RISM integral equation, solute-solvent

The general form of the (RISM) integral equation appropriate for treating solute-solvent interactions which has been derived in the literature is given by " ... 

In conclusion to these introductory notes, it is necessary to emphasize that the 3D-RISM theory of ion-molecular solutions provides an advanced description as compared to the reaction field continuum models of solvent. Treating the 3D-RISM integral equations is not much more computationally expensive than the electrostatic 3D boundary value prob-... 

The solute-solvent 3D-RISM equation complemented with the 3D-HNC closure was first obtained by Beglov and Roux [24] within the density functional method by reduction of the generalized closure of Chandler, McCoy, and Singer for nonuniform polyatomic systems [22]. In an alternative way, the 3D-RISM integral equation was derived by Kovalenko and Hirata [28] from the six-dimensional (6D), molecular OZ equation for a solute-solvent mixture of rigid molecules at infinite dilution. The latter is written as [21]... 

This equation expresses the friction coefficient in terms of the solute-solvent equilibrium correlation function Cu k)) and the nature of motions of solute (Fu k,t)) as well as solvent molecules (FA (/c,t)). The site-site direct correlation functions Cu k) can be obtained from the extended version of the RISM integral-equation theory [58, 59, 77], and the site-site intermediate scattering function F k,t) can be calculated as we have done in Sec. 5.3. For Fu k,t) the diffusion approximation is invoked since here we are interested in the diffusive motion of the solute ... 

MC and MD simulations could be replaced by other less expensive methods. We shall quote the use of the integral equation methods called RISM (reference interaction site model) explored by Ten-no et al. RISM methods give solute-solvent site-site correlations functions as MC and MD simulations do. The RISM procedure is inserted into an SCF QM cycle as for combined QM/MM methods based on simulations. We have no direct experience on computational costs of these procedures, but in general RISM calculations are faster by some orders of magnitude than MC simulations. This is paid for by a greater sensitivity of RISM to some features of the integral equation methods, such as the closure expressions one has to use. 

One important class of integral equation theories is based on the reference interaction site model (RISM) proposed by Chandler [77]. These RISM theories have been used to smdy the confonnation of small peptides in liquid water [78-80]. However, the approach is not appropriate for large molecular solutes such as proteins and nucleic acids. Because RISM is based on a reduction to site-site, solute-solvent radially symmetrical distribution functions, there is a loss of infonnation about the tliree-dimensional spatial organization of the solvent density around a macromolecular solute of irregular shape. To circumvent this limitation, extensions of RISM-like theories for tliree-dimensional space (3d-RISM) have been proposed [81,82],... 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

A different approach to mention here because it has some similarity to QM/MM is called RISM-SCF [5], It is based on a QM description of the solute, and makes use of some expressions of the integral equation of liquids (a physical approach that for reasons of space we cannot present here) to obtain in a simpler way the information encoded in the solvent distribution function used by MM and QM/MM methods. Both RISM-SCF and QM/MM use this information to define an effective Hamiltonian for the solute and both proceed step by step in improving the description of the solute electronic distribution and solvent distribution function, which in both methods are two coupled quantities. There is in this book a contribution by Sato dedicated to RISM-SCF to which the reader is referred. Sato also includes a mention of the 3D-RISM approach [6] which introduces important features in the physics of the model. In fact the simulation-based methods we have thus far mentioned use a spherically averaged radial distribution function, p(r) instead of a full position dependent function p(r) expression. For molecules of irregular shape and with groups of different polarity on the molecular periphery the examination of the averaged p(r) may lead to erroneous conclusions which have to be corrected in some way [7], The 3D version we have mentioned partly eliminates these artifacts. 

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