Singly and doubly charged clusters of (S2) metals. Ab initio and model calculations on Mg+n and Mgn++n(n ≤ 5)

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Singly and doubly charged clusters of (S2) metals. Ab

initio and model calculations on Mg+n and Mgn++n(n

≤ 5)

Geoffroy Durand, J.-P. Daudey, Jean-Paul Malrieu

To cite this version:

Geoffroy Durand, J.-P. Daudey, Jean-Paul Malrieu. Singly and doubly charged clusters of (S2) metals.

Ab initio and model calculations on Mg+n and Mgn++n(n

_{≤ 5). Journal de Physique, 1986, 47 (8),}

pp.1335-1346. �10.1051/jphys:019860047080133500�. �jpa-00210326�

Singly

and

_doubly

_charged

clusters

of

_(S2)

metals.

Ab initio

and model

calculations

on

_Mg+n

and

_Mgn++n

_{(n ~}

5)

G.

Durand,

J.-P.

_Daudey

and J.-P. Malrieu

Laboratoire de _Physique

_Quantique,

Université Paul-Sabatier, Unité Associée au C.N.R.S. n° _505,

118, route de _Narbonne, 31062 Toulouse Cedex, France

(Reçu le 7 _janvier 1986, accepté le 7 avril 1986)

Résumé. 2014 Des calculs ab initio

précis

incluant la corrélation électronique ont été effectués pour

Mg+2

et

_Mg++2.

Le deuxième

_système

_présente un minimum local. A l’aide d’un procédé de diabatisation, on a _{pu définir} un hamil-tonien modèle _{simple porté} par les

produits

des configurations

atomiques

s2,

s1 et s0. L’hamiltonien modèle inclut les effets de

_polarisation

instantanée (ainsi que les effets à trois

corps)

et la

_répulsion

coulombienne des

_charges

positives pour les agrégats doublement _chargés. Il _peut être considéré comme une

_{représentation}

de un ou deux

trous actifs dans une bande s

_polarisable.

Le modèle

_prédit

que la structure la _plus stable de

_Mg+3

et

_Mg++3

est linéaire, que les structures carrées et _{triangulaires} centrées sont _pratiquement

_{équivalentes}

pour

Mg+4

et

_Mg++4.

Pour

_Mg+5

et

_Mg++5

existent _également deux minima _quasiment

_{dégénérés :}

le tétraèdre centré et le carré centré. L’existence de telles structures a été confirmée par des calculs Hartree-Fock, par minimisation directe de l’énergie

et calcul de la matrice des constantes de force pour la conformation optimale, mais on _{démontre que dans l’approche}

du modèle à

_{particules indépendantes}

les _longueurs de liaisons sont trop longues et les structures centrées sont

trop stables.

_L’énergie

du minimum de

_Mg++5

est située à 1,71 eV au-dessus de la somme des

_énergies

des

_systèmes

Mg+3

et

_Mg+2.

Les _potentiels d’ionisation verticaux ont été calculés et on _{démontre que}

Mg+4, Mg+5

et

_Mg++5

peuvent évaporer un atome neutre.

Abstract 2014 Accurate ab initio CI calculations _on

Mg+2

and

_Mg++2

have been

_performed.

The second appears to

have a local minimum. Through a diabatization

_procedure,

it was

_possible

to define a

_simple

model Hamiltonian

spanned by products of atomic

s2,

s1 and s0

_{configurations.}

The model Hamiltonian includes instantaneous

pola-rization _{energies (and}

_three-body

effects) and Coulombic _repulsion of the holes in

_{doubly charged}

clusters, and may be considered as _treating one or two active holes in the _polarizable s band. The model

_predicts

linear

Mg+3

and

_Mg++3

structures and _{nearly degenerate} square and centred

equilateral

triangular structures for

_Mg+4

and

Mg++4; Mg+5

and

_Mg++5

both present nearly degenerate minima of _{centred tetrahedron and centred square}

structures. The existence of such structures has been confirmed _by UHF ab initio _(gradient + force) calculations;

it is shown however that

_{single-determinantal}

approaches overestimate the bond _lengths and _unduly favour the centred structures; the

Mg++5

minimum is 1.71 eV above the

_(Mg++3

+

Mg+2)

dissociation limit and 1.41 eV under the _{(3 Mg} + 2

_Mg+)

dissociation limit. The vertical Ionisation Potentials

_(IP)

have been calculated and it is shown that

_Mg+4,

_Mg+5

and

_Mg++5

may evaporate a neutral atom.

Classification

Physics Abstracts 36.40 -

31.20R - 31.20E

1. Introduction.

The atoms of column II of the

_periodic

table have

ground

state

_S2(lS)

_{configurations.}

The atoms _may be considered to

_keep

_essentially

this

_ground

state structure in small

clusters,

despite

some weak

hybri-dization with the

_{sp(3, 1 P)}

_{configuration.}

The elec-tronic

_population

of a neutral cluster of such elements

may thus be seen as an

(s)

full band. The

positive

ion

may

consequently

be treated as a

_single

hole

_problem,

delocalized on the different sites of the cluster

through

a

_single-hole

_operator. A

_doubly

_charged

cluster _may be considered as a two-hole

_system

in the s

_band,

and treated like a two-electron

problem,

including

correlation. The _{present paper proposes such}

sim-plified

models for

_singly

and

_doubly

charged

clusters of

_(s’)

atoms. For

_Mg

clusters the

diagonal

effective

energies

and interactions are extracted from accurate

MO-CI calculations on

Mng2+

(n = 1, 2)

and from the’

experimental Mg2

potential

curve.

The

_problem

received an increased interest from

the recent observation of an

Hg5 +

_{doubly charged}

cluster

_[1],

from electron

_impact

on a _{beam of mercury}

clusters. The _appearance at 22 eV of such a

_peak

in the mass

_spectrometer

_analysis

was

_{quite surprising}

since the Coulombic

_repulsion

between the two holes should

compel

such a small cluster to

_explode

into

two

_{singly charged}

_{fragments. Brechignac et}

al.

_[1]

suggested

that the

_{positive charge}

in this cluster

_might

be concentrated on the central atom of a _compact

cluster,

the

_{Hg+ +}

atom

_(the

ionization

_potential

of which is 29

_{eV) inducing}

_{stabilizing polarization}

forces

on the

surrounding

neutral atoms.

The

_present

work does not concern

_Hg

clusters

since the correct treatment of such a

_heavy

atom

requires

the inclusion of relativistic

effects;

their

treatment is in progress in our

_{laboratory through}

relativistic

_{pseudopotentials}

_{[2, 3],}

which will be

applied

further to this

_{precise problem through}

calculations of

_Hg2,

_{Hg2 ,}

_{H g2 I}

_potential

curves.

The

_present

_paper is limited to the isoelectronic

problem

of

_{Mg clusters,}

in order to see wether an

Mg25 +

cluster _may exist at least as a stable local

minimum. The

_question

is treated

_through

both model and ab initio MO

calculations,

which _compare

_fairly

well.

2. A

_simplified

model Hamiltonian for

_Mgn

and

Mgn +

clusters.

2.1 NEUTRAL INTERACTIONS. - Let

us assume that in the neutral

_ground

state, each atom

_essentially

keeps

an

(s2)

configuration.

This means that the cluster

is

_essentially

held

_{by dispersion}

forces. This also

means that the cluster should have a _compact

_form,

maximising

the number of nearest

_{neighbours through}

a

_packing

of tetrahedra. The dissociation energy of

Mgn

into n

_Mg

atoms should be

_directly

related to

the

_{number p}

of nearest

_{neighbours. Assuming}

an

additivity

of

_{repulsion energies}

and

_dispersion

forces

(i.e.

cancellation between

_{three-body repulsions}

and

three-body dispersive energies)

the dissociation energy

per atom

_De(n)

of the cluster should be

where

_De

is the dissociation energy of the dimer. This

_quantity

is rather weak for

_Mg2

_(De

= 0.05

eV)

and

_{equation (1) certainly}

underestimates the dis-sociation _energy of the cluster since the cohesive

energy of the metal is 1.1

eV,

much

larger

than

(12/2)

De/2,

which would be the

_asymptotic

limit of

equation (1).

This may be due to the involvement of the _p

_band,

to attractive interaction with second

neighbours and/or

an attractive balance between

many-body’§orces.

It is clear however that the disso-ciation energy

of Mgn

is much smaller than that of the

corresponding positive

cluster

_Mg,,,

which involves delocalization and

polarization

energies.

The

_precise

research of the

_geometry

_{and energy of the weak}

_Mgn

neutral cluster is not in the _scope of our

_model,

and we

simply

assume that its wave function is

domi-nated

_{by products}

of

_(s’)

atomic

_{configurations}

where A is an

_{antisymmetrizer,}

_l/Ji is either

_{an si si}

product

of 3s atomic

_orbitals,

or an

_angularly

cor-related function

The interaction energy between neutral atoms

_{4 i}

at distance _rij is

_supposed

to be

_equal

to the inter-action energy of the dimer

_{R(r;) = Rij}

and the total

cohesive _energy would be

2.2 MODEL HAMILTONIAN FOR THE SINGLY POSITIVE CLUSTER. - For the

positive

cluster

_Mg’

one _may

consider, according

to the Valence Bond

_(VB)

des-cription,

a series of n localized

_functions,

where the

hole concerns the various atoms. If the hole is on the

ith atom

where

_{ai is}

an annihilation

_operator

of a fl spin

electron

on the site i. Then the wave function of the cluster

is a linear combination of the

4Jt’s

The

matrix elements of the n x n model Hamiltonian

H are

_easily

obtained as follows. For the

_diagonal

elements,

one _may write

Rjk

represents

the instantaneous interaction between

neutral centres,

_supposed

to be

_equal

to that in the

corresponding

neutral

systems.

_Rit

is the interaction

between the

_positive

atom i and the neutral atom

_j.

Its determination will be

_given

below,

and

_IP,

is the lowest ionization

_potential

of the atom.

The

_off-diagonal

element

represents

the

_{hopping integral}

between atoms i

where F is the Fock _operator of the

_ground

state.

In fact

_{(see below)}

we have used effective

Fit

_integrals

which introduce some

_{reorganization}

effects.

For

_Mg’

the

_problem

reduces to a 2 x 2 matrix

which _generates two

_{roots 2 Eu+,}

_{2 Eg+}

and thus the functions

_R1+2

and

_{F 2}

are

_immediately

extracted from the

_knowledge

of the exact

_potential

curves of the diatomic cation

Two

_things

should be noticed :

i)

the method is

_identical,

at this _step, to a Diatom

in Molecules

_{(DIM) procedure [4];}

ii)

it includes

_polarization

forces since

_{R 2}

involves both a

_{purely repulsive}

_part and an

_al/2

r’,

polariza-tion _energy of the

_adjacent

atom

_by

the

_{positive charge.}

Since the

_singly

ionic cluster

_only

bears one

_positive

charge

the

_polarization

_energy

_keeps

an additive form.

2.3 MODEL HAMILTONIAN FOR THE DOUBLY CHARGED CLUSTERS. - For

a two-hole

cluster,

one _may

ima-gine

two

_types

of

_elementary

VB situations.

- The two holes

may be on different atoms,

generating

a

function l/Ji; +.

For Sz

= 0 _{one must} introduce two determinants :

Their number is thus

_{n(n -}

_1).

Their _{effective energy}

involves,

besides the _energy

_2IP1

needed to ionize

two atoms, the interactions between the

instanta-neously

neutral atoms, the interactions between the

positive

and the neutral atoms, and the Coulombic

repulsion

between the

_holes,

_Rit

+ which

_weakly

deviates from

_{riJ 1.}

The last term

_{6 pol}

is a correction

to the

_polarization

_{energy which is} not

additive,

since the field created

_by

the two holes on the neutral

polarizable

atom k must be combined

_vectorially.

Let us call

Eki = rkilrli

the field created

by

the hole i on the neutral atom k.

Then the

_polarization

energy is

Since

_Rkt

involved a term is

_{given by}

It

_gives

the

_three-body

correction to the

_polarization

energy.

- The two holes

may be on the same atom, which is now

s°,

while the others are neutral

(s’).

The

cor-responding

VB function is

The

_diagonal

_energy associated with this function is

IP2

is the double-ionization

_potential

R’ ’

is an effective interaction between

_Mg

and

Mg+

+,

and its

_practical

extraction will be

_given

below. One _may conceive two main _types of

_off-diagonal

interactions :

The

_{hopping integral corresponding}

to the

_jump

of

a hole from the

_{site j}

to the site k _may be taken

_equal

to its value for the

singly positive

cluster.

_Actually

where

_Ji

is the Coulombic _operator relative to the

_(3s)

atomic orbital of the atom _i; the three-centre

_integral

should be small

compared

to the monoelectronic

(essentially kinetic) integral.

One may

neglect

the bielectronic

_integrals

(cor-responding

to 2 e-

_jump)

such as

(even

when i _{= q} and _p =

k) ;

this

approximation

is usual in

chemistry,

in the various Zero Differential

Overlap (ZDO)

systematics [5, 6],

and in Solid State

Physics

in the Hubbard Hamiltonian

_[7]

which is the crudest version of this

_{approximation;}

the ZDO

approximation

consists in

_neglecting

the bielectronic interaction

_{involving overlap}

distributions.

from a

doubly charged

atom to an

(adjacent)

neutral

atom, this

jump

creating

a

_pair

of

_positively

_charged

atoms. In a

simple

model one _may write

In that

_precise

case the

_{integral i}

_{I Ji Ii>}

cannot be

neglected,

since it

_only

involves two centres, and

F å + =1=

F i; .

.

The two-hole model

_requires

the

_knowledge

of three

_{supplementary}

functions,

_Ri++,

_{R;) +}

and

_{F;) + .}

This information _may be obtained from the

_potential

curves of the

_{doubly charged}

diatom

Mg++.

In that

case one has two electrons in two

_orbitals,

and the

problem

is

_isomorphic

to the

_H2

_{problem. There}

are

four

_determinants

_{1 ab I, 1 ba I,}

_{1 a-a}

_I

_and

_{I bb}

_{where a}

and b are the two 3s AO’s of the two atoms A and B.

The matrix takes the form

The bicentric

_{exchange integrals Kab}

may be

neglected.

The

_triplet

state

(I ab I -

_ba

1)/.,/2,

of

_{3 Iu+}

sym-metry has an _energy

which determines

_{Rat +}

_(or

the deviation of the

repul-sion from its coulombic

_part).

The

_singlet

_’Z,,’

state

(I

a-a I -

bb

1)/.,,/2-

has an _energy

which determines the function

R 2 + .

The

_{(If 9 +)}

_ground

state _energy is

_given

_by

a second

degree equation

and the

_knowledge

of the

_{1 E ’}

_ground

state _energy

E(’,E 9 ’)

will determine the value of

_{F ab}

as a function

of the interatomic distance

2.4 DIATOMIC POTENTIAL CURVES OF

_Mg2,

_Mgi

AND

Mg2 + AND

r DEPENDENCE OF THE 6 PARAMETERS OF THE MODEL HAMILTONIAN. - The model involves

6

para-meters R,

R +, F +,

R + +,

R2+,

F 2 +.

The first is identical to the

_ground

state

_potential

_energy of the neutral dimer and rather than

_calculating

it,

a

function fitted to the

_{experimental spectroscopic}

parameters [8-9]

has been used

where A is a Morse function

involving

three

para-meters _{a1 a2 a3}

_(see

Table

_I).

The second

_{(R +)}

and third

_{(F + )}

functions were determined from accurate

MO-CI calculations on

Mg2+ ,

_using

a

_[3s

_2p

_ld]

basis

set. The

_2Eg

and

_2Eu

_potential

curves were in

agreement with a

_previous

calculation

_by

Stevens and

Krauss

_[10].

The

_resulting

functions have been fitted

as follows :

where A’

is a Morse

function,

_al the

polarizability

of neutral atoms

_{and f1 (r)}

a short _range

_screening

of the electric field. The three last

_parameters

come

from calculations on

Mg",

which were achieved

in the same basis set

_(and

_complete

_CI).

The main

problem

concerned the definition

_{of R2+,}

the energy of the

_{(I aa}

_{I -}

_bb

0/J2

state,

dissociating

into

Mg2+

+

Mg(3s2)

_(IP2

= 22.67

eV),

due to a curve

crossing

with a state

_dissociating

into

_Mg+(3p)

+

Mg+(3s),

at 19.72 eV above

Mg(3s2)

+

Mg(3s2),

i.e. 2.9 eV below the

_preceding

dissociation limit

_(IP2).

An ab initio diabatization

_{procedure [11]}

has been

applied

to that

_problem

in order to define

_nearly

diabatic

_12:+

and

_{1 Eg+}

excited states

_keeping

the

Mg(3s2)

+

Mg + + (3s0)

character from infinite

dis-tances to the relevant short range

region [12].

The

resulting

parameters

have been fitted as follows

where

R + +

involves a Coulombic

r-1

_repulsion,

a

Morse

_potential

and a

(screened) polarizability

_part,

a2

being

the

polarizability

of

Mg +

atoms.

R 2 +

has been fitted

_by

a

simple

Morse function.

Table Ia. - R

parameters

of

the model. M are Morse

_{functions :}

a2[(’ -

_{exp(- a3(r -}

ai)))2 - 1]

and fare

screening

functions :

1 -

a4

exp[-

a5(r -

a6 )2].

The

_{explicite representation}

of the

_polarization

energy in

equation

(26)

is not

_compulsory

for

_singly

charged

clusters. The determination of a

screening

function for the electric field was

_required

for a correct

estimate of the

_three-body

term in

_{equation (15).}

The

screening

function has been determined

_{by comparing}

the

_energies

of

_MgA

+

_MgB

_(3s

frozen)

in minimal and extended basis sets for

_MgA.

An

_analogous

procedure

has been used for

_MgA

+

_[MgB(+)

_(3s

frozen)].

The

_potential

curves used for the

parame-trization have been

_plotted

in

_{figure 1 a), c).}

The

ground

state

_potential

curve

of Mg+ +

is

repulsive

with

rab 1 behaviour,

but it _presents a local minimum for r = 2.93

A

which is due to

the

mixing

between the

«

pseudo-neutral »

(I

ab +

I b-ajl

configuration

with the

_doubly

ionic

_{( aa}

_{I +}

_bb

I)V2-

_{configuration}

(as

in

_{H2), through}

the

_hopping

_integral

F 2 +.

The

Fig. 1. -

a)

Experimental

Mg2

potential

curve. _b) Ab initio

Mg2 (2 Lu+

and

_{2 Lg+)}

_potential

curves. _c) Ab initio adiabatic

(-) and diabatic _(---)

_{Mg+ ’ potential}

curves.

parameters

have been

_plotted

as functions of the

interatomic distance in

_figure

2.

_They

have the

expect-ed

_{r-dependence.}

One should notice however that for short

_{r values,}

the

F 2 +

_integral

has a

_{larger amplitude}

than

_{F + ;}

this increase _may be due either to the effect of the hole

_(cf.

_Eq.

₍₂₀₎₎

or to the indirect role of the

Mg(3p)

+

_Mg(3s)

state which

_belongs

to the same

range of energy. Standard deviation from

experimental

data for the R

_parameter

of

_Mg2

and from ab initio

calculations for the

Rk+

and

_{Fk+ parameters}

_{of Mg2}

and

_{Mg+ +}

is less than

10- 3

hartrees.

3. Results.

3.1 VERTICAL IONIZATION POTENTIALS. -

Assuming

compact structures for the neutral clusters

_(i.e.

equi-lateral

_triangle

for

_Mg3,

tetrahedron for

_Mg4

and

_regular

_trigonal

_bipyramid

for

_M95

with r =

re(Mg2)

= 7.35 _a.u. interatomic

distances),

the vertical ionization

_potentials

have been calculated and

_they

_appear in table II for both

_single

and double ionizations. Notice the

_rapid

decrease of the ionization

potentials;

for a

_regular

linear chain or

_cycle

of n

atoms the ionization

_potential

would decrease as n-1

(as

shown from Huckel

_type

_analytic

derivations

_[13]).

If,

for a

_given

value of n, the cluster becomes more compact, the

_increasing

number of nearest

_neighbours,

and

_consequently

of

_{hopping integrals,}

results in a

lowering

of the ionization

_potential.

This increased

compactness

explains

the

_rapid

decrease of the calculated ionization

_potentials.

It may be

_interesting

to

_analyse

_{the space and}

_spin

symmetries

of the lowest states of the

ion,

as

_predicted

by

our

_model,

and

_compared

to the monoelectronic

picture,

which _suggests to

_empty

the

_{highest occupied}

MO

_(or

MO’s if a

_degeneracy

_occurs).

The lowest state of the

_triangular

_Mg+

is of course

degenerate, according

to the monoelectronic

_picture

(to

which our model Hamiltonian is here

_identical).

Fig.

2. -

r-dependence

of the various _parameters of the model Hamiltonian. a) diagonal terms _(for sake _of

compa-rison, the r-1 1

repulsion

has been subtracted from R +

_+).

b) off-diagonal

integrals.

Table II. - Vertical and adiabatic ionization

poten-tials

_{(in eV).}

(The

numbers in

_parentheses

are _{secondary minima.)}

A Jahn-Teller distortion must

_necessarily

occur.

In the same

_geometry

one _may

_expect

that the lowest state

_{of Mg3 +}

will be

_{of A1}

_symmetry.

It

_actually

is a

_triplet

3 Ai

state. The next _upper states

are

_degenerate

and would

_correspond

to a different space

part

E

with a

_{singlet spin}

_arrangement. The

lAi

associated

with the lower

_triplet

is

_higher

_{in energy, which is}

surprising

from an MO

_{filling point}

of view.

The tetrahedral

_Mg4+

lower state is

_{triply degenerate,}

as

_expected

from the monoelectronic

_picture.

The

doubly charged ground

state is

_{quintuply degenerate.}

For the

_{bipyramid configuration}

the

_Mg’

lower

state is

_2A1

with

_{larger charges}

on the comers of the

pyramids.

The

_{doubly charged}

lowest state is

_{t At}

and

_corresponds

to two holes in the same

_highest

MO;

two

_{degenerate triplet}

states are

_only

0.016 eV

above.

The occurrence of low

_lying

_open shell states,

sometimes

_{contradicting}

the MO

filling

rule,

exam-plifies

the

_importance

of the

_repulsion

between the

3.2 RELAXED GEOMETRIES AND ADIABATIC IONIZATION POTENTIALS. -

The results appear in table

III,

figu-res 3-4.

The

_Mg’

and

_{Mg2 +}

structures have

_already

been

given,

and one _may see

(Fig.1 c))

that the latter is a local

minimum in the

_{repulsive ground}

state

_potential

curve; this local minimum

might

accept

up to 6

vibrational

_levels,

the barrier

_being

0.35 eV.

The

_ground

state

_geometries

of

_Mg’

and

_{Mg3 +}

are both linear with

_longer

distances

(3.1

A)

in

_Mg’

than in

_{Mg3 +}

_(3.0

_{A) ;}

this is a

_general

trend in the

series. The states are of

2 E:

and

1 E:

_symmetry

in agreement with the orbital model

_(ionizations

from the

_{highest occupied}

_MO).

For both

_Mg4+

and

_{Mg4+ +}

two

_geometries

are in

competition, namely

the square and the centred

triangle.

Both are real minima on the

_potential

surfaces,

the former

_being

0.6 eV below for

_Mg:,

0.07 eV below for

_{Mg1 +.}

The nearest

_neighbour

distances are shorter in the centred

triangle

than in the

square,

especially

for the

doubly charged species

(Ar

= 0.2

A).

The states _are of

2A1

and

1 A1

symme-tries for the centred

_triangle,

_2B2g

and

_’A,,,

for the

square, in

agreement

with the MO

_picture.

For

_Mgs

and

_{Mg5 +}

two

_{nearly degenerate}

minima

were

found,

corresponding

to the centred tetrahedron

(more

stable

_by

0.02 eV in

_{Mg5+ )}

and centred _square

(preferred

in

_{Mg’ ’}

_by

0.09

eV). They

all are

_fully

symmetric,

and

_correspond

to ionizations of the

highest

valence MO in the delocalized MO

_picture.

Fig.

3. - Intrinsic

energies

of _Mg.,

_Mg:

and

_{Mg" +}

clusters

(model Hamitonian), relative to n

_{separated Mg}

atoms

for 0 neutral

_ground

state conformation. A and A, relaxed

conformations of the ions.

Table III. -

Energies

E (eV)

and nearest

_neighbour

distances a

(A)

ofm gg+ and Me

+

clusters

_(the

zero

of energy

Fig. 4. - Interatomic

distances between nearest _neighbours in _positive ions _{(relaxed conformations). V, V} UHF ab initio

calculation. A, A model

_{calculation. re Mg2 experimental}

_{re. dm} metallic nearest _neighbour distance.

It seems

_likely

that for

_larger

and

_larger

clusters the number of

_competing

minima will increase and their

exhaustive

research will become a real

_problem.

In

these structures, whose

topology

is

_similar,

the inter-atomic distances are almost identical.

For centred structures, the

_{positive charge}

tends to

be concentrated on the central atom; this

_charge

is

close to one for

doubly charged species;

this _may be

understood

_by

the decrease in electronic

_repulsions

when the central atom has no valence

electron,

_by

the

polarization

of the

_surrounding

neutral electrons and

_by

the role of cross-roads

_{played by}

the central

atom in the hole delocalization.

The relaxation

_{energies (differences}

between vertical and adiabatic ionization

_potentials)

are

_{quite large :}

There is no clear trend in this series.

The adiabatic ionization

_potentials

_{appear in} table II

They

decrease

_regularly

with n, and it is clear that one

is still

quite

far from the _asymptote, which would be the metallic values

_(one

and two extraction

_potentials

of the

_metal).

3.3

_EVAPORATION,

AGGREGATION AND

FRAGMENTA-TION PROCESSES. -

Figure

3

_gives

the intrinsic

_energies

of

_Mg.,

_Mg/

and

_Mg."

_taking

_{the energy of n}

_Mg

separated

atoms as the zero of _energy. This

_picture

makes _easy a direct discussion of the

_enthalpy

varia-tions in the processes

(neglecting

the zero vibration

_{contributions).}

The

aggregation

process

(a)

occurs from the relaxed

structure of

_{M ± 1}

_(k

=

1, 2),

toward _a relaxed

geo-metry

of Mgn+

(k

=

1, 2).

The decrease of the relaxed

geometry

energies

in

_figure

3 indicates that the

aggregation

process is

always exoenergetic.

We have

not

_analysed

the existence of barriers in these

pro-cesses, but it is

quite

clear -

as summarized in

figure

5

- that the

aggregation

of a neutral atom on the central

atom of the centred structure of

_{Mgn 1}

and

_{Mgn i}

should

_proceed

without

_difficulty,

i.e. in the same

symmetry

and with small _{rearrangements.} The

pro-cesses which

_change

the

_symmetry,

i.e. those

_involving

the _square

_geometries,

are more

_problematic

and the research of activation barriers should be per-formed.

The

_evaporation

_processes are

_{only possible}

from the vertical ionizations.

_They

are

_{only exoenergetic}

for

4+-+3+

+ 1 and

5+ -+ 4+

_(square)

+ 1 and for

5 + + -+ 4++

_(square)

+ 1. _{The energy} excess is

very low

(see Fig. 3)

and the existence of barriers is

possible

in view of the

_large

_geometry

_(and

_symmetry)

changes.

The

_{possible fragmentation}

of

_{Mg+ ’}

into

_Mg+

+

Mg+

has been

_analysed.

The

_{exoenergeticity}

is 1.72 eV when

_starting

from the lowest _{energy tetrahedral}

geometry

toward the relaxed

_geometry

_Mg’

and

Mg+

ions. For a

_{fragmentation pathway keeping}

C2,

symmetry,

the intersection between the relevant

A, (Mg 5 +

+)

and iB2(Mg3

+

_Mgi)

_potential

surfaces has its lowest energy 0.21 eV above the tetrahedron

Mg5 +

energy minimum. This

quantity

may be

Fig. 5. - Net

charges,

symmetries

of the clusters, and schematic views of

_{possible aggregation}

and

_evaporation

processes. ---+ allowed processes. --->

hypothetic

processes

(exoenergetic

but with

_possible

barriers).

Another

_{fragmentation}

_process,

has also been

_explored

Its

_{exoenergeticity}

is 1.09

_eV,

, lower than the

preceding

one. It is

quite

clear that the

doubly charged species

tend to

_explode

into

_equal

size

fragments,

due to the n

_dependence

of the

_stability

of

_Mgn

clusters.

The barrier for the extraction process of the central

atom

_{of Mg5 +}

is

_only

0.14 eV.

3. 4 OBSERVABILITY OF DOUBLY CHARGED CLUSTERS.

-8 The existence of local minima in the

potential

_energy

hyper-surface

of

_{Mgn+ +}

is not sufficient to insure the

observability

of these

clusters,

since these minima

are

significantly higher

than the result of a

fragmen-tation into

_Mgp

+

_{Mgq ( p}

+ q =

n).

Another factor

which _may

_prevent

the observation of the

_doubly

charged

clusters is the _great difference between the

geometry

of the neutral cluster and that of this local minimum. A vertical double ionization would not

lead to the

_{doubly charged}

cluster since it

_brings

an

important

energy excess and the _system could

_hardly

achieve the travel from the

_original

neutral cluster conformation to that of the

_{doubly charged}

minimum and _stay in that

region

which is

_only

surrounded

_by

weak barriers.

This statement is

_{schematically}

illustrated in

_figure

_6,

representing

for instance the

_{Mg (k+)}

(k

=

0, 1,

2)

potential

curves as functions of two _parameters

_only

r =

rAB =

rBc and 0 =

ABC.

It is

quite

clear that the vertical double ionization

_gives

a small

_probability

to

_stay

in the local minimum of the _upper surface. As a _very

_important

fact,

one should notice that in

all the cases studied

here,

the stable conformations of

Mg+

and

_{Mgn +}

are

_quite

similar. This means that an

ionization from

_Mg.’

to

_{Mg,n+ +}

would be

_nearly

vertical,

and would lead close to the bottom of the

Mg/+

structure,

giving

an

opportunity

for this

cluster to have a

_significant

lifetime. One must thus

assume that the

ejections

of the two electrons under the electron

_impact

are not simultaneous.

_During

the time

_separating

the first and second electron

emission,

the cluster

_Me

should rearrange toward the stable

M g,7 configuration,

which is an absolute

_minimum,

the second emission

_proceeding

without

_important

geometry

changes

towards the

_secondary

minimum of the

_{Me +}

_{potential hypersurface}

which holds the

Mgn+ +

cluster

_{(cf. Fig.}

_6).

These remarks are crucial for small

clusters,

and for

all cases where the

_charges

are

_quite

_{concentrated,}

inducing important

rearrangements

of the

_positive

cluster with

_respect

to the neutral one. For _very

_large

clusters the _{rearrangements}

_might

become small

(vide infra)

and creations of

_{doubly charged}

clusters

Fig.

6. - Schematic view of the lowest

potential

surfaces of

_Mg(,")

_(k = _{0, 1, 2) as} functions of r = rAB = ’Be and 0

= ABC.

4.

_Comparison

with ab initio SCF

calculations ;

metho-dological

discussion

4.1 RESULTS OF SCF CALCULATIONS. - Valence

only

[14]

SCF calculations have been

_performed

on both

singly

and

_{doubly charged}

_clusters,

_{using (3s, 2p)}

AO basis sets

_(i.e.

_deleting

the 3d AO involved in the MO-CI calculation on

Mg’

and

Mg2 +).

A UHF

(Unrestricted

Hartree

Fock)

formalism was used with

Sz = 1/2

for

_Mg’

and

_Sz

= 0 for

Mg’

+,

but for the latters the solutions were

_always

_pure

_singlets.

These calculations have been

_performed

to

_check;

i)

the occurrence of minima close to the structures

predicted by

our

_model;

_ii)

the

_{possible importance}

of p

AO’s which do not _appear

_explicitly

in our scheme.

The

_energies

and

_geometries

appear in table

III;

they

concern real minima obtained from a

gradient

program and force constants calculations.

They

con-firm that all the

_predicted

minima also appear in a monoelectronic

_approach.

This

_agreement

is not too

surprising

since,

for the studied

_minima,

the model

wave-functions were consistent with

_single

and double

ionizations from the

_{highest occupied}

MO.

_Significant

differences appear however :

i)

the ionization

_potentials

are

always

much

lowqr

than in our model. This is

_already

true for n =

1,

since the correlation of the

S2

_pair

in

_Mg

is _very

important, increasing

the ionization

_{potential by}

about 1

eV;

this defect is still

_acting

in the cluster UHF

calculation;

ii)

the relative _energy

_ordering

of the two minima for

_{Mg’, Mg4 +}

and

_Mg5+

is reversed with _respect to

our

_model;

one _may notice that for

_Mg4+

and

_Mg4++

the monoelectronic calculation favours the centred structure;

iii)

the bond interatomic distances are much

_larger

(see

Fig.

4).

These two

_points

of

disagreement

will be discussed later. The electronic

_populations

calculated

accord-ing

to the Mulliken criterion would indicate a more

even net

_charge

distribution;

they

also would indicate

a

_significant

_p

_population

on the central atom.

However a careful examination of the MO’s shows

that p, AO’s do not _appear with coefficients

_larger

than

0.15;

the

_{hybridization}

is therefore rather weak and the

_apparent

_p

_population

is

_essentially

due

to the

_overlap

_{of the p, AO’s of the central} atom with the s AO’s of the

surrounding

atoms, in a

_typical

artifact of Mulliken’s

_{population analysis.}

This is confirmed

_by

_{the fact that the Pc}

_diagonal

element of the

_density

matrix is

_negligible

₍

0.03

_eV).

4.2 COMPARISON BETWEEN MODEL AND ab initio SCF

CALCULATIONS. - The

discrepancies

between our

model and SCF results _may be

_easily

understood It is clear that ab initio SCF calculations introduce

many-body repulsive

effects which are not taken into

account in our model. However these

_many-body

effects are not _very

_large

and

_they

cannot

_explain

the differences in bond distances and in the relative

energy

orderings

of the

Mgi

and

Mg: ’

isomers.

i)

Regarding

the interatomic

distances,

one should

notice that in a cluster the

probability

for two

_adjacent

atoms to be both neutral is 1

- Ifor

_{singly charged}

n

clusters ai for

_{doubly charged}

clus-ters. In both cases this

probability

_goes

_rapidly

to one.

One _may therefore

_expect

that when n increases the

mean interatomic distances tend toward that of the

neutral cluster

_(i.e.

the metal for

_large

_n).

Since the monoelectronic

_picture

does not take into account

the

_dispersion

_{energy between neutral atoms,} it will let the cluster

_explode

and the interatomic distances will tend to

_infinity.

This failure will also occur in

_positive

clusters since the delocalization _{energy of the hole is}

bound

_by

kF +

_{(k being}

a

_scalar,

_depending

on the compactness of the

cluster)

as are the _upper or lower

limits of a band Since the

_repulsive

interactions ER

increase as n, the delocalization of the

hole,

included in the monoelectronic

_scheme,

cannot

_{fight efficiently}

against

the

_{increasing repulsions,}

and the

_positive

cluster is more and more

_loosely

bound with

_larger

and

This

defect,

which does not _appear in our

model,

is _very difficult to overcome in ab initio

calculation,

since it would

require

the calculation of

_dispersion

forces -

a task which is

_already

difficult for

_Mg2

[15].

ii)

Regarding

the relative

_energies

of centred versus

square structures in

_Mg+

and

_{Mg+ +,}

one should

notice that the Hartree-Fock

_{approximation only}

takes into account the static held In

_homogeneous

structures, as is the _square

_cluster,

the

_charge

is

_equally

spread,

the static field is weak

_everywhere;

on the contrary if the

_charge

is well concentrated on one

atom - as occurs for the central atom of the centred

triangle

- the static field is

important

on the

sur-rounding

atoms, and the HF

_{approximation}

is able to

take this effect into account.

Our model deals with instantaneous

_{repolarizations}

in all VB local structures, and therefore takes into

account an effect which would

only

_appear

through

double excitations in the MO scheme

_(change

of the hole from the

_{highest occupied}

MO to lower MO

_plus

one excitation to the outer

_{polarization space).}

These comments - which will be

_developed

in detail in a

forthcoming

_paper

[16]

-

explain

the

_preference

of the monoelectronic calculation for centred

struc-tures, as due to an unbalanced

_purely

static treatment

of

_polarization

effects.

5. Conclusion.

A careful extraction of information from the

_potential

curves of the diatoms

_Mg2,

Mg2

and

Mg2

+,

corres-ponding

to the states

_keeping

an sP +

_fl(p

_{+ q} =

4, 3,

2, p, q

2) character,

permits

to define a

simple

model

Hamiltonian of low size for the

_study

of

_singly

and

doubly charged

clusters. The Hamiltonian involves

only

6

_parameters

and its size is

n2

for

_{doubly charged}

clusters. The

_parameters

_{being analytic}

functions of the

interatomic distances

_{gradient techniques}

_{may be}

easily applied

Since the extraction of the _parameters has been done from calculations

_involving

basis sets and

CI,

they

should

_provide

some information for

_large

cluster

studies,

which are unattainable from direct MO-CI

calculations since

_{they rapidly require}

reductions of the basis set

_and/or

severe CI truncations. The

superio-rity

of our model over ab initio SCF calculations

appeared clearly

in the

_preceeding

discussion on both the internuclear distances

_(unreliable

in SCF

calcu-lations)

and the _energy

_ordering

of various minima

(unduly preferred

as centred structures in SCF

calcu-lations).

The

_present

work has shown stable structures for

Me

and

_{Mg,.+ +}

for n

5. It will be

_interesting

to

analyse

the

_{general topology}

of the

_potential

_energy

hypersurface(s), taking

benefit of some recent theo-retical advances

_[17J.

This would

_give

better insurance

regarding

the number of

minima,

a

_problem

which

will be of

_{increasing importance}

for

_larger

clusters. It will also be worthwile to

_analyse

the existence and

heights

of barriers for

_{aggregation, evaporation}

and

fragmentation

processes.

Extension to

_larger

clusters of

_Mg

atoms is in

progress. An isoelectronic series may be

treated,

namely

the

_Hg’

or

Hg/ +

clusters studied in

refe-rence

_[1].

The extraction of the basic parameters for this

_heavy

atom _{is in progress.}

_Finally

one should mention that a direct extension to rare _gas clusters is

easy to

_perform,

the number of AO

_being

then increased

to 3 n

_(and

the size of the matrix to

₍₃

n)2

for

_doubly

positive

clusters).

In that last case the

_problem

of

hybridization,

which _may lead to difficulties for

_large

values of n in

S2

_atoms, would be avoided

References

[1] BRECHIGNAC, C., BROYER, M., CAHUZAC, Ph., DELA-CRETAZ, G., LABASTIE, P. and WÖSTE, L., Chem.

Phys. Lett. 118 (1985) 174.

[2] ERMLER, W. C., LEE, Y. _{S., CHRISTIANSEN,} P. A. and

PITZER, K. S., Chem.

_Phys.

Lett. 81 ₍₁₉₈₁₎ 70.

CHRISTIANSEN, P. A., BALASUBRAMANIAN, K. and

PITZER, K. S., J. Chem. _Phys. 76 (1982) 5087.

[3] TEICHTEIL, Ch., PÉLISSIER, M. and SPIEGELMANN, F.,

Chem. _Phys. 81 (1983) 273, 283.

[4] KUNTZ, P. _J., Chem. _Phys. Lett. 16 (1972) 581 ; TULLY,

J. C., Modern Theoretical _Chemistry, Vol. 7,

G. A.

_{Segal ed.}

(New York, Plenum), 1977,

p.173.

[5] PARISER, R. and PARR, R. G., J. Chem. Phys. 21 (1953)

466, 767 ;

POPLE, J. A., Trans. Far. Soc. 49 ₍₁₉₅₃₎ 1375.

[6] POPLE, J. _{A., SANTRY,} D. P. and _SEGAL, G. A., J. Chem.

Phys. 43 _{(1965) 129;}

BAIRD, N. C. and _DEWAR, M. J. S., J. Chem. Phys. 50

(1969) 1262.

[7] HUBBARD, J., Proc. R. Soc. London A 276 (1963) 238.

[8] HUBER, K. P. and HERZBERG, G., Constants

_{of Diatomic}

Molecules _{(Van Nostrand,} N. _Y.) 1979.

[9] SCHEINGRABER, H. and _VIDAL, C. _R., J. Chem. _Phys. 66

(1977) 3694.

[10] STEVENS, W. J. and KRAUSS, M., J. Chem. _Phys. 67

(1977) 1977.

[11] SPIEGELMANN, F. and MALRIEU, J. _P., J. _Phys. B 17

(1984) 1259 ;

CIMIRAGLIA, R., MALRIEU, J. _{P., PERSICO,} M. and

SPIEGELMANN, F., J. _Phys. B 18 ₍₁₉₈₅₎ 3073.

[12] DURAND, G., BERNIER, A. and _{SPIEGELMANN, F.,} to be

published.

The effective

_coupling

between the

[Mg++

+

_Mg(3s2)]

and

_[Mg+(3s)

+

_Mg+(3p)]

configurations

is _{a 3s|F} |

3pz > hopping

the diabatic

_potential

curves to touch the adiabatic

ones in this

region.

[13] See for

_{example :}

SALEM, L., The Molecular _{Theory of}

Conjugated

Systems

(Benjamin,

N. Y.) 1965.

[14] DURAND, Ph. and _BARTHELAT, J. C., Chem. _Phys. Lett.

27 (1974) 191.

BARTHELAT, J. C. and _{DURAND, Ph.,} Gazzetta Chim. Ital. 108 (1978) 225.

[15] PURVIS, G. D. and BARTLETT, R. J., J. Chem. _Phys. 71

(1979) 548.

[16] MALRIEU, J. _{P., DURAND, G.,} DAUDEY, J. P., to be

published.

[17] LIOTARD, D., Symmetries and _{Properties of}

_non-rigid

molecules : A

_{Comprehensive}

_Survey.

_Proceedings

of an International

_Symposium,

_Paris, 1-7

_July

1982, J. Maruani and J. Serre _(eds), p. 323 (refe-rence therein).