Validity of the site-averaging approximation for modeling the dissociative chemisorption of H2 on Cu(111) surface: a quantum dynamics study on two potential energy surfaces.

Validity of the site-averaging approximation for modeling the dissociative chemisorption of H2 on Cu(111) surface: A quantum dynamics study on two potential energy surfaces Tianhui Liu, Bina Fu, and Dong H. Zhang Citation: The Journal of Chemical Physics 141, 194302 (2014); doi: 10.1063/1.4901894 View online: http://dx.doi.org/10.1063/1.4901894 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/19?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Site-specific dissociation dynamics of H2/D2 on Ag(111) and Co(0001) and the validity of the site-averaging model J. Chem. Phys. 143, 114706 (2015); 10.1063/1.4931040 The dissociative chemisorption of methane on Ni(111): The effects of molecular vibration and lattice motion J. Chem. Phys. 138, 174705 (2013); 10.1063/1.4802008 Reactive scattering of H 2 from Cu(100): Six-dimensional quantum dynamics results for reaction and scattering obtained with a new, accurately fitted potential-energy surface J. Chem. Phys. 121, 11379 (2004); 10.1063/1.1812743 Erratum: “Six-dimensional quantum dynamics of dissociative chemisorption of H 2 on Cu(100)” [J. Chem. Phys. 107, 3309 (1997)] J. Chem. Phys. 110, 2738 (1999); 10.1063/1.478000 Six-dimensional quantum dynamics of dissociative chemisorption of H 2 on Cu(100) J. Chem. Phys. 107, 3309 (1997); 10.1063/1.474682

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THE JOURNAL OF CHEMICAL PHYSICS 141, 194302 (2014)

Validity of the site-averaging approximation for modeling the dissociative chemisorption of H2 on Cu(111) surface: A quantum dynamics study on two potential energy surfaces Tianhui Liu, Bina Fu,a) and Dong H. Zhanga) State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People’s Republic of China

(Received 26 September 2014; accepted 3 November 2014; published online 18 November 2014) A new finding of the site-averaging approximation was recently reported on the dissociative chemisorption of the HCl/DCl+Au(111) surface reaction [T. Liu, B. Fu, and D. H. Zhang, J. Chem. Phys. 139, 184705 (2013); J. Chem. Phys. 140, 144701 (2014)]. Here, in order to investigate the dependence of new site-averaging approximation on the initial vibrational state of H2 as well as the PES for the dissociative chemisorption of H2 on Cu(111) surface at normal incidence, we carried out six-dimensional quantum dynamics calculations using the initial state-selected time-dependent wave packet approach, with H2 initially in its ground vibrational state and the first vibrational excited state. The corresponding four-dimensional site-specific dissociation probabilities are also calculated with H2 fixed at bridge, center, and top sites. These calculations are all performed based on two different potential energy surfaces (PESs). It is found that the site-averaging dissociation probability over 15 fixed sites obtained from four-dimensional quantum dynamics calculations can accurately reproduce the six-dimensional dissociation probability for H2 (v = 0) and (v = 1) on the two PESs. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4901894] I. INTRODUCTION

The dissociative chemisorption of molecular species on surfaces is of great importance to many industrial applications. It is a fundamental step in heterogeneous catalysis, and is often the rate-limiting step of the overall reaction. Tremendous efforts have been devoted to the understanding of the dissociative chemisorption dynamics both experimentally and theoretically in the last decades.1–15 Despite much progress achieved in the dissociative chemisorption studies, theoretically it is still challenging to investigate these processes at a full-dimensional quantum mechanical level, in particular for the polyatomic molecules dissociating on surfaces, due to the difficulties in constructing reliable potential energy surfaces (PESs) and developing quantum mechanical methodologies. Such full-dimensional quantum dynamics calculations were limited to the diatomic molecules such as H2 dissociating on metal surfaces,16–30 and the recent work of HCl/DCl dissociating on Au(111) surface.31–33 Early in 1995, Gross et al. carried out the first full-dimensional quantum dynamics calculation on non-activated dissociation of H2 on Pd(100) surface.25, 26 Kroes and co-workers demonstrated that electron-hole pair excitation need not be considered to obtain accurate results for activated dissociation of H2 on Pt(111) surface by performing full-dimensional quantum dynamics calculations.28 Recently, Jackson and Nave employed a fully quantum approach based on the Reaction Path Hamiltonian that includes all 15 molecular degrees of freedom and a) Authors to whom correspondence should be addressed. Electronic ad-

dresses: [email protected] and [email protected]

0021-9606/2014/141(19)/194302/8/$30.00

the effects of lattice motion to investigate the dissociative chemisorption of methane on a Ni(111) surface.34 Before this, reduced-dimensional models were employed in quantum dynamics studies of methane on Ni surfaces.35–38 Guo and co-workers carried out eight-dimensional quantum dynamics studies for CH4 dissociating on a Ni(111) surface,39 as well as six-dimensional (6D) quantum dynamics calculations of H2 O on Cu(111) and Ni(111) surfaces.40–43 Those PESs were fitted using the permutationally invariant polynomials44, 45 based on the density functional theory (DFT) calculations. In reduced-dimensional approaches,39–43, 46–50 the center of mass (COM) of the molecule was always fixed on a specific site during the reaction, i.e., the translational coordinates of the molecule in the xy plane were fixed. However, large differences were seen between the results using the fourdimensional (4D) site-specific and the full-dimensional (6D) calculations of diatomic dissociative chemisorption.17, 31, 32 Early in 1997, Dai and Light found that the dissociation probabilities from the 4D fixed-site quantum dynamics calculations are quite different from those from 6D calculations in the H2 +Cu(111) system.17 In addition, they averaged the 4D fixed-site dissociation probabilities for H2 initially in the ground rovibrational state (v = 0, j = 0) over three impact sites (bridge, center, and top) to obtain the site-averaging dissociation probability, which is similar in shape with the 6D dissociation probability, while the latter is shifted to higher energy by about 0.05 eV. They assumed this energy shift was caused by the zero point energy (ZPE) differences between the 4D and 6D calculations at the transition states, as the 6D calculation incudes the ZPE for the two lateral coordinates x and y. However, the shifted 4D site-averaging dissociation

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probability is substantially smaller than the 6D dissociation probability at high kinetic energy.17 Hence, the site-averaging approach proposed by them cannot reproduce the 6D probability. Jackson and co-workers have extensively studied the site and lattice effects for dissociative chemisorption of CH4 on various metal surfaces,34, 46 and their treatments within the sudden approximation have successfully reproduced the experimentally observed surface temperature dependence of dissociative sticking coefficients. However, the assumption made in the sudden approximation indicates that probabilities at different impact sites have the same energy dependence, but varying with the barrier height, which is not quite reasonable for some systems. They average over the results for many (X, Y) sites which cover the surface unit cell, sampling all possible impact sites on the surface. The consequence of the siteaveraging is shift of the probability curve along the energy axis. Recently, we carried out the 4D fixed-site and 6D quantum dynamics calculation for the HCl+Au(111) and DCl+Au(111) reactions,31, 32 based on the LFZ PES developed by neural network fitting to DFT energy points.33 A new finding for the site-averaging approximation was presented for both reactions for HCl/DCl initially in (v = 0, j = 0) state,31, 32 i.e., the 6D dissociation probability can essentially be reproduced without ZPE corrections by averaging the 4D site-specific dissociation probabilities over impact sites, as long as enough impact points are used in the averaging. If the validity of this new site-averaging approximation generally holds in other molecule-surface reactions, it should be a very promising approach to investigate the moleculesurface reactions at a full-dimensional quantum mechanical level. We can obtain the exact dissociation probabilities by averaging the site-specific results, instead of directly calculating the full-dimensional dissociation probabilities, which is much more time and memory consuming. In particular for the polyatomic molecules dissociating on surfaces, it is still very challenging to perform the full-dimensional quantum dynamics calculations. The H2 molecule dissociative chemisorption on surfaces such as Cu(111) is a prototype of diatomic molecule-surface reactions. It is very interesting to see if the validity of this new site-averaging approximation we observed holds in the H2 /Cu(111) system. In addition, further study is important to investigate the new site-averaging approximation for the reactions involving the vibrational excitation of the molecule reactant as well as the different PESs. In this work, we carry out the 4D site-specific and 6D quantum dynamics calculations for H2 dissociating on the Cu(111) surface, with H2 initially on (v = 0, j = 0) and (v = 1, j = 0) states, respectively, based on two different PESs. One early PES is the LEPS PES by Dai and Zhang,50 and another PES was developed using the specific reaction parameter approach to DFT points (SRP-DFT) by Kroes and co-workers.20 The new site-averaged approximation is investigated and discussed in detail for the title reaction. The paper is structured as follows: In Sec. II, we present the time-dependent wave packet method (TDWP) used in quantum dynamics calculation and briefly introduce the two

J. Chem. Phys. 141, 194302 (2014)

PESs we used. Detailed results and discussions are presented in Sec. III. A summary and conclusions are given in Sec. IV. II. THEORY A. The time-dependent wave packet method

The methodologies used in this study for the quantum dynamics calculations of the diatomic molecule dissociative chemisorption are similar to those used in our previous studies.31, 32 Briefly, a total of six degrees of freedom (6D), namely, X, Y, Z, r, θ , and φ, should be considered in the fulldimensional quantum dynamics calculations for the dissociative chemisorption of H2 on the corrugated, rigid Cu(111) surface, as shown in Fig. 1. The 6D Hamiltonian for the H2 +Cu(111) system is expressed as 1 ∂2 1 ∂2 jˆ2 Hˆ = − − + 2M ∂Z 2 2μ ∂r 2 2μr 2 1 ∂2 1 ∂2 1 2 cos α ∂ 2 + − − 2M sin2 α ∂x 2 sin2 α ∂x∂y sin2 α ∂y 2 +V (x, y, Z, r, θ, φ),

(1)

where M, μ, jˆ are the mass, reduced mass, and rotation operator of H2 , respectively. The x and y axes are parallel to the surface, Z is the perpendicular coordinate from the COM of H2 to the plane of the surface, r is the internuclear distance of H2 , θ is the polar angle, and φ is the azimuthal angle. The last term V (Z, r, x, y, θ, φ) is the interaction potential energy. The skewing angle α is the angle between the x and y coordinate axes as indicated in Fig. 1. For Cu(111), α equals 60◦ . The reference vibration eigenfunction φv (r) of H2 satisfies the equation ¯2 ∂ 2 − + V (r) φv (r) = εv φv (r), (2) 2μ ∂r 2 where V (r) is the 1D reference potential obtained from the total interaction potential V (x, y, Z, r, θ, φ) with other degrees of freedom fixed at specific values. H

ez

r Z θ

H

φ α =60

ex

Cu

ey FIG. 1. Molecular coordinates for the collision of H2 and Cu(111), where (x, y, Z) are the center-of-mass coordinates of H2 , r is internuclear distance of H2 , θ is the polar angle, and φ is the azimuthal angle. Here α is the skewing angle and equals 60◦ .


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The time-dependent wave function is expanded in terms of the translational basis of Z, φv (r), and angular momentum eigenfunctions Yjm (θ, φ) as (x, y, Z, r, θ, φ) = Fnvj m (t)uvn (Z)φv (r)Yjm (θ, φ)eikx x eiky y ,

center top

hcp

bridge

(3)

n,v,j,m

where kx = 2π nx /Lx and ky = 2π ny /Ly with Lx and Ly being the surface unit cell lattice constants. Because of the periodicity of the potential energy surface along x and y, we employed the superposition of periodic functions eikx x and eiky y to represent the wave functions of the x and y coordinates. The Z and r coordinates are efficiently represented by the sin-DVR basis functions,51, 52 while in the θ and φ direction, we used the direct product of two basis set, i.e., eimφ (m as an integer) and the Legendre polynomials, to represent the spherical harmonic functions by separating the φ-dependent term from jˆ2 .17, 53 The initial wave function (normal incidence) is chosen as (Z, r, x, y, θ, φ, t = 0) = G0 (Z)φv,j (r)Yjm (θ, φ),

(4)

where the wave packet G0 (Z) is chosen to be a standard Gaussian function (Z − Z0 )2 exp(−ik0 Z). (5) G0 (Z) = (π δ 2 )−1/4 exp − 2δ 2 The wave function is propagated using the split-operator method54 in which the exponential Schrödinger propagator is symmetrically split to propagate the wave packet.54 The timedependent wave function is absorbed at the edges of the grid to avoid boundary reflections.55 The initial state-selected total dissociation probability is obtained by projecting out the energy dependent reactive flux. + denotes the time-independent (TI) full scattering wave If ψiE function, where i and E are, respectively, initial state and energy labels, the total reaction probability from an initial state i can be obtained by the formula + + | Fˆ | ψiE . Pir = ψiE

(6)

In the above equation, Fˆ is the flux operator, defined as 1 Fˆ = [δ(ˆs − s0 )vˆs + vˆs δ(ˆs − s0 )], (7) 2 where s is the coordinate perpendicular to a surface located at s0 for flux evaluation, and vˆs is the velocity operator corresponding to the coordinate s. The full TI scattering wave + + | ψiE = 2π δ(E − E ). Using function is normalized as ψiE the expression in (7), Eq. (6) can be simplified to yield Pi (E) =

¯ + + Im(ψiE |ψiE )|s=s , 0 μr

where + = |ψiE

1 ai (E)

0

∞

ei(E−H )t/¯ |i (0)dt,

(8)

(9)

with ai (E) = φ iE | i (0) being the overlap between the initial wave packet i (0) and the energy-normalized asymptotic scattering function φ iE .

fcc

FIG. 2. The irreducible triangle of Cu(111) surface unit cell (shown in red triangle), which is approximated to be of C6v symmetry. The fcc and hcp sites are generalized to the center site. The atoms in the first, second, and third layers are shown in yellow, orange, and gray spheres, respectively. Note that the atoms in the third layer are right at the fcc site. Atoms in the fourth (fifth, sixth) layer are directly below the atoms in the first (second, third) layer, respectively.

Slightly different from the 4D calculations performed on the HCl/DCl+Au(111) system, the 4D site-specific quantum dynamics calculations for the H2 +Cu(111) reaction are carried out with the COM of H2 (x and y) fixed at the top, bridge, and center sites, respectively. As shown in Fig. 2, the Cu(111) surface was approximated to be of C6v symmetry by taking the potential at the hcp site to be the same as at the fcc site in the H2 +Cu(111) reaction. B. The two PESs

The two PESs were tested in the present work to investigate the new site-averaging approximation for the dissociative chemisorption of H2 on Cu(111). Although the Cu(111) surface is known to be of C3v symmetry, it was approximated to be of C6v symmetry in developing the two PESs of the reaction, with the potential energy on the hcp site the same as that on the fcc site. Fig. 2 shows the symmetric impact sites of Cu(111) surface unit cell, i.e., the bridge, top, hcp, and fcc sites. The hcp and fcc sites are generalized to be the center site under the approximation of C6v symmetry. One early PES,50 we used was a six-dimensional, corrugated PES constructed by Dai and Zhang as a LEPS function incorporating the DFT data of Hammer et al.56 (called the DZ PES). The DFT calculations employed a generalized gradient approximation (GGA) to compute the exchange-correlation energy.57 The reaction barrier height for the bridge site is lowest (0.72 eV), compared to those for the center site of 0.90 eV and the top site of 1.14 eV, indicating the dissociation on the fixed bridge site is much more favored. Interested readers can find more details of the PES in Refs. 50 and 58. Another six-dimensional PES was developed by Kroes and co-workers,20 where an implementation of the specific reaction parameter (SRP) approach to density functional theory was introduced.59 We called it the SRP PES henceforth. The PES was constructed using the corrugation reducing procedure (CRP) to a set of DFT energy points,60, 61 where the GGA was used to describe the electron exchange correlation effects, with the two most popular functionals, i.e., the


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J. Chem. Phys. 141, 194302 (2014)

SRP P W 91 RP BE V6D = xV6D + (1 − x)V6D ,

(10)

where the parameter x is obtained from the appropriate experimental data to produce the correct barrier height. In this PES, x equals 0.57. The barrier heights for the bridge site and top site on the SRP PES are 0.63 eV and 0.89 eV, respectively, while that for the center cite is highest (∼1.01 eV). More detailed description of the SRP PES can be found in Ref. 20. Basically, we can see that the two PESs were developed based on different approaches, and their resulting behavior such as the barrier heights are quite different . It was verified that the SRP PES is much more accurate, which can reproduce the experimental data with chemical accuracy in the kinetic energy region of [0.2, 0.9] eV.20 However, both of them were used in the current study due to the fact that we are interested in the dependence of the site-averaging approximation on the PES too. C. Computational details

The numerical parameters used in the dynamics calculations are given as follows: The two dimensional unit cell formed by x and y is covered by a 25×25 evenly spaced grid. For H2 initially in the ground rovibrational state (v = 0, j = 0), we used 127 and 40 sin-type DVR51, 52 points to describe Z and r coordinates, ranging from 1.0 to 16.0 bohrs and 0.5 to 6.0 bohrs, respectively. The orientation angle θ only needs 16 Legendre DVR points because of the symmetry and the maximum value of j (jmax ) equals 30. The azimuthal angle φ requires 61 evenly spaced Fourier grid points. The imaginary absorbing potentials are placed in the range of Z between 12.0 and 16.0 bohrs and r between 4.0 and 6.0 bohrs, respectively. The dissociation flux is calculated on the dividing surface of r = 3.5 bohrs. The time step for the propagation is 10 a.u. and we propagate the wave packets for 15 000 a.u. of time to converge the dissociation probabilities. The numeral parameters for H2 initially in the first vibrational excited state (v = 1, j = 0) are the same as those of the ground state, except the sin-type DVR points for r increases to 60 and jmax increases to 40. III. RESULTS AND DISCUSSIONS A. 4D and 6D results

The dissociation probabilities obtained from the 6D TDWP calculations based on the DZ PES, together with the 4D results for H2 fixed at the bridge, center, and top cites are shown in Fig. 3 as a function of the kinetic energy, with H2 initially in the ground rovibrational state (v = 0, j = 0) (Panel (a)) and initially in the first vibrational excited state (v = 1, j = 0) (Panel (b)), respectively. As shown in Fig. 3(a), the 4D site-specific and 6D dissociation probabilities for H2 initially in the ground state increase steadily as the kinetic energy increases, presenting the thresholds of roughly 0.5 eV, 0.65 eV,

1

(a) v=0 Dissociation Probability

PW9157 and the RPBE62 functionals, respectively. As a result, P W 91 two PESs based on the two functionals were obtained: V6D RP BE and V6D . The two PESs were then combined to generate a new PES, which can be described as

0.8

DZ PES

4D bridge site 4D center site 4D top site 6D

0.6

0.4

0.2

0 0.4

0.6

0.8 Kinetic Energy (eV)

1.0

1.2

0.4


0.8

1.0

1

(b) v=1 Dissociation Probability

194302-4

0.8

0.6

0.4

0.2

0 0.2

FIG. 3. (a) Comparison of dissociation probabilities for the scattering of H2 (v = 0, j = 0) from Cu(111) between the six-dimensional (6D) calculation and four-dimensional (4D) site-specific (bridge, center, and top) calculations on the DZ PES. (b) Same as (a), except for H2 (v = 1, j = 0).

and 0.9 eV for the results with H2 fixed at the bridge, center, and top sites, respectively, and the threshold of roughly 0.55 eV for the 6D dissociation probability. The dissociation probabilities for the bridge site are much larger than those for the top site and the 6D calculations in the entire energy region, and those for the center site in the low kinetic energy region just above the threshold, indicating the dissociation over the bridge site is much more favored. The dissociation probabilities for the center cite approach those for the bridge site in the kinetic energy above 0.9 eV. We note that the current 4D results on the DZ PES for H2 (v = 0, j = 0) are quite similar to those calculated by Dai and Zhang50 and Dai and Light,17 and our 6D results are also similar to those obtained by Dai and Light,17 except we calculated the results at higher collision energies. We can see the substantial threshold reduction from the corresponding results of H2 (v = 1, j = 0) in Fig. 3(b), compared to the results of H2 (v = 0, j = 0) shown in Fig. 3(a). The dissociation threshold for the bridge cite is lowest (∼0.2 eV), compared to those for both the center site and 6D calculations of roughly 0.25 eV, and those for the top site of roughly 0.5 eV, respectively, indicating the vibrational excitation of H2


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reduces the dissociation thresholds of bridge site and 6D calculations by about 0.3 eV, and those of center site and top site by about 0.4 eV. The dissociation probability of the bridge site first rises very quickly from the threshold, and remains nearly the constant (∼1.0) when the kinetic energy increases from 0.5 eV to 1.0 eV. The similar behavior is also seen from the results for the center site. The dissociation probability from the 6D calculations shows the similar threshold as the center site, but rise monotonically with increasing kinetic energy. On the whole, the 6D probability is quite different from the results calculated by reduced dimensional (4D) fixed-site approach, indicating the importance of full-dimensional calculations. Similarly, we calculated the 4D site-specific and 6D dissociation probabilities on the SRP PES, and show the results in Fig. 4. Overall, the dissociation probabilities as a function of kinetic energy on the SRP PES show much more oscillatory structures than those on the DZ PES, in particular for the curves of the center site and top site above 0.9 eV. These oscillations are possibly due to the fact that the SRP PES was developed for the dynamics investigations in kinetic energies lower than 0.9 eV, where the experimental data were reproduced with chemical accuracy.20 The 6D probability is quite

0.8

Dissociation Probability

(a) v=0 0.6

4D bridge site 4D center site 4D top site 6D

SRP PES

0.4

0.2

0 0.4

0.6


1.0

1.2

1


0.8

different from the three 4D site-specific results, which is also found on the DZ PES. As seen from Fig. 4(a), the dissociation thresholds of H2 initially in the (v = 0, j = 0) state are roughly at the kinetic energy of 0.5 eV, 0.75 eV, and 0.7 eV for the bridge, center, and top sites, respectively, in accord with the lowest barrier for the bridge site and highest barrier for the center site on the SRP PES. This is different from the DZ PES, on which the dissociation occurs over the top site with the highest barrier. The 6D dissociation probability rises steadily with increasing kinetic energy from the threshold of about 0.55 eV. The results of (v = 1, j = 0) in Fig. 4(b) show that the dissociation threshold for the top cite is similar with the 6D probability (∼0.2 eV), which is lower than that for the bridge site of roughly 0.3 eV and the center site of 0.35 eV. Thus, the vibrational excitation of H2 reactant is most efficient in reducing the threshold of the top site, compared to the rest of site-specific results and 6D results. B. Site-averaging approximation

Comparisons of the dissociation probability calculated using the 6D TDWP method and the site-averaging 4D results based on the DZ PES and SRP PES are made and shown below, respectively. Specifically, the site-averaging probabilities over 3 sites, 6 sites, and 15 sites as well as the 6D probabilities were calculated. The 3 site-averaging dissociation probability were obtained by averaging over the three high symmetric sites (bridge, center, and top), as shown in Fig. 2, with appropriate relative weights (3 for the bridge site, 2 for the center site, and 1 for the top site), which is the same as Dai and Light did.17 We can see from Fig. 5(a) that the current 6 sites consist of three midpoints of the top-center line, top-bridge line, and bridge-center line, and the original three fixed sites (bridge, center, and top). Similarly, the 9 midpoints of the two adjacent sites of the 6 sites mentioned above and the 6 fixed sites constitute the 15 sites, as shown in Fig. 5(b). The averaged dissociation probability over the 6 sites was obtained from the 4D fixed-site calculations with appropriate relative weights (3 for the bridge site, 2 for the center site, 1 for the top site, 6 for the sites on the boundary of the triangle, and 12 for the sites inside the triangle). The 15 site-averaging dissociation probability was calculated using the similar relative weights as the 6 site-averaging probability. The site-selecting routine

0.6

top

0.4

(a)

0.2

top

bridge

(b)

center

0 0.2

0.4

0.6 0.8 Kinetic Energy (eV)

1.0

FIG. 4. (a) Comparison of dissociation probabilities for the scattering of H2 (v = 0, j = 0) from Cu(111) between 6D calculation and 4D site-specific (bridge, center, and top) calculations on the SRP PES. (b) Same as (a), except for H2 (v = 1, j = 0).

bridge

center

FIG. 5. (a) The schematic of the distribution of 6 sites (3 black dots denote the bridge, center and top sites, and 3 red dots denote the midpoints of the adjacent sites). (b) The schematic of the distribution of 15 sites (9 additional sites correspond to the midpoints of the adjacent sites of the 6 sites shown above).


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1

Dissociation probability

(a) v=0 0.8

DZ PES

3 site-averaged

SRP PES 6 site-averaged 0.6

15 site-averaged 6D

0.4

0.2

0 0.4

0.6

0.8 Kinetic energy(eV)

1.0

1.2


0.8

1.0

1


0.8

0.6

0.4

0.2

0 0.2

0.4

FIG. 6. (a) Comparisons of six-dimensional dissociation probabilities and three site-averaging dissociation probabilities of H2 (v = 0, j = 0) based on the DZ PES, obtained by averaging the 4D dissociation probabilities over 3, 6, and 15 sites with appropriate relative weights. (b) The same with (a) except for H2 (v = 1, j = 0).

is the same as that of the HCl/Au(111) system,31 but the relative weights and the number of sites are different because the Cu(111) surface was approximated to be of C6v symmetry. The 3 site, 6 site, and 15 site-averaging dissociation probabilities, together with the 6D dissociation probability on the DZ PES are displayed in Fig. 6(a) for H2 initially in the ground rovibrational state (v = 0, j = 0), and in Fig. 6(b) for H2 initially in the first vibrational excited state (v = 1, j = 0), respectively. First, the 3 site-averaging dissociation probability for H2 (v = 0, j = 0) is substantially larger than the 6D probability in the entire energy region, although the shape of the 3 site-averaging curve somehow follows that of the 6D curve. Dai and Light contributed the differences between the 3 site-averaging and 6D dissociation probabilities to the differences in ZPE between six-dimensional and four-dimensional calculations at the transition states.17 However, as they have noted, the 6D dissociation probability was obviously larger than the shifted site-averaging dissociation probability at high kinetic energies.17 As a result, the explanation that ZPE differences lead to the energy shift in the six-dimensional results from the four-dimensional results is not convincing. Here, for

this reaction, we extend their site-averaging approach and examine the validity of the new site-averaging approximation we found in the HCl/DCl+Au(111) reaction.31, 32 It is clearly seen that the similarity between the 6 site-averaging and 6D dissociation probabilities is significantly improved, compared to the 3 site-averaging and 6D results. The overall behavior of the 6 site-averaging dissociation probability resembles that of the 6D dissociation probability, with the magnitude of the former slightly larger than the latter at the kinetic energy just above the threshold, then slightly lower in the medium energy region and finally higher in the kinetic energy region higher than 0.8 eV. Furthermore, as we see from the green and blue curves, the agreement between the 15 site-averaging probability and 6D probability is even better, in particular at the kinetic energy less than 0.8 eV, where the agreement is quite good. Another question is whether the validity of new siteaveraging approximation also holds in the vibrationally excite state of H2 . The results shown in Fig. 6(b) give us the answer. For the first time, the validity of the new site-averaging approximation in the molecule-surface reactions with the vibrationally excited molecule reactant is checked. We can see that the 3 site-averaging curve is much larger than the 6D curve at the kinetic energy lower than 0.6 eV, but the situation is significantly improved when more sites are included. The 6 site-averaging curve becomes close to the 6D curve, and the agreement between the 15 site-averaging and 6D curves is excellent and impressive. As a result, the 6D dissociation probabilities for the title reaction can be accurately reproduced by 15 site-averaging probabilities from 4D calculations with H2 initially in v = 0 and v = 1 states on the DZ PES. Next, we consider the validity of new site-averaging approximation on the SRP PES for the dissociative chemisorption of H2 on Cu(111). Figure 7 shows the results obtained by performing the same calculations as have been done in Fig. 6. For H2 initially in the ground rovibrational state, we can see in Fig. 7(a) large differences between the 3 siteaveraging dissociation probability and the 6D dissociation probability, with the magnitude of the former larger than the latter at the kinetic energies lower than 0.64 eV and smaller than the latter at kinetic energies above 0.64 eV. Obviously, the similarity between the 6 site-averaging and 6D dissociation probabilities is significantly improved, compared to the 3 site-averaging results and 6D results, for which the same trend is seen on the DZ PES. In addition, the agreement between the 15-site averaged probability and 6D probability is better than that between the 6 site averaged probability and 6D probability. In particular at the kinetic energy lower than 0.75 eV, the agreement is pretty good. It is also interesting that the site averaging scheme seems to wash out the oscillatory structures on the 4D site-specific probabilities on the SRP PES. For H2 initially in the first excited vibrational state, we can see from Fig. 7(b) that the 3 site-averaging probability is basically smaller than the 6D probability at the whole kinetic energy region. The 6 site-averaging probability becomes close to the 6D probability and the agreement between the 15 siteaveraging and 6D results is better, though the 6D probabilities are slightly larger than the 15 site-averaging results at the kinetic energy roughly above 0.5 eV, and slightly smaller just above the threshold.


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J. Chem. Phys. 141, 194302 (2014)

0.8


(a) v=0 0.6

3 site-averaged 6 site-averaged 15 site-averaged 6D

SRP PES

0.4

0.2

0 0.4

0.6


1.0

1.2

0.8


(b) v=1

abilities are also obtained by doing quantum dynamics calculations for H2 fixed at the bridge, center, and top sites, respectively. The validity of new site-averaging approximation essentially holds in the H2 +Cu(111) surface reaction both for H2 (v = 0, j = 0) and H2 (v = 1, j = 0) on the two PESs, i.e., the six-dimensional dissociation probability can be accurately obtained from the site-averaging results over 15 sites from the 4D site-specific calculations, without ZPE corrections, although with the approximation the full-dimensional results will not be exactly reproduced. Based on the validity of new sited-averaging approximation in HCl+Au(111), DCl+Au(111), and H2 +Cu(111) surface reactions, we are now confident that the validity of new site-averaging approximation, we observed should generally hold in many moleculesurface reactions. It is significant because it probably provides a good opportunity to investigate the dissociative chemisorption of many polyatomic molecules on metal surfaces at a fulldimensional quantum mechanical level, for which the direct full-dimensional calculations are currently very challenging.

0.6

ACKNOWLEDGMENTS 0.4

We thank Professor G. J. Kroes for sending us the SRP PES for this work. This work was supported by the National Natural Science Foundation of China (Grant Nos. 90921014 and 21303197), Ministry of Science and Technology of China (2013CB834601), and the Chinese Academy of Sciences.

0.2

0 0.2

1 R.

0.4


0.8

1.0

FIG. 7. (a) Comparisons of six-dimensional dissociation probabilities and three site-averaging dissociation probabilities of H2 (v = 0, j = 0) based on the SRP PES, obtained by averaging the 4D dissociation probabilities over 3, 6, and 15 sites with appropriate relative weights. (b) The same with (a) except for H2 (v = 1, j = 0).

Overall, the new site-averaging approximation has been verified to be efficient in the prototypical diatomic molecule dissociative chemisorption on the metal surface, i.e., H2 +Cu(111) surface reaction, for H2 initially in its ground rovibrational state and first excited vibrational state on two different PESs (the DZ PES and SRP PES). It is interesting that we can accurately obtain the 6D dissociation probabilities from the site-averaging 4D results without ZPE corrections, which should save lots of computational time and memory costs. IV. CONCLUSIONS

To summarize, the initial state-selected six-dimensional TDWP method has been employed to carry out the quantum dynamics calculations for the dissociative chemisorption of H2 on the rigid Cu(111) surface, with H2 initially in (v = 0, j = 0) and (v = 1, j = 0) states. The two different PESs, i.e., the DZ PES by Dai and Zhang, and the SRP PES by Kroes and co-workers, are used in the calculations. The corresponding four-dimensional site-specific dissociation prob-

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