Interdisciplinary Applied Mathematics

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Vlj = — (A/rf + (B/r)12,    (11.2)


where A = 0.37122 (kJ/mol)1/6— nm and B = 0.3428 (kJ/mol)1/12— nm. The total interaction energy between two SPC water molecules consists of the Lennard-Jones potential and the Coulombic potential based on classical electrostatics,


Vtotal = Vlj + Vc ,    (11.3)


where Vc between two molecules i and j is represented as the sum of Coulomb interactions acting among the charged points (i.e., the O and H atoms’ positions) in the following way:


(И-4)


O,H 1 ij


where r1J is the distance between site I of the molecule i and site J of the ij


molecule j.


The geometry is similar to the water geometry shown in Figure 11.2. However, the center    of    mass    coincides    with    the    oxygen    atom    for    the    SPC


model. The dipole moment of the SPC model is 2.27 D, compared to 1.85 D for the isolated molecule. The diffusion coefficient of the model is 3.6×10~5 cm2— s-1 at 300 K. The potential energy for liquid SPC water (Berendsen et al., 1987) at 300 K is —41.7 kJ/mol, and the density at 300 K and atmospheric pressure turns out to be 0.98 g — cm-3. Although the model behaves quite satisfactorily for most purposes, there is room for improvement with respect to density, radial distribution function, and diffusion constant. The properties of the SPC model are summarized in Table 11.3.


The SPC/E Model


The extended simple point charge (SPC/E) model is characterized by three point masses with O-H distance of 0.1 nm, H-O-H angle equal to 109.47°, charges on the oxygen and hydrogen equal to —0.8476 e and +0.4238 e, respectively, and with Lennard-Jones parameters of oxygen-oxygen interaction according to equation (11.2) (A = 0.37122 (kJ/mol)1/6— nm and B = 0.3428 (kJ/mol)1/12— nm) (Berendsen et al., 1987). The interaction between two SPC/E water molecules is represented by equations (11.2), (11.3) and (11.4). The SPC/E model has a dipole moment of 2.35 D. The diffusion constant is improved considerably compared to the SPC model. The agreement of the radial density distribution with experiment is somewhat better for the SPC/E model than for the SPC model (Berendsen et al., 1987).

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