CO Molecules on a NaCl(100) Surface: Structures, Energetics, and

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CO Molecules on a NaCl(100) Surface: Structures, Energetics and Vibrational Davydov Splittings at Various Coverages A. Daniel Boese, and Peter Saalfrank J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b03726 • Publication Date (Web): 20 May 2016 Downloaded from http://pubs.acs.org on May 24, 2016

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The Journal of Physical Chemistry

CO Molecules on a NaCl(100) Surface: Structures, Energetics and Vibrational Davydov Splittings at Various Coverages A. Daniel Boese1,2 and Peter Saalfrank1,∗ May 19, 2016 1

Universit¨at Potsdam, Institut f¨ ur Chemie, Karl-Liebknecht-Straße 24-25, D-14476 Potsdam-Golm, Germany 2 Karl Franzens Universit¨at Graz, Institut f¨ ur Chemie, Heinrichstraße 28, A-8010 Graz, Austria

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Corresponding author: [email protected] Abstract In this work, we study the adsorption of CO from low to high coverages at a defect-free NaCl(100) surface by means of cluster and periodic models, using highly accurate wavefunction-based QM:QM embedding as well as density functional theory. At low coverages, the most accurate methods predict a zero-point corrected adsorption energy of around 13 kJ/mol, and the CO molecules are found oriented perpendicular to the surface. At higher coverages, lower-energy phases with non-parallel/upright, tilted orientations emerge. Besides the well-known p(2×1)/antiparallel phase (T/A), we find other tilted phases (tilted/irregular, T/I and tilted/spiral, T/S) as local minima. Vibrational frequencies for CO adsorbed on NaCl(100) and Davydov splittings of the C-O stretch vibration are also determined. The IR spectra are characteristic fingerprints for the relative orientation of CO molecules, and may therefore be used as sensitive probes to distinguish parallel/upright from various tilted adsorption phases.

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1

Introduction

The accurate description of molecular systems by electronic structure methods has come a long way: Nowadays, we are capable of describing molecular energies of “chemically relevant” species with very high, sometimes so-called chemical accuracy of ∼ 1 kcal/mol or about 4 kJ/mol. This is one of the great achievements of modern quantum chemistry. Recently, this level of accuracy has also been reached for certain bulk periodic materials 1 . Molecules adsorbed at surfaces are literally in between the molecular and periodic solid-state world, posing, as such, qualitatively new challenges for electronic structure theory. These challenges are due to lacking periodicity in one direction (perpendicular to the surface), but also due to a great variety of bonding mechanisms (metallic, covalent, electrostatic, van der Waals), sometimes in combination, at the molecule-surface interface. The two main lines to reach high accuracy for adsorbate systems are wavefunctionbased ab initio methods (WaveFunction Theory, WFT), and Density Functional Theory (DFT), respectively. Among the WFT branch when applied to periodic systems, periodic Hartree-Fock theory corrected by correlation contributions (calculated by Møller-Plesset perturbation theory 2,3 , for example) is a promising tool to accurately treat adsorbates. In the periodic DFT world, on the other hand, dispersion-corrected functionals paved the way in particular towards systems exhibiting non-negligible van der Waals contributions to adsorbate binding – which seems the rule rather than the exception. The conceptually simplest of the latter class of approaches are DFT+D methods 4–6 , in which a semi-empirical dispersion term (D) is added to the DFT energy. For non-metallic surfaces besides periodic, also cluster models are useful, or a hybrid between the two. For the latter, typically a cluster is defined and treated on a high level of theory, then embedded in an environment which is described less rigorously. This approach can be very successful, as demonstrated recently in Ref. 7 for CO adsorbed on MgO(100). There, by a hybrid MP2:DFT+D cluster/periodic model and extensions of it, for low coverage chemical accuracy was reached. The adsorption of small molecules on ionic surfaces is important not only from a benchmarking point of view, but also for practical reasons. As an example, the vibrations of small species like CO on (ionic) surfaces can be used as sensitive probes for unravelling processes in heterogeneous catalysis 8–10 . The C-O vibration of CO is particularly useful, as this molecule is not only IR-active but it also has a strong chemical bond which cannot be easily broken. Further, CO readily adsorbs on many substrates.


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In the present work, we shall use accurate quantum chemical (“first principles”) methods to (re-) investigate the adsorption of CO molecules on an ideal, defect-free NaCl(100) surface. The availability of modern sensitive experimental probes and very accurate quantum chemistry methods makes this “old system” a worthwhile study object of current research. Numerous experimental facts are known on structures and vibrational features of CO/NaCl(100), typically for highcoverage phases 11–14 . From a dynamical point of view, one of the most striking features of CO/NaCl(100) is a particular long (∼ 4 ms) vibrational lifetime of the C-O stretch mode. This is due to a large mismatch between vibrational frequencies of the high-frequency, internal adsorbate mode and other (notably phonon) modes of the system 12,13,15 . In fact, the C-O vibration is so long-lived, that effcient “energy pooling” CO(v = 1) + CO(v = 1) → CO(v = 2) + CO(v = 0) can take

place on that substrate, which leads, after IR pumping of the CO(v = 1) vibra-

tional state, to highly vibrationally excited states (up to v = 15, see Ref. 15 and references therein). This effect is intimately related to excitonic coupling between CO dipoles, leading to Davydov splitting 16–18 in vibrational spectra at coverages high enough such that CO molecules can interact with each other. In fact, the excitonic coupling can in turn also be used as a probe for adsorbate structures at higher coverages. In this paper, we shall do precisely that, i.e., calculating structures of CO arrangements on NaCl(100) by accurate, state-of-the-art quantum chemistry, and predicting “fingerprints” of these structures based on computed (IR) vibrational spectra and excitonic splittings. We find structures where CO molecules are in parallel/upright orientation, however, at “higher” (but not necessarily full) coverages also lower-energy phases where molecules tilt towards the surface due to van der Waals (dispersion) interactions. These structures can be distinguished from their alternative, parallel/upright (P/U) structures by exciton-split IR spectra. Note that most structures predicted theoretically so far, were based on empirical force-fields 15,19–23 . A few exceptions exist, however, where cluster 24 and periodic first principles methods 25 have been used instead. Our present approach may offer some additional insight, by using state-of-the-art electronic structure theory. Also, in most studies so far only the monolayer phases were considered, while we also treat lower coverages over a wide range. Further, by using large unit cells also structures hitherto unknown did emerge. The present paper will furthermore try to set benchmarks and compare the accuracy and performance, of various models and methods. Specifically, a question addressed here is how ab initio theory (WFT) performs vs. dispersion-corrected


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density functional theory. We shall adopt three different model / method combinations and test them against each other: • Periodic supercell models will typcially be studied within DFT, using various exchange-correlation functionals and empirical dispersion (+D) corrections.

• Subtractive, quantum mechanical/quantum mechanical (QM:QM) hybrid methods with a “cluster” treated on a high (WFT) level of theory, then embedded in a periodic solid treated on a lower level, typically DFT. • As a third model we cut out “true” clusters from the periodic models, treating them as molecular models. Clusters were described by various electronic structure methods, both from WFT and DFT. These models and methods will be described in the next section, Sec. 2. In Sec.3 we will present and discuss results, first energies and structures (Sec.3.1), then vibrations including excitonic couplings (Sec.3.2). A final section 4 summarizes and concludes our work.

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Models and methods

2.1

Models

Both periodic calculations using supercell geometries, as well as (embedded) cluster calculations were carried out to describe the adsorption of CO molecules on the NaCl(100) surface. We use comparatively large quadratic unit cells, to allow for the treatment of a large range of coverages, and for structural flexibility. For the periodic calculations, we go from one to eight molecules on a 4×4 slab (corresponding to coverages θ between 1/8 and 1), and from one to 18 molecules on a 6 × 6 slab (corresponding to θ between 1/18 and 1). The coverage refers to avail-

able Na+ sites (on which CO adsorbs) of the surface layer, of which there are eight for the 4×4 and 18 for the 6×6 cell. Different layer thicknesses m giving rise to slabs 4×4×m (with m = 2 − 6) were built, and also a 6×6×4 cell was considered. Typical situations, for lowest coverages (and up to m = 4 at most), are shown in

Fig. 1. At θ = 1, all Na+ sites of the upper layer are covered. At intermediate coverages various structural models are possible for the same coverage, and several have been tried as will be described shortly. The cell models shown in the figure are the basis of periodic, supercell DFT(+D) calculations, as well as those of the QM:QM hybrid schemes and the “true” clusters for molecular model calculations. Even larger cells / clusters have been used occassionally. Details will follow below.


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4×4×2 Na16 Cl16 θ=1/8

4×4×3 Na24 Cl24 θ=1/8

4×4×4 Na32 Cl32 θ=1/8

6×6×4 Na64 Cl64 θ=1/18

Figure 1: Periodic surface models 4×4×m and 6×6×m for CO at low coverages, with increasing vertical cell size, m. Carbon is displayed in orange, oxygen in red, sodium in grey, and chloride in green. The coverage θ is indicated, and so is a formula Nan Cln , giving the composition of a repeating cell (without CO).

2.2 2.2.1

Electronic structure methods Periodic calculations

For the periodic DFT surface calculations, the VASP 26 program suite has been utilized. If not stated otherwise, an energy cutoff of 700 eV and standard VASP pseudopotentials 27 for C, O and Cl were applied while for Na, the semicore 2pstates have been treated as valence states. In all periodic surface calculations, a dipole/quadrupole correction perpendicular to the surface was adopted, using the ˚ has been utilized. built-in VASP routine. A sufficiently large vacuum gap of 15 A Total-energy calculations were carried out with a single k-point (Γ), or a 2×2×1 Monkhorst-Pack 28 k-point mesh, respectively. Differences between the two meshes in adsorption energies are very small (see below). The functionals according to Perdew-Becke-Ernzerhof (PBE, 29 ), the revised PBE (RPBE, 30 ), and the Heyd-Scuseria-Ernzerhof functionals (HSE, 31 ) have been used. PBE and RPBE are pure Generalized Gradient Approximation (GGA) functionals, while HSE is a hybrid functional containing exact exchange. Concerning the DFT dispersion correction, both the D2 scheme 4 with substituting the Na with the Ne term (to mimick Na+ 32 ), as well as the newer D3 correction 5 have been employed. To check methods and fix computational parameters the bulk NaCl crystal has been optimized in order to obtain interatomic distances. In these calculations, we


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used a larger cutoff of 800 eV, also testing the number of k-points until no further change in geometry was obtained. As a result, the theoretical (cubic) lattice ˚), 5.669 (PBE+D2), 5.581 (PBE+D3), 5.798 (RPBE+D2), constant a was (in A 5.609 (RPBE+D3), and 5.536 (HSE+D3), compared to the experimental value of 5.637 ˚ A 33 . All functionals yield reasonable values, with perhaps RPBE+D2 yielding a bit too large and HSE+D3 too small values. The interatomic distances obtained for the bulk with the respective method, were used in calculations for the adsorbate / surface supercell calculations below if not explicitly stated otherwise. In fact, we also performed calculations in which (some of the) surface atoms were allowed to relax.

2.2.2

QM:QM hybrid embedding calculations

We also performed QM:QM calculations, embedding clusters mimicking CO on NaCl into a periodic slab. A simple subtractive QM:QM scheme 7 was adopted, where the energies and gradients of the cluster are calculated twice: Once with the high-level (HL), and once with the low-level (LL) method, to give an energy: EQM:QM = EHL (cluster) − ELL (cluster) + ELL (periodic) = EHL (cluster) + ∆Ecorr (periodic)

.

(1) (2)

Here, the periodic surrounding is a perturbation (∆Ecorr (periodic)), as we just add this long-range, periodic correction to the high-level cluster calculation. For the HL cluster calculation, we employed Møller-Plesset perturbation theory in second order (MP2), utilizing different Dunning basis sets 34,35 : For the oxygen and carbon atoms of CO, the aug-cc-pVXZ basis sets were used while for chlorine, the aug-cc-pV(X+d)Z basis sets and for sodium the cc-pCVXZ basis sets were employed. For Na, O, and C only the [He] core shell was frozen when correlating the electrons in the MP2 calculations, and for Cl, the [Na] shell electrons were frozen. For the LL (periodic and cluster) calculations, we used the RPBE+D3 functional described before. This is because of experience gained for CO on MgO 7 , according to which PBE+D2 was found to underestimate the long-range dispersion by a large amount, and even BLYP+D2 (with a similar D2 coefficient than RPBE+D2) was underestimating the long-range corrections somewhat.

2.2.3

Cluster and “gas phase” calculations

“Pure” or “embedded” cluster calcuations with various forms of DFT or with MP2 were done with the TURBOMOLE suite 36 . We also performed highly accurate calculations for small isolated molecules (“gas phase calculations”), e.g., for vibraACS Paragon Plus Environment

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tional frequencies for CO and CO oligomers, using Coupled Cluster Single Doubles with perturbative Triples, CCSD(T) 37 , with the CFOUR program 38 . For CCSD(T) calculations Dunning’s basis sets 34,35 were used, while for DFT pure molecular models Jensen’s basis sets 39,40 were employed. For the latter basis sets, aug-pc-1 corresponds to a double-ζ basis set including diffuse functions, augpc-2 to a triple-ζ basis set, and so on. For very high accuracy for small molecules we also basis-set extrapolated post-Hartree-Fock methods (here CCSD(T)), using an empirical l−5 formula for Hartree-Fock 41 , and a l−3 formula for the correlation contribution 42 .

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Results and Discussion

3.1 3.1.1

Energies and geometries of CO on NaCl(100) Low coverage

(1) Periodic DFT calculations. In a first step periodic DFT calculations were performed for CO on NaCl(100) at low coverages. Specifically, 4×4×m (m = 2 − 6) supercell models and a 6×6×4

model were chosen, as shown (with the exception of 4×4×5 and 4×4×6) in Fig. 1.

One CO per cell gave rise to coverages of θ = 1/8 for 4×4 and θ = 1/18 for 6×6, respectively. In all cases CO adsorbs with the C-end pointing towards a Na+ ion. In Table 1, CO adsorption energies (which we define here as negative when adsorption is exoenergetic), De = E(free) − E(adsorbed) are displayed, when using

dispersion-corrected DFT. Here, E(free) and E(adsorbed) are electronic energies

for free and adsorbed CO, respectively. To obtain E(free), a single CO (without surface) was put in a large box and optimized, and also a naked NaCl surface (without CO) in supercells previously used for adsorbed species was considered, both in separate periodic DFT calculations. Again different functionals and dispersion corrections (D2 or D3) were adopted. A k-point mesh 2×2×1 was chosen for total-energy calculations, if not stated otherwise. Further, also the effects on surface relaxation were tested. For this purpose, in one class of calculations we performed geometry optimizations of C and O with surface structures obtained from bulk calculations as outlined above, without extra surface relaxation (“rigid”). Alternativley, all but the lowest two NaCl layers along the surface normal were allowed to relax (“relaxed”).


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periodic supercell model coverage θ = 1/8 method surface 4×4×2 4×4×3 4×4×4 4×4×5 4×4×6 PBE+D2 relaxed 21.5 21.9 21.9 22.3 22.0 rigid 21.5 22.0 22.1 22.5 22.1 a PBE+D2 rigid 21.6 22.1 22.2 22.7 22.3 PBE+D3 relaxed 20.3 21.3 21.2 21.3 21.3 rigid 20.3 21.3 21.4 21.5 21.5 RPBE+D2 relaxed 22.3 22.9 23.1 23.1 23.2 rigid 22.3 23.1 23.2 23.4 23.4 RPBE+D3 relaxed 22.3 23.1 22.9 23.2 23.3 rigid 22.3 23.2 23.3 23.4 23.4 a obtained when using a single k-point (Γ) only

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θ = 1/18 6×6×4 22.0 22.1 22.1 21.3 21.5 23.1 23.3 23.2 23.3

Table 1: Adsorption energy of a single CO in various supercells of NaCl, corresponding to low coverages. All energies in kJ/mol obtained from periodic DFT with different functionals and dispersion corrections, for relaxed and rigid surface models. A 2×2×1 k-point mesh and a plane-wave cutoff of 700 eV was used. From the table, the following trends can be seen: • The adsorption energies for the GGA functionals tested are between 20.3 and 23.4 kJ/mol. HSE+D3 (not shown) yields a bit smaller energies of 20.7 and 21.6 kJ/mol for the 4×4×2 and the 4×4×3 slabs, while we did not test any larger slab sizes with this method. Differences between PBE and RPBE are minor (∼ 0.1 kJ/mol). Different dispersion corrections (D2 vs. D3) have some effect for PBE (up to one kJ/mol, in an unpredictable direction), while RPBE+D2 and RPBE+D3 are almost identical. • Increasing the number of surface layers from m = 2 to m = 6 has an effect

of up to one kJ/mol on De , the latter slightly increasing with slab thick-

ness. Increasing the lateral extension of the unit cell (compare 4×4×4 with 6×6×4) has an even smaller effect, up to 0.3 kJ/mol at most. Since here also the coverage is changed (from 1/8 to 1/18), both coverages can indeed be considered as “low” in the sense that inter-adsorbate interactions are small. • The calculations are converged with respect to the choice of the k-point mesh: Choosing a single k-point (Γ) only, gives deviations in adsorption energies of at most 0.2 kJ/mol, in most cases even less. • Effects of surface relaxation are only 0.1-0.2 kJ/mol, keeping the surface

almost intact when a CO is adsorbed. In comparison, for CO on MgO, the surface relaxation was as much as 1.2 kJ/mol 7 . ACS Paragon Plus Environment

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All in all, however, results are comparable to CO/Mg(100), where 22.1 kJ/mol binding energy were obtained for the same k-point mesh, a 4×4×4 slab, and the PBE+D2 functional 7 . Here, our analogous calculation for CO/NaCl(100) yields 21.9 kJ/mol. The individual contributions to adsorption energies for CO are different for NaCl compared to MgO, however: Dispersion interactions of CO with the nearby Cl− are larger than with the O2− ion, somewhat compensated for by the smaller MgO lattice distance. The PBE+D2 DFT dispersion energy for CO/NaCl is 10.5 kJ/mol compared to 7.2 kJ/mol in an analogous calculation of CO on MgO. Hence for CO/NaCl, 21.9 kJ/mol-10.5 kJ/mol=10.4 kJ/mol are “DFT nondispersive”, mostly electrostatic contributions to the molecule-surface bond. For CO/MgO, this electrostatic contribution is 22.1 kJ/mol-7.2 kJ/mol=14.9 kJ/mol. The extra electrostatic stabilization of CO on MgO is coming from CO interacting with the larger formal charges of the MgO surface ions. (2) QM:QM embedding calculations. We also performed QM:QM calculations. For this purpose we embedded a CO(NaCl)9 cluster (a 3×3×2 cluster with a top Na5 Cl4 layer and a Na4 Cl5 below, a single CO on top of the central Na+ of the upper layer), into the 4×4×4 slab (see Ref. 7 for an analogous treatment for CO/MgO and pictures). We used the subtractive scheme of above, with the high-level cluster calculations done with MP2 and various Dunning basis sets, and the low-level calculations done by RPBE+D3 DFT as described before. These calculations serve as a test of the robustness and accuracy, of the earlier periodic DFT(+D) calculations. In Tab. 2, the basis set convergence of adsorption energies for the high-level part of the QM:QM embedding method is tested, and results of different periodic DFT calculations are summarized, for the 4×4×3 unit cell and θ = 1/8 (cf. Tab.1). Adsorption energies were determined as previously for the periodic DFT case. From the table we first note that there is, as usual 7 , a relatively strong dependence of the results on the basis set used for the HL (MP2) calculation. Assuming that the embedded calculations at the largest extrapolated basis set (abbreviated as (T,Q)Z ext.) are the best, they correspond to an adsorption energy of 20.0 kJ/mol, both for relaxed and unrelaxed surfaces. Na-C and C-O distances are 2.680 ˚ A and 1.132 ˚ A, respectively, for the relaxed surface. Compared to this “benchmark” for a single CO, one can conclude that adsorption energies are described best within pure periodic DFT calculations by PBE+D3 and HSE+D3, whereas the C-O distances are well described by HSE+D3 and the Na-C distances


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rigid relaxed method De De RNa−C embedded cluster calculations QM:QM/DZ 32.1 31.7 2.658 QM:QM/TZ 25.5 25.5 2.667 QM:QM/QZ 22.1 22.1 2.670 QM:QM/(D,T)Z ext. 23.8 23.7 2.658 QM:QM/(T,Q)Z ext. 20.0 20.0 2.680 periodic DFT, 4×4×3 slab PBE+D2 22.1 22.2 2.674 PBE+D3 21.4 21.5 2.678 RPBE+D2 23.2 23.3 2.722 RPBE+D3 23.2 23.4 2.874 HSE+D3 21.6 21.6 2.704

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RC−O 1.148 1.137 1.133 1.134 1.132 1.141 1.141 1.146 1.146 1.130

Table 2: Adsorption energies (in kJ/mol) of the QM:QM embedded (MP2/basis:PBE+D3/700eV, upper half of the table) and periodic DFT calculations (lower half of the table), for “rigid” and “relaxed” surface models. For the latter, the optimized Na-C and C-O distances (in ˚ A) are also shown. All periodic calculations are done at the Γ point only (which is why values for pure DFT calculations deviate marginally from Tab.1, where 2×2×1 kpoints were used instead), with a cutoff of 700 eV, and a 4×4×3 unit cell. Both the QM:QM and periodic DFT models correspond to a coverage of 1/8. Concerning the basis sets of the inner QM part, DZ, TZ and QZ are the acronyms for Dunning’s X-ζ basis sets mentioned in Sec.2.2.2, and (D,T)Z ext. and (T,Q)Z ext. indicate basis set extrapolations with the basis sets of double- and triple-ζ or triple- and quadruple-ζ quality, respectively. by PBE+D2 and PBE+D3. (3) Higher-level QM:QM embedding calculations. The accuracy of the QM:QM embedding scheme of above can be improved. First, by replacing MP2 as the “high level” method in QM:QM embedding calculations by an even more accurate method, such as CCSD(T). Second, the “cluster” defined in QM:QM schemes may not be large enough. In particular, long-range contributions could be missing when the cluster was chosen too small. To investigate the first point, we performed cluster calculations in full analogy to Ref. 7 , based on on CCSD(T) as the HL method. Accordingly, we start from the QM:QM/(T,Q)Z structure (cf. Tab.2), constructing three clusters of various 4+ size, namely NaCl4− 5 , Na5 Cl5 , and Na9 Cl5 . These were then embedded into pe-

riodic arrays of formal point charges (+1 and -1) and effective core potentials to


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mimick the NaCl surface, in analogy to the treatment for CO/MgO 7 . Then, both MP2 and CCSD(T) calculations were carried out for these clusters, to estimate the accuracy of MP2 against CCSD(T). Counterpoise corrections were included as in Ref. 7 to account for basis set superposition errors. For the (embedded) NaCl4− 5 cluster the difference between MP2 and CCSD(T) adsorption energies were found to be converged for the TZ basis set. Likewise, the difference between MP2 and CCSD(T) was the same for all three investigated clusters, namely -1.5 kJ/mol. This corrects our so far “best” adsorption energy of 20.0 kJ/mol (cf. Tab. 2), to a reduced value of 20.0 kJ/mol-1.5 kJ/mol = 18.5 kJ/mol. As for the second point of above and to investigate long-range corrections for QM:QM schemes, besides the previously studied Na9 Cl9 cluster, we also considered clusters Na25 Cl25 , Na49 Cl49 , Na81 Cl81 , and finally Na121 Cl121 . These were treated with various methods, e.g., MP2 and RPBE+D3. (All embedded in a 4×4×4 slab, the low-level method in Eq.(1) being RPBE+D3.) For MP2, we get a long-range correction ∆Ecorr (periodic) in Eq.(2) of 4.3 kJ/mol, whereas RPBE+D3 gives the slightly larger value of 5.0 kJ/mol. Thus, our most accurate value so far for the adsorption energy should be corrected by another 0.7 kJ/mol downward, to 18.5 kJ/mol-0.7 kJ/mol=17.8 kJ/mol. (4) Zero-point energy corrections. We also did normal mode analyses for adsorbed and “free” CO molecules, at various levels of theory, to get zero-point energy corrections. The most accurate estimate came from a QM:QM (MP2/TZ:RPBE+D3) embedding calculation, with the CO-Na9 Cl9 cluster embedded in a 4×4×4 cell as previously. We find that adsorbed CO on NaCl is destabilized by zero-point energy, by an amount of 4.6 kJ/mol. Thus, our “best estimate” for the zero-point corrected adsorption energy is 17.8 kJ/mol-4.6 kJ/mol=13.2 kJ/mol, corresponding to ∆Hads (0 K). This value is significantly smaller than the adsorption enthalpy of CO on MgO, partly due to the larger zero-point energy difference 7 . It should be noted, though, that according to periodic DFT calculations zero-point corrections could be somewhat lower also for CO/NaCl than estimated from the MP2 embedding calculations, in the order of 2-3.5 kJ/mol depending on method/basis. Considering also the error of the harmonic approximation, the (ZPE-corrected) ∆Hads (0 K) may easily be 1-2 kJ/mol larger than the ∼ 13 kJ/mol just given. (5) Comparison to experiment. At this point it is useful to compare to experiment. In Ref. 43 , for a coverage ACS Paragon Plus Environment

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of θ = 1/2 and at low temperatures, an adsorption enthalpy (absolute value) ∆Hads (T) of 14±1 kJ/mol was reported. In Ref. 44 , a value of 13 ±3 kJ/mol

was given, weakly coverage-dependent and at relatively low temperatures. In that reference, further analysis gave a ∆Hads (0 K) value of 12.6 ±3 kJ/mol. Hence, our theoretical values are “in the ballpark”. The CO-surface bond is thus relatively

weak, and can be characterized as being due to physisorption. About half of the bonding is due to dispersion.

3.1.2

Higher coverages

(1) Periodic DFT calculations: Parallel/upright phases. Next, we investigated coverage effects, again by using periodic DFT and slab models first. Always a 4×4×3 slab was adopted, and, like before, the two upper layers and the CO molecules were optimized. In Fig.2 we show the specific slab models considered, with coverages ranging from θ = 1/8 (one of the low-coverage cases of before), up to θ = 1. In that case each the eight Na+ ions of the upper layer is covered with one CO molecule. For cases with 2 ≤ 6 CO molecules per cell

various non-equivalent models are possible, some of them studied here. Specifically for N = 2 (θ = 1/4), N = 3 (θ = 3/8), and N = 4 (θ = 1/2) molecules per cell, two different arrangements denoted as N and N.d were realized, which differ from each other by the relative arrangement of CO molecules, as depicted in the figure (“d”=“diagonal” configuration). All optimized geometries shown have CO molecules in perfect “parallel/upright” (P/U) positions, with C pointing down to a Na+ ion. For θ = 1, this corresponds to the well-known p(1×1) structure of CO/NaCl(100), with one CO per cell, the latter containing one Na+ and one Cl− ion per layer 25,45 . Here we represent this configuration by a 4×4 cell with eight CO molecules in a cell consisting of eight Na+ and eight Cl− per layer. (Note that in our nomenclature, the p(1×1) would be √ √ 2 × 2.) Using the eleven configurations shown in Fig.2, various functionals and a single (Γ) k-point for the calculations, the energies gained when adding one more

CO molecule to the cell as listed in Tab. 3 were obtained. With an obvious notation, this adsorption energy was calculated as De = E((CO)N −1 @NaCl)+E(CO,

free)−E((CO)N @NaCl).

Comparing θ = 1/8 (low coverage) with higher coverages, we see that up to θ ∼ 1/2, the adsorption energies (without zero-point and temperature corrections)

depend only weakly on θ, in accordance with experiment 44 . At higher coverages,

the adsorption energies per additional CO become typically larger. Comparing ACS Paragon Plus Environment

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1 θ=1/8

4 θ=1/2

4.d θ=1/2

2 θ=1/4

2.d θ=1/4

5 θ=5/8

6 θ=3/4

3 θ=3/8

7 θ=7/8

3.d θ=3/8

8 θ=1

Figure 2: NaCl (4×4×3) cells with an increasing amount of CO on the surface: From a single CO molecule per cell (θ=1/8, N = 1 molecule) up to full coverage (θ=1, N = 8 molecules). The number of CO molecules and the coverage is given below the individual figures. In certain cases, several structural models (denoted as N and N.d) were considered. θ = 1/8 and θ = 1 (full coverage), for example, the adsorption energies for the latter are higher by 0.4 kJ/mol for PBE+D2. (This value can be larger depending on method, with the 5.5 kJ/mol for RPBE+D3 being an extreme case. The method dependence is non-negligible here, with an average extra stabilization of 2.2 kJ/mol for the five methods shown.) Qualitatively, all methods indicate that the CO molecules on the NaCl surface attract each other at high coverages, which explains the extra stabilization. Further, comparing D2 and D3 dispersion corrections suggests that the stabilization is more pronounced for the latter, in particular in case of RPBE. Comparing N and N.d configurations, the latter are typically more stable, probably because their average inter-adsorbate distances are smaller. A possible cause of the attraction is dispersive interaction between the molecules.

(2) Cluster calculations: Tilted phases. As mentioned, all of the values in Tab. 3 were obtained with the parallel/upright configurations. However, we also used a cluster approach to describe higher coverages of CO on NaCl(100). Here, we adopted surface slabs with only two layers, as three layers would cause a dipole moment. (This dipole moment is corrected in the periodic calculation, but cannot easily be corrected in the cluster model.) To ensure that the CO molecules do not exhibit too strong boundary effects, the lateral


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coverage PBE+D2 PBE+D3 RPBE+D2 RPBE+D3 HSE+D3

1 1/8 22.0 21.3 23.0 23.1 21.6

2 1/4 21.7 21.1 22.9 23.1 21.4

2.d 1/4 22.3 21.7 23.5 23.7 22.0

periodic slab 3 3.d 3/8 3/8 21.7 22.2 21.0 21.5 22.8 23.4 23.3 23.8 21.4 21.9

models: P/U 4 4.d 1/2 1/2 21.5 22.2 21.0 21.5 22.7 23.3 23.1 23.6 21.3 21.8

phases 5 6 5/8 3/4 22.1 22.5 21.9 23.4 23.6 25.1 27.2 28.3 22.3 23.6

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7 7/8 22.5 23.4 25.0 28.9 23.9

8 1 22.4 23.3 24.9 28.6 22.6

Table 3: Adsorption energy when placing a single additional CO molecule on NaCl (in kJ/mol), for different functionals and coverages representing the eleven structural models based on a 4×4×3 cell, as shown in Fig. 2. A single (Γ) k-point was used, and a cutoff of 700 eV. extension of the surface model was enlarged and no CO molecules are place at the boundary (outmost layers) of the cluster. We have used different such boundaryextended clusters, based on 6×6 motifs (corresponding to 4×4 in the periodic case), 8×8 motifs (corresponding to periodic 6×6), 10×10 motifs (corresponding to 8×8 periodically), and 12×12 motifs (corresponding to 10×10 periodically), respectively. For the 6×6×2 clusters, various coverages (with one to eight CO molecules per cluster) have been considered. Here we always placed CO molecules towards the center of the cluster, even at low coverages. For low coverages, then, these arrangements can be viewed as mimicking island formation. The situation 6×6 with eight CO molecules corresponds to full coverage (θ = 1) for a periodic 4×4 cell of above. 8×8, 10×10 and 12×12 motifs with N = 18, 32 and 50 molecules correspond also to θ = 1 – these larger clusters merely serve to allow for structural alternatives and detect larger regular patterns, should they arise. All of the models (except for N = 1 and N = 2, for the 6×6 cluster) considered are shown in Fig.3. More precisely, Fig.3 shows the structures for coverages θ ≥ 3/8 found when

starting from analogous clusters with CO molecules in perfect upright positions (as obtained from corresponding periodic DFT calculations), and then optimizing

them while freezing all NaCl surface atoms of the lower layer and those on the boundary NaCl units. Clearly, different minima were obtained in this case: The CO molecules started to tilt towards each other. Optimizations using smaller coverages (θ = 1/8 and θ = 1/4 in case of a 6×6 cluster), leaves the CO molecules in parallel/upright position. The tilt structures shown in Fig.4 for θ ≥ 3/8 were

obtained with PBE+D2, however, very similar structures are found with PBE-D3, B3LYP-D3, and MP2. We assign these tilted structures as being due to attractive

(van der Waals) dispersion interactions between the molecules. We will give nu-


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3 (θ = 3/8) 6×6 (4×4)

8 (θ = 1) 6×6 (4×4)

4 (θ = 1/2) 6×6 (4×4)

18 (θ = 1) 8×8 (6×6)

5 (θ = 5/8) 6×6 (4×4)

6 (θ = 3/4) 6×6 (4×4)

32 (θ = 1) 10×10 (8×8)

7 (θ = 7/8) 6×6 (4×4)

50 (θ = 1) 12×12 (10×10)

Figure 3: Optimized NaCl clusters with an increasing number of CO molecules. Shown are top views of the clusters, which consist of two NaCl layers each (i.e., n × n × 2) and which are surrounded by boundary substrate (Na, Cl) ions with no CO molecules. The numbers 3-50 indicate the number of CO molecules on the clusters, in this case all in a “tilt” configuration (see text). Formal coverages (without counting the boundary atoms) in brackets. The lateral cluster size n × n is given below the figures, with the size of a corresponding periodic cell (without boundary atoms) shown in brackets. Structures shown were obtained by cluster calculations on the PBE+D2/aug-pc-1 level of theory. merical evidence for this interpretation shortly. (3) Periodic DFT calculations: Tilted phases at monolayer coverage. We also searched for non-perpendicular phases within the periodic approach. For this purpose, we took the tilted, high-coverage (θ = 1) structures found in the cluster approach (in most cases) as a starting point for new optimizations including periodic boundary conditions. From the clusters, we removed the outer NaCl boundaries again, obtaining this way from a 6×6 cluster a 4×4 periodic cell, from a 8×8 cluster a 6×6 periodic cell and so forth, see the lower part of Fig.3. For the 4×4, a third NaCl layer was again included, leading to a periodic 4×4×3 unit cell, while for the larger lateral cells a periodic two-layer model n × n × 2 (n =6, 8, 10)

was usually adopted.


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8 T/I 4×4

18 T/I 6×6

32 T/I 8×8

32 T/A 8×8

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50 T/I 10×10

32 T/S 8×8

Figure 4: Optimized unit cells (top view) obtained from periodic DFT (PBE+D2), for CO on NaCl(100), coverage one. Different cell sizes (in the figure 4×4×3 or 8×8×2) were adopted. Only optimized, “tilted” arrangements are shown with “T/I” indicating the “tilted / irregular”, “T/A” standing for “tilted/antiparallel”, and “T/S” for “tilted/spiral” phases. Geometry re-optimization including relaxation of the topmost NaCl layer the periodic PBE+D2 level of theory lead now to different types of tilted, non-parallel structures which were lower in energy than the corresponding parallel/upright configurations. These motifs can roughly be classified in three different categories. • The first structural motif, called “tilted/irregular” (T/I) in the following,

is characterized by arrangements similar to those seen for clusters in the lower row of Fig.3. The periodic T/I unit cells obtained after geometry optimizations of the corresponding clusters, are shown in Fig.4, upper row, for increasing cell sizes 4×4 to 10×10. These T/I structures are characterized by parquet-like arrangements of small domains of ordered, tilt CO molecules. Within these small domains CO molecules are often pointing in one direction, in the next domain this direction being rotated azimutally by around 90 or 45◦ . Beyond domains no clear long-range order is visible. ACS Paragon Plus Environment

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• The second tilted phase found is the well-known “tilted/ antiparallel” (T/A)

phase of CO (θ = 1)/NaCl, called p(2×1)/antiparallel elsewhere 25,45 . To

find this structure, we started not from cluster structures but adapted the p(2×1)/antiparallel structure of Ref. 25 instead. In the T/A structure, the tilted CO molecules are oriented in a long-range ordered fashion: They form chains of CO molecules all tilt in one direction, with neighbour chains of CO molecules being tilted in opposite (“antiparallel”) direction. A pictorial representation, for the 8×8×2 unit cell, is shown in the second row of Fig.4. Note that this unit cell contains 32 CO molecules, rather than the smallest unit cell (called (2 × 1)) that can be defined which contains only two CO

molecules 25 . In agreement with Ref. 25 , we find the T/A structure to be most stable, for CO (θ = 1)/NaCl(100) at zero temperature (see below).

• Finally, we find with the large quadratic unit cells adopted here a third type

of tilted structures, when using manually modified initial starting structures derived from cluster models. Then we obtain what we will call “tilted/spiral” (T/S) in the following. These are periodic structures containing spirals, or concentric circles of tilted CO molecules. An example, for the 8×8×2

unit cell, is also shown in the second column of Fig.4 (“32 T/S”). Clearly, a “spiral-like” arrangement of CO molecules is seen. Spirals were found not only with 32 CO molecules in an 8×8×2 cell, but also with eight CO molecules in a 4×4×3 cell, and with 18 CO molecules in a 6×6×2 cell (not shown). For 50 CO molecules in a 10×10×2 cell, periodic DFT+D2 calculations did not converge for the T/S, but only for the T/I structure. Several test calculations in which the electronic structure method, the lattice relaxation protocol or the dipole correction in periodic DFT were altered, have been carried out. In all cases, non-perpendicular structures of the T/I and T/S types were found as (local) minima, in addition to the less stable parallel/upright and the more stable T/A arrangements. To quantify relative stabilities of various structures, we show in Tab. 4 the differences between several “tilted” configurations, using different periodic DFT methods and unit cells, all for θ = 1. The energy differences ∆E listed are given relative to the parallel/upright configuration, ∆E = (E(parallel/upright) − E(tilted))/N

,

(3)

per CO molecule (N is the number of molecules per cell). Thus, ∆E is a stabilization energy by tilting, per CO. From the table we note the following: • Tilted structures are lower in energy than the parallel/upright counterparts: ACS Paragon Plus Environment

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slab 4×4×3

N 8

6×6×2

18

8×8×2

32

10×10×2

50

structure T/I T/I+ZPE T/A T/A+ZPE T/S T/I T/I+ZPE T/S T/I T/I+ZPE T/A T/S T/I

periodic DFT method PBE+D2 PBE+D3 RPBE+D3 PBE 1.55 1.34 2.34 0.38 1.06 – – – 1.81 1.59 2.52 0.92 1.50 – – – 1.30 1.19 2.19 0.06 1.49 1.24 1.86 – 1.17 – – – 1.49 – – – 1.59 – 2.13 – 1.32 – – – 1.79 1.72 2.13 – 1.51 – 1.92 – 1.56 1.38 1.96 –

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HSE+D3 0.53 – 1.20 – – – – – – – – – –

Table 4: Differences in adsorption energies per CO molecule, ∆E of different tilted phases of CO on NaCl for coverage θ = 1, relative to the parallel/upright case, according to Eq.(3): “T/I”= “tilted/irregular”, “T/A”= “tilted/antiparallel”, “T/S”= “tilted/spiral”. All for various periodic DFT methods and cell sizes. ZPE indicates the zero point energy correction, N the number of CO molecules per cell. All values are in kJ/mol. Positive values indicate a higher stability of the tilted configuration, compared to P/U. All ∆E values are positive. Further, the antiparallel phase T/A is the lowest minimum, lower in energy than both the T/I and T/S phases. • On the PBE+D2 level of theory (for which we have the most complete data set), the extra stabilization energy per CO is around 1.5 kJ/mol-1.6

kJ/mol without zero-point energy (ZPE) corrections for the T/I phase, and around 1.8 kJ/mol for the T/A phase. ZPE corrections reduce these values slightly, by several tenths of kJ/mol. This stabilization by tilting is largely independent of the size of the chosen elementary cell, despite cells of different size allow for different CO arrangements. There are only small differences (0-0.2 kJ/mol depending on cell size, PBE+D2) between energies of the T/I and T/S configurations. The T/I motif seems slightly more stable than T/S. • As suggested earlier, tilting appears to be driven by dispersion. When dispersion is turned off, the difference between the parallel/upright and nonparallel configurations is strongly reduced. For the T/I phase, ∆E decreases from 1.55 (PBE+D2) and 1.34 (PBE+D3) to 0.38 kJ/mol (PBE) for the smallest unit cell (4×4×3). Along this line, the differences for RPBE+D3 are larger (up to 2.3 kJ/mol for T/I), while the HSE+D3 differences are smaller. ACS Paragon Plus Environment

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In Tab.5 we report geometrical parameters, namely the Na-C distance and the tilting angle ϑ (angle between CO molecular axis and surface normal), as obtained from periodic DFT, using the 4×4× 3 cells at coverage θ = 1, for the T/A and T/I phases, respectively. In the table, we give average values RNa−C and ϑ, because in practice we have a spread of values when using large unit cells with no symmetry ˚ to ±0.05 ˚ restrictions. The spread for bond lengths is tpyically ±0.01 A A depend-

ing on method and system, the spread for tilting angles around ±5◦ roughly. method PBE+D2 PBE PBE+D3 RPBE+D3 HSE+D3

T/A phase RNa−C (˚ A) ϑ (◦ ) 2.588 32.3 2.650 30.2 2.610 31.6 2.836 37.4 2.655 31.8

T/I phase RNa−C (˚ A) ϑ (◦ ) 2.582 33.0 2.661 31.1 2.594 32.9 2.829 36.8 2.605 33.0

Table 5: Average Na-C distances and tilting angles for T/A and T/I phases (4×4×3 cells, θ = 1), obtained with various periodic DFT methods.

From the table we first of all note that average PBE+D2 Na-C bond lengths for the T/A phase are 2.59 ˚ A, in good agreement with a recent experimental (LEED) value of 2.58 ± 0.08 ˚ A 25 . A tilting angle for the p(2×1) phase (corresponding to T/A) around 25◦ was predicted from IR measurements in Refs. 45,47 . In the recent

experimental (LEED) study gives a tilting angle of 28 ± 5◦ 25 . The CO molecules

are no longer right above the Na+ ions as for the U/P phase, but slightly displaced laterally. Our PBE+D2 tilting angles are around 32◦ . Hence also here the agreement is reasonable. Bond lengths and tilting angles obtained with other methods are found to be less reliable at least when compared to experiment. In particular, RPBE+D3 gives too long bond lengths and too large tilting angles. For the less stable tilted phases (T/I and T/S), interactomic distances and tilting angles ϑ are similar to those of T/A, however, the relative arrangement of tilted CO molecules differs. In Tab.5 we demonstrate the similarity of geometry parameters for the T/I phase.

Finally, the bond lengths of the respective parallel/upright structures exhibit distances from 2.680 to 2.943 ˚ A for PBE+D2 to RPBE+D3, respectively, i.e., they are about 0.1 ˚ A larger than for the tilted phases. (P/U values are not listed in the table). Computed tilting angles for the P/U phase deviate only up to 0.07 degrees


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from the ideal value (ϑ = 0◦ ). (4) Summary: Structures. As mentioned, tilted structures for CO/NaCl(100) are known since some time: For monolayer coverage (θ = 1), besides a high-temperature parallel/upright configuration (the p(1×1) phase, or P/U), at low temperatures (< 35 K) a second ordered phase (of p(2×1) symmetry) emerges with tilted, antiparallely oriented CO molecules – the p(2×1)/antiparallel structure, or T/A 23,25,45,46 . Here we find in agreement with previous work the T/A motif being most stable, at T = 0 and for θ = 1. At lower coverages (particularly for θ < 3/8) parallel/upright arrangements are more stable. In addition to the P/U and T/A configurations, we see various T/I phases when using large unit cells. For θ = 1, these are similar but not identical, to the “(2×1)/herringbone” structure found by periodic DFT+D and van-der-Waals DFT in Ref. 25 . According to Ref. 25 , the “(2×1)/herringbone” is higher in energy than the “p(2×1)/antiparallel” global minimum, similarly to what we found for our T/I structures. In our calculations using large unit cells, we were unable to find a (2×1)/herringbone structure. However, in addition to T/A and T/I phases, we get T/S (“spiral”) structures which have not been reported before to the best of our knowledge. They are similar in energy to T/I phases. The size of stable spirals depends in the end of the size of the chosen unit cell. In this context we note, perhaps related or not, that spiral structures were observed by Ertl and coworkers when studying reactions of CO with oxygen on Pt(100) 48 . We also note that in Ref. 49 , for the analogous (isoelectronic) N2 /NaCl(100) system, a transition from an ordered, tilted p(2×1) to another ordered structure with spiral structures (called “vortices” there) has been found by Monte Carlo simulations, when heating the surface above 25 K. For CO/NaCl, we see tilted structures / islands (T/I and T/S) not only at θ = 1 but already at lower coverages (but not too low coverages, when “upright” orientation becomes more favourable) and at 0 K. Based on our calculations and the findings of Ref. 25 , then, we expect that various tilted phases exist which are close in energy, and could therefore be populated at finite temperatures.

3.2

Vibrational analysis of CO on NaCl(100)

In Refs. 12,13,45,47,50 , IR spectroscopy has been used to characterize CO on NaCl(100). The role of vibrational exciton splitting has been emphasized. It is therefore inACS Paragon Plus Environment

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structive to study infrared spectroscopy and Davydov (excitonic) splittings for various structural models and coverages of CO/NaCl(100) from first principles.

3.2.1

“Single” CO: Free or adsorbed

Frequencies for the stretch vibration of a single CO molecule – free or adsorbed on NaCl(100) – obtained with different theoretical methods are listed in Table 6. Shown are the anharmonic, fundamental frequencies ω0 , the first two overtones ω1 and ω2 , the harmonic frequency ω0 (harm), and an anharmonicity constant 50 , δ = 2ω0 − ω1

.

(4)

All quantities are given in cm−1 (i.e., they are rather wavenumbers ν˜ = ω/2πc).

frequencies, anharmonicity method/model system ω0 ω1 ω2 ω0 (harm.) PBE+D2/aug-pc-1/mol. CO 2096 4166 6211 2129 PBE+D2/per. CO 2088 4150 6188 2124 RPBE+D3/per. CO 2066 4106 6121 2102 CCSD(T)/DZ/mol. CO 2070 4115 6134 2104 CCSD(T)/TZ/mol. CO 2110 4195 6254 2144 CCSD(T)/QZ/mol. CO 2126 4226 6300 2160 CCSD(T)/TQZa /mol. CO 2135 4244 6327 2163 experimentb CO 2143 4260 6350 2169 PBE+D2/per. (θ = 1/8) CO...NaCl 2103 4180 6232 2139 RPBE+D3/per. (θ = 1/8) CO...NaCl 2079 4132 6160 2114 experimentc (for θ = 1) CO...NaCl 2155 4274 – – a from an aug-cc-pV(T,Q)Z extrapolation using the l−3 and l−5 formulae b from Ref. 53 c for a coverage of 1, from Ref. 50

δ 26 26 26 25 25 26 26 26 26 26 –

Table 6: Vibrational frequencies for the stretch mode of CO, either free (CO, upper part of table) or adsorbed (CO...NaCl, lower part), obtained with DFT and CCSD(T), using either molecular (“mol.”) or periodic (“per.”) models. The fundamental frequency is ω0 , while ω1 and ω2 are 1st and 2nd overtones. The harmonic frequency is ω0 (harm), and δ an anharmonicity constant defined in Eq.(4). All frequencies are in cm−1 .

For the free molecule (“CO”), both a molecular quantum chemistry (“mol.”) ansatz has been tested (using CCSD(T) with increasing basis sets and and also PBE+D2/aug-pc-1), as well as a periodic (“per.”) approach. For the periodic ˚, with a planecalculations, a single CO was put in a cubic box of side length 20 A wave cutoff of 700 eV. For the (single) molecule adsorbed on NaCl (“CO...NaCl”), periodic DFT calculations using a 4×4×3 supercell (θ = 1/8), 700 eV cutoff, and ACS Paragon Plus Environment

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otherwise the same parameters as in previous sections were performed. For anharmonic analysis, vibrational energies were computed by solving a onedimensional nuclear Schr¨ odinger equation via a grid method from an anharmonic potential energy curve (along the normal mode coordinate), obtained with the respective electronic structure method. The harmonic frequencies for CCSD(T) and DFT without surface were determined by analytic second derivatives. Harmonic frequencies from periodic DFT were obtained by normal mode analysis with VASP. From the table we note that for “free CO”, both PBE+D2 and RPBE+D3 underestimate the fundamental experimental frequency by 30-40 (PBE+D2) and 70-80 (RPBE+D3) cm−1 , respectively, with the range referring to harmonic and anharmonic. The experimental (anharmonic) fundamental is at 2143 cm−1 53 . The error of CCSD(T) is much lower, being 10-20 cm−1 with aug-cc-pVQZ (denoted as CCSD(T)/QZ in the table). When extrapolating the basis set (TQZ in the table), the error is further reduced to 6-8 cm−1 . In fact it is known that the CCSD(T) (harmonic) frequency when pushed to the basis set limit, reproduces experiment to within 3 cm−1 51 . All differences become larger for the overtones. The surface imposes a blue-shift on harmonic and anharmonic corrected frequencies of 15 (PBE+D2) and 12 (RPBE+D3) cm−1 for fundamental frequencies; the anharmonicity does not change the shift. Obviously, this shift is not simply technical due to using periodic boundaries and plane-wave bases, but a real surface effect: Comparing for molecular CO the PBE+D2/aug-pc-1/molecular to the plane-wave calculation PBE+D2/periodic, it is seen that the harmonic frequency decreases (rather than increases), by 5 cm−1 . A surface-induced blue-shift of the CO stretch vibration of similar size as here, is also known for CO when adsorbed on MgO(100) 52 . In Ref. 52 , the origin of this shift has been analyzed theoretically in detail, with the result that besides the interaction of the CO multipole moments with the surface field, a so-called “wall effect” makes a decisive contribution: Accordingly, the Pauli repulsion between the C lone pair and surface oxygens increases when the CO molecules vibrates on the surface. Similar effects can be expected for CO/NaCl(100), however, here we do not further analyze these effects. In Tab.6, we also report an experimental value for adsorbed CO, of 2155 cm−1 . This value is from Refs. 45,50 and refers to the main peak at a coverage of θ = 1 (see below). In Ref. 50 , also a weak signal for the 0 → 2 transition (1st overtone) was

observed, at 4274 cm−1 . Compared to the experimental value for the gas phase

CO fundamental (2143 cm−1 , see Tab.6), then, experiment predicts a blue-shift on


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the NaCl surface in the order of 12 cm−1 for the main, 2155 cm−1 peak, whereas we find a blue-shift of 13 and 15 cm−1 , for RPBE+D3 and PBE+D2, respectively (for θ = 1/8). For the first overtone, the surface induces a comparable blue-shift according to experiment, and a somewhat larger blue-shift according to theory. We also note that the anharmonicity δ is similar for free and adsorbed CO, with all methods studied. Also the absolute values of δ are in good agreement with experiment. Concerning methods, we conclude that (i) DFT underestimates vibrational frequencies of CO compared to experiment and CCSD(T), but (ii) trends of anharmonicity and adsorption are well reproduced already on the DFT level of theory. From a physical point of view, the surface has only a moderate influence on the CO vibration, shifting it to the blue, due a relatively weak (physisorption) interaction.

3.2.2

Free CO aggregates: Davydov splitting

When two or more (CO) oscillators interact, the vibrational transitions (e.g., v = 0 → v = 1) split due to exciton (Davydov) coupling 55 . For a dimer, interacting via an operator Vˆ one obtains two different vibrational excitations rather than one. The excitation energies E+ and E− are in the simplest approximation (neglecting

dispersion shifts of the excited states) 55,56 :

E+ = h ¯ ω0 + ∆

(5)

E− = h ¯ ω0 − ∆ .

(6)

Here, ∆ = h10|Vˆ |01i is the excitonic shift, with |01i and |10i being vibrational

exciton states with one or the other of two molecules vibrationally excited to v = 1, while the other stays in v = 0. Further, in the point-dipole approximation 55 , ∆=

(1) (2) (1) ~ (2) ~ µ01 ~µ01 ~ (~µ01 R)(~ µ01 R) − 3 R3 R5

(7)

(n)

where ~ µ01 is the vibrational transition dipole moment for the v = 0 → v = 1 ex~ the distance vector between both molecules. citation of monomer n (=1,2), and R If both transition moments are arranged parallel (like in the parallel/upright configuration of CO on NaCl), they form a so-called H aggregate and 57 ∆=

µ201 R3

.

(8)

In this case the excitation to the lower state E− is forbidden, while the excitation

to E+ is enhanced: The transition dipole moments from the ground state (|ψG i =


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√ √ |00i) to excited states (ψ+ = 1/ 2(|01i + |10i) and ψ− = 1/ 2(|01i − |10i), are µG− = 0 √ 2µ01 µG+ =

(9) .

(10)

Then, only a single fundamental transition occurs with doubled intensity and blue˚ (the shifted by ∆. For two parallel, upright CO molecules at a distance of 3.99 A closest distance between two Na+ ions on NaCl(100)), and a vibrational transition dipole moment of µ01 = 3.50·10−31 Cm for free CO according to Ref. 58 , we estimate from Eq. (8) a blue-shift ∆ of ∼ 0.9 cm−1 for the vibrational fundamental. This corresponds to a vibrational level splitting

∆s = E+ − E− = 2∆

(11)

of 1.8 cm−1 . On a surface, the (transition) dipole can be enhanced 59 , however, probably not that much in case of weak interaction as here. When N CO molecules interact with each other, the vibrational excited state splits into N sublevels 56 . In Fig. 5, split vibrational levels of two, three (in a linear configuration) and five free CO molecules (in centered-square form) as a function of intermolecular distance, R, are displayed. R values between 2 and 6 ˚ A were considered, but only a narrower interval is shown. Calculations were performed with PBE+D2 (and D3) and with CCSD(T) from molecular models, using different basis sets. For trimer and pentamer only the CCSD(T)/aug-cc-pVDZ results are shown, for the dimer also CCSD(T)/aug-cc-pVTZ, and PBE+D2/aug-pc-1 and aug-pc-3. All CO molecules were in parallel/upright orientation. Vibrational energies are relative to the fundamental transition of the monomer. Also indicated are by vertical lines, shortest distances between metal ions of selected materials with rocksalt structure. For the dimer we find from Fig.5(a) that at NaCl-typical intermolecular distances (∼ 4 ˚ A), the splitting ∆s is between 2.5 (CCSD(T)/aug-cc-pVTZ) and 2.7 cm−1 (PBE+D2/aug-pc-2), i.e., in the expected range. The splitting increases ˚ (roughly with decreasing distance, R, reaching values of ∆s ∼ 10 cm−1 at 2.8 A

the Li-Li distance of a LiF crystal). According to the ideal Davydov point-dipole model, the splitting should be ∆0 = 2µ201 /R3 . In numerical practice, we find some deviations from this behaviour. Taking, for instance, CCSD(T)/aug-cc-pDZ values and a fitting interval R ∈ [2.0, 6.0]˚ A, we get ∆s ∝ R−3.9 . The point-dipole model is better fulfilled at larger intermolecular distances: Taking the interval ˚, we find a ∆s ∝ R−3 behaviour on the CCSD(T)/aug-cc-pDZ level, R ∈ [3.4, 6.0]A whose slope can be used to determine µ01 according to Eq.(8), as µ01 = 4.59·10−31 ACS Paragon Plus Environment

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(a) dimer LiF MgO

CaO LiCl

(b) trimer

NaCl

LiF MgO

10

(c) pentamer

NaCl

LiF MgO

4 2 0 -2 -4 -6

8

CCSD(T)/aDZ

6 4 2 0 -2 -4

R

-6 R

-8 3

4 3.5 R (Angstrom)

R

-8 4.5

vibrational shift (1/cm)

CCSD(T)/aDZ CCSD(T)/aTZ PBE+D2/apc3 PBE+D2/apc1

vibrational shift (1/cm)

8

-10

CaO LiCl

10

6 vibrational shift (1/cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-10

3

4 3.5 R (Angstrom)

4.5

12 10 8 6 4 2 0 -2 -4 -6 -8 -10 -12

CaO LiCl

NaCl

CCSD(T)/aDZ

R R 3

4 3.5 R (Angstrom)

Figure 5: Vibrational shifts as a function of distance R between CO molecules for the dimer (a), trimer (b) and pentamer (c), relative to the fundamental transition of the monomer. All C-O distances have been optimized, on the level of theories shown, at fixed R. The insets show top views of the (CO)n n-mers. Vertical lines mark distances imposed by lattice constants of several ionic crystals with rocksalt structure: The Li-Li distance in LiF is 2.85 ˚ A, ˚ ˚ ˚ Mg-Mg in MgO 2.98 A, Ca-Ca in CaO 3.40 A, Li-Li in LiCl 3.63 A, and Na-Na in NaCl 3.99 ˚ A. Symbols denote calculated values, lines serve only to guide the eye. For the pentamer, the second line from the top is degenerate. Abbreviations: apcn=aug-pc-n; aXZ=aug-cc-pVXZ. Cm. This is in fair agreement with the mentioned literature value µ01 = 3.50·10−31 Cm of Ref. 58 . Also, according to Eqs. (5) and (6) the ideal splitting should be symmetric for the dimer, around 0. In practice we find the splitting towards the lower frequency of Eq. (6) being larger than the one for the higher frequency (9). One further finds some small dependence on basis set (and dispersion model, not shown). For the linear trimer in Fig.??(b) a similar picture emerges with three vibrational transitions, however. A splitting range ∆s = Emax − Emin can be defined, where Emax and Emin are maximum and minimum frequencies. The splitting is more pronounced and about 30%-50% larger than for the dimer case. The central frequency is close to the original, unperturbed frequency. The pentamer splitting (Fig.5(c)) results in four frequencies (one degenerate), with the most two deviating frequencies 10%-170% larger than for the dimer (depending on method and distance). At the CCSD(T)/aug-cc-pVDZ level one finds ˚. From Fig. 5 the pen∆s = 5.5 cm−1 at a NaCl-typical distance of R = 4.0 A tamer resembles a solid the closest. At this level and R value, the shift is between


25

R

R

4.5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

+2.5 and -3.0 cm−1 , giving a a total splitting or “bandwidth”, ∆s of about 5.5 cm−1 . (PBE+D2/aug-pc-1, which is not shown, gives 5.8 cm−1 for comparison.) On the way towards even more realistic surface-like arrangements, we also did a calculation with two rather than one shells of CO molecules around a central CO, giving 25 CO molecules in total. The resulting bandwidth was about 10 cm−1 on the PBE+D2/aug-pc-1 level of theory in this case. These observations are roughly consistent with the simple exciton model which states, that for a two-dimensional “tight-binding like” (nearest-neighbour interaction) scenario, an exciton band is formed with a width ∆s ∼ 8∆ 56 , where ∆ is the dimer shift. In the next subsec-

tion, we will explicitly take the surface into account. Later, IR intensities will be considered.

3.2.3

Vibrational analysis of CO on NaCl(100): Higher coverages

(1) Frequencies from periodic DFT calculations: Parallel/upright phases. On a real surface, CO frequency shifts in particular at higher coverages are known for CO on MgO 54 , and also CO on NaCl 50 . Recall that the frequency of a “monomer” on NaCl (more precisely at low coverage θ = 1/8) imposed a blue-shift of the CO stretch upon adsorption according to Tab. 6. We now consider higher coverages. For this purpose calculations with several CO molecules on a periodic 4×4×3 slab were studied. The upper NaCl layer was completely relaxed, and the vibrational frequencies were determined by normal mode analysis. Coverages ranging from 1/8 to 1 were studied, corresponding to the periodic models 1, 2.d, 3.d, 4.d, 5, 6, 7, and 8 of Fig. 2, i.e., all CO molecules were in parallel/upright orientation in this case. The results are shown in Table 7. With N CO molecules per cell, N frequencies distributed over a splitting range (“band width”) ∆s = Emax − Emin are obtained. (Note that this is a Γ-point cal-

culation, i.e. ∆s refers to pure intra-cell vibrational splittings.) The splitting ∆s is comparable to the one obtained without the surface (cf. Fig. 5): For the gas phase dimer, ∆s = 2∆ was 2.7 cm−1 for the PBE+D2 molecular calculation. This value is to be compared to 3.0 cm−1 obtained on NaCl for periodic PBE+D2, within the 2.d periodic model of Fig.2. For the pentamer, the splitting was 5.8 cm−1 for PBE+D2 in the molecular case, compared to 4.9 cm−1 for PBE+D2/periodic, using the model 5 of Fig.2 with five CO per cell. For the largest coverage (θ = 1, eight CO molecules, model 8), the band width ∆s is 9.4 cm−1 with periodic PDE+D2. As a result, the total band width increases with coverage as expected. Comparing different functionals, the level splittings for a given coverage are ACS Paragon Plus Environment

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method PBE+D2

PBE+D3

RPBE+D3

HSE+D3

periodic P/U model 1 2.d 3.d 4.d 5 6 7 8 1 2.d 3.d 4.d 5 6 7 8 1 2.d 3.d 4.d 5 6 7 8 1 2.d 3.d 5 6

1 (θ) (1/8) (1/4) (3/8) (1/2) (5/8) (3/4) (7/8) (1)

2139 2139 2139 2139 2140 2140 2139 2138 2136 2135 2136 2136 2137 2136 2136 2135 2114 2111 2114 2114 2116 2116 2116 2115 2249 2252 2248 2251 2249

2

vibrational mode 3 4 5 6

7

8

2136 2135 2134 2136 2136 2133 2132

2134 2134 2136 2135 2133 2132

2134 2135 2135 2133 2132

2135 2134 2133 2132 2132 2131 2131 2131 2131 2129

2132 2133 2133 2133 2132 2130 2128

2133 2133 2133 2132 2130 2128

2132 2132 2131 2130 2128

2131 2130 2129 2129 2128 2127 2128 2127 2127 2125

2107 2112 2112 2112 2113 2112 2110

2112 2111 2112 2111 2111 2110

2111 2112 2111 2110 2109

2111 2111 2110 2110 2109 2108 2109 2109 2109 2107

2251 2247 2246 2147 2247 2246 2246 2145 2245 2245 2244 2243

Table 7: CO stretch frequencies in cm−1 for periodic 4×4×3 slabs with increasing coverage (θ) – see Fig. 2 for the selected “parallel/upright” (P/U) periodic models. Different functionals are used. For HSE+D3, only selected coverages (with N = 1, 2, 3, 5 and 6 molecules per cell) were considered. similar; the largest ∆s values are found for PBE+D3, while the lowest ones are found for HSE+D3, at least at low coverages. (2) Frequencies from periodic DFT calculations: Tilted phases for θ = 1. We also computed vibrational frequencies for the more stable tilted phases, from periodic DFT. Out of many examples, we show in Tab.8 results obtained from calculations using 4×4×3 slabs with coverage 1 (i.e., eight CO molecules per cell), for the “tilted/irregular” and “tilted/antiparallel” phases (cf. Fig.4). Frequencies of tilted phases are slightly red-shifted compared to the paralACS Paragon Plus Environment

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∆s – 3.0 4.5 4.6 4.9 6.4 8.0 9.4 – 3.2 2.8 3.8 5.9 7.5 8.7 9.8 – 3.2 3.6 3.0 5.4 6.2 7.9 8.5 – 0.9 2.4 5.1 6.3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

periodic model T/I T/A P/U

1 N 8 8 8

2

3

vibrational mode 4 5 6

7

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8

2130 2129 2129 2128 2127 2125 2122 2120 2135 2130 2130 2129 2127 2126 2125 2124 2138 2132 2132 2132 2131 2131 2131 2129

∆s 10.3 10.5 9.4

Table 8: CO stretch frequencies in cm−1 for 4×4×3 slabs of coverage 1 (N = 8 CO molecules per cell) for the “T/I” and “T/A” tilted phases. The parallel / upright (“P/U”) result from Tab.7 is shown for comparison. The PBE+D2 functional has been used in all cases. lel/upright θ = 1 phase of Tab. 7, also given in Tab.8 for comparison. The red-shift is ∼ 8 cm−1 for T/I, and ∼3-4 cm−1 for T/A, compared to P/U. A possible reason

for this shift are changed multipole moment interactions in comparison to the parallel configuration, either with the surface or of the CO molecules with each other, and / or a slightly changed “wall effect”. On the other hand, the “bandwidth” ∆s is only slightly affected by tilting (by ∼ 1 cm−1 ). (3) IR intensities from cluster calculations: Upright/parallel phases. For comparing to experimental IR spectra it is mandatory to not only compute frequencies, but IR intensities as well. Unfortunately we were unable to get reliable IR intensities in periodic calculations. Therefore, we determined IR intensities from cluster calculations instead. For this purpose we adopted the 6×6×2 clusters of Fig.3 (corresponding to 4×4 periodically). In a first step, we forced all CO molecules to be in parallel/upright orientation. We used models with one (coverage 1/8) to eight CO molecules (coverage 1) as before, giving cluster models 1-8 analogous to those of Fig.3 (there shown only for 3-8). As in Fig.3, CO molecules were always placed in the center of the cluster. The geometries of the adsorbates were constrained to be those found for the P/U structures from periodic PBE+D2 (using 4×4×3 slabs) in this case. This constraint was necessary as reoptimization of clusters would yield the tilted phases of Fig.3, at least for N ≥ 3. Then, frequencies and IR intensities were

determined with PBE+D2/aug-pc-1 cluster calculations in the double-harmonic approximation, as shown in Tab.9. All intensities there are given relative to the intensity of the N = 1 (θ = 1/8) peak. We can now directly compare the periodic calculations with the (constrained geometry) cluster calculations: The cluster with a single CO yields a harmonic


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1 quantity

N

(θ)

frequency

1

(1/8)

2

(1/4)

3

(3/8)

4

(1/2)

5

(5/8)

6

(3/4)

7

(7/8)

8

(1)

2

3

vibrational mode 4 5

6

7

8 ∆s

P/U cluster models

intensity frequency intensity frequency intensity frequency intensity frequency intensity frequency intensity frequency intensity frequency intensity a

2145 2139a 1.000 2146 1.890 2146 2.762 2145 3.619 2145 4.457 2145 5.218 2145 5.940 2145 6.486

– – 2143 0.005 2143 0.005 2149 0.006 2142 0.014 2142 0.010 2142 0.008 2142 0.067

2.8 2141 0.002 2142 0.002 2142 0.005 2142 0.003 2142 0.019 2142 0.012

4.2 2140 0.009 2141 0.000 2140 0.001 2140 0.001 2141 0.056

5.4 2139 0.020 2140 2137 0.003 0.008 2139 2138 2136 0.000 0.004 0.006 2139 2138 2138 2136 0.012 0.000 0.065 0.013

for a cluster without the periodic-structure constraint

Table 9: Vibrational analysis for an increasing number (N = 1 − 8, coverage θ) of CO molecules on 6×6×2 NaCl clusters. The structures are analogous to Fig.3, however, all CO molecules are forced into the parallel / upright position by using structures optimized in periodic calculations – otherwise the CO molecules tilt. (Except for N = 1 and 2; for N = 1, a test calculation with and without the periodic-structure constraint has been made, see the footnote). Both harmonic vibrational frequencies (in cm−1 ) and relative IR intensities are shown. The intensities are relative to the intensity of the single low-coverage peak (N = 1, θ = 1/8, in bold). Calculations were done on the PBE+D2/aug-pc-1 level of theory. frequency of 2145 cm−1 on the PBE+D2/aug-pc-1 level according to Tab. 9. The corresponding value using a periodic model with one molecule in a 4×4×3 slab model is 2139 cm−1 according to Tab.6. The cluster model is thus of similar accuracy compared to a plane-wave vs. aussian basis set calculation (∼ 6 cm−1 deviation) as observed for the free CO molecule (cf. Tab.4). The splittings ∆s obtained with either periodic or cluster calculations are in good agreement. They range from 2.8 to 9.0 cm−1 for the cluster models with coverages 1/8 to 1 (Tab.9), and from 3.0 to 9.4 cm−1 for the corresponding periodic models (Tab.7). Concerning (relative) IR intensities, we find for all N the highest-energy transition to be very intense, while other vibrations are weak. For instance, for the CO ACS Paragon Plus Environment

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dimer the intensity ratio between the higher- and lower-energy peak is ∼400. The

intensity of the monomer is about half of the most intense dimer peak. This is in agreement with the simplest form of the exciton model, where, for the dimer according to Eqs.(9) and (10), the higher-frequency transition is enhanced by a factor of two relative to the monomer while the lower-frequency is forbidden. For larger N the intensity of the highest-frequency peak increases roughly proportional to N according to Tab.9 while lower-lying transitions remain forbidden – in agreement with expectations from the simple exciton model 56 . Thus, for parallel/upright

adsorption, only the high-frequency end of the exciton band is IR-visible, and only a single CO frequency is expected to be seen in the IR spectrum. (4) IR intensities from cluster calculations: The tilted phases. From N = 3 CO molecules per 6×6 cluster (corresponding to 4×4 periodic), i.e. from a coverage of 3/8 onward, the CO molecules start to tilt towards each other when optimized as demonstrated in Fig.3. Since now the CO molecules are not parallel to each other anymore, frequencies and Davydov splittings change slightly as demonstrated already from the periodic calculations in Tab.8. However, also the IR intensities should differ from parallel/upright. (Relative) Intensities and frequencies arising from 6×6×2 clusters are displayed in Tab. 10. In the upper part of the table, geometries were those shown in Fig.3, fully optimized, for coverages 3/8 to 1. These models correspond to the T/I (or S) periodic models (so far only called so for θ = 1) – no clear assignment to either I or S is made here for simplicity. In addition, in the last two rows of the table we also consider the cluster analog of the periodic T/A model, by adopting a 6×6×2 cluster with eight CO molecules which were not optimized now but the T/A structure taken from the periodic calculation instead. In comparison to the values obtained in Tab. 9, the frequencies are further red-shifted. As an example, the highest-frequency peak for N = 3 shifts from 2145 cm−1 for the (constrained) P/U configuration of Tab.9), to 2132 cm−1 for the corresponding tilted (T/I(S)) phase of Tab.10. The corresponding red-shift is thus around 13 cm−1 in this case. About 6 cm−1 of the red-shift is due to the artifical geometry-constraints in the P/U configurations of Tab. 9. This can be seen from comparing a test calculation with unconstrained and constrained geometries for N = 1 (one molecule in upright orientation) in Tab.9, showing that releasing the geometry-constraint red-shifts the frequency from 2145 to 2139 cm−1 . We expect similar shifts also for N ≥ 3. However, there is also an additional, “physical”

red-shift due to tilting (on the order of 7 cm−1 ), similar to what was already found in the periodic calculations (cf. Tabs. 7 and 8). ACS Paragon Plus Environment

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1 quantity

N

(θ)

frequency intensity frequency intensity frequency intensity frequency intensity frequency intensity frequency intensity

3

(3/8)

4

(1/2)

5

(5/8)

6

(3/4)

7

(7/8)

8

(1)

frequency intensity

8

(1)

2

vibrational mode 3 4 5 6

7

8 ∆s

T/I (or T/S) cluster models 2124 0.071 2128 2123 0.037 0.175 2128 2125 2119 0.040 0.092 0.352 2128 2126 2125 2121 0.310 0.198 0.537 0.121 2128 2125 2124 2122 2120 0.043 0.095 0.007 0.118 0.319 2126 2126 2124 2122 2121 2120 0.783 0.064 0.360 0.315 0.111 0.059 T/A cluster model 2138 2133 2133 2132 2130 2130 2129 2128 1.992 3.788 0.177 0.010 0.010 0.041 0.202 0.539 2132 1.553 2131 3.000 2131 2.619 2132 3.460 2131 4.454 2131 3.466

2128 0.942 2128 0.029 2130 0.965 2128 0.045 2128 0.141 2127 0.874

Table 10: Vibrational analysis for an increasing number (N = 3 − 8, coverage θ) of CO molecules on a 6×6×2 NaCl cluster. All CO molecules are in a tilted configuration, either “T/I” (or “T/S”) as in Fig.3, or in the “T/A” configuration. All T/I(S) structures have been optimized in cluster calculations, on the PBE+D2/aug-pc-1 level of theory. Harmonic frequencies are in cm−1 , and intensities relative to the intensity of the N = 1 P/U peak of Tab. 9.

A second effect of tilting, again in agreement with the periodic calculations, is that also the level splittings become slightly larger: For full coverage (N = 8), for example, we have ∆s = 9.0 cm−1 in the parallel/upright phase (Tab. 9) but ∆s = 11.1 cm−1 in the tilted (T/I(S)) phase (Tab. 8). This may come as a surprise, since one would expect for tilted species a smaller Davydov splitting according to Eq.(7). However, at least for the T/I(S) structures the CO molecules are quasirandom oriented with respect to each other (quasi-random azimuthal angle), and also the intensities change which makes analysis more complex. One of the most striking effects of tilting is that the lower frequencies gain IR intensity, in expense of the highest-frequency mode. For instance, at θ = 1 the highest-frequency mode of the parallel/upright configuration has a relative intensity of about 6.5, all other signals being only 1 % of that value or less (Tab.9). In contrast for the tilted high-coverage phase, the relative intensity of the corresponding highest-frequency mode is around 3.5 according to Tab. 10, with several other satellite peaks appearing with considerable intensity. To facilitate analysis, ACS Paragon Plus Environment

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7.3 7.7 11.7 10.7 10.6 11.1

9.3


7 P/U

6 relative intensity

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5 4 3 T/A 2 T/I (S)

1 0

2120

2130 2140 CO vibrational frequency (1/cm)

2150

Figure 6: “IR spectra” for the parallel / upright and tilted / irregular (or spiral) and tilted / antiparallel phases of CO/NaCl(100), obtained from (constrained) clusters 6×6×2 with N = 8 CO molecules (θ = 1) each. The PBE+D2/aug-pc-1 level of theory was applied, and computed stick spectra are given with intensities relative to the N = 1 (θ = 1/8) P/U peak, cf. Tab. 9. we show in Fig. 6 “IR spectra” (in the form of stick spectra), estimated from 6×6×2 cluster calculations, for the parallel / phase as well as the tilted phases, tilted / irregular (or spiral), and tilted / antiparallel, all for θ = 1. We see an almost singular, intense peak at around 2145 cm−1 for the P/U phase, and more structured and less intense, red-shifted spectra for the T/I(S) and T/A phases. The T/A phase appears to be slightly less red-shifted than the T/I(S) phase. In fact, also according to the simple Davydov model tilting will soften selection rules Lower-energy transitions become partly allowed while higher-energy transitions loose intensity. In this case the transition dipole moment vectors from √ the ground state (|ψG i = |00i) to excited states (ψ+ = 1/ 2(|01i + |10i) and √ ψ− = 1/ 2(|01i − |10i) are µG− = ~ µG+ = ~

1 (2) √ ~µ(1) µ01 01 − ~ 2 1 (1) √ ~µ01 + ~µ(2) 01 2 (1)

(12) .

(13)

(2)

Only if the two transition dipole vectors ~µ01 and ~µ01 are parallel to each other (as in the “upright/parallel” model), the ideal cancellation (for µ ~ G− ) or enhancement (for ~µG+ ) occurs as in Eqs.(9) and (10), otherwise |~µG− | = µG− will be different √ from zero and |~ µG+ | = µG+ be different from 2µ01 . As an example, consider a ACS Paragon Plus Environment

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tilted dimer with both transition dipoles lying in the same plane (say, the (y,z) plane) but one monomer tilted by +ϑ with respect to the surface normal, the other one by −ϑ, similar to what is found in the T/A phase with ϑ ∼ 30◦ . Then √ √ √ √ µG− = 2µ01 sinϑ and µG+ = 2µ01 cosϑ, giving µG− = 22 µ01 and µG+ = 26 µ01 for ϑ = 30◦ . Given that intensities are proportional to |µif |2 , this would correspond to an intensity ratio I+ /I− of 3:1 in this case.

3.2.4

Comparison to experiment

At this point a comparison to experiment is useful. Experimentally (see, e.g., Refs. 45,50 ), one finds in the IR spectrum of CO/NaCl(100) an anharmonic CO frequency of 2155 cm−1 50 , cf. Tab. 6, at coverage θ = 1 and temperatures around 40 K. This single peak has been interpreted as being due the p(1×1) structure, where all molecules are in parallel/upright position. This interpretation is compatible with our findings. We also find a single peak for the parallel/upright θ = 1 phase (corresponding to p(1×1)) according to Tab. 9 and Fig.6, at 2145 cm−1 on the PBE+D2/aug-pc-1 level of theory. Given the harmonic approximation, the errors due to the cluster model and the (DFT) method, the accuracy of the theoretical frequency is fortuitously good. In addition, in experiment a minority peak is found at 2149 cm−1 for low temperatures 45,50 , below the assumed p(1×1) → (2×1) phase transition at around

35 K. At the same time, the majority peak at 2155 cm−1 slightly shifts to lower

frequencies, by less than 1 cm−1 . The emerging double-peak structure at low temperatures has been interpreted as being due to an exciton-split signal for the above-mentioned ordered (2×1) tilted structure 50 , with a splitting of about 6 cm−1 . Accordingly, the tilting molecules (i) slightly red-shift in their CO frequency due to the stronger interaction with the NaCl surface, and (ii) exciton-split states at even lower energies become possible due to the fact that the CO molecules are no longer “parallel/upright”. Also this interpretation is roughly consistent with our findings. We find a general red-shift of CO frequencies upon tilting, and splitted (rather than single) IR signals for tilted phases. Quantitatively, our simple cluster models are not in full agreement with experiment: The exciton splitting in the tilted phases seems a bit larger than experimentally (∆s ∼ 10 cm−1 in theory),

more fine-structured (due to symmetry-loss in the clusters), and also the overall

red-shift when going from P/U to T/I(S) phases is larger than experimentally. Again, this is partly due to artificial constraints for the P/U cluster geometries. Apart from that, the important role at low temperatures of tilting and the excitonsplit IR spectra emerging from there, is indisputable also from our first-principles calculations. ACS Paragon Plus Environment

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4

Summary and conclusions

We have investigated the adsorption of CO on NaCl(100) surfaces, using large unit cells and both WFT and DFT-based methods. The large unit cells allow us to consider both low- and high-coverage phases (up to monolayers), as well as more complicated adsorption patterns including island formation. We find that at low coverages CO preferentially adsorbs in parallel/upright fashion, while at higher coverages (≥ 3/8 in the 4×4 periodic model), tilted phases are more stable, making the all-parallel/upright phase only a local minimum. The most stable tilted phase found here also with large quadratic unit cells, for θ = 1 is the p(2×1)/antiparallel (“T/A”) structure proposed earlier by experiment 45 and theory 19,25 . In addition, albeit higher in energy, phases with “irregular” or “circularly” oriented CO molecules were found as stable structures. These may gain importance in particular at higher temperatures, when entropic effects come into play which were not considered here. Tilting a single molecule lowers the energy (without ZPE and temperature corrections) by about 1.5-1.8 kJ/mol according to PBE+D2 (periodic, θ = 1), the lower value for T/I(S), the higher for T/A structures. At low coverages, the adsorption energy at T = 0 K including zero-point corrections is found to be around 13 kJ/mol, when using the most accurate QM:QM embedding methods, in reasonable agreement with experiment. Without zeropoint corrections the binding energy is around 18 kJ/mol. Using gradient-corrected DFT+D typically overbinds by several kJ/mol, consistent with earlier work 19,25 . The inclusion of dispersion forces is mandatory, both for the molecule-surface interaction as well as for intermolecular interactions and tilting. Vibrational frequencies can be calculated to high accuracy with WFT methods, while gradient-corrected DFT+D gives larger deviations from experiment. However, even simple gradient-corrected DFT+D accounts for the Davydov vibrational exciton splittings. For an isolated, (parallel/upright) CO dimer separated by the NaCl-typical lowest distance of ∼ 4 ˚ A, a Davydov splitting ∆s = 2∆ of about 3 cm−1 is found. This coupling corresponds in a simple coupled degenerate two-level

model, to a Rabi oscillation time 56 , τ = π¯h/∆s of about 6 ps. This period sets also the stage for vibrational energy transfer rates during, e.g., vibrational energy pooling in CO/NaCl. When all CO molecules are in parallel/upright configuration, as in the so-called p(1×1) phase for θ = 1, only the highest-energy Davydov-split vibrational states is IR-visible. In tilted phases, side peaks several cm−1 red-shifted emerge while the main peak looses intensity. Again, this behaviour is in fair agreement with ACS Paragon Plus Environment

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experiment. We find this effect not only for θ = 1 but also for lower coverages, possibly including structures with island formation. In general, IR-spectroscopy should be highly sensitive tool to unravel structural details of CO on NaCl at various coverages. In particular, T/I phases might be distinguishable from possible T/S phases in this way. A further suggestion from our work is that tilted structures exist also at incomplete coverage. Further, at θ = 1 alternative structures to p(2×1) could be populated at finite temperatures. In particular, new T/I and T/S phases were predicted which could be identified by their characteristic vibrational spectra.

Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft through project Sa 547/9-1 is gratefully acknowledged.

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CO Molecules on a NaCl(100) Surface: Structures, Energetics, and

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