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3 QM/MM, Transition State Theory

3 QM/MM, Transition State Theory

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90



F.J. Keil



useful whenever one has to model a localised chemical reaction at an active site that

is influenced by an interacting larger environment. Recent advances in QM/MM

have been reviewed in various papers, for example [164–167]. In QM/MM simulations one wants to retain as much as possible the formalism of the methods that are

being combined and to introduce well-defined coupling terms. The entire system is

divided into the inner QM region that is treated quantum-mechanically and the

outer MM region that is described by a force field. There is not one single QM/MM

method, and the multitude of different implementations can be characterised by

several main distinctions. For example, the so-called subtractive methods apply the

QM approach to the active site and the MM method to the entire system, including

the active site. The MM contribution for the active site has to be subtracted:

tot

acts

E ẳ Eacts

QM ỵ EMM À EMM :



(29)



The advantage of this approach is that it allows one, in a simple way, to combine

different QM schemes and MM schemes. The disadvantage is that the active site has

to be calculated by MM which might be difficult for complex electronic structures.

The additive scheme applies the MM only to the environment of the active site, and

a coupling term has to be introduced for the two regions:

coupl

envir

E ẳ Eacts

QM ỵ EMM ỵ EQM=MM :



(30)



The coupling terms normally include bonded terms across the QM/MM boundary, non-bonded vdW terms and electrostatic terms. A further problem is the treatment of the QM/MM boundary. The choice of the QM region is usually made by

chemical intuition. This region can be enlarged step-wise, and its sensitivity to the

QM/MM results can be checked. Standard QM/MM applications employ a fixed

QM/MM partitioning where the boundary between the QM and MM regions is

defined once and for all at the outset. Park and Heyden [168] have derived a mixedresolution Hamiltonian and an explicit symplectic integrator for conservative mixedresolution systems that allow for a dynamic change in resolution of selected groups

of atoms during MD simulation. The so-called adaptive partitioning of the Lagrangian (APL) method permits a simulation with accuracy comparable to an atomistic

one at the computational cost of a coarse-grained one.

DFT is the workhorse for the QM part. For extensive QM/MM MD simulations

one has to refer to semi-empirical methods. Linear scaling local correlation methods have also been used [169]. For the MM part proper force fields have to be

employed, e.g. GROMOS [105], AMBER [101], OPLS [103]. The electrostatic

coupling between the QM charge density and the MM charge model can be done in

various ways [170]. Mostly, electrostatic embedding is employed which allows for

the polarisation of the QM region since the QM calculation is performed in the

presence of the MM charge model, whereby the MM point charges are included as

one-electron terms in the QM Hamiltonian. The treatment of the QM/MM boundary

can be executed in various ways. Most schemes give nearly the same results as long



Multiscale Modelling in Computational Heterogeneous Catalysis



91



as the charges at the QM/MM boundary are carefully treated [171]. Introduction of

dangling hydrogen bonds or treating the frontier functional group as a pseudo-atom

with an effective one-electron potential are the most common approaches.

Nowadays many QM and MM software packages offer QM/MM capabilities.

ChemShell (www.chemshell.org) software is an example of a modular QM/MM

implementation.

Finding the transition states in high-dimensional spaces is a challenging problem. Transition states are first-order saddle points. The algorithms for finding firstorder saddle points on one spin PES can be divided into two groups: (1) approaches

based on interpolation between a reactant and a product minimum and (2) those

using only local information. A combination of both algorithms is probably the

most efficient way of finding first-order saddle points. Interpolation methods

generate a sequence of approximate MEP by interpolating between a reactant and

a product state. The highest energy configuration along an MEP is a first-order

saddle point. Both reactant and product states must be known so that these methods

cannot reveal unexpected chemical pathways with multiple intermediates. Furthermore, if multiple pathways exist, only that nearest to the interpolated guess will be

found [10]. The interpolation algorithms convert a saddle point search in configuration space to a minimisation problem in discretised path space. Minimisation

problems in path space can easily handle large numbers of low-frequency modes,

a significant challenge for most local surface walking algorithms. Interpolation

algorithms include, for example, nudged elastic band (NEB) [172] and the string

method [173]. These methods initiate the search for a transition state by assuming

that the MEP is a straight line in multidimensional space connecting the reactant

and product states. Peters et al. [10] have shown that the growing string method, an

interpolation method that does not require an initial guess for the initial pathway,

needs significantly fewer gradient calculations to find the saddle point than the NEB

and the string method.

Local surface-walking algorithms explore the PES using local gradient and

usually second derivative information. These methods can be initiated anywhere on the PES. These algorithms perform poorly for systems with several

low-frequency vibrational modes or for searches started far from a transition state.

Furthermore, even if a transition state is found it is possible that it does not

connect reactant and product states. Therefore, it is recommendable to employ

an interpolation algorithm like the growing string method to generate a starting

point for the local surface walking algorithm. Two of the most used algorithms

of this type are the P-RFO method by Baker [174] and the dimer method by

Henkelman and Jo´nsson [175] or its improved version by Heyden et al. [11]. The

latter method is available in some commercial program packages like VASP [65]

or QChem [13].

The reaction rate constants are mostly calculated based on the harmonic TST.

Comprehensive review of this subject was presented by H€anggi et al. [176]. The

rate coefficients for elementary reactions on a catalyst surface are obtained by

conventional TST in the following way:



92



F.J. Keil







kB T QTS Tị

Eỵ

exp

kTST Tị ẳ

;

h QR ðTÞ

RT



(31)



where kB is Boltzmann’s constant, h Planck’s constant, T is the absolute temperature

and E+ is the difference in electronic energies between the transition state and the

reactant state, respectively. The partition functions of the transition state, QTS(T), and

the reactant state may be calculated, for example, like this:

!



 pffiffiffi 

0:5 3nÀ6

Y

p

2pMkB T

T3

1



oel ;

Àyvj=T

s yA yB y C

h2

j 1Àe



(32)



where M is the molecular mass, yi are the moments of inertia, the yvj the normal

modes and oel the electronic energy. A transmission coefficient can also be introduced which has the general form like this [19]:

gTị ẳ GTịkTịgTị;



(33)



where G(T) arises from dynamical recrossing. It takes into account that some

trajectories that cross the dividing surface in the direction of products recross and

return to the reactant region. G(T) is smaller than one. k(T) arises from quantum

mechanical tunnelling. k(T) is greater or equal to one. g(T) takes deviations of the

equilibrium distribution in phase space into account. g(T) can be either less than or

greater than one. In conventional TST g(T) is set equal to one. Further developments of TST may be found in papers by Truhlar’s group [7–9].

There are important examples where the harmonic/rigid-rotor approximation to

TST fails in describing the reaction kinetics. Even worse, simulations based on the

static approach can sometimes lead to completely incorrect prediction of the

reaction mechanism. For example, in catalytic transformations of short alkanes,

entropy plays an important role. During the reaction the mobility of the reactants

varies according to the strength of their interactions with the zeolite, leading to a

substantial entropy contribution to the free-energy reaction barrier. Entropy can

even stabilise some otherwise unstable reaction intermediates, opening unexpected

alternative reaction channels competing with the mechanism deduced from a static

TST search. Therefore, one has to explore the free-energy surface and not just of the

PES in configuration space. Bucko and Hafner [177] have shown that the static

approach, corrected for dynamical effects within harmonic TST, is insufficient for

describing reactions including weakly bound adsorption complexes such as hydrocarbon conversion reactions. The most important reasons for this failure were found

to be as follows. (1) An adsorption complex identified by static total-energy

minimisation is not a proper representation of the reactant state. Hence the work

needed to create an adsorption complex represents in some cases an important

contribution to the free-energy barrier. This contribution is not taken into account in

harmonic TST, which is based on the analysis of the energy surface in the vicinity

of stationary points only. (2) The static approach does not account for reaction



Multiscale Modelling in Computational Heterogeneous Catalysis



93



intermediates which are not potential-energy minima, hence it does not allow for

changes in the reaction mechanism induced by thermal fluctuations.

Bucko and Hafner [177] employed transition path sampling techniques [178,

179] to overcome the static TST problems (see also [196]).

Other general aspects for computing catalytic reactions are discussed by

Raimondeau and Vlachos [38, 184], Berendsen [180] and Broadbelt and Snurr [181].



3 Applications: From the Active Centre to the Chemical

Reactor

An example of a heterogeneous catalytic reaction will be presented, which, to the best

of my knowledge, for the first time describes the complete picture consistently from

the active centre to the chemical reactor [182, 183]. Previous simulations of heterogeneous catalytic reactions were summarised by Raimondeau and Vlachos [184],

Broadbelt and Snurr [181], Vlachos [185] and Santiso and Gubbins [36]. Furthermore,

results on simulations of heterogeneous catalytic reactions may be found in books by

van Santen and Sautet (eds.) [40] and van Santen and Neurock [41]. Christensen and

Norskov [186] describe some computational investigations on trends of reactivity of

catalyst surfaces.

Molecular simulation of heterogeneous catalytic reactors, which is a multiscale

problem (Fig. 5) initiated by quantum chemical calculations, may be used in combination with TST to obtain intrinsic kinetic data and to elucidate the reaction

mechanism.

In [182, 183] the alkylation of benzene with ethene over H-ZSM-5 was investigated. The electronic energies and vibrational frequencies of each stationary point

along the reaction coordinate were calculated for the two mechanisms presented in

Fig. 6.

Adsorption from the gas phase into the zeolite leads to a decrease of the potential

of each molecule (Fig. 7).

macroscopic

m



mesoscopic

mm



μm

microscopic

nm



active center



Fig. 5 Orders of magnitude in heterogeneous catalytic reactors



94



F.J. Keil

One-step alkylation of benzene:

H2C

H

Si



CH2



H



O



O



Al



Si



CH2



+ H2C



+



O



Si



ZOH



O



Al



Si



co-adsorption



ZOH(C2H4,C6H6)

H2C



CH2

H



H

O



Si



O



Al



k1



Si



O



Si



k-1



O



Al



Si



reaction



ZOH (C8H10)

H



H

O



Si



O



Al



Si



Si



O



O



Al



desorption



+



Si



Two-step alkylation of benzene:

CH2



H2C

H

Si



H



O



O



Al



CH2



+ H2 C



Si



O



Si



O



Al



Si



adsorption



ZOH

H2C



CH2



C



O



Si



Al



C

Si



O



Al



O



k1



Si



k-1



O



Si



Al



CH3



H

O



CH3



H



H



O



H

Si

CH3



H



H



C



Si



+



Si



O



Al



reaction 1



O



H

Si



adsorption



ZOH2H5(C6H6)

H

Si



O



Al



CH3

C

O



H



H

Si



Si



O



Al



O



Si



reaction 2



ZOH(C8H10)

H



H

Si



O



Al



O



Si



Si



O



Al



O



Si



+



desorption



Fig. 6 Elementary steps involved in the alkylation of benzene



This is caused by dispersion interaction between the carbon atoms of the reactant

and the oxygen atoms of the zeolite framework. Interaction of the reactants with the

Brønsted acid site further reduces the potential. The intrinsic energy barrier for

the forward reaction is the difference between the bottom of the well for the coadsorption of A and B at the active site and the top of the transition state. First, the



Energy



Multiscale Modelling in Computational Heterogeneous Catalysis



95



Transition

structure

A(g), B(g)



A(ads, O),

B(ads, O)



C(g)

E I1



A(ads, H+),

B(ads, H+)



E I-1



C(ads, O)

C(ads, H+)



(1)



(2)



(3)



(4)



(6)



(5)



(7)



Reaction coordinate



Fig. 7 Schematic energy diagram for a zeolite-catalysed one-step reaction A + B ! C. (1)

Reactants in the gas phase. (2) Adsorption of A and B in the zeolite channels. (3) Co-adsorption

of A and B on the Brønsted acid site. (4) Formation of the transition state. (5) Product adsorption

on the Brønsted site. (6) Product adsorption in the channel system. (7) Product in the gas phase



reaction mechanism for the alkylation was elucidated using DFT applied to cluster

representations of the active site. Second, the MP2:DFT hybrid approach suggested

by Tuma and Sauer [193] was employed. Structure optimisation of all stationary

points within the full ZSM-5 unit cell using DFT with periodic boundary conditions

has been the first step in this approach. Stationary points were characterised by

harmonic frequencies obtained by diagonalisation of the full dynamical matrices

(absence of imaginary frequencies for minima and presence of exactly one imaginary frequency for all transition structures). To confirm that the transition states

were connected to the correct energy minima, each transition state was perturbed

slightly along the reaction coordinate in the reactant and product direction. The perturbed geometries were used as starting geometries for energy minimisation. The

second step was the calculation of the high-level correction; that is the difference

between MP2 and DFT single-point adsorption energies and energy barriers, respectively, for clusters of progressively larger size. These clusters were cut out from

the periodic DFT-optimised structure. The size-dependent high-level correction

[187]

DECịhigh ẳ DECịMP2 DECịDFT



(34)



was then extrapolated to the periodic structure (S). This periodic model limit,

~

DEðSÞ

high ; is added to the plane-wave DFT energy for the periodic structure,

DE(S)DFT, to get an estimate of the MP2 energy for the full periodic system,



96



F.J. Keil



~

~

~

DESị

MP2 ẳ DESịDFT ỵ DESịhigh ;



(35)



where the tilde is used to discriminate energies which were obtained through fitting

and/or extrapolation, respectively, from those obtained directly from quantum

chemical calculations. The third step was the extrapolation of the MP2 energy to

the CBS limit. Additionally, single-point coupled cluster calculations [CCSD(T)]

were conducted to account for higher order correlation effects. Thermodynamic

contributions arising from finite temperatures were taken into account by calculation of partition functions within the harmonic approximation assuming separability of electronic translational, rotational and vibrational terms. The first step in

linking the outcome of a quantum chemical calculation (i.e. the PES) to a rate of

reaction is the calculation of rate coefficients for elementary steps. For the alkylation of benzene the barrier crossing is an infrequent event because the condition

Ebarr/RT ) 1 holds. This allows the use of classical TST. Therefore, the intrinsic

rate constants were calculated according to (31)/(32).

The next step is the modelling of multi-component adsorption and diffusion

inside the pores (Fig. 8).

This sequential strategy is possible if the length and time scales of the problem

are well separated, and if suitable models for linking the levels of modelling are

available. The adsorption isotherms for ethene, ethane, benzene, ethylbenzene and

hydrogen in MFI (all silica form) were determined by “configurational-bias Monte

Carlo” (CBMC) simulations in the grand canonical ensemble at a variety of temperatures. The outcome of these simulations depends on the quality of the force

fields. A united atom force field was employed that was parametrised against adsorption isotherms in zeolites. The simulated isotherms were fitted to a three-site

Langmuir expression:

qi; sat; A bi; A fi qi;sat; B bi; B fi qi; sat; C bi; C fi





;

1 þ bi; A fi

1 þ bi; B fi

1 þ bi; C fi



qi fi ị ẳ



(36)



C

B



C

B

gas phase

T, p



A



A



bou

nda

ry



A



B



lay

er



A



A

C



+



B

C



A

9

1



8

2



3



4

5



porous

support



A



7

B



C

6



Fig. 8 Adsorption and diffusion inside the pores



1) Diffusion through the boundary layer

2) Adsorption at the pore entrance

3) Diffusion of reactants inside the pores

4) Adsorption at active centers

5) Reaction of the surface

6) Desorption of products

7) Diffusion of products inside pores

8) Desorption from the pore exit

9) Diffusion of the products through the

boundary layer



Multiscale Modelling in Computational Heterogeneous Catalysis



97



where qi,sat,X denotes the saturation capacity of species i on site X, bi,X is the affinity

constant and fi is the gas phase fugacity of species i. The continuum level of an

entire pellet is calculated according to the partial differential equation

@qi

1 1 @ 2

1

z Ni ị ỵ ni r;

¼À 2

r z @z

r

@t



i ¼ 1; 2; . . . ; n



(37)



where qi is the loading of species i, r is the zeolite framework density, z the

diffusion path, Ni the molar flux of species i, ni the stoichiometric coefficient and

r is the rate of reaction. As the composition of reactants and products along the

pores changes continuously due to reaction, the adsorption equilibria for arbitrary

compositions inside the pores need to be calculated. This was achieved by means of

ideal adsorbed solution theory (IAST) [194] which requires only the pure component isotherm data as input. These isotherms are taken from CBMC simulations.

The suitability of the IAST has to be checked by a limited number of MC simulations of the multi-component adsorption equilibria. Therefore, IAST is a linking

model for MC adsorption results.

MD simulations were carried out in a rigid zeolite framework for a variety

of loadings and temperatures, employing the same force field as was used in the

CBMC simulations. From these data, self-diffusivities and Maxwell–Stefan diffusivities were extracted. These diffusivities were used in the Ni terms of (37). Again

one has to check the validity of the Maxwell–Stefan approach by means of some

multi-component MD simulations. As the composition of the molecular mixture

inside the pores changes along the pore and with time, one has also to refer to a

linking model for diffusivities, as a complete calculation by MD would be by far too

time consuming. The Maxwell–Stefan approach serves as a linker. For details

see [182].

The rate of reaction of the one-step mechanism (see Fig. 6) is given by

r

¼ r~ ẳ kf qEỵB;Hỵ kr qEB;Hỵ ;

r



(38)



where kf and kr are the rate coefficients for the forward and reverse reaction of

ethene and benzene to form ethylbenzene, respectively, qEỵB;Hỵ is the amount of coadsorbed “ethene + benzene” at the active sites and qEB;Hỵ is the amount of adsorbed ethylbenzene at the active sites. As benzene and ethylbenzene are mostly sited

in the zeolite cages whilst the ethene molecules can move into the channels and the

cages, the assumptions made for Langmuir–Hinshelwood kinetics are not fulfilled.

For qEỵB;Hỵ and qEB,H analytical expressions are needed in order to calculate them

from species loadings qE, qB, and qEB. As was outlined in [182], such terms may be

obtained from MC simulations of the multi-component adsorption isotherms. As

benzene and ethylbenzene are mostly sited inside the cages, fitting to Langmuir–

Hinshelwood kinetics is not possible.

To conclude, by employing suitable linking models based solely on molecular data, one can obtain results on a macro level, i.e. composition profiles inside



98



F.J. Keil

Conventional analysis: D = const.

1

2 bar



effectiveness factor



10 bar



conventional



T = 653 K

pB / pE = 5



multiscale model



0.1

10-7



10-6



10-5



particle radius



Fig. 9 Effectiveness factor



catalyst pellets and reactors. The pellet (37) is solved, for example, by a finite

difference approach, and a reactor plug flow model was used for the entire reactor.

The results are presented in [182, 183]. One result is discussed here. Figure 9 shows

plots of the effectiveness factor vs particle radius. The bulk phase composition has a

benzene/ethylene molar ratio of five. It can be seen that the conventional results

obtained for a fixed value of an effective diffusivity deviates considerably from

detailed simulations based on real local compositions and rates along the pores, in

particular for higher pressures.



4 Outlook

Over the last few years there has been considerable progress in multiscale modelling

of catalytic processes, although there is still considerable room for improvement on

all levels. One of the weakest points is the harmonic TST. There is increasing

evidence that this fails for reactions involving loosely bound reactant and/or

transition states where entropy makes a significant contribution to the free-energy

reaction barrier. The free energy of activation may be derived by free-energy

integration schemes such as the Blue-Moon ensemble technique in combination

with constrained ab initio MD simulations. There is a need for accurate experimental

kinetic data to assess the reliability of modelling approaches for predicting effective



Multiscale Modelling in Computational Heterogeneous Catalysis



99



reaction rates for zeolite-catalysed reactions. The problem of vdW forces with DFT

calculations is in part solved. In heterogeneous catalysis, only for very few reactions

is the mechanism known in detail. Well understood are ammonia synthesis and the

hydrogenation of ethylene. The industrial Envinox process has also been investigated in great detail [188–191]. On the force field level there is a need for force fields

that can describe interactions between hydrocarbons and Brønsted acid sites. Such

force fields have to be parametrised against accurate quantum mechanical calculations because experimental data are hard to obtain. Reliable approaches are required

to calculate diffusion coefficients for slowly diffusing molecules such as aromatics

which are difficult to obtain directly from classical MD simulations. On the continuum level, the description of multi-component diffusion can be improved by

accounting for strong adsorption sites by using, for example, the effective medium

approximation by Coppens and co-workers [192]. Therefore, modelling of catalysed

heterogeneous reactions will be an active field of research in the future.



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