Intermolecular correlations, self-consistent

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

Very recently, detailed further analyses of the translation and reorientation dynamics of SCW have been reported showing consistency with our results in the limit of very low-density supercritical states for water [47], The dynamical behavior of SCW in this study is also compared to that of supercritical benzene showing that the density dependence of the self-diffusion coefficient and rotational dynamics of SCW is smaller than that of supercritical benzene because SCW is capable of maintaining stronger degrees of structural correlations and orientational anisotropy than benzene, which tends to lose intermolecular correlations at a much faster rate upon decreasing density [47],... 

With the success of these calculations for isolated molecules, we began a systematic series of supermolecule calculations. As discussed previously, these are ab initio molecular orbital calculations over a cluster of nuclear centers representing two or more molecules. Self-consistent field calculations include all the electrostatic, penetration, exchange, and induction portions of the intermolecular interaction energy, but do not treat the dispersion effects which can be treated by the post Hartree-Fock techniques for electron correlation [91]. The major problems of basis set superposition errors (BSSE) [82] are primarily associated with the calculation of the energy. 

In this substection we will shortly discuss the computational methods used for calculation of the spin-spin coupling constants. Two main approaches available are ab initio theory from Hartree-Fock (or self-consistent field SCF) technique to its correlated extensions, and density function theory (DFT), where the electron density, instead of the wave function, is the fundamental quantity. The discussion here is limited to the methods actually used for calculation of the intermolecular spin-spin coupling constants, i. e. multiconfigurational self consistent field (MCSCF) theory, coupled cluster (CC) theory and density functional theory (DFT). For example, the second order polarization propagator method (SOPPA) approach is not... 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

Most popular in the ab initio calculation of intermolecular potentials is the so-called supermolecule method, because it allows the use of standard computer programs for electronic structure calculations. This method automatically includes all the electrostatic, penetration and exchange effects. If the calculations are performed at the SCF (self-consistent field) level the induction effects are included, too, but the dispersion energy is not. The latter, which is an intermolecular electron correlation effect, can be obtained by configuration interaction (Cl), coupled cluster (CC) calculations or many-body perturbation theory (MBPT). These calculations are all plagued... 

Now the expression (19) is an uncoupled formulation of the polarizability. We can replace it by a polarizability derived from coupled Hartree-Fock perturbation theory, which is more accurate, because it takes account of the reorganisation of the electron distribution in a self-consistent manner. Better still would be to evaluate the monomer polarizability by a method that takes account of electron correlation as well . But whatever the level of calculation, we can once again perform a much better calculation of the monomer property than is possible for the dimer. In this way we arrive at a description of the induction energy that is far more accurate than we can obtain through either intermolecular perturbation theory, where the perturbation is treated in an uncoupled fashion, or from a supermolecule calculation, where the size of the basis is limited by the need to perform calculations at a large number of points on the potential energy surface. 

The above physical features imply that a fully self-consistent treatment of intramolecular and intermolecular pair correlations is more important for star polymers than linear chains, and the concept of ideality is expected to be of much less utility even at high melt densities. The treatment of star polymers within a self-consistent PRISM formalism has been very recently pursued by Grayce and Schweizer. Here we give a brief description of some of the essential theoretical modifications... 

The self-consistently determined intermolecular pair correlations, gij r), and collective partial structure factors, S-jik), also display several unique physical features due to the star-branched architecture. An example for the pair correlations is shown in Figure 38 for a meltlike... 

Intramolecular correlations are handled in different approximate manners in the various BGY approaches. Taylor and Lipson treat pair correlations on the same chain as input to the theory in a manner similar to PRISM theory. In contrast, the formulations of Eu and Gan, " and also Attard, yield closed integral equations for both the intra- and intermolecular pair distribution functions. Thus, in a sense the intra- and intermolecular pair correlations are treated on an equal footing, and a self-consistent integral equation theory is naturally obtained. Eu and Gan have recently presented a comparison between their BGY approach and self-consistent and non-self-consistent PRISM theory, in both general conceptual terms and within the context of numerical predictions for specific model hard-core systems. For the jointed hardcore chain model studies, the theory of Eu and Gan appears quantitatively superior to PRISM predictions, particularly for the equation of state. ... 

The question of the effects of attractive tail potentials on macro-molecular structure, and the appropriate closure(s), require much further study. Advances in the closure question may also result in more thermodynamically consistent theories, which is important for questions such as the equation of state, excess properties of multicomponent systems, and construction of constant pressure (not volume) theories. Other basic theoretical issues relate to the potential importance of corrections to site preaveraging, and the development of tractable schemes to systematically compute such corrections. The issue of self-consistency of intramolecular and intermolecular correlations is rather unsettled from the point of view of both how important these effects are (the answer is problem-specific), and how to best go about construeting a computationally convenient theory to treat them. 

The standard approximation for the tJi k) has been to invoke the Flory ideality hypothesis. Although this is not on as firm a footing as in the melt case, it is a good first approximation, and is useful because the self-consistent determination of intramolecular and intermolecular correlations is a demanding task in polymer blends. The monomer structure is input to the theory through the single chain structure factors, something which is difficult to do in other liquid state theories. 

In Eqs. (19), (24), and (30), self-consistent PRISM theory is formulated in a general manner to allow for the modeling of polymer mixtures and polymer models containing an arbitrary number of interaction sites. A range of polymers of various complexities have been analyzed using PRISM theory. Piitz et al. [60] studied isotactic and syndiotactic polypropylene, head-to-head syndiotactic polypropylene, poly(ethylene propylene), and polyisobutylene using PRISM theory and MD simulations. In Figs. 8a, 8b we plotted the six independent pair correlation functions between intermolecular sites for isotactic polypropylene (iPP). 

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