , i . e . , the vector p o t e n t i a l term r e q u i r e d by the molecular Aharonov-Bohm e f f e c t has not been i n cluded i n our wave f u n c t i o n s d e s c r i b i n g the nuclear motion. We expect these s t r u c t u r e s t o be h i g h l y f l u x i o n a l and pseudorotating about a 60° bond angle a t moderate temperatures. I f the molecule i s frozen i n a r a r e gas matrix at low temperatures, e.g., at 5°K, the p r e d i c t e d v i b r a t i o n a l frequencies f o r the A i and B states are r e l e v a n t . Under these experimental c o n d i t i o n s the molecules should not have access to the f u l l region about the c o n i c a l i n t e r s e c t i o n (60°). The small value of the asymmetric s t r e t c h demons t r a t e s that low temperatures are required to i s o l a t e c e r t a i n forms. ESR s p e c t r a l s t u d i e s on the s i m i l a r Group IA trimers N a (63) and K (64) show that i s o l a t e d C v s t r u c t u r e s are observed at temperatures l e s s than o r equal to 5°K. Based on our semiq u a n t i t a t i v e comparison o f the a l k a l i and Group IB surfaces i t should be p o s s i b l e to i s o l a t e n o n - f l u x i o n a l forms of the Group IB trimers i n low temperature matrices. The predominant feature of the Group IB trimer surface i s the low b a r r i e r connecting the B and A i s t r u c t u r e s about the c o n i c a l i n t e r s e c t i o n . T h i s has s i g n i f i c a n t r a m i f i c a t i o n s 2

2

2

2

2

2

2

2

2

2

2

2

3

2

2

2

2

s

g

2

2

2

3

3

2

2

2

2


12.

RICHTSMEIER ET AL.


Table 7.

Frequency

Small Group IA and IB

V i b r a t i o n a l Frequencies f o r Group IB Trimers

(cm "*")

a

^(\) 2

U> ( B ) 2

2

2

U) ( B ) 3

V

a

o)

1

2 A

2

I>

Cu

Ag

Au

445

264

276

138

36

57

163

57

143

450

273

290

237

168

168

102

49

_b

460

275

285

= symmetric s t r e t c h , o> = bend, U)^ = asymmetric s t r e t c h .

Îmaginary

C

201

Clusters

2

frequency.

Symmetric s t r e t c h at D~,

geometry.


202

M E T A L BONDING A N D

INTERACTIONS

with regard to the s p e c t r a l f e a t u r e s expected f o r these molecules and w i l l p l a y a key r o l e i f they have s u f f i c i e n t i n t e r n a l energy to access the d i f f e r e n t s t a t e s i n the r e g i o n of the c o n i c a l i n t e r section. I t i s i n t e r e s t i n g to note that the energy of the upper surface i s r e l a t i v e l y h i g h compared to the l o w e s t - l y i n g s u r f a c e except i n the r e g i o n about 9 = 60°. The c o n i c a l i n t e r s e c t i o n a c t u a l l y moderates the two energies so that they i n t e r s e c t at 0 - 60°. As an example, the energy of the B2 s t a t e f o r Au3 i s ~25 kcal/mol above the energy of the A i s t a t e i n the r e g i o n of the A i minimum. 2

2

2


Summary The general s t r u c t u r a l f e a t u r e of the Group IA and Group IB trimers i s the f l u x i o n a l nature of the p o t e n t i a l energy s u r f a c e c h a r a c t e r i z i n g these molecules. Although s p e c i f i c s t r u c t u r a l forms may be i s o l a t e d i n m a t r i c e s a t low temperatures, most gas phase t r i m e r s (except those produced i n very c o l d supersonic expansions) w i l l probably be q u i t e f l u x i o n a l and pseudorotate between the v a r i o u s A i and B2 geometries. T h i s adds compli c a t i o n to the treatment of the v i b r a t i o n a l l e v e l s and the s p e c t r o scopy of these molecules. These s m a l l metal trimers appear to be q u i t e simple a t f i r s t glance y e t t h e i r s t r u c t u r a l f e a t u r e s suggest that they a r e q u i t e complicated and w i l l o f f e r new i n s i g h t s about bound but h i g h l y f l u x i o n a l molecules. They represent a new but extremely important form of chemical s p e c i e s . 2

2

Acknowledgement T h i s work was supported, i n p a r t , by N a t i o n a l Science Found a t i o n Grants CHE-7905985 (David A. Dixon) and CHE-7909075 (James L. Gole). David A. Dixon i s an A l f r e d P. Sloan Founda t i o n Fellow (1977-78), a Camille and Henry Dreyfus Teacher Scholar (1978-1983), and a Dupont Young F a c u l t y Grantee (1978). Literature Cited 1.

2.

3. 4. 5.

a)

Gerber, W. H.; Schümacher, E. J. Chem. Phys. 1978, 69, 1692. b) Dunning, T. H., J r . J. Chem. Phys. 1980, 73, 2304. a) Konowalow, D. D.; Olson, M. L. J. Chem. Phys. 1979, 71, 459. b) Ermler, W. C.; Lee, Y. S.; P i t z e r , K. S. J. Chem. Phys. 1979, 70, 293. M a r t i n , R. L.; Davidson, E. R. Mol. Phys. 1978, 35, 1713. Gole, J . L.; C h i l d s , R. H.; Dixon, D. A.; Eades, R. A. J . Chem. Phys. 1980, 72, 6368. Shepard, R.; Jordan, K. D.; Simons, J . J. Chem. Phys. 1978, 69, 1788.


12.


6.

Eades, R. A.; Scanlon, K.; Overend, J . ; Dixon, D. A. unpublished r e s u l t s . a) S i n f e l t , J . H. Acc. Chem. Res. 1977, 10, 15. b) Robinson, A. L. Science, 1974, 185, 772. c) M u e t t e r t i e s , E. L.; Rhodin, T. N.; Band, E.; Bruker, C. F.; P r e t z e r , W. R. Chem. Rev. 1979, 79, 91. d) Stwalley, W. C.; Koch, M. E. O p t i c a l Engineering 1980, 19, 71. Hoffman, R.; Lipscomb, W. N. J. Chem. Phys. 1962, 36, 3489. a) Kunz, P. J. "Atom-Molecule C o l l i s i o n Theory"; Ed. R. B. Bernstein, Plenum, New York, 1979; p. 79 b) T u l l y , J . C. "Modern T h e o r e t i c a l Chemistry: SemiE m p i r i c a l Methods of E l e c t r o n i c S t r u c t u r e C a l c u l a t i o n s " ; Ed. G. A. Segal, Plenum, New York, 1977; V o l . 7B, p. 173. Schaeffer, H. F., Ed.; "Modern T h e o r e t i c a l Chemistry. Methods of E l e c t r o n i c S t r u c t u r e Theory"; Plenum, New York, 1977; p. 111. Johnson, K. H. Annu. Rev. Phys. Chem. 1975, 26, 39. Pople, J . A.; Beveridge, D. L. "Approximate Molecular O r b i t a l Theory"; McGraw-Hill, New York, 1970. Halgren, T. A.; Lipscomb, W. N. J. Chem. Phys. 1973, 58, 1569. See Konowalow, D. D.; Rosenkrantz, M. E. "The E l e c t r o n i c Structure and Spectra o f L i g h t Alkali Diatomic Molecules and t h e i r Molecular Spectra"; ACS Symp. S e r i e s , 1981. See f o r example Wahl, A. C.; Das, G. "Modern T h e o r e t i c a l Chemistry Methods of E l e c t r o n i c S t r u c t u r e Theory"; Ed. H. F. Schaefer, I I I , Plenum, New York, 1977; Chap. 3. See Jordan, K. D.; Kurtz, H. A. "Theory of Metal Atom-Water I n t e r a c t i o n s and Alkali H a l i d e Dimers"; ACS Symp. S e r i e s , 1981. Dixon, D. A.; Gole, J. L.; Jordan, K. D. J. Chem. Phys. 1977, 66, 567. Eades, R. A.; Dixon, D. A.; Gole, J . L. unpublished r e s u l t s . Dixon, D. A.; Eades, R. A.; T r u h l a r , D. G. J. Phys. 1979, B 12, 2741. S h a v i t t I . "Modern T h e o r e t i c a l Chemistry. Methods of E l e c t r o n i c S t r u c t u r e Theory"; Ed. H. F. Schaeffer, I I I , Plenum, New York, 1977; Chap. 6. Goddard, W. A., I I I ; Dunning, T. H., Jr.; Hunt, W. J . ; Hay, P. J. Acc. Chem. Res. 1973, 6, 368. Koopmans, T. P h y s i c a (Utrecht) 1934, 1, 104. Eades, R. A.; Dixon, D. A. J. Chem. Phys. 1980, 72, 3309. a) Dupuis, M.; Rys, J.; King, H. F. J. Chem. Phys. 1976, 65, 111. b) King, H. F.; Dupuis, M.; Rys, J. Nat. Resour. Comput. Chem. Software Cat. 1980, V o l . 1, Prog. No. QHO2 (HONDO). Rothenberg, S.; Kollman, P.; Schwartz, M. E.; Hayes, E. F.; A l l e n , L. C. I n t . J. Quant. Chem. Symp. 1970, 3, 715. Huzinaga, S. J. Chem. Phys. 1965, 42, 1293.

7.


8. 9.

10.

11. 12. 13. 14.

15.

16.

17. 18. 19. 20.

21. 22. 23. 24.

25. 26.

Small

Group

IA

and

IB

Clusters


203

204 27.

28. 29.

30. 31. 32.


33.

34. 35.

36. 37. 38. 39. 40. 41.

42. 43. 44. 45. 46. 47. 48.

49. 50. 51. 52. 53.

METAL


Dunning, T. H., J r . ; Hay, P. J. "Modern T h e o r e t i c a l Chemistry. Methods of E l e c t r o n i c S t r u c t u r e Theory"; Ed. H. F. Schaeffer, I I I , Plenum, New York, 1977; Chap. 1. Glasstone, S.; L a i d l e r , K. J . ; E y r i n g , H. "The Theory of Rate Processes"; McGraw-Hill, New York, 1941. a) Cashion, J. K.; Herschbach, D. R. J. Chem. Phys. 1964, 40, 2358. b) Sato, S. J. Chem. Phys. 1955, 23, 592, 2465. Rumer, G. Göttingen Nachr. 1932, 377. Gelb, A.; Jordan, K. D.; S i l b e y , R. Chem. Phys. 1975, 9, 175. T a y l o r , H. S.; E y r i n g , H.; Sherman, A. J. Chem. Phys. 1933, 1, 68. a) Issacson, A. D.; Muckerman, J. T. J. Chem. Phys. 1980, 73, 1729. b) F a i s t , M. B.; Muckerman, J. T. 1979, 71, 225, 233. P i t z e r , K. S. A c c t s . Chem. Res., 1979, 12, 271. Huber, K. P.; Herzberg, G. "Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules"; Van Nostrand Reinhold, New York, 1979. Olson, M. L.; Konowalow, D. D. Chem. Phys. 1977, 21, 333 and J . Chem. Phys. 1979, 71, 450. Koch, M. E.; Stwalley, W. C.; C o l l i n s , C. B. Phys. Rev. L e t t . 1979, 42, 1052. Moore, C. E. "Atomic Energy L e v e l s " ; NBS C i r c u l a r No. 467 (Government, Washington, D.C.) V o l s . 1 and 2. H i r s c h f e l d e r , J.O.; C u r t i s s , C. F.; B i r d , R. B. "Molecular Theory of Gases and L i q u i d s " ; Wiley, New York, 1954. Helbing, R. K. B.; Rothe, E. W. J . Chem. Phys. 1968, 48, 3945. a) P r i t c h a r d , D.; Carter, G., Chu, F. Y.; Kleppner, D. Phys. Rev. A 1970, 2, 1922. b) P r i t c h a r d , D.; Chu, F. Y. Phys. Rev. A 1970, 2, 1932 Jahn, H. A.; Teller, E. Proc. Roy. Soc. London, Ser. A 1937, 161, 220. Mead, C. A.; T r u h l a r , D. G. J. Chem. Phys. 1979, 70, 2284. Mead, C. A. Chem. Phys. 1980, 49, 23. Davidson, E. R. J. Am. Chem. Soc. 1977, 99, 397. Kendrick, J . ; Hillier, I . H. Mol. Phys. 1977, 33, 635. Bagus, P. S.; d e l Conde, G.; Davies, D. W. Faraday Discuss. Chem. Soc. 1977, 62, 321. Pulay, P. "Modern T h e o r e t i c a l Chemistry. A p p l i c a t i o n s of E l e c t r o n i c S t r u c t u r e Theory"; Ed. H. F. Schaeffer, III, Plenum, New York, 1977; Chap. 4. Simons, J . Annu. Rev. Phys. Chem. 1977, 28, 15. Anderson, E.; Simons, J . J. Chem. Phys. 1976, 64, 4548. Richtsmeier, S.; Dixon, D. A.; Gole, J. L. unpublished results. Herrmann, A.; Schümacher, E.; Wöste, L. J. Chem. Phys. 1978, 68, 2327. Eaker, C. W.; Parr, C. A. J . Chem. Phys. 1976, 65, 5155.


12.

RICHTSMEIER ET AL.

54.

Dixon, D. A.; Stevens, R. M.; Herschbach, D. R. Faraday Discuss. J . Chem. Soc. 1977, 62, 110. Davidson, E. R.; Borden, W. T. J. Am. Chem. Soc. 1977, 99, 2054. Companion, A. L. Chem. Phys. L e t t . 1978, 56, 500. Pickup, B. T. Proc. Roy. Soc. (London) A 1973, 333, 69. Whitehead, J . C.; G r i c e , R. Mol Phys. 1973, 26, 267. Hart, G. A.; Goodfriend, P. L. Mol. Phys. 1975, 29, 1109. Pickup, B. T.; Byers Brown, W. Mol. Phys. 1972, 23, 1189. Richtsmeier, S. C.; Gole, J. L.; Dixon, D. A. Proc. Nat. Acad. S c i . , USA 1980, 77, 5611. H i l p e r t , K.; G i n g e r i c h , K. A. Ber. Bunsen ges. Phys. Chem. 1980, 84, 739. Lindsay, D. M.; Herschbach, D. R.; Kwiram, A. L. Mol. Phys. 1976, 32, 1199; 1980, 39, 529. Thompson, G. A.; Lindsay, D. M. J. Chem. Phys. 1981, 74, 959.

55. 56. 57. 58. 59. 60. 61.


62. 63. 64.

RECEIVED August 26,

Small Group

IA and IB Clusters

1981.


205

Metal Bonding and Interactions in High Temperature Systems

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