A4 Lagrangian-Laplace mechanics of the O:H-O coupled oscillators.
A4.1 Derivatives for the short-range interactions
With the known Coulomb potential and the measured segmental length dx and phonon stiffness x[74],
parameters in the L-J (EL0, dL0) and the Morse (EH0, ) potentials can be mathematized. Table A1.1 lists
the expressions of the zeroth to third derivatives of the respective Taylor series. Table A1.2 gives the
corresponding values (energies) of the zeroth- to third-order items evolution with the pressure. It confirms
that the harmonic approximation is suitable because the 3rd item is much smaller than the 2nd item.
Table A1.1 The zeroth- to third-derivative of the L-J and Morse potentials
Derivative
L-J potential
Morse potential
Derivatives
Vx0(Ex0)
EL0
EH0 (3.97 eV)[81]
Ex0
Vx
0
0
dx0
Vx+ Vc
0
0
dx0+ux
Vx = kx
2
72EL0 d L0
2 2 EH0

Vx
3
 1512EL0 d L0
 6 3 EH0
Note: With the measured frequency x and calculated kc, Eq (8) in the main text gives the kx. Ex is
obtainable by the given parameters and the x and dx at each quasi-equilibrium sites.
Table A4.2 The values for the first four items of the Taylor series of the L-J and the Morse potentials.
Nonlinear contribution of the 3rd term is negligibly small.
Calculated energy (eV)
P
L-J potential
(GPa)
Morse potential
2nd
3rd
(×10 )
(×10 )
16.8102
0.1063
10
0th
1st
0
0.0625
0
5
2nd
3rd
(×10 )
(×10-3)
0.7465
0.0102
3.6447
0.6387
0.0085
0.9904
3.3859
0.5300
0.0066
2.9185
0.4391
3.1875
0.4247
0.0049
1.9033
0.2212
3.0450
0.3271
0.0034
0th
1st
10.1750
3.9700
0
8.2883
2.7002
0.1458
4.7185
15
0.1755
20
0.1919
-3
-3
-3
30
0.2477
0.6599
0.0397
2.6290
0.1880
0.0016
40
0.2498
0.2432
0.0089
2.1285
0.1022
0.0007
50
0.2165
0.0967
0.0024
1.6465
0.0581
0.0003
60
0.1605
0.0697
0.0017
1.1595
0.0626
0.0005
A4.2 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 kL  kC  mL  a , kH  kC  mH  b , kC mL  c , kC mH  d ,
kC ΔH  Δ L   VC  f P  mL  e , kC Δ H  Δ L   VC  f P  mH 
f , this equation becomes,
 d 2u L
 2  auL  cuH  e  0
 dt
 2
 d uH
 buH  duL  f  0

 dt 2
(A4.1)
Assuming that the initial displacements uL(0)=uH(0)=0, and the initial velocities duL dt
  L0 ,
t 0
duH dt t  0   H0 . Eq. (S1) can be reorganized based on Laplace transformation,


2

 s  a U L  cU H   L0  e s

2

 dU L  s  b U H   H0  f s


(A4.2)
where UL and UH are the Laplacians of the uL and uH, respectively, with


0
0
U L  U L s    uL t e  st dt , U H  U H s    u H t e  st dt
where, s is a complex variable. Introducing  L 

a  b2  4cd
a  b 2  
2 , we obtain the solutions to Eq. (S2)，
and  H  a  b  2   , where
1
1

U L  AL s 2   2  BL s 2   2

L
H

1
1
U  A
 BH 2
H
H 2
2

s L
s   H2

(A4.3)
where
AL 
AH 
c H0  b L0   L0 L2
 H2   L2
a H0  d L0   H0 L2
 H2   L2
; BL  
; BH  
c H0  b L0   L0 H2
 H2   L2
a H0  d L0   H0 H2
 H2   L2
;
.
These parameters denote the vibrational amplitudes.
An inverse Laplace transformation of Eq. (S3) results in Eq.(5) in the main manuscript, and the
correlation between the frequency and force constants:
1

2
1  1
m k  m k  m  m k  m k  m k  m  m k 2  4m m k 2  


L H
L
H C
L H
H L
L
H C
L H C 
 H, L 2πc  2m m  H L

 H L


2
k H, L  2π 2 m H, L c 2  L2   H2  k C  2π 2 m H, L c 2  L2   H2  m H, L k C2 m L, H






and,

k H, L  kC
1
H, L 
2πc
mH, L


2 2
2
 k H, L  4π c mH, LH, L  kC
(A4.4)
by delaminating H(kL) and L(kH) that make no contribution to the cross terms.