Electron correlations and chemical bonds in aggregates of monovalent elements. A poor man's description

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Electron correlations and chemical bonds in aggregates of monovalent elements. A poor man’s description

J. Friedel

To cite this version:

J. Friedel. Electron correlations and chemical bonds in aggregates of monovalent elements. A poor man’s description. Journal de Physique, 1987, 48 (1), pp.93-106. �10.1051/jphys:0198700480109300�.

�jpa-00210427�

Electron correlations and chemical bonds

in aggregates of monovalent elements. A _poor man’s description

J. Friedel

Université Paris-Sud, Laboratoire de Physique des Solides Bât 510, ⁹¹⁴⁰⁵ Orsay, Cedex France

(Requ ^{le 16} juin 1986, accepté ^{le 26} septembre 1986)

Résumé.

A l’aide d’un simple ^{modèle de} Hubbard dans la limite des faibles corrélations, ^nous analysons le rôle des interactions et des corrélations électroniques ^{sur la nature} des liaisons dans les agrégats monovalents. Nous discutons la validité de cette approximation simple.

Abstract.

Using ^a simple Hubbard model in the weak correlation limit, ^we analyse the role of electron interactions and correlations ^{on the nature of} bonding ⁱⁿ monovalent aggregates. ^We discuss the validity ^{of this} simple approximation.

Classification

Physics ^Abstracts

31.20P

71.00

Introduction.

The Hubbard model [1, 2] ^{has been} developed ^to analyse ^cases where electron correlations are too strong

to be treated only ^{as a small} correction in the effective

potential ^{of otherwise} delocalised independent ^elec-

trons, ^{and for} aggregates ^too large ^to be treated by

more exact configurations interaction techniques.

This treatment of electron correlations is compen- sated by ^{a more} approximate description ^{of wave} functions, ^{based on} atomic averages. Thus the ^wave functions are based on a LCAO approximation:

where a ) ^{are a reduced set of atomic orbitals on sites}

a. The kinetic energies ^are expressed ^{in terms of}

intersite transfer integrals

and electron interactions by ^atomic averages :

etc.

In the simplest ^{Hubbard model, one} neglects ^non orthogonality of atomic functions on different sites

one restricts t to nearest neighbours ^{a, b and} only

consider intraatomic Coulomb interactions U.

In the limit of I U /t I ^small [3], ^a development ^will give for the total energy of ^an aggregate

E=A+Bt+CU+DU2/t+0(U3/t2). ⁽⁶⁾

The term A is essentially ^{of atomic} origin [4]. ^{Bt is the} bonding ^{due to electron} delocalisation. CU is the Coulomb repulsive ^term computed in the Hartree Fock limit for uncorrelated delocalised electrons [5] : ^their

random distribution over lattice sites, ^with only ^the

Pauli exclusion principle ^{on atomic} orbitals, ^{leads to} charge fluctuations responsible ^{for this term} (Fig. 1).

Finally ^{the term} DUZ/t corresponds ^{to the first devia-}

tion from perfect ^{Hartree Fock disorder} [3] : ^the

interaction U tends to decrease the charge fluctuations

by ^a correlation of interactomic electronic jumps. ^{In this} perturbation limit, it is the first term due to electron

Fig. ^1.

Charge fluctuations in the Hartree Fock limit.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480109300

correlations ; ^{it is} clearly ^{of a} less local nature than the Kohn Sham correction [1, 6]. ^{One can} finally ^remark

that the terms in Bt and DU2/t ^are necessarily negative

while CU is positive.

The purpose of this paper is ^{to test} this limit for

problems ^{of small} aggregates ^{where the} physical proper- ties are generally ^{known and more refined} theoretical

treatments at hand. We shall see that this « poor man’s treatment » gives simple and reasonable results, ^but

that the exact values of the parameters involved, ^and notably U, ^{are not} easy to extract from first principles.

1. Dimers versus close packed aggregates.

We first recall and slightly ^correct earlier results on the relative stability ^{of a close} packed aggregate ^{and of a}

collection of dimers [7], ^thus setting the model and its

approximations.

1.1 DIMERS.

For a dimer of hydrogen ^{atoms with}

valence electrons in Is orbitals I a ) and l b ) , ^{we can}

write for the singlet ^and triplet ^states respectively

where

or

Because the Hamiltonian is preserved ^{when the two} nuclei a, b ^are permuted, 141) ^{must be} symmetrical ^or antisymmetrical ^{in this} permutation. ^{There are therefore two} types ^of singlet ^{states :}

while

With the Hamiltonian written with obvious notations

the energies ^{of these various states are} given by

where S, t, ^{U and u are} given by equations (2) ^to (5)

and

If one neglects S, j, v, ^the preceeding equations give

where

is the repulsive ^Coulomb energy between the two

neutral atoms on a and b.

In the simplest ^Hubbard approximation, ^one neglects

u with respect ^{to U. The most} stable (singlet) ^{state has}

then the energy

In the limit I U It I ^{4, a} development ⁱⁿ U/t ^{is valid.}

It gives, ^{with t 0 for s} orbitals, ^an energy per ^atom

The two first terms in this development ^{are of atomic} origin ; ^the following ^{ones are due to electron de-} localisation ; ^{the terms} linear in t and U are obtained in the delocalized Hartree Fock limit, ^{and the term in}

U2/t is the first deviation from delocalized Hartree

Fock, ^{due to} electron correlations. They could have been computed directly.

More precisely, ^{in the} delocalized Hartree Fock

limit, the electrons are assumed to move independently

from each atom (A = 03BC ) , ^with opposite spins ^{in the} singlet ^state.

The term in t gives the increase in kinetic energy per atom, and thus per electron, ^{due to this} delocalization ;

due to the virial theorem, ^{this is} also, ^{for the Coulomb}

forces involved, ^the corresponsing lowering ^{to total}

energy.

The term in 1 U takes into account the intraatomic 4

interactions between electrons of opposite spins ^{in the}

same limit, ^{where the} probability ^{that a} charge ^fluctu-

ation of two electrons on the same site is, per site,

1 1 1 2X2 ^4’

The term in U2/32 t ^{can be} considered as due to a

virtual excitation of bonding ^{states into} antibonding

ones (Fig. 2), owing ^{to the} V12 interactions. Single

excitations correspond ^to matrix elements such as

with

they ^vanish by symmetry. ^The matrix element for double excitation is

the excitation energy - 4 t leads then, in the limit of small U’s, ^{to a} contribution per ^alom

The equilibrium distance r2 between the two protons

can be deduced from the variation with this distance of R and t. We can assume

JOURNAL DE PHYSIQUE. -T. 48, ^N. 1, JANVIER 1987

Fig. ^2.

Singlet ^and triplet ^{states of the} H2 ^molecule.

where direct estimates show that [7]

Then writing

one finds, by minimizing E2 :

with

One sees that the correlation term in U2 both stabilizes the dimer and increases the bond length.

1.2 CLOSE PACKED AGGREGATES. - A development

similar to (24) ^{can be} given ^{for a} (macroscopic)

aggregate where each of the N atoms is surrounded by

n equidistant ^nearest neighbours [3, 7].

In the Hubbard limit, the energy pir ^{atom reads}

where the effective Is band width is related to its second moment and can be written (n, ^{number of} neighbours)

These expressions have numerical factors deduced

from a rectangular band, ^{with constant} density ^of

states ; but they ^{can be shown to remain} good approxi-

mations for more realistic band shapes.’

More precisely, ^{the band} term -1 w ₄ ^{comes from the}

fact that the broadening of the 1s band is essentially symmetrical in energy, because its first moment is zero ; with one electron per atom, the lower half of the band is occupied, ^{with an} average energy per electron

4 w. The increase of w with BIn ^is equivalent ^to computing ^the kinetic energy assuming ^{a Brownian}

motion of the electron from site to site.

The term in 1 U is the same as for the dimer because 4

the charge fluctuations are the same per ^atom for dimers and for close packed aggregates.

The term in U2/16 w takes into account virtual collisions of electrons from the occupied ^{into the} unoccupied part ^{of the 1s} band, ^{with an} average energy of excitation w and a probability ^of charge fluctuations in the initial and final states 1 B2 x 1 - 1. ^{2 16}

Using ^the variations (25), (26) of t and R with the interatomic distance r, ^one finally ^obtains

where w is given by (32) ^and

1.3 DIMERS VERSUS CLOSE PACKED AGGREGATES. - The preceding analysis ^{can be} directly ^{extended to the}

valence s electrons of all monovalent elements, ^with similar values of q/p.

Equations (30) ^and (35) ^{show that, for small values}

of I U/t2 , ^{close packed} aggregates ^have interatomic distances somewhat larger ^than dimers ; ^{but the corre-}

lation correction expands ^{dimers more than close}

packed aggregates.

A similar balance is observed for the energy, when

comparing equations (24) ^to (31) ^or (29) ^to (33).

Qualitatively, ^the Hartree Fock term due to delocalisa- tion stabilises the close packed aggregates, ^{while the} correlation term favours dimers. As the repulsive ^term

also favours the dimers, the critical value of I U/t2 I

above which dimers are preferred ^{and below which}

close packed aggregates ^{are more stable somewhat} depends ^{on the value of} q/p. ^{Table I} gives ^{this critical}

Table I.

Critical ^value of I Uc/t2 versus q/p.

value I Uc/ t2 l. ^{It is seen that,} for the reasonable value

q/p ^{=.--!z 2.5,} UC/t2 ^{=-= 2. This is much smaller than the}

critical value 4 above which the development ⁱⁿ I U /t I used here is no longer ^valid.

This simple ^minded analysis is coherent with observa- tion for monovalent elements if U Uc ^{for all alkali}

metals, ^while U:> Uc ^{for H, where the stable state is}

made of dimers weakly coupled by dispersion ^forces neglected ^here.

Indeed dimers seem to be prefered ^{for all elements}

with strong electronegativity (H, ^{0, N,} halogens) ^while

close packed aggregates ^are prefered ^{in all other}

elements. It is obvious that a similar analysis ^{can be}

extended in principle ^{to other} types of valence electrons

( p, d ) ^{with similar} qualitative conclusions [7]. ^This

remark therefore suggests that U increases with elec-

tronegativity, ^a point ^{we will come back to at the end of}

this paper.

2. Dimers versus trimers in monovalent elements.

The preceding discussion can be extended to comparing

dimers with small close packed aggregates, ^{and more}

specifically with trimers.

The situation for monovalent trimers is indeed well studied theoretically by interaction configurations techniques [8, 9] ; ^{a number of} experimental results fit with their conclusions, notably ^from astrophysical ^data [10], and from studies in vapours [11] ^and jets [12].

These studies conclude again ^{to a} difference of be- haviour between H and alkali metals.

Close packed H3 ^{trimers are} unstable ; only ^the

situation of a dimer H2 weakly coupled ^at long range with an H atom is stable. Contrarywise H+ ^{are stable as}

close packed equilateral triangles.

For alkali metals, neutral trimers exist as isocele

triangles, ^{with a summit} angle ^somewhat larger ^than 7T 1i°

The question ^{to be discussed} here is how much of this behaviour can be described using ^{the same} simple

Hubbard model.

2.1 H TRIMERS.

We first compare, for H trimers,

three simple geometries (Fig. 3).

In an independent electron molecular orbitals Hartree

approximation, ^{we can use wave functions of the} type

For configurations ^of figures ^{3a, a, b, c, the one}

electron energy ^states satisfy respectively

Hence

E ⁼ Eo ^{+ 2 t and E =} Eo - t (doubly degenerate) ^{for the} equilateral triangle.

E ⁼ Eo ^{± t} J2 ^{and E =} Eo for the linear chain.

E ⁼ Eo ^{± t and E =} Eo for the dimer + atom.

These spectra ^are pictured figure ^4.

In the corresponding Hartree Fock approximation, ^the ground ^{state of} H3 ^{can be written as} (cf. Appendix A)

Fig. ^3.

^Three simple configurations ^{for trimers :} a) equila-

teral triangles ; b) ^linear chains ; c) ^{dimer + atom.}

Fig. ^4.

One electron energy spectrum ^{in the limit} U ⁼ 0 for the configurations ^of figure ^3.

where a refers to the molecular orbital with lowest

energy, f3 ^{to the next} molecular orbital state and

I ± ) ^{refer to the} sign ^{of the} spin. ^{With that} approxi- mation, ^{the total} energies ^of H3 ^{are :}

to be compared ^with (Eq. (23))

For H3’ , ^one obtains in a similar way

to be compared ^with

It has been assumed here that each atom remains neutral and thus that each pair ^{of atoms} repels ^{with the}

same energy ^R (r) ^{as for a} dimer. This point, ^{which is} obviously ^{not exact for} H3’ , ^is discussed in § 2.4 below.

The computation, ^{detailed in} Appendix A, ^{is cohe-}

rent with the fact that, ^{in such a} linear combination of atomic orbitals approximation, ^the exchange ^{hole of}

the Hartree Fock approximation (38) is of atomic dimension. As a result the term in U in the _{energy of} reduces to the sum of Coulomb interactions between « + ) and I a - )

states, and between I (3 + ) and I a - ) ^{states, of} antiparallel spins, ^{while the term in U} in the energy of

H3 ^{is that} between I a ) ^{states of} opposite spins.

Because of the asymmetry ^of the trimers considered,

the charge distribution of each one-electron state is not

uniform, ^{as for a dimer or a close} packed aggregate. ^As

a result, ^{the linear term in U now} depends on the

geometry of the trimer.

The correlation terms in U2/t ^{can be} computed treating ⁱⁿ perturbation ^the configuration interaction.

For H3, ^{this can be written} formally

where the summation extends ^over excited states

I .p f ) , of ^{energy Ef 3 A somewhat long but} straightfor-

ward computation (Appendix B) gives ^{Y =} U2/xt ^with

x = 18 for the equilateral triangle ^{and x} = 17.3 for the chain.

Inclusion of this term gives finally ^the following developments ^for H3 :

with

Development (40), ^{valid for} H3 , shows that the

bonding ^{term in t due to} delocalization strongly ^stabil-

ises the equilateral triangle. ^{This is due to the} large bonding energy of the most bonding ^{state in this} configuration (Fig. 4a). ^{The Coulomb term in U also}

stabilises the equilateral triangle. ^{As the terms in}

U2/t, ^{which are not} given here, ^{have similar coef-} ficients, ^{it is seen that the} equilateral triangle configura-

tion is preferred ^for any value of I Ult I ^small enough

for development (40) ^{to be valid.}

Development (42), valid for H3, ^{shows that the}

delocalization term in t tends to stabilize the equilateral triangle, ^{but the terms in U and in} U2/t ^{tend to stabilize}

the dimer. There should therefore be a critical ratio of

I U / t I ^{below which the} equilateral triangle ^is preferred

and above which the dimer is preferred. Because the

repulsive ^{term in R favours the} dimer, ^{this critical value}

should increase with the ratio q/p. ^More precisely, minimizing ^the energy with respect ^{to r} gives

with

and

Comparison ^with (29), (30) shows that for

q/p ^{= 2.5, the critical} value I U/t2 I ^{is again very near}

to 2. Thus the Hubbard model leads to the same

conclusions for the relative stability ^{of a} closepacked

trimer versus a dimer as for the relative stability ^of closepacked macroscopic aggregates ^versus dimers, ^i.e.

large ^{values of} I U/t2 I ^{lead to dimers, with the same}

critical value of I V / t2 I. ^{It is therefore} coherent with this conclusion to think again that U > Uc ^{for H and}

U Uc for alkali metals, with l Ve/ t2 1== ^2.

2.2 NEUTRAL ALKALI TRIMERS.

For alkali trimers where I U/t2 I ^{is expected to be} small and the equila-

teral triangle ^{to be more} stable than the dimer or the linear chain, inspection ^of figure ^{4a shows} that, ^{in the}

limit U -+ 0, ^{one electron} occupies ^a degenerate ^state

( ^{EJ3 =} Ey ). The trimer is then expected ^{to lower its}

energy still further by ^a spontaneous distortion.

In the simplest such Jahn Teller distortion, ^the equilateral triangle ^{becomes an isocele} triangle ^{with a}

summit angle 0 # I. ₃ Treating 0 - -7jr- ₃ ^{as a} perturbation

to second order, ^{one can show that the} equilibrium

state of lowest energy corresponds to 03B8 > 3’ ₃ ^as

schematically ^{shown in} figure ^{5. The critical} angle depends ^{on the ratio} q/p ^{and on the value} of I U/t2 I ’

However for q/p ^{= 2.5 and} vanishingly ^{small values of} U, ^60° Oc ^{90°; and it is not very sensitive to U} when, ^{as assumed} here, I U/t2 2. This is in _agree-

ment with more exact computations ^{and the few} experimental ^evidence gathered ^{so far.}

More precisely, let t’ and t" be the transfer integrals

involved in the isocele triangle pictured ⁱⁿ figure ^{6. If}

we write

the summit angle 8 ^is larger than 03C0 ^{if T"} > T’.

Fig. ^5.

Jahn Teller distortions from the equilateral triangle.

Fig. ^6.

^Isocele triangle.

The corresponding one-electron wave functions satisfy, ^{for U =} 0, ^the equation

with roots

One note that

the lifting ^{of the} degeneracy ^is qualitatively ^that

of figure ^5.

Thus the total Hartree energy is, in the limit U - 0,

To simplify ^the discussion, ^we shall consider two

types of distortion from the initial equilateral triangle.

We shall consider them as independent, ^{and thus} neglect their weak coupling :

a) ^a pure dilatation of the equilateral triangle, ^with

T’ ⁼ T ". The equilibrium condition for z’ ⁼ T" ⁼ 0 leads to

D) a snear like moae With

A development ^of (47) ^then gives (cf. Appendix C)

If we write

we have

with

With (45) ^and (49), ^this gives ^to second order term in T’:

Putting ^this expression ⁱⁿ (50), ^{we see} that, ^as in any Jahn Teller effect, the linear term :t tT’ lifts the

degeneracy, decreasing the energy in ^a symmetrical

way whatever the sign ^{of T’,} thus of T" - T’. The terms in T’2, ^{which are} altogether positive ( t 0 ) , ^stabilise

the values of T’, thus the total distortion to a finite

r’2 t

value. Finally ^{the term in ±} 12 ₁₂ stabilizes preferentially

the a’ branch where the third electron is on the E J3

state, thus the distortion with T’ 0 _or, according ^to

The equilibrium distortion is given by minimizing

Ea3’. ^{This gives}

With q/p == ^{2.5, this} gives

Hence

As pr =-= 1.4 ^for hydrogen [13] ^{and is} larger ^{for other} elements, it is seen that 0 deviates from 60° by ^{a rather}

small amount.

Finally ^the correction U in the Hartree Fock approxi-

mation can be deduced from computing (Appendix A)

where ’lLaa ^and ’lLal3 ^{are given by} (A.6), (A.13).

From (46), ^{one deduces}

and

Hence

with, for the shear mode

one obtains

The Hartree Fock energy then becomes, ^{for the}

a’ branch

Minimizing ^with respect ^{to T’ shows that} the correction in U slightly ^increase the Jahn Teller distortion, ^which becomes

The effect is not large ^{in the} limit I Ult 2 ^con-

sidered here.

2.3 INTERATOMIC COULOMB TERMS.

We have im-

plicitly ^{assumed so far} that, ^{in all} configurations, ^atoms keep neutral. This is generally ^{the case, and when not}

so, does not change ^our conclusions.

First, in neutral aggregates with alternate structures

i.e. where closed circuits with only ^{even number of}

interatomic jumps ^{can be drawn, a} half filled band leads to all atoms being ^neutral. Particular cases in our

discussion are the dimer and the linear chain. For such cases, if

are the one electron wave functions, they satisfy

To each solution 10 (E) ) ^with coefficients an ^corres-

ponds ^a solution I qi’ ( - E ) ) with coefficients

( -1 ) n an where n is the parity ^{of the site n. The sum}

of the contributions an 2 of the lower and upper hand halves of the band on the electronic density ^{on each site}

n are thus equal. ^{As for a full} (non degenerate s) ^band,

the electronic density ^{for a full band} gives 2 electrons

on each site, ^a half filled band as here has necessarily

one electron on each atomic site. This can be checked

directly.

Still in neutral aggregates, ^{the use of Bloch states for} the equilateral triangle shows that again ^{each atom has}

the same electronic charge, ^that is neutral. This is not so for the choice of stationary ^{wave functions} (A3)

made in Appendix A. However because of the degener-

acy of the partially occupied ^states /3, y with energy

Ep, y ⁼ Eo ^{+ t, one} should make an average between the contributions of the 8 ^state (as ^{assumed in the} computation) and of the y state ; these add to a total of

one electron per site, while the terms in U and

U2 computed ⁱⁿ appendices ^{A and B are} independent

of the choice of basis.

Now the Jahn Teller distortion lifts the degeneracy

between the /3 ^{and y} states, ^{and creates} ionic charges ^on

the atoms of the neutral trimer. These are different for

0 > 3 ^{but not very large. Thus in} the limit of weak distortions where the contributions of T’ and T " are

1 1 1

negligible, ^{the ionic} charges ^{are ±} 3’ ^{3 +} 1, =+= 6 ^{6 6 on atoms}

a, b, c ^{for e ’} "3. The effect of these charges ^{has been}

3

taken into account in computing ^the intra-atomic term in U, ^{but not the} interatomic terms in R. This correction

changes the overall stability ^{of the} (distorted) ^trimer by

a small term

r =1.5 atomic units, ^{this is} of the order of 1.5 eV.

Because of its symmetry for 0 / §, ^its dependence ^{on r}

will not alter the symmetry ^{of the} energy splitting ^for

small distortions nor the fundamental asymmetry ^for

0 a § ₃ ^{shown in figure 5. But this} dependence ^{on r will}

somewhat shift the equilibrium distortion.

Finally ⁱⁿ H’ , ^{the ionic} charge ^is equally distributed in the equilateral triangle. ^The ensuing interatomic Coulomb repulsion ^is largest ^{for that} configuration,

where it introduces a positive correction e = 6 eV. e2 3r

From the estimates below for H, this is about

U. Thus the equilateral triangle ^{is still} preferred ^for H

in all the range I U /t 14 where the perturbation

scheme used here is valid.

3. Value of U ; validity ^{of the} Hubbard model.

If we take the Hubbard model at its face value, ^{U can}

be computed ^from (3). ^{For 1s wave functions of} H, ^this gives ^{U = 17 eV.} Applying (24) ^to compute ^{the cohe-} sive energy - ( E2 - ² Eo) ^{of H2,} which is 4.4 eV, ^we find I U / t2 1== 2. This is nearly ^{of the} right ^{order of} magnitude, ^{as the} preceding discussion suggests

UI t2 > ^2.

It also justifies ^{the use of a} perturbation scheme valid

for I UI t2 4.

There are however a number of difficulties.

a) First, ^as pointed ^{out in} (21), if the interatomic interaction u is kept ⁱⁿ computing R, ^{it should be} kept everywhere ; ^{this amount to} replace ^U by ^{U - u for}

dimers. This is physically ^{obvious, as} the energy involved in electron correlations is there the difference in energy between ^two electrons on the same or on

different atoms. In more extended aggregates, ^{the role} of u would be more complex ; however, ^{for a} half-filled band and weak correlations, ^one expects ^the charge

fluctuations to be mostly with very short wavelength,

thus U - u to replace also U in the most effective

charge fluctuations. However a direct computation ^{of u}

gives e 2 e2 ^{with r} = 1.4 atomic units for H2, ^{thus u == 19}

r

eV, ^even larger ^{than U !}

b) ^The qualitative solution of this difficulty ^{can be}

taken as due to the fact that we have used too restricted

a basis of wave functions.

Indeed virtual intra-atomic excitations are known to

reduce both u and U.

a. For two electrons sitting ^on different atoms, ^the

possibility of the electrons to be virtually ^excited by

their interaction from their ground ^{1s states to excited} (mostly 2p) ^{states creates for} H2 ^{a van der Waals}

attraction which is very effective because of the small-

ness of the distance between protons [141. ^{For two}

alkali atoms, ^{where the van} der Waals interaction is

smaller, ^{one can take into account a reduction in u due}

to the polarization of ^{their inner shells.} Finally ^for aggregates larger ^than dimers, the correlation between valence electrons screens out their long range interac-

tions ; ^for large enough aggregates, ^{this effect can be}

taken into account by using, ⁱⁿ computing u, ^a dielectric

constant which takes into account this screening by

valence electrons.

In the same way, intratomic correlations, ^{which can}

be described in terms of virtual excitations (mostly

Is --+ 2s for H) lower the value of U for two electrons

sitting ^{on the same atom} [15]. ^The ensuing ^effective

value Uc ^{can be} directly ^{measured on free atoms M} by

the reaction

which involves the energy

where I is the first ionization potential ^{and A the}

electron affinity. ^{For H, this is} Ue ^{= 12.8 eV} ( ^{U = 17} eV).

b. In an effective ^Hubbard model, ^{one is then led to}

use U, ^{=1- A instead of U and still} neglect ^u (or ^{use a}

small value ue Ue). Figure 7 shows that Uc ^{varies in a}

rather regular way in the periodic table, in much the

same way ^as Pauling’s electronegativity. ^{This is not} surprising, ^as this is defined as proportional ^to

I + A and I >> A. It is thus clear from this table that

electronegative ^elements (rare gases excluded) ^have large ^{values of} UC. ^{This is in} qualitative agreement ^with the observation that dimers are more stable than

closepacked aggregates ^{for the most} electronegative elements, namely H, 0, N+ ^{and halogens} [13] ; ^more precisely, ^{H has a value of} Ue ^{more than 2 to 3 times}

those of alkali metals, ^{while the} corresponding ^transfer integrals t ^{are not} expected ^to vary ^so strongly. ^{One can}

also note in figure ⁷ that of all elements forming closepacked aggregates, ^the divalent metals have the

largest ^{values of} Ue’ suggesting ^{that for these metals}

electron correlations play a stronger role. This is also in

qualitative agreement ^{with recent} theoretical analysis [16].

c) ^This effective Hubbard model with Ue ^{=- I - A}

and ue - 0 however does not fit quantitatively any

Fig. ^7.

^{Values of} Ue ^{in the} periodic table. In parenthesis

values where ^A is inknown. T transitional elements ; R rare

earth elements. Underlined: elements where dimers ^are

preferred.

experiment. This is well known for transitional metals

[3, 17], ^{where the use of such an} effective Hubbard model requires ^{the use of U = 2 to 2.5} eV, ^{and not} Ue = ^{6-8 eV as} pictured figure 7. In the same way the

use of Ue ^from figure ^{7 and of the} experimental

cohesive energies [18] ^{would lead} to I U/t2 ^{2 for H}

and > 2 for alkali metals, ⁱⁿ complete contradiction with the conclusion of this _paper.

In fact, ^the argument ^{which led to the} replacement ^of

U by Ue ^{=1- A is} incomplete [19]. ^{This is sketched in} Appendix ^{D :}

-

Because of the electron transfers produced by t,

the inter and intraatomic correlations which lower u

and U are less effective than assumed.

-

On the contrary, ^the virtual excitations produce supplementary ^{channels of} interatomic transfers, ^thus increase the effective transfer integrals.

The various corrections, together ^{with an account of}

the nonorthogonality ^{of the atomic functions} (i.e.

S = (a I b) ::F 0) ^are taken into account in more refined

computations for very small aggregates.

If one wants to keep ^a simple Hubbard model which

can be extended to larger aggregates, ^{one is therefore} bound to use an effective value to U which, ^if probably varying qualitatively ^like Ue ^{= I - A, is not} quantita- tively given by ^it.

A related study for 4-atomic Hubbard clusters has been done independently by ^Y. Ishii and S. Sugano [20]. ^{It leads to} similar conclusions.

Appendix ^A

TRIMERS IN THE HARTREE FOCK LIMIT.

a) ^{One electron states.}

Let I a ) , l f3 ), l y ) ^be

the three electron states linear, combinations (36) ^of

atomic states la), (6), )c) :

satisfying equations ^{such as}

Figure ⁴ gives the energy spectra Ea, E J3’ Ey ^{for the}

three simple geometry ^{of the} equilateral triangle, ^the

linear chain and the dimer + atom. The corresponding

values of the coefficients can be taken as

for the equilateral triangle,

for the linear chain.

In fact, ^{because of their} energy degenerescence,

states /3 ^and y ^are equally populated ^{in the} equilateral triangle, leading ^{to the} neutrality ^{of each atom here as}

for the linear chain. The choice (A.3) ^{is the most}

natural for studying ^{the Jahn Teller effect discussed}

below.

b) ^{Ground state.}

^For H3’, ^the corresponding ^Hartree

Fock ground ^{state is the} singlet ^{state built with the one}

electron state « a of lowest energy Ea.

For H3, The Hartree Fock ground state l «/Ii) ^is

built with the one electron states I a ) and l 03B2 ) ^of

lower energies Ea ^and EB.

^{It has the form (38) : it is}

thus antisymmetrical ^{and is} easily ^{checked to be an} eigenfunction of Sz = 03A3siz _z ^{and of}

with eigenvalues respectively equal

c) Coulomb and exchange interactions.

For H3’, ^the quantity ^to compute ^is

Using (A.1), ^{this is}

Thus X = 1 ^{3 U for the} equilateral triangle ^{and X} = 3 for ⁸

the linear chain.

For H3, ^the quantity ^to compute ^is

Developing (38) ^as

we can write

where

It is easy ^to show that

with U.. given by (A.5) ^and

Using (A.1) ^{one obtains} (A.6) ^and

Finally

with (A.3), (A.4), ^this gives ^{X =} _C/ ^{for the} equilateral triangle and X = 5 U for the linear chain.

8

Appendix ^B

CORRELATION TERM IN H3.

In computing ^the second order perturbation ^{term Y} given by (41), ^{the excited states can be taken as}

follows :

One easily deduces, with for instance

Using (A.1), (A.3), (A.4), ^the expressions ^{of table}

B.I can be computed ^{for the two} configurations ^consid-

ered

Tables B.II and B.III then give ^the coefficients of Y for the two configurations considered.

Table B. I.

Values of integrals for ^two configurations.

Table B. II.

Coefficient of ^Y for equilateral triangle.

Table B. III.

Coefficients of ^Y for the linear chain.

From these, ^one deduces that Y ⁼ U2/xt, ^where

x = 18 for the equilateral triangle, ^{and -} = 21 ⁼

17.3 for the linear chain

Appendix ^C

JAHN TELLER EFFECT FOR NEUTRAL TRIMERS.

Let

t’, t ", t"’ be the transfer integrals ^{for the most} general triangle (Fig. 8). ^{The one electron} energies satisfy, ^for

U ⁼ 0, ^the equation

Fig. ^8.

^Most general triangular shape ^{for a trimer.}

If we write

a development in successive powers of the T’s gives ^for

the fundamental state Ea ^{and the} initially ^{excited states}

E /3, ’Y ^{the new energies}

where

The developments ^{of the text are obtained for}

T fft = T It.

Appendix ^D

ROLE OF ELECTRON DELOCALIZATION ON THE INTRA- ATOMIC CORRELATIONS.

We show in this appendix that, in the delocalised limit, ^{the decrease of U due to}

virtual intra-atomic correlations is less than for an

isolated ion, but that this is compensated by ^{an increase}

in transfer integrals.

Taking hydrogen ^{as an} example, ^{we can write the}

lower singlet ^{state in terms of one electron wave} function I a(i) ), I b(j)) ^where now a ) ^con-

tains not only ^the ground state a’ ) (ls) ^{but an}

excited state a" ) (say ^{2s) :}

The total energy for a H- ion then reads

with

where Eo + 4 is the energy of the excited 2s state.

Also

where

Also

A development in small a " gives

Minimizing ^{E with} respect to a " gives

and

For a dimer in the limit U t, ^{we can} write in the Hubbard approximation

where

where

A development in small a " gives Ed (a " ) ^{which can}

be similarly ^{minimized with} respect ^to a ". The complex

result simplifies somewhat if we neglect differences

between t, t’, ^{t ". Then at} equilibrium.

This formula shows clearly ^{that if we} put

as deduced from the energy of the ion, the energy of the dimer is not simply ^obtained by replacing ^U by

Ue in the formula

Indeed, in the limit of I Ult l ^{small, and if one} neglectec t ^with respect ^to V, ^one would obtain for the correction to the term U

In 2" a ^{term equal to}

1 ^{(U- Ue) , thus} half that deduced from the ion. This reduction is due to the delocalization of the electrons

produced by t :0 0, ^{which lowers the effect} of intra- atomic correlations.

However a larger correction, working ⁱⁿ opposite direction, ^{comes from the} replacement ^{of V} by

V + t in the corrective term. This comes from the increase in the average ^transfer integral ^{due to the}

virtual excitations, ^{indeed from the} possibility of cross

Is # 2s interatomic transfers. This correction of t (and

not of U) complicates ^the development ^{in U and t made}

in the text. The approximate ^formulae developed ^here

assume anyway I tld ¹ but I U/t 4. ^{As the} promotion energy 4 ^{is less than the} ionization potential I, ^{which as} pointed ^{out in the text} is of order U, ^the discussion of this Appendix ^is only ^{valid for a limited}

range of values of I U/t I .

Note added in proof : ^the discussion in 2.2 can be

simplified and extended by minimizing directly E3 given by (50) ^{with respect to} r’ and r". The same

qualitative conclusions hold, ^{with an} explicit coupling

between dilation and shear modes.

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