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.
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.
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.