Modelling the adsorption of short alkanes in protonated chabazite: The impact of dispersion forces and temperature

Graphical abstract

Highlights

► Alkane adsorption in chabazite is modelled using electronic structure theory. ► Finite temperature effects are estimated by molecular dynamics simulations. ► An extrapolation mechanism to finite temperature is proposed. ► Results are critically compared to experimental data.

Abstract

The adsorption of alkanes in a protonated zeolite has been investigated at different levels of theory. At the lowest level we use density-functional theory (DFT) based on semi-local (gradient-corrected) functionals which account only for the interaction of the molecule with the acid site. To describe the van der Waals (vdW) interactions between the saturated molecule and the inner wall of the zeolite we use (i) semi-empirical pair interactions, (ii) calculations using a non-local correlation functional designed to include vdW interactions, and (iii) an approach based on calculations of the dynamical response function within the random-phase approximation (RPA). The effect of finite temperature on the adsorption properties has been studied by performing molecular dynamics (MD) simulations based on forces derived from DFT plus semi-empirical vdW corrections. The simulations demonstrate that even at room temperature the binding of the molecule to the acid site is frequently broken such that only the vdW interaction between the alkane and the zeolite remains. The finite temperature adsorption energy is calculated as the ensemble average over a sufficiently long molecular dynamics run, it is significantly reduced compared to the T = 0 K limit. At a higher level of theory where MD simulations would be prohibitively expensive we propose a simple scheme based on the averaging over the adsorption energies in the acid and in the purely siliceous zeolite to account for temperature effects. With these corrections we find an excellent agreement between the RPA predictions and experiment.

1. Introduction

Acid zeolites are widely used in industry as catalysts for hydrocarbon conversion reactions. Zeolites are not only very efficient catalysts, the interactions between the hydrocarbon molecules and the inner walls of their cavities also permit to design catalysts with a very good selectivity. The diameter and shape of the pore structure of the zeolites and the interactions of molecules with the zeolite framework is also of decisive importance for their use as molecular sieves.

The simplest way to test the interaction between molecular reactants and the zeolite is to investigate their adsorption properties. For alkanes (saturated hydrocarbon molecules) experiment has demonstrated that the zero-coverage adsorption energy increases roughly linearly with the length of the molecular chain. In an acid zeolite one of the terminal CH₃ groups forms a weak chemical bond to the acid site while the remaining part of the molecule interacts with the zeolite framework through dispersion forces (van der Waals interactions). With increasing length of the alkane chain the adsorption energy increases by a constant increment for each additional CH₂ group. The contribution of the van der Waals (vdW) interactions to the adsorption energy depends on the specific channel and pore geometries of the material. This picture has been confirmed by almost constant differences between the adsorption energies of alkanes in purely siliceous, protonated and Na-exchanged zeolite, independent of the chain length. Eder and Lercher [1] were able to relate the experimentally observed increase in the adsorption energy with increased alkane chain length to the framework density, but an understanding at molecular level cannot be obtained from experiments.

Many theoretical investigations of the adsorption and diffusion of alkanes in zeolites (calculation of adsorption isotherms, diffusion coefficients, …) use molecular dynamics simulations based on empirical force fields describing the interactions between the molecules and between the molecules and the zeolite [2]. These techniques even permit predictions about the size- and shape-selectivity of special zeolite structures [3].

Investigations of processes where chemical bonds may be broken and re-formed require, however, a higher level of theory. Stronger chemical bonds between reactants are described with good accuracy by density functional theory (DFT). DFT is based on the Hohenberg–Kohn–Sham theorem which states that the ground-state energy of a many-electron system may be expressed as a function of the one-electron density. However, DFT cannot describe the weak dynamical electron–electron correlations which cause the vdW interactions. DFT calculations of the adsorption of alkanes in a zeolite based on semi-local exchange–correlation functionals describe the interaction of the molecules with the acid site with reasonable accuracy, but not the increase of the adsorption energy with increasing chain length [4].

Post-DFT corrections designed to account for dispersion forces have been proposed by Grimme [5]. The corrections are expressed as semi-empirical pair interactions, the force-field parameters are determined by comparison with high-level quantum-chemical calculations for a large molecular training set. The easy implementation and low computational costs made it the method of choice in many studies. At a higher level of theory, the semi-local correlation functional of DFT has been replaced by a non-local functional designed to produce the vdW forces arising from long-range non-local electronic correlations [6]. The non-local part of the functional was constructed using a simplifying approximation to the dielectric function of the system in the Random Phase Approximation (RPA). The approach has been extended to the calculation of the Hellmann–Feynman forces acting on the atoms, permitting fully selfconsistent calculations of the atomistic geometry including the influence of the vdW forces. Recently full many-body calculations of the total energy in the RPA in connection with the Adiabatic-Connection Fluctuation–Dissipation Theorem (RPA-ACFDT) [7,8] have been performed. These calculations include the vdW energy seamlessly and accurately, but can be performed only at a fixed atomic geometry.

Very recently we have investigated the adsorption of small alkanes in Na-exchanged chabazite using fully periodic calculations at all three levels of theory [9]: (i) The force-field approach of Grimme correcting DFT calculations with the gradient-corrected exchange–correlation functional of Perdew, Burke and Ernzerhof [10] – PBE-d. (ii) The non-local “vdW-functional” of Dion et al. [6], and (iii) the RPA-ACFDT. The results of this study have demonstrated that a comparison with experiment is meaningful only if the influence of finite temperature is taken into account. Bučko et al. [11] performed MD simulations of the adsorption of propane in protonated chabazite using the PBE-d method and demonstrated that at elevated temperatures the weak bond between the alkane and the acid cite is frequently broken. On average at 300 K the alkane remains only about two thirds of the time close to the active site. During the remaining time the molecule moves rather freely through the cavity and interacts with the inner wall of the zeolite through dispersion forces. We have used this result to approximate the adsorption energy of all alkanes in Na-exchanged chabazite at T = 300 K by the average of the T = 0 K adsorption energies in Na-chabazite and in purely siliceous chabazite in the proportion of 2 to 1. This simple and admittedly rather rough approach suggests that the RPA-ACDFT (i.e. the highest level of theory) describes alkane adsorption rather well [9]. Here we extend these studies to the adsorption of methane, ethane, and propane in protonated chabazite and we perform a thorough investigation of temperature effects by performing MD simulations for all three alkanes.

Our paper is organized as follows. After the introduction we provide a very brief summary of the underlying theory and computational methods. It is followed by a section on 0 K-energy minimization calculations, where we describe calculations of adsorption energies at different levels of theory. In the section on MD simulations we investigate the finite temperature effects on adsorption energies and propose an extrapolation mechanism to correct 0 K adsorption energies for those finite temperature effects. In the discussion we try to compare our results to experiment. At the end of the paper we give our conclusions.

2. Theoretical and computational setup

2.1. Theoretical description of the bonding between the alkane and protonated chabazite

The bonding between an alkane and an acid zeolite consists of two contributions. The first contribution stems from the interaction with the acid site and the second contribution originates from the vdW-interactions between the alkane and the zeolite wall. Both interactions present different challenges for the methodology applied.

To create an active site one Si atom in the zeolite structure is substituted by an Al atom. Since Al has one valence electron less than Si a local charge deficit is created. To compensate this charge deficit a H atom binds to one of the activated O atoms and donates its electron to the framework. This leaves a positively charged Brønstedt acid site which can now form a weak bond with the alkane. The character of the bond with the acid site is best illustrated by the difference electron density (= electron density distribution within the alkane-zeolite complex minus the electron densities of the clean zeolite and the isolated alkane molecule, recalculated at their geometries in the adsorption complex) shown in Fig. 1. The alkane molecule is polarized in the electrostatic field of the acid site and a bonding charge is accumulated between the terminal methyl group of the molecule and the acid proton. This charge redistribution is well described within density functional theory. In this work, we will use the generalized gradient approximation in the parameterization of Perdew, Burke and Ernzerhof (PBE) [10]. vdW-interactions arise from dynamical dipole–dipole interactions. A spontaneous charge fluctuation in one part of the system induces a dipole at some distant region of space. Such dynamical correlations between electrons are not described in DFT. Different approaches have been proposed to correct DFT calculations for the neglect of dispersion forces. In this work, we focus (i) on the semi-empirical PBE-d approach of Grimme [5], (ii) the non-local vdW-density-functional of Dion et al. [6], and (iii) the calculation of the dielectric response function in the RPA, combined with the ACFDT for the calculation of the total energy [7,8].

Fig. 1.

Contour plot of the electron density redistribution (difference electron density) induced by the binding of a propane molecule at an acid site in chabazite. Si atoms of the framework are shown in yellow, O atoms in red, the Al atom in dark grey. The H atoms of the acid site and of the alkane are shown in white, carbon atoms in light grey. Red constant-density surfaces surround electron depleted regions and blue surfaces surround regions of increased electron density. Isodensity surfaces are drawn at ± 2 × 10⁻³ electrons/ų. The accumulation of charge in the region between the terminal CH₃ group of the molecule and the acid proton reflects the formation of a weak chemical bond. Note that while both the alkane molecule and the O–H group of the acid site are quite strongly polarized, no charge redistribution is found on the chabazite framework. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

A semi-empirical correction is provided by adding an energy $E^{\mathit{disp}}$ described by a sum over pair interactions between atoms located at a distance $R_{\mathit{ij}}$, with a distance-dependence characteristic for dipole–dipole interactions

$$E^{\mathit{DFT}+d}=E^{\mathit{DFT}}+E^{\mathit{disp}}$$ (1)

with

$$E^{\mathit{disp}}=s_6{\displaystyle \sum _{i=1}^{N_{\mathit{at}}-1}}{\displaystyle \sum _{j=i+1}^{N_{\mathit{at}}}}\frac{C_6^{\mathit{ij}}}{R_{\mathit{ij}}^6}{f}_{\mathit{dmp}}(R_{\mathit{ij}})\text{.}$$ (2)

The approach was proposed by Grimme and named DFT + d [5]. In our calculations we will use PBE as DFT functional and the originally proposed C₆-factors, which describe the polarizabilities of the different nuclei. At short interatomic distances the divergence of the C₆/R⁶ potentials has to be eliminated by a damping function $f_{\mathit{dmp}}$. For all further details we refer the reader to the Supplementary Material and to the original work of Grimme [5].

Another strategy to capture non-local correlation effects is via a modification of the exchange–correlation functional. In the vdW-density-functional a semi-local exchange-functional from an appropriately chosen version of the generalized gradient approximation (GGA) [13,14,16] is combined with the local short-range part of the correlation-functional treated within the local density approximation (LDA), and a long-range non-local correlation functional accounting for the vdW effects,

$$E_{\mathit{xc}}=E_x^{\mathit{GGA}}[n(\mathbf{r})\text{,}\nabla n(\mathbf{r})]+E_c^{\mathit{LDA}}[n(\mathbf{r})]+E_c^{\mathit{LR}}[n(\mathbf{r})\text{,}n({\mathbf{r}}^{\prime })]$$ (3)

with

$$E_c^{\mathit{LR}}=\frac{1}{2}\int d\mathbf{r}d{\mathbf{r}}^{\prime }n(\mathbf{r})\mathrm{\Phi }(\mathbf{r}\text{,}{\mathbf{r}}^{\prime })n({\mathbf{r}}^{\prime })\text{,}$$ (4)

where the kernel $\mathrm{\Phi }(\mathbf{r}\text{,}{\mathbf{r}}^{\prime })$ is a function of the electron densities and their gradients at sites $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$ – for all details we refer to Dion et al. [6]. An overview of recent applications of the vdW-DF functional has been presented by Langreth et al. [15,17].

In the random phase approximation (RPA) the exchange–correlation energy is given by

$$E_{\mathit{xc}}=E_x^{\mathit{HF}}([{\varphi }_i])+E_c^{\mathit{RPA}}({\chi }^{\mathit{KS}}([{\varphi }_i]))$$ (5)

where $E_x^{\mathit{HF}}$ denotes the Hartree–Fock exchange energy of the many-electron system with orbitals $[{\varphi }_i]$ and $E_c^{\mathit{RPA}}$ is the RPA correlation contribution which is a functional of the Kohn–Sham response function of the non-interacting electrons ${\chi }^{\mathit{KS}}([{\varphi }_i])$ . The correlation energy is calculated from the frequency-dependent dielectric function of the system via a coupling-constant integration. After performing the integration the correlation energy $E_c^{\mathit{RPA}}({\chi }^{\mathit{KS}}([{\varphi }_i]))$ is given by

$$E_c^{\mathit{RPA}}({\chi }^{\mathit{KS}}([{\varphi }_i]))=-{\int }_0^{\infty }\frac{d\omega }{2\pi }\mathit{Tr}\left\{\ln [1-\nu {\chi }^{\mathit{KS}}(i\omega )]+\nu {\chi }^{\mathit{KS}}(i\omega )]\right\}$$ (6)

in terms of the response-function ${\chi }^{\mathit{KS}}$ of the non-interacting system.

In principle the evaluation of the dielectric function requires a summation over all occupied and empty eigenstates. In practical calculations the number of plane waves in the basis set (and with it the number of eigenstates) is finite. To correct for this inaccuracy we use an extrapolation-mechanism that has been well established for the free electron gas (see, e.g., Ref. [7]),

$$E_c(E_{\mathit{cut}}^{\chi })=E_c^{\infty }+\frac{A}{{\left(E_{\mathit{cut}}^{\chi }\right)}^{3/2}}\text{.}$$ (7)

In this expression $E_{\mathit{cut}}^{\chi }$ denotes the energy cut off for the calculations and $E_c^{\infty }$ are the corresponding energies for an infinite energy cut-off. Still a large dependency of total energies on this energy cut-off can be observed, which vanishes when energy differences are considered [7].

While the self-consistent calculation of RPA energies is in principle possible [18], most often preconverged PBE wave-functions are used in calculations for extended systems. Ren et al. [19] proposed to consider the RPA correlation energies as a correction to self-consistent Hartree–Fock calculations (RPA-HF). Using this approach they were able to minimize errors in calculations.

2.2. Molecular dynamics simulations

According to the established practice, adsorption energies are calculated as the difference between the total energy of the adsorbate–substrate complex determined by searching the minimum on the potential energy surface at T = 0 K, and the sum of the energies of the adsorbate in the gas-phase and of the clean substrate, determined also at T = 0 K. Experimentally measured adsorption energies are differences in the free enthalpies at finite temperature. As long as the adsorption energies are very large compared to thermal energies (approximately $3/2k_{B}T$ per degree of freedom), finite temperature effects are modest. For alkanes in acid zeolites, however, even at room temperature the thermal energy is comparable to the strength of the binding to the acid site. Hence the binding to the active site may be broken. This allows the molecule to move inside the cavity and to access energetically less favorable configurations on the potential energy surface. vdW binding to the inner wall of the zeolite in different geometries leads to the formation of multiple, energetically almost degenerate local minima on the potential energy surface.

These effects may be described by performing isothermal molecular dynamics (MD) simulations for alkane molecules in the cavity of the zeolite. The adsorption energies at temperature T are given by

$$\mathrm{\Delta }{E}_{\mathit{ads}}(T)=\langle E_{\mathit{zeo}+\mathit{alk}}{\rangle }_T-(\langle E_{\mathit{zeo}}{\rangle }_T+\langle E_{\mathit{alk}}{\rangle }_T)\text{,}$$ (8)

where $\langle E_x{\rangle }_T$ is the ensemble average of the internal energy at temperature T in system x (either the clean zeolite, the isolated alkane, or the zeolite–alkane complex) from separate MD simulations. Temperature was controlled using an Andersen thermostat [20] with the temperature of the heat bath set to T = 300 K and a collision probability of 0.015. The time-increment was 1 fs, the length of each MD run was about 150 ps. The MD simulations were used to calculate the ensemble average for the internal energy and to investigate the geometry of the adsorption system by monitoring the distribution of the distance of the carbon atom in the terminal CH₃ group and the acid proton and of the shortest distance between a carbon atom of the alkane and a Si atom of the zeolite framework.

2.3. Computational setup

Throughout our work we used the latest release of the Vienna Ab-Initio Simulation Package [21,22], which is a plane-wave code based on DFT and the PAW method [23,24] for describing the electron–ion interaction. The calculation of exact exchange energies is described in Refs. [25,26] and the method for the calculation of the response function is explained in Refs. [27,28]. The selfconsistent calculations using the vdW-DFT functional were performed as described by Gulans et al. [12].

For all calculations we used a cut-off energy of 400 eV and only one k-point centered at the Γ-point of the unit cell. Careful convergence checks have demonstrated that higher cut-off energies and larger k-point sets leave the results largely unchanged. The long-range cutoff for the pair interactions in the PBE-d method was set to 18 Å, but results did not seem to be especially sensitive to this parameter. RPA calculations were performed using PBE wave-functions. For the calculation of the adiabatic coupling integral we chose a maximum energy of $E_{\mathit{cut}}^{\chi }=200\text{eV}$ and extrapolated to $E_c^{\infty }$ in the way described in Ref. [7], which seems sufficient for our calculations.

The details of structural relaxation and MD simulations will be given in the corresponding sections.

2.4. Structure of the zeolite

In this work, we studied the adsorption of short alkanes in chabazite, a zeolite with 12 tetrahedral (T) sites per rhombohedral unit cell (space group $R\overline{3}m$). All T sites are crystallographically equivalent, each tetrahedral atom is coordinated by four crystallographically inequivalent oxygen atom labelled O(1) to O(4). As in our earlier investigation of Na-exchanged chabazite [9] we have used the lattice constants optimized by Bučko et al. [11] at the PBE-d level which are in very good agreement with experiment (a = 9.34 (9.29) Å, α = 94.1 (93.9)°, experimental values in parentheses). The binding within the zeolite framework is of iono-covalent character, vdW corrections have only a minimal influence on the crystal structure. To create an acid site, one of the Si atoms is replaced by an Al atom and the electron-deficit on the framework is compensated by attaching a hydrogen atom to an oxygen site next to Al. Hence in chabazite there are four distinct possible locations of the acid site. With only one Al/Si substitution per unit cell we create a high-silica chabazite with a Si/Al ratio of 11.

3. Adsorption energies in the T = 0 K limit

3.1. Structural relaxation

The potential energy surface of an alkane bound to an acid site in a zeolite is very flat, with many local minima. To find the global energy minimum we followed the strategy described in our previous work [9]. To systematically explore configuration space, we started with the adsorption of methane were the structural optimization depends only on the distance between the carbon atom and the acid proton and the orientation of the CH₄ tetrahedron relative to the framework. To find the equilibrium adsorption configuration for ethane one hydrogen atom was replaced by a CH₃-group and the system was relaxed. The procedure was repeated by substituting another hydrogen atom by a methyl group and the configuration with the lowest energy was adopted. The same approach was applied for finding the optimal adsorption structure for propane, replacing in turn one of the four hydrogen atoms in the CH₃ group not bound to the acid site by another methyl group.

For the structural optimization we used a combination of a quasi Newtonian algorithm, damped molecular dynamics and a conjugate gradient algorithm. We considered the structures to be converged when all forces were smaller than 0.015 eV/Å. This is more accurate than most other calculations in similar systems, but due to the flat potential energy surface close to the minimum we considered this to be necessary.

The structural optimization was performed at the PBE, PBE-d and vdW-DF levels of theory where the forces acting on the atoms can be calculated using the Hellmann–Feynman theorem (adding within the PBE-d method the semi-empirical pair forces). One major point of interest is the impact of the vdW forces on the bond between the alkane and the acid site, as described by the distance between the hydrogen atom and the closest carbon atom of the alkane (C₁). The C₁–H bond lengths, averaged over all four possible locations of the acid proton are given in Table 1 for all three alkanes, calculated using the three different methods. Without vdW corrections, at the PBE level the bond length varies between 2.27 and 2.33 Å. The semi-empirical vdW corrections at the PBE-d level lead to shorter C₁–H distances between 2.16 and 2.25 Å. In contrast the vdW functional leads to strongly increased bond lengths between 2.47 and 2.55 Å. The elongation of the bond lengths relative to the PBE level is unexpected, since we found similar bond lengths in PBE and vdW-DF calculations for Na-exchanged chabazite.

Table 1.

Distances between the acid proton and the closest carbon atom of the alkane adsorbed in chabazite (in Å), averaged over the four possible locations of the acid proton and calculated at the PBE, PBE-d, and vdW-DF levels of theory.

	PBE	PBE-d	vdW-DF
CH4	2.28	2.16	2.55
C2H6	2.27	2.22	2.47
C3H8	2.33	2.25	2.50

The second important point is the interaction between the alkane and the inner wall of the zeolite, which is of pure vdW character. The adsorption in purely siliceous chabazite has already been studied in our previous work [9]. At the PBE level, the adsorption energy is very low, it increases from −2.36 kJ/mol for methane to −4.17 kJ/mol for propane. The weak interaction with the framework corresponds to a location of the molecule almost in the center of the cavity, the shortest C–Si distances are about 4.3 ± 0.05 Å. Significantly larger adsorption energies were found using the PBE-d and vdW-DF methods (see Table 2), increasing with each additional CH_x group by an increment which is about half as large as the binding of methane. Within the PBE-d approach the larger adsorption energies correlate with C–Si distances contracted by about 0.1 Å, whereas with the vdW-DF we find almost the same C–Si distances as without vdW forces.

Table 2.

Adsorption energies (in kJ/mol) of methane, ethane and propane in acid chabazite, as obtained from fully self-consistent calculations at the PBE, PBE-d, and vdW-DF levels of theory. RPA and RPA-HF adsorption energies were obtained from calculations using PBE-d geometries.

Si-chabazite	CH4	C2H6	C3H8
PBEa	−2.36	−2.73	−4.17
PBE-da	−18.30	−28.14	−40.05
vdW-DFa	−33.18	−48.51	−68.22
RPAa	−13.20	−18.65	−26.30
RPA-HFa	−14.48	−22.39	−30.85
Protonated chabazite
PBE
	−6.73	−8.44	−7.48
	−3.76	−5.11	−3.74
	−7.04	−7.58	−8.28
	−10.29	−12.83	−10.64
Avg.	−6.96	−8.49	−7.54
PBE-d
	−32.46	−47.38	−59.31
	−32.32	−43.42	−59.44
	−27.78	−40.33	−50.20
	−34.76	−45.98	−58.87
Avg.	−31.83	−44.28	−56.96
vdW-DF
	−38.39	−58.52	−81.02
	−37.92	−58.40	−74.50
	−34.85	−50.46	−67.84
	−41.75	−59.10	−78.95
Avg.	−38.28	−56.62	−75.58
RPA
	−19.69	−27.65	−34.96
	−17.16	−21.65	−27.21
	−16.46	−25.41	−26.46
	−26.00	−31.51	−36.92
Avg.	−19.83	−26.56	−31.39
RPA-HF
	−28.17	−38.82	−47.64
	−26.08	−32.40	−43.83
	−22.40	−33.27	−35.59
	−32.65	−40.18	−47.63
Avg.	−27.33	−36.17	−43.67

^aFrom Ref. [9].

For all alkanes except methane the adsorption configuration is determined by two competing trends. The first is to maximize the strength of the bond between a terminal CH₃ group of the alkane and the acid site, and the second the tendency to form strong vdW interactions between the remaining CH_x groups and the wall of the cavity. As an example the optimized adsorption configurations for propane at all four different acid sites are displayed in Figs. 2 and 3. A summary of all C₁–H and C_x–Si (x = 2,3) distances for methane, ethane and propane at all four acid site can be found in the Supplementary Material.

Fig. 2.

Equilibrium adsorption geometry of propane bound to Brønstedt acid sites at the (a) O(1) and (b) O(2) atom. Distances in Å have been calculated using PBE, PBE-d, and vdW-DF (from top to bottom). Si atoms of the framework are shown in yellow, O atoms in red, the Al atom in dark grey. The H atoms of the acid site and of the alkane are shown in white, carbon atoms in light grey. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3.

Equilibrium adsorption geometry of propane bound to Brønstedt acid sites at the (a) O(3) and (b) O(4) atom. Distances in Å have been calculated using PBE, PBE-d, and vdW-DF (from top to bottom). See Fig. 2.

At site O (1) the C₁–H distance is contracted with PBE-d compared to PBE, the C₂–Si distance becomes much shorter at the expense of a larger C₃-distance from the framework. With vdW-DF the distance from the acid proton is even slightly increased and the binding of the remaining CH_x group to the framework is re-arranged similarly as with PBE-d. At site O(2) the vdW corrections within PBE-d leave the distance from the acid site unchanged, but shorten the distances between the remaining C atoms and Si atoms of the framework.

The O–H group of the Brønstedt acid site at the O(3) atom is oriented towards the center of a six-membered ring of the chabazite structure. If the binding between the acid site and the molecule is optimized, the propane chain stretches right across the largest cavity of the zeolite. This leads to larger distances between the alkane and the cavity walls. For the acid proton attached to the O(1), O(2) or O(4) atoms the minimum distance between a framework Si atom and a C atom ranges between 3.88/3.64/3.97 Å and 4.06/3.87/4.07 Å (PBE/PBE-d/vdW-DF), but measures 4.23/4.08/4.31 Å for configuration O(3). For all configurations the vdW-DF functional leads to significantly increased C₁–H distances. A similar increase has also been found for the interaction between an alkane and an Na extra-framework atom, but the effect is stronger for the Brønsted acid sites.

3.2. Energetics of the optimized adsorption configurations

Our calculated adsorption energies are summarized in Table 2, including those for purely siliceous chabazite. In the absence of an acid site PBE leads to a very low adsorption energy of −3.1 ± 1.0 kJ/mol for all three alkanes. With PBE-d the adsorption energy of methane increases to −18.3 kJ/mol and further by an increment of about −10 kJ/mol for each additional CH_x group. The vdW-DF predicts an even larger adsorption energy of −33.2 kJ/mol for methane and a larger increment of approximately −17.5 kJ/mol.

For protonated chabazite the adsorption energy is still very low at the PBE level, −7.7 ± 0.8 kJ/mol averaged over all four acid sites. The difference between the four sites reflects their different acid strengths. vdW corrections at the PBE-d level increase the averaged adsorption energy of methane to −31.8 kJ/mol, and by an almost constant increment of −12.5 kJ/mol for the longer alkanes. The increase caused by the dispersion forces is largest for the O(2) site providing the weakest binding at the PBE level and smallest for the O(3) site where the geometry prohibits a more efficient optimization of the interactions with the cavity. Even stronger vdW effects are predicted by the vdW-DF: the adsorption energy of methane increases to −38.3 kJ/mol, the increment per additional CH_x group to about −18.7 kJ/mol. Again the increase compared to the PBE level is smallest for the O(3) site.

3.3. Adsorption energies from the RPA

Within the RPA or the RPA-HF calculations no forces acting in the atoms are available, a structural optimization is not possible. Therefore we had to rely on structures relaxed at a lower level of theory. Test calculations produced very similar RPA total energies for the PBE, PBE-d and vdW-DF structures. For that reason we calculated RPA adsorption energies for the PBE-d structure.

RPA calculations based on the PBE orbitals lead to a averaged adsorption energy for methane of −19.8 kJ/mol and an increment per additional CH_x group of about −5.8 kJ/mol. Both values are considerably lower than obtained at the PBE-d and vdW-DF levels of theory. The methane adsorption energy is increased by −6.6 kJ/mol compared to purely siliceous chabazite, the increment remains approximately constant.

Larger values (methane adsorption energy −27.3 kJ/mol, average increment −8.2 kJ/mol) are obtained in RPA calculations based on orbitals and exchange energies from selfconsistent Hartree–Fock calculations (RPA-HF). In this case the adsorption energy of methane in purely siliceous chabazite is only −14.5 kJ/mol, while the increment for longer alkanes is the same. The comparison of the RPA and RPA-HF result shows that the choice of the orbitals and exchange energy influences both the dispersion forces and the interaction between adsorbate and acid site.

4. Molecular dynamics simulations

Experimental alkane adsorption energies are measured at elevated temperatures between 300 and 600 K and should not directly be compared with the zero-temperature calculations. To address this issue we performed molecular dynamics calculations at T = 300 K for all three alkanes adsorbed in a protonated chabazite with the acid proton attached to the O(4) atom. In experiment typically the zero-coverage adsorption enthalpy is measured. To achieve conditions similar to those in low-pressure adsorption measurements we doubled the cell volume as described in Ref. [11]. The simulations were performed for an acid zeolite with only one Al/Si substitution site in the doubled cell. The forces on the atoms are calculated using PBE-d.

Fig. 4 shows the variation of the distance between the acid proton and the nearest carbon atom of the alkane and of the minimum distance between a carbon atom and a Si atom of the framework. The time-dependence of the C₁–H distance shows that during most of the time the alkane remains attached to the acid site, with the C₁–H distance fluctuating between 2 and 3 Å. For short intervals, however, the bond to the acid site is broken and the molecule moves to rather large distances. The bond breaking occurs more frequently for the smaller molecules methane and ethane and the molecule moves farther away from the active than the larger propane. In their simulations Bučko et al. [11] considered propane to be adsorbed to the Brønsted site if the C₁–H distance is smaller than 3 Å. Using this criterion we find that at T = 300 K the probability that the molecule is bound to the acid site is 0.61 for methane, 0.60 for ethane and 0.78 for propane. Fig. 5 shows the distribution of the C₁–H distances for all three molecules. While the width of the main peak is almost the same for all three species, it is evident that with increasing length of the alkane the largest distance from the acid site is limited by steric effects, the length of the chain in relation to the diameter of the cavity.

Fig. 4.

Variation of the distance between the acid proton and a carbon atom of the alkane (C₁–H distance, in red) and the minimum distance between a carbon atom and a Si atom of the framework (C–Si distance, in black) during molecular dynamics simulations at T = 300 K for (a) methane, (b) ethane and (c) propane in protonated chabazite cf. text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5.

Distributions of the minimum C₁–H (a) and C–Si (b) distances for methane, ethane and propane in chabazite at T = 300 K cf. text.

The minimum distance between a carbon atom of the alkane and a silicon atom of the framework reflects the vdW interactions between adsorbate and zeolite. For all three molecules this distance fluctuates around about 4 ± 0.5 Å. As demonstrated in Fig. 5(b) the distribution of the C–Si distances is narrowest for propane. For this molecule two or even three CH_x groups interact with the framework. Combined with the large size of the molecule this limits its mobility. The distribution is widened in both directions for ethane. The vdW interactions between molecule and cavity wall are weaker, the molecule is smaller, together this accounts for the broadening of the distribution to larger distances. Because only two bonds between the CH₃ groups and the framework have to be optimized, it is easier to find a low-energy configuration at a close distance from the framework. Such a configuration with the ethane molecule in close contact with the wall of the cavity at a place far from the acid site is shown in Fig. 6. For methane we find a longer tail in the probability distribution for larger Si–C distances, because after breaking the bond to the acid site the small molecule spends more time close to the center of the cavity before forming a vdW bond to the framework. The adsorption energy of the alkane molecule at T = 300 K was calculated according to

$$\mathrm{\Delta }{E}_{\mathit{ads}}(T)=\langle E_{\mathit{zeo}+\mathit{alk}}{\rangle }_T-(\langle E_{\mathit{zeo}}{\rangle }_T+\langle E_{\mathit{alk}}{\rangle }_T)$$ (9)

as the difference between the ensemble averages of chabazite with an adsorbed alkane, and the averages for the alkane molecule in the gas-phase and for the clean zeolite. From this procedure we obtain values of −23.91 kJ/mol for methane, −34.87 kJ/mol for ethane and −45.53 kJ/mol for propane. Our value for propane is in very good agreement with the result of Bučko et al., who calculated a value of 44 kJ/mol [11].

Fig. 6.

Configuration of an ethane molecule in close contact with the chabazite framework at a large distance from the acid site. For the explanation of the symbols, see Fig. 2 cf. text.

Our finite-temperature values are reduced by 10.85, 11.11, and 13.34 kJ/mol compared to the T = 0 K limit, i.e. by approximately the same amount, independent of the length of the alkane. The increase of the adsorption energy from methane to ethane is −9.8/−11.1/−11.0 kJ/mol, from ethane to propane  − 11.9/−12.9/−10.7 kJ/mol for purely siliceous chabazite at T = 0 K, protonated chabazite [acid group at site O(4)] at T = 0 K, and protonated chabazite at T = 300 K. This analysis demonstrates that the increment per additional CH_x group (which is caused by the vdW interactions with the framework) is almost independent of the presence of Brønsted sites and also almost independent of temperature. The temperature-dependence of the adsorption energy stems mostly from the fact that with increasing thermal energy the bond with the acid site is more frequently broken.

To perform ab initio MD simulations at a higher level of theory for the description of dispersion forces is out of question, the computational effort would be much to high. In our previous work [9] we have followed Bučko et al. [11] to estimate the temperature effect on adsorption energy calculated using the vdW-DF or within the RPA. Bučko et al. had noted that a proton transfer between the acid site at propane can take place only if the C₁–H distance is smaller than 3 Å. Their MD simulations demonstrated that at T = 300 K the probability to find the molecule within this distance is 0.68. Hence the adsorption energy of alkanes in Na-exchanged chabazite was approximated by the adsorption energies in purely siliceous and Na-exchanged chabazite (both calculated at T = 0 K), averaged in a proportion of 1–2. Admittedly the choice of a maximum distance of 3 Åis a bit arbitrary, so we decided to use a less ambiguous approach. The adsorption energy at 300 K is approximated again by an average of the adsorption energies in purely siliceous ($E_{\mathit{ads}}^{\mathit{Si}-\mathit{Chab}}$) and protonated ($E_{\mathit{ads}}^{H-\mathit{Chab}}$) chabazite,

$$E_{\mathit{ads}}^{300K}=P^{300}(R_{H-C_1}<x)E_{\mathit{ads}}^{H-\mathit{Chab}}+(1-P^{300}(R_{H-C_1}<x))E_{\mathit{ads}}^{\mathit{Si}-\mathit{Chab}}\text{,}$$ (10)

where the weighting factors are determined by $P^{300}(R_{H''–''C_1}<x)$, the probability to find the C₁ atom of the alkane within a distance of at most x Å from the acid proton. The value of x is determined by fitting the estimated $E_{\mathit{ads}}^{300K}$ for all alkanes to the values from the MD-simulations using the PBE-d approach. The optimized value is x = 2.5 Å, and this leads to values of $P^{300}(R_{H-C_1}<x)$ = 0.34, 0.32 and 0.43 for methane, ethane and propane. These weighting factors derived from the MD simulations based on the PBE-d method will also be used with the adsorption energies calculated at other levels of theory. The adsorption energies at T = 300 K calculated using this approach are summarized in Table 3 and displayed in Fig. 7.

Table 3.

Adsorption energies at T = 300 K (in kJ/mol) for methane, ethane and propane in protonated chabazite, estimated using the proposed averaging scheme, compared to experiment. At the PBE-d level, the averages are also compared with the results of the MD simulations.

	CH4	C2H6	C3H8
PBE	−3.92	−4.70	5.62
PBE-d	−22.90	−33.30	−47.32
vdW-DF	−34.91	−51.11	−71.38
RPA	−15.45	−19.69	−28.49
RPA-HF	−18.85	−26.80	−36.21
MD-PBE-d	−23.91	−34.87	−45.53
Expa	−21.6	−34.1	−46.8
Expb	−20.4	−30.8	−37.3

^aRef. [29]

^bRef. [31]

Fig. 7.

Adsorption energies for methane, ethane and propane in protonated chabazite at 300 K, as calculated at different levels of theory and compared to experiment (full squares – Ref. [30], open squares – Ref. [31]).

5. Discussion

To our knowledge alkane adsorption in chabazite has been investigated in the early 1970s by Barrer and Davis [31] for protonated chabazite, and in 1997 and 2008 by Denayer et al. [29,30] for Na-exchanged chabazite. Both studies have been performed for chabazite with a rather low Si/Al ratio between 3 and 4, leading to 3–4 active sites (extra-framework Na or Brønsted sites) per unit cell. To a first approximation one would expect that the adsorption energies in protonated and Na-chabazite differ only by a constant offset, reflecting the different adsorption strength of the active sites. This relation holds quite well for methane and ethane, but for propane the difference is distinctly larger. This is probably due to the much larger size of the Na atoms which restricts the mobility of the larger propane molecule which is forced to remain close to the active site and this leads to a larger adsorption energy.

Our studies have been performed for chabazite with a high Si/Al ratio, which makes comparison with experiment difficult. The thermal motion of the molecules underlies almost not steric hindrance and therefore we think that results obtained by Barrer and Davies correspond more closely to our calculated adsorption energies. Best agreement with experiment is achieved at the RPA-HF level of theory, only for ethane theory underestimates the adsorption energy by 1–2 kJ/mol. RPA underestimates the adsorption energies by 5–8 kJ/mol. The PBE-d values (note the good agreement between the simple estimate and the MD simulations) agree with the experiments of Denayer et al. for Na-chabazite, but are larger than the values of Davies and Barrer for protonated chabazite by up to 8 kJ/mol. The van der Waals functional/vdW-DF) strongly overestimates all adsorption energies.

6. Conclusions

We have investigated the adsorption of short alkanes in protonated chabazite at different levels of theory, ranging from DFT calculations with semi-empirical dispersion corrections (PBE-d) over the non-local van der Waals correlation functional (vdW-DF) to calculations of the correlation energy via the random phase approximation in conjunction with the adiabatic-coupling density-fluctuation theorem (RPA-ACFDT). Temperature effects have been studied using isothermal molecular dynamics simulations at the PBE-d level. The results of the MD simulations have been used to estimate temperature effects at a higher level of theory via weighted averages of the adsorption energies in purely siliceous and protonated chabazite.

Our results demonstrate that due the weak interaction of the alkane with the acid site the bond between a terminal CH₃ group and the acid proton is frequently broken with increasing temperature, enabling the molecule to diffuse through the cavity of the zeolite and to optimize the vdW interactions with the framework. With increasing size of the molecule, steric effects tend to restrict thermal motions to the region around the acid site. The weighted averages of the adsorption energies calculated at the RPA-HF level (i.e. with selfconsistent Hartree–Fock orbitals and exchange energies) lead to excellent agreement with the values measured at T = 300 K. PBE-d leads to reasonable, albeit slightly too large values, while the vdW-DF approach grossly overestimates the adsorption strength.