This note discusses a statistical mechanical trajectory method for the determination of normal mode frequencies and degeneracies of molecular svstems. The method is a well-known formal element of condensed matter theory ( I ) . Compared to common lratrix diagonalization methods on a practical level, the trajectory method is much less efficient for those cases in which diap-onalization is feasible, and, in fact, is much less used. In physical chemistry courses, therefore, it would he inappropriate to replace the usual general discussion of harmonic motion based on coordinate transformations which produce a separable Hamiltonian (2) by a discussion of the trajectory method. However, the trajectory method illustrates an interesting array of pedagogically valuable tonics. and this makes it auite suitable for swecial . . pruhlems or projrct i for ambitious students. 'l'hese pedayogicdlv interwtinr features i n r l~~d r :(1) rmvhasis of the basir vie; that statistical mechanics is mechanics plus statistics (3); (2) emphasis of the concepts of spectral range and resolution which are basic to spectroscopy (4); (3) illustration of the practical importance of classical mechanics in chemistry (5); (4) demonstration of numerical methods for integrating differential equations (4); (5) demonstration of numerical methods for constructing finite samnles from wrobabilitv distributions such as t h e~a x w e l l -~o l t z m a n n distribution of atomic velocities ( 4 ) ; this could he coupled with consideration of stochastic dynamical methods (6); (6) demonstration of general features of interatomic forces, including anharmonic iterations (26) and their influence on the approach of a dvnamical svstem to thermal equilibrium (7); (7) illustration of the roie of time correlation functions in statistical mechanics (la, 8). Additionally, the trajectory method permits substantial flexibility of implementation. Therefore, not all points above need be emphasized, and students could be given the freedom t o adopt either primitive or sophisticated approaches to any of these aspects of the method.The formula that we discuss is based on a swectral analvsis of classical trajectories. Suppose that, by some method, we have obtained a segment of classical trajectory of duration T. The velocities of all the atoms are known on a grid of N time woints sewarated bv At; the x component of the velocity of the ith aiom at time n h t is oj,(n~i). For notational convenience we take N = 21 + 1. The uj,(nAt) is treated as primary data and is Fourier analyzed to obtainThe spectral resolution is Aw = 2n/r, and the analysis is constructed so that the spectral range is from -1Aw to lAw. The formula inverse to eq 1 is In terms of these quantities the vibrational density of states, g b ) , can he estimated from = (2/Aw) 2 ( l~~, ( k~w ) l~/ 2~l u ;~( h~~w ) I~)i,"=r>.. The outside sum is over the 3Nvelocity coordinates; k covers the integers 0, . . . , 1, and h' ranges over integers -1, . . . , 1.Note that for a system in thermal equilihrium the denominator in eq...
Progress in understanding liquid ethylene carbonate (EC) and propylene carbonate (PC) on the basis of molecular simulation, emphasizing simple models of interatomic forces, is reviewed. Results on the bulk liquids are examined from the perspective of anticipated applications to materials for electrical energy storage devices. Preliminary results on electrochemical double-layer capacitors based on carbon nanotube forests and on model solid-electrolyte interphase (SEI) layers of lithium ion batteries are considered as examples. The basic results discussed suggest that an empirically parameterized, non-polarizable force field can reproduce experimental structural, thermodynamic, and dielectric properties of EC and PC liquids with acceptable accuracy. More sophisticated force fields might include molecular polarizability and Buckingham-model description of inter-atomic overlap repulsions as extensions to Lennard-Jones models of van der Waals interactions. Simple approaches should be similarly successful also for applications to organic molecular ions in EC/PC solutions, but the important case of Li + deserves special attention because of the particularly strong interactions of that small ion with neighboring solvent molecules. To treat the Li + ions in liquid EC/PC solutions, we identify interaction models defined by empirically scaled partial charges for ion-solvent interactions. The empirical adjustments use more basic inputs, electronic structure calculations and ab initio molecular dynamics simulations, and also experimental results on
The quasi-chemical organization of the potential distribution theorem -molecular quasi-chemical theory (QCT) -enables practical calculations and also provides a conceptual framework for molecular hydration phenomena. QCT can be viewed from multiple perspectives: (a) As a way to regularize an ill-conditioned statistical thermodynamic problem; (b) As an introduction of and emphasis on the neighborship characteristics of a solute of interest; (c) Or as a way to include accurate electronic structure descriptions of near-neighbor interactions in defensible statistical thermodynamics by clearly defining neighborship clusters. The theory has been applied to solutes of a wide range of chemical complexity, ranging from ions that interact with water with both long-ranged and chemically intricate short-ranged interactions, to solutes that interact with water solely through traditional van der Waals interations, and including water itself. The solutes range in variety from monoatomic ions to chemically heterogeneous macromolecules. A notable feature of QCT is that in applying the theory to this range of solutes, the theory itself provides guidance on the necessary approximations and simplifications that can facilitate the calculations. In this Perspective, we develop these ideas and document them with examples that reveal the insights that can be extracted using the QCT formulation.
simulations emphasize some differences: the asymmetries of bulk solution inner-shell structures are moderated compared with clusters, but still present; and inner hydration shells fill to slightly higher average coordination numbers in bulk solution than in clusters.
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