Pair Potential Procedure

This technique has widely been applied in a series of papers by Clementi and coworkers for the description of the solvation of amino acids, peptides as well as of RNA, DNA and their constituents (for reviews see 161 l62 ). The interactions of some cations with these types of molecules were also described 163 166 . Pair potentials between small model molecules and the cations Li+ 62), Na+ 167), K+ l68), NII4+ I68), Mg2+ 98 and Ca2 + 98) were developed in order to describe ion-ionophore interactions169 . 

Within the frame of this approach the interaction energy of two molecules is described as the sum of pairwise interactions. It is assumed that each atom of the host interacts with each atom of the guest independently  

Different forms of the pair potentials e can be used. In most cases a three parameter function of the following type was applied  

Aij5 Bjj and CF are constants for a given pair of atoms and qj and are atomic net charges of the atoms i and j. The first two terms correspond to the Lennard-Jones potential and the third term to the electrostatic point charge — point charge interaction. 

Atoms of the same kind in similar chemical environments can be grouped in the same classes, i.e. they are forced to have the same constants. By this procedure a compromise between flexibility and accuracy can be made. Depending on the purpose of the calculations different class assignments and thus different sets of constants can be fitted on the basis of the same set of ab initio interaction energies. 

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Friedman (1971). This approach is described as the hypernetted chain procedure and in it ion-ion pair potentials are expressed as the sum of four terms. These are ... 

An alternative to the GB, COSMO, and Poisson electrostatic calculations is to model the solution to the Poisson equation in terms of pair potentials between solute atoms this procedure is based on the physical picture that the solvent screens the intra-solute Coulombic interactions of the solute, except for the critical descreening of one part of the solute from the solvent by another part of this solute. This descreening can be modeled in an average way to a certain level of accuracy by pairwise functions of atomic positions.18, M 65 One can obtain quite accurate solvation energies in this way, and it has recently been shown that this algorithm provides a satisfactory alternative to more expensive explicit-solvent simulations even for the demanding cases of 10-base-pair duplexes of DNA and RNA in water.66... 

The first option involves the obtainement of an analytic function that reproduces the interaction energy between couples of molecules which has been calculated by solving the Schrodinger equation usually by means of an ab initio method. The advantge of this possibility is that information about any potential energy hypersurface point can be obtained from the calculation whereas experimentally this is not always possible. The practical procedure in order to build up an ab initio pair potential for the interaction between two molecules a and P can be divided in four steps. 

An interesting combined use of discrete molecular and continuum techniques was demonstrated by Floris et al.181,182 They used the PCM to develop effective pair potentials and then applied these to molecular dynamics simulations of metal ion hydration. Another approach to such systems is to do an ab initio cluster calculation for the first hydration shell, which would typically involve four to eight water molecules, and then to depict the remainder of the solvent as a continuum. This was done by Sanchez Marcos et al. for a group of five cations 183 the continuum model was that developed by Rivail, Rinaldi et al.14,108-112 (Section III.2.ii). Their results are compared in Table 14 with those of Floris et al.,139 who used a similar procedure but PCM-based. In... 

Our experimental set-up (described in ref. 7), allows us to record steady state absorption and emission spectra over a wide range of densities (10 5 to 20 at/nm3) in the Ar supercritical domain (Tc = 150.8 K, Pc = 49 bar). Representative absorption and emission spectra are shown in figure 1. These spectra could be reproduced with a good accuracy by means of equilibrium MD simulations performed with a standard procedure [8], In these simulations, the NO X-Ar and Ar-Ar interaction potentials were taken from the literature [9], We extracted an analytical NO A-Ar pair potential by an iterative fit of the experimental spectra, valid for the whole supercritical domain. 

In contrast to the pair potential model where the interaction energy hypersurface is approximated by an additive procedure using a simple mathematical function, the basis of the model of Gresh et. al. is an energy partitioning scheme. 

The two most commonly used procedures for the division of the pair potentials into the reference and the perturbation parts have been proposed by Barker and Henderson (BH) (19) and Chandler, Weeks, and Anderson (CWA) (23). Both procedures have been found to be equally satisfactory and the BH scheme is used here because of its simplicity. The division of the pair potential into repulsive and attractive parts, according to the BH scheme, is... 

Assuming the pair potential known, the radial distribution function for two-dimensional systems can be calculated using the two-dimensional version of the Ornstein-Zernike equation, Eq. (22), and one of the closure relations. Although Eq. (22) does not relate one to one the radial distribution function with the pair potential, one might attempt to invert the procedure to get u(r) from the experimental values for g(r). Thus, by taking the Fourier-Bessel (FB) transform [43,44] of Eq. (22) an expression for c(k) is obtained in terms of the FB transform of the measured total correlation function, i.e. 

Figure 16 show the effective pair potential for the systems in Fig. 15, calculated from the Orstein-Zernike equation with three different closure relations, following the procedure of deconvoluting the experimental g(r). In both cases, the effective pair potential exhibits an attractive component, independent of the closure relation, which is not consistent with the DLVO potential. As pointed out, the DLVO potential has a short-range first mini-... 

BINOL-phosphates as efficient Brpnsted acid catalysts in the enantios-elective Strecker reaction shows that C-nucleophiles can be applied in the chiral ion-pair catalysis procedure. This, in turn, not only increases the diversity of possible transformations of this catalyst but also shows the great potential chiral Brpnsted acids in asymmetric catalysis. 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

The free energy T of the molecular fluid consists of the ideal and non-ideal contributions, J [p = T lp] + J "[p]. First the molecular DFT approach replaces the whole intermolecular interaction by the sum of all pair interactions which appear via the pair potential 4>. That procedure fixes the free energy of the system as functional of both the pair potential < (X, X2) and the pair density p< (xi, X2),... 

In the previous Subsection we have introduced three PCM versions differing in the technique adopted for the calculation of AGei, and there are several others which consider different models for the solvent, such as liquid crystals, confined liquids, etc.. Most of such PCM procedures use an approach based on pair-potential effective energies for the calculation of Gdis-However, there are other PCM versions based on a reaction field approach. 

The computational procedure now in use in the PCM scheme is similar to the one adopted for the dispersion term according to the pair potential method the relevant formulas may be found in Floris et al. (1991) and in Tomasi and Persico (1994). 

The first term represehts the repulsive branch and the second term represents the attractive branch of the interaction potential between two atoms. By performing a non linear least square fit procedure the parameters (Al, ai, Ai A2, a2, A2) of the empirical pair potential are determined. In the fit procedure we have used the binding energy values of Au-dimer calculated at various interatomic distances by RDFT. The estimated points by RDFT and the fitted function are shown in Fig. 1. The potential parameters for the gold dimer interaction are determined as Ai = 1222.86345, A2 = -3.93623329, Ai = 2.94056151, A2 = -1.30223862, ai = 0.806351693, 2 = 0.216139972. In these parameters energy is in eV, and distance is in... 

Similar expressions apply to the other types of pairs, and all of them contain the zeroth-order modified Bessel function of the second kind K0(xr), or, simply, the Bessel K0 function. In Figure 1 we show a graph of the smoothly decreasing nonoscillatory Bessel K0 function, which can subsequently be contrasted with the pair potentials w(r) that emerge only after the three free energy terms (polyion self-energy, counterion transfer, and direct pair interaction) are added and minimized, a procedure that involves determination of the functions 0(r) and Q(r). 

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