Water, liquid radial distribution function

Fig. 55. Radial distribution function for liquid water synthesized as a mixture of ice I, ice II and ice III (from Ref. 75>)...

In short, our S-MC/QM methodology uses structures generated by MC simulation to perform QM supermolecular calculations of the solute and all the solvent molecules up to a certain solvation shell. As the wave-function is properly anti-symmetrized over the entire system, CIS calculations include the dispersive interaction[35]. The solvation shells are obtained from the MC simulation using the radial distribution function. This has been used to treat solvatochromic shifts of several systems, such as benzene in CCI4, cyclohexane, water and liquid benzene[29, 37] formaldehyde in water(28, 38] pyrimidine in water and in CCl4(31] acetone in water[39] methyl-acetamide in water[40] etc. 

Figure 13.1 Oxygen"-oxygen radial distribution function, gooM, in liquid water determined by various X-ray scattering experiments (reproduced with permission from [1] copyright American Institute of Physics).

The solvation structure around a molecule is commonly described by a pair correlation function (PCF) or radial distribution function g(r). This function represents the probability of finding a specific particle (atom) at a distance r from the atom being studied. Figure 4.32 shows the PCF of oxygen-oxygen and hydrogen-oxygen in liquid water. 

The local structure in liquids can be measured by X-ray diffraction and described by either a radial distribution function or the pair correlation function. In particular, the oxygen-oxygen pair correlation function or reduced radial distribution function for water, goo(r) Fig. 11.74, can be obtained from... 

Figure 7.7 The radial distribution function ko( ) for oxygen atoms about a K+ ion in liquid water. See Rempe et al. (2004). The dashed curve is the contribution to Ko(r) from the nearest four oxygen atoms, and the dashed-dot curve is the contribution from the 5 and 6 nearest oxygen atoms. Notice the lack of definition obtained from the Ko(r) solely because a minimum separating a from a 2 mean hydration shell is indistinct.

For liquids near surfaces the radial distribution functions become asymmetrical and are generally more difficult to handle. The situation is again relatively simple for molecularly flat surfaces when only density variations normal to the surface have to be considered. This is, for instance, the case for spherically symmetrical molecules for rods, or molecules with asymmetrical interaction (water), the situation is again more complicated. In that case one can Introduce a (linear) distribution function g[z) as... 

Many of the effective pair potentials used to model liquid water have been adjusted to ensure a reasonable representation of measured radial distribution functions, especially qo- sense this scaling is understandable as most simula-... 

Probably the most notable work on the structure in liquid water based upon experimental data has been that of Soper and co-workers [6,8,10,30,46,55]. He has considered water under both ambient and high temperature and pressure conditions. He has employed both the spherical harmonic reconstruction technique [8,46] and empirical potential structure refinement [6,10] to extract estimates for the pair distribution function for water from site-site radial distribution functions. Both approaches must deal with the fact that the three g p(r) available from neutron scattering experiments provide an incomplete set of information for determining the six-dimensional pair distribution function. Noise in the experimental data introduces further complications, particularly in the former technique. Nonetheless, Soper has been able to extract the principal features in the pair (spatial) distribution function. Of most significance here is the fact that his findings are in qualitative agreement with those discussed above. 

The potential function used in MD calculations of liquid water is the so-called SPC potential—that is, a semiempirical potential yielding the almost correct radial distribution function for oxygen atoms at room temperatures. However, we emphasize that what we want to see is characters that are independent of detailed tunings of the system such as potential functions, bond lengths, bond angles, and even masses of atoms. The properties depending sensitively on them are attributed to system specificities and are outside our present concern. 

Figure 16. The power spectrum density for molecules for which rigid rotators are replaced by point particles, keeping the radial distribution function of liquid water unchanged.

Fig. 4. (a) The radial distribution functions goH r), goo r), and gHH r)) of flexible TIP3P liquid water calculated using the NHC-RESPA and ISO-NHC-RESPA methods, (b) The error in the distribution functions as a function of time, where the exact distributions are those that have been generated from a long run with a small time step. The legends report the large time step... 

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