Chapter 4 Results and Discussions
4.1 Instantaneous normal mode of density of state (DOS)
In the body frame of a rigid SPC/E water molecule, the principal moments of inertia with respect to the x-, y-, and z-axis are Ix 7.41102, Iy 3.29102 and
10 1
07 .
1
z
I mÅ2 , respectively. Here, the m is mass of a water molecule.
Before we discuss the rotational INM DOSs of the OH and the OH-axes, we need to mention first that the rotational INM DOS of the three principal axes of water molecules at T=300K have been presented in Ref. [4]. The rotational INM DOSs of the principal x-, y- and z-axis are, generally, a broad spectrum with a peak at about 500, 830 and 530 cm1, respectively. We use Eqs. (2.3.30) and (2.3.31) in Chapter two to define the rotational INM DOS of the OH and OH-axes and, in those equations, the value of cos2 is about 31 and that of sin2 is about 32. The results of the rotational INM DOS of the OH- and the
OH-axes are shown in Fig. 3. So, the rotational INM DOS of the OH- axis, more like that of the y-axis, has a peak at a position near 700 cm1 and the rotational INM DOS of the
OH- axis, more similar as that of the x- axis, has a peak at a position near 550 cm1 and a shoulder at 950 cm1, which is due to the contribution of the y-axis.
In terms of the VP analysis, in Fig.4, we have calculated the normalized rotational INM spectra of the OH- and the OH-axes for various VGs. For both rotational axes, the
real-frequency lobe of the rotational DOS INM has a blue shift in average frequency with increasing the asphericity of the VG. Also, the fraction of the imaginary-frequency rotational
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INMs of a VG is found to decrease with the increase of the asphericity. The blue shift in the rotational INM DOSs for different VGs can be explained by the fact that the number of the hydrogen bonds increases in a trend with the asphericity of a VG. Besides, the higher VG shoulder appear for OH-axis.
Similar as the VP analysis, we classify water molecules according to their H-bond configurations. The rotational INM DOS of the OH- and the OH-axes of SPC/E liquid water at T=300K are shown in Figs. 5 and 6 for ten H-bond configurations, respectively.
Essentially, the rotational INM DOS depends on the number of the donated H-bonds of a molecule. Even for the D3A2 configuration with three donor H-bonds, the INM DOS is similar as those of water molecules with two donor H-bonds. We find that the real-frequency lobe of the rotational INM DOS has a blue shift in the average frequency and the fraction of the imaginary-frequency rotational INMs decreases with the number of the donated H-bonds associated with a water molecule. Besides, the real-frequency are more wide with two acceptor of the rotational INM DOS of the OH-axis. The wide is towards blue shift. The acceptor number can slightly affect the rotational INM DOS of the OH-axis evidently for the D2A3. Because the Oxygen is heavy that difficultly affect the rotational of water molecules.
4.2 Angular velocity autocorrelation function (AVAF) and Power spectrum
Calculated with MD simulations, the results of the normalized AVAFs with the instantaneous rotational axis along the OH-, OH-, and z-axes are shown in Fig.7. The AVAF along the z-axis decays slower than the others two axes. The AVAF along the OH-axis
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is similar as that along the OH-axis.
By classifying the water molecules generated by our simulations by the VP analysis, we also calculate the AVAFs of molecules in each of the four VGs. The calculated results of the normalized AVAFs along the OH- and the OH-axes for the four VGs are shown in Fig. 9 and 10. Indicated by our results, the normalized AVAFs of VG IV behave like an
underdamped oscillation. As the asphericity of the VPs of water molecules decreases, the oscillating amplitudes of the normalized AVAFs become smaller and the AVAFs of the VG I are closes to the overdamped behavior. Generally, for each VG, the oscillations of the AVAFs decay slower for the OH axis than for the OH-axis.
Similar as for the VP analysis, we can perform the analysis of the H-bond configuration for water molecules in our simulations and calculate the AVAFs for molecules in each
subensemble of the H-bond configuration. The simulated results of the normalized AVAFs along the OH- and the OH-axis for ten H-bond configuration are shown in Figs. 13 and 14, respectively. Generally, the AVAFs of molecules with the same number of H-bond donors behave similarly and they are presented in the same panel in Figs. 13 and 14. Specially, the AVAFs of the D3A2 configuration along the OH- and the OH-axes have similar behaviors as those of the H-bond configurations with two donors.
In the relations between the AVAFs analyzed by the VP and the H-bond configuration, we find that the AVASF of VG IV is similar in behavior as those of the H-bond
configurations with two donors and the AVAF of VG I is similar as those of the molecules with one donor H-bond. In a general conclusion, the more the number of donated H-bonds associated with a molecule is, the more like the underdamped behavior the AVAG of the molecule.
For these AVAFs, we can get their power spectrum from Eq.(2.3.13). In Fig. 8, we
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compare the power spectrum of the normalized AVAF and the rotational DOS of the real-frequency INMs, with both spectra being normalized, for the OH-, OH- and z axis.
The peak positions of both spectra for the three rotational axes are almost coincident. At the zero frequency, the power spectra of the AVAFs are not zero and give the self-diffusion[15]
coefficients of the rotational motions of a single water molecule along the three axes in the long-time limit and the estimated self-diffusion coefficients are 0.0001.
Figs. 11 and 12 show the comparisons of the power spectra DOH() and DOH() with the rotational DOS of the real-frequency INMs for the four VGs, respectively. The peak positions of both spectra for the four VG are almost coincident. We can take note as the aspericity of the VG increases, the value of the power spectrum at zero frequency decreases.
Also, Figs. 15and 16 show the comparisons of the power spectra DOH() and ()
OH
D with the rotational DOS of the real-frequency INMs for the ten H-bond configuration, respectively. The peak positions of both spectra for the ten H-bond configuration are almost coincident. As the number of the donated H-bond decreases, the self-diffusion coefficients of the rotational motions of water molecules along both the OH- and OH- axes are
expected to increase.
To get the short-time limit, we calculate the
t from Eq.(2.3.15) and (2.3.17) with the effective spectra. In Fig. 17 shows the effective spectra Deff()/kBT, withOH ,OH
obtained by the power spectra of the AVAFs and the corresponding spectra under the stable-INM approximation. For a principal axis, the effective spectrum Deff() is a linear combination of the DOSs, which are weighted by the inverse of the moment of inertia, of the others two principal axes. However, for the OH- and OH- axes, the effective spectra, DOHeff ()or DOHeff ()
, are composed of the DOSs and weighted by the
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inverse of the moment of inertia of three principal axes and the different fractions for of the x- and y-axes. So, the comparisons indicate that the effective spectra are similar in shape as those under the stable-INM approximation.
Figs. 18 and 19 show the comparisons of the effective spectra DOHeff ()/kBT and T
k
DOHeff()/ B with those under the stable INM approximation for the four VGs, respectively.
Also, Figs.20 and 21 show the comparisons of the effective spectra DOHeff ()/kBT and T
k
DOHeff()/ B with those under the stable INM approximation for molecules in the ten H-bond configurations, respectively.