Manuscript - submitted to J. Mol. Spectrosc. Ab Initio Calculations of Torsionally Mediated Hyperfine Splittings in E States of Acetaldehyde Li-Hong Xu, E.M. Reid, B. Guislain – UNB, Canada J.T. Hougen – NIST, USA E.A. Alekseev, I. Krapivin – Kharkov, Ukraine Manuscript - submitted to J. Mol. Spectrosc.

“Historical” Overview CH3OH has ~70 kHz hyperfine splittings (Lamb dip) Torsion-mediated spin-rotation effect for E state Not: Pg (torsional angular momentum) But: e±ing (torsional function) Three recent publications: [A] J. Chem. Phys. 143 (2015) 044304-1-12, L.H. Coudert, et al. “Spin-torsion effects in the hyperfine structure of methanol”. [B] J. Chem. Phys. 145 (2016) 024307-1-20, S.P Belov, et al. “Torsionally mediated spin-rotation hyperfine splittings at moderate to high J values in methanol”. [C] J. Chem. Phys. 145 (2016) 244301-1-15, B. Lankhaar, et al. “Hyperfine interactions and internal rotation in methanol”. MW transition (Lamb dip) nt=0, Q27, K = 0 ¬ 1 E2 71 kHz 20 kHz our work

A Logical Question – What Is Next? nt = 2, 63 ¬ 53 E Vadim’s labelling is K=3 IK’s notation is K=2E Does this phenomenon exist in other methyl rotor containing molecules, i.e. CH3CHO About 120 transitions (50-150 GHz) have been looked at by the Kharkov group. No evidence of HF splittings This talk How to obtain HFS constants and predict the HF splittings for CH3CHO? - scale to rotational constants (A, B, C) from CH3OH - ab initio, calculate spin-rotation constants, roadmap from Coudert et. al. J. Chem. Phys. 143 (2015) 044304 Compare with experiment If calculations show splittings 3-4X smaller than that of CH3OH, we expect no splittings will be seen in experiment Where to predict? nt = 2, E 62 ¬ 52 nt = 0, 63 ¬ 53 A

Comparison of Ab Initio Levels: Spin-Rotation Constants for CH3OH Ab Initio Values C2xx C2yy C2xy Ref. (3 out of 32 constants) MP2/6-311+G(3df,2p) 1.741 -1.697 1.715 [D] CCSD(T)/pVTZ 1.666 -1.662 1.665 [A] CCSD(T)/aug-cc-pVTZ 1.660 -1.673 1.668 [C] Experimental Values 1.748 -1.732 1.547 [B] Ref. [A] J. Chem. Phys. 143 (2015) 044304-1-12, Coudert, et al. [B] J. Chem. Phys. 145 (2016) 024307-1-20, Belov, et al. [C] J. Chem. Phys. 145 (2016) 244301-1-15, Lankhaar, et al. [D] Present work Spin-rotation constants can be calculated reasonably well by ab initio CCSD(T)/aug-cc-pVTZ 1.660 -1.673 -1.668 [C] Overall signs changed to agree with ours Cxy, signs changed to agree with expt. The sign of Cxy for expt can not be determined

Comparison of Spin-Rotation Constants for CH3OH and CH3CHO Based on CH3OH results, we have calculated spin-rotation constants for CH3CHO at MP2 level (recognizing additional e’s, double bond…) Three E Spin-Rotation Constants (kHz) MP2 CH3OH CH3CHO Ratio C2xx 1.741 0.581 3 C2yy -1.697 -0.616 2.8 C2xy -1.715 -0.596 2.9 Rotation Constants (cm-1) CH3OH CH3CHO A 4.2537 1.8849 2.3 B 0.8102 0.3487 2.3 C 0.7925 0.3032 2.6 We have used three coefficients (C2xx C2yy C2xy) to calculate hyperfine doublet splittings for CH3CHO F 27.6468 7.5671 3.6 r 0.8102 0.3286 2.5 Coudert’s xx,yy,zz

CH3OH: Obs. & Calc. Doublet Splittings JCP 145 (2016) 024307 symbols: obs. lines: calc. (Solid filled points: excluded from the fit) Kharkov nt = 0, DJ = 0 K = 0  -1 NNOV nt = 0, DJ = 1 K = -1  -2 NNOV nt = 0, DJ = 0 K = 2  -1 CH3OH: observed doublet splittings range from 25 – 75 kHz 75/”3” = 25 kHz near or below the current sub-Doppler (Lamb dip) resolution Doublet splitting size Nearly linear relation in J Initial model did not include green and violet two series, but predicted exactly where they should be when they were measured NNOV nt = 0, DJ = 0 K = 3  -2 NNOV nt = 0, DJ = 0 K = -2  1 K = -2, -1, 0, 1, 2, 3 J = 13 - 34 55 data pts

HF Doublet Splittings in kHz Q (K = 0 ¬ -1) Q (K = -1 ¬ 0) CH3OH CH3CHO decreasing 3 times Go higher J? CH3CHO using ab initio ABC three parameters Red: doublet splitting, Green: Intensity CH3CHO HF splittings are about 3 times smaller than that of CH3OH For CH3CHO, we might need to go higher J in order to see observable splittings, except small intensity drops down even more at higher J

CH3CHO: Comparison With Experiment Frequency Transition HF Splitting (kHz) Transition Branch type (MHz) nt J   J  calc obs Ka Kc  Ka Ka 105813.013 0 25 14  26 12 0.01 none +7 19  +6 21 c-type P 105963.808 0 25 15  26 13 0.02 none -7 18  -6 20 c-type P 110903.605 0 23 6  22 9 1.1 none -3 21  +4 18 b-type R 111757.490 0 32 10  31 12 2.0 none +5 28  +6 26 c-type R 108307.403 1 12 3  12 1 5.6 none -1 11  0 12 b-type Q 108521.902 1 17 5  16 7 5.4 none -2 15  -3 13 c-type R KaKc: Upper-state asymmetric-rotor quantum numbers on the left, lower-state on the right, in the notation of IK, Ref. [19]. Expt Simulated Doppler profile Expt after subtraction of Doppler profile Lorentz approximation of the observed Lamb dip

Concluding Remarks CH3OH Ab Initio C2xx C2yy C2xy MP2/6-311+G(3df,2p) 1.741 -1.697 1.715 CCSD(T)/pVTZ 1.666 -1.662 1.665 CCSD(T)/aug-cc-pVTZ 1.660 -1.673 1.668 Expt 1.748 -1.732 1.547 Spin-rotation constants can be calculated reasonably well by ab initio even at MP2 level Ab initio CH3CHO spin-rotation constants are about 3X smaller than that of CH3OH values, consistent with the ratio of their rotational constants CH3CHO theoretical results of six E doublet splittings agree well with the absence of observed splittings or broadenings in Lamb-dip traces for these lines CH3OH CH3CHO c2xx 1.741 0.581 c2yy -1.697 -0.616 c2xy -1.715 -0.596 nt = 2, E 62 ¬ 52

HF Doublet Splittings in kHz small splitting, good intensity “good” splitting, small intensity CH3CHO using ab initio ABC three parameters, vt=0, (C2yy C2xx C2xy) Black: doublet splitting, Red: Transition frequency, Green: Intensity small splitting, good intensity

Level HF Splittings in kHz (C2yy C2xx C2xy) CH3CHO nt = 0 t is an index in order of energy J t = 1 2 3 4 5 6 7 21 8.557 8.950 0.297 2.840 4.999 0.152 3.121 20 7.928 8.407 0.796 2.523 4.799 0.167 2.616 19 7.294 7.873 1.232 2.229 4.546 0.164 2.147 nt = 1 21 6.531 6.760 1.731 0.401 5.651 0.736 2.923 20 6.000 6.267 2.132 0.091 5.511 0.526 2.311 19 5.469 5.778 2.482 -0.173 5.308 0.348 1.778 nt = 2 21 6.547 8.835 -0.114 7.349 0.848 2.659 0.148 20 5.946 8.408 0.277 7.010 0.903 2.371 0.219 19 5.371 7.988 0.607 6.593 0.953 1.934 0.176 CH3OH nt = 0 J K = -1 0 +1 +2 -2 +3 -3 21 34.176 18.007 1.274 3.474 21.572 0.031 0.716 20 32.723 16.307 1.196 3.049 19.450 0.029 0.563 19 31.246 14.630 1.116 2.565 17.290 0.025 0.438 Selection rules: trsA1 « trsA2 and DF = DJ CH3CHO using ab initio ABC three parameters

Spin-Rotation Coupling Constants Reference: Coudert et al. J. Chem. Phys. 143 (2015) 044304 bg = xx, yy, zz, xz z/x/y => a/b/c nuclear spin - molecular rotation tensor [C] (MHz) Compute S-R tensor using G09 bg = xy and yz Three coefficients (C2xx C2yy C2xy) were used in fitting methanol expt. doublet splittings

CH3OH vs CH3CHO E Spin-Rotation Constants (kHz) (ab initio calculation at MP2 level) CH3OH + higher order CH3CHO + higher order c1zz 0.111253(56) c2zz -0.205173(56) c1zz 2.63715(92) c2zz 0.80585(92) c1yy -0.02318(63) c2yy -1.69667(63) c1yy 0.0199(11) c2yy -0.6157(11) c1xx 0.01489(66) c2xx 1.74144(66) c1xx -0.47088(61) c2xx 0.58077(61) c1xy -0.0212(21) c2xy -1.7149(21) c1xy 0.1687(20) c2xy -0.5962(20) c1xz 2.95364(32) c2xz 0.06895(32) c1xz -1.00712(94) c2xz -0.88136(94) c1yz -3.23302(32) c2yz -0.11558(32) c1yz 1.5984(10) c2yz 1.0478(10) Rotation Constants CH3OH CH3CHO A 4.2537 1.8849 B 0.8102 0.3487 C 0.7925 0.3032 r 0.8102 0.3286 F 27.6468 7.5671 Coudert’s xx,yy,zz Follow our CH3OH approach, we have used three coefficients (C2xx C2yy C2xy) to calculate hyperfine doublet splittings for CH3CHO

trsA1 = tsA1rA1 or tsA2rA2 Physics Considerations for CH3OH Nuclear spins: IC = IO = 0 and IH = ½ CH3OH has no nuclei with electric quadrupoles The four H nuclei are all magnetic dipoles Where are the magnets & magnetic fields? CH3: I1, I2, I3 and OH: I4 Overall rotation J (Jx, Jy, Jz) Internal rotation Pg Group-theoretically allowed spin-rotation interaction operators srA1 = sA1rA1 or sA2rA2 Skip many details Þ Torsion-Mediated Spin-Rotation Operators Deg. E proton spin operators (IE  I1 + e∓i2/3 I2 + e±i2/3 I3) E torsional function (e±ig are tE±) Overall rotation (rA1 or rA2) O H4 H2 H3 C H1 Requiring the E torsional wavefunction which agrees with doublets observed in E species HFS in E species, J dependent Spin-Spin: (43)9/2 = 54 Spin-Rotation: 123 = 36 Spin-Torsion (Pa): 121 = 12 trsA1 = tsA1rA1 or tsA2rA2 sEtE = tsA1 + tsA2 + tsE

A schematic display of the torsion-mediated spin-rotation operators. Fitting Hamiltonian A schematic display of the torsion-mediated spin-rotation operators. <tsrA1|Hsr(-)|tsrA1> = - < tsrA2|Hsr(-)|tsrA2> IEA1: tsA1 IEA2: tsA2 Skip many details (group theory, operator equivalents, lab-fixed axes  molecule-fixed axes), and jump to the quantitative treatment, fit and prediction. torsionally mediated spin-rotation operators, by the slightly cumbersome notation IEA1 and IEA2, where the first subscript E is to remind us that they are related to the nuclear-spin operators of species E in Eq. (2), while the additional subscripts A1 and A2 indicate their actual symmetry in G6 H_sr(+): The matrix element of this operator on both sides of the equality in Eq. (15a) is +(1/2)C1[F(F+1) – J(J+1) – I(I+1)]. (Details of the calculations in this section are given in the supplementary material.) Matrix elements of the type shown in Eq. (15a) cannot explain the observed splitting patterns under discussion here unless unexpectedly large changes (of the order of 50 or 100 kHz) are postulated for the C1 values in the upper and lower states of the transition. See supplemental material at [URL will be inserted by AIP] for more details on the calculations in this paper.   The other 3 are expected to be small: Jz2 low K {Jx, Jz} small Dab & Dbc {Jy, Jz} CH3OH vt=012 fit Dab=-0.0038095(38) Dbc=0.538(12)x10-3 Dac=5.177(29)x10-2 torsional variation of the torsional function e±nig

Part II - Physics Considerations for CH3OH Nuclear spins: IC = IO = 0 and IH = ½ CH3OH has no nuclei with electric quadrupoles The four H atoms are all magnetic dipoles Where are the magnets & magnetic fields? Magnets: Magnetic fields: CH3: I1, I2, I3 and OH: I4 Overall rotation J (Jx, Jy, Jz) (12 Cartesian components) Internal rotation Pg How many bilinear operators with at least one I component? O H4 H2 H3 C H1 Spin-Spin: (43)9/2 = 54 Spin-Rotation: 123 = 36 Spin-Torsion (Pa): 121 = 12 Without symmetry: there are 102 (= 54 + 36 + 12) bilinear operators with at least one I component. Without including torsional variation spin-spin, spin-rotation, spin-torsion With symmetry: consider one J & one I (I1x Jy, etc.). We keep no spin-spin, no spin-torsion, only spin-rotation.

Rotation: Jy is rA1 and Jx , Jz are rA2 Group-Theoretically Allowed Spin-Rotation Interaction Operators srA1 = sA1rA1 or sA2rA2 Rotation: Jy is rA1 and Jx , Jz are rA2 Spin: (I1 , I2 , I3 ), I4 12 proton spin operators I4y is sA1 and I4x , I4z are sA2 5 operators Easy part = non-degenerate A spin operators: not helpful! Iy is sA1 and Ix , Iz are sA2 (IA  I1 + I2 + I3) 5 operators O H4 H2 H3 C H1 I1, I2, I3 regrouping: Using constant coefficients chosen from the three cube root IE should take minus-plus and plus-minus signs torsion-mediated spin-rotation operators Summary on symmetry: spin-spin and spin-torsion operators are ignored spin-rotation operators are divided into two types: srA1 from sA1rA1 or sA2rA2 10 terms Spin-Rotation strA1 from tsA1 rA1 or tsA2 rA2 9 terms Torsion-Mediated Spin-Rotation (Þ 6 for DJ = 0 hf mixing) The red part gives us hope. It involves tE functions, which don’t exist in “ordinary” molecules, and which therefore might lead to some unexpectedly large splittings in trE states in methanol. ots of unity Hard part = degenerate E spin operators (IE  I1 + e-i2/3 I2 + e+i2/3 I3) (IE+)x  (I1 + e-i2/3 I2 + e+i2/3 I3)x (IE)x  (I1 + e+i2/3 I2 + e-i2/3 I3)x (IE+)y  (I1 + e-i2/3 I2 + e+i2/3 I3)y (IE)y  (I1 + e+i2/3 I2 + e-i2/3 I3)y (IE+)z  (I1 + e-i2/3 I2 + e+i2/3 I3)z (IE)z  (I1 + e+i2/3 I2 + e-i2/3 I3)z This is where we introduce the torsional variation, because the torsional function e±ig are tE± sEtE = tsA1 + tsA2 + tsE one tsA1 and one tsA2 torsion-spin operator from each sE pair (9 terms) Cannot get srA1