18 MOLECULE NEAR METAL SURFACE I. THEORIES OF ADSORPTION AND PROPERTIES OF SURFACE LAYERS BEHAVIOUR OF A MOLECULE NEAR A METAL SURFACE BY K. HUANG* and G. WYLLIE Received 18th November, 1949 An account is given of the energy changes associated with electron transfer to or from a molecule a t distances from the metal surface such that quantum- mechanical tunnel effect may be neglected. A self-consistent treatment of the image field is used t o find the distribution of charge on a large adsorbed mole- cule. The modification in electronic structure of an adsorbed molecule in conse- quence of the formation of a weak covalent bond with the metal is treated in terms of a very simple model. A possible semi-empirical approach to treatment of the catalytically active surface is examined.It is of interest in a number of applications to consider the interaction between a molecule and a metal surface. At large distances, this inter- action is of the same nature as the van der Waals’ interaction between molecules, being attractive in character. This attractive potential has been discussed by a number of authors, especially Lennard- Jones, and Margenau and Pollard.2 The short-range interaction is of a more com- plicated character, and has only received a satisfactory discussion for the case in which the short-range forces are due to electron exchange and are purely repulsive.3 * Now a t Department of Theoretical Physics, Liverpool University. Lennard-Jones, Trans. Faraday Soc., 1932, 28, 334. Margenau and Pollard, Physic. Rev., 1941, 60, 128.Pollard, ibid., 1941, 60, 578.K. HUANG AND G. WYLLIE 19 Experimentally, it is usual to distinguish between physical adsorption and chemisorption, according as the binding energy between the adsorbed molecule and the solid surface is small or large. In physical adsorption the binding energy is of the order of the cohesive energy of a molec- ular crystal : in chemisorption it is of the order of the energy of an ionic or homopolar chemical bond. In the past, chemisorption has fre- quently been called " activated adsorption," since, in many cases it occurs comparatively slowly, and the temperature dependence of the rate corresponds to the necessity of a thermal activation energy. It seems, however, that chemisorption occurs instantly in those cases where experiments have been carried out with thoroughly clean metal surfaces (e.g.hydrogen on tungsten4), so it is at least possible that the observed activation characterizes the displacement of a previously chemisorbed layer rather than the simple process of adsorption. It is generally observed that heterogeneous catalysis is accompanied by chemisorption of at least one of the reactants on the catalyst surface. It is thus of interest to enquire what changes in molecular structure may be expected to accompany chemisorption. The object of the present paper is to outline the relevant considerations and draw some qualitative conclusions. I. A number of workers (e.g. Nyrop,5 Dowden,%') have suggested electron transfer in one direction or the other as the essential process in catalytic action. We shall consider first of all the energy changes associated with such transfer, where the separation of molecule and metal is such that we may neglect the influence of quantum-mechanical tunnel effect on this energy.Suppose that the work function, which is both the first ionization potential and the electron affinity of the metal, is x, that the electron affinity of the free molecule is A and that its first ionization potential is I. Then if molecule and metal are a t a great distance from each other, the work done in transferring an electron from metal to molecule is and from molecule to metal Typical values of the quantities concerned are x = z to 5 eV, A = o to I eV, I = g to 12 eV, so that in general W+ > W- > 0. Let us suppose an electron to have been transferred in one direction or other.There now exists an electrostatic attraction between metal and ionized molecule. We discuss first the simple case in which the mole- cule consists of a single atom. The ion will polarize under the influence of the charge induced by itself on the metal surface, and so long as its interaction with that charge is sufficiently described by the image field, the electrostatic energy of the system when the ion is at a distance Y from the surface is given by W - = x - A , w+ = I - x. where K is the polarizability of the ion in E.S.U. The approximation of the image field breaks down for static charges under two conditions, (i) if the surface charge density on the metal ap-- proaches the value of one electron per surface atom ; (ii) if the value of Y diminishes below the effective diameter of a metal atom or of the ion, whlchever is larger.The value of e 2 / 4 ~ is 2 eV for Y = 24A ; the second term in E, is gener- ally smaller. Thus, if W- has the rather small value of 2 eV, it is possible Roberts, Some Problems in Adsorption (Cambridge University Press, 1939). Nyrop, The Catalytic Action of Surfaces (Copenhagen, 1937). 6 Dowden, Research, 1948, I, 239. Nature, 1949, 164, 51.2 0 MOLECULE NEAR METAL SURFACE to form a negative ion near the surface a t such a distance that the exchange interaction is negligible, without expenditure of energy. The ion will, of course, only be in equilibrium at such a distance if there is already some sort of obstructing layer (e.g. of previously adsorbed atoms) between it and the metal surface.A larger molecule cannot be characterized electrostatically by a point charge and point dipole when i t is a t a distance from the surface compar- able with its own dimensions. We shall treat in some detail the case of a relatively large conjugated system, in which the charge transferred to the molecule is small compared with the total charge of the mobile electrons in the normal molecule, using a method developed by Coulson and his co-workers.* In a molecule of this type there are vacant molecular orbitals separated by only a small energy interval from the highest orbitals occupied by the mobile electrons in the ground state of the normal molecule. The charge transferred from the metal will enter such vacant orbitals.The molecular orbitals of the mobile electrons are constxucted as linear combinations of atomic orbitals #, s = I , . . . N ; we make the usual simplifying assumption that only one orbital on each atom need be considered. Coulson has defined polarizability coefficients (1) where q, is the number of mobile electrons on the atom s and ut is the Coulomb integral for the atom t . Under the condition mentioned above, i.e. that the fractional change in the q, is small, we may in a first approxim- ation treat these coefficients as having the same values in the adsorbed as in the normal molecule. In the final configuration of the molecule near the metal surface, there is an electrostatic field arising from the charge distribution on the molecule and its electrical image in the surface.This potential enters into the Coulomb integrals us, causing a change Au, as compared with the values in the normal molecule. If then E be the average number of electrons per atom (concerned in mobile orbital formation) transferred to the molecule, and E + Aq, the actual excess on atom s, we have from (I) in a first approximation where the repetition of the dummy index t implied summation. according to the atomic orbital 4,. Aq, = A%, - * ( 2 ) In the model used, the charge E + Aqr must be supposed distributed Thus we have the charge distribution - ( E + &s)e I +(y - Vr) I where r, is the position vector of the sth nucleus, and this together with its electrical image gives rise to a potential distribution which is every- where proportional to E + Aqs.The potential occurs linearly in the integrand of the Coulomb integral at, so we have the strictly linear relation, AM: = a t u ( ~ + A4u). * - (3) The a,, depend on the form of wave function involved and the geometrical configuration of the system. An example of their calculation €or a par- ticular simple case is given in the Appendix A. Eqn. (2) and (3) together determine Aq, and Au, for given E, for they can be combined to give the systems of linear equations and Longnet-Higgins and Coulson, Trans. Faruduy SOC., 1947, 43, 87 ; Proc. Roy. SOC. A , 1947, 191, 39.K. HUANG AND G. WYLLIE 21 These may be more neatly expressed in terms of the column matrices {A?} {Aa) and the square matrices {x} {a}. Then we have {E - na}{Aq} = {na>{~), - (44 {E - an){Aa) = {a>{€>, ( 5 4 where {E} is the diagonal unit matrix and { E } is E times the unit column matrix {I}.If the molecule is, as discussed above, too far from the metal surface for tunnel effect to be of importance, E is simply an integral multiple of the reciprocal of the number of atoms sharing the mobile orbitals (e.g. 6 in benzene). If however, the molecule is close to the metal surface but not too much distorted, the above calculation may give a good description of the charge distribution except in those atoms which are directly bound to the surface. In this case, E is to be determined by the consideration that the highest occupied level in the molecule must now coincide in energy with the highest occupied level in the metal. As the energy e0 of the highest filled level in the molecule is a function of the Coulomb integrals a,, Coulson has introduced the derivatives 3e0/3a, which we shall denote by the row matrix where ga = 34,,/3aa.There are two physically interesting possibilities. = (41, 5 2 - - * 6, * - * 5,) Then in first approximation we must have at equilibrium. It follows from (5a) that I - x = {(}{E - a X } - ~ { a } { c ) . - (7) (8) 1 - X and hence E = This in conjunction with (4a) gives the charge distribution in the adsorbed molecule. 11. In the above calculation we have treated the image field in a self-consistent manner ; in particular we have considered the electrons of the molecule as moving in the field of the electrical image of the average electron distribution rather than in the field of the image of the “in- stantaneous ” electron configuration (we are here concerned with the fictitious classical configuration used in setting up the Hamiltonian of the system).It appears, however, from experiments on the emission of electrons from metal surfaces that electrons of energy sufficient to escape from the metal move when near the surface in a field which is at least very closely approximated by the image field. This is the result of the correlation between the movements of the electron considered and those of the other electrons in the metal, which can be quantitatively expressed in terms of the density matrix of the metallic electrons.** 10 The correlation is less important the higher the kinetic energy of the electron. Crudely speaking, the conduction electrons of the metal no longer have time to adapt themselves to its motion.Thus the “in- stantaneous ” rather than the average electrical image will be of importance only for the molecular orbitals of high energy (and so of low average kinetic energy). These are precisely the orbitals which are responsible for chemical binding. However, the difference between this and a self- consistent treatment can only become significant when the molecule is dealt with by a more precise method than that of linear combination of atomic orbitals used in Wigner, ibid., 1934, 46, {&{E - anl-1{4{o* I. 9 Wigner and Seitz, Physic. Rev., 1934, 46, 509. l o Herring and Nichols, Rev. Mod. Physics, 1949, 21, 185. 1002.2 2 MOLECULE NEAR METAL SURFACE Bosworth l1 has attempted to treat the adsorption of a hydrogen atom to a metal surface by a variational method, regarding the image potential as a perturbation.He assumes a perturbed wave junction of the form and chooses the coefficients so as to minimize the energy of the system for a given distance R of the nucleus from the surface. The difficulty of the calculation is that the integral of the Hamiltonian diverges at the surface unless the image field is correctly cut off. Bosworth evades the difficulty by assuming the analytic form, - e2/4x, for the potential energy of the image force for positive and negative x , then integrating straight through the surface and taking the principal value of the resulting improper integral. In any case, a variation calculation which neglects exchange effects can in these circumstances claim little accuracy.A treatment of the exchange interaction between an atom and a metal surface by the Heitler-London method has been given by Pollard who obtains the energy of interaction in terms of a series of integrals involving the density matrix of the metal and atom. He imposes, however, a restriction on the wave function of the system which is only justified by the supposition that the electrons of the metal are in pairs of opposite spin with identical space wave-functions. By doing so, he denies the possibility of covalent bond formation, which is essentially associated with spin exchange, and in consequence his results are only valuable for the repulsive exchange force in physical adsorption, which was the problem in which he was interested.An immediate corollary of Pollard's work l a is that it is not possible by the Heitler-London method to predict important binding by exchange forces between an adsorbed radical and the free conduction electrons. It is necessary in this model that an electron of the metal should have a localized wave-function, more nearly approximating an atomic orbital than a Bloch wave, for formation of a covalent link. On the other hand, it is evident fIom consideration of the usual LCAO molecular-orbital treatment of the addition of an atom to a linear chain of similar atoms, each contributing one binding electron, that in this model it is just the lowering in energy of all the electron waves by the altered boundary con dition that gives rise to the binding of the new atom.A distinction between the two approximations in a given case may be made by magnetic determinations, as one would expect a stronger suppression of para- magnetism i f the Heitler-London picture is in better accord with the physical facts. In order to have some indication of the effects of a covalent link be- tween molecule and metal, we have used the LCAO molecular-orbital method to discuss the simplest model which can be expected to give relevant results. We consider two linear chains of identical atoms, coupled end to end, and examine the electron distribution in one of them when the number of atoms in the other tends to infinity. This constitutes a reduced model of a polyolefine free radical interacting end on with a metal surface.The calculation will merely be outlined here, details being given in Appendix B. For convenience, we consider the number of atoms in each chain to be even, zm in the first, zv in the second, and allow Y to tend to infinity. The secular equation is set up in the usual way, and with some manipula- tion reduces to the form (cos 8 + sin 8 cot zm8)(cos 8 + sin 8 cot zv8) = 0. Evidently this equation has a root 8, in each interval between successive singularities of cot zv8, and two roots in any such interval which also llBosworth, Proc. Roy. SOC., New South Wales, 1941, 74, 538. ?P = a1 $1' + a2 $28 + a3 * 2 m This device is more ingenious than valid. Pollard, Physic. Rev., 1939, 56, 324.K. HUANG AND G. WYLLIE 23 contains a zero of sin (2m + 1)8.The regularity of this distribution is sufficient to ensure that i f f(8) is a function analytic in the interval a < B < b , Examination of the coefficients in the orbitals of the whole system for varying values of 0, shows that we may regard the orbitals of the whole system, which are regularly spaced in 8, as divided into bands centred on the levels for the short chain by itself (corresponding to the zeros of sin (zm + 1)0), the electronic charge in an orbital diminishing sharply with increasing separation in energy from the centre of its band (for the short chain of 2m atoms). The electronic charge on a given atom due t o all the orbitals in the band corresponding to a single level of the short chain may then be evalu- ated by integration in the limiting case.After some approximation, we find for the number of electrons on the sth atom, counting from the free end of the chain, contributed by the Zth band :{I - (1 - P)-+ exp [- ir( - ) '1 SV'I - 8 2 sin2 [~ir/(2m + 111 2m + I 2nz + I $1 2m + I 241 ) I 1 9 2s (I - a2) sin [zZn/(zun + I)] x cos [-(zir - where and 8 is the ratio of the exchange integral between the two atoms at the coupled ends of the chains to the exchange integral between neighbouring atoms in either chain. The approximations used are valid for small 6 ; for sufficiently small 6 the above rather clumsy expiession is quite well approximated by = k2 - 2 cos2 [Z~/(2m + I)] + P cos [2Z~/(2m + I)], irS2 IT 2sIir 2m A{ + I I - exp [ - (2m + 6 4 sin4 (-)I zm + I cos 7-} 2w + I The corresponding expression for the free chain is I (I -cos-).2 S Z n 2.312 + I 'Lm +- I It thus appears that the effect of coupling to a long similar chain is to smooth out the charge variations in each orbital towards the coupled end of the short chain, the effect being most marked for the occupied levels of higher energy ( I approaching m). The importance of geometrical factors for this type of binding is indicated by the high power of 6 appear- ing in the exponential, 6 being itself a very steeply varying function of the atomic separation. A similar calculation carried through for the case that the two chains consist of different atomic species proved the inadequacy of the simple LCAO method. The difficulty is of the same type as arises in the dis- cussion of heterocyclic compounds.8 Unfortunately, it is much easier to obtain a reasonable approximation to the charge distribution, by methods of the sort discussed above, than to obtain a sufficient approximation to the energy of the system to be able to choose reliably between different, a prior2 reasonable, molecular configurations. In order to approach this latter problem, i t will be neces- sary to consider a specific system and use a much more empirical method.However, we may, from the point of view developed here, estimate what might be a profitable approach. The heterogeneous catalytic reactions which have been most ex- tensively studied under well-controlled conditions are the ortho-para- hydrogen conversion, hydrogen-deuterium exchange, and various hydro- genation reactions catalyzed by transition metals, especially of the nickel,24 MOLECULE NEAR METAL SURFACE tungsten, palladium, platinum family.These metals have high work functions which are nevertheless considerably lower than the first ionization potentials of the molecules concerned. It is therefore not possible to account €or their catalytic effectiveness in terms of simple electron transfer in either direction, which we have discussed in 9 I. They have unfilled d levels, which lend themselves, according to the Pauling picture of directed valence l3 to the formation of complex bond- systems. They are strongly paramagnetic, and the paramagnetism of the powdered metals is strongly influenced by the chemisorption of a catalytic pois0n.1~ A preliminary account of experimental work on the relation between catalytic properties and the electronic band structure of the catalyst is given by Couper and Eley.15 It thus appears probable that a representation of electronic conditions at the catalytic metal surface in terms of Pauling's resonating bond theory of the metal structure, with the formation of largely covalent bonds between definite metal atoms and the adsorbed reactants, should give a reasonable physical approximation to the real situation. The authors are indebted to Dr.D. D. Eley, Prof. C. A. Coulson and Prof. N. F. Mott for illuminating discussion of various aspects of the subject- matter of this note. One of us (G. W.) is indebted to the Anglc-Iranian Oil Co. Ltd., for a grant which made this work possible. Appendix A.-Calculation of {a) in a Concrete Case.As an example, we shall consider an ideal model most suited for our treat- ment. It consists of a linear molecule perpendicular to the metal surface. The interatomic distance and the distance of the first atom from the metal surface is assumed to be both equal to d. Metal The atomic orbitals are the usual p function which we assume to be given by the Slater type of function : Since the rigorous integration t o obtain a,, in the manner we have described above is not possible analytically for s i t , we shall suppose that the distri- bution I (b I2 which consists of the two well-known symmetric loops can be re- placed by two points at the charge-centres of the loops. The distance of the charge centre from the symmetry plane is given by * f (13) - I 5 1 _ - - 8 * a,,' This value leads immediately to the following expression for a,, Pauling, Proc.Roy. SOC. A , 1949, 196, 343. Dilke, Eley and Maxted, Nature, 1948, 161, 804. l5 Couper and Eley, ibid., 1949. 164, 578.K. HUANG AND G. WYLLIE 25 For s = t , we have the contribution from the image and the density itself : the former contribution can be obtained with the similar approximation as before For the contribution of the density itself the above approximation obviously breaks down. But we shall find the rigorous integratian can be carried through in this case. In accordance of the procedure we have described above (uJ2 is given by the following integral : The angular parts as they stand are not convenient for integration, but they may be readily converted to the desirable form.The integrand is seen only t o in- volve the relative angles between the fixed p-function axis and the two vectors Y and Y‘, and the integrations over the angles are just uniform integration over all directions of Y. Clearly we may replace them by integrating over all direc- tions of the p-function axis and that of Y‘, whereas regarding Y as fixed in direction. The integral becomes then where the 4 integration of Y is carried out. and the final result is All the successive integrations can be carried out by elementary means, 21 I0 (a8,) = -ape2 ; . (16) (15) and (16) combined give : Appendix B;-Consider a chain of 2m + 2~ similar atoms, of which the first 2m are evenly spaced, while the successive ZY are also evenly spaced with the same interval but the space between the 2mth and ( z n + 1)th atoms is different from the others.Let the wave function for a valence electron in a single atom separated from the rest be t,4 the corresponding energy E,, and the exchange integral between neighbouring atoms a t the normal separation be /3, while that between the 2mth and (2m + 1)th atoms is Sp. Then if the wave function of an electron in the chain be represented by a linear combination of the available wave functions of lowest energy of the different atoms Y, = + a&h2 . . ., and if we neglect all interactions except that of nearest neighbours, the energies E(= E , + e) of the states available for the electrons are given by the secular equation : = o26 MOLECULE NEAR METAL SURFACE where all the elements not on the three middle diagonal lines are zero.we substitute B = 2,3 cos 8, this equation reduces to If now sin ( 2 ~ + 1)s sin (2m + I)8 = 6 2 , sin ZYS sin 2mB which may be written in the form (cot 8 + sin 8 cot 2 m 8 ) ( ~ 0 ~ 8 + sin 8 cot 2r8) = S2. Evidently this equation has a root 8, in each interval between successive singular- ities of cot zr8, and two roots in those of these intervals which also contain a zero of sin (zm + 1)O. The regularity of this distribution is sufficient to ensure that, if f ( 0 ) is a function analytic in the interval a Q 8 < b Again, for a given value oi 0, the coefficients a, are determined by the set of linear equations : - 2 cos ea, + a2 = o a, - 2a2 cos B + a3 = o a,,,-, - 2uZm cos 8 + SU~,,,+~ = o 6a2,,, - 2 ~ ~ , , + ~ cos 8 + u ~ ~ + ~ = o Q2m+2r-1 - zazm+zr cos 6 = 0.Hence S=I J where A , = sin so, where s = I, 2, . . . zm. A,,,, = (6 sin 8)-I[sin (2m+1)8 sin no-82 sin 2m8 sin ( n - I ) B ] n = I , . . . ZY. The denominator in a, reduces to I sin 2m8 cos (2m + 1)s - S2 sin 2m8 sin (zm + 1 ) B If Y is large, this may be approximated by 4 + r/sin2 e p - 2 sin2 (2m + I)e - z cos 8 sin (zm + I ) e sin 2me + S2 sin2 zn~ejtt, where 4 is of the same order of magnitude as m. The factor to be multiplied by r/sin2 8 is a positive definite form, so for very large Y the denominator in a, tends to (r[a2 + sin2 (2m + I ) 8 sin-2 8(6-2 - 2 cos2 8 + P cos 28) + sin (2m + I ) 8 cos (zm + I ) 8 sin-2 8 sin 2 d ( 1 - S Z ) ] ) , . If, then, we suppose 8 to be small, we see that the coefficients of the atomic wave functions in the group I , .. . 2m will be small unless sin ( 2 m + I ) 8 is also small. Thus for the first zm atoms we may regard the almost uniformly spaced energy levels of the system as divided into bands centred on the levels for the 2112 chain by itself (the zeros of sin (2m + I ) 8 ) , with the degree of occupation of the levels in any band falling off with distance from the centre. Within a given band, then, the electronic charge on the sth atom (s < 2 m ) due to a given orbital may be approximated by sin2 s8 r(S2 + a, sin2 (2m + I ) 8 + u2 sin (2m + I ) 8 cos (zm + 1)6'K. HUANG AND G. WYLLIE 27 where a,, a, are constant, since the circular functions of 8, 28 vary only slowly with 8 in comparison with those of (2w + I)@. The total electronic charge on this atom due to orbitals within the band is then the sum of all the similar terms for which 8 lies between (I - 4) w and (I + Q)w, where w = w/(zm + I ) , I being some integer. Then, as observed above, if we make Y tend to infinity, this sum can be replaced by the integral, (2wz '+ 1)n .(::; 62 + a, sin2 8' + u2 sin e' cos 8'1 where we have replaced (2w + 1)8 by 8' in the above expression. If s is so small that sin2 [s8'/(2m + I)] may be taken as constant, the integral can be evaluated at once, giving sin2 sZw/(m + 4) in precise agreement with the expression for the charge on the sth atom due to the corresponding orbital in the isolated 2w chain. For larger s, however, the integral cannot be so simply evaluated. If we replace sin 8' by x and approximate t o the quadratic denominator by A eBa(s-a)* extending the range of integration in x from - io to + 00, and choosing the constants A , B, a to give the correct minimum value of the denominator a t the correct place, and the correct value of the integral for small s. we obtain for the charge on the sth atom due to the Zth band sin2 [s8'/(2m + I)] d8' x cos [-(zT 2 s - I - - sz sin A)]}, z m + I 241 2m + 1 where In 2zn $1 L- 6 - 2 - 2 cos2 - + 62 cos ~ 2113 + I 2m + I' El. H. W i l l s Physical Laboratory, Uqziversity of Bristol.