On the electronic structure of Hgn and Hg+ n aggregates

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On the electronic structure of Hgn and Hg+ n aggregates

P. Joyes, R.J. Tarento

To cite this version:

P. Joyes, R.J. Tarento. On the electronic structure of Hgn and Hg+ n aggregates. Journal de Physique,

1989, 50 (18), pp.2673-2681. �10.1051/jphys:0198900500180267300�. �jpa-00211092�

On the electronic structure of Hgn ^and Hg+n aggregates

P. Joyes (1) and R. J. Tarento (1,2)

(1) Laboratoire de Physique ^des Solides, ^Bât. 510, Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France

(2) Laboratoire de Physique ^des Matériaux, ¹ place ^Aristide Briand, ⁹²²⁹⁵ Meudon, ^France (Reçu ^{le 19 mai} 1989, révisé le 14 juin 1989, accepté ^{le 16} juin 1989)

Résumé.

Nous étudions les agrégats ^{de mercure où un} changement ^{dans la nature} de la liaison

chimique apparaît ^aux faibles nombres d’atomes. Dans une première partie, ^nous considérons les

agrégats ^neutres où la transition métal ~ Van der Waals apparaît pour nc ~ 15, ^{résultat en bon} accord avec un autre travail théorique. ^{Dans une deuxième} partie, ^{nous étudions} l’énergie

d’ionisation I. Pour n ~ 50, ^un modèle de type métallique donne le bon ordre de grandeur. ^Pour

n ~ 10, ^{ce modèle ne convient} plus ^{et nous} présentons ^{une autre} méthode où les résultats

expérimentaux, pour n ~ 4, ^sont utilisés pour fixer les paramètres. ^{Pour 4 n ~} 7, l’accord avec

l’expérience ^est bon. Nous examinons aussi le problème de la localisation de la charge positive ^en

excès dans Hg+n.

Abstract.

We study small mercury aggregates in which the nature of the chemical bonding changes ^with respect of the bulk. In a first part, ^{we consider neutral} aggregates ^{for which we find} that the transition from the metallic to the Van der Waals bonding ^occurs for nc ~ 15 in close agreement ^{with a} previous theoretical work. In a second part ^we study ^the ionization energy I. For n ~ 50, ^a simple metallic model gives ^the experimental ^{order of} magnitude. This model is

no longer valid in the n ~ 10 range for which ^we present another calculation. In this study ^{we fix}

our parameters by using ^the experimental values for n ~ 4. For 4 n ~ 7, ^our results are in agreement ^with experiment. ^We also examine the localization of the excess positive charge.

Classification

Physics ^Abstracts

36.40 - 31.20

1. Introduction.

Mercury ^{is one of the} divalent elements where a change ^{in the nature} of the chemical bonding

can be expected when the average connectivity ^varies (here, connectivity ^{means the number}

of nearest neighbours per atom). This has first been observed in the bulk phase, ^{when the} liquid ^is expanded, ^{at a} density of about 8-9 g/cm3, ^a metal-insulator transition is observed [1].

This conclusion was obtained by electrical conduction, thermopower, Hall coefficient and

Knight shift studies (Refs. given ⁱⁿ [1]). Several theoretical works have been devoted to this

question ^{and the} general interpretation ^is that, ^{as the} density decreases, ^a gap appears ^at the Fermi level, between the s and the p bands, ^{which causes} the insulator behaviour [2, 3].

Another situation where the connectivity is lowered with respect ^to the bulk appears for

Hgn aggregates. ^There again, ^{when n} decreases, ^numerous interesting physical ^{effects have}

been observed : we will mention three of them.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180267300

2674

a) One knows that in metallic aggregates (alkalis, ^{noble or} transition metals) the ionization

potential I(R) ^{as well as} the electronic affinity AE(R) ^{exhibits R-1} dependence.

R is the radius of the aggregate [4-7]

(W is the bulk work-function). For mercury it has been shown [8] ^that 7(/?) followed relation

(1) only ^for large ^sizes (n -- 70). ^{In this} study ^R is related to n, the number of atoms, by

R

ro ⁿ 1/3 where ro is ^an atomic radius. For smaller sizes, ^the experimental ^{values are} larger

than the predictions of relation (1) by ^{more than one eV} in the range n -- 10. Let ^us notice that in the analysis of reference [8] the factor 3/8 of relation (1) ^is replaced by 1/2 but this does not alter the general conclusion.

The previous observation has been interpreted ^{as due to a} change in the chemical bonding

which would occur in a progressive way for ne S ^70. For n ::-.> nc ^{the nature of the} bonding ^is

the same as in the bulk and for n s nc the metallic character is progressively ^{lost in a transition}

towards a Van der Waals bonding (as ^{in rare} gases). ^The physical explanation ^{of this}

transition is the same as for the bulk metal-insulator transition mentioned before. As the

connectivity decreases, the s and p bands separate and there is no longer any covalent bonding

because in the s band the bonding ^{and the} antibonding ^{levels are} complete.

b) ^Another experimental observation tends to confirm this conclusion. In metallic aggregates ^{it is} usually observed that the average interatomic distance d(R) decreases with size [9, 10]. Simple models have been proposed ^to interpret this behaviour [11, 12]. They ^lead

to a relation between d and Z (the average connectivity) :

where the subscript ^{oo refers to the bulk} phase ^and where p and q ^are coefficients which give

the dependence ^on the interatomic distance r of the repulsion energy between atomic ^cores

(a exp (- pr ) ) and of the electronic energy (a - exp (- qr ) ). ^As for the metallic elements

p

q, the application ^of (3) in the whole size range gives ^the expected decrease of

d with Z. Mercury ^{does not} follow this general behaviour since d (R ) increases from the bulk to the dimer (d. = 3.03 Â, d(Hg2)

^{3.36 Â} [13]). This effect can be interpreted ^{as a}

transition to Van der Waals bonding ^{as n} decreases since it is known that in this type ^of bonding ^an increase of d(R) ^is expected ^{as R decreases} [14].

An experimental study ^of d (R ) for mercury has ^not yet been made. Such a study might

reveal the two regimes discussed before : decrease with R before the Van der Waals transition and increase after it.

c) ^Another interesting experimental result has been given by ^the study of the inner shell autoionization spectrum [15, 16]. The atomic transition which is studied is

where the d state is either d5/2 ^or d3/2. When this kind of excitation occurs in Hgn ^{it first} keeps

its atomic character but, since the excitation energy is larger ^than the ionization energy of the aggregate, ionization occurs after a time of about 10-13 s. Let us notice that ionization occurs

for both d5/2 ^and d3/2 excitations for n :::. 2 but it occurs only ^{for the} d5/2 excitation in the atom.

Two regimes ^are observed. For Az 12, ^the excitation energy is inversely proportional ^to

n in a way which is characteristic of ^an excitonic behaviour. As an exciton is created one may

conclude that in this range of n values the bonding is of Van der Waals type. ^For 12 rx n s 40 the previous 1/n dependence ^{is no} longer followed which shows that a deviation from a Van der Waals bonding in the whole aggregate begins ^to appear. It has also been observed that the shape ^{of the} experimental ^{lines is} markedly asymmetric ^for aggregates larger ^{than n -} 20 which is an illustration of a gradual ^{evolution towards} the bulk character.

The three experimental results mentioned above lead us to the conclusion that a change ⁱⁿ

the nature of the chemical bonding ^{occurs in} Hgn ^{at a} value nc of about ^{some tens} of atoms.

The fact that all the experiments ^{do not} give exactly ^{the same} value for nc ^{seems to} be less

important, ^{the main} point being the existence of a transition.

Let us notice that, ^once the covalent bonding begins ^to appear, the electronic properties ^are

still far from the bulk ones. The study of the evolution with size of each of them will be done

separately ^{and is not} in the scope of this paper.

We will limit ourselves to band structure considerations in Hgn ^{and will} study (Sect. 2) ^the

decrease of the mixing of the s and p bands ^as the size decreases. We use a tight binding

method whose results agree for small sizes with previous calculations [16] and has the

advantage ^of being easily extendable to aggregates ^{with n}

100.

In section 3, ^we will examine Hg’ ^{ions and} particularly the localization of the excess singly positive charge ^{in these} aggregates. ^{Let us discuss} briefly ^this question.

According ^{to the} general laws ^of electrostatics when ^a macroscopic conducting ^{medium is} charged, ^the charges ^{tend to localize at the surface} of the medium where they adopt ^the spatial distribution which minimizes their electrostatic repulsion energy. This kind of model has been extended to metallic aggregates. ^For example, it has been used for determining ^the

critical number of atoms, nP (X) above which Xn + aggregates (X ^{is a} given element) begin ^to

be stable. For copper the agreement between theoretical results and experiments ^is satisfactory : n;ri (Cu )theo

¹⁴ [17] ; n;ri (Cu )exp

^{20 [18]. Let us} notice that the foremen- tioned experimental value has been recently ^confirmed by ^{results on} silver and gold :

n 3j(Ag)

^{22 ;} n 3 (Au)

¹⁷ [19] ^{which are of the same order of} magnitude.

As a change ^{in the nature of} bonding ^{seems to occur} in neutral aggregates, ^one may wonder whether a similar change ^{could occur in ionized} aggregates. ^{In this} spirit, ^a discussion of a

possible « Van der Waals » structure of Hgn + ^is given ⁱⁿ [20] ^{where the}

²⁺

charge ^is

localised on one central site. This model allows us to understand why particles ^{such as}

Hg 5 2+ do exist whereas a calculation of their stability by ^a metallic model leads to the opposite

conclusion. Let us notice that the problem ^{of the} charge localization in ionized divalent aggregates ^{is a} general question also studied in [21] ^for Mgn + ^and [22] ^for Ben ^{+ .}

2. Neutral Hgn.

For determining ^the binding energy per ^atom Ec and also the ionization energy of

Hgn aggregates ^{we use the} following Hamiltonian which describes the coupling ^{between an s}

and a p bands

where the notation is the same as in reference [3] ^{and where} e, and Ep ^{are the} energies ^{of the}

atomic 6s and 6p ^{levels ;} ts, tp ^{and y2 are} the intersite hopping integrals ^s

^s,

p -+ p and ^s

p and y 1 is the intrasite hopping integral ^s

p.

2676

The method of resolution of (4) is detailed in [12]. ^{In this} work, it is shown that the energies

of the three bands Ep(i ) (doubly degenerate), ESP(i ) ^and EPS(i ) (singly degenerate) ^are given by :

where ^e (i ) ^{are the n} (non-dimensioned) eigenvalues (1 * 1 * n ) of the matrix obtained from 4 by letting _Ep

-S=tp= ^{yl =} y2 = 0 and ts=l.

Our calculations have been performed ^on three cuboctahedral : shapes ^{with n} = 13, ^{55 and} 147, ^with _{ep -} e,

6.1 eV, ts

^0.6 eV, tp

^{0.5 eV, Y2}

^{0.2 eV, YI}

^{0. These} shapes

have already been used for the study ^of Hgn aggregates [16, 23]. ^Our parameters ^{are of the}

same order of magnitude ^{as those} given ⁱⁿ [23]. They ^are justified ⁱⁿ particular by ^{the fact} they give ^{the correct} bulk limit for Ec ^{as we will see} below. Let us also note that in our calculation there is ^one parameter less than in reference [23] ^{since we have} only ^one p --+ p integral

instead of two.

One of the advantages ^of equations (5) ^and (6) ^{is to allow the} study ^of large aggregates ^since

we only ^{need to} know the n 6 (i ) values. The method can also be extended to the bulk by using ⁱⁿ (5) ^and (6) ^the e (i ) values deduced from periodic ^fcc boundary conditions. We then obtain E,

^0.52 eV, ^a value which is close to the measured cohesive energy per ^atom

-

0.5 eV [20]. ^{Our other} Ec ^{values are} given in table 1 they ^are very similar ^to those obtained in [16], 0.20 and 0.28 eV instead of 0.20 and 0.27 eV for n = 13 and 55, respectively. ^{We also} give the values of the energy difference between the one-electron Fermi level and the 6s atomic level. We observe that this quantity increases with n.

An evaluation of the binding energy in ^a Van der Waals bonding ^{is :}

This evaluation, ^{made in} [16], gives 0.14 eV and 0.20 eV for n

13 and 55 respectively.

A comparison ^{with the} Ec values of table 1 would lead to the conclusion that the covalent

bonding ^{is more} stable than the Van der Waals bonding, ^even for n = 13. However we have not taken into account the reduction of the covalent energy due ^to the electronic correlations.

One knows that the larger ^the reduction, the smaller _n, as it has been shown by using ^the

Gutzwiller technique ^{for an s band} [24, 25]. ^{If we} adopt the estimate made in [16] ^{of the}

electronic correlation effect (reduction of the cohesive energy of 0.07 eV and 0.03 eV for

n

13 and 55 respectively) ^{we are led to} corrected covalent energies of 0.13 eV and 0.25 eV for n = 13 and 55 respectively. ^{Now we see} that the covalent energy is smaller than

Ev,dw ^{for n}

^{13. This means that for n} smaller than about 15 there appears ^a transition from the metallic to the Van der Waals bonding.

3. Ionized Hg§ .

3.1 LARGE SIZES, n > 50.

For relatively large ^sizes (n > 50) in all the models the electronic structure is assumed to be of the same nature as in the bulk (hybridization ^between

the s and p bands). ^{Then the} metallic ionization energy is given by ^a formula like equation (1)

where the 3 e2/8 ^{R term} expresses the potential energy of the Hg’ aggregate ^{which is} globally

charged. ^However equation (1) which is valid for alkalis [5, 6] ^must be modified before being

applied ^{to our} problem.

Indeed, ^{in the case of} alkalis, ^the one-electron Fermi energy of the ^s band is, for any n, close to the es atomic energy. In particular when n --+ oo, the one-electron Fermi energy (of

a cc lattice) tends towards es (symmetric band). ^{This is not the case for} Hgn ^{since the one-}

electron Fermi energy, EF, ^{increases with n} (see ^Tab. I). ^{We have to} take this into account by writing, instead of (1)

In table 1 ^we give the values obtained for I(R) ^{with R}

ro n1/3,

ro

1.51 Â (average

distance in the bulk) ^{and the} experimental values when available. We see that the agreement between experiment ^and theory ^{is rather} good ^{for n - 55. On the} contrary ^{there is a difference} between experiment ^{and the} theoretical result for n - 13. This discrepancy ^was expected ^since

in this range of ⁿ values the metal-like structure is no longer ^valid.

Table I . - Size dependence of the cohesive _energy (Ee), ^{the Fermi level} (El,) (origin of ^the energies : the 6s atomic level) ^and the ionization potentiel (I) ⁱⁿ Hgn. The values obtained in

reference [16] ^are given ⁱⁿ parentheses.

3.2 SMALL SIZES, n 10. The fact that neutral Hgn ^{exhibits a} changing in the chemical bond prompts ^an intriguing question concerning ^the charge localization in ionized

H ’ . Indeed, ^{for n} 10, ^{as the structure} of the neutral aggregate is Van der Waals-like, ^one

may think that ^as in Van der Waals ionized aggregates (Xe’ [26, 27]) ^the positive ^{hole is}

localized on one (or ^{on a} few) central site.

In this range of n values the ^s band is completely separated from the p band. For example,

our calculation of section 2 gives ^a gap of about 2 eV for n

13. Therefore we think that it is

justified ^to describe mercury ionized aggregates by only considering the external 6s levels.

This assumption, ^{which can also be} applied ^to aggregates of other divalent elements such as

Mg [21], ^{is well} justified ^{in the case of} Hg where the energy difference between 6s and 6p

levels is large (- ^{6, 1} eV) ⁱⁿ comparison ^to 2.7 eV for Be or Mg [28].

The main features of our model for Hg’ aggregates with n , 8 are the following :

the wave function is an expansion ^{on a set of} atomic determinants. Each determinant describes n - 1 electrons, ^see figure ^{1 for the case n}

^{3 ;}

-

for each n value, ^we study ^{two kinds of} shapes. Firstly, geometries where all the distances are equal ^and short di = dl. ^{In these} shapes ^the positive charge is shared between all the sites which are almost equivalent : the hole is delocalized. Secondly, geometries ^{with two} possible ^distances dl and d2 > dl where, ^{as we will} see, the hole is mainly ^{localized on a}

limited number of sites (2 ^or 3) ;

2678

-

the parameters used in this calculation are given in table II. They ^{have the} following physical meanings.

Fig. 1.

The three determinant wave functions for Hg3+ ^with equal interatomic distances.

Table II.

Values o f the parameters (in eV) ^used for small ionized cluster, ^their meaning is given in ^{the text.}

. Ev ^{and U are atomic} energies.

They ^are determined from the atomic single and double ionization energies.

2022 3 (d) is the transfer energy between ^two adjacent sites. We will ^use /3 (d2) = 1 ^{2 (dl).}

2022 ER ^{is the} repulsion energy between ^two neutral atoms with distance dl. ^{We assume that} ER ^is negligible ^{for d}

d2.

. Epol is the attractive polarization energy between ^an ionized and a neutral site with distance dl. ^{We also assume that} Epol ^is negligible ^{for d}

d2.

Our assumptions ^on the variations with d of /3, ER ^and Epol ^are justified by the fact that

/3 ^{decreases more} slowly than the other two parameters when d increases. The variations with distance of /3 ^and ER ^have already been mentioned in section 1.

By using ^these parameters, ^we obtain the analytical energies given table III for

n

2, ^{3 and 4.}

The ionization energy I is deduced from these expressions by using :

Table III. - Energies of ^the localized and delocalized states versus the size.

where E(Hgn) is obtained from equation (7) by considering compact shapes ^and EYdW(2)

^{0.07 eV.} By fitting ^{9 to the} experimental results and by keeping the stablest shapes

we have determined 0, ER ^and Epol ^{(Tab. II) then we} have used these parameters ^for calculating the other ionization potential given in table IV. For each n value we give ⁱⁿ figure 2 the stablest shape ^of Hg’ ^{in the} application ^of (9). ^{We also} give in table IV the weight

of the localized configurations, ^{i.e. the} probability of localization of the hole on the central sites drawn as circles in figure ^2.

Fig. ^2.

The stablest shapes ^of Hgn . We have drawn as circles the sites which are distant from each

other by dl distances. The hole is more localized on these sites than on others (solid circles).

2680

Let us not that the value obtained for the parameter {3 ^{is of the same order of} magnitude ^as

the value used in the one-electron calculation of the previous sections. There is not a strict

equality because in this part ^{of the} study ^we give ^{a more} precise description ^{of the} quantuum

structure by considering energies ER ^and Epol (repulsive and attractive respectively) ^which

were not included before.

In table IV we can see that for n

4 the agreement ^with experiment ^is good and that the

charge ^{tends to localize on some} central sites. This characteristic is similar to the

Xen ^behaviour [26].

Table IV. Size dependence o f the experimental and theoretical ionization potential, ^{and the} weight o f the ^localized configuration for Hg£ . (The ^labels o f the shape ^are defined in Fig. 2).

Conclusion.

In this paper ^we have studied neutral Hgn aggregates ^{for which a} transition from ^a metal-like

bonding ^{to a Van der Waals} bonding is obtained at about nc - ^{15. We have} applied ^{our model}

to Hg+ aggregates ^{with a} relatively large ^{number of atoms} (n 50 ) and have obtained ^a

satisfactory agreement ^with experiment.

For smaller Hg’ (n:5 10 ) ^{a method is} developed ^where only 6s orbitals are taken into account. This method allows one to study ^the possible localization of the excess positive elementary charge ^{for n . 4. In a} way similar ^to Xe’ aggregates ^{the excess} charge ^{tends to}

localise on a part of the sites. Our study is limited to n

7 where the previous effect still appears.

The model can be extended to Hgn + aggregates ^{where a} precise theoretical description ^has already ^been published ^for Hg2 + [30]. ^{Our work on this} question is in progress.

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