I am currently studying the textbook Infrared and Raman Spectroscopy, 2nd edition, by Peter Larkin. In a section entitled Symmetry: Infrared and Raman Active Vibrations, the author says the following:

In a molecule with a center of symmetry, vibrations that retain the center of symmetry are IR inactive and may be Raman active. Such vibrations, as shown in Fig. 2.12, generate a change in the polarizability during the vibration but no change in a dipole moment. Conversely, vibrations that do not retain the center of symmetry are Raman inactive, but may be IR active since a change in the dipole moment may occur.

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For molecules without a center of symmetry, some vibrations can be active in both the IR and Raman spectra. Molecules that do not have a center of symmetry may have other suitable symmetry elements so that some vibrations will be active only in Raman or only in the IR. Good examples of this are the in-phase (symmetric) stretches of inorganic nitrate and sulfate shown in Fig. 2.13. These are Raman active and IR inactive. Here, neither molecule has a center of symmetry but the negative oxygen atoms move radially simultaneously resulting in no dipole moment change. Another example is the 1,3,5 trisubstituted benzene where the C-Radial in-phase stretch is Raman active and IR inactive.

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What is the precise meaning of "in-phase" (and "out-of-phase") in this context? I understand the concept of phase in the context of waves (physics), but their use here, as related to molecules and vibrational spectroscopy, is unclear to me. I would greatly appreciate it if people would please take the time to explain this, including the associated mathematics, so that I have a more substantive understanding of what this means in this context.

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    $\begingroup$ It is just a different way of calling symmetric (in phase) and asymmetric (out of phase) vibration modes. One may see "in phase" vibrations this way: take in every moment an t-y(n) plot, where t is time, and y1,y2,yn is the bond length between the central atom and the surrounding ones. In a in-phase (Symmetric) vibration mode, the approximately harmonic vibrations for the plotted bonds show the same phase (ie: maxima at the same t, minima at the same t) $\endgroup$ – The_Vinz May 16 at 18:10
  • $\begingroup$ @The_Vinz What you're describing is not clear to me. $\endgroup$ – The Pointer May 16 at 18:50
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    $\begingroup$ Take $\ce{CO2}$ as an example. The symmetric vibration has the $\ce{C=O}$ bonds moving in-phase with each other: when one stretches out, the other stretches out at the same time. For the antisymmetric stretch, it is the opposite, so as one bond stretches out, the other compresses. $\endgroup$ – Tyberius May 16 at 20:40
  • $\begingroup$ @Tyberius Ahh, so that's all it means? Sounds simpler than I thought. $\endgroup$ – The Pointer May 16 at 20:47
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    $\begingroup$ Yes, at a physical level it's just that simple. The_Vinz gives a pretty decent overview of the math and what the phase is actually referring to (the offset in time between when one bond reaches it max/min length and when the other does). $\endgroup$ – Tyberius May 16 at 20:54

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