Wavelet Algorithm for Solving Integral Equations of Molecular Liquids. A Test for the Reference Interaction Site Model GENNADY N. CHUEV, 1 MAXIM V. FEDOROV 1,2 1 Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Moscow Region, 142290, Russia 2 Theory and Computation Group, Centre for Synthesis and Chemical Biology, Conway Institute of Biomolecular and Biomedical Research, Department of Chemistry, University College Dublin, Belfield, Dublin 4, Ireland Received 9 December 2003; Accepted 26 March 2004 DOI 10.1002/jcc.20068 Published online in Wiley InterScience (www.interscience.wiley.com). Abstract: A new efficient method is developed for solving integral equations based on the reference interaction site model (RISM) of molecular liquids. The method proposes the expansion of site–site correlation functions into the wavelet series and further calculations of the approximating coefficients. To solve the integral equations we have applied the hybrid scheme in which the coarse part of the solution is calculated by wavelets with the use of the Newton–Raphson procedure, while the fine part is evaluated by the direct iterations. The Coifman 2 basis set is employed for the wavelet treatment of the coarse solution. This wavelet basis set provides compact and accurate approximation of site–site correlation functions so that the number of basis functions and the amplitude of the fine part of solution decrease sufficiently with respect to those obtained by the conventional scheme. The efficiency of the method is tested by calculations of SPC/E model of water. The results indicated that the total CPU time to obtain solution by the proposed procedure reduces to 20% of that required for the conventional hybrid method. © 2004 Wiley Periodicals, Inc. J Comput Chem 25: 1369 –1377, 2004 Key words: integral equations; wavelet basis set; reference interaction model; water; hybrid scheme Introduction Computer simulations of molecular liquids are a very powerful and useful instrument; unfortunately, they are limited by high compu- tational costs and extensive computational resources. The integral equation theory 1–28 provides an alternative technique with low computational costs required to treat molecular effects in liquids. At present there are several main approaches based on integral equations. The first one is the reference interaction site model (RISM) pioneered by Chandler and Andersen 1 and then extended to the dipolar liquids by the XRISM treatment. 4,5 The theory is based on calculations of radial distribution functions (RDF) via the site–site Ornstein–Zernike (SSOZ) integral equation. The theory has been successfully applied to calculate the structural and ther- modynamic properties of various chemical and biological sys- tems. 3,6 Recently, three-dimensional (3D) extensions of the RISM theory have been developed to obtain 3D correlation functions of interaction sites of solvent molecules around a solute of arbitrary shape. 7–18 The 3D RISM theory has been used to evaluate in details solvation structure for bulk water, 7,8,11 polar and nonpolar molecules in water, 15 complex organic molecular ions in a polar solvent, 14 and ion pairs in an aqueous electrolyte solution. 17,18 The molecular Ornstein–Zernike (MOZ) theory is another method to calculate 3D solvation structure in molecular liquids. The MOZ theory treats the orientation dependence of intermolecular interac- tions through the rotational invariant expansions of interaction potentials and correlation functions. 19 The recent MOZ calcula- tions 20 –24 have indicated that the theory is able to reproduce the thermodynamic, dielectric, and structural properties for aprotic solvents. Concerning the numerical schemes for the solving integral equations, we note that most of the current methods are based on the Fourier transform of integral equations and apply discretization of r - and k -spaces by choosing a grid with a large number of points. As a result, the problem reduces to calculations of a huge Correspondence to: G. N. Chuev; e-mail: genchuev@rambler.ru Contract/grant sponsors: Russian Foundation for Basic Research and the Centre for Synthesis and Chemical Biology. © 2004 Wiley Periodicals, Inc.