Crossing Electronic States in the Franck–Condon Zone of Carbon

Letter pubs.acs.org/JPCL

Crossing Electronic States in the Franck−Condon Zone of Carbon Dioxide: A Five-Fold Closed Seam of Conical and Glancing Intersections Sergy Yu. Grebenshchikov* and Raffaele Borrelli‡ Department of Chemistry, Technical University of Munich, Lichtenbergstr. 4, 85747 Garching, Germany S Supporting Information *

ABSTRACT: Intersections of electronic states are recognized as natural molecular hubs routing initial photoexcitation toward specific products. In this Letter, we investigate a five-fold intersection between the 2,31A′ and 1,2,31A″ electronic states of CO2 responsible for the ultraviolet absorption of the molecule between 120 and 160 nm. It is demonstrated that the intersection is not an isolated point but a seam with the following properties: (1) The seam comprises both conical and glancing intersections. (2) The seam is a closed loop traced by the concerted displacements of two CO bonds. (3) The seam is constrained to linear geometries and confined to the CO bond plane. (4) The seam passes directly through the Franck−Condon zone. Topography and properties of the intersection seam are validated using assignment of the electronic wave functions and illustrated using circulation of the derivative couplings along closed contours. The possibility of observation of strongly curved seams is briefly discussed. SECTION: Spectroscopy, Photochemistry, and Excited States

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ltraviolet (UV) photodissociation of carbon dioxide1 is of considerable importance for atmospheric, planetary, and interstellar chemistry and has received much attention of the experimentalists. Two diffuse bands, centered near 67 750 (148 nm) and 75 000 cm−1 (133 nm), are particularly well-studied, with the measurements covering the absorption profiles of irregular vibronic bands as well as the photofragment distributions in the singlet and triplet dissociation channels.2−4 Theoretical studies5−7 related the absorption maxima to the lowest excited valence states 11Σ−u and 11Δu (the low energy band at 148 nm) and 11Πg (the high energy 133 nm band). Recently, the vibronic origin of the diffuse bands was established by solving the electron−nuclear Schrö dinger equation for the first six interacting singlet electronic states using full dimensional global potential energy surfaces (PESs) computed at a high level of ab initio theory.8 The quality of ab initio PESs is manifested in the near spectroscopic accuracy of the calculated absorption spectra. The study demonstrated the significance of three strong nonadiabatic interactions in the Franck−Condon (FC) zone of the near-linear molecule: the rotoelectronic Renner−Teller (RT) effect in the orbitally doubly degenerate 1Δu and 1Πg states, the conical intersections (CIs) between 1Δu and 1Πg, and the Herzberg−Teller effect in the transition dipole moments of 1Σ−u , 1Πg, and 1Δu with X̃ Σ+g . Carbon dioxide is photoexcited directly into the region of multiple degeneracies, with all excited states linked through a network of couplings created at RT intersections and CIs. In this Letter, we investigate the topography of the intersecting states and demonstrate that all intersections are © 2012 American Chemical Society

connected into a single seam that stretches over a substantial range of bond distances in the linear molecule. Recently, Levi et al. reported an extensive ab initio study of a seam of two-state CIs in methylamine.9,10 The authors discovered a CI seam induced by a motion of a single proton in a well-specified molecular plane, conjectured that the seam should be closed, and gave heuristic arguments supporting this conjecture. Below we present direct numerical evidence of an intersection seam that comprises five optically active electronic states, is induced by atomic motion along one line, passes directly through the FC zone, and forms a closed loop. Adiabatic energies of the first three 1A′ and the first three 1A″ electronic states, correlating at linearity with X̃ 1Σ+g , 11Πg, 11Σ−u , and 11Δu, are calculated on a dense 3D grid of CO bond distances R1,2 and the bond angle αOCO. Details of the calculations, carried out using MOLPRO package,11 are available as Supporting Information. Whereas PESs for the quantum dynamical calculations are constructed at the MRCI level of theory, the present work is mainly based on CASSCF calculations. The difference between two sets is quantitative but not qualitative: A closed intersection seam is found at both levels. (See the Supporting Information.) Analysis of crossings between calculated adiabatic states is indispensable in constructing an adequate quasi-diabatic Received: August 31, 2012 Accepted: October 15, 2012 Published: October 15, 2012 3223

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Figure 2. Seam of conical and glancing intersections in the plane of the two CO bond distances R1 and R2 and αOCO = 179.8°. CASSCF results are shown with red (A′ symmetry) and blue (A″ symmetry) solid lines. MRCI results are shown with black (A′) and green (A″) dashed-dotted lines. Also shown are two branches of the glancing intersection between near degenerate 1Σ−u and 1Δu states (straight solid blue lines). Γ1,2,3 marks projections of the closed contours for angular NACMEs onto the (R1,R2) plane. Contours Γ1,2 are located in the plane of the bend and are normal to the (R1,R2) plane. Γ1 is centered at (R1,R2, αOCO) = (2.20a0, 2.33a0, 179.8°); Γ2 is centered at (2.10a0, 2.80a0,179.8°); Γ3 lies in the (R1,R2) plane and is centered at (2.15a0, 2.50a0, 179.8°). Solid squares indicate seam points obtained using NACMEs along the contours Γ3. Gray blob indicates the FC point. Figure 1. Cuts through ab initio PESs along (a) OCO bond angle (R1 = R2 = 2.2 a0) and (b) one CO bond length (R2 = 2.2 a0, αOCO = 177°). In panel c, CASSCF potentials for the diabatic 1Πg (blue) and 1Δu/1Σ−u (red) states crossing along the closed seam are shown in the (R1,R2) plane (αOCO = 179°). The lowest shown contour is 8.0 eV, and the contour spacing is 0.25 eV.

calculations. At the MRCI level, the A′ and A″ seams are virtually indistinguishable (green and black dashed-dotted lines in Figure 2). A remarkable property of the calculated seam is that it traces out a closed isolated loop; this is valid for both CASSCF and MRCI calculations. A CI seam can be viewed as a line of magnetic field flux whose vector potential An modifies the Hamiltonian for nuclear motion in the adiabatic electronic state |n⟩ (molecular Aharonov−Bohm effect14). A closed loop in Figure 2 is akin to the field of a toroidal magnet, with the equipotential lines of An encircling the seam in the bending plane perpendicular to the (R1,R2) plane. A closed strongly curved seam has several implications. “Inconvenient” is that the tuning mode keeps changing its direction as the seam is traversed or that the popular local diabatization15 has to give way to a global scheme. On the other hand, closed seams suggest that new interference phenomena should become observable (see below). Closed intersection seams were mentioned in several studies. For example, a triangular seam, formed by three crossing straight seams, has been reported for cyclic ozone.16 A strongly curved intersection arch has been discovered in methylamine and assumed to be a part of a closed seam.9,10 Our CO2 study might be considered to be a numerical demonstration of the validity of this assumption. If the state 1Σ−u (A″ symmetry block) is also considered, then the Πg/Δu seam in Figure 2 has to be augmented with Πg/Σ−u and Δu/Σ−u seams. The resulting picture is simplified by an accidental near-degeneracy between Δu and Σ−u : The Δu/Σ−u energy gap at linear geometries falls consistently below 300 cm−1, which is at the limit of the ab initio accuracy. Consequently, the Πg/Σ−u seam (not shown in Figure 2) is virtually indistinguishable from the Πg/Δu seam. If the states Δu and Σ−u are assumed distinct, then they cross along two straight lines cutting through the closed loop (blue solid lines parallel to R1 and R2 in Figure 2). The intersection is glancing because coupling via the bending mode is allowed only in the second

representation for dynamics calculations. Two types of intersections with regard to their symmetry properties are identified in CO2. (See Figure 1a,b.) RT intersections1 occur between one A′ and one A″ state which become degenerate at linearity (αOCO = 180°). Among the six calculated states, two such pairs are present, corresponding to the orbitally degenerate 1Δu and 1Πg states (Figure 1a). The intersecting states remain degenerate as long as αOCO = 180°, and the “degeneracy manifold” is the whole (R1,R2) plane. Conical intersections occur inside A′ or A″ symmetry blocks and stem from the 1Πg/1Δu crossing discovered by Knowles et al.5,6 in linear CO2 with bond distances R1* = R2* ≈ 2.2a0 (Figure 1b). At this geometry, the A′ as well as A″ components of 1Πg and 1Δu intersect, giving rise to two coinciding degeneracies in the (R1,R2) plane, one involving 2,31A′ and the other involving 1,21A″ or 2,31A″ states. The intersections are conical because the degeneracies are lifted linearly along two orthogonal directions, 12 the bending angle α OCO (symmetry breaking coupling mode) and a combination of the two collinear stretches (tuning mode which for CO2 is defined only locally). The A′ and A″ CIs occur not at one point (R*1 ,R*2 ) but along a line FCI(R*1 ,R*2 ) = 0 in the (R1,R2) plane: The degeneracy manifold is a 1D seam,12,13 CASSCF approximations to which are depicted in Figure 2 for the states 2,31A′ (red solid line) and 1,31A″ (blue solid line). Results for both symmetries converge toward one single line of a four-fold crossing because the A′ and A″ states are almost RT-degenerate. The residual discrepancy reflects the limited accuracy of CASSCF 3224

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order.15,17 If the states Δu and Σ−u are assumed degenerate, then the straight line seams dissolve into the (R1,R2) plane, whereas the Πg/Σ−u and Πg/Δu seams become identical. Counting two A′ states, the seam in Figure 2 therefore represents a five-fold degeneracy. Topography of the five-fold seam implies the following diabatic Hamiltonian near linearity15,17

H(1,2) = TRT

⎛ VΠ′ αQ u ⎞ ⎜ ⎟ ⎜ αQ u VΔ′ ⎟ ⎜ ⎟ ⎜ VΣΔ′′ βQ u γQ u2 ⎟ +⎜ ⎟ ⎜ βQ u VΠ′′ βQ u ⎟ ⎜ ⎟ ⎜ γQ u2 βQ u VΣΔ′′⎟⎟ ⎜ ⎝ ⎠

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Figure 3. (a−c) CASSCF energies of the states 21A′ and 11A″ (blue), 21A″ (green), and 31A′ and 31A″ (red) versus CO bond distance. The second bond distance is fixed at 1.96 a0 (a), 2.00 a0 (b), and 2.40 a0 (c). Bond angle is αOCO = 179.8°. The state 21A″ is artificially shifted by 400 cm−1 to lower energies to distinguish it from other states. Variation of the diabatic quantum numbers ⟨L2z ⟩ along the same cuts are shown in (d−f) for A′ states and in (g−i) for A″ states (quantum number axis is placed on the right).

Here TRT is the 5 × 5 matrix of the rotoelectronic kinetic energy.18 Diabatic states are labeled by the projection of the electronic orbital momentum L2z on the molecular figure axis and by symmetry. Deviations from linearity are described using the coordinate Qu ∼ sin (αOCO) for which the lowest symmetry allowed order is kept. The potential matrix consists of a 2 × 2 block of A′ and a 3 × 3 block of A″ states. The diabatic energies VΔΣ″ of Δu and Σ−u are chosen degenerate; tiny adiabatic splitting can be recovered by adding off-diagonal second-order terms ∼R1R2. Because of orbital degeneracy, VΠ′ = VΠ″ and VΔ′ = VΣΔ″, and the five-fold intersection seam is defined via VΠ′ = VΔ′ = VΣΔ′′ = VΠ′′

state. For example, the adiabatic states 21A′ (blue line) and 31A′ (red line) in Figure 3d preserve their labels ⟨L2z ⟩ = 1 (Πg state) and 4 (Δu state) along the cut in R1. This cut, taken at R2 = 1.96a0, does not cross the seam but almost touches it at 2.70 a0, and the observed avoided crossing is a mere curvature effect.21 For R2 = 2.00a0, the cut passes twice through the seam, and the adiabatic states 2,31A′ exchange their Πg and Δu characters twice (Figure 3e). This means that the avoided crossing region in Vn(R1) hides two densely spaced intersection points. Further extension of R2 drives the two intersections farther apart and makes them equally identifiable in Vn(R1) and ⟨L2z ⟩ (Figure 3f). Crossings between A″ states (Figure 3g−i) involve reorderings of three quantum numbers ⟨L2z ⟩ = 0, 1, and 4, and the diabatic assignments are generally more difficult to follow. Along the tangent’ cut in panel g, the state 11A″ (blue line) keeps its Πg character without any crossings. The states 21A″ (green line) and 31A″ (red line) swap their labels ⟨L2z ⟩ = 0 and 4 at R1 = 2.71a0. This is an example of the glancing intersection between 1Σ−u and 1Δu. In the cuts (f,i), crossings are encountered between all three pairs Πg/Δu, Πg/Σ−u , and Σ−u / Δu, and each integer ⟨L2z ⟩ is passed over several states. However, because 1Σ−u and 1Δu are degenerate, the sides ⟨L2z ⟩ = 0 and ⟨L2z ⟩ = 4 of the three-state diagrams can be glued together giving precisely the two-state diagrams of Figure 3d−f. The seam of CIs can also be validated using topological tests probing properties of electronic states transported along closed contours in the space of nuclear coordinates.19,20,22 For real Hamiltonians, one commonly seeks to detect the sign change in the adiabatic state, either directly (see for example refs 12, 14, 16, and 23) or  as in this work  using the circulation of interstate nonadiabatic coupling matrix elements, NACMEs f nm (Q) = ⟨n |∂/∂Q|m⟩. (See, for example, refs 12 and 23−25.) To this end, a series of circular contours Γi in internal coordinates was constructed and parametrized using polar coordinates (ρ,φ). Examples in Figure 2 include two contours normal to the seam plane (Γ1 and Γ2) and one lying in the seam plane (Γ3). For each contour, angular NACMEs f nm(φ) and circulations

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Diabatic potentials VΠ,Δ at linearity can be constructed by interpolation of the appropriately relabeled ab initio energies. The resulting smooth diabatic PESs are depicted in Figure 1c. An extended description of the contour maps of the MRCI PESs of the five crossing electronic states near linearity is given in the Supporting Information. Up to now, we deliberately avoided the discussion of how the closed isolated seam in Figure 2 has been determined. Detection of degeneracies in the ab initio states can be far from straightforward.16,19,20 Three different methods were used to validate our result, each giving a different perspective on the properties of the crossing states and the closed seam. The basis of Figure 2 forms the seam found from the numerical solution of eq 2 on a dense grid of CO bond distances and αOCO ≈ 180°. Examples of adiabatic potentials Vn(R1) for three fixed R2 values are shown in Figure 3a−c. The set of points at which the adiabatic potentials touched (cf. Figure 3b,c) was defined as an intersection seam. States are presumed to switch diabatic labels upon passing the touching point. Several approximations to the seam have been constructed in this way, all very much alike, for αOCO ranging between 179 and 179.99°. This approach, avoiding explicit diabatic assignments, becomes precarious for closed or strongly curved seams. Example is given in Figure 3a,b. In both frames, the bond length R1 = 2.7a0 is perceived as an approximate intersection point, but only in panel b do the diabatic states actually cross. We re-examined the crossing seam and assigned each electronic state |n⟩ with an expectation value ⟨L2z ⟩ = ⟨n |L2z |n⟩. Electronic assignments for states 2,31A′ are summarized in Figure 3d−f along the same cuts as in panels a−c. Color coding is also taken from panels a−c: Line of a particular color depicts evolution of the diabatic quantum number ⟨L2z ⟩ for a particular adiabatic 3225

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∫0

2π

⟨n|

∂ | m⟩ d φ ∂φ

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crosses the flux line, then NACMEs near crossings behave like delta functions;10 positions of the delta peaks indicate points belonging to the seam. (f 23(φ) for the A′ states is shown in Figure 4a.) Seam points located using Γ3 contours with different centers and radii are shown in Figure 2 with solid squares. This work demonstrates, using symmetry arguments and direct ab initio computations, that the intersection seam of the first five excited singlet electronic states of CO2 is constrained to linear geometries and forms a closed loop in the (R1,R2) plane. The seam passes through the FC zone (see Figure 2) and is directly accessible via UV excitation. The Herzberg− Teller and RT interactions, active in the FC zone, also perturb the vibronic bands, making a detailed analysis of nonadiabatic effects in the UV spectrum of CO2 a challenging task.8 CIs are believed to provide a universal mechanism for ultrafast relaxation, and possibilities of a direct optical detection of a CI are being actively explored (see, e.g., refs 27−29). What options are available for detecting a complete closed intersection seam? The ability to map out the coherent motion around the seam loop appears to be crucial but achievable for CO2 only in the vacuum UV range because the potential difference along the calculated seam reaches 3.0 eV. At longer wavelengths, one might attempt to probe a curved seam locally. CO2 excited with UV light of wavelength λph ≥ 120 nm dissociates mainly into the spin-allowed channel O(1D) + CO(X1Σ+). Figures 1b and 2 demonstrate that as the CO bond elongates, the molecule passes through the curved seam twice and thus encounters two CIs separated by a distance R of the order of the seam curvature radius. At the first encounter, the initial excitation bifurcates into two components following the upper (A↑) and the lower (A↓) adiabats, with the lower subpacket further excited in the bending mode due to the geometric phase effect.30 At the second encounter, the split components recombine into ∼[A↑ exp(ik↑R) + A↓ exp(ik↓R)] and evolve toward the common product channel. Because bending vibration (the coupling mode) is active in transitions at both CIs, it is the oscillation patterns in rotational distributions of CO(X1Σ+), which reveal the phase shifts exp(iΔkR) induced by the seam curvature R. There is growing evidence that a coherent passage through two CIs leads to measurable even− odd population oscillations in the fragment channels.27,30 Such “two double-slit” intramolecular interferometers are promising objects of quantum dynamical studies.

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were evaluated using MOLPRO for pairs of electronic states |n⟩ and |m⟩. Representative NACMEs for contours Γ1,2,3 are shown in Figure 4. Visualizing the intersection seam as a closed

Figure 4. Ab initio angular NACMEs f nm(φ) along the closed contours versus polar angle φ. In panel a, single-valued NACMEs are shown: f13(φ) (states 11A″ and 31A″, contour Γ2, green line); f 23(φ) (states 21A′ and 31A′, contour Γ1, red line); f 23(φ) (states 21A′ and 31A′, contour Γ3, blue line). In panel b, double-valued NACMEs are shown: f12(φ) (states 11A″ and 21A″, contour Γ2, green line); f 23(φ) (states 21A″ and 31A″, contour Γ1, red line).

magnetic flux line with a vector potential An = ⟨n |∂/∂φ|n⟩ helps us to understand qualitatively the results. Indeed, the values of Θnm(Γ) are related to the geometric phases ∮ ΓAdl acquired by the states along the contour Γ.14,22 For two isolated states, zero geometric phase for either |n⟩ or |m⟩ implies zero Θnm(Γ); geometric phase of ±π for both states implies Θnm(Γ) = π.25 Because lines of the vector potential of a toroidal magnet are concentric circles centered on the flux line, any contour lying in the plane of the seam (like Γ3) should give vanishing ∮ ΓAndl and vanishing Θnm(Γ3). Ab initio values along such contours are Θ23(Γ3) ≈ 10−3 for states 2,3A′ and Θnm(Γ3) ≤ 10−1 for any pair of A″ states. If contours run along vector potential lines (like Γ1 and Γ2), then ∮ ΓAndl does not vanish. For 2,3A′, Θ23(Γ1) ≈ 2.90 and Θ23(Γ2) ≈ 3.14, that is, reasonably close to π, and the calculated NACMEs are single-valued functions of φ (see Figure 4a). This demonstrates the local validity of the found seam for the two intersecting A′ states. For A″ states, angular NACMEs and Θnm(Γ1,2) accurately predict three-state intersections. Indeed, NACME f13(φ) for the states 11A″ and 31A″ is a single-valued function of ϕ (see Figure 4a) with Θ13(Γ1) ≈ 2.90. In contrast, NACMEs f12(φ) and f 23(φ) in Figure 4b are both double-valued with f(φ + 2π) = −f(φ). This induced geometric phase effect of Han and Yarkony25 is a manifestation of the Πg/Σ−u and Σ−u /Δu intersections simultaneously enclosed by the contours Γ1,2.26 Whereas canonical contours Γ1 or Γ2 provide excellent local tests, Baer and coworkers successfully used the zero circulation paths Γ3 to locate the seam globally.9,10 If such a contour

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ASSOCIATED CONTENT

S Supporting Information *

Summary of the ab initio calculations and the discussion of the MRCI PESs of the five crossing electronic states. This material is available free of charge via Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address ‡

Dipartimento di Scienze Agrarie, Forestali e Alimentari, University of Torino, Via Leonardo da Vinci 44, 10095 Grugliasco (TO), Italy.

Notes

The authors declare no competing financial interest. 3226

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(22) Berry, M. V. Quantal Phase Factors Accompanying Adiabatic Changes. Proc. R. Soc. London, Ser. A 1984, 392, 45−57. (23) Baer, M. Beyond Born Oppenheimer: Electronic Non-Adiabatic Coupling Terms and Conical Intersections; Wiley: Hoboken, NJ, 2006. (24) Mebel, A. M.; Baer, M.; Lin, S. H. Probing the Nature of Surface Intersection by ab initio Calculations of the Nonadiabatic Coupling Matrix Elements: A Conical Intersection Due to Bending Motion in C2H. J. Chem. Phys. 2000, 112, 10703−10706. (25) Han, S.; Yarkony, D. R. Nonadiabatic Processes Involving Three Electronic States. I. Branch Cuts and Linked Pairs of Conical Intersections. J. Chem. Phys. 2003, 119, 5058−5068. (26) Manolopoulos, D.; Child, M. S. Cyclic Phases at an n-Fold Degeneracy. Phys. Rev. Lett. 1999, 82, 2223−2227. (27) Dixon, R. N.; Hwang, D. W.; Yang, X. F.; Harich, S.; Lin, J. J.; Yang, X. Chemical “Double Slits”: Dynamical Interference of Photodissociation Pathways in Water. Science 1999, 285, 1249−1253. (28) Ashfold, M. N. R.; Cronin, B.; Devine, A. L.; Dixon, R. N.; Nix, M. G. D. The Role of πσ* Excited States in the Photodissociation of Heteroaromatic Molecules. Science 2006, 312, 1637−1640. (29) Lim, J. S.; Kim, S. K. Experimental Probing of Conical Intersection Dynamics in the Photodissociation of Thioanisole. Nat. Chem. 2010, 2, 627−632. (30) Abe, M.; Ohtsuki, Y.; Fujimura, Y.; Lan, Z.; Domcke, W. Geometric Phase Effects in the Coherent Control of the Branching Ratio of Photodissociation Products of Phenol. J. Chem. Phys. 2006, 124, 224316−224322.

ACKNOWLEDGMENTS Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. We thank W. Domcke for useful discussions.

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REFERENCES

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