Combined Experimental and Ab Initio Multireference Configuration

Article pubs.acs.org/JPCC

Combined Experimental and Ab Initio Multireference Configuration Interaction Study of the Resonant Inelastic X‑ray Scattering Spectrum of CO2 Dimitrios Maganas,† Paw Kristiansen,‡,§ Laurent-Claudius Duda,§ Axel Knop-Gericke,‡ Serena DeBeer,†,∥ Robert Schlögl,†,‡ and Frank Neese*,† †

Max-Planck Institute for Chemical Energy Conversion, Stiftstrasse 34−36, D−45470 Mülheim an der Ruhr, Germany Inorganic Chemistry Department, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4−6, 14195 Berlin, Germany § Department of Physics and Astronomy, Division of Molecular and Condensed Matter Physics, Uppsala University, Box 516, S-751 20 Uppsala, Sweden ∥ Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, United States ‡

ABSTRACT: The fundamental problem of the symmetry breaking in the resonant inelastic X-ray scattering (RIXS) of the CO2 gas molecule is studied. The measurements were performed under catalytically relevant conditions within an in-house constructed reaction cell. The experimental RIXS plane is constructed from a sequence of resonances, covering the near-edge X-ray absorption fine structure (NEXAFS) spectrum up to 539 eV. The spectra show significant sensitivity with respect to the excitation frequency. The NEXAFS absorption spectrum, as well as the corresponding RIXS spectra, is interpreted with the aid of multireference configuration interaction (MRCI) theory. In this framework, the configuration interaction space spans the space of the intermediate and final states with single and single−double excitations. The dynamic character of the RIXS spectra is investigated by considering the electronic−nuclear vibrational coupling with the bending and antisymmetric stretching vibrations in the important intermediate excited states. In addition, the vibronic coupling mechanism involving the Renner−Teller effect and the core−hole localization pseudo-Jahn−Teller effect of the intermediate states is fully considered. The physical origin of the observed spectral features is discussed qualitatively and quantitatively in terms of individual core-to-valence excitations and valence-to-core decays, respectively. The computational protocol presented here, based on multireference wave function ab initio theory, serves as an important reference for future theoretical and experimental applications of RIXS spectroscopy.

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INTRODUCTION Analysis of electronic spectra of chemical systems requires an unambiguous relationship between property and structure. A vast number of spectroscopic techniques are currently employed for such purposes, spanning the full wavelength region of optical light up to the hard X-ray energy scales. In this context, the core electron-based spectroscopies have special features. They are element-specific and probe the electronic structure around the core hole. In fact, X-ray absorption and X-ray emission core electron spectroscopies (XAS and XES, respectively) have been widely used to probe oxidation and spin states, as well as local environments in transition-metal complexes and biological systems.1−6 In addition, when emission is treated as a secondorder process, in resonance with absorption, the dynamic nature of the electron and structural properties of the system under investigation can be addressed.7 This is the course of the resonant core electron techniques, such as the inelastic X-ray scattering (RIXS; referred to also as resonant X-ray emission, RXES, and/or resonant X-ray Raman scattering, XRR). These techniques are employed to study stationary and dynamic © 2014 American Chemical Society

properties of matter, probing electronic and nuclear−vibrational loss processes. In recent years, there has been an increasing interest in applying resonant X-ray techniques for the analysis of crystalline and bulk materials, including transition-metal complexes, metalloproteins, and small gaseous molecules.7−39 From a theoretical point of view, it is certainly desirable to translate the experimental observables into electronic structure information.40,41 To this end, it is important to establish theoretical methodologies with significant predictive performance. However, in the case of resonant core electron spectroscopies, it is anything but trivial to calculate the experimental spectroscopic response of a material. This is a complicated task, which requires inclusion of electron correlation and electronic relaxation effects as well as state interference, multiplet structure treatments, and in the case of transition-metal systems, spin− orbit coupling effects. Among many approaches in the field,7 Received: June 6, 2014 Revised: August 2, 2014 Published: August 11, 2014 20163

dx.doi.org/10.1021/jp505628y | J. Phys. Chem. C 2014, 118, 20163−20175

The Journal of Physical Chemistry C

Article

provided excellent results in the prediction of vibrationally resolved RIXS spectra of many small molecules.9,47−50 The pure quantum chemical description of such a problem would require optimization of the vibrational states for each electronically excited state. This is, however, a computationally demanding task of extreme complexity. Alternatively, the so-called “classical model” approach can be applied.51 In this approach the electronic transitions occur vertically between the initial− intermediate and intermediate−final PESs. Averaging over all possible structures along the involved PESs provides the line shapes of transitions. Although in this approach concepts such as the lifetime−vibrational interference is not explicitly treated, it has been shown that those line shapes are usually close to the envelopes of the fine structure that would be obtained from a pure quantum chemical description.51 Recently, this model has been successfully utilized to interpret the temperature dependence of the nonresonance X-ray scattering spectrum of the CO2 gas molecule.52 CO2 is widely appreciated as an example of frequencydependent symmetry breaking.11,12,14,15 Apart from its environmental role in nature as a greenhouse gas,53 there is an increasing interest about its role in catalytic reactions.54,55 In fact, one of the most important and successful industrial processes in the context of selective oxidation is the silver-catalyzed epoxidation of ethylene with molecular oxygen.56−58 The important product of this process is ethylene oxide, which is used in the production of ethylene glycol, polyester fabrics, surfactants, and detergents. The principal challenge associated with this epoxidation process is the prevention of an undesired competitive reaction, which involves complete oxidation of ethylene to carbon dioxide.55 Xray spectroscopy has been employed to monitor and characterize several steps of this important reaction mechanism.59,60 We have recently developed a reaction cell that allows in situ XAS and RIXS investigations of the partial epoxidation of ethylene over a Ag catalyst at 1 atm and up to 250 °C. When operating under catalytically relevant conditions, new experimental challenges arise; for instance, at higher pressures gas concentrations can be large enough to induce spectral distortions.18 In this paper, we focus on the resonance emission spectroscopy of CO2. The RIXS spectrum of this molecule is understood in detail.14 In fact, numerous systematic theoretical studies have revealed the ability of the RIXS spectroscopy to probe the symmetry breaking and restoration in the gas phase by coupling electronic and vibrational excitations.14,46,61 Hence, the CO2 RIXS spectrum is an ideal test case for (a) understanding the changes that occur in the spectra under catalytically relevant conditions and (b) investigating the implementation of a multiconfigurational CI approach for the interpretation of these spectra. We present the experimental CO2 oxygen K-edge near-edge Xray absorption fine structure (NEXAFS) spectrum and the twodimensional 1s2p CO2 RIXS plane recorded along the incident photon frequency range of the CO2 oxygen K-edge NEXAFS spectrum. Importantly, the data is obtained in a real reaction cell and compared to spectra obtained under ideal conditions. The observed experimental features are then studied by multireference configuration interaction theory in which frequencydependent symmetry breaking is modeled by a generalized protocol that includes vibronic coupling and accounts for spectral distortions. Hence, the present contribution represents a new approach, in terms of both the experimental setup and the level of theory, which is applied for the spectral interpretation.

these problem can be most adequately treated with multireference ab initio wave function methodologies.17,42 In the framework of the ab initio wave function theory, the task is to determine the necessary low-lying states reached by experiment and construct the multiplet structure of the final states on an equal footing through a multiconfigurational treatment. Such an approach would most conveniently rely on the complete active space self-consistent field (CASSCF) technique. However, in RIXS spectroscopy, both core and valence electrons must be correlated; hence, the resulting active space, as well as the required states to be treated within the CASSCF module, becomes prohibitively large. Alternatively, one may resort to restricted active space configuration interaction methods, such as the restricted active space self-consistent field (RASSCF) or restricted active space configuration interaction (RASCI). It must be highlighted that several calculation protocols based on large-scale configuration interaction (CI), CASSCF, and RASSCF wave functions with fully separate state-optimized orbitals for X-ray spectroscopy were already available in the 1980s, thus including explicitly core hole relaxation into the orbitals on which the CI wave functions were expanded.43−45 However, it is important to note that it is usually not sufficient to simply obtain a CASSCF or RASSCF wave function that is averaged over chosen states. Therefore, one needs either to obtain better orbitals at the CASSCF (or RASSCF) step or to repair the deficiencies of the orbitals in a post-CASSCF calculation. Recently, the RASSCF/RASPT2 protocol has performed well for the calculation of 2p3d RIXS spectra of Ni(II) transition-metal complexes.17,34 In an analogous fashion, in uncontracted multireference configuration interaction (MRCI) treatments the orbitals are relaxed in the framework of a correlation calculation. Here, one performs single and double excitations relative to all reference determinants. Upon diagonalization of the nonrelativistic Hamiltonian to all orders, one obtains a fully relaxed electronic structure and realistic charge and spin distributions. However, even in this approach the straightforward diagonalization leads to size-inconsistent results; hence, the quality of the treatment degrades for larger molecules. In addition, for core electron spectroscopies, it is necessary to include all the doubly occupied orbitals; hence, the number of the required single and double excitations for the calculation of the final state multiplet structure becomes extremely large. Despite these practical difficulties, at the present time, MRCI provides the most attractive and transparent method of dealing with problems in the field of resonant inelastic X-ray scattering. In addition, the cross section of an X-ray scattering event is in general strongly photon energy dependent. This is due to the fact that the core excited intermediate state has a lifetime which is comparable with the absorption bandwidth. Thus, in line with the general theory of the resonance Raman, as the energy of the incident radiation approaches resonance with a particular electronically excited state, the vibronic levels of that state begin to dominate the intensity profile. Therefore, the coupling between electronic and nuclear motions described over appropriate electronic potential energy surfaces (PESs) is a relevant concept for the interpretation of X-ray scattering spectra. Because the sum-over-states formulation of the absorption cross section is very difficult to evaluate exactly for molecules with many vibrational degrees of freedom, expert groups in the field have chosen to solve the problem in the time domain using density functional theory (DFT). In fact, a simulation protocol has been developed for RIXS spectra based on a wavepacket time-dependent approach.9,46 This method has 20164



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THEORY RIXS Spectroscopy. During the RIXS experiment, the electronic states that give rise to the edge of an absorption spectrum are resonantly excited states that subsequently decay. For an O atom, the radiative decay with the highest probability after 1s core hole creation refers to a 2p-to-1s transition. The spectroscopy is therefore denoted as 1s2p RIXS. Alternatively, the process can be viewed as an inelastic scattering of the incident photon at the O atom. Theoretically, both absorption and inelastic scattering are described by the Kramers−Heisenberg relation. For the absorption, after averaging over different molecular orientations and polarizations, the cross section σA(Eex), where Eex is the incident photon energy, can be written in SGS units as 4π σA(Eex ) = Eex ∑ 3ℏc F

determined variationally through the solution of the MRCI eigenvalue problem: HIJ ΨMRCI = E ΨMRCI

with HIJ = ⟨ΦI|H̑ BO|ΦJ⟩ and Ψ = ΣiCi|Φi⟩. In the majority of MRCI calculations, the excited CSFs are considered up to double excitations. Thus, in the course of an uncontracted MRCI, the excited configurations are constructed by performing single and double excitations relative to each CSF in reference space. Thus, the MRCI wave function can be written as ΨMRCI =

ρ=x ,y,z

(EFI − Eex )2 + Γ

(1)

Analogous to Raman scattering, the X-ray scattering is described by second-order time-dependent perturbation theory. In its most fundamental form, one obtains the polarizability tensor for which the inelastic scattering is given by

∑ V

⟨F |mρ|V ⟩⟨V |mλ|I ⟩ EVI − Eex − i Γ

+

9ℏ4c 4

∑ ∑ F

(5)

EXPERIMENTAL PROCEDURES X-ray Absorption and X-ray Resonant Inelastic Scattering Measurements. The CO2 molecule was investigated systematically by both X-ray absorption spectroscopy and resonant inelastic X-ray scattering (RIXS). The O- K XAS and RIXS measurements were performed at beamline U41-PGM64 of BESSY II in Berlin, Germany. This is an undulator beamline that delivers about 1013 photons/s to the end station when tuned to the O- K-edge with a resolution of 170 meV, which was the beamline resolution used for both the XAS and RIXS measurements. XAS were recorded by means of total fluorescence yield (TFY) collected by a photodiode. The RIXS measurements were performed using a grazing incidence Rowland spectrometer65 operating at a resolution of 450 meV. All RIXS spectra were performed at a 90° scattering angle with inplane incident X-ray polarization. Detection (see also ref 62) was performed in a standard “polarization-blind” manner using a grazing-incidence grating and a position-sensitive MCP in singlephoton-counting mode (by optical read-out). The CO2 gas was put in the focus of the beam by means of a flow cell that separated the 7 sccm flowing CO2 at 1 atm from the UHV with a 100 nm thick silica nitride membrane. The outlet gas of the cell was monitored continuously by a quadruple mass spectrometer to ensure that the setup was leak-free during the measurements. Computational Strategy. All calculations were performed with the ORCA suite of quantum chemistry programs.63 Large core- and valence-augmented correlation consistent aug-ccpCVXZ (X = D, T, Q) all-electron basis sets were used throughout.66 Scalar relativistic effects were treated explicity by employing the implemented second-order Douglas−Kroll−Hess (DKH)67−69 corrections. For these calculations the corresponding relativistically recontracted aug-cc-pCVXZ basis set was used. Initial orbitals for the MRCI calculations are obtained from complete active space self-consistent field calculations. The chosen active space contains 12 electrons in 10 orbitals (CASSCF(12, 10)) and involves all bonding and antibonding C−O σ and π valence orbitals as well as the dominant Rydberg type orbitals (see discussion below). The chosen orbitals are

EVF + Esc − i Γ

In eqs 1 and 2, |I⟩, |F⟩, and |V⟩ denote initial, final, and intermediate states, respectively. The energies Eex and Esc correspond to the incident and scattered radiation, respectively, while EFI, EVI, and EVF are the transition energies between the initial−final, initial−intermediate, and intermediate−final states, respectively. The parameter Γ is a phenomenological lifetime controlling the line width, c the speed of light (∼137 in atomic units), and 4π the solid angle; m is a component of the electric dipole transition operator (m = −Σiri + ΣAZARA), where ri is the position operator for the ith electron and A sums over nuclei with charges ZA at positions RA. For scattering spectroscopies, the polarizability tensor αρλ(Eex,Esc) is described by the Kramers− Heisenerg−Dirac (KDH) expression formula62 (eq 2). The resonance scattering cross section σRIXS(Eex,Esc), averaged over all orientations of the molecule and integrated over all directions and polarizations of scattered radiation, is 8πEsc3Eex

ivab

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⟨F |mλ|V ⟩⟨V |mρ|I ⟩ (2)

σRIXS(Eex , Esc) =

iva

The three terms consist of configurations I with spin-coupling v, which have zero, one, and two particles located in external orbitals (a,b). The number of |ΦI⟩’s can grow enormously and quickly leads to intractable computational problems. One possibility to truncate the list of |ΦI⟩’s consists of including only those excited CSFs that interact appreciably with |0⟩ through the use of perturbation theory.63 Further details on the MRCI module and the way it is implemented in the ORCA computational package are discussed elsewhere.63

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αρλ(Eex , Esc) =

∑ Civ|Φiv⟩ + ∑ Civa|Φiva⟩ + ∑ Civab|Φivab⟩ iv

|⟨F |mρ|I ⟩|2 Γ

∑

(4) MRCI

|αρλ(Eex , Esc)|2

ρ,λ=x ,y ,z

(3)

The first term in eq 2 describes a resonant scattering process which is a two-photon process involving virtual absorption to the entire manifold of the core-excited intermediate state |V⟩ followed by emission to the final state |F⟩. The second term in eq 2 describes the nonresonant process. Equations 1−3 indicate that the contributions from all the intermediate states have to be summed before squaring. Because the individual terms can be positive or negative, counterintuitive interference contributions may arise. MRCI. In the MRCI treatment, one starts from a multiconfigurational zeroth-order wave function which is written as a linear combination of the zeroth-order wave function and excited configuration state functions (CSFs) |ΦI⟩. The coefficients with which the CSFs enter into the MRCI wave function are 20165



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optimized in a state average fashion of 20 singlet and triplet states. These orbitals are used in the subsequent MRCI calculations. To calculate the RIXS spectrum, the abovementioned orbital space is extended in both directions to include the core region as well as a sufficiently large number of virtual orbitals. This results in an active space consisting of 22 electrons in 20 orbitals. The reference starting configurations are defined individually to span all the one hole−one particle configurations within the chosen active space. Single (CIS) or single- and double- excitations (CISD) relative to the chosen configurations were performed for the MRCI. The number of the configuration state functions that were included in the variational space were controlled by the selecion threshold Tsel = 10−6 Eh. The problem was thus simplified to approximately 50 000 and 800 000 CSFs for the MRCI/CIS and MRCI/CISD calculations, respectively. Only initial configurations with weight larger than 10−5 in any of the roots (Tpre = 10−5) were considered after diagonalization of the given initial reference space. In addition, the Davidson correction was applied to approximately account for the effect of the unlinked quadrupole substitutions (MRCI +Q). Excitations were performed from localized orbitals according to the Pipek−Mezey localization scheme.70 Core orbital relaxation was not considered in the MRCI calculations. The potential effect of this simplification is analyzed in detail below. The dynamic nature of the observed resonant emission spectra was considered by treating the vibronic coupling between the intermediate states reached by X-ray absorption with the vibrational antisymmetric stretching and bending modes. For these calculations, MRCI/CIS (22,20) potential energy curves along the bending and antisymmetric stretching vibrational modes were constructed and plotted in diagrams of energy versus dimensionless normal mode coordinates Qmode. Qmode relates to the vector of atomic Cartesian displacements δX by the following relationship:

Figure 1. Black: experimental CO2 NEXAFS TFY recorded at 1 atm (this work). Red: the same spectrum after edge background subtraction (red dotted line). Blue: low-pressure cell detection reported by Gunnelin et al.13−15 Green: X-ray Raman (XRS) detection reported by Sakko et al.71 Black dotted lines indicate the three characteristic absorption maxima centered at 534.8, 536.0, and 538.1 eV.

resolved photoion yield spectroscopic technique (ARPIS) refers to ionized CO2+. This spectrum might reflect the same final states observed in an XAS experiment close to the ionization limit, but for the low-lying core excited states, this cannot be ensured. Overall it seems that the spectra reported by Gunnelin et al. and Sakko et al. are likely to be the least distorted and thus are chosen to be compared with the calculated spectra. Three signals are observed in the pre-edge region of the O Kedge TFY spectrum. The highest intensity signal is located at about 535.0 eV, with a shoulder being observed at 536.0−536.5 eV and a third feature at about 538.0 eV (as indicated by the vertical black dotted lines in Figure 1).73 In Figure 2, the NEXAFS spectrum of CO2 along with the observed RIXS plane is shown. The RIXS map is based upon 18 raw spectra (shown in Figure 2, right panel) and interpolated for about 100 incident energies, spanning the incident energy range of 534−539 eV. Furthermore, the RIXS map qualitatively can be divided in three well-distinguished incident energy resonance regions. The first region ranges between 534.0 and 536.0 eV incident energies, with two emission features at 523.5 and 527.5 eV having comparable intensity. The second region ranges between 536.0 and 537.5 eV incident energies, featuring a broad emission signal centered at 525.5 eV. Finally, the third region between 537.5 and 539 eV incident energies contains two emission features. The signal with the highest intensity is located at 528.0 eV, while a second weak signal is located at 524.0 eV. In an effort to obtain further insight regarding the number of the observed resonant emission features as well as their relative intensities, interpolated slices along the incident photon energy axis were taken for nine resonance energy positions between 535.0 and 539.0 eV (Figure 3).

3N

Q mode =

⎛ ωmode ⎞1/2 ⎜ ⎟ ∑ Lk modeδXk Mk ⎝ ℏ ⎠ k=1

where ωmode is the vibrational frequency of the respective mode, {Lkmode} the orthogonal matrix obtained upon numerical diagonalization of the mass-weighted Hessian matrix, and M the vector of atomic masses.

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RESULTS AND ANALYSIS Experiment. In Figure 1, the experimental O K-absorption spectrum of CO2 obtained using total fluorescence yield mode is presented (black line). The shape of the spectrum does not change upon edge background subtraction (red dotted line). Thus, the energy positions of the absorption features are accurately reflected using TFY. However, the corresponding spectral weights may become distorted for nondiluted samples (viz. 1 atm of CO2). In fact, weak features (above 538 eV) are greatly enhanced over strong absorption lines (resonance at 534.6 eV) because of saturation effects. On the other hand, this effect emphasizes the shoulder at 536.0 eV, which is found to be an interesting transition region for the RIXS spectra, as is discussed bellow. For comparison, we also include a XAS spectrum reported by Gunnelin et al. from a low-pressure cell (a few torr) recorded using a photocurrent from an electrode in the cell,14,15 as well as a X-ray Raman scattering (XRS) spectrum reported by Sakko et al.52,71 Both spectra are less influenced by saturation because of the lower total cross section for the measured signal. In another study by Adachi et al.,72 the angle20166



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Figure 2. Left, top panel: experimental CO2 NEXAFS spectrum. Left, bottom panel: interpolated CO2 RIXS plane. The observed signal region is indicated by red. The green arrows point to the three visible features of the NEXAFS spectrum centered at 535.0, 536.5, and 538.5 eV used to calculate the corresponding resonant emission spectra. Note that the RIXS plane plots the incident photon energy on the x-axis and the scattered photon energy on the y-axis. Right panel: Experimental resonant emission spectra used to generate the interpolated RIXS plane. The incident energies are given to the right (scattered photon energy now on x-axis). The black line indicates elastic scattering.

Figure 3. Experimental CO2 NEXAFS spectrum (TFY) (left panel). Arrows indicate several incident photon energy hvin positions of the RIXS plane with the corresponding interpolated resonance emission spectra (middle panel). In the right-hand panel the respective calculated resonance emission spectra at the MRCI/CISD level are shown.

Electronic Structure and Geometric Properties. The linear closed-shell CO2 molecule in D∞h symmetry features a totally symmetric 1 Σ g+ ground state with the electron configuration 1σ2g 1σ2u2σ2g 3σ2g 2σ2u4σ2g 3σ2u1π4u1π4g 2π0u5σ0g 4σ0u (Figure 4). The O K-edge XAS process involves the O 1s core electron excitations from the 1σ2g and 1σ2u occupied core orbitals to the unoccupied 2π0u, 5σ0g , and 4σ0u orbitals. Furthermore, as is shown in Figure 4, according to dipole selection rules only the u → g (g → u) excitations are allowed. Therefore, in a simple one-electron

picture and according to principle group theoretical considerations, the absorption spectrum will be dominated by the 1σ2g → 2π0u(1Πu), 1σ2u → 5σ0g (1Σu), and 1σ2g → 4σ0u(1Σu) single electron excitations. Under resonant conditions, the intermediate states 1 Πu and 1Σ+u reached by absorption will subsequently emit to form the multiplet structure of the final states. An example is given in Figure 4 for the 2π1u1π2g,u → 1σ1g valenceto-core electron decay. It should be noted that under D∞h symmetry the final states involving the 2π1u(valence)2 → 1σ1u, 20167



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Figure 4. CO2 MOs and term symbols arising from single, electron core excitations (black long arrows). The core excited electrons, as well as the resonant decay electrons, are indicated by blue and red arrows, respectively. Red long arrows and the corresponding term symbols reflect an example of resonant emission with symmetry breaking from the 1Πu intermediate state. The indicated orbital occupation number refers to the 1Σ+g ground state.

Figure 5. MRCI/CIS scan along the antisymmetric stretching vasym (left panel) and bending vbend (right panel) vibrational modes of the important core excited intermediate states: 1Πu, 11Σ+g,u and 21Σ+g,u, indicated with black, red, and blue lines, respectively. Gray lines indicate n1Σ+g,u states, (n > 2).

5σ1g (valence)2 → 1σ1g , and 4σ1u(valence)2 → 1σ1u excitations are not formally accessible, as the corresponding absorption excitations are parity-forbidden. The CO2 molecule, however, presents the first example of symmetry breaking probed by resonant inelastic X-ray scattering spectroscopy.11,12,61,74−76 It is well-known that the de-excitation mechanism of the core excited + intermediate states of CO2 with symmetries 1Πu and 1Σg,u proceeds via a 2-fold vibronic coupling mechanism.14,61,74 The intermediate Πu state is subject to the Renner−Teller effect (RTE); thus, it is allowed to couple with the in- and the out-ofplane bending vibrational modes. Therefore, the molecular symmetry is reduced along the bending interconversion pathway D∞h → C2v. Furthermore, owing to the near degeneracy of the core orbitals 1σ2g and 1σ2u, the core excited intermediate states of symmetry Σ+u and Σ+g or Πu and Πg will form quasi-degenerate doublets and doublets of doublets states, respectively. These states have the correct symmetry to couple with the antisymmetric stretching vibrational mode. This process is known as “core hole localization”, suggesting that the symmetry of the molecule can be additionally reduced via a pseudo-Jahn− Teller (PJT) process along the antisymmetric stretching interconversion pathway D∞h → C∞h. It should be mentioned that within such a symmetry-reducing mechanism parity-

forbidden electronic transitions could borrow intensity from parity-allowed electronic transitions.61 In fact, the absorptionforbidden transitions 1σ2u → 2π0u(1Πu), 1σ2g → 5σ0g (1Σg), and 1σ2u → 4σ0u(1Σg) are mixed into the dipole-allowed 1σ2g → 2π0u(1Πu), 1σ2u → 5σ0g (1Σu), and 1σ2g → 4σ0u(1Σu) via vibronic coupling. Whether or not the high symmetry of a molecule will be reduced in the course of a resonant emission experiment depends on (a) the strength of the vibronic coupling and (b) the lifetime of the core excited intermediate state relative to the period of the respective vibration. Thus, understanding the observed resonant emission spectral features requires theoretical approaches that can treat the interaction between the intermediate and final state multiplets. This problem has been a subject of a numerous experimental and theoretical studies.9,12,14,15,46,49,61,76−81 However, to the best of our knowledge, no treatment of the CO2 RIXS spectrum on the basis of the two-fold vibronic coupling mechanism has been reported at the level of the rigorous MRCI method. Vibronic Coupling Mechanism. Figure 5 shows the MRCI/CIS generated potential energy surfaces along the bending and stretching vibrational modes, respectively. For the Renner Teller problem at hand,82 in which the vibronic coupling 20168



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the D∞h → C2v interconversion pathway and the resonance emission will be observed from a bent CO2 structure (Figure 5, right panel). However, as the resonance incident energies shift toward the first manifold of 1Σ+g,u states (Figure 5, left panel), distortions along the antisymmetric stretching mode will also occur. Thus, the resonant emission will be dominated by bent structures with unequal CO bond lengths. Finally, upon shifting the resonant incident energy in the region of the second manifold 1Σ+g,u, every distorting force is canceled and ideally the resonance emission will occur from an undistorted linear structure. In the following sections, the mechanism described above will be used in the framework of MRCI theory to evaluate the experimentally observed RIXS spectra. X-ray Absorption Spectroscopy. The O K-edge NEXAFS spectrum of CO2 is calculated by applying the MRCI/CIS and MRCI/CISD methods as shown in Figure 6. Both techniques provide reasonable intensity distributions and energy positions for the important experimental features. These calculated spectra agree well with previous calculations based on the particle−hole methods.15,71 For comparison, the corresponding B3LYP/augcc-pCVTZ TD-DFT spectrum is provided in Figure 6. As can be seen, the relative energy position of the electronic states dominating the main signal as well as the shoulder are heavily underestimated within the TD-DFT scheme. They show up as accidentally degenerate. This is in benefit of the signal dominated by the Rydberg states located at 538 eV. In fact, these states are shifted at lower energies in a way which also accidentally matches the correct energy position of the experimental spectrum. It appears that in this level of theory one can obtain good agreement between theory and experiment only by considering vibrational excitations and properly accounting for anisotropic broadening.71 Nevertheless, the absorption event in the sudden approximation is mainly dominated by electronic excitations, which should be treated explicitly before any extra correction is applied. In this concept it can be concluded that, as expected, MRCI/CISD provides the best agreement with experiment. Furthermore, we compare the energy shift required to ensure that the highest intensity features of the experimental and calculated spectra with the three techniques are located at the same position. This shift reflects the remaining discrepancies of the applied methods to accurately predict the absolute transition energy. These discrepancies arise from (a) shortcomings in the description of the core region of the electronically excited intermediate state, (b) limitations of the basis set, and (c) shortcomings in the accurate modeling of spin-free (scalar) relativistic effects. As expected, the largest shift (14 eV) is observed for the B3LYP/TD-DFT calculation. This value drops down to about 8 eV for MRCI/CIS and is finally reduced to only 4.2 eV in MRCI/CISD. Investigation of the basis set dependence of the results implies that the largest part of the remaining discrepancy is probably due to the lack of electronic relaxation (e.g., the use of nonrelaxed core orbitals) in the intermediate state. For DFT XAS and nonresonant XES methodologies, it is a common practice to treat such a shift as an element-specific property.1,2,4,6 On the other hand, orbital optimization has been attempted to some extent in a study of the intermediate states of O2 in using the CASSCF/MRPT2 method.85 Recently, RASSCF/RASPT2 has been used to describe RIXS spectra of transition-metal complexes (Ni(II) and Fe(II),(III)).17 It appears that excellent calculated intensities and energy distributions of the RIXS spectra can be achieved by neglecting the electronic relaxation effect, provided that the multiplet structure of the final

operator is considered up to fourth order, the energies of the degenerate coupling states are given by 1 E1,2(ρ) = ± (K 0 ± g )ρ2 + jρ 4 2

(6)

Here, K0 is the force constant for the bending mode, while g and j are the quadratic and fourth-order coupling constants, respectively. At this level of approximation, the contributions from other excited states to the vibronic coupling mechanism are omitted.83,84 In addition, another source of symmetry breaking involves the quasidegenerate core excited intermediate states of gerade and ungerade symmetry |Πg,u⟩ and |Σ+g,u⟩ which can couple with the antisymmetric stretching vibrational mode through the PJT effect.61,79,82 Such perturbation removes the degeneracy of the coupled states by lowering the molecular symmetry along the stretching vibration mode Qasym as given by eq 7: 1⎛ F2 ⎞ 2 1 ⎛ F4 ⎞ 4 E1,2(Q asym) = ± ⎜K 0′ ± ⎟Q asym ± Δ ∓ ⎜ 3 ⎟Q asym 2⎝ Δ⎠ 4⎝Δ ⎠ (7)

Here, we assume that the two states have similar primary force constants K0′ , while F is the pseudo Jahn−Teller coupling constant and 2Δ is the energy gap between the two states in the undistorted configuration. Estimates of the vibronic coupling constants can be obtained by fitting of the potential surfaces of the |Πu⟩ state along the bending mode with eq 6 and the corresponding potential energy surface for the |Πg,u⟩ and |Σ+g,u⟩ states along the antisymmetric stretching mode with eq 7. The fit parameters are presented in Table 1. Table 1. Vibronic Coupling Parameters for |Πu⟩, |Πg,u⟩, 1|Σ+g,u⟩, and 2|Σ+g,u⟩ States mode/constants Qbend mode/constants Qasym

states

K0 (eV)

g (eV)

|Πu⟩ 0.81 1.18 states K′0 (eV) |Πg,u⟩ 1|Σ+g,u⟩ 2|Σ+g,u⟩

0.25 0.26 0.24

j (eV) 7.13 × 10−3 F (eV) 8.5 × 10−3 1.2 × 10−2 7.2 × 10−3

ΔERTE (eV) 1.63 ΔEPJTE (eV) − 1.61 −

As expected, the RTE along the bending mode is strong, with a RT stabilization energy of ΔERT = 1.63 eV resulting in dynamic instability toward bent structures. The vibronic coupling is strongly dominated by the quadratic term g = 1.18 eV, whereas small contributions from higher-order terms are also observed, i.e., j = 7.1 × 10−3 eV. Analogous to the RTE for the bending mode, the PJTE along the antisymmetric stretching mode is + significant for the 11 Σ g,u intermediate states. The PJT stabilization energy for this state is ΔEPJT = 1.61 eV, indicating strong distortions along the stretching mode. The vibronic coupling constant is estimated to be F = 0.012 eV. On the other hand, as can be seen in Figure 5 (right panel), the intermediate states |Πg,u⟩ and n1Σ+g,u , (n > 1), reached by resonance with incident energies around 535 and 538 eV, appear to be only very weakly coupled. In fact, the vibronic coupling interaction will not lead to any new minima in the adiabatic potential energy surface. Therefore, these states are not sensitive to any distorting force and will retain the linear symmetry of the molecule. Furthermore, the above evaluated vibronic coupling mechanism can be used to explain the observed experimental RIXS spectra. As has been discussed previously, by reaching resonance with the 1Πu states, CO2 will undergo molecular distortions along 20169



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Figure 6. Black lines indicate the TEY NEXAFS O K-edge spectrum of CO2. Red lines represent the corresponding calculated MRCI/CIS (left panel), MRCI/CISD (right panel), and B3LYP/TDDFT (bottom panel) spectra. Red stick lines correspond to the contributing states. All calculated spectra were energy shifted. A constant line shape broadening of 1.5 eV was applied.

and 2% 21Σg(1σ2u → 4σ0u) (540 eV) through a mechanism which mixes symmetry-allowed with symmetry-forbidden transitions as discussed above. Such mixing turns out to be possible in the framework of double excitations of the MRCI/CISD method. Spin-forbidden triplet states were found to contribute less than 1% in preliminary spin−orbit coupling (SOC) corrected MRCI/ CIS calculations. Therefore, spin−orbit coupling corrections were not applied in the calculated spectra although they can be treated at any excitation scheme in our MRCI program. Resonant X-ray Emission Spectroscopy. RIXS Intensity Mechanism. Using the methodology described in the computational details section, the calculated RIXS spectra are evaluated from coupled stretching D∞h → C∞h and bending D∞h → C2v potential energy surfaces, constructed along the corresponding bending and antisymmetric stretching vibrations. In this way both three and four degrees of freedom of the CO2 molecule in the linear and bent structures enter into the multireference treatment of the resonant emission final states |F⟩. It is in general expected that the absorption event within the sudden approximation occur mainly from the ground-state equilibrium geometry of CO2. Under resonant inelastic scattering conditions, however, it is possible that the excitations can occur from distorted molecules even at ambient condition if the detection time scale allows this. For the calculation of the RIXS spectra we have indeed considered the most general case in which the resonant emission occurs for an average of structures identified along the ground and excited PESs. As can be seen in Figure 3, very good agreement is observed between the experimental and the MRCI/CISD calculated RIXS profile spectra at incident energies located between 534 and 539 eV. Furthermore, the calculated spectra are analyzed in terms of the dominating geometric structures. In Figure 8, such analysis is

state is correctly described and that dynamic correlation is taken into account at least to some degree.47,50,81,85,86 Deconvolution of the calculated spectra can be performed in terms of states that are dominated by 1s-to-valence singleelectron excitations. As can be seen in Figure 7, the experimental spectrum is dominated by the expected 59% 1Πu(1σ2g → 2π0u) at 534.8 eV, 29% 11Σu(1σ2u → 5σ0g ) at 536.0 eV, and 2% 21Σu(1σ2g → 4σ0u) at 540 eV. In addition, notable contributions are observed from the parity-forbidden states 8% 1Πg(1σ2u → 2π0u) (536 eV)

Figure 7. Deconvoluted MRCI/CISD XAS O K-edge calculated spectrum. As in Figure 6, the black line corresponds to the experimental TEY NEXAFS O K-edge spectrum of CO2. The red line represents the corresponding MRCI/CISD calculated spectrum. Red stick lines correspond to the contributing states. Blue, green, cyan, purple, and yellow lines correspond to the individual spectra of the contributing states. All calculated spectra were energy shifted. A constant line shape broadening of 1.5 eV was applied. 20170



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Figure 8. Experimental (black line) versus calculated MRCI/CISD (red line) RIXS spectra. The experimental spectra were taken at incident energies 535.0 eV (left panel), 537.0 eV (middle panel), and 539.0 eV (right panel). The calculated resonant emission spectra represent averaged profiles over all geometries identified along the corresponding potential energy surfaces presented in Figure 5.

valence to core 2π1u4σ1g → 1σ1g electron decay. This analysis is in agreement with the expectations for the resonant emission spectrum at the NEXAFS resonant incident energy of 535 eV.14 Furthermore, for the calculations of the complicated resonance emission spectrum observed at an incident energy of 536.5 eV, it seems reasonable to consider the situation in which the incident energy shifts off-resonance with the 1Πu state and reaches resonance with the corresponding 11Σ+g,u states. This is expected to significantly influence the structure of the CO2 molecule. In fact, the calculated resonant emission spectrum along the coupled bending−stretching potential energy surface reveals that the experimentally observed RIXS spectrum reflects static CO2 geometries that are significantly distorted. For the best fit of the calculated spectrum with respect to experiment, presented in Figure 9 (middle, left), the molecule is still bent (AO−C−O = 167.7°) while the C−O bond lengths differ significantly (1.33 and 1.01 Å, respectively). Deconvolution of this calculated RIXS + spectrum reveals contributions from both 1Πu and 11Σg,u intermediate states. In particular the spectrum is dominated by 58% 2π1u1π2g → 1σ1g 2% 2π1u1π2u → 1σ1g and 40% 5σ1g 1π2g → 1σ1u electron decays. Furthermore, shifting the resonance incident energy to 538−539 eV, the RIXS spectrum is dominated by the Rydberg 21Σ+g,u or Rydberg-like states n1Σ+g,u. These states, as described above, remain unaffected under the action of the bending vibrational mode (Figure 5). In fact, the calculated RIXS spectrum (Figure 9, bottom, left) reflect static geometries with linear structure and slightly distorted bond lengths (best fit, 1.23 and 1.11 Å). Deconvolution of this spectrum reveals that the feature at 528 eV of the spectrum is dominated by states corresponding to 18% 4σ1u1π2g → 1σ1g and 62% nσ1u1π2g → 1σ1g valence-to-core electron decays. In addition, the lower intensity feature at 524 eV is dominated by states with 15% 4σ1u1π2u → 1σ1g and 6% nσ1u1π2u → 1σ1g character.

performed for three representative RIXS spectra observed at 535, 537, and 539 eV. In fact, the RIXS spectra observed at incident energies around 535 eV are dominated by bent structures because of resonance with 1Πu intermediate state. On the other hand, for incident energies around 539 eV, the corresponding RIXS spectra are rather dominated by stretched structures because of resonance with the n1Σ+g,u states (n > 1) . Finally, the multisignal spectrum observed at incident energies around 537 eV is dominated by both bent and stretched structures because of coupled resonance between 1Πu and 11Σ+g,u states. Electronic Structure Analysis from Static Geometries. Furthermore, the individual calculated spectra corresponding to static CO2 geometries are analyzed. Although, as discussed above, the correct band shape of the RIXS spectra is reproduced when the average of structures is considered along the corresponding PESs, such analysis is still useful as it provides information about the individual electron decays involved in the resonant emission processes. Figure 9 provides the calculated resonance emission spectra originating from static geometries, which were identified to better reproduce their experimental counterparts observed at emission energies 535.0, 536.5, and 538.5 eV. In this context, the physical origin of the observed spectral features is discussed qualitatively and quantitatively. In particular, the calculated resonant emission spectrum corresponding to electron decay from the 1Πu state is presented in Figure 9 (top, left). This spectrum reflects bent structures. Indeed the best fit to experiment is observed for a molecule with AO−C−O = 167.7° and almost equal C−O bond lengths (1.24 and 1.21 Å, respectively). Deconvolution of the calculated spectrum shows that the signal at 528 eV corresponds to states that are dominated by valence-to-core 2π1u1π2g → 1σ1g electron decay, thus leading to final states with 1σ2g {...}1π1g 2π1u electron configuration (|F1π1g ,2π1u⟩). In addition, the signal at 524 eV is dominated by formally forbidden states corresponding to 21% 2π1u1π2u → 1σ1g and 21% 2π1u3σ2u → 1σ1g electron decays. Likewise, the resulting final states |F1π1u,2π1u⟩ and |F3σ1u,2π1u⟩ will obtain 1σ2g {...}1π1u1π2g 2π1u and 1σ2g {...}3σ1u1π2u1π2g 2π1u electron configurations. Finally, the shoulder at 521 eV is dominated by states |F4σ1g ,2π1u⟩ with 1σ2g {...}4σ1g 3σ1u1π2u1π2g 2π1u electron configuration corresponding to 5%

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CONCLUSIONS In summary, we have presented experimental CO2 RIXS spectra recorded under catalytically relevant conditions within a reaction cell constructed in-house. The spectra are collected at a sequence of incident photon energies spanning the absorption region of the corresponding O K-edge XAS spectrum. The observed 20171



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Figure 9. Experimental (black line) versus calculated MRCI/CISD (red line) RIXS spectra. The experimental spectra were taken at incident energies 535 eV (top panel), 536.5 eV (middle panel), and 538.0 eV (bottom panel). The calculated resonant emission spectra correspond to static geometries, and they originate from |Πu⟩ 1|Σ+g ⟩ and n|Σ+g ⟩ (n > 2) intermediate states reached by X-ray absorption (Figures 5 and 6). Colored lines deconvolute the spectra in terms of the dominating resonance emission states (presented as red sticks). The corresponding molecular geometries are shown to the right of the spectra.

experiment. It is demonstrated that the observed RIXS spectra for resonant energies between 533−535 eV are dominated by bent molecules. In addition, for incident energies ranging between 535−537 eV, the observed RIXS spectra correspond to highly distorted structures along both stretching and bending pathways. Finally, for higher resonant energies, linear molecules that are only slightly stretched dominate the RIXS spectra. From a more general perspective, RIXS spectroscopy in combination with ab initio protocols provides a sensitive and accurate probe of the structure and the electronic properties of molecules. Although the applicability of this approach is presently limited to small molecules, it provides a solid foundation for future applications of RIXS spectroscopy, in

features show significant sensitivity with respect to the incident photon energy. To interpret the experimental data, ab initio calculations at the level of multireference configuration interaction were performed. Specifically, we have constructed potential energy surfaces along the dominant bending and antisymmetric stretching vibrational modes. In agreement with previous studies, it was found that the Renner−Teller and the core hole localization pseudo-Jahn− Teller effects couple the |Πu⟩ and |Σ+g,u⟩ symmetric core excited states with the bending vibrational and the antisymmetric stretching vibrational modes, respectively. The RIXS spectra were evaluated along the corresponding predominant vibrational modes, resulting in satisfactory agreement between theory and 20172



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particular to the field of transition metals and heterogeneous catalysis.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support of this work by the Max Planck Society, as well as, the European Community’s Seventh Framework Program (FP7/2007-2013, grant agreement no. 312284). Helmholtz-Zentrum-Berlin, in particular the electron storage ring BESSY II, is acknowledged for provision of synchrotron radiation at the beamline U41-PGM. The reviewers of the manuscript are thanked for their constructive comments.

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