Transition metal bonding functions and their application in adsorptions and catalytic reactions

Journal of Molecular Catalysis, 64 (1991) 53-84

53

Transition metal bonding functions and their application in adsorptions and catalytic reactions Part I: Theoretical model K. H. Huang P.O. Box 276, Departmat of Chemistry and Institute of Physical Chemistry, Xiamen University, Xiamen 361005 (China) (Received January 22, 1990; accepted July 2, 1990)

Abstract A new method has been proposed for rapidly and chemically-intuitively giving correct information on the relative abilities or relative data (binding energy, bond structure, bond strength, vibration frequencies of surface species, etc.) of chemisorptions, dissociation and reactions on various transition metals, and the effects caused by the Metal-Promoter (or -Support) Interactions. In order to achieve this purpose, the LCAO method is ilrst used to derive the wave function of the mono-transition-metal atom, represented by the combination of molecular orbital (M.O.) bands, where each M.O. is described as the linear combination of s and d orbitals. Second, based on various electronic spectra before and after adsorption, we assume that the adsorption and reaction occur chiefly on the valence band around the Fermi level. The valence band consists of three M.O. Groups (MOGs): the vacant MOG at the bottom of the s band, the vacant or fractional halfoccupied MOG of the d band and the occupied MOG at the top of the occupied d band near the Fermi level, denoted as ry(Mi, Vs), ?P(Mi, Vd) and ?P(Mi, d&, respectively. Assuming that the M.O.‘s energy distribution is even, the concept of mean energy and mean chemisorption binding energy is employed, and the three MOGs can be simplified as three representative M.0.s. According to the principle of Bloch energy band formation, the concept of d-s band overlap and the probability theory, some simple formulas with the parameters of metal band structure such as the width of the d band, the atomic orbital effective exponents and the total electron number of the s and d orbitals have been derived to calculate coefficients of the s and d orbitals of the three representative M.0.s. The interaction between metal and adsorbate is characterized by bonding functions which depend on three factors: the overlap integral between the wave function of three representative M.0.s and the adsorbate, the thermodynamic potential and the ability of electron transfer. The bonding functions D, A, B and AI3 have been proposed to characterize the bonds involved in metal electron donation, metal electron acceptance, d electron back-donation and u-r coordination, respectively. Our model involves intuitive chemical localized bonds and represents the delocalized effects of energy bands. Its advantages (relatively realistic, intuition, rapidity, convenience and practice) and drawbacks (the inability to obtain the absolute amounts of surface bond strength, electron charge distribution and detailed energy levels) were discussed in comparison with the EHMO, CNDO, XbSW, LDF, GVB and EMT methods.

Introduction Transition metal-supported catalysts and transition metal catalysts have been widely used in industrial process and fundamental research, such as

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54

hydrogenation, dehydrogenation, hydrogenolysis, isomerization, aromatization, oxidation, ammonia synthesis, methanol synthesis, Fischer-Tropsch synthesis etc. With the advances of modern surface science and chemistry, many new experimental and theoretical methods are available to tackle this very important field. The effects of the d orbital or d band on the adsorption and catalytic properties of metals were recognized and emphasized early in the 1950s. For example, to investigate the correlation of percentage d-character (Pauling) and catalytic activity [ 1, 21; to characterize the parallel changes in d band vacancies of metal alloy and chemisorption and catalytic power [3], a simple idea was proposed, that the electron of the adsorbate entered ‘holes’ in the d band and paired with metal d electron to form predominantly adsorption covalent bond; however the wave function composition and energy levels involved in the d hole have not yet been determined. Deeper insight into the phenomena of chemisorption on transition metals has been approached stepwise using quantum-mechanical theory. As is well known, chemists are more accustomed to the language of the valence bond, and consider a chemisorption covalent bond is formed by overlap of one orbital of adsorbate with another orbital of metal atom to form a localized two-center covalent bond. However, in metals, energy bands (or molecular orbital bands) exist in which the valence states are quasi-continuous molecular orbitals (M.0.s) from the ground state up. Thus, how to represent interaction of the localized orbitals of adsorbate with the quasi-continuous M.0.s constitutes a difllcult problem. There are two basic quantum-mechanical computational models for approaching this problem [4]. The first basic model considers the metal as a slab of inflnite in the two directions of the surface plane. This allows band theory technique to be applied to the problem. It is called the extended surface model or the semi-infinite model. The Grimley ‘surface bond’ (tight bonding) method [5], Anderson-Newns Hamiltonian method [6] and its analogues [ 7-91, and the Shustorovich-Baetzold-Mutterties method for explaining the effect of the width of d band [lo], ascribe to the extended surface model. The dominant methodology for quantitatively implementing the extended surface model is the local density function (LDF) theory and its variants. The key point is how to describe the quantum-mechanical coupling between the adsorbate orbitals and the metal wave functions involved in the band structure. The second basic model method is called the local cluster model, in which there are a wider variety of currently employed methods, including EHMO, CNDO, X,-SW, LDF [ 111, ab initio method such as Hartree-Fock (HF) theory [ 12 1, Multi-configuration-Self-Consistent-Field (MC-SCF’) method such as the Generalized Valence Bond (GVB)-Perfect Pairing (PP) [ 13, 141 and GVB-Configuration Interaction (CI) [2, 15, 161, Effective-Medium-Theory (EMT) [ 17-l 91, and Bond-Order-Conservation Morse-Potential method (BOC-MP) [ZO-221. Every method has its advantages and disadvantages. For example, the ionization potentials of 5~, 1~ and 4a M.0.s of the free CO molecule have

55

been calculated by the HF, GVB-PP, GVB-CI methods [ 161 and the LDF method [ 231. Compared with experiment [4], the HF results provide a qualitatively incorrect description, because it shows that the CO 1~ level is higher than CO 50 for the gaseous CO molecule, which is not in agreement with experiment. The ionization spectrum from the LDF method is qualitatively similar to experiment, although quantitatively the agreement is very poor. However, GVB gives good agreement both qualitatively and quantitatively, especially with the GVB-CI method. The language used in GVB is not that of molecular orbitals but that of valence bonds. Messmer [4] indicated that using GVB language for describing interactions and transferability of orbitals (and concepts) from one system to other is economic, so it is particularly attractive. On the other hand, the EMT method has been successful in computing the hydrogen adsorption sites, bond lengths and vibration frequencies on some simple planes of Ni and W [24]; its results are also in good agreement with experiment. EMT assumes the embedding adsorbed atom in an inhomogeneous host, the primary effect of the inhomogeneous environment is included by replacing it with a homogeneous electron gas of a density equal to that of the host at the atom site. In terms of the selections of configurations and parameters, GVB and EMT methods can approach their results in good agreement with experiment, while with the LDF method, its results are usually not so good quantitatively. However, the LDF method can provide some very useful basic concepts or models for explaining the electronic spectra of metals or various adsorbate-derived electronic spectra. For example, from the Anderson-Newns model of chemisorption [ 6b, 111, the calculations of atomic adsorption on jellium [25, 261 and embedded cluster calculations [ 2 71, a virtual band of states or resonances are formed just at the upper part of the broad metal bands. These empty levels above Fermi level can be measured by Bremsstrahhmg Spectroscopy or Inverse UPS (I-UPS) [28]. Moreover, a broad peak in the range of O-3 eV below the Fermi level has been measured by a variety of electron emission spectra such as Field Emission Energy Distribution (FEED) spectroscopy, Ultraviolet Photoelectron Spectroscopy (UPS) and Surface Penning Ionization Electron Spectroscopy (SPIES) [28]. Gumhalter et al. [29] employed and extended the Anderson-Newns Hamiltonian model, arrived an tmifled interpretation of the electronic spectra (UPS, SPIES, XPS, EELS etc.) of CO adsorbate-derived resonances, noted as CO 2#-derived resonances (CO 2nX is equal to CO 277 noted in our paper, similarly hereinafter). His model includes both initial-state chemical effects (i.e. back-donation induced bonding) and final-state dynamic screening (relaxation) effects, i.e. these features are not only to the spectra of CO 2?r*-derived levels themselves, but also show up in the spectra of adsorbate core levels and core-to-valence transitions. Backgrmmd and purpose of our transition metal bonding function WLooTel Although considerable progress in chemisorption theory of transition metals by quantum-mechanical methods has been achieved in the past two

56

decades, as briefly reviewed in the above paragraph, their applications are still in the primary stage, and theory still lags behind experiment. It is necessary to develop the old methods and propose new effective methods to answer various questions concerning the nature of chemisorption, catalytic reactions and the effects of Metal-Support (or -Promoter) Interactions on transition metals. For example, because the hydrogen atom is the simplest adsorbate, almost every theoretical method has been used to characterize hydrogen adsorption on metals [6b, 7, 10, 11, 24, 30-341. It is obvious that the adsorption binding energy would intimately relate to its substrate structure. But the many theories described above never answer such an important question, that although the Group VIII metals exist in quite different electronic structures and band structures among themselves, why is the adsorption of hydrogen rather similar on Group VIII metals? as reviewed by Nieuwenhuys [ 351. The differences in initial heat (Q) and sticking probability (S’P) for H on the various single crystal surfaces of a Group VIII metal are larger than the differences in Q and SP among similar surfaces of the various Group VIII metals [35]. Our model (the detailed results will be presented in Part V of this series of papers) indicates it might be caused by direct participation of both vacant or half-occupied s and d metal orbitals around the Fermi level in H bonding. We will show that, since the orbital symmetry and energy level of the H atom can match with the s orbital of Group VIII metals, during metal changes from left to right or from bottom to top in this group, the bonding function between metal d orbital and Hls orbital will be decreased, but the bonding function between metal s orbital and Hls orbital will simultaneously be increased. In H adsorption, these two opposite effects approach the same order, so that the total bonding effect of H adsorption on various Group VIII metals is rather similar. The LDF method assumes that the hydrogen-metal bond is formed primarily with the d orbitals and that the s, p electrons serve primarily to renormalize the parameter so that the adatom-d orbital is most effective, and s,p electrons do not significantly contribute directly to the bonding [ 71. The self-consistent model of hydrogen by Newns [6b] considers the coupling between hydrogen orbital and the d orbitals of the nearest neighbor metal atoms, and does not consider the direct coupling between the metal s orbital and the Hls orbital. Similar treatments were undertaken by Varma and Wilson [ 71, Muscat and Newns [ 111, and Muscat [ 341. Muscat used a model depicting embedding a cluster of transition metal atoms (mullln tin) at the surface of an essentially jellylike medium [34]; the calculated results of hydrogen adsorption energies decrease in the order: Fe > Co > Ni, or Ru > Rh > Pd, these results are not in agreement with experiment, which were shown in Figs. 4 and 5 of [34], respectively. Similar incorrect results were reported in [7 1. On the other hand, using the GVB method to calculate the adsorption of hydrogen atom on a small cluster, the conclusion that direct bonding with s, p electrons of the metal atom is important in M-H bond formation was made 131 I. The EMT model considers that the d orbit& are of primary effect for H bonding, and s, p electrons provide a homogeneous electron gas density

57

effect on H bonding. Both GVB and EMF methods provide better calculated results than the LDF method. It must be noted, however, that the achievements of cluster models including GVB, EMT, EHMO, CNDO and XdSW methods depend largely on the selections of configurations and parameters by the Self-Consistent method or trial and error method. The arbitrariness of these selections may result in an incorrect conclusion. In short, the present situation of quantum-mechanical methods still cannot resolve the problem of the simplest adsorbate, H, and many adsorption spectra and catalytic behavior on various catalysts still have not yet been explained, which will be discussed here. All of these factors demonstrate the need to develop the old methods and propose some new ones. We attempt here to establish a transition metal bonding function model, which can rapidly give correct answers concerning the relative abilities or relative data (binding energy, bond structure, bond strength, vibration frequencies, etc.) of chemisorptions, dissociations and reactions on various transition metals and the effects caused by Metal-Promoter Interaction and Metal-Support Interaction. Our model uses a very simple computational method and some reliable parameters such as the width of the d band (w,), atomic orbital effective exponents, metal exit work function (v), and the total number of s and d electrons of the metal atom; these initial parameters are not altered arbitrarily during the entire process, thus avoiding the error or mistake of arbitrary parameter selection. Our model is based on many modern experimental facts, in particular on various electronic spectra including adsorbate-derived resonances [29]. On the other hand, we shall also adapt correct concepts or methods from all other theories as much as possible. The metal atom wave function of our model is derived from the concept of metal energy M.O. bands, so the electron-delocalized effect of the metal is represented in the metal atom function. However, if the metal atom wave functions bond with adsorbates or interact with the reaction intermediates, we will employ the localized valence bond language, which is a much more convenient and economical concept for treating the problems of chemisorption and reaction, and is more readily accepted by chemists. An outline of our model

In order to achieve our purpose as described above, we will first present the wave function of the mono-transition-metal atom as a combination of M.O. bands, where each M.O. is described as the linear combination of s orbital and d orbitals. Second, based on various electronic spectra, we assume that adsorption and reaction occur chiefly on the valence band around the Fermi level. The valence band consists of three M.O. Groups (MOGs): the vacant MOG at the bottom of the s band, the vacant or fractional halfoccupied MOG of the d band, which is located between the s-band and the occupied d-band, and the occupied MOG at the top of the occupied d-band near the Fermi level. The three MOG’s are denoted as ?@Mi, Vs), ?KMi, Vd) and ry(Mi, d,,,), respectively. The MOG of ‘ly(Mi, d,,,,) is equivalent to the MOG of the top of the metal broad band, where a broad peak below O-3

58

eV has been observed by FEED, UPS and SPIES [28] after adsorption. On the other hand, after adsorption, a broad peak in the range of 3-5 eV above the Fermi level has been measured by the Inverse Photo-Emission (IPE) spectra, which belong to the empty M.0.s of adsorbate-derived resonances located on MOGs of 1yrMi,Vs) and ?QvIi, Vd) [ 291. In terms of the concept proposed by Varma and Wilson [7] and the similar concept proposed by SBM [lo], each MOG can be simplified to one representative M.O., i.e. WMi, Vs), ?@Mi, Vd) and VMi, d,,,,) can be transformed into three corresponding representative M.0.s. The simplification is based on the following reasoning. We assume the energy level distributions and the chemisorption binding energy distributions are equal within each MOG, i.e. the separation of energy levels and the separation of the chemisorption binding energies between any two neighbor M.0.s are almost the same. In this condition, we use a representative M.O., whose energy level or whose chemisorption binding energy is equal to the mean energy or the mean chemisorption binding energy of the MOG, and then the total energy or the total chemisorption binding energy of the MOG is equal to the energy or the chemisorption binding energy of the representative M.O. (its value is the mean energy or the mean chemisorption binding energy of the MOG) multiplied by the total number of M.0.s in this MOG. In mathematics, it is equivalent to say that all M.0.s of the MOG are degenerated to one representative M.O., in other words, all M.0.s of the MOG have the same energy. We will employ this simplified assumption to convert formula (G5) to (G8) and (G9) (see below). After simplification, we obtained three representative M.O.s, each M.O. having its corresponding coefficients of s and d orbitals. The square of the s and d coefficients are equivalent to the density functions of the s and d orbitals, respectively. The next step is to use a completely new method to calculate these density functions. According to the principle of Bloch energy band formation, the concept of d-s band overlap and probability theory, some simple formulas (using the metal band structure parameters such as W,, the atomic orbital effective exponents and the total electron number of s and d orbit&) can be derived to calculate the coefficients of s and d orbitals for the three representative M.0.s. In brief, with three representative formulas and using some reliable and easily found parameters, we can easily calculate the wave function of the mono-metal atom. Our model belongs to the extended surface model, because we will use the metal band structure to derive the wave function of the mono-metal atom. However, where the wave functions of the metal atom interact with adsorbate or the reaction intermediates, we will use the cluster model and valence bond language. In catalytic reactions, we deal with the breaking and forming of bonds in molecules. The steps of adsorption, desorption, dissociation and surface reactions are considered to be localized over specified surface atoms. This model is intuitively reasonable and has been verified by many experimental methods. The localized model can be discussed in terms of u-r coordinative activation of u bonds (e.g. C-H, H-H, C-C, etc.) and

59

r bonds (e.g. C=O, C=C, N=N, N=O, etc.) originally proposed by Dewar [ 361 and Chatt and Duncanson [ 371, and in terms of the multi-coordination model of metal clusters to form polynuclear metal complexes reviewed by Mutterties [38, 391, e.g. the incline-mounting model of chemisorption of N2 on the six-metal-atom cluster of the o-Fe(l1 1) plane, w~-v~,v~(T~), proposed by Huang [ 401. The concepts of catalytic mechanisms of homogeneous coordination complexes [ 411 can be used to explain surface reactions. For example, the c&insertion of adsorbed a-olefins or M = CO into the metal-all@ (M-R) bond on coordination complexes is similar to the mechanisms of Fischer-Tropsch synthesis, in which =CH2 inserts into the M-R bond to form higher molecular weight alkyls, and through p-elimination, producing olefins, as proposed and reviewed by Bell [ 421.Based on the above reasoning, our model attempts to establish the metal bonding functions ‘A’ and ‘B’ for characterizing (+ and rr bonds, respectively. Recently, Ishi et al. [28] addressed the question of the inadequacy of Blyholder’s back-donation concept, because from recent measurements by ARUPS [44], I-UPS [45], HREELS [46] and NEXAFS [47], it seems more likely that the adsorbed CO 2# orbital is unoccupied in the neutral ground state in both strong and weak adsorption systems. Gumhalter et al. [29] proposed that the hybridization of the CO 271rorbital with the substrate s, p, d band produces the CO 2#-derived resonances; even if the center of CO 271E-derivedresonances lies above the Fermi level, its wing may still extend below Fermi level, allowing fractional charge transfer into the resonance (i.e. back-donation into the resonance). Moreover, the occupied and unoccupied parts of the resonance (which may be thought of as a fractionally occupied localized band of states) may be come a source of intra-resonance dynamic relaxation and polarization processes which are largely localized on the adsorbate. Based on the experimental facts and the valence band structure described above, we have revised the conventional d back-donation concept and extend the concept of CO 2r-derived resonances [29] to propose our new concept of d back-donation. The details wiII be reported in Part II of this series of papers. Subsequently, we wiII show that, with our new concept instead of the conventional concept proposed originalIy by Blyholder, the language of the valence bond and of conventional u-r coordination still can be employed. We propose that the interaction between metal and adsorbate is characterized by bonding functions which involve three factors: the overlap integral between the wave function of the valence band (or three representative M.0.s) and the adsorbate, the thermodynamic potential and the ability of electron transfer. The wave functions of adsorbate M.0.s are represented by their corresponding atomic orbital probe (see details below); this choice is to avoid the error that the M.O. calculation might affect the conclusion. Based on the model described above and using the language of the valence bond, bonding functions for D, A, B and AB wiIl be proposed to characterize the bonds involved in metal electron donation, metal electron acceptance, d electron back-donation and u-r coordination, respectively.

60

In the next section, we begin to outline of our model. We divide it into the following sections: (1) How to derive the wave function of mono-transitionmetal atom as the combination of the M.O. bands involving the s-band, d band and occupied d band, with each M.O. represented by the linear combination of metal s and d orbit&. After derivation, the new concept of atomic energy band layer structure will be used instead of the conventional concept of the metal atomic orbital layer structure; (2) How to characterize the valence band and express them as three representative M.O.s, each M.O. described as a linear combination of metal s and d orbitals; and how to use the metal band parameters (band structure) to estimate the coefficients of s and d orbitals of the corresponding M.O.; (3) To propose the concept of transition metal bonding functions and the reference atomic orbital probe in order to characterize various surface bonds; (4) To describe briefly the calculation method of our model; (5) To discuss the advantages and defects of our model. The wave function of mono-transition-metal band representation Representation LCAO method

of molecular

orbital

atom and its M.O.

bands of transition

metal

by the

We assume that a small transition metal particle with n atoms consists of n s A.0.s (atomic orbitals) and 5n d A.0.s in the outer shell. For a first approximation, we neglect the effect of higher vacant p A.0.s. According the principles of the M.O. method of LCAO, the interactions among these 6n A.0.s yield 6n M.0.s. The wave functions of the M.0.s are represented as PI, *z., “‘7 ?PEn,in order of decreasing energy level.

(2) (3)

61

lp4 =

. . . . . .

(4)

where c$(s,~), +(d,J represent the s A.O. and d A.O. of the ith metal atom, respectively; a(& shlk)represents the contribution coefficient of s A.O. of the kth atom for the ith M.O.; u(i, dhlyoj) represents the contribution coefficient of dj A.O. of the kth atom for the ith M.O. The A.O. Slater exponent (pi = Zi/n - 8) represents the effective potential energy of atomic nucleus of ith A.O. The p= and j.&dvalues of 3d, 4d, 5d transition metal atoms are shown in Fig. 1. The pd values of various transition metals periodically increase with the number of d electrons in the same row, i.e. the energy levels of d A.0.s decrease with the d electrons. This is almost parallel with the energy levels of the top of d bands on the metals, which decrease with the d electron numbers [7, lo] (see Fig. 2). Approximately, we assume, the 6n M.0.s are separated by three M.O. bands. The first band consists of PI, ?Pz, - * - to ?&, its average energy being proportional to the energy of the s A.O. of a free atom, i.e. (E, +E, + . . . +&)/ X=IC~E,~, where El, ES, . . . E,, are the energies of M.0.s PI, !Pz, - 9 . and Pz, respectively. ES0 is the energy of the s A.O. of a free atom. The second band consists of ?Pz+l, ?Pz+2, ..a to ?Pu;its average energy is proportional to the energy of the d A.O. of a free atom, i.e. (E,,, +Ez+2 + . . . +I$,)/ y -X =&Ed’, where Ed0 is the energy of d A.O. of a free atom. The third band consists of PY+1, 1yy+2, . - . 1y,,; its average energy is proportional to the sum of the energy of d A.O. of a free atom and electron-pairing energy, i.e. (E,,, +Eu+z + . . . + E&67% - y = k3(Edo+ Epo), where E,’ is the electronpairing energy. At absolute zero (0 K), the M.0.s of the second band are vacant or half-occupied, and the M.0.s of the third band are electron-occupied orbitals. Usually, the Fermi level is situated somewhere between the second and the third bands. For the M.0.s of the first band located at a field of lower potential (higher energy level), and the M.0.s of the second and third bands located at a field of higher potential (lower energy levels), the wave functions of M.0.s can‘be rewritten from (1)-(6n) as follows: First M.O. band, E, +E, f. . . +Ez=k,ESox ?P,,=YP~+P~+~~~+?Pz

(A)

62

1.6 -

2.4 -

2

t

I

d'

d’

d”

Fig. 1. Periodic changes transition metals.

dd

d”

in the effective

d6

atomic

d5

dn

exponents

of s and d orbitak

on various

(1)’

63

o-

‘\.zL, 0

-2-

Ce”t,e

\

o---cl -4-

‘L:;>:+‘ d

-6-

-8

\

\

Fermi

.

Level

----A o\,,~

E

T Ep CJ

d Bottom

\.\,_*

I I SC .I

I v

, Cr

ld.a/

I l Mn Fe co

d ED

I Ni

2. Periodic changes in the d band parameters for 3d metals (the 4d and 56 metals show analogous trends (the data for this graph were derived from the band cakulation by Watson [7] and illustrated by Shustorovich et al. [lo]).

Fig.

y3=

......

p*= ......

(3)’ (4)’

64

?P s+2=“‘“’

(x+2)’

?Px+3=““”

(x+3)’

ThirdM.0.

u(Y+l,

u(Y +

band,

E,+,+E,+2+...+E,=k,(6n_Z/)(E,'+E,')

~M2b#hM2,

h‘s)+

1, SMn)&Mm /-d +

$l&+l,

j$lu(Y

&2,j)+j’j(&2,

r&d)+

+ 1, &vu.04(&z, /-‘d

(Y+1Y

The total wave function of the crystal metal particle is the sum of (A), (B) and (C), i.e.:

65

Representation of the wave function of mono-transition-metal atom by MO. band-s The contribution of the ith metal atom (Mi) to the total wave function of the metal particle is !P&, i.e., the sum of all terms involving the Mi atom in formulas (1)’ to (6n)‘: TMi

=

(*phi

+

( ~JL&~

+

( ppd(occ))Mi

where

(~pJMi

=

-ia@

+

+

1,

j$l{a(~+

+a(x+2, (~,docc)h

= My

sMiMi) +

1,

a@

2,

sMMi) +

* * * +

a&,

hfili)hNsMi,

14)

&i,.fl

. * - +a(%

dhli,j)+

hi,

.Oht@Mi,

+ 1, &,

+ 46%

hi,

.7>+ a(Y + 2, hi, .i)Mj(d~i,

WI

14

+ 1, SMOf a@ + 2, SMJ + * * . + a@% &IWM~,

+ j$,lb(Y f...

+

1-4

31

(E3)

I&)

From (El), (E2), (E3), substituting into (E), we obtain:

+,=$+ila(k

dMi,&d!!(dm,pd) ,.&,M.O. band

1

kc~+la(k sMi)&(sMi,

f

6n +

k F+

Assuming

l&us>

5 1 jgla(k

&i,

.Mj’j(dm

pd)

pd(Occ)

M*“-

band

@‘I

where &(dhli, CL~) = c+&l(dMi,CL~)forj = 1 to 5, aI = 1

(Eb)

From (Ea), substituting into (E’), we obtain:

k-u+1 j-1

(E”)

PMi is defined as the wave function of mono-metal-atom Mi. This new mono-atomic function relates to the band structure of the metal. The square of the coefficient of the s or d A.O. is the s% or d% probability represented in the corresponding M.O. band. In the classical concept, the wave function of the free atom is considered as the linear combination of A.O.s, whereas in the new concept of the wave function of any mono-metal atom in this paper, it is considered as the linear combination of M.O. bands. The classical concept of the atomic shell layer structure is converted into a new concept of energy-band-layer structure here. According to the symmetry of lattice matrix, q&T + la+ 777&+nQ = !P&

(Et> where E, b, c are the vectors of lattice constants and 1, m, n, are artyintegers. The formul~(EJ indicates that the wave function ‘u,, of the Mi atom appears repeatedly on any lattice points with the same symmetry. In other words, the coefficients of the fOITrmla(E”) for ?& are the statistical vahes of interactions among all metal atoms, due to the delocalization of the energy band. The energy band calculation based on lattice symmetry is then possible for estimating the values of this coefficient. The adsorption and surface reaction chiefly proceed on the outer valence layer of the atom, if by the new concept deduced here, i.e. they proceed

67

on the valence band around the Fermi level of the mono-transition-metal atom function. Thus, in the following paragraph, we will explain how to represent the wave functions related to the valence band of mono-transitionmetal atom and how to estimate the coefficients of wave function by the metal band structure. The use of band structure to estimate the coefficients of s and d wave functions on the valence band around the Fermi level Correlation between metal energy band structure (Zq classical theory) and the MO. band structure (by LCAO theor&

band

Based on the classical metal band theory, pm can also be expressed by Bloch functions:

where U,Jr) is a function, generally depending on the wave vector k. -- k is periodic
the coencients

of s and d wave

Many facts indicate that the percentage functions of s and d A.0.s in the M.O. bands around the Fermi level are very important for the understanding of adsorption and surface reactions over transition metals. Because the result of band structure calculations is in better agreement with experiment, we

68

are interested in how to use the characteristic data of band structure, such as the d-band width and the relative potential levels of s and d bands, to estimate the coefficients of s and d functions in M.O. bands around the Fermi level. It is obvious that when the relative potential difference between the s band and the d band is held constant, i.e. pd- c~s does not change, the overlap of the d band by the s band would be increased by increasing the d band width (w,). On the other hand, if the W, is held constant, the overlap of the d band by the s band decreases with increasing /Jo- p,, that is when the average potential level of the d band is farther away from that of the s band, the overlap between them would be less. The overlap between the d band and s band around the Fermi level causes the percentage function of s orbital at the bottom of the s-band, P(s, Vs), to decrease and simultaneously the percentage function of the d orbital of the same band, P(d, Vs), is increased from zero to a certain value. Thus, we assume: Ws, Vs) = K&I - /0%

(Gl)

P(d, Vs) = 1 -P(s,

(G3)

Vs)

where K is a proportional constant and P(s, Vs) and P(d, Vs) are the s and d percentage functions at the bottom of the s band respectively. The atomic function @(n, 1, A) can be represented by the well-known Slater A.0.s of the following form: @Xn, 1, A)= rl, Jo, cp)N,, Irn-‘-S

exp(-&aH)

(G3)

where the A.O. exponent (EL)and the effective principle quantum number (n - 6) are specified by Slater for any A.O. of any atom by a simple Slater recipe [49]. If we use the concept of band overlap by the Bloch theorem to obtain the wave functions around the Fermi level, then the effective A.O. exponent would be represented by a single p value for a specified band. This is because each band has a characteristic p value exhibiting a characteristic effective potential field, and any atomic function of (G3) overlapping on these band would approximately present the same p value as that of the priority A.O., while the portion Yl, A(0, q)N,, lrn-l-* still retains its characteristic form. From the concept of Bloch band overlap, the mono-metal atom wave function corresponding to the vacant M.0.s at the bottom of the s-band, denoted by Vs band, can be represented by the s and d percentage functions P(s, Vs) and P(d, Vs): q(Mi, Vs) =P(s, Vs) @(SW, ~2 +P(d, Vs) Q(dMi, A)

(G4)

where the summation, defined as those contributing M.0.s at the bottom of the s-band, was overlapped with the ‘d band’. It is obvious that we consider s and d bands as the LCAO of s and d A.0.s separately, but the combination of s and d bands in (G4) is done by means of overlap functions. The approach is quite different from the LCAO theory in formula (E”), in which we consider s-s, d-d, and s-d interactions simultaneously.

69

Based on formula (E”), the part of the mono-transition-metal atom wave function corresponding to the vacant M.0.s at the bottom of the s-band around the Fermi level can be expressed as follows: ?P(Mi, Vs)(LCAO representation) =k ~_,,u(k, s&$(s~~, c$) + k=J-n’ 5 &@k j=l

dMi,j)&(dMi, ICL~)

(G5)

where n’ is the number of vacant M.0.s at the bottom of the s band. According to LCAO theory, the sum of the square of the corresponding coefficients of s and d orbit& in (G5) are the probabilities for Snding s orbital and d orbitals at the bottom of the s band, denoted as (s%)“, and (d%)vS, respectively, i.e.: 2 (s%hs=

i 4% k = .T - n’

%d

I

c-331

I 2

5 (d%>vs

=

,_$_,,

jFl

crja(k,

hi,

.d

(G7)

We would emphasis that the base wave functions of LCAO (G5) are +(s& CL,)and &(dMi, CLd),which are the A.0.s of a free transition-metal atom; while the base wave functions of (G4) (the band-overlap representation) are @(s,~, pS) and @(dMi, pu,),which are not the A.0.s of a free atom. However, according to the principle of quantum chemistry, it is possible to find two equivalent representations by selecting two series of different base wave functions. We will define the (G4) representation as an approximate result of (G5) LCAO representation, if two approximation conditions are satisfied. The first approximation condition is that the energies of all n’ M.0.s at the bottom of the s-band are assumed to be the same; the second approximation condition is that the effective electronic field using the concept of Bloch band-overlap can be used instead of the electron correlation term l&, .i.e. @(dwi, p,) can be used instead of &(dMi, &. This means that when the d orbital overlaps on the bottom of the s band, the electronic interactions between the d orbital and all A.0.s l/rti term, can be approximately neglected if we use the effective potential field of the s band, EL,,considered as the effective atomic exponent of the d orbital, instead of ,.&dapplied in a free atom. The condition that the energies of all n’ M.0.s have the same value results in the corresponding coefficients of n’ M.0.s in (G5) being equal with each other, thus: (G8)

W-0 substituting (G8), (G9) into (G5), and considering the second approximation condition, i.e. 4(dMi, pd) was changed to @(dMi, &, we obtain: 1I’(Mi, Vs)=n’o(z,

sMi>@ff%i,

IIs)+5n’a(~,

hi,

I>@C&,

I&J

Wa)

On the other hand, (G6) and (G7) are transformed to (G6a) and (G7a), respectively.

From (G6a) and (G7a), we can further denote the coefficients of (G5) as the s and d percentage functions: a(*, s& = I(S%)vs11+2’l-l’2

(G6b)

C&(2?, dMi,I?)= l(d%~~1’/215’?X’l-1n

(G7b)

Substituting (G6b) and (G7b) into (G5a), ly(Mi, Vs) =In’l”21(s%)vs11n~SMi, C(S)+ 1~‘I’/21(d%XI,11nRd~i, c$)

(G5b)

Let P(s, Vs) = jn’11’21(s%)v,1”2

(G6c)

P(d, Vs) = 15n’1”21(d%h$‘2

(G7c) and substitute (G6c) and (G7c) into (G5b); then the wave function can be represented as the s and d percentage functions as follows: WMi, Vs) =P(s, Vs)@@~s, /-k) +P(d, Vs)qdMi, FLS)

(G5c) where (G5c) equals (G4). Thus the result of the LCAO method is in agreement with that of the Bloch band-overlap method under the above approximations. By using the band overlap concept and the similar approximation methods as described above, the M.0.s of the d-band around the Fermi level of metal atom can be written as follows: (1) For vacant or half-occupied M.0.s of the d band around the Fermi level, defined as the Vd band, WMi, Vd) =JYst Vs)Rk,

k) +J’(d, Vd)QXIdMi,/&

(HI)

where P(s, Vd) and P(d, Vd) are the s and d percentage functions at the Vd band, respectively. (2) For occupied M.0.s of the d band around the Fermi level, defined as: ly(Mi, d,,,) =P(s, d&@@,i,

PJ +J’(d> &,,)@(dMi, k)

038)

71

where P(s, d& and P(d, d,,,,) are the percentage functions of s and d orbitals, respectively, at the d,,, band. If denoted by the similar form of (G6c) and (G~c), we find: P(s, Vd) = In#nl(s%)vdlln

(H1a)

P(d, Vd) = (5n#2((d%)v$n

(H1b)

where ?zd is the number of the vacant and half-occupied M.0.s of d band around the Fermi level. p(s, do,,) =

l~d,,~“2)(s%)&,11/2

WW

J’(d,

15~,,c11”l@W~c11~

W2bl

do,,)

=

where nd,, is the number of the occupied M.0.s of the d band around the Fermi level. Let the sum of s and d percentage functions on the Vd band and d,,, band equal 1, i.e.: J’(s, Vd) +P(d, Vd) +p(s, a,,) +P(d, d.xc) = 1

(K1)

The values of the above percentage functions can be estimated by the following probability analysis. Because the M.0.s of the d band around the Fermi level are near the M.0.s of the bottom side of the s-band, the probability of presenting the s and d A.0.s on the top of the d-band is proportional to 1 -P(s, Vs) and 5 -P(d, Vs), respectively. This is caused by the overlap between the s-band and the d-band. On the other hand, the occupied and unoccupied states of the M.0.s on the top of the d band around the Fermi level are proportional to n,+ d and (12 -ns+d), reSpeCtiVdy, where ns+d is the total electron number of s and d A.O.s, with a maximum number of 12. Based on the above analysis, we obtain: P(S,

Vd)=K’(l

-P(S,

P(d, Vd)=K’(5-P(d, p(s, do,,) = K’(l -p(s, P(d, d,,J = K’(5 -P(d,

Vs))(12-%+,)

(K2)

Vs)(12-Y&+,)

(K3)

Vs)>%+d vs>>%+d

(K4) o(5)

On combining the equations (Kl), (K2), (K3), (K4), (K5), (Gl) and (G2), and solving, we obtain: K’ = l/60

cK6)

The valence band consists of the vacant valence band and the occupied valence band. The vacant valence band is defined as involving the vacant and the half-vacant M.0.s around the Fermi level, whose wave functions are sum of the P(Mi, Vs) shown in eqn. (G4) and the !P(M& Vd) shown in eqn. (Hl). The occupied valence band is defined as ?P(Mi, d&, shown in eqn. (H2). The percentage function of the s A.0.s of the vacant band is the sum of P(s, Vs) and P(s, Vd). According to (Gl), (K3) and (K6), we obtain:

72

P(s, Vs) +P(s, Vd) =K(/.L~- /-@I$

+ l/60(1 -P(s,

Vs))(12 -%+d)

(K7)

The proportional constant K can be obtained from Ni, by substituting the known values of W,= 4.64 eV [7], P(s, Vs) +P(s, Vd) =0.60 calculated by the band structure [18, 191, ,.&d-,.&=1.42 and ?&+d=lO, we obtain: K= 1.9578 The calculation

results

of s and d percentage

function

and discussions

Using the values of K and K’, and the W, data of various metals [7], according to eqns. (Gl), (G2), (K2)-(K5), the distribution of percentage function of s and d orbital on various valence M.O. bands around the Fermi level can be calculated for 3d, 4d, 5d transition metals. The results are listed in Table la, b and c, respectively. On comparing the results of Table la, b and c with those obtained by other methods, we can see the above model is successful and possesses some advantages. For example: (1) our results show that Ni has the greatest number of s-electrons per atom in the valence band, approximately 50% and 140% more than that for Pd and Pt, respectively. This agrees with energy band calculations and experimental results [50, 511; (2) the data of Fig. 2 were derived from the band calculations by Watson cited in [ 71, and were given in [ lo], which shows the band structure of 3d transition metals. The Fermi level is found near the bottom of the d-band for SC 3d’, then gradually changes location to near the top of the d-band across the 3d periodic row, it thus parallels the decrease of the d percentage function across the 3d periodic row, as shown in Table la; (3) in our model, the effects of both d and s in the valence band have been accounted for, so it can be used to characterize the adsorption and surface reactions on transition metals more perfectly than that by other models, which consider only the d band. We will discuss this aspect in more details in subsequent papers; (4) our model is based on the parameter from band structure calculations which has been proven with good theoretical and experimental agreement. In the process of calculation, we do not adjust or select any parameter for fitting the experimental data, thus avoiding the problem of arbitrariness of parameter selection in certain ranges which might be incurred with EHMO, CNDO methods; (5) using the concept of the Bloch theory in our model, the electron delocalized effect can be more perfectly characterized than by atomic cluster models, such as the EHMO, CNDO and X&?W methods. The band structure calculation shows that the delocalized effect causes some of the d energy levels to increase by several eV, but the cluster model usually cannot represent such a big effect. For example, the ionization potentials calculated by the X~LSWmethod are approximately 2 eV greater than the corresponding average bulk work function for 13-atom cube-octahedral clusters [52]. Poorer results

73 TABLE 1 (a) Percentage the 3d period Metal PL, Fd Pd-PLs

Wd(ev) PCS,vs> P(d, Vs> PCs,Vd) P(d, Vd> PCs, dew) PC& do,,)

@)

functions

of s A.O. and d A.O. in valence

M.0.s

around

the Fermi level on

SC

Ti

V

Cr

Mn

Fe

co

Ni

0.8108 1.0000 0.1892 5.61 0.0660 0.9340 0.1401 0.6099 0.0467 0.2033

0.8514 1.2167 0.3653 6.55 0.1092 0.8908 0.1188 0.5479 0.0594 0.2739

0.8919 1.4333 0.5414 7.45 0.1423 0.8577 0.1001 0.4832 0.0715 0.3452

0.7973 1.5333 0.7360 7.71 0.1869 0.8131 0.0813 0.4187 0.0913 0.4187

0.9730 1.8667 0.8937 6.42 0.2725 0.7275 0.0606 0.3560 0.0849 0.4985

1.0135 2.0833 1.0698 5.79 0.3617 0.6383 0.0425 0.2908 0.0851 0.5816

1.0541 2.3000 1.459 5.32 0.4585 0.5415 0.0271 0.2229 0.0812 0.6688

1.0946 2.5167 1.4221 4.64 0.6000 0.4000 0.0133 0.1533 0.0667 0.7667

Percentage

functions

of s A.O. and d A.O. in valence

M.0.s

around the Fermi level on

the 4d period Metal PS l-h Al-Ps

wd(ev) PCS,vs> P(d, Vs) PCs,Vd) PC4 Vd) PC%&cc,,) P(d, &cc)

Y

Zr

Nb

MO

Tc

Ru

Rh

Pd

0.7500 0.8108 0.0608 6.83 0.0174 0.9826 0.1474 0.6026 0.0491 0.2009

0.7875 0.9865 0.1990 8.45 0.0461 0.9539 0.1272 0.5395 0.0636 0.2697

0.7000 1.0676 0.3676 9.48 0.0731 0.9269 0.1081 0.4725 0.0772 0.3394

0.7375 1.2432 0.5057 10.15 0.0975 0.9025 0.0902 0.4098 0.0902 0.4098

0.9000 1.5135 0.6135 9.76 0.1231 0.8769 0.0731 0.3436 0.1032 0.4810

0.8125 1.5946 0.7821 9.04 0.1694 0.8306 0.0554 0.2780 0.1107 0.5559

0.8500 1.7703 0.9203 7.60 0.2371 0.7629 0.0381 0.2119 0.1144 0.6356

0.8875 1.8514 0.9639 5.86 0.3220 0.6780 0.0226 0.1441 0.1130 0.7203

in valence

M.0.s

(c) Percentage period Metal PCLO

Fd pd-,h Wd(eV)

m, vs> P(d, Vs> PCs,W PC& Vd) PCs, &cc) PC& &cc)

functions

of s and d A.0.s

around the Fermi level on the 5d

La

Hf

Ta

W

Re

OS

Ir

Pt

0.7143 0.7500 0.0357 8.32 0.0084 0.9916 0.1487 0.6013 0.0496 0.2004

0.7500 0.9125 0.1625 8.60 0.0370 0.9630 0.1284 0.5383 0.0642 0.2691

0.7857 1.0750 0.2893 11.42 0.0496 0.9504 0.1109 0.4724 0.0792 0.3375

0.8214 1.2375 0.4160 11.95 0.0682 0.9318 0.0932 0.4068 0.0932 0.4068

0.8571 1.4000 0.5429 12.02 0.0884 0.9116 0.0759 0.3407 0.1064 0.4770

0.8929 1.5625 0.6696 11.41 0.1149 0.8851 0.0590 0.2743 0.1180 0.5487

0.9286 1.7250 0.7964 10.07 0.1548 0.8452 0.0423 0.2077 0.1268 0.6232

0.8452 1.8000 0.9548 8.17 0.2288 0.7712 0.0257 0.1410 0.1285 0.7048

74

than this may occur on using the EHMO and CNDO methods. The double orbital Slater exponents were used in the EHMO method for calculation of some Ti, Cr, Ni, Fe metal clusters [53], but these fits were made only with an overlap criterion, and did not give further explanation. In our opinion, this is because the single Slater exponent cannot characterize the large delocalized effect for forming the band structure. By using two effective Slater experiments for characterizing the s and d bands, its result would be better. Because our models based on the result of the band structure calculation and the double exponents were used for the s and d bands, the delocalized effect can then be more accurately characterized. The bonding

functions

The wave function

of surface

of the valence

metal atom band of sueace

mommetal

atoms

In the above paragraphs, the formula of !P(IvH)is the wave function of any mono-metal-atom. Because the valence of the surface metal atom is unsaturated, the wave function of the surface atom for adsorption and surface reaction can be represented as a portion of ?P(Mi), i.e. ?P(Mi) (surface) = cr?#(Mi), where the proportional constant (Ydepends on the exposed state of the surface atom. The wave function of the vacant valence band of any mono-metal atom is represented by eqns. (G4) and (Hl), i.e.: ?P(Mi, V)= ?P(Mi, Vs)+ !P(Mi, Vd)=P(s,

Vs>@(%i, &+P(d,

+P(s, Vd)@(sMi, cud)+P(d, Vd)Rd,i,

k)

Vs)@(&i,

k) (H3)

and the wave function of the occupied valence d-band near the Fermi level is represented by eqn. (H2), i.e.: ly(Mi, 4,,) =P(s, &cc)@&,

k) +P(d> &,,)@(d,i,

k)

(H2)

By the same reasoning, the wave functions of the surface atom for adsorption and the surface reactions, are (H3) and (H2) multiplied by the proportional constant (Y. In actual application, we use the model only for comparing the properties under the same surface exposed state, i.e. under equal cyvalues. Bonding types caused by the valence band interacting with the adsorbate The adsorption and surface reactions occurring between the surface

atom and adsorbate (or intermediate) can be considered as the wave functions of (H3) and (H2) interacting with the adsorbate (or intermediate) orbitals. According to the different situations of electron transfer, there are several types of bonding which might occur: (1) If the valence band M.0.s interact with a vacant orbital of adsorbate to form a new M.O., and two electrons of metal donate to the new M.O. for bonding. This is called metal-donating bond, denoted by D, as shown in Fig. 3a.

75 Metal

(Mi)

surface

Adsorbate

bond (A)

Fig. 3. Various bonds formed by interactions between the valence band and adsorbate.

(2) If the valence band M.0.s of metal interact with an A.O. or M.O. of adsorbate to form a new M.O., each side furnishes one electron for bonding. When the Fermi level of the metal is higher than the energy level of the adsorbate, the electron transfer direction is from metal to adsorbate, and the bond is called a metal-donating radical bond, denoted by DR, as shown in Fig. 3b. (3) For the same conditions as in Da, but with the Fermi level of the metal lower than the energy level of adsorbate, the electron transfer is from adsorbate to metal, and the bond is called a metal-accepting radical bond, denoted by AR, as shown in Fig. 3c. (4) If a vacant valence band M.O. reacts with an occupied orbital to form a new M.O., and the bonding electrons were furnished by adsorbate, the bond is called a metal-accepting bond, denoted by A, as shown in Fig. 3d. (5) If an electron-occupied M.O. of the d band interacts with a vacant orbital (ti or 7jL) of adsorbate derived to form a new M.O., and the occupied orbital donates electrons for bonding, it is called a metal back-donating bond, denoted by B, as shown in Fig. 3e.

76

me relative bonds

criteria

for

the feasibility

of f-&g

various

surface

Our model in this paper is to establish the relative criteria for comparing the properties of adsorption and surface reaction among various metal catalysts at equal conditions. Three factors have been proposed as the relative criteria for the feasibility of forming the adsorption bonds and surface bond types described above. The first factor is the overlap integral between the wave function of surface atom and the wave function of the adsorbate. According to the type of bond forming, we calculate the overlap integral between the appropriate wave function of the metal surface atom and the corresponding wave function of adsorbate; the ability to form the corresponding bond is proportional to its overlap integral. The second is a thermodynamic factor for forming the required bond. Forming bonds of D, DR and B is more feasible thermodynamically when the average energy levels of the valence band are higher than the level of adsorbate, i.e. the abilities to form these bonds are inversely proportional to the average potential energies of the s band and d band, I_LS and pd7 respectively. However for A and AR bonds, the inverse case would occur, i.e. the thermodynamic feasibility of forming A and AR bonds is proportional to pS and ,_&d for the s band and d band respectively. The third factor is an electronic factor. D and DR bonds require electron transfer from the valence band of the metal to the adsorbate, i.e. the abilities to form D and Da bonds are inversely proportional to the exit work function U. For the d back-donation bond, B, the bonding function is proportional to the d band width wd. The bonds A and AR require the metal to accept electron from adsorbate, thus the abilities to form A and AR bonds are proportional to the exit work function U. According to the rules of probability [54], let the event El occur ml times out of n, alternatives, and let Ez occur m2 times out of a second distinct set of n2 alternatives. Evidently their probabilities are P1 =m,/n, and P2 =m2/n2, for El and E2 events respectively. If one trial is made out of n, and one trial out of the independent set n2, we can seek the probability that the combined event El and E2 occurs. The total number of ways in which selection may be made from the two groups regarded as successful is m1m2. Hence, the probability is: P = m,m,ln,n,

= PIP2

&I>

The three factors for the relative criteria for forming surface bonds can be considered as three distinct and approximately independent events. The wave overlap integral depends on the symmetry matching and the energy level matching between the wave functions of the metal valence band and that of the adsorbate; greater overlap integral would be obtained as their energy levels approach each other. The situation is quite different wlth the second factor - the thermodynamic energy stability, which requires higher energy levels of the metal valence band for forming D, DR and B bonds and

77

lower ones for forming A and AR bonds. In EHMO calculation, the nondiagonal terms of the secular equation, det(H-ES) = 0, were computed from the Woslberg-Helmhotz equation: H, = 3KS,(Hii + Hti)

W)

Using our method, the forming bond strength is proportional to %, which combines the two factors (events) of S, and )(H, +H,), where S, is the overlap integral, the first independent factor, identical to that we proposed, and the second independent factor is the thermodynamic factor, i.e. the stabilizing energy factor, which is considered as the average of Hii and Hti Because we don’t wish to resolve the secular equation directly, the semiempirical formula (L2) is not used; instead, to emphasize the nature of the forming bonds, we establish the relative thermodynamic energy criterion for a specilled adsorbate as described above, where the Hii of adsorbate is kept constant. On the other hand, the special feature for metal catalysts compared with non-metal catalysts is to exhibit the high electron delocalized effect, thus the electron state of the forming bond depends not only on the surface atom but also on the electronic state of the metal bulk, i.e. the work function U. It can also be considered as a distinct and approximately independent event. The above three factors are simultaneously necessary for forming the surface bonds. Therefore we represent the total result as the product of the three factors, based on the reasoning from eqn. (Ll). Bonding functions fcyr characterizing various surface bonds Based on the above analysis, the bonding functions for various metal surface bonds are proposed as follows: (1) Metal donation bonding functions of D and DR D =K,(U,/u){(l/&P(s, +(lIc~Z(s,

Vs)S[@&,

Vd)S(@&,

A), Xl + (l//-@(d,

/-& X)+(ll/-G?d,

vs)S(@(d~~ /-& X)

Vd)S(@(dM,, /-&, X)1

(L3)

where S(Q>,X) is the overlap integral between @ and the orbital of adsorbate X, KD is proportional constant, and for non-dimensional purposes, U has been divided by V, (the work function of Pt). The D, bonding function has the same form as eqn. (L3), but with a different proportional constant of Kn,. (2) Metal acceptor bonding functions of A and AR A= KA(~&){&Y~,

Vs)S((@&,

+ w&s, Vd)S(Rs,i,

/J.& X) + E*SP(d,Vs)S(Q(&i,

cud),X) + pZ(d,

Vd)S(@(d,i,

cud),X)1

A), X) (L4)

The AR bonding function has the same form as eqn. (L4), but with a different proportional constant of KAR. (3) Metal d-back-donation bonding function B = KG%/WPt)(l/&S(Rd~i,

pd),

WY4

L)

(L5)

Here for the non-dimensional formula IV, is divided by IV, (the d band width of platinum), and PH = 1 is not denoted in these equations. (4) Metal U-T bonding function AB If the surface bond formed is a u-n- bond, it can be considered as two independent events occurring simultaneously; one is the acceptance by the vacant orbital of the metal valence band of two electrons from the adsorbate to form a u bond, i.e. to form a metal-acceptance bond A; another is the back-donation of d occupied electrons of the d-occupied band to the antibonding orbital (8 or 7jr) to form a r bond, i.e. to form a metal d-backdonation bond B. According to eqn. (Ll), the bonding function of the u-r bond is the A bonding function multiplied by the B bonding function.

Relation between the mono-surface-metal atom model and multisurface atoms model. The reference atomic orbital (A.O. “probe”) How to use the mono-su@xce-metal atom surface atoms interacting with adsorbate

model

to characterize

multi-

In spite of great efforts in M.O. calculation of cluster models relating to the adsorption and surface reaction on metal clusters, difllculties are still present, such as: (1) the absence of information regarding the detailed stereochemical location of adsorbate on the surface; (2) the arbitrariness of parameter selection for fitting the experimental result; (3) some problems about the approximate formula for calculating H, since the metal system, the non-metal system and the bonds of different nature all use the same equation. Inverse conclusions might be made if the parameters or adsorption structure models were changed. The wave function of surface mono-metal atom (notice: it is not single) given by eqns. (H3) and (H2) characterizes the electron configuration of the surface metal atom, including the effects from all other metal atoms, by using the band structure model. When the adsorption and surface reaction occur on multi-metal atoms, we can consider the event as several bonds simultaneously forming on the multi-metal atoms, any bond with any corresponding atom acting as an independent event with its corresponding bonding function. According to probability analysis, we can use the product of all bonding functions to characterize the relative abilities of adsorption and reaction of the multi-metal-atoms. Our model can be applied when information regarding the detailed stereochemical location of adsorbate on the surface is absent. For comparing the behaviour of any two metal catalysts, the errors due to arbitrary selection of atom-atom distances in the calculations of overlap integrals are almost in equal order for the same kind of surface bond. Our model appears relatively correct, because our results are compared under the same conditions, such as at the same kind of bond nature, the same metal crystal surface and its exposed state, the same mechanisms of adsorption and reaction, and the same reaction conditions, etc. This is quite different from the usual EHMO,

79

CNDO models, in which the comparison of results is made by comparison of the absolute amounts. For example, the absorption energy is calculated by the net change of total energy after adsorption compared with the initial total energy of metal cluster + unadsorbed adsorbate, in which the calculation errors due to arbitrariness of the selection of atom-atom distances might be greater than the adsorption energy, therefore the calculated results cannot supply the correct answer, even of relative correctness. Whereas in our model, the bonding functions are represented as a product equation, in which the errors caused by parameter selections do not influence the relative order of correctness, within a certain wide range, in the usual cases. Reference A.0. (A.O. probe) As described in the introduction, the existence of many experimental results and theoretical concepts enable the discussion of bond cleavage, bond formation and the activation of (Tand r-bonds during adsorption and surface reaction. Based on known experimental data and the known structure of adsorbates for characterizing and comparing the bonding abilities of various transition metal catalysts, we can select some reference A.O.s, whose symmetry of wave function is similar with the M.O. of the adsorbate which participates in the bond formation, and whose energy is of the same order as the participating M.O. of the adsorbate. Then the overlap integral between metal and reference A.O. (denoted X in the above paragraph) approximately represent the overlap integral of the surface bond. For the same reasons as described above, this should not influence the relative order of bonding functions, which we will show in subsequent papers by detailed calculations.

Calculation

method

The Mulliken method [ 491 was used for the calculation of overlap integral S for a pair of overlapping A.0.s X, and X,,: SK,

Xb, P, 0 =/X,X,

dv

WI

P = *G-b + dRIaH

CL71

t = (PL3- l-bJ&-& + PLJ

WI

where R is the internuclear distance between a and b. The Slater A.0.s are used in this paper. Some master formulas for integrals of the d Slater A.O. have been derived using Mulliken’s method. The data of metallic radii and exit work function are listed in Table 2, while the data of j.& p,, and w, are shown in Tables la, b and c for 3d, 4d and 5d metals, respectively. The general procedure for the calculation of metal bonding functions is: 1. List the parameters: j_~,and pd (A.O. effective exponents for metal s and d orbitals and for the reference A.O. probe of adsorbate M.O.), RM

80 TABLE

2

Metallic radii and exit work functions SC

R (A) U (ev)

R

@I

U (eV)

R

(&

U (ev)

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Zn

1.606 1.448 1.311 1.249 1.334 1.241 1.253 1.246 1.275 1.330 4.1 4.5 5.0 5.15 4.65 4.33 3.5 4.33 4.3 4.5 Y

Zr

Nb

MO

Tc

Ru

Rh

Pd

Ag

Cd

1.776 3.10

1.590 4.05

1.429 4.30

1.363 4.60

1.352 4.61

1.325 4.71

1.345 4.98

1.376 5.12

1.440 4.26

1.485 4.22

La

Hf

Ta

W

Re

OS

Ir

Pt

Au

Hg

1.873 3.5

1.564 3.9

1.430 4.25

1.371 4.55

1.371 4.96

1.338 4.83

1.357 5.27

1.373 5.65

1.440 5.1

1.500 4.49

(radius of metal atom), R, (radius of adsorbate atom), W, (width of d band), U (exit work function), ns+d (the total number of s and d electrons of metal atom). 2. Calculate the various overlap integrals between metal A.O. and the reference A.O. probe. 3. Calculate the coefficients of s and d orbitals of the three representative M.0.s for the valence band, by the equations (Gl), (G2), (K2)-(K5). 4. Calculate the metal bonding functions of D, A, B and AB, by the eqns. (L3), (L4), (L5) and (L4)x(L5), respectively. The calculation of binding functions are conducted by a compiled program which is simply and rapidly compared with EHMO, CNDO, X&W and other quantum-mechanical computational methods. In general, the calculation of all metal bonding functions for a metal-adsorbate interaction takes only several minutes.

The advantages model

and drawbacks

of the metal bonding

function

The advantages of our model are: (1) ‘Relative Reality’. It should be emphasized that the metal bonding

functions are proposed to describe the relative bonding abilities of metals to donate, accept and back-donate electrons to the standard A.O. probe. It can be used in a manner similar to that of the electronegativity of elements, and the standard electrode potentials can be used to treat chemical reactions. The process of cognition is not one step; one useful way is by recognizing the relative reality to approach the absolute reality. All current quantummechanical methods characterizing the nature of adsorption are approximate methods, in which many assumptions and simplifications are employed; our

81

method is no exception. Although many methods make use of advanced mathematical methods, some important factors still cannot be included in their treatment, and assumptions or simplifications are made which are far from reality. For example, the LDF method neglects that metals s orbitals directly contribute to M-H bonding, which we have discussed above; LDF also assumes that adsorption proceeds on all M.0.s of the d band, which is convenient for using the band theoretical technique, but in fact adsorption chiefly proceeds on the valence band, as verified by various electronic spectra. Many methods such as EHMO, CNDO, Xc+SW etc., are satisfied by using the self-consistent method, believing this is based on the main principle of quantum-mechanical methods, and pay less attention of the arbitrariness of parameter and configuration selections, which may provide a negative result of reality. Our model attempts to preserve reality as much as possible. For example, we take the reality of the metal structure-band structure as important background data for our model, employing the reliable parameters of band structure such as W,, U, ns+d etc. and use as a basis various electronic spectra of the metal and the changes in these spectra due to adsorption, to propose adsorption and reactions on the valence band around the Fermi level. We also use the band theoretical technique, but employ it in calculating W, (calculated by other authors), and use the mean energy concept [7] to simplify separately all the levels of the bottom of the s band, the vacant or half-occupied d band and the top of the occupied d band into three different representative M.0.s. This is different from treatment by the LDF method, because all the M.0.s in our case primarily contribute to the adsorption bond. Moreover, in order to keep to reality, we never change the parameters during the entire process, and use the Reference Atomic Orbital Probe of adsorbate M.O., to avoid having an error of M.O. calculation influence the correct conclusion. (2) Intuiticrn,rapidity and convenience. In the 1930s the localized and delocalized models were disparate, and if someone attempted to couple the delocalized orbital with localized orbital, he might be advised to: “think deeply about the uncertainty principle!“. However, recent experimental and theoretical advances have generated much constructive common ground whereby localized and delocalized retinues are mixed, the paramount issue being the degree of mixing [ 10, 551. In our model, we first use the delocalized mode in the wave function of the mono-metal atom, to show how we can use the band structure concept to estimate the coefficients of s and d orbitals of band M.0.s; we then compare the results obtained by the LCAO method and by the band technique, to show that the mix concept is reasonable (note: we represent the wave function in a form which can be easily understood by chemist readers rather than with a simplified unitary matrix). Second, we use the wave function of mono-metal atom to interact with the M.O. orbital with the language of the valence bond, which is intuitive and to which chemists are more accustomed. When we discuss the mechanisms of adsorption and reaction on a metal surface, we use the language of the bond forming and breaking in a molecule or reaction intermediate. Based on experimental

82

facts and our model, we have revised the conventional d back-donation concept and extend the CO 271F-derived resonance concept to propose a new concept, arriving a unified interpretation of the electronic resonance spectra and adsorbate vibration spectroscopy, which will be reported in the next paper. The new concept is very important for understanding the mechanism of CO dissociation, in which the r bond of C-O is weakened while simultaneously the M-C bond is strengthened. The other outstanding advantages of our model are its rapid calculation and convenient application. For example, calculation of the various bonding functions for a metal adsorbate takes only several minutes; for employing our model in determining the effects of Metal-Promoter (or -Support) Interactions, the changes in metal valence states are considered to be caused by promoter or support, and are parallel with the changes in the A.O.‘s effective exponents, thus their effects can be easily tackled by our model. Although the A.O. effective exponents are not very suitable for representing the valence states quantitatively, we need only the relative effects. It is convenient for our model to treat this case. In particular, current experimental methods such as XPS, AES, ESR etc. can be easily applied for determining the metal valence states. Thus it is convenient and rapid to apply our model to understand the effects of Metal-Promoter-Support Interactions. (3) From 2%~~ to Practice. In a subsequent series papers, we will use our model in adsorption and catalytic reactions, for example the characterization of CO adsorption vibration spectra, the structure of CO species adsorbed on various transition metals, characterization of the hydrogen atom adsorption binding energy and adsorption vibration spectra on some transition metals, and comparison of our results with those obtained by other current methods such as LDF, EMT etc.; discussions of the CO dissociation mechanisms, answers to questions as to why Ru, Fe and Co usually exhibit the properties of Fischer-Tropsch synthesis, Ni promotes CO hydrogenation to methane, and Pd, Cu direct CO hydrogenation to methanol; and disclosure of the nature of the effects of Strong Metal-Support (-Promoter) Interactions. By using our model to select appropriate promoters or propose special catalyst preparation methods for achieving some special surface structures, we have successfully developed four new high effective catalysts such as one with high selectivity for ethylene oxide synthesis [ 561, low temperature-low pressure-high activity Mn(II)-promoted iron catalysts for ammonia synthesis [57, 581, a highly effective, low energy consuming catalyst for the synthesis of methyl formate from exhaust gas containing CO and methanol [59], and high activity-low pressure Cu-based catalysts for methanol synthesis. The disadvantage of our model is its inability to obtain the absolute amounts of surface bond strength, electron charge distribution and the details of energy levels, which can be obtained from other methods. We also use other advanced methods for remedying the defects of our model. In our experience, our model can be used to select rapidly and correctly important information on primary features, which is very helpful in selecting the configurations and parameters needed in employing other advanced methods.

83

Acknowledgement

This research has been supported by the Chinese National Science Foundation, China Petroleum-Chemical Corporation and the International Research Cooperation Fund of the European Economic Communities(EEC)CIl-0113-B for the author’s stay at Universite Catholique de Louvain, Belgium, in 1989 in cooperation with Prof. B. Delmon. The initial work was carried out in 1983 when the author was a visiting scholar at the Department of Chemical Engineering, University of California, Berkeley. He wishes to thank Profs. Alexis T. Bell and B. Delmon for their invitations and for comments on this paper, and Prof. K. Tamaru for his helpful suggestions and comments. He should also acknowledge Dr. E. Andreta (EEC) for his encouragement during the completion of this work. Helpful discussions with Prof. C. T. Au and Mr. Y. L. Xiong are gratefully acknowledged. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 16 17 18 19 20 21 22 23 24 25 26 27 28

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