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Notes: The results of the Brownian dynamics simulation of a hydrocarbon chain in a membrane bilayer described in the preceding paper are used to analyze the 13C NMR T1 relaxation in lipid bilayer vesicles. The analysis shows that the frequency dependence of the relaxation does not arise from gauche–trans isomerization or from axial rotation of the entire lipid molecule. However, a model in which fast axial rotation (D(parallel)≈2×1010 s−1) and slow noncollective diffusive director fluctuations (D⊥≈1–2×108 s−1) are superimposed on the internal motions quantitatively accounts for both the magnitude and frequency dependence of the T1 data. An effective viscosity for the interior of the bilayer in the range of 1 cp, and a director order parameter of 0.5–0.7 are required to fit the NMR data. Collective effects do not appear necessary for explaining the NMR T1 data in vesicles, although they may be important for multilamellar dispersions.

Notes: The potential energy function about the C–C single bond for the ground state 1,3-butadiene has been derived from ab initio calculations at both the Hartree–Fock (HF) level with 6-31G, 6-31G*, and 6-311G** basis sets and the second-order Møller–Plesset perturbation (MP2) level with 6-31G* basis set with the complete geometry optimizations at each of 15 fixed CCCC dihedral angles; the total energies and optimized geometries for the s-trans, gauche, and s-cis conformers were also determined at MP2 level with 6-311G* basis set and the third-order Møller–Plesset perturbation (MP3) level with 6-31G* basis set. The second stable conformer of the butadiene is predicted to be a gauche structure from all the calculations with a CCCC dihedral angle between 35° and 40° and a barrier of 0.5–1.0 kcal/mol to the s-cis transition state, and the theoretical torsional potentials are in good agreement with the experimental potential function of trans–gauche–gauche case derived by Durig et al.; by contrast, the theoretical torsional components differ significantly from the experimental results obtained from a trans–cis model. Vibrational frequencies and force field for s-trans and gauche conformers of 1,3-butadiene are determined at the Hartree–Fock and MP2 levels with 6-31G, 6-31G*, 6-311G, and 6-311G* basis sets. The mean absolute percentage deviations of the calculated frequencies from the experimental values (not corrected for anharmonicity) are ∼10%–13% and 3%–6% for the Hartree–Fock and MP2 methods, respectively. The effects of polarization functions and electron correlation on the force fields are studied, and the additivity of correlation and d function effects are discussed. Comparisons are made with other force fields, including experimental and previous ab initio results.

Notes: In the classical time-dependent Hartree approximation (TDH), the dynamics of a single molecule is approximated by that of a "field'' (each field being N "copies'' of the molecule which are transparent to one another while interacting with the system via a scaled force). It is shown that when some molecules are represented by a field of copies, while other molecules are represented normally, the average kinetic energy of the system increases linearly with the number of copies and diverges in the limit of large N. Nevertheless, the TDH method with appropriate energy scaling can serve as a useful means of enhancing the configurational sampling for problems involving coupled systems with disparate numbers of degrees of freedom.

Notes: Basis set and polarization function effects on the ground state optimized geometries and harmonic frequencies at the second-order Møller–Plesset perturbation (MP2) level have been studied for 11 small molecules (one or two heavy atoms) containing one or two carbon atoms, as well as for propene, propane, isobutene, acetaldehyde, methyl ether, cis- and trans-1,2-difuoroethylene. A series of basis sets ranging in quality from 4-21G to 6-311G** have been used for the small systems; for the larger systems 6-31G and 6-31G* basis sets were compared. In addition, three modified 6-31G basiss sets in which d basis functions are added to certain (but not all) heavy atoms were introduced to study the effect of polarization functions in systems containing heteroatoms. It was found that the inclusion of d functions in basis sets is important for calculating the equilibrium geometries, especially for CC and CX (X=N, O, and F) bonds. For vibrational frequencies, however, addition of d functions to basis sets often does not produce a significant improvement; for many alkenes and alkanes MP2/6-31G (MP2 with a 6-31G basis set) and MP2/4-21G calculations give good results for the frequencies that are comparable to those obtained with MP2/6-31G* and MP2/4-21G*, respectively. For molecules containing heteroatoms, the MP2/6-31G (MP2/4-21G) frequencies are generally rather close to the MP2/6-31G* (MP2/4-21G*) results except for the vibrations involving CX or XH stretching, for which the MP2/6-31G(MP2/4-21G) values are usually too low. Such deficiencies can be removed by addition of d basis functions to one of the atoms involved in CX or XH bonding. It is suggested that such basis sets with limited polarization functions can be usefully applied to larger molecules. Some experimental frequencies which are not consistent with the ab initio values are discussed and reassignments are proposed.

Notes: The free energy, energy, and entropy of solvation, relative to the pure liquid, are analyzed. By a coupling parameter integration it is shown that only averages over the solute–solvent interaction energy contribute to the free energy and that the solvent–solvent interaction term, which contributes the so-called cavity (solvent reorganization) term to the energy, is cancelled exactly by a corresponding term in the entropy. These terms exist even in the infinite dilution limit since they arise from the derivative of the free energy with respect to the solute density. Following the approach of Garisto et al. [J. Chem. Phys. 79, 6294 (1983)], the site–site Ornstein–Zernike integral equations and HNC closures are used to determine the derivatives of the distribution functions with respect to the density. This makes it possible to calculate the energetic and entropic contributions to the solvation free energy in the infinite dilution limit. The method is applied to pure solvent and to infinitely dilute aqueous solutions of cations, anions and neutral Lennard-Jones particles. The results are in agreement with numerical calculations of the thermodynamic quantities by use of finite difference values for the temperature derivatives. A simple empirical relation for the charge dependence of the solvation free energy is observed; it is shown for the case of an ion in a dipolar solvent, as typified by aqueous electrolyte solutions, that the free energy of solvation varies quadratically with the charge and is very nearly equal to one-half the solute–solvent portion of the solvation energy. Some discussion of the relation of the present results to entropy–enthalpy compensation and to computer simulations is given.

Notes: A method is presented that uses integral equation theory to determine analytic temperature derivatives of the radial distribution functions. It is illustrated by studying the solvation thermodynamics of monatomic solutes in aqueous solution. The results agree well with the density derivative method developed previously [Yu and Karplus, J. Chem. Phys. 89, 2366 (1988)]. An expression for the solvation enthalpy is derived which allows direct comparison with experimental and isobaric–isothermal (NPT) ensemble simulation data. Satisfactory agreement with experiment is found for pure water and for the aqueous solvation of monovalent ions. Simple equations that exploit the site–site HNC closures are given for the decomposition of the potential of mean force into its enthalpic (or energetic) and entropic components. Since the extended RISM (HNC-RISM) theory yields an incorrect (trivial) value of the dielectric constant, two different ways to correct for the asymptotic behavior of the solute–solute potential of mean force are compared. They lead to similar results but the method in which the solvent dielectric constant is modified from the outset can be applied more generally.The interactions between nonpolar and between polar solutes in water are decomposed into enthalpic and entropic contributions. This is difficult to do by computer simulations because of the lack of precision in such calculations. The association of nonpolar solutes in water is found to have comparable enthalpic and entropic contributions; this result disagrees with the usual description of an entropy-dominated hydrophobic interaction. For ions, the somewhat surprising result is that the association of like-charged species is enthalpy driven while for oppositely charged ions entropic effects are dominant. The process of bringing two like-charged ions together leads to higher local charge density; the more favorable solvation enthalpy arising from this increase in charge density (q2 dependence) more than compensates for the Coulombic repulsion. For oppositely charged ions, association leads to a partial charge neutralization in which the favorable Coulombic attraction is overwhelmed by the loss of stabilizing solvation enthalpy. The entropic increase is due to the greater freedom of the surrounding water molecules resulting from the partial charge neutralization.

Notes: A theoretical force field for in-plane vibrations of benzene has been determined from ab initio calculations at both the Hartree–Fock level with 4-21G, 6-31G, and 6-31G* basis sets and the MP2 level with 4-21G and 6-31G basis sets. The average error of the calculated frequencies at the MP2 level is between 2% and 3%. The reliability of the force field and vibrational frequency predictions of the calculations are analyzed. All diagonal stretching force constants obtained at the MP2 level are in quantitative agreement with Ozkabak–Goodman experimental force field, while the diagonal force constants involving ring deformation and CH rock are somewhat overestimated by the theory. Most of the off-diagonal force constants agree with the Ozkabak–Goodman results in sign but there are some significant quantitative differences in magnitude. Comparisons are made with other force fields, including results obtained by scaling ab initio calculations or introducing modified Hamiltonians. A simple extrapolation method for introducing correlation corrections into Hartree–Fock force constants gives excellent results for benzene.

Notes: The effective electronic donor/acceptor coupling (HDA) for a bridged electron transfer system is defined. An expression for HDA in terms of the bridge Green's function is developed for systems represented by a tight-binding Hamiltonian. HDA is computed exactly for two systems and the existence of a dimensionless parameter which determines whether the effective coupling oscillates or decays with increasing donor/acceptor distance is demonstrated. A numerical technique for computing HDA is developed and shown to be significantly more powerful than conventional techniques.

Notes: Inelastic neutron scattering spectra are calculated from harmonic and damped harmonic models of the internal dynamics of a small protein, the bovine pancreatic trypsin inhibitor (BPTI). Numerical Fourier transformation of the intermediate scattering function Fvibinc (q, t) is used to calculate the inelastic scattering. This permits the inclusion of multiphonon scattering and frictional damping effects. Although for a typical experimental configuration, the multiphonon contribution does not significantly alter the form of the scattering at frequencies below about 30 cm−1, it does have a significant effect on the scattering intensity at higher frequencies. Frictional damping is introduced into the harmonic model by assuming that each mode acts as an independent damped Langevin oscillator. With this model and the assumption that the lowest frequency modes are overdamped while the higher frequency modes are underdamped, improved agreement with the experimental BPTI powder results is obtained. The measured scattering from BPTI in solution shows increased intensity at frequencies below 50 cm−1 relative to the powder results. The solution scattering profile can be reproduced approximately by the addition of overdamped Langevin oscillator normal modes to the dynamic model in best agreement with the powder data. Several other aspects of neutron scattering from proteins are examined. Anisotropy in the harmonic resolution broadened scattering is demonstrated. Spectra calculated assuming classical equations of motion are shown to agree with those calculated with the full quantum-mechanical dynamical model. Translational diffusion broadening is found to be small compared to the instrumental resolution broadening for the range of scattering wave vectors of interest. The contribution of the coherent scattering to the measured intensity is calculated for the case of a partially hydrated protein. Under typical experimental conditions, the measured cross sections are dominated by the incoherent scattering and the self part of the coherent scattering, a result that justifies the comparison of experimental data with calculated incoherent scattering spectra.

Notes: In molecular dynamics simulations of flexible molecules, the treatment of motions that take place on different time scales (e.g., bond stretching and bond angle bending vs dihedral angle librations and overall translation and rotation) or of forces that vary differently with distance (e.g., bonding forces vs electrostatic forces) is a major consideration. The most rapidly varying quantity limits the integration time step, while the most slowly varying quantity usually determines the simulation time required. The reversible multiple time-step methods introduced by Tuckerman, Berne, and Martyna [J. Chem. Phys. 97, 1990 (1992)] allow one to use different integration time steps in the same simulation so as to treat the time development of the slow and fast motions most effectively. As a first step toward extending multiple time-step methods to macromolecules, we investigate their utility for determining the dynamics of n-alkanes as neat liquids and in aqueous solution. Because the dynamics is done in Cartesian rather than internal coordinates, the different time steps are implemented in terms of hard and soft forces acting on individual atoms; e.g., hard forces are contributed by bond stretching (and bond angle bending) terms and soft forces by the remaining intramolecular and intermolecular (van der Waals and electrostatic) terms in the additive empirical potential function used to describe these systems. Integrations with and without a Nose–Hoover thermostat are performed. Comparisons are made with ordinary single time step molecular dynamics and dynamics with constraints applied to the bonds. The stability of the integration methods and the dynamic properties are considered. It is shown that stable integrations that yield agreement with standard dynamics can be achieved with a saving of up to a factor of 5 in computer time.