Table 2.
Fourier expansion coefficientsa for ab initio calculated elements of the symmetry-adapted spin-rotation hyperfine tensor in kHz for the methyl and hydroxyl hydrogensb in CH3CHO.
| Ea,Ebc | C1 | C2 | C4 × 101 | C5 × 101 | C7 × 102 | C8 × 102 | σ × 1033 |
|---|---|---|---|---|---|---|---|
| H654aa | −2.668541(60) | 0.864234(60) | −1.08354(60) | 0.44347(60) | −0.8347(60) | 0.3299(60) | 0.5 |
| H654bb | 0.4481511(98) | 0.5711596(98) | −0.211404(95) | 0.091996(95) | −0.16789(96) | 0.04523(96) | 0.1 |
| H654cc | 0.0056904(59) | −0.6150126(73) | 0.269017(63) | −0.131423(93) | 0.18017(90) | −0.07179(64) | 0.07 |
| H654ab | 1.046126(26) | −0.902223(28) | 0.55623(26) | −0.34093(27) | 0.4585(27) | −0.2548(26) | 0.2 |
| Ea,Ebd | C1 | C2 | C4 × 101 | C5 × 101 | C7 × 102 | C8 × 102 | σ × 103 |
| H654bc | −0.1988570(78) | −0.5978437(72) | 0.233788(74) | −0.111522(74) | 0.16445(72) | −0.05854(76) | 0.07 |
| H654ac | −1.546908(19) | 1.006859(19) | −0.69420(22) | 0.39498(22) | −0.5217(19) | 0.2713(19) | 0.08 |
| A1e | C0 | C3 × 101 | C6 × 102 | C9 × 103 | σ × 103 | ||
| H654aa | −4.4080943(43) | −0.306366(60) | −0.39178(60) | −0.5152(60) | 0.05 | ||
| H654bb | −0.60920230(46) | 0.0565405(56) | 0.050860(70) | 0.05137(60) | 0.005 | ||
| H654cc | −0.51296483(53) | 0.0041913(92) | 0.007482(58) | 0.03878(85) | 0.005 | ||
| H654ab | −0.33233713(57) | −0.4509870(64) | −0.064574(66) | −0.01392(77) | 0.004 | ||
| H1aa | −4.4691361(37) | −1.006584(51) | −0.46249(51) | −0.4415(51) | 0.04 | ||
| H1bb | 1.12733677(77) | −0.080874(11) | −0.07824(11) | −0.0750(11) | 0.009 | ||
| H1cc | −0.71453571(55) | −0.1215039(76) | −0.018540(76) | 0.02373(76) | 0.007 | ||
| H1ab | 0.3546764(23) | −0.519036(32) | −0.22522(32) | −0.1980(32) | 0.03 | ||
| A2f | C3 × 101 | ||||||
| H654bc | −0.02669(24) | 0.2 | |||||
| H654ac | 0.30515(47) | 0.3 | |||||
| H1bc | Small | ||||||
| H1ac | Small |
The last two letters in symbols like H654aa, H1aa, etc. indicate tensor components labeled by the a,b,c principal axes of the molecule. Off-diagonal tensor elements labeled ab, bc, or ac are actually the averages (1/2)(ab+ba), etc. Numbers in parentheses are one standard uncertainty in the last two digits (type A, k = 1) [13]; σ is the overall standard deviation of the fit in kHz. Some expansion coefficients cannot be well determined because the corresponding spin-rotation hyperfine tensor components fall below a printout cut-off in Gaussian for some values of the internal rotation angle.
H654 and H1 indicate spin-rotation constants for the three methyl H atoms and for the hydroxyl H atom, respectively.
Spin-rotation tensor elements of species E. The Ea and Eb components are expanded as C1 cosσ + C2 cos2σ + C4 cos4σ + C5 cos5σ and – C1 sinα + C2 sin2α – C4 sin4α + C5 sin5α, respectively.
Spin-rotation tensor elements of species E. Ea and Eb components are expanded as C1 sinα + C2 sin2α + C4 sin4α + C5 sin5α and C1 cosα – C2 cos2α + C4 cos4α – C5 cos5α, respectively.
Spin-rotation tensor elements of species A1, with Fourier expansions in C3n cos3nα.
Spin-rotation tensor elements of species A2, with Fourier expansions in C3n sin3nα.