3 QM/MM, Transition State Theory
Tải bản đầy đủ - 0trang
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
Q¼
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
ỵ
ỵ
;
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.
References
1. Kohn W, Sham LS (1965) Self-consistent equations including exchange and correlation
effects. Phys Rev A 140:1133–1138
2. Kohanoff J (2006) Electronic structure calculations for solids and molecules: theory and
computational methods. Cambridge University Press, Cambridge
3. Martin RM (2008) Electronic structure: basic theory and practice methods. Cambridge
University Press, Cambridge
4. Szabo A, Ostlund NS (1996) Modern quantum chemistry: an introduction to advanced
electronic structure theory. Dover Publications, Mineola, NY
5. Helgaker T, Jørgensen P, Olsen J (2002) Molecular electronic theory. John Wiley & Sons,
New York
6. Shavitt I, Bartlett RJ (2009) Many-body methods in chemistry and physics: MBPT and
coupled cluster theory. Cambridge University Press, Cambridge
7. Truhlar DG, Steckler R, Gordon MS (1987) Potential energy surfaces for polyatomic
reaction dynamics. Chem Rev 87:217–236
8. Pu J, Gao J, Truhlar DG (2006) Multidimensional tunnelling, recrossing, and the transmission coefficient for enzymatic reactions. Chem Rev 106:3140–3169
9. Gao J, Truhlar DG (2002) Quantum mechanical methods for enzyme kinetics. Ann Rev Phys
Chem 53:467–505
10. Peters B, Heyden A, Bell AT, Chakraborty A (2004) A growing string method for determining transition states: comparison to the nudged elastic band and string methods. J Chem Phys
120:7877–7886
11. Heyden A, Bell AT, Keil FJ (2005) Efficient methods for finding transition states in chemical
reactions: comparison of improved dimer method and partitioned rational function optimization method. J Chem Phys 123:224101-1/14
12. Kresse G, Hafner J (1993) Ab initio molecular dynamics for liquid metals. Phys Rev B
47:558–561
13. Shao Y et al (2006) Advances in methods and algorithms in a modern quantum chemistry
program package. Phys Chem Chem Phys 8:3172–3191
100
F.J. Keil
14. Banerjee A, Adams N, Simons J, Shepard R (1985) Search for stationary points on surfaces.
J Phys Chem 89:52–57
15. McQuarrie DA (2000) Statistical mechanics. University Science Books, Sausalito
16. Hill TL (1987) An introduction to statistical mechanics. Dover Publications, Mineola, NY
17. Chandler D (1987) Introduction to modern statistical mechanics. Oxford University Press,
Oxford
18. Tuckerman ME (2010) Statistical mechanics: theory and molecular simulation. Oxford
University Press, Oxford
19. Truhlar DG, Garrett BC, Klippenstein SJ (1996) Current status of transition-state theory.
J Phys Chem 100:12771–12800
20. Zener C (1932) Non-adiabatic crossing of energy levels. Proc Roy Soc London Ser A
132:696–702
21. Frenkel D, Smit B (2001) Understanding molecular simulation, 2nd edn. Academic, San
Diego
22. Landau DP, Binder K (2009) A guide to Monte Carlo simulations in statistical physics, 3rd
edn. Cambridge University Press, Cambridge
23. Allen MP, Tildesley DJ (1989) Computer simulation of liquids. Oxford University Press,
Oxford
24. Newman MEJ, Barkema GT (1999) Monte Carlo methods in statistical physics. Oxford
University Press, Oxford
25. Binder K, Heermann DW (2010) Monte Carlo simulation in statistical physics: an introduction. Springer, Heidelberg
26. Rapaport DC (2004) The art of molecular dynamics simulation, 2nd edn. Cambridge
University Press, Cambridge
27. Griebel M, Knapek S, Zumbusch G (2009) Numerical simulation in molecular dynamics:
numerics, algorithms, parallelization, applications. Springer, Heidelberg
28. Tuckerman ME, Martyna GJ (2000) Understanding modern molecular dynamics: techniques
and applications. J Phys Chem B 104:159–178
29. Versteeg HK, Malalasekera W (1995) An introduction to computational fluid dynamics.
Prentice Hall, Upper Saddle River, NJ
30. Lomax H, Pulliam TH, Zingg DW (2001) Fundamentals of computational fluid dynamics.
Springer, Heidelberg
31. Laney CB (1998) Computational gasdynamics. Cambridge University Press, Cambridge
32. Grotendorst J, Attig N, Bl€
ugel S, Marx D (2009) Multiscale simulation methods in molecular
sciences. J€ulich Supercomputing Centre, J€
ulich
33. Sherwood P, Brooks BR, Sansom MSP (2008) Multiscale methods for macromolecular
simulations. Curr Opin Struct Biol 18:630–640
34. Koci P, Novak V, Stepanek F, Marek M, Kubicek M (2010) Multi-scale modelling of
reaction and transport in porous catalysts. Chem Eng Sci 65:412–419
35. Lynbarsev A, Tu YQ, Laaksonen A (2009) Hierarchical multiscale modelling scheme from
first principles to mesoscale. J Comput Theor Nanosci 6:951–959
36. Santiso EE, Gubbins KE (2004) Multi-scale molecular modeling of chemical reactivity. Mol
Simul 30:699–748
37. Starrost F, Carter EA (2002) Modelling the full monty: baring the nature of surfaces across
time and space. Surf Sci 500:323–346
38. Vlachos DG (2005) A review of multiscale analysis: examples from systems biology
materials engineering, and other fluid-surface interacting systems. Adv Chem Eng 30:1–61
39. Baeurle SA (2009) Multiscale modeling of polymer materials using field-theoretic methodologies: a survey about recent developments. J Math Chem 46:363–426
40. van Santen RA, Neurock M (2006) Molecular heterogeneous catalysis. Wiley-VCH, Weinheim
41. van Santen RA, Sautet P (2009) Computational methods in catalysis and materials science.
Wiley-VCH, Weinheim