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A quantum-mechanical model of heterogeneous catalysis
Volume 197, number 1,2
CHEMICAL PHYSICS LETTERS
A quantum-mechanical
4 September 1992
model of heterogeneous catalysis *
V.Z. Kresin and W.A. Lester Jr. ’ Chemical Sciences Division, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA
Received 25 February 1992; in final form 26 June 1992
A quantum-mechanical model for heterogeneous catalytic reactions is developed based on the reaction Hamiltonian method developed by the authors. It is shown that the presence of the surface leads to additional channels of reaction. These are found to dominate the exponential smallness of the reaction probability of the direct channel producing large reaction probabilities for surface-catalyzed reactions. The dependence of catalytic reaction probability on reactant dissociation energy and vibrational frequencies, and the leakage of the electronic wavefunction out of the surface is discussed.
1. Introduction The origin of catalytic activity of surfaces continues to attract a lot of interest. Although there are several theoretical approaches to such heterogeneous reactions, perhaps the most commonly employed is transition state theory which describes the mechanism of catalytic activity as the creation of an alternative pathway with a lower energy barrier (activation energy) for the reaction; see, e.g., refs. [ l-31. The approach has proved fruitful, but, nevertheless, is phenomenological. In addition, because of large computational requirements, one is only now seeing the advent of ab initio approaches with the promise of a detailed molecular description of major regularities of catalytic dynamics (see, e.g., ref. [ 41). As a result, the search for quantum-mechanical models that are able to describe the effect of the catalyst (surface) as the cause for increased reaction rate retains high interest. This paper is concerned with the quantum-meCorrespondence to: W.A. Lester Jr., Chemical Sciences Division, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA. * This work was supported in part by the Director, Offtce of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the US Department of Energy, under Contract No. DE-AC03-76SF00098. ’ Also at Department of Chemistry, University of California, Berkeley, CA 94720, USA.
chanical description of catalytic surface reactions. Typically, a transition from reactans+products (R-P) occurring at a surface consists of two steps: ( 1) chemisorption and (2) chemical reaction (R+ P ) . In this paper, we are mainly concerned with the dynamics of the reactive step. The approach developed below is a generalization of the reaction Hamiltonian (RH) method described by the authors and their collaborators [ 5,6]. In the RH method, a chemical reaction is treated as a quantum transition R-P governed by a special reaction Hamiltonian. This Hamiltonian, in second quantization representation, has the form fiD= 1 HEa:ai. i,f
(1)
Here a:, ai are creation (annihilation) operators for the final (products) and initial (reactants) states. The matrix element Hgmay be written in the form H:aHHe’Fg,
(2)
where F?=(&l$i)
(3)
is the Franck-Condon (FC) factor between nuclear wavefunctions gf and @icorresponding to initial (R) and final (P) states in a diabatic representation, and He’ is an electronic factor,
0009-2614/92/Q 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
1
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where
Here {rM,R,} are the sets of electronic and nuclear coordinates of the molecules, V is the total potential energy, and A and & are electronic wavefunctions in a diabatic representation; for a detailed description, see refs. [ 3,4]. Note that R. denotes the equilibrium configuration. The state-to-state description of chemical reactions contained in refs. [ 3,4] is based on quantum transition theory with the Hamiltonian defined by eqs. ( 1) and (2). If the reaction occurs on a single potential energy surface and the spacing between electronic terms is relatively large, the reaction probability is described in lowest order by the golden rule, otherwise one can include higher terms. The approach is similar to Bardeen’s treatment of tunneling [ 71 and the subsequent development of the tunneling Hamiltonian [ 8 1. If one is interested in the relative product vibrational distributions, then the problem is reduced to the evaluation of the corresponding FC factors. Note that the calculation of the FC factors requires the evaluation of nuclear wavefunctions, i.e. those for internal and relative degrees of freedom (see, e.g., ref. [ 91). The method has been applied to photodissociation of C2Nz [ 61, and to the exchange reactionsOH+D+OD+HandClI+D-Cl+ID [5]. It is important to note also that the present stateto-state description of a chemical reaction leads to strong dependence of the dynamics (and product energy distributions) on the initial state of the reactants. It will be shown below that the presence of the catalyst leads to additional reaction channels that proceed through an excited state rather than the ground state. In this paper, we focus on reactions that occur at a surface. The model to be described enables one to understand from a quantum-mechanical viewpoint the role of the surface in catalytic activity. It is known that the surface provides an adsorption site which is the essential important step preceding the chemical reaction. In this paper, we are concerned with the direct impact of the surface on subsequent chemical dynamics. Based on the decomposition of the total Hamiltonian presented in 2
LETTERS
4 September
1992
section 2, see eq. (7 ), we introduce two sets of channels of surface reaction: direct and catalytic. The probabilities of R-+P transitions for both sets are evaluated below.
2. Theory 2.1. Hamiltonian. Direct and catalytic channels Consider the reaction A+BC+BAC.
(6)
We assume for convenience that MA3 MB, M, (the generalization to arbitrary masses is straightforward and will be given elsewhere). For simplicity we ignore quantum state changes of molecule A caused by the reaction. We study reaction (6) in the presence of a surface with the aim of understanding from a quantum-mechanical perspective the origin of catalytic activity; we consider here a simple model which describes such activity. The presence of the surface leads to a surface-molecule interaction and therefore an additional term in the total Hamiltonian, so that Ej,,, =& +AD+AC ,
(7)
where fio corresponds to isolated reactants and products, AD, the term describing direct contributions, is defined by eqs. ( 1) and (2), and AC, the term connected with catalytic contributions, describes the interaction of the surface with the molecule. The chemical reaction (6) can be described as proceeding through various channels. One, the direct channel, is similar to the gas-phase process where the reaction is governed by the Hamiltonian fiD (see ref. [ 41) and eqs. ( 1) and (2) with amplitude TD=H:=ilDFy,
(8)
where 1, =H$, and Fy is a FC factor (3). The catalytic channels involve the total Hamiltonian (7 ), including AC. The reaction dynamics depends strongly on the precursor, i.e. on the quantum state of the adsorbant. There are two types of adsorption. One is molecular adsorption; examples are adsorption of ethylene [ 1,3 1, CO [ I,2 1, and NO [ 2 ] on metal surfaces and, at low temperatures, O2 and
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N2 [ I]. The other case is atomic adsorption, i.e. dissociation of reactant BC at the surface. The adsorption step results in a transition of BC from a bound to a continuum state with a resultant drastic increase in the FC factor; see below. It is useful to distinguish different types of biomolecular surface reactions based on the precursor adsorption step. Although less prevalent, adsorption of both molecules is realistic. For example, the reaction CO+NO-+COI+N [2] may occur through such a channel. CO oxidation may also occur in a similar way, if oxygen adsorption is molecular. Note that the Eley-Rideal mechanism, see, e.g., refs. [ 1,2], which describes the case of one molecule adsorbed and the other in the gas phase fits this category. We next consider a simple quantum-mechanical model of a surface reaction. We shall evaluate and compare amplitudes for direct and catalytic channels. A detailed analysis of several specific reactions will be given elsewhere. 2.2. Direct channel According to eq. (8), the direct reaction amplitude D is proportional to Ffi. Considering the system of eq. (6 ) the nuclear wavefunction &can be written (in the harmonic approximation) as the product &=Q(Q, MQzMq3)
2 Q*=AX, -AX,,
(10)
where AX,, AX1 are Cartesian displacements; the set q3 describes bending motion. We assume, for simplicity, that M,=M,. The FC factor has the form
where $3 =hr(~c)$kib@rot
be described elsewhere. After simple, but tedious calculations involving transformation to Cartesian coordinates and subsequent multidimensional integration, we arrive at te following expression for the direct reaction amplitude, eq. ( 8), TD=Ab(L4i/L4) exp[ - (&/A)2]
.
(13)
Here Ai= (fi//.&&)1’2is the vibrational amplitude of the reactant diatom, ~1 is the reduced mass, A= (A: +A$)‘/2, A2= (fi/pJ22)‘/2, Q2 is the frequency of the symmetric mode BAC (MB = MC ) . (Note that product ABC presents no fundamental difficulty for this approach. The present choice is for convenience only. ) The quantity A is the corresponding reduced mass, and &=pi -pi; pb and& are the equilibrium distances between the atoms in the initial (i) and final (f ) states. In addition, Jb=exp( -E,,/AB, )A, where ,l contains electronic and normalization factors (see eqs. (8) and (4)). One can see directly from eq. ( 13 ) that the direct channel amplitude contains a small exponential factor exp[ - (Ap/A)2]. The shift & is of the order of the bond length, whereas A is of the order of the vibrational amplitude. For example, for the hydrogenation of ethylene 40% 2.6 A, A x 0.15 A, so that &/ A x 17, leading to a negligibly small direct reaction probability.
(9)
where Q,, Q2 are normal modes; see, e.g., ref. [ 111,
Q, =A& +AX*,
4 September 1992
(12)
is a nuclear wavefunction for the R state.
For simplicity we ignore rotational and bending degrees of freedom in the present development and, furthermore, assume that the initial and final vibrational states correspond to Ui= r+= 0 (vi and vfare the initial and final vibrational quantum numbers). The more general treatment is straightforward and will
2.3. Catalytic channels ( 1) Molecular adsorption. Let us now analyze the catalytic channels. The appearance of these channels is caused by the presence of the surface. One can show that the amplitude of this channel does not contain a small exponential factor, as follows. The catalytic channels involve the total Hamiltonian ( 7 ) , including AC. For present purposes we assume that A and BC are molecular but the internal structure of A is not changed. This channel is described by the amplitude Tg=
F HgHg(Ek-Ei)-’
)
(14)
where k is a set of intermediate (virtual) states (see, e.g., ref. [lo], ch. VI). From the Born-Oppenheimer representation of the wavefunction, we obtain 3
Volume 197, number
Tc= c A;FgFz(
CHEMICAL
1,2
AEki)-’
,
PHYSICS
(15)
k
where Ak =He’L fk In .
(16)
4 September
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1992
tion of B and C. The transition amplitudes for A + B + C-t BAC is described by the matrix element TfC, where f is the final state and c is similar to the k states, but in this case c corresponds to a real and not a virtual process. One can write
Note p’ is defined by eq. (5), and
T”=AF
A,G= s 9~R(Rs)Acq~‘~i(R,)dR,dr,dr,I.,=,,,
Assuming that mB = mc, the hydrogenation case, then, for example, the transition to the final vibrational state with V,, = 1, J',= V,,= 0 is described after FC evaluation by the amplitude
(17) where I# (j=i, k) is the total (surface and adsorbates) electronic wavefunction. Eq. ( 15) contains a set of intermediate states k. Let us separate the most interesting part of the k spectrum, where the intermediate states belong to the continuum. As a result, the transition i-+f (R+P; see eqs. ( 14 ) and ( 15 ) ) can be treated as a two-step process. The first step is a transition i-k from the initial bound state i of BC to continuum k governed by the operator fi’. The next step (k-+f) is provided by the reaction Hamiltonian fiD. It is important to emphasize that the k states are virtual states and, in accordance with quantum transition theory, the transitions to such states do not require conservation of energy (see, e.g., ref. [ lo] ). For this reason virtual transitions can be provided by the time-independent interaction AC. The concept of virtual transitions reflects, in a convenient way, the fact that the eigenstates of the total Hamiltonian, including AC, differ from the eigenstates in the absence of the surface and contain a noticeable admixture of the final states @r. The amplitude of the i+f transition has the form T’&(m)-‘F&‘fi,
Tfc=xfexp( -f),
f=E,,/Q
.
(20)
We will not write out an explicit expression for the electronic constant )2”. It is important to note that the amplitude in this case does not contain an exponentially small factor. (3 ) FC factors. Let us evaluate the FC factors. We consider for concreteness the molecular adsorption model. Note that the amplitude for the case when one of the reactants is adsorbed atomically contains similar FC factors. The initial and final wavefunctions @iand ef are described by eqs. (9) and ( 12). The intermediate state is the dissociative state of the diatomic system. The nuclear wavefunction can be written @k =
&&-,~t,(&)
.
(21)
Here &( R,) describes the center-of-mass motion of AB (cf. eq. ( 12)). The wavefunction &re, can be written in the form
(18)
where Kcorresponds to the continuum. One can see that the evaluation of the vibrational distribution is reduced to the calculation of FC factors. (2) Atomic adsorption. For atomic adsorption, the adsorption step corresponds to a transition from the bound BC state to the isolated B and C atoms at the surface. Unlike molecular adsorption the dissociative state of the BC system is real, not virtual. The hydrogenation of ethylene provides an example of such a reaction. In our model (here we neglect surface irregularities and local distortions), one can separate variables and consider the translational mo4
(19)
fc *
XAi( -(Fyi3
(p-p,,E))exp(ip.z,).
(22)
Here Ai is the Airy function, which provides a proper description near the turning point pt as well as semiclassical behavior far from pt (see, e.g., ref. [lo] ), F= - &(R) /aR, and ek is the electronic term for the &th state. Based on eqs. ( 15 ) and ( 2 1)) we arrive, after some manipulations, at the result (23)
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where D is the dissociation energy, 6 --I = d,- ’ + A, =aAi where a=const.; see eq. (16). Unlike eq. ( 13), eq. (23) does not contain the exponentially small term exp [ - ( A~/YI)~]. The smallness of TD reflects the negligible overlap of the nuclear wavefunctions with the bound states & and @iisee eqs. ( 3 ) , ( 8 ), and ( 13 ) . If we consider the overlap of fir and & (@iand &), where & corresponds to a dissociative continuum state (see eqs. ( 15) and (23) ), the situation differs in a drastic way. The presence of the surface leads to additional channels through the intermediate states &that dominate the exponential smallness of the direct channel. This is a quantum-mechanical manifestation of the catalytic activity of the surface. Eq. (23 ), which gives the amplitude of catalytic reaction and should be compared with eq. ( 13), is the major result of this paper. Consider the function &.+ which enters the righthand side of eq. (23) through D (see eq. (16)); it is defined by eq. ( 16 ) and can be written in the form
4 September 1992
o;‘,
where d,., is the surface dipole and r. of the molecule from the surface. The leakage out of the surface leads to an surface dipole, and, correspondingly, in the value of 1,. 2.4. Unimolecular surface reaction
This class represents a different interesting case. Here the surface-molecule interaction leads to molecular dissociation. The process is similar to gasphase indirect photodissociation; see, e.g., refs. [ 11,121. The surface provides the transition to the predissociative state which undergoes a radiationless transition to the final dissociative state. The predissociative state is real and characterized by a finite lifetime. The transition amplitude from a bound predissociative state to the final continuum state is given by, cf. refs. [ 5,12 1, T,, =;s&,
(24) Here p,,(r,)
is the electronic
Iv,l(r,)‘,PN=14(Rs)(*,
and
density, pel( rs) =
fki=!hVili~R=m
.
The functions v/i and v/k are t?kCtrOIIiC WaVefUnCtions describing the initial and intermediate states of the molecule. The other functions in eq. (24) have similar meaning. The sets r,, R, and rM, RM are electronic and nuclear coordinates and vsM is a sum of the Coulomb terms, namely VSM= C e*lr,-rM S,M + 1 Ze*IR,-r, S,M
I-’ I-‘+...
.
(25)
After some manipulations, we obtain the following expression for Aki: Ati=
I drM.hi(rMMrM 1 ,
where L(rM)=Jdrs
&Irs-rMI-’
(26) and Ap=pe,(rs)
-PN(&.).
One can see that the catalytic effect on the surface reaction strongly depends on the leakage of the electronic states out of the surface. In the dipole approximation we obtain
is the distance increase in the increase in the to an increase
.
(28)
This FC factor is similar to those of eqs. ( 15 ) and ( 19) and does not contain an exponentially small factor.
3. Conclusions We have developed a quantum model of heterogeneous catalysis. Our approach is based on quantum transition theory and follows from a generalization of our reaction Hamiltonian method [ 4,5]. The role of the surface is twofold. First, we are dealing with the adsorption of a molecule on a solid surface. For example, Pt is a good absorbant for the ethylene molecule (see, e.g., ref. [3] ). Second, the catalyst directly affects the chemical dynamics. The dynamics depends strongly on the initial quantum state of the reactants. The surface provides the transition from the bound to the excited continuum state of the reactant BC. For the case of molecular adsorption, virtual transitions make possible participation of such states. There are many factors, of course, which affect the total picture, such as the effect of temperature, change 5
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in a local state caused by the adsorption, etc. These problems will be considered elsewhere. In a subsequent publication, we shall follow a density matrix formalism in order to consider in a more rigorous way very strong surface-molecule interactions. The main results can be summarized as follows: ( 1) The interaction between reactants and surface leads to an increase in reaction rate owing to an increase in a Franck-Condon factor. ( 2 ) Comparison of the reaction rate for direct and catalytic channels has been carried out for the generalized reaction A+BC-+BAC. The reactant BC enters the reaction step in an excited continuum state. For molecular adsorption the surface-reactant interaction leads to additional, virtual channels of reaction. The contribution of additional, intermediate states leads to a drastic increase in reaction rate. (3 ) The reaction amplitude T for molecular adsorption depends on the dissociation energy D and vibrational frequencies Q ( TaG/D).Leakage of the electronic wavefunction out of the surface is also found to be an important factor.
Acknowledgement
The authors are grateful to Professor A. Bell and Professor P. de Gennes for fruitful discussions. This work was supported in part by the Director, Office of Energy Research, Office of Basic Energy Science,
4 September I992
Chemical Sciences Division of the US Department of Energy, under Contract No. DE-AC03-76SF00098.
References [ 1 ] G. Bond, Heterogeneous catalysis. Principles and applications (Clarendon Press, Oxford, 1987); R. Gasser, An introduction to chemisorption and catalysis by metals (Clarendon Press, Oxford, 1985 ) [ 21 S. Lombard0 and A. Bell, in: Surface Sci. Rept. I3 ( 199 1) 1. [3] S. Morrison, The chemical physics of surfaces (Plenum Press, New York, 1990). [4] P. Siegbahn, L. Petterson and U. Wahlgren, J. Chem. Phys. 94 (1991) 4024. [ 51V.Z. Kresin and W.A. Lester Jr., Chem. Phys. 90 ( 1984) 335. [ 61 C.E. Dateo, V.Z. Kresin, M. Dupuis and W.A. Lester Jr., J. Chem. Phys. 86 (1987) 2639. [ 7 ] J. Bardeen, Phys. Rev. Letters 6 ( 196 1) 57. [8] M.H. Cohen, L. Falicov and J. Phillips, Phys. Rev. Letters 8 (1962) 316. [9] V.Z. Kresin and W.A. Lester Jr., J. Phys. Chem. 86 (1982) 2182. [ IO] L. Landau and E. Lifshitz, Quantum mechanics (Pergamon Press, Oxford, 1979). [ 111 L. Landau and E. Lifshitz, Mechanics (Pergamon Press, Oxford). [ 121 J.-A. Beswick and J. Durup, in: Chemical photophysics (CNRS, Paris, 1979) p. 385. [ 131 V.Z. Kresin and W.A. Lester Jr., in: Advances in photochemistry, eds. D. Volman, K. Gollnick and G. Hammond (Wiley, New York, 1986) p. 95.
CHEMICAL PHYSICS LETTERS
A quantum-mechanical
4 September 1992
model of heterogeneous catalysis *
V.Z. Kresin and W.A. Lester Jr. ’ Chemical Sciences Division, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA
Received 25 February 1992; in final form 26 June 1992
A quantum-mechanical model for heterogeneous catalytic reactions is developed based on the reaction Hamiltonian method developed by the authors. It is shown that the presence of the surface leads to additional channels of reaction. These are found to dominate the exponential smallness of the reaction probability of the direct channel producing large reaction probabilities for surface-catalyzed reactions. The dependence of catalytic reaction probability on reactant dissociation energy and vibrational frequencies, and the leakage of the electronic wavefunction out of the surface is discussed.
1. Introduction The origin of catalytic activity of surfaces continues to attract a lot of interest. Although there are several theoretical approaches to such heterogeneous reactions, perhaps the most commonly employed is transition state theory which describes the mechanism of catalytic activity as the creation of an alternative pathway with a lower energy barrier (activation energy) for the reaction; see, e.g., refs. [ l-31. The approach has proved fruitful, but, nevertheless, is phenomenological. In addition, because of large computational requirements, one is only now seeing the advent of ab initio approaches with the promise of a detailed molecular description of major regularities of catalytic dynamics (see, e.g., ref. [ 41). As a result, the search for quantum-mechanical models that are able to describe the effect of the catalyst (surface) as the cause for increased reaction rate retains high interest. This paper is concerned with the quantum-meCorrespondence to: W.A. Lester Jr., Chemical Sciences Division, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA. * This work was supported in part by the Director, Offtce of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the US Department of Energy, under Contract No. DE-AC03-76SF00098. ’ Also at Department of Chemistry, University of California, Berkeley, CA 94720, USA.
chanical description of catalytic surface reactions. Typically, a transition from reactans+products (R-P) occurring at a surface consists of two steps: ( 1) chemisorption and (2) chemical reaction (R+ P ) . In this paper, we are mainly concerned with the dynamics of the reactive step. The approach developed below is a generalization of the reaction Hamiltonian (RH) method described by the authors and their collaborators [ 5,6]. In the RH method, a chemical reaction is treated as a quantum transition R-P governed by a special reaction Hamiltonian. This Hamiltonian, in second quantization representation, has the form fiD= 1 HEa:ai. i,f
(1)
Here a:, ai are creation (annihilation) operators for the final (products) and initial (reactants) states. The matrix element Hgmay be written in the form H:aHHe’Fg,
(2)
where F?=(&l$i)
(3)
is the Franck-Condon (FC) factor between nuclear wavefunctions gf and @icorresponding to initial (R) and final (P) states in a diabatic representation, and He’ is an electronic factor,
0009-2614/92/Q 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
1
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1,2
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PHYSICS
where
Here {rM,R,} are the sets of electronic and nuclear coordinates of the molecules, V is the total potential energy, and A and & are electronic wavefunctions in a diabatic representation; for a detailed description, see refs. [ 3,4]. Note that R. denotes the equilibrium configuration. The state-to-state description of chemical reactions contained in refs. [ 3,4] is based on quantum transition theory with the Hamiltonian defined by eqs. ( 1) and (2). If the reaction occurs on a single potential energy surface and the spacing between electronic terms is relatively large, the reaction probability is described in lowest order by the golden rule, otherwise one can include higher terms. The approach is similar to Bardeen’s treatment of tunneling [ 71 and the subsequent development of the tunneling Hamiltonian [ 8 1. If one is interested in the relative product vibrational distributions, then the problem is reduced to the evaluation of the corresponding FC factors. Note that the calculation of the FC factors requires the evaluation of nuclear wavefunctions, i.e. those for internal and relative degrees of freedom (see, e.g., ref. [ 91). The method has been applied to photodissociation of C2Nz [ 61, and to the exchange reactionsOH+D+OD+HandClI+D-Cl+ID [5]. It is important to note also that the present stateto-state description of a chemical reaction leads to strong dependence of the dynamics (and product energy distributions) on the initial state of the reactants. It will be shown below that the presence of the catalyst leads to additional reaction channels that proceed through an excited state rather than the ground state. In this paper, we focus on reactions that occur at a surface. The model to be described enables one to understand from a quantum-mechanical viewpoint the role of the surface in catalytic activity. It is known that the surface provides an adsorption site which is the essential important step preceding the chemical reaction. In this paper, we are concerned with the direct impact of the surface on subsequent chemical dynamics. Based on the decomposition of the total Hamiltonian presented in 2
LETTERS
4 September
1992
section 2, see eq. (7 ), we introduce two sets of channels of surface reaction: direct and catalytic. The probabilities of R-+P transitions for both sets are evaluated below.
2. Theory 2.1. Hamiltonian. Direct and catalytic channels Consider the reaction A+BC+BAC.
(6)
We assume for convenience that MA3 MB, M, (the generalization to arbitrary masses is straightforward and will be given elsewhere). For simplicity we ignore quantum state changes of molecule A caused by the reaction. We study reaction (6) in the presence of a surface with the aim of understanding from a quantum-mechanical perspective the origin of catalytic activity; we consider here a simple model which describes such activity. The presence of the surface leads to a surface-molecule interaction and therefore an additional term in the total Hamiltonian, so that Ej,,, =& +AD+AC ,
(7)
where fio corresponds to isolated reactants and products, AD, the term describing direct contributions, is defined by eqs. ( 1) and (2), and AC, the term connected with catalytic contributions, describes the interaction of the surface with the molecule. The chemical reaction (6) can be described as proceeding through various channels. One, the direct channel, is similar to the gas-phase process where the reaction is governed by the Hamiltonian fiD (see ref. [ 41) and eqs. ( 1) and (2) with amplitude TD=H:=ilDFy,
(8)
where 1, =H$, and Fy is a FC factor (3). The catalytic channels involve the total Hamiltonian (7 ), including AC. The reaction dynamics depends strongly on the precursor, i.e. on the quantum state of the adsorbant. There are two types of adsorption. One is molecular adsorption; examples are adsorption of ethylene [ 1,3 1, CO [ I,2 1, and NO [ 2 ] on metal surfaces and, at low temperatures, O2 and
CHEMICAL PHYSICS LETTERS
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N2 [ I]. The other case is atomic adsorption, i.e. dissociation of reactant BC at the surface. The adsorption step results in a transition of BC from a bound to a continuum state with a resultant drastic increase in the FC factor; see below. It is useful to distinguish different types of biomolecular surface reactions based on the precursor adsorption step. Although less prevalent, adsorption of both molecules is realistic. For example, the reaction CO+NO-+COI+N [2] may occur through such a channel. CO oxidation may also occur in a similar way, if oxygen adsorption is molecular. Note that the Eley-Rideal mechanism, see, e.g., refs. [ 1,2], which describes the case of one molecule adsorbed and the other in the gas phase fits this category. We next consider a simple quantum-mechanical model of a surface reaction. We shall evaluate and compare amplitudes for direct and catalytic channels. A detailed analysis of several specific reactions will be given elsewhere. 2.2. Direct channel According to eq. (8), the direct reaction amplitude D is proportional to Ffi. Considering the system of eq. (6 ) the nuclear wavefunction &can be written (in the harmonic approximation) as the product &=Q(Q, MQzMq3)
2 Q*=AX, -AX,,
(10)
where AX,, AX1 are Cartesian displacements; the set q3 describes bending motion. We assume, for simplicity, that M,=M,. The FC factor has the form
where $3 =hr(~c)$kib@rot
be described elsewhere. After simple, but tedious calculations involving transformation to Cartesian coordinates and subsequent multidimensional integration, we arrive at te following expression for the direct reaction amplitude, eq. ( 8), TD=Ab(L4i/L4) exp[ - (&/A)2]
.
(13)
Here Ai= (fi//.&&)1’2is the vibrational amplitude of the reactant diatom, ~1 is the reduced mass, A= (A: +A$)‘/2, A2= (fi/pJ22)‘/2, Q2 is the frequency of the symmetric mode BAC (MB = MC ) . (Note that product ABC presents no fundamental difficulty for this approach. The present choice is for convenience only. ) The quantity A is the corresponding reduced mass, and &=pi -pi; pb and& are the equilibrium distances between the atoms in the initial (i) and final (f ) states. In addition, Jb=exp( -E,,/AB, )A, where ,l contains electronic and normalization factors (see eqs. (8) and (4)). One can see directly from eq. ( 13 ) that the direct channel amplitude contains a small exponential factor exp[ - (Ap/A)2]. The shift & is of the order of the bond length, whereas A is of the order of the vibrational amplitude. For example, for the hydrogenation of ethylene 40% 2.6 A, A x 0.15 A, so that &/ A x 17, leading to a negligibly small direct reaction probability.
(9)
where Q,, Q2 are normal modes; see, e.g., ref. [ 111,
Q, =A& +AX*,
4 September 1992
(12)
is a nuclear wavefunction for the R state.
For simplicity we ignore rotational and bending degrees of freedom in the present development and, furthermore, assume that the initial and final vibrational states correspond to Ui= r+= 0 (vi and vfare the initial and final vibrational quantum numbers). The more general treatment is straightforward and will
2.3. Catalytic channels ( 1) Molecular adsorption. Let us now analyze the catalytic channels. The appearance of these channels is caused by the presence of the surface. One can show that the amplitude of this channel does not contain a small exponential factor, as follows. The catalytic channels involve the total Hamiltonian ( 7 ) , including AC. For present purposes we assume that A and BC are molecular but the internal structure of A is not changed. This channel is described by the amplitude Tg=
F HgHg(Ek-Ei)-’
)
(14)
where k is a set of intermediate (virtual) states (see, e.g., ref. [lo], ch. VI). From the Born-Oppenheimer representation of the wavefunction, we obtain 3
Volume 197, number
Tc= c A;FgFz(
CHEMICAL
1,2
AEki)-’
,
PHYSICS
(15)
k
where Ak =He’L fk In .
(16)
4 September
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1992
tion of B and C. The transition amplitudes for A + B + C-t BAC is described by the matrix element TfC, where f is the final state and c is similar to the k states, but in this case c corresponds to a real and not a virtual process. One can write
Note p’ is defined by eq. (5), and
T”=AF
A,G= s 9~R(Rs)Acq~‘~i(R,)dR,dr,dr,I.,=,,,
Assuming that mB = mc, the hydrogenation case, then, for example, the transition to the final vibrational state with V,, = 1, J',= V,,= 0 is described after FC evaluation by the amplitude
(17) where I# (j=i, k) is the total (surface and adsorbates) electronic wavefunction. Eq. ( 15) contains a set of intermediate states k. Let us separate the most interesting part of the k spectrum, where the intermediate states belong to the continuum. As a result, the transition i-+f (R+P; see eqs. ( 14 ) and ( 15 ) ) can be treated as a two-step process. The first step is a transition i-k from the initial bound state i of BC to continuum k governed by the operator fi’. The next step (k-+f) is provided by the reaction Hamiltonian fiD. It is important to emphasize that the k states are virtual states and, in accordance with quantum transition theory, the transitions to such states do not require conservation of energy (see, e.g., ref. [ lo] ). For this reason virtual transitions can be provided by the time-independent interaction AC. The concept of virtual transitions reflects, in a convenient way, the fact that the eigenstates of the total Hamiltonian, including AC, differ from the eigenstates in the absence of the surface and contain a noticeable admixture of the final states @r. The amplitude of the i+f transition has the form T’&(m)-‘F&‘fi,
Tfc=xfexp( -f),
f=E,,/Q
.
(20)
We will not write out an explicit expression for the electronic constant )2”. It is important to note that the amplitude in this case does not contain an exponentially small factor. (3 ) FC factors. Let us evaluate the FC factors. We consider for concreteness the molecular adsorption model. Note that the amplitude for the case when one of the reactants is adsorbed atomically contains similar FC factors. The initial and final wavefunctions @iand ef are described by eqs. (9) and ( 12). The intermediate state is the dissociative state of the diatomic system. The nuclear wavefunction can be written @k =
&&-,~t,(&)
.
(21)
Here &( R,) describes the center-of-mass motion of AB (cf. eq. ( 12)). The wavefunction &re, can be written in the form
(18)
where Kcorresponds to the continuum. One can see that the evaluation of the vibrational distribution is reduced to the calculation of FC factors. (2) Atomic adsorption. For atomic adsorption, the adsorption step corresponds to a transition from the bound BC state to the isolated B and C atoms at the surface. Unlike molecular adsorption the dissociative state of the BC system is real, not virtual. The hydrogenation of ethylene provides an example of such a reaction. In our model (here we neglect surface irregularities and local distortions), one can separate variables and consider the translational mo4
(19)
fc *
XAi( -(Fyi3
(p-p,,E))exp(ip.z,).
(22)
Here Ai is the Airy function, which provides a proper description near the turning point pt as well as semiclassical behavior far from pt (see, e.g., ref. [lo] ), F= - &(R) /aR, and ek is the electronic term for the &th state. Based on eqs. ( 15 ) and ( 2 1)) we arrive, after some manipulations, at the result (23)
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where D is the dissociation energy, 6 --I = d,- ’ + A, =aAi where a=const.; see eq. (16). Unlike eq. ( 13), eq. (23) does not contain the exponentially small term exp [ - ( A~/YI)~]. The smallness of TD reflects the negligible overlap of the nuclear wavefunctions with the bound states & and @iisee eqs. ( 3 ) , ( 8 ), and ( 13 ) . If we consider the overlap of fir and & (@iand &), where & corresponds to a dissociative continuum state (see eqs. ( 15) and (23) ), the situation differs in a drastic way. The presence of the surface leads to additional channels through the intermediate states &that dominate the exponential smallness of the direct channel. This is a quantum-mechanical manifestation of the catalytic activity of the surface. Eq. (23 ), which gives the amplitude of catalytic reaction and should be compared with eq. ( 13), is the major result of this paper. Consider the function &.+ which enters the righthand side of eq. (23) through D (see eq. (16)); it is defined by eq. ( 16 ) and can be written in the form
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o;‘,
where d,., is the surface dipole and r. of the molecule from the surface. The leakage out of the surface leads to an surface dipole, and, correspondingly, in the value of 1,. 2.4. Unimolecular surface reaction
This class represents a different interesting case. Here the surface-molecule interaction leads to molecular dissociation. The process is similar to gasphase indirect photodissociation; see, e.g., refs. [ 11,121. The surface provides the transition to the predissociative state which undergoes a radiationless transition to the final dissociative state. The predissociative state is real and characterized by a finite lifetime. The transition amplitude from a bound predissociative state to the final continuum state is given by, cf. refs. [ 5,12 1, T,, =;s&,
(24) Here p,,(r,)
is the electronic
Iv,l(r,)‘,PN=14(Rs)(*,
and
density, pel( rs) =
fki=!hVili~R=m
.
The functions v/i and v/k are t?kCtrOIIiC WaVefUnCtions describing the initial and intermediate states of the molecule. The other functions in eq. (24) have similar meaning. The sets r,, R, and rM, RM are electronic and nuclear coordinates and vsM is a sum of the Coulomb terms, namely VSM= C e*lr,-rM S,M + 1 Ze*IR,-r, S,M
I-’ I-‘+...
.
(25)
After some manipulations, we obtain the following expression for Aki: Ati=
I drM.hi(rMMrM 1 ,
where L(rM)=Jdrs
&Irs-rMI-’
(26) and Ap=pe,(rs)
-PN(&.).
One can see that the catalytic effect on the surface reaction strongly depends on the leakage of the electronic states out of the surface. In the dipole approximation we obtain
is the distance increase in the increase in the to an increase
.
(28)
This FC factor is similar to those of eqs. ( 15 ) and ( 19) and does not contain an exponentially small factor.
3. Conclusions We have developed a quantum model of heterogeneous catalysis. Our approach is based on quantum transition theory and follows from a generalization of our reaction Hamiltonian method [ 4,5]. The role of the surface is twofold. First, we are dealing with the adsorption of a molecule on a solid surface. For example, Pt is a good absorbant for the ethylene molecule (see, e.g., ref. [3] ). Second, the catalyst directly affects the chemical dynamics. The dynamics depends strongly on the initial quantum state of the reactants. The surface provides the transition from the bound to the excited continuum state of the reactant BC. For the case of molecular adsorption, virtual transitions make possible participation of such states. There are many factors, of course, which affect the total picture, such as the effect of temperature, change 5
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in a local state caused by the adsorption, etc. These problems will be considered elsewhere. In a subsequent publication, we shall follow a density matrix formalism in order to consider in a more rigorous way very strong surface-molecule interactions. The main results can be summarized as follows: ( 1) The interaction between reactants and surface leads to an increase in reaction rate owing to an increase in a Franck-Condon factor. ( 2 ) Comparison of the reaction rate for direct and catalytic channels has been carried out for the generalized reaction A+BC-+BAC. The reactant BC enters the reaction step in an excited continuum state. For molecular adsorption the surface-reactant interaction leads to additional, virtual channels of reaction. The contribution of additional, intermediate states leads to a drastic increase in reaction rate. (3 ) The reaction amplitude T for molecular adsorption depends on the dissociation energy D and vibrational frequencies Q ( TaG/D).Leakage of the electronic wavefunction out of the surface is also found to be an important factor.
Acknowledgement
The authors are grateful to Professor A. Bell and Professor P. de Gennes for fruitful discussions. This work was supported in part by the Director, Office of Energy Research, Office of Basic Energy Science,
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Chemical Sciences Division of the US Department of Energy, under Contract No. DE-AC03-76SF00098.
References [ 1 ] G. Bond, Heterogeneous catalysis. Principles and applications (Clarendon Press, Oxford, 1987); R. Gasser, An introduction to chemisorption and catalysis by metals (Clarendon Press, Oxford, 1985 ) [ 21 S. Lombard0 and A. Bell, in: Surface Sci. Rept. I3 ( 199 1) 1. [3] S. Morrison, The chemical physics of surfaces (Plenum Press, New York, 1990). [4] P. Siegbahn, L. Petterson and U. Wahlgren, J. Chem. Phys. 94 (1991) 4024. [ 51V.Z. Kresin and W.A. Lester Jr., Chem. Phys. 90 ( 1984) 335. [ 61 C.E. Dateo, V.Z. Kresin, M. Dupuis and W.A. Lester Jr., J. Chem. Phys. 86 (1987) 2639. [ 7 ] J. Bardeen, Phys. Rev. Letters 6 ( 196 1) 57. [8] M.H. Cohen, L. Falicov and J. Phillips, Phys. Rev. Letters 8 (1962) 316. [9] V.Z. Kresin and W.A. Lester Jr., J. Phys. Chem. 86 (1982) 2182. [ IO] L. Landau and E. Lifshitz, Quantum mechanics (Pergamon Press, Oxford, 1979). [ 111 L. Landau and E. Lifshitz, Mechanics (Pergamon Press, Oxford). [ 121 J.-A. Beswick and J. Durup, in: Chemical photophysics (CNRS, Paris, 1979) p. 385. [ 131 V.Z. Kresin and W.A. Lester Jr., in: Advances in photochemistry, eds. D. Volman, K. Gollnick and G. Hammond (Wiley, New York, 1986) p. 95.