^{1,a)}

### Abstract

The absorption spectrum of CO2 in the wavelength range 120–160 nm is analyzed by means of quantum mechanical calculations performed using vibronically coupled potential energy surfaces of five singlet valence electronic states and the coordinate dependent transition dipole moment vectors. The thermally averaged spectrum, calculated for T = 190 K via Boltzmann averaging of optical transitions from many initial rotational states, accurately reproduces the experimental spectral envelope, consisting of a low and a high energy band, the positions of the absorption maxima, their FWHMs, peak intensities, and frequencies of diffuse structures in each band. Contributions of the vibronic interactions due to Renner-Teller coupling, conical intersections, and the Herzberg-Teller effect are isolated and the calculated bands are assigned in terms of adiabatic electronic states. Finally, diffuse structures in the calculated bands are vibronically assigned using wave functions of the underlying resonance states. It is demonstrated that the main progressions in the high energy band correspond to consecutive excitations of the pseudorotational motion along the closed loop of the CI seam, and progressions differ in the number of nodes along the radial mode perpendicular to the closed seam. Irregularity of the diffuse peaks in the low energy band is interpreted as a manifestation of the carbene-type “cyclic” OCO minimum.

Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

I. INTRODUCTION

II. QUANTUM MECHANICAL CALCULATIONS

A. Molecular Hamiltonian

B. The initial state and the absorptionspectrum

C. Numerical details

III. RESULTS

A. Electronic assignments

B. Vibronic assignments of the high energy band

C. Vibronic assignments of the low energy band

D. Vibrational assignments in the state

IV. SUMMARY AND CONCLUDING REMARKS

### Key Topics

- Absorption spectra
- 44.0
- Carbon dioxide
- 33.0
- Excited states
- 22.0
- Wave functions
- 17.0
- Quantum optics
- 12.0

## Figures

(a) Experimental absorption cross section of CO2 (T = 195 K; in cm2) as a function of photon energy (in 103 cm−1). Numbers on the cross section axis denote powers of ten. The photon energy range studied in this work is indicated with two dashed lines. (b) Absorption spectrum, calculated at T = 200 K (blue line, vertically shifted by 0.9 × 10−18 cm2), compared with the experimental spectrum (black line). Electronic assignment in terms of adiabatic states is given. Experimental data are from Ref. 17 . Theoretical spectrum is shifted by 400 cm−1 to lower energies.

(a) Experimental absorption cross section of CO2 (T = 195 K; in cm2) as a function of photon energy (in 103 cm−1). Numbers on the cross section axis denote powers of ten. The photon energy range studied in this work is indicated with two dashed lines. (b) Absorption spectrum, calculated at T = 200 K (blue line, vertically shifted by 0.9 × 10−18 cm2), compared with the experimental spectrum (black line). Electronic assignment in terms of adiabatic states is given. Experimental data are from Ref. 17 . Theoretical spectrum is shifted by 400 cm−1 to lower energies.

Cuts through the PESs of the adiabatic states along (a) OCO angle (fixed coordinates are R 1 = 2.2 a 0 and R 2 = 2.3 a 0) and (b) one CO bond distance (fixed coordinates are R CO = 2.3 a 0 and αOCO = 175°). The thick black line in both panels is the ground electronic state ; excited A′ and A′′ states are shown with thin black and brown lines, respectively. Red lines and arrows illustrate the non-vertical excitation process and the effect of the TDM on the initial vibrational wave function ψ0 in . ψ0 is shown in both panels with a dashed line, and the shapes of the initial state in the optical transitions via y and x components of the TDM, μ y ψ0 (a) and μ x ψ0 (b) are shown with solid lines. Arrows, indicating excitations, pass through the maximum of ψ0 and the maxima of μ y ψ0 and μ x ψ0.

Cuts through the PESs of the adiabatic states along (a) OCO angle (fixed coordinates are R 1 = 2.2 a 0 and R 2 = 2.3 a 0) and (b) one CO bond distance (fixed coordinates are R CO = 2.3 a 0 and αOCO = 175°). The thick black line in both panels is the ground electronic state ; excited A′ and A′′ states are shown with thin black and brown lines, respectively. Red lines and arrows illustrate the non-vertical excitation process and the effect of the TDM on the initial vibrational wave function ψ0 in . ψ0 is shown in both panels with a dashed line, and the shapes of the initial state in the optical transitions via y and x components of the TDM, μ y ψ0 (a) and μ x ψ0 (b) are shown with solid lines. Arrows, indicating excitations, pass through the maximum of ψ0 and the maxima of μ y ψ0 and μ x ψ0.

Spatial components μ x (a) and (d), μ y (b) and (e), and μ z (c) and (f) of the ab initio TDM vectors for the adiabatic states 21 A′ (purple), 31 A′ (brown), 11 A′′ (black), 21 A′′ (purple), and 31 A′′ (brown) from the ground state . A′ states are shown with solid, A′′ states with open symbols. Dependences on the CO bond distance are shown in (a)–(c) [fixed coordinates are R 2 = 2.2 a 0 and αOCO = 179°] and dependences on the OCO angle are shown in (d)–(f) [fixed coordinates are R 1 = 2.2 a 0 and R 2 = 2.3 a 0]. TDMs are measured in atomic units. The coordinate axes are sketched in (b): y is directed along the molecular figure axis, and z is normal to y in the molecular plane, and x is normal to the molecular plane.

Spatial components μ x (a) and (d), μ y (b) and (e), and μ z (c) and (f) of the ab initio TDM vectors for the adiabatic states 21 A′ (purple), 31 A′ (brown), 11 A′′ (black), 21 A′′ (purple), and 31 A′′ (brown) from the ground state . A′ states are shown with solid, A′′ states with open symbols. Dependences on the CO bond distance are shown in (a)–(c) [fixed coordinates are R 2 = 2.2 a 0 and αOCO = 179°] and dependences on the OCO angle are shown in (d)–(f) [fixed coordinates are R 1 = 2.2 a 0 and R 2 = 2.3 a 0]. TDMs are measured in atomic units. The coordinate axes are sketched in (b): y is directed along the molecular figure axis, and z is normal to y in the molecular plane, and x is normal to the molecular plane.

(a) The absorption spectrum σ(E ph|J i = 0) for a single initial state J i = 0 (blue) compared with the spectrum of Ref. 17 measured at T = 195 K (black). Thin sticks indicate the major experimental peaks separated into a strong (brown) and a weak (blue) progression in the high energy band, and a strong progression in the low energy band (black). Their positions are taken from Table X of Ref. 15 augmented with data from Ref. 17 . Panels (b)–(d) show decomposition of the calculated spectrum and vibronic assignments: (b) σ(E ph|J i = 0) spectrum (thin gray line) broken down into two low resolution components stemming from excitations of the bent (21 A′, 11 A′′, thick black line) and linear (31 A′, 31 A′′, thick blue line) adiabatic states. Two spectra at the bottom are the parallel (σ∥, thick green line) and the perpendicular (σ⊥, thick red line) components of σ(E ph|J i = 0). (c) Decomposition of σ∥ (green) in terms of resonance states of the Hamiltonian Eq. (11) . Gray sticks mark resonance positions (every 3d calculated state is shown). Black line is the sum of Lorentzians in Eq. (16) . (d) Vibronic assignment of σ∥ (green) in terms of the most intense resonances shown at the bottom of the panel with thin sticks. Vertical shift of the calculated spectra is 0.9 × 10−18 cm2 in panels (a) and (b), and 0.6 × 10−18 cm2 in panels (c) and (d).

(a) The absorption spectrum σ(E ph|J i = 0) for a single initial state J i = 0 (blue) compared with the spectrum of Ref. 17 measured at T = 195 K (black). Thin sticks indicate the major experimental peaks separated into a strong (brown) and a weak (blue) progression in the high energy band, and a strong progression in the low energy band (black). Their positions are taken from Table X of Ref. 15 augmented with data from Ref. 17 . Panels (b)–(d) show decomposition of the calculated spectrum and vibronic assignments: (b) σ(E ph|J i = 0) spectrum (thin gray line) broken down into two low resolution components stemming from excitations of the bent (21 A′, 11 A′′, thick black line) and linear (31 A′, 31 A′′, thick blue line) adiabatic states. Two spectra at the bottom are the parallel (σ∥, thick green line) and the perpendicular (σ⊥, thick red line) components of σ(E ph|J i = 0). (c) Decomposition of σ∥ (green) in terms of resonance states of the Hamiltonian Eq. (11) . Gray sticks mark resonance positions (every 3d calculated state is shown). Black line is the sum of Lorentzians in Eq. (16) . (d) Vibronic assignment of σ∥ (green) in terms of the most intense resonances shown at the bottom of the panel with thin sticks. Vertical shift of the calculated spectra is 0.9 × 10−18 cm2 in panels (a) and (b), and 0.6 × 10−18 cm2 in panels (c) and (d).

Left and right columns depict 3A′ adiabatic components of the 3D wave functions of resonance states in progressions (v ϕ, 0, 0) and (v ϕ, 0, 1), respectively. The probability density distribution is projected onto the (R 1, R 2) plane. Shaded black indicates regions of high density. 57 Middle column depicts eigenstates ψ2D(R 1, R 2) calculated in the 2D single state model. All states belong to progression (v ϕ, 0)2D; red contours represent positive, blue contours negative values of ψ2D. In all panels, a 2D contour map of the 3A′ adiabatic potential for αOCO = 179° (gray lines) and the CI seam (black thick line) are sketched. Axis tic labels are in a 0. States marked are mixed with states shown in Fig. 6 . The top middle panel shows author's drawing of a Mongolian hat.

Left and right columns depict 3A′ adiabatic components of the 3D wave functions of resonance states in progressions (v ϕ, 0, 0) and (v ϕ, 0, 1), respectively. The probability density distribution is projected onto the (R 1, R 2) plane. Shaded black indicates regions of high density. 57 Middle column depicts eigenstates ψ2D(R 1, R 2) calculated in the 2D single state model. All states belong to progression (v ϕ, 0)2D; red contours represent positive, blue contours negative values of ψ2D. In all panels, a 2D contour map of the 3A′ adiabatic potential for αOCO = 179° (gray lines) and the CI seam (black thick line) are sketched. Axis tic labels are in a 0. States marked are mixed with states shown in Fig. 6 . The top middle panel shows author's drawing of a Mongolian hat.

(Upper panels) 3A′ adiabatic components of the 3D wave functions of resonance states in the progression (v ϕ, 1, 0). The state (3, 1, 0) could not be found. Lower panels depict eigenstates ψ2D(R 1, R 2) calculated in the 2D single state model and belonging to the progression (v ϕ, 1)2D. The layout of all wave functions is the same as in Fig. 5 . The 2D states are mixed with states (v ϕ + 4, 0)2D; counterparts of those marked are shown in Fig. 5 .

(Upper panels) 3A′ adiabatic components of the 3D wave functions of resonance states in the progression (v ϕ, 1, 0). The state (3, 1, 0) could not be found. Lower panels depict eigenstates ψ2D(R 1, R 2) calculated in the 2D single state model and belonging to the progression (v ϕ, 1)2D. The layout of all wave functions is the same as in Fig. 5 . The 2D states are mixed with states (v ϕ + 4, 0)2D; counterparts of those marked are shown in Fig. 5 .

(Upper panels) Frequencies (a) in the major experimental progression, (b) in the progression (v ϕ, 0, 0) in the full calculations, and (c) in the progression (v ϕ, 0)2D in the 2D model. In (b), frequencies are lifted by 50 cm−1; in (c), the frequency scale is omitted. Arrows emphasize the positions of dips in the progressions. (Lower panels) Potential of the 31 A′ state (d) at αOCO = 179° along the ab initio CI seam and (e) across the Mongolian hat top. Angular coordinate ϕ along the seam in (d) is chosen such that ϕ = 180° for R 1 = R 2 = 2.24 a 0; the “radial” coordinate in (e) runs along the antisymmetric stretch R −, and the origin R − = 0 is the point R 1 = R 2 = 2.5 a 0. Potential curves are raised through the zero point energies of “missing” coordinates, ℏω r /2 and ℏωϕ/2 in (d) and (e), respectively. Vibrational ladder in the 2D model is indicated in (d) and (e). Energy spacings forming two dips in the progression in panel (c) are highlighted with gray and yellow.

(Upper panels) Frequencies (a) in the major experimental progression, (b) in the progression (v ϕ, 0, 0) in the full calculations, and (c) in the progression (v ϕ, 0)2D in the 2D model. In (b), frequencies are lifted by 50 cm−1; in (c), the frequency scale is omitted. Arrows emphasize the positions of dips in the progressions. (Lower panels) Potential of the 31 A′ state (d) at αOCO = 179° along the ab initio CI seam and (e) across the Mongolian hat top. Angular coordinate ϕ along the seam in (d) is chosen such that ϕ = 180° for R 1 = R 2 = 2.24 a 0; the “radial” coordinate in (e) runs along the antisymmetric stretch R −, and the origin R − = 0 is the point R 1 = R 2 = 2.5 a 0. Potential curves are raised through the zero point energies of “missing” coordinates, ℏω r /2 and ℏωϕ/2 in (d) and (e), respectively. Vibrational ladder in the 2D model is indicated in (d) and (e). Energy spacings forming two dips in the progression in panel (c) are highlighted with gray and yellow.

Shown left of the vertical dashed line are frequencies in the pure progressions of eigenstates in which the 21 A′ adiabatic component (purple, blue, and brown lines and symbols) or the 11 A′′ adiabatic component (red lines and symbols) are most populated. Open black symbols are the two experimental vibrational progressions (triangles and squares), as well as the frequency shift between their band maxima (circles) taken from Table 3 of Ref. 19 . Shown right of the vertical dashed line are energy intervals between adjacent diffuse peaks in the experimental spectrum of Ref. 17 (black open circles) and between most intense resonance states (green solid circles). Note that the frequency scale changes in the upper part of the figure.

Shown left of the vertical dashed line are frequencies in the pure progressions of eigenstates in which the 21 A′ adiabatic component (purple, blue, and brown lines and symbols) or the 11 A′′ adiabatic component (red lines and symbols) are most populated. Open black symbols are the two experimental vibrational progressions (triangles and squares), as well as the frequency shift between their band maxima (circles) taken from Table 3 of Ref. 19 . Shown right of the vertical dashed line are energy intervals between adjacent diffuse peaks in the experimental spectrum of Ref. 17 (black open circles) and between most intense resonance states (green solid circles). Note that the frequency scale changes in the upper part of the figure.

2A′ adiabatic component of the resonance state with E 0 = 8.3795 eV and Γ0 = 62 cm−1. The probability density distribution is projected onto the (R 1, R 2) plane in the left panel and onto the (αOCO, R 2) plane in the right panel. Shaded black indicates regions of high density. 57 The 2D contour maps of the 21 A′ adiabatic potential in the (R 1, R 2) plane and in the (αOCO, R 2) plane are sketched in the left and right panels, respectively. Thick black lines trace out the probability density buildup away from the linear FC region.

2A′ adiabatic component of the resonance state with E 0 = 8.3795 eV and Γ0 = 62 cm−1. The probability density distribution is projected onto the (R 1, R 2) plane in the left panel and onto the (αOCO, R 2) plane in the right panel. Shaded black indicates regions of high density. 57 The 2D contour maps of the 21 A′ adiabatic potential in the (R 1, R 2) plane and in the (αOCO, R 2) plane are sketched in the left and right panels, respectively. Thick black lines trace out the probability density buildup away from the linear FC region.

2A′ adiabatic components of the 3D wave functions of states in the progression (v s , 0, 0) (upper panels) and (0, 0, v a ) (lower panels) shown against the 2D contour maps of the 21 A′ adiabatic potential in the (R 1, R 2) plane. 57

2A′ adiabatic components of the 3D wave functions of states in the progression (v s , 0, 0) (upper panels) and (0, 0, v a ) (lower panels) shown against the 2D contour maps of the 21 A′ adiabatic potential in the (R 1, R 2) plane. 57

2A′ adiabatic components of the 3D wave functions of states in the progression (0, v b , 0) shown against the 2D contour maps of the 21 A′ adiabatic potential in the (αOCO, R 2) plane. In the left lower frame, the ground vibrational state in the carbene-type OCO minimum is shown and labeled (0, 0, 0) c . 57

2A′ adiabatic components of the 3D wave functions of states in the progression (0, v b , 0) shown against the 2D contour maps of the 21 A′ adiabatic potential in the (αOCO, R 2) plane. In the left lower frame, the ground vibrational state in the carbene-type OCO minimum is shown and labeled (0, 0, 0) c . 57

Absorption spectrum of the state and assignments of the three progressions (v s , 0, v a ) in which the most intense absorption lines are found.

Absorption spectrum of the state and assignments of the three progressions (v s , 0, v a ) in which the most intense absorption lines are found.

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