Magnetic pin-torsion Coupling in Methanol L. H. Coudert, 1 C. Gutle, 1 T. R. Huet, 2 and Jens-Uwe Grabow 3 1 Laboratoire Interuniveristaire des ystèmes Atmosphériques, UMR 7583 CNR - Universités Paris Est et Paris Diderot, Créteil, France 2 Laboratoire PhLAM, UMR 8523 CNR - Université de Lille 1, Villeneuve d Ascq, France 3 Institut für Physikalische Chemie und Elektrochemie, Gottfried-Wilhelm-Leibniz-Universität, Hannover, Germany June 24, 2015 Coudert et al. pin-torsion methanol 69 th IM 1 / 20
Overview Coudert et al. pin-torsion methanol 69 th IM 2 / 20
Overview 1 Magnetic hyperfine couplings in a non-rigid molecule Coudert et al. pin-torsion methanol 69 th IM 2 / 20
Overview 1 Magnetic hyperfine couplings in a non-rigid molecule 2 pin-torsion effects in methanol Coudert et al. pin-torsion methanol 69 th IM 2 / 20
Overview 1 Magnetic hyperfine couplings in a non-rigid molecule 2 pin-torsion effects in methanol 3 Analysis Coudert et al. pin-torsion methanol 69 th IM 2 / 20
Magnetic hyperfine coupling in 1 Σ molecules The spin-spin coupling 1 E = µ N 2 i g i j>i g j (I i I j ) 3(I i r ij )(I j r ij )/r ij 2 r ij 3 The spin-rotation coupling 1 E R = eµ N c g i I i i j i Z j r ji (v j γ i v i ) r ij 3 1 Gunther-Mohr, Townes, and van Vleck, Phys. Rev. 94 (1954) 1191 Coudert et al. pin-torsion methanol 69 th IM 3 / 20
Magnetic hyperfine coupling in 1 Σ molecules The two contributions to spin-rotation 1 v i = ω r }{{} i αλ σ }{{} i Rot. Torsion where σ i 0 if i top 1 Heuvel & Dymanus, J. Mol. pec. 45 (1973) 282 & J. Mol. pec. 47 (1973) 363 Coudert et al. pin-torsion methanol 69 th IM 4 / 20
Magnetic hyperfine coupling in 1 Σ molecules The two contributions to spin-rotation 1 v i = ω r }{{} i αλ σ }{{} i Rot. Torsion where σ i 0 if i top In methanol σ i = { rchi for i = 1 to 3 0 for i = 4 1 Heuvel & Dymanus, J. Mol. pec. 45 (1973) 282 & J. Mol. pec. 47 (1973) 363 Coudert et al. pin-torsion methanol 69 th IM 4 / 20
The 3 J = 0 magnetic coupling hyperfine operators pin-spin 1 H ss = pin-rotation 1 H sr = pin-torsion 2 H st = 4 F (I i, I j, J) ij (α) i<j 4 (I i J)E i (α) i=1 4 (I i J)G i (α) i=1 1 Thaddeus, Krisher, and Loubser, J. Chem. Phys. 40 (1964) 257 2 Heuvel & Dymanus, J. Mol. pec. 45 (1973) 282 & J. Mol. pec. 47 (1973) 363 Coudert et al. pin-torsion methanol 69 th IM 5 / 20
The 3 rovibrational coupling operators Operator Expression Method ij (α) 1 E i (α) 1 G i (α) 2 2 βγ Rij βγ (α)j βj γ J(2J 1)(J + 1)(2J + 3) βγ Ci βγ (α)j βj γ J(J + 1) {ρj z + p α, β J βd i β (α)} 2J(J + 1) tructure Ab initio First order nuclear contribution only 1 Thaddeus, Krisher, and Loubser, J. Chem. Phys. 40 (1964) 257 2 Heuvel & Dymanus, J. Mol. pec. 45 (1973) 282 & J. Mol. pec. 47 (1973) 363 Coudert et al. pin-torsion methanol 69 th IM 6 / 20
pin-rotation coupling tensor calculation Cβγ i (α), with 1 i 4, β, γ = x, y, z, and 0 α 360 with ACE2 2 at the CCD(T) 3 level with pvtz 4 basis set. 2 ACE2, tanton, Gauss, et al., 1992 3 Raghavachari, Trucks, Pople, & Head-Goreon, Chem. Phys. Lett. 157 (1989) 479 4 Dunning, J. Chem. Phys. 90 (1989) 1007 Coudert et al. pin-torsion methanol 69 th IM 7 / 20
ymmetry considerations Using the G 6 symmetry group 1 Hyperfine operators H sr = H sr (A 1 ) O sr (A 1 ) + H h sr(a 1 ) O h sr(a 1 ) + H sr (E a ) O sr (E a ) + H sr (E b ) O sr (E b ) H sr (A 1 ) = (I 1 + I 2 + I 3 ) J, Hsr(A h 1 ) = I 4 J H sr (E a ) = 1 2 (2I 1 I 2 I 3 ) J, 3 H sr (E b ) = 2 (I 2 I 3 ) J Rovibrational operators O sr (A 1 ) = 1 3 (E1 + E 2 + E 3 ), O h sr(a 1 ) = E 4 O sr (E a ) = 1 3 (2E1 E 2 E 3 ), O sr (E b ) = 1 3 (E 2 E 3 ) 1 Hougen, Kleiner, & Godefroid, J. Mol. pec. 163 (1994) 559 Coudert et al. pin-torsion methanol 69 th IM 8 / 20
Accounting for the torsion For an n non-degenerate A 1 or A 2 level 1 3 O sr (A 1 ) and O h sr(a 1 ) This leads to the hyperfine spin-rotation coupling Hamiltonian H Hfs = C sr H sr (A 1 ) + C sr,h H h sr(a 1 ) where C sr = n O sr (A 1 ) n and C sr,h = n O h sr(a 1 ) n 1 Wolf, Williams, & Weatherly, J. Chem. Phys. 47 (1967) 5101 2 Coudert & López, J. Mol. pec. 239 (2006) 135 3 Tudorie, Coudert, Huet, Jegouso, & edes, J. Chem. Phys. 134 (2011) 074314 Coudert et al. pin-torsion methanol 69 th IM 9 / 20
Accounting for the torsion For an n non-degenerate A 1 or A 2 level 1 3 imilar O sr (A results 1 ) and for Othe sr(a h 1 ) spin-torsion and spin-spin couplings. This leads to the hyperfine spin-rotation coupling Hamiltonian Doubly degenerate E-type H Hfs levels = Cshould sr H sr (Abe 1 ) treated + C sr,h Hinsr(A h 1 ) the same way. where C sr = n O sr (A 1 ) n and C sr,h = n Osr(A h 1 ) n 1 Wolf, Williams, & Weatherly, J. Chem. Phys. 47 (1967) 5101 2 Coudert & López, J. Mol. pec. 239 (2006) 135 3 Tudorie, Coudert, Huet, Jegouso, & edes, J. Chem. Phys. 134 (2011) 074314 Coudert et al. pin-torsion methanol 69 th IM 9 / 20
ummary Level ymmetry pin-rotation pin-torsion pin-spin N A 1 /A 2 E C sr & C sr,h ame + C sr A Coupling constants C st & Cst,h ame + C st A C ss Hyperfine coupling operators & C ss,h ame + C ss A 6 & Css,h A 10 A 1 /A 2 H sr (A 1 ) & H h sr(a 1 ) H ss (A 1 ) & H h ss(a 1 ) 4 E ame + H sr (E a ) & H sr (E b ) ame + H ss (E a ), 10 H h ss(e a ), H ss (E b ), & H h ss(e b ) Coudert et al. pin-torsion methanol 69 th IM 10 / 20
Hyperfine coupling constants numerical evaluation ymmetry adapted operators for spin-torsion with O st (Γ) = {ρj z + p α, β J β Γ β (α)} 2J(J + 1) A1 β (α) = { s 0 β + s 3 β cos 3α for β = x, z sin 3α for β = y s 3 β Coudert et al. pin-torsion methanol 69 th IM 11 / 20
Hyperfine coupling constants numerical evaluation ymmetry adapted operators for spin-torsion with O st (Γ) = {ρj z + p α, β J β Γ β (α)} 2J(J + 1) A1 β (α) = { s 0 β + s 3 β cos 3α for β = x, z sin 3α for β = y s 3 β O st (A 1 ) O st (E a ) O h st(a 1 ) s 0 x 7.74826 s 1 x 12.01180 s 0,h x 53.46121 s 3 x 0.14834 s 2 x 0.69876 s 3,h x 0.43484 s 3 y 0.06417 s 1 y 8.61518 s 3,h y 0.22573 s 2 y 0.15320 s 0 z 27.51335 s 1 z 2.09658 s 0,h z 59.98014 s 3 z 0.08271 s 2 z 0.51433 s 3,h z 0.18517 Coudert et al. pin-torsion methanol 69 th IM 11 / 20
Hyperfine coupling constants numerical evaluation ymmetry adapted operators for spin-torsion with O st (Γ) = {ρj z + p α, β J β Γ β (α)} 2J(J + 1) A1 β (α) = { s 0 β + s 3 β cos 3α for β = x, z sin 3α for β = y s 3 β O st (A 1 ) O st (E a ) O h st(a 1 ) 92 constants arise s 0 x 7.74826 s 1 x 12.01180 s 0,h x 53.46121 s 3 x 0.14834 s 2 x 0.69876 s 3,h x 0.43484 s 3 y 0.06417 s 1 y 8.61518 s 3,h y 0.22573 s 2 y 0.15320 s 0 z 27.51335 s 1 z 2.09658 s 0,h z 59.98014 s 3 z 0.08271 s 2 z 0.51433 s 3,h z 0.18517 Coudert et al. pin-torsion methanol 69 th IM 11 / 20
Hyperfine coupling constants values Rotation-torsion energies and eigenvectors obtained from a fit of the high-resolution data 1 Level 1 01 A 2 0.71 1.80 4.22 2.17 0.0 0.0 1 11 A 2 6.68 7.94 2.11 1.04 3.31 7.22 1 10 A 1 6.64 6.95 2.11 1.13 3.31 7.22 C sr C sr,h C ss C ss,h C st C st,h 2 02 A 1 0.71 1.80 1.01 0.52 0.0 0.0 2 12 A 1 2.71 4.18 0.50 0.27 1.11 2.40 2 11 A 2 2.68 3.19 0.50 0.25 1.10 2.40 3 03 A 2 0.71 1.81 0.47 0.24 0.0 0.0 3 13 A 2 1.72 3.24 0.35 0.19 0.55 1.20 3 12 A 1 1.69 2.25 0.35 0.18 0.55 1.20 1 Xu, Fisher, Lees, hi, Hougen, Pearson, Drouin, Blake, & Braakman, J. Mol. pec. 251 (2008) 305 Coudert et al. pin-torsion methanol 69 th IM 12 / 20
Hyperfine coupling constants values-continued C st for a J k E-type levels with J = 6 and 6 k 6. Coudert et al. pin-torsion methanol 69 th IM 13 / 20
Hyperfine coupling constants values-continued Calculated hyperfine pattern for the 1 01 A 2 0 00 A 1 transition at 48 372 MHz and the 1 0 E 0 0 E transition at 48 376 MHz 1 1 Tudorie, Coudert, Huet, Jegouso, & edes, J. Chem. Phys. 134 (2011) 074314 Coudert et al. pin-torsion methanol 69 th IM 14 / 20
Hyperfine coupling constants values-continued Calculated hyperfine pattern for the 1 01 A 2 0 00 A 1 transition at 48 372 MHz and the 1 0 E 0 0 E transition at 48 376 MHz 1 ymmetry adapted hyperfine functions are used so that the total nuclear spin-rotation-torsion function obeys the Pauli exclusion principle. 1 Tudorie, Coudert, Huet, Jegouso, & edes, J. Chem. Phys. 134 (2011) 074314 Coudert et al. pin-torsion methanol 69 th IM 14 / 20
Analysis: experimental data set Transition F pectrometer Hyp 2 11 A 2 2 12 A 1 2502.8 Ref. 1 & Lille 3 12 A 1 3 13 A 2 5005.3 Ref. 1 & Lille D 5 15 A 2 6 06 A 1 6668.5 Hannover & Lille D 9 1 E 8 2 E 9936.2 Hannover & Lille 4 32 A 1 5 23 A 2 9978.7 Lille 2 0 E 3 1 E 12178.6 Hannover & Lille D 6 15 A 2 6 16 A 1 17513.3 Ref. 1 D 2 1 E 3 0 E 19967.4 Hannover 3 2 E 3 1 E 24928.7 Hannover D 4 2 E 4 1 E 24933.5 Hannover D 2 2 E 2 1 E 24934.4 Hannover 5 2 E 5 1 E 24959.1 Hannover D 6 2 E 6 1 E 25018.1 Hannover D 1 01 A 2 0 00 A 1 48372.4 Ref. 1 M 1 Heuvel & Dymanus, J. Mol. pec. 45 (1973) 282 Coudert et al. pin-torsion methanol 69 th IM 15 / 20
Analysis results: observed & calculated hfs. patterns Coudert et al. pin-torsion methanol 69 th IM 16 / 20
Analysis results: observed & calculated hfs. patterns C sr Dominated by spin-spin coupling = 0.71 Csr,h = 1.80 C ss = 4.22 Css,h = 2.17 Coudert et al. pin-torsion methanol 69 th IM 16 / 20
Analysis results: observed & calculated hfs. patterns Coudert et al. pin-torsion methanol 69 th IM 17 / 20
Analysis results: observed & calculated hfs. patterns C sr Dominated by spin-rotation from the hydroxyl group = 1.69 Csr,h = 2.25 C ss = 0.35 Css,h = 0.18 C sr = 1.72 C sr,h = 3.24 C ss = 0.35 C ss,h = 0.19 Coudert et al. pin-torsion methanol 69 th IM 17 / 20
Analysis results: observed & calculated hfs. patterns Coudert et al. pin-torsion methanol 69 th IM 18 / 20
Analysis results: observed & calculated hfs. patterns Predicted spin-rotation splitting from the methyl group was too large Coudert et al. pin-torsion methanol 69 th IM 18 / 20
Analysis results: observed & calculated hfs. patterns Predicted spin-rotation splitting from the methyl group was too large Also dominated by spin-rotation from the hydroxyl group Coudert et al. pin-torsion methanol 69 th IM 18 / 20
Analysis results: observed & calculated hfs. patterns For lines displaying a doublet structure, it was assumed that spin-rotation from the hydroxyl group is dominant. Predicted spin-rotation splitting from the methyl group was too large Also dominated by spin-rotation from the hydroxyl group Coudert et al. pin-torsion methanol 69 th IM 18 / 20
Analysis results: parameters (khz) Parameter No fitting Fit I Fit II c 0 zz 12.626 2.094(974) 1.766(910) c 0,h zz 12.877 25.802(1060) 28.535(1055) s 0,h zz 59.980 0.0 1 69.290(12625) 12 center frequencies TD (unitless) 1.2 1.1 1 Constrained value. Coudert et al. pin-torsion methanol 69 th IM 19 / 20
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