液体的内压和内能的统计热力学理论

利用分布函数理论导出了液体的内能和内压公式.液体的内压和过剩内能可以表示成体积的幂级数形式,其中的系数可以用多体相互作用势和多体径向分布函数表出,它们仅仅与温度有关.讨论了液体仅存在第n次多体相互作用势情形的内压和过剩内能的表达式,结果与Egelstaff的微扰理论结果具有相同的形式,不仅给出了相应参数的表达式而且适用于多体相互作用较强的情形.定义了物性参数α(T)和m,得到的液体过剩内能和内压的表达式与Frank实验结果具有相同的形式,其结果不仅给出了参数α(T)和m的表达式,而且指出了Frank的过剩内能和内压公式只适用于参数α(T)和m与体积无关的液体.     

Alternative expressions for energies and forces due to angle bending and torsional energy

We have derived alternative expressions for computing the energies and forces associated with angle bending and torsional energy terms commonly used in molecular mechanics and molecular dynamics computer programs. Our expressions address the problems of singularities that are intrinsic in popular angle energy functions and that occur from other chain rule derivations of force expressions. Most chain rule derivations of expressions for Cartesian forces due to angle energies make use of relations such as where ? is a bond or torsion angle, E(?) is energy, and ?/?x represents a derivative with respect to some Cartesian coordinate. This expression leads to singularities from the middle term, ?1/sin ?, when ? is 0 or π. This is a problem that prevents the use of torsional energy expressions that have phase angles, ?°, other than 0 or π, such as in E(?) = κ[1 + cos(n? ? phsi;°)]. Our derivations make use of a different, but equivalent, form of the chain rule: This form still possesses singularities for the bond angle forces since the last factor is undefined when ? is 0 or π. However, the alternate form may be used to great advantage for the torsional angle forces where no such problem arises. The new expressions are necessary if one desires the use of torsional energy expressions with general phase angles. Even for energy expressions in common use, i.e., with phase angles of 0 or π, our force expressions are as computationally efficient as the standard ones. The new expressions are applicable to all molecular simulations that employ restrained, or phase-shifted, torsional angle energy expressions.     

Matrix elements of N-particle explicitly correlated Gaussian basis functions with complex exponential parameters

In this work we present analytical expressions for Hamiltonian matrix elements with spherically symmetric, explicitly correlated Gaussian basis functions with complex exponential parameters for an arbitrary number of particles. The expressions are derived using the formalism of matrix differential calculus. In addition, we present expressions for the energy gradient that includes derivatives of the Hamiltonian integrals with respect to the exponential parameters. The gradient is used in the variational optimization of the parameters. All the expressions are presented in the matrix form suitable for both numerical implementation and theoretical analysis. The energy and gradient formulas have been programmed and used to calculate ground and excited states of the He atom using an approach that does not involve the Born-Oppenheimer approximation.     

Approximate analytic expression for the stability ratio of colloidal dispersions

Approximate analytic expressions are derived for the stability ratios of dispersions of spherical colloidal particles with and without viscous interactions between particles on the basis of the DLVO theory of the potential energy of the electrostatic and van der Waals interactions between two approaching particles. The obtained approximate stability expressions agree with exact numerical results with negligible errors and are applicable irrespective of the magnitude of the potential maximum unlike the previous approximate stability expressions, which are applicable only when the potential maximum is much greater than the thermal energy.     

The Calculation of Interfacial Tension and Electrostatic Free Energy of Spherical Ionic Micelles with High Surface Potentials

Based on extended Langmuir's method on the dressed micelles, approximate expressions for the calculation of interfacial tension and electrostatic free energy of spherical ionic micelles with high surface potentials have been presented. These expressions are derived from nonlinear Poisson‐Boltzmann equation. The present formulae for the calculation of interfacial tension and electrostatic free energy of spherical ionic micelles are in quite good agreement with Hayter's results.     

Density functional perturbational orbital theory of spin polarization in electronic systems. I. Formalism

A perturbational approach is presented for the general analysis of spin-polarization effect on electronic structures and energies within spin-density functional formalism. Explicit expressions for the changes in Kohn-Sham [Phys. Rev. 140, 1133 (1965)] orbital energies and coefficients as well as for the change in total electronic energy are derived upon using the local spin density and self-interaction-corrected exchange-correlation functionals. The application of the method for atoms provides analytical expressions for the exchange splitting energy and spin-polarization energy. The atomic exchange parameters are obtained from the expressions for the elements with Z=1-92 and they match well with Stoner exchange parameters for 3d metal elements.     

Multiple impacts and energy transfer in a three-body system for noncollinear collisions

Summary Within the impulsive framework, the energy transfer processes in collisions of atoms with diatomic molecules are considered. In the case of noncollinear collisions involving multiple impacts between the particles, analytic expressions for the amount of the collision energy transferred to the internal degrees of freedom of the molecule have been derived. The limiting cases of these expressions are the well-known Mahan (a single impact) and Mahan-Shin (collinear collisions) formulas. The efficiency of energy transfer in collisions of cesium halide molecules with xenon atoms has been computed as an example; the results obtained agree well with the data of accurate trajectory calculations.     

Closed-form expressions of quantum electron transfer rate based on the stationary-phase approximation

Closed-form rate expressions are derived on the basis of the stationary-phase approximation for the Fermi golden rule expression of the quantum electron-transfer (ET) rate. First, on the basis of approximate solutions of the stationary-phase points near DeltaG = 0, -lambda, and lambda, where DeltaG is the reaction free energy and lambda is the reorganization energy, three closed-form rate expressions are derived, which are respectively valid near each value of DeltaG. Numerical tests for a model Ohmic spectral density with an exponential cutoff demonstrate good performance of the derived expressions in the respective regions of their validity. In particular, the expression near DeltaG = -lambda, which differs from the semiclassical approximation only by a prefactor quadratic in DeltaG, works substantially better than the latter. Then, a unified formula is suggested, which interpolates the three approximate expressions and serves as a good approximation in all three regions. We have also demonstrated that the interpolation formula can serve as a good quantitative means for understanding the temperature dependence of the quantum ET rate.     

Stress and heat flux for arbitrary multibody potentials: a unified framework

A two-step unified framework for the evaluation of continuum field expressions from molecular simulations for arbitrary interatomic potentials is presented. First, pointwise continuum fields are obtained using a generalization of the Irving-Kirkwood procedure to arbitrary multibody potentials. Two ambiguities associated with the original Irving-Kirkwood procedure (which was limited to pair potential interactions) are addressed in its generalization. The first ambiguity is due to the nonuniqueness of the decomposition of the force on an atom as a sum of central forces, which is a result of the nonuniqueness of the potential energy representation in terms of distances between the particles. This is in turn related to the shape space of the system. The second ambiguity is due to the nonuniqueness of the energy decomposition between particles. The latter can be completely avoided through an alternate derivation for the energy balance. It is found that the expressions for the specific internal energy and the heat flux obtained through the alternate derivation are quite different from the original Irving-Kirkwood procedure and appear to be more physically reasonable. Next, in the second step of the unified framework, spatial averaging is applied to the pointwise field to obtain the corresponding macroscopic quantities. These lead to expressions suitable for computation in molecular dynamics simulations. It is shown that the important commonly-used microscopic definitions for the stress tensor and heat flux vector are recovered in this process as special cases (generalized to arbitrary multibody potentials). Several numerical experiments are conducted to compare the new expression for the specific internal energy with the original one.     

On the non-existence of maxima in variational computations containing non-linear parameters

The homogeneity properties of the kinetic and potential energy operators are used to obtain expressions for the second derivatives of the energy expectation value. These are used to demonstrate that in atoms as well as in molecules in the neighborhood of the equilibrium geometry the variational energy cannot have maxima with respect to the non-linear parameters.     

Adsorption free energy of variable-charge nanoparticles to a charged surface in relation to the change of the average chemical state of the particles

Variable-charge nanoparticles such as proteins and humics can adsorb strongly to charged macroscopic surfaces such as silica and iron oxide minerals. To model the adsorption of variable-charge particles to charged surfaces, one has to be able to calculate the adsorption free energy involved. It has been shown that the change in the free energy of variable-charge particles is related to the change in their average chemical state upon adsorption, which is commonly described using surface complexation models. In this work, expressions for the free-energy change in variable-charge particles due to changes in chemical binding are derived for three ion-binding models (i.e., the Langmuir, Langmuir-Freundlich, and NICA models) and for changes due to nonspecific binding for the Donnan model. The expressions for the adsorption free energy of the variable-charge particles to a charged surface are derived on the basis of the equality of the (electro)chemical potential of the particles in the bulk solution and adsorption phase. The expressions derived are general in the sense that they account for the competition between charge-determining ions that bind chemically to the particles, and they also apply in case of the formation of chemical bonds between particle ligands and surface sites. The derived expressions can be applied in the future to model the adsorption of variable-charge nanoparticles to charged surfaces. The results obtained for the NICA-Donnan model make it possible to apply this advanced surface complexation model to describe the adsorption of humics to minerals.     

Expressions for the stress and elasticity tensors for angle-dependent potentials

The stress and elasticity tensors for interatomic potentials that depend explicitly on bond bending and dihedral angles are derived by taking strain derivatives of the free energy. The resulting expressions can be used in Monte Carlo and molecular dynamics simulations in the canonical and microcanonical ensembles. These expressions are particularly useful at low temperatures where it is difficult to obtain results using the fluctuation formula of Parrinello and Rahman [J. Chem. Phys. 76, 2662 (1982)]. Local elastic constants within heterogeneous and composite materials can also be calculated as a function of temperature using this method. As an example, the stress and elasticity tensors are derived for the second-generation reactive empirical bond-order potential. This potential energy function was used because it has been used extensively in computer simulations of hydrocarbon materials, including carbon nanotubes, and because it is one of the few potential energy functions that can model chemical reactions. To validate the accuracy of the derived expressions, the elastic constants for diamond and graphite and the Young's Modulus of a (10,10) single-wall carbon nanotube are all calculated at T = 0 K using this potential and compared with previously published data and results obtained using other potentials.     

A constrained variational approach for energy derivatives in Fock-space multireference coupled-cluster theory

In this paper, we present a formulation based on constrained variational approach to enable efficient computation of energy derivatives using Fock-space multireference coupled-cluster theory. Adopting conventional normal ordered exponential with Bloch projection approach, we present a method of deriving equations when general incomplete model spaces are used. Essential simplifications arise when effective Hamiltonian definition becomes explicit as in the case of complete model spaces or some special quasicomplete model spaces. We apply the method to derive explicit generic expressions upto third-order energy derivatives for [0,1], [1,0], and [1,1] Fock-space sectors. Specific diagrammatic expressions for zeroth-order Lagrange multiplier equations for [0,1], [1,0], and [1,1] sectors are presented.     

Fast force field expressions for computer simulations

It is the aim of this work to develop analytical force field expressions that can be rapidly evaluated by computers. The new expressions approximate the energy hypersurfaces as described by the usual force fields. The energies and the derivatives of the energy expressions, i. e. the forces acting on the atoms, in most cases can be very quickly calculated as functions of one squared distance of two atoms per interaction, avoiding slow operations like cosine, square root etc. Formulae and algorithms are given to calculate the parameters needed from those of the AMBER force field.     

Higher molecular-deformation derivatives of the configuration-interaction energy

We derive expressions for the first through fourth derivatives of the configuration-interaction (CI) electronic energy with respect to molecular deformation. By using unitary exponential parameterizations of the wavefunction's orbital and configuration amplitude response together with a power-series expansion of the geometry dependence of the hamiltonian, a computationally attractive expression for the CI energy derivatives is obtained. The use of so-called direct methods in evaluating the CI derivatives is discussed as are the relative efforts involved in using our CI-based energy-derivative expressions and those which we obtained earlier for derivatives of the multiconfigurational self-consistent-field energy. The power-series expansion of the geometry dependence of the hamiltonian that we have derived may be used for evaluating molecular-deformation derivatives for any approximate wavefunction constructed from a set of orthonormal orbitals.     

Solvation thermodynamics: theory and applications

Potential distribution and coupling parameter theories are combined to interrelate previous solvation thermodynamic results and derive several new expressions for the solvent reorganization energy at both constant volume and constant pressure. We further demonstrate that the usual decomposition of the chemical potential into noncompensating energetic and entropic contributions may be extended to obtain a Gaussian fluctuation approximation for the chemical potential plus an exact cumulant expansion for the remainder. These exact expressions are further related to approximate first-order thermodynamic perturbation theory predictions and used to obtain a coupling-parameter integral expression for the sum of all higher-order terms in the perturbation series. The results are compared with the experimental global solvation thermodynamic functions for xenon dissolved in n-hexane and water (under ambient conditions). These comparisons imply that the constant-volume solvent reorganization energy has a magnitude of at most approximately kT in both experimental solutions. The results are used to extract numerical values of the solute-solvent mean interaction energy and associated fluctuation entropy directly from experimental solvation thermodynamic measurements.     

Surface structure and energy spectra of localized states

A simple model for surface reconstruction is developed utilizing the next-nearest-neighbor approximation within the framework of molecular-orbital theory. Exact energy expressions for this model and various special cases are derived, and they are illustrated by numerical results. The influence of a very weak surface deformation on the energy spectrum of surface states is also discussed.     

Phase relationships in a ternary solution composed of a solvent and two monodisperse homologous polymers

Exact expressions are derived for the spinodal, critical conditions, and separation factor of a ternary solution consisting of a pure solvent and two monodisperse homologous polymers in which the Flory-Huggins interaction parameter χ depends separately on the concentrations of the polymer components. The results allow one to see the difference from previous expressions obtained with χ depending on the total concentration of the polymer, and are expected to be useful for experimental determination of the Gibbs free energy of ternary solutions.     

Exact analytical expressions for the potential of electrical double layer interactions for a sphere-plate system

Exact, closed-form analytical expressions are presented for evaluating the potential energy of electrical double layer (EDL) interactions between a sphere and an infinite flat plate for three different types of interactions: constant potential, constant charge, and an intermediate case as given by the linear superposition approximation (LSA). By taking advantage of the simpler sphere-plate geometry, simplifying assumptions used in the original Derjaguin approximation (DA) for sphere-sphere interaction are avoided, yielding expressions that are more accurate and applicable over the full range of κa. These analytical expressions are significant improvements over the existing equations in the literature that are valid only for large κa because the new equations facilitate the modeling of EDL interactions between nanoscale particles and surfaces over a wide range of ionic strength.