1. 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.

2. “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) , L.H. Coudert, et al. “Spin-torsion effects in the hyperfine structure of methanol”.
[B] J. Chem. Phys. 145 (2016) , 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) , 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

3. 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 ( 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)
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

4. 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) [D]
CCSD(T)/pVTZ [A]
CCSD(T)/aug-cc-pVTZ [C]
Experimental Values [B]
Ref. [A] J. Chem. Phys. 143 (2015) , Coudert, et al. [B] J. Chem. Phys. 145 (2016) , Belov, et al. [C] J. Chem. Phys. 145 (2016) , Lankhaar, et al. [D] Present work
Spin-rotation constants can be calculated reasonably well by ab initio
CCSD(T)/aug-cc-pVTZ [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

5. 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)
MP CH3OH CH3CHO Ratio
C2xx
C2yy
C2xy
Rotation Constants (cm-1)
CH3OH CH3CHO A B C
We have used three coefficients (C2xx C2yy C2xy) to calculate hyperfine doublet splittings for CH3CHO F r
Coudert’s xx,yy,zz

6. CH3OH: Obs. & Calc. Doublet Splittings
JCP 145 (2016)
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 =
55 data pts

7. 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

8. CH3CHO: Comparison With Experiment
Frequency Transition HF Splitting (kHz) Transition Branch type
(MHz) nt J   J  calc obs Ka Kc  Ka Ka
 none  c-type P
 none  c-type P
 none  b-type R
 none  c-type R
 none  b-type Q
 none  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

9. Concluding Remarks CH3OH
Ab Initio C2xx C2yy C2xy
MP2/6-311+G(3df,2p)
CCSD(T)/pVTZ
CCSD(T)/aug-cc-pVTZ
Expt
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
c2yy
c2xy
nt = 2, E
62 ¬ 52

10. 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

11. Level HF Splittings in kHz
(C2yy C2xx C2xy)
CH3CHO
nt = 0 t is an index in order of energy
J t =
nt = 1
nt = 2
CH3OH nt = 0
J K =
Selection rules:
trsA1 « trsA2 and DF = DJ
CH3CHO using ab initio ABC three parameters

12. Spin-Rotation Coupling Constants
Reference: Coudert et al. J. Chem. Phys. 143 (2015)
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

13. CH3OH vs CH3CHO
E Spin-Rotation Constants (kHz) (ab initio calculation at MP2 level)
CH3OH + higher order CH3CHO + higher order
c1zz (56) c2zz (56) c1zz (92) c2zz (92)
c1yy (63) c2yy (63) c1yy (11) c2yy (11)
c1xx (66) c2xx (66) c1xx (61) c2xx (61)
c1xy (21) c2xy (21) c1xy (20) c2xy (20)
c1xz (32) c2xz (32) c1xz (94) c2xz (94)
c1yz (32) c2yz (32) c1yz (10) c2yz (10)
Rotation Constants
CH3OH CH3CHO A B C r F
Coudert’s xx,yy,zz
Follow our CH3OH approach, we have used three coefficients (C2xx C2yy C2xy) to calculate hyperfine doublet splittings for CH3CHO

14. 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: I 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

15. 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:
Jz low K
{Jx, Jz} small Dab & Dbc
{Jy, Jz}
CH3OH vt=012 fit
Dab= (38)
Dbc=0.538(12)x10-3
Dac=5.177(29)x10-2
torsional variation of the torsional function e±nig

16. 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: I 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 (= ) 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.

17. 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 ), I proton spin operators
I4y is sA1 and I4x , I4z are sA operators
Easy part = non-degenerate A spin operators: not helpful!
Iy is sA1 and Ix , Iz are sA2 (IA  I1 + I2 + I3) 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