Correlation functions from simulations

If the density of solute species is finite, the total correlation functions in Eq. (70) can be easily sampled from simulations and thereafter the PMF can be computed straightforwardly. However, in the infinitely dilute limit of solute density, e.g., only two solute molecules are immersed in solvent, it is almost impossible to direcdy sample the solute—solute total correlation functions from simulation, and the solute-solute correlation function can then be calculated through the integral equation theories with or without the incorporation of simulation for computing the solute—solvent correlation functions. 

Fig. 7.2 Comparison of experimental dotted line) and simulated solid line) neutron total correlation function T(r) of 45S5. The difference of the two dash line) and eight important (out of total fifteen) partial pair correlation functions from simulations are also shown. Qmax of 59 A (the maximum Q value from experiment) and Lorch window function were used in neutron broadening of simulated structures [21]...

Fig. 26. The OO pair correlation function from the Monte Carlo simulation of liquid water 0 BNS potential, Rowlinson potential,--------(from Ref. 72>)...

The self-correlation functions from the Stockmayer simulation are closer to Gaussians than those from the modified Stockmayer simulation. 

Fig. 3.58. Pair correlation functions from Monte Carlo simulation for Charged soft sphere model for LiCI liquid at 883 Kand 28.3 cm mol" .

Experimentalists often rely on motional models, based on hydrodynamics, in order to interpret their liquid state spectra. MD simulations, can be considered as model-free in the sense that they do not assume the molecular motion to be in any specific regime. MD can be used to evaluate the motional models and even replace them. MD simulations can be used to calculate both the correlation times and the whole correlation functions. This is useful in those cases when correlation times cannot be deduced from measurements of other isotopes in the same molecule or when there is no method available at all. Correlation functions give information about intermolecular interactions and reveal cases when several motional modes are contributing to relaxation mechanisms at slightly different time scales. This can be observed as multiple decay rates. Time correlation functions from MD simulations can be Fourier transformed to power spectra if needed to provide line shapes and frequencies. 

The so-called product reactant Ornstein-Zernike approach (PROZA) for these systems was developed by Kalyuzhnyi, Stell, Blum, and others [46-54], The theory is based on Wertheim s multidensity Ornstein-Zernike (WOZ) integral equation formalism [55] and yields the monomer-monomer pair correlation functions, from which the thermodynamic properties of the model fluid can be obtained. Based on the MSA closure an analytical theory has been developed which yields good agreement with computer simulations for short polyelectrolyte chains [44, 56], The theory has been recently compared with experimental data for the osmotic pressure by Zhang and coworkers [57], In the present paper we also show some preliminary results for an extension of this model in which the solvent is now treated explicitly as a separate species. In this first calculation the solvent molecules are modelled as two fused charged hard spheres of unequal radii as shown in Fig. 1 [45],... 

The Kirkwood-Buff theory of solutions was originally formulated to obtain thermodynamic quantities from molecular distribution functions. This formulation is useful whenever distribution functions are available either from analytical calculations or from computer simulations. The inversion procedure of the same theory reverses the role of the thermodynamic and molecular quantities, i.e., it allows the evaluation of integrals over the pair correlation functions from thermodynamic quantities. These integrals Gy, referred to as the Kirkwood-Buff integrals (KBIs), were found useful in the study of mixtures on... 

This paper is divided as follows we first discuss the MC procedure and introduce the analysis of time correlation function for simulation of solute-solvent structures. Next we give results for the radial distribution function and consider the case of protic and aprotic solvents. The quantum mechanical analysis is made next. The paper is closed with a summary of the most important achievements and a few suggestions derived from our ongoing research. 

In the past, comparisons between NMR relaxation and MD simulations have concentrated on internal motions, since these often involve sub-nanosecond time scales that could be examined with limited computer resources. In this approach, overall rotational motion is removed by an rms fitting procedure (for example, on backbone atoms in regular secondary structure), and computing time-correlation functions from the result. Typical results are shown in the upper panel of Figure 8.1 similar plots have been presented many times before [4,12,10,11]. Many backbone vectors are like Thr 49, and decay in less than 0.1 ns to a plateau value which can be identified as the order parameter for that vector. Most regions of regular secondary structure resemble this, although there can be exceptions, and there is potentially important information in the decay rates and plateau values that are obtained. 

In Sections 2 and 3, we set up a formalism for dealing with the dynamics of dense fluids at the molecular level. We begin in Section 2 by focusing attention on the phase space density correlation function from which the space-time correlation functions of interest in scattering experiments and computer simulations can be obtained. The phase space correlation function obeys a kinetic equation that is characterized by a memory function, or generalized collision kernel, that describes all the effects of particle interactions. The memory function plays the role of an effective one-body potential and one can regard its presence as a renormalization of the motions of the particles. 

Chuev GN, Vyalov I, Georgj N Extraction of atom-atom bridge and direct correlation functions from molecular simulations a test for ambient water, Chem Phys Lett 561 175—178, 2013. 

A direct way to calculate the time correlation function from the values saved during the simulation run is just to literally implement its definition. Suppose we have M+ values of. 4(0 and B t), obtained at the regular time intervals mSt, where m is an integer running from 1 to M, stored in the... 

A review of field quantities and formal results is followed by discussion of equilibrium theories and simulations for polar liquids. Relaxation processes are considered primarily in terms of correlation function treatments starting with models of behavior in simple systems and going on to cooperative behavior in more complex systems as a function of temperature with some reference to polymers and molecular crystals. Other aspects discussed include correlation functions from transient birefringence ion-solvent interaction effects in electrolytes and relaxation in mixtures of polar liquids. 

Kierlik and Rosinberg [68,98,99] were the first to apply Wertheim s theory in the form of a free energy functional to produce a DFT for non-associating polyatomic molecules. As input to the theory, they estimated the cavity correlation function from a first-order functional Taylor series around the homogeneous result [99]. Results were in good agreement with molecular simulation results for hard sphere chains. 

Reptation quantum Monte Carlo (RQMC) [15,16] allows pure sampling to be done directly, albeit in common with DMC, with a bias introduced by the time-step (large, but controllable in DMC e.g. [17]) and the fixed-node approach (small, but not controllable e.g. [18]). Property estimation in this manner is free from population-control bias that plagues calculation of properties in diffusion Monte Carlo (e.g. [19]). Inverse Laplace transforms of the imaginary time correlation functions allow simulation of dynamic structure factors and other properties of physical interest. 

Many authors have since then applied Monte Carlo (MC) and molecular dynamics (MD) simulations to molten salts. The simulations yielded the partial pair correlation functions, from which the inter-ionic distances and coordination numbers were deduced, as shown in Table 3.7. Generally the interionic distances were... 

VER in liquid O 2 is far too slow to be studied directly by nonequilibrium simulations. The force-correlation function, equation (C3.5.2), was computed from an equilibrium simulation of rigid O2. The VER rate constant given in equation (C3.5.3) is proportional to the Fourier transfonn of the force-correlation function at the Oj frequency. Fiowever, there are two significant practical difficulties. First, the Fourier transfonn, denoted [Pg.3041]

If the Bath relaxation constant, t, is greater than O.I ps, you should be able to calculate dynamic properties, like time correlation functions and diffusion constants, from data in the SNP and/or CSV files (see Collecting Averages from Simulations on page 85). 

Theoretical results of similar quality have been obtained for thermodynamics and the structure of adsorbed fluid in matrices with m = M = 8, see Figs. 8 and 9, respectively. However, at a high matrix density = 0.273) we observe that the fluid structure, in spite of qualitatively similar behavior to simulations, is described inaccurately (Fig. 10(a)). On the other hand, the fluid-matrix correlations from the theory agree better with simulations in the case m = M = S (Fig. 10(b)). Very similar conclusions have been obtained in the case of matrices made of 16 hard sphere beads. As an example, we present the distribution functions from the theory and simulation in Fig. 11. It is worth mentioning that the fluid density obtained via GCMC simulations has been used as an input for all theoretical calculations. 

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

Fig. 1.15. Translational and angular velocity correlation functions for nitrogen. MD simulation data from [70], T = 122 K, densities are indicated in the figure. Reduced units for time t = (e/cr2), for density p" = p

---