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Missing short-range interactions in the hydrogen bond of compressed ice

Combining the Lagrangian-Laplace mechanics and the known pressure dependence of the length-stiffness relaxation dynamics, we have determined the critical, yet often-overlooked, short-range interactions in the hydrogen bond of compressed ice. This approach has enabled determination of the force constant, cohesive energy, potential energy of the vdW and the covalent segment at each quasi-equilibrium state as well as their pressure dependence. Evidencing the essentiality of the inter-electron-pair Coulomb repulsion and the segmental strength disparity in determining the asymmetric H-bond relaxation dynamics and the anomalous properties of ice, results confirmed that compression shortens and stiffens the OH bond and meanwhile lengthens and softens the covalent bond.

Water and ice has attracted much attention because of its anomalous performance relating to issues from galaxy to geology, climate, biology, and to our daily lives [1][2][3][4][5][6][7]. As the building unit, the hydrogen bond (O:H-O) [8] relaxes in different manners under the change of environment conditions, which determines the anomalous properties of water and ice. Contributions have been made experimentally [9][10][11], computationally [12][13][14][15][16], and theoretically [17,18] to the understanding of water and ice based on the polarizable or non-polarizable models [19,20], including the TIPnP (n varies from 1 to 5) series [19][20][21][22][23]. Using ab initio density functional theory and molecular dynamics calculations, one is able to reproduce some of the anomalies demonstrated by compressed ice with limited knowledge about the nature of the inter-and intra-molecular interactions [24].
The objective of this work is to explore analytically the energy relaxation dynamics of the segmented O:H-O bond of ice under compression based on the Lagrangian-Laplace mechanics [25][26][27]. With the known length-stiffness relaxation dynamics of the O:H-O bond under compression [3][4][5][6] as input, we have been able to determine the force constants, the potential well depths, and the cohesive energies of each part of the O:H-O bond as well as their pressure dependence.
A linear hydrogen bond is assumed for simplicity because the O:H-O bond angle in ice is valued at 170 ± 4° [28]. By averaging the surrounding background interactions of H 2 O molecules and protons and the nuclear quantum effect on fluctuations [29], we focus on the short-range interactions in this O:H-O bond with H being the coordination origin. As illustrated in Figure 1, the van der Waals force is limited to the O:H bond (denoted L) [30], the exchange interaction is within the H-O polar-covalent bond (denoted H) [31], and the Coulomb repulsion (denoted C) applies between the electron pairs attached to the adjacent oxygen ions, see Supplementary information (SI) [32].
Because of the short-range nature of the interactions, only the solid lines in the shaded area in The Lagrangian L = T -V consists of the total kinetic energy T and the total potential energy V. Q i denotes the generalized non-conservative forces. Here, it is the pressure f P . The time-dependent represents the generalized variables, denoting the deviating displacements from the equilibrium position of the springs L and H here, i.e. u L and u H . The kinetic energy T consists of two terms, as the H is taken as the coordination origin, The potential energy V is composed of three terms [32]: the van der Waals interaction , and the Coulomb repulsion Here, d C0 = d L0 +d H0 is the distance between the adjacent oxygen ions at equilibrium. d C = d L +d H denotes that distance at quasi-equilibrium. u C = u L +Δ L -u H +Δ H shows the change of the distance between the two oxygen ions at quasi-equilibrium. The u L and u H are assumed to be of the opposite sign because of the O:H and H-O dislocate in the same direction [24]. A harmonic approximation of the potentials by omitting the higher-order terms in their Taylor's series yields, where V x (d x0 ), commonly denoted E x0 , is the potential well depths (n = 0 terms) of the respective bond. Noting that the Coulomb potential never has an equilibrium point where the repulsion force is 0, we can then expand this potential at quasi-equilibrium point. Therefore, the terms of n = 1 is the force equaling 0 for the L and H segments at equilibrium, while equaling C C u V ′ − for the C spring at quasi-equilibrium. Here, C V ′ denotes the first order derivative at the quasi-equilibrium position, i.e.
( ) . The coefficients of the n = 2 terms, or the curvatures of the respective potentials, denote the force constants, i.e., for harmonic oscillators. The n ≥ 3 terms are the high-order nonlinear contributions that are insignificant, as it will be shown.
Substituting Eqs (2) and (3) into (1) leads to the coupled Lagrangian equation, A Laplace transformation [32] turns out solutions to Eq (4),  The k L (ω H ) and the k H (ω L ) remains almost constant.
The force constant due to Coulomb repulsion is, at equilibrium. Here, ε r is the relative dielectric constant of ice, equaling to 3.2. ε 0 = 8.85×10 -12 F/m, is the vacuum dielectric constant. The ： q = 2e for the electron lone pair, andq = 0.2e or so, is the effective charge referring to our density functional theory optimizations [32]. In this situation, the k C equals to 0.17 eV/Å 2 at 0 GPa. The ω L and ω H dependence of the k L and the k H , in Figure 2, shows that the k L increases from 1.44 to 5.70 eV/Å 2 while the k H increases from 21.60 to 42.51 eV/Å 2 with their respective frequency.
The k L (ω H ) and the k H (ω L ) remains, however, almost constant. Therefore, Eq. (6) can be simplified as, With the measured ω L = 237.42 cm -1 and ω H = 3326.14 cm -1 for the ice-VIII phase under the atmospheric pressure [3][4][5][6], Eq (7) [3][4][5][6]24]. At 60 GPa, the k L = 10.03 eV/Å 2 and k H = 11.16 eV/Å 2 , the E L recovers slightly, see Table   1. Results indicate that the nature of the interaction within the segment remains though the length and force constant approaches to equality, which means that the sp 3 -hybridized oxygen could hardly be de-hybridized by compression.     [1]. This allows the inclusion of the ultra-short-range interactions discussed here.

Missing short-range interactions in the hydrogen bond of compressed ice
(2) Electron pair repulsion DFT derived charge distribution in the cross-section of a unit cell of the ice-VIII phase [2]. Red colour represents for electron pairs and the blue for O 2cores. The electron pairs are strongly and eccentrically localized.

Potentials for the H-bond ultra-short-range interactions
The short-range interactions include the van der Waals force limited to the O:H bond [3], the exchange interaction in the H-O polar-covalent bond [4], and the Coulomb repulsion between the lone and shared electron pairs attached to the oxygen ions.
S1) where V L0 and V H0 , commonly denoted E L0 and E H0 , are the potential well depths of the van der Waals bond and the covalent bond, respectively. r x and d x0 (x= L, H, and C) denote the interionic distances (corresponding the length of the springs) at arbitrary position and at equilibrium, respectively. a is a parameter controlling the width of the potential well. e r is the relative dielectric constant of ice, equaling to 3.2. e 0 = 8.85×10 -12 F/m, is the vacuum dielectric constant. ： q andq denote the charges on respective oxygen ion in two segmented bonds. With the known Coulomb potential and the measured length-stiffness relaxation parameters [2], the L-J and Morse potentials may be mathematized. Table S1 lists the expressions of the zeroth-to third-derivative of the Taylor series for L-J and Morse potentials. Table S2 gives the corresponding values (energies) of the zeroth-to third-order items evoluting with the pressure. It confirms that the harmonic approximation is suitable because the 3rd item is much smaller than the 2nd item. Table S1 The zeroth-to third-derivative of the L-J and Morse potentials Derivative L-J potential Morse potential

Lagrangian-Laplace solution
With the Lagrangian approximation, the vibration equations for O:H-O hydrogen bond can be deduced as shown in Eq.(4) in the main text.
Letting ( ) . These parameters denote the vibrational amplitudes. An inverse Laplace transformation of Eq. (S4) results in Eq.(5) in the main manuscript, and the correlation between the frequency and force constants:  [2].

H-O covalent bond energy[2]
From the P-dependent critical temperature T C for the ice VII-VIII phase transition [2], we can obtain the H-O bond energy. The relative shift in ∆T C (P) = T C (P 0 ) -T C (P), H-O bond length (d H ) and energy (E H ) are correlated as follows [2]: where l can take either form of the l L or l H expressed in Fig. S2 caption.
As P>P 0 , only the segment 0 d d H > p l satisfies the measured T C -P relation that shows the pressure-depressed T C . Assuming s 0 = π×(0.053 nm) 2 (the cross-section area of the H-O bond), we obtained E H0 = 3.97 eV from fitting to the measured pressure-dependent T C for VII-VIII phase transition [2].

Pressure-dependent force constants and cohesive energies
The Lagrangian-Laplace derived force constants and cohesive energies can be formulated as: