Coriolis perturbations in the 3ν4 level of the à state of formaldehyde Barratt Park, University of Göttingen / MPI for Biophysical Chemistry 72nd International Symposium on Molecular Spectroscopy Champaign-Urbana, Illinois, Tuesday, June 20, 2017 The A-state is historically special because of its key role in helping us understand the electronic spectroscopy and photochemistry of polyatomic molecules. Recently it’s been brought back into the limelight over the past decade, because it has served as the prototype molecule for elucidation of the roaming mechanism. Coriolis interactions are directly related to the PES and force field and can therefore profoundly influence the dynamics. Chuang et al. J. Chem. Phys. 87 3855 (1987)

The Coriolis Effect , Hvr Hv where 𝐼 𝛼𝛽 ′ = 𝐼 𝛼𝛽 −2 𝐐 † 𝜻 𝛼 𝜻 𝛽 𝐐, and 𝐺 𝛼 = 𝐐 † 𝜻 (𝛼) 𝐏 𝒉 21 = 𝐉 † ( 𝐈 (𝑒) ) −1 𝐆+c.c. 𝜐 4 𝜐 6 + a-axis rotation Γ 𝒉 𝜐 4 : 𝜐 6 (𝑎) =Γ 𝐽 𝑎 ×Γ 𝜐 4 ×Γ 𝜐 6 𝜐 6 torque = A 2 × B 1 × B 2 = A 1 𝒉 𝜐 4 : 𝜐 6 (𝑎) = 𝐴𝐽 𝑎 𝜁 46 𝑎 𝑄 4 𝑃 6 + 𝑄 6 𝑃 4 +c.c.

The 𝜐 4 : 𝜐 6 interaction in the à state. The 𝜐 4 and 𝜐 6 fundamental frequencies in the X̃ and à states 𝜐 4 (cm−1) 𝜐 6 (cm−1) X̃ state 1167 1249 à state 124 894 𝜐 4 and 𝜐 6 are near degenerate in the X̃ state and interact strongly ( 𝜉 𝜐 4 : 𝜐 6 (𝐴) =10.41 cm−1). In the à state, a vibronic distortion depresses the 𝜐 4 frequency, so the analogous 𝜐 4 : 𝜐 6 interaction does not occur. Instead, there is a 3 𝜐 4 : 𝜐 6 interaction. Prior to our work, the strength of this interaction was not known, and the precise 𝜐 6 fundamental frequency had not been reported. Energy (cm−1) Out of plane deformation angle (degrees) P. Jensen, P. R. Bunker, J. Mol. Spectrosc. 94, 114 (1982).

LIF Spectra 6 0 1 4 0 3 4 0 1 6 0 1 𝑇 𝑟𝑜𝑡 ≈3 K 𝑇 𝑟𝑜𝑡 ≈6.5 K Wavenumber (cm−1) Wavenumber (cm−1) 6 0 1 𝑇 𝑟𝑜𝑡 ≈3 K 𝑇 𝑟𝑜𝑡 ≈6.5 K Wavenumber (cm−1) 4 0 1 6 0 1

Existing data on à 61, 43, and 4161 Job et al. [1] (1969) High resolution absorption spectrum of 43 at 𝑇=200–400 K. 863 assigned lines (𝐽≤25, 𝐾 𝑎 ≤6) Assignments are only made up to 𝐾 𝑎 =6 because higher-lying 𝐾 𝑎 manifolds were severely perturbed. Ramsay and Till [2] (1979) High-resolution magnetic rotation study of the 43 region. Includes 120 lines from the interacting 61:43 system (𝐽≤18, 𝐾 𝑎 =4–10) Our work shows many of the vibrational assignments to be nominally incorrect. Our work [3] (2016) Jet-cooled high-resolution LIF spectrum 56 assigned lines from 61, 43, and 4161 (𝐽≤5, 𝐾 𝑎 ≤2) Previously, the origin and rotational constants of 61 and 4161 had not been precisely known. This information provided the missing “key” to working out perturbed structure of 43. We discovered not only an a-axis Coriolis interaction between 61 and 43, but also a c-axis Coriolis interaction between 4161 and 43. [1] V. A. Job, V. Sethuraman, K. K. Innes, J. Mol. Spectrosc. 30, 365 (1969) [2] D. A. Ramsay, S. M. Till, Can. J. Phys. 57, 1224 (1979) [3] G. B. Park, B. C. Krüger, S. Meyer, D. Schwarzer, T. Schäfer, J. Chem. Phys. 144, 194308 (2016)

Reduced Term Value Plot Ka < 6 Region Perturbed Region (Ka ≥ 6) 4161 43 (0th order) 43 (eigs.) 61 (0th order) 61 (eigs.) 61 43 4161 Ka = 10 Ka = 4 Ka = 5 9 10 3 Ka = 5 9 2,e 4 8 9 2,f T − 1.06 J(J + 1) / cm−1 T − 1.06 J(J + 1) / cm−1 1,e 1,f 4 8 7 3 0,e 8 2,f 2,e 3 7 1,f 6 1,e 0,f 7 2,f 2,e 5 1,f 6 1,e 0,f 6 J(J + 1) J(J + 1) A plot of the term energy (T), reduced by 𝐵 𝐽(𝐽+1) is shown, as a function of 𝐽(𝐽+1) for low- 𝐾 𝑎 manifolds of the 61, 43 and 4161 levels. Our fit to the combined data set is shown as curves. Only very low J and Ka rotational levels for 61 and 4161 are available from our work. However, these data points allow accurate determination of the vibrational band origins and rotational constants, allowing us to predict the higher-lying rotational structure. Reduced term value plot (continued) of the higher-lying 𝐾 𝑎 manifolds. In this region, there are large Coriolis perturbations. For 𝐾 𝑎 ≤7, the zero-order 43 levels are higher in energy than the interacting 61 levels. However, at 𝐾 𝑎 =8, 61 overtakes 43, leading to a reversal in the ordering of the levels. For 𝐾 𝑎 ≤6, the zero-order 43 levels lie below the interacting 𝐾 𝑎 −1 levels of 4161. However, at 𝐾 𝑎 =7, the ordering reverses, leading to a reversal in the direction of the c-axis Coriolis interaction. For these reasons, it was impossible for Job et al. to fit 43 levels with 𝐾 𝑎 ≥7 to an effective Hamiltonian that does not take Coriolis interactions explicitly into account.

Fit Results (in cm−1) Our work: Park et al., J. Chem. Phys. 144, 194308 (2016). Constrained. From Henke et al., J. Chem. Phys. 76, 1327 (1982). From Job et al., J. Mol. Spectrosc. 30, 365 (1969).

Evaluation of deperturbed fit constants Deperturbed A constant and ΔK constant of 43

Evaluation of deperturbed fit constants Deperturbed A constant of 61 A constants and estimates of 𝛼 6 𝐴 ≡−𝜕𝐴/𝜕 𝑣 6 (in cm−1). 𝑛 𝐴( 4 𝑛 6 1 ) 𝐴( 4 𝑛 6 0 ) 𝐴 4 𝑛 6 0 −𝐴( 4 𝑛 6 1 ) 9.2642a 8.9519b −0.3123 1 9.090a 8.75194b −0.3381 2 8.9022b 8.618b −0.2842 a. Our work: J. Chem. Phys. 144, 194308 (2016). b. Annu. Rev. Phys. Chem. 34, 31 (1983).

Evaluation of deperturbed fit constants c-axis Coriolis constant Comparison of 𝜉 (𝐶) strengths for two sets of Ã-state levels Perturbation Strength (cm−1) 𝜉 4 3 : 4 1 6 1 (𝐶) 0.197(3)a 𝜉 2 1 4 3 : 2 1 4 1 6 1 (𝐶) 0.2b a. Our work: Park et al., J. Chem. Phys. 144, 194308 (2016). b. Apel and Lee, J. Chem. Phys. 85, 1261 (1986).

Evaluation of deperturbed fit constants a-axis Coriolis constant X 𝜉 4 1 : 6 1 (𝐴) A 𝜉 4 3 : 6 1 (𝐴) 10.4144 cm−1 3.182 cm−1

Evaluation of deperturbed fit constants 𝜐 4 is highly anharmonic in the à state. From the one-dimensional double well model of Coon et al. [1] | 4 3 eig =−0.523 | 4 1 0 +0.689 | 4 3 0 +0.489 | 4 5 0 +… The 𝜉 4 3 : 6 1 (𝐴) interaction is probably dominated by the 𝜁 46 (𝐴) term in the vibration-rotation Hamiltonian. 61 interacts with 43 via this term due to the large contribution of | 4 1 0 to the anharmonic 43 eigenstate. X 𝜉 4 1 : 6 1 (𝐴) A 𝜉 4 3 : 6 1 (𝐴) 10.4144 cm−1 3.182 cm−1 Energy (cm−1) 𝑯= 𝑇( 6 1 ) 𝐴𝐽 𝑎 𝜁 46 𝑎 𝑄 4 𝑃 6 + 𝑄 6 𝑃 4 0 𝑇( 4 1 ) 𝑯 vib 𝑇( 4 3 ) Out of plane deformation angle (degrees) [1] J. Mol. Spectrosc. 20, 107 (1966).

The pattern repeats… 2 1 4 3 fit parameters [1] Param Value (cm−1) 𝜐 0 30340.077(1) 𝐴 8.2191(1) 𝐵 1.10603(6) 𝐶 1.00965(8) ∆ 𝐾 × 10 4 −14.35(6) ∆ 𝐽𝐾 × 10 5 10.6(2) ∆ 𝐽 × 10 6 4.9(1) 𝛿 𝐾 × 10 5 55(2) 𝛿 𝐽 × 10 7 −3(1) 𝐻 𝐾 × 10 7 2.9(2) 𝐻 𝐾𝐽 × 10 9 5(3) 𝐻 𝐽𝐾 × 10 9 −0.6(4) 𝐻 𝐽 × 10 10 −2(3) 𝜙 𝐾 × 10 8 10(1) 𝜙 𝐾𝐽 × 10 10 9.4(7) 𝜙 𝐽 × 10 10 −8(3) The depression of Δ 𝐾 ( 2 1 4 3 ) from its expected value is similar to that of Δ 𝐾 ( 4 3 ). This strongly suggests a 2 1 4 3 : 2 1 6 1 a-axis Coriolis interaction with a similar magnitude 𝜉 2 1 4 3 : 2 1 6 1 (𝐴) ≈ 𝜉 4 3 : 6 1 𝐴 =3.2 cm−1. “Using genetic algorithms for fitting rotational constants of the 2 0 1 4 0 3 band, good agreement between the simulation and the measured spectra is achieved over a wide range of the spectrum. However, the region between 30390 and 30410 cm−1 had to be excluded from the simulation, which might indicate the presence of perturbations.” [1] [1] M. Motsch et al., J. Mol. Spectrosc. 252, 25 (2008)

Thank you! Co-workers: Bastian Krüger Sven Meyer Dirk Schwarzer Tim Schäfer Special thanks: Alec Wodke The entire Wodtke group Bob Field Alexander von Humboldt Foundation

Extra Slides

Conclusions We have analyzed and fit the lowest-lying set of Coriolis interactions in the à state of formaldehyde. There are expected to be many higher-lying perturbations that arise from the same terms in the vibration-rotation Hamiltonian. Our work provides a blueprint for the analysis of these interactions. Our work will contribute to the understanding of how rotation-vibration interaction plays a role in important dynamical processes (such as pre-dissociation) in the à state. We underscore the importance of understanding trends in spectroscopic constants. Anomalies can be used to pinpoint the specific perturbation mechanism.

Vibration-Rotation Hamiltonian Tn = Classical Kinetic Energy Translation Rotation Vibration Vib-Rot (Coriolis, Centrifugal Distortion, …) The Hamiltonian can be written as a sum of kinetic and potential energy contributions. For many multi-body problems, it is nice to be able to separate the classical kinetic energy into three parts: overall translation of the center of mass, rotation of the system, and vibration within the rotating frame. However, it is not possible to achieve a rigorous separation of rotational and vibrational degrees of freedom. This leads to a fourth term: the so-called vibration-rotation term, which leads to effects such as Coriolis interaction and centrifugal distortion.

Vibration-Rotation Hamiltonian velocity of ith nucleus velocity due to vibration within molecular reference frame velocity of c.o.m. rotation of molecular frame b ☉c a

Vibration-Rotation Hamiltonian

Vibration-Rotation Hamiltonian Eckart Conditions 1.) 2.) 1.) 1.) 2.)

Vibration-Rotation Hamiltonian Eckart Conditions 1.) 2.) 1.) 1.) 2.) Ttrans Trot Tvib Tvib-rot

Vibration-Rotation Hamiltonian Ttrans Trot Tvib Tvib-rot Tvib-rot in terms of Meal/Polo cross product matrices Trot in terms of I Tensor Tvib in Normal Coords 𝜻 (𝛼)

Vibration-Rotation Hamiltonian To do QM, we prefer a Hamiltonian written in momenta rather than velocities. The momentum Pi conjugate to Qi is . 𝜕 𝑇 𝑛 𝜕 𝐐 =𝐏= 𝐐 + 𝛼 𝜔 𝛼 𝜻 (𝛼)† 𝐐 I think you should skip the next three slides. Just say the definitions of P and J and then give the result (slide 17). The angular momentum Ja about the a-axis is . 𝐼 𝛼𝛽 ′ ≡ 𝐼 𝛼𝛽 −2 𝐐 † 𝜻 (𝛼) 𝜻 (𝛽) 𝐐 Ga 𝛚= 𝐈 ′ −1 𝐉−𝐆 Next, do a bit of algebra to eliminate 𝐐 and 𝛚….

Vibration-Rotation Hamiltonian , Hvr Hv where 𝐼 𝛼𝛽 ′ = 𝐼 𝛼𝛽 −2 𝐐 † 𝜻 𝛼 𝜻 𝛽 𝐐, and 𝐺 𝛼 = 𝐐 † 𝜻 (𝛼) 𝐏 expand: 𝐈 ′ −1 ≡𝝁 𝜇= 𝐈 𝑒 −1 + 𝑘 𝜕𝜇 𝜕 𝑄 𝑘 𝑒 𝑄 𝑘 +… 𝐻 𝑣𝑟 = 𝐉−𝐆 † 𝐈 𝑒 −1 𝐉−𝐆 + 𝐉−𝐆 † 𝑘 𝜕𝝁 𝜕 𝑄 𝑘 𝑄 𝑘 𝐉−𝐆 +… “First-order” Rot-Vib Hamiltonian

Vibration-Rotation Hamiltonian , Hvr Hv 𝐻 𝑣𝑟 = 𝐉−𝐆 † 𝐈 𝑒 −1 𝐉−𝐆 + 𝐉−𝐆 † 𝑘 𝜕𝝁 𝜕 𝑄 𝑘 𝑄 𝑘 𝐉−𝐆 +… = 𝐉 † 𝐈 𝑒 −1 𝐉+ 𝐆 † 𝐈 𝑒 −1 𝐉+ 𝐉 † 𝐈 𝑒 −1 𝐆 + 𝐆 † 𝐈 𝑒 −1 𝐆+ 𝜕𝝁 𝜕𝐐 𝐐𝐉 † 𝐉+… Rigid Rotor Coriolis Vib. Term ?? Centrifugal Dist. Higher Order Terms Strategy: expand H into terms, classified according to power of (Q,P) and J h20 = Harmonic Oscillator (lowest terms in Hv) h02 = J†(I)-1J = Rigid Rotor h21 = J† (I)-1G (+c.c.) = 1st Order Coriolis effect h12 and h22 will give rise to Centrifugal distortion

The 𝜐 4 : 𝜐 6 Coriolis interaction in the formaldehyde X state 𝒉 21 = 𝐉 † ( 𝐈 (𝑒) ) −1 𝐆+c.c. 𝐺 𝛼 = 𝐐 † 𝜻 (𝛼) 𝐏 Selection Rules: Γ 𝒉 𝜐 4 : 𝜐 6 (𝑎) =Γ 𝐽 𝑎 ×Γ 𝜐 4 ×Γ 𝜐 6 𝐽 𝑎 𝐽𝐾𝑀 =𝐾 𝐽𝐾𝑀 Δ𝐾=0 𝑄 𝑘 ∝ 1 2 𝑎 𝑘 † + 𝑎 𝑘 𝑃 𝑘 ∝ 𝑖 2 𝑎 𝑘 † − 𝑎 𝑘 = A 2 × B 1 × B 2 = A 1 ∆ 𝑣 4 =±1 𝒉 𝜐 4 : 𝜐 6 (𝑎) = 𝐴𝐽 𝑎 𝜁 46 𝑎 𝑄 4 𝑃 6 + 𝑄 6 𝑃 4 +c.c. ∆ 𝑣 6 =±1 G4 𝐸 (12) 𝐸 ∗ (12) ∗ C2v 𝐶 2𝑎 𝜎 𝑎𝑏 𝜎 𝑏𝑐 D2 𝑅 0 𝑅 𝑎 𝜋 𝑅 𝑐 𝜋 𝑅 𝑏 𝜋 𝜓 𝑟𝑜𝑡 𝜓 𝑣𝑖𝑏 𝜓 𝑛𝑢𝑐 A1 1 1 : 𝑇 𝑎 ee 𝜐 1 , 𝜐 2 , 𝜐 3 Ortho A2 −1 −1 : 𝐽 𝑎 eo B1 𝑇 𝑐 , 𝐽 𝑏 oo 𝜐 4 Para B2 𝑇 𝑏 , 𝐽 𝑐 oe 𝜐 5 , 𝜐 6 a c ⊗ b

The 𝜐 4 : 𝜐 6 Coriolis interaction in the formaldehyde X state 𝒉 21 = 𝐉 † ( 𝐈 (𝑒) ) −1 𝐆+c.c. 𝐺 𝛼 = 𝐐 † 𝜻 (𝛼) 𝐏 Selection Rules: Γ 𝒉 𝜐 4 : 𝜐 6 (𝑎) =Γ 𝐽 𝑎 ×Γ 𝜐 4 ×Γ 𝜐 6 𝐽 𝑎 𝐽𝐾𝑀 =𝐾 𝐽𝐾𝑀 Δ𝐾=0 𝑄 𝑘 ∝ 1 2 𝑎 𝑘 † + 𝑎 𝑘 𝑃 𝑘 ∝ 𝑖 2 𝑎 𝑘 † − 𝑎 𝑘 = A 2 × B 1 × B 2 = A 1 ∆ 𝑣 4 =±1 𝒉 𝜐 4 : 𝜐 6 (𝑎) = 𝐴𝐽 𝑎 𝜁 46 𝑎 𝑄 4 𝑃 6 + 𝑄 6 𝑃 4 +c.c. ∆ 𝑣 6 =±1 “Resonant” Matrix Elements: 𝜐 4 𝜐 6 + a-axis rotation 𝜐 6 torque 𝑣 4 +1, 𝑣 6 −1,𝐽𝐾𝑀 𝒉 𝜐 4 : 𝜐 6 (𝑎) 𝑣 4 , 𝑣 6 ,𝐽𝐾𝑀 ≠0

The 𝜐 4 : 𝜐 6 Coriolis interaction in the formaldehyde X state Spectroscopic Hamiltonian for the interacting 41 and 61 levels of the formaldehyde X̃ state, from Ref [1]. (All parameters in cm−1 units.) [1] A. Perrin, F. Keller, and J.-M. Flaud, J. Mol. Spectrosc. 221, 192 (2003)