Fig. 1. Mode-specific dynamic contribution of zero-point vibration to intermolecular H-bond length. Left panel shows vibration schemes of twelve modes together with their approximate vibration periods reproduced from dynamic simulations (left corners, unit is fs) and calculated fundamental frequencies (italic figures at lower right corners, in unit cm⁻¹). Right panel indicates variance of R_O…H when all modes are vibrating with ZPVE and when only one of twelve modes is vibrating with its ZVPE. Curves for single modes are plotted within one corresponding vibration period, and different colour curves correspond to different colours in left figure (top right corner of each mode). The black curve for all modes is plotted within the greatest vibration mode period. Contributions from individual modes are preliminarily classed into three groups, given they might respectively dominate three geometric determinants for R_O…H as indicated by the water-dimer model in the right panel. Colour codes as red, green and blue are correspondingly assigned, and coloured horizontal and vertical bars in the right panel indicate approximate temporal range and spatial effects on R_O…H of modes in each group. Dash line crossing the vertical bars indicates equilibrium H-bond length.

Fig. 2. Blending effect of modes in each group on H-bond geometrical determinants and R_O…H variance as well as corresponding cumulative time-averaging effects. (a) Comparison of R_O…O fluctuation induced by modes 2 and 4 with that by all ZPVs, and their geometric projections on R_O…H with other two determinants keeping in equilibrium constants. (b) Comparison of ∠O⋯O–H fluctuation induced by modes 5, 6, 7, and 8 with that by all ZPVs, and their geometric projection on R_O…H after subtracting the R_O…O dynamic influence and keeping R_O–H in equilibrium constant. (c) Comparison of R_O–H fluctuation induced by modes 9 and 11 with that by all ZPVs, and their geometric projection on R_O…H after subtracting both R_O…O and ∠O⋯O–H dynamic influences. Sum curve sums up data in sole mode case while collective curve takes data from another coupled simulation where all highlighted modes are simultaneously excited. Grey background curves in panels (a), (b), and (c) are for all-ZPV ΔR_O…H respectively induced by variances of all determinants, without only R_O…O variances and without R_O…O or ∠O⋯O–H variance. (d) Curves coloured in red gradation are extrapolated estimations of the cumulative time-averaged R_O…O from modes 2 and 4, when initial vibration phase of two modes is set to be 0, π/2, π, and 3π/2, separately, to form 16 phase combinations. The dotted line and value in brown are for referencing of the experimental results. The grey line and the value in parentheses are from a faster all-ZPV simulation adopting smaller triple-ζ basis sets (TZ) for estimating the convergence behaviour. Curves coloured in blue gradation is for R_O–H from modes 9 and 11. Black curves and approximately converged values are for all-ZPV case and R_O…H temporal domains sensitive to R_O…O and R_O–H variances are shaded by corresponding colours.

Fig. 3. All-ZPV real-space distributions of LCPs between water molecules and fluctuations of electron density at LCPs (ρ_C) as well as total energy density at LCPs (H_C) with respect to R_O…H. For all snapshots, ∇²ρ_C > 0 is preserved. Crossing of dashed lines denote the equilibrium ρ_C and H_C. Correspondingly divided regions shaded in yellow are for possibly strengthened H-bonds with both shorter R_O…H and larger or both shorter R_O…H and more negative H_C. Grey shaded regions are for possibly weekend ones in the opposite conditions. Inset shows real-space distribution of H-bond LCPs during vibration and qualitatively describes its time evolution. Model of equilibrium water dimer is drawn and the equilibrium bond path connecting H-bonding oxygen and hydrogen atomsis denoted by grey dotted line. The black dot denoting BCP_Eq is the H-bond BCP in equilibrium conformation. During plotting LCPs, oxygen atom in proton–donor molecule is fixed, while the line connecting the two oxygen atoms and the plane determined with the extra upright hydrogen atom in the proton–donor molecule coincide for all snapshot views. As shown by the colour triangle, all data points are coloured according to contributions to ΔR_O…H from dynamic variances of three geometric determinants. There are 315 data points for each distribution pattern.

Fig. 4. All-ZPV fluctuations of interaction energy and decomposed components in water dimer with respect to simulation time and R_O…H. Left panel shows values of each term and percentages of E_ster, E_ind, and E_disp in E_int, with respect to simulation time. The horizontal black dashed line in upper part denotes the equilibrium E_int. Term percentages at equilibrium structure (0 fs) and two other exemplified moments with E_int equal to equilibrium value aremarked and their positions are highlighted by vertical dotted lines. Region with slant lines in the left panel is for repulsive effect, and is respectively indicated by positive energy values and negative percentages. Right panel shows term distribution with respect to R_O…H. Dashed lines within are for artificial manipulation to change R_O…H under frozen monomeric geometries and H-bond angle.