D. Tranca, F.J. Keil

437

Journal of Chemical Technology and Metallurgy, 48, 5, 2013, 437-444

AB INITIO MOLECULAR DYNAMICS STUDIES OF CHEMICAL REACTIONS

D. Tranca, F.J. Keil

Hamburg University of Technology, Institute of Chemical Reaction Engineering, Eissendorfer Str. 38, 21073 Hamburg, Germany

ABSTRACT

The present paper describes the dynamics of hexane cracking in ZSM-5 and methanol oxidation on vanadium beta zeolite (V-BEA) by means of ab-initio Molecular Dynamics (AIMD) and transition path sampling (TPS).

Keywords: ab-initio Molecular Dynamics, transition path sampling, hexane cracking, methanol oxidation.

Received 25 May 2013Accepted 29 July 2013

INTRODUCTION

Empirical force fields allow to simulate very large molecular systems (>100 K atoms) by generating tra-jectories of several nanoseconds. They are of limited usefulness because they quite often cannot be trans-ferred to other systems. In particular, empirical force fields cannot account for strong variations in the struc-tural and electronic properties. In chemically complex systems, many types of atoms and molecules give rise to extremely many different interactions making para-metrization of empirical force fields a formidable task. Furthermore, in case a specific system is understood by means of empirical force fields, changing a single spe-cies provokes often enormous efforts to parametrize the new potentials needed.

Another approach employed is transition path sampling (TPS). This method does not require fore-knowledge of reaction mechanism, and, therefore, is suitable for studying complex dynamical structures of high-dimensional systems. The dynamics of many such systems involve rare but important transitions between long-lived states, for example reactants and products of a chemical reaction which are separated by a barrier. In

this case, the system spends the bulk of its time fluctu-ating within stable states, so that transitions occur only rarely. In the frame of TPS it is sufficient to specify the reactants and products of a transition. A set of markedly different reactive trajectories is investigated from which the reaction mechanisms can be found.

Ab initio molecular dynamics (AIMD) where the forces acting on the nuclei are computed on-the-fly from electronic structure calculations, can overcome these limitations. There are various levels of sophistication available for executing AIMD calculations. In the pre-sent paper the Born-Oppenheimer molecular dynamics (BO-MD) and transition path sampling (TPS) will be employed.

MethodsThe fundamental idea of every ab-initio molecular

dynamics approach is to compute the forces acting on the nuclei from electronic structure calculations on-the-fly as the molecular trajectory is generated. The electronic variables, therefore, are not integrated out beforehand but are considered as active degrees of freedom. To date, density functional theory (DFT) is the most commonly electronic structure theory [1,2], but any other quantum

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chemical approach can be used as well. The mathemati-cal problem to be solved in AIMD is the evaluation of expectation values O of arbitrary operators ),(ˆ PRO with respect to the Boltzmann distribution [3]:

( , )

( , )

ˆ ( , )ˆE

E

d d O eO

d d e

b

b

−

−= ∫

∫

R P

R P

R P R P

R P (1)

where R and P are the nuclear positions and momenta, while

B B = 1/k T (k b ≡ Boltzmann constant, T ≡ absolute temperature). The total energy is given by

)(21

2R

PV

MEEE

N

I I

Ipotkin +=+= ∑

= (2)

where the first term on the right-hand side denotes the kinetic energy of the nuclei and the second term the potential energy. The term N indicates the number of nuclei and MI the respective masses. Based on Birk-hoff’s ergodicity hypothesis, one can solve Eq. (1) as the temporal average:

1ˆ ˆlim ( ( ), ( ))t

O dtO t tt→∞

= ∫ R P (3)

Birkhoff’s theorem says that, in the absence of extra conserved quantities, the time average of any observable over the phase space trajectory of a system point that starts anywhere on the energy surface will be equal to the microcanonical average of the same observable [3]. By employing Newton’s equation of motion, the nuclei are treated only classical instead of quantum mechanical. Usually this is a minor approximation, except for light atoms, like hydrogen, or low temperatures. Similar to Monte Carlo [4] also Molecular Dynamics [5] employs importance sampling by preferentially visiting phase space of low energy. A relation between molecular simulations and thermodynamics is given by the equi-partition theorem [3]:

2

1

1 32 2

N

I BI

M Nk T=

=∑ IR (4)

where R is the time derivative of the nuclear positions. The other terms have the same meaning as in Eq. (2). The interatomic forces ( )

II RF V= −∇ R are determined “on-the-fly” using DFT methods. Therefore, AIMD does not depend on adjustable parameters, but only on R. Finding the ground state electronic eigenfunctions ˆ( ) | ({ }; ) | ( )o el i o nucV H Vψ ψ= +R r R R

of the Hamiltonian related to the respective molecular system involves a lot of computational costs. The Born-Oppenheimer approximation (separating the electronic and nuclear motions) is mostly employed. Then, the potential energy may be written as:

ˆ( ) | ({ }; ) | ( )o el i o nucV H Vψ ψ= +R r R R (5)

where ˆ ({ }; )el iH r R is the electronic Hamiltonian that depends on the electronic coordinates {ri} and para-metrically on the nuclei coordinates R. As the nuclei have masses that are about three orders of magnitude larger than the electrons, the Born-Oppenheimer ap-proximation is justified, as the electrons are nearly instantaneously in equilibrium with the much heavier nuclei. The term Vnuc(R) represents the nuclear repul-sion. Using density functional theory (DFT) the potential energy, calculated “on-the-fly”, is given by the following expression [6]:

( ) [{ }; ] [{ [ ( )]}; ] ( )KSi i nucV E E Vψ ψ r= = +R R r R R

(6)

KS ≡ Kohn-Shamr(r) ≡ ground state electron density

iψ ≡ Kohn-Sham orbitalIn the Born-Oppenheimer Molecular Dynamics

(BOMD) the potential energy E[{ iψ };R] is minimized at every MD step with respect to { iψ (r)} under the constraint

( ) | ( )i j ijψ ψ d=r r (7)

This leads to the following Lagrangian:

20

1

{ } ,

1({ }; , )2

min [{ }; ] ( | )i

N

B i II

i ij i j iji j

L M

Eψ

ψ

ψ ψ ψ d=

= −

− + Λ −

∑

∑

IR R R

R

(8)

By solving the corresponding Euler-Lagrange equations

I I

d L Ldt

∂ ∂=

∂ ∂R R

(9a)

| |d L Ldt ψ ψ

∂ ∂=

∂ ∂ (9b)

D. Tranca, F.J. Keil

439

one obtains the associated equations of motion:

{ } { | }min [{ }; ]

|

|2 |

|

ii j ij

I I RI i

ij i jij

iij j

i ji

M R E

E

E

ψ ψ ψ dψ

ψ ψ

ψ d ψd ψ

=

= −∇ =

∂ ∂

− + Λ −∂ ∂

∂− − Λ ∂

∑

∑ ∑

I I

I

R

R R

R

(10a)

ˆ0 | | || ij j e i ij j

j ji

E Hd ψ ψ ψd ψ

≤ − + Λ = − + Λ∑ ∑

(10b)

The first term on the right-hand side of Eq. (10a) is the Hellmann-Feynman force [7], the second term is the Pulay force [8] which is a constraint force owing to the orthonormality constraint (Eq. (7)). The Pulay force is non-vanishing if and only if the basis functions jφ

explicitly depend on R. The last term originates from the fact that there is always an implicit dependence on the atomic positions through the expansion coefficients cij(R) within the common linear combination of atomic orbitals jφ :

( ) ( )i ij jj

cψ φ= ∑R R (11)

The last term on the right-hand side of Eq. (10a) vanishes whenever jφ (R) is an eigenfunctions of the Hamiltonian within the subspace spanned by the basis set. In the BOMD approximation, the electronic and nuclear subsystems are fully decoupled from each other. Thus, the electronic structure part is reduced to solving a time-independent quantum problem by solv-ing the Schrödinger equation employing Hartree-Fock approach, density functional theory (DFT) or any other quantum chemical method, concurrently to propagat-ing the nuclei via classical Molecular Dynamics. This procedure will be used for hexane cracking.

A further approach is the transition path sampling (TPS) which is based on a random walk in the space of reactive trajectories [9,10]. A reactive trajectory is a sequence of states, described by the reaction coordinates and their conjugate momenta, generated by molecular dynamics simulations, connecting reactants (A) and products (B). States A and B are distinguished on the basis of suitable geometry parameters ( )φ that change significantly during the reaction.

In analogy to Monte Carlo simulations, a random walk in reactive trajectory space is realized via trial moves, called "shooting" and "shifting". The number of states t along each trajectory is fixed in TPS. In a "shoot-ing" move, a state i on the past trajectory is selected randomly and the corresponding momenta are modified by adding a small Boltzmann-distributed noise (Fig. 2).

The shooting move is accepted if it still connects A and B; otherwise it is rejected.

In a “shifting” move, a state i from a region typical for A (or B) is chosen, all states before (after) i on the trajectory are discarded and the same number of states is added at the other end of trajectory by continued in-tegration of the equations of motion (Fig. 3).

A path in TPS is defined as a discretized sequence

A

B

A

B

reactive trajectories non-reactive trajectories

A A

B B{ , }x pi i { , }x pi i

{ ; }x p+ pi i id

shooting move integration forward and backward

A

B

shifting move

Fig. 1. Reactive and non-reactive trajectories.

Fig. 2. In a shooting move for Newtonian dynamics a new path is created from an old one by perturbing the momenta at a randomly selected time slice.

Fig. 3. In a shifting move for Newtonian dynamics a new path is created from an old one by translating the initial point of the trajectory forward in time or backward in time.

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of states 0 t 2t[x ,x ,x ,...,x ] tD D≡x in which consecutive states, or time slices, are separated by a small time incre-ment Dt. A specific time slice x = {r, r} consists of the positions and momenta of all particles in the entire system.

The path length t is a fixed time which is chosen a priori. But this is not necessary. The probability P[x] of a path x to occur in the path ensemble depends on the distribution of initial conditions and on the equations of motion of the underlying dynamics.

The path probability (Markovian dynamics) can be expressed as follows:

∏

−D

=D+D →=

1/

0)1(0 )(/)()(][

t

ititi ZxxpxP

ttrx (12)

r(x0) ≡ distribution of initial conditions (phase space density, for example, canonical: r0 = exp(-bH(x0)), microcanonical: r0 = d(E-H(x0))r(xiDt) → x(i+1)Dt ≡ short time probability to go from xiDt to x(i+1)Dt in a time DtZ(t) ≡ normalization factor

Z(t) = [ ];D P x D∫ ∫x x summation over all pathways x The transition path ensemble (TPE) is defined as

the subset of trajectories that connect states A and B:1

0( ) ( ) ( ) [ ] ( )AB AB A BP Z h x P h xtt−=x x

0( ) ( ) [ ] ( )AB A BZ D h x P h xtt = ∫ x x (13)

00

1 if 1 if( ) ( )

0 otherwise 0 otherwiseA B

x A x Bh x h x t

t

∈ ∈ = =

The constraint hA(x0)hB(xt) ensures that paths that do

not connect A and B have zero weight in this ensemble. ZAB includes only A-to-B paths (Fig. 4).

In TPS no dividing surface is defined. For deter-ministic dynamics, the transition probability is a delta function

( ) [ ( )]t t t t t t tp x x x f xd+D +D D→ = − (14)

fDt(xt) ≡ short-time propagator, for example, in MD the Verlet algorithm.

Substituting of Eq. (14) and Eq. (12) in Eq. (13) yields:

10 0

/ 1

( 1)0

[ ] ( ) ( ) ( )

( ) [ ( )]

AB AB A Bt

i t i ii

P Z x h x h

x x f x tt

t

t r

d

−

D −

+ D=

=

− D∏

x (15)

0 0 0( ) ( ) ( ) ( )AB A BZ dx x h x h xtt r= ∫

Integrations over the states along the path have been carried out at all times except zero.

The path sampling techniques aim to generate trajectories x according to the probability distribution function Eq. (15) by applying a Markov chain MC scheme. Random walks through the trajectory space are created by accepting or rejecting them according to the Metropolis criterion.

- A first path x(o) belonging to the path ensemble (connecting A and B) is generated.

- From this existing path, a new trial path x(n) is cre-ated with generating probability Pgen[x

(o) → x(n)] where the superscripts “o” and “n” stand for “old” and “new”.

- One then accepts the newly generated trial path with a probability Pacc[x

(o) →x(n)].To maintain the desired path ensemble distribution

of Eq. (4), it is sufficient to obey the detailed balance condition:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

[ ] [ ] [ ]

[ ] [ ] [ ]

o o n o nAB gen acc

n n o n oAB gen acc

P P P

P P P

→ → =

= → →

x x x x x

x x x x x (16)

Because x(o) belongs to the transition path ensemble, hA(x(o)) = hB(xt

(o)) = 1.Using Eq. (15), the ratio of the acceptance prob-

abilities yields:

( ) ( )( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )

[ ][ ] [ ]( ) ( )[ ] [ ] [ ]

n oo n ngenn nacc

A o Bn o o o nacc gen

PP Ph hP P Pt

→→=

→ →

x xx x xx xx x x x x

(17) where P[x] is given by Eq. (12), and Pgen depends on the algorithm that generates a new path from an old one. A common way to fulfill condition (17) is by applying the

hA = 1 hB = 1

oldnew

Fig. 4. Trajectories from A to B.

D. Tranca, F.J. Keil

441

Metropolis rule:

( ) ( )( )( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )( )( ) ( )

( ) ( ) ( )

[ ][ ][ ] min 1, ( ) ( )[ ] [ ]

[ ][ ]( ) ( ) min 1,[ ] [ ]

n ongeno n n n

acc A o B o o ngen

n ongenn n

A o B o o ngen

PPP h hP P

PPh hP P

t

t

→→ = =

→ →

= →

x xxx x x xx x x

x xxx xx x x

(18)The steps of the shooting move are:- The first step in the shooting procedure is to select

a time slice )(ot ′x on the old path x(o), with 0 ≤ t’ ≤ t.

this slice is changed by changing the momenta of all or some of the atoms (Fig. 5).

- A new path then evolves automatically by integrat-ing the equations of motion backward and forward in time from the modified state )(n

t ′x until t = 0 and t = t,

respectively. The TPS will be employed for simulating the oxidation of methanol to formaldehyde on vanadium beta zeolite.

Applicationsa) Hexane cracking on ZSM-6 and Faujasite zeolites

Hexane may be cracked at various bondings (C1-C2, C2-C3, C3-C4) [15]. Adsorption and cracking on ZSM-5 and zeolite Y (FAU) have been calculated by a QM/MM approach [16]. The H-MFI and Y zeolites are represented by a T276 and a T400 cluster size, respectively, and the zeolite structures were taken from X-ray diffraction data[17]. For the ZSM-5 a Si atom was replaced by an Al atom at the T12 site, and the resulting negative charge was compensated by a proton bonded to one of the neighboring framework oxygens. The QM/MM implementation (QChem) [18] follows an electro-static embedding scheme. The atoms in the QM part (30 atoms) were fully relaxed, whilst the MM part (802 atoms for ZSM-5, 1176 atoms for FAU) was kept in its crystal-lographic positions. The B97 6-31G* functional was

A

Bpt'

p't'

x'o xo

Fig. 5. Shooting move.

Fig. 6. Snapshots of C2-C3 clearage of hexane on ZSM-5.

Journal of Chemical Technology and Metallurgy, 48, 5, 2013

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employed completed by dispersion corrections accord-ing to Grimme [11]. Transition state (TS) calculations were executed using the dispersion corrected range-separated wB97X-D 6-311G* functional [12]. Once the TS is known, the trajectories along the reaction paths are calculated. In Fig. 6 there are snapshots shown of how some products are formed starting from the TS C2-C3. The acidic hydrogen is shared by C2-C3 fragments for the first 336 fs. After 336 fs, the hydrogen can be trans-ferred either to the C2 or C3 atoms. When it is transferred to C3, C4H10 is formed. This leaves the C2 fragment as C2H5

+ which can either deprotonate and form C2H4 or form the alkoxide intermediate structure. At about 469 fs, C2H4 is formed. The second trajectory reveals similar sharing of the acidic H for more than 324 fs, but then the H is transferred to the C2 structure forming C2H6 after 338 fs. This leaves the C4 fragment C4H9

+ which can either deprotonate and form C4H8 isomers or form the alkoxide intermediate structure. The C4H8 isomers are formed after 677 fs. The alkoxide structures are formed after 624 fs. Fig. 7 shows the C3-C4 cracking process in ZSM-5. The molecule splits into C3H8, which is a stable structure, and C3H7, which has a proton in access. The C3H6 molecule is formed after approximately 800 fs. Over this period of time C3H8 moves 4.5 Å away from the initial TS, while C3H6 diffuses 1.5 Å away from the initial TS. The dotted line shows the movement of the alkoxide structure (the line is nearly horizontal). The alkoxide is stabilized near the zeolite acid site starting from 650 fs. The dotted line shows the motion of cracked C3-C4 atoms. The formation of the C-H-C intermediate

is formed up to 400 fs. From 0 to 400 fs, a very small movement is observed in this region (C-H-C intermedi-ate state), but the system is cracked after 400 fs.

Only a few results are presented above. One ob-serves that the BOMD approach gives automatically intermediate molecules.

b) Methanol oxidationThe DFT calculations were carried out using the

VASP program [19], where the electronic wave functions have been expanded into plane waves up to an energy cut-off of 500 eV, and a projected-augmented-wave (PAW) scheme [20] has been used in order to describe the interactions between the valence electrons and the nuclei (ions). The crystal structure of vanadium beta zeolite consisted of a tetragonal unit cell with fixed edge lengths of a = 12.7 Å, b = 12.7 Å, and c = 26.77 Å. The vanadium beta zeolite (V-BEA) contains nine different tetrahedral sites (T-sites). The most stable adsorption has been proven to be on the T1 site [13]. The reaction calculations done were for this T1 site. The transition states were localized using the improved dimer method [14]. In order to compute the adsorption energies, the total electronic energies for the methanol in vacuum, formaldehyde and H2 molecules had also to be computed. Dispersion corrections were introduced. Starting from the transition state (TS) about 1500 tra-jections were calculated employing the transition path sampling method [21].

In the scheme below the steps of formaldehyde formation are shown for closed shell simulations: Si

O

VSi O O Si Si O O Si

Si

O

V

Si

O

V

Si

O

V

Si

O

V

1.81.22

O HCH3

OCH3

H CH2O

H

OCH3

O O

O

O

O

2.13

1.8

2.41

2.7

2.7

2.7Si O O Si Si O O Si

Si O O Si + CH O2

H

H

H

The vanadium atom is single-bonded to three O-Si Fig. 7. C3-C4 cracking process in ZSM-5. Distances away from the transition state as a function of time.

D. Tranca, F.J. Keil

443

groups, and double-bonded to an O atom. The H-atom belonging to CH3OH points towards the V-O-Si group (d(O-H) = 1.22 Å. The adsorption energy for this structure is -47.51 [kJ/mol] (zero-point energy (ZPE) corrected: -41.63 [kJ/mol]). In the first step of the reac-tion, an interaction between the proton belonging to the CH3OH group and the V-O-Si site is observed. This in-teraction leads to the formation of a methoxide structure and a H-OSi group. A subsequent interaction between the CH3O and the vanadium, leads to the formation of the CH2O product. The geometric details of the reaction are presented in Fig. 8.

This is only one example of two further reaction schemes.

CONCLUSIONS

Employing ab initio Molecular Dynamics (AIMD) and transition path sampling (TPS) enables the detection of intermediates in reaction mechanisms without prior

knowledge of the reactions. This gains an edge over conventional approaches.

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