Determination of Critical Indices by “Slow” Spectroscopy: NMR Shifts by Statistical Thermodynamics and Density Functional Theory Calculations Vytautas Balevicius,* ,† Vytautas Juozapas Balevicius, † Kestutis Aidas, † and Hartmut Fuess ‡ Faculty of Physics, Vilnius UniVersity, LT-10222 Vilnius, Lithuania, and Institute of Materials Science, UniVersity of Technology Darmstadt, D-64287 Darmstadt, Germany ReceiVed: August 24, 2006; In Final Form: NoVember 27, 2006 The temperature dependencies of NMR shifts in the critical region of two coexisting phases have been simulated using statistical thermodynamics and graph-theory consideration of equilibrium processes of molecular association. Microparameters of magnetic screening of various water and water/pyridine structures used in the statistical averaging have been evaluated by density functional theory calculations (PBE1PBE and B3PW91 functionals in the 6-311++G** basis set). The gauge-including atomic orbital (GIAO) approach has been applied to ensure gauge invariance of the results. Solvent effects were taken into account by a polarized continuum model (PCM). NMR shifts “order parameters” (Δδ ) |δ + - δ - |) and “diameters” (φδ ) |(δ + + δ - )/2 - δ C |, where δ + , δ - , and δ C are the chemical shifts of coexisting phases and at the critical point respectively) have been calculated in each case close to the lower critical solution point (T L ) and processed using linear regression analysis of Δδ ∼|T - T L | and φδ ∼|T - T L | in the log-log plot. It has been shown that the critical index  can be evaluated with high precision from the slope of Δδ ) f(T - T L ) at any realistic set of model input parameters. The slope of diameter has been found to depend on both input  and R values. The obtained φδ slopes (0.58-0.63) are very close to 2 values. The results are discussed within the concept of complete scaling. Results of simulation are compared and supported by experimental NMR data for water/2,6-lutidine, acetic anhydride/n-heptane, and acetic anhydride/cyclohexane systems. I. Introduction Critical indices and critical amplitudes are those crucial and universal parameters that reflect most peculiarities of physical systems close to their phase transition point. 1,2 Among the most intriguing findings in this field in recent years were the discovery of crossover between the solvophobic and ionic critical re- gimes, 2,3 multiple critical points, 4,5 and nonequilibrium restruc- turing phenomena. 6,7 Despite very precise and exhaustive studies, several open questions persist, and some conclusions are even controversial. Hence, new convenient experimental ways to determine various critical parameters based on “easily measurable” quanti- ties are of high scientific and practical value. A concept to establish a relationship “critical parameter” S “easily measur- able quantity” requests a theoretical background. Such a concept would, first of all, support the existence of correlations of this kind, and furthermore, it should determine physical margins of their validity. Most spectral parameters (vibrational frequencies, band intensities, NMR chemical shifts, etc.) can be considered as those easily measurable quantities. Exceptional cases may exist, e.g., analysis of complicated vibrational band shapes in systems with very strong H-bonds, 8 or NMR spectra of nuclei of low natural abundance, when a huge amount of data has to be sampled and processed. 9 Nevertheless NMR chemical shifts are indeed parameters that can be measured with very high accuracy, even in the region of critical fluctuations. 10-14 NMR shifts of nuclei directly involved in the most important interactions, e.g., hydrogen bonding, dipole-dipole aggregation, clustering, etc. that fate the major structural features of the system, are extremely sensitive to temperature and composition changes. Thus they can be considered as the main source of information studying critical phenomena. NMR is, however, in a dynamical sense a “slow” spectroscopy. This means that measured NMR shifts are averaged over numerous spatial manifolds in the system (various associates, clusters, conformations, etc.), and hence, some features and parameters of critical behavior are “hidden” in this averaging procedure. The main challenge could be formulated as “what are the chances to extract certain informa- tion on critical parameters applying physical techniques, that on the one hand are slow in time scale of molecular processes, but on the other hand, provide a huge and precise set of necessary data to be collected in short time experiments?” A suitable way to determine critical indices based on the temperature-dependence of NMR chemical shift is proposed in the present work. Statistical thermodynamics and graph theory considerations were applied to describe processes of chemical equilibrium of association of molecules, and the quantum chemical density functional theory was used to evaluate individual properties (magnetic screenings) of various molecular structures. II. Graph Theory and Thermodynamics of Molecular Association The mathematical methods of topology and graph theory are widely used solving very sophisticated cases of molecular structure features, kinetics of chemical reactions, chemical equilibrium, etc. 15,16 Complicated molecular processes, chemical * Corresponding author phone: 370 5 2366 040; fax: 370 5 2366 003; e-mail: vytautas.balevicius@ff.vu.lt. † Vilnius University. ‡ University of Technology Darmstadt. 2523 J. Phys. Chem. B 2007, 111, 2523-2532 10.1021/jp065477x CCC: $37.00 © 2007 American Chemical Society Published on Web 02/20/2007