GEOMETRICAL RELAXATION OF MODEL METALLIC CLUSTERS

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GEOMETRICAL RELAXATION OF MODEL

METALLIC CLUSTERS

A. Julg, G. del Re, M. Bourg, V. Barone

To cite this version:

A. Julg, G. del Re, M. Bourg, V. Barone. GEOMETRICAL RELAXATION OF MODEL

GEOMETRICAL RELAXATION

OF

MODEL METALLIC CLUSTERS

A. JULG

Laboratoire de Chimie Quantique, Universit6 de Provence, Place V.-Hugo, 13331 Marseille Cedex 3, France

G. DEL RE (*)

U.E.R. de Physique, UniversitC de Provence M. BOURG

U.E.R. de Physique, Universitk de Provence, Place V.-Hugo, 13331 Marseille Cedex 3, France

and V. BARONE

Cattedra di Chimica Teorica dell' Universitii, via Mezzocannone 4, 80134 Napoli, Italy

R6sum6.

-

Une forme itkrative simple de la mCthode de Huckel simulant un prockdk SCF avec optimisation de la gComCtrie a Btk utiliske pour ktudier la forme des agrBgats finis d'atomes monovalents qui correspondent B la lirnite B des chaines r6guli&res,

B

des rCseaux carrks, B des rBseaux cubiques. Les rksultats sont discutks en termes de liaisons alternkes et d'effets d'arrondissement pour des tailles croissantes des agrkgats.

Abstract.

-

A simple iterative form of the Huckel method simulating an SCF procedure with optimization of geometry is used to study the geometrical distortions of simple regular one- two- and three-dimensional lattices brought about by the existence of boundaries. The results are discussed in connection with bond alternation and rounding-off effects at the corners and edges.

1. Introduction.

-

We report. here the results of a first attempt to study by quantum-mechanical methods the small changes in geometry that clusters undergo when their sizes increase, and the distor- tions due to boundaries.

We have applied an iterative version of the MO-LCAO Hiickel method to treat increasing numbers of atoms (up to the 4 x 4 x 4 cube) so to obtain some indication of the validity of our results in the real-crystal limit: although such diverse numbers of centres have been indicated as 13 and 40 for clusters actually simulating solids [I, 2, 31, we have found that the number of atoms for which some degree of convergency is obtained strongly depends on the type of cluster under study.

The total energy of the system (closed shell ground state) is given by

(2) where Hc" is the matrix corresponding to the effective Hamiltonian without electron repulsion and

V,,,,

is the core repulsion energy. The electron populations and the bond orders (q and 1) are

In the above formulas the subscripts rand s refer to the basis orbitals and to atoms a s well, since there is only one basis orbital per atom: j refers to molecular orbitals, and (occ.) means that onlv occupied orbitals are considered.

2- Description and discussion of the method-

-

_{Consider now an alternative to the current}

We have treated atoms with a _{single valence procedure for the determination of minimum energy}

electron (alkali metal atoms) with an orthogonal _{geometries which consists in adopting an initial set} basis set of one s-like orbital per atom. _{of bond orders 1% and determining, for every (r, s)}

Let U be matrix defined the eigenva- pair, a sequence of distances which correspond to lue equation _{the minimum energy for every given bond order : at}

. -

H U = U E ( U + U = l ) . (1) the end the entire iteration procedure, if it conver- ges, a set of bond orders and a set of distances will

(*) Permanent address: Cattedra di Chimica Teorica dell' be found which, when inserted into H give at the

Universiti, via Mezzocannone 4, 80134 Napoli, Italy. same time an SCF solution of (1) and the geometry

C2-30 A. JULG, G. DEL RE, M. BOURG AND V. BARONNE

corresponding to the least energy. A semi-empirical simulation of this procedure requires a formula giving the dependence on the bond orders of the off-diagonal elements of the H matrix correspon- ding to the least energy for any given set of bond orders. Such a formula can be introduced on the basis of empirical relationships between distances and bond orders.

The effective core repulsion V,,,, has been discussed in the preceding paper [4]. If the orbital energies associated with eq. (1) are denoted by E ,

the final formula for the total energy is

N

1

E = 2 2 ei --(rr

I

r r ) + - ~ ' l : , ( r v

I

ss) (4)

4 2 r,,

the net charge Q, being assumed to vanish because our model systems, involve little or no charge separation.

Strictly speaking, the above considerations refer to a closed shell but in our approximation the energy is given by the same equation even when the highest occupied level is degenerate or the number of electrons is odd.

The present method implies complete delocaliza- tion (metallic character) when, as in our case, each atom is linked to two or more neighbours by the same A 0 and with the same bond parameter (prS). Thus a system of completely delocalized MO's corresponding to Bloch waves can be associated to a linear chain of equal equidistant atoms with a single atomic orbital (a or T ) per atom. Defects and interruptions of periodicity cause a loss in delocali- zation, as is shown by the values of the bond orders

l,s, which in such circumstances are no longer equal.

The differences in bond orders suggest that the corresponding bond distances, which had been supposed to be equal, are actually different. This is confirmed by experimental evidence [5].

The possibility of different lengths for the same bond suggests that one should allow for distance dependence in the bond parameters

6,

[6]. The following functional dependence will be adopted here :

(k is a positive constant, Ro a reference distance and

po

the corresponding bond parameter).

In conjugated molecules, where a relationship between the bond order 1, and the equilibrium distance R , was found long ago, a linear formula of the type

seems to be satisfactory.

We shall adopt here the same kind of relationship with the restriction that nearest-neighbour approxi- mation is introduced everywhere.

+ The bond parameters of nearest-neighbour pairs.

are given by :

3. Choice of parameters and model systems.

-

The aim of the present work is to investigate the geometrical relaxation due to boundaries in metals at a highly simplified level. Therefore, the choice of the parameters is not critical. However, to ensure a minimum of realism, the metal under consideration has been taken to be lithium.

The parameters a and b of eq. (6) have been obtained by noting that a bond distance 2.66

A

corresponds to Liz 171 (bond order= 1) and an average bond order 8 / v 3 [8] corresponds to the distance 3.05 of the bulk metal. Thus

On the basis of overlap we have taken k = 0.906 in eq. (7), subject to revision.

The model systems we have studied fall into three classes : linear chains, two-dimensional arrays and three-dimensional lattices. All have been taken at zero order to consist of equal atoms at equal distances from their nearest neighbours, the itera- tive procedure described above being the only way in which differences in those distances are intro- duced.

The high symmetry of the systems studied causes some difficulties of principle, because, except in the case of linear chains, degenerate energy levels are present that will disappear as soon as the symmetry is suppressed by the slightest perturba- tion. In our cases, it is most likely that the symmetric arrangement is one of high energy ;

however, it should correspond to the average with respect to the symmetric vibration. Thus the best occupation scheme to be adopted under this assumption corresponds to Hund's rule : the dege- nerate orbital that would be associated with the lower energy under an infinitesimal distortion is not the same for distortion in the opposite sense, so that the correct average is probably obtained by putting the degenerate orbitals on the same footing.

4. Results and discussion.

-

The results are collected in figures 1-4. A brief discussion is give11 here.

FIG. 2. - Simple squares.

FIG. 3. - Centred squares.

FIG. 4. - External contours of simple and centred squares.

4.1 LINEAR CHAINS. - According to the proce- dure used in these computations, all linear chains have alternating short and long bonds, the shortest bond being the terminal ones. The alternancy becomes regular with the fifth atom from either ;

and in a typical even chain the values of the bond lenghts in A are found to be 2.68

-

3.05

-

2.70

-

3.03

-

...

3.03

-

2.71

-

3.03

-

2.70

-

3.05

-

2.68, the three dots representing a section of the chain where the lenghts take the same values as those immediately preceding and following the dots.

Even chains have a binary symmetry axis at the centre of a bond, so that bonds alternate all along the chain, odd chains must have two equal bonds on either side of the central atom, and therefore the alternancy is so to say damped towards the centre of long odd chains.

The results for a semi-infinite chain correspond to the odd-chain case (I) those for the infinite chain

to the even-chain one [PI. The average bond order for odd chains is 0.662 81 (55 atoms), that for even chains is 0.606 96 (50 atoms). The average between them is 0.634 5, i.e. practically the value obtained by Baldock (0.636 6) for the infinite chain.

4.2 PARALLEL CHAINS.

-

When two equal chains are placed one next to the other, and corresponding atoms linked to each other, distor- tions in the direction perpendicular to the chain are to be expected and are found by our theoretical method.

The alternancy of bonds along each chain is partly lost, although paths as long or longer than one chain can be found where short and long bonds alternate. Alternancy is also found for closed paths in the graphs representing the systems at hand, and is a consequence of the alternancy property which all our systems share. Indeed, the formation of drops, which is the most striking feature of two-chain systems, follows a rule analogous to the 4 n rule well-known in the theory of aromatic hydrocarbons : in pairs of even chains all cycles having 4 n corners (n being any integer) have alternately long and short sides, and &membered cycles are particularly favoured (although 6- membered cycles are also recognizable when the chain lengths are not multiple of 4) : in pairs of odd chains it is the 4 n + 2 cycles that have alternating long and short sides.

The case of more than two parallel chains is illustrated in figure 1.

4.3 SQUARES.

-

We have studied two types of square clusters which we shall call simple and centred squares, figures 2 and 3. They differ by the arrangements of the bonds with respect to the sides. The centred square array corresponds to a simple-square lattice terminated by (1.0) sides.

In a first stage, we have required that the system should have maximum symmetry : that of the square for n x n clusters and the corresponding centred squares and rectangles for n x m clusters.

In the simple squares symmetry imposes very strong conditions. Already in the 3 x 3 square the corners come closer to the centre than in a regular square ; but no tendency to form a circle develops, because the rounding process converges, as is seen in figure 2, to a square with slightly rounded corners and sides. This is another way ,to say that the average lengths of the bonds of the central chains tends to a fixed value (2.99

A

for the four central bonds of a square with an odd number of atoms and 2.97

A

for the even case) and the atoms at the centre tend to adopt a regular-square configuration.

The distances along the perimeter tend to become equal when the sizes increases. The same happens for the central 2 x 2 or 3 x 3 square of even and odd squares : this gives the (symmetric) limit for an infinite number of centres. The average bond orders in the even and in the odd cases (0.42 and 0.38 respectively) give a mean value 0.40 very close to that obtained by Baldock for the infinite case

C2-32 A. JULG, G. DEL RE, M. BOURG AND V. BARONNE

In the centred squares (Fig. 3), the rounding-off phenomenon is much less important than in simple squares.

4.4 CUBES.

-

The cube of side 2 has the same length of the edge as the square (2.93

A).

This distance seems to correspond to boundary effects, because the elementary cubes of the centres of larger edges : in the 4 x 4 x 4 cubic arrangement the central cube has an edge of 3.03 and the shortest bond lenght (between the two central sites of an outer edge) is 2.93

A.

In the cubes the same effect is present as in the squares, namely a tendency of the corners to get closer to the centres, and a tendency of the faces to become slightly convex.

The cubes appear to be stable towards changes in the initial geometry : the cubes with an odd number of atoms take up a lower symmetry arrangement when the upper degenerate occupied levels are

filled according to Hund's rule, but the effects just mentioned remained there.

5. Conclusion.

-

Although the above results are to be taken cum grano salis when the distances are

fairly large, the general consistency of the results and the agreement with known facts in very small clusters [l] suggest very strongly that the general trends can serve as the basis for reliable predic- tions. Calculations of the cluster diffraction pat- terns produced by the undistorted and the distorted

4 x 4 cubes (in collaboration with Pr. Larroque of Toulouse) suggest that experimental testes are feasible. Further work is in progress in this direction for f. c. and b. c. lattices.

Note added in proofs : The topic of this communication has already been exposed in a part of an article sent in July 1976 to the Philosophical Magazine. Here the parameters a and b (rel. 6) are

more satisfactory in an experimental point of view.

References

[ I ] MESSMER, R. P., Chem. Phys. Lett. 11 (1971) 5. r51 Tables of interatomic distances, edited - - by L. E. Sutton [2] MESSMER, R. P., TUCKER, C. W. Jr., JOHNSON, K. H., (London, The Chemical Society) 1958.

Chem. Phys. Lett. 36 (1975) 423. [6] MULLIKEN, R. S., 3. Phys. Chem. 56 (1952) 295.

[7] BAETZOLD, R. C., 3. Chem. Phys. 55 (1971) 4355. C3I FAESSAERT, D. J. M. and VAN DER AVOIRD, A.,

sU,.f.

s c i . 55 181 BALDOCK, G. K., Proc. Phys. Sci. A 66 (1953) 2.

" (1976) 291. [9] TSUJI, M., HUSINAGA, S. and HASINO, T., Rev. Mod. Phys.

32 (1960) 425.