Volume 17 1, number 1,2 CHEMICAL PHYSICS LETTERS 27 July 1990

Quantum mechanical reactive scattering by a multiconfigurational time-dependent self-consistent field (MCTDSCF) approach

Audrey Dell Hammerich, Ronnie Kosloff Department of Physical Chemistry and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University,

Jerusalem 91904, Israel

Mark A. Ratner Department of Chemistry, Northwestern University. Evanston, IL 60208, USA

Received 22 February 1990; in final form 9 May 1990

The major obstacle to the description of systems containing a large number of degrees of freedom is the exponential increase of computational time and effort with dimensionality. A strategy is presented to overcome this obstacle as well as the shortcoming of the omission of correlations, while still maintaining the simplicity and strengths of a mean-field description, based upon iden- tifying the crucial dynamical correlations and incorporating them with multiconfigurations. The collinear reactive scattering of H + Hz illustrates the techniques involved and their adaptability, flexibility, and breadth of applicability. MCTDSCF simulations, constructed from time-dependent variational principles, are compared with the numerically exact solution of the Schrijdinger equation, agreement is found.

1. Introduction

A variety of theoretical approaches are applied to describe, model, and simulate basic dynamical phenomena [ l-201. Yet, there currently exists no sufficiently accurate and feasible methodology, shown to be valid for all regimes of dynamics, that is applicable to systems containing a large number of degrees of freedom. The major obstacle is exponential growth of computational time and effort with dimensionality. In order to be potentially successful, any treatment must therefore seek to simplify the many-body problem by incorporating manageable approximations which neither neglect nor obscure the relevant chemistry and physics.

In many areas, mean-field approximations are proven procedures [ 21-251 for overcoming the scaling of effort with dimensionality. Their key attribute is the reduction of dimensionality by decomposing one D-di- mensional problem into D one-dimensional problems. This is amply demonstrated in molecular dynamics by applications of the self-consistent field (SCF) approximation [ 26,271. However, simple mean-field theories, static SCF, are not broadly applicable to the whole range of molecular dynamics phenomena. In addition to neglecting all correlations amongst the various degrees of freedom, SCF methods have coordinate system rep- resentations which are energy dependent, and their static nature precludes simulating dynamical processes.

The time-dependent analogue to quantum SCF, the time-dependent self-consistent field (TDSCF) method [ 28 1, unlike static SCF, is applicable to dynamic processes and addresses part of the correlations by incor- porating SCF in an explicitly time-dependent formulation. TDSCF introduces an effective time-dependent po- tential under the influence of which energy may be transferred between different modes of the system, yielding

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time correlations between these modes. Until somewhat recently, TDSCF was emphasized more in nuclear than molecular problems [ 291 and while encouraging, its applicability in understanding molecular dynamics is still in its infancy [ 30-391. Though the approach can indeed treat a broader range of problems, it nevertheless is an inadequate theoretical tool for simulating many dynamical systems of interest. Most notably, it can omit important correlations. Without such correlations, TDSCF methods are generally unable to correctly describe situations when the wavefunction bifurcates into two or more parts, as in reactive scattering and nonadiabatic transitions.

2. MCTDSCF approach

2.1. Objectives

A simple way to account for the important omitted correlations in TDSCF is to systematically correct by the addition of configurations, producing a multiconfigurational time-dependent self-consistent field (MCTDSCF) description [ 34,35,38,39]. This procedure does not increase the dimensionality of the problem and the computational effort does not depend too critically upon the number of configurations. The main idea behind the multiconfiguration improvement is to produce a more flexible description by incorporating the physically relevant correlations. While MCTDSCF is a very promising and intuitively appealing approach, its possibilities have barely been explored [ 34,35,38-401,

The principal impetus to a multiconfigurational treatment is to give more flexibility to the wavefunction describing the system while at the same time rendering the general many-body problem tractable. The main assumption of the simple TDSCF approach is the decomposition of the one D-dimensional problem into D one-dimensional ones by appealing to a mean-field treatment where the total wavefunction is represented by a Hartree product of D single “particle” wavefunctions. Thus, rather than attempt to solve the time-dependent Schriidinger equation for the often intractable one D-dimensional equation of motion of the exact Y, the wave- function is approximated (via TDSCF) as the product

!P( I, 2, . ..) QI)=,&NjJ) *

In the multiconfigurational approach correlations are introduced into the right-hand side of eq. ( 1) by in- cluding more than one Hartree product in the wavefunction. For each configuration, the D one-dimensional equations of motion are then variationally solved for the $(j, t) employing the MCTDSCF approximation for

ul

where M configurations are included in eq. (2 ).

2.2. Mathematical formulation

Assuming that the initial state of the system described by the multiconfigurational wavefunction given in eq. (2) is fixed and known, its time evolution can be ascertained by appealing to the McLachlan variational principle [ 4 1 ] which requires that the functional 1, where

I= J H . . . ifig-l?!Pr(ifig-H!P)dl d2...dD, --oo

(3)

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be stationary with respect to arbitrary variations of !i? Minimization of the space integral leads to a set of equa- tions of motion for the @,(J t). Without additional information this set is indeterminate. The nature of the conditions or constraints placed upon the wavefunction and how they are implemented as the system evolves in time are determined by the physically relevant correlations being introduced.

A condition arises naturally when the state of the system is identified at its asymptotic limits: a configuration for reactants and another orthogonal configuration for products. In order to generalize the idea of orthogonal configurations, one spatial dimension is chosen on which orthogonality is to be invoked. For clarity, denote this spatial dimension as D and the wavefunction on this space as x,,, (D, f ). Then the wavefunction of eq. (2 ) can be equivalently written as

(4)

Imposing normalization on the D- 1 spaces containing the 9, and orthogonality on the Dth space

(9mCi,~)lbnti,t))=l, (xmU4f)lxn(D,t)>=O, m#n,

along with total normalization for the wavefunction, leads to

(5)

(6)

(Note that the above normalization conditions are introduced purely for notational simplicity.) This choice for the wavefunction is related to a projection operator P which determines the correlations in the Dth space. This operator is a sum of elementary orthogonal projectors

klxn = &2ni^x, 3 (7)

which effects a partitioning of the Dth space into the direct sum of M subspaces. For greater generality, the Mth elementary projector is defined on the complementary subspace of the other M- 1 projectors

M-l

P,=f- c Pm. (8) PI=,

Hence the amplitudes ( ,ym Ix,> of eq. (6) are the probabilities associated with the projectors p,. With the desired correlations included in the state description and with a projection operator defined which

determines these correlations, one can return to the SchrGdinger equation and its variational solution. There is great latitude in the manner by which MCTDSCF equations of motion can be derived. Not only is there a choice in the space upon which the projection is carried out and the specific projection to be used, there is also flexibility in implementation of the projection. In particular, two cases can here be identified which differ ac- cording to how and when the projection is accomplished.

In the first case, the equations of motion for the projected wavefunctions are directly obtained. Using the previously defined projection operator, the time-dependent Schriidinger equation can be written as

where ti= f is the sum of projections, which reduces to the M equations in terms of the elementary projectors

For each of the above equations a functional can be defined. The resulting variational solution yields the MCTDSCF equations of motion for the multicontigurational wavefunction given in eq. (4) assuming that en-

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ergy is conserved via the x motion. Thus for the nth configuration, its D one-dimensional equations of motion are

(lib)

Analogous equations for the projected motion exist for the other configurations. In the second case, the constraints are directly applied to the functional of eq. (3). The equations of motion

so derived are identical for the @ degrees of freedom. Defining a matrix of overlaps

the equations of motion for the x degrees of freedom are

D-l

-ifi c (hn(k t) I&(k t) > P.di t) xm(D, t) , k=l >

(12)

(13)

where the prime denotes omission of the kth term from the product in eq. ( 12 ). As the evolution in the equa- tions for the x degrees of freedom is unprojected, the projections are performed after integrating the equations of motion. Though this latter case is not variationally equivalent to explicitly employing a time-dependent pro- jection operator, it does provide a simple procedure for evolving the projection in time. Applications of both cases (projection prior to integrating the equations of motion and post-integration projection) will be illustrated.

3. Numerical results and discussion

For an initial assessment of the viability of an MCTDSCF approximation, a physical system is chosen where only two configurations in eq. ( 1) need be considered. The process examined is the collinear scattering of a hydrogen molecule by molecular hydrogen. This system has been exhaustively investigated and has a reliable potential energy surface (ref. [ 42 1, with parametrization of ref. [ 431). Furthermore it displays purely quantum effects. The ability of a formalism to incorporate the relevant correlations in such a scattering problem is a stringent test of the approximation and is a precursor to extension of the methodology to higher dimensions and multisurfaces.

The multiconfigurational approach affords great freedom in the choice of Hamiltonians, wavefunctions, ini- tial conditions, projections, and in implementation. In order to explore this flexibility, the two methods for implementation introduced in section 2.2, denoted as MCTDSCF-1 and MCTDSCF-2 respectively, were em- ployed in reactive scattering simulations at several representative collision energies. These simulations are com- pared with both the numerically exact propagation obtained via a Chebychev expansion of the evolution op- erator [ 441 and with the single configuration simple TDSCF approximation.

The calculations were performed using bond coordinates. In these coordinates coupling between modes is found in the kinetic energy operator as well as in the potential operator. Computation time would be saved

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if the potential could be written as a sum of products of potential terms of the different modes. The LSTH potential [ 42,431 used in this work does not have this form and as a result the potential evaluation became the most time consuming part of the calculations. In all calculations spatial derivatives were calculated using the fast Fourier transform method [ 201. For conciseness, only simulations and data with the initial coniditions summarized in table 1 are reported.

The equations of motion were integrated by a scheme which employs a low order polynomial approximation to the evolution operator [ 45 1. This scheme has the very attractive feature of allowing for variable time steps during the propagation. Considerable time savings can result from using this option as the largest eigenvalue determines the upper limit to the time step in a fixed time step propagation such as second-order differencing (SOD). In contrast, the time step of a variable method can be large in asymptotic regions where the potential variations are small and small in the interaction region where the potential varies significantly. This method has been shown to be equivalent to the short iterative Lanczos procedure, demonstrated to offer an accurate and flexible alternative to other existing schemes for propagating the time-dependent SchrSdinger equation

[461. For this scattering application, the polynomial approximation was not only an alternative propagation method,

it proved to be an imperative choice for stability as SOD does nol work. The difficulty stems from the SOD requirement of calculating matrix elements between successive time intervals [ 20,46,47], for example

(fi)=(Y(&At)l@Y(t)), (14)

and the multiconfigurational self-consistent Hamiltonian. These two conditions result in complex expectation values, even for observables!

Fig. 1 portrays significant time frames of the evolving dynamics for the highest collision energy reported. The leftmost panel displays the numerically exact simulation and serves as the reference by which the ap- proximate methods can be assessed. Starting at 0 au with the initial state on the right, denoted by the lighter contour lines, the wavepacket spreads while traversing the entrance channel. The frame at 800 au evidences a slight extension of the y bond coordinate over its initial value. The initial state is not an eigenstate of the Hamiltonian, not even asymptotically. Hence the wavepacket oscillates, undergoing small excursions between the inner and outer vibrational potential walls with a period determined by the initial “translational” mo- mentum, as it moves down the entrance channel. Upon entering the interaction region at 1400 au the wave- function collides with the repulsive wall of the potential creating interference between incoming and reflected parts. Subsequent motion is consistent with the classical bobsled effect [ 16,481 giving rise to an oscillatory pattern for the reactively scattered part of the wavefunction. Some tunneling is observed at late times when

Table I Propagation parameters

( 1 )initial state Morse oscillator eigenfimction in y coordinate

0.99x, 0.99 v=Q 0.01 v= 1

0.01x2 1.00 v=2

Gaussian wavepacket in 1 coordinate centered at 9.0 with width 0.5 and initial momenta, corresponding to the three collision energies simulated, of

0,: -6.0, -5.0, -4.0 h: -5.9, -4.9, -3.9

(2) potential: LSTH potential surface [42,43] truncated at dissociation limit: &=a.1 74474

(3) grid: dx=dy=O.15 in bond coordinates

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EXACT

1600

MCTDSCF-1 MCTDSCF-2

27 July 1990

TDSCF

i/ / j / ! 1 I/

1 pL . ^_

L

. . _ ~-~~~~:~~~~~::~:~~~~:~~~~~~~~~~~

.-.._........_. - .._..............

._.._........_.. - ..___....................

Fig. 1. A comparison of scattering simulations derived from the numerically exact result with those from the multiconfiguration pre- (MCTDSCF-1) and post- (MCTDSCF-2) integration projection approaches and from the single configuration (TDSCF) approxima- tion at a total energy of -0.1440 au. The bond coordinate axes are all 11.1 au except for the 15.6 au axes at time frame 3100.

the reaction has almost terminated. Note the suggestion of a resonance state at 3 100 au where a small portion of the wavefunction resides along the symmetric stretch coordinate. About one-third of its amplitude has pen- etrated into the classically forbidden potential region with energy in excess of the average total energy.

Comparing the exact result to both the single configuration and multiconfiguration propagations, one is struck by the total inability of the simple TDSCF approximation to yield any reacted products. Apparently, corre-

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lations introduced by the time-dependent potential at the single configuration level are an ineffective mech- anism for transferring energy between the modes and the contracted state is instead reflected off the inner and outer potential walls about the turning point. It is well known that SCF methods work best for coordinate sys- tems specifically chosen so that the modes are as uncorrelated and separable (in either the spatial or frequency domain) as possible. This physical separability restriction is generally eased in TDSCF methods by the time dependence of the mean fields. In the case of a scattering event the more important correlations are dynamic due to the changes in interatomic distances which naturally accompany the reaction. The last panel of fig. 1 shows that the time dependence of the TDSCF mean fields is unable to represent the essential features of these correlations so that the energy exchanged between the modes is insufficient for reaction. This failure has been also observed in Jacobi coordinates in which the only coupling between the two modes comes from the potential terms. Nevertheless, the short time dynamics are well reproduced. Only at 1200 au when the exact wave- function begins to sample the reactive channel does the overlap of the TDSCF state with the exact state start to deviate from a value of near unity (fig. 2). This is perfectly in accord with the majority of TDSCF inves- tigations which are generally successful when applied to basically single channel dynamics as in the various studies of such “half reactions” as predissociation and photodissociation.

The results for the two multiconfiguration approximations given in the middle two panels of fig. 1 are in stark contrast to the simple TDSCF approximation. Whether the projection is performed prior to integrating the equations of motion (MCTDSCF- 1) or as a post-integration procedure (MCTDSCF-2), the consideration of only two configurations captures the essential aspects of the relevant correlations between the modes al- lowing for the energy transfer necessary to describe and simulate this scattering event. The entrance channel dynamics, where a correctly more contracted wavefunction is evolving, exhibit better agreement with the exact result than does the TDSCF description. While a more quantitative comparison of the MCTDSCF-1 and -2 methods will be given later, clearly the post-integration projection displays dynamics which are qualitatively in better agreement with the exact dynamics. Even such fine details as the wavefunction skewing at 800 au and the greater extent of delocalization of the reactively scattered as well as backscattered portions of the wave- function are well reproduced. However, there do exist differences between the exact and MCTDSCF propa- gations. In particular, the bimodal distribution of that portion of the wavefunction in the reactive exit channel is absent. With only two configurations, it may be expecting too much of an approximate theory to be able to

. \ _ ;0.1440 SC__

0.40 I I I I Fig. 2. Overlaps of the exact propagation with the post-integra- 0 850 1700 2550 3400 tion projection approximation (MCTDSCF-2). The dashed line

represents the modulus of the overlap and the solid line the over- time lap of the mod&.

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Table 2 Branching ratios

Energy EXACT MCTDSCF-1 MCTDSCF-2 TDSCF

-0.1549 23.8 0 17.6 -0.1500 53.5 52.9 39.4 -0.1440 75.0 84.5 75.5 0

represent bifurcation into two distinct channels as well as an additional correlated separation. The multiconfigurational approaches were also examined at two lower collision energies and compared with

the exact propagation. The post-integration projected MCTDSCF approximation again showed the better qual- itative agreement. At the lowest collision energy, where tunneling appears to be an important mechanism, the preprojection procedure (MCTDSCF-1 ) gave no reactively scattered products. Yet it showed better quanti- tative agreement at an intermediate energy with respect to the branching ratio. Table 2 summarizes the branch- ing ratios obtained at the three different collision energies. The high result for MCTDSCF- 1 is consistent with the greater percentage of its wavefunction which reaches the saddle point before collision with the repulsive wall. The incoming part of the wave interferes with the part reflected off the potential, positioning amplitude in the saddle region into the exit channel. Unlike the exact and MCTDSCF-2 results, here the wavefunction is not rather sharply peaked but displays high probability throughout a large region of configuration space. The dynamics of the MCTDSCF-2 wavefunction at the intermediate energy which gives a lower than actual branch- ing ratio behaves in an opposite manner to that given above. However, there is an important difference. Qual-

Fig. 3. “Quantum” trajectory representations of the scattering simulations. The heavy solid line is for theexact propagation while the dotted and dashed lines denote the MCTDSCF-1 and MCTDSCF-2 approximations respectively. The light solid line, which is displaced by 0.25 au, is a nonreactive case, TDSCF at high and MCTDSCF-1 at low energies. x average

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itatively the approximate state is in excellent agreement with the exact wavefunction as far as spatial extent and even the position of peak maxima and other fine details are concerned. The lack of agreement is solely in intensity. At later times, the approximations at lower energies may also be suffering from an inability to properly represent tunneling. The transfer of amplitude from one configuration to another is governed by the specific choice of projection, including its time dependence, and by the overlap of 9, and &. While the addition of more configurations will always improve the approximations, the incorporation of another configuration on the x spatial dimension should definitely be considered.

More insight into the nature of the approximations can be gained by inspecting other average properties. One way to visualize the quantum molecular dynamics is to average the bond distances and form a “quantum” trajectory. Fig. 3 represents the resultant trajectories for the scattering simulations in fig. 1. For all of the ap- proximations the classical turning point is well estimated, exhibiting the expected decrease with increasing en- ergy, and the entrance channel dynamics are well reproduced.

The success of the multiconfiguration approach depends on the ability of the different configurations to cap- ture the essence of the dynamical behavior. Fig. 4 displays examples of the two approximations, MCTDSCF- 1 and MCTDSCF-2, to the dynamical correlation. By construction the wavefunctions 1, and ,Q are orthogonal

MCTDSCF-1

Re(dhxl + +2x2)

MCTDSCF-2

Fig. 4. Decomposition of the multiconfiguration wavefunctions into their respective configurations for the -0.1440 au simulations at 2300 au. Shown are the real parts ofeach separate configuration and their sum. The MCTDSCF-1 wavefunction is constructively formed from spatially separated configurations while the post-integration projected MCTDSCF-2 wavefunction exhibits a great mixing of its constituent configurations whose orthogonality is maintained by phase mismatch.

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(eqs. (4) and (5 ) ). The manner in which this orthogonality is constructed is a direct indication of the cor- relations incorporated by the methods. The dominant dynamical correlation in reactive scattering is the split- ting of the wavefunction into reactive and nonreactive parts. The MCTDSCF-1 method constructs a spatial orthogonality (fig. 4 left panels) which addresses the observation that the two chemical channels are spatially separated from each other. On the other hand in the MCTDSCF-2 method the orthogonality is constructed through phase relations (fig. 4 right panels) which means a momentum-type correlation in which the chemical channels are identified through the direction of the outgoing velocity. As the momentum changes sign during the collision the projection operator which is associated with the correlation has to be time dependent. The probability for reaction of the two methods compared to the exact value is presented in table 2.

4. Conclusions

Quantum mechanical treatment of reactive scattering poses difficult challenges for approximate theories. The combination of a continuum spectrum in both reactant and product channels with differing asymptotic co- ordinates are the important obstacles. These difficulties plague application of mean-field methods because of the need to describe the splitting of the wavefunction into distinct chemical channels. This aspect of reactive scattering requires that at least one configuration be included for each differing chemical channel. The lack of configurational flexibility is the reason for the failure of simple (single configuration) TDSCF to describe a typical bimolecular reaction.

Multiconfiguration TDSCF is an appropriate generalization of simple TDSCF to deal with situations with several reactive channels. Formally, one can construct MCTDSCF equations from time-dependent variational principles. One then has the possibility of selecting as many configurations as required, and of implementing the multiconfiguration treatment in a variety of ways. It is appropriate to select physically motivated projection operators to describe the differing configurations. Different projections describe different correlations, and cer- tain projeclions will clearly be more appropriate than others; the more appropriate projections will lead to a more rapid convergence of the description as the number of configurations is increased.

Considering that the wavefunction evolves in time from the reactant through the collision region to the final state (a superposition of backscattering and product channels), the physically more reasonable choice for a projection operator is a time-dependent projector, which separates the outgoing reacted wavepacket from the backscattered component. Such a dynamical projection operator has been implemented and applied to the H + H2 reaction. The resulting calculations, here denoted as MCTDSCF-2, have better qualitative and quantitative features than either the simple TDSCF or static projections, at least at this simple two-configuration level.

The methods described here are intended to be used in large and more complex dynamical encounters. It is therefore important to examine the scaling of the method with dimensionality. Eq. (4) suggests that the method scales as M+ 1 where M is the number of configurations multiplied by D- 1 where D is the number of degrees of freedom of the system. This growth in computation is slow compared to the full dynamics which scale as the power of D. An alternative very promising MCTDSCF approach has been formulated by Meyer, Manthe, and Cederbaum [ 401. Their formulation scales as D2; in their scheme more correlations are taken into account, the cost of more elaborate computational effort. In higher dimensional calculations, a product form for the potential is a time saving necessity for computational feasibility. Bond coordinates with pair po- tential construct an efficient representation.

Examination of figs. l-4 shows that the simple multiconfiguration methods proposed here qualitatively de- scribe the nature of the H + Hz reactive collision. Scattering of the packet and its self interference near the sad- dle point, the multi-peaked product density, the closer approach to the turning point at higher energies, and the overall reactivity patterns are appropriately described even by the very simple two-configuration model employed here. These features are not reproduced in simple TDSCF (again in bond coordinates as well as Jacobi coordinates), which does not develop any component in the reactive channel.

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A number of important technical points can be made. In particular we note that second-order differencing schemes for the time propagation, which can be useful for exact dynamics, become problematic in the MCTDSCF context due to complex valued averages, that can lead to divergence. The multicontigurations and the mixed potential and kinetic terms make the use of the split operator propagation scheme very difficult. Alternative integration schemes based on a low order polynomial approximation of the evolution operator, as employed here, have a considerable advantage over other propagation methods.

Perhaps most important, the use of time-dependent projectors to define an MCTDSCF approximation to chemicalIy reactive scattering seems to overcome the two difficulties that simple TDSCF has with such systems; firstly it permits description of many channel scattering, as will be necessary for any true reactive process with quantum yield less than unity. Secondly, it permits physically based definition, on the basis of a time-dependent projector, of precisely which channels will be used to construct the configurations. Thirdly, it contains dy- namical corrections absent in simple TDSCF. Finally, insight is gained by watching the dynamical evolution of the wavefunction, something that time-independent formalisms simply cannot do.

MCTDSCF methods should share many of the advantages of simple TDSCF (conservation of norm and energy, general applicability to weak and strong coupling limits, classical and semiclassical limits, good ac- curacy) and like TDSCF methods possess coordinate dependence. The relatively slow growth of computational effort with increase of the system’s dimensionality is a great advantage of these methods, indicating consid- erable promise for their use in dynamical calculations of quantum mechanical encounters.

Acknowledgement

This research was supported by a grant from the GIF, the German-Israeli Foundation for Scientific Research and Development. The Fritz Haber Research Center for Molecular Dynamics is supported by the Minerva Gesellschaft ftir die Forschung, GmbH Munich, Federal Republic of Germany. ADH gratefully acknowledges receipt of a Lady Davis postdoctoral fellowship. MR thanks the Chemical Division of the NSF for partial support.

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