Correlation functions, direct molecular

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. 

Zhao SL, WuJZ An efficient method for accurate evaluation of the site-site direct correlation functions of molecular fluids. Mol Phys 109(21) 2553—2564, 2011a. 

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. 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

Perhaps the simplest two-site cooperative systems are small molecules having two binding sites for protons, such as dicarboxylic acids and diamines. Despite their molecular simplicity, most of these molecules do not conform with the modelistic assumptions made in this chapter. Therefore, their theoretical treatment is much more intricate. The main reasons for this are (1) there is, in general, a continuous range of macrostates (2) the direct and indirect correlations are both strong and intertwined, so that factorization of the correlation function is impossible. In addition, as with any real biochemical system, the solvent can have a major effect on the binding properties of these molecules. 

The molecular approach, adopted throughout this book, starts from the statistical mechanical formulation of the problem. The interaction free energies are identified as correlation functions in the probability sense. As such, there is no reason to assume that these correlations are either short-range or additive. The main difference between direct and indirect correlations is that the former depend only on the interactions between the ligands. The latter depend on the maimer in which ligands affect the partition function of the adsorbent molecule (and, in general, of the solvent as well). The argument is essentially the same as that for the difference between the intermolecular potential and the potential of the mean force in liquids. 

This paper presents a brief review of the use of explicitly correlated wave functions in molecular quantum chemistry computations. This review is restricted mainly to the direct variational approaches. Special attention will be given to two-electron molecular systems. The possible direction of extending the use of the correlated wave function for three- and four-electron molecular systems as well as the accuracy of the results will be discussed. 

Here, H and C are symmetric matrices whose elements are the partial total hap(r) and direct cap,(r) pair correlation functions a,ft = A,B) W is the matrix of intramolecular correlation functions wap r) that characterize the conformation of a macromolecule and its sequence distribution and p is the average number density of units in the system. Equation 17 is complemented by the closure relation corresponding to the so-called molecular Percus-... 

The first term, the kinetic energy, is difficult to calculate directly from the density, and it is for this reason that the molecular orbitals mentioned above are introduced a very good approximation to the kinetic energy corresponding to the density can be calculated from the orbitals as it would be in HF theory. This approach does not however yield the exact kinetic energy because it assumes that the electrons in each orbital do not interact with electrons in other orbitals. The exact exchange-correlation functional must therefore contain a corrective term to incorporate the effect of electronic interactions on their kinetic energy. In practice, such a term is not explicitly included in common functionals. 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

The physical origin of correlation energy is in the nature of the Hartree-Fock equations. The inter-electronic interaction is represented by coulombic and exchange terms each electron has a direct interaction with the average charge of all the others obtained by squaring the one-electron functions (the molecular orbitals), but an exchange interaction only with elections of the same... 

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