Key words: effective Hamiltonian; catalysis; oxygen adsorption; transition metal oxides; metal porphyrines.
The transition metal compounds (TMC) are widely used as catalysts in many practically important processes such as oxygenation, CO insertion, isomerization of hydrocarbons etc []. The overall performance of catalysis can be limited by different stages of the process like transport of reagents and products or by the chemical transformation itself to be completed in the coordination sphere of transition metal atom. Even in the case when the diffusion is the limiting stage in the overall process, a catalyst must interact with reactants in a way which ensures the high rate of the chemical stage of transformation of reactants to products which is a limiting step in the absense of catalyst. For that reason the key problem in quantum chemical understanding of the TMC catalysis is that of disclosing the factors controlling the form of potential energy surface (PES) of reactants/products complex with a catalyst (catalytic complex or CC) as compared to that of free reactants.
The construction of PES is a complicated problem and the complications have twofold origin. First, the TMC based catalysts (CTMC's) are usually rather large systems: it is true both for solid state heterogeneous catalysts, for enzymes, and for their biomimetic models. Second, an essential feature of electronic structure of the TMC's (including CTMC's which comprise catalyst and reactants/products molecules attached to the latter) is the d-shell which is usually believed to serve as electron donor or acceptor for the reactants []. Also, interactions between reactants and catalyst can change not only charge but also the spin states of the both getting around the spin conservation rules restricting the reaction []. High correlation of electrons in the d-shell of the transition metal atom in complexes leads to the problems for the self consistent field (SCF) based methods when the latter are applied to spin and symmetry properties of the ground state of the TMC's as well as to d-d electron spectra. The situation in CTMC's is even more complicated since there not only d-electrons but also those in the cleaving or forming bonds in the reactants/products ligands must be treated with account of correlations. The density functional theory based methods encounter the problems analogous to those of other one-electron approaches. For example, they predict wrong ground state of nickel oxide. Here it goes not about purely numerical problem, but on reproducing qualitative features of electronic structure of TMC. Incidentally strongly correlated calculations of the whole catalytic system are hardly available even in one point of the configuration space; the construction of the potential energy surfaces (PES's) is really incredible task. At the same time the correlated description is necessary only for a small part of the whole system i.e. for the reaction center. The situation is typical for applying hybrid methods when different parts of the system are described with different levels of approximation (usually the combination of quantum and molecular mechanical approaches is used).
It is important to address not purely computational problem of describing TMC catalysis but also the problem of ''understanding'' the latter. The analogous reasons moved K. Ruedenberg to formulate a dichotomy between the high precision numerical calculation and clearness of qualitative description []. Indeed in order to reach quantitative accuracy we use model Hamiltonians as much as exact as it is possible and try to solve the corresponding eigenvalue/eigenfunction problem as precise as possible. This way sometimes leads us to numerical exactness but very rarely to qualitative understanding. On the other hand understanding means employing absolutely different tools i.e. the phenomenological Hamiltonians and approximate wave functions related to them. The most renown example of phenomenological Hamiltonian is of course the Heisenberg Hamiltonian which contains single parameter J, and almost no electrons - they are replaced by spins. An example somewhat closer to our topic is the phenomenological Hamiltonian of the crystal field theory [], which just neglects all the electrons except those in the d-shell and replaces all possible effects of the surrounding by several parameters whose exact number depends on the symmetry of the complex. Everyone knows that these phenomenological approaches may be easily questioned: why the number of parameters is that; why their numerical values are those, and so on. But still, they do give insight into the nature of phenomena.
It seems that this dichotomy can not be overcome but with use of a method which in fact lifts the contradiction between the model (even ab initio) and phenomenological approach. That is the method of effective Hamiltonian. It is supposed to bridge the gap between a model Hamiltonian which is assumed to be exact (it may be the ab initio one or not, but for the purpose at hand it is considered as exact one) and a phenomenological Hamiltonian reflecting the principal qualitative features of the process or object under consideration. The main features of the effective Hamiltonian approach as applied to rationalization of the phenomenological approaches are
1) a reduced number of electron variables;
2) principal calculability of the parameters;
3) its form coincides with that of the phenomenological Hamiltonian.
This general scheme has been recently implemented in the effective crystal field (ECF) method [] which represents the hybrid approach aimed to describe the electronic spectra of d-d excitations in the transition metal compounds (TMC). It exploits the group function formalism and the Löwdin partitioning method to separate the electronic variables relevant to the d-shell and the ligands respectively. It allows to take into account both electron correlations in the d-shell and Coulomb and resonance interactions between the d-shell and the ligands. The calculations of a wide range of the electronic spectra of TMC's have shown that the ECF method allows to describe the d-d transitions in a satisfactory agreement (within 1000 cm-1) with experiment IJQCECF,ZhFizKhim,KhimFiz.
The ECF approach resulted in a good agreement between the calculation
results and experimental data and high speed of calculation. At the same
time there are essential restrictions imposed on the problems which can be
solved by the ECF method. They arise from two features of the ECF approach:
the electron correlations in ligands are totally neglected and the number of
electrons in d-subsystem must be a good quantum number (be approximately
constant) to use correctly the perturbation expansion in the effective
Hamiltonian method. Obviously, such restrictions do not allow to use the
ECF method for description of catalytic processes (or, more commonly, to
processes accompanied by significant modifications in electronic structure
of ligands). The purpose of this work is to parallel the derivation led to
the ECF method in order to bridge the gap between the model and
phenomenological description of catalysis by transition metal complexes and
to bypass the difficulties cited above. To do so we, first, describe a
phenomenological Hamiltonian relevant for description of catalysis by TMC's
[,,,]. First according to
ChemPhys,DAN,CatMod,OrgReact the form of the phenomenological Hamiltonian
is the sum of the Hamiltonians for the components, i.e. the reactants
and catalyst and of their interaction:
| (0) |
If the interaction between the catalyst and reactants/products vanishes the
eigenfunction of the total Hamiltonian become FreactiÙFcatk and their eigenvalues are simply sums of the eigenvalues of the
subsystem Hamiltonians:
| (0) |
| (0) |
| (0) |
| (0) |
So that the phenomenological Hamiltonian described above does its work: it gives a phenomenological description of transformation of reactants coordinated by a catalyst. The above estimate for the amplitudes Eq. ( 5) allows to establish correlations between physical properties of free catalyst and its activity, which from our point of view justifies the phenomenological approach to catalysis proposed in ChemPhys,DAN,CatMod,OrgReact. The following sections are aimed to construct a hybrid approach taking into account the electron correlations in the d-shell and some part of ligands on equal footing.
The general way to perform derivation of effective Hamiltonian for the
partitioning interacting subsystems of a complex system is the combination
of the McWeeny group function technique [] with the Löwdin
partition technique []. Let H be the Hamiltonian of the
system consisting of two parts A and B. Therefore, H can be
represented as a sum of the Hamiltonians for bare A and B subsystems and
the operator of interaction between subsystems:
| (0) |
| (0) |
| (0) |
As it is clear from analysis of the desirable features of the phenomenological Hamiltonian given in the Introduction the electron correlations must be covered in both the d-shell and to some extent in the ligands. The hybrid scheme assumes dividing the whole system into parts. The subsystem requiring highly correlated quantum chemical method consists of the d-shell of transition metal atom and of some electronic states of the ligands. The choice of the ligand variables can be done in a unified way.
Dividing a TMC in two subsystems: the d-shell and the ligands (the s-
and p-orbitals of the transition metal atom are ascribed to the ligands)
is proven to be natural for the case when the electronic structure of the
ligands are chemically intact. This separation is implemented in the ECF
method, where the Hamiltonian is presented by a sum of the bare Hamiltonians
of the subsystems and operators of Coulomb and resonance interaction between
them:
| (0) |
| (0) |
| (0) |
| (0) |
A common way to single out the one-electronic states of the ligands responsible for chemical transformation is to select several highest occupied and lowest unoccupied molecular orbitals (MO's) of the reactants. These MO's are regarded as r (reactive) subsystem. The examples of choosing such MO's are given in the next Section. Procedures allowing to select these MOs automatically or in a special way are implemented as well.
The selection of the r-subsystem leads to dividing the model (''ab
initio'') electronic Hamiltonian H which differs from that given by
Eq. (9). The d-shell and r-subsystem are now to be combined
in the (dÅr)-subsystem for which electron correlations must be taken
into account as completely as possible. The rest of the TMC (l\ominus r-subsystem) is described by a low-level (SCF-based) method. Therefore, for
these subsystems
| (0) |
| (0) |
| (0) |
| (0) |
The effective Hamiltonian Eq. (14) depends on the wave function of
the (l\ominus r)-subsystem. We use as an approximation that this subsystem
is constructed from the MOs of the l-subsystem obtained by the SCF
procedure with the effective Hamiltonian Eq. (10) excluding
those ascribed to the r-subsystem. The effective Hamiltonian for the (dÅr)-subsystem can be written as a sum of one-electron and
two-electron contributions []:
| (0) |
| (0) |
| (0) |
| (0) |
The next contribution describes one-electron transfers between the d-shell
and the orbitals active in the chemical transformation. It changes the
number of electrons in the d-shell and in the r-subsystem:
| (0) |
| (0) |
| (0) |
Two-electron contributions can be also further subdivided:
| (0) |
| (0) |
The number of one-electron states included in the (dÅr)-subsystem is relatively small (typically, less than ten). Such size of the space allows to use the method of full configuration interaction to solve the eigenproblem for the effective Hamiltonian Eq. (14). We used the unitary group analysis [,,] to obtain the weights of the spin-adapted configurations in the ground and excited states of TMC's.
The energy of the combined system can be readily determined using the wave
functions of the subsystems. Let us denote by Yl\ominus r the
single determinant wave function of the (l\ominus r)-subsystem and by YdÅrk the k-th eigenfunction of the effective Hamiltonian
Eq. (14). Then, the electronic energy of the k-th state of the
combined system is approximated by
| (0) |
The method described in the previous Section is implemented in a program suit CATALYST. It is mainly directed towards analysis of electronic structure of catalytic TMC's. In the present work we describe some first applications of the above formalism to description of the evolution of electronic structure in a series of processes: the isomerisation of quadricyclane to norbornadiene and adsorption of atomic oxygen on the surfaces of transition metal oxides and metal porphyrines.
The reaction of isomerisation of quadricyclane to norbornadiene is the re-arrangement of four-membered ring into two double bonds. Without catalyst this reaction is forbidden by symmetry according to the Woodward-Hoffmann rules []. More precisely, the HOMO and LUMO (having the b1 and b2 symmetries, respectively) of quadricyclane and norbornadiene intersect in the course of transformation. This reaction is studied experimentally [,]. It was observed that some metal porhyrines are the catalysts of the isomerisation of quadricyclane to norbornadiene.
The interpretation of the experimental data on this reaction was made using Mango-Schachtschneider (MS) theory []. It was demonstrated that the MS theory predicts the catalytic activity of Mn-phtalocyanine (Mn-pc) and inactivity of Co-tetraphenylporphyrine (Co-tpp) in contradiction with experiment []. The MS theory states that the orbital symmetry can be preserved by transfer of an electron pair from the HOMO of the reagent to unoccupied orbital of TMC catalyst of the same symmetry and, vice versa, of an electron pair from the occupied orbital of catalyst to the LUMO of the reagent. The dxz and dyz orbitals of the Co-tpp (ground state is doublet 2A1g) having the symmetry of intersecting MOs of the reactants are fully occupied that leads to inactivity of Co-tpp in this reaction according to MS theory (but not to experiment). At the same time in Mn-pc these orbitals can take part in the redistribution of electrons between catalyst and reagents but experiment reveals inactivity of Mn-pc in the reaction of isomerisation of quadricyclane to norbornadiene.
In works [,] another approach to analysis of catalytic activity of TMC's has been proposed. The catalytic activity of Co-tpp can be explained by the entanglement of the complex ground state 2A1 with the direct products of the 4A2g or 4B2g quartet states of Co-tpp and the 3A2 triplet state of the system quadricyclane-norbornadiene. This contribution can be obtained from the excited states of both the d-shell of the catalyst and of the reactant. An interaction of these terms is ultimately caused by one-electron transfers between the subsystems ChemPhys,DAN.
This model can be quantified by using the method described above. We studied the chemical reaction path from quadricyclane to norbornadiene in the coordination sphere of Co-tpp. The distance between the cobalt atom and the center of the four-membered ring was fixed at 2.0 Å . The intersecting MOs of the b1 and b2 symmetry (which are HOMO and LUMO for quadricyclane and norbornadiene) were included in the r-subsystem. The energies of excitation of the singlet ground state 1A1 to the triplet state 3A2 approximately equals to 7 eV both for quadricyclane and norbornadiene at their respective equilibrium geometries. Close to the transition state the 1A1 and 3A2 terms are almost degenerate. Our calculations have shown that the ground state of the catalytic complex is the doublet 2A1 which is mainly the direct product of the ground states of the catalyst and reactants. The overall weight of the states with charge transfer is about 1% along the whole reaction path. The doublet direct product of the low-lying quartet 4A2g of the Co-tpp and the 3A2 triplet of the quadricyclane-norbornadiene system contributes to the ground state. When the equilibrium geometric structures of quadricyclane and norbornadiene are under consideration the weight of this state is negligeably small. At the same time, in the vicinity of the barrier of reaction this doublet became the major admixture to the leading ground state configuration. Its amplitude exceeds 0.15. Therefore, the concept [] loading the important role upon the excited states of the catalyst and reagents (and also the states with charge transfer) for the catalytic activity of the Co-tpp is in agreement with the numerical results obtained using the CATALYST package.
Oxygen is the commonest oxidative agent. The catalytic reactions of
oxidation are extensively used for detoxication of organic substances and of
carbon oxide. The transition metal oxides or related materials are the
widely used oxydation catalysts. The mechanism of oxidation on these
catalysts is, however, not well understood. At the same time it was shown
that the activation energy of the process of the oxidation of dihydrogen is
proportional to the oxygen binding energy to the catalyst surface
BoreskovKK. On the basis of this dependence it was concluded []
that the rate-determining step in the surface oxidation of H2 is
|
The program package CATALYST allows to elucidate the state of oxygen atoms on the surface of transition metal oxides. The important characteristics of the oxygen atom adsorbed is its spin state. The method gives the amplitudes of the configurations (characterized by their Young tableaus) in the wave function for the combined system (d-shell and 2 orbitals of the oxygen atom). Only in the case when the both oxygen orbitals are singly occupied the state of the oxygen in the configuration can not be easily defined. Moreover, in this case one Young tableau does not allow to determine the weights of singlet and triplet oxygen states. We must consider two corresponding Young tableaus and solve the system of two inhomogeneous linear equations. The required subduction coefficients are taken from Ref. []. The calculations were carried out using the cluster model of the solid catalyst. Transition metal oxide with the rock salt lattice type was modelled by a cluster containing 125 ions (62 ions of transition metal and 63 ions of oxygen). The r-subsystem was constructed from singly occupied px and py orbitals directed parallel to the surface. The varied characteristics of the adsorption were the distance from the oxygen adsorbed to the transition metal ion and the distance from the transition metal ion to the surface. The dependencies of the ground state spin on these geometry parameters are plotted in the maps (see Figs. 1-3) for the cobalt, nickel, and iron oxides (dotted lines denote approximate borders between different spin states).
For cobalt (II) oxide this map is relatively simple. The major part of the map (including large Co-Oads distances) corresponds to the overall sextet state. The main contribution to the ground state is the direct product of the quartet state of the d-shell (this contribution naturally corresponds to the ground state of the transition metal ion on the surface of oxide) and the triplet state of atomic oxygen. It can be noticed that the weight of the ionic configurations with electron transferred from the d-shell to oxygen increases with decreasing the distance Co-Oads. The total charge on the oxygen atom is zero for large r(Co-Oads) and reaches -0.42 for r(Co-Oads)=1.5 Å and r(Co-surf)=0.0 Å . The growth of the r(Co-surf) to 0.3 Å leads to increase of the absolute charge on the oxygen atom by merely 0.01. When r(Co-Oads)=1.4 Å the ground state of the catalytic complex switches to quartet with leading contribution from the direct product of the doublet state of the Co ion on the oxide surface and triplet state of the oxygen atom. The charge on the oxygen atom in this state is -0.46. The contribution from the configurations with singlet oxygen is negligibly small. It can be stated that the value of r(Co-surf) is not significant for defining the electronic state of adsorbed oxygen atom on the surface of cobalt oxide.
For nickel (II) oxide the spin-states map is similar to that for cobalt (II) oxide. The quintet ground state is characteristic for the most part of the map. The main contribution to this state comes from the triplet configuration of oxide and the triplet configuration of atomic oxygen. For small distances r(Ni-Oads) the ground state of catalytic complex is the triplet (direct product of the singlet state of oxide and the triplet state of oxygen is the main contribution). It can be noticed that the absolute charge on the oxygen adsorbed on the nickel oxide (-0.37 for r(Ni-Oads)=1.4 Å ) is remarkably smaller than that for the oxygen adsorbed on the cobalt oxide.
The spin map for the oxygen adsorption on the iron (II) oxide is much reacher than those for cobalt (II) and nickel (II) oxides. In this case the exit of the metal atom from the surface plane affects the overall spin state and the state of oxygen in the expansion. If the distance r(Fe-Oads) ³ 2.1 Å then the ground state of the catalytic complex is triplet (obtained from the quintet state of FeO and the triplet state of atomic oxygen). For small distances r(Fe-surf) the ground state is degenerate (33+34). When both r(Fe-Oads) and r(Fe-surf) are large enough the ground state is non-degenerate (35). The absolute charge on the oxygen atom for these triplet states does not exceed 0.01. In the range of r(Fe-Oads) between 1.5 Å and 2.0 Å the ground state of the catalytic complex is quintet. In the small region for the r(Fe-Oads) distance close to 1.5 Å and the r(Fe-surf) distance close to zero, the quintet ground state (51) is non-degenerate and mainly formed by the triplet states of the iron oxide and atomic oxygen. The charge on the oxygen atom in this state is rather large and equals to -0.53. Another quintet state (52+53) is doubly degenerate. The decrease of the distance r(Fe-Oads) from 2.0 Å to 1.5 Å results in increase of the absolute charge on the oxygen atom from 0.05 to 0.51. Moreover, the contribution to the ground state from the states with singlet oxygen exceeds 10% for r(Fe-Oads)=1.5 Å . If r(Fe-Oads) £ 1.4 Å the ground state of the complex is doubly degenerate triplet (31+32). It is interesting that the absolute charge on the atomic oxygen in this state (-0.47 for r(Fe-Oads)=1.4 Å ) is lower even than for the quintet state with greater distance r(Fe-Oads). At the same time the contribution from the configurations with singlet oxygen is greater than for the quintet states.
The conclusion can be drawn that the state of non-charged oxygen adsorbed on the surface of cobalt, nickel and iron oxides is predominantly triplet. The essential contribution of the singlet oxygen is observed only for the iron (II) oxide for small distances r(Fe-Oads). At the same time the configurations with negatively charged oxygen essentially contribute to the ground state of the catalytic complex. In the case of iron (II) oxide these configurations have the most weight in the ground state. It is experimentally well established that CoO and NiO are good catalysts of the oxidation by atomic oxygen. As it is shown the singlet oxygen is not formed on their surfaces. Therefore, the important role in the catalytic process is given to the negatively charged states of the oxygen in accordance with the experiment-based conclusions [,].
The transition metal porphyrines can be also used as the catalysts of the oxidation. It is interesting to compare the spin maps of the atomic oxygen coordinated to the transition metal atom of the metalloporphyrine and to the surface ion of an oxide. The formal difference is in the environment of the transition metal atom ((l\ominus r)-subsystem). We carried out the comparison on the example of cobalt, iron and manganese compounds. The spin maps for the atomic oxygen adsorbed on these metal porphyrines are shown in Figs. 4-6. When we look at the states map for the CoP=O we see that it is largely occupied with the states where the oxygen atom is largely in its triplet and the radical-anion states. Only at unrealistically short Co-O distances the states with a remarkable contribution of singlet oxygen appear. What is particularly important that the borders between the areas are sharp so that very small variation of the Co-O separation drastically changes the state of the coordinated oxygen atom. This feature is characteristic for all maps. The map for FeP=O is somewhat richer, and the area occupied by the states with significant contribution of singlet oxygen atom is large. However there is no place where O2- dianion can be found with noticeable weight though it is frequently attracted to explain things in chemistry of monooxygenase enzymes and their biomimetic models Biomim.
The most interesting from that point of view is the map for MnP=O. It is rather simple with only one state seen. However everywhere we can see a noticeable weight of singlet oxygen atom. The pictures obtained for two ends of this short range Mn, Fe, and Co, however, fairly correspond to a well known observation that all these metal porphyrines may serve as catalysts for a variety of oxygenation processes. However, CoP is known direct the process toward radical pattern: many oxygenation products, chain reaction etc, but MnP directs reaction to single insertion product whether it goes about monoxygenation of olefines or alkanes. FeP occupies in this respect an intermediate position.
The reactions in the coordination sphere of transition metal atoms are of common use. The development of the quantum chemical methods calculating the electronic states and the energies of the catalytic complexes is very important. There are many problems in the construction of such approaches due to necessity to take into account electron correlations in the d-shell and in the reactants simultaneously. We constructed such a method which uses different levels of approximation for reactive part of system and the environment (ligands). The method is realized as a program package.
This approach was applied to a series of experimental situations. The catalytic activity of the Co-tpp in the reaction of isomerization of quadricyclane to norbornadiene is explained by admixture of the excited states of quadricyclane-norbornadiene system and Co-tpp and affirmed by numerical calculations. The states of the oxygen adsorbed on the transition metal oxides are calculated and the experimental conclusion is confirmed about important role of the states with electron transfer from the d-shell to the oxygen.
This work has been supported by the INTAS through the grant 94-4089. It is a part of the PhD thesis of A.M.T. One of us (A.M.T.) acknowledges financial support from the Haldor Topsø e A/S.