Controls
Live readouts
Numerical checks
The science
Atoms of a molecule vibrate about their equilibrium positions. In the harmonic approximation the potential energy is a quadratic form in the internal coordinates (two O-H bond lengths and the H-O-H angle) with force constants k_r, k_rr, k_alpha, k_r_alpha. Transforming to Cartesian coordinates gives a Hessian H = B^T F B; the mass-weighted Hessian M^(-1/2) H M^(-1/2) has eigenvalues omega^2 and eigenvectors that are the normal modes (Wilson GF method). Wavenumber is nu = omega / (2 pi c).
Water has three modes: symmetric stretch, bend and asymmetric stretch. All change the molecular dipole, so all are infrared active. Heavier D lowers the stretch wavenumbers by about 1/sqrt(2) and the bend slightly less because the oxygen also moves.
Simplifications: harmonic model with literature force constants (k_r = 845.4 N/m, k_rr = -10.1 N/m, k_alpha/r^2 = 76.1 N/m, k_r_alpha/r = 22.8 N/m). Real water is anharmonic, so the observed fundamentals (3657, 1595 and 3756 per cm) are 2 to 7 percent lower than the harmonic values shown; gas phase only, no rotation, no hydrogen bonding. The drawn displacement is exaggerated threefold for visibility and the playback is slowed by about 1e13.