Angular velocity autocorrelation function

Shown in Fig. 6 are the AVAFs along the x-, y-, and z-axis calculated by MD simulations. At the short-time scale, the AVAF along the y-axis decays faster than those along the x- and z-axis. Less attenuation is found in the AVAF along the z-axis and the AVAFs along the x- and z-axis are quite similar.

We also use the VP analysis in eq. (III.2) to discuss the AVAFs. In Figs. 8(a), 8(b), and 8(c), the simulated results of the AVAFs along x-, y-, and z-axis for the four VGs are shown. It is found that at the short-time scale, the AVAF becomes faster decaying with increasing the asphericity. This is consistent with the blue shift in the INM rotational DOSs for different VGs.

C. Short-time behavior of

C_μ⁽¹⁾

(

t

) and

^Cμ(2)

⁽

^t

⁾

In this section, we focus on the short-time behaviors of the first and the second order orientational correlation functions (^Cμ^(l)

⁽

^t

⁾

, l = 1 or 2,

μ

= x, y, or z) for t smaller than 0.2ps. For these OCFs under the second cumulant approximation given in eq. (II.8),

ψ

(t) can be calculated by three different ways. Two of these are calculated by the total INM DOS and the real-frequency INM DOS; the other is by the power spectrum of the AVAF.

Figs. 9(a), 9(b), and 9(c) show the ^D_μeff

( ω )

^kb^T , with μ = x, y, z, of the AVAF power spectrum and those of the stable INMs for the four VGs. The shape and the area for the ^{D k}b^T

eff

( ω )

μ spectrum of the z-axis are found to be similar to those for

the spectrum of the x-axis and are larger than those for the spectrum of the y-axis. In Fig. 4, Fig. 5, the DOS of the x-axis is similar to that of the z-axis. D_μ^eff

⁽ ω ⁾

is a combination of the DOSs of the others two axes divided by the corresponding moment of inertia. This is the reason why the shapes of

D

^eff_X (

ω

)

k

_b

T

_and

T k

D

_Z^eff(

ω

) _b are similar and the area of

D

_Y^eff(

ω

)

k

_b

T

is the smallest. In addition to

the spectrum shape and area, ^{D k}b^T eff

( ω )

μ is 0 at ω = 0, but the power spectrum of AVAF is not. Also, as the asphericity of the VG increases, the value of ^{D k}b^T

eff

( ω )

μ

at ω = 0 is decreased. This difference in the spectrum values at ω = 0 makes the OCF

)

)

(

( t

C_μ^l calculated by the real-frequency INM and the power spectrum of AVAF different.

In Figs. 10 and 11, the three fitting lines in the short-time scale (within about 20 fs) can exactly fit the MD simulation results. Among these fittings, the best one for the longest time is the ^Cμ^(l)

⁽

^t

⁾

calculated by the power spectrum of AVAF, because the AVAF is calculated by the MD simulation and

D

_η(

ω

) in eq. (II.12) is obtained from the power spectrum of Ω_η(t) by the inverse Fourier cosine transformation. For this reason, the ^Cμ^(l)

⁽

^t

⁾

calculated by the power spectrum of AVAF has the longest

accurate fitting time than other. The ^Cμ^(l)

⁽

^t

⁾

calculated by the total INM DOS provides an accurate estimation within t < 20 fs, but fails at longer times. Eq. (II.15) point out that the total INM DOS is larger than that of the real-frequency INMs such that D_μ^eff

⁽ ω ⁾

in eq. (II.8) will decay faster, and the ^Cμ^(l)

⁽

^t

⁾

calculated by the real-frequency INMs decays slowly than the results of MD simulation. On the other hand, comparing the results of the x-, y-, and z-axis, we can see the four lines of

y-axis decay slowly than others, and the z-axis is fastest. Because the area of

T

k

D

_Y^eff(

ω

) _b is small than others, in eq. (II.8), (II.8-2), and (II.12), tell us when the

area of D_μ^eff

⁽ ω ⁾

increase, the decay speed of ^Cμ^(l)

⁽

^t

⁾

will increase too.

As t < 0.2ps, both the ^Cμ^(l)

⁽

^t

⁾

of the real-frequency INMs and the power spectrum of AVAF have good predictions, but after t = 0.2ps, they have different behaviors. We have discussed previously that the ^{D k}b^T

eff

( ω )

μ at ω = 0 for the real-frequency INMs and the power spectrum of AVAF are distinct. Because, when t > 0, ^Dμ^eff

^{( 0 )}

will be more important influence on ^Cμ^(l)

⁽

^t

⁾

, so as time increase, the ^Cμ^(l)

⁽

^t

⁾

of the power spectrum of AVAF will decay faster than the real-frequency INMs. Besides, comparing ^Cμ⁽¹⁾

⁽

^t

⁾

and ^Cμ⁽²⁾

⁽

^t

⁾

, the real-frequency INMs have a better qualitatively correct behavior than the power spectrum of AVAF in ^Cμ⁽²⁾

⁽

^t

⁾

, but when t < 35fs, the

power spectrum of AVAF have longer accurate fitting time either ^Cμ⁽¹⁾

⁽

^t

⁾

or ^Cμ⁽²⁾

⁽

^t

⁾

. By VP analysis, in Figs. 12(a), (b), (c) and Figs. 13(a), (b), (c), we can observe that as VG

ξ

increases, the number of bumps increases. This is very reasonable, because we know that as

ξ

increases, the average value of H-bond length decreases. The H-bond will restrict water molecular motions, like inside a cage. When the H-bond length decreases, the restricting force will be stronger and make the rebound in molecular motion occur. So, the bumps come from the reversion or palliation of rotational motion in water molecule. That is the reason why the bumps increase with

ξ

increase. From VP analysis, there is another appearance. When

ξ

increasing, the accurate fitting time will be extended in the ^Cμ^(l)

⁽

^t

⁾

of the real-frequency INMs and the power spectrum of AVAF. This may reveal that the numbers of H-bond affect the

accuracy of prediction by the real-frequency INMs.

From the above discussions, we find that the short-time behavior of ^Cμξ^(l)

⁽

^t

⁾

is a Gaussian decay within t < 20fs and, even for different VGs, can be accurately fit by the three kinds of method. When t > 0.2ps, these methods do not provide correct predictions. Therefore, we have to use another theory to describe their behaviors at the long-time scales.

D. Long-time behavior of

C_μ⁽¹⁾

(

t

) and

C_μ⁽²⁾

(

t

)

In the previous section, we use the total, real-frequency INMs and the power spectrum of AVAF to fit the short-time behavior of the OCF. But, for the long-time scales, these methods do not provide correct predictions. By eq. (II.18), the long-time behavior of ^Cμ^(l)

⁽

^t

⁾

is expected to be a single-exponential decay. Thus, we change to fitting parameters for different rotational axes and VGs in Table III and Table IV. The linear fitting function f_μ^(l)

(

t

)

for the

μ

-th principal axis and for VG

ξ

are written,

where a_μ^(l) is a constant,

τ

^D(lμ⁾ is the rotational-diffusion decay time, l is the index for the l-th order Legendre polynomial, and

D is the rotational diffusion coefficient for

_μ the

μ

-th principal axis.

In Fig. 14 and Fig. 15,

^ln

^Cμ^(l)

⁽

^t

⁾

is well fit by the linear function ^fμ^(l)

⁽

^t

⁾

so that the long-time behavior of

^ln

^Cμ^(l)

⁽

^t

⁾

is linear in time for each axis. For ^fμ⁽²⁾

⁽

^t

⁾

, although the

τ

^D(lμ⁾ values of the three axis are different but the three lines are very

close to parallel. For ^fμ⁽¹⁾

⁽

^t

⁾

, the y-axis decays faster than the x-axis instead of parallel to each other.

By VP analysis, in Figs. 16(a), (b), (c) and Figs. 17(a), (b), (c), the

^ln

^Cμξ⁽¹⁾

⁽

^t

⁾

and

)

(

ln

C_μξ⁽²⁾ t , with

μ

= x-, y-, or z-axis, also can be well fit by the linear function

)

)

(

( t

f_μξ^l for each VG

ξ

. In Table III, and IV, we can find that the changes of

ξ

_and

) (l D_μξ

τ

are not related and, thus, these lines are almost parallel. According to these

results, the long-time behavior of ^Cμξ^(l)

⁽

^t

⁾

calculated by MD simulation is an exponential decay. At the short-time scales, we know the bumps of water molecule motion increase when VG

ξ

increases. These bumps will decrease the decay speed.

Therefore, in the long-time scale, the sequence of the ^Cμξ^(l)

⁽

^t

⁾

from MD simulation in different VG

ξ

is reasonable. Moreover, these nearly parallel lines show that the decay rates in long-time scale are independent of their VGs.

In this section, we have already discussed the long-time behavior for different axes and the four VGs. We know that the long-time behavior is an exponential decay when t > 2ps. But, we find that the intermediate time scale behavior (0.2ps. < t < 2ps) is

fitting function to fit the intermediate time-scale behavior of ^Cμ^(l)

⁽

^t

⁾

and the discussions are given in the next section.

E. Intermediate time scale of

^Cμ⁽¹⁾

⁽

^t

⁾ and

^Cμ⁽²⁾

⁽

^t

⁾

Although the intermediate time behavior is not described by a single Gaussian decay or a single exponential decay, we think that the transition process from the short-time scale to the long-time scale must have parts of the residual memory.

Therefore, we use a fitting function that is composed of two kinds in mathematical formula. In Ref. [16], they use a combination of two terms of power law and multiplied by exponential function to fit the intermediate time-scale behavior of the orientational correlation function of water in the regime of supercooled liquids. Thus we assume the transition process is a power-law-like behavior.

Before constructing the fitting function in the intermediate time scale, we make

)

To analyze the intermediate-time behavior (within 0.2ps < t < 2ps), we first extend the fitting function ^Fμ^(l)

⁽

^t

⁾

for the long time behavior from the range t > 2ps to t > 0.2ps. The results for their comparison with the simulated ^Cμ^(l)

⁽

^t

⁾

are shown in the left-hand panels in Fig. 18, and Fig. 19. We find that the single exponential decay

for the long time behavior is not fit well within 0.2ps < t < 2ps. So, we define

which is a power-law function multiplied by a Gaussian function. In the fitting function,

s and

^(l)

p are parameters of the power-law function and

^(l)

τ

_i^(l) is the rotational-diffusion decay time in the intermediate-time scale.

For each VG, the corresponding fitting function is given as

].

axes. In the previous section, we know that the bumps of water molecule motion will decrease the decay rate. For different VGs, we can clearly observe that the relation between

ξ

and the decay rate in Figs. 20(a), (b), (c) and Figs. 21(a), (b), (c).

From these results, we also tabulate these fitting parameters for different axes and VGs in Table V and Table VI, respectively. We find that the appearance time of bumps will be extended when

ξ

increase. This result is consistent with the previous one, especially for the y-axis, the geometry make IY smaller than other two axes, therefore, the initial motion memory keeps in a longer period. Besides, ^Cμ⁽¹⁾

⁽

^t

⁾

decays more slowly than ^Cμ⁽²⁾

⁽

^t

⁾

. Obviously, in Fig. 20(b),

Δ

^Cμξ⁽¹⁾

⁽

^t

⁾

in VG III and

IV have inflection points.

The fitting functions provide good predictions to us in the intermediate-time behavior. This result indicates that the OCF after getting rid of the H-bond restrictions in the intermediate-time scale is a combination of the residual memory of the Gaussian decay and the exponential decay extending into a long-time scale.

Chapter V Conclusions

By MD simulations with the SPC/E model, we have investigated the behaviors of the OCF of water molecules at different time scales. For examining the effects of interaction force, especially the H-bonds between molecules, we use the VP analysis to distinguish the local structures of molecules in a configuration.

In chapter II, we have derived the short-time behavior of the OCF, which is a Gaussian decay. We show that the behavior of the OCF within t < 0.02ps can be well described with three kinds of approximation: the total and the real-frequency INM spectra and the power spectrum of the AVAF. For t > 0.02ps, some inflection points in the OCF occur, with the number of inflection points determined by the strengths of the H-bonds attaching to a molecule and, therefore, the H-bond strengths determined by the local structures of the molecule. Thus, the local structures of water molecules have an enormous impact in the short-time behavior of the OCF.

According to the Debye theory for the rotational diffusion in molecular liquids, the long-time behavior of the OCF is a single-exponential decay with the rotational-diffusion decay time given as ⁽⁾ =

[

( +1) _μ

]

⁻¹

τ

_D^lμ

l l D

, where l = 1, 2 and Dμ. is the rotational-diffusion coefficient. Thus, the ratio

τ

_D⁽¹_μ⁾/

τ

⁽_D²_μ⁾ of the decay times for the OCFs with l=1 and 2 in the long time limit is expected to be three. For my MD simulation results, the logarithm of the OCFs with l=1 and 2 in long-time limit are

indeed well. Fit with linear functions; however, given in Table III and Table IV, the ratio of the slopes of the linear functions for l=1 and 2 is about 1.6 ~ 2.2. This phenomenon has been discussed in paper [21], in which similar MD simulation results of the SPC/E model are reported. In this paper, the difference between the simulated ratio and the prediction by the Debye theory is explained by the rearrangements of the H-bond network by constantly breaking and forming H-bonds through the reorientation of water molecules. Therefore, the Debye diffusive model is unsuitable for describing the reorientational dynamics in water and the authors of the paper introduced the extended jump model (EJM) for describing the reorientations in liquid water. Although the Debye theory can not predict accurately the rotational-diffusion decay time, the long-time behaviors of the OCFs still follow a single-exponential decay.

For investigating the intermediate-time behavior of the OCF, I introduce

Δ

^Cμ^(l)

⁽

^t

⁾

, which is the difference between the OCF obtained by the MD simulation and the single-exponential decay in the long time limit. In general, the behaviors of

Δ

^Cμ^(l)

⁽

^t

⁾

for molecules with different characteristics of local structures are all well fit by a power-law function multiplied by a Gaussian function, only with different coefficients in the fitting function, which are resulted from the H-bond effect and the three different moments of inertia of the anistopic water molecule. For a special case, the H-bonds attaching to molecules with highly aspherical local structures cause an inflection occurring in the

Δ

^Cμ^(l)

⁽

^t

⁾

of the y-axis.

Tables

TABLE I. Parameters for water model: SPC/E.

r

OH

(Å)

∠

HOH (deg)

10⁻3

×

M

(kcalŹ²/mol)

N (kcalÅ⁶/mol)

q

O

(e)

qH

(e)

1.0 109.47 629.4 625.5 -0.8476 0.4238

TABLE II. Averaged number fraction χ =ξ Nξ N and imaginary mode fractions ~f_ν of Voronoi group (VG) ξ (ξ= I, II, III, and IV) in liquid water at ρ=^1.0gcm⁻³ and T = 300 K. ξ is the asphericity range of each VG. N_ξ is the averaged number of molecules in VG ξ and N is the total number of molecules. ^~fν⁽ν =^X,Y,Z) is the imaginary-mode fraction of the normalized rotational INM spectrum of molecular ν axis.

VG I VG II VG III VG IV

ξ

below-1.46 1.46-1.72 1.72-1.98 1.98-above

χ

ξ 0.015975 0.672295 0.296647 0.015083

f

X

~ 0.224 0.104 0.051 0.035

f

Y

~ 0.128 0.073 0.047 0.04

f

Z

~ 0.104 0.043 0.021 0.014

TABLE III. Parameters of linear fitting functions for lnC1(t) of different rotational axes and VGs at long-time scales, where a_μξ^(l) is a constant for the displacement, _D^(l)

τ

μξ is the rotational-diffusion decay time, l is the index for the l-th order Legendre polynomial, D_μ is the rotational diffusion coefficient for the μ-th principal axis, and ΔE is relative error determine by

∑=

I

2-13.5 -0.41466 5.1334 0.012431

II

2-19.5 -0.31058 5.39 0.013393

III

2-20 -0.20931 5.3447 0.014476

IV

2-16 -0.12938 5.2408 0.02275

X-axis

Total 2-19.5 -0.27859 5.372 0.013285

I

2-12.5 -0.65265 4.983 0.023495

II

2-22 -0.1796 4.59 0.003832

III

2-17 -0.04856 4.73485 0.024630

IV

2-18 -0.04921 4.97018 0.015825

Y-axis

Total 2-26 -0.13867 4.6425 0.005449

I

2-6.3 -0.73726 3.145 0.010381

II

2-14 -0.28617 3.183 0.003904

III

2-16 -0.2209 3.459 0.019859

IV

2-12.5 -0.12185 3.465 0.030653

Z-axis

Total 2-14 -0.26403 3.267 0.006578

TABLE IV. Same as TABLE III. expect for lnC2(t).

lnC

2

(t)

Group Fit range (ps)

) 2 (

a

μξ

τ

D⁽²_μξ⁾

ΔE

I

2-5 -0.99868 2.144 0.007341

II

2-10 -0.83313 2.5414 0.012281

III

2-9 -0.482 2.39297 0.007292

IV

2-7 -0.38072 2.45441 0.008944

X-axis

Total 2-9 -0.6985 2.4655 0.009524

I

2-7 -1.1752 2.71311 0.010248

II

2-10 -0.58509 2.6698 0.011055

III

2-10 -0.26394 2.63012 0.003636

IV

2-9 -0.17546 2.7748 0.010203

Y-axis

Total 2-10 -0.47688 2.6572 0.007014

I

2-5.5 -1.4677 1.8421 0.018488

II

2-7 -0.9694 2.0476 0.006449

III

2-8 -0.61506 2.043 0.01

IV

2-6 -0.38348 2.0326 0.01055

Z-axis

Total 2-7 -0.82459 2.0225 0.005519

TABLE V. List

τ

_D⁽¹_μξ⁾ /

τ

_D⁽²_μξ⁾ in different VGs and axes

VG I II III IV Total

X-axis 2.394 2.121 2.233 2.135 2.179

Y-axis 1.837 1.72 1.8 1.791 1.747

) 2 ( ) 1 (_μξ

/ τ

_μξ

τ

_{D D}

Z-axis 1.707 1.554 1.693 1.705 1.615

TABLE VI. Parameters of fitting functions for C1(t) of different rotational axes and VGs at intermediate-time scales, s_ξ^(l) and p_ξ^(l) are parameters of the power-law function and

τ

_i^(l_ξ⁾ is the rotational-diffusion decay time in the intermediate-time scale, and ΔE is relative error.

C

₁

(t) S

⁽¹⁾

ξ

) 1

P

( ξ

) 1 (

τ

_iξ

ΔE

I

0.06272 -0.4995 1.8048 0.02145

II

0.08891 -0.34087 2.6646 0.01406

III

0.07697 -0.21321 2.824 0.01434

IV

0.04605 -0.13292 2.7607 0.03312

X-axis

total

0.08438 -0.31067 2.7214 0.01298

I

0.08933 -0.54971 2.4162 0.03423

II

0.02934 -0.53187 1.192 0.00994

III

0.01244 0.75345 1.716 0.00945

IV

0.03901 0.45948 3.1403 0.03308

Y-axis

total

0.02027 -0.54095 1.32125 0.017416

I

0.07586 -0.61966 1.5869 0.0144

II

0.04068 -0.59235 1.5872 0.00936

III

0.0796 -0.21275 2.7537 0.02959

IV

0.0525 0.04045 2.7691 0.04265

Z-axis

total

0.0459 -0.5143 2.275 0.00946

TABLE VII. Same as TABLE V. expect for C2(t).

C

₂

(t) S

⁽²⁾

ξ

) 2

P

( ξ

) 2 (

τ

_iξ

ΔE

I

0.04447 -0.78868 1.0746 0.01407

II

0.111064 -0.49973 1.7694 0.021268

III

0.06711 -0.49496 1.6281 0.012534

IV

0.05486 -0.43906 1.7854 0.012842

X-axis

total

0.09304 -0.50442 1.6986 0.014977

I

0.06238 -0.69042 1.436 0.021751

II

0.06572 -0.63096 1.8978 0.021229

III

0.03369 -0.52678 1.7006 0.007105

IV

0.0212 -0.41489 1.968 0.02108

Y-axis

total

0.05525 -0.61313 1.838 0.012398

I

0.0468 -0.78145 1.0335 0.047407

II

0.08742 -0.60683 1.4358 0.011412

III

0.07708 -0.54317 1.6044 0.019173

IV

0.03185 -0.66527 1.1346 0.020189

Z-axis

total

0.07838 -0.60293 1.4299 0.00972

Figures

Fig. 1. Body frame of SPC/E water molecule. The origin is located at the center of

mass. Oxygen atom is located on the positive x axis and the line joining the two hydrogen atoms is parallel to the y axis. The OH bond distance is

r

_OH =1

A

& and the bond angle

∠

HOH

= 109 . 47 °

.

0 2 4 6 8

r (Angstrom)

0 0.5 1 1.5

2 2.5 3 3.5

g

_OO

(r) g

OH

(r) g

_HH

(r)

Fig. 2. The radial distribution functions of SPC/E liquid water at mass density

ρ=1.0 g/cm³ and temperature T=300K. The sold, dash and dot-dash lines stand for gOO(r), gOH(r) and gHH(r), respectively.

0 2 4 6 8 0

1 2 3 4

g OO (r)

VG I VG II VG III VG IV

0 2 4 6 8

0 1 2

g OH (r)

0 2 4 6 8

r (Angstrom)

0 1 2

g HH (r)

Fig. 3. The g

OO(r), gOH(r) and gHH(r) radial distribution functions for each Voronoi group (VG). The solid, dot, dot-dash and dash lines are for VG I, II, III and IV, respectively.

-400 -200 0 200 400 600 800 1000

ω (cm ^-1 )

0 0.0005 0.001 0.0015 0.002 0.0025

D( ω)

Total Translation

Rotation along x axis Rotation along y axis Rotation along z axis

Fig. 4. INM DOS of SPC/E liquid water at T=300K. The solid line is the total DOS.

The dot line is the translational contribution. The dash, dot-dash and dot-dot-dash lines are the contributions of rotations along the x, y and z axes, respectively.

-600 0 -300 0 300 600 900 1200

Fig. 5. Normalized rotational INM DOS for molecules in each VG subensemble. In

each panel, the solid, dash, dot-dash and dot lines are for VG I, II, III and IV, respectively. The panel (a), (b) and (c) are for the contributions with rotational axis along the x, y and z principal axes, respectively.

0 0.05 0.1 0.15 0.2 0.25 0.3

t (ps)

-0.5 0 0.5 1

Ω (t)

x axis y axis z axis

Fig. 6. Normalized angular velocity autocorrelation functions (AVAFs) of SPC/E

liquid water. The solid, dash and dot-dash lines are for the angular velocity vectors along the x, y and z principal axes, respectively.

0 200 400 600 800 1000 0

0.001 0.002 0.003

D( ω)

Real INMs AVAF

0 200 400 600 800 1000

0 0.001 0.002 0.003

D( ω)

0 200 400 600 800 1000

ω (cm ^-1 )

0 0.001 0.002 0.003

D( ω)

(a) x axis

(b) y axis

(c) z axis

Fig. 7. Comparison between the real-frequency rotational INM DOS (solid lines) and

the power spectrum of the normalized AVAF (dash lines).

0 0.05 0.1 0.15 0.2 0.25 dot-dash lines are for the VG I, II, III and IV, respectively.

0 0.05 0.1 0.15 0.2 0.25

0 500 1000 1500 0

0.05 0.1

D

x

eff

( ω) / k

B

T

0 500 1000 1500

0 0.05 0.1

Real INM AVAF

0 500 1000 1500

ω (cm

^-1

)

0 0.05 0.1

D

x

eff

( ω) / k

B

T

0 500 1000 1500

ω (cm

^-1

)

0 0.05 0.1

(a) VG I (b) VG II

(d) VG IV (c) VG III

Fig. 9(a). Comparison of the effective AVAF power spectrum (dash lines) of the x

axis with the stable-INM approximation (solid lines). The panel (a), (b), (c) and (d) are for VG I, II, III and IV, respectively.

0 500 1000 1500 0

0.05 0.1

D

y

eff

( ω) / k

B

T

0 500 1000 1500

0 0.05 0.1

Real INM AVAF

0 500 1000 1500

ω (cm

^-1

)

0 0.05 0.1

D

y

eff

( ω) / k

B

T

0 500 1000 1500

ω (cm

^-1

)

0 0.05 0.1

(a) VG I (b) VG II

(c) VG III (d) VG IV

Fig. 9(b). Same as Fig. 9(a) except for the y axis.

0 500 1000 1500 0

0.05 0.1

D

z

eff

( ω) / k

B

T

0 500 1000 1500

0 0.05 0.1

Real INM AVAF

0 500 1000 1500

ω (cm

^-1

)

0 0.05 0.1

D

z

eff

( ω) / k

B

T

0 500 1000 1500

ω (cm

^-1

)

0 0.05 0.1

(a) VG I (b) VG II

(c) VG III (d) VG IV

Fig. 9(c). Same as Fig. 9(a) except for the z axis.

0 0.05 0.1 0.15 0.2 0.9

1

C

x(1)

(t)

MD All INM AVAF Real INM

0 0.05 0.1 0.15 0.2

0.9 1

C

y

(1)

(t)

0 0.05 0.1 0.15 0.2

t (ps)

0.8 0.9 1

C

z

(1)

(t)

Fig. 10. Comparison of the MD simulation results (solid lines) for the short-time

behavior of C⁽¹⁾(t) with the second cumulant approximation given in eq. (II.8). The dash, dot and dot-dash lines are the results for

ψ (t) calculated with the DOS of total

INMs the DOS of real-frequency INMs and the power spectrum of AVAF,

respectively.

0 0.05 0.1 0.15 0.2 0.7

0.8 0.9 1

C

x

(2)

(t)

MD All INM AVAF Real INM

0 0.05 0.1 0.15 0.2

0.7 0.8 0.9 1

C

y

(2)

(t)

0 0.05 0.1 0.15 0.2

t (ps)

0.6 0.7 0.8 0.9 1

C

z

(2)

(t)

Fig. 11. Same as Fig. 10 except for C

⁽²⁾(t).

0 0.05 0.1 0.15 0.2 0.75

0.8 0.85 0.9 0.95 1

C

x

(1)

(t)

MD All INM AVAF Real INM

0 0.05 0.1 0.15 0.2

0.85 0.9 0.95 1

0 0.05 0.1 0.15 0.2

t (ps)

0.9 0.95 1

C

x

(1)

(t)

0 0.05 0.1 0.15 0.2

t (ps)

0.9 0.95 1

(b) VG II (a) VG I

(c) VG III (d) VG IV

Fig. 12(a). Same as Fig. 10 excpet for

C_x⁽¹⁾(t)of each VG.

0 0.05 0.1 0.15 0.2 0.7

0.8 0.9 1

C

y

(1)

(t)

MD All INM AVAF Real INM

0 0.05 0.1 0.15 0.2

0.85 0.9 0.95 1

0 0.05 0.1 0.15 0.2

t (ps)

0.9 0.95 1

C

y

(1)

(t)

0 0.05 0.1 0.15 0.2

t (ps)

0.9 0.95 1

(b) VG II (a) VG I

(d) VG IV (c) VG III

Fig. 12(b). Same as Fig. 10 excpet for

C⁽_y¹⁾(t)of each VG.

0 0.05 0.1 0.15 0.2 0.6

0.7 0.8 0.9 1

C

z

(1)

(t)

MD All INM AVreal INMreal

0 0.05 0.1 0.15 0.2

0.8 0.85 0.9 0.95 1

0 0.05 0.1 0.15 0.2

t (ps)

0.85 0.9 0.95 1

C

z

(1)

(t)

0 0.05 0.1 0.15 0.2

t (ps)

0.85 0.9 0.95 1

(b) VG II (a) VG I

(d) VG IV (c) VG III

Fig. 12(c). Same as Fig. 10 excpet for

C_z⁽¹⁾(t)of each VG.

0 0.05 0.1 0.15 0.2 0.4

0.6 0.8 1

C

x

(2)

(t)

MD INM AVreal INMreal

0 0.05 0.1 0.15 0.2

0.6 0.7 0.8 0.9 1

0 0.05 0.1 0.15 0.2

t (ps)

0.7 0.8 0.9 1

C

x

(2)

(t)

0 0.05 0.1 0.15 0.2

t (ps)

0.7 0.8 0.9 1

(b) VG II (a) VG I

(d) VG IV (c) VG III

Fig. 13(a). Same as Fig. 10 excpet for

C_x⁽²⁾(t)of each VG.

0 0.05 0.1 0.15 0.2

0 0.05 0.1 0.15 0.2 0.4

0.6 0.8 1

C

z

(2)

(t)

MD All INM AVreal INMreal

0 0.05 0.1 0.15 0.2

0.6 0.7 0.8 0.9 1

0 0.05 0.1 0.15 0.2

t (ps)

0.6 0.7 0.8 0.9 1

C

z

(2)

(t)

0 0.05 0.1 0.15 0.2

t (ps)

0.7 0.8 0.9 1

(b) VG II (a) VG I

(d) VG IV (c) VG III

Fig. 13(c). Same as Fig. 10 excpet for

C_z⁽²⁾(t)of each VG.

0 5 10 15 20

t (ps)

-5 -4 -3 -2 -1 0

ln C

(1)

(t)

Fitting line x axis y axis z axis

Fig. 14. The long-time behaviors of

C⁽_x¹⁾(t) (solid line), C_y⁽¹⁾(t) (dash line) and C⁽_z¹⁾(t) (dot-dash line). The dot lines are the results of fitting with the Debye model.

0 2 4 6 8 10

t (ps)

-5 -4 -3 -2 -1 0

ln C

(2)

(t)

Fitting line x axis y axis z axis

Fig. 15. The long-time behaviors of

C_x⁽²⁾(t) (solid line), C_y⁽²⁾(t) (dash line) and C_z⁽²⁾(t) (dot-dash line). The dot lines are the results of fitting with the Debye model.

0 5 10

t (ps)

-2 -1 0

ln C

x

(1)

(t)

VG I VG II VG III VG IV Fitting line

Fig. 16(a) The long-time behaviors of

C_x⁽¹⁾(t) for VG I (black dash line), VG II (violet dash line), VG III (green dash line) and VG IV (blue dash line). The dot lines are the results of fitting with the Debye model.

0 5 10 15

t (ps)

-3 -2 -1 0

ln C

y

(1)

(t)

VG I VG II VG III VG IV Fitting line

Fig. 16(b) Same as Fig. 16(a) except for

C⁽_y¹⁾(t).

0 2 4 6 8 10

t (ps)

-3 -2 -1 0

ln C

z

(1)

(t)

VG I VG II VG III VG IV Fitting line

Fig. 16(c) Same as Fig. 16(a) except for

C⁽_z¹⁾(t).

0 1 2 3 4 5 6 7

t (ps)

-3 -2 -1 0

ln C

x

(2)

(t)

VG I VG II VG III VG IV Fitting line

Fig. 17(a) The long-time behaviors of

C_x⁽²⁾(t) for VG I (black dash line), VG II (violet dash line), VG III (green dash line) and VG IV (blue dash line). The dot lines are the results of fitting with the Debye model.

0 2 4 6 8 10

t (ps)

-4 -3 -2 -1 0

ln C

(2)

(t)

VG I VG II VG III VG IV Fitting line

Fig. 17(b) Same as Fig. 17(a) except for

C⁽_y²⁾(t).

0 1 2 3 4 5 6

t (ps)

-4 -3 -2 -1 0

ln C

z

(2)

(t)

VG I VG II VG III VG IV Fitting line

Fig. 17(c) Same as Fig. 17(a) except for

C_z⁽²⁾(t).

0 1 2 3 4 5 on the left, the solid lines are the results of MD simulations and the dot lines are the Debye model for the long-time behavior F_μ⁽¹⁾

(

t

)

(

μ

=

x or

,

y z

). In the diagram on the fitting results with the model given in eq.(IV.4).

0 1 2 3

0 1 2 3 4 5 VG IV. The line meanings are the same as those in Fig. 18.

0 1 2 3 4 5 VG IV. The line meanings are the same as those in Fig. 18.

0 1 2 3 4 5 VG IV. The line meanings are the same as those in Fig. 18.

0 1 2 3 VG IV. The line meanings are the same as those in Fig. 18.

0 1 2 3 VG IV. The line meanings are the same as those in Fig. 18.

0 1 2 3 VG IV. The line meanings are the same as those in Fig. 18.

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