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Vibrational modes of ammonia
The molecular symmetry of ammonia is C3v. The number of vibrational modes can be found by using the formula 3N-6 for non-linear molecule. So it has six vibrational modes. It has E, C3 and σv symmetry operations. The motion of the nitrogen atom and the three hydrogen atoms in relation to one another give birth to these modes. All three hydrogen atoms travel symmetrically along the molecule's axis, either in the direction of the nitrogen atom or away from it. This mode is known as symmetric stretch (v₁). The symmetry in the N-H bond stretching is reflected in this stretching motion. Of the three vibrational modes, this one has the highest frequency. In Bending (ν₂) configuration, the nitrogen atom stays still while the three hydrogen atoms move in different directions from one another. Changes in the bond angles result from this action, which includes the molecule bending out of plane. Because of how the hydrogen atoms move like an umbrella, this mode is often referred to as the "umbrella mode". Asymmetric Stretch (ν₃) is also named as asymmetric stretching mode. While one atom approaches the nitrogen atom, two hydrogen atoms move apart in this mode.

Ammonia has four atoms. Each atom is associated with three vectors. NH3 has three irreducible representation A1, A2 and E. Total modes of vibration includes the vibrational, rotational and translational modes.

Total modes = 3A1 + A2 + 4E

Rotational modes = A2 + E

Translational modes = A1 + E

Vibrational modes = Total modes - Rotational modes - Translational modes = 3A1 + A2 + 4E - A2 - E - A1 - E = 2A1 + 2E

Miscellaneous possible references
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Re NMR coupling constant sign: ADB + Lovering 1962: see ADB article

Re NMR coupling constant sign:

Re NMR coupling constant sign:

Re NMR coupling constant sign:

Re NMR coupling constant sign: A. D. Buckingham, E. E. Burnell, C. A. de Lange and A. J. Rest (1968) Molecular Physics 14:2, 105-109, "N.M.R. studies of 3,3,3-trifluoropropyne dissolved in different nematic liquid crystals" https://www.tandfonline.com/doi/abs/10.1080/00268976800100111

Re NMR coupling constant sign: T. R. Krugh and R. A. Bernheim (1970) "Anisotropies and Absolute Signs of the Indirect Spin–Spin Coupling Constants in 13CH3F" J. Chem. Phys. 52, 4942 (1970); https://doi.org/10.1063/1.1672729 Ref.3 = Bernheim and Lavery 1966

Re NMR coupling constant: Roger S. Macomber (1998) "A Complete Introduction To Modern NMR Spectroscopy" Chap.9 Some factors that influence the sign and magnitude of J: Second-order (strong) coupling effects" https://archive.org/details/ACompleteIntroductionToModernNMRSpectroscopyByRogerSMacomber/page/n145/mode/2up

Re NMR coupling constant sign:

Re NMR coupling constant sign: Gerhard Hägele (2017) Some introductory remarks to “In memoriam of Prof.Harry R. Hudson”, Phosphorus, Sulfur, and Silicon and the Related Elements, 192:6, 578-580, https://doi.org/10.1080/10426507.2017.1300034 quote = "the sign of coupling constant 2JPH was settled by double resonance"

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Re Paul Martin Sr. and polio: https://www.theglobeandmail.com/canada/article-rhesus-monkeys-pizza-boxes-and-armed-robbery-canadas-history-shows/, https://www.pembrokeobserver.com/news/local-news/paul-martins-struggle-to-introduce-a-polio-vaccination

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Et en français:

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Re Wilbur K. Howard: