20076
J. Phys. Chem. 1996, 100, 20076-20088
Charge Sensitivity and Bond-Order Analysis of Reactivity Trends in Allyl-[MoO3] Chemisorption Systems: Two-Reactant Approach R. F. Nalewajski* K. Gumin´ ski Department of Theoretical Chemistry, Jagiellonian UniVersity, R. Ingardena 3, 30-060 Cracow, Poland
A. Michalak Department of Computational Methods in Chemistry, Jagiellonian UniVersity, R. Ingardena 3, 30-060 Cracow, Poland ReceiVed: July 28, 1996X
The molecular charge responses and bond-order changes due to the chemisorption are reported for model allyl-[MoO3] (010)-surface structures. Charge rearrangement patterns are determined from the charge sensitivity analysis (CSA) in the atomic resolution. The bond-order analysis is based upon the valence indices calculated within the one-determinantal (Kohn-Sham, KS) difference approach, comparing bonding patterns in a molecule and in the separated reactants limit, respectively. Polarization (P) and charge transfer (CT) stages of adsorption are examined separately; the overall (CT+P) patterns are also reported to examine the relative importance of these effects in the chemisorption systems considered. These charge sensitivity criteria are calculated from the semiempirical hardness matrix for M ) (adsorbate|substrate) reactive systems including a large two-layer surface cluster. Rough estimates of the electrostatic (ES), P, and CT contributions to the interaction energy are also available in the CSA approach; they provide an approximate energetical hierarchy among the alternative chemisorption arrangements. These predictions are compared with the corresponding KS results for small surface clusters. The set of collective charge displacements, which diagonalize the interreactant part of the hardness matrix, are used to interpret the isoelectronic Fukui function for the allyl f [MoO3] CT. The mechanism of selective allyl oxidation to acrolein is examined in more detail. It is shown that the selective allyl oxidation can be rationalized in a concerted bond-breaking-bond-forming mechanism, conjectured from a combination of the CSA charge rearrangements for large clusters and the KS bond multiplicity data for the smaller (active site) chemisorption complexes.
1. Introduction Chemisorption systems of heterogeneous catalysis are challenging for theoretical investigations. Due to known limitations of a tractable size of molecular systems in rigorous quantum mechanical calculations, the cluster representations of the infinite substrate are usually adopted. In such investigations only small active site models can be used, which can prove inadequate for large adsorbates. In infinite surface systems the moderating action of the crystal reminder (R, reservoir) upon a given subsystem M ) [adsorbate (A)-active site (B)] can also influence the course of surface reactions. Moreover, a deeper understanding of such processes also requires a detailed analysis of various geometrical factors, such as a relaxation of the adsorbate, surface reconstruction, alternative mutual arrangements of A relative to B, and the transition-state structures along admissible reaction pathways. The electronic conditions, e.g., an overall reduction/oxidation state of M, measured by the average number of electrons, N, which can be changed by the charge transfer ∆N from the reservoir, can also control the actual direction and specificity of catalytic processes. Even for fixed N in the surface cluster (isoelectronic processes) an interpretation of a reaction mechanism calls for additional information about the relative importance of the mutual polarization of reactants A and B in M and of the isoelectronic (internal) charge transfer (CT) between them. This variety of molecular charge reconstruction processes for both closed and open M can be probed in semiempirical charge sensitivity analysis (CSA).1-5 In the X
Abstract published in AdVance ACS Abstracts, November 15, 1996.
S0022-3654(96)02291-5 CCC: $12.00
CSA model the external effects such as those of promoters, supports, temperature, and external electric field are phenomenologically included through the variations of the chemical potential µ due to the environment of M. An example of such a full CSA treatment of chemisorption systems has recently been reported for large toluene-[V2O5] clusters.6 It is the purpose of the present work to extend this analysis to the allyl-[(010)MoO3] complexes, which were the subjects of recent theoretical investigations.7,8 Changes in atomic charges of reactants can be related to bond weakening/strengthening effects only approximately, since the overall bond-order results from a subtle interplay between covalent and ionic components. More direct, quantum mechanical measure of the bond multiplicity are needed to predict reactivity/activation patterns in surface reactions involving large adsorbates. It has been recently demonstrated that useful absolute bond-order indicators can be generated by comparing bonding patterns in a molecule and in the separated atom limit,8-12 respectively, generated within the one-determinantal [Hartree-Fock (HF) and Kohn-Sham (KS)] difference approach in the framework of the two-electron density matrix. From such absolute bond orders one can predict relative changes in the bonding pattern due to the chemisorption, by subtracting the bond orders in the separated reactants limit, M0 ) A 0 + B 0, from those in M. However, these supplementary calculations can be performed only for much smaller (saturated) clusters involving only the active site region of the surface. The molecular systems examined in this work include allyl (adsorbate) and the [MoO3] (010)-surface cluster. This choice © 1996 American Chemical Society
Allyl-[MoO3] Chemisorption Systems of a chemisorption system has been made to study the catalytic reaction of the selective oxidation of allyl to acrolein. As shown by Grzybowska et al.,13 MoO3 is a highly efficient and selective catalyst for this process. It has also been found experimentally that the (010)-surface represents the cut of the crystal that is active in this reaction.14,15 The allyl oxidation has also been postulated as the second stage of the selective oxidation of propylene on molybdenum oxide catalysts.16,17 Large two-layer surface clusters will be used in the present CSA, to include effects due to the crystal environment of a small active site region. 2. Theoretical Background 2.1. Summary of the AIM-Resolved CSA. In the reactive system M ) A-B consisting of the acidic (A) and basic (B) reactants one usually discusses various stages of their interaction.5,18 For example, the M0 ) A0 + B0 state of separated reactants is often used as the reference for defining the P and CT changes in both reactants, due to the presence of the reaction partner. The P stage is defined by the intrareactant equilibria of the mutually closed, interacting reactants in the closed system M(+) ) (A|B), relative to yet another reference state of interacting, rigid reactants: M+ ) (A0|B0). Similarly, the CT stage in the closed reactive system is determined by the equilibrium in M ) (A:B), representing mutually open reactants (global equilibrium state), relative to M(+). To include the presence of the reservoir in chemisorption systems, one has to consider the open reactive systems M(*) ) (R:A:B), standing for the open M in contact with the macroscopic reservoir, R, or M(†) ) (RA:A|B:RB), denoting the mutually closed reactants in contact with their respective reservoirs. The basic purpose of the theory of chemical reactivity is to predict reactivity trends, e.g., charge redistribution patterns, from the relevant characteristics of reactants. It has been demonstrated previously that the separated reactant data, which ignore the reactant interaction present at short distances, cannot adequately represent the system of large interacting reactants.1,2,5,6 In the consistent two-reactant (quadratic) approach1,2,5,6 one therefore uses the first- and second-order properties of interacting reactants in M+ to determine their mutual polarization in M(+), and their properties in M(†) to determine the interreactant CT in M. In the atoms-in-molecules (AIM) description1-6 of a molecular system M consisting of m atoms, the global equilibrium energy of a given molecular system is expressed as a function of the number of electrons N and the AIM-resolved vector v ) (V1, V2, ..., Vm) representing the external potential V(r) at nuclear positions: E ) E(N,v). Similarly, the nonequilibrium electron distributions define the energy function EAIM(N,v), in which the global number of electrons is replaced by the AIM electron population vector N ) (N1, N2, ..., Nm). The CSA canonical open system characteristics for the fixed external potential are the global chemical potential, µ ) ∂E/∂N, AIM chemical potentials (electron population gradient), µ ) ∂EAIM/∂N, global hardness, η ) ∂2E/∂N2 ) ∂µ/∂N, and AIM hardness matrix (electron population Hessian), η ) ∂2E/∂N∂N ) ∂µ/∂N ≈ {γi,j}, modeled19 as the valence-shell electron repulsion tensor (twoelectron repulsion integrals from the Pariser20 and Ohno21,22 formulas). These AIM chemical potential and external potential vectors, together with the hardness matrix define all the system chemical potential and charge responses that are of interest in the theory of chemical reactivity: the global softness, S ) η-1 ) ∂N/∂µ, the AIM softness matrix, σ ) η-1 ) ∂N/∂µ, the local AIM softnesses, s ) ∂N/∂µ, the AIM Fukui function (FF) indices, f ) ∂N/∂N ) s/S ) sη, and the linear response (negative internal softness) matrix,23 β ) (∂N/∂v)N.
J. Phys. Chem., Vol. 100, No. 51, 1996 20077 The displacements (perturbations) of the global equilibrium in the system as a whole, [dN, dv], are related to the corresponding responses in the global chemical potential and AIM electron populations, [dµ, dN], by the following transformation:
[
[dN, dv] ) [dµ, dN]
]
0
1 -η ≡ [dµ, dN] H
1T
(1)
The inverse transformation involves the global hardness, Fukui function indices, and linear response matrix:
[dµ, dN] ) [dN, dv]
[ ]
η f ≡ [dN, dv] H-1 fT β
(2)
The dN ) 0 process corresponds to the global equilibrium in M, while dN * 0 displacement refers to the ground state in M(*). Let us consider now the reactant resolution in M(†). The CSA determines chemically interesting responses of atomic electron populations in A and B, dNA ) (dNa, dNa′, ...) and dNB ) (dNb, dNb′, ...), to typical perturbations created by the presence of the other reactant and the reservoirs, RA, RB: a displacement in the reactant average numbers of electrons, dNA ) dNA 1AT, dNB ) dNB 1TB, and changes in the external potential on both reactants, dvA, dvB. The generalizations of eqs 1 and 2 for such a reactant resolution case, which relate the vector of displacements of the intrareactant equilibria, [dNA, dNB, dvA, dvB], to the corresponding response vector, [dµA, dµB, dNA, dNB], are
[dNA, dNB, dvA, dvB] ) [dµA, dµB, dNA, dNB]
(
0 0 1A 0B 0 0 0A 1B T A,A T 0 -ηA,B 1A A -η 0BT 1BT -ηB,A -ηB,B
≡ [dµA, dµB, dNA, dNB] H h
(3)
[dµA, dµB, dNA, dNB] ) [dNA, dNB, dvA, dvB]
(
ηA,A ηB,A
ηA,B ηB,B
)
fA,A fA,B B,B fB,A f
(fA,A)T (fB,A)T βA,A βA,B (fA,B)T (fB,B)T βB,A βB,B
≡ [dNA, dNB, dvA, dvB]H h -1
)
(4)
here the condensed hardness matrix elements {ηX,Y} are defined in terms of derivatives of the equilibrium chemical potentials of reactants with respect to the global electron populations,
{ ( )}
η(A|B) ≡ ηX,Y )
∂µY ∂NX
(5)
NY
the blocks of the Fukui function matrix f(A|B) ) {fX,Y} in the reactant resolution are
fX,Y )
( ) ∂NY ∂NX
, (X, Y) ) (A, B)
(6)
NY
and the corresponding reactant resolved blocks of the linear response matrix β(A|B) ) {βX,Y} are defined by the derivatives
20078 J. Phys. Chem., Vol. 100, No. 51, 1996
βX,Y )
( ) ∂NY ∂vX
NX
Nalewajski and Michalak
,NY, (X, Y) ) (A, B)
(7)
Putting dNA ) dNB ) 0 gives the intrareactant polarization in M(+), while dNA * 0 and/or dNB * 0 determines the equilibrium state of the mutually closed reactants in M(†). In the latter case dµX ) dµRx, X ) A, B. Consider now the physisorption state (before the interreactant CT) of M(+), corresponding to the purely external potential perturbation dv(+) ≡ [dNA ) 0, dNB ) 0, dvA, dvB]. The associated atomic population displacements dNX(+) ≡ dNX[dv(+)], X ) A, B (from eq 4), define the P pattern of charge rearrangements in the chemisorption system and thus the polarized AIM electron populations NX(+) ) NX0 + dNX(+). Equation 4 also determines the new levels of the chemical potential in both reactants, µX(+) ) µX0 + dµX(+), where dµX(+) ≡ dµX[dv(+)], and thus the effective chemical potential difference for the interre(+) (+) actant CT: µ(+) CT ) µA - µB . The optimum amount of CT, (+) NCT[dv ] ) dNA ) - dNB, which equalizes the chemical potential throughout M, is then obtained from the familiar expression in the quadratic approximation:1-5 NCT ) - µ(+) CT / (+) (+) (+) (+) η(+) ; here η ) η + η 2η represents the in situ CT CT A,A B,B A,B hardness of the polarized reactants, defined by the condensed (+) hardness matrix η(A|B)[dv(+)] ) {ηX,Y }. Finally, the CT patterns of the AIM charge reconstruction are
dNXCT
) dNX[NCT] )
NCTfXCT,
X ) A, B
(8)
Here the isoelectronic (in situ) FF vector for the B f A CT includes both diagonal and off-diagonal components,1,2,5,6
fCT ) (fACT, fBCT) ) (fA,A, -fB,B) + (fA,B, -fB,A) diagonal off-diagonal
(9)
A,B - fB,B. where fACT ≡ fA,A - fB,A and fCT B ≡ f The CT and P processes in reactive systems can be given alternative collective mode interpretations, in the reference frames of the populational normal modes (PNM), minimum energy coordinates, externally and internally decoupled charge redistribution channels, etc.1-6,24 Of particular interest in the theory of chemical reactivity is the partitioning of the hardness tensor of M into its internal (intrareactant) and external (interreactant) parts:
(
)
(
)
A,A 0A,B A,A ηA,B η ) ηi + ηe, ηi ≡ ηB,A B,B , ηe ≡ 0 B,A B,B (10) 0 η η 0 The first matrix (ηi) defines the PNM of the uncoupled reactants, called the internally decoupled modes (IDM), as its eigenvectors Ui,
ηi Ui ) hi(diag.) Ui
(11)
while the second, interaction matrix similarly defines its own eigenvectors Ue, called the interreactant modes (IRM):
ηe Ue ) he(diag.) Ue
(12)
with the eigenvalues he) {h1, h2, ..., hm}. The IRM set reflects the charge coupling between reactants; it is particularly attractive for interpreting processes in chemisorption systems involving adsorbate (small subsystem) and substrate/cluster (large subsystem), since it naturally distinguishes between modes involving the adsorbate, the number of which is equal to that of the constituent atoms in the adsorbate, and a very large number of modes solely responsible for the intrasubstrate polarization, with no participation of the adsorbed molecule.1,2,5 The isoelectronic
FF vector of eq 9 can be uniquely partitioned into the IRM contributions on specified atoms: IRM
f iCT )
∑R ωi,RCT
(13)
with the mode contributions ωCT i,R reflecting a shift in the electron population of atom i through the mode R, per unit interreactant CT. We conclude this short CSA outline with a few remarks on the method estimates of the electrostatic (EES), polarization (EP), and charge transfer (ECT) interaction energies. The relevant expressions in the point-charge approximation of the perturbing external potential indices dv ) (dvA, dvB), B
A
b
a
dva ≈ - ∑qb/|Ra - Rb|, dvb ≈ - ∑qa/|Ra - Rb| (14) due to the net charge distributions qX ) NX - ZX, X ) A, B, are1, 2 A
B
a
b
EES ≈ ∑∑qaqb/|Ra - Rb|
(15)
EP ≈ 1/2 dvβ (A|B) dvT
(16)
(+) 2 (+) ECT ≈ - 1/2[µCT ] /ηCT
(17)
2.2. Summary of Quantum Mechanical Bond Multiplicities from the One-Determinantal, Finite-Difference Approach. In a series of recent articles8,10-12 a new quantum mechanical, finite-difference approach to classical problems of the chemical valence and bond multiplicity has been developed within the one-determinantal (HF or KS) description of molecular systems. These indices are formulated in terms of the atomic and diatomic contributions (ionic and covalent) to the molecular (M) expectation value of the displacements in the bond order (Pˆ ) or density (Fˆ ) operators in a molecule, relative to the reference state of the separated atoms limit (SAL), identified by the superscript(0). The overall absolute valence in the spin nonpolarized approximation [restricted HF (RHF) or the local density approximation (LDA) theories] and its partitioning into the total atomic and diatomic contributions are defined in the following way: AIM
V ) - 1/2 〈∆Pˆ 〉 ≡ Tr P ∆P ≡
∑a
AIM
Va + ∑∑Vab (18) a
