Hydrogen-impurity binding energy in vanadium and niobium

HAL Id: jpa-00211057

https://hal.archives-ouvertes.fr/jpa-00211057

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Hydrogen-impurity binding energy in vanadium and

niobium

A. Mokrani, C. Demangeat

To cite this version:

Hydrogen-impurity binding

energy in

vanadium and niobium

A. Mokrani and C.

_Demangeat

IPCMS, UM _380046, Université Louis Pasteur, 4 rue Blaise Pascal, 67070

_Strasbourg,

France

(Reçu

le 25

janvier 1989,

accepté

sous

_forme

le 24 avril

1989)

Résumé. 2014

L’énergie

d’interaction

H-défaut,

dans les métaux de

transition,

est _{estimée par la}

méthode des

_opérateurs

de Green dans le schéma des liaisons fortes. Cette

_énergie

_{électronique}

(ou chimique)

est la somme de _quatre termes :

_i)

_{contribution des états liés introduits par}

l’hydrogène,

ii)

contribution de la bande,

iii)

terme d’interaction électron-électron neutre

_(sans

transfert de

_charge

entre le métal et

_{l’hydrogène)}

et

_iv)

un terme

_{proportionnel}

au transfert de

charge.

Nous avons choisi comme matrice le vanadium et le niobium et comme

_impuretées

substitutionnelles les deux éléments à

_gauche

et les deux éléments à droite de la matrice dans le tableau

_périodique.

Une forte

_répulsion

est observée entre deux

_hydrogènes

en

_premiers

voisins dans le vanadium et dans le niobium ; la

_position

la

_plus

stable

_correspond

à une

_paire

H-H en

quatrièmes

voisins. Nous avons observé une attraction de

_{l’hydrogène}

par les

impuretés

substitutionnelles situées à

_gauche

de la matrice et une

_répulsion

_{par celles situées à droite de la}

matrice.

Abstract. 2014 H-H and H-substitutional

_impurity

interaction energy are estimated

_by

the Green operator method

_developed

in the

_tight

_binding

_{approximation.}

This electronic

_{(or chemical)}

energy is

split

in four terms :

_i)

the bound states

_(introduced

_by

the

_hydrogen)

_{contribution,}

ii)

the band structure contribution,

iii)

the electron-electron interaction without

_charge

transfer and

_iv)

the

_charge

transfer

_(between

matrix and

_impurity)

contribution. The calculations are

done for the transition metal matrix vanadium and niobium. The substitutional

_impurities

considered are the two elements located at the left and at the

_right

in the same row as the matrix in the

_periodic

table.

_Strong

H-H

_repulsion

is observed when the

_hydrogen

atoms are at first nearest

neighbouring positions ;

the

_stability

is obtained for

_hydrogen

interstitials at fourth nearest

neighbouring

positions.

We have observed attraction of the

_{hydrogen by}

the substitutional

impurities

located at the left of the matrix in the

_periodic

table and

_{repulsion by}

those located at

the

_right.

Classification

Physics

Abstracts 71.55

1. Introduction.

Up

to now, the ab initio determination of the electronic structure of

_light

interstitials in transition metals has attracted

_only

few authors. We can notice the

_pioneering

work of

Yussouff and Zeller

_[1]

followed

_by

the one of Klein and Pickett

[2].

More

_recently

Akai et al.

[3]

and

_Oppeneer

and Lodder

_[4]

have

_proposed

models for the determination of

_respectively

hyperfine

field and de

_{Haas-van-Alphen scattering.}

This last calculation seems to be most

promising

because it takes into account the influence of

_zero-point

motion of

_hydrogen

isotopes.

On the other hand there is a considerable amount of

_{semi-empirical}

work devoted to this

problem

(see

for

_{exemple Demangeat}

_[5]

and Refs. there

_in).

Once the electronic structure

calculation of the

_single

interstitial and substitutional has been

_performed

one can

_try

to

estimate the

_binding

_{energy of}

_pair

of

_hydrogen

atoms or

_{pair consisting}

of an

_hydrogen

and a

substitutional

_impurity.

In

fact,

the

_hydrogen

interstitial

_put

in the lattice has three effects : the first one is the

modification of the

_screening

clouds around

it,

the second is the modification of the stress

field exerted on the

_neighbouring

metal atoms of the

_impurity

and the third one is the

_large

vibrations exhibited

_by

the H atom around its

_equilibrium

_position.

All these effects can be

taken into account

_through

the « effective medium »

theory

[6, 7],

which was

_applied

to

H-substitutional

_impurity

in vanadium and niobium

_by

Jena et al.

_[8].

The effect of the true

inhomogeneous

environment on the atom is simulated

_by

that of an effective

_homogeneous

medium. The basic idea is to calculate the

_binding

_{energy of} an atom immersed in an

homogenous

electron gas which has a

_{density equal}

to _{the average value of the host metal}

density

around the

_impurity

site. Interaction with the metal core

_electrons,

_{hybridization}

between H s-electrons and transition metal d-electrons and a correction for the

_{inhomogeneity}

of a real metal are also introduced. One of the

_advantage

of the effective medium

_theory

is the fact that the

_binding

_{energy is calculated without the}

_separation

into the electronic

_part

and the elastic one. Puska and Nieminen

_[9]

included also the

_{hydrogen zero-point}

motion

_by

considering

the

_{quantum-mechanical}

nature of the

_{hydrogenic impurity.}

Their model is based

on the formulation

_{given by Sugimoto}

and Fukai

_{[10] :}

_{the total energy} of the combined

light-impurity-metal

system

is the sum of the lattice energy of the host metal and the

_impurity

energy

eigenvalue.

All these

_energies

are functions of the

_{displacements}

of host atoms. The main difference between Puska and Nieminen and

_Sugimoto

and Fukai models is the construction of the

_potentiel

_{energy field}

_{« experienced » by}

the

_impurity

atom. The first authors use the effective medium to _{express this}

_potentiel

whereas the second

_adopt

the double

_Born-Mayer

form for this

_potentiel.

From this

_{potentiel they}

deduce the Kanzaki forces

_[11]

and the

_{displacements}

of the metallic atoms. This

_procedure

is made until self

consistency.

Other methods cannot take all the effects into account and a

_splitting

is made

into an electronic

(chemical)

_part

and elastic

_part.

Along

the lines

_{proposed by}

Einstein and Schrieffer

_[12]

there have been calculations for H-H in Pd

_[13]

in Fe

_[14]

in

_V,

Nb

_[15]

in Ni

_[16].

In all these calculations the effect of electron-electron term has been

_neglected.

Also calculations relative to H-substitutional

binding

energy in Pd

[13]

in

V,

Nb

[15]

in Ni

[17]

have been

performed.

In the

_following

the model

_developed

for the electronic structure of a

_{single hydrogen}

in

vanadium and in niobium is outlined in section 2.

_Only

the electronic

_part

of the

_binding

energy between

hydrogen

and another

_point

defect is

_reported

here.

_However,

we have

recently presented

[18]

the elastic

_binding

_{energy between} a

_pair

of

_hydrogen

in Pd.

Unfortunatly,

the models

_reported

in references

_[9]

and

_[18]

_rely

on the lattice Green function of the pure metal and not on the Green function of the

alloy

as it should be

_[19]

in the case of

interstitial

_alloys.

_Also,

in the same manner, the zero

_point

motion _energy must be related to

the force constant of the

_alloy

and not

_only

to the force constant _{of the pure metal} as it has

been done in

_[9].

A

_formalism,

for the determination of these force constants, has been

proposed recently

[20]

but no numerical estimation has been

performed

yet.

In a recent

_study

Mokrani et al.

_[21]

have derived the field of forces around

_hydrogen

in a bcc transition metal

these non electronic terms remains to be

_correctly

formulated and thus we have restricted

ourselves to the derivation of the electronic terms.

The chemical

_binding

_{energy of} a

_pair

of

_hydrogen

atoms in V and Nb is described in section 3. The results are

_compared

to Switendick

criterion

_[22]

relative to the minimum distance between the

_hydrogen

atoms. The section 4 is devoted to the determination of defects

_consisting

of an

_hydrogen

atom and substitutional

_impurity

of transition metal

_type.

Conclusion and outlooks are

_given

in the last section.

2. Electronic structure of a

_{single hydrogen}

atom in bcc transition metals.

The theoretical framework of our model is formed

_by

a

_{tight binding}

Green function method

for

_hydrogen

interstitial

_{[15, 23].}

In this section we will focus on two

_{points :}

₁₎

the _group

theoretical

_analysis

of the extended

_potentiel,

₂₎

the Slater-Koster fit used for the

determination of the band structure _{of pure metal and the}

_hopping

_integrals

between the

hydrogen

and the metallic atoms.

The band structure _{of pure transition metal is described}

_by

an

_spd

Slater-Koster

_(S-K)

fit to

a first

_principle

calculations

_[24].

The metallic

_spd

orbitals are labelled

_[Rm>,

where R is the metallic

_site,

m the orbital

_symmetry

(m

= _{s, x,}

y, z, xy, yz, zx,

xz -

yZ,

3 z2 -

r2).

Let T

being

the tetrahedral

_{position occupied by}

the

_hydrogen

atom and

_{[Ts )}

the

_{corresponding}

extra s

orbital. The

_perturbed

Hamiltonien

_HT,

for one

_spin,

is

_{given by :}

where

_Ho

_{is the pure metal}

Hamiltonian,

hT

is the

_perturbation

introduced

_by

_hydrogen

on the

interstitial site T :

where

_ET

_{is the energy level introduced}

_by

the

_hydrogen

on

T, f3T’R

is the

_{hopping integral}

between

_{(Ts ]}

and

_[Rm)

and

_{V Tm’ (R)}

is the matrix element of the

_{hydrogen perturbation}

VT

concerning

the « d » orbitals of the metallic atoms centred on the nearest

_neighbouring

sites R of the

_hydrogen

interstitial T. The difference between

_(2.1)

and the

_simplified

Hamiltonien of reference

_[23]

is in the

_expression

of the matrix element

_{V Tm’ (R).}

The

hydrogen

perturbation

VT

is

_expressed

in

_spherical

form :

where the radial

_part

V

₍

_{r - T 1 )}

_depends

on the distance between the interstitial site T and its nearest

_neighbouring

sites R and

_{Yo’(r ’-^}

_T)

is the

_spherical

harmonic

_{corresponding}

to the

s orbital of the

_hydrogen

atom. While

_decomposing

the « d » orbital

_[Rm>

as

_product

of

radial

_part

_F2(r)

and cubic harmonic

_{d’"(r) (site}

R is taken as the

origin)

the matrix element

V Tm’ (R)

is

_{given by :}

these matrix elements are

_expressed

_{(appendix A)}

in terms of the direction cosines

for L =

0, 2,

4

respectively.

The difference between these

integrals

and the usual one

_[24]

is

that,

in these

_integrals

the radial

_part

is of « d »

_type.

The matrix elements

V"(R)

are

reported

in table 1.

Table 1. - Matrix elements

_{o f}

the

_hydrogen

interstitial

_perturbation

on its nearest

_neighbouring

metallic atoms.

_(l,

m,

n)

are the direction cosines

_{of TR}

(R

is one

_{of the first}

neighbour

of T).

The labels

_(1,

2,

3, 4,

5)

denote

_respectively

the « d » orbital

_symmetries

_(xy,

yz,

zx,

x2 _y2,

3 z2 -

r).

To determine

_lhe

_phase

shifts

_ZT(E)

introduced

_by

the

_hydrogen

interstitial

_T,

we use the

Green

_operator

method. The

_density

of state modification for one

_spin

is

_{given by :}

Tr and

Tr(l)

mean trace in N dimensions

_(N

is the host atomic

_number)

and trace in the

pure Hamiltonian

Ho

and the

perturbed

Hamiltonian

HT

respectively.

The well known

Dyson

relation between the host and the

_perturbed

Green

_operators

used in the substitutional

impurity

case cannot be

_applied

here because of the difference in the number of sites in the host and the

_perturbed

_system.

We define a new reference

_system

with

_(N

+

_{1 )}

dimensions

corresponding

to _{the pure metal and} an isolated

_hydrogen

interstitial T. The

_{corresponding}

Hamiltonian is the

_{following :}

Let

GT’

the Green

_operator

associated to this Hamiltonian. It is related to

_GT(E)

_by

the

following Dyson

equation

in

_(N

+

_{1 )}

dimensions :

The intrasite matrix element on the interstitial site T of

G T

is

_{given by :}

where :

The

_L1f(E)

and

_rf(E)

_quantities

are defined as follows :

AT’(R)

is a function of the matrix elements of the host Green

_operator

_G°(E)

and of the

hopping integrals

between the

_hydrogen

interstitial T and its nearest

_neighbouring

metallic

atoms.

Through

group theoretical

arguments,

A’(R),

for different sites R and different orbital

symmetries

m, can be

_expressed

in terms of irreducible

_quantities

_[25].

The

_phase

shift

_ZT(E),

_{given by}

the

_integration

of

_(2.6)

from - oo to

_E,

takes the

_following

form :

where

_{nOm (R, E)}

is the local

_density

of states _{of the pure metal for the}

_symmetry

m. The energy level

_Et

of the bound state introduced

_by

the

_hydrogen

interstitial is determined

_by

the

_{following equation :}

The pure metal

hopping integrals

are deduced

_by

_{Papaconstantopoulos [26]}

_through

an

adjustable

set of

_parameters

determined

_by

the

_fitting

to APW band structure calculations. There are substantial differences

_depending

on how many bands above the Fermi level _you

level because the

_binding

_{energy between}

_point

defects is related

_only

to

_occupied

states. The

hopping integrals

/3

Kr

between the

_hydrogen

and the metal are taken from Slater-Koster fit

(using

an orthonormalized

_basis)

to APW band structure of the

_{dihydrides VH2}

and

NbH2

in

_CaF2

structure

_[27].

In

fact,

the

_{hydrogen-metal}

distance for

_hydrogen

at tetrahedral site T in bcc V and Nb metals is little different from that in the

_{dihydrides VH2}

and

NbH2.

The correction of the distance has been made

_through

a _{power law described}

_by

Harrisson

_[28].

where

_dTR

is the

_{hydrogen-metal}

distance and 1 is the orbital momentum

_{corresponding}

to the

orbital of

_symmetry

m. We can also use the

_{exponentiel decreasing}

law

_[29],

but we have to

introduce one

_parameter.

To

_perform

the numerical calculation we have to determine the unknown

_quantities

ET

and the matrix elements

_{V Tm’ (R).}

_{The energy level}

_ET

introduced

_by

the

_hydrogen

is related to the number of electrons

_NT

in the « s » orbital

_{by :}

where

_EH

= - 0.5 _a.u. is the atomic level of an isolated

_hydrogen

and

_UT

is the Coulomb

effective

_integral

of the

_hydrogen

_{which may be taken from}

_[30].

_Concerning

the

V Tm’ (R)

term on nearest

_neighbour

R of the

_hydrogen

atom we restrict ourselves to the most

important

component

(ssu).

The number of electrons in the s orbital of the

_hydrogen

is

_given

by :

The first term is the

_filling

of the « s » bound state on the site

_T,

at _{the energy}

El

and the second term is the contribution of the band. The band

_position

of the pure metal

was fixed such as to

_give

the

_experimental

value for the work function of Vanadium and Niobium

_[31].

We have to

_solve,

in terms of

_(sgcr),

_equation

_(2.10)

and Friedel’s

_screening

rule :

In both vanadium and niobium

host,

we find that the

_hydrogen

interstitial introduces a

bound state below the metal conduction bands. The local

_density

of state on the

_hydrogen

interstitial site is illustrated

_by

the

_figure

1.

3. H-H interaction energy.

The _{purpose of this}

_paragraph

is to calculate the electronic

_(chemical)

_{interaction energy}

between an

_hydrogen

atom located at tetrahedral site T and another at tetrahedral site

Ti (i

= 1 -

5 ) (Fig. 2).

This _{energy is}

given by :

The first term is the monoelectronic _{energy in} Hartree-Fock

_{approximation,}

the second term is the electron-electron _energy contribution and the third term is the ion-ion

contribution. The monoelectronic interaction is

_expressed

as a function of the

_density

of states

Fig.

1. - Local

density

of states on the

_hydrogen

interstitial in vanadium.

Fig.

2. - Different tetrahedral

_{positions occupied by}

the two

_hydrogen

interstitials.

_T,(!

= 1,

5 )

is the

i -th

_neighbouring

of the interstitial T. The distance between the two

_hydrogen

atoms is

_{given by}

dTTi

=

B/27

a

where a is the lattice constant of the matrix.

hydrogens

An-n’. (E) :

Due to the Friedel sum

_rule,

the

_integration

of

_(3.2)

_by

_parts

_gives

the

_{following expression}

for the monoelectronic energy :

where the Z’s are the

_phase

shifts for the

_pair

of

_hydrogen

atoms at T and

_Ti

_(ZTT,

_(E))

and for isolated

_single

_hydrogen

at _p =

T,

Ti

(Zp(E».

The electron-electron contribution to the H-H interaction _{energy is} defined as follows :

where

_{Èel-el (T,}

_Ti)

is the electron-electron interaction _{energy of the}

_system

with two

interacting hydrogen

interstitials,

Eel-el (p )

is the _{interaction energy of the}

_{single hydrogen}

system

and

_Eeol-el

_{is the interaction energy of the host metal taken} as reference. We note that all these

_energies

are of the Coulombic

_form,

and can be

_expressed

as an

_integral

over the

space of the

product

of the electronic

_density

_nel(r)

and the electronic

_potential

z,el(r)

for each

_system.

where the electronic

_potential

is related to the electronic

_density

as follows :

and the electronic

_density

is related to the Green

_operator

matrix element

_{by :}

3.1 ELECTRONIC STRUCTURE OF TWO HYDROGENS SYSTEM. - In order to describe the electronic structure of the

_system

with two

_{interacting hydrogen}

_{interstitials,}

we

_apply

the

Green

_operator

method

_developed

in the

_{tight binding approximation.}

The Hamiltonian of the metal

_{perturbed by}

the two

_hydrogen

interstitials is

_expressed

as follows :

where

_{6 VTT,}

is the direct interaction between the two

_hydrogens.

This term is due to the

rearrangement

which is necessary to

_satisfy

the Friedel sum rule. In our calculation we retain

only

the terms _up to the first order in

_{& Vu,}

and the

remainder,

we define the

_following

Hamiltonian

_{corresponding}

to the two

_hydrogens

system

without the direct interaction :

The associated Green

_operator

is

_{given by :}

From the

_Dyson

relation :

we show that the

phase

shift

ZTT, (E),

appearing

in

(3.3),

is the sum of the

_phase

shift

1ZTT, (E)

corresponding

to the Hamiltonian

_Hzwr,

and a modification

_SZTT,

_(E)

due to the direct

interaction

_{SV TI; :}

The

_phase

shift

_{ZTI. 1 (E)}

is obtained

_{by solving}

the

_{following Dyson}

_equation

in

(N

+

2 )

dimensions :

the

_prime

stands for isolated interstitials without interaction with the

metal ;

the

correspond-ing

Hamiltonian is

_{given by :}

and its associated Green

_{operator :}

The

_density

of state modification due to the indirect interaction between the two

_hydrogen

interstitials is defined

_{by :}

where

Tr (2)

means trace in

_(N

+

₂₎

dimensions. We

_expand

these traces in the base of the atomic orbitals centred on the different sites of the

_respective

system,

then

_(3.16)

becomes :

where the second summation is over all metallic sites R and orbitals m =

(s, p, d ).

The

interasite matrix

elements,

obtained

_{by solving}

the

_Dyson

_{equation (3.13),}

are

_expressed

as a

where :

The

_quantity

_D;’(E)

is

_{given by}

_(2.9)

in which we add the H-H interaction term

rl’(E) :

The interaction terms are defined as follows :

The

_phase

shift

_ZTT,

_(E)

introduced

_by

the two

_hydrogens

_(without

direct

_{interaction),}

_up to

the energy level

E,

is

_{given by}

the

_integration

of

_{(3.17) :}

This

_expression

can be written as a sum of the

_{single-hydrogen phase}

shifts

_ZT(E)

and

ZT. (E)

and the modification

_âZTI.

_(E)

due to the indirect interaction between the two

hydrogen

interstitials :

where

_{Z (E)(p =}

_T,

_{Ti )}

is

_{given by (2.14)}

and the H-H contribution

_{by :}

3.2 ELECTRON-ELECTRON INTERACTION ENERGY. - As for the

_phase

shift

_(3.12),

the

electronic

_density

of the

_{interacting hydrogen}

system

can be

_split

in two

_{parts :}

The first term is the electronic

_{density corresponding}

to the

GTT’(E)

_(without

direct

After

_inserting

_(3.25)

into

_(3.4)

we obtain the

_{following splitting}

of the electron-electron

interaction _{energy :}

where

and

the second term of the

_right

hand side of

_(3.29)

has been

_neglected

because we restrict ourselves

_only

to terms _up to first order in

_{8 vTI.}

_(r).

The first term is identical to the

_{corresponding}

one in the monoelectronic _energy.

_Thus,

these terms cancel in the final form _{of the H-H interaction energy}

_[25].

The electron-electron interaction energy

(3.28) corresponding

to the Hamiltonian

HTI.

i

can be

_split

into two terms

_{(Appendix B) :}

the first term _{is the neutral energy}

_{corresponding}

to the two isolated

_hydrogens

_system

and the second is the

_charge

transfer contribution. The ion-ion energy is restricted to the Coulombic interaction between the two

_protons

of the two

_hydrogen

atoms at T and

Ti.

When we add this energy to the neutral term, we obtain the

_{following repulsion}

_{energy :}

The

_charge

_{transfer interaction energy is}

_{given by}

the relation

_(B.12)

of the

_{appendix B.}

3.3 RESULTS AND DISCUSSIONS. -

The numerical calculation of the H-H interaction energy has been done in the vanadium and niobium matrix. The energy levels

ElI)

and

E 1(2)

of the bound states introduced

_by

the two

_hydrogen

interstitials

_{(Fig. 3)}

are determined

by

the Green

_operator

_{poles given by}

the

_following

_{equation :}

The different

_parts

_{of this energy} are

_represented

in

_figure

4. The bound state contribution

given by

the

_splitting

of the bound states

is found attractive for all

_positions

of the two

_{interstitials,}

whereas the band contribution

given by

the

_integration

of

_(3.24)

from the bottom

_Ebottom

of the band to the Fermi level

Fig.

3. -

Bound states _{energy levels introduced}

_by

the

_hydrogen

atoms at tetrahedral interstitial sites in vanadium. The label 0

_correspond

to a

_{single hydrogen}

atom at T whereas the labels

_{i (i}

=

1-5 )

correspond

to

_pair

of

_hydrogen

atoms at T and

_Ti

_(Ti

is the i -th

_neighbouring

site of

_T).

Eb

is the bottom of the conduction band.

is found

_repulsive

for all

_positions

of the two interstitial atoms. The

_repulsive

energy

(3.31)

decreases

_rapidly

whith the distance between the two

_hydrogens.

The

_charge

_{transfer energy} oscillates versus the H-H distance. When we

_bring

_together

all these

_energies

we observe a

minimum when the two

_hydrogens

are at fourth nearest

_{neighbouring positions.}

This result is in

_good

_agreement

with Switendick criterion

_[22].

4. H-substitutional

_impurity

interaction energy.

In this

_paragraph

we

_report

on the H-substitutional

_impurity

interaction. We use the same

formalism as for the H-H interaction. The substitutional

_impurities

are chosen from those

located in the same row as the host atom in the

_periodic

table. The

_{perturbation potentiel}

introduced is assumed to be localised on the substitutional site and

_affecting

_only

the « s » and « d » orbitals.

The hamiltonian of the metal

_{perturbed by}

the

_hydrogen

atom at interstitial

_position

and

substitutional

_impurity

is

_{given by :}

where

_Ho

_{is the pure} metal

Hamiltonian,

_hP

is

_given

in

_{(2.2) ;}

the third term is the local

perturbation

at the substitutional

_impurity

and

_{8 VST is}

the direct interaction between the two

defects.

To determine the matrix elements

_VS

of the localised

_potential

we have assumed

_spherical

symmetry

so that we have

Vd

for m = d. The matrix element

Vs

is assumed to be

proportional

Vd.

V s

= a

V S ,

with a = 1/2. No bound state is introduced

by

the substitutional

impurity.

As for the H-H interaction the

_binding

_{energy between the} interstitial and the substitutional

Fig.

4. - H-H interaction

energies

AE(T, Ti)

between two

_hydrogen

interstitials at tetrahedral

position

We show

that,

up to first order in

_SVST,

the contributions

_arising

from the direct interaction

cancel each other.

Therefore,

the Hamiltonian used to calculate

_{DE (S, T)}

is the

_{following :}

The S-H interaction _{energy is}

_{given by :}

The first term _{is the monoelectronic energy and} can be

_splitted

into the bound state

contribution

_{given by}

the shift _{of the energy level of the}

_{single hydrogen}

bound state

(Ee )

due the substitutional

_impurity

_(El)

and the band contribution :

The electron-electron term is determined in the same _way as for the H-H interaction and is

the sum of a neutral term and of a

_charge

transfer contribution.

A detailed derivation can be found in reference

_[25].

It can be seen from table II that all the three terms

_appearing

in the

_binding

_energy

DE(S, T),

are of

_comparable

order of

magnitude.

Therefore

_they

are not in

_quantitative

agreement

with our

_previous

calculation

[15]

where

_only

one electron contributions have been taken into account. The

_comparison

with the

_{semi-empirical}

effective medium of Jena et al.

[8]

shows a clear

_discrepancy

between the two results. This

_discrepancy

_{may be related} to the

rough approximation

used in the effective medium

_theory

for the « d »

_part

of the transition metal matrix. On the other

hand,

Shirley

and Hall

_[35]

have

used,

for the chemical

_binding

energy between H and substitutional

impurity,

the

asymptotic

limit of the formulation derived

by Demangeat

et al.

_[37]

for the

_binding

_{energy between carbon and}

_impurities

in aFe. In

_fact,

this model is

_only

valid when bound states are not extracted from the conduction band. Reasonable

_agreement

is found with the few

_experimental

results

_[36].

Table II. - H-substitutional

_impurity

_{interaction energy in} vanadium and niobium

_(meV).

Table III. - This table shows the H-substitutional

impurities

interaction

_energies

_(meV)

in

vanadium and niobium.

_(a)

_corresponds

to results

_of

_purely

electronic

_origin

whereas

_(b)

is

for

calculations

_for

which electronic and elastic terms are taken into account ;

_comparison

is

made with a

_few

experimental

results

(c).

5. Conclusion.

We have

derived,

within the

_{tight binding approximation}

of the Green

_operator

_method,

a

general

formulation for the electronic

_part

of the

_binding

_{energy between}

_hydrogen

and another

_point

defect in a bcc transition metal. This

_binding

_{energy takes into} account, in a

general

way, without

resorting phenomenological

model,

the two

_particles

contribution. We have written down a

_general

formulation for this term which

is,

for the

_{hydrogen perturbation}

extending

up to nearest

_neighbouring

metallic atoms of the

_hydrogen

interstial,

numerically

untractable. We have therefore restricted ourselves to the most

_important

intrasite Coulomb

terms so that the

_leading

contribution has been

_fully

taken into account.

In the case of

_binding

_{energy between} two

_Hydrogen

atoms in a bcc transition matrix we

have

_explicitly

shown that the two

_particle

interaction is not

_negligible

in

_comparison

with the

one electron contribution.

_{Nevertheless,}

the result obtained

_{by taking}

into account the two

particle

contribution remains in

_qualitative

_agreement

with our

previous

result

[15]

whereas

only

one electron contribution were

_present.

These results are in accord with Switendick

criterion

_[22]

which states that in metal

_hydride

solid solution two

_Hydrogen

atoms cannot

come closer than 2.1

Â.

_However,

the values obtained are

_greater

than those

_expected

from

the

_experimental

results. These differences may be related to the

_neglect

of the elastic

_binding

energy and the

rough

approximation

used for the calculation of the two

_particle

interaction.

In the case of

_binding

_{energy between} one

_Hydrogen

atom and a substitutional

impurity,

Our results are

_{qualitatively}

and

_{quantitatively}

different from those

_{reported by}

Jena et al.

_[8]

in the effective medium

_{approximation.}

We have found

_trapping

for the H-substitutional

_impurity

interaction when the defect is located to the left of

V,

Nb in the

_periodic

table and

_repulsion

when the defect is located to the

_right.

These results are in

_agreement

with the few

_experimental

results.

le A.I. 2013

Expansion of

the

_{product of}

the

_spherical

harmonics

_Y2

_Y2m

in linear hination

_of

the the

_spherical

harmonics

_Yoo,

_Y2’

and

_Y4.

Appendix

A.

The purpose of this

appendix

is to determine the matrix elements

_V"(R)

_(2.4)

of the

hydrogen perturbation

of the « d » orbitals centred on the first

_neighbour

R of the interstitial

impurity.

We transform the cubic harmonics

_{dm(r )}

into

_spherical

harmonics

_{Y2 (r -}

_T)

_by

the

following unitary

transformation :

where

D (2)

are the transformation matrix of « d » cubic harmonics into the

_spherical

harmonics. The

_product

of two

_spherical

harmonics can be

_decomposed

as follows :

Expansion

of the

_product

of the

_spherical

harmonics « d »

(A.2)

are

_given

in the table A.I.

By

insering (A.1), (A.2), (A.3)

in

_(2.4)

we obtain the

_{following expression}

of the matrix

elements of the

_hydrogen

_{perturbation :}

We define the two centers

_integral

S8U,

spcr,

sdo, by :

for L =

0, 2,

4

respectively.

All the matrix elements

V’, (R)

_are

given

in the table I.

Appendix

B.

To determine the electron-electron _{interaction energy defined in}

_(3.28)

we write the

electronic densities

_n¥ri

_(r)

and

_n"(r)

_(p

=

T,

Ti)

_{as a} function of the intersite matrix

elements of the Green

_operator

GTT’(E)

and

_GP(E)

where

_{cp x (r )}

is the atomic orbital centred on the site À :

The.summations on À and k ’ are over all the metallic sites R and on the interstitial site

p in

(B.la)

and on the two interstitial sites T and

_Ti

in

_(B.lb).

By

inserting

the

_Dyson

relation

_(2.8)

in

_(B.la)

and

_(3.13)

in

_(B.lb),

the

_perturbed

electronic densities can be

decomposed

as a sum of the pure metallic

_density

n 0 el (r),

the

electronic

_density

introduced

_by

isolated neutral

_hydrogen

atoms

_{dn; (r)}

and the

_charge

transfer contribution

_An’(r)

in case of one

_hydrogen

or

An’ (r)

in case of two

_{hydrogens :}

If

_nH(r)

is the electronic

_density

of an isolated

_hydrogen

atom at site _p, the neutral

modification of the electronic densities of the

_perturbed

_systems

is :

With this

_{decomposition}

of the electronic

_densities,

the electron-electron interaction _energy

where the first term is the neutral interaction _{energy and is}

_expressed

as the Coulombic

interaction between the electrons of the two

_hydrogen

atoms on T and

_{Ti :}

where

_{vP (r)}

is the Coulombic

_potential

of an isolated

_hydrogen

atom associated to

ni!(r ).

The

_charge

transfer term takes the

_following

form :

In the

_following,

we introduce the

_density

matrix modification for one

_hydrogen

_{system :}

Then the first

_integral

in

_expression

_(B.6)

takes the

_following

form :

where

_{IRw’f’l’,}

is the fourth centre Coulombic

_{integral :}

In all our calculation we restrict ourself to the intrasite

_{integrals :}

between the « d » orbitals for the metallic sites

_(A

= R noted

Uj

and between the _{« s »}

orbitals

_(A

=

T,

Ti

noted

U1).

These Coulombic

integrals

_can be related to those defined in

[34]

by

transforming

the cubic harmonics into the

_spherical

harmonics.

These

_quantities

are the « d » electron number on the metallic site R in the host metal

(B.lla),

in the

_{single hydrogen}

_system

_(B.llb)

and in the two

_hydrogen

_system

_(B.llc).

We

determine all the

_integrals

_present

in

_(B.6)

in the same _way as for the first

_integral

_(B.8).

Finally

we obtain the

_following

result :

In the numerical calculation the summation over R is restricted to the nearest

_neighbouring

shell of the

_hydrogen

atom.

References

[1]

YUSSOUFF M. and ZELLER R., Recent

Developments

in Condensed Matter, Eds. J. T. Devreese,

L. F. Lemnens, V. E. Vandoren and J.

_Vonroyen

_(Plenum

Press, New

_York)

1981, p. 135.

[2]

KLEIN B. M. and PICKETT W. E., Electronic Structure and

_Properties

of

_Hydrogen

in Metals, Eds. P. Jena and C. B. Satterthwaite

_(Plenum

Press, New

_York)

1983,

p. 277.

[3]

AKAI M., AKAI H. and KANAMORI _J., J.

_Phys.

Soc.

_Jpn

56

₍₁₉₈₇₎

1064.

[4]

OPPENEER P. M. and LODDER A., J.

Phys.

F 18

₍₁₉₈₈₎

869.

[5]

DEMANGEAT

C.,

Electronic Band Structure and its

_Application,

Lect. Notes

_Phys.

_(Springer

Verlag)

283

₍₁₉₈₇₎

146.

[6]

NORSKOV J. K. and LANG N. D.,

Phys.

Rev. B 21

₍₁₉₈₀₎

2131.

[7]

STOTT M. J. and ZAREMBA _E.,

_Phys.

Rev. B 22

₍₁₉₈₀₎

1564.

[8]

JENA P., NIEMINEN R. M., PUSKA M. J. and MANNINEN M.,

Phys.

Rev. B 31

₍₁₉₈₅₎

7612.

[9]

PUSKA M. J. and NIEMINEN R. M.,

Phys.

Rev. B 29

₍₁₉₈₄₎

5382.

[10]

SUGIMOTO H. and FUKAI Y.,

Phys.

Rev. B 22

₍₁₉₈₀₎

670.

[11]

KANZAKI H., J.

_Phys.

Chem. Solids 2

₍₁₉₅₇₎

24.

[12]

EINSTEIN T. L. and SCHRIEFFER J. R.,

Phys.

Rev. B 27

₍₁₉₇₃₎

3629.

[13]

DEMANGEAT

C.,

KHAN M. A., MORAITIS G. and PARLEBAS J.

C.,

J.

_Phys.

France 41

₍₁₉₈₀₎

1001.

[14]

MINOT C. and DEMANGEAT

_C.,

J. Chem.

_Phys.

86

₍₁₉₈₇₎

2161.

[15]

MOKRANI A., DEMANGEAT C. and MORAITIS G., J. Less. Common. Met. 130

₍₁₉₈₇₎

209.

[16]

KHALIFEH J. _M., Philos. mag. B 58

(1988)

111.

[17]

KHALIFEH J. M., J.

_Phys.

F 18

₍₁₉₈₈₎

1527.

[18]

KHALIFEH J. M., MORAITIS G. and DEMANGEAT C.,

Phys.

Lett. 93A

₍₁₉₈₃₎

235.

[19]

TEWARY V. _K., Adv.

_Phys.

22

₍₁₉₇₃₎

757.

[20]

MORAITIS

G.,

KHALIFEH J. and DEMANGEAT

C.,

J. Less Common. Met. 101

₍₁₉₈₄₎

203.

[21]

MOKRANI A., DEMANGEAT C. and MORAITIS

_{G., Phys.}

Lett. A

₍₁₉₈₉₎

submitted.

[22]

SWITENDICK A. C., Z.

_Phys.

Chem. Neue

_Folge

117

₍₁₉₇₉₎

89.

[23]

KHALIFEH J. M. and DEMANGEAT C., Philos.

_Mag.

B 47

₍₁₉₈₃₎

191.

[24]

NUSSBAUM A., Solid State

_Phys.

18

₍₁₉₆₆₎

165.

[25]

MOKRANI A., Thesis

Strasbourg

(1988).

[26]

PAPACONSTANTOPOULOS D. A., Handbook of the Band Structure of Elemental Solids

_(Plenum

Press, New

_York)

1986.

[27]

PAPACONSTANTOPOULOS D. A. and LAUFER P. M., J. Less Common. Met. 130

₍₁₉₈₇₎

229.

[28]

HARRISSON W. A., Electronic Structure and the

_Properties

of Solids

_(Freeman,

San

_Francisco)

1980, p. 504.

[29]

ANDERSEN O. K. and JEPSEN O.,

Phys.

Rev. Lett. 53

₍₁₉₈₄₎

2571.

[31]

MICHAELSON H. B., J.

_Appl.

_Phys.

48

₍₁₉₇₇₎

4729.

[32]

GAUTIER F., MORAITIS G. and PARLEBAS J.

_C.,

J.

_Phys.

F 6

₍₁₉₇₆₎

381.

[33]

MORAITIS G. and DEMANGEAT C.,

Phys.

Lett. A 83

₍₁₉₈₁₎

460.

[34]

SLATER J.

C.,

Quantum

_Theory

of Molecules and Solids VI & V2

_(1965).

[35]

SHIRLEY A. I. and HALL C. K., Acta Met. 32

₍₁₉₈₄₎

49.

[36]

TANAKA S. and KIMURA H., Trans.

_Jpn

Inst. Met. 20

₍₁₉₇₉₎

647.