Download presentation
Presentation is loading. Please wait.
Published byHelena Hall Modified over 6 years ago
1
Coriolis perturbations in the 3Ξ½4 level of the AΜ state of formaldehyde
Barratt Park, University of GΓΆttingen / 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 itβs 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 (1987)
2
The Coriolis Effect , Hvr Hv
where πΌ πΌπ½ β² = πΌ πΌπ½ β2 π β π» πΌ π» π½ π, and πΊ πΌ = π β π» (πΌ) π π 21 = π β ( π (π) ) β1 π+c.c. π 4 π 6 + a-axis rotation Ξ π π 4 : π 6 (π) =Ξ π½ π ΓΞ π 4 ΓΞ π 6 π 6 torque = A 2 Γ B 1 Γ B 2 = A 1 π π 4 : π 6 (π) = π΄π½ π π 46 π π 4 π 6 + π 6 π 4 +c.c.
3
The π 4 : π 6 interaction in the AΜ state.
The π 4 and π 6 fundamental frequencies in the XΜ and AΜ states π 4 (cmβ1) π 6 (cmβ1) XΜ state 1167 1249 AΜ state 124 894 π 4 and π 6 are near degenerate in the XΜ state and interact strongly ( π π 4 : π 6 (π΄) =10.41 cmβ1). In the AΜ state, a vibronic distortion depresses the π 4 frequency, so the analogous π 4 : π 6 interaction does not occur. Instead, there is a 3 π 4 : π 6 interaction. Prior to our work, the strength of this interaction was not known, and the precise π 6 fundamental frequency had not been reported. Energy (cmβ1) Out of plane deformation angle (degrees) P. Jensen, P. R. Bunker, J. Mol. Spectrosc. 94, 114 (1982).
4
LIF Spectra 6 0 1 4 0 3 π πππ‘ β3 K π πππ‘ β6.5 K Wavenumber (cmβ1) Wavenumber (cmβ1) 6 0 1 π πππ‘ β3 K π πππ‘ β6.5 K Wavenumber (cmβ1)
5
Existing data on AΜ 61, 43, and 4161 Job et al. [1] (1969)
High resolution absorption spectrum of 43 at π=200β400 K. 863 assigned lines (π½β€25, πΎ π β€6) Assignments are only made up to πΎ π =6 because higher-lying πΎ π 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 (π½β€18, πΎ π =4β10) 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 (π½β€5, πΎ π β€2) Previously, the origin and rotational constants of 61 and 4161 had not been precisely known. This information provided the missing βkeyβ 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. KrΓΌger, S. Meyer, D. Schwarzer, T. SchΓ€fer, J. Chem. Phys. 144, (2016)
6
Reduced Term Value Plot
Ka < 6 Region Perturbed Region (Ka β₯ 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 β 1.06 J(J + 1) / cmβ1 T β 1.06 J(J + 1) / cmβ1 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 π΅ π½(π½+1) is shown, as a function of π½(π½+1) for low- πΎ π 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 πΎ π manifolds. In this region, there are large Coriolis perturbations. For πΎ π β€7, the zero-order 43 levels are higher in energy than the interacting 61 levels. However, at πΎ π =8, 61 overtakes 43, leading to a reversal in the ordering of the levels. For πΎ π β€6, the zero-order 43 levels lie below the interacting πΎ π β1 levels of However, at πΎ π =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 πΎ π β₯7 to an effective Hamiltonian that does not take Coriolis interactions explicitly into account.
7
Fit Results (in cmβ1) Our work: Park et al., J. Chem. Phys. 144, (2016). Constrained. From Henke et al., J. Chem. Phys. 76, 1327 (1982). From Job et al., J. Mol. Spectrosc. 30, 365 (1969).
8
Evaluation of deperturbed fit constants
Deperturbed A constant and ΞK constant of 43
9
Evaluation of deperturbed fit constants
Deperturbed A constant of 61 A constants and estimates of πΌ 6 π΄ β‘βππ΄/π π£ 6 (in cmβ1). π π΄( 4 π 6 1 ) π΄( 4 π 6 0 ) π΄ 4 π βπ΄( 4 π 6 1 ) 9.2642a 8.9519b β0.3123 1 9.090a b β0.3381 2 8.9022b 8.618b β0.2842 a. Our work: J. Chem. Phys. 144, (2016). b. Annu. Rev. Phys. Chem. 34, 31 (1983).
10
Evaluation of deperturbed fit constants
c-axis Coriolis constant Comparison of π (πΆ) strengths for two sets of AΜ-state levels Perturbation Strength (cmβ1) π 4 3 : (πΆ) 0.197(3)a π : (πΆ) 0.2b a. Our work: Park et al., J. Chem. Phys. 144, (2016). b. Apel and Lee, J. Chem. Phys. 85, 1261 (1986).
11
Evaluation of deperturbed fit constants
a-axis Coriolis constant X π 4 1 : 6 1 (π΄) A π 4 3 : 6 1 (π΄) cmβ1 3.182 cmβ1
12
Evaluation of deperturbed fit constants
π 4 is highly anharmonic in the AΜ state. From the one-dimensional double well model of Coon et al. [1] | eig =β0.523 | | | β¦ The π 4 3 : 6 1 (π΄) interaction is probably dominated by the π 46 (π΄) term in the vibration-rotation Hamiltonian. 61 interacts with 43 via this term due to the large contribution of | to the anharmonic 43 eigenstate. X π 4 1 : 6 1 (π΄) A π 4 3 : 6 1 (π΄) cmβ1 3.182 cmβ1 Energy (cmβ1) π―= π( 6 1 ) π΄π½ π π 46 π π 4 π 6 + π 6 π π( 4 1 ) π― vib π( 4 3 ) Out of plane deformation angle (degrees) [1] J. Mol. Spectrosc. 20, 107 (1966).
13
The pattern repeatsβ¦ fit parameters [1] Param Value (cmβ1) π 0 (1) π΄ 8.2191(1) π΅ (6) πΆ (8) β πΎ Γ 10 4 β14.35(6) β π½πΎ Γ 10 5 10.6(2) β π½ Γ 10 6 4.9(1) πΏ πΎ Γ 10 5 55(2) πΏ π½ Γ 10 7 β3(1) π» πΎ Γ 10 7 2.9(2) π» πΎπ½ Γ 10 9 5(3) π» π½πΎ Γ 10 9 β0.6(4) π» π½ Γ 10 10 β2(3) π πΎ Γ 10 8 10(1) π πΎπ½ Γ 10 10 9.4(7) π π½ Γ 10 10 β8(3) The depression of Ξ πΎ ( ) from its expected value is similar to that of Ξ πΎ ( 4 3 ). This strongly suggests a : a-axis Coriolis interaction with a similar magnitude π : (π΄) β π 4 3 : π΄ =3.2 cmβ1. βUsing genetic algorithms for fitting rotational constants of the band, good agreement between the simulation and the measured spectra is achieved over a wide range of the spectrum. However, the region between and cmβ1 had to be excluded from the simulation, which might indicate the presence of perturbations.β [1] [1] M. Motsch et al., J. Mol. Spectrosc. 252, 25 (2008)
14
Thank you! Co-workers: Bastian KrΓΌger Sven Meyer Dirk Schwarzer Tim SchΓ€fer Special thanks: Alec Wodke The entire Wodtke group Bob Field Alexander von Humboldt Foundation
15
Extra Slides
16
Conclusions We have analyzed and fit the lowest-lying set of Coriolis interactions in the AΜ 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 AΜ state. We underscore the importance of understanding trends in spectroscopic constants. Anomalies can be used to pinpoint the specific perturbation mechanism.
17
Vibration-Rotation Hamiltonian
Tn = Classical Kinetic Energy Translation Rotation Vibration Vib-Rot (Coriolis, Centrifugal Distortion, β¦) 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.
18
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 βc a
19
Vibration-Rotation Hamiltonian
20
Vibration-Rotation Hamiltonian
Eckart Conditions 1.) 2.) 1.) 1.) 2.)
21
Vibration-Rotation Hamiltonian
Eckart Conditions 1.) 2.) 1.) 1.) 2.) Ttrans Trot Tvib Tvib-rot
22
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 π» (πΌ)
23
Vibration-Rotation Hamiltonian
To do QM, we prefer a Hamiltonian written in momenta rather than velocities. The momentum Pi conjugate to Qi is π π π π π =π= π + πΌ π πΌ π» (πΌ)β π 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 πΌ πΌπ½ β² β‘ πΌ πΌπ½ β2 π β π» (πΌ) π» (π½) π Ga π= π β² β1 πβπ Next, do a bit of algebra to eliminate π and πβ¦.
24
Vibration-Rotation Hamiltonian
, Hvr Hv where πΌ πΌπ½ β² = πΌ πΌπ½ β2 π β π» πΌ π» π½ π, and πΊ πΌ = π β π» (πΌ) π expand: π β² β1 β‘π π= π π β1 + π ππ π π π π π π +β¦ π» π£π = πβπ β π π β1 πβπ + πβπ β π ππ π π π π π πβπ +β¦ βFirst-orderβ Rot-Vib Hamiltonian
25
Vibration-Rotation Hamiltonian
, Hvr Hv π» π£π = πβπ β π π β1 πβπ + πβπ β π ππ π π π π π πβπ +β¦ = π β π π β1 π+ π β π π β1 π+ π β π π β1 π + π β π π β1 π+ ππ ππ ππ β π+β¦ 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 = Jβ (I)-1J = Rigid Rotor h21 = Jβ (I)-1G (+c.c.) = 1st Order Coriolis effect h12 and h22 will give rise to Centrifugal distortion
26
The π 4 : π 6 Coriolis interaction in the formaldehyde X state
π 21 = π β ( π (π) ) β1 π+c.c. πΊ πΌ = π β π» (πΌ) π Selection Rules: Ξ π π 4 : π 6 (π) =Ξ π½ π ΓΞ π 4 ΓΞ π 6 π½ π π½πΎπ =πΎ π½πΎπ ΞπΎ=0 π π β π π β + π π π π β π π π β β π π = A 2 Γ B 1 Γ B 2 = A 1 β π£ 4 =Β±1 π π 4 : π 6 (π) = π΄π½ π π 46 π π 4 π 6 + π 6 π 4 +c.c. β π£ 6 =Β±1 G4 πΈ (12) πΈ β (12) β C2v πΆ 2π π ππ π ππ D2 π
0 π
π π π
π π π
π π π πππ‘ π π£ππ π ππ’π A1 1 1 : π π ee π 1 , π 2 , π 3 Ortho A2 β1 β1 : π½ π eo B1 π π , π½ π oo π 4 Para B2 π π , π½ π oe π 5 , π 6 a c β b
27
The π 4 : π 6 Coriolis interaction in the formaldehyde X state
π 21 = π β ( π (π) ) β1 π+c.c. πΊ πΌ = π β π» (πΌ) π Selection Rules: Ξ π π 4 : π 6 (π) =Ξ π½ π ΓΞ π 4 ΓΞ π 6 π½ π π½πΎπ =πΎ π½πΎπ ΞπΎ=0 π π β π π β + π π π π β π π π β β π π = A 2 Γ B 1 Γ B 2 = A 1 β π£ 4 =Β±1 π π 4 : π 6 (π) = π΄π½ π π 46 π π 4 π 6 + π 6 π 4 +c.c. β π£ 6 =Β±1 βResonantβ Matrix Elements: π 4 π 6 + a-axis rotation π 6 torque π£ 4 +1, π£ 6 β1,π½πΎπ π π 4 : π 6 (π) π£ 4 , π£ 6 ,π½πΎπ β 0
28
The π 4 : π 6 Coriolis interaction in the formaldehyde X state
Spectroscopic Hamiltonian for the interacting 41 and 61 levels of the formaldehyde XΜ state, from Ref [1]. (All parameters in cmβ1 units.) [1] A. Perrin, F. Keller, and J.-M. Flaud, J. Mol. Spectrosc. 221, 192 (2003)
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.