o-p H2 conversion on noble metals

HAL Id: jpa-00246451

https://hal.archives-ouvertes.fr/jpa-00246451

Submitted on 1 Jan 1991

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.

o-p H2 conversion on noble metals

E. Ilisca

To cite this version:

E. Ilisca. o-p H2 conversion on noble metals. Journal de Physique I, EDP Sciences, 1991, 1 (12),

pp.1785-1807. �10.1051/jp1:1991240�. �jpa-00246451�

Classification

Physics

Abstracts

82.65J 31.30G 73.20A

o-p H~ conversion

_on

noble metals

E. Ilisca

Universitd Paris 7, Laboratoire de

Magndtisme

des Surfaces, 2 Place Jussieu, 75251Pads Cedex 05, France

(Received

29 May 1991,

accepted

9

September 1991)

Rkswnk. Los

expdriences

de «EELS» _sur de

l'hydrogdne

molkculaire

physisorbb

I basse

tempdrature

_sur des surfaces

d'Ag(lll)

ont ddmontrd _que la vitesse de conversion

ortho-para

dtait trds

rapide

(de l'ordre d'l _ou

2min).

Ces

expdfiences

contredisent les anciennes thdories toutes basdes _{sur un}

catalyseur magndtique.

Nous examinons une

large

famille de processus,

opdrant

_sun des surfaces

mdtalliques

_non

magndtiques

et _non dissociatives,

qui

convertissent les moldcules

d'hydrogdne

_en bmettant des

paires

blectron-trou.

L'importance

des btats

blectroniques mbtalliques

et moldculaires dans les mbcanismes de conversion est ici pour la

premidre

fois

reconnue. Le mkcanisme trouvb le

plus

efficace est _un processus dbnomrnb Coulomb-Contact qui

induit le saut virtuel, aller et retour, d'un Electron du mbtal _sur la moldcule. Sa vitesse de rdaction est _en accord _avec

l'expbrience.

Abstract.

Electron-energy-loss experiments

^of

H2 Physisorbed

at low temperatures on a

Ag(I II)

surface have indicated that the o-p H2 conversion rate is _very fast

(zz1-2min),

in contradiction with the usual belief that the

catalyst

must be

magnetic

to be efficient. We examine

a

large family

of processes valid for non-magnetic and non-dissociative metals which

physically

convert the

hydrogen

molecules _on the basis of the emission of metal electron-hole

pairs.

The

importance

of metal and molecule electron states in the processes is for the first time

emphasized.

The most efficient mechanism found is the Coulomb-Contact _one based _{on a} virtual

charge

transfer, back and forth, from the metal to the molecule,

having

_a rate in agreement with

experiment.

1. Indoducfion.

Since

Wigner

in

1933,

it has been established that the

physical ortho-para

conversion of

hydrogen

molecules _occur

through

_a

magnetic

nucleus

uncoupling

induced

by

_a

nearby

electron

spin ill. Experimentally,

all

catalysts

_were built up

by dispersing paramaguetic impurities

_{on a}

diamagnetic

_support

[2]. Astonishingly

in

1982,

fast o-p

H~

conversion _was observed _{on pure} noble metal clean surfaces

[3-5],

in contradistinction with the usual belief.

All studies

performed

at low temperatures

(T

= 10-25

K)

concluded at pure

H~ physisorption

but

display

different conversion patterns. ^For

Ag polycristalline

^film and

Ag(I

I

I) surface,

ortho molecules

disappear

within the first few minutes of initial _exposure

[3]

whereas _on

Cu(100)

surfaces the conversion rate was estimated to be less than

19b/min [4].

^To our

knowledge,

the electron energy loss

high

resolution studies of

H~

adsorbed _on

Ag surfaces, performed by

Avouris _et al.

[3], provide

the

primary

observation of _o-p

H2

conversion

observed _on

(I)

_a

single crystal, (it)

_a metal without

chemisorption,

and

(iii)

_a

non-magnetic catalyst.

The

only

theoretical attempt to model o-p

H~

conversion _{on a} clean metal surface failed since the

best,

_among the

investigated

_processes, leads to a conversion time of the order of 20 h

[6].

The purpose of this

study

is to

display

altemative mechanisms which

rely

on a

catalyst magnetic ground

state. In these processes the ortho nuclei

angular

momenta and energy are

dissipated by

the emission of

(e-h) pairs

at the metal surface and then carried away to the bulk. Short accounts of this

family

of _processes have

already

been

published [7, 8J.

Section II describes the

physical

model. The two

interacting

_{systems are} the metal and molecule electrons with the nuclear protons. ^{The transition}

paths

_are defined for one-step as

well _as for

two-step

processes. Section III is devoted to

one-step

processes where the two o-p selection

rules,

which

pertain

_on the molecule nuclear

spin

^{and rotation} states, _are

simultaneously

satisfied. The

Wigner dipolar

^{and the}

hyperfine

contact mechanisms involve _a

magnetic

excitation of the metal whereas the orbital

coupling

is

performed by

electron momentum transfer. Section IV details the two-step _processes ^{where the} two o-p selection rules are

successively

satisfied. These _are still first order

time-dependent

mechanisms but take

advantage

of

non-diagonal couplings

to build virtual intermediate states. We

distinguish

ionic

and neutral virtual states and denote

by

Coulomb-Contact and

Exchange-Contact

the

corresponding

mechanisms. Section V compares the o-p conversion rate relative

strengths

of the different _processes and summarizes their main features.

2. Wave functions and dansition rates.

2. I THE _{NUCLEAR SYSTEM.} We consider _a

H~

molecule

physisorbed

_on a metal

surface,

at

low

temperatures (T=10-50K).

The nuclear system involves the

spin

and

position

coordinates of the molecular protons, ^denoted

by

_a and b in the

following.

It is described

by

_a

set of ortho

(L odd,

I

=

I)

and para

(L

_even, 1

=

0)

states, where _as usual L and I denote the

rotational and nuclear

spin angular

momenta of the

H2

molecule. The

corresponding

eigenstates (Lmi)

and

(Im;)

are

represented by spherical

harmonics and

by angular

momentum addition of two nuclear

spins 1/2.

We therefore _{assume, as} usual in the

physisorption regime,

that the Pauli

principle

remains

operative

within the

hydrogen molecule,

in

spite

of the overall

antisymmetrization

of the electron system.

Moreover,

at low

temperatures

^{and in the framework of the}

experimental investigation,

the conversion process

originates primarily

from the L

= I

- L

=

0 transition of energy s~~ ^{14.7 mev}

(we

shall restrict _our

study

to this

case).

The molecule is assumed to be adsorbed at _a distance d from the metal surface.

2.2 THE _{ELECTRON SYSTEM.} The two

H~

^electrons occupy the

«~(ls) spin

orbitals

denoted

by

_g

(and _#,

_a bar _on the

top

^{of the}

spin

orbital will

indicate,

from _{now on, a}

spin down).

The

corresponding ground

state is

denoted,

_as

usual, by ~I)

with

eigenstate (g@(.

^{In the conversion} process a metal _or molecule electron

might

be

promoted

to the molecular

antibonding

excited _state

«~(

ls

)

denoted

by

_u in the

following.

If

only

_one electron

already occupies

the

bonding

orbital g, then the two-electron states

~I]

will be

represented by

the _wave functions _gu

(..., while,

if there _are

already

two electrons _g, the

resulting

state is

the _resonance

~I]

_described

_by

_the three-electron Slater determinants

(g@u(

^and

(g#R( [9].

The metal in its

ground

and initial states is described

by

a conduction band which is assumed to be

completely

filled up ^to the Fermi level

(small temperature

^effects are

neglected).

It is

composed

of N

doubly degenerate

one-electron Bloch states denoted k

(and k)

_:

_(... kk.. (.

^{The electron}

^system

^{is thus described in its}

ground

and

coupled

metal-

molecule state

by

_a Slater determinant of 2 N ₊ 2 one-electron _{states :}

Ill

=

(g#.. kE.. (1)

which represents its initial and

spin-singlet

state. We consider small

energetic

excitations where _one electron is transferred from _a state k below to a state X above the Fermi level. The hole k

couples

to the electron _{x to} build e-h

pair

states which

might

be either

magnetic

_{or non}

magnetic

_ones. We denote

by ~[kxJ

=

(kg ^fix (11

_the

_{spin singlet}

one, whereas

~[kxJ

is _one of the 3

triplet magnetic

substates _m

=1, 0,

-1:

[kxJ, (kg

₊

ix (/$

(ki(.

The final state of the electron system will thus be denoted

by

either

Sj

or

Tj

:

Sri

= g

# [kx

_or

by

_:

Trj

=

gg

³

jkx (2)

according

to the

spin multiplicity

of the

corresponding

e-h

pair

excitation.

In the

two-step

processes the metal-molecule

complex

_goes

through

_a virtual state denoted

by

6~ or

T~, owing

to its

spin multiplicity.

If the electron

jump arises,

from the metal to the

molecule,

the intermediate state built from

Hi

and

A4~+ is ionic

js~j

=

i121j

_x

2~c1

=

jgg.. iikui (3)

and

similarly by interchange

of k and _x, I and

3,

S and T. When the electron

jump

arises within the molecular space

(a)

_or

simultaneously

in the metal and molecule spaces

(b)

the intermediate has neutral

components.

^{We shall therefore consider different virtual}

paths, depending

_on the process and characterized

respectively by

the virtual intermediate states

T~)

=

~[~I(

_{x ~M~]}

=

(~[guJ

^kk..

^(4a)

16~> =

~1311

_x

3~4~1= j _ii

gu..

Fe

₊

[Pa.. kx..

-(1(ga.. ^ix

₊

lgR.. kg

₊

(#u.. ^fix

+

(#u.. kg (J) ^(4b)

where ~M~, ~A4~ denote the metal

ground singlet

and excited

triplet

states

respectively

while

~Mj+ the excited doublet of the metal ionized with _a hole.

2.3 MOLECULE _STATES. The

bonding

and

antibonding

molecular orbitals _g and _{u are}

taken from Hartree-Fock orbitals

[10J

and

developed

in

spherical

harmonics _:

g(r)

= AI

j _~~~~

Rj (r) ^Y~ (a ₎

_Y

_{(f) (5)}

A similar

expansion

is

performed

for _u with summation _on I

odd,

to

satisfy

the

parity requirement. Following

_a

procedure justified by

Harris

[I lJ

_we retain

only

^{the lower} rank in each _case, since the series

expansions

_are

rapidly converging,

and obtain

g(r)= .~ ([l+A(r-b(Je-~'~~~' [l+A(r+b)Je-~~~+~~) (6)

b r

JOURNAL DE PHYSIQUE T I,M 12, DtCEMBRE _{)W) 70}

where b is half the intemudear distance _:

ab/2.

For the

antibonding u(r)

fonction _we have _:

u(r)

=

BRj (r) Y~(a) Y~ (f) (7)

The

procedure, Rj(r)

and numerical values for the constants _are

given

in

Appendix

A.

2A METAL _STATES. The metal electron _states k and _{x are} assumed to

belong

either to _a

surface _{or to a} bulk band which _crosses the Fermi level _s~ in _a thin slab of width 2 s~~ = 29.4 mev around _s~. The

general

form of their _wave functions _are choosen _{as :}

ll'k(r)

₌

e'"

'~ $i

(z) (8)

#~(z

_m

zo)

₌ a e~ ^~~~ _cos

(k~

_{z + ~fi}

(8a)

#r~

(z

_w

zo)

₌ ^b e

~~~

(8b)

where k

=

(kj, k~ ),

r =

(p, z)

_are written in

cylindrical

coordinates relative to _an axis Iz

perpendicular

to the metal

surface,

with

origin

I at the molecular _center located _{at a} distance d from the

edge

of the

jellium background (half

_a

perpendicular interlayer spacing

above the first atomic

layer)

_as

represented

in

figure

_{I. zo} denotes the

position

of the

matching plane

where the metal and _vacuum functions _are matched. _a,

b,

~fi and y~ are defined

by

the

matching procedure

and normalization

requirement (y~

=

0 for _a bulk

band).

_{y~ is} calculated

from the work function

4l, assuming

_a

simple rectangular

surface

potential:

_y~=

(2 4l)~'~.

The noble metal surface bands have been observed

by angular

resolved UV

photoemission [12, 13]. They

_are located in the L gap where the

nearly-free

electron like s-p

bands _are

split

at the Brillouin _zone

boundary.

As the d bands _are

relatively

far

(

= 3.7 eV below

[14, 15J),

^{the surface bands} are known to be

fairly

well

reproduced

qithin the

nearly-free

two band model

[16, 17J.

M ~

H~

^rneta

~

i z

b

d

Fig.

I.

H2-Metal

model geometry

together

with the

Ag(I

II surface _state wave-function. J and M _are

respectively

the Jellium and

Matching planes.

_n

= 1, ^2... indicate the first atomic

planes.

In this _{paper we} shall

mainly

be concerned with the

Ag(I

I

I)

surface states which

give

the most efficient _o-p rate. It will allow _us to compare the different _processes and to

identify

the most efficient _one.

However,

_a

~uualitative

discussion will also be

given

for the relative

Cu(100)

surface and bulk band

efficiency.

The

Ag(ill)

surface band

dispersion

has been

measured

by

inverse

photoemission

below

(m~=

0.7

m) [18J

and above

(m~ =m) [19]

the

Fermi level. We shall therefore _{assume an} average effective _{mass of m~}

= 0.8 _m, in the close

vicinity

of the Fermi energy, which leads _{to a} surface

density

of states of

0.12eV-~

states/surface

ion.

Matching

and numerical values of the different _wave functions _are

given

in

Appendix

A.

2.5 THE TRANSITION RATES. We consider transitions which involve _a simultaneous

change

in the electron and nuclear systems with total energy conservation. The electron system

jumps

from the initial

ground

state

ii )

=

($)

to _a final excited state

f)

=

(Sr)

_or

(Tr) by

promoting

a Fermi electron k of energy s~ ^to a

higher

_{energy E~} ^and different momentum X,

leaving

_a hole behind whereas the nuclear

system

starts in _an initial ortho state

(o,I

=

if

=

I,

L

= I _m

mi)

and relaxes in _a final _para state p

)

=

00).

The

correspond- ing ortho-para H~

^transition rate relative _to one-step processes can thus be written _{as :}

~o

~ p ^"

~/ _{i I (f'P}

_~

~'°

ii)

^~ ~

(~X

~k

~op) (~)

k,x i,f

The _energy and

angular

momentum released

by

the

hydrogen

molecules

during

the

conversion process are transferred _to the metal surface

through

electron excitations which

dissipate quickly

in the

bulk,

when carried _away

by

the emission of electron-hole

pairs.

3C denotes the electron-nucleus Hamiltonian interaction and _we shall consider

successively

_:

(I)

_a

dipole-dipole coupling W, (it)

_a

hyperfine

contact

Y,

and

(iii)

_an electron orbital momentum-

nuclear

spin coupling

O.

The

two-step

_process ^introduce virtual intermediate state

(v)

=

(6~)

or

T~)

^{where the} molecular

antibonding

_{u state} is excited. This _u state presents ^two

advantages

_: it has _a stronger contact with the protons ^and

larger overlap

with the metal states than the

bonding

_g state, which is _more concentrated at the molecular _center. The Coulomb interactions C among metal and molecule electrons

produce

this virtual excitation while

changing

the

parity

of the

H~ rotation,

whereas the

hyperfine

contact Y induces _a

corresponding parity

transition among the

H~

^nuclear

spins.

The

ortho-para

transition rate, relative _to

two-step

_processes then takes the form _:

P~~~

₌

~/ _{£ £ ii f, pi C(}

v,

q) (v,

_q

(Y( I, o;i)

₊

if, pi _Y(

_v,

q) iv,

_q

(C( I, o,1) _(~/

k-x,,t

/A~. (Ex

~k

Sop) (1°)

where

A=E~-E,

is the virtual transition _energy and

(q)

one of the

(L= lmi

1

=

0)

and

(L

= 0 1

=

1m,)

u

eigenstates.

We shall

successively

consider :

(I)

the

Exchange-Contact

XY process where the metal and molecule remain neutral in their intermediate virtual states, and

(it)

the Coulomb-Contact UY process characterized

by

_an

ionic intermediate and virtual state. From _a formal

point

of view we shall

adopt, throughout

the _paper, irreducible tensors in

spherical

basis

[20J

which present many

advantages

compared

to Cartesian _tensors.

3.

One-step

conversion processes.

3. I THE WIGNER _{DIPOLAR PRocEss.} The

dipolar

_process,

originally suggested by Wigner

in 1933

[lJ

relies _on the

ability

of _an electron

spin

located within _a

magnetic impurity

to

produce

_an

inhomogeneous magnetic

field which _{exerts a} torque on the 2

proton

^nuclear

spins

of a

nearby hydrogen

molecule. The

uncoupling

of the nuclear

spins,

which breaks the nuclear Pauli

antisymmetrization

and leads to the conversion process, has been described _{as a}

dephasing

of the nuclear

spin processions [21].

In the

Wigner theory

this

dephasing

_can

only

occur if the

impurity ground

state is

magnetic,

that

is,

contains _a

non-vanishing

electron

spin.

In the

following

_we

enlarge

this

theory, replacing

the electron

spin

_average

by

_a transition from _an initial _non

magnetic ground

state to final

magnetic

excited states of the metal.

3.I.I Hamiltonian. The

antisymmetric part

^of the

dipolar interaction,

between the

spin s(a )

of _a metal electron _a and the nuclear

spins

I of the

hydrogen

_{protons a} and

b,

_can be

written in tensorial form _{as :}

JCd(a )

= P

I

^~~~

_{(S~(" )}

x

i~)~. iT~(a) T~(b)i _{(I I)}

where the nuclear

spin

difference

operators

i~ ₌

I~(a) -I~(b)

and the

position

tensor:

7*(r)

=

Y~(@,

_~fi

)/r~+

_~.

_r(r,

_@,

_{4 )}

_is defined

by

the distance vector between electron _a and

proton

p=a or b. The

strength

of the

dipole-dipole

interaction amounts to p =

8.1 10~ ^~ _a.u.

Performing

_a limited series

expansion

in the internuclear vector ah and

retaining

the

leading

term, we obtain _:

T~(b) T~(a)

=

ab~ V~T~ (12)

where

V~T~= (15)~/~ (T~x _{e~)~ (13)}

is obtained

by applying

the Gradient Formula of Irreducible Tensor

operators,

^while

e~ is the unit tensor in

spherical

^basis

[20].

Inserting (12)

in

(11), using

the

recoupling properties

of four

angular

momenta, and

summing

_over all

(metal

and

molecular)

electrons _a, the

Wigner

Hamiltonian _can be written in the

simple

form _:

H~

₌

£

2 _p

(3

_{ar )~/~}N~

Ii ~(a _{) (14)}

a

where the nuclear tensor;

N~

= (i~ x ah )~

(15)

pertains

_on the nuclear

spin

and rotational

degrees

of

freedom, inducing

the _o

- p

transition,

whereas the electron tensor _:

fi2

_{(Jm3 1)2}

(16)

~ X S

operates on the

spin

and

position (relative

to the molecule center

I) degrees

of freedom of electron _a.

3.1.2 Matrix elements and conversion rate. The matrix elements for the

dipolar

Hamiltonian _are

simply

worked out. For the

hydrogen

nuclear _{ones, we} find _:

(p[N([ _o,I)

=

(- )~ C(l12( mim,)$8~~~~

_~

(17) (3)

whereas the electronic _{ones can} be reduced to _one electron matrix elements _:

($[E([T~~)=- ~~~~C(312(p-m,m)(x[T(_~[k) (18)

(2)

where _m

corresponds

to _a

particular magnetic

substate of the emitted e,h

pair,

k and _x

being respectively

the hole and electron _wave functions which

belong

to the metal surface band

defined

by (8)

in the close

vicinity

of the Fermi level. In order to obtain _a

simple analytic

formula for the conversion rate, we evaluate the electron matrix element in the

approximation ki

=

0,

which leads to :

ix Tii k)

=

~?

~

((

^{" ~~'~}

yv1~ (i)1~ (19)

Then, bringing together (17), (18), (19)

and

inserting

in the conversion rate

(9),

_we obtain _:

P~~(lf~

=

? _~

(~ j)~ ^~~(~'~~~ _#(I)

_j~

Nj(e~) _e~~. (20) (15 )

The

Wigner

conversion rate is thus found to be

proportional

to the _square of the

product

of the electron

density

at the molecule

center

_{and the}

density

of states at the Fermi

level,

both

relative _to the metal surface band herewith considered. For the

Ag(I

II

)

surface

band,

and the numerical values

given

in

Appendix A,

_we estimate the conversion time

(r

=

P ~) relative to the

Wigner dipolar coupling

W at _a metal-molecule distance d

= 4.5 b _:

r~~( Xj

m 6.5 _x

10~

_s

= 2 months

(21)

3.2 THE _{HYPERFINE CONTACT PRocEss.} The

hyperfine

contact process arises from the

difference in the electron-nucleus _contact interaction between the _two

hydrogen protons.

^If the electron is _a metal _one, it will be defined _{as a} direct metal-molecule contact. But there is also another

mechanism,

where the metal electron

overlapping

the molecular electronic cloud

can be

substituted,

in the

antisymmetrization building

_up,

by

a

hydrogen

electron which has stronger contact with the protons, since it has _a

(ls)

character. This _process, hereafter defined

as an indirect contact, was introduced

by Buckingham

et al.

[22]

in 1971 in the

analysis

of chemical shifts in

paramagnetic

mixtures and has

already

been found to be _more efficient than the

dipolar

_one in the o-p conversion of

H~

molecules adsorbed _on metal oxides

[23].

The

hyperfine

contact Hamiltonian is written _{as a} difference in the _contact interaction of the two

protons

^with any metal _or molecule electron _:

Hc

=

~

(~ _£

i _S

(« )I (ha (au ) (22)

where _p and I _are defined _as

previously (below Eq. (ll))

and

~pa )

represents ^the Dirac

operator 8(r~ r~),

_p

= a, b. The orbital _average is then

performed

and

expressed

_{as :}

lTrlHcl ii

=

~

)~~~~~

pi «14~(b)

4~

(a)1 (23)

where the function 4l contains

products

of metal

k,

_x and molecule g electron _wave functions _: 4~

=

kx lklg)

_gx

kg lg

_x

) (24)

and _« is the

singlet-triplet

electron

spin

operator defined

by 100( «( ₍₁

p

)

=

(- )~.

The

first term in 4l refers _{to a} direct contact of the e-h

pair

with the protons ^{whereas the second} and third _ones result from the metal-molecule electron

overlap. By expanding

the function 4~

around the molecular center, as

performed

above

(Eq. (12))

_:

4l(b)

4l

(a)

=

ab V4l

(1)

₊

(25)

and

keeping only

the first linear and

leading

term we may express the orbital average

(23)

_{as :}

T~ H~ S~)

₌ ^{~ " ~~~~~~ ~}

(i

_«

(ah

8

(26)

where the function & is

composed

of two

parts

^related to the direct and indirect contact _: 8

=

8d

₊

8i~~ (27a)

8~

₌ ~k

VIX

_{+ X}

V~k) (27b)

Bind _" g

(a)j jk gj VIX

+

vIk lg _IX ) (27C)

After calculation of all necessary matrix

elements,

summation _over all azimuthal quantum numbers and

angles,

and

inserting

in

(9),

_we obtain the conversion _rate relative _to the

hyperfine

contact process denoted in the

following by

Y:

P~~( ^Y)

=

~~ _"

(p

_{ah )~}

8(

^~

N/(e~)

e~~

(28)

It is of interest to compare the relative

strengths

of the direct and indirect contact processes Y~ and

(~~.

When the e-h

pair belongs

to the surface band

(8)

_we obtain _:

Pop( Yind)

_g

(a) j ~

g

i

^~

l'~p(Yd) ^#(~) (29)

From

(6)

_we find _g

(a)

=

0,476

while the

overlap

between the surface _{state wave} function and the molecular orbital _can be

simply

estimated

by noticing

that the molecule orbital

being

_more

sharply

localized _:

1#

_1g

)

₌

V'(i)

_g

(r)

dr

= 6 v~

(1) (30)

where the

integration

has been

performed

_over half _a molecular _space, while _an exact numerical

integration gives

_:

( # g)

= 5

~ (I).

Substitution in

(28) gives

^thus _:

P~~( (~~)

=

6P~~(Y~).

It is therefore remarkable

that, although

the metal-molecule

overlap

remains

small,

the contact process is enhanced

by

the overall

antisyrrJmetrization

of the electron system which allows the contact to occur within the adsorbed molecule. A similar result has

already

been obtained in the _case of d-electrons

oxyde catalysts [23].

The conversion rate relative to the

hyperfine

indirect contact process is found to have _a similar structure _as

Wigner's

:

~8

~3

l'op(

_{l~>nd) ~}

_f (R~bl'v)~ _{# (l)} ~'~~(EF)

_Eop

(31)

It is however found to be about two orders of

magnitude

faster

P~~((~~)

= 128

P~~(Xj.

The numerical estimate relative to the

Ag(I

I

I)

surface

band,

at d

= 4.5

b,

leads for the direct and indirect contact processes Y to the conversion times _:

r~~( Y~)

₌ 3 _x

10~

_s

= 3

days (32)

r~~( (~~)

= 5 _x

10~

_s

= 14 hours

(33)

3.3 THE _{HYPERFINE ORBITAL PROCESS.}

3.3.I Hamiltonian. Let _us recall that the

coupling

of _an electron of _{momentum p} with _a

magnetic

field _can be

represented by

aHamiltonian _:

H

=

e/mA

_p

(34)

when the

magnetic

field arises from _a proton nuclear

spin I,

located at a distance _r from _an electron _a, the

corresponding potential

vector is written _:

A(a )

=

~°

p ~

I _x

~. (35)

4ar r

As V A

=

0,

_we have A p = p A and the Hamiltonian is written _:

H(

_a

)

=

f

^{~ ~}

(36)

h

r~

where _p

m 8.I

x10~~

a.u. When the

magnetic

field arises from the two

hydrogen proton

nuclear

spins,

^if we define the nuclear

spin

difference I

=

I(a) I(b)

and

perform

_a similar

expansion

around the molecule center of _mass, the relevant

antisymmetric

Hamiltonian takes the form _:

Ho(«)

=

~ i.

((ab.V)~xpj. ⁽³⁷⁾

2h r

By applying

the

gradient

formula

[20]

and

switching

to irreducible tensors,

(37)

becomes _:

Ho(«)

=

~j~ ^{(3 )i'2111}

x

abi

x

j( ^~

x

pi j

^°

(38)

After the

uncoupling

and

recoupling

of four

angular

momenta, we separate the nuclear from the electron _{tensors, sum over} all electrons

a of the

system

^{and obtain} :

Ho

=

£ £ q.

^{~Q/ EJ}

(

_«

(39)

M

= (i~ x

ab~Y (39a)

El

=

~~

x V~ ^~

(39b)

r~

q

= 6 _p

[3 ar(2 j

+

1)]~°

^{~ J}

(39c)

j j

⁰

where the notation

( ) corresponds

to the usual 6

j-symbol. Computation

of the _« 6

j

» leads to : a~ ^{= p}

j(4 j

₊

1) «/5ji'2

3.3.2 Matrix elements. The nuclear matrix elements _are similar _to those of section III-A

(see (17)).

^{After summation} of their

products,

_over the azimuth

quantum numbers,

_we

obtain :

£ IF Ii _{Oil) (Oil} k

v

PI

~

8j,k 8

~, v

(-

~

~

(4°)

m, ml

where _o;i is _one ortho substate _: II m;

mi),

and the 2 N ₊ 2 electron matrix elements can be reduced to one-electron elements _{as :}

sf) £ w[(a)) S)

⁼

^{(2)~/~(X (El k) (41)}

Bringing together (39), (40)

and

(41),

the

resulting

conversion rate, relative to the orbital process denoted

O,

_can be written _:

4

w~(ab)~

_R ^~

z z j

+

IX II ~l _'~

~

~~

_{~~ ~~~~} ~~~~

p~~~(O)

₌

15 fi

~~ ~~

~~

~~

~

~

The electron matrix elements and the above _{sum are} calculated in

Appendix

B. The

summation _over the electron band states

k,

_{X can} be

decomposed

into two parts. ^The wave

vectors

moduli,

which

correspond

to the emitted

(e-h) pairs,

remain almost constant:

k

= x =

k~

where the band intersects the Fermi level. Thus the _sum of Dirac functions reduces to e~~

N~(e~).

The orientations of the _wave vectors

k,

_X are then

averaged

in _a

plane parallel

to the surface

leading

to the

analytic expression

of the conversion rate :

~ _{~ ~~}~~

Po

~~

~

p(o )

= ~ ~ ^a

2

cop N2(e~)

i

(k~) (43)

where the average

(I(k~) ),

defined

by (B13),

contains all the characteristics of the involved electron band

through

the

Laplace

transform

L(A )

of the metal

layer

electron

density (87).

In the

particular

_case of the

Ag(ill)

surface band

L(A )

takes the form

(B14), leading

to a

conversion

time,

relative _to the orbital _{process :}

r~~(O ⁾

₌ ⁵ x

10~

s = 6

days (44)

It is worth

noting

that this

long

time arises from the small value of

k~

= 0.074. The orbital process

depends

_on the

ability

of the electron bands to transfer the molecular nucleus

angular

momenta to the metal surface. The surface band considered above _can at most transfer

2

k~

which is small. A much

larger

momentum transfer could be

promoted by

the

Cu(100)

surface

band,

observed in the

vicinity

of the

lf point

of the Surface Brillouin Zone

[24J.

In this

case

k~

= 0.66 and the _average

I(k~)

is found to be increased

by

two orders of

magnitude.

However,

the

corresponding

effective _mass

being

decreased

by

_a

factor10,

the rate remains

even smaller. A side consequence of the small

angular

momentum

transfer,

within the

Ag(I

I

I)

surface

band,

is that the distance

dependence

of the _rate is erased.

Contrary

to all the processes, considered

here,

the orbital rate is found to be

weakly dependent

_on the metal- molecule distance.

4.

Two-step

contact-Coulomb processes.

In this section _we

investigate

the

major

_processes in which the

hyperfine

contact Y induces the nuclear

spin singlet-triplet

_o-p

transition,

while the

corresponding

rotational transition

being performed by

the Coulomb interaction _among the electrons _: C. Note that Y induces

simultaneously

_a

singlet-triplet

transition in the electron

system.

^{We shall}

distinguish

processes which involve _one electron virtual

jump

from those which involve two. In the former XY processes the metal and molecule

exchange

_an electron and remain neutral in their intermediate virtual

step

^{whereas in the later} UY process a metal electron is

virtually

transferred to the molecule. The _reverse process where _one molecule electron is transferred to the metal is found to be less efficient.

4.I NEUTRAL INTERMEDIATE STATES: THE XY _PRocEss. We extend the mechanism

developed

for oxides

[25]

to the _case of metal

catalysts.

Two channels _are open

according

to the

spin

manifold of the intermediate virtual _state. If the Coulomb interaction C _acts first it involves _a

singlet-singlet

^transition where _one metal electron k

jumps

to the

antibonding

molecular electron u while _a

bonding

molecular electron is transferred to a metal excited state x. In _a second and virtual

step

the excited molecular electron _u relaxes to its

ground bonding

state g. If the

hyperfine

contact acts

first,

it

promotes

a molecular g electron to _{a u} state

by

_a simultaneous

singlet-triplet

transition in the electron and nuclear

systems.

^The second Coulomb step

brings

_a metal Fermi electron k in the molecular

ground

while

transferring

the

antibonding virtually

excited electron _{u to a} metal electron excited state X. The two channels of this XY process are

represented

in

figure

2. The

symbol

X

emphasizes

the two-electron

molecule-metal

cross-jump.

We choose the _energy of the intermediate molecular state

~I]

_{at the same} intemudear distance _as in the

~I(

_one, since the transition is virtual.

(a)

~

~

5i _c

u

,

Ti

yi _~

^X

@ ~', lb

g i

(b)

5i

u

~

^{~ ~}

Ti

v

c

^~ X

l~C _~ ~

,

i ^g

metal molecule metal

initial final

Fig.

2. The two-step ^XY process, in the

singlet

channel

(a)

and

triplet

channel ~b). ^{The first}

(resp.

second)

step ^is

represented by

_a full

(resp. dashed)

line. C denotes the Coulomb interaction and Y the

hyperfine

contact one.

The matrix elements of the intra-molecular

hyperfine

_contact

(22),

_over the molecular

orbitals,

_are

easily

written _{as :}

(~I) _Y( ~I]

=

( (2)~'~ g(a) u(a)

I _«

(45)

where

(

= 3A _x 10~ ^~ _a-u- and with the values

given

in

Appendix (A),

the orbital

product

at the

proton

a:

g(a)u(a)

=

0.358b~~

_The

(2N+2)

electron matrix

elements,

for the Coulomb

interaction,

_are reduced _to 2-electron _ones. For the _two different

spin

channels _we obtain

js~ (C( S;j

=

(3)~°jXU

_ICI

_{gk) (~~~)}

(T~(C( T~)

=

(gX(C( kU) ~~~~~

where

$, _T~,

S~, T~ are

given by (1), (2), (4). _By defining

the rotational transition of the

symmetrized

molecular orbitals

product

_:

[guim

=

IL

=

°lg(I) u(2)

₊

g(2) u(I)(

L

=

i

ml

,

(47)

the

exchange strength

between the metal electron-hole

pair

and the excited molecule

Jm (kx )

=

(kx

_c

lguJm)

,

(48) inserting (45-48)

in

(10),

and

summing

_over all

quantum

states, _we obtain the o-p conversion rate, relative to the XY process as :

Po~p(XY)

= 12

«<~g~(a) u~(a) EopN2(e~)/fi £ _jJ~/Aj2)~~ (49)

where A

=

E(~I] ₎ E(~I/ )

and the brakets

( )

~~ indicate _an orientation average over the

band states _wave vectors k and _x described

by

their

eigenfunctions (8).

The Contact-

Exchange

_process: XY presents the essential features of the

one-step

processes to be

proportional (I)

to the number of metal excited

pair

states estimated from the _area of the intersection between _a

parabolic

band and the two

parallel planes

of

energies

_e~±

s~~

N~(s~) _s~~

and

(it)

to the fourth _power of the band _vacuum tails which interact with the molecule electron

e~~

^~'~ _The

reason is that the

exchange integral J~(kx )

has its essential

content in the close

vicinity

of the molecular _center

I,

_one bohr

around, J~

remains thus almost

proportional

to the metal electron

density

at I:

#(1)

_(~.

However, apart

^from this

scaling factor,

the Coulomb interaction is

quite

_strong and the XY process is found to be much

stronger

^{than the best}

one-step

process

(~~.

From

(30)

and

(45)

_{we can} write

P~~(XY)

~ ~

=

[6

x10

(J~)/AJ (50)

Pop(fnd)

where the molecular excitation _energy A

= 12 eV. We shall not

give

_a detailed calculation of the

exchange integrals J~,

^since the one-electron

jumps

are found to be _more

efficient,

but

rather

give

_a

qualitative

estimate. In the Coulomb

integral (g(

I _u

(2) (1/rj~( k(I )

_X

(2 ))

the stronger localization arises from the molecular orbital _g which locates electron

(I)

almost _on I. Then

replacing

_rj~

by

_r~ we obtain

(g(I)(k(I)) (u(2) ii /r~( x(2)).

For the

Ag(I ii)

surface

band,

with the numerical values

given

in

Appendix A,

_we obtain _an

overlap

(g(k)

₌ 0.035 and from

Appendix

C _a Coulomb

integral

of 0.6eV which leads for J to 2 _x

10~~

_eV.

Inserting

these values in to the ratio

(50)

leads to _a relative

efficiency

for the

two-step

^reaction

path

of 110. The conversion time is thus

strongly reduced, by

two orders of

magnitude

_:

r~~(XY)

₌ ^{7 min}

⁽⁵¹⁾

We have

already

_seen that the direct contact process arises from the metal electrons while the faster indirect _one mixes metal and molecule electrons

through

their

overlap

to reach the

protons.

In the

exchange-contact

mechanism the metal electrons _are

brought by

the Coulomb

interaction in molecular states which have

strong

contacts with the

protons

^nuclear

spins.

4.2 CHARGE-TRANSFER INTERMEDIATE STATES THE UY PROCESS. We now consider the

family

of _processes where _one metal Fermi electron k is

virtually

transferred to the

antibonding

molecular orbital _u and returns to a metal excited state X. The molecular

structure of this intermediate

Hi

ion can be

depicted by

its

approximate ground

state

~I], _{(g#u (,}

which is known to be _a resonance

(centered

at 2.3 eV above

~I(

and rather

broad).

This _resonance dominates the

scattering

_cross section

experiments

in EELS

experiments

in the gas

[9].

^On a metal

surface,

the

antibonding

_u state is

pushed

downwards

by

the attraction of the

image

force. In the

chemisorption

_process it _even starts to be filled

[26J.

In the

physisorption regime,

_we

might reasonably

consider that this

~I]

_{state becomes}

almost _a bound _state in the

vicinity

of the metal _vacuum level. The _two

steps

^{which link this} intermediate virtual state to the initial and final _{ones are}

provided by

the Coulomb interaction among the electrons C and their

hyperfine

contact with the

hydrogen

nuclear

spins

Y. If the

C-interaction acts

first,

_as it is

spin conserving

the virtual state is _a

singlet

_{: S~} whereas if it is the Y-interaction _we have _a virtual

triplet

: T~ ^whose

eigenstates

_are

given by (3).

The two

singlet

and

triplet

channels of this

family

of _{processes are}

represented

in

figure

3. The matrix elements of the

hyperfine

contact Y

given by (22)

between the

eigenstates (S~)

^and

T~) ⁽²⁾

are

easily

obtained _{as :}

iT~j Yj s~j

=

tu(a)jx(a)

_{+ x}

(b)j/2

I. _«

(52)

whereas between

$ ) (

I

)

and

T~)

_we obtain _:

(T~ Y( Si)

=

(u (a)[k(a)

₊

k(b)J/ Ii

«

(53)

where

k(a) [resp.

_X, uJ ^{denotes the} value of the

eigenfunction

k

[resp,

_x,

u]

at the molecular proton a and _{« are} the

singlet-triplet

electron

spin equivalent

operators ^defined as

previously by

their matrix elements. The

(2

N ₊

2)

electron Coulomb matrix elements C

=

I~

~ p

I/r~p

can be reduced _to two-electron Coulomb _c

=

I/rj~

matrix

elements, leading

to _:

lsvlcl Si)

=

/121ug

ICI

kg) lug

_ICI

gk)

₊

+

lUklcl kk)

₊

IK«kl2lUKlCl kK) lUKlCl ")1) (54)

jT~jcj T~j

₌

2jugjcj xg) jug (c( gx)

+

+

lUklcl xk) lUklcl kx)

+

IK«kl2lUKlCl xK) lUKlCl Kx)J (55)

If _we insert the metal and molecule wavefunctions

(8)

and

(5-7)

into

(54)

and

(55)

to examine the relative orders of

magnitude

of the different Coulomb matrix elements _{we can} conclude

that

(I)

^the Coulomb

integrals containing

3 molecular wavefunctions and _one metal

wavefunction _are much

larger

than those which contain 3 metal and _one molecular-ones.

(The physical

_reason is that the molecular orbitals _{are more}

strongly

localized than the metal band

states)

and

(it)

the Coulomb

integrals

of the form

jug (c( gk)

contribute

negligibly

_as the molecular orbitals g and _{u are}

orthogonal.

We shall therefore

neglect

in the

following

all but the first

leading

term in

(54)

and

(55)

which become _:

(Sv

^C

S;

"

(2 )~~ jug

_c

kg (56a)

( Tf

C _TV

" 2

(

_{Ug C Xg}

) (56b)

Then

performing

the rotational _averages

IL

=

o

ix (a)

_{+ x}

(b)

L

=

o

j

=

2 _x

(1) (5?a)

(L

₌

I,

_{m i}

(k(a)

₊

k(b)

L

=

I, m1)

=

2

k(1) (57b)

the rotational transition _:

(L=0(u(

L=

I,mi)

_=u~.

₍₅₈₎

(al

5

u

~ Tr

v x

i _flk

-+_ g

(b)

~' Tv

~

~~

+_ g

me

IQ mo(ecu(e mete(

initial I trial

Fig.

3. The two-step UY process, in the

singlet

channel

(a)

and

triplet

channel ~b). The Coulomb step is the first

(resp. second)

_one in

(a) (resp. (b)),

and reversed for the

hyperfine

contact one.

The electron and nucleus

spin

transition _:

(S= lm~,1=0(I.«( S=0,1= lm;) =8~,,~~ (59)

we obtain _:

(Tf,

_P

Y(

Sv> q (Sv> q

Cl $., °d)

₊

(Tf>P

C _TV> q (TV> q

Y( $> °11)

"

=

(

^2~'~ _u

(a)

8

~, ~

ll'(I) (u~

_g c

ll'g ) (60)

where _we have assumed that the electron states

belong

to the _same band described

by

the

eigenfunctions ll'given by (8). Inserting (60)

^into the conversion rate

(10)

relative to this UY process, and

summing

_over the different electron states which contribute to the rate, we

obtain

Po ~p(uy)

=

22

33

«<2

_e~~

N2(e~)/fi £ v~(i) ju~

_g

cl v~gj /Aj~)~ ⁽⁶¹⁾

where A is the virtual excitation _energy almost

equal

to the work function 4~ and the brakets

( _)~~

indicate _an orientation _{average over} the band state _wave vectors k and _x.

As all the

preceding

_processes

(except

the orbital

one),

the Contact-Coulomb UY process is

proportional

to the

pair-density

of states :

N~(e~)

_e~~ and to the fourth power of the band state tails

e~~

^Y~ _It is also

proportional

to the square of the

non-diagonal

metal-molecule

Coulomb

integral

_:

U~

₌

(u~

_{g c}

ll'g).

In the

following

_we restrict _our consideration to _an