Comparison of group IA and group IB homonuclear clusters

Comparison of group IA and group IB homonuclear clusters

379 Surface Science 156 (1985) 379-385 North-Holland, Amsterdam COMPARISON CLUSTERS J. FLAD, OF GROUP IA AND GROUP IB HOMONUCLEAR G. IGEL-MANN, H...

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379

Surface Science 156 (1985) 379-385 North-Holland, Amsterdam

COMPARISON CLUSTERS J. FLAD,

OF GROUP IA AND GROUP IB HOMONUCLEAR

G. IGEL-MANN,

H. PREUSS

and H. STOLL

Institut ftir Theoretische Chemie, Uniuersitiit Stuttgart, Pfaffenwaldring Rep. of Germany Received

10 July 1984; accepted

for publication

55, D - 7000 Stuttgart SO, Fed.

20 July 1984

Results of pseudopotential calculations are compared and Cu. Ag clusters with up to four atoms.

for neutral

and singly ionized

Li, Na, K

1. Method We have recently performed valence-only calculations for group IA and IB clusters [1,2]. While it is difficult to maintain the same level of accuracy for different atoms of a group in all-electron calculations, valence-only results should be well suited for a comparative study of atomic and molecular properties within a group. In our calculations, core-valence interaction is described by semilocal pseudopotentials of the form

In (l), Qx is the core charge, P/ is the projector on angular symmetry I around core X, the parameters Bt and p; are adjusted to atomic energies (cf. refs. [1,2] for details). Which cores should be used for group IA and IB atoms? The choice Q,, = 1 would appear to be natural, at least for the alkalis. But, compared to experiment, large bond-length errors arise for alkali dimers (Ar, = 0.4 au for K,, e.g.), with pseudopotentials (1) adjusted to atomic SCF energies; agreement is only obtained if comparison is made to frozen-core all-electron calculations. The last statement is not even true for group IB dimers; here errors arise already at the frozen-core SCF level (Ar, = -0.3 au for Cu,, e.g.). Which are the reasons for these errors? The error in the IB group is due to the large size of the X+(d”) cores; deviations of the core-core interaction from the point-charge approximation cannot be neglected. The error mentioned for group IA molecules is due to the neglect of core polarization. We 0039-6028/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

380

J. Flud el al. / Group IA and group IB homonuclear clusters

have shown [1,2] that both errors can be effectively corrected for by (i) introducing overlap corrections into the core-core interaction energy, which are determined from frozen-core SCF calculations for X+-X+ ion pairs, and (ii) by adding to the pseudopotentials (1) core-polarization potentials of the form

Here 0~~is the dipole polarizability of core X, and fA is the field generated at the site of core X by the valence electrons and the surrounding cores. The interaction of the valence electrons is described, in our calculations, by the ab initio Hartree-Fock SCF method, with a density-functional approximation for the (valence) correlation energy. The SCF equations are solved using finite Gaussian-type (GTO) basis sets; the basis sets are atom-optimized. of the size 4~2~. It has been shown [1,2] that with this approach an accuracy is obtained of 0.1 au for bond lengths, 0.1 eV for dissociation energies, and 5 cm--’ for vibrational frequencies of group IA and IB dimers. Let us now turn to the results for three- and four-atomic clusters.

2. Structures Of the various cluster structures studied by us, the following ones proved to be lowest in energy: and equilateral triangle (D3,,) for Xl; a Jahn-Teller distorted triangle (with C,, symmetry) for X,; a (planar) rhombus (DZh) for both Xi and X,. The electronic ground states are singlets (doublets) for even (odd) electron number. In the X, case, the (static) Jahn-Teller distortion is to an “obtuse” isosceles triangle (apex angle y > 60”, 2B, ground state), with a single exception: for Li, an acute triangle (*A, state) is lower in energy, in our calculations, but the energy difference to the obtuse one is extremely small here (0.01 eV). It is remarkable that the same equilibrium structures are found for group IA and IB clusters, although the bulk crystal structure is different for Li, Na, K (bee) from that for Cu, Ag (fee). This indicates that the structure of small metal clusters is not directly connected to the symmetry of the crystal lattice. Our calculations do not provide evidence for the stability of linear geometries for three- and four-atomic Cu and Ag clusters discussed by some workers in the past (cf. e.g. ref. (31). The bond lengths r, of the X2 clusters, as well as the nearest-neighbour distances d in the crystal, are quite different for group IA and IB (v,(Kz) = l&~(Cu,)), and also quite different within each group (r,(K,) = l.Sr,(Li,)). The ratio d/r, is similar, however, 1.13-1.20 for Li, Na, K; 1.15-1.16 for Cu, Ag. The results depicted in figs. 1 and 2 are given, therefore, in units of the X, bond lengths.

J. Flad et al. / Group IA and group IB homonuclear clusters

381

Fig. 1 shows the minimal X-X distances rmin in the clusters. These distances are not too different, as expected, from r,(X,) for the neutral clusters. The distances for the ionized clusters are consistently larger than for the corresponding neutral ones, although the differences become smaller, of course, for larger number of atoms. In particular for the alkalis, the Xl bond lengths are in remarkable agreement with the bulk nearest-neighbour distances. While the results for Na and K are rather similar in terms of r,(X,), the Li clusters exhibit smaller relative distances, probably due to the larger p contributions (although the effect may be exaggerated for X, in our calculations - cf. the discussion in ref. [l]). Fig. 2 shows average X-X distances ? between neighbouring atoms in the clusters. It is seen that 7 approaches the bulk values more rapidly than rminr but this effect is more pronounced for group IA than IB. In any case, a distinct alternancy between clusters with even and odd number n of valence electrons is observed; for n odd, the clusters are less strongly bound and the distances are larger. Again, the (relative) Y values are markedly smaller in Li than in Na and K clusters. Note that, for all four-atomic clusters considered, the bulk sequence of 7 is correctly reproduced. Let us now discuss bond angles. The apex angle y of the (obtuse) triangle

(a)

P

r

mln

1.2

. 0.

(b) mln

1.2

1.1

1.0

n

2

3

L

Fig. 1. Minimal X-X distances r,,,,“, in units of X, bond lengths, for X, ( -) and Xz (- -) clusters (X = Li (m), Na (O), K (A) in (a); X = Cu (0). Ag (0) in (b)). Bulk values are indicated by horizontal lines.

J. Flud et al. / Group IA and group IB homonuclear clusters

382

(b)

(8) 1.2

1.2

1.1

1.1

1.0

1.c

n

n

,

2

3

L

Fig. 2. Average X-X bond distances 7. in units of X, - -) clusters (X = Li (m), Na (0). K (A) in (a); X = assumed to be 3 for X,, X-T, and 5 for X,. Xi r = ‘,(X2) is included). Bulk values are indicated by

2

3

i,

bond lengths, for X,, (-) and X,: (-C (0). Ag (0) in (b)). The number of bonds is (the shorter diagonal of the rhombus with horizontal lines.

varies from 78’ to 95” from Li to K, and from 76” to 84” from Cu to Ag. The changes are large, because the potential surface is extremely flat with respect to y. y increases with increasing atomic number, which is consistent with the corresponding decrease of the energy difference between the obtuse triangle and the linear geometry (cf. below). The apex angle of the rhombus, on the other hand, is nearly constant for all X, clusters (50” to 54” for Li to K, 56” for Cu and Ag). because it is essentially fixed by the bonds along the sides and the shorter diagonal.

3. Energies The dissociation energies of the X, clusters show a considerable variation within group IA and IB (D,(Li,) = 2.0D,(K,)), and between the groups (D,(Cu,)= 3.8D,(K,)), but the ratio of the cohesive energy per atom of the bulk crystal to that of the dimer is not too different in all cases (3.1-3.6 for Li to K; 3.5-3.6 for Cu, Ag). The energies, depicted in figs. 3 and 4, are given, therefore, in units of the corresponding X, energies. Fig. 3 shows binding energies per valence electron. The non-monotonous behaviour of this quantity with cluster size is apparent; clusters with an even

J. Flad et al. / Group IA and group IB homonuclear clusters

383

number of valence electrons are favoured. The trend is similar for group IA and IB, although the alternancy is more pronounced in the latter case. The X, energies are lower, in our calculations, than the X, ones for all X with the exception of Li. The experimental results by Hilpert and Gingerich [4] lead to the same conclusion for Na, and Cu, (but not for Ag,). The X, energies are only 30-4056 of the bulk values, still rather similar to the dimer energies. More similar to the bulk are the results for the ionized clusters Xz ,but, as for bond lengths, this similarity is stronger for the alkali group than for Cu and Ag. The enhancement of the bulk cohesive energy from Li, Na to K (and Cu, Ag) does not show up in our calculated values for X, and XT clusters. Fig. 4 shows binding energies per bond. For these quantities, the approach to the bulk values is, as expected, much faster than for binding energies per atom (or valence electron), but more so for X n than for Xl. The curves for the ionized clusters exhibit a characteristic difference between group IA and IB: for Cu and Ag, the dimer ions have smaller dissociation energies than the neutral dimers, while the reverse is true for Li, Na, K. Let us now consider energy differences AE between different structures of a given cluster. For the linear and obtuse-triangle geometries of X,, AE de-

(9)

Ec

E

(b)

c

4

0

0 0

3.0-

2.0-

l.On

n

I

I

I

I

I

t

2

3

4

2

3

4

Fig. 3. Binding energies per valence electron E, (with respect to n X or (n - 1)X + X’ ). in units of and X: (- - -) clusters (X = Li (m), Na (0). K (A) in (a); X = Cu D,(X,)/Z for X, ( -) (III), Ag (0) in (b)). Experimental values for X, [4] are indicated by (non-connected) points, bulk values by horizontal lines.

1.c

G.!

n 0.

2

?

L

) and Xi (---- - -) Fig. 4. Binding energies per bond E,, in units of 13,(Xz). for X,, (clusters (X = Li (W), Na (O), K (A) in (a); X = C’u (cl), A&O) in (b)). The number of bonds is assumed to be 3 for X,, X;; 5 for X,, X; ; 6 for the bulk crystal. Experimental values [4j are indicated by (non-connected) points. bulk values by horizontat lines.

creases from 0.23 to WI@5 for Li to K, and from 0.05 to 0.02 for Cu to Ag (in units of the X, binding energies). This example shows that the malleability of the clusters increases when going down a column of the periodic table. but more so for group IA than IB. An example, where A E increases with atomic number, is the *A,- 2B, separation of X,.

4. Ionization potentials Adiabatic ionization potentials IP (in units of atomic ionization energies) are given in fig. 5. Again, the odd-even alternancy is apparent. Our results are surprisingly similar within each of the two groups considered, but while there is a slight decrease of IP from X to X, for the alkalis, there is an increase for Cu and Ag. This X/X, characteristic has also been observed experimentally [S&j. The aiternancy from X, to X, is more pronounced for group IB than IA clusters, but in any case, the approach to the bulk work function is slow. The bulk work function is larger (in terms of atomic LPs) for group IB than IA metals, and this property seems to be reflected by our cluster results for X,.

J. Ffad et al. / Group IA and group IB homonueiear clusrers

(a)

IP

‘*O# x

385 (b)

IP

l.O-

A

U

t

Y 0.5-

0.5. n

1

2

3

L

Fig. 5. Adiabatic ionization potentials IP, in units of atomic (e), K (A) in (a); X = Cu (a), Ag (0) in (b)). Experimental bulk work Sunctions by horizontal lines.

1

I

,

2

3

n 4

IPs, for X, clusters (X = Li (m), Na values [5,4] are indicated by points,

Acknowledgment

We are grateful to the Deutsche Forschungsgemeinschaft support.

for financial

References 111J. Fiad, G. Igel, M. Dolg, H. Stall and H. Preuss, Chem. Phys. 75 (1983) 331.

PI

J. Flad, G. Igel, H. Preuss and H. Stolt, Ber. Bunsenges. Physik. Chem. 88 (1984) 241; J. Flad, G. Igel-Mann, H. Preuss and H. Stolt, Chem. Phys. 90 (1984) 257. 131 C. Bachmann, J. Demuynck and A. Veillard, Faraday Symp. Chem. Sot. 14 (1980) 170. [41 K. Hilpert and K.A. Ginger&h, Ber. Bunsenges. Physik. Chem. 84 (1980) 739; K. Hilpert, private communication. E. Schumacher and L. Wiiste, J. Chem. Phys. 68 (1978) 2327. (51 A. Herrmann, 161D.E. Powers, S.G. Hansen, M.E. Geusic, D.L. Michalopoulos and R.E. Smalley, J. Chem. Phys. 78 (1983) 2866; M.D. Morse, J.B. Hopkins, P.R.R. Langridge-Smith and R.E. Smalley, J. Chem. Phys. 79 (1983) 5316.