THE ELECTRONIC STRUCTURE OF ADSORBED ATOMS AND OTHER SURFACE DEFECTS

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THE ELECTRONIC STRUCTURE OF ADSORBED ATOMS AND OTHER SURFACE DEFECTS

T. Grimley

To cite this version:

T. Grimley. THE ELECTRONIC STRUCTURE OF ADSORBED ATOMS AND OTHER SURFACE DEFECTS. Journal de Physique Colloques, 1970, 31 (C1), pp.C1-85-C1-92.

�10.1051/jphyscol:1970115�. �jpa-00213746�

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JOURNAL DE PHYSIQUE Colloque C 1, supplément au no 4, Tome 31, Avril 1970, page C 1

-

85

THE ELECTRONIC STRUCTURE OF AD SORBED ATOMS AND OTHER SURFACE DEFECTS

by T. B. GRIMLEY

Résumé. - Un niveau atomique qui tombe dans les bandes d'énergie d'un cristal devient un niveau virtuel quand l'atome est adsorbé. Le nombre d'électrons dans ce niveau virtuel dépend des positions relatives du niveau virtuel et du niveau de Fermi. Mais dans l'approximation de Hartree- Fock, le remplissage des niveaux et leurs énergies sont reliés par les interactions Coulombiennes entre électrons. Par conséquent, il y a un problème de self-consistence à résoudre pour que la structure électronique d'atome adsorbé soit découverte. Ce problème.est discuté en ce qui concerne les atomes alcalins, l'hydrogène, et les atomes de transition adsorbés par un métal. La théorie de la structure électronique d'un atome d'impureté, et d'une lacune dans la surface d'un cristal est briè- vement présentée.

Abstract. - A discrete atomic level which falls within the energy bands of crystal persists in the adsorbed atom as a virtual level with fractional occupancy determined by the level's position with respect to the Fermi level. But in the Hartree-Fock theory of atoms, the orbital energies, and their occupancies are related through the Coulomb and exchange interactions of the electrons. Conse- quently, the position of a virtual level with respect to the Fermi level cannot be known until its occupancy is known, and vice versa. There is therefore a self-consistent problem to be solved before the electronic structure of the adsorbed atom (i. e., its level densities, and occupancies) can be determined. The solutions to this problem are discussed for alkali, hydrogen, and transition series atoms on metals. The theory of the electronic structure of impurity atoms, and vacancies in a crystal surface is briefly presented.

1. Introduction. - First of al1 we need t o be clear what we mean by the electronic structure of a surface defect. Consider for example, an atom adsorbed on a solid. Only the combined system, atom plus solid, can be investigated either theoretically or experimentally, but if we imagine the coupling between the atom and the solid reduced steadily t o zero, certain features of the electronic structure of the combined system will change steadily t o become the electronic structure of the free atom, and certain other features will turn into the electronic structure of the original solid.

The former constitute the electronic structure of the adsorbed atom. Several theoretical studies of adsorption by metals 11, 41, have used models in which the wave functions of the combined system are simply expressed as linear combinations of free atom, and free metal wave functions, and the idea that the adsorbed atom has a definite electronic structure (different from that of the free atom) arises quite naturally from such a description ^(l). A simple example may be useful here.

Consider an atom with one valency electron in a level with energy EA falling within the quasi-continuous

(1) It is not necessary to adopt this description. An investigation of the ammonia molecule in the Hartree-Fock approximation using cc one-centre expansions ⁾⁾of the Hartree-Fock molecular orbitals [SI does not explicitly recognize that ammonia is formed from nitrogen and hydrogen atoms, but it gives just as good a description of the molecule as an investigation using

« four-centre expansions » which does recognize the constituent atoms.

spectrum of levels for electrons in the metal. The level structure of the metal is described by its level density pZ)(c), the free atom simply has a discrete level at EA, and the coupled system, atom plus metal, is described by a level density ~ ( 8 ) . We want to know how t o associate a contribution t o P(E) with the adsorbed atom. T o d o this we only need t o realize that the atomic state at E, is not a stationary state of the combined system ; it decays t o zero once the atom- metal coupling is switched on, and so has a finite lifetime, and therefore acquires a certain width. Thus we can say that the original discrete atomic level a t EA persists in the combined system as a virtual level described by a level density function PA(&). The situation is similar to that encountered in treating the excited states of al1 isolated atoms and molecules ; in quantum theory an « isolated » atom is always coupled t o the zero-point motion of the electromagnetic field, and al1 excited states have a natural level width.

In Our system, the virtual level is filled to the Fermi level cF (Fig. l), and this defines an occupation number or occupancy, nA according t o

The level density p,, and its occupancy nA charac- terize the electronic structure of the adsorbed atom.

Reducing the atom-metal coupling t o zero causes pA to degenerate to a d-function a t the atomic level E,, and n, ^-tO or 1 according as EA is above or below E,.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970115

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T. B. GRIMLEY

FIG. 1. - A partially filled virtual level in an adsorbed atom.

As well as turning the free atom level into a virtual level, the atom-metal coupling aIso perturbs the level density of the metal ; + p , =

+

6 p M , and the level density in the coupled system is the sum

The change 6 p M describes a response of the metal to the adsorbed atom.

It is clear from equation (1) that the adsorption of an atom will generally result in changes in the occupancies of its orbitals, from occupancies zero or unity for each spin direction to fractional values. This has important consequences because in the Hartree- Fock theory, the treatment of the Coulomb interactions of the electrons relates the orbital energies and occupancies. 1 give only two examples. Firstly, the 3s orbital energy in Na ; ls2 2s' 2p6 3s is - 0.182 1 Har- tree (') but in Na- ; ls2 2s' 2p6 3s2it i~ - 0.012 5 Har- tree [6]. Secondly, the 3d orbital energy in Ni in the 3d8 4s2 configuration is - 0.558 Hartree, but in the 3d9 4s configuration it is

-

0.345 Hartree [7].

2. An alkali atom on tungsten. - Consider the interaction of a Na atom with the (1 10) face of tungsten. This is the high work function face, and the atomic 3s level is about 0.03 Hartree above the Fermi evel ⁽³⁾so in the first instance we expect the atom to lose its valency electron to the metal. However, because the discrete atoinic level is turned into a virtual level with part of its density below ^E,, the electron transfer is not complete, and in the adsorbed atom the 3s orbital is partially occupied. We assume (see

3

2.1) equal occupancy nA by spin

1

and spin

4

electrons so that the Coulomb repulsion between electrons raises the 3s orbital energy from the free atom value EA to a value ^{8 ,} in the adsorbed atom depending expli- citly on nA ;

(2) 1 H a r t r e e 2 7 . 2 e V = 4 . 3 6 x 1 0 - 1 8 J .

(3) This is based on Schmidt and Gomer's [SI value of 5.85 eV for the work function. The correct value appears to be in doubt, but for the present discussion it is only necessary that the work function be high enough to ensure that the occupancy of the virtual level is small.

where J is the Coulomb energy of electrons in a doubly occupied 3s orbital cp,,

J ^E0.17 Hartree for Na. Raising the atomic level in this way raises the virtual level, and its occupancy there- fore falls. Evidently a self-consistent solution for nA

and ^{E A}has to be sought. However, it is fairly certain

that, for Na on the (1 10) face of tungsten, nA will be rather small ; the adatom charge will be nearer

+

1 than zero.

But there is another aspect of the electronic structure of the adsorbed atom which we have glossed over in the above discussion. Equation (2) cannot be used until the 3s orbital cp, is specified, since this fixes EA and J. But in the Hartree-Fock theory, the orbitals themselves as well as their energies depend on their occupancies. One has different 3s orbitals for Na and Na-, and different 3d orbitals for Ni in the 3d8 4r2 and 3d9 4s configurations. Consequently, having found the occupation number n, using the free atom 3s orbital, which is calculated for occupation numbers

we ought now to recalculate the 3s orbital using the adatom occupation numbers nAt ⁼n , ~ = nA. The result will be to raise EA, and reduce the value of J.

The details of the atom-metal coupling will also be altered. These changes in turn affect H A so the 3s orbital has to be calculated once more. However, present models of adsorption are such great simplifications of a complex problem that the pursuit of self-consistency along these lines would be an extravagance. But there may be some justification for taking the first step, and making one recalculation of the 3s orbital.

This would mean solving the Hartree-Fock equations for a Na ion with the outer electron configuration

SPA SPA

so that for example, a 3s electron « sees >>

an extra Coulomb potential

due to the partial occupancy of the orbital with both spin

1

and spin J electrons. Evidently such a recalcu- lation is harder to justify the smaller n,.

2.1. DETERMINATION OF THE OCCUPANCY.

-

The occupancy of the orbital cpA in the adsorbed atom can only be determined theoretically by treating the atom- metal coupling. The simplest starting point is a mode1 Hamiltonian of the type first used by Anderson [9]

in the theory of dilute alloys :

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THE ELECTRONIC STRUCTURE OF ADSORBED ATOMS C l - 8 7 H~~~

22

are ~~~~i~~ creation and destruc- a self-consistent solution. These are illustrated in tion operators for electrons in the atomic orbital 9, figure for Na ^Onthe (11°) face tungsten. The two

A "+ A

with spin o which is either or J, nA, = CA, cAU is the

"+

" ^A corresponding number operator, and cko, ckU and nk, have the same meanings for the metal orbital cp,.

EA and .sk are orbital energies, and J is the Coulomb repulsion energy (3). The first two terms in (4) describe the free atom, the third describes the free metal, and the fourth is the coupling between them. This is simply a sharing of the electrons between the atom and the metal.

By treating the Hamiltonian (4) in the Hartree-Fock approximation we find as expected, that n,,, the occu- ,pation number of the orbital cp, for electrons with

spin o, can be calculated from

"A*

(5) ^FIG.^2.

-

Self-consistency plots of AT and for sodium

n~~ = adsorbed on tungsten.

- w

The formula for PAU, the virtual level density for eiec- curves are mirror images in the diagonal plane so trons with spin o is their one intersection has fiAt = nAL = a,, and there

r

is, after all, only one virtual level with small occupancy Pau = (6) like that in figure 1. Unlike the free atom, the adsorbed

n

(

(E

-

^&Au

-

a)'

+ r Z )

atom is non-magnetic. This important conclusion

depends only on the assumption that, even when where is the orbital energy for spin a electrons in

the adsorbed atom HA-, = O

and a ( ~ ) and ï ( ~ ) are the real, and negative imaginary parts of a function q ( ~ ) :

The atom-metal coupling is contained only in q, and according to equation (6) this function determines, through ï, the width of the virtual level, and through a, its position. The latter is defined as the root ^E,, Say, of the equation

and it is displaced from the orbital energy E,, by the level shift function a.

Looking back to equations (1) and (2) we see that we are now allowing different virtual levels for f and

4

spins. The Hamiltonian (4) certainly allows this ; solutions with nAT # nAl may exist (they exist for the free atom !) and if they do, the adsorbed atom is paramagnetic like the free atom, but with a smaller magnetic moment. It seems that the adsorption of hydrogen by metals may involve such solutions [4], (see also § 3). Equations (5), (8) cover al1 possibilities.

We use them to calculate for al1 allowed values of n~~ in (7), namely O

<

n,l 6 1, and also nAl for all allowed n ~ ? . There are thus two relations between nAr and nA1 to be satisfied simultaneously to get

so that E,, has its lowest value, p,, still has most of its density above EF so that n ~ ,

<

1, even at this point.

Since EA > EF this is a reasonable assumption, but it could be vitiated if the level shift function a assumed such large values below EF that the lowest value of E,,

is displaced to below E,.

Values of n, which have been reported in several theoretical investigations of the adsorption of alkali atoms by tungsten are given in Table 1. Generally, magnetic solutions have not been sought, but the small- ness of the nA-values, makes it almost certain that none exist. Wojciechowski's [3] work is closely connected with Mulliken's 1101 theory of donor-acceptor com- plexes, and the very small nA-values he reports are perhaps due to his neglect of anionic charge transfer

Valency orbital occupancies for some alkali atoms on tungsten

"A

System

ref.[11] ref.[3] ref.[12]

- - -

Na-Wloo

. . . . .

0.018 Na-Wllo

. . . . . . .

0.013 Na-Wll1

. . . . . . .

0.019

K-Wloo

. . .

0.235 0.023 0.325 K-Wllo

. . . . . . . .

0.165 0.019

K-Wlll

. . . . . . . .

0.260 0.024 C S - W ~ ~ ~

. . .

0.22 0.009 C S - W ~ , ~ .

. . .

0.13 0.008 CS-W~~,,.

. . .

0.18 0.009

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C i - 8 8 T. B. GRIMLEY states in the wave function. In terms of Our model

Hamiltonian (4), this neglect is described by restric- ting the summation over k in the atom-metal coupling term (the fourth term in

fi)

to metal orbitals with energies greater than CF.

3. A hydrogen atom on a metal. ^-Although hydrogen is the simplest atom, it is not the simplest adsorbate because, for some metals, it seems that the Hartree-Fock ground state of the Haqiltonian (4) might be the magnetic state with unequal occupancies of the 1s atomic orbital for electrons with f and J spins.

The reasons for this are the particularly strong depen- dence of the 1s orbital energy on its occupancy, the fact that the orbital is only singly occupied in the free atom, and that the ionization potential is so large.

Thus, the 1s orbital energy in H ; 1s is

-

0.5 Hartree, but in H - ; ls2 it is

-

0.044 Hartree, so in Our model EA = - 0.5 Hartree, and J = 0.456 Hartree ^(4).

Consequently, although the singly occupied level is far below e,, the doubly occupied level is above it (see Fig. 3), and we do not therefore, in the first ins- tance, expect any electron transfer at al1 between the

FIG. 3.

-

1s orbital energies in H and H-.

atom and the metal. An equivalent argument is to say that the ionization potential of hydrogen is greater than the work function of the metal, and the electron affinity (0.0278 Hartree) less than it.

The slightest atom-metal coupling turns the discrete levels into virtual levels whose occupancy falls sharply from 1 to O as they pass through the Fermi level. It is therefore possible to suppose that, in the weak coupling limit, there is only a partial electron transfer between the atom and the metal so that the adsorbed atom has equal (fractional) numbers ^GA of spin f and spin J electrons, the value of n, being fixed by the requirement that the orbital energy (2) be at c,. In fact this non-magnetic adsorbed atom does not form spontaneously ; in the weak coupling limit the magnetic state is the ground state. We can see this by examining equations (5)-(8) in the limit q + O. p,, is a 6-function

at &A#, and so n,, is O or 1 according as &A, is above o r below 8,. If, for example, nAl = O, pAT is a 8-function at EA. If is below e,, and hence occupied so nAt = 1.

Increasing nAl raises eA1 above EA according to equation (7), and the level empties (n,? = O) as it goes through 8,. nA1 behaves similarly as a function of n,?, and figure 4a can be drawn showing the two relations between nAt and nAl which have to be satisfied simultaneously to get a self-consistent solution. There are three intersections, the outer two describe the magnetic state, the inner one the non-magnetic state. Although it is clear on quite general grounds (from the varia- tional aspect of the Hartree-Fock energy for example) that the magnetic state has the lower energy, an actual energy calculation is trivial. We require the forma- tion energy AW (negative for stability) of the non- magnetic adsorbed atom from the free atom and metal.

We take Our energy zero for electrons at the Fermi level, so in equation (2) the orbital energy is zero and n, = - EA/J. The Hartree-Fock energy of the adsorbed atom is simply

(because the orbital energy is zero), and since that of the free atom is just EA, we have

A W =

-

JnA - EA =

-

EA(l

-

^n,).

Since EA is negative, and O < n, < 1, AW > O and the non-magnetic adsorbed atom does not form spon- taneously in the weak coupling lirnit. Of course A W= O in this limit for the magnetic state because now the electronic structure of the adsorbed atom is just the same as that of the free atom.

With a small but finite atom-metal coupling, the sharp corners in figure 4a are rounded off, and we go to figure 4b, but the magnetic state is still the ground state. There are two virtual levels, one near EA the

(4) To use this value is to imply that some recalculation of

the 1s orbital aiong the lines mentioned in 8 2 has been made; FE. 4. - Self-consistency plots of and ~ A J for hydrogenon the value of J calculated from eauation (3) . , for the hydrogen 1s - - a metal. (a) Zero atom-metal coupling, (6) weak coupling, (c)

orbital is 0.625 Hartree. strong coupling.

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THE ELECTRONIC STRUCTURE OF ADSORBED ATOMS C l - 8 9 other near EA i- J, the former is nearly full, the latter

is nearly empty (see Fig. 5). Increasing the atom- metal coupling causes the outer (magnetic) intersections to contract on to the inner one, and when the

FIG-. 5. - The two virtual levels of adsorbed hydrogen for weak atom-metal coupling.

coupling is strong enough only the inner intersection remains (Fig. 4c). Unlike the free atom, the adsorbed atom is now non-magnetic. Newns [4] has used the experimentally rneasured binding energies of hydrogen on titanium, chromium and nickel to estimate the strengths of the atom-metal couplings, and he finds that, in al1 cases, it is just strong enough to ensure that only the non-magnetic state exists. The absence of the magnetic state reflects the « chemical » nature of the atom-metal bond ; a considerable reorganiza- tion of the electrons, involving a pairing up of their spins, has taken place to give the observed binding energies of about 0.11 Hartree. n,-values reported by Newns [4] are 0.70, 0.61 and 0.51 for titanium, chromium and nickel severally. These are certainly large enough to justify recalculating the 1s orbital using the adsorbed atom occupation number as rnentioned in $ 2 .

4. A transition atom on a metal. - The ordinary Hartree-Fock theory of the electronic structures of the free atoms of the fourth row transition eleinents gives us the familiar picture of an incomplete 3d subshell with orbital energy below that of the 4s subshell.

Scandium is an example ; Eîd =

-

0.344 Hartree, E,, = - 0.210 Hartree [13]. But in spite of the appea- rance to the contrary in figure 6, there are no vacant

FIG. 6. - 3d and 4s orbital energies, and occupancies in scandium.

one-eIectron levels in the atom below E,. If we transfer a spin f electron from the metal to one of the four vacant d orbitals, Coulomb (J) and exchange (K) interactions of the two d electrons raise the d orbital energy by an amount U = J

-

K so that it now lies above ^8,. The 4s level is affected to a far lesser extent by this changed occupancy of the 3d level, and in a first analysis of the electronic structure of the adsorbed atom, we shall assume that the 4s orbital stays doubly occupied. This means ignoring the undoubted exis- tence, in a proper theory, of an « s + d transformer » action by the metal. The situation is now similar to that with adsorbed hydrogen, and in the weak coupling limit we can draw figure 7a to show the self-consistent occupancies nit. and nzt of a pair of d orbitals in the adsorbed atom. It should be particularly noted that both spins are t in the lowest energy states (Hund's rule) so that al1 three intersections in figure 7a describe magnetic states. As before with hydrogen, the two outer intersections belong to the ground state ; they correspond to two of the ten components of the '0 ground state of the free atom. Again we can verify by direct calculation that the adsorbed atom with the electronic structure defined by the inner intersection, n I t = n,t = n say, with n determined so as t o bring the 3d orbital energy to the Fermi level

does not form spontaneously in the weak coupling limit.

FIG. 7. - Self-consistency plots of ni? and nzt for a scandium atom on a metal. (a) Zero atom-metal coupling, (b) weak

coupling.

At the most, only two of the five d orbitals can be equally involved in the atom-metal coupling, so with a finite but small coupling, a graphical representation of the self-consistency conditions becomes very diffi- cult. The situation is however quite simple if one d orbital,

1

3 z 2

-

r2

>

say, dominates the atom-metal coupling so that the other four orbitals can still be treated in the first approximation as belonging to a 4-fold degenerate discrete level. Let n i t be the occupancy of the

1

3 z2 - r2

>

orbital, so that nZt refers to any one of the other four. Then a small atom-metal coupling only rounds off the sharp corners of the graph of n I t versus nzt, and so takes us to figure 7b.

There are still three intersections, and we know that the outer two are still energy minima, and the inner one

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C l - 9 0 T. B. GRIMLEY an energy maximum. But the outer two do not any

more give states of equal energy because the occupancy n,? of the orbital involved in the atom-metal coupling is not the same at the two intersections. At one, n I t

=

1 and n Z T = O, at the other, nit ^E0, nZt = 1. Which of these gives the ground state (a magnetic doublet) depends on the details of the atom- metal coupling, and in particular on the electronic structure of the metal for energies near E ~ . The two possible electronic structures of the adsorbed atom are depicted in figure 8. There is either a single virtual

FIG. 8. - TWO possible electronic structures of adsorbed scan- dium. Only the d levels are shown.

level near E3, and a discrete 4-fold level near E,,+ U, or the virtual level is near E3,

+

U , end the Cfold level (singly occupied) is near E,,. In the former case the adsorbed atom carries a small positive charge, in the latter a small negative charge. In the strong coupling limit the ground state would be n , ^@1, nz ⁼1 except that, when some critical coupling is exceeded, the non-magnetic state n , = rii,, nZî = n z l = O will become the ground state. This reminds us again that strong atom-metal bonds will always involve spin pairing in the adsorbed atom.

The above discussion is sufficient to show that a study of the electronic structures, and binding energies of atoms of the transition elements on solid surfaces is a rich field. Serious work is now in progress.

5. Impurity atoms and vacancies in the surface. -

Impurity atoms in the surface can be investigated in the same way as adsorbed atoms, but vacancies cannot, because a free vacancy does not exist. Vacancies are handled by supposing that they simply contribute an extra term in the potential energy of an electron in an otherwise perfect crystal with a surface, and of course impurity atoms can also be handled this way.

The Hamiltonian operator for this model in its simplest form is

A *+ *

= &k ;ka

+

vkl Cka Cl<r

ku klu (10)

with only spin-independent level shifts (k = I), and scattering terms (k # 1) introduced by the defect potential.

To make progress we go over to Wannier's representation, and express the orbitals cpk of the defect-free

crystal in terms of point Wannier functions am localized on the lattice points m,

We suppose also that the defect potential is localized on the defect lattice point m ⁼O in the sense that, in the Wannier representation, it has only one finite matrix element, Voo = V say. It is now easy to calculate the contribution, Ap(e), which the defect makes to the level density of the system. If we define (cf. equation (8))

GO(&) = a(&)

-

ig(&) = lim

C ¹

^{v U k ~}

^1'

S.+O E

+

^is^-^{c k Y}

If the equation

has a root, ^{E ,}say, in one of the bands of crystal levels, there is a « resonance », i. e. a virtual level at ^E, due to the defect. However, thelevel density Ap may actually be negative at ^{E ,}; the virtual level is then for holes not electrons. If equation (12) has a root which falls outside the crystal bands then (like any similar root of equation (9)) it describes a real discrete level going with the defect. Such a level does not have occupancy exactly unity even when it falls below ^{E ,} ; these discrete levels have « tails » in the energy bands.

The orbitals of the defect crystal are also expressed approximately in terms of the point Wannier functions, so the occupancies n, of these functions are altered by the defect. If we define

"

(b - if) (b*

-

if*)

Anm ⁼- -

V - a + i g (1 3) gives the alteration in nm. This formula predicts the familiar Friedel oscillations [14] of An, as a function of m, but for a surface defect there is a definite aniso- tropy in the long range behaviour. The disturbance (13) may be classed as a response of the solid to the defect in its surface ; an adsorbed atom elicits a similar response [15]. Because our knowledge of one-electron wave functions in the surface region of a crystal is at present so inadequate, few applications of equations (1 1)-(13) to surface defects have been made, and we cannot therefore comment on the usefulness of the simple Hamiltonian (10) in this field. Of course, the limitations of the localized model with a single para- meter V , well-known in the theory of dilute alloys, will naturally be present here also.

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THE ELECTRONIC STRUCTURE OF ADSORBED ATOMS

J. FRIEDEL. - 1 agree to the usefulness of your model for a qualitative discussion of some of the main problems involved in chemisorption. But one should be careful in using it for quantitave analysis, in most cases. Thus :

1) The model is certainly sensible in cases like transitional or rare earth atoms adsorbed on « normal ⁾⁾ metals, as one knows that the same type of approxima- tion works fairly well when the same impurities are dissolved in those matrices. Anderson's model hamiltonian is then known to work reasonably well, thus the « virtual)) level can be described by a simple width. Even there, it is not sure that crystal field effects are not important for degenerate d states. Also, you neglect in your treatment the fact that the charge trans- ferred from the adsorbed atom to the metal is locally concentrated near the adsorbed atom. Its electro- static interaction with the adsorbed atom and the scattering of the conduction electrons should really be al1 treated selfconsistently. However these are details.

2) In other cases you have mentioned, 1 am more sceptical, because 1 believe that the « virtual level » is so broad and complex in form that it is not really useful to introduce the concept.

In particular, for transitional impurities on transitional metals, 1 expect overlap of atomic d function effects (i. e. molecular orbital effects) to broaden the excess density of states due to adsorption considera- bly, and in a complex way, showing highly directional effects. For hydrogen adatoms, having myself first studied the possibility of a magnetic state when H is dissolved in a metal, it would be unfair for me to reproach you to have considered the problem. Howe- ver I would now consider the possibility to be highly unlikely, again because the s-s interactions involved in the chemisorption on the valence electrons of the metal should lead to very broad virtual levels. A further problem here is that we do not know whether the proton can stay on the surface without being' automa- tically absorbed into the metal.

Even if an adsorbed atom is not magnetic, it would be of interest to study the way it might become magnetic when going away from the metal. Your model might there provide a good starting point for this old problem.

M. GRIMLEY. - Whenever the binding energy of the atom to the surface is large, the virtual level density p,, in equation ( 6 ) is expected to be broad, and to have a good deal of structure, but the occupancy of the atomic orbital is still given by equation (5), and the model does not, in my opinion, cease to be useful just because p,, ceases to have a simple form. 1 agree however that we have to be careful in using this simple model for a quantitative analysis of chemisorption.

Overlap between ^cp, and the set cp,, which is neglected

JSSION

in Anderson's model, is an essential feature of che- mica1 bonding. Inclusion of this overlap makes the function q(ej more complicated, but it does not otherwise affect the self-consistency condition. Of course the interpretation of n,, is altered ; it becomes the number known in quantum chemistry as the net atomic population for spin a.

M. J. BEEBY.

-

1 wish to question whether one can hope to get a correct description of the binding of an adatom to a surface using Hartree-Fock theory when the atom is not strongly bound to the surface.

Surely a molecular orbitals calculation is required.

M. GRIMLEY. - The theory I described is a simple version of the self-consistent field molecular orbital theory ; it uses a minimal basis set on the atom, and includes only one Coulomb integral. By allowing nAr # nAl it certainly affords a correct description of the electronic structure when the atom-metal separa- tion is large. However, you are right to question its application t o weakly bound adatoms. To discuss these correctly we shall need to include some «no- bond » terms in the Hamiltonian. By « no-bond » terms 1 mean products of Fermion operators al1 belonging to the metal or al1 belonging to the atom.

M. A. A. MARADUDIN.

-

A few years ago Pliskin and Eischens observed a peak in the absorption spectrum of hydrogen adsorbed, 1 believe, on platinum.

The frequency of-this peak shifted downward by a factor of about J2, thus suggesting that this peak has its origin in the vibration of the adsorbed atom against the metal surface. 1 would welcome your comments regarding the feasibility of calculating by your methods the energy of interaction of a hydrogen atom with a metal surface as a function of the distance of the atom from the surface. In particular, I am interested in the possibility of calculating the curvature of this function in the vicinity of its minimum.

M. GRIMLEY. - Binding energy calculations are, of course very difficult. Here we need, amongst other things, an accurate knowledge of the wave functions, and self-consistent potentials of electrons near the surface of a semi-infinite metal. Then we could calculate the interaction, and overlap matrices (overlap cannot be ignored in a binding energy calculation), and so find the function q(e) for al1 distances of the atom from the surface. This determines the potential energy curve, and so the force constant for small vibrations of the atom against the (fixed) surface. 1 think that this calculation is feasible for a n alkali metal, but not for a transition metal. The potential energy curve itself may not be very good, but expe- rience with simple systems shows that the force constant is a good deal easier to calculate than the binding energy. I imagine you had this fact in mind.

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C l - 9 2 T. B. GRIMLEY

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